Quantum Plasmonic Sensors
Changhyoup Lee, Benjamin Lawrie, Raphael Pooser, Kwang-Geol Lee, Carsten Rockstuhl, Mark Tame
QQuantum Plasmonic Sensors
Changhyoup Lee,
1, 2, ∗ Benjamin Lawrie,
3, 4
Raphael Pooser, Kwang-Geol Lee, Carsten Rockstuhl,
1, 6, 7 and Mark Tame Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany Quantum Universe Center, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA Quantum Information Science Group, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA Department of Physics, Hanyang University, Seoul 04763, Republic of Korea Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany Max Planck School of Photonics, Germany Department of Physics, Stellenbosch University, Stellenbosch 7602, South Africa (Dated: December 4, 2020)The extraordinary sensitivity of plasmonic sensors is well known in the optics and photonics community.These sensors exploit simultaneously the enhancement and the localization of electromagnetic fields close tothe interface between a metal and a dielectric. This enables, for example, the design of integrated biochemicalsensors at scales far below the di ff raction limit. Despite their practical realization and successful commercial-ization, the sensitivity and associated precision of plasmonic sensors are starting to reach their fundamentalclassical limit given by quantum fluctuations of light – known as the shot-noise limit. To improve the sensingperformance of these sensors beyond the classical limit, quantum resources are increasingly being employed.This area of research has become known as ‘quantum plasmonic sensing’ and it has experienced substantialactivity in recent years for applications in chemical and biological sensing. This review aims to cover both plas-monic and quantum techniques for sensing, and shows how they have been merged to enhance the performanceof plasmonic sensors beyond traditional methods. We discuss the general framework developed for quantumplasmonic sensing in recent years, covering the basic theory behind the advancements made, and describe theimportant works that made these advancements. We also describe several key works in detail, highlighting theirmotivation, the working principles behind them, and their future impact. The intention of the review is to set afoundation for a burgeoning field of research that is currently being explored out of intellectual curiosity and fora wide range of practical applications in biochemistry, medicine, and pharmaceutical research. CONTENTS
I. Introduction 2II. Plasmonic sensors 2A. Surface plasmon polaritons 3B. Surface plasmon resonance sensing 51. Angular interrogation 62. Spectral interrogation 63. Limit of detection 6C. Intensity vs phase sensing 7D. Localized surface plasmon resonance sensing 81. Sensitivity and figure of merit 92. Plasmon-enhanced fluorescence and Ramanscattering 10E. Surface plasmon resonance imaging 12III. Quantum sensors 12A. Parameter Estimation Theory: Cramér-Raobound 131. Single-parameter estimation 132. Multiparameter estimation 15B. Shot-noise limited sensing 151. Shot-noise limited intensity sensing 152. Shot-noise limited phase sensing 16C. Sub-shot-noise intensity sensing 171. Quantum-enhanced intensity sensing 17 ∗ [email protected]
2. Multiparameter or multimode intensitysensing 183. Quantum noise reduction in intensitymeasurements 19D. Sub-shot-noise phase sensing 221. Phase sensing in Mach-Zehnderinterferometers 222. Phase sensing with NOON states 233. Single-mode phase sensing 254. Multiple-phase sensing 265. Quantum sensing with SU(1,1)interferometers 27E. Quantum sensing beyond the Cramér-Rao bound 29IV. Quantum-enhanced plasmonic sensors 30A. Quantum plasmonic intensity sensing 311. Intensity sensing with discrete variable states 312. Intensity sensing with continuous variablestates 343. Intensity sensing robust to thermal noise 38B. Quantum plasmonic phase sensing 391. Phase sensing with discrete variable states 392. Phase sensing with continuous variable states 43C. Quantum plasmonic sensing based onemitter-plasmon coupling 43V. Perspective and outlook 44Appendix 45A. Multiparameter QFIM 45 a r X i v : . [ qu a n t - ph ] D ec B. Calculation of multiparameter QFIM 45C. CR bound in lossy MZIs 46Acknowledgments 46Glossary 46References 47
I. INTRODUCTION
Optical sensors are used in many areas of science and tech-nology – notable examples include gyroscopes for naviga-tion [1], accelerometers for monitoring structural deforma-tions [2] and seismic wave activity [3], electrochemical de-vices for measuring renewable energy storage [4], and densitysensors for analyzing seawater in climate change research [5].Of particular interest to the biological and chemical sciencesare sensors that measure the presence and behavior of bac-teria, viruses, and proteins [6–9]. Here, highly sensitive opti-cal sensors provide a deeper understanding of the biochemicalprocesses involved in a given scenario in a non-invasive label-free manner, enabling, for example, drug development for im-proved health care [10]. Plasmonic systems based on metalnanostructures have long been used as a basis for optical sen-sors in this context [11–15]. The surface of a metal supports aquasiparticle known as a surface plasmon (SP) [16, 17], whichenables electromagnetic field confinement below the di ff rac-tion limit [18–20] and allows enhanced sensing compared toconventional optical sensors [21–28]. Indeed, since the 1990s,surface plasmon resonance (SPR) sensors have been commer-cialized by several companies [24, 29–32]; they are now a vi-tal tool for studying biomolecular interactions for fundamen-tal and applied sciences [33, 34].Despite their successful commercialization, the high sen-sitivity and precision of plasmonic sensors are beginning toreach a fundamental limit given by quantum fluctuations oflight – known as the shot-noise limit (SNL) [35, 36]. Inthe past few years, improved sensitivity and precision beyondthe SNL has been shown to be achievable for practical opti-cal sensors using concepts from quantum metrology [37–46].Several impressive experiments have demonstrated the basicworking principles of quantum metrology using various typesof quantum techniques in bulk optics [47–51], integrated op-tics [52], and biological systems [42, 53]. It is natural to askwhether these techniques can also be applied to plasmonicsensors to improve their performance. This question has beenfurther motivated by recent advances in our understanding ofthe fundamental theory underlying quantum plasmonic sys-tems, which has recently been extensively developed and ex-perimentally studied [54–66]. Advances in quantum plasmon-ics have enabled research groups to pursue the application ofquantum metrology principles to plasmonic sensors in just thepast few years. This area of research has become known as‘quantum plasmonic sensing’. As the community moves fur-ther into this growing area, it is now an ideal time to look backto review and consolidate the past research achievements, aswell as to look forward to identify future applications in the biochemical sciences and potential technologies beyond.The interested reader can find a multitude of review articleson plasmonic sensors that have already been written from var-ious perspectives over the last few decades [11–14, 21, 23–25, 27, 28, 34, 67]. This review, on the other hand, aimsto comprehensively introduce for the first time the generalframework developed for quantum plasmonic sensors in re-cent years and covers the basic theory behind the framework.While the goal is to present a broad overview of the researchlandscape, we also focus on several key works, where wehighlight the motivation, provide details of the working prin-ciples, and discuss the future impact. We have endeavored toinclude all relevant works in this exciting and growing multi-disciplinary area of research.This review is aimed at researchers working in the broadfields of plasmonics, biochemistry, photonics, quantum op-tics, and quantum information science. In order for it to beaccessible and beneficial to readers from all these fields, thephilosophy has been to structure the review into three distinctsections: In section II, we introduce conventional plasmonicsensing, where we describe the di ff erent types of plasmonicsensors that exist in research and industry based on the prin-ciples of classical physics, along with their sensing perfor-mance. In section III, we introduce quantum sensing, wherewe describe the tools and concepts that have taken conven-tional sensing from the classical to quantum regime, whicho ff ers a significantly improved sensing performance. In sec-tion IV, we then bring the concepts of sections II and III to-gether, and review recent work that has focused on integrat-ing quantum sensing techniques with conventional plasmonicsensors. We describe how this approach has opened up a newroute for improving the performance of plasmonic sensors. II. PLASMONIC SENSORS
The sensitivity of a sensor, including that of a plasmonicsensor, is generally defined as S y = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d y d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (1)where x represents an implicit parameter to be estimated froma measurement of an explicit quantity y [23]. The sensitiv-ity can thus be understood as the extent to which the explicitparameter y changes for a given change of the implicit param-eter x . Such a relation between x and y strongly depends onthe physical system used to encode the parameter to be es-timated, but it also depends on the physical quantities that x and y represent. As will be shown in the following sections, ina chosen plasmonic setting, various physical parameters y canbe measured to estimate an individual parameter x at stake. Anappropriate sensor, therefore, needs to be considered in a waysuch that the sensitivity expressed in eq 1 takes rather largevalues. Plasmonic sensors, which exploit the interaction oflight with the materials to be sensed at the interface betweena metal and a dielectric in the sensing process, cope with thisrequirement very well. In addition to a high sensitivity, an-other appealing feature of plasmonic systems is the capabilityto confine the light to a spatial domain well below the di ff rac-tion limit, which is not possible with conventional photonicsystems [18–20]. Therefore, plasmonic sensors are known toenable compact sub-di ff raction-limited sensing with high sen-sitivity. This also allows the measurement of tiny quantitiesof analytes, which is a feature that has been prompting theirdevelopment now for quite some time.In this section, we explain the physical basis of plasmonicsensing with typical basic structures. We describe how plas-monic features can be designed to optimize the sensitivity. Wealso introduce a few plasmonic sensors that have attracted in-tensive interest from various scientific communities. The plas-monic sensors in this section are considered to be operatedwith classical light, but in following sections we will considerhow quantum states of light and quantum measurements canbe combined with such plasmonic sensors to further improvetheir performance. A. Surface plasmon polaritons
According to established notation, a plasmon is a chargedensity oscillation in a metal. A plasmon polariton is a hybridexcitation where an electromagnetic field is coupled to a plas-mon. The additional term ‘surface’ expresses that such an ex-citation is confined to the interface between a metal and a di-electric. To understand the appearance and the basic physicalproperties of surface plasmon polaritons (SPPs), we start byconsidering an interface between a bulk dielectric and a bulkmetal, as shown in Figure 1(a). Here, ‘bulk’ simply impliesthat the size of the material is much larger than the wavelengthof light in all directions. That is, we consider the dielectric andmetal as semi-infinite half spaces.For a given material distribution in space, various electro-magnetic modes can be found by solving Maxwell’s equa-tions. The ‘macroscopic’ Maxwell equations in the time do-main are written as [68] ∇ · D ( r , t ) = ρ f ( r , t ) , (2) ∇ · B ( r , t ) = , (3) ∇ × E ( r , t ) = − ∂ B ( r , t ) ∂ t , (4) ∇ × H ( r , t ) = J f ( r , t ) + ∂ D ( r , t ) ∂ t , (5)where ρ f ( r , t ) is the free charge density and J f ( r , t ) is the freecurrent density, respectively. Note that E ( r , t ) and B ( r , t ) areexperimentally observable fields, while D ( r , t ) and H ( r , t ) areauxiliary fields introduced to capture the response of the ma-terials.At the interface between the two materials that we wish toconsider, all the fields have to satisfy specific conditions thatcan be derived from Maxwell’s equations by evaluating themin a small volume embracing the interface. These interfaceconditions can be written as [68] n × ( E m − E d ) = , (6) n · ( B m − B d ) = , (7) n · ( D m − D d ) = σ s , (8) n × ( H m − H d ) = j s , (9)where n is the normal vector characterizing the interface. Thesubscript for the field denotes the medium where the field is FIG. 1. (a) Electric field profile of surface plasma waves, the quantaof which are called SPPs, at the interface between a bulk dielectricand a bulk metal. (b) Dispersion relation for a SPP at the interfacebetween the metal and dielectric – see eq 12 – and a freely propa-gating field mode in the dielectric [ k || , max = √ (cid:15) d ( ω/ c )]. The fre-quency ω sp denotes the surface plasma frequency – the frequency atwhich surface electrons collectively oscillate. defined around the considered interface, and σ s and j s are theunbounded (free) surface charge and current density betweenthe two media, respectively.To solve the Maxwell equations, the relationship betweenthe observable and auxiliary fields needs to be specified. Thisis done by constitutive relations that are given for the elec-tric and magnetic fields in the time domain by D ( r , t ) = (cid:15) E ( r , t ) + P ( r , t ) and H ( r , t ) = B ( r , t ) /µ − M ( r , t ). Theelectric polarization P ( r , t ) and the magnetization M ( r , t ) candepend, in general, on both the electric and magnetic field.In the time domain and while restricting ourselves to a lin-ear system governed by response theory, the dependency ofthe polarization (magnetization) can usually be expressed bya convolution of the electric (magnetic) field with some mate-rial specific response function.In the context of this review it is fully su ffi cient to considerisotropic, homogeneous, non-magnetic materials [ M = H ( r , t ) = B ( r , t ) /µ ] without electro-magnetic couplingin each region, with no free charge ( σ s =
0) and no free cur-rent ( j s = E ( r , t ) = E ( r , ω ) e − i ω t . The convolutionin the time domain will be a product in the frequency domainand the constitutive relations collapse to some algebraic equa-tions. The response of the individual bound charges and cur-rents in the bulk materials to an electromagnetic field are thencharacterized by the relative electric permittivity in the fre-quency domain, i.e., (cid:15) d for the dielectric and (cid:15) m ( ω ) for themetal, and we write D ( r , ω ) = (cid:15) (cid:15) j ( ω ) E ( r , ω ) in the respec-tive material j . Note that the permittivity of any material gen-erally depends on the frequency, but in the typical range offrequency of interest in optics, the permittivity of the dielec-tric (cid:15) d can be assumed to be constant with a real positive valueto a good approximation. For the metal, the electric permittiv-ity can be described by the Drude model, written as [17] (cid:15) m ( ω ) = − ω ω + i γω , (10)where γ denotes a damping factor and ω p is the plasma fre-quency. We thus have (cid:15) m ( ω ) = (cid:15) (cid:48) m ( ω ) + i (cid:15) (cid:48)(cid:48) m ( ω ). One can seethat when γ < (cid:113) ω − ω , the real part of the permittivity isnegative, i.e., (cid:15) (cid:48) m ( ω ) <
0, which applies to typical metals (e.g.,gold and silver) at optical frequencies. An extended Drudemodel can also be found for the metal, for example in ref 69,which fits the empirically measured data for the dispersion inthe case of gold in the range of wavelength between 500 nmand 1 µ m, as reported in ref 70.In the structure considered above, various electromagneticmodes can be found by considering the wave equation in thefrequency domain, ∇ E ( r , ω ) + ω c (cid:15) ( r , ω ) E ( r , ω ) = | k | = k of the spatial part E ( r , ω ) = E ( r ) e i k · r to the frequency ω ofthe temporal part e − i ω t , given by k = ω c √ (cid:15) d [straight linein Figure 1(b) as the permittivity of the dielectric is assumedto be non-dispersive], where c is the speed of light in vac-uum, k = (cid:113) k x + k y + k z and the k i are the components of thewavevector k for the mode. In Figure 1(b) we consider theparticular case of the modes propagating in a direction set bythe plane of the interface, i.e., the x - y plane, so that k z = k || = (cid:113) k x + k y = k . These modes are the elementary so-lutions of the wave equation in free space. On the other hand,there exist propagating modes that are bound to the metal-dielectric interface as solutions to the wave equation. Theseare hybridized modes, in that they correspond to a couplingbetween the electromagnetic field and longitudinal plasma os-cillations, i.e., density oscillations of electrons in the conduc-tion band of the metal. The modes are called surface plasmawaves [lower curve in Figure 1(b)]. In the limit of large k || , themodes have a corresponding small wavelength and an electro-static approach can be used. The modes can thus be obtainedfor the electric potential Φ as a solution to the Laplace equa-tion ∇ Φ = (cid:15) m ( ω ) + (cid:15) d =
0. Substituting the Drude modelof eq 10 into (cid:15) m ( ω ) with γ = ω sp = ω p / √ + (cid:15) d . The quanta ofthese surface plasma waves are called SPs [17] and were firstpredicted by Ritchie [71]. In the limit of small k || , the modeshave a corresponding long wavelength, where charge is trans-ported over a considerable distance during the plasma oscil-lation. The resulting current sets up additional electromag-netic fields that interact back again with the electrons duringtheir oscillation, causing retardation. In this regime, the waveequation needs to be used to obtain solutions for the modes.The quanta of these surface plasma waves are called SPPs andcontain SPs in the limit k || → ∞ . The word ‘polariton’ em-phasizes the joint interaction between the matter part of theexcitation (the electron plasma oscillation, or plasmon) andthe light part of the excitation (the electromagnetic field, orphoton) [54]. In this section, we will use a classical descrip-tion of SPPs, i.e., surface plasma waves, and show how theyenable classical plasmonic sensing, but we keep the quantizedname, SPP, as is regularly done in the literature. This also hasthe benefit that it anticipates their use in the quantum regime,which we cover in detail in section IV.From the wave equation, the electric field of the confined SPP mode at the interface can be written as [17] E j ( r , t ) = E j ( r ) e i ( k (cid:107) x − ω t ) e − κ j | z | , (11)where E j ( r ) represents a vectorial field profile for the mode,the subscript j denotes either dielectric (d) or metal (m) andwe are considering propagation in the x direction, i.e., k x = k || and k y =
0. The vectorial field profile, E j ( r ), corresponds toa field with directionality in the x and z direction, i.e., a trans-verse magnetic field (in the y direction). This is due to theimaginary part of the conductivity associated with the Drudemodel being positive [72]. The dispersion relation for theparallel-to-interface wavenumber is written as [17] k SPP (cid:107) ( ω ) = ω c (cid:115) (cid:15) d (cid:15) m ( ω ) (cid:15) d + (cid:15) m ( ω ) . (12)This dispersion relation is valid for both real and com-plex (cid:15) m ( ω ), i.e., for metals with and without attenuation.When ω approaches the surface plasma frequency ω sp , thedenominator (cid:15) d + (cid:15) m ( ω ) approaches zero, so the wavenum-ber k SPP (cid:107) ( ω ) diverges. This is another indication that SPs canbe understood as the limiting case of SPPs for large wavenum-ber.In terms of field confinement, the electrons at the metal sur-face strongly bind the electromagnetic field of the SPP to theinterface, resulting in a huge enhancement of the electromag-netic field near the surface. The field given in eq 11 in the z direction is generally ‘sub-wavelength’ confined, as the fieldfalls o ff as e − κ j | z | , where κ j = (cid:113) ( k SPP || ) − k (cid:15) j and k = ω c .On the other hand, the confinement of the field in the x - y plane is no longer limited as it would be in a bulk mate-rial, i.e., limited by the usual three-dimensional di ff ractionlimit [18, 19, 54]. The field given in eq 11 possesses a planewave component with respect to a single wavevector k || andin the present discussion has an infinite spatial extent in the y direction. While convenient for mathematical modeling, anactual SPP field will be laterally confined in the y directionand for a fixed frequency ω it must be made from a sum ofplane waves via Fourier synthesis, each wave with di ff erent k x and k y components. The corresponding spatial extent ofsuch a confined field can be smaller than that allowed in thebulk dielectric. This leads to a SPP to be described as ‘sub-di ff raction’ confined on the surface, with the field obeying amore compact two-dimensional di ff raction limit for the ge-ometry [19]. Such a confinement in space on the surface (sub-di ff raction) and perpendicular to the surface (sub-wavelength)is the inherent feature that enables SPP modes to be highlysensitive to the optical properties of the dielectric medium.When the metal-dielectric interface is illuminated with lightfrom the dielectric region, the dispersion curve of the ra-diation mode, k in , does not cross the dispersion curve ofthe SPP, k SPP || , for a given permittivity (cid:15) d in the dielectricfor any frequency ω [see Figure 1(b) for the extreme casewhere k in || = k in = ω c √ (cid:15) d ]. In other words, the parallel-to-interface wavevector k in (cid:107) ( ω ) of the incident light can neverbe equal to k SPP (cid:107) ( ω ) of eq 12 for a fixed frequency. Thismeans that any light directly incident on the metal cannot ex-cite SPPs. A novel scheme is therefore required to satisfy FIG. 2. The Kretschmann configuration for plasmonic sensing. (a) The system is composed of three layers, where a thin metallic film issandwiched by a prism with permittivity (cid:15) p and an analyte with permittivity (cid:15) a . (b) Two examples of an analyte are shown. In (i) a ligandmolecule binds to a receptor on the metal surface modifying the local permittivity (cid:15) a . In (ii) a biomolecule is present near the surface resultingin a change of (cid:15) a . (c) Dispersion relation for light and SPPs in the system. The label ‘SPP(a)’ and ‘SPP(p)’ denote the SPP mode at theanalyte-metal interface and prism-metal interface, respectively. The corresponding surface plasma frequencies are given as ω (a)sp and ω (p)sp . Theincident light in the prism region (straight line) cannot cross the dispersion curve for ‘SPP(p)’, whereas the evanescent field on the oppositeside of the prism (the same straight line) can cross the dispersion curve for ‘SPP(a)’. the excitation condition k in (cid:107) ( ω ) = k SPP (cid:107) ( ω ). To this end, var-ious schemes have been demonstrated for the excitation ofSPPs at the interface between a bulk metal and a bulk dielec-tric, e.g., a prism [73, 74], a grating [16], a randomly roughsurface [75, 76], or a scanning near-field probe with sub-wavelength aperture [77]. The most widely used scheme is theprism setup, with two typical configurations: the Otto configu-ration [73] and the Kretschmann configuration [74]. The for-mer configuration requires dedicated techniques in practice,while the latter configuration has led to many successful ap-plications in plasmonic sensing [10], including commercial-ized versions. Therefore, we will focus on the Kretschmannconfiguration as a plasmonic sensing platform and elaborateon various assessments of its sensing performance in the nextsection. B. Surface plasmon resonance sensing
SPPs at the metal-dielectric interface can only be excitedwhen the excitation condition is satisfied, i.e., the modematching condition k in (cid:107) ( ω ) = k SPP (cid:107) ( ω ). However, realizing theexcitation condition is practically not easy because the dis-persion relation of eq 12 is sensitive to the optical propertiesof the interfaced medium, i.e., the permittivity of the dielec-tric (cid:15) d . Thus, when the aforementioned schemes are used toexcite SPPs for a given structure, the system parameters needto be finely tuned and carefully stabilized in a controlled man-ner to satisfy the resonance condition. This might be undesir-able and impractical from a general point of view, but para-doxically it is the desired feature for sensing in general. Thisis the basic principle of SPR sensing.We consider the most paradigmatic setup for the excitationof SPPs, the so-called Kretschmann configuration, that con-sists of three layers: the first layer is a prism ( (cid:15) = (cid:15) p ), thesecond layer is a metal film [ (cid:15) = (cid:15) m ( ω )], and the third layer is a medium of an analyte ( (cid:15) = (cid:15) a ), as shown in Figure 2(a). Ex-amples of the analyte medium are given in Figure 2(b). Sup-pose that light is injected towards the metal interface from theprism region below, with an incident angle θ in . The parallel-to-interface wavenumber of light impinging on the metal filmis given as k (cid:107) ( ω ) = √ (cid:15) p ( ω/ c ) sin θ in and as mentioned, it isalways smaller than the wavenumber of a SPP at the interfacebetween the prism and the metal film for a fixed frequency ω ,i.e., SPPs cannot be excited via direct illumination. How-ever, when the incident angle is greater than the critical an-gle, this leads to total internal reflection and subsequently itcauses the excitation of an evanescent field on the oppositeside of the prism. The parallel-to-interface wavenumber ofthe evanescent field is still the same as the incident one, as itis a conserved quantity because of the translation invarianceof the interface. When the thickness of the metal film is small,the decaying tale of the evanescent field can reach the inter-face between the metal film and the analyte region, where thewavenumber of the SPP is given by k SPP (cid:107) ( ω ) with (cid:15) d = (cid:15) a beingsmaller than (cid:15) p . Consequently, for a SPP at the metal-analyteinterface, the resonant condition k (cid:107) ( ω ) = k SPP (cid:107) ( ω ) can be metfor a specific incidence angle called the resonance angle θ res .This is shown in Figure 2(c) as a crossing point (star) betweenthe dispersion curves. The SPP at the metal-analyte interfaceis thus excited via so-called evanescent field coupling. Fora full resonant excitation of SPPs, the spatial mode of theevanescent field should have a significant overlap with thatof the SPP. Furthermore, a finite beam width of the incominglight and the radiative and damped nature of the SPP modeneed to be considered to include practical aspects involved inthe conversion process [78].When all the coupling conditions are fulfilled, the total in-ternal reflection is attenuated, an e ff ect known as attenuatedtotal internal reflection (ATR). Observing that the reflectancedrops down from near-unity to zero, one can indirectly cer-tify that the incident light is converted into a SPP mode. Tosee this in theory, one can solve Maxwell’s equations for thethree-layer system and obtain the reflection coe ffi cient writtenas [16] r spp = e i k d r + r e i k d r r + , (13)where d is the thickness of the metal film, r uv = (cid:16) k u (cid:15) u − k v (cid:15) v (cid:17) (cid:46) (cid:16) k u (cid:15) u + k v (cid:15) v (cid:17) for u , v ∈ { , , } , k u = √ (cid:15) u ( ω/ c )[1 − ( (cid:15) /(cid:15) u ) sin θ in ] / denotes the normal-to-surface componentof the wave vector in the u th layer and (cid:15) u is the respectivepermittivity. The resonant excitation of SPPs can be identifiedby a dip in the reflectance R spp = | r spp | measured in terms ofthe incident angle θ in , the wavelength λ , or frequency ω [seeinset of Figure 2(a)]. The reflectance dip is highly sensitiveto the permittivity (cid:15) a of an analyte, so analysis of the reflectedlight enables the estimation of various kinds of implicit pa-rameters that characterize the optical properties of an analyte,or its relevant e ff ects. For example, SPR sensors can be usedto characterize kinetic parameters [79–82], such as the equi-librium constant, dissociation constant, and association con-stant. They can also be used for sensing of structural or mate-rial parameters [23, 83, 84], such as the thickness of adsorbedmolecules and their refractive index. Interestingly, these ex-amples can be understood by a single phenomenological andmacroscopic parameter – a refractive index ( n a = √ (cid:15) a ) thatchanges. Therefore, for the sake of simplicity, but withoutloss of generality, we will focus on refractive index sensing.
1. Angular interrogation
When light is converted to a SPP in a prism setup such asthe Kretschmann configuration, the reflectance is minimized,which ensures the resonant excitation of a SPP. By varying theincidence angle θ in for a fixed frequency ω of light in the prismsetup, one can find the reflectance dip at the resonant incidentangle θ res , at which the parallel-to-interface wavenumber ofthe exciting evanescent field is equal to that of the SPP. Math-ematically we have k n p sin θ res = k (cid:115) n (cid:15) (cid:48) m ( ω ) n + (cid:15) (cid:48) m ( ω ) , (14)where n a(p) = √ (cid:15) a(p) denotes the refractive index of the an-alyte (prism). It is clear to see that the resonance angle θ res changes with the refractive index n a of the analyte, i.e., thereflectance curve shifts when the refractive index n a changes.The sensitivity of the resonance angle with respect to changesin n a can be obtained from eq 14 and is written as [22] S θ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d θ res d n a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:15) (cid:48) m ( ω ) (cid:112) − (cid:15) (cid:48) m (cid:2) (cid:15) (cid:48) m ( ω ) + n (cid:3) (cid:113) (cid:15) (cid:48) m ( ω ) (cid:104) n − n (cid:105) − n n . (15)The angular sensitivity S θ is given in degrees per refractiveindex unit (RIU) and monotonically increases with decreas-ing wavelength [see Figure 3(a)]. On the other hand, it di-verges in the short wavelength regime because the sensitiv-ity S θ becomes singular when (cid:15) (cid:48) m ( ω ) = n n / (cid:104) n − n (cid:105) . This indicates that the sensitivity of the Kretschmann configurationusing gold is better than that using silver in the typical rangeof wavelengths of interest in optics [see Figure 3(a)] becausethe plasma frequency of gold is smaller than that of silver.Furthermore, a smaller contrast between n a and n p is helpfulfor increasing the angular sensitivity, so for example, usingBK7-glass is better than using SF14-glass for a given analytewith n a = .
32. As can be seen from Figure 3(a), typical sensi-tivities of a Kretschmann plasmonic sensor using the angularinterrogation method are in the range 10-10 degrees / RIU.
2. Spectral interrogation
The reflectance curve can also be measured as a functionof wavelength (or frequency) of the incident light that is in-jected into the prism setup with a fixed incidence angle. In thisscenario, the reflectance dip is obtained at a resonance wave-length λ res , which satisfies the resonant condition of eq 14,and shifts with the change of the refractive index n a of an ana-lyte. The spectral sensitivity can be obtained in the same wayas above and is written as [22] S λ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d λ res d n a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = [ (cid:15) (cid:48) m ( ω )] n (cid:12)(cid:12)(cid:12)(cid:12) d (cid:15) (cid:48) m ( ω )d λ res (cid:12)(cid:12)(cid:12)(cid:12) + (cid:2) (cid:15) (cid:48) m ( ω ) + n (cid:3) (cid:15) (cid:48) m ( ω ) d n p d λ res n a n p . (16)Like the angular sensitivity S θ , the spectral sensitivity S λ ex-hibits a singularity but in the limit of long wavelengths, i.e.,the spectral sensitivity S λ monotonically increases with thewavelength [see Figure 3(b)]. However, the spectral sensi-tivity S λ increases with | (cid:15) (cid:48) m ( ω ) | , so the Kretschmann config-uration using silver is more sensitive than setups using gold,which is contrary to the angular sensitivity S θ . Here again, us-ing BK7-glass is better than using SF14-glass for a given an-alyte with n a = .
32 since the smaller the contrast between n p and n a is, the more sensitive the Kretschmann configurationsetup is to the change of n a . As can be seen from Figure 3(b),typical sensitivities of a Kretschmann plasmonic sensor us-ing the spectral interrogation method are in the range 10 -10 nm / RIU.
3. Limit of detection
The higher the sensitivity S y of a sensor, the more sensi-tively an explicit observable parameter y changes with respectto the change of an implicit parameter x of an analyte. How-ever, the noise inevitably involved in the measurement limitsthe minimum detectable range ∆ y min of the parameter y beingmeasured, even though the sensor may be highly sensitive. Anoverall figure of merit for sensing quality needs to take intoaccount both the sensitivity S y and the minimum detectablerange ∆ y min , or equivalently the value of the noise level. Thisleads to the definition of a ‘limit of detection’ (LOD) [23, 27],also known as the resolution [35] of the sensor, written asLOD = ∆ y min S y . (17)Current state-of-the-art classical plasmonic sensors canachieve a minimum LOD of ∼ − -10 − RIU, which covers J . Homola et al . / Sensors and Actuators B
54 (1999) 16–2418 considerably with wavelength. If an angular SPR sens-ing device operates far from the cut-off, the sensitivitycan be approximately expressed as: S P ! ! S P ! + = ! ( n p2 − n a2 ) (5)Therefore, the sensitivity at longer wavelengths de-pends mainly on the refractive index contrast betweenthe prism and the analyte and increases with decreasingcontrast. Apparently, the influence of the optical con-stants of the metal supporting SPW is more pro-nounced at short wavelengths near the singularity insensitivity. As in systems which use metals with lower " " mr " the singularity occurs at longer wavelengths, thesensitivity of structures using gold is higher than that ofsystems using silver. Fig. 4. The angular sensitivity versus the wavelength for system:prism coupler (SF14 or BK7 glass)–metal (gold or silver)–analyte( n a = Sensiti ! ity : wa ! elength interrogation For configurations of prism-based SPR sensors inwhich the angle of incidence is fixed and a shift in theresonant wavelength due to variations in the refractiveindex of analyte is to be measured, the spectral sensitiv-ity can be obtained by differentiating Eq. (3) in and n a which yields: S P = d d n a = " mr2 n a3 d " mr d + ( " mr + n a2 ) " mr d n p d n a n p (6) Fig. 5. Sensitivity of the prism coupler-based SPR sensor using thewavelength interrogation as a function of the wavelength for system,prism coupler (SF14 or BK7 glass)–metal (gold or silver)–analyte( n a = J . Homola et al . / Sensors and Actuators B
54 (1999) 16–2418 considerably with wavelength. If an angular SPR sens-ing device operates far from the cut-off, the sensitivitycan be approximately expressed as: S P ! ! S P ! + = ! ( n p2 − n a2 ) (5)Therefore, the sensitivity at longer wavelengths de-pends mainly on the refractive index contrast betweenthe prism and the analyte and increases with decreasingcontrast. Apparently, the influence of the optical con-stants of the metal supporting SPW is more pro-nounced at short wavelengths near the singularity insensitivity. As in systems which use metals with lower " " mr " the singularity occurs at longer wavelengths, thesensitivity of structures using gold is higher than that ofsystems using silver. Fig. 4. The angular sensitivity versus the wavelength for system:prism coupler (SF14 or BK7 glass)–metal (gold or silver)–analyte( n a = Sensiti ! ity : wa ! elength interrogation For configurations of prism-based SPR sensors inwhich the angle of incidence is fixed and a shift in theresonant wavelength due to variations in the refractiveindex of analyte is to be measured, the spectral sensitiv-ity can be obtained by differentiating Eq. (3) in and n a which yields: S P = d d n a = " mr2 n a3 d " mr d + ( " mr + n a2 ) " mr d n p d n a n p (6) Fig. 5. Sensitivity of the prism coupler-based SPR sensor using thewavelength interrogation as a function of the wavelength for system,prism coupler (SF14 or BK7 glass)–metal (gold or silver)–analyte( n a = (a) (b) FIG. 3. The angular sensitivity in (a) and the spectral sensitivity in (b) are investigated as a function of the wavelength injected into the prismcoupler. Example setups consider glass (SF14 or BK7), metal (gold and silver) and analyte ( n a = . a wide range of optical designs, interrogation methods, andoperating wavelengths [35]. Examples include the detectionof nucleic acids identifying specific bacterial pathogens [85]and the monitoring of protein multilayer systems [86]. Al-though we focus on the refractive index change for the sensi-tivity, the LOD may be given in other units better suited to thespecific application, depending on the quantity x being mea-sured. Other commonly used units for the LOD in biochem-ical plasmonic sensors are ng / mL and nM, which correspondto measuring concentrations of substances. For consistencywe will continue to use RIU, as it covers these cases also upto a functional relation. More generally, the above formulain eq 17 clearly shows that a good sensor requires the LODto be reduced via the enhancement of the sensitivity S y anda reduction of the noise ∆ y min . As will be discussed in sec-tion IV, the LOD can be significantly reduced when the quan-tum resources described in section III are exploited togetherwith plasmonic systems. In general, the sensitivity of the plas-monic sensor is not modified in the quantum scenario as it de-pends mainly on the physical setup; it is the reduction of thenoise ∆ y min , for a given integration time and intensity, that isthe crucial feature quantum sensing provides. In the literature,LOD, resolution, and sensitivity have often been interchange-ably used in the sense that a small LOD (or resolution) impliesa large sensitivity for a fixed ∆ y min and vice versa. For a clas-sical plasmonic sensor, the main and inevitable contribution tothe noise stems from ‘shot noise’, as will be discussed in de-tail in section III. On the other hand, for a quantum plasmonicsensor, the noise can be reduced below that of the shot noise.The noise reduction o ff ered by the use of quantum re-sources opens up a route to reducing the LOD below whatis possible classically, which enables the precision of a plas-monic sensor to be improved so that it can detect smallerchanges in the implicit parameter x of an analyte. Such animprovement is beneficial in many applications of plasmonicsensors, for instance the detection of pathogens in small quan-tities in the early stages of a disease [33], or the contamination of food and water by minute amounts of a substance [6]. C. Intensity vs phase sensing
When SPPs are excited at the resonance angle in theKretschmann configuration, the total internal reflection oflight is maximally attenuated. Around the resonance a steepcurve in the intensity of the reflected light is exhibited as theincident angle is varied, as shown in the inset to Figure 2(a).The change of the intensity is often analyzed to infer the re-fractive index of an analyte medium or other relevant prop-erties. In angular interrogation, the shift in the resonanceangle corresponding to the minimum of the reflected inten-sity is measured. Another intensity-based sensing approach,known as ‘intensity modulation’, is to fix the incident angleat the steepest point of the reflection curve (o ff -resonance),i.e., the inflection point, and monitor the change in the re-flected intensity as the entire resonance curve shifts. Simi-lar LODs to angular interrogation can be achieved using thismethod [35]. On the other hand, the phase of the reflectedlight also changes across the resonance curve and abruptly atthe resonance angle, which can be detected in an interfero-metric setup. This indicates that detecting the phase or phasechange of the reflected light can be exploited as an alternativesensing method to intensity sensing. Such behavior has moti-vated the study of various phase-sensitive plasmonic or SPRsensors linked with an interferometric system to measure arelative phase shift [87–89], including multi-pass interferom-eters [90], imaging interferometers [91, 92], Mach-Zehnderinterferometers (MZIs) [93–97], and heterodyne interferome-ters [98, 99].Aside from the issue of noise in the measurement, a nat-ural question arises: Which one is more sensitive, intensitysensing or phase sensing? There have been many studiescomparing these two canonical sensing schemes. A numberof works have shown that the sensitivity of phase sensing FIG. 4. MNP supporting a LSP excitation when illuminated with an external electric field. (a) The MNP supports a LSP which is damped byinternal ohmic loss and external radiation into the far field. (b) An analyte, consisting of a ligand binding to a receptor, changes the backgroundpermittivity of the MNP. This causes a change in the resonance position of the system with respect to the wavelength of the exciting field. schemes is a few orders of magnitude higher than schemesbased on the detection of the intensity change [94, 99–112].Although contradictory observations may be found in the lit-erature, e.g., ref 113, their paradoxical conclusion has beenrebutted by other studies, for instance those mentioned in thededicated discussion by Kabashin et al. [114], who report aKretschmann plasmonic sensor with a LOD of 10 − usingphase sensing with measurement noise included, represent-ing an improvement in the LOD of two orders of magnitudecompared to intensity sensing.Regardless of which one is more sensitive, intensity andphase are the canonical conjugate variables for light. One canalmost always decompose plasmonic sensors, or even moregenerally, photonic sensors into the two types: phase sensorsand intensity sensors (often called amplitude sensors). Eachtype of plasmonic sensor works di ff erently and consequentlyo ff ers distinct functionalities and respective advantages [114].Such a distinction is also used to categorize quantum opti-cal sensors, where useful quantum properties are di ff erent de-pending on the type of sensing being performed, as will bediscussed in section III. Thus, for quantum plasmonic sen-sors, intensity and phase sensing will also be treated sepa-rately in section IV. One may wish to have a combined ver-sion, but again they are complementary variables, so both pa-rameters cannot be precisely estimated simultaneously due tothe Heisenberg principle. This is also discussed in section III. D. Localized surface plasmon resonance sensing
The SPPs discussed above are propagating surface wavesand exist at a planar interface between a dielectric and a metal.On the other hand, non-propagating SPs can also be foundin spatially localized metallic structures with a size compara-ble to or smaller than the wavelength of light. In this case,the conduction electrons oscillate coherently with a frequencythat depends on the size and the shape of the metallic struc-ture, the density and the e ff ective mass of the electrons, andthe medium in which the structure is embedded [115]. Theoscillation of the electron density is called a localized surfaceplasmon (LSP) [116, 117]. Typical examples include metal-lic nanoparticles (MNPs) [118, 119], where the excited LSPswith discrete resonance modes lead to sharp spectral absorp-tion and scattering profiles, but also strong electromagnetic near-field enhancement in the proximity of metallic structures.Since LSPs can be excited in metallic structures with a sizesmaller than the di ff raction limit, they have often been usedfor super-resolution imaging [120–124].As the simplest example, consider a single spherical MNPwith a radius r that is embedded in a medium with a dielectricconstant (cid:15) d , as shown in Figure 4(a). When the MNP is illu-minated by a single-mode electromagnetic field whose wave-length is much longer than the size of the MNP, i.e., λ (cid:29) r ,the electrostatic (or non-retarded) approximation can be ap-plied. This approximation allows the problem to be solvedusing Laplace’s equation ∇ Φ ( r ) = Φ ( r ). This is known as the quasi-staticapproximation as the spatial dependency of the field is gov-erned by the same equation as in the electrostatic case but thefields continue to oscillate in time at the high frequencies cor-responding to the visible part of the spectrum. The solution ofLaplace’s equation inside and outside the MNP reads Φ in ( r , θ, φ ) = ∞ (cid:88) l = l (cid:88) m = − l a lm r l Y ml ( θ, φ ) , ≤ r ≤ r , (18) Φ out ( r , θ, φ ) = ∞ (cid:88) l = l (cid:88) m = − l b lm r − ( l + Y ml ( θ, φ ) , r ≤ r , (19)where θ ( φ ) represents the polar (azimuthal) angle in a spher-ical coordinate system, Y ml ( θ, φ ) is a spherical harmonic func-tion of degree l and order m , and the a lm and b lm are ampli-tudes. The interface conditions of eqs 6 and 8 impose the con-tinuities of ∂ Φ /∂ r for the tangential components and (cid:15) i ∂ Φ /∂ r for the normal components across the MNP’s interface at r = r . They lead to the explicit expression for the resonance fre-quencies of the LSPs, written as (cid:15) (cid:48) m ( ω ) (cid:15) d + l + l = , (20)where l denotes the mode index of angular momentum of theLSP [125]. Neglecting damping for simplicity, i.e., using theDrude permittivity of eq 10 with γ =
0, one can obtain theresonance frequencies of LSPs, written as ω l = ω p (cid:34) l (cid:15) d ( l + + l (cid:35) / . (21)For small spheres, the dipolar excitation of LSPs is the mostdominant, i.e., l =
1, so that eq 20 gives the resonant conditionfor the dipolar LSP excitation as (cid:15) (cid:48) m ( ω ) = − (cid:15) d , leading to theresonant frequency ω = ω p (1 + (cid:15) d ) − / using eq 21, knownas the Fröhlich condition. If the background permittivity (cid:15) d is modified, for instance by a ligand binding to a receptor, asshown in Figure 4(b), then the resonance position shifts by ∆ ω (correspondingly by ∆ λ for wavelength). Thus, the resonanceposition can be used to sense a change in the environment ofthe MNP.Such resonant features of the dipolar LSP excitation canalso be seen in the expression of the scattering and absorptioncross sections written as [126] σ sca ( ω ) = ω (cid:15) V c [ (cid:15) (cid:48) m ( ω ) − (cid:15) d ] + [ (cid:15) (cid:48)(cid:48) m ( ω )] [ (cid:15) (cid:48) m ( ω ) + (cid:15) d ] + [ (cid:15) (cid:48)(cid:48) m ( ω )] , (22) σ abs ( ω ) = ω(cid:15) / Vc (cid:15) (cid:48)(cid:48) m ( ω )[ (cid:15) (cid:48) m ( ω ) + (cid:15) d ] + [ (cid:15) (cid:48)(cid:48) m ( ω )] , (23)where V is the particle volume. Note that LSPs can be reso-nantly excited at the frequencies ω l regardless of the wavevec-tor, i.e., the resonance is independent of the illumination di-rection due to the full spherical symmetry of the MNP [126].This is in contrast to the excitation of SPPs, which are onlyexcited when both the frequency and the wavenumber of theincident light equal those of the SPP for a given structure.For larger spheres or in situations where the spheres are ex-cited by some emitter in close proximity, higher-order modeswith l ≥ ff erentmaterials, or non-spherical / non-ellipsoidal MNPs, can be con-sidered, but one should use numerical electromagnetic meth-ods such as finite-di ff erence time-domain and finite-elementmethods to obtain approximate solutions, since an analyticalsolution for arbitrary structures cannot be found [129–132].Complex structures are known to exhibit intriguing featuresand the potential for interesting optical applications in plas-monic sensing [133–140]. In section IV A 2 we will discussquantum plasmonic sensing using LSPs at various types ofnanostructures.Interestingly, in the limit of a very large MNP, i.e., l → ∞ ,with the electrostatic approximation ( r (cid:28) λ ) holding, the dis-persion relation of eq 20 and the resonant frequency of eq 21lead to the relations (cid:15) (cid:48) m ( ω ) = − (cid:15) d and ω l →∞ = ω p (1 + (cid:15) d ) − / ,respectively. These are equal to those for SPs at the interfacebetween a dielectric and a metal. Such a consideration clearly FIG. 5. (Top) The scattered light, i.e., the intensity I , from metallicnanostructures shifts with the refractive index change ∆ n a , conse-quently causing a shift of the resonance wavelength ∆ λ res . A minutechange of the resonance wavelength can only be resolved by a sensoro ff ering a high sensitivity of S λ, LSP . The FOM is defined as the ratioof S λ, LSP to the full width at half maximum Γ λ , when the latter can bewell defined. When this is not the case, an alternative version needsto be used, the FOM ∗ , which is defined as the ratio of the relativeintensity change | d I / I | (Bottom) to the refractive index change d n a of the analyte. identifies the inherently di ff erent nature of LSPs compared toSPPs. SPPs are a hybrid mode comprised of an electromag-netic field and electron oscillations, while LSPs are the exci-tation of electron oscillations for a given illumination by anelectromagnetic field. In other words, SPP modes are boundsolutions to the wave equation and once excited they existwithout needing reference to the field that caused them, whileLSP modes are responsive modes that are highly damped andcan couple back into the far field, therefore they are inherentlytransient for a certain timescale when an external driving fieldis illuminating the MNP.
1. Sensitivity and figure of merit
The features of LSPs described above can be exploited forsensing. In particular, the spectral sensitivity can be definedas S λ, LSP = | d λ res / d n a | for refractive index sensing. Sensi-tivities of LSPs at single MNP of various shapes are on theorder of 10 -10 nm / RIU [141]. However, due to the sharpresonance peaks of LSPs [142], most LSP sensors employa particular figure of merit, FOM, to quantify the ability ofthe sensor to resolve small refractive index changes, definedas [143] FOM = S λ, LSP Γ λ , (24)where Γ λ is the resonance linewidth (or full width at half max-imum). The FOM is mainly used to compare the sensing po-tential of various sensing schemes. From eq 24, it can be seenthat a good LSP sensor requires the FOM to be enhanced byreducing the linewidth and increasing the sensitivity.When the resonance spectrum cannot be modeled by a sim-ple Lorentzian shape, e.g., in complex plasmonic structures0 FIG. 6. MNP enhancement of fluorescence. (a) Fluorophores are excited by an external field and emit into the far field. (b) Fluorophores closeto the surface of a MNP have their emission quenched by coupling to LSPs whose higher-order modes are strongly damped. (c) Fluorophoresat a distance from the surface of a MNP couple their emission mainly into the dipolar LSP mode which is not as strongly damped and thefluorescence is therefore enhanced. such as metamaterials [144], the linewidth Γ λ is ill definedand an alternative figure of merit needs to be defined. In thiscase, one can use the FOM ∗ defined as [145]FOM ∗ = max λ (cid:12)(cid:12)(cid:12)(cid:12) d I d n a (cid:12)(cid:12)(cid:12)(cid:12) I = max λ S λ, LSP × (cid:12)(cid:12)(cid:12) d I d λ (cid:12)(cid:12)(cid:12) I . (25)The FOM ∗ accommodates the relative intensitychange | d I / I | = | ( I ( λ + ∆ λ ) − I ( λ )) / I ( λ ) | with respectto the change of the refractive index d n a for the optimalwavelength λ that maximizes the quantity | d I / d n a | / I , asillustrated in Figure 5.LSP sensors are an interesting approach for measuring re-fractive index changes, as they provide an opportunity to di-rectly measure molecular binding events [11]. However, theFOM for LSP sensors using spherical MNPs is lower than thatof SPP sensors by about an order of magnitude [141], with acorresponding di ff erence in the LOD. For a comparable FOM(or FOM ∗ ), LSPs at non-spherical MNPs or as unit cells ofmetamaterials need to be considered [146].
2. Plasmon-enhanced fluorescence and Raman scattering
For LSPs, in addition to the conventional colorimetric de-tection scheme, i.e., measuring the resonance shift due tothe change of the refractive index, other sensing principleshave widely been studied and used. Two representatives areplasmon-enhanced fluorescence (PEF) sensing / imaging [147,148] and surface enhanced Raman scattering (SERS) [149].Here, the plasmon merely enhances the signal of the Ramanor fluorescence signal, but is itself not used for sensor trans-duction. Despite this, the enhancement of the signal by theplasmon can improve the signal-to-noise ratio (SNR) whichis basically proportional to the signal intensity, and thus theLOD. Moreover, the sub-di ff raction scale of the supportingmetal nanostructure provides increased spatial resolution thatcan be used for imaging. A variety of biochemical sensingapplications based on PEF and SERS can be found in relevantreview papers [28, 34, 148, 150]. This section explores thestate-of-the-art for classical PEF and SERS sensing modali-ties that are now being re-envisioned as quantum plasmonicsensors, as will be discussed in more detail in section IV C. Although fluorescence signals enable low-background de-tection by filtering the excitation beam, the fluorescence ofnanometer sized fluorophores is ine ffi cient due to the weaklight-matter interactions that originate from a large mismatchbetween the physical size of the fluorophores and the wave-length of visible light, as shown in Figure 6(a). This canlead to photobleaching when a high excitation power is usedto extract a measurable signal. In practice, the fluores-cence of fluorophores can either be enhanced or quencheddepending on the absorption and scattering characteristics ofa MNP. A largely enhanced electric field can be formed atmetallic nanostructures when the incident wavelength of lightmatches with their LSP resonance. This enhanced electricfield can excite fluorophores placed within the evanescent de-cay depth of the LSP. This can be e ffi cient when the absorp-tion spectrum of the fluorophore matches with the LSP res-onance. The enhancement in the excitation rate of a fluo-rophore with an absorption dipole moment d is given by κ = | d · E LSP | / | d · E inc | , where E LSP and E inc are the electricfield at the fluorophore position with and without the plas-monic structure, respectively. [151]In addition to the enhancement of the excitation of the flu-orophore, LSPs can also modify the fluorophore emission dy-namics. The typical fluorescence lifetime of an isolated andexcited fluorophore is on the order of a few ns up to tens ofns. This has to be compared to the almost instantaneous de-cay of an excited LSP mode in metal nanostructures whenthe external driving field is switched o ff [152, 153]. Whena fluorophore is coupled to a metallic nanostructure, the ex-cited state of the fluorophore decays faster while transferringthe excited energy to the metallic nanostructure. This e ff ecthappens if the emission wavelength of the fluorophore is res-onant with the LSP modes. When the fluorophore is lessthan ∼ FIG. 7. (a) Schematic of an experimental setup used to enhancethe fluorescence of a single terrylene molecule using a gold nanoparticle (AuNP). Terrylene molecules are embedded in a thin crys-talline p-terphenyl film (20 nm) and are illuminated under total inter-nal reflection. The same objective collects fluorescence of individ-ual molecules while a AuNP attached to a glass fiber tip is scannedacross. (b) Near-field fluorescence image using a AuNP (80 nm indiameter) attached tip. (c) Cross-sectional profile of the white dashedline in (b). The inset shows a SEM image of the AuNP attached tothe tapered-fiber tip. (d) Fluorescence decay lifetime of a terrylenemolecule with (black) and without (red) employing a AuNP (80 nmin diameter). The inset shows a plasmon spectrum of a AuNP (black),the fluorescence spectrum of a single terrylene molecule (red), andthe excitation laser line (green). Adapted and reprinted from ref 122with permission from American Chemical Society and from ref 155with permission from Optical Society of America. moderate-distance case can be understood by the plasmoni-cally enhanced local density of states, which is proportionalto | E LSP | . The dipolar mode of the LSP can radiate into thefar-field ( γ r , LSP ), but it also su ff ers from nonradiative Ohmiclosses ( γ nr , LSP ). In general, γ r , LSP and γ nr , LSP are much largerthan the radiative decay rate ( γ r ) and nonradiative decay rate( γ nr ) of the bare fluorophore, without the plasmonic struc-ture. Therefore, when the excited state of the fluorophoredecays mostly into LSP modes, i.e., the fluorescence decaylifetime reduction is significant, the quantum e ffi ciency (QE)of the fluorophore coupled to the LSP can be obtained ina simplified form of η coupled ∼ γ r , LSP / ( γ r , LSP + γ nr , LSP ). Ifthe intrinsic QE of the fluorophore [ η = γ r / ( γ r + γ nr )] ishigh, the plasmon will reduce the QE of the hybrid system.However, when the original fluorophore QE is very low ( ∼ afew percent), the QE can be highly enhanced by coupling toLSPs. This enhancement in the emission process, when com-bined with the excitation enhancement, leads to a huge fluo-rescence enhancement, over a thousand times, becoming pos-sible [156]. The fluorescence enhancement, well below thesaturation limit of the excited population of the fluorophore, isgiven by κη coupled /η . It has been shown that the fluorescenceof a vertically aligned terrylene single molecule is enhancedby 30 times when coupled to a spherical gold nanoparticle with κ ∼ η coupled ∼ .
3, and η ∼ . V ∼ r [see eq 23], while the scattering isproportional to V ∼ r [see eq 22], where r is the diame-ter of the MNP [162]. As a result, for a smaller MNP, theradiating power decreases faster than the absorption, thus re-ducing the QE. Therefore, small nanostructures ( <
20 nm)are usually used to quench fluorescence [163]. Larger parti-cles can act as e ffi cient scattering centers. In this case, theexcited energy of the fluorophore can be radiated into the farfield more e ffi ciently with the help of LSPs that act as an an-tenna at optical frequencies [164, 165]. When the diameterof a spherical MNP becomes too large ( >
100 nm), the dipo-lar field approximation does not hold anymore, and higher-order modes start to contribute significantly, reducing the QEand the radiation e ffi ciency [166–168]. In addition, the LSPmode volume becomes larger for a larger MNP, reducing theexcitation field enhancement. Thus, in PEF-based imaging,the spatial resolution is better when a smaller nanostructureis used [122]. PEF can be more e ffi cient for other shapesof metallic nanostructures, including rods [169], stars [170],dimers [171], and arrays [172]. Sharp structures can allowa higher field enhancement compared to single spheres, andnanogaps between MNPs can support even higher field en-hancement together with a high QE [67, 173]. The enhanceddetection sensitivity by PEF has been demonstrated for detect-ing many di ff erent bio-samples [28, 34, 148, 150]. For exam-ple, silver-nanoparticle-assisted PEF has been applied for thedetection of streptavidin and human IgE, reaching a LOD of0.25 ng / mL [174]. The hot-spots formed in a gold nanorod ar-ray have been employed for detecting single-strand DNA witha LOD of 10 pM [175]. Although the modification of the fluo-rescence signal is significant in the given scenarios in this sec-tion, they can still be described in the weak-coupling regime.In the strong-coupling regime between the emitter and plas-monic modes of metallic systems, however, other exotic func-tionalities of quantum plasmonic sensing schemes can be uti-lized, as will be described in section IV C.The LSP-induced local field enhancement can also con-tribute to SERS. The overall enhancement in the SERS sig-nal is attributed to the combined e ff ects of the enhanced fieldintensity as in PEF and the chemical enhancement due to thecharge transfer between the metallic nanostructure and the tar-get molecule [176, 177]. Here, the local field enhancementis frequently considered as the dominating factor for the en-hanced SERS. In a back-of-the-envelope estimation, the en-hancement of the SERS signal, thanks to the supporting plas-monic structure, is proportional to the fourth power of theelectric field at the spatial location where the molecules areplaced. Therefore, a major aim is frequently to tune the geo-metrical properties of the plasmonic structure such that a plas-mon resonance is supported at the frequency used in the ex-periments. Also, the field enhancement has to occur in thespatial region where the molecules are placed.2 FIG. 8. (a) Seminal spatially resolved SPR imaging of dimyristoylphosphatidic acid monolayer utilizing a Kretschmann readout scheme ata fixed angle of incidence of 47.2 ◦ . Reprinted from ref 159 with permission from Springer Nature. (b) An early demonstration of spatiallyresolved biomolecular sensing with Kretschmann-based SPR imaging schemes demonstrating that only appropriately functionalized gold padsgive a response to single strand DNA. Reprinted from ref 160 with permission from American Chemical Society. (c) a schematic representationof SPR imaging with a smartphone camera. Reprinted from ref 161 with permission from Springer Nature. E. Surface plasmon resonance imaging
Improving the sensitivity of plasmonic sensors can enablethe detection of smaller concentrations of materials of inter-est in the same integration time, or it can enable the detectionof the same concentration of materials in a shorter integrationtime. However, when a sensor needs to detect many di ff erentmolecules, most plasmonic sensors are operated in an itera-tive fashion, resulting in slow measurements. SPR imagingattempts to reduce measurement times by parallelizing SPRsensors on a single chip. Whereas conventional LSP and SPPsensors utilize angle-resolved or spectrally-resolved measure-ments, SPR imaging systems utilize intensity-resolved mea-surements: they sit at the inflection point of a plasmonic reso-nance and monitor spatially-resolved changes in intensity foran array of plasmonic sensors, or for a continuous plasmonicsensor. This concept was first introduced in 1988 as SP mi-croscopy [178]. In general, the ability to spatially map varia-tions in the index of refraction with SPR imaging is a powerfultool for fundamental materials science, chemistry and biol-ogy. When arrays of sensors are functionalized to detect dif-ferent molecules, SPR imaging can enable substantial scalingof SPR sensors for applied sensing of enzyme-substrate in-teractions, DNA hybridization, antibody-antigen binding, andprotein interaction dynamics [160, 179–183].Figure 8(a) illustrates early SPR imaging of monolay-ers of dimyristoylphosphatidic acid with a spatially resolvedKretschmann sensor [159]. This work, along with otherearly research[178], illustrated the potential of SPR imagingfor sub-di ff raction-limited, high sensitivity imaging of low-contrast samples. Since then, substantial technical develop-ment has gone into the improvement of SPR imaging systemsfor parallelized detection of di ff erent molecular systems. Asshown in Figure 8(b), SPR imaging of gold pads functional-ized with appropriate probes allows for the spatially selectivedetection of target DNA that is complementary to the func-tionalization group [160]. By functionalizing each elementof the plasmonic sensor array di ff erently, it is therefore pos-sible to achieve highly beneficial scaling for the detection oflarge numbers of di ff erent molecules. Figure 8(c) illustratesthe degree to which SPR imaging platforms can be deployedin the field through integration with ubiquitous imaging tech- nologies like smartphone cameras [161]. However, a commonchallenge in SPR imaging is that o ff -the-shelf imaging sys-tems introduce substantial noise compared with single pixeldetectors. While it is possible to reach the fundamental classi-cal limit given by the shot noise with appropriate experimentaldesigns, further improvements could be made possible withquantum imaging schemes, as will be discussed in section IV. III. QUANTUM SENSORS
Photonic devices, such as the plasmonic sensors introducedin the previous section, exploit light as a probe for their op-eration. While plasmonic sensors o ff er an improved sensitiv-ity compared to other types of photonic devices, they share acommon problem – random fluctuations in the measured sig-nal due to the statistical nature of light. The origin of thesefluctuations can be derived using a classical theory to someextent by considering light to be made from discrete parti-cles [184]. However, once the wave properties of light areincluded, the fluctuations can only be understood from a morefundamental quantum theory [185]. When a coherent state oflight – thought of as the quantum state that most closely de-scribes the light from a laser [186] – is employed in a sensor,the noise in the signal arises from the fact that the coherentstate consists of photon number states with a weighting thatfollows a Poisson distribution. This noise will obviously af-fect the LOD for a given plasmonic sensor, as mentioned insection II B 3, limiting its resolution. However, if the noisecan be reduced, then the sensor can provide a better resolution,enabling more precise measurements to be made [187]. Re-ducing this noise involves the use of quantum noise reduction,or squeezing. To understand how the noise, commonly knownas ‘shot noise’, is present in plasmonic sensors and to see howit can be reduced by using specialized quantum techniques,we kick o ff this section by o ff ering a brief overview of param-eter estimation theory. This is a theory that has been widelyconsidered in the field of quantum metrology [188–190], andit is vital for determining fundamental bounds on how well pa-rameters can be estimated in both the classical and quantumregime. After an introduction to parameter estimation the-ory, we look at a few paradigmatic examples of classical op-tical sensors and corresponding quantum sensors, while leav-3ing more extensive details to other review articles dedicated toquantum metrology and imaging [37–40, 43, 44, 46, 191]. A. Parameter Estimation Theory: Cramér-Rao bound
1. Single-parameter estimation
The complete process for the estimation of a parameter insensing can be divided into four key steps: (i) Input statepreparation; (ii) Interaction for parameter encoding; (iii) Mea-surement; and (iv) Estimation based on the measurement out-comes, as depicted in Figure 9. In the final step, the esti-mation of a parameter is made over multiple repetitions ofa measurement or time-integration. The number of repeatedmeasurements determines the finite size ν of a sample. Eachsample with a size ν is used by an estimator ˆ x to yield anestimate x est of the parameter, and the quality of the estima-tor ˆ x can be assessed by several statistical features. The ex-pectation value of the estimate, (cid:104) x est (cid:105) , is compared with theparameter’s true value x from the underlying population be-ing sampled. Their di ff erence indicates a bias of the estima-tor and can be interpreted as an estimation accuracy. Whenthe di ff erence is zero, i.e., (cid:104) x est (cid:105) − x =
0, the estimator issaid to be unbiased and equivalently the accuracy is perfect.Here, (cid:104) .. (cid:105) denotes the average over all possible configurationsof the sample, which strictly speaking can only be theoreti-cally considered, but it may be well approximated by an ac-tual repetition of sampling in an experiment, i.e., collectinga large number of samples. Another quantity of great impor-tance is the mean-squared-error (MSE) defined as MSE[ ˆ x ] = (cid:104) ( x est − x ) (cid:105) = (cid:104) ( x est − (cid:104) x est (cid:105) ) (cid:105) + ( (cid:104) x est (cid:105) − x ) , where the firstterm is the variance ∆ x of the estimator and the second termis the squared bias of the estimator. The MSE and the varianceare equivalent when an estimator is unbiased. The variance isoften called an estimation uncertainty or interpreted as an es-timation precision. The latter implies that the estimate x est would vary over the repetition of an identical and independentsampling, or equivalently, over di ff erent configurations of thesample with a size ν . In this review, we take the square root ofthe variance, ∆ x est , as the estimation uncertainty or estimationprecision, as is often done in experiments. In the literature, ∆ x est is sometimes called sensitivity / resolution with the inter-pretation that a large ∆ x est limits the capability of a sensor tosense / resolve a minute change of an observable quantity. Ina qualitative sense, these terms can be used interchangeablyaccording to their close relation, but in a quantitative sense, ∆ x est is the uncertainty of an estimator according to parame-ter estimation theory, which we introduce in this section. TheLOD is then simply given by ∆ x est divided by the sensitivity. FIG. 9. The process for the estimation of a parameter in sensing isdivided into four key steps: Preparation, Interaction, Measurementand Estimation.
Let us consider a simple example of the above definitions,where one aims to estimate the parameter given by the popu-lation mean, whose true value is µ , by using the sample meanas an estimator, i.e., ˆ x = ¯ x = (cid:80) ν j = x j /ν . The value obtainedby a sample with a size ν is then x est and because the samplemean is an unbiased estimator for any population [192], wehave (cid:104) x est (cid:105) = µ . However, x est for each sample will vary andthe variance of the sample mean is given by ∆ x = σ x /ν ,where σ x is the population variance, which holds for anysize ν without requiring the approximation imposed by thecentral limit theorem [190]. It is clear that the standard devi-ation ∆ x est is inversely proportional to √ ν when the samplemean is taken to estimate the population mean, i.e., the esti-mation becomes more precise as the sample size ν increases.Besides the statistical scaling with ν , the population variance, σ x , also plays an important role in reducing the standard de-viation ∆ x est , which is related to the role of the quantum re-source in quantum metrology, as we will see in this section.Now consider a more general estimation scenario where anunbiased estimator ˆ x is used to estimate a true value x . Inthis case, the MSE is simply equal to the variance ∆ x , andit is known that a lower bound to the standard deviation ∆ x est exists and is given by ∆ x est ≥ √ ν F ( x ) , (26)which is called the Cramér-Rao (CR) inequality [188, 192].Here F ( x ) denotes the Fisher information (FI) defined as [193] F ( x ) = (cid:90) dy p ( y | x ) (cid:32) ∂ p ( y | x ) ∂ x (cid:33) , (27)with p ( y | x ) being an underlying conditional probability den-sity of obtaining the measurement outcome within the interval y and y + dy when the true value is x (for a finite distributionwe have (cid:82) dy → (cid:80) y ). The FI represents a sort of measureof the amount of information that a measurement outcome y carries on average about the true value x . A higher value ofthe FI is obviously more advantageous for obtaining a betterestimation precision. The lower bound of eq 26, called the CRbound, is asymptotically saturable in the limit ν → ∞ whenthe maximum-likelihood estimator is employed [192, 193].Importantly, the FI of eq 27 depends on the physical sce-nario, i.e., the probe state, the parameter encoding, and themeasurement for a fixed true value x . This implies that the FImay be increased further by changing the physical scenario,for instance, by going from a classical scenario to a quantumscenario. Thus, a further lower bound to eq 26 can exist andbe achieved when a more optimal scenario is chosen. The CRinequality can be developed to [194, 195] ∆ x est ≥ √ ν F ( x ) ≥ √ ν H , (28)where H represents the quantum Fisher information (QFI), de-fined as H = max { ˆ Π y } F ( x ) . (29)The QFI is essentially the maximized FI over all possi-ble quantum measurements, or put more formally, over4all positive-operator valued measures (POVMs) { ˆ Π y } , suchthat ˆ Π y ≥ (cid:82) dy ˆ Π y = (completeness). Here,the y correspond to the possible outcomes from the quantummeasurement, similar to the outcomes y in the classical casein eq 27, with p ( y | x ) = Tr( ˆ ρ x ˆ Π y ) and ˆ ρ x as the parameter en-coded state. When the parameter encoding is caused by aunitary process, the QFI is independent of the value x (seeexamples discussed in section III D), since the same type ofunitary process could be implicitly included as part of the op-timal measurement [196], such that the value x can be tunedto a certain value for which F is fully maximized to be H .This is not the case when the parameter is encoded through anon-unitary process, i.e., the QFI is given as a function of x (see examples discussed in section III C). In either case, theQFI still depends on the probe state and the encoding processof the parameter, implying that the QFI can be maximizedby using an optimal probe state for a given encoding process.The lowest bound given by the maximized QFI over all inputstates is often called the ultimate quantum limit. In this sense,a number of studies have investigated the optimal probe statesin various sensing or estimation scenarios [38, 39, 41, 43].The lower bound of eq 28 is called the quantum Cramér-Rao (QCR) bound and it is regarded as the fundamental limitin the estimation precision for a given input state and encod-ing process. It can be shown that for a probe state with theparameter x encoded, given by the state ˆ ρ x , the optimal mea-surement with the POVM { ˆ Π y } that reaches the QCR boundhas to satisfy the following two conditions [38, 194]:Im[Tr( ˆ ρ x ˆ Π y ˆ L x )] = , (30)ˆ Π / y ˆ ρ / x Tr( ˆ ρ x ˆ Π y ) = ˆ Π / y ˆ L y ˆ ρ / x Tr( ˆ ρ x ˆ Π y ˆ L y ) , (31)where ˆ L x is the symplectic logarithmic derivative (SLD) op-erator defined such that ∂ ˆ ρ x ∂ x = (cid:16) ˆ ρ x ˆ L x + ˆ L x ˆ ρ x (cid:17) . (32)The above two conditions are rather cryptic, however they canbe fulfilled if a measurement setup with { ˆ Π y } is constructedby a set of projection operators over the eigenbasis of the SLDoperator ˆ L x [38, 194]. This is a necessary and su ffi cient con-dition to reach the QCR bound for a full-rank state of ˆ ρ x , forwhich the SLD operator is unique. On the other hand, fora rank-deficient state the optimal measurement satisfying theabove two conditions is not unique [194].The QFI, H , of eq 29 can be written in terms of the SLDoperator by H = Tr( ˆ ρ x ˆ L x ) . (33)For a parameter-encoded state with a given spectral decompo-sition, i.e., ˆ ρ x = (cid:80) n p n | ψ n (cid:105) (cid:104) ψ n | with (cid:104) ψ n | ψ m (cid:105) = δ n , m , the SLDoperator can be written as [38, 194, 197]ˆ L x = (cid:88) n , m (cid:104) ψ m | ∂ x ˆ ρ x | ψ n (cid:105) p n + p m | ψ m (cid:105) (cid:104) ψ n | , (34)where the summation is taken over n , m for which p n + p m (cid:44) ρ x is pure, i.e., ˆ ρ x = | ψ x (cid:105) (cid:104) ψ x | for some | ψ x (cid:105) , the SLD operator is given by ˆ L x = ∂ x ˆ ρ x , which simplifieseq 33 to H = (cid:104) (cid:104) ∂ x ψ x | ∂ x ψ x (cid:105) + (cid:104) ∂ x ψ x | ψ x (cid:105) (cid:105) , (35)where | ∂ x ψ x (cid:105) ≡ ∂ x | ψ x (cid:105) . When the parameter is encodedthrough a unitary process, i.e., ˆ ρ x = e ix ˆ G ˆ ρ e − ix ˆ G , where ˆ ρ is the initial probe state and ˆ G is a generator of the param-eter x , one can show that H = (cid:104) ( ∆ ˆ G ) (cid:105) , where (cid:104) ( ∆ ˆ G ) (cid:105) = (cid:104) ˆ G (cid:105) − (cid:104) ˆ G (cid:105) is the variance of the generator with respect tothe probe state that undergoes parameter encoding.The evaluation of the QFI determines the lowest estima-tion uncertainty ∆ x est (or the highest precision) for a givenparameter encoding and probe state. As we shall show in sec-tion III B, when a random feature obeying the Poisson dis-tribution with a mean of N dominates the estimation uncer-tainty, the lower bound is called the SNL, where ∆ x est scaleswith N − / .In interferometric sensing, the minimum estimation uncer-tainty achievable by a coherent probe state (the quantum statethat represents light from a laser) is called the standard quan-tum limit (SQL), where N represents the mean photon numberof the state. We will show in section III B that ∆ x est scaleswith N − / [39, 41, 43]. The term SQL was introduced byCaves in his early papers in the sense that it is the limit of stan-dard interferometers made of standard devices without quan-tum squeezing [187, 198, 199]. In the literature, the termsSNL and SQL have mostly been used in intensity sensing andphase sensing, respectively. However, they can sometimes beconsidered as synonymous because of the same fundamentalprocess that causes them.On the other hand, when an optimal quantum probe state isused that maximizes the QFI for a given parameter encoding,we will show in section III D that the estimation uncertaintyis reduced so that it scales with N − [39, 41, 43]. The associ-ated minimum is called the Heisenberg limit (HL) or ultimatequantum limit, and scaling of N − is often called Heisenbergscaling .An important note is due about the above mentioned scal-ings and their comparison. The size of the sample is assumedto be fixed in both the classical and quantum cases, i.e., ν corresponds to a constant number of repetitions, or probes,and therefore the integration time of the measurement is fixed.One could consider increasing ν for the classical case in orderto achieve a similar scaling in precision to the quantum case.However, this may not be practical due to time constraints ofthe system being sensed, which is especially relevant for a dy-namic biological or biochemical system.Furthermore, the scaling of the precision for the SNL is N − / using a classical probe state, but for a quantum probestate, for example in the interferometric setting, the precisionis the HL with scaling N − . This means that one could obtainthe same precision of a quantum probe state with mean photonnumber N using a classical probe state with the mean photonnumber N increased to N . However, this second approachto leveling the classical and quantum scaling may also not befeasible in a given sensor. First, the sensor may be at its phys-ical limit in terms of the power being used to achieve a highprecision, due to the type of materials the sensor is made fromand its construction. Any higher power may cause structural5changes and distort the response of the sensor, introducingadditional sources of noise proportional to the intensity [35].Second, and perhaps more importantly from the context ofnon-invasive sensing, is that the analyte itself may have a dam-age threshold for the power it can tolerate. This is particularlythe case when monitoring small quantities of biological sys-tems [42], for which plasmonic sensors are regularly used anda high precision is required with a limited optical power.The above considerations apply to any case where there is agap between the scaling of the precision for the classical SNLand the quantum case, which may or may not have Heisen-berg scaling. Thus, there is a clear benefit to using a quantumapproach to improve the precision of a plasmonic sensor.
2. Multiparameter estimation
The formulation introduced above applies when a single pa-rameter is estimated, but estimation of multiple parameters areoften of interest in diverse areas of science and technologysuch as phase-contrast imaging [200] and gravitational-waveastronomy [201]. An extended theory for multiparameter es-timation is thus required [202, 203]. Consider the problem ofestimating a set of multiple parameters x = ( x , x , · · · , x M ) T from the measurement results y that have been drawn from aconditional probability density p ( y | x ). The M × M covari-ance matrix Cov( x est ) = (cid:104) ( x est − (cid:104) x est (cid:105) )( x est − (cid:104) x est (cid:105) ) T (cid:105) of anyunbiased estimator ˆ x is found to be bounded by the so-calledFisher information matrix (FIM) [38, 189], written asCov( x est ) ≥ F − ( x ) ν , (36)where the FIM, F ( x ), is defined by[ F ( x )] jk = (cid:90) d y p ( y | x ) ∂ p ( y | x ) ∂ x j ∂ p ( y | x ) ∂ x k . (37)The multiparameter CR inequality of eq 36 is satisfiedwhen Cov( x est ) − F − ( x ) /ν is a positive semi-definite ma-trix [38, 189, 204]. The CR bound can always be saturated bya maximum likelihood method in the limit of large ν [205].As in single parameter estimation, the CR inequality of eq 36for multiparameter estimation can be further reduced in thequantum regime, leading to the multiparameter QCR inequal-ity written as [38, 189]Cov( x est ) ≥ F − ( x ) ν ≥ H − ν , (38)where the details of the quantum Fisher information matrix(QFIM), H , are given in Appendix A. The inequality in eq 38means that in the matrix sense n T Cov( x est ) n ≥ n T F − ( x ) n ν ≥ n T H − n ν , (39)for arbitrary M -dimensional real vectors n [206]. This can beexploited in the case when a global parameter ˜ x = (cid:80) j n j x j ,defined as a linear combination of multiple parameters is ofinterest. We then have ˜ x est = n · x est and ( ∆ ˜ x est ) = Var( ˜ x est ) = n T Cov( x est ) n , which is of interest in distributed sensing and will be discussed further in section III D 4. Examples in-clude relative phase estimation in a two-mode interferome-ter, where n = (1 , −
1) [207], or the average phase estimationin an M -mode interferometer, where n = (1 , · · · , / M [208,209].In the next sections, we use the introduced framework ofthe CR and QCR bound for single and multi-parameter es-timation to give some basic examples of the SNL and SQLin optical probing schemes, distinguishing between intensityand phase sensing. The individual types of quantum sensorwe discuss here will be connected to the corresponding typesof plasmonic sensor already discussed in section II, eventuallyleading to the advances that will be discussed in section IV. B. Shot-noise limited sensing
To understand where and how the shot noise appears, we fo-cus on the two most canonical types of photonic sensing [210–212]: Intensity sensing and phase sensing, as discussed forplasmonic sensors in section II C. They are distinguished bythe physical quantity that is being monitored and analyzed toidentify the optical properties of an analyte. The goal of in-tensity sensing is to measure a change of the intensity of lightwhen it passes through an analyte, whose properties under in-spection are encoded in the change of intensity. On the otherhand, the goal of phase sensing is to read out a change of thephase of the outgoing light from an analyte. These two kindsof sensing schemes lead to di ff erent sensitivities in plasmonicsensors, as described in section II C. Here, we focus on howthe estimation precision is determined according to the sens-ing type.
1. Shot-noise limited intensity sensing
Intensity sensing can be modeled as the estimation of thetransmittance T of a beam splitter (BS), as shown in Fig-ure 10(a). A typical scheme to estimate the transmittance ofthe BS is to inject light with an intensity I i and then to mea-sure the intensity I t of the transmitted light. The ratio of theincident intensity to the transmitted intensity can be used toestimate the transmittance, e.g., the estimated transmittancecan be obtained by T est = ¯ I t / I i , where ¯ I t = (cid:80) ν j = I t , j /ν for ν repetitions of the measurement, with each measurement yield-ing I t , j . To take into account the e ff ect of losses that furtherdecreases the measured value I t , j on top of the transmission,one needs to replace the divisor I i by η I i , with the transmis-sion e ffi ciency η ∈ [0 , T est = ¯ I t /η I i . The estimationuncertainty is thus given by ∆ T est = ∆ ¯ I t η I i = ∆ I t √ νη I i , (40)where ∆ I t denotes the standard deviation of the transmitted in-tensity based on the underlying distribution [190], which de-pends on the type of light being injected.As mentioned, the classical source of light that has mostwidely been considered and employed in standard photonicsensing is modeled quantum mechanically by the coherent6 FIG. 10. (a) Single-mode intensity sensing, where the transmittanceof a BS models a transmissive object and is estimated by the ratio ofthe intensity of the transmitted light to that of the incident light. (b)Two-mode phase sensing, where a relative phase between two armsis estimated from a measurement analyzing the interference at theoutput ports. state [185]. It is also used for setting the classical bench-mark or SQL [37, 39]. The coherent state is formally givenas the displaced vacuum state | α (cid:105) = ˆ D ( α ) | (cid:105) , where ˆ D ( α ) = exp[ α ˆ a † − α ∗ ˆ a ], with a displacement parameter α ∈ C andthe operators ˆ a and ˆ a † representing the annihilation and cre-ation operators for the optical mode [185]. The displaced vac-uum state can be projected into the Hilbert space spanned bythe Fock states and is written as a superposition of photonnumber states, | n (cid:105) , weighted by a Poisson distribution, p ( n ).When the coherent state is used as an input state in Fig-ure 10(a), i.e., | Ψ (cid:105) in = | α (cid:105) , and it passes through the BSwith transmittance T , whose value is unknown and is to beestimated, the outgoing state remains as a coherent state andis written as | Ψ ( T ) (cid:105) out = | √ T ηα (cid:105) , with η characterizingthe e ff ect of loss [185]. An intensity measurement is de-scribed quantum mechanically by the projectors onto the pho-ton number state, i.e., | n (cid:105) (cid:104) n | , where we omit any geomet-ric factors associated with photonic modes and detection forsimplicity. The underlying conditional probability of findingthe output state | Ψ ( T ) (cid:105) out in the state | n (cid:105) is then p ( n | T ) = out (cid:104) Ψ | n (cid:105)(cid:104) n | Ψ (cid:105) out = e − T η N ( T η N ) n / n !, where N is the averagephoton number of the input probe state, i.e., N = (cid:104) ˆ n (cid:105) in = I i = (cid:104) α | ˆ n | α (cid:105) = | α | = (cid:80) n np ( n ), with ˆ n = ˆ a † ˆ a = (cid:80) ∞ i = n | n (cid:105) (cid:104) n | .This leads to the standard deviation of the transmitted inten-sity ∆ I t = (cid:104) ( ∆ ˆ n ) (cid:105) / = ( (cid:104) ˆ n (cid:105) out − (cid:104) ˆ n (cid:105) ) / = √ T η N , finallygiving the estimation uncertainty of eq 40 for the specific caseof the coherent state input as ∆ T est = (cid:115) T νη N . (41)This equation is the result of using the sample mean ( ¯ I t ) asan estimator. One can also show that the CR bound, ∆ T CR ,defined in eq 26 for the intensity measurement in the abovescenario is the same as eq 41 [213]. This implies that thesample mean chosen above is an optimal estimator when the intensity is measured. It is interesting to note that the QCRbound, ∆ T QCR , defined in eq 28 is also the same as eq 41,meaning that the measurement of the intensity is the op-timal measurement in this scenario, even if quantum mea-surements were allowed. One more relevant observation isthat the uncertainty analyzed by a linear error propagationmodel, ∆ T est = (cid:12)(cid:12)(cid:12)(cid:12) ∂ (cid:104) ˆ n (cid:105) out ∂ T (cid:12)(cid:12)(cid:12)(cid:12) − (cid:104) ( ∆ ˆ n ) (cid:105) / , turns out to be the sameas eq 41.From eq 41 it is clear that the estimation uncertainty scaleswith the inverse square root of the intensity, i.e., N − / ,where N is the mean of the underlying Poisson photon numberdistribution of the input coherent state. The right hand side ofeq 41 is the SNL and it can also be called the SQL in intensitysensing. Therefore, the estimation uncertainty or precision forintensity sensing using a coherent state of light is shot-noiselimited.
2. Shot-noise limited phase sensing
The paradigmatic scenario for phase sensing is in the formof a MZI, as shown in Figure 10(b). Here, the goal is theestimation of the relative phase di ff erence between the twopaths. Consider that a coherent state | α (cid:105) is fed into mode a ofthe first BS, and the vacuum is assumed to be in mode b , i.e.,the input state is | Ψ (cid:105) in = | α (cid:105) a | (cid:105) b . The BS operator can bewritten as ˆ B ( τ, θ ) = exp[ τ e i θ ˆ a † ˆ b − τ e − i θ ˆ a ˆ b † ] and transformsthe operators ˆ a and ˆ b as [214–216]ˆ B ˆ a ˆ B † = √ T ˆ a − e i θ √ − T ˆ b , (42)ˆ B ˆ b ˆ B † = e − i θ √ − T ˆ a + √ T ˆ b , (43)where T = cos τ represents the transmittance of the BS and θ is an associated phase shift which we choose to be θ = π/ T , the outgoing state undergoes a relative phase shift thatis induced by the operator ˆ U ( φ ) = exp[ i φ (ˆ a † ˆ a − ˆ b † ˆ b )], whichtransforms the operators as ˆ U ˆ a ˆ U † = e − i φ/ ˆ a and ˆ U ˆ b ˆ U † = e i φ/ ˆ b . The state finally exits out of the second BS with trans-mission T . For generality, we also include optical loss occur-ring between the two BSs, with the amount of loss character-ized by channel transmission e ffi ciencies η a and η b for the twomodes. The output state is then written by [185] | Ψ ( φ ) (cid:105) out = | α out ( φ ) (cid:105) a | β out ( φ ) (cid:105) b , (44)where α out ( φ ) = T √ η a α e i φ − (1 − T ) √ η b α e − i φ and β out ( φ ) = i √ T (1 − T ) √ η a α e i φ + i √ T (1 − T ) √ η b α e − i φ . Suppose thatwe perform an intensity measurement at the two outputsand analyze the measurement results to estimate the relativephase φ . When the optimal estimator is chosen among manykinds of unbiased estimators, the estimation uncertainty isgiven by the CR bound, asymptotically saturable by the max-imum likelihood method in the limit of a large sample size.It can be minimized by optimizing the transmittance T of theBSs, which can be shown to be T opt = √ η b / ( √ η a + √ η b ), for7which the CR bound in the vicinity of φ = ∆ φ CR ≈ √ ν √ η a + √ η b √ η a η b √ N , (45)where N = | α | and ∆ φ CR is independent of the phase of thecoherent state input. This is known as the standard interfer-ometric limit (SIL) ∆ φ SIL [217, 218], which applies when anintensity measurement is used at the outputs of the second BS.It is clear that ∆ φ CR of eq 45 scales with N − / , also called theSQL. When φ increases and approaches π/
2, the above opti-mal value of T opt deviates, but the scaling with N is still kept.When η a = η b = η , T opt = /
2, i.e., a 50:50 BS is the optimalchoice in the above classical scenario, for which the CR boundreads ∆ φ CR = ( νη N ) − / , regardless of φ . The QCR bound ofeq 28 can also be calculated using eq 35 for the probe statewhich we take as the state just before the parameter encoding,i.e., the state present in between the first BS and the phaseshifter in the MZI, which is written as | Ψ (cid:105) prob = | α prob (cid:105) | β prob (cid:105) ,with α prob = √ T η a α and β prob = i (cid:112) (1 − T ) η b α . The QFIis given by H = | α prob | + | β prob | = [ T η a + (1 − T ) η b ] N ,clearly showing that the QCR bound, ∆ φ QCR ≥ ( ν H ) / , isshot-noise-limited regardless of the value of T and type ofmeasurement. C. Sub-shot-noise intensity sensing
Probing an analyte with a coherent state leads to the SQL(or SNL), where the estimation uncertainty scales with N − / .This is because a coherent state consists of discretized quan-tum particles (photons) that populate the energy levels of aspatial mode with a certain distribution, resulting in Poisso-nian photon number statistics on average over finite temporalintervals. As discussed in section III A, the QCR bound onthe estimation uncertainty is dependent on the probe state thatbecomes encoded with the information of a parameter to beestimated. An important question naturally arises: Can wefurther reduce the QCR bound below the SQL by optimiz-ing the probe state? If this is possible, then we can also ask:What would be the optimal state to achieve the ultimate un-certainty bound? To address these questions, we now brieflyshow how intensity sensing and phase sensing can be furtherimproved with the help of quantum states of light within theframework of the QCR bound. For more details, the latestreview articles devoted to quantum metrology can be con-sulted [41, 43, 44]. The techniques introduced here will thenbe used in section IV, where we describe recent work on im-proving plasmonic sensors using quantum resources.
1. Quantum-enhanced intensity sensing
When a sample mean is used in intensity sensing as de-scribed above, the crucial quantity that a ff ects the estimationuncertainty is ∆ I t . Is there a state that minimizes ∆ I t ? Toanswer this, we need to look at photon number distributions.Individual photons undergo a Bernoulli process when pass-ing through a BS, i.e., each photon is either transmitted with aprobability T or reflected with a probability (1 − T ). Given the incident photon number distribution p in ( m ), the output photondistribution p out ( n | T ) can be written as p out ( n | T ) = ∞ (cid:88) m = n (cid:32) mn (cid:33) ( η T ) n (1 − η T ) m − n p in ( m ) . (46)This leads to the variance of the transmitted intensity given as( ∆ I t ) = ∞ (cid:88) n = n p out ( n | T ) − ( ∞ (cid:88) m = np out ( n | T )) = T η (1 − T η ) (cid:104) ˆ n (cid:105) + ( T η ) (cid:104) ( ∆ ˆ n ) (cid:105) , (47)where (cid:104) ˆ n (cid:105) and (cid:104) ( ∆ ˆ n ) (cid:105) are the average and the variance of thephoton number of the incident state, respectively. For a fixedaverage photon number (cid:104) ˆ n (cid:105) = N , and given η and T , we thenhave that the smaller the photon number variance (cid:104) ( ∆ ˆ n ) (cid:105) ofthe probe state, the smaller the variance ( ∆ I t ) , thus reducingthe estimation uncertainty. The state with the smallest photonnumber variance is the Fock state | N (cid:105) = ( N !) − / (ˆ a † ) N | (cid:105) , forwhich (cid:104) ( ∆ ˆ n ) (cid:105) = ∆ T est = (cid:115) T (1 − η T ) νη N . (48)This clearly shows a quantum enhancement by a factor of (1 − η T ) in comparison with eq 41 obtained using a coherent stateas a probe, although it still scales with N − / . Substitutingthe probability p out ( n | T ) of eq 46 with p in ( m ) = δ m , N into thediscretized form for the FI of eq 27, one can show that theCR bound is the same as eq 48. For the outgoing state writ-ten as ˆ ρ T = (cid:80) n p out ( n | T ) | n (cid:105) (cid:104) n | , the QCR bound can also becalculated, consequently showing that the QCR bound is thesame as the CR bound because ∂ T | n (cid:105) = | α (cid:105) and | N (cid:105) lead to Poissonand binomial photon number statistics in the transmitted light,respectively, as shown in Figs. 11(a) and (b). The varianceof the binomial distribution, σ = η T (1 − η T ) N , is smallerthan that of the Poisson distribution, σ = η T N , while their
FIG. 11. Intensity sensing using classical and quantum states. (a) Acoherent state is used to measure the transmission T . The measuredpopulation distribution is Poissonian. (b) The photon number stateis used to measure the transmission T . The measured populationdistribution is binomial, with a smaller standard deviation, σ B , thanthe coherent state, σ P . (a)(b) (c) FIG. 12. (a) Precision ratio of an absorption spectroscopy measurement using single photons ( ∆ T est , q ) to the SNL ( ∆ T est , c ) for di ff erent types ofhaemoglobin: (left) oxyhaemoglobin (HbO2) and (right) carboxyhaemoglobin (HbCO). The absorption parameter is experimentally estimatedat individual wavelengths ranging from 790 to 808 nm. Reprinted from ref 219 under CC-BY license. (b) Precision ratio of a quantumpolarimetry measurement using single photons to the SNL for sucrose solution. The optical activity of sucrose solution with concentration C = / ml is experimentally estimated at input polarization angles in a range from − ◦ to 100 ◦ in steps of 10 ◦ . Reprinted from ref 213with permission from IOP Publishing. (c) (inset) The absorbance a ∈ [0 , ∞ ) of a medium is experimentally measured in the presence of otherlosses, such as surface losses γ and copropagation loss β . The FI is analyzed, as a function of the length of a Pockels cell modulator, for anabsorbance measurement with a two-mode squeezed vacuum state input and a coincidence counting scheme (upper). The latter is comparedto the case using a single-mode thermal state of light with and without dark counts (lower). Reprinted from ref 220 under CC-BY-4.0 license. mean values are equal, i.e., µ B = µ P = η T N . As can be seenfrom the above simple formulation, it is known that the pho-ton number state is the optimal state that minimizes the QCRbound, i.e., attaining the ultimate quantum limit in intensitysensing [221, 222].Intensity, or equivalently loss parameter sensing, has beenconsidered with various probe states [45]. The optimal strate-gies of estimating a general one-parameter quantum processwere studied in terms of the Kraus representation [223] andlater the ultimate quantum bound on the estimation uncer-tainty in loss parameter sensing was derived [222], which iseq 48. Gaussian probe states (a special class of quantum stateswhose Wigner function is a Gaussian function [46]) were con-sidered in intensity parameter sensing [222], and their sensingperformances were shown to be improved by the use of a Kerrnonlinearity [224]. It was shown that the ultimate bound isachievable only by the Fock state probe among all single-mode quantum probe states, including non-Gaussian probestates [221]. The optimality of a Fock state probe in inten-sity sensing has been exploited in various types of sensing,such as in absorption spectroscopy to analyze the organic dyemolecule dibenzanthanthrene [225] or haemoglobin [see Fig-ure 12(a)] [219]. Other relevant studies include quantum po-larimetry to measure the optical rotation occurring in chiral media [see Figure 12(b)] [213] and an experiment to measurethe absorbance of a lossy medium [see Figure 12(c)] [220]. Ofparticular interest to this review is that the Fock state probe hasrecently been used in plasmonic sensing [226], whose detailswill be discussed in section IV A.
2. Multiparameter or multimode intensity sensing
Single-mode intensity sensing can be improved by usingancillary modes [227]. It has been shown that entanglementcan further improve the estimation of an unknown damp-ing constant in an interferometric setup with specific probestates and measurements considered [228]. Entangled Gaus-sian probes were shown to lead to a better strategy for discrim-inating lossy bosonic channels in the presence of a thermalenvironment [229]. The estimation of a loss parameter canalso be made simultaneously with the estimation of tempera-ture [230] or phase [231]. More generally, the estimation ofmultiple loss parameters needs to be considered with regardsto several applications, such as in image sensing [232], wherethe precise measurement of an image at individual pixels of anamplitude mask corresponds to the problem of the precise es-timation of multiple loss parameters. Increasing the resolution9of thermal electromagnetic sources can also be formulated asloss parameter estimation [233–237]. Other examples of im-portance are to probe the change of intensity parameters overdi ff erent frequencies [219] or temporal modes [42].A recent study has derived the ultimate quantum boundto the estimation uncertainty of multiple intensity parame-ters and identified the optimal schemes reaching the ultimatequantum bound [238]. Consider the problem of the estima-tion of K transmittivities { T k } Kk = of K transmissive channelsusing what is called an ‘ancilla-assisted entangled parallel’strategy, as illustrated in Figure 13(a). The probe states aresent through signal modes (S) representing the K transmis-sive channels modeled by K BSs, whereas ancillary modes(A) are used to transmit states entangled with the probe statesand are kept lossless. To make the scenario more general,multimode features at individual channels are taken into ac-count, so that M k modes are assumed to be used for probingthe k th transmittance. The overall probe state can thus be writ- FIG. 13. Multiple intensity parameter sensing. (a) General ancilla-assisted strategy: Each signal mode (red) is sent to a fictitious BSwith unknown transmittivity T j in the j th mode, while an ancilla (yel-low) is kept lossless. The composite output state of the signal modesand the ancilla mode is measured to estimate the vector T consistingof all transmittivities. Virtual environment modes (green) are alsoconsidered and assumed to be in the vacuum state before enteringthe BSs. Reprinted from ref 238 with permission from AmericanPhysical Society. (b) Left: Circular dichroism (CD) is experimen-tally observed by measuring the intensity-di ff erence of the transmit-ted left-circularly polarized (LCP) light and right-circularly polarized(RCP) light upon propagation through a chiral medium. Right: Gen-eral ancilla-assisted CD sensing scheme, similar to (a), but with addi-tional losses η L and η R for the respective LCP and RCP modes. Thetransmittivities T L and T R represent the transmittivities of the LCPand RCP modes through a chiral medium, respectively. Reprintedfrom ref 239. ten as a joint pure state | Ψ (cid:105) SA , with energy constraints on theindividual channels, i.e., (cid:104) Ψ | ˆ n k | Ψ (cid:105) SA = N k for k = , · · · , K ,where ˆ n k = (cid:80) M k m = ˆ a † k , m ˆ a k , m . It was shown that the maximumQFIM H can be written as [239] H = diag (cid:32) η N T (1 − η T ) , · · · , N K η K T K (1 − η K T K ) (cid:33) , (49)where η k is an additional but unwanted loss rate for the k thchannel. Such an ultimate quantum bound exhibits a quan-tum enhancement in comparison with the QFIM for a productcoherent (PC) state probe written as H = diag (cid:32) η N T , · · · , η K N K T K (cid:33) . (50)It has been shown that the above maximum QFIM of eq 49 canbe achieved by the product probe | Ψ (cid:105) = (cid:78) Kk = | NDS (cid:105) k [238],where | NDS (cid:105) k is the so-called number-diagonal-signal (NDS)state with the energy constraint of N k on the signal mode [240,241], defined as | NDS (cid:105) k = (cid:88) n k ≥ (cid:112) p ( n k ) | n k (cid:105) S | φ n k (cid:105) A , (51)where | n k (cid:105) S = | n (cid:105) k · · · | n M k (cid:105) k is an M k -mode number statebasis at the k th signal channel, {| φ n k (cid:105) A } is an orthonormal ba-sis at the k th ancillary channel, and p ( n k ) is the probabilitydistribution of n k . The signal energy constraint on k th sig-nal mode then reads as N k = (cid:80) ∞ m k = m k p ( m k ) with p ( m k ) = (cid:80) n + ··· + n Mk = m k p ( n k ).According to the formulation introduced in section III A,one can find the optimal measurement setting that reaches theultimate multiparameter quantum bound. It has been shownthat the optimal setting is the joint measurement with theSchmidt bases, i.e., the basis | φ n k (cid:105) on the ancillary modesand the number basis | n k (cid:105) S on the signal modes [238]. Thisleads the FIM to be equal to the QFIM of eq 49 for any { T k } .Note that the multiparameter QCR bounds given by eq 49 canbe simultaneously achieved since the associated SLDs com-mute [189].The above formulation has recently been applied to one no-table example, called circular dichrosim (CD) sensing, whichis shown in Figure 13(b). Typical CD sensing schemes aimto estimate the di ff erence of the transmittivities between left-and right-circular polarization through a chiral medium com-posed of either chiral molecules [242, 243] or chiral nanopho-tonic structures [244–246]. By modeling CD sensing quantummechanically [see Figure 13(b)] and using eq 39, Ioannou etal. identified the ultimate quantum limit on the estimation un-certainty of an estimate ( T L − T R ) and investigated the opti-mality in terms of various quantum state inputs and quantummeasurements [239].
3. Quantum noise reduction in intensity measurements
The most widely exploited scheme for measuring intensitychanges in both the classical and quantum regime is reference-assisted transmission or absorption spectroscopy. Here, thesignal mode encodes a single intensity parameter of an object0 (a) (b)
FIG. 14. (a) (Top) Twin beam [i.e., the TMSV state of eq 53] is generated via a SPDC process. One mode of the twin beam passes througha weakly absorbing object and arrives at a CCD camera array, while the other mode is directly sent to another area of the CCD array. Asubtraction is performed between the two noisy images, realizing the intensity-di ff erence measurement scheme, whereby an image of the objectis constructed. (Bottom) For two example sets (upper and lower row) of the objects, three kinds of schemes are used: (left) di ff erential-intensitymeasurement using a TMSV state, (middle) di ff erential-intensity measurement using a PC state, and (right) direct-intensity measurement usingsingle-mode coherent state. Reprinted from ref 232 with permission from Springer Nature. (b) (Top) The parametric process in the double-lambda scheme converts two pump (P) photons into one probe (Pr) photon and one conjugate (C) photon, generating a bright twin beam, i.e.,the TMSD state of eq 54. In the experiment, a bright pump beam couples with the probe and conjugate fields through a hot Rb vapor cellinvolving the double-lambda scheme. The spatially correlated bright twin beams are generated with cylindrical symmetry over a range ofangles ∆ θ around the axis of the pump beam. When the spatial field profile of the probe seed is modified by an amplitude mask, an entangledimage can be generated between the probe and conjugate outgoing field. (Bottom left) Image of the probe seed entering a hot Rb vaporcell. (Bottom middle) Image of the outgoing probe beam. (Bottom right) Image of the outgoing conjugate beam, clearly showing anti-spatialcorrelation with the image of the probe beam. Reprinted from ref 247 with permission from American Association for the Advancement ofScience. and a reference mode is kept unchanged. Examples includequantum imaging [248], quantum illumination [249, 250] andquantum sensing [251]. In particular, an intensity-di ff erencemeasurement between the signal and reference output modeshas often been performed in order to remove common excessnoise between the two modes [252]. This is a technique thathas recently been used in classical plasmonic sensing to reachthe SNL [35, 36, 94].The ultimate aim of using quantum probe states is to reducethe noise below the SNL. The noise reduction of the intensity-di ff erence measurement of two modes a and b can be quanti-fied by the so-called noise reduction factor (NRF) [253–256],defined by the ratio of the variance of the photon number dif-ference between the signal and reference mode to that of co-herent states with matching average photon number, writtenas σ = (cid:104) [ ∆ (ˆ n b − ˆ n a )] (cid:105)(cid:104) ˆ n a (cid:105) + (cid:104) ˆ n b (cid:105) . (52)It can be shown that σ ≥ σ < ff erence is minimal, i.e., the optimal probe state generally written in the form of the photon-number-correlated state | Ψ (cid:105) ab = (cid:80) n w n | n , n (cid:105) with any weights { w n } ,for which σ =
0. An example is the twin Fock (TF)state | TF (cid:105) = | N , N (cid:105) with σ (TF) =
0. For a PC state | α, β (cid:105) ,on the other hand, σ (PC) =
1, which sets the SNL.Another useful photon-number-correlated state leadingto σ = | TMSV (cid:105) = ˆ S ( ξ ) | , (cid:105) = ∞ (cid:88) n = c n | n , n (cid:105) , (53)where ˆ S ( ξ ) = exp[ ξ ∗ ˆ a ˆ b − ξ ˆ a † ˆ b † ] denotes the two-modesqueezing operator with ξ ∈ C and c n = ( − e i θ s tanh r ) n / cosh r for ξ = re i θ s . It is clear from eq 53 that the TMSV state ex-hibits a strong quantum correlation in the photon number be-tween the two modes, leading to σ (TMSV) =
0, whereas theindividual modes follow thermal statistics, i.e., (cid:104) ( ∆ ˆ n ) (cid:105) / (cid:104) ˆ n (cid:105) = N + N = sinh r . This strong photon number cor-relation has been used in various applications such as quan-tum ellipsometry [260, 261], absorption / transmission mea-surements [262–266], quantum sensing [267], quantum illu-mination [250], quantum radiometry [268, 269], and quan-tum gravity tests [270]. Figure 14(a) presents an experimen-tal demonstration of a quantum imaging technique using thenonclassical spatial correlation of TMSV states for a weaklyabsorbing object, in direct comparison with the experimen-tal images obtained using classical light [232]. The TMSVstate is normally regarded as a weak-field probe since the av-erage photon number N is small, i.e., small r , due to the weaknonlinearity of the material used for realizing the two-modesqueezing operation [271].When sensors operating in the high-intensity regime aremore preferable or favorable, one can employ a brighter en-tangled state at the expense of decreasing the quantum cor-relation. For example, the state produced by performing thetwo-mode squeezing operation on a vacuum state in one modeand coherent state in another (whose intensity can be high).This state is formally called a two-mode squeezed displaced(TMSD) state and has been investigated from various perspec-tives [272–274]. The TMSD state can be written as | TMSD (cid:105) = ˆ S ( ξ ) | α (cid:105) | (cid:105) , (54)for which (cid:104) ˆ n a (cid:105) = sinh r + | α | cosh r and (cid:104) ˆ n b (cid:105) = sinh r + | α | sinh r and σ (TMSD) = | α | | α | + + | α | ) sinh r . (55)It is clear that for a given small r , the intensity of the indi-vidual modes can be increased by cranking up the pump laserpower (i.e., | α | ) incident to the nonlinear medium that pro-duces the TMSD state. A large | α | increases the NRF, but theNRF is always less than unity, implying that the quantum cor-relation of the TMSD state is always stronger than that of theclassical case and can thus be exploited in quantum imaging orsensing. In an experiment, the TMSD state can be generatedvia a four-wave mixing (FWM) process in a double-lambdaconfiguration provided by a Rb vapor cell [275–277], wherea NRF ≈ − ≈
10 log (1 − . µ W of optical power has been exploited in quantum sens-ing [278, 279], imaging [247, 280], and quantum plasmon-ics [55]. The application to quantum plasmonic sensing, inparticular, will be described in more detail in section IV A 2.Figure 14(b) presents a demonstration of entangling spatialinformation of the probe beam with the conjugate beam by aFWM scheme that generates a TMSD state [247].An interesting comparison can be made among the afore-mentioned quantum states of light in quantum imaging orintensity parameter sensing scenarios. Consider a setup,where the object with transmittance T is placed in the sig-nal mode a , and the signal and reference modes undergo in-e ffi cient transmission channels with factors η a and η b , respec-tively, which include the detection e ffi ciencies of the two de-tectors [281]. The intensity-di ff erence measurement opera-tor ˆ n − = ˆ b † out ˆ b out − ˆ a † out ˆ a out can be written in terms of inputoperators using the Heisenberg picture [185]. Here, states are static and operators contain the relevant dynamics. In thiscase, we can model the change of the transmittance and lossesin the modes as fictitious BSs. This enables the output NRFto be calculated for the respective probe states [185]. For ex-ample, the input-output relation given byˆ a out = (cid:112) G η a T ˆ a in + (cid:112) ( G − η a T ˆ b † in + (cid:112) (1 − η a ) T ˆ a bath + (cid:112) (1 − T )ˆ a obj , (56)ˆ b out = (cid:112) G η b ˆ b in + (cid:112) ( G − η b ˆ a † in + (cid:112) (1 − η b )ˆ b bath (57)can be applied to ˆ n − and the expectation value taken with re-spect to the initial vacuum state | (cid:105) | (cid:105) for the TMSV stateprobe, or the initial displaced state | α (cid:105) | (cid:105) for the TMSDstate probe. Here, ˆ a bath (ˆ b bath ) is the input operator of a fic-titious BS describing a lossy channel of the signal (reference)mode [185], ˆ a obj denotes the virtual input operator of the ob-ject, and G = cosh r with r being the squeezing parameterand θ s = π assumed. The bath and object modes are taken tobe in the vacuum initially.The NRFs of the respective output states can thus be writtenas σ (TF)out = − T η + η T η a + η b , (58) σ (TMSV)out = + G | α | ( T η a − η b ) − ( T η + η ) T η a + η b , (59) σ (TMSD)out ≈ + G − (cid:104) G ( T η a − η b ) − η (cid:105) GT η a + ( G − η b , (60)where the limit | α | (cid:29) σ out <
1. The latter holds for σ (TF)out at any value of T ,implying that the use of the TF state always achieves the sub-shot-noise measurement. The other two states, on the other FIG. 15. Comparison of the noise reduction for di ff erent quantumstates: TF, TMSV and TMSD. The parameters chosen are G = . | α | =
10 and η a = η b = .
8. The starred versions (TF ∗ , TMSV ∗ and TMSD ∗ ) correspond to the case when T η a = η b . Inset: Generaltwo-mode scheme using an input state that consists of the signal andreference mode to estimate the intensity parameter T in the presenceof losses η a,b for each mode. Here, the transmissive object with T and linear losses are described by fictitious BSs. G ( T η a − η b ) is less than T η + η and η , respectively. These conditionscan be simply satisfied when T η a is close to η b , motivating aconsideration of the ideal case that T η a = η b = η , for whichthe NRFs of eqs 58 to 60 become σ (TF)out = − η, (61) σ (TMSV)out = − η, (62) σ (TMSD)out ≈ − η ( G − G − . (63)These NRFs are shown in Figure 15 as the starred versions. Asexpected, a sub-shot-noise measurement can be achieved withall the considered states at any value of T . This has been ex-perimentally demonstrated in various studies [276, 283, 284],including the imaging of a weakly absorbing object, forwhich η b = η a and T ≈ ff erence measurement [232], defined bySNR = |(cid:104) ˆ n b (cid:105) − (cid:104) ˆ n a (cid:105)|(cid:104) [ ∆ (ˆ n b − ˆ n a )] (cid:105) / . (64)When symmetric probes are used (such as the TF, andTMSV states) and a lossless reference channel is assumed,i.e., (cid:104) ˆ n a (cid:105) in = (cid:104) ˆ n b (cid:105) in = N and (cid:104) ( ∆ ˆ n a ) (cid:105) in = (cid:104) ( ∆ ˆ n b ) (cid:105) in , and η b =
1, the SNR can be rewritten asSNR = ( η b − η a T ) N (cid:112) N (cid:2) ( T η a − η b ) Q M + T η a η b ( σ in − + ( T η a + η b ) (cid:3) , (65)where Q M = (cid:104) ( ∆ ˆ n ) (cid:105) in / (cid:104) ˆ n (cid:105) in − σ in is the NRF for the input state [253–256]. The Mandel Q-parameter is equal to or greater than zerofor all classical light, so non-classical light can be identifiedby Q M <
0. As mentioned previously, the NRF σ in ≥ σ in < Q M and σ in . One finds that the minimumvalues of Q M and σ in are − | Ψ in (cid:105) = | N (cid:105) a | N (cid:105) b .The quantum enhancement in the SNR can be quantified bythe ratio of the SNR for the particular quantum state probe tothat for the balanced PC probe state, written by R SNR = SNR q SNR c = (cid:115) T η a + η b ( T η a − η b ) Q M + T η a η b ( σ in − + ( T η a + η b ) . (66)Note that that quantum enhancement with R SNR > T η a − η b ) Q M + T η a η b ( σ in − <
0, so aquantum probe with − ≤ Q M < ≤ σ in < Q M = − σ in =
0, which are the minimum bound for the respectiveparameters. In this optimal case, R SNR > T and η a,b .The use of the TMSV state of eq 53, for which Q M = N and σ in =
0, provides a quantum enhancement onlywhen ( T η a − η b ) N < T η a η b with respect to the balancedPC state with Q M = σ in =
1. This indicates thatprobing with a TMSV state is more advantageous as N de-creases. When η a ≈ η b and T ≈
1, or η a T ≈ η b , the useof a TMSV state becomes more useful even with high N .Such an enhancement has been observed in quantum imag-ing experiments [232, 251]. One can also find other usefulstates with σ in = Q M <
0, for example, pair coher-ent states [288] or finite-dimensional photon-number entan-gled states [289]. These states have all been considered inquantum plasmonic sensing, as we will discuss in detail insection IV A.
D. Sub-shot-noise phase sensing
1. Phase sensing in Mach-Zehnder interferometers
The shot-noise limited interferometric phase sensor de-scribed in section III B can be improved by making use ofquantum resources. The original idea proposed by Caves,who largely contributed to the advent of the field of quan-tum metrology, is to inject a squeezed vacuum state | ξ (cid:105) intothe input mode b of the first BS in the MZI shown in Fig-ure 10(b), where previously the input state in mode b wasassumed to be the vacuum in the shot-noise limited clas-sical sensor [187]. The squeezed vacuum state is definedas | ξ (cid:105) = ˆ S ( ξ ) | (cid:105) , where the single-mode squeezing operatoris represented by ˆ S ( ξ ) = exp( ξ ∗ ˆ a − ξ ˆ a † ) and ξ = re i θ s ,with r ≥ x θ s / = (cid:104) ˆ x θ s / (cid:105) , withˆ x θ s / = ( e − i θ s / ˆ a + e i θ s / ˆ a † ) / √
2. Here, θ s / | Ψ (cid:105) in = | α (cid:105) a | ξ (cid:105) b , with the total averagephoton number of the two modes being N = | α | + sinh r . Inthe absence of loss ( η a = η b = | Ψ (cid:105) prob = ˆ B ( π/ , π/ | Ψ (cid:105) in that probes the phase information, resultingin | Ψ ( φ ) (cid:105) = ˆ U ( φ ) | Ψ (cid:105) prob before the measurement. For such alossless case, Pezzé and Smerzi showed that the QCR boundto the estimation uncertainty of the relative phase φ is writtenby [291] ∆ φ QCR = √ ν (cid:113) | α | e r + sinh r , (67)where the optimal phase condition between the displacementand squeezing parameters is set, i.e., cos(2 θ c − θ s ) =
1. Notethat the QCR bound ∆ φ QCR is the same for any true value of φ ,and it can reach Heisenberg scaling ∆ φ QCR ∝ N − for N (cid:29) | α | (cid:39) sinh r (cid:39) N /
2. It can be shown that ∆ φ QCR ≈ e − r / √ ν N when | α | (cid:29) sinh r [187] and ∆ φ QCR ≈ e − r / (cid:112) ν N (4 | α | +
1) when | α | (cid:28) sinh r [291]. These casesshow that too small or too large amounts of squeezing can-not achieve Heisenberg scaling, but they do provide sub-shot-noise sensing due to the additional factor that multi-plies N − / .Interestingly, the squeezed vacuum state considered abovehas been shown to be the optimal state that can be inserted intomode b of the lossless MZI when the coherent state is insertedinto mode a , as it minimizes the estimation uncertainty of therelative phase φ [292]. This indicates that the initial idea pro-posed by Caves turns out to be a good choice as a modificationfor better phase sensing in a MZI setup. The original goal herewas to improve classical gravitational wave detectors, where astrong classical field is used in one mode of the interferomet-ric setup [293]. This has now led to the latest improvementsto the LIGO detector [294] and the Virgo detector [295]. Fur-thermore, the QCR bound of eq 67 is not only theoretical, butpractically attainable by photon-number-counting detectors atthe two output ports of the MZI [291], or a parity detectionscheme at the output port a of the MZI [296].When a coherent state does not need to be used in mode a ofthe MZI, the optimal input state for phase estimation using theMZI setup is two single-mode squeezed vacuum states withanti-phases, i.e., | Ψ (cid:105) opt = ˆ S a ( − r ) ˆ S b ( r ) | , (cid:105) with r ∈ R [297],for which the QCR bound reads ∆ φ QCR = / √ ν N ( N + N = r being the total average photon number.The MZI phase sensing setup has sometimes been con-sidered with a single phase shift in mode a , i.e., ˆ U ( φ ) = e i φ ˆ a † ˆ a [298, 299]. However, in this setting particular care mustbe made in quantifying the extent to which the estimation un-certainty is reduced, since an optical phase can only be definedin a relative sense with respect to a reference phase [300].Hence, a sensing analysis with a single phase shift is validonly when a reference beam with a certain phase is assumed,whose resource also needs to be counted when determiningthe dependence of the uncertainty on N .In the case of loss in both arms of the MZI, the calculationof the QCR bound becomes complicated due to the probabilis-tic nature of photon loss that needs to be taken into account.In this case, it is more convenient to use the relation betweenthe QFIM H and the Bures distance D for the infinitesimallyclose states ˆ ρ φ and ˆ ρ φ + d φ [194, 301, 302]. Using the relevantformulation introduced in Appendix B, one can thus derivethe QCR bound for the case η a = η b = η , written as [303, 304] ∆ φ QCR = √ ν (cid:113) | α | η (1 − η ) + e − r η + η sinh r . (68)It is clear that ∆ φ QCR increases with loss, i.e., as η de-creases, whereas the lossless QCR bound of eq 67 is recov-ered when η =
1. It can also be shown that the Heisenbergscaling ∆ φ QCR ∝ N − promised for the lossless cases starts todeteriorate as η decreases.The QCR bound ∆ φ QCR of eq 68 has been shown to beachievable by a linear optical setup using a weak local oscilla-tor field and photon counting [304]. Other types of measure-ment schemes have been investigated, for example, homodynedetection with a measurement operator ˆ M = ˆ x φ HD , a single-mode intensity measurement with ˆ M = ˆ a † ˆ a , an intensity-di ff erence measurement with ˆ M = ˆ b † ˆ b − ˆ a † ˆ a , and a parity measurement with ˆ M = ( − ˆ a † ˆ a [303, 305]. In the case of ho-modyne detection (see Appendix C), it was shown that this of-fers a nearly optimal measurement scheme and approaches theQCR bound of eq 68 in the large power regime. Apart fromlinear photon loss, di ff erent noise models have also been stud-ied, for example, such as phase drift and thermal noise [303].
2. Phase sensing with NOON states
In addition to exploiting continuous variable states such asthe squeezed vacuum state in phase sensing, discrete variablestates or definite-photon-number states have also been con-sidered as probe states for phase sensing [308]. The most fa-mous state is the so-called NOON state, defined as | NOON (cid:105) = √ ( | N , (cid:105) + | , N (cid:105) ), where N photons are found in eithermode a or b [309–311], exhibiting N -particle maximal en-tanglement [312]. The generation of NOON states is verydi ffi cult with current technology, and they have been limitedto N = (cid:104) ( ∆ ˆ n − ) (cid:105) of the photon number di ff erence between the two modesis maximal among the states with N photons in total, con-sequently maximizing the QFI written by H = (cid:104) ( ∆ G ) (cid:105) ,where ˆ G = ˆ n − / ∆ φ QCR = / √ ν N , clearly manifestinga Heisenberg scaling enabled by the maximal variance of thephoton number di ff erence of the probe state. It is interest-ing to note that injecting a coherent state and a squeezed vac-uum state into the two input ports of the first BS in a MZIgenerates an e ff ective NOON state, as it has a large over-lap with the NOON state, i.e., ˆ B ( π/ , π/ | α, ξ (cid:105) ≈ | NOON (cid:105) for sinh r = | α | = N / N (cid:29) (cid:104) ( ∆ ˆ n − ) (cid:105) , similar to the NOON state probe, inorder to achieve sub-shot-noise sensing.The QCR bound for the NOON state has been shown tobe achievable by the measurement of the observable ˆ A N = | , N (cid:105) (cid:104) N , | + | N , (cid:105) (cid:104) , N | [50], or parity detection [309]. Onemajor obstacle in using the NOON state for phase sensingfrom a practical perspective is that it is extremely sensitiveto photon loss, since the loss of a single or a few photonsdrastically changes the state’s photon number distribution andits variance (cid:104) ( ∆ ˆ n − ) (cid:105) quickly decreases. The estimation un-certainty associated with the NOON state thus becomes largeeven with a moderate amount of photon loss [315–318]. Suchan extreme sensitivity to photon loss can be alleviated by ex-ploiting partial entanglement at the expense of a degradedperformance, while still achieving sub-shot-noise limited be-havior [319]. Hence, adding other photon-number compo-nents (e.g., | k , N − k (cid:105) for k < N ) into the NOON state helpsphase sensing with NOON states become more robust to lossand provide sub-shot-noise sensing even in the presence ofloss. In particular, in phase sensing with the definite-photon-number state | Ψ (cid:105) prob = (cid:80) Nk = c k | k , N − k (cid:105) with N particles dis-tributed over the two modes, one can find the optimal distri-bution { c k } that maximizes the QFI and consequently enablesone to beat the SQL or SNL at any loss level [217, 320].The latter has been experimentally demonstrated with op-4 (a)(b) FIG. 16. (a) (left) Linear optical network to demonstrate optimal phase estimation in the presence of loss. Initial BSs prepare the optimal probestate depending on the amount of loss measured via a loss monitor in a proof-of-principle demonstration. (right) Phase estimation uncertaintiesmeasured using two-photon optimal states (blue circles), NOON (red squares) states, and weak-coherent states to set the SIL regime (graydiamonds), for five phase values φ = , ± . , ± . ff erent transmittivities η . Horizontal linesrepresent the theoretical CR bounds for individual input states in the presence of loss. Reprinted from ref 306 with permission from SpringerNature. (b) (left) Two-photon NOON state undergoes a phase rotation in a collinear interferometer. The signal and idler photons are separatedvia a PBS and measured by superconducting nanowire detectors. The high e ffi ciency and visibility in the experiment meant that post-selectionwas not needed, leading to the demonstration of unconditional quantum enhancement. (right) Experimentally measured standard deviation ofthe estimated phases with error bars determined via the standard bootstrapping technique. The theoretical SNL (purple line) and CR bound forphase sensing with NOON states (orange line) are shown with 95% confidence regions. Reprinted from ref 307 with permission from SpringerNature. timally engineered definite-photon-number states [306] [seeFigure 16(a)], and has been considered theoretically for quan-tum plasmonic sensing, as will be discussed in more detail insection IV B 1. The unconditional quantum enhancement inphase estimation precision with NOON state has also been ex-perimentally demonstrated in a nearly lossless scenario withsignificantly increased e ffi ciency and visibility [307] [see Fig-ure 16(b)].The most commonly used detection scheme for phase sens-ing with NOON states is the N -fold coincidence detectionscheme that counts the number of events when N photons aredetected simultaneously [321]. This allows one to investigatethe N -fold detection modulation with the phase φ , leading tothe probability of a coincidence detection given by p coin ( φ ) = f N [1 + V N cos( N φ )]2 , (69)where f N is the proportion of the input state that causes an N -fold coincidence detection event, and V N is the N -photon vis-ibility. The CR bound for this scheme can be calculated withthe underlying probabilities p coin ( φ ) for the detection event and 1 − p coin ( φ ) for the no detection event, and is written as ∆ φ coin = (cid:112) p coin ( φ ) [1 − p coin ( φ )] f N V N N | sin( N φ ) | ≤ f N V N N | sin( N φ ) | , (70)where the upper bound takes into account the worst case of theprobability p coin ( φ ) = /
2. For comparison with the classicalbenchmark, the CR bound ∆ φ coin in eq 70 can be comparedwith ∆ φ SIL of eq 45, leading to an inequality ∆ φ coin < ∆ φ SIL for a sensor to be so-called super-sensitive [322, 323]. Theinequality can be specified, when | sin( N φ ) | =
1, as1 < f N V N N ˜ η , (71)where ˜ η = (cid:104) √ η a η b / ( √ η a + √ η b ) (cid:105) . For the threshold vis-ibility defined as V (th) N = (cid:113) ˜ η/ f N N , the above inequalitycan be written as V N > V (th) N , i.e., only a measured visibil-ity higher than the threshold visibility demonstrates ‘super-sensitivity’ [322]. The latter can be considered as a gen-uine criterion for the quantum enhancement of the precisionin sensing experiments with NOON states instead of ‘super-resolution’, which has sometimes been misinterpreted, since5it can also be produced by only classical light and projectivemeasurements [322]. The above theory will be used in sec-tion IV B when describing quantum plasmonic sensors thathave considered the use of NOON states.A simple example of NOON state phase sensing uses thetwo-photon NOON state √ ( | , (cid:105) + | , (cid:105) ). It can be read-ily created by exploiting Hong-Ou-Mandel (HOM) interfer-ence [324]. The HOM interference occurs when two singlephotons are injected into a lossless 50 /
50 BS, i.e., the out-put state is written as ˆ B ( π/ , π/ | , (cid:105) = ( | , (cid:105) + | , (cid:105) ) / √ | , (cid:105) is absentdue to destructive interference caused by the indistinguish-able paths leading to the same output state | , (cid:105) . When thetwo-photon NOON state undergoes a relative phase shift de-scribed by the operator ˆ U ( φ ), the outgoing state can be writ-ten up to a global phase as ( | , (cid:105) + e − i φ | , (cid:105) ) / √ φ -modulation isobserved due to the multiplied constant factor of N = /
50 BS and detecting the coincidence of single pho-tons at the output using single-photon detectors. The visibilityof the two-photon measurement signal (as φ is modulated) isthen V = η a η b / ( η + η ), which enables the super-sensitivityinequality to be reduced to approximately ( η a + η b ) / > . | N (cid:105) | N (cid:105) incident into a MZI [326], the photon-number correlatedstate [318, 327–332], the TMSV state with parity detec-tion [333], Schrödinger’s cat states [334], and multi-headedcoherent states [335, 336]. More details of these states andtheir performance can be found in ref 308.
3. Single-mode phase sensing
Instead of two-mode phase sensing, single-mode phasesensing has also been investigated for the sake of simplic-ity. Here, the implicit assumption is that the phase of areference mode is set. The latter is enabled by employ-ing phase-sensitive detection schemes, e.g., homodyne detec-tion [337, 338] or general-dyne measurement [339]. Thus,the phase information in single-mode phase sensing cannotbe extracted from a probe state by phase-insensitive detectionschemes, such as photon-number counting, i.e., ∂ φ p ( n | φ ) = | Ψ (cid:105) in undergoesa phase shift described by the operator ˆ U ( φ ) = e i φ ˆ a † ˆ a , asshown in Figure 17(a). As in the intensity sensing case, onemay consider states having a small uncertainty in their phasefor single-mode phase sensing. One typical quantum statewhose phase uncertainty can be reduced below the SNL isthe squeezed vacuum state, | ξ (cid:105) . The phase shift operator ˆ U ( φ ) FIG. 17. Single-mode phase sensing. (a) A coherent state or asqueezed vacuum state are used to measure the phase φ . (b) Thephase space representation for the states. The coherent state has anestimation precision of ∆ φ coh for the phase φ , whereas the squeezedvacuum state has a smaller estimation precision of ∆ φ sq . The ini-tial phase of the squeezed state, | ξ (cid:105) , is θ s = π and the phase a ff ectedsqueezed state (cid:12)(cid:12)(cid:12) ξ e i φ (cid:69) rotates in phase space counter-clockwise by theangle φ . rotates a state in phase space about the origin and the outputstate is | Ψ (cid:105) out = ˆ U ( φ ) | ξ (cid:105) = | ξ e i φ (cid:105) , for which the QCR boundis obtained as [340, 341] ∆ φ QCR = √ ν (cid:112) N + N ) , (72)where N = sinh r represents the average photon number ofthe initial squeezed vacuum state. It is clear that the QCRbound of eq 72 scales with N − , not only reaching Heisenbergscaling [340, 341] but also beating the SQL of ∆ φ = / √ ν N that is obtained for a single-mode coherent state input. Sucha quantum enhancement in phase sensing is enabled by thesmaller phase uncertainty of the squeezed vacuum state ascompared to the coherent state with the same energy, as shownin Figure 17(b). One also finds that the photon number statethat is the optimal state for single-mode intensity sensing can-not encode any phase information since the photon numberstate exhibits a full phase uncertainty [185].Generally speaking, to measure the change of a particularphysical quantity being induced by a parameter x with bet-ter precision or lower estimation uncertainty, the probe stateshowing the least uncertainty in that particular parameter isthe most useful. A more analytical understanding can bemade through the relation between the QFI and the fidelitybetween two infinitesimally close states ˆ ρ ( x ) and ˆ ρ ( x + d x ) fora given parameter x , i.e., F Q ( x ) = lim d x → { −F [ ˆ ρ ( x ) , ˆ ρ ( x + d x )] } / (d x ) [342, 343] (see also Appendix B). That is, the es-timation capability is related to the ability to distinguish twoinfinitesimally close states ˆ ρ ( x ) and ˆ ρ ( x + d x ) and a better dis-tinction can be made with the least uncertainty in x of theprobe state.One can show that the QCR bound of eq 72 is reached byhomodyne detection, which measures the quadrature variableof ˆ x θ HD = ( e − i θ HD ˆ a + e i θ HD ˆ a † ) / √ θ HD depending on the value of r , θ s and φ , [345]. The optimality of homodyne detection holds inthe absence of loss, i.e., probing with a pure squeezed vac-uum state, whereas a realistic squeezed state of light involvesan inevitable thermal photon contribution [346], for whichthe QCR bound can only be obtained by performing an ex-otic measurement with projectors over the eigenstates of the6SLD operator. It has been shown that the SLD operator inthe case of probing with a squeezed thermal state can take theform of ˆ x ˆ p + ˆ p ˆ x , followed by Gaussian unitary operations,where ˆ x = ˆ x θ HD = and ˆ p = ˆ x θ HD = π/ are the quadrature variableoperators [342, 345].
4. Multiple-phase sensing
Single-phase estimation can be extended to estimating mul-tiple phases, θ = ( θ , θ , · · · , θ d ) T . This is relevant to applica-tions such as phase imaging [200], which measure phase con-trast and interference or gravitational wave detectors [201],which measure multiple parameters. It is also relevant toquantum plasmonic imaging, which will be discussed in moredetail in section IV A 2. The estimation uncertainty of the to-tal of all phases is governed by the covariance matrix Cov( θ )and lower bounded by the QCR bound, as given in eq 38.The uncertainty of the total of all phases estimated can bequantified by the sum of the individual uncertainties: | ∆ θ | = (cid:80) j ( ∆ θ j ) = Tr[Cov( θ )]. This quantity is often compared us-ing three typical cases: (i) a scheme using optimal classicalstate inputs, setting the classical benchmark or SQL, (ii) ascheme that estimates the phases individually by using opti-mal separable quantum state inputs, which is called ‘individ-ual estimation’ or a ‘local strategy’, and (iii) a scheme thatestimates all the phases simultaneously by using optimal en-tangled state inputs and a collective measurement (when nec-essary), which is called ‘simultaneous estimation’ or a ‘globalstrategy’. The comparison of these three cases aims to ad-dress the following questions: Can schemes using quantumresources beat the SQL in multi-parameter estimation? Is asimultaneous estimation approach beneficial as compared toan optimal (maybe quantum) individual estimation approach?Relevant studies attempting to answer these questions have re-cently been started from various points of view, arising fromthe fact that multiparameter estimation is non-trivial, and de-pends on the kind of parameters estimated, as well as type ofsensing scenario [203].In one of the earliest works on this topic, Humphreys etal. showed that the quantum enhancement in the precision ofsimultaneous estimation of multiple phases can be obtainedby a coherent superposition of N photons among d modes,in which individual phases θ j = , ··· , d are encoded [347]. Ithas been shown that the considered entangled state is opti-mal among the class of definite-photon-number states, with N being the total number of photons, and the advantage over in-dividual estimation scaling with O ( d ). The particular optimalmeasurements identified in ref 347 have recently been gener-alized in ref 206, which derived the necessary and su ffi cientconditions for projective measurements to saturate the multi-parameter QCR bound in the case of pure probe states.Among N -particle photonic states, Holland-Burnettstates [326] and NOON states have been shown to be theoptimal states for simultaneous multiple-phase estimationwhen d = d improvementover individual estimation, as shown in ref 347, but the useof Gaussian states o ff ers no more than a factor of 2 improve-ment [349]. Furthermore, when both definite-photon-number states and indefinite-photon-number states are all consideredtogether, the precision given by the QCR bound for simulta-neous estimation can be obtained or even further enhancedthrough an individual estimation strategy using propermode-separable input states [207]. It was found that a largeparticle-number variance within each mode plays a crucialrole in improving the QCR bound of multiple-phase estima-tion. This comparable or even better sensing performanceprovided by individual schemes suggests experimentallymore favorable settings for multiple-phase estimation.Moreover, the ultimate quantum limit for simultaneousmultiple-phase estimation is only achievable by an optimalstate that manipulates entanglement among both particles andmodes [350]. A more conclusive mathematical proof hasbeen provided by Proctor et al. , showing that entanglementin simultaneous estimation leads to no fundamental precisionenhancement over individual estimation when the generatorsof the multiple parameters commute and yields no more thana factor of 2 enhancement even when the generators do notcommute [351].Another interesting but more challenging scenario is to es-timate multiple phases that are governed by non-commutinggenerators, e.g., the three components of a magnetic fieldin terms of the spatial coordinates [353]. For this scenario,a three-fold improvement over individual estimation strate-gies has been shown to be achievable by simultaneous es-timation schemes using permutationally invariant quantumstates [354], including the superposition of three Greenberger-Horne-Zeilinger-type states, each of which is known to at-tain the QCR bound of estimating a magnetic field alignedalong one of the specific axes [355]. The work in ref 354 alsoshowed that too much entanglement is detrimental for achiev-ing a Heisenberg scaling in terms of the total number of parti-cles N . In general, for non-commuting generators, there exitsa tradeo ff for the precisions among the individual estimators,i.e., more precise estimation of one parameter leads to lessprecise estimation of the others [231, 356, 357]. Hou et al. have recently derived the minimal tradeo ff for the precisionof simultaneous estimation of a three-dimensional magneticfield, finally leading to the identification of the ultimate quan-tum limit [358].Entanglement appears to lie at the heart of multiparam-eter estimation and has been studied from several perspec-tives, but here again, in the estimation of multiple phases,entanglement does not always guarantee a quantum enhance-ment [207, 350]. However, entanglement becomes a moresignificant factor in estimating a global parameter that is com-posed of multiple parameters to be encoded across multiplemodes or locations, e.g., distributed sensing or networkedsensing. The global phase parameter that is often consideredis a linear combination of multiple phases, written as ˜ θ = (cid:80) j n j θ j , with positive n j and normalization (cid:80) j n j =
1. Forthis, eq 39 needs to be used. The e ff ect of quantum entangle-ment in the uncertainty ∆ ˜ θ of distributed sensing has sparkedsome interest. Recent work suggests useful schemes usingquantum entanglement to gain an enhancement over schemesthat measure the { θ j } individually and then compute a globalparameter ˜ θ via classical communication.Along the lines mentioned above, Proctor et al. have shownthat entanglement can significantly enhance the precision only7 (b) (c)(a) FIG. 18. (a) (left) An experimental set-up for distributed phase sensing with M =
4. A displaced squeezed state input is split into four identicaland entangled probes through a BS network (BSN). The multiple-phase-encoded probe state is measured with homodyne detection (HD) set-ups, from which the average phase of multiple phases is estimated. (right) The precision of the estimated average phase for di ff erent averagenumbers N of photons per sample are compared between the entangled scheme ( σ e ) and the separable scheme ( σ s ). Their theoretical precision(solid lines) as well as the SQL (dashed line) are also presented. Reprinted from ref 208 with permission from Springer Nature. (b) (left) Aphase-squeezed state injected into two variable BSs configures the continuous-variable multipartite entangled probe to estimate the averagedisplacement. An individual displacement on the squeezed phase quadrature variable is induced by an electro-optic modulator (EOM) drivenby radio-frequency (RF) fields. The output state is measured by homodyne detection, leading to the estimate of the average displacement.(right) The variance of the estimated values as a function of the transmittivity of the second BS (VBS2) is compared among di ff erent cases:entangled sensors (circles and black curves for experiment and theory), classical separable sensors (triangles and green curves for experimentand theory) and the SQL (black orizontal dotted line). Adapted and reprinted from ref 352 with permission from American Physical Society. when global parameters are of interest in estimation [351].Ge et al. considered a linear optical network and separa-ble quantum inputs to show that injecting separable quantumstates into only a few modes out of all the modes achieves aquantum enhancement [359], resulting in the Heisenberg scal-ing 1 / Nd in terms of both the constrained photon number N and the number of parameters d . Particularly, TF states havebeen shown to be useful quantum states achieving the Heisen-berg scaling ∝ / Nd , showing a quantum improvement overan individual estimation approach with the precision scaledas 1 / N √ d . Zhuang et al. have proposed theoretical schemesthat use a squeezed vacuum input being injected into a BSarray to estimate a linear combination of displacement param-eters, or that of multiple phases [360]. The proposed schemeswere shown to achieve Heisenberg scaling in the absence ofloss, and have recently been experimentally demonstrated forphase parameters [208] [see Figure 18(a)] and for displace-ment parameters [352] [see Figure 18(b)]. These proposedschemes can be enhanced by continuous-variable error correc-tion to reinstate the Heisenberg scaling, at least up to moderatevalues of d , even in the presence of loss or decoherence [361].The schemes have all used homodyne detectors, but single-photon detectors can also be used with an anti-squeezing op-eration that transforms the initial squeezed vacuum state intothe vacuum, which achieves Heisenberg scaling in distributed phase sensing [362].In the special case that Gaussian states are employed asan input into an array of BSs to estimate the average of in-dependent phase shifts, Oh et al. have identified the opti-mal scheme that exploits partially entangled Gaussian probestates, as maximally entangled probe states are rather detri-mental [209].As is evident from the above-mentioned recent work, manystudies appear to show that entanglement may not be theonly quantity that determines the characteristic behavior ofthe multiple-phase quantum sensors under investigation. Insuch a sense, a new operational concept called multiparame-ter squeezing has recently been suggested to identify metro-logically useful states and optimal estimation strategies [366].This can be seen as a counterpart to the NRF that characterizesquantum enhancement in quantum noise reduction in intensitymeasurements. This is highly relevant to quantum plasmonicimaging.
5. Quantum sensing with SU(1,1) interferometers
Conventional MZIs like those illustrated in Figure 10(b)consist of two inputs and two outputs, and can be describedin a group-theoretical framework as SU(2) [328], the special8
FIG. 19. (a-c): SU(1,1) nonlinear interferometers with direct intensity readout, with dual homodyne readout, and with a truncated layoutincluding dual homodyne readout after a single nonlinear amplifier. A pump laser is illustrated in green, probe and conjugate fields arerepresented in red and blue, and BSs are illustrated to represent loss in the interferometer, with transmission η . A transducer is shown insidethe interferometer that imparts the sensors phase φ onto the probe. Reprinted from ref 363 with permission from American Physical Society.(d): Precision (normalized here by the mean photon number | α | , and presented as the variance rather than the standard deviation of φ ) asa function of the gain of the second nonlinear amplifier in a nonlinear interferometer with direct intensity readout. The gain of the firstnonlinear amplifier is set to G =
2, and the internal transmissions are set to η ai = η bi = η i =
1. The blue, yellow, green, and orange linesrepresent η ae = η be = η e = η e for a nonlinear interferometer with asymmetric gain between the twononlinear amplifiers. Increased gain r in the second nonlinear amplifier reduces the sensitivity to external losses 1 − η e . Reprinted fromref 364 with permission from American Physical Society. (f): Example of a sensor based on a truncated nonlinear interferometer. The Rbvapor cell serves as a nonlinear amplifier and the transducer imparts the sensors phase onto the probe or the probe’s local oscillator. Reprintedfrom ref 365 with permission from American Physical Society. unitary group of degree 2. The SU(2) group, which is a simpleLie group, is equivalent to the rotation group in three dimen-sions, which has the nice feature that it allows one to visualizethe operations of BSs and phase shifters as rotations in three-dimensional space. This is true for both classical and quantumoptical readout fields. When the BSs in linear MZIs, or otherinterferometers like Fabry-Perot interferometers, are replacedby nonlinear amplifiers like four-wave mixers or optical para-metric amplifiers, as shown in Figs. 19(a) and (b), the result-ing nonlinear interferometer can be described by the SU(1,1)group [328], another type of simple Lie-group where the oper-ations can be visualized as rotations and Lorentz boosts. Sim-ilarly, truncated nonlinear interferometers composed of onenonlinear amplifier followed by dual homodyne detectors, asshown in Figs. 19(c) and (f), can also be described by theSU(1,1) group [365, 367, 368]. Describing interferometricsensors with such a group-theoretical approach allows for astraightforward analysis of the quantum states that optimizean interferometericic estimation precision. Both SU(2) and SU(1,1) interferometers can achieve a phase precision with ascaling approaching 1 / N for N photons, but SU(2) interfer-ometers require a squeezed vacuum state to be injected intothe second channel of the first BS in order to achieve the HLwhen the first channel is fed by a coherent state of light, asdiscussed in section III D 1. In contrast, SU(1,1) interferome-ters can be operated with vacuum, coherent state, or combinedcoherent state and squeezed vacuum state inputs, and they of-fer an enhancement in precision that scales with the gain of thenonlinear amplifier [328, 369, 370]. However, because at leastone of the nonlinear amplifiers in a conventional SU(1,1) non-linear interferometer relies on phase-sensitive amplification, itis essential to maintain a near-zero phase di ff erence within theinterferometer to maintain the precision advantages [371].As discussed in section III D 1, the main obstacle to prac-tical quantum phase sensing with squeezed states of light isprotecting the vulnerable quantum advantage against losses,as described in eq 68. Indeed, in the presence of losses,SU(2) interferometers with a squeezed vacuum input asymp-9totically approach the sensitivity of classical SU(2) interfer-ometers with the same mean photon number [372]. As a re-sult, the development of detectors with near-unity detectione ffi ciency has been a critical research topic for years. Whilesome progress has been made with Bayesian parameter esti-mation as a tool for mitigating the e ff ect of losses on squeezedSU(2) interferometers [291], SU(1,1) interferometers o ff er analternative approach that is more robust against losses in somecases. Their dependence on loss can be broken down into de-pendencies on internal and external losses, represented hereby (1 − η i ) and (1 − η e ), respectively. Here, η i and η e are as-sumed to be balanced for both arms of the interferometer, i.e. η ai = η bi = η i and η ae = η be = η e , as shown in Figures 19(a)-(c). For a SU(1,1) interferometer with direct intensity readout,as illustrated in Figure 19(a), external losses, or losses afterthe second nonlinear amplifier largely represent losses due tofiltering before detection and losses due to imperfect detectore ffi ciency. Internal losses include losses within the nonlinearamplifiers as well as losses from optical interactions with asensor in the interferometer. In the limit of η i =
1, and whenthe interferometer is operated in a balanced mode where thetwo nonlinear amplifiers have identical gain and loss, with acoherent state used in each input port, the external loss modi-fies the precision ∆ φ of the SU(1,1) interferometer as [371] ∆ φ e = η − / ∆ φ, (73)In other words, unlike SU(2) interferometers, external loss ina balanced SU(1,1) interferometer does not change the func-tional description of the precision except for the addition ofa prefactor because of the second nonlinear amplifier [373].In fact, because the second nonlinear amplifier is a phase-sensitive amplifier that can exhibit noiseless amplification, in-creasing its gain can compensate for external loss. The e ff ectof external loss in an unbalanced SU(1,1) interferometer isshown in Figure 19(d) [363]. Here, the gain of the first non-linear amplifier is set to 2, internal losses are neglected, andthe gain of the second nonlinear amplifier is varied. In thecase of no external loss, the precision is almost independentof gain. On the other hand, in the limit of arbitrarily large gain,the interferometeric precision becomes insensitive to externalloss [363]. Figure 19(e) expresses this insensitivity to externalloss more directly by plotting the interferometeric precision,normalized by the SNL, as a function of η e for three di ff erentgain ratios, r / r . As the gain of the second amplifier, r , is in-creased relatively to that of the first, r , the precision becomesincreasingly insensitive to external loss.However, for the purposes of plasmonic sensing, internallosses that occur on a plasmonic sensing element within onearm of the interferometer must be considered. When η e is setequal to unity and the internal transmission η i is varied (setting η bi = η ai = η i such that any additional loss from the sensor ismatched), for bright coherent inputs | α (cid:105) and | β (cid:105) seeding thefirst nonlinear amplifier and N i photons inside the balancedinterferometer, the precision ∆ φ of the SU(1,1) interferometerbecomes [371] ∆ φ i = (cid:34) + − η i η i N i | α | + | β | (cid:35) / ∆ φ, (74)This suggests that balanced SU(1,1) interferometers are robustagainst small levels of loss, but as with conventional squeezed interferometers, increasing loss will push the precision to theSNL.Truncated nonlinear interferometers that include only a sin-gle nonlinear amplifier followed by dual homodyne detection,as illustrated in Figs. 19(c) and (f), provide the same phaseprecision as conventional nonlinear interferometers, but theyo ff er some substantial advantages. First, they o ff er a simplerexperimental design by removing the phase-sensitive nonlin-ear amplifier from the nonlinear interferometer. Second, theytypically exhibit reduced loss (and thus a greater quantum en-hancement) because the nonlinear amplifier itself exhibits atradeo ff between gain and loss that always introduces someloss into the measurement. Finally, truncated nonlinear in-terferometers can use arbitrarily high power local oscillators.This is critical because the SNL for such a measurement isdefined by the sum of the powers of the two-mode squeezedstate output from the interferometer and the local oscillators.Thus, it is possible to arbitrarily increase the power of the lo-cal oscillators, thereby increasing the SNR of the measure-ment, without introducing excess power to the sensor, andwhile taking advantage of the quantum noise reduction in atwo-mode squeezed state. No truncated nonlinear interfero-metric plasmonic sensor has been demonstrated in the liter-ature to date, but a growing literature has described the pre-cision of this approach for photosensitive sensing applica-tions [365, 367, 368]. E. Quantum sensing beyond the Cramér-Rao bound
For phase estimation using a pure probe state in a single-mode setup with ˆ U ( φ ) = e i φ ˆ a † ˆ a or in a two-mode setupwith ˆ U ( φ ) = e i φ (ˆ a † ˆ a − ˆ b † ˆ b ) , the QFIs of eq 35 are given as H = (cid:104) ( ∆ ˆ a † ˆ a ) (cid:105) and H = (cid:104) [ ∆ (ˆ a † ˆ a − ˆ b † ˆ b )] (cid:105) , respectively, where (cid:104) .. (cid:105) denotes the average over the probe state. This indicates thatphase estimation becomes more precise by increasing the pho-ton number variance of the probe state in a single-mode setupand the variance of the photon number di ff erence of the two-mode probe state in a two-mode setup, respectively. A naturalquestion is then, how large can these variances be increasedby?Rivas and Luis [374] have suggested an estimation strat-egy in a single-mode setup using a superposition of a vacuumand a squeezed vacuum state. The considered probe statehas been shown to achieve an arbitrary scaling in precision,i.e., ∆ φ ∝ N − k , where N is the mean photon number and k is an arbitrary value with k >
1. Such an alluring analysissparked an intensive debate in the literature because it seemedto beat the Heisenberg scaling ( ∆ φ ∝ N − ) that is regardedas the fundamental ultimate limit that can never be beaten byany other physical strategy [375]. The state studied by Rivasand Luis is not the only example and states with various pho-ton number distributions have been considered as candidatesfor increasing the QFI, for example, the SSW state [376], theSS state [377], Dowling’s model [378], and the small-peakmodel [379]. When the maximal photon number of a probestate is upper bounded, the so-called ON state—a superpo-sition of vacuum and a Fock state—is known to lead to themaximal photon number variance [380]. One can show thatthe use of the ON state in single-mode phase estimation can0indefinitely increase the QFI, while keeping the average pho-ton number fixed. The ON state is also known to be useful inquantum computation [381], and a version with 18 excitationshas been realized in the harmonic motion of a single trappedion [382].On the other hand, when the maximal photon number of aprobe state is unbounded [383], more bizarre photon numberstatistics can be found, as discussed in ref 380. For exam-ple, phase estimation using a probe state with a Borel photonnumber distribution [384, 385] can lead to sub-Heisenbergscaling ∆ φ ∝ N − / , calculated by the QCR bound [380].Some heavy-tailed and sub-exponential distributions exhibit adiverging or even an infinite variance [386]. Particular interesthas been paid to the Riemann-Zeta distribution as an exampleshowing an infinite QFI, leading to completely precise phaseestimation without uncertainty in a two-mode scheme [387].This last example is rather mysterious. Together with the othersub-Heisenberg strategies mentioned above, these examplesneed to be justified in order to certify if their precisions areachievable in practice, putting aside the question of how onemight generate those photon number distributions in the firstplace.Apart from exploiting exotic photon number statistics, sub-Heisenberg-limited precision can also be achieved by usingnonlinear e ff ects in many-body systems [388–394]. The non-linear e ff ects in atomic ensemble systems have experimentallydemonstrated sub-Heisenberg-scaling [395]. However, a care-ful examination of the total resources is needed, as this deter-mines how the precision scales.Over the last decade, the debate has been devoted to addressone simple question: Can the HL or scaling be beaten? [309,328, 333, 396–398] Fortunately, a conclusive answer has fi-nally been proved [375, 396, 399–405], and shows that theoverall scaling should properly include the amount of re-sources required to obtain a priori probability distribution ofthe unknown parameter and the number of measurements re-peated to achieve the asymptotic QCR bound. With such anaccounting of the total resources, one can show that the afore-mentioned scenarios all turn out to be Heisenberg scaling-limited.In particular, when the likelihood function being used forthe QCR bounds is highly non-Gaussian and the sample sizeis small, Ziv-Zakai (ZZ) bounds are known to be more ap-propriate [406–410]. The quantum version of the ZZ boundhas been derived, showing that the MSE would be higher andthus tighter than a corresponding QCR bound [399]. UsingZZ bounds, Giovannetti and Maccone have proved that sub-Heisenberg strategies are ine ff ective [400]. When a smallamount of prior information is given, no sub-Heisenberg scal-ing is achievable, i.e., sub-Heisenberg scaling requires a largeamount of prior information. When a large amount of priorinformation is given, however, one can just guess a randomvalue based on the prior distribution without performing anymeasurement. A random guess with a large amount of priorinformation has been proven to achieve a comparable preci-sion to the corresponding ZZ bound. It has also been shownthat the maximum gain analyzed by the ZZ bound with a uni-form prior is about 1.73 with respect to the initial uncertaintygiven in the prior distribution of the true value. Consideringa finite amount of prior information, Górecki et al. have re- cently showed that the HL needs to be corrected by an addi-tional constant factor of π [411].To investigate the practical achievable precision with a fi-nite amount of prior information and a limited number ofmeasurements, which often nullifies QCR bounds in practice,Bayesian approaches have been considered in various situa-tions [405, 412–415]. In this respect, it has been shown thatquantum sensors can be enhanced by machine learning [416]or calibrated by neural networks [417]. IV. QUANTUM-ENHANCED PLASMONIC SENSORS
As seen in section II, plasmonic structures provide sub-di ff raction sensing with high sensitivity [21–28], whereasfrom section III it is clear that quantum resources provide theability to reduce the noise and estimation uncertainty belowthe SNL or SQL [37–46]. Individual sensing techniques havebeen developed for plasmonic sensing and quantum sensingindependently from each other over the last few decades,which has led to the establishment of separate scientific fieldsin academia and industry. Research is now being devoted tocombining the techniques of plasmonic sensing and quantumsensing with the aim of providing a new breed of plasmonicsensors with high sensitivity and high precision at scales be-low the di ff raction limit. These new types of sensors are called‘quantum plasmonic sensors’ and they provide a sensing per-formance that cannot be obtained solely by either classicalplasmonic sensors or conventional quantum optical sensors.In this section, we review recent studies that have exploitedquantum resources to improve the sensing performance ofplasmonic sensors.The decomposition of a general sensing procedure dis-cussed in section III A (see also Figure 9) allows us to clas-sify four di ff erent kinds of sensor [418]: First, a sensor iscalled a ‘classical sensor’ if a classical state input, an ordi-nary optical transducer, and a classical measurement schemeare used. Second, when the classical input and measurementare replaced by a quantum state input and a quantum measure-ment, this introduces quantum observables, and the sensor iscalled a ‘quantum sensor’, enabling the potential for sub-shot-noise sensing. Third, when the ordinary optical transducerin a classical sensor is replaced by a plasmonic transducer,but the state and measurement remain classical, the sensor iscalled a ‘plasmonic sensor’, enabling sub-di ff raction sensingwith high sensitivity, but ultimately shot-noise limited. Fi-nally, a ‘quantum plasmonic sensor’ is formed when a plas-monic structure is employed for the transducer, and a quan-tum state input and a quantum measurement are used. In thiscase, the plasmonic structure provides a high sensitivity viathe sub-di ff raction confinement of light [18–20] , while thequantum input state and measurement provide the potentialfor sub-shot-noise sensing. The four kinds of sensors are illus-trated in Figure 20, with the hope that quantum plasmonic sen-sors are expected to achieve combined benefits that have onlybeen obtained individually by plasmonic sensors or quantumsensors.More generally, the quantum properties of a plasmonic sys-tem can also play a role in sensing, e.g., electron tunnel-ing [419, 420] or quantum size e ff ects [421–423]. In this case,1 FIG. 20. The four di ff erent types of photonic sensors, classifieddepending on the type of physical system used for the probe state,measurement scheme, and transducer encoding the parameter to besensed. The default type is the classical sensor (bottom left), whichcan be upgraded by moving in either the horizontal or vertical di-rection to become a plasmonic sensor (bottom right) or a quantumsensor (top left). Such an upgrade involves the individual benefits ofeither type, however, the benefits of both types can be achieved by anultimate type of photonic sensor, called a quantum plasmonic sensor(top right). the quantum plasmonic sensors do not necessarily use exter-nal quantum resources, such as input states or measurements,but exploit inherent quantum e ff ects stemming from the plas-monic systems themselves. A careful analysis of whether thequantum e ff ects give rise to sub-shot-noise sensing or simplyimprove the sensitivity must be performed. While the meth-ods presented in this review are invaluable for such analyses,in the following sections we first focus on recent studies thatfit into the above introduced classification of quantum plas-monic sensors, where the plasmonic structure provides a highsensitivity and quantum input states and measurements pro-vide the capability for sub-shot-noise sensing, thus improvingthe LOD. We then briefly review other approaches to quantumplasmonic sensors at the end of the section that do not fit intothis classification. A. Quantum plasmonic intensity sensing
The first method we describe is the use of a plasmonic trans-ducer that encodes parameter information into light where theintensity of the light is dependent on the parameter to be esti-mated. Below, we review ‘intensity-sensitive’ plasmonic sen-sors that use quantum states of light and quantum measure-ments.
1. Intensity sensing with discrete variable states
As introduced in section II B, the ATR prism setup can beused in such a way that the intensity of the reflected light from the prism is modulated depending on the refractive indexof an optical sample under characterization. To use the the-ory of intensity parameter sensing discussed in section III B 1,we describe the reflection from the prism as the transmissionthrough the prism setup, i.e., | r spp | = T , where r spp is givenby eq 13.A practical and useful intensity-parameter SPR sensingscheme is the two-mode scheme shown in the inset of Fig-ure 15, but where the BS with transmittance T is replaced bya prism setup. A relevant quantum theory of the ATR prismsetup [424, 425] can be used. Within the ATR prism setup,the two-mode scheme can be considered in terms of a sig-nal mode that passes through the prism setup ( a ) and an idlermode that is kept as a reference ( b ). In this regard, Lee etal. theoretically studied the two-mode scheme with a partic-ular type of two-mode state [426] called a twin-mode (TM)state | Ψ twin (cid:105) = (cid:80) c n , m | n (cid:105) a | m (cid:105) b with | c n , m | = | c m , n | . Such astate has the following symmetric properties: (cid:104) ˆ a † ˆ a (cid:105) = (cid:104) ˆ b † ˆ b (cid:105) and (cid:104) ( ∆ ˆ a † ˆ a ) (cid:105) = (cid:104) ( ∆ ˆ b † ˆ b ) (cid:105) , which covers path-symmetricstates with c n , m = c ∗ m , n e − i γ [296, 427]. In this two-modeplasmonic sensing scheme, an intensity-di ff erence measure-ment of the observable ˆ n − = ˆ b † out ˆ b out − ˆ a † out ˆ a out is considered,i.e., the intensity of the transmitted light of the signal mode(reflected from the prism) is compared with the intensity ofthe reference mode. The measured intensity-di ff erence can beinserted into an appropriate estimator together with eq 13 inorder to estimate the refractive index of an analyte.As a classical benchmark, the balanced PC state in-put | α (cid:105) a | α (cid:105) b can be considered. When the average photonnumber of the individual modes is restricted by N for boththe TM state input and the PC state input, i.e., (cid:104) ˆ a † in ˆ a in (cid:105) = (cid:104) ˆ b † in ˆ b in (cid:105) = N , the intensity-di ff erence signal for both statesis the same, i.e., (cid:104) ˆ n − (cid:105) = N ( η b − η a T ) with non-ideal chan-nel transmittance η a,b , which includes the detection e ffi ciency,as modeled in Figure 15. However, the associated measure-ment noise (cid:104) ( ∆ ˆ n − ) (cid:105) is di ff erent, leading to di ff erent SNRs.By comparing the SNR of the TM state with that of the PCstate, one finds that their SNR ratio R SNR = SNR TM / SNR PC can be written as eq 66, where the Mandel-Q parameter Q M and the NRF σ of eq 52 play important roles in intensity-sensitive SPR sensing. As discussed in section III C 3, a quan-tum probe with − ≤ Q M < ≤ σ in < R SNR >
1. The optimal state thatmaximizes the ratio R SNR is the TF state, which exhibits anunconditional quantum enhancement regardless of the valuesof N , T and η a,b . The TMSV state of eq 53, on the other hand,provides a conditional quantum enhancement that depends onthe values of N , T and η a,b . In particular, when η a ≈ η b and T ≈
1, or η a T ≈ η b , the ratio R SNR for the TMSV statecase becomes almost the same as the case of using a TF stateinput.As mentioned in section III, the Fock state input is knownto be the optimal state reaching the ultimate quantum limitin the estimation uncertainty in both single parameter (sec-tion III C 1) and multiparameter (sections III C 2 and III C 3)estimation scenarios that monitor the change of the intensityof the transmitted light through an optical sample [221, 238,239]. However, Fock states with high photon number N (cid:29) APD
PPKTP
SMF SPDC Type II Q W P H W P H W P B a n d p a ss fi l t e r Lens Coincidence APD P B S P r i s m G o l d M i rr o r SignalIdler I r i s P B S A n a l y t e ✓ in
Acrylic boxSMF: single mode fiberHWP: half-wave plateQWP: quarter-wave platePBS: polarizing beam splitter
FIG. 21. Schematic of the prism setup probing the concentration of BSA with heralded single photons. The photons of a photon pair generatedby a PPKTP nonlinear crystal are sent to the ATR prism setup and to a detector of the idler channel, respectively. The detection of the idlerphoton heralds that the signal photon has been injected into the ATR prism setup. The number of transmitted signal photons are then countedfor a given analyte and an incident angle that can vary. Reprinted from ref 226 with permission from Optical Society of America. nology [428–430]. Only a few novel schemes have been sug-gested for the generation of large Fock states, but the fidelityis limited [431–435]. Alternatively, one can use N singlephotons to achieve the same limit that would be obtained bythe Fock state with | N (cid:105) . This can be seen by the fact thatthe individual single photons in the Fock state | N (cid:105) undergoan independent Bernoulli sampling. Such an equivalenceallows the use of single photons for various quantum sen-sors [213, 219, 220, 225], including a plasmonic sensor [226]which we now introduce.The experimental quantum plasmonic sensing setup con-sidered in ref 226 is shown in Figure 21. Single-photon pairsare generated via SPDC in a nonlinear periodically poledpotassium-titanyl-phosphate (PPKTP) crystal, where the de-tection of the idler photon heralds that the signal photon hasbeen injected into the prism setup. The detection of N idlerphotons thus corresponds to the use of N signal photons in thesignal channel, i.e., equivalent to the use of the Fock state | N (cid:105) in the sense of estimation uncertainty. The experimental esti-mation uncertainty is measured by the repetition of an identi-cal experimental observation a thousand times, which yieldsa distribution of the estimated total transmittance through thewhole setup, T total = N t / N , where N t is the number of trans-mitted and detected signal photons out of N injected heraldedsingle photons for each repetition. The e ff ective transmittancethrough the prism setup can be obtained by normalizing thetotal transmittance by a normalization factor N , i.e., T prism = T total / N . In the experiment, the normalization factor is de- fined as the average transmittance measured with an air ana-lyte at an individual incident angle θ in , i.e., N ( θ in ) = (cid:104) T air ( θ in ) (cid:105) which leads to T prism ( θ in ) = T total ( θ in ) / (cid:104) T air ( θ in ) (cid:105) . The use ofair is because the light entering the prism is o ff -resonant fromthe plasmonic excitation across the entire range of incidentangles considered. This approach enables the elimination ofthe e ff ect of an incident angle-dependent misalignment whenscanning through a wide range of incident angles.The above quantum plasmonic sensor aims to estimate therefractive index of the blood protein of bovine serum albu-min (BSA) in aqueous solution whose concentration C is con-trolled in this proof-of-principle demonstration. From themeasurement of the average transmittance (cid:104) T prism (cid:105) , the refrac-tive index of the analyte is estimated according to the theorymodel of the reflectance [see eq 13] by setting (cid:104) T prism (cid:105) = R spp .The analyte is placed on the opposite side of the gold film(see the inset in Figure 21) and its refractive index a ff ects theresonant condition of SPPs, changing the resonance angle sat-isfying eq 12 or equivalently the intensity of the transmittedlight for a fixed incident angle, i.e., (cid:104) T prism (cid:105) .The overall behavior of the measurements are presented inFigs. 22(a) and (b). The average transmittance (cid:104) T prism ( θ in ) (cid:105) measured in the experiment when varying the incident angle θ in is shown in Figure 22(a) for deionized water, i.e., C = C = (a) BSA 2%DI water
Histogram at ✓ in = 67
25% steps. (d) The estimation uncertainty of the refractive index, given as the error bar at each pointin (c), corresponding to the theoretical expectation for the classical and quantum scenarios, demonstrating a 10 ∼
20% quantum enhancement.Reprinted from ref 226 with permission from Optical Society of America. pend on the incident angle, which modulates the transmittanceof the setup and thereby the output photon number statistics.Note that the size of the error bars represents the estimationuncertainty of the e ff ective transmittance for N single-photoninputs. The corresponding classical benchmark is defined bythe case of using a coherent state with the same input poweras N single photons. The experimentally measured quan-tum noise is thus compared with the classical benchmark, asshown in Figure 22(b). Here, the measured uncertainty is in-terpreted in terms of the total transmittance (cid:104) T total (cid:105) that con-sists of the e ff ective transmittance through the prism setupand loss including the detection e ffi ciency. Such a compari-son clearly exhibits a quantum enhancement in the experimentand is in good agreement with the ultimate quantum limit in-troduced in section III C 1.For a specific estimation of the refractive index, the in-cidence angle is set to θ in = . ◦ as an example and themeasurement is repeated for each concentration in a rangefrom 0% to 2% in 0 .
25% steps. From the repeated mea-surements, the refractive indices are estimated using eq 13(i.e., the reflectance R spp ). The relation between the refrac-tive index and the concentration of BSA solution can be de-termined by the slope of the linear fitting function, yield-ing d (cid:104) n BSA (cid:105) / d C = (1 . ± . × − [see Figure 22(c)],which is in good agreement with the previously measuredvalue of 1 . × − [436]. The quantum property of lightplays no role in this relation of the mean values, which can be thought of as more of a calibration, but it plays an im-portant role in reducing the estimation uncertainty of the re-fractive index. The latter is demonstrated in the experimentby repeating the estimation, which produces a distribution ofthe estimated refractive indices. The estimation uncertaintieswith varying n BSA are shown in Figure 22(d), which evidentlydemonstrates a 10 ∼
20% quantum enhancement in com-parison with the classical benchmark. As described in sec-tion II B 3, this enhancement directly a ff ects the LOD of thesensor and improves its sensing performance.As discussed in section III C 1 and shown in Figure 22(b),the quantum enhancement in the intensity-sensitive sensor de-pends on the total transmittance of the whole sensing setup.Therefore, the quantum enhancement can be further increasedby improving the total transmittivity of the setup by minimiz-ing channel loss and maximizing the detection e ffi ciency.A tapered fiber based quantum SPR sensing scheme hasalso been demonstrated recently for measuring BSA concen-tration [437] and salinity [438]. In these works, a taperedhetero-core structure was fabricated, which is composed oftwo multi-mode fibers and a single-mode fiber, coated witha 50 nm thick gold film [see Figure 23(a)]. Heralded singlephotons were used as inputs, prepared by SPDC using a 2mm thick type-II metaborate crystal ( β -BaB2O , BBO) andby applying a coincident detection scheme. Based on the longevanescent field decay of the single-mode fiber and by reduc-ing the diameter of the fiber, the authors showed that it was4 FIG. 23. (a) Schematic of the tapered hetero-core fiber quantum sen-sor. (b) Experimentally obtained variance in the total transmittanceof the system shown together with the theoretical values for classical(red) and quantum (blue) cases. Reprinted from ref 438 with permis-sion from Elsevier. possible to reach the theoretically predicted estimation uncer-tainty below the SNL, as shown in Figure 23(b).
2. Intensity sensing with continuous variable states
Sensors that exploit quantum noise reduction, or squeezedlight [187], have seen renewed interest in recent years as agrowing number of devices that utilize optical readout - fromgravitational wave detection to ultra-trace plasmonic sens-ing at the nanoscale. They have approached their ultimatelimits of detection as defined by the Heisenberg uncertaintyprinciple. At this limit, the noise is dominated by back ac-tion and the quantum statistics of light, leading to the HL,or the SNL when a coherent state of light is used. Simul-taneously, many devices, including plasmonic sensors, havereached tolerance thresholds in which power in the readoutfield can no longer be increased without increasing the noisedue to back action or thermal e ff ects. Beyond these limits,squeezed light can be used to further improve the precision inthese platforms. In recent years a growing number of sen-sors based on quantum noise reduction have been demon-strated [55, 232, 278, 280, 283, 439–447].As discussed in section III C 3, intensity-di ff erence noisereduction allows one to replace shot-noise limited sensors thatoperate in a di ff erential configuration in order to reject com-mon mode noise and go below the SNL with a version thatexploits quantum correlations present in two-mode squeezedstates. One expedient method to obtain intensity-di ff erencesqueezing is via FWM [275–277], which is based on a third-order optical nonlinearity. An example setup is shown in Fig-ure 24. The noise in this amplifier can be derived in the in-teraction frame of the Heisenberg picture. In this frame, theinteraction Hamiltonian for the case with degenerate pumpingfields is H = i (cid:126) χ (3) ˆ a pr ˆ a c ˆ a † p ˆ a † p + H.C. , (75)where χ (3) is the nonlinear coe ffi cient, ˆ a p is the pump fieldoperator, ˆ a pr is the input probe field’s operator, and ˆ a c is athird ‘conjugate’ field which is parametrically amplified fromthe vacuum. The energy level diagram describing this systemis shown in Figure 24.In many experiments the pump field is powerful relativeto the probe and is undepleted. In this simplified scenario the FIG. 24. FWM in rubidium vapor, simplified schematic for quan-tum noise reduction. A pump laser (P) is used to derive a probe (Pr)field by frequency-shifting the pump in an acousto-optic modulator(AOM). The double-pass configuration ensures the probe is not dis-placed relative to the pump as the AOM frequency is selected. Theprobe is o ff set from the pump by 3 GHz, or approximately equal tothe hyperfine ground state splitting at the D1 line (795 nm). Thepump and probe are jointly detuned from resonance by approxi-mately 0.8 GHz. They overlap at a small angle (0.3 ◦ ) inside theRb vapor cell. The resulting probe and conjugate fields are separatedfrom the pump by a polarizing beam splitter (PBS) and producesa TMSD state. When incident on a balanced detector, the intensity-di ff erence between the probe and conjugate (C) shows quantum noisereduction, visible on a spectrum analyzer (SA). The inset shows theenergy levels in Rb associated with a double Λ system. The excitedstate hyperfine splittings are much smaller than the ground state, andare blurred with respect to each other in the Doppler-broadened va-por due to heating to approximately 120 ◦ C. As two pump photonsare absorbed, coherence between the hyperfine levels ensures thatwhen the probe field stimulates emission, a third field must be emit-ted from the virtual level illustrated by the upper dashed line in orderto maintain energy conservation. field operators take the form of eqs 56 and 57 in section III C 3.This leads to a TMSD state and intensity-di ff erence noise re-duction as given in eq 63. The important point about eq 63is that the noise is less than one shot-noise unit for η > G >
1. Thus, using two-mode squeezing in lieu of a tradi-tional reference subtraction on a balanced detector will yield a - - - - - η N R F ( d B ) FIG. 25. Quantum noise reduction as a function of transmission η on both the probe and conjugate fields ( η a = η b = η ) for nonlineargain G =
4. This accounts for imperfect detection e ffi ciency of thetwo fields on identical detectors. Here, the noise reduction, or ‘noisepower’ is given by 10 log (1 − P / P ) dB, where P is the shot-noise, P is the noise from the TMSD state, and P / P is the NRF σ . η ( G − / (2 G −
1) over the shot-noise lim-ited measurement [247, 275, 284]. The NRF, or total ‘noisepower’ contained within a power spectral measurement, as afunction of transmission on both the probe mode (a) and con-jugate mode (b), with η a = η b = η and G = FIG. 26. (a) Optical setup showing the FWM experiment with ab-breviations DMD: digital micromirror device, PBS: polarizing beamsplitter, AOM: acousto-optic modulator, BD: beam dump, SA: spec-trum analyzer, Rb: rubidium, λ/
2: half wave plate and ND: neutraldensity. One beam passes through the EOT medium while the otheris attenuated with the neutral density filter in order to balance lossesand maximize the noise reduction, or squeezing. Inset: an SEM im-age of a zoomed in subarray of the nanostructures. (b) Upper right:the image of the probe beam imprinted by the AOM before the plas-monic medium. Lower right: the transmitted image after EOT. Onlythe intensity has been attenuated while the number of spatial modespresent remains the same. images in the probe and conjugate fields that exhibit quantumnoise reduction. The plasmonic medium transmits the probeimage while the conjugate image is attenuated by a neutraldensity filter. The LSPs confined to the edges of the trian-gular holes transmit all of the spatial information containedin the image, unlike the case for SPPs. Figure 26(b) showsthe incident probe image imprinted by the AOM (top) and thetransmitted probe image after the EOT (bottom).The transmission is strongly dependent on polarization dueto the shape of the nanostructures, such that when the po-larization overlaps with an edge of the triangles, a LSP isexcited, resulting in transmission of multiple spatial modes.Figure 27(a) and (b) show the transmission and noise poweras a function of polarization, respectively. As the polariza-
FIG. 27. Extraordinary optical transmission of squeezed light.(a) Single color transmission for the hole array for a polarizationof 0 − ◦ , and (b) The relative noise intensity and SNL for therebalanced probe and conjugate after the probe has passed throughthe EOT medium. (c) The measured squeezing as a function of trans-mission through the hole array and through a variable neutral densityfilter. The noise based on the model in eq (63) is shown in black. Theerror bars in (a) and some error bars in (b) are smaller than the sym-bols. Reprinted from ref 55 with permission from American PhysicalSociety. FIG. 28. Experimental setup for plasmonic sensing in the Kretschmann configuration using FWM to provide the signal probe and referencebeams, with abbreviations: SA, spectrum analyzer; PBS, polarizing beam splitter; ND, neutral density filter; P, pump beam; Pr, probe beam;C, conjugate beam; AOM, acousto-optic modulator. The probe passes through a prism coated with a plasmonic thin film on the hypoteneuse,while the conjugate beam serves as a reference after balancing its intensity with that of the transmitted probe. Di ff erential detection (A-B)reveals a signal with transmission-dependent NRF. Reprinted from ref 283 with permission from American Chemical Society. tion becomes incident with the base edge of the triangles, thenoise power is well below the SNL, resulting in a quantumnoise reduction. Figure 27(c) shows the quantum noise reduc-tion (squeezing) as a function of transmission. The e ff ectsof losses in SPP waveguides [425] and scattering in metalnanoparticle arrays [289] have been previously treated as ef-fective BSs. A BS model also matches the experimental data,as the theoretical noise reduction as a function of BS trans-mission in Figure 27(c) shows. This experimentally demon-strates the capability for LSP-mediated EOT to transmit quan-tum images while conserving macroscopic quantum informa-tion, such as quantum noise reduction.Exploiting the plasmons’ propensity to conserve quantuminformation means that they can serve not only as goodnanoscale quantum information platforms, but also as quan-tum sensors which make use of squeezing. SPPs have beenused extensively to detect trace biochemical compounds inthe Kretschmann configuration [74]. State-of-the-art classicalSPR sensors utilize di ff erential detection with a reference fieldthat does not interact with the SPP in order to eliminate noisepresent in the probe laser [36, 94, 450]. Many of these sensorsare now only limited by the SNL[35, 36, 94], and quantumsensors are required for further improvements in precision.A plasmonic SPP sensor in the Kretschmann configurationcan use quantum noise reduction to supercede the precisionof state-of-the-art classical device. Figure 28 shows an ex-perimental setup using FWM to produce a TMSD state and aKretschmann sensor to obtain a precision below the SNL.A 43.5 nm thick gold film was deposited on a prism, andindex matching oils were deposited on the film in order tomeasure a shift in the plasmonic resonance as a function ofthe refractive index. The intensity-di ff erence is measured atthe two output channels A and B, with a ND filter placed onthe conjugate mode (A) to equal the transmittivity of mode Band reduce the noise in the measurement signal to a minimum, as discussed in section III C 3. An acousto-optic modulator(AOM) was used to place a modulation on the probe mode(B). This leads to a peak in the measurement signal at thedriving frequency 1.5 MHz when viewed on a spectrum ana-lyzer, as shown in Figure 29(c) and (d) for two example cases.The height of the peak is proportional to the magnitude of theamplitude modulation on the probe field and due to the conju-gate field being unmodulated the peak is also proportional tothe magnitude of the intensity-di ff erence signal. On the otherhand, the sidebands of the signal (either side of the peak) cor-respond to the noise floor (the variance) of the measurementsignal, as there is minimal amplitude modulation of the probe(and thus of the intensity-di ff erence measurement signal) atfrequencies away from the AOM resonance. Using a spectrumanalyzer to give a power spectrum of the measurement signalin this way allows the intensity-di ff erence (proportional to thepeak at resonance) and the intensity-di ff erence noise (the side-bands) to be obtained from the same plot. Thus, both the sig-nal amplitude and noise can be read o ff simultaneously, allow-ing an immediate characterization of the SNR of the system.An additional feature is that technical noise at low frequen-cies (such as laser or vibration fluctuations) can e ff ectively beeliminated in a manner similar to lock-in detection, allowingthe sensor’s precision to be brought close to the theoreticallimit.Figure 29(a) shows the resulting resonance curve of the sig-nal as a function of the angle of incidence and the refractiveindex. Every data point corresponds to a quantum noise re-duction, as shown in Figure 29(b). These points are obtainedby comparing the sidebands of the signal on the spectrum ana-lyzer, as shown in Figure 29(c) and (d) for two example cases,where the peak occurs at 1.5 MHz due to the frequency setby the AOM. A maximum noise reduction of 4.6 dB is ob-tained on the left-hand shoulder of Figure 29(b). It is notablethat a large amount of squeezing (noise reduction) is observed7 FIG. 29. Resonance curves and noise reduction for plasmonic sensing in the Kretschmann configuration with squeezed light. (a) Plasmonicresonance as a function of relative angle of incidence (0 ◦ corresponds to on-resonance for n = .
3) and index of refraction (green: n = = = = = = ff erencebetween the black and red sidebands. Squeezing decreases as transmission decreases, but virtually all data points are squeezed. A maximumof 4.6 dB below the SNL is observed in the left shoulder of the peaks in (a). A minimum of 0.3 dB is observed in the n = = = near the inflection points in Figure 29(b), meaning that onecan choose to operate this SPR sensor at a single position[36],where a large quantum e ff ect is still observed. At all pointson a transmission vs index curve, a higher SNR would beachieved than is possible with the classical version of this sen-sor. Comparing the quantum sensor with a classical sensorin which the incident power may be turned up indefinitely,the quantum sensor still compares favorably. The quantumlight source used here can be operated at equivalent opticalpowers, and theoretically the amount of quantum noise re-duction does not depend on the incident optical power [seeeq (63) in section III C 3]. On the other hand, the classicalsensor cannot be used at powers beyond the point of thermalmodulation[451] of the plasmon or the damage threshold ofphoto-sensitive ligands[452, 453]. These thresholds are eas-ily within the power capabilities of the squeezed light sourceused here and many others. Notably, it is also possible to usethis configuration without amplitude modulation to obtain aquantum-modulated signal at DC [447].Similar sensing configurations have been used to detect ul-trasonic pressure waves [454] and establish long range quan-tum plasmonic networks [455]. In the case of ultrasonic mea-surements, the index of refraction shift is caused by local vari-ations in air pressure above the plasmonic nanosurface at a fre-quency of 199 kHz. Using LSPs and EOT, a change in trans-mission serves as a transduction of index of refraction shiftsonto optical intensity modulation. The magnitude of the inten-sity modulation (representing the magnitude of the refractiveindex modulation ∆ n ) can then be detected with noise belowthe SNL using a TMSD probe state. Figure 30 shows the ex-perimental setup. A chamber with an ultrasonic transducer isused to provide pressure waves at the surface of a plasmonicthin film, which consists of triangular hole arrays. One of thetwo twin beams passes through the EOT sensor while the other continues in free space and acts as a reference. The resultingdi ff erence measurement contains the modulation signal, pro-vided by the function generator to the ultrasonic transducer,on top of a reduced noise background.Figure 31 shows the measured signal (and noise floor) asthe magnitude of the change in refractive index ∆ n increases.The classical signal (coherent state probing) is shown in blue(i) and the TMSD state signal is shown in red (ii). The sig-nals are normalized by the average shot-noise of the classicalcase, shown as a light blue line. Clearly the TMSD state hasreduced noise compared to the classical case: the minimumnoise levels show a di ff erence of 4 dB. The SNL ratio for theTMSD state is higher than the classical case as ∆ n decreases.For very small index shifts, the TMSD version of the sensoris therefore able to discern index shifts that are smaller thanthe classical sensor, enabling a detectable shift ∆ n of O (10 − )RIU. The minimum resolution of the refractive index modula-tion can be determined at a 99% confidence level in the SNR.In the quantum case this gives ∆ n min = . × − RIU withthe TMSD state and ∆ n min = . × − RIU with classicalcoherent states. This demonstrates a quantum enhancement of56% in the precision of the sensor.As described in section II E, SPR imaging enables highlyparallelized, label-free, real-time, high precision biochemi-cal sensing[160, 178–183]. However, all of the quantumplasmonic sensors described in this review thus far are sin-gle pixel sensors. Quantum plasmonic sensors that utilizemulti-spatial-mode quantum correlations in entangled opti-cal readout fields are increasingly a plausible idea. Early re-search that focused on the transduction of quantum imagesby EOT showed that SPP-mediated EOT processes do notmaintain spatial information, thus limiting the potential forEOT-based spatially resolved quantum sensors[456]. How-ever, LSP-mediated EOT processes do maintain spatial infor-8
FIG. 30. Ultrasonic pressure wave sensing. A FWM configura-tion is used to generate entangled probe and conjugate beams in aTMSD state. The probe beam serves as a probe for the plasmonicsensor while the conjugate beam serves as a quantum-correlated ref-erence. SA: spectrum analyzer; BD: beam dump; FG: function gen-erator. Reprinted from ref 454 with permission from Optical Societyof America. mation, as already discussed in this section and shown in Fig-ure 26 [55]. Kretschmann-style SPR sensors also maintainspatial information, as discussed in section II E.Furthermore, as discussed in section III C 2 and high-lighted in recent reviews [248, 457, 458], substantial workhas gone into the development of quantum imaging plat-forms for other applications. For SPR imaging sensorsthat are robust to higher power optical excitation, discretevariable entanglement is less beneficial, but higher powermulti-spatial mode squeezing could be leveraged to enablequantum-enhanced SPR imaging. As mentioned above, andas shown in Figure 26, plasmonic images exhibiting intensity-di ff erence squeezing can be e ffi ciently transduced by LSP-mediated EOT processes. The same e ff ect has been shownmore recently for a combination of phase-sum and intensity-di ff erence entanglement [455]. Despite the demonstrationof the plasmonic transduction of multi-spatial-mode quan-tum states of light, there have been no reports of quantum-enhanced SPR imaging. A key ingredient for such an ad-vance is the multi-pixel readout of an array of plasmonic sen-sors that can benefit from quantum correlations in the read-out field. A growing number of research e ff orts address ex-actly this need. For instance, researchers are now utiliz-ing electron-multiplying CCD cameras for spatio-temporallyresolved readout of multi-spatial-mode squeezed states oflight [459–461], and preliminary research has shown thatcompressive imaging techniques can be used for squeezedlight imaging with single pixel detectors [280]. Together,multi-spatial-mode quantum light sources, multi-pixel plas-monic sensors, and readout schemes for characterizing multi-spatial-mode quantum signals provide the necessary buildingblocks for quantum-enhanced SPR imaging.
3. Intensity sensing robust to thermal noise
The strong photon number correlation possessed by theTMSV state of eq 53, with a NRF of σ =
0, has also been used
FIG. 31. Measured signal while linearly ramping the driving volt-age of the ultrasonic transducer, thus increasing the change of re-fractive index of air ( ∆ n ), when probing with coherent states, trace(i), and with twin beams (as a TMSD state), trace (ii). The lighter-weight lines represent the shot-noise (i) and squeezed noise floors(ii). Reprinted from ref 454 withe permission from Optical Societyof America. to probe an array of gold nanoparticles and measure the refrac-tive index change of a glycerin-water solution [462]. The two-mode scheme shown in Figure 15 was used to carry out quan-tum spectroscopy with the input state | Ψ (cid:105) in = | TMSV (cid:105) , wherethe measurement is a coincidence detection [see Figure 32(a)].The glycerin-water solution surrounds a plasmonic transducerthat consists of a hexagonal array of spherical MNPs with adiameter of 130 nm and lattice period of 1.1 µ m, as shownin Figure 32(c). The idler mode of the TMSV state is sentthrough the sample, while the signal mode is sent to a di ff rac-tion grating to investigate the spectral response using the co-incidence counting that follows. The sensing performance ofsuch a quantum scheme is compared with conventional spec-troscopy that uses a classical probe state generated from alamp, as shown in Figure 32(b).The authors directly compared quantum and classical spec-troscopy methods in terms of their robustness to thermal noiseinfluencing the detection. The thermal noise is artificially real-ized by a lamp inside the monochromater part [see Figs. 32(a)and (b)], which induces additional photon counts N s in themeasurement of the signal photons S s . The noise level is con-trolled by changing the brightness of the lamp, i.e., N s = ,2 × , and 7 × c / s are considered in the experiment.The e ff ect of thermal noise is examined with respect to thedistinguishability of the two resonance spectra measured fortwo concentrations C of glycerin-water solution on top of thearray of MNPs.The results obtained by quantum and classical spectroscopyfor the two concentrations under the three noise conditions arepresented in Figure 33. The red dots represent the spectra for C =
40% and black dots for C = ff erent forthe quantum and classical spectroscopy schemes; the curvesfor the quantum case are more distinguishable than those forthe classical case. This is clearly seen for the case when thenoise photocount N s is 70 times larger than the signal photo-9 Ref. [29], the number of coincidences between two APDs isproportional to the spectral function of the sample T ð ω Þ atthe frequency of the idler photon, R ð ω s Þ ∼ T ð ω p − ω s Þ : (2)In contrast to the conventional transmission spectros-copy, the spectral selection is performed in the channelwithout the structure under test. Following the analogy withimaging experiments, we refer to the method as quantum(ghost) spectroscopy.The remarkable feature of the quantum spectroscopyis its robustness against the environmental noise, whichallows one to conduct measurements at extremely low-photon fluxes. The noise typically includes backgroundoptical noise and electronic noise of the APDs. Let usdenote the number of SPDC photocounts as S s;i ¼ η s;i P ,where P is the number of photon pairs, produced by the SPDC source, and η s;i is the quantum efficiency of thesignal and idler channel, respectively. The number ofcoincidences is proportional to the probability of jointdetection by the APDs in two channels: R ð ω s Þ ¼ η s η i P: (3)Assuming that noise photocounts of the two APDsare uncorrelated, the number of noise coincidences R N isgiven by accidental overlap of photocounts within the timewindow of the coincidence circuit Δ t . It also includescomponents due to accidental overlap of SPDC and noisephotocounts: R N ¼ N s N i Δ t þ S s N i Δ t þ N s S i Δ t; (4)where N s and N i are the number of noise photocounts ofAPDs in signal and idler channels, respectively. The totalnumber of coincidences is given by R Total ¼ R ð ω s Þ þ R N : (5)Even though N s , N i ≫ S s , S i , the number of noisecoincidences R N is strongly suppressed compared to R ð ω s Þ because of a narrow coincidence time window Δ t . Thus, thespectral response of the structure can be revealed from R Total even under excessive noise situations.
III. COMPARISON TO THE CONVENTIONALTRANSMISSION SPECTROSCOPY
In conventional transmission spectroscopy, the sample isplaced between the light source and the monochromator;the transmission spectrum is obtained directly from APDphotocounts [see Fig. 1(b)]. The signal-to-noise ratio(SNR) for the transmission spectroscopy is given bySNR T ¼ η s P=N s . For the quantum spectroscopy, theSNR is given by SNR Q ¼ R ð ω s Þ =R N ¼ η i η s P=N s N i Δ t ,see Eqs. (3) and (4), where it is assumed that N s N i ≫ S s N i ; N s S i . The ratio of the two SNRs is given bySNR Q = SNR T ¼ η i =N i Δ t: (6)The advantage in the SNR for the quantum spectroscopyis provided by the use of high-quantum efficiency, low-noise APD in the idler channel, and the narrow coincidencetime window Δ t . IV. THE PLASMONIC ARRAY SENSOR
We use an array of metal nanoparticles which has anarrow Fano-type plasmon resonance, due to diffractivecoupling of localized surface plasmons [34 – ’ s anomaly. Laser 407 nm M
Coincidence circuit
BBO crystals UVM GRNPBSAPD APDMLL sampleRG
MonochromatorNoise PH SMFCouplerHalogen lamp L Sample L GRAPDM MonochromatorNoisePHCounter signalidler (a)(b)
FIG. 1. (a) Quantum spectroscopy setup. Diode laser at 407 nmis used as a pump for a set of three BBO crystals cut for type-ISPDC; the UV mirror (UVM) reflects the pump beam andtransmits the SPDC signal; a nonpolarizing beam splitter (NPBS)splits the SPDC signal into two arms; in one arm, there is amonochromator, which consists of a mirror (M), a diffractiongrating (GR), a pinhole, and an APD. In another arm, there is asample, which represents an array of gold nanoparticles as-sembled into a cell. The SPDC signal is focused onto the sampleby achromatic lenses (L), filtered by a red glass filter (RG), anddetected by the APD. Counts of the two APDs are sent to acoincidence circuit; the source of the noise is provided by theexternal APD, illuminated by the lamp (not shown), whose signalis mixed electronically with the signal of the APD in the signalchannel. (b) Transmission spectroscopy setup. A halogen lamp isused as a source of the broadband light; the light after a single modefiber (SMF) is collimated by a coupler, focused by lenses onto thesample, filtered by the monochromator, and detected by the APD.
KALASHNIKOV et al.
PHYS. REV. X If the anomaly wavelength is close to the plasmonresonance of individual nanoparticles, then the collectiveplasmonic mode is excited. Excitation of the mode resultsin a narrow Fano-type resonance in the transmissionspectrum of the array. The position and shape of theFano resonance depend on the shape of the nanoparticles,distance between them, and refractive index of the sur-rounding medium. High sensitivity to the local change ofthe refractive index, as well as the high-quality (Q) factorof the nanoparticle array resonance, constitutes the basisfor its sensing applications [37,41].The nanoparticle array is fabricated using a combinedmethod of nanosphere lithography with femtosecond laser-induced transfer, which allows production of large-scaleperiodic arrays of spherical nanoparticles [37]. A hexago-nal array with dimensions of approximately × mm isfabricatedwithaparticlediameterof130nmandahexagonallattice period of . μ m [see Figs. 2(a) and 2(b)]. Thefabricated particles are partially embedded (around two-thirds) into a polydimethylsiloxane (PDMS) substrate, witha part (around one-third) above the substrate surface (thisis accessible to the local environment). With the samplecovered by a second PDMS layer, the nanoparticles are in ahomogeneous surrounding and the array provides a Fanoresonance centered at 806 nm with a quality factor Q ≈ ,whereQisdefinedastheratiooftheresonancewavelengthtothe resonance width, defined from the fit using the Fanoformula (see details below). The refractive-index sensitivity is measured by removingthe cover PDMS layer and adding testing glycerin-watersolutions with different concentrations on top of the array.The sensitivity is calculated as a ratio of the resonance shiftin nm to the change of the refractive index of the testingsolution. V. EXPERIMENTAL SETUP
The quantum spectroscopy setup is shown in Fig. 1(a).Three bulk BBO crystals (Dayoptics) with thicknesses of0.3 mm, 0.5 mm, and 0.5 mm are pumped by a continuous-wave vertically polarized diode laser at 407 nm (OmicronPhoxX-405-60). The BBOs are cut for type-I collinearSPDC. One of the BBOs is set for the frequency-degenerateregime, while two others are detuned from the degenerateregime by tilting their optical axis [26]. The resulting SPDClight source has a spectral width of around 250 nm and isshown in Fig. 3. The pump is eliminated by a UV mirror,and the SPDC is split by a nonpolarizing beam splitter(NPBS). In one arm of the NPBS, we inserted a home-builtdiffraction grating monochromator (Thorlabs GR-1205)with resolution 1.5 nm. The light after the monochromatoris detected by an avalanche photodiode (APD, PerkinElmer SPCM-AQRH-14FC).The sample is attached to a glass substrate and mountedin another arm of the NPBS in the focal plane of twoconfocal achromatic lenses with focus length f ¼ mm.A mechanical translation stage (not shown) is used toposition the sample. The focused light has a diameter of μ m on the sample surface. The sample area with thearray is surrounded by PDMS walls and covered by a thinglass plate forming a cell. The refractive index of themedium surrounding the array can be changed by addingdifferent solvents inside the cell. The SPDC light passesthrough the broadband red glass filter (RG), and then it iscoupled into the multimode fiber and detected by the APD (a)(b) FIG. 2. Images of the plasmonic nanoparticle array on a PDMSsubstrate. (a) Dark-field microscope image at × magnification.(b) Scanning electron microscope (SEM) image. FIG. 3. Spectrum of coincidences of SPDC from three BBOcrystals. The error bars are the standard deviations. QUANTUM SPECTROSCOPY OF PLASMONIC NANOSTRUCTURES PHYS. REV. X (c) μ m1 μ m UVM: UV mirrorNPBS: nonpolarizing beam splitterM: mirrorGR: gratingL: achromatic lensRG: red glass filterPH: pinhole
FIG. 32. (a) Schematic of quantum spectroscopy with a twin beam (i.e., the TMSV state of eq 53) generated from a SPDC source (BBOcrystals), where coincidence photon counting is carried out by two APDs (avalanche photodiodes). A monochromator placed in the signalchannel consists of a mirror (M), a di ff raction grating (GR), a pinhole (PH) and an APD, and selects a particular single mode of interest underinvestigation, realizing spectroscopy. (b) Schematic of conventional spectroscopy with classical light generated from a Halogen lamp, wherea single APD is used together with the same monochromator from the quantum spectroscopy scheme. (c) The MNP array under investigation:Dark-field microscope image (top) and scanning electron microscope image (bottom). Adapted and reprinted from ref 462 under CC-BY-3.0license. count S s [Figs. 33(e) for classical and (f) for quantum]. Suchrobustness against the thermal noise can be attributed to thestrong photon number correlation of the TMSV state, whichis absent in the classical scheme. B. Quantum plasmonic phase sensing
In addition to measuring changes in the refractive index ofan optical sample using intensity sensing, it is also possibleto use phase sensing, as outlined in section II C. In the quan-tum regime this provides a complementary setting for quan-tum plasmonic sensing. In order to carry out quantum phasesensing using plasmonics, the physical geometry of the sys-tem needs to be modified so that changes in the refractive in-dex are picked up in the phase of the optical signal, rather thanthe transmission amplitude. In this case, a metal nanowire isone example that can be considered and it provides a compactsetting in which to perform sensing below the di ff raction limitdue to the highly confined field [18, 19]. A nanowire also hasthe potential to be modified into more advanced nanophotoniccircuitry for complex functionality [20, 463] and has alreadybeen considered for classical phase sensing [21].
1. Phase sensing with discrete variable states
The theoretical work by Lee et al. [418] first studied the useof a nanowire in quantum phase sensing and the model con-sidered is shown in Figure 34(a). Here, a source produces atwo-mode quantum state of light, | Ψ in (cid:105) , and a silver nanowireis placed in one of the modes as a probe ( p ), while the othermode is used as a reference ( r ). A biological medium is con-sidered to surround the nanowire and for a given change in adynamical quantity, such as concentration, a change in the re-fractive index, n , occurs which changes the relative phase be-tween the probe and reference modes by φ ( n ) = (cid:96)β ( n ), where (cid:96) is the length of the nanowire and β is the propagation constantin the nanowire, which is a function of n . A measurement ofthe quantum state is then performed in order to obtain an es-timate of the phase and therefore an estimate of the refractiveindex. The dynamical quantity in the biological medium, suchas concentration, can then be estimated from the refractive in-dex.In the ideal case when there is no loss, a NOON state iswell known to be an optimal state for quantum phase sens-ing [51, 53, 313], as discussed in section III D 2, allowing oneto reach the HL for the precision. It can be used as the two-mode quantum state produced by the source in Figure 34(a),i.e., | Ψ in (cid:105) = √ ( | N , (cid:105) p,r + | , N (cid:105) p,r ), where N denotes the num-ber of photons in a given mode. The relative phase in thenanowire is picked up by the first term in the state, | N , (cid:105) →
70 times larger than the signal in the monochromator[see Fig. 5(f)]. The only effect of the noise is a slightdecrease in the accuracy of the resonance measurements dueto the contribution of noise coincidences. However, the difference between the resonance minima remains the sameas in the case of low noise. This shows that quantumspectroscopy has high resistance to noise and can be usedwith extremely low-photon fluxes, which are not possible
FIG. 5. Transmission spectrum of the nanoparticle array with different concentrations of glycerin-water solution on top, measured atdifferent noise conditions. Red circles correspond to 40% glycerin concentration and black squares are assigned to 50% glycerinconcentration. Results in (a), (c), and (e) are obtained by the conventional transmission spectroscopy with different noise levels of photocounts = s, × photocounts = s, and × photocounts = s, respectively. Results in (b), (d), and (f) are obtained by thequantum spectroscopy with different noise levels: photocounts = s, × photocounts = s, and × photocounts = s, respectively.The signal in the channel with the monochromator is the same for all the experiments in (a)-(f) and is equal to photocounts = s. Theerror bars are the standard deviations. QUANTUM SPECTROSCOPY OF PLASMONIC NANOSTRUCTURES PHYS. REV. X Classical Quantum N o i s e FIG. 33. Transmission spectra measured by conventional classical (left) and quantum (right) spectroscopy under di ff erent noise conditions[ N s = (top), 2 × (middle), and 7 × c / s (bottom)]. The measurement is performed for water-glycerin solutions deposited on anarray of MNPs with di ff erent concentrations [40% (red dots) and 50% (black dots)]. The photon flux of 10 c / s is kept in the monochromatorfor all the experiments and the error bars denote the standard deviations of the measured signals. Adapted and reprinted from ref 462 underCC-BY-3.0 license. e iN φ ( n ) | N , (cid:105) . The resulting state is measured using the opti-mal quantum observable ˆ A = | , N (cid:105)(cid:104) N , | + | N , (cid:105)(cid:104) , N | (seesection III D 2). This leads to a measurement signal (cid:104) ˆ A (cid:105) = A cos[ N φ ( n )] and a corresponding precision in the estima-tion of the refractive index of ∆ n = N (cid:12)(cid:12)(cid:12)(cid:12) d φ d n (cid:12)(cid:12)(cid:12)(cid:12) − , which is theHL for a single-shot measurement, i.e., ν =
1, as discussed insection III D 2. In contrast, the best classical strategy uses thetwo-mode classical coherent state | Ψ in (cid:105) = (cid:12)(cid:12)(cid:12) α/ √ (cid:69) p (cid:12)(cid:12)(cid:12) α/ √ (cid:69) r produced by the source, with | α | = N , which picks up therelative phase in the first term: (cid:12)(cid:12)(cid:12) α/ √ (cid:69) p → (cid:12)(cid:12)(cid:12) e i φ ( n ) α/ √ (cid:69) p .The optimal measurement consists of a BS and measure-ment of the observable ˆ M = ˆ I p − ˆ I r , which is the intensity-di ff erence, as exploited in section III C 3. This leads to a measurement signal (cid:104) ˆ M (cid:105) = M cos[ φ ( n )] and a correspond-ing precision ∆ n = √ N (cid:12)(cid:12)(cid:12)(cid:12) d φ d n (cid:12)(cid:12)(cid:12)(cid:12) − , which is the SNL or SQL.The NOON state therefore provides a √ N improvement inthe precision compared to the classical case for a given ν . Thesignals (cid:104) ˆ A (cid:105) and (cid:104) ˆ M (cid:105) , and their associated precisions are shownin Figures 34(b) and (c) for N =
4, where they are comparedwith the case in which a conventional dielectic nanowire isused. The results clearly show the benefit of using a plas-monic nanowire with a quantum source and measurement forphase sensing.The improvement in the phase sensing precision comesfrom two distinct factors that are revealed by writing the pre-cision (in this case the LOD) as ∆ n = ∆ φ (cid:12)(cid:12)(cid:12)(cid:12) d φ d n (cid:12)(cid:12)(cid:12)(cid:12) − , where ∆ φ = FIG. 34. Quantum plasmonic phase sensing using a nanowire. (a) Two-mode scheme with a quantum source, plasmonic nanowire andmeasurement. (b) Measured signal for the optimal observable used ( ˆ O = ˆ M for classical (C) or ˆ A for quantum (Q)) for a dielectric andplasmonic nanowire. (c) The precision of the refractive index ∆ n for classical (C) and quantum (Q) states with dielectric and plasmonicnanowires. The photon number on average is N =
4, the nanowire radius is 50 nm, the propagation length of the nanowire is 4 µ m and thefree-space wavelength is λ =
810 nm. Reprinted from ref 418 with permission from American Chemical Society. ∆ O (cid:12)(cid:12)(cid:12)(cid:12) d (cid:104) ˆ O (cid:105) d φ (cid:12)(cid:12)(cid:12)(cid:12) − is the precision of the phase estimation and ˆ O isthe observable in the measurement ( ˆ O = ˆ A or ˆ M ). ∆ φ ob-viously depends on the observable ˆ O and state | Ψ in (cid:105) , but in-terestingly not on the nanowire properties, and is thereforequantum in origin. On the other hand, (cid:12)(cid:12)(cid:12)(cid:12) d φ d n (cid:12)(cid:12)(cid:12)(cid:12) represents thesensitivity, S φ , which depends on the length and propagationconstant [ φ ( n ) = (cid:96)β ( n )], i.e., the nanowire properties, and istherefore classical in origin. Both classical and quantum fac-tors are needed to obtain a good precision, however, it is thequantum factor that improves the precision beyond the SQL,as discussed in section II B 3.In a more realistic scenario, losses in the nanowire must beconsidered. In this case, even for moderate losses, the NOONstate is no longer the optimal quantum state. A more generalanalysis using the QFI, H , must then be undertaken in orderto find the optimal state to use. In this case, a more generalstate | Ψ in (cid:105) = (cid:80) Nn = c n | n , N − n (cid:105) p , r can be considered [320],with the coe ffi cients c n optimized in order to maximize H and thus minimize the QCR bound ∆ φ min = (max { c n } H ) − / ,which in turn minimizes the estimation uncertainty ∆ n . In Lee et al. [418] it was found that plasmonic sensors using theseoptimized states provide a precision beyond the SIL, which isthe SNL of standard interferometers when loss is present [320]and discussed in section III B 2, even for a moderate amountof loss in a nanowire. Results were obtained for arbitrary N ,where the precision of the optimized states was shown to al-ways be an improvement over the SIL precision.While the use of the QFI enables an optimization of thequantum state used by the source, it is not clear what the opti-mal measurement is that allows one to reach the lower boundin the estimation uncertainty of ∆ φ min , i.e., the QCR boundintroduced in section III A. Further work in this direction willbe needed to uncover practical measurement schemes for dis- crete variable quantum phase sensing in plasmonics. Further-more, the use of di ff erent types of plasmonic material [464]for the nanowire could be considered in order to reduce losswhile maintaining the high field confinement, including theuse of metamaterials [146, 465–467], graphene [468] and hy-brid material systems [469–471].The use of a plasmonic nanowire for phase sensing has re-cently been experimentally investigated for the simple caseof a two-photon NOON state ( N =
2) generated by paramet-ric down-conversion [325], as shown in Figure 35(a). Thetwo-mode state used takes the form | Ψ in (cid:105) = √ ( | H , (cid:105) , + | , V (cid:105) , ), where | H (cid:105) ( | V (cid:105) ) corresponds to two horizontal(vertical) polarized photons in a given spatial mode. The po-larization dependence of the photons enables the two spatialmodes to be combined into a single spatial mode in an opti-cal fiber, while maintaining an e ff ective ‘two-mode’ setting– both polarization ‘modes’ co-propagate in the same spa-tial mode. The vertical (horizontal) polarization representsthe probe (reference) mode. The optical fiber is then taperedand a silver nanowire attached to the end, which enables thee ffi cient excitation of SPPs and is shown in Figure 35(b) and(c). The output of the signal is collected by a microscope ob-jective and sent to a measurement stage with optical elementsand single-photon detectors for the measurement of the phase.In the experiment, the phase was modified using a liquidcrystal phase retarder outside the nanowire in order to gain anunderstanding of the performance of the nanowire for phasesensing in a controlled manner. While not a direct demonstra-tion, the approach is equivalent to the case that the phase ispicked up directly in the nanowire, as phase shift operationscommute with the loss accumulated in the nanowire [41]. Amore direct demonstration is yet to be performed, however,this is more challenging due to the requirement of a ligand2 FIG. 35. Experimental investigation of quantum plasmonic phase sensing using a silver nanowire. (a) Generation of polarization encodedNOON state ( N =
2) via parametric down-conversion. The two spatial modes of the state (1 and 2) are combined into a single mode (4), whilemaintaining the two-mode scenario in the polarization degree of freedom. The entangled photons in the single spatial mode are injected intoan optical fiber. (b) The fiber is tapered with a silver nanowire fixed to the end. The out-coupled light is collected and sent to a measurementstage. (c) Scanning electron microscope image of the tapered fiber and nanowire. (d) Measurement signal for the two-photon NOON state(red) and classical case (blue). The y-axis on the left (right) is for the NOON state (classical case). Adapted and reprinted from ref 325 withpermission from Optical Society of America. coating along one of the axes of the nanowire in order foronly one polarization mode to interact with the biochemicalsubstance being sensed. Fortunately, this can be achieved in anumber of ways, for example using nanografting [472, 473].By modifying the phase of the probe mode, then convertingthe probe and reference modes back into spatial modes andmeasuring the coincidences, a measurement equivalent to thatof the observable ˆ A can be performed. The signal of this mea-surement is shown in Figure 35(d) as the red curve, along withthe corresponding classical signal as the blue curve, equiv-alent to the observable ˆ M . The NOON state signal clearlyshows an oscillation that occurs over a phase twice as smallas the classical case, as expected – a phenomenon known as‘super resolution’ [322].The coincidence counts in Figure 35(d) can be normalizedto give a probability of a coincidence [53], according to eq 69in section III D 2, i.e., p coin ( φ ) = f (1 + V cos 2 φ ) /
2, where f represents the total proportion of photons that lead to a two-photon coincidence and V is the two-photon visibility. Thiscoincidence probability can then be used to infer the uncer-tainty in the measurement of the phase, ∆ φ coin . In this re-alistic and lossy setting, the SIL for the precision in estima-tion of the phase is given by [217, 323] ∆ φ SIL = / √ N η , which is simply the SNL of standard interferometers in thecase of no loss multiplied by the factor 1 / √ η , where η is thetotal loss factor. In order to show an improvement, the un-certainty in the experiment must satisfy ∆ φ coin < ∆ φ SIL , af-ter which it can be called ‘super sensitive’ [323]. Using thenormalized probability function p coin ( φ ), the bound can be re-formulated as [325] 1 < f V /η , which can be inverted togive a threshold visibility of V (th)2 = (cid:113) η/ f . A visibilityabove this value shows an improvement in the precision be-yond the SIL. In the absence of loss, f = η = V (th)2 = − / ≈ . V ≈ . V (th)2 rises. It was concluded that evenif V = ffi ciencies at the various components inthe setup, while still accommodating for loss in the plasmonicnanowire. Thus, the precision could be improved beyond theSIL with further experimental improvements.Various challenges must be overcome for a complete ex-3perimental demonstration of quantum phase sensing using aplasmonic system and discrete variable quantum states. Re-ducing losses in the components of the setup, implementingthe phase change at the nanowire itself, as well as the use ofmore robust-to-loss states other than the NOON state are allavenues for further study to address the challenges. Thesewould enable the realization of practical nanoscale quantumplasmonic interferometric sensors.
2. Phase sensing with continuous variable states
Despite the substantial developments in classical interfero-metric plasmonic sensing and the progress in interferometricquantum plasmonic sensing with discrete variable quantumstates described above, very little progress has been made inthe development of squeezed interferometric plasmonic sen-sors. One reason for this is the increased loss in interferomet-ric sensors. Intensity-based plasmonic sensors are most sensi-tive at the inflection point of the plasmon absorption spectrum,so the detrimental e ff ect of the plasmon absorption can be mit-igated somewhat. Phase-based plasmonic sensors are mostsensitive at the maximum of the plasmon absorption spec-trum [114], so as described in section II C, little squeezingwould remain in a squeezed interferometric plasmonic sen-sor after accounting for plasmonic absorption. Nevertheless,some progress has been made in the characterization of plas-mons exhibiting squeezing in the phase-sum quadrature, no-tably in a report that characterized the two-mode squeezingin intensity-di ff erence and phase-sum quadratures for LSPssupported in triangular nanoapertures [455]. Further, substan-tial progress has been made in the development of squeezedinterferometric sensors for other applications [478]. More-over, as described in section III D 5, nonlinear interferome-ters o ff er favorable scaling with loss compared with conven-tional squeezed light readouts, and could plausibly be inte-grated with plasmonic sensors in order to obtain a quantumadvantage. Truncated nonlinear interferometers in particu-lar o ff er favorable loss scaling and the ability to use a highpower local oscillator in order to improve shot-noise limitedprecision without concern for photochemical or photothermale ff ects [365, 367, 368]. Many opportunities still exist to de-velop loss-resilient squeezed interferometric plasmonic sen-sors based on lower loss plasmonic sensor designs and newapproaches to SU(2) and SU(1,1) interferometry. C. Quantum plasmonic sensing based on emitter-plasmoncoupling
Outside the field of plasmonic sensing, much attention hasbeen paid to the plasmonic control of emitters [59, 479–482],including semiconductor quantum dots [483] and color cen-ters [484] in the weak and strong coupling regimes. It is in-creasingly evident that these e ff ects can be leveraged for alter-native approaches to quantum plasmonic sensing, building onthe PEF and SERS sensors described in section II D 2. Theoptical field at metallic nanostructures can be significantlyenhanced via the excitation of LSPs, enabling strong cou-pling with matter in the vicinity of metallic structures [54]. The e ff ects that originate from strong coupling are inher-ently nonclassical [479], and therefore a quantitative analy-sis and understanding of the various phenomena measured inexperiments require a full-quantum mechanical description,whereby the plasmonic nanostructures supporting LSP exci-tations can be treated as a lossy and small-mode-volume res-onator in cavity quantum electrodynamics [57, 479]. A resultof this is that strong coupling is very sensitive to various struc-tural and material parameters of an analyte being placed inthe proximity of a metallic structure, which has led to severaltypes of quantum plasmonic sensors.In one of the first examples of a quantum sensor based onemitter-plasmon coupling, a coupled system of a quantum dotand metallic nanorod was studied as a means to sense a smallchange in the local refractive index around the system in thenear-infrared regime [474], as shown in Figure 36(a). Thequantum phase-dependent changes in the coherent exciton-plasmon coupling provide the capability of detecting and dis-tinguishing adsorption or detachment of target molecules. Asimilar structure has also been considered for optical detectionand recognition of single biological molecules [485]. Adsorp-tion of a specific molecule to the nanorod results in the ultra-fast upheaval of coherent dynamics of the system, that turnso ff the blockage of energy transfer between the quantum dotand the nanorod. The emission of the system is thus stronglymodified depending on the adsorption event. Measuring theemission pattern from a quantum dot and metallic nanoshellsystem would also enable the remote detection of the spatialcoordinates and movement of biological molecules or nanos-tructures [475], as shown in Figure 36(b). Refractive indexsensing is also possible by analysing the Rabi-splitting spec-trum of the exciton-plasmon strong coupling in the gap be-tween a plasmonic nanorod and plasmonic nanowire [486].In another study, a full quantum mechanical theory wasdeveloped to model how a small amount of absorbing Trini-trotoluene molecules influences the spectrum of a graphenespaser based on a graphene flake with quantum dot emit-ters [487]. The study of the emission spectrum from anemitter-plasmon quantum dynamical system for sensing pur-poses is a promising direction for future work, although acareful analysis is required in order to determine whether thequantum e ff ects being exploited provide a quantum advantagein terms of sub-shot-noise sensing, an improvement in the sen-sitivity, or both.A recent theory proposal has suggested the use of quan-tum dots as quantum labels bonded to individual antibody-antigen-antibody complexes being placed inside or close to ananoplasmonic dimer [476], as shown in Figure 36(c). As thesurface density of the analyte-emitter complexes changes, theextinction cross section spectrum is modified, which not onlycauses spectrum shifting, but also brings multiple resonancepeaks due to the strong coupling between the emitters and LSPat the nanoplasmonic dimer. Through a statistical study ofmultiple analyte-emitter complexes for the theoretical simula-tion of realistic conditions, the proposed splitting-type sens-ing approach using quantum emitter labels has been shownto achieve a nearly 15-fold sensitivity enhancement in com-parison with conventional shifting-type label-free plasmonicsensors. Such a study demonstrates the potential of quantumplasmonic sensors to detect a single analyte, which is usually4 FIG. 36. Quantum plasmonic sensors based on emitter-plasmon coupling. (a) A single quantum nanosensor in a to-the-end configuration,where a gold nanorod and quantum dot are functionalized. Reprinted from ref [474] with permission from IOP Publishing. (b) Illustrationof quantum detection and ranging. The antenna includes a quantum dot and a metallic nanoshell. F ( r , t ) and G ( r (cid:48) , t (cid:48) ) refer to two di ff erenttime-dependent emission intensity patterns of an optically active nanoscale system at two di ff erent locations. Reprinted from ref 475 withpermission from AIP Publishing. (c) An illustration of a strong-coupling immunoassay setup. A gold hemisphere nanodimer cavity capturesan immunoassay complex in the proximity of the plasmonic hotspot. Reprinted from ref 476 with permission from American Chemical Society.(d) A nanofiber with dark-field heterodyne illumination. Nanoparticles in a droplet of ultrapure water are detected when entering the probebeam waist next to the nanofiber. Reprinted from ref 477 with permission from Springer Nature. challenging since the size of an analyte (typically <
100 nm)is far smaller than the optical wavelength of the light excitingthe system.Metallic nanorods have been used in a recent experiment,where an optical nanofiber is immersed in a droplet of watercontaining nanoparticles, such as silica nanospheres or goldnanorods, and biomolecules, as shown in Figure 36(d). Het-erodyne interferometry was used together with a dark-fieldillumination approach [477]. The experiment demonstrateda quantum-noise limited precision of evanescent single-molecule biosensing, allowing a reduction of four orders ofmagnitude in the optical intensity that is required to maintainstate-of-the-art sensitivity.In a related theoretical study, the fraction of the total spon-taneous emission energy from an emitter coupled to SPPs,called the β -factor [488], has been investigated for two ni-trogen vacancy centers (NVCs) in diamond placed in a plas-monic waveguide [489]. It was shown that a maximally en-tangled state of NVCs and a product NVC state provide theoptimal estimation of the β -factor at initial times and at longtimes, respectively.Finally, the use of color center emitters for quantum sens-ing of magnetic [490, 491] and electric [492] fields has ex-perienced significant attention in the past few years. The useof a plasmonic system as a mediator of the sensing signal isan interesting development. A recent experiment has demon-strated the use of a plasmonic groove waveguide interactingwith multiple NVCs, which are positioned at the end of thegroove waveguide milled in a thick gold film [493]. The goldfilm carries the microwave control signal for the NVCs, whilethe groove waveguide acts as a fluorescence collector. V. PERSPECTIVE AND OUTLOOK
In this review we covered the background and latest devel-opments in the emerging field of quantum plasmonic sens-ing. This is a research field that sits at the interface betweenplasmonic sensing and quantum metrology – the former pro-vides researchers working in the field with decades of exten-sive knowledge in classical optical sensing across a wide arrayof plasmonic systems, their commercial development and ap-plications, while the latter provides new concepts and methodsintensively developed in recent years that enable the perfor-mance of the sensors to be improved by exploiting quantummechanical features in various quantum systems.The review started with a discussion of conventional plas-monic sensors and their basic working principles. The excita-tion of SPPs was shown to enable enhanced sensing comparedto standard optical sensors due to their high electromagneticfield confinement. Two main types of plasmonic sensing werediscussed – intensity and phase sensing – with the benefits ofeach described as depending on the specific application. Thiswas followed by a discussion of estimation theory and the lim-its of sensing in plasmonic systems using classical light, i.e.,the SNL. We then showed how the use of quantum states oflight and quantum measurements enables sensing beyond thisSNL, where we elaborated on single- and two-mode sensingschemes while reviewing recent works on multimode or mul-tiple parameter sensing. Recent works combining these quan-tum sensing techniques with plasmonic sensing were then re-viewed. We covered the basic theory behind the work andhighlighted its motivation in relation to applications, includ-ing biosensing, monitoring chemical reactions and ultrasound5sensing. It was shown that despite the presence of loss in plas-monic systems, one can use techniques from quantum sensingto obtain improvements in sensing performance.The field of quantum plasmonic sensing has grown steadilyover the last five years, with researchers studying many waysin which to combine plasmonic and quantum sensing. How-ever, much work still needs to be done to bring quantum plas-monic sensors to the same level of maturity as their classicalcounterparts, including successful commercialization. Newresults from the field of quantum sensing for dealing withloss [43–45, 238, 305], including error correction [361, 494–497] and novel resource states [41, 498–500], may lead toa widening in the range of plasmonic systems that can beused for quantum sensing and their related applications. Fur-thermore, non-standard material systems, including metama-terials [146, 465–467], graphene [468, 501] and more exotictwo-dimensional materials [502, 503] may help reduce lossand further improve the level of precision in applications,such as monitoring chemical and biological reactions [24],food safety [504], pathogen detection [6, 8], and environ-mental monitoring [7]. Hybrid systems that exploit electro-optic [469–471], nonlinear [505], nonlocal [506], quantumsize e ff ects [60, 421] and electron tunelling [61] could alsoo ff er additional quantum functionality. Another interestingdirection is the incorporation of quantum emitters into plas-monic systems, such as quantum dots [59, 481] and color cen-ters [484], which may bring further applications, such as mag-netic [490, 491] and electric field sensing [492], and molec-ular spectroscopy [507]. There is also the natural connectionbetween sensing and imaging [123], and it remains to be seenhow the techniques developed for quantum plasmonic sens-ing can be translated to plasmonic imaging [120, 121, 124]in order to improve aspects such as image resolution, featureextraction and pattern recognition for applications in the lifesciences and medicine.The field of quantum plasmonic sensing is likely to developinto a rich subfield of optical science and it is an exciting timeto enter this emerging research field. Multidisciplinary col-laboration from researchers working in plasmonics, quantuminformation science, material physics, chemistry, biology andmedicine will advance this field significantly, bringing withthem new opportunities for sensing in science and industry.For these prospective studies, this review will provide a help-ful guidance and inspire novel avenues of research. APPENDIXAppendix A: Multiparameter QFIM
The QFI matrix, H , in eq 38 is defined by[ H ] jk = Tr[ ˆ ρ x ( ˆ L j ˆ L k + ˆ L k ˆ L j ) / , (A1)with ˆ L j being a symmetric logarithmic derivative operator as-sociated with j th parameter x j [194]. Here, F − and H − areunderstood as the inverse on their support if the matrices aresingular, i.e., not invertible [359].For single parameter estimation, the optimal measurementsetting reaching the QCR bound given by eq 28 always ex- ists and can be constructed with projectors onto the eigen-vectors of the SLD operator [38, 41, 194]. For multiparam-eter estimation, on the other hand, the QCR bound has beenknown to be attained only if the SLD operators commute,i.e., [ ˆ L j , ˆ L k ] = j , k , which is valid even when thegenerators do not commute, i.e., [ ˆ G j ( x j ) , ˆ G k ( x k )] (cid:44)
0. Thisis the su ffi cient condition for the saturability of the multipa-rameter QCR bound, in which case the optimal set of POVMscan be constructed over the common eigenbasis of the com-muting SLD operators [354]. A weaker, but necessary andsu ffi cient condition for the saturability can be found for purestates ˆ ρ x = | Ψ ( x ) (cid:105) (cid:104) Ψ ( x ) | , written as [202, 508] (cid:104) Ψ ( x ) | ( ˆ L j ˆ L k − ˆ L k ˆ L j ) | Ψ ( x ) (cid:105) = , (A2)for all j , k . The condition of eq A2 implies the commu-tation relation among the SLD operators only on averagewith respect to the state | Ψ ( x ) (cid:105) . For commuting genera-tors ˆ G j ( x j ) associated with the evolution of a pure probe state,i.e., [ ˆ G j ( x j ) , ˆ G k ( x k )] = j , k , eq A2 is satisfied, so thatthe multiparameter QCR bound can be saturated [508]. Appendix B: Calculation of multiparameter QFIM
Generally for M parameters φ = { φ , · · · , φ M } , the relationis given by [203] (cid:88) j , k H jk d φ j d φ k = D ( ˆ ρ φ , ˆ ρ φ + d φ ) , (B1)where the Bures distance can be written in terms of the quan-tum fidelity as D ( ˆ ρ φ , ˆ ρ φ + d φ ) = (cid:20) − (cid:113) F ( ˆ ρ φ , ˆ ρ φ + d φ ) (cid:21) (B2)and the quantum fidelity F is defined as [509, 510] F ( ˆ ρ φ , ˆ ρ φ + d φ ) = (cid:32) Tr (cid:113) (cid:112) ˆ ρ φ ˆ ρ φ + d φ (cid:112) ˆ ρ φ (cid:33) . (B3)Thus, the calculation of the quantum fidelity leads to the cal-culation of QFIM for any pair ( φ j , φ k ).In particular, the input state | Ψ (cid:105) in = | α (cid:105) a | ξ (cid:105) a considered inthe above MZI is called a Gaussian state as its characteristicWigner function can be described by a Gaussian distribution.Because the BS operation ˆ B , the phase operation ˆ U ( φ ), andlinear photonic loss channel, including ine ffi cient detectors,are a Gaussian map, the output state ˆ ρ out is kept in the formof a Gaussian state. This enables us to describe the outputstate ˆ ρ out in terms of the first-order moment vector d , and thesecond-order moment matrix V [511, 512]. The displacementvector d is defined as d j = Tr[ ˆ ρ φ ˆ Q j ], whereas the covariancematrix V is defined by V jk = Tr[ ˆ ρ φ { ˆ Q j − d j , ˆ Q k − d k } / { ˆ A , ˆ B } ≡ ˆ A ˆ B + ˆ B ˆ A . Here, ˆ Q denotes a quadratureoperator vector for a two-mode continuous variable quantumsystem and is written as ˆ Q = ( ˆ x , ˆ p , ˆ x , ˆ p ) T , satisfying thecanonical commutation relation, [ ˆ Q j , ˆ Q k ] = i Ω jk , where Ω = (cid:32) − (cid:33) × I (B4)6and I n is the n × n identity matrix.Due to the input state remaining a Gaussian state through-out the evolution, the analytical form of the quantum fidelitycan be found for the displacement vectors and covariance ma-trices [343, 513], so one can readily calculate the quantumfidelity. For two Gaussian states ˆ ρ j = , with d j = , and V j = , ,the quantum fidelity can be calculated via [343, 513] F ( ˆ ρ , ˆ ρ ) = exp (cid:104) − δ d T ( V + V ) − δ d (cid:105) ( √ Γ + √ Λ ) − (cid:113) ( √ Γ + √ Λ ) − ∆ , (B5)where ∆ = det( V + V ), Γ =
16 det( Ω V Ω V − I / Λ =
16 det( V + i Ω / V + i Ω / δ d = d − d .Using eqs B1, B2 and B5, one can calculate the QFIM H generally for multiple parameters φ = { φ , · · · , φ M } .In the case when only a single parameter φ encodedby ˆ U ( φ ) is of interest, the QFI is calculated by H = (cid:104) − (cid:112) F ( ˆ ρ φ , ˆ ρ φ + d φ ) (cid:105) / d φ . Appendix C: CR bound in lossy MZIs
Homodyne detection is particularly useful, as the measure-ment bases are represented by Gaussian states, i.e., the POVMelement is written as ˆ Π x = | x φ HD (cid:105)(cid:104) x φ HD | , where | x φ HD (cid:105) denotesa quadrature variable state. Projection of the reduced outputstate ˆ ρ out,a = Tr b [ ˆ ρ out ] into the measurement basis | x φ HD (cid:105) leadsto the probability density function of the quadrature variableoutcomes { x } , which is written as p ( x | φ ) = (cid:104) x φ HD | ˆ ρ out,a | x φ HD (cid:105) . (C1)It can be shown that the probability density function of eq C1follows a Gaussian distribution for the measurement out-comes, leading to homodyne detection often being called aGaussian measurement [512, 514]. Hence, the probabilitydensity function of eq C1 can be described by the first-ordermoment vector ˜ d and the second-order moment covariancematrix ˜ V , for which the FI can be calculated via [190, 515] F ( φ ) = ∂ ˜ d T ∂φ ˜ V − ∂ ˜ d ∂φ +
12 Tr (cid:34) ˜ V − ∂ ˜ V ∂φ ˜ V − ∂ ˜ V ∂φ (cid:35) . (C2)For an optimally chosen homodyne angle φ HD with respect tothe phase φ being estimated, one can show that the CR boundreads [303] ∆ φ CR = √ ν (cid:115) | α | e r + − ηη | α | . (C3) ACKNOWLEDGMENTS
At KIT, this work was partially supported by the Volkswa-gen Foundation and by the VIRTMAT project. At ORNL, BLwas supported by the U. S. Department of Energy, O ffi ce ofScience, Basic Energy Sciences, Materials Sciences and En-gineering Division. At HYU, KGL was supported by the Ba-sic Science Research Program through the National Research Foundation (NRF) of Korea and funded by the Ministry ofScience and ICT (Grants No. 2020R1A2C1010014) and Insti-tute of Information & Communications Technology Planning& Evaluation (IITP) grant funded by the Korea government(MSIT) (No. 2019-0-00296). At SU, MST was supported bythe South African National Research Foundation, the Councilfor Scientific and Industrial Research National Laser Centreand the South African Research Chair Initiative of the Depart-ment of Science and Innovation and National Research Foun-dation. GLOSSARY • Bias - The di ff erence between an estimate x est and thetrue value x of the parameter being estimated on aver-age, i.e., ¯ x est − x . An unbiased estimator has a bias ofzero. • Cramér-Rao bound - The lower bound of the Cramér-Rao inequality. • Cramér-Rao inequality - The inequality that the stan-dard deviation of an unbiased estimator of a parametershould obey for a given measurement setting. The in-equality reads ∆ x ≥ / √ ν F ( x ), where ∆ x is the stan-dard deviation, ν is the number of measurements in asample and F ( x ) is the Fisher information. The equalityholds when the optimal unbiased estimator is chosen. Itcan also be asymptotically saturated in the limit of large ν when maximum-likelihood-method is employed as anestimator although it is not optimal. • Estimate - The value x est obtained from the estimatorˆ x ( y , ... y ν ) for a given sample (single set of data ob-served). It is also known as a point estimate. • Estimation accuracy - The interpretation of the bias¯ x est − x . The accuracy becomes better as the bias de-creases. • Estimation precision - The interpretation of the standarddeviation, ∆ x , of the estimate x est , each taken from asample made up of a finite set of ν measurements of theparameter x . The smaller ∆ x is, the better the precisionis. It is commonly known as the fluctuation or uncer-tainty in estimation, or can sometimes be interpreted asthe resolution. It can also be understood as the repro-ducibility. • Estimation uncertainty - The quantity directly givenby the standard deviation ∆ x . The smaller ∆ x is, thesmaller the uncertainty is. • Estimator - A rule, ˆ x , that yields an estimate x est of anunknown parameter x from an underlying probabilitydistribution based on the observed data y from a sampleof ν measurements, y ≡ ( y , ... y ν ). For example, ˆ x canbe expressed by some function f , i.e., ˆ x = f ( y ). Themost commonly used estimator is the sample mean.7 • Extraordinary optical transmission - the transmission oflight through a structured medium, where the transmis-sion is due to the excitation of surface plasmons. Asimilar structured medium that does not support surfaceplasmons has a reduced transmission. • Fisher information - A quantification of the amount ofinformation about a parameter x contained in the mea-surement results for a given measurement setting. Weexpress it by F ( x ). • Heisenberg limit - The ultimate quantum limit achiev-able using the optimal quantum state and measurement,or the precision scaled with N − that is often the caseof interferometeric sensing. A scaling of N − is calledHeisenberg scaling, where N is the average number ofparticles in the resource. • Limit of detection - An overall figure of merit for sens-ing quality that takes into account both the sensitivity S y and the minimum detectable range ∆ y min , or equiva-lently the value of the noise level. This is often inter-preted as a resolution. • Mean-squared-error - The average square distance be-tween the estimator ˆ x ( y , ... y ν ) and true value of x as thedata y , ... y ν varies according to the underlying proba-bility distribution. It is a measure of average closenessof an estimator ˆ x to the true value x , and depends onboth the standard deviation ∆ x and the bias ¯ x est − x . Ifthe estimator is unbiased, then the mean-square erroris equivalent to ∆ x , allowing to refer to the standarddeviation ∆ x as the ‘estimation error’, ‘estimation pre-cision’, or simply ‘precision’. • Multi-parameter estimation - This is categorized intothree kinds:1. Individual estimation - Multiple parameters( x , x , · · · , x n ) are estimated individually. Thisscheme is also called a ‘local strategy’.2. Simultaneous estimation - Multiple parame-ters ( x , x , · · · , x n ) are estimated simultaneously.This scheme is also called a ‘global strategy’.3. Distributed estimation - A global parameter X ,which is a function of multiple parameters, i.e., X = f ( x , x , · · · , x n ), is estimated. • Noise reduction factor - In an intensity di ff erence mea-surement, it is the ratio of the variance of the pho-ton number di ff erence between the signal and referencemode for a given quantum state to that of coherent stateswith matching average photon number. • N -mode - A sensing scheme where N spa-tial / temporal / angular / spectral modes – the eigenvectorsof the wave equation – are used. Here, N can be“single”, “two”, or “multi”. • Quantum Cramér-Rao bound - The lower bound of thequantum Cramér-Rao inequality. • Quantum Cramér-Rao inequality - The same as Cramér-Rao inequality but with the Fisher information replacedby quantum Fisher information, i.e., ∆ x ≥ / √ ν H . • Quantum Fisher information - The maximized Fisherinformation over all possible physical measurement set-tings { ˆ Π } , i.e., H = max { ˆ Π } F ( x ). • Sensitivity - The derivative of a quantity y being mon-itored, which is used to obtain an estimation of someparameter x that changes with y . Mathematically, thesensitivity is S y = | d y / d x | . By increasing the sensitiv-ity a sensor becomes more sensitive to changes in x . • Shot noise - Pure noise originating from the underly-ing statistical properties of a resource to be analyzedor a measurement involving the random fluctuation ofelectric signals. It has nothing to do with fundamentalnoise, which in an ideal theory model can be assumed tobe completely absent. The term originally comes fromelectronics, where a current consists of a stream of dis-crete charges, i.e., electrons. The charges are randomlydistributed in space and time, thereby following a Pois-son distribution. The same random feature is present ina coherent state of light that consists of discretized par-ticles, i.e., photons, that follow a Poisson distribution inphoton number. This leads to the shot noise when thecoherent state’s intensity, i.e., photon number, is mea-sured. • Shot-noise limit - The best possible precision obtainedusing the classical resource of a coherent state. 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