Mechanical systems in the quantum regime
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Mechanical systems in the quantum regime
Menno Poot , Herre S. J. van der Zant Kavli Institute of Nanoscience, Delft University of Technology, P. O. B. 5046, 2600GA Delft, The Netherlands
Abstract
Mechanical systems are ideal candidates for studying quantum behavior of macroscopic objects. To this end, a mechanicalresonator has to be cooled to its ground state and its position has to be measured with great accuracy. Currently, variousroutes to reach these goals are being explored. In this review, we discuss di ff erent techniques for sensitive positiondetection and we give an overview of the cooling techniques that are being employed. The latter include sideband coolingand active feedback cooling. The basic concepts that are important when measuring on mechanical systems with highaccuracy and / or at very low temperatures, such as thermal and quantum noise, linear response theory, and backaction,are explained. From this, the quantum limit on linear position detection is obtained and the sensitivities that have beenachieved in recent opto and nanoelectromechanical experiments are compared to this limit. The mechanical resonatorsthat are used in the experiments range from meter-sized gravitational wave detectors to nanomechanical systems that canonly be read out using mesoscopic devices such as single-electron transistors or superconducting quantum interferencedevices. A special class of nanomechanical systems are bottom-up fabricated carbon-based devices, which have very highfrequencies and yet a large zero-point motion, making them ideal for reaching the quantum regime. The mechanics ofsome of the di ff erent mechanical systems at the nanoscale is studied. We conclude this review with an outlook of howstate-of-the-art mechanical resonators can be improved to study quantum mechanicsKeywords: Quantum-electromechanical systems; QEMS; Nano-electromechanical systems; NEMS; Optomechanics;Quantum-limited displacement detection; Macroscopic quantum mechanical e ff ects; Active feedback cooling; Sidebandcooling Contents1 Introduction 3 Present address: Department of Electrical Engineering, Yale University, New Haven, CT 06520, USA
Email addresses: [email protected] (Menno Poot), [email protected] (Herre S. J. van der Zant)
Preprint submitted to Physics Reports October 13, 2011
Backaction and cooling 25 mechanical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 v ( t )
60C Square-law detection 61 . Introduction Mechanics is probably the most well-known branch of physics as everyone encounters it in every-day life. It describesa wide range of e ff ects: from the motion of galaxies and planets on a large scale, the vibrations of a bridge inducedby tra ffi c or wind, the stability of a riding bicycle, to the trajectories of electrons in an old-fashioned television on amicroscopic scale. In the early days of physics, mainly objects that could be seen or touched were studied. Until thebeginning of the twentieth century it was thought that the three laws of motion obtained by Newton described the dynamicsof mechanical systems completely. However, the development of better telescopes and microscopes enabled the study ofmechanical systems on both much larger and smaller length scales. In the early 1900s, the rapid developments that led tothe theory of special and general relativity and quantum mechanics showed that the laws of classical mechanics were notthe whole truth.Relativistic corrections turn out to be important for objects with large masses or with velocities approaching the speedof light. It is therefore an important factor in astrophysics, where one studies the dynamics of heavy objects like galaxiesand black holes or the bending of light by the curvature of space-time. When the masses and velocities of the objectsinvolved are made smaller and smaller, the relativistic corrections eventually vanish and one obtains the classical laws ofmotion [1].Quantum mechanics, on the other hand, is particularly well suited to describe the mechanics of objects at the other endof the length-scale range, i.e., (sub)atomic objects. In the beginning of the twentieth century, quantum theory successfullyexplained the photo-electric e ff ect, black-body radiation, and the atomic emission spectra. Quantum mechanics is di ff erentfrom classical and relativistic mechanics in the sense that objects are no longer described by a definite position, but by awavefunction. This wavefunction evolves deterministically according to the Schr¨odinger equation and its absolute valuesquared should be interpreted as the position probability-density function, the so-called Born rule [2]. To find the objectat a particular location one has to measure its position. This process, however, inevitably disturbs the evolution of thewavefunction [3, 4].Quantum mechanics does not only describe processes at the (sub)atomic scale successfully, but it also explains themicroscopic origin of many macroscopic e ff ects such as the electronic properties of solids, superfluidity and so on. Unlikein relativity where one can simply take the limit m , v →
0, in quantum mechanics it is still not entirely clear how thetransition from quantum mechanics to classical mechanics exactly happens [5, 6]. Although Ehrenfest’s theorem impliesthat the expectation values of the momentum and potential energy obey Newton’s second law [2], the classical laws ofmotion cannot in all cases be recovered by simply taking ~ → gedanken experiment. Superpositions of small objects are readily observed, agood example are the singlet and triplet spin states in a molecule, but this becomes increasingly di ffi cult for larger andlarger systems mainly due to decoherence [10, 11]. Superpositions have for example been created using the circulatingcurrent in superconducting quantum interference devices (SQUIDs) [12] and with fullerenes [13, 14].Nanomechanical devices [15, 16] are interesting candidates to further increase the size of systems that can be putin a superposition [11, 17]. These systems are the logical continuation of micromechanical devices which are madeusing integrated-circuit technology, but then on a much smaller, nanometer scale. Micro-electromechanical systems(MEMS) are currently widely used as, for example, accelerometers in airbags, pressure sensors, and in projectors. Whenscaling these devices down to the nanometer scale, their resonance frequencies increase and at the same time their massdecreases. From an application point of view nano-electromechanical systems (NEMS) may enable single-atom mass-sensing, mechanical computing and e ffi cient signal processing in the radio-frequency and microwave bands. From ascientific point of view these devices are interesting as they can be cooled to temperatures so low that the resonator isnearly always in its quantum-mechanical ground state. Figure 1 shows some examples of miniature mechanical resonators.In the last decade, many groups have pursued demonstration the quantum limit of motion. The rapid progress inthe development of sensitive optical techniques and mesoscopic electronics have led to detectors that have sensitivitiesthat are approaching the quantum limit on position detection [3]. Moreover, very recently the long-sought quantummechanical behavior of mechanical resonators has become reality in two di ff erent experiments indicating that the quantumregime of mechanical motion has now been entered: Selected by Science magazine as research breakthrough [21] ofthe year 2010, the groups of Cleland and Martinis demonstrated quantum mechanical behavior of a 6 GHz mechanicalresonator by coupling it to a superconducting flux qubit [22]. At dilution refrigerator temperatures the first ten energystates of the harmonic mechanical oscillator, including the ground state, could be probed. The measurement itself isperformed on the superconducting qubit which acts as a two-level quantum system whose response changes when the3 a) m
200 m m
10 m m
10 m m (c) (b)(d) Figure 1: Di ff erent types of MEMS / NEMS used at Delft University of Technology. (a) Piezoresistive cantilever. A deflection of the cantilever changesthe electrical resistance between the two sides. This resonator geometry can be used for mass detection. (b) Bistable buckled beam. Blue and redindicates the two stable positions of the beam, which can be used to encode digital information [18]. (c) A beam resonator that is embedded in the loopof a dc SQUID (red) [19]. A magnetic field couples the position of the beam to the magnetic flux through the loop. (d) Suspended carbon nanotube(white) as a flexural resonator [20]. occupation of the mechanical resonator state changes. A large coupling between the two quantum systems is achievedby using the piezoelectric properties of the mechanical resonator material. In a di ff erent approach Teufel et al. [23] use asuperconducting microwave cavity to actively cool a mechanical drum resonator to such low temperatures that it is in thequantum mechanical ground state for most of the time. The drum resonator is integrated in the superconducting resonantcircuit to provide strong phonon-photon coupling and the measurements use concepts developed for optical cavities toachieve e ffi cient cooling.These two breakthrough experiments not only show that non-classical behavior can be encoded in the motion of amechanical resonator but also open a new exciting research field involving quantum states of motion. In this review wewill summarize the main theoretical and experimental discoveries that have led to the demonstration of quantum motion.Our main focus will be on the mechanics, i.e., on the mechanical properties of resonators and on the di ff erent optical andelectronic detection schemes that have been developed to measure their displacements. Since measuring always meansthat the detector has to be coupled to the resonator, we will also discuss the coupling and the consequences it may haveon the measurement itself. Furthermore, the advantages and disadvantages of the various approaches will be discussed. The concept of the quantum limit on mechanical motion detection and its implications became relevant in the 1970swhen more and more sensitive gravitational wave detectors [24] were designed (see for example Refs. [25] and [26]),raising questions on the (im)possibility to violate Heisenberg uncertainty principle [2, 3, 27]. Nowadays, these issues areimportant when measuring micro- and nanomechanical devices with very sensitive detectors or at very low temperatures.There are two important considerations when approaching the quantum limit. First, the thermal occupation is impor-tant, which is defined in Sec. 3.3 as n = ( k B T R / h f R ) − / f R is the resonance frequency and T R is the resonatortemperature. A value for n that is below 1 indicates that the resonator is in its ground state most of the time (see Sec. 3.3for a more detailed description of the thermal occupation). Second, the zero-point motion sets the ultimate limit on theresonator displacement. At high temperatures and in the absence of actuation, Brownian motion determines the resonatorposition. As temperature decreases, the displacement decreases to the point that the quantum regime is reached ( n . u = p ~ / π m f R with m the resonator mass. Toobserve the zero-point motion, one has to detect it. This sounds trivial, but it turns out that a measurement on a quantumsystem inevitably disturbs it. Quantum mechanics sets a limit to the precision of continuously measuring the position ofthe resonator, the standard quantum limit. A detector at that limit is therefore called quantum-limited. Detection of thezero-point motion thus requires resonant frequencies higher than about 1 GHz (where n = Table 1 provides an overview of the di ff erent groups that have performed experiments with mechanical resonators thatapproach the quantum limit in position-detection sensitivity or that have been cooled to a low resonator temperature. Themicro- and nanomechanical devices listed in this table are made using so-called top-down fabrication techniques, whichare also employed in the semiconductor industry. Di ff erent groups use di ff erent types of resonators: doubly clampedbeams, singly clamped cantilevers, radial breathing modes of silica microtoroids, membranes, micromirrors and macro-scopic bars. We have indicated the resonator mass, which ranges from 20 ag to 1000 kg, and the resonator frequency,which ranges from about 10 Hz to a few GHz. From these numbers the zero-point motion u has been calculated; fortop-down devices it is generally on the order of femtometers.These top-down nanoscale structures listed in Table 1 (e.g. beams, cantilevers and microtoroids) are made by etchingparts of a larger structure, for example a thin film on a substrate, or by depositing material (evaporating, sputtering) on aresist mask that is subsequently removed in a lift-o ff process. In both cases, patterning of resist is needed, which is doneusing optical or electron-beam lithography. State-of-the-art top-down fabricated devices have thicknesses and widths ofless than 100 nanometer.A major drawback of making smaller resonators to increase their frequency, is that the quality factor decreases [16].The quality factor is a measure for the dissipation in the system. A low quality factor or “Q-factor” means a largedissipation, and this is an unwanted property for resonators in the quantum regime. The associated decoherence ofquantum states then limits the time for performing operations with these states. For example, in the experiments ofRef. [22] where Q is on the order of a few hundred, the time for the manipulation of quantum states is limited to only 6 ns.The decrease in Q-factor with device dimensions is often attributed to the increase in surface-to-volume ratio [16, 71,72]. An explanation comes from the fabrication which may introduce defects at the surface during the micro-machiningprocesses that are involved. These defects provide channels for dissipation, resulting in the low Q-factor.Having this in mind, a di ff erent approach is to use the small structures that nature gives us, to build or assemblemechanical resonators. Bottom-up devices are expected not to su ff er from excessive damping, as their surface can bedefect-free at the atomic scale. Examples are inorganic nanowires , carbon nanotubes and few-layer graphene. The lasttwo are examples of carbon-based materials. Using these bottom-up materials, mechanical devices with true nanometerdimensions can be made with the hope that surface defects can be eliminated. High Q-values are therefore expected forthese devices, which, combined with their low mass, make them ideal building blocks for nano-electromechanical systems. Table 2 shows the properties of mechanical resonators that have been made so far using bottom-up fabricated devices.Their frequencies are high: by choosing the right device geometry resonances in the UHF band (300 MHz - 3 GHz) arereadily made, as Table 2 shows. When comparing the quality factors and zero-point motion of these devices, it is clearthat the nanowire performance is more or less comparable to top-down fabricated devices since their thickness is of theorder of 10-100 nm, about the size of the smallest top-down fabricated devices.Due to their low mass m and high strength (see the mechanical properties listed in Table 3 in Sec. 2) the frequencies ofcarbon-based resonators are high and their zero-point motion u large. Note, that in Table 1 u was given in femtometer,whereas in Table 2 it is given in picometer. Figure 2 illustrates this point more clearly. The quantum regime with a large u and small n is positioned in the upper right corner. When cooled to dilution refrigerator temperatures, carbon-basedresonators would therefore be in the ground-state while exhibiting relatively large amplitude zero-point fluctuations.Position detectors for bottom-up NEMS, however, are not yet as sophisticated as those for the larger top-down coun-terparts. Consequently, non-driven motion at cryogenic temperatures (i.e., Brownian or zero-point motion) nor activecooling has been reported for carbon-based NEMS. The devices always need to be actuated to yield a measurable re-sponse. This is at least in part due to their small size, which makes the coupling to the detector small as well. Neverthelessimpressive progress in understanding the electromechanical properties of bottom-up resonators has been made in recentyears using so-called self-detecting schemes. In these schemes, the nanotube both acts as the actuator and detector of itsown motion. Ultra-high quality factors have now been demonstrated for carbon nanotube resonators at low temperatures[81] as well as a strong coupling between electron transport and mechanical motion [106, 82]. In this Report we pay special attention to the mechanics of bottom-up materials as this has not been reviewed in suchdetail as the mechanics of the silicon-based top-down devices. In particular we will focus on the carbon-based materialsas they have extraordinary mechanical and electrical properties. Sometimes “nanowire” is also used for top-down fabricated devices. Here, this term is used exclusively for grown wires. able 1: Overview of recent key experiments with micro- and nanomechanical resonators in chronological order. Several types of resonators anddetection methods are used by di ff erent groups in the field and are measured at di ff erent temperatures T . The table shows the resonance frequency f R ,quality factor Q and the mass m of the resonator. From this, the zero-point motion u is calculated. Group Resonator f R (MHz) Q T (K) m (kg) u (fm) Ref. . · −
15 [28]2 UCSB GaAs beam 117 1700 0.030 2 . · − / Au beam 19.7 35000 0.035 9 . · −
21 [30]4 LMU Si / Au cantilever 7 . · − . · −
12 [31]5 Schwab SiN / Al beam 21.9 120000 0.030 6 . · −
24 [32]6 LKB Paris Si micromirror 0.815 10000 295 1 . · − . · − [33]7 UCSB AFM cantilever 1 . · − . · − / TiO beam 0.278 9000 295 9 . · − . · − . · − [36]10 MPI-QO Silica toroid 57.8 2890 300 1 . · − / Au cantilever 127 900 295 5 . · −
36 [38]12 LIGO Micromirror 1 . · − . · − . · − [39]13 LIGO Micromirror 1 . · − . · − . · −
37 [41]15 JILA Gold beam 43.1 5000 0.25 2 . · − . · − . · −
100 [43]17 NIST Si cantilever 7 . · − . · − . · − . · − [45]19 Harris SiN membrane 0.134 1100000 294 3 . · − . · − . · − . · − . · − . · −
133 [50]24 Roukes SiC / Au beam 428 2500 22 5 . · −
20 [51]25 IBM Si cantilever 4 . · − . · −
29 [52]26 Delft AlGaSb beam 2.00 18000 0.020 6 . · − . · − . · . · − [53]28 AURIGA Al bar 9 . · − . · . · − [53]29 Alberta Si cantilever 1040 18 295 2 . · −
20 [54]30 Tang Si beam 8.87 1850 295 1 . · −
27 [55]31 JILA Al beam 1.53 300000 0.050 6 . · −
30 [56]32 JILA Al beam 1.53 10000 0.050 6 . · −
30 [56]33 LMU SiN beam 8.90 150000 295 1 . · −
23 [57]34 Queensland Silica toroid 6.272 545 300 3 . · − . · − [58]35 Tang Si cantilever 13.86 4500 295 4 . · −
37 [59]36 Vienna Si cantilever 0.945 30000 5.3 4 . · − . · − . · − . · − . · −
300 2 . · . · − [64]41 Painter Silica double tor. 8.53 4070 300 1 . · − . · −
27 [66]43 MPI-QO / LMU SiN beam 8.07 10000 300 4 . · −
15 [67]44 Schwab SiN / Al beam 6.30 1000000 0.020 2 . · −
25 [68]45 MPI-QO / LMU SiN beam 8.30 30000 300 3 . · −
16 [69]46 UCSB AlN FBAR 6170 260 0.025 2 . · − . · − . · − able 2: Overview of recent experiments with bottom-up resonators. Several types of resonators are used: carbon nanotubes (CNT), nanowires (NW),single-layer graphene (SLG), and few-layer graphene (FLG) or graphene oxide (FLGO) sheets. The table shows the resonance frequency f R and qualityfactors at room and cryogenic temperature ( Q RT and Q cryo resp.). T min is the lowest temperature at which the resonator is measured. m is the mass ofthe resonator and ℓ is its length. From these data, the zero-point motion u is calculated. Group Type f R (MHz) Q RT Q cryo T min (K) ℓ ( µ m) m (kg) u (pm) Ref.
Cornell CNT 55 80 300 1.75 7 . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − . · − NW 59 2200 300 2.5 2 . · − . · − . · − . · − . · − . · − . · − . · − . · −
24 215 10 3.3 1 . · − op-downnanotubesnanotubesnanodrums Figure 2: Comparison of the resonance frequency f R and zero-point motion u of top-down (Table 1) and two kinds of bottom-up devices: Suspendedcarbon nanotubes [20, 81, 106] and graphene nanodrums [107]. The resonance frequencies and zero-point motion are much larger for the bottom-updevices. (a) (b) (d)(c)diamond graphite C carbon nanotube Figure 3: The structure of the di ff erent allotropes of carbon. (a) Diamond has two intertwined face-centered cubic lattices. (b) Graphite consists ofstacked planes of hexagonally ordered carbon atoms. A single plane is called a “graphene sheet”. (c) A C buckyball molecule. (d) A single-walledcarbon nanotube, which can be viewed as a graphene sheet that has been rolled up and sewn together. (c) and (d) Reprinted from Nanomedicine: Nan-otechnology, Biology and Medicine,
3, M. Foldvari, M. Bagonluri, Carbon nanotubes as functional excipients for nanomedicines: I. pharmaceuticalproperties, 173–182, Copyright (2008), with permission from Elsevier.
Carbon exists in many di ff erent forms, ranging from amorphous coal to crystalline graphite and diamond. Diamondhas a face-centered cubic structure as shown in Fig. 3a and is one of the hardest materials known. Its Young’s modulus(Table 3) is extremely high: about 1 TPa. Graphite has a very di ff erent crystal structure: it consists of stacked planes ofcarbon atoms in a hexagonal arrangement (Fig. 3b). Its Young’s modulus for in-plane stress is nearly as high as that ofdiamond, but it is much lower for out-of-plane stress, as Table 3 will indicate. This di ff erence is caused by the nature ofthe bonds holding the carbon atoms together. Atoms in one of the planes are covalently bonded to each other, whereasdi ff erent planes are held together by the much weaker van der Waals force.Graphite and diamond were already known for millennia, but in the last decades novel allotropes of carbon werediscovered. First, in 1985 C molecules, called Buckminsterfullerenes or “buckyballs”, were synthesized [108]. Then inthe early 1990s carbon nanotubes were discovered [109]. These consist of cylinders of hexagonally ordered carbon atoms;similar to what one would get if one were to take a single layer of graphite and role it up into a cylinder. In 2004 anotherallotrope called “graphene” was identified [110, 111]. This is a single layer of graphite, which is, unlike a nanotube, flat.Graphene is usually deposited onto a substrate using mechanical exfoliation, and high-quality sheets of mm-size havebeen made using this technique [112]. Another way of making graphene devices is to grow it directly on a substrate,and in a semi-industrial process meter-sized sheets have been reported [113]. Although truly two-dimensional structuresare not energetically stable [114], graphene can exist due to fact that it contains ripples that stabilize its atomically thinstructure [115, 116, 117, 118]. Figure 3 shows the structure of the four di ff erent carbon allotropes.Although nanomechanical devices have been made out of diamond using top-down fabrication techniques [119, 120,71, 121], in this Review we focus on bottom-up fabricated carbon-based NEMS made from few-layer graphene or sus-pended carbon nanotubes. 8 .4. Outline of this review This Review consists of this introduction, three main Sections, one on nanomechanics, one on backaction and cooling,and the last one focusses on di ff erent types of detectors. These are supplemented with a final Section summarizing someprospects and future directions in the field of quantum electromechanical systems or QEMS in short. In Section 2 we willdiscuss mechanics at the nanoscale. From continuum mechanics, the general equations of motion are obtained. Theseare illustrated by studying the dynamics of a number of nanomechanical devices, such as beam and string-like resonators,buckled beams and suspended carbon nanotubes. We will demonstrate that the dynamics of a particular vibrational modeof the resonator is that of a harmonic oscillator. Another important point is that in nanoscale devices, tension is a crucialproperty that must be included in the analysis. As we will show it also provides a unique tool for the tuning of resonatorswhen their thickness is on the atomic to nanometer scale.In Section 3 we discuss the properties of the (quantum) harmonic oscillator and we study the e ff ects of backaction. Ameasurement always influences the measured object itself and this is called backaction. It has important consequences forlinear detection schemes, and we will show that backaction ultimately limits their position resolution to what is knownas the standard quantum limit. This limit will be derived in several ways. Backaction, however, can also be used toones advantage as it provides a way to cool resonator modes. This can be done by backaction alone (self-cooling).Alternatively, the resonator temperature can be lowered by adopting active cooling protocols. An overview of the twomost popular protocols will be given, including a summary of the main achievements.In general, two distinct approaches for position detection of mechanical oscillators are used: optical and electrical. InSec. 4, we give an overview of the di ff erent detection schemes. We will discuss the use of optical cavities in various forms,optical waveguides and their analogues in solid state devices in which electrons play the role of photons. In addition, wewill summarize the concepts behind capacitive and inductive actuation and detection as well as the self-detection schemesthat are used in bottom-up NEMS devices. Special attention will be paid to the achieved position resolution and to thelimitations that prevent the standard quantum limit to be reached. Furthermore, we explain the mechanisms behind thebackaction and, when possible, quantify the coupling between resonator and detector.9 . Mechanics at the nanoscale Nearly, all objects, including macroscopic and nanometre-sized systems, have particular frequencies at which theycan resonate when actuated. These frequencies are called the eigenfrequencies or normal frequencies; when actuated atan eigenfrequency, i.e. on resonance, the amplitude of vibration can become very large and this can have catastrophicconsequences. Examples include the breaking of a glass by sound waves and the collapse of the Tacoma bridge whichwas set in motion by the wind flowing around it.Continuum mechanics provides the tools to calculate these frequencies. For simple geometries such as cantilevers,doubly-clamped beams or thin plates, the frequencies of flexural or torsional modes can be calculated analytically. How-ever, in many NEMS experiments more complicated geometries are used, such as microtoroids [122], suspended photoniccrystal structures [62, 123], or film bulk acoustic resonators (FBARs) [22, 124] and the vibrational mode may have morea complicated shape than the simple flexural or torsional motion. To calculate the eigenfrequencies and mode shapes inthese cases one has to rely on numerical calculations. Popular implementations include the software packages ANSYSand COMSOL which are based on the finite-element method. These packages have additional advantages as they providea means to model other properties of the nanomechanical system: This includes electrostatic interactions (see e.g. Refs.[125, 126, 57]), thermal e ff ects [127, 122], and electromagnetic (optical) properties [55, 62, 128].In this Section, we will summarize some main results of continuum mechanics needed to describe the experimentsdiscussed in this Report. We will review analytical expressions for the eigenfrequencies of simple structures such asbeams, buckled beams and strings. This material has been described in several textbooks (see e.g. the book by A. Cleland[15]). Less attention has been devoted to thin beams or plates with nanometer-sized cross sections made from, for example,carbon nanotubes or graphene. These resonators are in a di ff erent regime than top-down devices that generally have largersizes. In particular, the deflection of these nano-resonators can exceed their thickness or radius so that tension-inducednonlinear e ff ects start to play a role. We derive the equations for describing these nonlinear e ff ects in nanobeams andshow that the induced tension can be used to tune the frequencies over a large range. We end this section with a discussionon the mechanics of (layered) graphene resonators that can be viewed as miniature drums. To describe the motion of mechanical objects, the dynamics of all particles (i.e., atoms and electrons) which make upthe oscillator should, in principle, be taken into account. It is, however, known that for large, macroscopic objects this isunnecessary and that materials can be accurately described as a continuum with the mechanical behavior captured by a fewparameters such as the elasticity tensor. Molecular dynamics simulations [129, 130, 131, 132, 133, 134] and experiments[20, 135, 107] demonstrate that even for nanometer-sized objects continuum mechanics is, with some modifications, stillapplicable. This means that the dynamics of the individual particles is irrelevant when one talks about deflections anddeformations; the microscopic details do, however, determine the material properties and therefore also the values ofmacroscopic quantities like the Young’s modulus or the Poisson ratio.The basis of continuum mechanics lies in the relations between strain and stress in a material. The strain tells how thematerial is deformed with respect to its relaxed state. After the deformation of the material, the part that was originallyat position x is displaced by u to its new location x + u . The strain describes how much an infinitesimal line segment iselongated by the deformation u ( x , y , z ) and is given by [136]: γ i j = ∂ u i ∂ x j + ∂ u j ∂ x i + ∂ u m ∂ x i ∂ u m ∂ x j ! . (1)This definition shows that strain is symmetric under a reversal of the indices, i.e., γ i j = γ ji . The diagonal elements (i.e., i = j ) of the first two terms are the normal strains, whereas the o ff -diagonal elements ( i , j ) are the shear strains. Eq. 1 isexact, but the last, non-linear term is only relevant when the deformations are large [15] and will not be considered in thiswork.To deform a material external forces have to be applied, which in turn give rise to forces inside the material. When thematerial is thought of as composed of small elements, each element feels the force exerted on its faces by the neighboringelements. The magnitude and direction of the force depend not only on the location of the element in the material but alsoon the orientation of its faces (see Fig. 4a). The force δ F on a small area δ A of the element is given by: δ F i = σ i j n j δ A , (2) In this Report, the so-called Einstein notation [136] for the elements of vectors and tensors is employed. When indices appear on one side of anequal sign only, one sums over them, without explicitly writing the summation sign. For example, x i = R ij x j reads as x i = P j = R ij x j . The index runsover the three cartesian coordinates ( x , y , z ), where x = x , x = y and x = z . Finally, the symbols ˆx i ( ˆx , ˆy and ˆz ) denote the unit vectors in the fixedrectangular coordinate system (which form a basis) so that a vector r can be expressed as r = r i ˆx i . igure 4: (a) Visualization of the stress tensor on a cubic element ∆ V . The force per unit area is the inner product of the stress tensor and the normalvector of the surface n . (b) Deformation of a plate under plane stress. The original plate (dotted) is deformed by the stress σ xx . where n is the vector perpendicular to the surface and σ is the stress tensor. Now consider an element of the material withmass ∆ m and volume ∆ V . When it is moving with a speed v = ˙u , its momentum ∆ p is: ∆ p ≡ Z ∆ m v d m = Z ∆ V ρ v d V , (3)where ρ is the mass density. The rate of change of momentum equals the sum of the forces working on the element. Theseforces include the stress σ at the surface and body forces F b that act on the volume element ∆ V :d ∆ p d t = Z ∆ V F b d V + Z δ ( ∆ V ) σ d A . (4)Examples of body forces are gravity, with F b = ρ g ˆz and the electrostatic force q E , where g is the gravitational acceleration, q is the charge density, and E is the local electric field. Using the Green-Gauss theorem, the integral over the boundary ofthe element can be converted into an integral over the volume: R δ ( ∆ V ) σ d A = R ∆ V ∂σ i j /∂ x i · ˆx i d V . Eq. 4 should hold for any element because so far nothing has been specified about the shape or size of the element. This then yields Cauchy’sfirst law of motion [136]: ρ ¨ u j = ∂ σ i j ∂ x i + F b , j . (5)A similar analysis for the angular momentum yields Cauchy’s second law of motion: σ i j = σ ji . With these equations (andboundary conditions) the stress distribution can be calculated for a given applied force profile F b ( x , y , z ). The stress tensor gives the forces acting inside the material, whereas the strain tensor describes the local deformationof the material. These two quantities are, of course, related to each other. When the deformations are not too large, thestress and strain tensor are related linearly via the the fourth-rank elasticity tensor E : σ i j = E i jkl γ kl (6)The properties of the stress and strain tensor imply that E i jkl = E jikl = E i jlk = E kli j , so E has at most 21 independentelements out of a total of 3 × × × =
81 elements. This makes it possible to express Eq. 6 in a convenient matrixrepresentation: σ xx σ yy σ zz σ xz σ yz σ xy = E xxxx E xxyy E xxzz E xxxz E xxyz E xxxy E xxyy E yyyy E yyzz E yyxz E yyyz E yyyx E xxzz E yyzz E zzzz E zzzx E zzzy E zzxy E xxxz E yyxz E zzzx E xzzx E xzzy E yxxz E xxyz E yyyz E zzzy E xzzy E yxxz E xyyz E xxxy E yyyx E zzxy E yxxz E xyyz E xyyx γ xx γ yy γ zz γ xz γ yz γ xy , (7)or in short hand notation : [ σ ] = [ E ][ γ ]. The inverse of the elasticity tensor is called the compliance tensor C , whichexpresses the strain in terms of the stress: γ i j = C i jkl σ kl , or [ γ ] = [ C ][ σ ] . (8)The number of independent elements of E is further reduced when the crystal structure of the material has symmetries[15, 137, 138]. The most drastic example is an isotropic material, whose properties are the same in all directions. In this Note that there are three di ff erent notations for the elasticity tensor: E is the actual tensor with elements E ijkl . Finally, the elements can also bewritten in a matrix [ E ]. able 3: Mechanical properties of materials that are used in nanomechanical devices. Most materials have a density ρ around 3 · kg / m and aYoung’s modulus of the order of 10 GPa. The carbon-based materials graphite and diamond are slightly lighter, but much sti ff er. Compiled from Refs.[15] and [140]. Material ρ (10 kg / m ) E (GPa) ν Silicon 2.33 130.2 0.28Si N (crystaline) 2.65 85.0 0.09SiO (amorphous) 2.20 ∼
80 0.17Diamond 3.51 992.2 0.14Graphite (in-plane) 2.20 920 0.052Graphite (out-of-plane) 2.20 33 0.076Aluminum 2.70 63.1 0.36Gold 19.30 43.0 0.46Platinum 21.50 136.3 0.42Niobium 8.57 151.5 0.35GaAs 5.32 85.3 0.31InAs 5.68 51.4 0.35case, only two independent parameters remain: the Young’s modulus E and Poisson’s ratio ν . The compliance matrix isin this case given by: [ C ] = / E − ν/ E − ν/ E − ν/ E / E − ν/ E − ν/ E − ν/ E / E / G / G
00 0 0 0 0 1 / G G = E + ν , (9)where G is the shear modulus. By inverting [ C ], the elasticity matrix is obtained:[ E ] = + ν )(1 − ν ) E (1 − ν ) E ν E ν E ν E (1 − ν ) E ν E ν E ν E (1 − ν ) 0 0 00 0 0 G G
00 0 0 0 0 G . (10)When a plane stress σ xx is applied, a material will be stretched in the x-direction, as illustrated in Fig. 4b. The resultingstrain γ xx = σ xx / E induces stress in the y- and z-directions, which are nulled by a negative strain (i.e., contraction) inthese directions that is ν times smaller than the strain in the x-direction. This follows directly from the structure ofthe compliance matrix. In the opposite situation where a plane strain is applied, the stress can directly be calculatedusing the elasticity matrix, Eq. 10 [136]. The Young’s modulus and Poisson’s ratio of materials that are frequently usedfor nanomechanical devices are indicated in Table 3. The Young’s modulus of most semiconductors (Si, GaAs, InAs),insulators (Si N , SiC, SiO ) and metals is of the order of 100 GPa. This is much larger than the values for soft materialssuch as polymers (typically between 0.1 and 1 GPa) which have also been used for nanomechanical devices [139], butstill smaller than that of diamond and graphite. Their Young’s modulus of slightly less than 1 TPa combined with a lowmass density makes these carbon-based materials ideal to build high-frequency resonators. Also, the large spread in themass density of the metals should be noted.For non-isotropic materials, the Young’s modulus and Poisson’s ratio depend on the direction of the applied stressand are defined as: E i = / C iiii and ν i j = − C ii j j / C iiii ( i , j ) [141]. Graphite is highly anisotropic, and its mechanicalproperties are important for studying many carbon-based nanomechanical devices. It consists of layers of carbon atomsthat are stacked on top of each other (see Fig. 3b and 5b) with an inter-layer spacing c = .
335 nm [142]. The individuallayers are called graphene sheets. The unit cell of graphene consists of a hexagon with a carbon atom on each corner, asillustrated in Fig. 5a. Each of the six carbon atoms lies in three di ff erent unit cells; a single unit cell thus contains twocarbon atoms. The sides of the hexagon have a length d cc = .
14 nm, so that the unit cell has an area of 5 . · − m ,and the two-dimensional mass density is ρ d = . · − kg / m . The graphene planes are not located exactly above eachother but every other layer is shifted by half the unit cell, or equivalently, it is rotated by 60 o around an axis through one12 igure 5: (a) The unit cell of graphene with the dimensions indicated. a is the length of the two translation vectors a , = a [ ± , √ , d cc is thedistance between two carbon atoms. The area of the unit cell is √ a = . · − m . (b) Bending of a few-layer graphene sheet. The equilibriumdistance between the graphene layers is c = .
335 nm [142]. of the carbon atoms. Three of the six atoms are on top of the atoms in the other layer and the other three are located atthe center of the hexagon below them. The six-fold rotational symmetry ensures that the elastic properties are the samewhen looking in any direction along the planes, i.e., they are isotropic in those directions [137, 138]. On the other hand,the mechanical properties for deformations perpendicular to the planes are quite di ff erent. It is therefore convenient tointroduce the in- and out-of-plane Young’s modulus, E (cid:31) and E ⊥ respectively, and the corresponding Poisson’s ratios ν (cid:31) and ν ⊥ . They are defined such that the compliance matrix is given by :[ C ] = / E (cid:31) − ν (cid:31) / E (cid:31) − ν ⊥ / E ⊥ − ν (cid:31) / E (cid:31) / E (cid:31) − ν ⊥ / E ⊥ − ν ⊥ / E ⊥ − ν ⊥ / E ⊥ / E ⊥ / G (cid:31) / G (cid:31)
00 0 0 0 0 1 / G ⊥ , (11)where E (cid:31) = .
92 TPa, E ⊥ =
33 GPa, G (cid:31) = . G ⊥ = .
44 TPa, ν (cid:31) = .
052 and ν ⊥ = .
076 [137]. The largein-plane sti ff ness is the reason that carbon nanotubes and graphene have very high Young’s moduli of about 1 TPa, whichmakes them one of the strongest materials known. Note, that these six elastic constants are not independent as the in-planeshear modulus is given by G (cid:31) = ( E − E ) / In the previous Section the relation between the stress and strain in a material was given. Here, we focus on the energyneeded to deform the material. From this, the equations of motion are derived. For small deformations, the potentialenergy U depends quadratically on the strain. It should be invariant under coordinate transformations [136], leading to U = Z V U ′ d V with U ′ = E i jkl γ i j γ kl , (12)where U ′ is the potential energy density. From this definition it follows that the stress is given by σ i j = ∂ U ′ /∂γ i j . For anisotropic material Eq. 12 reduces to [138]: U ′ = E + ν (cid:18) γ i j + ν − ν γ kk (cid:19) . (13)Although Eq. 12 is valid for any mechanical system in the linear regime, it is not straightforward to analyze a system thisway. Therefore, we focus on two simple and frequently used geometries where the equation of motion can be obtainedwithout too much e ff ort, namely plates and beams. Note that this definition is slightly di ff erent from the conventional definition of the Poisson’s ratio in an anisotropic material that was given on page12. The values of the elastic constants depend on the quality of the graphite samples. Therefore, slightly di ff erent values can be found in the literature.Compare, for example, the data in Refs. [143] and [144] with the values in Ref. [137] a) h (b) n T TF(c) zy x
Figure 6: (a) Bending of a plate with thickness h . The top part of the plate is extended, whereas the bottom is compressed. The black dashed lineindicates the neutral plane. (b) The normal vector of the top surface of a slightly deflected plate makes a small angle with the unit vector ˆz (dotted). (c)If tension T is present in a plate, a net vertical force F results when the displacement profile has a finite curvature. A plate is a thin object that is long and wide, i.e., h ≪ ℓ . w . When a torque is applied to it, it bends, as illustratedin Fig. 6a. The top part, which was initially at z = h /
2, is extended whereas the bottom part of the beam, originally at z = − h /
2, is compressed. There is a plane through the plate where the longitudinal strain is zero: the so-called neutralplane. The vertical displacement of this plane is indicated by u ( x , y ) and for small deflections it lies midway throughthe plate [138], which we take at z =
0. Because of the small deflection and small thickness this is called the thin-plateapproximation.Consider the top (or bottom) face of the plate that is shown in Fig. 6b: because there is no material above (below) thatsurface, there cannot be a normal force F n at this surface (except at the clamping points). In other words, the perpendicularstress components vanishes: F n , i = σ i j n j =
0. For a thin plate, the normal vector n at the top and bottom face points in thez-direction i.e., n = ± ˆz to first order in the displacement u or, equivalently, in the radius of curvature R − c . The conditionfor vanishing stress thus becomes: σ xz = σ yz = σ zz = u x = − z · ∂ u /∂ x , γ xx = − z · ∂ u /∂ x u y = − z · ∂ u /∂ y , γ yy = − z · ∂ u /∂ y u z = u , γ zz = z ν/ (1 − ν ) · ∇ u γ xz = γ yz = , γ xy = − z · ∂ u /∂ x ∂ y . (14)The vertical displacement of the material u z is thus equal to the deflection u for every value of z . Besides this verticaldisplacement there is also a horizontal displacement ( u x and u y ) induced when u changes. In that case, the materialdisplaces in di ff erent directions above and below the neutral plane (Fig. 6a) as expressed by the proportionality with z .Moreover, the strain components averaged over the thickness h , γ i j ( x , y ), are all zero, as the contributions above and belowthe neutral plane cancel each other.To proceed, Eq. 14 is inserted into Eq. 13 and the integration over z in Eq. 12 is carried out. This yields the energyneeded to bend the plate: U B = Eh − ν ) Z Z ∂ u ∂ x + ∂ u ∂ y ! + − ν ) ( ∂ u ∂ x ∂ y ) − ∂ u ∂ x ∂ u ∂ y d x d y (15)It is possible that the plate is not only bent, but that it is also under a longitudinal tension T = R σ d z (positive fortensile tension, negative for compressive tension). The tension is tangential to the surface and from Fig. 6c it is clear thatthe longitudinal tension results in a restoring force in the z-direction when the plate is bent, i.e., when ∂ u /∂ x ,
0. Thetension deforms the plate, as indicated in Fig. 4b. The displacement results in a strain field γ αβ = ( ∂ u β /∂ x α + ∂ u α /∂ x β + ∂ u /∂ x α · ∂ u /∂ x β ), where Greek indices run over the x and y coordinate only. Using Eq. 12 the work done by applying thetension is calculated. The resulting stretching energy is: U T = Z Z γ αβ T αβ d x d y , where T αβ = Z h / − h / σ αβ d z ≡ h σ αβ . (16)The equation of motion for the vertical deflection of the plate is obtained when the variation of the total potential energy U = U B + U T + U F is considered ( U F = − RR Fu d x d y includes the e ff ect of an external force per unit area F in the14 x y +h/2-h/2-w/2 +w/2 zx y r (b)(a) Figure 7: Cross-sections of a rectangular beam with thickness h and width w (a) and a cylindrical beam with radius r (b). z-direction) for an arbitrary variation in the displacement u → u + δ u . This yields the equation of motion for the plate[138]: ρ h ∂ u ∂ t + D ∇ − ∂∂ x α T αβ ∂∂ x β ! u ( x , y ) = F ( x , y ) , (17)where ρ is the mass density of the material . The first term in Eq. 17 is the inertial term and the last term is theexternal force acting on the plate. The term in brackets is the force resulting from the deformation of the plate. Here, D = Eh / − ν ) is the so-called bending rigidity of the plate, the prefactor in Eq. 15 that quantifies how much energyit costs to bend a unit area of the plate. The tension makes Eq. 17 nonlinear as a displacement of the plate elongates itand thereby induces tension, i.e. T αβ = T αβ [ u ( x ′ , y ′ )]( x , y ). However, for small deformations the displacement-inducedtension is small and will be overwhelmed by the bending rigidity or by tension induced by the clamping. Then the tensionis independent of u and Eq. 17 is linear. Table 1 shows that many nanomechanical devices are doubly-clamped beams or cantilevers instead of plates i.e., theyhave a width that is much smaller then their length. For a beam, the normal components of the stress on the sides shouldalso vanish, i.e., σ yy = σ xy = y can be done directly. This implies that the tension T αβ is only in the x-direction. Moreover, the material displaces in the x-direction so that T = T xx is independent of x .Combining all of this yields the equation of motion in the “thin-beam” approximation: the Euler-Bernoulli equation withtension: ρ A ∂ u ∂ t + D ∂ u ∂ x − T ∂ u ∂ x = F , (18)where the crosssection area A equals wh for a rectangular beam and for a cylindrical beam A = π r (see Fig. 7). Thestructure of Eq. 18 is similar to that of the the equation of motion for a plate, Eq. 17. The first term is the accelerationand the term on the right-hand side is the external force. The restoring force due to the bending sti ff ness depends onthe fourth order derivative of the displacement w.r.t. x . For the tension, this is a second-order derivative. The bendingrigidity D = Eh w / − ν ) can be written as the product of the Young’s modulus and the second moment of inertia D = EI / (1 − ν ), where the small correction (1 − ν ) is often omitted [15]. For a rectangular beam the second moment ofinertia is I = h w /
12; for a (solid) cylinder with radius r it is I = π r / − Nm for a carbon nanotube (Sec. 2.4.4) or as large as 10 − Nm for a millimetre-sized mirror [33]. A cantilever is a beam that is clamped on one side ( x =
0) and free on the other side ( x = ℓ ). Because a cantileveris not fixed on the, say, right side, the tension is zero apart from more exotic cases where electrostatic forces [74, 80] orsurface tension [145, 146] acts on the free end. With T =
0, the eigenmodes u n and (angular) eigenfrequencies ω n satisfy: ω n ρ Au n = D ∂ u n ∂ x . (19)The solution to this equation is a linear combination of the regular and hyperbolic sine and cosine functions (sin( kx ),cos( kx ), sinh( kx ), and cosh( kx ) resp.), where k = ω D /ρ A . Their coe ffi cients are determined by the boundary conditions. In principle, the first term should read ρ h ∂ u /∂ t + ρ h ∂/∂ t ( ∂ u /∂ x + ∂ u /∂ y ) /
12 as the material is also moving in the x and y direction. Thesecorrections are, however, negligible when h /ℓ ≪ Note that the units of the bending rigidity, tension and external force are di ff erent from the case of a plate due to the integration over the y-coordinate. D is given in Nm instead of Nm( = J) and T is now in N instead of N / m. The external force F is given per unit length instead of per unit area. Fromthe context it should be clear what the meaning of the di ff erent symbols is. able 4: Normalized eigenfrequencies and average mode deflection µ n for the first 5 flexural eigenmodes for cantilevers and doubly-clamped beams. µ n = ℓ − R ℓ ξ n ( x ) d x indicates the displacement of the mode averaged along the length of the resonator per unit deflection. This number is importantto calculate the detection e ffi ciency for detectors that couple over the whole length of the beam (p26). α n and β n are the solutions of Eq. 21 and 18respectively which determine the eigenfrequencies. For large n the solutions approach α n → ( n + / π and β n → ( n + / π . Mode Cantilever Beam n α n f n / f µ n β n f n / f µ n u (0) = u ′ (0) =
0, where ′ denotes di ff eren-tiation with respect to x . At the free end, the force in the z-direction and the torque vanish, so u ′′ ( ℓ ) = u ′′′ ( ℓ ) = ffi cients. This system always has a trivial solution u n = ffi cients are zero. There are, however, certain values k = k n where one of the four boundary conditionsis automatically satisfied. These values correspond to the eigenmodes of the flexural resonator. Using the other threeboundary conditions, three of the coe ffi cients are expressed in the fourth one (which we call c ). For a cantilever wedefine α n ≡ k n ℓ and the n-th eigenmode is: u n ( x ) = c sin( α n x ℓ ) − sinh( α n x ℓ ) − sin( α n ) + sinh( α n )cos( α n ) + cosh( α n ) (cid:20) cos( α n x ℓ ) − cosh( α n x ℓ ) (cid:21)! . (20)Note, that if u n ( x ) is an eigenmode of the cantilever, then c · u n ( x ) is one as well, for every value of c . In the following partwe will therefore use the normalized eigenfunctions ξ n , which satisfy ℓ − R ℓ ξ n d x =
1. The eigenfrequencies are givenby: cos( α n ) cosh( α n ) + = , ω n = π f n α n ℓ − p D /ρ A . (21)This equation can only be solved numerically. The first few solutions α n are indicated in Table 4 and Fig. 8b shows thecorresponding mode shapes. Unlike in a string under tension, a type of resonator that will be discussed in the next section,which has f n / f = ( n + µ n = ℓ − R ℓ ξ n d x , decreases with increasing mode number n . This is because a part of the cantilever is movingupwards and an other part is moving downwards.The analysis for the flexural eigenmodes of a doubly-clamped beam, which does not have a free end but which isclamped at both x = x = ℓ , closely follows that for a cantilever. The di ff erence is that the boundary conditions at x = ℓ are now u ( ℓ ) = u ′ ( ℓ ) =
0. With β n ≡ k n ℓ this yields: u n ( x ) = c sin( β n x ℓ ) − sinh( β n x ℓ ) − sin( β n ) + sinh( β n )cos( β n ) + cosh( β n ) (cid:20) cos( β n x ℓ ) − cosh( β n x ℓ ) (cid:21)! , (22)and cos( β n ) cosh( β n ) − = , ω n = π f n = β n ℓ − p D /ρ A . (23)Table 4 shows that β n > α n , so that the eigenfrequencies of a clamped-clamped beam are higher than that of a cantileverwith the same dimensions. This is because the additional clamping makes it sti ff er. Moreover, due to symmetry µ n vanishes for the odd modes of a beam. Figure 8 shows the first three mode shapes.Equations 21 and 23 show that when the resonator is made shorter while its transverse dimensions are kept the same,the eigenfrequencies increase due to the ℓ − term in the characteristic frequency scale ω ch ≡ ℓ − p D /ρ A [147, 148]. Onthe other hand, the crosssectional area and the bending rigidity depend on the size of the resonator. The exact scaling is ω ch ∝ h /ℓ and ω ch ∝ r /ℓ for a rectangular and cylindrical resonator respectively . When making all dimensions of aresonator smaller [38], the resonance frequencies thus increase inversely proportionally with the size. This scaling is thereason that nanomechanical resonators can have very high resonance frequencies of more than 1 GHz [28]. So far we have not considered the e ff ect of tension in the examples of flexural resonators. In certain types of materialsthere can be so much stress that the bending rigidity hardly contributes to the restoring force. The resonator is then astring under tension instead of a beam. Eq. 18 shows that this is the case when T ≫ D /ℓ , or when this is rewritten inthe (longitudinal) strain γ , γ ≫ ( h /ℓ ) /
12. Tension is thus more important in resonators with a large aspect ratio ℓ/ h .16 a)(b) (c)(d) n = 0n = 1n = 2 Figure 8: Schematic of a singly-clamped cantilever (a) and a doubly-clamped beam (c). (b) and (d) show the shape of the first three ( n = , ,
2) flexuralmodes of these flexural resonators. This shape is calculated using Eqs. 20 and 22 respectively.
The eigenfrequencies of a string under tension are f n = p T /ρ A × ( n + / ℓ and the corresponding modeshapes are ξ n ( x ) = √ π nx /ℓ ).Tension in the resonator arises when materials with di ff erent thermal expansion coe ffi cients or di ff erent lattice con-stants are used. An example of the former is given by Regal et al. [50], where a 50 µ m long aluminum beam resonatoris annealed at 150 to 350 o C, thereby increasing the resonance frequency from 237 kHz to 2 . ff erent lattice constants. If the layer with the resonator has a smaller latticeconstant that the layer underneath it, then the resonator is strained. By engineering the heterostucture, di ff erent amountsof tension can be induced in the resonator [149, 150]. Another way of inducing tension is by placing the resonator on aflexible substrate that can be bent. As the top part of the substrate is elongated, the resonator becomes strained. With thistechnique, the resonance frequency of a resonator has been tuned by more than a factor 5 [151]. The most commonly usedhigh-stress material is silicon nitride. Under the appropriate growing conditions thin films with stresses of ∼ ff ects are important. These systems will be treated in Sec. 2.4.4.Although high Q -values seem to be a general observation for strained beams, the mechanism behind the increaseis not completely clear. Recall that the Q-factor is proportional to the ratio between the energy stored in the oscillatorand the energy dissipated per cycle. Supported by experiments, the authors in Ref [72] demonstrate that stress doesnot substantially change the dissipation rate in strained beam mechanical oscillators but rather significantly increases theelastic energy stored in the resonator. They argue that the microscopic origin of the damping lies in localized defect statesin the material. In the abovementioned experiments by Regal et al. the quality factor of an aluminum beam resonator wasincreased by a factor 50 by annealing it [50]. Although the induced tension also increased the resonance frequency by afactor 10, the quality factor increased more. This means that in this case the annealing did not just increase the energy inthe resonator, but it actually reduced the damping rate ( γ R = ω R / Q ), which is the ratio of the resonance frequency ω R andthe quality factor. Also in this system the physical mechanism behind this remarkable increase is still unclear. In the case of a string resonator the tension is tensile, i.e., the resonator is elongated. It is also possible that theresonator is compressed. In that case, the tension is negative. When the tension exceeds a critical value, it is energeticallyfavorable for the beam to have a non-zero flexural displacement; the beam buckles. Buckled beams form an importantclass of (nano)mechanical resonators, and can be formed when the resonator consists of layers of di ff erent materials[155, 156, 19], or using thermal expansion [18].The starting point for the analysis of buckled beams is the Euler-Bernoulli beam equation (Eq. 18) with no externalforce, i.e., F =
0. The boundary conditions are the same as for a doubly-clamped beam: u (0) = u ( ℓ ) = u ′ (0) = u ′ ( ℓ ) = ℓ ) and that of the clamped beam ( ℓ ) are di ff erent when the clampingpoint exert a longitudinal tension on the beam. This is the so-called residual tension, T = EA γ , with γ = ( ℓ − ℓ ) /ℓ .17 igure 9: (a) The distance between the clamping points ℓ of a doubly clamped beam di ff ers from the length of the free beam ℓ . If the induced tensionis large enough, it is energetically favorable for the beam to displace, releasing strain energy at the cost of bending energy. This is called buckling. Dueto the displacement u ( x ) the length of the beam is extended to L ( t ). (b) The total potential energy U T + U B for a beam below (dotted) and above (solid)the buckling threshold. In the former case the potential energy only has a single minimum at u max =
0, whereas the latter has two minima at non-zerodeflection. (c) An example of the calculated eigenfrequencies of a 50 µ m-long beam with D = . · − J [19]. The mode shapes at the position ofthe dots are shown.
The second, positive, contribution comes from the stretching of the beam when it flexures. The resulting length of theflexed beam is denoted by L = R ℓ { + ( ∂ u /∂ x ) } / d x . Combining both e ff ects gives for small ( ∂ u /∂ x ≪
1) deflections: T ≈ T + EA ℓ Z ℓ ∂ u ∂ x ! d x . (24)In the absence of driving, both the displacement and the tension are time-independent and the static deflection u dc satisfies: D ∂ u dc ∂ x − T dc ∂ u dc ∂ x = , with T dc = T + EA ℓ Z ℓ ∂ u dc ∂ x ! d x . (25)This only has a non-trivial solution u dc , T dc = n T c , where T c = − π D / L is the critical tension at which thebeam buckles and n is an integer. The solution is then u dc ( x ) = u max [1 − cos(2 π nx /ℓ )] /
2. When an initially unstrainedbeam is compressed slightly, work is done and the energy stored in U T increases. The potential energy has a singleminimum around u max = T is mademore negative than T c , it becomes energetically favorable for the beam to convert a part of U T into the bending energy U B . The potential energy now has two minima at non-zero static displacements, as illustrated by the solid line in Fig. 9b.The beam buckles to a displacement that keeps the tension exactly at T c (for n = u max = ± ℓ/π · ([ T c − T ] / EA ) / for T ≤ T c <
0. Note, that in the absence of astatic force, the beam does not have a preferential direction of buckling.To find the eigenmodes of the buckled beam, we do not only focus on the static deflection, but we also include thedynamics of the displacement. When driving the modes of the beam, the total deflection u is the sum of the static ( u dc )and an oscillating part ( u ac ). The time-dependent displacement satisfies: ρ A ∂ u ∂ t + D ∂ u ac ∂ x − T dc ∂ u ac ∂ x = T ac ∂ u dc ∂ x , (26)with: T ac = EA ℓ Z ℓ ∂ u dc ∂ x ∂ u ac ∂ x d x . (27)Note, that both sides of Eq. 26 are linear in u ac and that the static displacement u dc thus acts as an e ff ective spring constantfor ac motion, as indicated by the r.h.s. of Eq. 26. The eigenfrequencies of buckled beams were calculated by Nayfeh et al. [157]. As an example the eigenfrequencies and modes for the beam used in Ref. [19] are shown in Fig. 9b.At zero buckling the frequency of the fundamental mode is zero as the potential energy is quartic in the displacement:the quadratic terms in U B and U T cancel each other exactly. The frequency of the fundamental mode increases with18ncreasing buckling due to the contribution of T ac . The first higher mode has an eigenfrequency ω / π = .
44 MHz and isindependent of u max , as the mode is anti-symmetric around the node, giving T ac =
0. When u max is increased to 0 . µ m,the two lowest modes cross and the fundamental mode is higher in frequency than the first odd mode. Thus, when thelength of the beam, bending rigidity and the buckling displacement are known, the eigenfrequencies and modeshapes of abuckled beam can be calculated.
In Sec. 2.4.2 we showed that tension can overwhelm the influence of bending rigidity in nanomechanical deviceswith high aspect ratios. For these thin wires the displacement can be of the order of the resonator thickness and in thiscase nonlinear e ff ects connected with the deflection-induced tension become important. Suspended carbon nanotubesresonators are prototypical examples where the induced tension can be so large that the tubes are tuned from bending(beam-like) to tension dominated (string-like) [20, 158].The induced tension is a key feature of thin resonators as it can be used to electrically tune their frequency over a largerange. In this Subsection we derive the equations for the frequency tuning and give some typical numbers for single-wallCNTs. We will start with considering the device geometry of a suspended CNT resonator and derive some of the basicequations describing the electrostatics of the problem. The analysis is, however, also applicable to other thin string-likeresonators such as multi-wall CNTs, suspended graphene nanoribbons, or long suspended nanowires made from inorganicmaterials (see Table 2).Fig. 10a shows an atomic force microscope image of a suspended single-wall carbon nanotube in a three terminalgeometry [159]. The nanotube is connected to source and drain electrodes, enabling electrical transport measurements.The tube is suspended above a gate electrode at a distance h g , which cannot only be used to change the electrostaticpotential on the tube, but also to drive the resonator [160] and to induce tension. An electrostatic force can be applied tothe tube by applying a voltage V g between a gate electrode (Fig. 10b) and the nanotube. The potential energy dependson the capacitance between the tube and the gate C g , and is U F = − C g V /
2. The potential energy depends on the distancebetween the gate and the tube h g − u ( x ) via the gate capacitance, which means that there is a force acting acting on thenanotube. The electrostatic potential energy is written as: U F = − R ℓ c g ( x ) V / x , where c g ( x ) is the capacitance perunit length, and the potential energy equals by definition U F = − R ℓ Fu d x . The electrostatic force per unit length is thus F ( t ) = ∂ c g /∂ uV .To calculate the displacement dependence of the capacitance, we first consider the spatial profile of the electrostaticpotential. Under the assumption that the screening e ff ect of the source and drain electrodes is negligible, the tube isviewed as an infinitely long grounded solid cylinder, suspended above a conducting plate at an electrostatic potential φ ( z = = V g . The potential profile for u = φ ( y , z ) = V g − V g h g / r ) ln (cid:20) z + q h − r (cid:21) + y (cid:20) z − q h − r (cid:21) + y . (28)The field lines associated with this potential are shown in Fig. 10b. The deflection of the nanotube is included by replacing h g with h g − u . After dividing the induced charge by the gate voltage, the capacitance per unit length c g ( x ) is obtained : c g ( x ) = πǫ arccosh (cid:16) [ h g − u ( x )] / r (cid:17) ≈ πǫ arccosh (cid:16) h g / r (cid:17) + πǫ q h − r arccosh (cid:16) h g / r (cid:17) u ( x ) . (29)The approximation in Eq. 29 is allowed because the displacement u is typically much smaller than h g . This in contrast totop-down fabricated devices, where higher order terms can be important and electrostatic softening of the spring constantmight occur [163, 57]. This softening is thus due to nonlinear capacitance terms, which should be contrasted to thenonlinearities due to the displacement-induced tension that we will discuss further on in this Section. For thin resonatorsunder high built-in tension, however, the nonlinear capacitance terms can become relevant again. This has been nicelyillustrated in graphene resonators, which show a small downward shift in the frequency around zero-gate voltage [104].We will not consider these e ff ects in the following, but we finally note that the electrical softening is closely related to theoptical spring that will be discussed in Sec. 3.5.3. The modes are classified by their shape and not by the ordering of eigenfrequencies. The fundamental mode is the mode without a node. This is the potential energy for the tube, in contrast to the energy stored in the capacitor: + C g V /
2. The di ff erence in the sign is because the voltagesource performs work when the capacitance changes [161], which should also be taken into account. This expression might appear di ff erent from those in Refs [135] and [158], but note that arccosh( x ) = ln( x + √ x − ≈ ln(2 x ) for x ≫ a n o t u b e D r a i n S o u r c e T r e n c h V g x h g z y(a) (b) Figure 10: (a) an AFM image of a suspended nanotube device connected to the source and drain electrode. The tube is suspended above the trench only.The suspended part of this device is ℓ = . µ m long and the radius of the tube is r = . h g changes, the gate capacitance C g changes. The electrostatic force per unit length is now: F ( t ) = ∂ c g ∂ u V ( t ) = πǫ V ( t ) q h − r arccosh (cid:16) h g / r (cid:17) . (30)Often, the gate voltage consists of two parts: a static part V dcg and a time-dependent part V acg cos( ω t ) to drive the nano-tube at frequency f = ω/ π . The experimental condition V acg ≪ V dcg ensures that terms proportional to ( V acg ) arenegligible. The force is then the sum of a static and driving contribution: F = F dc + F ac cos( ω t ), with F dc = πǫ / ( h − r ) / arccosh (cid:16) h g / r (cid:17) · ( V dcg ) and F ac( t ) = πǫ / ( h − r ) / arccosh (cid:16) h g / r (cid:17) · V dcg V acg ( t ).The bending mode vibrations are described by the Euler-Bernoulli beam equation with tension included, i.e., Eq. 18.When the amplitude of the oscillation u ac is small compared to the larger of the tube’s radius and the static displacement,terms proportional to u are negligible and the tube is in the linear regime . Similar to the analysis presented for thebuckled beams, the equation of motion (Cf. Eq. 18) is separated: D ∂ u dc ∂ x − T dc ∂ u dc ∂ x = F dc , (31) − ω ρ Au ac + i ωγ u ac + D ∂ u ac ∂ x − T dc ∂ u ac ∂ x − T ac ∂ u dc ∂ x = F ac . (32)The first equation describes the static displacement of the tube that is induced by the dc gate voltage. This equation isindependent of u ac . On the other hand, the ac displacement, given by Eq. 32, depends on the static displacement and alsoon the static tension T dc . Similar to the case of the buckled beam, the static tension has two contributions, as indicatedby Eq. 25: The first one is the residual tension due to the clamping as the length of the suspended part is not necessarilyequal to the length when it would not be clamped. For example, a nanotube resonator could be strained during the growthprocess or lay slightly curved on the substrate before suspending it. The second contribution is the displacement-inducedtension: The gate electrode pulls the resonator towards it, thereby elongating it. Moreover, the oscillator experiences atime-dependent variation in its length and T ac is the part of the tension that is linear in u ac . Both e ff ects are included inEq. 32. As the tension contains the static displacement, it has to be solved self-consistently [158, 135, 159] with Eq. 31to find the static displacement. The resulting static tension, ac tension and dc displacement are then inserted into Eq. 32to find the eigenfrequencies ω n and the response function u ac ( x , ω ) [164].To analyze the system of Eqs. 24, 31 and 32, it is useful to take a closer look at their scaling behavior [135]. Inthis section, we use the convention that primed variables indicate scaled (dimensionless) variables. An obvious way tonormalize the coordinate x is to divide it by the tube length: x ′ = x /ℓ , so that the equation for the static displacement Eq.31 becomes: ∂ u dc ∂ x ′ − ℓ T dc D ∂ u dc ∂ x ′ = ℓ F dc D ≡ l dc . (33)On the right hand side, a natural length scale for the static displacement, l dc , appears. However, scaling the displacementwith l dc is not handy because l dc equals zero at zero gate voltage. Therefore u dc (and u ac ) are scaled by the radius of thetube: u ′ dc = u dc / r . Moreover, the tension has become dimensionless, resulting in an equation for the static displacementwhere the number of parameters has been reduced from 5 to 2: ∂ u ′ dc ∂ x ′ − T ′ dc ∂ u ′ dc ∂ x ′ = l ′ dc , (34) This refers to the dynamical behavior. The static displacement is actually nonlinear when the tube is in the strong bending limit [158]. b) (c)(a) T’ = 0 dc T’ = 10 dc Figure 11: (a) static displacement profiles for T ′ dc = T ′ dc = (bottom). (b,c) The calculated static displacement at the center of thenanotube (b) and the corresponding tension (c) for various value of the residual tension T ′ . The limits for the tension for small and large static forcesare indicated.Figure 12: Gate-voltage tuning of the eigenfrequencies of the first (a) and second (b) flexural mode of a 1 µ m-long suspended carbon nanotube (seeTable 5). The residual tensions are T = − , , +
230 pN for the solid, dashed and dotted lines respectively, which corresponds to the dimensionlessvalues T ′ = − . , , + where T ′ dc = ℓ T dc D = T ′ + Ar I Z ∂ u ′ dc ∂ x ′ ! d x ′ , T ′ = ℓ T D , and l ′ dc = l dc / r . (35)The definition of T ′ dc with the ℓ dependence shows that tension becomes more and more important when the length of thedevice increases. Figure 11a shows the dc displacement profiles for di ff erent values of the static tension. In the case wherethe bending rigidity dominates (top panel) the profile is rounded at the edge, whereas for high tension (lower panel) theprofile is much sharper. The tension and center deflection are calculated by self-consistently solving Eq. 31 with Eq. 35,and are plotted in Fig. 11b and c. Two di ff erent slopes can be distinguished in the double-logarithmic plot of Fig. 11b.These correspond to the weak and strong bending regime [158], where u dc is proportional to F dc and F / dc repectively.The two regimes cross at T ′∗ dc = √ ≈ . , l ′∗ dc = · / ≈ l ′ dc = l ′∗ dc is calledthe cross-over voltage, V ∗ g . AFM measurements of the gate-induced displacement of multi-walled carbon nanotubes haveconfirmed this scaling behavior experimentally [135].To calculate the gate-tuning of the resonance frequency f R ( V dcg ), the scaling analysis is also applied to the equation forthe ac displacement, Eq. 32. One immediately finds the length scale l ac = ℓ F ac / D for the ac force and T ′ ac = T ac ℓ / EI .As in Sec. 2.4.1, ω ch = ( D /ρ A ) / /ℓ is again the characteristic frequency scale for the bending mode vibrations. Next,one has to solve Eqs. 34 and 35 to obtain the static tension and dc displacement. Then, the boundary conditions areimposed to find the resonance frequencies. Figure 12a shows the calculated eigenfrequencies, plotted against the staticpulling force for di ff erent residual tensions. The higher the residual tension is, the higher the resonance frequencies are ata given V dcg . The value T ′ = − . ≈ T c indicates that the thin resonator is close to buckling; this is visible by the nearlyvanishing resonance frequency of the first mode at low V dcg . When the static force is increased, all resonance frequenciesincrease and the di ff erences between the curves due to the di ff erent residual tensions become smaller.To relate the dimensionless quantities in this Subsection to physical ones, the system dimensions are needed. Table5 shows the estimated sizes and the calculated values of several parameters for two typical nanotube devices, one with alength ℓ = µ m and one with a length ℓ =
200 nm. Note, that with our definition of u n , the mass and spring constantappearing in the zero-point motion u (Eq. 51) and the equipartition theorem (Eq. 54) are equal to the total mass m and k R = m ω n respectively. There is no need to introduce an e ff ective mass; see Sec. 3.1. The table indicates that for a longdevice tension is important and the di ff erence between f R and f is large. For a short resonator, the mechanical propertiesare mainly determined by the bending rigidity and f R ≈ f . The higher value of V ∗ g indicates that a larger gate voltage has21 able 5: Data for two suspended carbon nanotubes with di ff erent lengths ℓ . The values of the parameters are calculated with a nanotube radius r = . h g =
500 nm. The resonance frequency f R , tension T dc and static displacement u dc( ℓ/
2) are evaluatedfor a gate voltage V dcg = m , f and D are the mass, resonance frequency of the nanotube without tension and the bendingrigidity respectively. Furthermore, k dc and k R = m ω R are the static and dynamic spring constants [165, 166, 167], C g is the capacitance to the gate and u the zero-point motion (see Sec. 1 for its definition and the discussion in Sec. 3). ℓ µ m m − kg D − Nm f f R k R − N / m u C g ∂ C g /∂ u / nm F dc ℓ
21 4.2 pN l dc 4412 7.1 nm u dc( ℓ/
2) 4.1 0.018 nm k dc − N / m T dc . · − nN T ′ dc
73 1 . · − V ∗ g l ′ dc 2 . · ff erence between a nanobeam resonator with a high frequency and one with a lowfrequency that is tuned using a gate voltage to the same frequency as the former [159]. The current associated with themotion in mixing experiments (see Sec. 4.3.3) [73, 20, 76, 78, 82, 83] is proportional to the length-averaged displacementamplitude µ n u n ( ω ); it is thus proportional to the length scale l ac that determines the amplitude. As a consequence, themechanical signal drops rapidly with decreasing length, making the measurement of single-walled carbon nanotubes with f > ℓ . . µ m and r = × smallercompared to a f =
100 MHz device with ℓ ≈ . µ m. The latter tube can also operate at 1 GHz frequency by tuning itwith a gate-induced tension of T ′ dc = · . In this case the signal decreases too, but only by a factor of 10. A tension of T ′ dc = · corresponds to a strain of about 0 . So far, we have only considered one-dimensional resonators such as beams and strings. An example of a two dimen-sional resonator is a graphene nanodrum. This device consists of a hole that is etched in a substrate and that is covered bya (few-layer) graphene flake. Surprisingly, an one-atom layer can be suspended and holes with a diameter of 100 µ m havebeen reported [171]. A much smaller version of these devices (a nanodrum) has been used to study the (im)permeabilityof graphene to gases [100] and to measure the bending rigidity of and tension in the flake using an atomic force micro-scope [107, 172]. In the latter experiments, an atomic force microscope tip is used to apply a force F tip to the flake asillustrated in Fig. 13a. This versatile technique has also been applied to other geometries [173, 174] and nanomaterials[175, 176, 177]. The point ( r , θ ) where the force is applied can be varied and the resulting deflection of the nanodrumis measured. The restoring force that opposes the applied force has several contributions. First of all, there is the bendingrigidity of the flake D , and secondly, tension may be present in the flake.Since graphite is highly anisotropic, the analysis of the bending rigidity of an isotropic material in Sec. 2.3 has to begeneralized. Using the compliance tensor in Eq. 11 the rigidity for bending along the sheets (see Fig. 5b) is calculated: D = E (cid:31) h / − ν (cid:31) ) , (36)which only contains the in-plane elastic constants. However, when the number of graphene layers becomes small, correc-tions to Eq. 36 have to be made. Consider the situation in Fig. 5b where a few-layer graphene sheet is bent with a radius The analysis is in principle also valid for other layers two-dimensional membranes; for a thin isotropic membrane Eq. 38 and the derivationfollowing it are still valid if the appropriate bending rigidity D is taken.
22f curvature R c . In the continuum case, the bending energy is given by U B /ℓ W ≡ D / R c = R h / − h / E (cid:31) ( z / R c ) d z , whentaking ν = N , the continuum approximation in the z-direction isno longer valid and the stress is located only at the position of the sheets z i = c ( i − [ N + / c = .
335 nm is theinter-layer spacing (see p. 12). The integral over z is replaced by a sum and the bending rigidity becomes: D N = E (cid:31) h − ν (cid:31) ) N − N , (37)where the thickness is set by the number of layers h = Nc , which reduced to Eq. 36 in the limit N → ∞ . According to Eq.37 the bending rigidity vanishes for a single layer. However, molecular dynamics simulations have shown that a singlelayer of graphene still has a finite bending rigidity: of the order of one eV [117, 178, 179] (compare this to the valuefor a double layer ( N =
2) calculated with Eq. 37: D =
54 eV). The rigidity of a single layer comes from the fact thatelectrons in the delocalized π -orbitals, located below and above the sheet, repel each other when the sheet is bent [179].The deflection of the nanodrum satisfies the equation for the deflection of a plate, Eq. 17, and when the force appliedby the AFM tip is assumed to be located at a single point ( x , y ), one gets [138, 180, 181, 182]: D ∇ − ∂∂ x α T αβ ∂∂ x β ! u ( x , y ; x , y ) = F tip δ ( x − x ) δ ( y − y ) . (38)Here, the ∇ -operator and the partial derivatives ∂/∂ x i are working in the xy-plane only, as the z-dependence is absorbedin the bending rigidity (see Sec. 2.3) and where the tension T αβ = R h σ αβ d z . This equation is di ffi cult to solve in itsmost general form, but fortunately some simplifications can be made: The tension tensor can have both normal andshear components. It is, however, always possible to find two orthogonal directions where the shear components are zero[136]. When we assume that the tension is uniform then these directions are independent of position, so without loss ofgenerality the x and y-axis are taken along the principle directions of the tension. When the di ff erence in tension in the xand y direction, ∆ T = ( T xx − T yy ) /
2, is small, first the solution for isotropic tension ( T αβ ≈ T δ αβ ) is obtained and then thecorrection due to the finite ∆ T can be calculated [165]. Here, we will focus on the situation where ∆ T =
0. For a circularmembrane, it is convenient to use polar coordinates and the equation for the displacement reads : (cid:16) D ∇ − T ∇ (cid:17) u ( r , θ ; r , θ ) = F tip r δ ( r − r ) δ ( θ − θ ) . (39)The solution is written as: u ( r , θ ; r , θ ) = ∞ X m = R m ( r ; r ) cos( m θ − m θ ) . (40)Inserting this into Eq. 39 yields for the radial coe ffi cients: R ( r ; r ) = A I ( λ r / R ) + B K ( λ r / R ) + C ln( r / R ) + D + R ( p )0 ( r ; r ) , (41) R m ( r ; r ) = A m I m ( λ r / R ) + B m K m ( λ r / R ) + C m ( r / R ) − m + D m ( r / R ) m + R ( p ) m ( r ; r ) ( m > , (42)where I m and K m are the Bessel functions of the first and second kind respectively, and λ = p T R / D is a dimensionlessparameter that indicates the importance of the tension in comparison with the bending rigidity of the flake.The flake with radius R is clamped at the edge of the circular hole so the boundary conditions are u ( R ) = u / d r | r = R =
0. Furthermore, the deflection at the center is finite and smooth (i.e., d u / d r | r = = ffi cients { A m , B m , C m , D m } can be calculated analytically. Figure 13b shows the deflection profiles calculated where the force isapplied at di ff erent distances r from the center. The deflection of the flake is clearly reduced when the AFM tip ismoved away from center of the nanodrum. This indicates that its local compliance k − f ( r , θ ) = ∂ u ( r , θ ; r , θ ) /∂ F tip decreases. As the tension is assumed to be isotropic, k − f is independent of θ and its radial profile, shown in Fig. 13cfor di ff erent values of the tension, contains all the information. In analogy with the displacement profile of a bendingand tension-dominated carbon nanotube (Fig. 11a), the profile is rounded at the edge of the hole for vanishing tension( λ = λ → ∞ ) the compliance profile becomes much shaper at the edge and diverges at ∇ = ∂ ∂ x + ∂ ∂ y = ∂ ∂ r + r ∂∂ r + r ∂ ∂θ and ∇ = ∂ ∂ x + ∂ ∂ y = ∂ ∂ r + r ∂ ∂ r − r ∂ ∂ r + r ∂∂ r + r ∂ ∂θ + r ∂ ∂θ + r ∂ ∂ r ∂θ − r ∂ ∂ r ∂θ . In the limit λ → ∞ , the ∇ term in Eq. 39 vanishes and only a second order di ff erential equation remains. Therefore, the boundary conditionsd u / d r | r = , R = igure 13: (a) Schematic overview of the nanodrum. A few-layer graphene flake is suspended over a circular hole with radius R . A force is applied at thepoint ( r , θ ) using an AFM tip. This results in a deflection of the nanodrum (b) Colormaps of the calculated deflection profile (Eq. 39) of a nanodrumwith vanishing tension. The force is applied at the location of the cross and the color scale is identical in all four panels: white corresponds to a largedeflection and dark gray to no deflection. (c) The calculated radial compliance profile for di ff erent values of the tension, with λ = TR / D . the center for a point force. In practice the tip has a finite radius of curvature which prevents that the spring constantof the flake k f ( r =
0) vanishes [172]. By comparing the experimentally measured profile with the calculated ones, thevalues for D and T can be determined [107]. Knowing the tension (and in principle also the bending rigidity, but thisis negligible for thin flakes) an estimate of the resonance frequency can be made. The fundamental eigenfrequency of acircular drumhead is f = p T /ρ d / π R · ν , where ν ≈ . T ∼ . / m for exfoliated single layer graphene flakes [172].With this value and the value for the two-dimensional mass density of single-layer graphene, ρ d = . · − kg / m (seepage 12) we obtain a resonance frequency of 100 MHz for a nanodrum with a diameter of 2 R = µ m. By decreasing thesize of the hole, the frequency can be increased. For a hole with a diameter of 500 nm, the frequency already exceeds 1GHz. The latter resonator has a mass of only m = · − kg and a very large zero-point motion of u = . . Backaction and cooling In this Section we consider the influence of the detector, the so-called backaction and we discuss ways to cool theresonator to the ground state. To describe these e ff ects in detail concepts such as thermal noise, Brownian motion, and thee ff ective resonator temperature are introduced. Next we will discuss detector backaction and show that it plays an impor-tant role, especially when continuously probing the resonator properties. Backaction is therefore often an unwanted, butunavoidable element in measurements on mechanical systems. It determines how precise the position can be monitored;ultimately, quantum mechanics poses a limit on continuous linear detection, the standard quantum limit. We will derivethis limit in di ff erent ways. The standard quantum limit can be circumvented by employing di ff erent detection schemes,such as square-law detection and backaction evading measurements. These will however not be discussed in depth in thisReview.Backaction does not just impose limits, it can also work to one’s advantage: It can squeeze the resonator motion[183, 184, 185, 186, 187], couple and synchronize multiple resonators [188, 189], and backaction can cool the resonator aswe will show. Besides this self-cooling by backaction, other active cooling schemes have been developed. In this Sectionwe will discuss two di ff erent cooling schemes, namely active feedback cooling and sideband cooling. It is interesting tonote that, at present, cooling has only been performed on top-down devices because the experiments to observe thermalmotion of bottom-up devices are more challenging. Before starting the discussion of backaction, first an important point has to be addressed. In the previous Sectionwe have given the equations of motion for several nanomechanical systems. Solving these equations gives the frequencyand the displacement profile of a particular mode. An important conclusion of describing small displacements aroundthe equilibrium position of NEMS is that each mode can be viewed as a harmonic oscillator. To show this, we start withexpanding the displacement in the basis formed by the eigenfunctions ξ n [191, 192, 159, 164]: u ( x , t ) = X n u ( n ) ( t ) ξ n ( x ) . (43)Inserting this into the equation of motion and taking the inner product with ξ n yields the displacement of mode n . Forexample, for the nanobeams discussed in Sec. 2.4.4, Eq. 32 yields: (cid:16) ω ′ n − ω ′ + i ω ′ γ ′ (cid:17) u ( n )ac = l ac µ n ; µ n = Z ξ n ( x ′ ) d x ′ . (44)The left hand side shows that the frequency response of each mode is equal to the response function of a damped drivenharmonic oscillator. The same conclusion is reached for the other examples: all their modes can be describes as harmonicoscillators. The mathematical reason behind this is that the equation of motion for small deformations of a mechanicalsystem can be written in the form m e ff ¨ u ( r , t ) = − γ ˙ u ( r , t ) + L [ u ( r , t )] for some Hermitian operator L . Its eigenfunctionsare the mechanical modes and these form a complete orthogonal basis. After expanding the displacement in this basis, aset of uncoupled harmonic oscillators results. An important question that one should ask after the transformation from thespatial modes to the harmonic oscillators is: What is the e ff ective mass m e ff of the oscillator? This question might seemtrivial at first, but the concept of the e ff ective mass has given rise to much confusion in the past years. From the equationof motion it follows that the e ff ective mass of mode n equals m e ff , n = R V ρξ n d V . The e ff ective mass thus depends on thenormalization of the basis functions, or equivalently on the definition of the mode displacement u n . In this Review wehave adopted the convention that the basis functions ξ n are orthonormal, i.e. ℓ − R ℓ ξ n d x =
1. In this case the e ff ectivemass equals the total mass of the system (i.e., m e ff , n = m ). On the other hand, one can also use di ff erent normalizations,for example using the average displacement ( ℓ − R ℓ ξ n d x =
1) or using the maximum displacement (max ξ n = ff ective mass di ff ers from the total mass and it depends on the exact mode profile. To illustrate the confusionthis may create, consider the flexural modes of a beam resonator: With the latter two definitions, every mode has adi ff erent e ff ective mass. Moreover, tuning a flexural resonator from the bending to stretching-dominated regime (Sec.2.4.4) changes the e ff ective mass of its modes. These complications are avoided by using orthonormal basis functions,where ℓ − R ℓ ξ n d x = ff erent coupled modes in nano and micromechanical systems[193, 194, 195, 164, 124, 196], most experimental and theoretical work focuses on a single mode only and the harmonicoscillator describes the dynamics of the entire mechanical system. In this case, we make no distinction between the It can be shown that these functions form a basis for functions that satisfy the homogeneous boundary conditions [190, 159]. For simplicity it is assumed that the mass density is constant. x is given by u ( n ) ξ n ( x ) and not by u ( n ) itself.Before reviewing the properties of the classical and quantum harmonic oscillator, we stress, however, that knowledgeabout the displacement profile remains important when analyzing NEMS experiments. In particular, di ff erent detectorsor driving forces may couple di ff erently to the displacement profile and could detect therefore di ff erent modes. In mostcases both the driving and detection mechanisms couple to the average displacement of the resonator. In this case, anti-symmetric modes are not visible in nanomechanical experiments as these have a vanishing value of the length-averageddisplacement µ n (see Eq. 44 and Table 4). An example is a nanotube resonator with frequency mixing readout (Secs. 2.4.4and 4.3.3) that is either coupled to a back gate or to a local gate; the former couples uniformly to the whole nanotube, sothat the detected signal is proportional to µ n and consequently only symmetric modes can be detected. A local gate may,depending on its position, couple to all modes. The harmonic oscillator is probably the most extensively studied system in physics. Nearly everything that returnsto its equilibrium position after being displaced can be described by a harmonic oscillator. Examples range from thesuspension of a car, tra ffi c-induced vibrations of a bridge, and the voltage in an electrical LC network, to light in anoptical cavity. For large amplitudes, the oscillator can become nonlinear. We will not study that situation in this Report,but instead we refer the reader to the large body of literature on this subject, see e.g. Refs. [197, 198]. We will nowproceed with describing the classical and quantum harmonic oscillator in more detail and reviewing their basic properties. In a harmonic oscillator, the potential energy depends quadratically on the displacement u from the equilibrium posi-tion: V ( u ) = k R u , (45)where k R is the spring constant. The parabolic shape of the potential results in a force that is proportional to the displace-ment. When damping and a driving force F ( t ) are included, the equation of motion reads: m ¨ u = − k R u − m γ R ˙ u + F ( t ) , (46)for a resonator with mass m and damping rate γ R . When the oscillator is displaced and released, it will oscillate atfrequency ω R with a slowly decreasing amplitude due to the damping. The quality factor Q = ω R /γ R indicates how manytimes the resonator moves back and forth before its energy has decreased by a factor e.The harmonic oscillator responds linearly to an applied force; in other words, it is a linear system. Any linear systemis characterized by its impulse response or Green’s function [199]. For the harmonic oscillator, the impulse response h HO ( t ), is the solution to Eq. 46 with F ( t ) = k R δ ( ω R t ): h HO ( t ) = sin( ω R t ) e − ω Rt Q Θ ( ω R t ) , (47)where Θ ( t ) is the Heaviside stepfunction. The impulse response function describes how the resonator reacts to a kick attime t = t =
0. The resonator then oscillates back and forth with a period 2 π/ω R and these oscillations slowly die out dueto the dissipation. With the impulse-response function the time evolution of the displacement for a force with arbitrarytime-dependence F ( t ) can be obtained directly: u ( t ) = h HO ( t ) ⊗ F ( t ) / k R , (48)where the symbol ⊗ denotes convolution.In many experiments the oscillator is driven with a periodic force F ( t ) = F cos( ω t ). After a short ( ∼ γ − R ) transient,the resonator oscillates with the same frequency as the driving force. This motion is not necessary in-phase with thedriving signal. Both the amplitude and phase of the motion are quantified by the transfer function H HO ( ω ) that is obtainedby taking the Fourier transformation of the equation of motion, Eq. 46: H HO ( ω ) = k R u ( ω ) F ( ω ) = ω R ω R − ω + i ωω R / Q . (49) This is the Green’s function for a high-Q resonator. For lower Q-values, the resonator oscillates at a slightly lower frequency ω ′ R = ω R p − (1 / Q ) and the impulse response of an underdamped oscillator (i.e., one that has Q > /
2) is: h HO ( t ) = sin( ω ′ R t ) exp( − ω R t / Q ) · [1 − (2 Q ) − ] − / Θ ( t ). Anoverdamped resonator ( Q < /
2) returns to u = ff erent impulse response. Throughout this Review it is assumedthat Q ≫ ω ′ R ≈ ω R and h HO is given by Eq. 47. Note, that Eq. 49 is valid for all (positive) values of Q . By convention [199], the Fourier transformation is defined as: X ( ω ) = F [ x ( t )] = R + ∞−∞ x ( t ) exp( − i ω t ) d t so that the inverse transformation is givenby: x ( t ) = F − [ X ( ω )] = π R + ∞−∞ X ( ω ) exp( + i ω t ) d ω . igure 14: (a) The Green’s function h HO ( t ) and (b) frequency response H HO ( ω ) of a harmonic oscillator with Q =
10. (c) Eigenenergies E n (dashed)and the corresponding wave functions ψ n ( u ) (solid) of the harmonic oscillator for n = .. The magnitude | H HO ( ω ) | and phase ∠ H HO ( ω ) are plotted in Fig. 14b. When the driving frequency is far below theresonance frequency, the oscillator adiabatically follows the applied force: u ( t ) = F ( t ) / k R ⇔ H HO =
1, and so both | H HO | and ∠ H HO are small. The small motion is then almost in phase with the driving signal. The amplitude grows whensweeping the frequency towards the natural frequency. Exactly on resonance, the amplitude has its maximum | H HO | = Q .The phase response shows that at the resonance frequency, the displacement lags the driving force by − π/
2. When furtherincreasing the driving frequency, the oscillator can no longer follow the driving force: the amplitude drops and the lagapproaches − π . The motion is then 180 o out of phase with the applied force. The width of the resonance peak is relatedto the damping: the full width at half maximum of the resonance equals γ R = ω R / Q . In quantum mechanics the harmonic oscillator is described by the Hamiltonian ˆ H = ˆ p / m + m ω R ˆ u [2], as theclassical displacement coordinate u and momentum p = m ˙ u have to be replaced by the operators ˆ u and ˆ p = − i ~ · ∂/∂ u .The displacement is described by a wave function ψ ( u ) that satisfies the time-independent Schr¨odinger equation:ˆ H ψ = − ~ m ∂ ψ∂ u + m ω R ˆ u ψ = E ψ. (50)This equation is solved by introducing the creation and annihilation operators: ˆ a † = ( m ω R ˆ u − i ˆ p ) / √ m ~ ω R and ˆ a = ( m ω R ˆ u + i ˆ p ) / √ m ~ ω R respectively. The Hamiltonian then becomes ˆ H = ~ ω R (ˆ n + ), where ˆ n = ˆ a † ˆ a is the numberoperator that counts the number of phonons in the oscillator. The eigenenergies are E n = ~ ω R ( n + ), with eigenstates | n i . The corresponding wave functions ψ n ( u ) are plotted in Fig. 14c. The lowest ( n =
0) eigenstate has a non-zeroenergy E = ~ ω R , the so-called zero-point energy. Even when the oscillator relaxes completely, it still moves around thepotential minimum at u =
0. The probability density of finding the resonator at position u , is given by | ψ ( u ) | when theresonator is in the ground state. The zero-point motion u is the standard deviation of this probability density: u ≡ Z ∞∞ u | ψ ( u ) | d u ! / = h | ˆ u | i / = s ~ m ω R . (51)The zero-point motion is an important length scale that determines the quantum limit on continuous linear position mea-surement and is also related to the e ff ective resonator temperature as the following Sections will show. In the previous Section, Sec. 3.2.2 it was shown that a resonator always moves because it contains at least the zero-point energy. In practise, except at the lowest temperatures, the zero-point motion is overwhelmed by thermal noise.Thermal noise is generated by the environment of the resonator. As an example, consider a resonator in air. At roomtemperature, the air molecules have an average velocity of about 500 m / s. The molecules randomly hit the resonator andevery collision gives the resonator a kick. These kicks occur independently of each other, so the resonator experiencesa stochastic force F n ( t ). Other thermal noise sources are phonons in the substrate that couple to the resonator via theclamping points, fluctuating amounts of charge on nearby impurities and so on. The environment of the resonator isthus a source of random fluctuations on the oscillator. The force noise can be described by an autocorrelation function27
25 50-404 -202 0 25 50 0 0.9 1 1.1 f/f R t ω R A u t o - c o rr e l a t i on D i s p l ace m e n t R /u
AA rms2
R /u uu rms2 R φφ u(t)/u rms A(t)/u rms φ (t) (b)(a) (c) t ω R PS D ( a . u . ) S uu Figure 15: (a) Simulated time-trace of the displacement (blue), amplitude (black) and phase (red) of a resonator that is driven by Gaussian white noisefor Q =
50. The phase ϕ is in radians and both the displacement u and amplitude A are normalized by the root-mean-square displacement h u i / . Thecorresponding auto-correlation functions and the displacement-noise power spectral density are shown in (b) and (c) respectively. The area under thecurve in (c) equals the variance of the displacement. R F n F n ( t ) = E [ F n ( t ′ ) F n ( t ′ + t )] (the symbol E denotes the expectation value) or by its power spectral density (PSD) S F n F n ( ω ) [200]. For white noise, the latter is independent of frequency and F n ( t ) has an infinite variance. For a given F n ( t ) the realized displacement is easily found using the Green’s function, i.e., with Eq. 48. Figure 15a shows a simulatedtime-trace of this so-called Brownian motion. The resonator oscillates back and forth with frequency ω R while its phaseand amplitude vary on a much longer timescale. The displacement can be written as u ( t ) = A ( t ) cos[ ω R t + ϕ ( t )] (seeAppendix A) and the time-traces of the amplitude A and phase ϕ are plotted in Fig. 15a as well. Figure 15b shows thecalculated autocorrelation functions of the displacement, amplitude and phase. The displacement autocorrelation R uu ( t )displays oscillations with period 2 π/ω R , whereas R AA and R ϕϕ do not contain these rapid oscillations. All three functionsfall o ff at timescales ∼ Q /ω R . Note, that R AA ( t ) does not go to zero for long times, because E [ A ] > A ( t ) is alwayspositive.The displacement PSD is proportional to the force noise PSD and is given by [200]: S uu ( ω ) = k − R | H HO ( ω ) | S F n F n . (52)When S F n F n is white in the bandwidth of the resonator, which is typically assumed, S uu has the characteristic bell shapeshown in Fig. 15c. Moreover, the PSD can be used to obtain the variance of the displacement: h u i = π Z ∞ S uu ( ω ) d ω = Q ω R k R S F n F n = π Q f R k R S F n F n . (53)The force noise PSD is related to the temperature, and in equilibrium the resonator temperature equals the environmentaltemperature. The equipartition theorem [201] relates the variance of the displacement to the equilibrium temperature:12 k R h u i = m h ˙ u i = k B T . (54)The thermal energy k B T is distributed equally between the potential energy and the kinetic energy. By combining Eqs. 53and 54, a relation between the force noise PSD and the properties of the resonator is found: S F n F n ( ω ) = k B T m ω R / Q . (55)This so-called fluctuation-dissipation theorem [201, 202, 203] shows that on one hand the force noise PSD can directly beobtained from the resonator properties and temperature, without knowing its microscopic origin. On the other hand, theforce noise determines the dissipation (i.e., quality factor) of the resonator.At equilibrium, the temperature of the resonator is proportional to the variance of its Brownian motion as indicated byEq. 54. However, out of equilibrium the force noise is no longer given by Eq. 55, and the resonator temperature can be The engineering convention for the single-sided power spectral density, S XX ( ω ) = S XX ( ω ) + S XX ( − ω ), is used. Here, S XX ( ω ) = F [ R XX ] isthe double-sided PSD and R XX is its autocorrelation function. The variance of X is given by h X i = R XX (0) = (2 π ) − · R ∞−∞ S XX ( ω ) d ω = (2 π ) − · R ∞ S XX ( ω ) d ω . This peak shape is often said to be Lorentzian, although formally that is not correct. The peak is proportional to [( ω R − ω ) + ω R / Q ] − , whichcan be approximated by ω − R / [( ω R − ω ) + ω R / Q ] for ω ≈ ω R . The latter is indeed a Lorentzian, but the approximation is only valid for frequenciesclose to the resonance frequency of a high-Q resonator. igure 16: The resonator temperature T R plotted against the environmental temperature T . At low temperatures T the resonator temperature saturatesat the zero-point energy: T R = ~ ω R / k B and at high temperatures T R = T (dashed line). The insets show the occupation probability P n of the energylevels at k B T / ~ ω R = . , / ln 2 ≈ .
44 and 2.0. di ff erent from T . The e ff ective resonator temperature T R is defined as: T R ≡ k R h u i k B = k R k B Z ∞ S uu ( ω ) d ω π = k R k B Z ∞ | H R | S FF ( ω ) d ω π , (56)which yields T R = T in equilibrium. When the force noise is larger than that of Eq. 55, the e ff ective resonator temperatureis higher than the environmental temperature. When h u i is smaller than its equilibrium value, T R < T . Equation 56 showsthat the resonator temperature can be obtained from the experimental displacement noise PSD: The resonator temperatureis proportional to the area under the curve (Fig. 15c). Note that the suggestive notation d ω/ π = d f is used in Eq. 56 asin an measurement typically the real frequency f is on the horizontal axis and not the angular frequency ω .When the resonator is cooled to very low temperatures where k B T ∼ ~ ω R , the classical description breaks down asthe quantized energy-level structure (Fig. 14c) becomes important. Semi-classically, the thermal and quantum e ff ects arecombined by replacing the force noise of Eq. 55 with the Callen and Welton equation [204]: S F n F n ( ω ) = m ω Q · ~ ω coth ~ ω k B T ! , (57)so that for Q ≫ h u i = u · coth ~ ω R k B T ! ⇔ T = ~ ω R k B ln − h u i + u h u i − u . (58)Then by inserting the first part of Eq. 58 in Eq. 56 the resonator temperature is obtained . The dependence of T R onthe environmental temperature is shown in Fig. 16. At high temperatures ( k B T ≫ ~ ω R ) the resonator temperature is thetemperature of the environment: T R = T . At zero temperature the resonator temperature is determined by the quantumfluctuations: T R = ~ ω R / k B .In thermal equilibrium, the energy levels of a harmonic oscillator have occupation probabilities that are given by [201]: P n = (cid:18) e − ~ ω Rk B T (cid:19) n (cid:18) − e − ~ ω Rk B T (cid:19) . (59)The average thermal occupation is n = P ∞ n = nP n = [exp( ~ ω R / k B T ) − − [201], which equals k B T R / ~ ω R − . The insetsin Fig. 16 show the occupation probabilities at three di ff erent temperatures T . At low temperature the resonator is in theground state most of the time. At k B T = ln 2 ~ ω R the probability finding the resonator in the ground state is exactly 50%and the average occupation is n =
1. At any non-zero temperature, there is always a finite probability to find the resonatorin an excited state. With the statement that “the resonator is cooled to its ground state” one actually means n . ff erent cooling techniques, but now we will focus onthe detection of the resonator position, in particular on the role of the detector, backaction and the standard quantum limit. Note, that some authors use Eq. 58 instead of Eq. 56 as the definition of T R , which implies that when h u i = u , T R =
0. This in contrast to thedefinition used here where T R = ~ ω R / k B for h u i = u . In the latter case T R is not the actual temperature, but it is a measure for the (quantum orthermal) fluctuations. .4. Backaction and quantum limits on position detection Since the discovery of the Heisenberg uncertainty principle in 1927 it is known that quantum mechanics imposeslimitations on the uncertainty with which quantities can be measured. This was first discovered for single measurementsof conjugate variables, such as the position u and momentum p of a particle, or the components σ x , σ y and σ z of the spinof a spin-1 / µ i is | c i | when | ψ i = P i c i | µ i i was the expansion of the original state in the basis of eigenstates of the operator ˆ µ , withˆ µ | µ i i = µ i | µ i i . At the same time, the wave function collapses into the state corresponding to the measured value µ i : | ψ i → | µ i i . To obtain the probability of a certain outcome of a measurement of a di ff erent quantity ν , the state | µ i i hasto be expanded in the basis | ν i i . The uncertainties in µ and ν satisfy: ∆ µ · ∆ ν ≥ (cid:12)(cid:12)(cid:12) [ ˆ µ, ˆ ν ] / i (cid:12)(cid:12)(cid:12) , where [ ˆ µ, ˆ ν ] = ˆ µ ˆ ν − ˆ ν ˆ µ is thecommutator of ˆ µ and ˆ ν . For ˆ µ = ˆ u and ˆ ν = ˆ p , this yield the Heisenberg uncertainty principle for position and momentum: ∆ u · ∆ p ≥ ~ /
2. Note that quantum mechanics does not forbid to determine the position with arbitrary accuracy in a singlemeasurement.Most measurements are, however, not single, strong measurements, but weak continuous measurements instead [205].As a measurement of the position disturbs the momentum of the resonator, a subsequent measurement of the positionafter a time ∆ t inevitably is influenced by the previous measurement. This backaction is therefore important in continuouslinear displacement detectors. In an experiment, backaction results in three di ff erent e ff ects on the resonator: a frequencyshift, a change in damping, and a change in the resonator temperature. If the resonator temperature is lower than the bathtemperature, backaction has led to self-cooling, whereas a higher temperature indicates that there is a net energy flowfrom the detector to the resonator.A way to circumvent backaction is to perform measurements on the position squared . As we show in Appendix C,such a square-law detector probes the energy states of the resonator and these are not disturbed by the measurement itselfas the energy operator equals, and hence commutes with, the Hamiltonian of the system. Therefore, this measurementscheme is called a quantum non-demolition (QND) or a backaction evading (BE or BAE) measurement [4].In the following section, continuous (linear) detectors and their backaction are discussed in detail and the quantumlimits on position detection are explored. The analysis presented is largely based on the work by Clerk and co-workers[206, 207]. We start with an analysis of generic linear detectors and what the e ff ects of backaction are. Then we discussthree di ff erent routes to arrive at the quantum limit: the Haus-Caves derivation, the power-spectral density method, andthe optimal estimator approach. These all have di ff erent ranges of applicability and rigor but lead to the same conclusion:the quantum limited resolution for continuously monitoring the resonator position is approximately the zero-point motion u . The sensitivity and resolution of a detector will be important concepts in the following discussion. These notions,however, sometimes lead to confusion and, before defining them mathematically, we will first discuss the similaritiesdi ff erences between them. Both are a measure of the imprecision with which the position is measured. A noisy detectorhas a bad sensitivity and a bad position resolution, whereas a good detector has a good sensitivity and resolution. Coun-terintuitively, this means that the latter detector has lower sensitivity values , although one often says that it has a highersensitivity . This is the first point that leads to confusion. The second is the di ff erence between sensitivity and resolution.This can be understood as follows: Consider a resonator that is standing still and that is coupled to a detector that in-evitably introduces noise in its output signal. When measuring the position for a short period, the inferred position hasa large uncertainty due to the detector noise. By measuring longer, the noise averages out and the uncertainty decreases.This uncertainty is the resolution of the detector and is measured in units of length. It thus depends on the duration of themeasurement ∆ t . For white detector noise, the resolution improves (i.e. its value decreases) as 1 / √ ∆ t . The proportionalityconstant between the resolution and the measuring time is the sensitivity, and it has the units of m / √ Hz. In the discussionof Fig. 18 it will be shown that the sensitivity is easily extracted from the noise spectrum of the detector output.
A continuous linear position detector gives an output that depends linearly on the current and past position of theresonator. Figure 17 shows the scheme of a generic linear detector. Note again, that the analysis is for a generic continuouslinear detector and quantum e ff ects only come into play in Sec. 3.4.2. We will discuss all its elements step by step. Theoutput signal of the detector is related to the displacement by: v ( t ) = A λ v ( t ) ⊗ u ( t ) + v n ( t ) . (60)Here, λ v is the responsivity of the detector, with λ v ( t ) = t < v n is the detector noise and A isa dimensionless coupling strength. The coupling between the resonator and the detector is an important element when Continuous linear detectors are usually (implicitly) assumed to be time-invariant [208]. Unless stated otherwise, this is also assumed in this Report.Examples where the linear detector is not time-invariant are frequency-converting, stroboscopic and quadrature measurements [208, 4, 3, 27]. R AuF n v n vF BA resonator coupling detector φ det,n φ det λ v λ F measurementdevice A backaction Figure 17: The detector is coupled (via A ) to the resonator. Its output v ( t ) depends linearly on u ( t ) with a response function λ v but also contains detectornoise v n . The detector exerts a backaction force F BA = A Φ det on the resonator that contains a stochastic part F BA , n and a linear response to u ( t ). Bothcontributions add up with the thermal or quantum force noise F n . The resonator displacement u is obtained via the transfer function H R = H HO / k .Figure 18: The resonator displacement u is measured by the detector. The detector also adds imprecision noise v n . The sum of this noise and the physicaldisplacement v is recorded. (a) shows schematically time traces of the resonator displacement u , imprecision noise v n and the apparent displacement v .Often, one is interested in the noise spectrum of the detector output, especially when the resonator displacement is its Brownian motion. This can bedone using a spectrum analyzer and (b) shows that from a resulting spectrum the resonance frequency f R , the quality factor Q , the detector noise floor S u n u n and the force noise S F n F n are readily extracted. The signal-to-noise ratio is the height of the resonance peak divided by the height of the noisefloor. It is therefore advantageous to have a large quality factor as this leads to a larger signal-to noise ratio. discussing di ff erent detectors in Sec. 4. The choice of the coupling A is slightly arbitrary as it could also be incorporatedin λ v . However, in most detectors it is possible to make a distinction between the coupling to the resonator and the output.Di ff erent types of detector will be discussed extensively in Sec. 4, but in the case of an optical interferometer, v representsthe number of photons arriving at the photon counter, in a single-electron transistor it is the current through the island,and in a dc superconducting-interference-device detector it represents the output voltage. The di ff erence between u , v and v n is illustrated in Fig. 18a. Note, that often the detector responds instantaneously to the displacement. In that case A λ v ( t ) = ∂ v /∂ u δ ( t ) and A λ ( ω ) = ∂ v /∂ u . The frequency response of the resonator is then flat .To calculate the sensitivity of the detector, the noise at the output of the detector, v n , is referred back to the inputusing the known response function λ v and gain A . This yields the displacement noise u n at the detector input. Theequivalent input noise PSD is S u n u n = S v n v n / | A λ v | . This power spectral density is an important parameter that characterizesthe detector. In nanoelectromechanical experiments, this noise floor is usually determined by the classical noise in theelectronics of the measurements setup. In optical experiments, however, the noise floor can be shot-noise limited; quantummechanics now sets the imprecision of the experiment. We will discuss the quantum limit in the next subsection. Notethat for a flat frequency response u n is simply given by v n / A λ v .The detector does not only add noise to the measured signal, it also exerts a force F BA ( t ) on the resonator. This is theso-called backaction force. Backaction, in its most general definition, is the influence of a measurement or detector on anobject. The detector backaction is a force on the resonator. This can be seen from the following argument: when thereis no coupling between the resonator and the detector, i.e., A =
0, the Hamiltonian describing the total system is the sumof that of the Hamiltonian of the oscillator and of the detector Hamiltonian. When these are coupled ( A , H int = − A Φ det u . The backaction force is then F BA = − ∂ H int /∂ u = A Φ det [206]. Thebackaction force is thus also proportional to the detector-resonator coupling A . The quantity Φ det is related to one of theinternal variables of the detector; for example in an optical cavity it is proportional to the number of photons, in a SET tothe electron occupation, and in a SQUID to the circulating current.The backaction force has three di ff erent contributions: • A deterministic force that is independent of the displacement. This changes the equilibrium position of the resonator. When the output of the detector has a delay time τ , A λ v ( t ) = ∂ v /∂ u δ ( t − τ ) and A λ v ( ω ) = ∂ v /∂ u exp( − i ωτ ). igure 19: (a) The position resolution of the detector in units of the amplitude of the Brownian motion h u i i / (classical limit) for di ff erent couplingstrengths A . The solid line is the total resolution; the dotted line is the contribution of the backaction force noise (Eq. B.5) and the dashed line thecontribution of the imprecision noise (Eq. B.6). The total resolution is optimized at A = A opt ≈ . Q = , λ v = λ F = S v n v n · k = S Φ det , n Φ det , n = S F n F n and S Φ det , n v n = Without loss of generality it can thus be set to zero and it is not considered here. • A force that responds linearly to the displacement: F BA , u = A λ F ( t ) ⊗ Au ( t ). This changes the e ff ective resonatorresponse from H R to H ′ R , where H ′− R = H − R + A λ F ( ω ). An example of this force is the optical spring that we willencounter in Sec. 3.5.3. This part of the backaction can lead to cooling, see Sec. 3.5.1 [32]. • A stochastic force F BA , n ≡ A Φ det , n that is caused by the fluctuations in the detector, Φ det , n . Note, that this forcenoise and the imprecision noise v n can be correlated, i.e., S Φ det , n v n ,
0. In any case, this contribution tries to heat theresonator since it adds up to the original thermal force noise..In summary, this whole process of action and backaction is the one shown schematically in Fig. 17: The resonator positionis coupled to the input of the detector, which adds imprecision noise v n and exerts force noise on the resonator. For smallcoupling A , the statistical properties of Φ det , n and v n are independent of the resonator displacement [206, 207].One way to analyze the data is to measure the detector output using a spectrum analyzer. This way, informationabout the resonance frequency, quality factor, and imprecision and force noise PSD can easily be obtained, as illustratedin Fig. 18b. However, in a linear detection scheme one is usually interested in measuring the resonator position asaccurate as possible, without perturbing the resonator considerably. Using optimal-control and estimation theory, the bestestimate ˆ u for the resonator displacement in the absence of the detector, u i , is found. The resolution of the detector is ∆ u = (cid:16) E [ u i − ˆ u ] (cid:17) / , as explained in detail in Appendix B. It quantifies the di ff erence between the displacement that theresonator would have had when it was not measured and the one reconstructed from the detector output. By rewritingthe resolution as ∆ u = (cid:16) E [( u − ˆ u ) + ( u i − u )] (cid:17) / it becomes clear that there are two contributions [209]: the first one,which we name ∆ u n , indicates how well the realized displacement is reconstructed from the detector output, whereasthe second term, ∆ u BA , quantifies the di ff erence between u i and the realized displacement, i.e., how much the motion isperturbed. Heuristically, we can understand that the first term is due to the imprecision of the detector, whereas the secondis due to the backaction.The resolution is plotted in Fig. 19a as a function of the coupling strength A . In experiments this coupling strength isan important parameter and Sec. 4 we will give typical numbers for the di ff erent detection schemes. We now discuss thegeneral features of Fig. 19 in more detail. With a low coupling ∆ u is large (i.e., the detector has a low resolution) becauseof the large imprecision noise contribution ∆ u n (dashed line) of the detector. The backaction contribution is very small.An increase of the coupling reduces ∆ u because by increasing the coupling, the mechanical signal becomes larger whereas S v n v n remains the same, so the relative contribution of the imprecision noise decreases. The increase of the coupling alsoraises the backaction contribution, which still remains small. The resolution improves with increasing A up to the pointwhere the optimal value A = A opt is reached. A further increase of A makes the backaction force noise dominant, drivingthe resonator significantly, thus yielding a higher ∆ u . The system analysis of the linear detector discussed above is valid for any – quantum limited or not – linear detector.An elegant way of deriving the quantum limit of a continuous linear position detector was given by Haus and Mullen [210]and was extended by Caves [208]. They consider the situation where the input and output signal of the detector are carried32y single bosonic modes, ˆ a u and ˆ a v respectively. When the “photon number gain” of the detector is G = h ˆ a † v ˆ a v i / h ˆ a † u ˆ a u i , onemight think that the modes are related to each other by ˆ a v = √ G ˆ a u . This is, however, not valid as this gives [ˆ a v , ˆ a † v ] = G instead of the correct value, 1 [208, 207]. The actual relation is ˆ a v = √ G ˆ a u + ˆ v n . Here, ˆ v n represents the noise addedby the amplifier. This operator has a vanishing expectation value ( h ˆ v n i =
0) and is uncorrelated with the input signal([ˆ a u , ˆ v n ] = [ˆ a † u , ˆ v n ] = a v , ˆ a † v ] = v n , ˆ v † n ] = − G for the commutator and, more importantly, ∆ a v ≥ G ∆ a u + | G − | for the noise in the number of quanta of the output mode [207]. The first term is the amplifiedinput signal (i.e., the resonator motion) and the second one is the noise added by the amplifier. In the limit of large gain ( G ≫ ∆ ( a eqvu ) ≡ ∆ a v / G − ∆ a u ≥ . This means that a quantum-limiteddetector adds at least half a vibrational quantum of noise to the signal.As pointed out in Ref. [207] most practical detectors cannot easily be coupled to a single bosonic mode that carriesthe information of the resonator to the detector, because there is also a mode that travels from the detector towards theresonator. Therefore, the linear-system analysis at the beginning of this Section is used to further explore the quantumlimits on continuous linear position detection. In the previous discussion, no constraints were enforced on the detector noises Φ det , n and v n . If both noise contributionscould be made small enough, the resolution would be arbitrarily good. This is unfortunately not possible. It can be shownthat the power spectral densities must satisfy : S v n v n ( ω ) · S Φ det , n F det , n ( ω ) − (cid:12)(cid:12)(cid:12) S Φ det , n v n ( ω ) (cid:12)(cid:12)(cid:12) ≥ | ~ λ v ( ω ) | , (61)or, equivalently, when this is referred to the input: S u n u n ( ω ) · S F BA , n F BA , n ( ω ) − (cid:12)(cid:12)(cid:12) S F BA , n u n ( ω ) (cid:12)(cid:12)(cid:12) ≥ ~ . (62)These constraints enforce the quantum limit of the linear detector and should be considered as the continuous-detectorequivalent of the Heisenberg uncertainty principle: Accurately measuring the position results severe force noise and viceversa.Clerk et al. [207] continue now by finding the gain where the total added noise at the input, i.e., S u n u n ( ω ) + | H ′ R | S F BA , n F BA , n ( ω ), is minimized. As they already point out, this is not entirely correct because at every frequency adi ff erent optimal gain is required. Usually, only the optimal gain at ω = ω is used and then the magnitude of the signaland detector noise are equal at that frequency. In that case, the imprecision noise and the backaction-induced displacementprovide exactly half of the total added noise [207, 30, 66]. However, by optimizing the total resolution , the true optimalgain is found, see Fig. 20. The resolution is optimized at A = .
87 and reaches a value of 0 .
81 times the zero-pointmotion.All three methods (the Haus-Caves derivation, the total added noise at the input, and the optimal estimator) indicatethat the detector adds about the same amount of noise as the zero-point fluctuations of the resonator itself.
To prepare a nanomechanical system in the ground state, the thermal occupation of its normal modes should beminimal. The most direct approach is to mount an ultra-high frequency ( f R > T <
50 mK) so that n ≤
1. Such a resonator will, however, have a very small zero-point motion and the readout of tinyhigh-frequency signals at millikelvin temperatures is di ffi cult. An alternative approach is to perform the experiments withlower-frequency resonators and / or at higher temperatures. The thermal occupation is then higher than one and coolingtechniques have to be used to reduce the temperature of the resonator T R well below the environmental temperature T .Figure 21 and Table 6 show how recent experiments are approaching the limit n ≤ In the opposite limit where the detector does not have any net gain, i.e., G =
1, no additional noise is required by quantum mechanics. Here, it is assumed that the measurement of v ( t ) does not result in an additional force noise on the resonator and that the detector has a large powergain. For more details, see Ref. [207]. able 6: Overview of resonator temperature and cooling of recent experiments with micro- and nanomechanical resonators. The table shows theresonance frequency f R , the temperature of the environment T , the minimum resonator temperature T min R and the corresponding number of quanta. Thenumbers of the first column correspond to the experiments listed in Table 1. f R (MHz) T (K) T min R (K) ¯n Cooling method Factor Ref. . · . . · −
295 18 5 . · Photothermal 16.3 [31]5 22 0.030 0.035 33 Backaction [32]6 0 .
81 295 10 2 . · Sideband 29.3 [33]7 0 .
013 295 0.14 2 . · Feedback 2 . · [34]8 0 .
28 295 10 7 . · Sideband 29.3 [35]9 0 .
81 295 5.0 1 . · Feedback 58.6 [36]10 58 300 11 4 . · Sideband 27.1 [37]11 127 295 295 4 . · [38]12 1 . · −
295 0.80 9 . · Sideband 366.5 [39]13 1 . · −
295 6 . · − . · Sideband 4 . · [40]14 0 .
55 300 175 6 . · Photothermal 1.7 [41]15 43 0.25 1.0 483 [42]16 2 . · − . · − . · Feedback 85.7 [43]17 7 . · −
295 45 1 . · Sideband 6.5 [44]18 0 .
71 295 295 8 . · [45]19 0 .
13 294 6 . · − . · Sideband 4 . · [46]20 0 .
56 35 0.29 1 . · Sideband 120.0 [47]21 8 . · −
300 0.070 1 . · Feedback 4 . · [48]22 74 295 21 5 . · Sideband 14.0 [49]23 0 .
24 0.040 0.017 1 . · Sideband 2.3 [50]24 428 22 22 1 . · [51]25 5 . · − . · [52]26 2 . . · − . · − . · Feedback 2 . · [53]28 9 . · − . · − . · Feedback 2 . · [53]29 1 . ·
295 295 5 . · [54]30 8 . . · [55]31 1 . . . . · [57]34 6 . . · Feedback 5.1 [58]35 13 . . · [59]36 0 .
95 5.3 1 . · −
29 Sideband 4 . · [60]37 65 1.65 0.20 64 Sideband 8.2 [61]38 8 . . · Sideband 49.0 [62]39 119 1.4 0.21 37 Sideband 6.6 [63]40 1 . · −
300 1 . · −
237 Feedback 2 . · [64]41 8 . . · Sideband 23.9 [65]42 1 .
04 0.015 0.130 2 . · Sideband [66]43 8 . . · [67]44 6 . . · − . . . · [69]46 6 . · .
07 [22]47 2 . . . · − .
34 Sideband 113.8 [23]34 igure 20: The quantum limit for a continuous linear position detector using the optimal estimation method. The resolution of the detector in units of thezero-point motion is plotted for di ff erent coupling strengths A . The solid line is the total resolution, ∆ u ; the dotted and dashed line are the contributionof backaction, ∆ u BA , and imprecision noise, ∆ u n , respectively. The inset shows that the total resolution depends on the cross-correlation coe ffi cient S F BA , n u BA , n / ( S F BA , n F BA , n · S u n u n ) / . The main panel is calculated with an optimal cross-correlation of 0.36. For all values of A the total added noise isslightly less than u . To measure the displacement, the resonator is coupled to a detector. As shown in Sec. 3.4, this influences the resonator,and, in particular, this backaction adds force noise and it can damp the motion, which can lead to cooling. The dampingrate of the resonator is then increased from its intrinsic value γ = ω / Q to γ R = γ + γ BA , where γ BA = − ω A Im[ λ F ] /ω is the damping induced by the detector [206]. The resonator temperature is [32, 56, 212]: T R = γ T + γ BA T BA γ + γ BA . (63)The resonator temperature is thus the weighted average of the environmental temperature T and the so-called backactiontemperature of the detector T BA . For strong resonator-detector coupling ( γ BA ≫ γ ) the e ff ective temperature is T R = T BA .When this is below the environmental temperature, the resonator is cooled by the backaction. Eq. 56 shows that thebackaction temperature is determined by the force noise exerted on the resonator: T BA = S F BA , n F BA , n / k B m γ BA . Both S F BA , n F BA , n and γ BA are proportional to A so that T BA is independent of the resonator-detector coupling A . In other words,the backaction temperature is an intrinsic property of the detector.Although it might not be immediately clear, this cooling mechanism corresponds to the usual notion of cooling:Cooling is done by coupling something to something else that is colder. In the case of backaction cooling the cold objectis the detector. Because the resonator is not actively cooled, but only brought into contact with the detector, backactioncooling is therefore also called “self cooling” or “passive feedback cooling”. It was shown in Sec. 3.3 that the resonator temperature is proportional to its random motion. By reducing that motionthe resonator gets cooled. One way to do this is using feedback. When the position of the resonator is measured and fedback to it via an external feedback loop, the motion can be amplified or suppressed. Feedback control was already usedto regulate the motion of soft cantilevers [213, 214, 215] for magnetic resonance force microscopy [216], when it wasrealized that this technique can also be used to cool a mechanical resonator towards its ground state [217]. Actually, thelowest resonator temperature to date ( T R = . µ K, see Table 6) has been reached using this cooling method [64].Feedback systems are usually analyzed within the linear system representation. Figure 22 shows a schematic ofthe process. The resonator, with frequency ω / π and Q-factor Q , is driven by the thermal force noise F n ( t ) and itsdisplacement u ( t ) is detected. The detector output contains not only the displacement but also imprecision noise u n ( t ). The apparent position v is the signal at the output of the detector and this is thus the sum of the physical displacement andthe detector noise: v = u + u n . The information contained in v is used to apply a force F FB to the resonator that damps itsthermal motion . The relation between the feedback force and the apparent position is described by the linear system or This assumes that the e ff ective spring constant k − k A Re[ λ F ] does not change considerably. As shown in Sec. 3.5.2, it is in general much moredi ffi cult to alter the spring constant than to alter the damping. In this section we assume that the detector has unit gain, i.e., A λ v ( ω ) =
1. Then the imprecision noise at the output, v n , and the noise referred to theinput, u n , are the same. In principle, also the backaction force noise of the detector has to be included [213, 53, 218]. Although this term was never important in activefeedback cooling experiments so far, the e ff ect can be included by using the damping and resonator temperature of the coupled resonator and detectorinstead of the intrinsic ones of the resonator alone. igure 21: Overview of mechanical resonators with low temperature or occupation, compiled from Table 6. The top panel shows the experiments withthe lowest occupation numbers. These are reached using conventional cooling (gray) and active feedback and sideband cooling (black). The dotted lineis located at n =
1. As discussed on page 29, once the thermal occupation is below this value, the resonator is cooled to the ground state. The bottompanel shows the starting temperature T (gray) and the final temperature T min R (black) achieved by groups that have actively cooled their resonator below100 mK. Note the diagonal line in the LIGO experiment which is due to a strong optical spring e ff ect (see Sec. 3.5.3), so both the resonator temperatureand frequency change when the laser power is increased [40]. filter with transfer function H FB ( ω ). The output of the filter is multiplied by a selectable gain g . One can think of it asa knob to crank up the gain of an amplifier. This forms a closed-loop system [199, 213] with the following equations ofmotion: m ¨ u ( t ) + m ω ˙ u ( t ) / Q + m ω u ( t ) = F n ( t ) + F FB ( t ) , (64) F FB ( t ) = m ω g · h FB ( t ) ⊗ [ u ( t ) + u n ( t )] . (65)The presence of the feedback results in a di ff erent displacement for a given thermal noise realization F n ( t ). The feedbackthus changes the resonator response from H R to the closed-loop transfer function H ′ R , given by: H ′− R = H − R − gk H FB , or H ′ R = k − − (cid:16) ωω (cid:17) + iQ ωω − gH FB . (66)Comparing this with the response of the resonator itself, cf. Eq. 49, shows that the real part of H FB modifies theresonance frequency from ω to ω R = ω p − g Re[ H FB ( ω )], whereas the imaginary part alters the damping rate from γ to γ R = γ − g ω Im[ H FB /ω ]. Using this closed-loop transfer function, the PSDs of the physical (i.e., the real resonatordisplacement) and observed displacement (i.e., the detector output) are obtained: S uu ( ω ) = S F n F n / ( m ω ) + g | H FB | S u n u n (cid:12)(cid:12)(cid:12)(cid:12) − (cid:16) ωω (cid:17) + iQ ωω − gH FB ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) , (67) S vv ( ω ) = S F n F n / ( m ω ) + (cid:12)(cid:12)(cid:12)(cid:12) − (cid:16) ωω (cid:17) + iQ ωω (cid:12)(cid:12)(cid:12)(cid:12) S u n u n (cid:12)(cid:12)(cid:12)(cid:12) − (cid:16) ωω (cid:17) + iQ ωω − gH FB ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) . (68)The resonator displacement PSD S uu shows that the resonator indeed responds to the force noise with the modified transferfunction H ′ R instead of H R . The force noise that drives the resonator (i.e., the numerator of the right-hand side of Eq. 67) The gain g and filter H FB are defined such that | H FB ( ω ) | =
1. Both g and H FB are dimensionless. R H FB gk uF n u n vF FB ResonatorFeedbackDetector
Figure 22: Linear system representation of the active feedback cooling scheme. The resonator displacement u is converted by detector to its outputsignal v . This adds imprecision noise u n and the sum of this noise and the physical displacement is the signal that is measured, for example using aspectrum analyzer. This signal v is also fed back to the resonator to attenuate the Brownian motion. In the feedback loop a filter with response H FB anda variable gain g · k are included. The resulting feedback force F FB adds up with the (thermal) force noise F n . The resonator’s response to the appliedforces is determined by its transfer function H R = H HO / k . still contains the original contribution S F n F n , but now it also has a contribution due to the imprecision noise that is fed backto the resonator. Since the latter is always positive, the feedback loop adds additional force noise to the resonator. Theapparent position PSD is also modified: It is not simply the sum of S uu and S u n u n because the feedback creates correlationsbetween the imprecision noise u n ( t ) and the actual resonator position u ( t ).So far, the analysis was general for any linear feedback system and di ff erent implementations of the feedback filter H FB are possible, each with their advantages and drawbacks. Using optimal control theory, the best feedback filter can inprinciple be found [213]. In practise, often simpler, but therefore suboptimal, filters are used. The PSDs of the true andapparent resonator displacement are plotted in Fig. 23 for the two simplest feedback schemes: • Velocity-proportional feedback where the measured displacement is used to apply a velocity-dependent force on theresonator with h FB ( t ) = − ω − · ∂/∂ t , H FB = − i ω/ω . In this case the damping rate increases from γ to γ · (1 + gQ )as indicated by Eq. 66. Figure 23a shows that at low gains, S uu is lowered and thus that the resonator is cooled.However, when the gain is increased further, the tails of S uu start to rise as the detector noise (the second term inthe numerator of Eq. 67) is fed back into the resonator. Above a certain value g = g min , too much detector noise isfed back to the resonator and the resonator temperature increases again. Figure 24 shows the resonator temperatureas a function of the feedback gain. The minimum resonator temperature in the limit g ≫ Q − is [43]: T R , min = s m ω Tk B Q S u n u n = T √ SNR , for g = g min = √ SNR / Q . (69)The minimum resonator temperature is thus set by the signal-to-noise ratio (SNR ≡ S uu ( ω ) g = / S u n u n ) of the originalthermal noise peak and the detector noise floor, as was illustrated in Fig. 18b. Finally, note that, unlike for backactioncooling, the resonator temperature does not saturate at a certain value when g → ∞ . Eventually more and morenoise is added and the resonator temperature keeps on increasing with increasing gain. • Displacement-proportional feedback where the displacement is directly fed back to the resonator, which is charac-terized by h FB ( t ) = − δ ( t ) , H FB = −
1. This changes the spring constant from m ω to m ω (1 + g ) and the resonancefrequency to ω · (1 + g ) / . The sti ff ening of the resonator reduces its thermal motion and hence its tempera-ture, but to achieve the same cooling factor as with the velocity-proportional feedback the gain should be Q timeslarger. This, however, also feeds back much more detector noise to the resonator in the usual situation where Q ≫
1. Cooling can therefore only be achieved when the SNR is large. Figure 23b shows that only heating in-stead of cooling is achieved for the choice of the parameters used to perform the calculation. For high-Q resonatorsvelocity-proportional feedback is superior to displacement-proportional feedback.It is important to note that the feedback creates correlations between the resonator displacement and the detectorimprecision noise, which lead to a substantial change in the shape of the noise spectra. Figure 23 shows calculated noisepower spectral densities for di ff erent gains. Without feedback (i.e., g =
0) the spectrum of the apparent motion is simplythe sum of that of the harmonic oscillator, S uu , and a constant background level due to the noise, S u n u n . The Figure showsthat, even though the peaks in S uu only become broader or shift in frequency, the peaks in S vv becomes distorted whenthe feedback is applied. The resonator temperature is no longer simply given by the area under the peak in S vv due tothe abovementioned correlations between u n and u . For large gains it is even possible that the peak in the spectrum S vv changes into a dip (Fig. 23a). A way to circumvent this problem of determining the resonator temperature, is to use a37 igure 23: Feedback cooling of a resonator with Q =
100 using velocity-proportional (a) and displacement-proportional (b) feedback. The top panelsshow the PSD of the observed displacement (i.e., that of the detector output) S vv and the bottom panels show the real (physical) displacement PSD, S uu .The gain is stepped from g = g = . S u n u n = and all PSDs are scaled by S F n F n / ( m ω ) . second detector to measure the resonator motion [58]. The noise of the second detector is not correlated with that of theresonator and the measured PSD is again the sum of a constant background and the resonator motion.Active feedback cooling experiments have mainly been done on resonators in the kHz range, where one can simplymeasure the position, di ff erentiate and feed the resulting signal back to the resonator. Cohadon et al. demonstrated thefirst active cooling of a mirror using feedback control [219]. They use a high-finesse cavity with a coated plano-convexresonator as the end mirror. A feedback force is applied using a 500 mW laser beam. An acousto-optical modulator isused to control the exerted radiation pressure on the resonator. Next, Kleckner and Bouwmeester actively cooled a 12.5kHz AFM cantilever from room temperature to 0 .
135 K [34]. A tiny plane mirror attached to the cantilever served as themovable end mirror of the cavity and again a second high-power laser was used to apply the velocity proportional force.Arcizet et al. cooled a millimetre-scale resonator in an optical cavity to 5 K by applying an electrostatic feedback forceon the resonator [36]. By using a piezo element to apply feedback to an ultrasoft silicon cantilever cantilever coolingfrom 2.2 K to 5 mK was demonstrated by Poggio and coworkers [43]. Feedback cooling has also been demonstrated withthe gravitational-wave detectors AURIGA [53] and LIGO [64]. In the former experiment the 2 ton heavy detector wascooled from 4 K to 0.17 mK. The motion of the bar is measured capacitively using a resonant electric circuit coupled toa SQUID amplifier. The output of the amplifier is put through a low-pass filter to create the π/ Figure 24: Resonator temperature for velocity-proportional feedback vs feedback gain g for a resonator with Q = k S u n u n / S F n F n = which gives a signal-to-noise ratio of 10 and the solid line is for k S u n u n / S F n F n =
1, with SNR = . For low gains thetwo curves overlap. When g & .
1, the detector noise already starts to heat the resonator for the low-SNR curve, whereas the high-SNR curve stilldecreases. As predicted by Eq. 69 the minimum temperature is lower in the latter case, and this occurs at a higher gain. igure 25: (a) schematic overview of an optical cavity. The cavity is driven via an input laser with frequency ω d . The left mirror is fixed, but theright mirror is a mechanical resonator that can move. The resonator displacement u determines the length of the cavity and thus the cavity resonancefrequency ω c . A part of the circulating power is transmitted by the left mirror to a detector. The linewidth of the cavity is κ . From T. J. Kippenberg, K. J.Vahala, Science 321 (2008) 1172–1176. Reprinted with permission from AAAS. (b) When the cavity is driven on its resonance, the intensity inside thecavity is largest, a detuning reduces the intensity. A displacement of the resonator shifts the cavity resonance (purple) and changes the intensity of thelight inside the cavity (orange and purple dots). (c) The displacement dependence of the radiation pressure F rad . When the resonator oscillates, the forcereacts with a delay due to a finite value of κ as indicated by the ellipsoidal trajectories. velocity proportional feedback and then injected into the electronic circuit. In the latter experiment the center-of-massmotion of the four mirrors of a Fabry-Perot cavity with 4 km long arms is reduced from room temperature to only 1 . µ k.This is done by adjusting the servo control of the mirrors, which creates a velocity-proportional feedback force in the rightfrequency range without a ff ecting the very high signal-to-noise ratio.When the resonator frequencies are high, say above 1 MHz, delays in the feedback circuit start to play a role. Theforce is then applied when the resonator has already advanced and a purely velocity-proportional feedback will have adisplacement-proportional component, degrading the cooling performance. The e ff ect of a delay is even more dramaticwhen it equals half the resonator period, so that the Brownian motion is actually amplified instead of attenuated. Further-more, the bandwidth (or sampling speed in the case of a digital filter) of H FB should be at least a few times ω R , which isoften not an issue for ω R . Another commonly used cooling technique is sideband cooling [221, 222, 223]. In this technique the resonator isembedded in an optical [33, 61, 37, 49, 35, 39, 46, 47, 60, 63, 62] or microwave cavity [44, 50, 56, 68, 23]. Thesedetection schemes will be discussed in detail in Sec. 4, but here a brief introduction is given. Figure 25a shows a schematicdrawing of an optical cavity where the right mirror is the mechanical resonator. Both mirrors have a low transmissionso that a photon is reflected many times before it can go out through the left mirror, and then toward the detector. Sucha cavity has many di ff erent optical eigenmodes, but here we focus on a single one and denote its resonance frequencyby ω c . In analogy with the Q-factor and linewidth γ R of a mechanical resonator, the cavity has an optical Q-factor Q opt and linewidth κ . A laser sends light with frequency ω d into the cavity. This frequency can be di ff erent from the cavityresonance frequency ω c ; the light is then detuned. Because the resonance frequency of the cavity is determined by thecavity length, a displacement of the resonator changes ω c . As illustrated in Fig. 25b this leads to a change in the intensity(and phase) of the light in the cavity, which results in a change in the detector output. Optical cavities used this way arevery sensitive position detectors for two reasons: first, it enhances the intensity of the light by a factor Q opt , and secondlyit makes the intensity depend strongly on the displacement [222].Each photon in the cavity carries a momentum p ph = ~ ω d / c whose direction is reversed when it reflects o ff the mirror.Here c is the speed of light. The resonator thus experiences a kick of 2 p ph every time a photon reflects, the so-called39adiation pressure. A single round trip of a photon in a cavity of length L takes a time 2 L / c , whereas the average time thata photon spends in the cavity is κ − . The total transfer of momentum per photon is thus p tot = ~ ω c /κ L . The total forceexerted on the resonator is also proportional to the number of photons present in the cavity, n c , and equals F rad = ~ ω c n c / L .Note, that n c is proportional to κ − and to the input power.The radiation pressure depends on the displacement of the resonator. This is illustrated in Fig. 25c: A small change in u changes the cavity frequency ω c , which in turn leads to a proportional change in radiation pressure. This thus changesthe e ff ective spring constant of the resonator to k R = k − ∂ F rad /∂ u ; the so-called optical spring [224, 225, 226, 227, 228,39, 229, 230]. Similar to displacement-proportional feedback this can cool the resonator [39, 64, 231] and can even lead tobistability of the resonator position [225]. A much stronger cooling e ff ect is, however, the fact that the number of photonsdoes not respond immediately to a change in displacement, but that they can only slowly leak out of the cavity at a rate ∼ κ . This was first realized [224] and demonstrated [232] by Braginskiˇı and coworkers. When κ ≫ ω , a change in thedisplacement changes the number of photons instantaneously and F rad follows the orange curve in Fig. 25c. However, afinite value of κ causes a delay in the response of the radiation pressure as indicated by the ellipsoids in Fig. 25c. In thecase of red-detuned driving ( ω d < ω c ; the ellipse is traversed counterclockwise) work is done by the resonator so that itloses energy, whereas for blue-detuned driving ( ω d > ω c ; clockwise trajectory) the resonator gains energy. The increaseddamping for red detuning cools the resonator, as the backaction temperature associated with the detector is very low[233, 234, 223]. This process is called “dynamical backaction” and the cooling mechanism is called “sideband cooling”because the cavity is driven o ff -resonance, i.e., on a sideband [236]. Note, that sideband cooling can also be describedin the language of backaction cooling (Sec. 3.5.1) as the optical force responds with a delay to the displacement. This isthus a detector response λ v (Fig. 17) with a real (the optical spring) and an imaginary part (the delay).The ultimate limit on the resonator temperature that can be reached with sideband cooling has been studied using theradiation-pressure Hamiltonian in the rotating-wave approximation [233, 234]:ˆ H = ~ ∆ ˆ c † ˆ c + ~ ω ˆ a † ˆ a + ~ G OM ˆ c † ˆ c (ˆ a † + ˆ a ) . (70)Here, G OM = u ∂ω c /∂ u is the optomechanical coupling rate and ˆ c † (ˆ c ) is the creation (annihilation) operator for a cavityphoton (from (to) a photon with frequency ω d ). ∆ = ω d − ω c is the detuning of the laser light with respect to the cavityresonance frequency. The quantum mechanical picture of sideband cooling is that a phonon together with a red-detuneddriving photon can excite a photon in the cavity. This is a likely process because it up-converts the red-detuned drivingphoton to a frequency closer to the cavity resonance. This removal of phonons cools the resonator. The opposite processis also possible: a cavity photon can emit a phonon and a lower-frequency photon, thereby heating the resonator. Therate of these two processes depends on the density-of-states of the cavity at ω d + ω and ω d − ω respectively. If thedetuning is at exactly at the mechanical frequency ∆ = − ω and the cavity linewidth is small, the lowest temperatures areobtained. The process is analogues to the doppler cooling of cold atoms. In the good-cavity limit ( ω m & κ ) the lowestresonator occupation is n min = ( κ/ ω ) ≪ ffi ciently high powers. The cooling powershould be large enough to remove the heat coming from the environment to reach the ground state and the cooling poweris proportional to the input power [223]. Table 6 and Fig. 21 show the final thermal occupation numbers that have beenreached up to now.The e ff ects of sideband cooling (i.e., frequency shift, change in damping, and cooling) were first considered in thecontext of gravitational wave detectors, where the e ff ect was more-or-less viewed as a technical point with limited ap-plications, see Refs. [224, 232, 238] and references therein. The first experiments with the aim of cooling mechanicalresonators towards the ground state were reported in 2006 by Arcizet [33] and Gigan [35] cooled resonators with frequen-cies of a few hundred kHz by modest factors of 30 and 10 respectively. These measurements were done using free-spaceoptical cavities which were in the unresolved sideband regime (i.e., where ω R ≪ κ ). Schliesser and co-workers used amircotoroidal resonator vibrating at 58 MHz as cavity (see Sec. 4.1.3) and demonstrated cooling to 11 K [37], whichcorresponds to an occupation number of n = T R = . n ∼ is therefore higher thanthe abovementioned experiments by Schliesser. A major step forward was made in 2008 by the Kippenberg group, whowere the first to reach the resolved sideband regime [49]. Their 73.5 MHz microtoroidal resonator has a cavity line widthof only 3.2 MHz, placing this device deep in the good-cavity limit. The resonator was cooled to ∼
19 K.It was realized that a way to further cool resonators is to start at a lower temperature, so less phonons have to berefrigerated away to reach the groundstate. This approach was pursued by the group of Lehnert using superconductingstripline resonators (Sec. 4.1.5). Their first experiment was in the unresolved-sideband regime [50], but this was soon A recent proposal uses a displacement-dependent cavity damping instead of the usual displacement-dependent cavity frequency [235]. ff ort has been done to place optical cavities at cryogenic temperatures. Gr¨oblacher et al. precooled their microresonatorto 5 K and subsequently cooled the 1 MHz resonator to 1.3 mK, where n =
32 [60]. Around the same time, Schliesser etal. cooled a resonator placed in a cryostat from 1.7 K to 0.2 K where n =
63. Very recently, the first actively cooled devicewith an occupation number smaller than one was demonstrated by Teufel et al. [23]. They employed sideband cooling ina resonant superconducting circuit.
In the previous Subsections backaction cooling, active feedback cooling, and sideband cooling have been discussed.Although it might appear that these methods are unrelated, the converse is true: From the linear system representation ofbackaction (Fig. 17) and that of active feedback cooling (Fig. 22), it is clear that they are closely related. In the formercase the delayed force, needed to achieve the largest cooling factors, is caused directly by the detector, whereas in thelatter it is actively exerted by the experimenter. The coupling A and the gain g play the same role in the respective pictures,and so do the detector force response λ F and the feedback filter H FB . Also, the lowest temperature that can be reached isdetermined by the noise, Φ det , n and u n respectively. Note, however, that for backaction cooling the resonator approaches T BA when the coupling A → ∞ , whereas in the case of active feedback cooling the resonator temperature diverges for g → ∞ , and the minimum temperature occurs at a finite value of the feedback gain. It was also shown that one candescribe sideband cooling in a similar fashion: namely as the (delayed) response of the optical force to the displacement,and the same is true for bolometric forces where the delayed force is due to the finite heat capacity [31, 41, 211, 239].Finally, we note that there are cooling mechanisms in non-cavity systems, such as in superconducting single-electrontransistors [32, 240] and double dots [241] that are formally identical to sideband cooling.Another important point is illustrated in the experiments done by Teufel and coworkers (lines 31, 32 in Tables 1 and 6).When cooling the resonator, in this case using sideband cooling [56], the quality factor decreases. The resolution of thedetector then degrades when the sensitivity stays the same because the resonator bandwidth γ R increases. In other words,one has less time to average-out the imprecision noise. The opposite occurs when the damping of the resonator is reduced:the resolution is improved, but the resonator temperature increases significantly [66]. In particular, this happens in a cavityoptomechanical system for a blue-detuned laser drive, but this e ff ect has also been observed in single-electron transistors[32], superconducting interference devices [242] and in many other detectors. In this regime energy is transferred fromthe cavity to the mechanical system. The oscillation amplitude grows and the resonance becomes sharper as the resonatorenergy increases. When the coupling is strong enough, the damping rate can vanish ( γ + γ BA =
0) or even becomenegative. Beyond this instability the resonator exhibits self oscillations that are only bound by nonlinearities in either theresonator or the detector [224], and this can lead to complex nonlinear dynamics. Theoretical work on the classical e ff ectsin this regime has been documented in Refs. [243, 244, 245, 246]. Experimentally this instability has been observed ina variety of micro and nanomechanical systems, including microtoroidal resonators [247, 248, 37], Fabry-Perot cavities[33, 228, 40, 239], superconducting stripline resonators [56] and microspheres [249]. Devices that are self oscillatingcan be useful from an application point of view as ultra-high frequency, low phase-noise oscillators [51], or as memoryelements [250]. 41 igure 26: Resolution (top panel) and sensitivity (bottom panel) of the experiments that are listed in Table 7. Experiments with optical (electrical)detection are shown in gray (black) respectively. The solid line indicates the standard quantum limit. As discussed in Sec. 3, detectors can have animprecision resolution below the quantum limit at the price of a large backaction force noise. There is a trend that the resolution and sensitivity degrades(their value increases) with increasing resonator frequency.
4. Detection methods
In this section we discuss the main detection methods for mechanical motion in nanoscale systems or in mechanicaldevices which aim at reaching the quantum regime. Most methods involve a linear displacement detector, i.e., the outputof the detector depends linearly on the displacement of the resonator. As shown in the previous sections, such schemesinevitably introduce backaction on the resonator position, which consequently leads to the fundamental limit on thesensitivity set by quantum mechanics. We will only briefly comment on quadratic (square-law) detectors, as they havenot been studied to the same extent as linear detectors. In recent years, tremendous improvement in the sensitivity hasbeen obtained using a variety of di ff erent detection methods. Using optical cavities, displacements sensitivities as small as10 − m / √ Hz (see Table 7) have been reached; using mesoscopic electromechanical devices the sensitivity can be as goodas 10 − m / √ Hz. Figure 26a shows the resolution ∆ u n due to the imprecision noise S u n u n of the experiments listed in theTable. (For the exact definition of ∆ u n in this context we refer to Eq. B.7 in Appendix B). Note, that this quantity does notinclude the e ff ects of backaction force noise and can therefore be substantially smaller than the standard quantum limit asthe Figure shows. This is most clearly demonstrated by the relatively low-frequency resonators that are read-out optically[36, 33, 43, 53, 60]. Although the backaction force noise is present to ensure the SQL and thereby heating the resonator;at room temperature, this contribution is masked by the much larger thermal motion. Also in solid-state devices at lowtemperature, ∆ u n < u has been achieved [32, 66, 68] and there an increase in resonator temperature due to backaction isseen [32].The most straightforward method to analyze the data measured in an experiment is to record the output of the lineardetector as a function of time; for a photodetector in a cavity experiment (Sec. 4.1) this would be an output current and fora dc SQUID position detector (Sec. 4.5) this is the voltage over the SQUID. This signal can be fed to a spectrum analyzerand from the measured spectrum, the resonance frequency and Q-factor can directly be obtained in the linear response.From the thermal noise spectra (cf. Fig. 18b) other parameters like S u n u n , S F tot F tot can be obtained as illustrated in theFigure.An important issue in measuring the resonator dynamics is the available bandwidth of the setup. In particular thisholds for the solid-state devices that often have a high impedance ( ≫ Ω ). The combination of high impedances andunavoidable stray capacitances can lead to RC times that are smaller then the resonator period 1 / f R . There are several waysto circumvent this problem: frequency mixing, impedance matching using tank circuits, or using low-impedance devicessuch as superconducting quantum interference devices (SQUIDs) and microwave striplines. Yet another way relies onself-detection which yield dc information about the vibrational motion, e.g. rectification and spectroscopy measurements.These (non-linear) methods are mainly used in bottom-up devices.In the following sections, we will discuss di ff erent detection schemes focusing on the resonator-detector couplingand on the backaction mechanism. As we concentrate on detection schemes aimed at the quantum regime, we will, forexample, not consider piezo-resistive detection schemes, which are very common in MEMS and for applications, but42 able 7: Overview of the sensitivity and resolution of recent experiments. The table shows the type of detector that was used, and the displacement-imprecision noise S u n u n from which the resolution ∆ u n is calculated. The ratio of the resolution to the zero-point motion u is also stated. The numbersof the first column correspond to the experiments listed in Table 1 and Table 6. Detector S u n u n fm / √ Hz ∆ u n (fm) ∆ u n / u Ref. . . . . · . ·
415 [31]5 SET 0 .
35 5 . .
25 [32]6 Opt. cav. 4 . · − . · − .
62 [33]7 Opt. cav. 100 38 7 . . · − . · − .
62 [36]10 Opt. cav. 3 . · − .
53 5 . . ·
506 [38]12 Opt. cav. 0 . .
06 8 . .
03 9 . · − .
04 [40]14 Opt. cav. 1 . · . ·
778 [41]15 APC 2 . . ·
271 2 . . · − . · − . .
54 49 39 [46]20 Opt. cav. [47]21 Opt. cav. 45 2 . . · − .
05 0 .
42 [49]23 Stripline 200 2 . ·
19 [50]24 Magn. mot. 12 . . ·
339 [51]25 QPC 1 . ·
588 20 [52]26 SQUID 10 132 51 [19]27 Capacitive 3 . · − . · − .
34 [53]28 Capacitive 3 . · − . · − .
42 [53]29 Opt. cav. 1 . · . · . · [54]30 Opt. trans. 72 6 . ·
232 [55]31 Stripline 600 1 . ·
57 [56]32 Stripline 45 696 23 [56]33 Capacitive 2 . · . · . · [57]34 Opt. cav. 1 . · − . . ·
76 [59]36 Opt. cav. 0 .
03 0 .
18 0 . . · − .
34 7 . .
04 11 . . .
26 61 1 . · [63]40 Opt. cav. 1 . · − [64]41 Opt. cav. 0 .
07 4 . . . .
57 [66]43 Opt. cav. 0 .
64 23 1 . . . . . .
76 [69]46 Qubit [22]47 SQUID 2 . . .
23 92 23 [23]43arely used for nanoscale experiments [38, 251]. The main disadvantage of this method is that it requires a large currentthat is dissipated in the resonator.
Table 8: Overview of the coupling factors reached in optical and microwave cavity experiments. As explained in the text g OM = ∂ω c /∂ u quantifieshow much the cavity frequency changes with the displacement and G OM = g OM × u is the vacuum coupling rate. The cavity frequency ω c / π liesin the GHz range for the microwave cavities and is ∼ THz for the optical cavities. κ is the cavity linewidth. In the experiment on the last line, asuperconducting qubit is used instead of a cavity. In this case, the value of vacuum Rabi rate is listed as an estimate of G OM [22]. Group g OM π MHznm ! G OM π (Hz) ω c π (GHz) κ π (MHz) L OM (m) Ref.
JILA 1 . · − . · − [50]JILA 4 . · − . · − [56]JILA 3 . · − · − [66]Schwab 7 . · − · − [252]Schwab 8 . · − . · − [68]Aalto 1.0 16.5 7.64 7 . · − [253]NIST 56 226 7.47 0.17 133 · − [254]NIST 49 198 7.54 0.20 154 · − [23]Vienna 1 . ·
206 2 . ·
120 2 . · − [35]LKB Paris 1 . ·
75 2 . · . · − [36]LKB Paris 1 . · . · . · − [45]Harris 2.1 2.68 2 . · . · . · − [60]MPI-QO 1 . ·
599 3 . ·
19 27 . · − [61]Painter 1 . · . · . ·
646 1 . · − [62]Painter 3 . · . · . ·
113 5 . · − [65]Vienna 11 2.70 2 . · . · − [237]MPI-QO / LMU 10 145 1 . · . · − [67]Cornell 9 . · . · / LMU 40 660 3 . · . · − [69]MPI-QO 1 . · . · . ·
15 32 . · − [255]MPI-QO 2 . · . · . · [256]Cornell 6 . · . · . · . · . · − [230]UCSB 6 . · F . The light that eventually comes out of the cavity contains the displacementsignal, enabling position detection. Also, when the light interacts with the resonator, the photons exert a backaction forceon the resonator; the radiation pressure that was introduced in Sec. 3.5.3. Di ff erent types of optical cavities exist that wewill treat separately in the next Subsections. Cavities cannot only be realized in the optical domain where the frequenciesare hundreds of THz, but they can also be made with superconducting resonant circuits that operate in the microwavefrequency range (GHz). At the end of this section we will introduce these microwave cavities and compare them to theiroptical equivalents.An important figure of merit for cavity-optomechanical systems is the coupling strength, which is usually defined as g OM = ∂ω c /∂ u . It indicates how much the cavity frequency shifts per unit motion. Table 8 shows the coupling strength,the cavity frequency, its damping rate, and some derived quantities for recent experiments with optical and microwavecavities: The optomechanical coupling length is defined as L OM = ω c / g OM , which equals the physical cavity length inthe case of a Fabry-P´erot cavity. L OM is a convenient quantity as the displacement normalized by it, directly gives therelative change in cavity frequency: u / L OM = ( ω c ( u ) − ω c (0)) /ω c (0). To obtain a high optomechanical coupling it is thusadvantageous to have a small cavity. In the experiments reported in Refs. [35, 36, 45] the Fabry-P´erot cavity length isonly a few millimeters. The coupling rate that appears in the cavity-resonator Hamiltonian (Eq. 70) has been introducedas G OM = g OM u , which indicates how much the cavity moves due to the zero-point motion of the resonator and this44 b)(a) 60 µm mechanical RBMoptical WGM (d)(c) Figure 27: Di ff erent implementations of optical cavities for position detection. (a) Schematic overview of an optical cavity with a movable membranein it. The cavity consists of two fixed mirrors at a distance L . The membrane at position x can be located at di ff erent intensities of the standing wavein the cavity (green). The membrane (inset) is made of 50 nm thick silicon nitride and is held by a silicon substrate. The membrane reflectivity is r c = .
42. Reprinted by permission from Macmillan Publishers Ltd: Nature 452, 72–75, copyright 2008. (b) Top: scanning electron micrograph ofa microtoroid. Bottom: schematic illustration of the optical whispering gallery mode (WGM) and the mechanical radial breathing mode (RBM). TheWGM encircles the toroid and is mainly located at the rim. In RBM vibrations the rim of the toroid moves outward, thereby elongating the WGM cavitylength. Adapted from Refs. [261] and [122]. Reprinted by permission from Macmillan Publishers Ltd: Nature 421, 925–928, copyright 2003. (c) Artistimpression of a double-disk cavity. The cavity consists of two silica disks separated by a layer of amorphous silicon. Similar to the microtoroid in panel(c), WMGs exist in each of the two disks. They from bonded and antibonded modes (modes with even and odd parity respectively). The mechanicalmode of interest is that where the gap between the two disks changes. Reprinted figure with permission from Q. Lin et al , Phys. Rev. Lett. 103(2009) 103601. Copyright 2009 by the American Physical Society. (d) The setup used to study the mechanics and the optical properties of the zippercavity. Near-infrared (NIR) laser light goes through an erbium-doped fiber amplifier (EDFA) and a variable optical attenuator (VOA). The polarizationis adjusted in the fiber-polarization controller (FPC) and the light is split in two paths: one through a fiber Mach-Zehnder interferometer (MZI) andthe other through a tapered fiber that couples the light to the zipper cavity. Finally, the photons in both paths are detected with photodetectors (PD).Reprinted by permission from Macmillan Publishers Ltd: Nature 459, 550–555, copyright 2009. coupling rate is sometimes called the vacuum coupling rate [259]. The role of this quantity becomes clear when oneconsiders a quantum state in which both the mechanical resonator and the cavity are in an eigenstate of the individualsubsystems. In the uncoupled system (i.e., for G OM =
0) the system remains in this state since it is also an eigenstate ofthe total Hamiltonian. However, this is no longer the case in the coupled system. The state will then oscillate betweenstates with more energy in the mechanical resonator and states with more photons in the cavity. The rate at which thishappens is proportional to G OM . Note that this is very similar to Rabi oscillations in a two-level system coupled to a cavity[260]. Furthermore, the Hamiltonian in Eq. 70 shows that the optomechanical coupling is related to the backaction: theforce exerted by a single photon in the cavity is ~ g OM , which can be as large as 13 fN for the on-chip cavities that we willencounter in Sec. 4.1.4. Finally, we note that the coupling strengths g OM and G OM are solely determined by the devicegeometry and are independent of the optical and mechanical Q-factors. A schematic picture of a typical Fabry-P´erot setup has already been shown in Fig. 25a and was discussed in Sec. 3.5.3.Briefly, the cavity consists of two mirrors, one of which can move, whereas the other one is fixed. When the moveablemirror is displaced, the cavity length changes and therefore also the cavity resonance wavelength. An important propertyof the cavity is the reflectivity of the mirrors. A large reflectivity means that the optical quality factor is high, i.e., thelight bounces back and forth many times so that the interaction between mechanical motion and the photon is stronglyenhanced. A small fraction of the light is allowed to come out of the cavity and is guided to a photo-detector. These candetect individual photons and therefore the position detection sensitivity can be shot noise limited.There are two main implementations to measure the light that comes out of the cavity. The first method, shown in Fig.25a, uses a polarized beam splitter in combination with a λ/ The quality factor indicate the number of oscillations of the optical field before it leaves the cavity or before it is dissipated. In optics, one usuallyspeaks about the finesse F of the cavity, which is the number of reflections of the field before it is lost. The finesse is related to the optical quality factor: F ≡ ∆ ω c /κ = Q × ∆ ω c /ω c . Here, ∆ ω c is the free spectral range of the cavity, i.e. the distance between subsequent cavity resonances ( ∆ ω c = π c / L ).Typical numbers for F range from 10 to 10 for high-quality cavities [222]. − m / √ Hz [64].For a Fabry-P´erot cavity the coupling constant is g OM = ω c / L , where L is the cavity length [122] as explained inSec. 4.1. For a cm-long cavity and light with a visible wavelength this gives g OM ∼ π ·
10 MHz / nm (see Table 8). In thepresence of a strong coherent pump laser, the e ff ective coupling rate is enhanced by the square root of the number photonsin the cavity √ n c . The optomechanical strong coupling regime where G OM · n / c ≫ ω R , κ c was first reached in a 25 mmlong Fabry-P´erot cavity [237], as evidenced by the mode splitting [263] of the mechanical resonator and the detuning ofthe cavity. Strong coupling has now also been demonstrated in other optomechanical systems [254, 255, 264].Fabry-P´erot cavities are very sensitive position detectors and have already been used for a while to approach the quan-tum limit on position detection, see for example Ref. [209]. Their properties (shot-noise limited sensitivity, backactionetc.) are therefore well known. The most sensitive implementation of the Fabry-P´erot interferometer reaches the impres-sive sensitivity of 10 − m / √ Hz for the motion of the di ff erential motion of the two mirrors forming a 0.25 mm longcavity with a finesse of 230 000 [45]. Currently, a lot of e ff ort is put in reaching true strong coupling where G OM > κ, ω R and in the observation of quantum backaction [265, 266]. Another use of cavities in optomechanics, is the setup pioneered by the group of Harris, where a flexible membrane ispositioned inside a rigid cavity [46]. A schematic drawing of this setup is shown in Fig. 27a. One of the advantages of thisimplementation is that the mechanical resonator and the mirrors of the cavity are separated, so that a high-quality cavitycan be made without degrading the mechanical properties of the resonator. The membrane, with a low (field) reflectivity r c , can be positioned at di ff erent locations x inside that cavity, i.e., at nodes or at anti-nodes of the standing light-fieldwaves. Depending on where the membrane is placed, the coupling is di ff erent: At an anti-node, the membrane stronglyinteracts with the cavity, whereas at a node, it does not. In a way this system can be viewed as two coupled cavities.For a particular optical mode, the light is predominantly on one side of the membrane exerting a radiation pressure onit from that side. From the above it is clear that the cavity frequency ω c should be a periodic function of the membraneposition x m and so is the optomechanical coupling. Using a one-dimensional model [46] the coupling strength is obtained: g OM ≈ π | r c | c / ( L λ ) sin(4 π x m /λ ) assuming | r c | ≪
1. This optomechanical system thus exhibits a tuneable coupling.At a node of the optical field ( x m = n λ/ ∼ ff erent optical cavity modes occur and near these avoided crossingsthe cavity frequency depends strongly on the position of the membrane squared (i.e., square-law position detection).In the realization of Ref. [267] the tilting results in an increase in the coupling from 30 kHz / nm to &
30 MHz / nm ,which might enable direct measurements of the quantization of the membrane’s energy (see the discussion in Sec. 3 andAppendix C) [268, 269]. As we have discussed before, it is advantageous to have a cavity with a high finesse (or the related optical qualityfactor, see footnote 33) in combination with a high mechanical Q-factor. In the experiments described in Sec. 4.1.2 thiswas done by physically separating the mirrors from the mechanical resonator. A di ff erent approach is to use microtoroids(see Fig. 27b), which can have optical Q-factors in excess of 10 or equivalently F > [261]. Light can be coupledinto these devices via free-space evanescent coupling by positioning a tapered fiber close ( ∼ µ m) to it [270]. The lighttravels around the outer edge of the toroid in, what is called, a whispering-gallery mode (WGM) (see Fig. 27b, lowerleft panel). The light in this mode is strongly coupled to the mechanical vibrations of the toroid, in particular to its radialbreathing mode (RBM). In this mode, the toroid expands and retracts in the radial direction, thereby changing its diameterslightly in time. The mechanical RBM frequencies are of the order of 10 to 100 MHz; the mechanical quality factor canbe as high as 32000 and depends on the exact device geometry [271]. The coupling length is in this case the toroid radius, L OM = R [122], and the coupling constant g OM = ω c / R can reach 2 π ·
10 GHz / nm (see Table 8). This high value can beunderstood from the much smaller dimensions of the toroid cavity: The cm-long Fabry-P´erot cavities are now replaced aby toroid with a circumference of a few hundred µ m.A very similar system are silica microspheres [270, 249, 63], where again the whispering-gallery mode couplesstrongly to the mechanical breathing modes with a frequency of the order of 100 MHz for a typical diameter of ∼ µ m.The optical quality factor of these devices exceeds 10 and typical values for the mechanical Q-factor are 10 . Light iscoupled into the WGM by focussing a laser beam close to the sphere. Similar to the fiber taper, the light enters the spherevia the evanescent field and this causes a detectable phase shift in the transmitted light [63].46ooling has been extensively studied in the microtoroid systems [37, 49, 58] and an advantage is that they can beintegrated and precooled to helium temperatures in cryostats [61, 63]. In the latter experiments, thermal occupationnumbers as low as n =
70 and 37 have been achieved respectively. The small scale of these devices also bears anotheradvantage in that it can be coupled to other mechanical resonator by placing them in the near vicinity of the toroid.An example is described in Refs. [67, 69], in which the flexural modes of a nanomechanical SiN string are probed viathe toroid with a position sensitivity that is two times below the standard quantum limit. The coupling g OM decreasesexponentially with the distance between the string and the toroid and a maximum value of g OM = π ×
10 MHz / nmhas been reported. Finally, it was demonstrated that microtoroid can be actuated electrostatically using gradient forces,making it an optoelectromechanical system [272]. A recent review summarizes the achievements in cavity optomechanicswith whispering-gallery modes [122]. A clever way to further increase the optomechanical coupling has been realized by Painter and co-workers [62, 123,65]. The basic idea is as follows: When placing two optical waveguides in close vicinity (submicron scale), symmetricand anti-symmetric optical modes form with a mode volume of the order of λ . The optical coupling length, L OM , can beviewed as the length scale over which a photon’s momentum is transferred, which, in this case, is reduced to a length scaleof the order of the optical wavelength, λ . Thus, the coupling can be estimated to be g OM ∼ ω c /λ . Since the wavelengthis about 1 . µ m, the coupling can be at least an order of magnitude larger than in the micro-cavities discussed above (seeTable 8). Two implementations have been built. In one version [127, 65], a pair of silica (or SiN) disks separated bynanometre-scale gaps was used as shown in Fig. 27c. The coupling depends on the separation between the disks andfor an air-gap of 138 nm, the coupling was found to be g OM = π ·
33 GHz / nm. E ffi cient cooling of the mechanicalmode was achieved [65], and static and dynamic mechanical wavelength routing was demonstrated [231]. Furthermore,Wiederhecker et al. demonstrated attractive and repulsive forces between the disks [127] and optomechanical tuning ofthe cavity modes over more than 30 nm [230] in a similar device.An even larger coupling is obtained with two stoichiometric silicon nitride ladder structures with a photonic crystalstructure (“zipper cavities”) as illustrated in Fig. 27d. With a separation of 120 nm between the two waveguides, acoupling of g OM = π ·
123 GHz has been achieved [62]. The strong coupling yields a large optical spring e ff ect (seeSec. 3.5.3), where the resonance frequency is mainly determined by the laser field instead of by the structural propertiesof the resonator. This e ff ect shows up as a change in resonance frequency if the input power and detuning are changed.Interestingly, the optical spring only acts on the di ff erential motion of the beams. The common-mode vibrations arenot a ff ected by the light field. By varying the detuning of the driving light, Lin et al. could shift the di ff erential modethrough the resonance frequency of the common mode. The observed Fano-like lineshape indicates coherent mixing ofthe mechanical excitations [196].Another advantage of optomechanical crystals is the control over the location of the optical and mechanical modes[123]. Since the spatial extent of modes di ff ers, the coupling strength can be engineered. Thus, the simultaneous con-finement of optical (photonic crystals) and mechanical modes (phononic crystal) leads to strong, controllable light-matterinteractions [273]. Finally, note that in on-chip optomechanical devices the optical gradient force is typically much largerthan the photon pressure [55, 274, 65, 127, 275]. Instead of using an optical cavity, electromagnetic waves can also be confined in a superconducting microwave cavity[276]. The electromagnetic cavity mode is sensitive to capacitive changes and this principle has been used to measurenanomechanical motion [50, 66, 68]. The device used in the first realization of such a transmission-line position detector[50] is shown in Fig. 28.The working principle is as follows: In the superconducting transmission line with an inductance and capacitance perunit length, radiation propagates. On one side microwaves are injected via a capacitively coupled feedline; the other endcan be open or shorted to ground (so-called λ/ λ/ ; in the experiments, the dissipation is determined by the couplingto the feedline and not by internal losses (i.e., they are overcoupled) [50]. An overview of the key parameters of thesedevices is given in Table 8. Large optomechanical coupling is obtained for thin, long doubly clamped beams that arepositioned as close as possible near the ground plane. A coupling of g OM = π ·
84 kHz / nm is reported [68] for a 17047 d)(a) u (c) (b) Figure 28: Illustration of the stripline resonators used by Regal et al. (a) Chip with 6 meandering stripline resonators (pink) and the straight feedline(green). The λ/ ff erent length, which allows for frequency multiplexing. (b) The resonators are coupled to the feedline bya capacitive elbow coupler. (c) and (d) zooms of the resonator. The 50 µ m long aluminum beam resonator is under tensile stress which is induced byannealing at an elevated temperature. A movement of the beam changes the capacitance of the stripline to the ground plane (blue) and thereby the cavityresonance frequency. Reprinted by permission from Macmillan Publishers Ltd: Nature Physics 4, 555–560, copyright 2008. nm wide and 140 nm thick beam, that is formed from 60 nm of stoichiometric, high-stress silicon nitride and 80 nm ofaluminum. The resonator is located 75 nm from the gate electrode. Further substantial improvements in the couplingstrength are di ffi cult with this particular geometry and fabrication technology. Sulkko et al. [253] used a focussed ionbeam to create a very small gap of about 10 nm between a mechanical resonator and a resonant microwave circuit andobtained g OM = π · / nm, thus increasing the coupling considerably (see Table 8).An important di ff erence with optical cavities is that there are no single photon detectors available for microwave fre-quencies. In optical systems these do exist, and the detection of the light in itself is quantum-limited. Present commercialmicrowave amplifiers, however, always add substantially more noise than required by quantum mechanics since they arenot shot-noise limited. This problem can largely be overcome by using a Josephson parametric amplifier. Teufel et al. [66] have made a nearly shot-noise-limited microwave interferometer and demonstrated nanomechanical motion detec-tion with an imprecision below the standard quantum limit. On the other hand, similar to their optical equivalents, thephoton-pressure backaction of the stripline position-detectors on the mechanical resonator has been observed [56] andeven backaction-evading measurements have been reported [252]. In the latter experiment, a single-quadrature measure-ment of motion with a sensitivity of four times the zero-point motion has been demonstrated. Sideband cooling has beenperformed in a series of experiments [56, 68, 252]. Very recently Teufel et al. were the first ones to demonstrate groundstate cooling using a superconducting LC resonator [23], in which a movable membrane both acts as the capacitor of theLC circuit and as the mechanical oscillator (“drum resonator” geometry). They obtained a thermal occupation of n = . ff ers from the abovementioned superconducting cavities in that it consists of a stripline resonator withlower frequency (100 MHz) and lower quality factor (234). In the second experiment, the resonator is an open-endedcoaxial cable with a resonance frequency of 11 GHz and a Q of 80. Mechanical motion can also be detected by measuring the transmission of electrons or photons through devices whichembed a movable part. The motion of the mechanical resonator modulates the electron or photon transmission and the48 µm (a)(b) e- (b) (c) Figure 29: (a) Readout using the light transmitted from one waveguide (left) to another one (right). When one of the cantilevers moves, the transmissionof the light changes. This is a non-interferometric optical method to detect mechanical motion. Reprinted by permission from Macmillan PublishersLtd: Nature Nanotechnology 4, 377–382, copyright 2009. (b) An atomic-point-contact displacement detector. A tunnel contact between the beam and anearby electrode is made using electromigration. When the Au beam is far from the electrode, the tunneling probability for electrons (see inset) is lowand the resistance is high. Closer to the electrode, the transmission increases and the resistance is lower. Reprinted figure with permission from N. E.Flowers-Jacobs, D. R. Schmidt, K. W. Lehnert, Intrinsic noise properties of atomic point contact displacement detectors, Phys. Rev. Lett. 98 (9) (2007)096804. Copyright 2007 by the American Physical Society. (c) Setup to measure multi-walled nanotube vibrations via field emission. Electrons areemitted by applying a voltage V A = −
500 to −
900 V between the nanotube and a nearby (3 cm) screen. The electrons hit the screen at a certain positionand show up a bright spot. The insets at the bottom show the spots for a silicon-carbide resonator that is driven o ff -resonance (left) and on resonance(right). In the latter case the spot is blurred due to the motion of the resonator. Adapted from Refs. [278] and [90]. Reprinted with permission fromS. T. Purcell et al. Tuning of nanotube mechanical resonances by electric field pulling, Phys. Rev. Lett. 89 (27) (2002) 276103. Copyright 2002 by theAmerican Physical Society and S. Perisanu et al.
Appl. Phys. Lett. 90 (4) (2007) 043113. Copyright 2007, American Institute of Physics. output signal contains spectral information about the resonator dynamics. Below we will discuss in detail two basicconfigurations of this detection scheme: optical waveguides and electron tunneling devices, including electron shuttles.
As mentioned above, light can be guided on a chip using optical waveguides. These are pieces of transparent material(e.g. Si, SiO or SiN) which are carved out of the underlying substrate. Light can propagate through these waveguidesand can be directed to any place on the chip. Just outside the waveguide the electromagnetic field is not zero and decaysexponentially with the distance from the waveguide. When two waveguides are placed close together, light can be trans-mitted from one to the other . The transmission between the waveguides depends on the distance between them and theycan be employed as a position detector. The method is also compatible with operation in water [280], which is importantfor (bio)sensor applications, and it has also been used to make optoelectronic switches [281]. Li et al. [59] have usedthe transmission-modulation principle to measure the mechanical motion of cantilever structures made in a silicon-on-insulator platform with high sensitivity. In particular, they placed the suspended ends of two waveguides facing each otherat a distance of 200 nm as shown in Fig. 29a. The transmission of photons through the gap depends on the distance andthe exact position of the cantilevers (misalignment). The thermal motion of the cantilevers modulates the transmissionand the transmitted light is fed into a photodiode. The spectrum clearly reveals multiple resonances of the cantilevers witha sensitivity of 40 fm / √ Hz at room temperature. An advantage of this method over optical cavities is that the detectionscheme allows for transduction of nanomechanical motion over a wide range of optical frequencies, instead of workingonly at certain well-defined wavelengths.
When two metallic electrodes are placed close (up to a few nanometer) to each other, electrons can tunnel throughthe gap between them. With a voltage applied between both electrodes, a net current flows that depends exponentiallyon their distance. This exponential distance dependence can be used to measure displacements of mechanical resonators.Flowers-Jacobs et al. [42] have measured the displacement of the doubly-clamped gold beam shown in Fig. 29b usingan atomic point contact (APC) made by electromigration [282]. Since the tunneling resistance between the APC and thebeam is high, a tank circuit is used for impedance matching to 50 Ω high-frequency amplifiers. The Brownian motionof the beam is observed at a temperature of 250 mK with a shot-noise limited imprecision of 2 . / √ Hz. From the A related, but di ff erent method is demonstrated in Ref. [279], where photons are transferred from one waveguide to another, which runs parallel tothe former. The coupling between the waveguides depends on the distance between them (i.e., on the displacement) and photons that enter the secondwaveguide are “lost” and not detected at end of the first waveguide. . et al. used a similar technique where an STM tipwas positioned above a MEMS resonator [283]. The current modulation due to the resonator motion was down-mixed(Sec. 4.3.3) to overcome bandwidth limitations. Based on these methods an interesting new type of detector is proposed[284]. By incorporating the movable tunnel junction into a loop threaded by a magnetic flux, the APC can be used tomeasure the momentum instead of the position of the resonator. Also, other more sophisticated schemes are envisionedfor detecting entanglement in the mechanical quantum oscillator [285].Another method is based on field emission of electrons from a vibrating tip. Consider, for instance, a multi-walledcarbon nanotube [278] mounted under vacuum in a field emission setup (see Fig. 29c). A large voltage is applied betweenthe carbon nanotube and an observation screen which lights up at the position where electrons are impinging. Electronsare accelerated from the tip of the nanotube to the screen by the electric field. Motion of the nanotube cantilever resultsin a small blurring of the spot at the screen. The thermal motion of the nanotube already blurs the spot a bit, but whenthe vibration amplitude is enlarged by applying a RF driving signal on nearby electrodes the blurring of the spot becomesmore pronounced on resonance. The electric field also pulls on the nanotube, thereby increasing the resonance frequency[278]. The method has been used to build a nanotube radio [74], a mass sensor with atomic resolution [77], and high Q silicon carbide nanowire resonators at room temperature [90].In more complicated circuits shuttling of electrons has been considered [286, 125, 287]. These devices operate asfollows: there are two electrodes with a movable metallic island in between. When the resonator is closer to one of theelectrodes, the transmission between the island and this electrode increases, allowing for electrons to jump on the island.The electrostatic force drives the island with the negatively-charged electrons to the other electrode. This way, they aretransported to the other electrode by the resonator (mechanical transport of electrons). In steady state this shuttling occursat the resonance frequency of the oscillating island. Interestingly, the voltage applied between the two electrodes canamplify this motion and lead to large amplitude oscillations; an example of an electomechanical instability . This subjecthas been extensively reviewed by Gorelik et al. [288] and we refer to this work for further reading. In many nano-electromechanical devices, movable metallic parts form capacitances with nearby metallic electrodesor ground planes. Displacement of the mechanical structure inevitable leads to a change in these capacitances whichcan be detected electrically if they are large enough. Likewise, the motion of the mechanical resonator can be actuatedby applying voltages between the di ff erent electrodes as we have discussed in Sec. 2.4.4. This capacitive actuation andread-out of mechanical motion has been applied in several ways. Here, we will discuss three popular methods. One of the first mesoscopic devices that was used for position detection was the single-electron transistor (SET) [29].This device consists of a metallic island that is connected by tunnel contacts to a source and drain electrode. The totalcapacitance of the island is so small that the energy required to add one electron surpasses the thermal energy [289] andelectrons can only enter and leave the island one by one (sequential tunneling). The resulting single-electron currentthrough the island is very sensitive to the electrostatic environment. Thus, if a nearby resonator is capacitively coupled tothe island, and it has a di ff erent electrostatic potential, the device can be used as a displacement sensor, as illustrated in Fig.30. The first realization of such a device was made by Knobel and Cleland [29], where the SET measured the position ofa 116 MHz beam resonator with a sensitivity of 2 fm / √ Hz. This value corresponds to a position resolution a factor of 100above the standard quantum limit, as is indicated in Table 7. The SET was operated as a mixer, a technique that we willdiscuss in more detail in Sec. 4.3.3. Subsequent work by LaHaye et al. [30] demonstrated, using a superconducting radio-frequency SET, a sensitivity of four times the standard quantum limit. The role of the RF SET is to provide impedancematching for the high-frequency resonator signal (19 . et al. [32] observed the backaction of the superconductingSET on the mechanical resonator; a change in the capacitance leads to a change in the average charge (“occupation”)[290], which in turn leads to a change in the electrostatic force the resonator experiences. Depending on the bias conditions(gate voltage) of the SET, shifts in the resonance frequency and the damping rate were observed. Thus the backactionforce on the detector, characterized by the function λ F that was introduced in Sec. 3.4.1, is bias dependent for thistype of detector [240, 212, 291, 292]. The e ff ective resonator temperature has been measured for di ff erent couplingstrengths between the resonator and the SET. In the experiment this is done by varying the voltage di ff erence betweenthe resonator and the SET. For small couplings, the resonator temperature equals that of the sample chip i.e., close to This shuttling instability can thus be viewed as an example of backaction induced self-oscillations as discussed in the previous Section. Whenpursuing the quantum limit in these devices, people refer to the ability to transport exactly one charge carrier per cycle from source to drain; thezero-point motion does not play a crucial role in achieving this. lectrostatic potential c u rr e n t (c)(d) s ou r c ed r a i n ga t e (a)(b) electrostatic potential c u rr e n t Figure 30: (a) Illustration of the dependence of the current through a SET on the electrostatic potential. The current is almost zero in the Coulombblockade regime and displays so-called Coulomb peaks where the blockade is absent. The resonator motion modulates the potential (horizontal arrow)and this leads to a proportional change in the current. (b) The device used in the Schwab group. The superconducting single-electron transistor (SSET)is made from aluminum and is connected to a source and drain electrode. The potential of the SET can be varied using a gate electrode and is alsoinfluenced by the position of the beam resonator. From M. D. LaHaye et al. , Science 304 (2004) 74–77. Reprinted with permission from AAAS. (c)Illustration of self-detection via rectification. Motion of the resonator leads to a decrease in the average current on top of the Coulomb peak, whereasdeep in the blockade region it leads to a (small) increase. (d) SEM picture of a ultra-high quality factor nanotube resonator. From G. A. Steele et al. ,Science 325 (2009) 1103–1107. Reprinted with permission from AAAS. the base temperature of the dilution refrigerator. For higher couplings, the resonator temperature is set by the backactiontemperature of the SET, and is raised to about 200 mK.It should also be possible to use a SET to sense the vibrations of a suspended carbon nanotube, but the small capacitivecoupling between the nanotube and the SET island makes this a challenging task [293]. It is more advantageous to use thesuspended carbon nanotube itself as a self-detecting SET. Using current rectification and frequency mixing, informationabout the driven motion of the suspended nanotube has been obtained [81, 82, 106]. In these experiments, a strongcoupling between mechanical motion and the charge on the nanotube has been observed. The strong coupling is dueto the electrostatic force generated by individual electrons tunneling onto the nanotube. For the devices used in theseexperiments, the change in equilibrium position of the nanotube after adding a single electron easily surpasses the zero-point motion. Typically, the single-electron tunnel rate is much larger than the resonance frequency, indicating that thebackaction is determined by the average number of electrons (“occupation”) on the nanotube. This backaction leadsto frequency shifts and changes in damping as a function of gate voltage. To be more specific, the damping increasesdramatically with the amount of current flowing through as the electron tunneling produces a large stochastic backactionforce.In Refs. [81, 106] readout using current rectification is employed. While the nanotube motion is actuated by a RFsignal on a nearby antenna, the detected signal is at DC. The key to understand this is the notion that nanotube motione ff ectively translates into an oscillating gate voltage, which smears out the sharp features of the SET current, as illustratedin Fig. 30c [295]. The technique is of special interest as it constitutes a square-law detector (see Appendix C). Moreover,it allows for the motion detection with small currents, enabling the observation of ultra high Q-factors, exceeding 100,000at millikelvin temperatures. The low dissipation enables the observation of single-electron tuning and frequency tuningoscillations, analogous to the Coulomb oscillations in the SET current. Recently, this self-detecting rectification schemehas also been employed for a thin aluminum beam [296]. A quantum-point contact (QPC) is a narrow constriction in a two-dimensional electron gas, whose conductance can beadjusted using electrostatic potentials [297]: Every time another channel for electrons becomes available, the conductanceincreases by one conductance quantum 2 e / h . In practice these sharp steps are smoothed by temperature and when theQPC is biased near such a step it is very sensitive to changes in the electrostatic potential. If mechanical vibrationsmodulate the electrostatic fields, the QPC can also be used as a position detector. Cleland et al. have used this principle to The rectification and mixing measuring method have in common that they both rely on the nonlinearity of the current-voltage or current-gate-voltagecharacteristics [294]; the experimental implementation is, however, di ff erent. It should also be noted that rectification is more commonly used to detectdisplacement in mechanical resonators. For example, in the experiments of Ref. [74, 77] (Sec. 4.2) the dc tunnel current contains information about thevibrating carbon nanotubes since a time-dependent displacement changes the source-drain distance and thereby the averaged current. / √ Hz [298].In this case the QPC is part of the beam and frequency mixing (Sec. 4.3.3) is used to detect the displacement through thevoltage across the QPC. The signal is amplified through the piezoelectric e ff ect in GaAs that will be discussed in moredetail in Sec. 4.4.A similar sensitivity has been achieved in a completely di ff erent setup by Poggio and coworkers [52]. Now a 5 kHzmicromechanical resonator is hanging above a QPC that is located on a di ff erent substrate. Thermal noise specta havebeen recorded and the authors have mapped out the transduction factor as a function of cantilever position relative to theQPC. Furthermore, they observe that the cantilever Q-factor is not a ff ected by the QPC source-drain current, indicatingweak coupling and therefore negligible backaction.A QPC can also be used to probe vibrational modes of the host crystal itself [299]. This substrate is a truly three-dimensional resonator that consists of on the order of 10 atoms. Strictly speaking, the latter example does not constituteof a capacitive detector as the transduction is through the piezoelectric e ff ect; the motion directly influences the source-drain voltage and not the conductance of the QPC. Frequency mixing has been adopted in top-down solid-state devices as a versatile technique to convert high-frequencymotion into a low-frequency signal [298, 29, 300]. It is most often used in combination with capacitive detection tech-niques, although it is also used in combination with electron tunneling (Sec. 4.2) and piezoelectric resonators (Sec. 4.4).The basic principle is as follows. Consider the generic relation between the input (i.e., the displacement u ) and the detec-tor output v introduced in Sec. 3: v ( ω ) = λ v ( ω ) u ( ω ). By modulating the transduction λ v at a frequency f LO (often calledthe local oscillator frequency), the displacement at frequency f R is converted into a signal with frequencies f R + f LO and f R − f LO . The latter component at the di ff erence frequency can be chosen to be at a frequency far below the resonatorfrequency: for example in the kHz range. The signal at this frequency is not a ff ected by the RC time of the measurementsetup and can therefore be measured straightforwardly.The technique has become of particular interest for detecting vibrational motion of bottom-up devices. Sazonova etal. [73] were the first to apply frequency mixing to suspended carbon nanotube resonators. They observed multiple gate-tunable resonances with Q-factors on the order of 100 at room temperature. Subsequently Witkamp et al. [20] identifiedthe bending mode vibrations of a carbon nanotube. Nowadays the technique has been employed by many groups, notonly restricted to carbon nanotubes [76, 78, 126, 82, 83, 85], but also applied to suspended graphene sheets [103, 104],free-hanging semiconducting nanowires [94, 96] and charge-density-wave sheets [105]. Furthermore, several variationsto the original mixing scheme have been implemented, including FM [85] and AM modulation [126].A more detailed understanding of this self-detecting method can be obtained by considering, for example, the sus-pended carbon nanotube of Fig. 10. The conductance of a semiconducting nanotube depends on the induced charge,which is the product of the gate voltage and the gate capacitance, i.e. G = G ( C g V g ). As we have shown before, thegate capacitance is position dependent and therefore the conductance varies in time with the resonator frequency. Innanotube mixing experiments, the current is measured, which is the product of the conductance and the bias voltage: I = V b ∂ G /∂ u · u , from which we identify λ v = V b ∂ G /∂ u . The modulation of the transduction can thus be done byapplying an ac bias voltage: V b = V ac cos( ω LO t ), as explained in Fig. 31. In these experiments backaction comes againfrom the electrons flowing through the nanotube [82]. A di ff erent mechanism to transduce a displacement into an electrical signal uses piezoelectricity. In a piezoelectricmaterial a stress induces an electric polarization, or equivalently, an electric displacement D [137]. This results in avoltage di ff erence across the resonator that can be used to infer the displacement. Piezoelectric resonators are di ff erentfrom most other methods discussed in this Section as no separate detectors are needed; they detect their own motioninstead. Moveover, they are also di ff erent from the other self-detecting resonators (e.g. the suspended carbon nanotubesand graphene resonators that were described above) since no external signal needs to be applied in order to measurethe displacement (for example, in the nanotube experiments of Sections 4.3.1 and 4.3.3 one has to apply a source-drainvoltage to transduce the displacement). Still, from the discussion in Sec. 3.4 it is clear that backaction forces should alsoact on a piezoelectric resonators when measuring the generated voltage. This happens through the converse piezoelectrice ff ect, where an electric field generates a strain in the material. Referring back to the definition of stress and strain in Sec.2, the direct and converse piezoelectric e ff ect are [137]: D i = d i jk σ jk , γ i j = d i jk E k . (71)Here d is the third-rank tensor with the piezoelectric coe ffi cients. Similar to the properties of the elasticity tensor, manyof the elements d i jk are zero, or related to other elements, depending on the symmetries of the crystal structure of thematerial. A necessary requirement to have at least one nonzero element is that the material’s unit cell does not have an52 ? lock-inamplifier inputref. out PCB back-gate DS V gac V gdc f Δ f Δ f ??? fff f ΔΔ ff Δ f+- Δ f ?? ff ΔΔ fff+- i v - c on v . ~ ? HPF
Figure 31: Schematic overview of the measurement electronics used by Witkamp et al. for position detection of a suspended nanotube using frequencymixing [20]. A radio frequency (RF) generator applies an ac voltage to the back-gate electrode to drive the suspended nanotube. A dc gate voltage isadded via a bias-T (indicated by the “ + ”). The same generator is used to generate the ac bias voltage by mixing its output with the reference outputof the lock-in amplifier. At the source electrode the voltage has spectral components at f + ∆ f and f − ∆ f , whereas the gate voltage is oscillating atfrequency f . The nanotube mixes both signals, which results in an output current at the drain electrode with spectral components at ∆ f , f + ∆ f , f − ∆ f ,2 f + ∆ f and 2 f − ∆ f . The ∆ f part of the current flowing through the nanotube is converted into a voltage and is measured with a lock-in amplifier. Theprinted circuit board (PCB) with the sample, bias-T and 50 Ω terminator is located inside the vacuum chamber of a probe station. inversion center. For most piezoelectric materials, including GaAs and AlN, the piezoelectric coe ffi cients are of the orderof pC / N and they can have both positive and negative values.Piezoelectric displacement detection of small structures has first been employed with AFM cantilevers [301, 302, 303],before extending the technique to the nanomechanical domain. Tang et al. used this detection scheme to detect themotion of GaAs / AlGaAs beams [304]. The beams are made asymmetric to ensure that the piezoelectric signal is notnulled by the opposite stresses at both sides of the neutral plane (Sec. 2.3). Cleland et al. used piezoelectricity to detectthe motion of GaAs beams with an integrated QPC in it. There, a deflection of the beam induces in-plane stress and theresulting out-of-plain field D acts as an e ff ective gate voltage. This then changes the current through the QPC [298].An important consideration is that piezoelectric materials cannot generate large currents. The resonator should thereforebe connected to a high-impedance load, such as the gate electrode of the QPC [298], to the gate of a SET [305], or ahigh-impedance amplifier [304].Mahboob and Yamaguchi used the two-dimensional electron gas in a flexural beam resonator made of a GaAs / AlGaAsheterostructure to measure the displacement and they demonstrated parametric amplification by modulating the beam’sresonance frequency using the converse piezo-electric e ff ect [306]. The converse piezo-electric e ff ect was also used totune the frequency of a doubly-clamped zinc-oxide nanowire by Zhu and coworkers [94]. Okamoto et al. found anenhancement of backaction e ff ects due to excitation of carriers in an piezoelectric GaAs resonator [307]. Many of theabovementioned e ff ects were combined by Masmanidis and coworkers who used a three-layer structure, consisting ofp-doped, intrinsically doped, and n-doped GaAs to demonstrate piezoelectric actuation, frequency tuning, and nanome-chanical bit operations [308]. By electrically shifting the carriers with respect to the neutral plane, the strength of theactuation could be adjusted. Finally, by capacitively coupling a bulk AlN resonator to a qubit, O’Connell et al. obtainedthe high coupling rates required to perform quantum operations on a mechanical resonator [22]. This experiment wasalready mentioned in the Introduction and will be discussed in more detail in Sec. 5.2. Superconducting quantum interference devices (SQUIDs) are well known for detecting small magnetic signals, suchas those generated by our brains. These devices consist of a superconducting ring in which one or more Josephsonjunctions are incorporated [309]. The voltage across the loop depends on the amount of magnetic flux that threads theloop, Φ mag . This flux-dependence has been employed to sense the motion of a cantilever with a small magnetic particleattached to it [310].It is, however, also possible to detect displacements by incorporating a mechanical resonator in the SQUID loop[311, 312, 313, 314, 315, 316, 317, 186, 318, 319, 320]: In the presence of a constant magnetic field, a change in the In this experiment the measured signal contained both piezoelectric and piezoresistive components. SQUIDs are also used indirectly for position detection, where the SQUID is used as low-noise voltage amplifier that amplifies the signal generatedby capacitive detection [53]. In this section, we only focus on direct position detection with SQUIDs. I B I F Φ mag uB I/ - J B I/ + J B Figure 32: Schematic overview of a dc SQUID position detector, including the suspended beam resonator and measurement setup. The SQUID loopis indicated in brown. The output voltage of the SQUID depends on the bias current I B that is sent through the SQUID and also on the amount of flux Φ mag through the loop. A magnetic field B transduces a beam displacement u into a change in magnetic flux and subsequently in a change in the outputvoltage V . The flux Φ mag is fine-tuned with a stripline current I F . Backaction results from the circulating current in the SQUID loop J . Reprinted figurewith permission from M. Poot et al. , Phys. Rev. Lett. 105 (2010) 207203. Copyright 2010 by the American Physical Society. resonator displacement changes the loop area and thereby the flux through the loop. This is illustrated in Fig. 32. Recently,Etaki et al. have used a dc SQUID as a sensitive detector of the position of an integrated mechanical resonator. Theydetected the driven and thermal motion of a 2-MHz buckled-beam resonator with femtometre resolution at millikelvintemperatures [19] and employed active feedback cooling to cool the resonator to 20 mK [70]. In the present experiments,the sensitivity is limited by the cryogenic amplifier and not yet by the, in principle quantum limited SQUID itself.Backaction has also been observed, leading to tunable shifts in the resonance frequency and damping of the resonator[242]. Di ff erent from the backaction in the capacitive readout schemes, the backaction of the SQUID has an inductivecharacter as it is caused by the Lorentz force generated by the current circulating in the loop of the SQUID. This currentalso runs through the resonator (Fig. 32) and the magnetic field that couples displacement and flux, generates a Lorentzforce on the resonator. A fast and relatively easy way to actuate and read-out nanomechanical motion in clamped-clamped resonators is themagnetomotive technique [321, 322, 28, 194]. Nowadays, this method is mainly used to determine the vibrational fre-quencies and to characterize dynamic properties such as the Q-factor and nonlinear behavior. It is, however, of limiteduse for QEMS experiments and we will therefore only briefly describe the mechanism: An ac current I ac is sent throughthe conducting (part of the) beam which is placed in a strong static magnetic field. The ac current causes an ac Lorentzforce that drives the beam µ n B ℓ I ac . At the same time, motion of the beam induces a time-varying voltage (Faraday’s law)at the driving frequency V em f = µ n B ℓ ˙ u . By sweeping the driving frequency, the resonance can be found by measuring thefrequency response of the voltage over the beam. The required large magnetic field is typically generated using supercon-ducting magnets in a cryogenic environment, but recently room temperature magnetomotive actuation and detection hasbeen demonstrated with a 2 T permanent magnet [323]. An important feature of this technique is that observed mechanicalmodes can be distinguished from electronic resonances by varying the magnetic field. Since both the driving force andthe detector signal are proportional to B , the mechanical signal scales as B [324]. At low temperatures, suspended quantum dots may reveal information about their vibrational states: in the parameterregime dominated by Coulomb blockade physics, the quantum mechanical level spectrum of the confined electronic sys-tem can be characterized by transport measurements. The levels show up as steps in the current-voltage characteristics.A more accurate measurement involves the recording of a so-called stability diagram in which the di ff erential conduc-tance dI / dV plotted as function of gate voltage V G and source-drain voltage V . Now the excitations (due to electronicor vibrational degrees of freedom) show up as lines, whose energy can directly be read out [325] as illustrated in Fig.33. In the case of vibrational states these are also called vibrational side bands and when the harmonics of a particularvibration are excited, the lines form a spectrum with equidistant spacing. Thus, electron tunneling through suspendedquantum dots can excite vibrational modes and these modes can then be detected as steps in the current-voltage character-istics. Their observation involves, however, two important considerations: the energy resolution and the electron-vibroncoupling. Concerning the energy resolution of transport spectroscopy measurements one must have ~ ω R > k B T so thatmeasurements are always performed in the quantum limit of the mechanical mode.Secondly, the electron-vibron coupling must be high enough; it can be characterized by a dimensionless parameter54 ate voltage S ou r ce - d r a i n v o l t a g e g = 0.5 ev g = 0 ev g = 4 ev I > 0I < 0I = 0 I = 0 hf /e R phononblockade(a) (b) (c) Figure 33: Stability diagrams for zero (a), intermediate (b), and large (c) electron-vibron coupling g ev . A stability diagram is a grayscale plot of thedi ff erential conductance plotted against the gate and source-drain voltage. Coulomb peaks (Fig. 30) appear here as dark lines, whereas regions with lowdi ff erential conductance d I / d V are white. (a) shows a regular stability diagram without signatures of the mechanical resonator. There are regions wherethe current is blocked and regions where current flows. These are separated by the high-conductance lines (dark). In (b) a finite electron-vibron couplingis present and mechanical excitations appear as lines parallel to the original lines of high conductance (vibrational sidebands). The source-drain voltagewhere these lines cross the Coulomb diamond edges, equals h f R / e as indicated. For large g ev the current is blocked at low source-drain voltage as shownin (c) and vibrational excitations can only been seen at high bias voltage. g ev = ( ∆ u / u ) , where ∆ u is the shift of the resonator position induced by adding one elementary charge, and u isthe zero point motion of the mechanical oscillator [326]. The parameter g ev determines the step height in the current-voltage characteristic. We can consider three regimes in describing the influence of vibrational modes on transport: theweak electron-vibron coupling regime with g ev ≪
1, the intermediate regime with 0 . < g ev < g ev ≫
1) limit. The boundaries of the intermediate regime are, however, somewhat arbitrary. Figure 33 shows calculatedstability diagrams for three di ff erent values of the electron-vibron coupling. In the weak coupling regime (see panel(a)) only the regular Coulomb step is present (no side bands are visible) and consequently vibrational modes cannot bedetected in a transport experiment. Only for su ffi ciently large electron-vibron coupling, one or multiple so-called Franck-Condon steps can be observed in the current, which show up as lines in the stability diagram (the intermediate regime,panel (b)). This was first demonstrated in molecular junctions exciting C radial breathing modes [327], and later insuspended carbon nanotubes probing the radial breathing mode [328], or the longitudinal, stretching modes [329]. Inthe strong electron-vibron coupling limit, the vibrational induced excitations are only seen for larger bias voltages as theheight of the first steps is exponentially suppressed. In panel (c) this is clearly visible: the grey lines representing thevibrational excitations are only present for high source-drain voltages. Importantly, the suppression holds for any gatevoltage and as a result the current at low bias is suppressed in the whole gate range [330, 331]. Degeneracy points in theCoulomb diamonds are no longer visible in the stability diagrams and one speaks of phonon blockade of transport. Phononblockade has been observed in suspended quantum dots embedded in a freestanding GaAs / AlGaAs membrane [332] andsubsequently in suspended carbon nanotube quantum dots [333].Recently, is has been shown that inelastic electron tunneling spectroscopy (IETS) can also be used to gain informa-tion about vibrational modes in suspended quantum dots. H¨uttel et al. [334] observed a harmonic excitation spectrumconnected to the longitudinal stretching modes of a suspended carbon nanotube in the Coulomb-blockaded regime, whiletemperature only allows the observation of a single excitation. The non-equilibrium occupation of the modes is explainedby the pumping via electronic states, revealing a subtle interplay between electronic and vibrational degrees of freedom. Actually, this is the shift in the location of the minimum of the potential energy of the resonator. The tunneling of the electron is assumed to takeplace on a timescale that is much faster than the resonator can react. igure 34: A mechanical resonator coupled to a Josephson qubit. The mechanical resonator (left) is cooled cryogenically to its ground state andmanipulation of its quantum state at the single phonon level has been demonstrated. The quantum properties of the resonator are probed by thesuperconducting qubit which is positioned on the top side of the right panel. Reprinted by permission from Macmillan Publishers Ltd: Nature 464,697–703, copyright 2010.
5. Prospects
So far, we have mainly concentrated on the limits of (linear) displacement detection. As discussed before, thereare other ways of demonstrating quantum behavior. For example, square-law detection would directly probe the energyeigenstates of the mechanical resonator (Sec. 3 and App. C). Another approach is to couple a mechanical quantumoscillator to another quantum system such as the well-studied Josephson qubit [335]: the state of the mechanical resonatorchanges the state of the qubit, which can then be probed to provide information about the mechanical states. A generalproblem in such schemes is that the coupling needs to be su ffi ciently strong so that the exchange of quantum states occursbefore decoherence sets in. In April 2010, the first experimental realization of a coupled quantum system involving amechanical resonator (see Fig. 34) has been reported by the groups of A. Cleland and J. Martinis [22]. They demonstratedthe superposition and coherent control of the quantum states of a mechanical resonator. The key aspect of their experimentis the use of a piezoelectric material for the resonator which boosts the coupling; it would be extremely hard to reachsimilar coupling strengths by using electrostatic forces alone (see also Table 8). We will briefly come back to this groundbreaking experiment in the second part of this Section when discussing coupling mechanical systems to other quantumsystems from a more general point of view (see Sec. 5.2).Also concerning the linear detection schemes many challenges lie ahead. Strong coupling, ground-state preparationand detection imprecision below the quantum limit have all been reached now. These achievements set the stage forfurther studies on controlling and detecting non-classical states of mechanical motion; eventually one would like to havea mechanical resonator in its ground state which is strongly coupled to a single photon so that quantum state of singlephonons and photons can be exchanged. Large coupling strengths and high Q-factors are necessary ingredients for reach-ing this goal. It is not a priori clear which detector scheme will be the best suited. The optical and superconducting cavitieshave the advantage that the underlying concepts are well known and have studied in detail; for detection schemes basedon other (mesoscopic) devices (e.g. superconducting SQUIDs, carbon-based resonators) the understanding is clearly notat the same level. For example, the backaction mechanisms in these cases are not understood in all details as these cou-pled detector-resonator systems cannot be mapped directly onto the Hamiltonian describing cavity dynamics (Eq. 70).More theoretical and experimental research is needed to elucidate on the underlying mechanisms and physics. In the nextsubsection, we will briefly discuss some of the issues and challenges of linear detectors in more detail. It is clear that in the coming years more e ffi cient read-out and cooling techniques will be employed. For optical systemschallenges lie in increasing the coupling strength and to construct optical setups that can be incorporated in dilutionrefrigerators. In that respect, the recently developed on-chip optic experiments are promising. In the gravitational wavecommunity the next generation of LIGO will use squeezed states to further improve the displacement sensitivity [336].For the electronic systems, reduction of detector noise is a major issue which can be achieved by the implementationof quantum-limited amplifiers (SQUIDs [337, 338], point contacts [206, 339, 340] or Josephson parametric amplifiers[341, 66]). Furthermore, nonlinear (quantum) e ff ects (see e.g. Refs. [342, 343, 344]) have so far received less attentionand this may become an interesting research line.For cooling experiments high signal-to-noise ratios are important as well as high Q-factors. Since Q-factors are limitedby material properties, mechanical oscillators of new materials (preferably in a crystalline form) will have to be fabricated56 ieldlines substratesingle spinin diamond tipmicrowavesfor EPRa surface electron spinto optics cantilever Figure 35: Left: a cantilever resonator containing normal metal rings to study persistent currents. In a magnetic field, the currents produce a torque onthe cantilever which shifts the cantilever frequency. By measuring the resonance frequency, the persistent currents could be studied with unprecedentedsensitivity. From A. C. Bleszynski-Jayich et al. , Science 326 (2009) 272–275. Reprinted with permission from AAAS. Right: Schematic representationof the diamond-based scanning spin microscope proposed in Refs. [345, 346, 347]. The single spin of a NV defect in a diamond tip is a sensitivemagnetometer for the local magnetic field with nanoscale spatial resolution. Magnetic fields near the surface shift the electron spin resonance (EPR)frequency of the NV center, which can for example be detected by exciting the EPR transition with a microwave field and monitoring the change inphotoluminescence of the probe spin. Reprinted with permission from C. L. Degen, Appl. Phys. Lett. 92 (24) (2008) 243111. Copyright 2008,American Institute of Physics. and characterized. We should add here that the understanding of the mechanisms controlling dissipation in mechanicalresonators is also still open. For example, the role of tension in elevating the Q-factor is not completely understood. Thetemperature dependence of the Q-factor is another open-standing problem; in some cases a very strong dependence isobserved, while in other cases the dependence is weak. What limits dissipation and what is the role of microscopic defectsuch as two-level fluctuators [16, 71, 72]? Furthermore, for low-mass resonators new e ff ects may start to play a role suchas the influence of adsorbents on the resonator surface and nonlinear damping terms associated with the induced tensionintroduced in Sec. 2.4.4.The push for refining detection schemes will undoubtedly lead to the construction of better sensors, which eventuallywill be quantum-limited. These will lead to new applications aimed at both solving fundamental science questions aswell as the development of commercial products. An example of the former direction is the recent detection of persistentcurrents in metal rings fabricated on nanoscale cantilevers [348], illustrated in Fig. 35a. The precision is one order ofmagnitude better than previous experiments on persistent currents, which are mainly based on SQUIDs. It is also expectedthat the newly developed optics techniques (i.e., on-chip optics) will find their way in novel applications in controlling,stopping and storing light [349, 350]. Another promising example is the detection of single spins using magnetic resonanceforce microscopy analogous to the magnetic resonance imaging. Single electron spins have already been detected [351];the next step is the detection of nuclear spins which requires a thousand fold increase in the sensitivity. An interestingrecent proposal (Fig. 35b) is the use of NV centers so that operation at room temperature becomes feasible [347]. Ifsuccessful, this would yield a revolution in imaging: Many chemical elements carry a nuclear magnetic moment, so that asensitive enough detector can determine their identity and arrangement in more complex molecules. Finally, NEMS mayfind an application in mass sensing [352, 353, 78, 77, 76, 354, 97, 355]: the state-of-the art is that single gold atoms canbe measured (see Sec. 4). The goal is to be able to detect masses with resolution better than 1 Dalton (the mass of ahydrogen atom) so that each element can be identified making it a mass spectrometer [356]. In most of this review we have treated optical systems with movable parts as stand-alone systems. Interesting newquantum systems can be built when coupling optical set-ups to other quantum systems such as an ensemble of atoms ina Bose-Einstein condensate. The coupled system would form a hybrid quantum system in which hybrid strong couplingwould enable the creation of atom-oscillator entanglement and quantum state transfer. A theoretical proposal appeared in2007 [357] and in 2008 the interaction between a Bose-Einstein condensate and the optical field in cavity was studied ex-perimentally [358] demonstrating strong backaction dynamics. Subsequent theoretical work showed that strong couplingbetween a mechanical resonator and a single atom should be feasible [359]. Strong coupling with a single atom has indeedbeen observed in the blue-detuned cavity regime: Experiments with a single, trapped Mg + ion interacting with two laserbeams showed the stimulated emission of centre-of-mass phonons [360]. At high driving, coherent oscillating behavior isobserved, which can be viewed as the mechanical analogue to an optical laser, i.e., a phonon laser. In another approach,the coupling between vibrations of a micromechanical oscillator and the motion of Bose-condensed atoms on a chip aremediated by surface forces experienced by the atoms placed at one micron from the mechanical structure [361]. As illus-trated in Fig. 36, the Bose-Einstein condensate probes the vibration of a cantilever. By adjusting the magnetic trappingpotential, the discrete eigenmodes of the condensate can be tuned, which in turn tunes the coupling to the cantilever.Systems of many resonators coupled to each other via atoms in an optical lattice are envisioned [362].57 igure 36: Bose-Einstein condensate (BEC) on an atom-chip couples to a cantilever. The cantilever motion modulates the potential of the atoms, therebycoupling the cantilever position to atomic motion. Left: schematic overview of the setup. The atoms of the BEC detect the cantilever vibrations whichcan independently measured via the readout laser. Right: photograph of the atom chip. Reprinted figures with permission from D. Hunger et al. , Phys.Rev. Lett. 104 (2010) 143002. Copyright 2010 by the American Physical Society A di ff erent approach to create entangled states of mechanical motion by coupling a mechanical quantum oscillator toa solid state device. In the experiment of the group of Cleland and Martinis [22], a Josephson qubit served as a two-levelsystem probing the mechanical states of a 3 ng FBAR resonator (c.f. Fig. 34). With a mechanical frequency of 6 GHz,a mechanical Q-factor of about 100 and a coupling of 62 MHz, there was just enough time to perform a few quantumoperations on the mechanical resonator before the quantum state decohered.If it would be possible to increase the Q-factor while maintaining a strong coupling, quantum mechanical manipulationof long-lived phonon states comes into reach. For example, with a Q-factor of 10 and a frequency of 1 GHz, themechanical state would survive for 100 µ s before it significantly decoheres. This time is typically much longer than thecoherence time of a superconducting qubit, indicating that quantum information can be stored in the mechanical motionand transferred back when needed. The long phonon life times should also be contrasted to the short-lived states in opticalcavities: with a frequency of several hundreds of THz and a state-of-the-art optical Q of 10 , the photon lifetime is afew ns. Also in this case the mechanical states can be used as storage or delay unit for quantum information if e ffi cientexchange of photons and phonons can be achieved (i.e., strong coupling). From a more general point of view and as a cleardirection for the future, one can envision the construction of hybrid systems involving quantum mechanical oscillatorsthat exploit and combine the strengths of the individual quantum systems. The mechanical quantum resonators wouldthen serve as reservoirs of long-lived states with the advantage that they can be well coupled to a variety of other quantumsystems.As a final note, let us return to the crossover between quantum and classical systems as discussed in the Introduction.The quantum experiments of Refs. [22, 23] were performed on mechanical oscillators with a mass of 3 ng and 48 pgrespectively. This means that a mechanical object consisting of about 10 atoms can still behave quantum mechanically.The crossover has thus not been reached in experiments and continues therefore to be a subject of future studies. Againthis aspect shows that the field of mechanical systems in the quantum regime is still largely unexplored: It will still rapidlydevelop in the years to come bringing many exciting new experiments and discoveries.58 . Acknowledgements We thank Andreas H¨uttel and Daniel Schmid for their suggestions about the manuscript. We are indebted to SamirEtaki, Benoit Witkamp, Yaroslav Blanter, Francois Konschelle, Miles Blencowe, Jack Harris, and Hong Tang for thediscussions on a wide range of the topics covered in this Report, and for their suggested improvements. This work wassupported by FOM, NWO (VICI grant), NanoNed, and the EU FP7 STREP projects QNEMS and RODIN.59 . Complex Green’s function and displacement
One is often interested in knowing the amplitude A ( t ) and phase ϕ ( t ) of the displacement when it is written as u ( t ) = A ( t ) cos( ω R t + ϕ ( t )). Since this is only one equation for two functions, the amplitude and phase are not uniquely defined.Consider, for example, the solution A ( t ) = u ( t ) / cos( ω R t ) and ϕ ( t ) =
0. Already for a pure sine wave u ( t ) = sin ( ω R t ) thisresults in a rapidly varying amplitude. The usual notion of the amplitude and phase are that when the signal is close tosinusoidal, the amplitude and phase are that of the sine wave. This can be realized using the complex displacement, whichis defined as u c ( t ) ≡ A c exp( i ω R t ) with the requirement that Re[ u c ( t )] = u ( t ). A convenient way of implementing this, isusing the complex extension of the resonator Green’s function, cf. Eq. 47: u c ( t ) = h c ( t ) ⊗ F ( t ) / k R , h c ( t ) = − ie i ω R t · e − ω Rt Q Θ ( t ) , (A.1)so that Re[ h c ] = h HO . The amplitude and phase of the resonator displacement are in this case given by the modulus andargument of the complex amplitude: A ( t ) = | A c | and ϕ ( t ) = ∠ A c respectively.Another notion that is often used in the literature is that of quadratures. Now the displacement is written as u ( t ) = U ( t ) cos( ω R t ) + P ( t ) sin( ω R t ). Here U is the in-phase component and P is the out-of-phase component (sometimes calledquadrature). The quadrature representation is related to the amplitude-and-phase representation by: U = A cos ϕ , P = A sin ϕ , and by A = U + P , tan ϕ = P / U . The quadratures are also readily calculated using the complex Green’sfunction: U = Re[ u c ] and P = Im[ u c ]. Finally, we note that if the displacement is in the in-phase quadrature, i.e. u ( t ) = A cos( ω R t ), then the velocity d u / d t ≈ − ω R A sin( ω R t ) is in the out-of-phase quadrature. B. Optimal filtering of v ( t ) In the presence of both position (imprecision) and force noise, one wants to reconstruct the resonator motion in theabsence of the detector, u i ( t ) = h R ( t ) ⊗ F n ( t ), as good as possible from the measured time trace v ( t ) of the detector output.This is done by finding the estimator ˆ u = g ( t ) ⊗ v ( t ) that minimizes the resolution squared: ∆ u = E [( u i − ˆ u ) ]. Usingthe autocorrelation functions and converting these into noise PSDs using the Wiener-Khinchin theorem [200, 207] theresolution is written as: ∆ u = R u i u i (0) − R u i ˆ u (0) + R ˆ u ˆ u (0) = π Z ∞−∞ h S u i u i ( ω ) − G ( ω ) S u i v ( ω ) + | G ( ω ) | S vv i d ω. (B.1)Minimizing this w.r.t. G , yields the optimal filter G opt = S vu i / S vv [200], where: S vu i = S ∗ u i v = AH R ( H ′ R ) ∗ λ ∗ v S F n F n , (B.2) S vv = S ∗ vv = A | H ′ R | | λ v | (cid:16) S F n F n + A S Φ det , n Φ det , n (cid:17) + S v n v n + A Re h λ v ( H ′ R ) ∗ S Φ det , n v n i , (B.3)so that the squared resolution is given by: ∆ u = π Z ∞ h S u i u i ( ω ) − | G opt ( ω ) | S vv i d ω. (B.4)Depending on the properties of the detector and the coupling A , two important limits can be distinguished: • The detector exerts backaction force noise, but the displacement noise is negligible: S v n v n =
0. In this case theintegral in Eq. B.4 is easily solved and one finds: ∆ u BA = h u i i · + S F n F n A S Φ det , n Φ det , n − = h u i i · + S F n F n S F BA , n F BA , n − . (B.5)The resolution thus increases (gets worse) with increasing A as the influence of the detector, i.e. the force noise F BA , n ( t ), on the resonator motion grows. For small values of A this goes as ∆ u BA ∝ A . • The detector adds displacement imprecision noise whereas the backaction noise is very small, i.e. S Φ det , n Φ det , n = ∆ u n = h u i i · J (SNR , Q ) , with SNR = Q k R S F n F n S u n u n , (B.6)60 -2 -3 -2 -1 frequency S u u n n f /Q R Δ u n J ( S NR , Q ) S vvv / | A | λ SNR Q = Q = (a) (b) Q = Figure B.37: Plots of the function J (SNR , Q ) for three di ff erent values of Q . The right side of the plot where SNR ≫ ≪
1) themechanical signal is buried in the imprecision noise and the error is as large as the resonator signal giving J =
1. (b) An alternative definition of ∆ u n (cf. Eq. B.7) is the area of the gray rectangle indicated in the noise spectrum. under the assumption that the detector noise PSD referred to the detector input S u n u n = S v n v n / A | λ v | is white. J (SNR , Q ) is a function (see the plots in Fig. B.37a) that depends on the quality factor and the signal-to-noise ratio,SNR. J tends to 1 when the signal-to-noise ratio is well below unity. In that case, the signal contains so much noisethat it hardly contains information about the displacement and using the signal v ( t ) is not going to give much moreinformation then just assuming that the resonator is at u =
0. The average error that is made in the latter case is h u i i / .When SNR → ∞ the function J (SNR , Q ) goes to zero as Q / (2 / SNR / ). The resolution improves with increasingcoupling as the resonator signal is amplified more and more with respect to the noise floor S v n v n .For practical purposes it is convenient to use a slightly di ff erent definition of the resolution [291, 30, 19] that doesnot involve the signal-to-noise ratio dependent estimator: ∆ u n ≡ S u n u n π f R Q = h u i i SNR , (B.7)which is readily extracted from the measured noise spectra as indicated in Fig. B.37b. This definition is basedon the fact that one can measure the position during a time ∼ Q / f R before the resonator has forgotten its initialamplitude and phase (see also Fig. 15). Note again, that this is the imprecision noise of the detector which does nottake the e ff ect of backaction noise into account. C. Square-law detection
In a u -squared detector the output of the detector depends quadratically on the displacement. In analogy with Eq. 60the detector output is now v ( t ) = A λ (2) v ( t ) ⊗ u ( t ) + v (2) n ( t ), where the superscript indicates that this is the response of thesquare-law detector. A resonator oscillating with frequency f R , results in a detector output with frequency components at2 f R and at dc. As with every detector, the output also contains imprecision noise, v (2) n . Again, by increasing the couplingstrength A the signal-to-noise ratio can be improved. In Sec. 3.4 it was shown that for a linear detector this leads to anincreased backaction on the resonator, so it is interesting to see if the square-law detector has backaction. The backactionof a detector can vanish when the commutator of the quantity that is measured ( ˆ u = u (ˆ a † + ˆ a ) ) at di ff erent times is zero[4]. In the Heisenberg representation [2] we have [ ˆ u ( t ) , ˆ u ( t )] = iu sin( ω R { t − t } ) · ( ˆ u ( t ) ˆ u ( t ) + ˆ u ( t ) ˆ u ( t )) , f R , but only the dc component (see e.g. Ref. [81]). The operator corresponding to the u -detection can bewritten as ˆ u = u (ˆ a + ˆ a † + ˆ a ˆ a † + ˆ a † ˆ a ). The first two terms oscillate at twice the resonator frequency and when theyare discarded, one finds that this detector detects ˆ a ˆ a † + ˆ a † ˆ a = n +
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