Squeezing-enhanced Raman spectroscopy
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Squeezing-enhanced Raman spectroscopy
Yoad Michael , Leon Bello , Michael Rosenbluh and Avi Pe ’ er * The sensitivity of classical Raman spectroscopy methods, such as coherent anti-stokes Raman spectroscopy (CARS) or stimulatedRaman spectroscopy (SRS), is ultimately limited by shot-noise from the stimulating fi elds. We present the complete theoreticalanalysis of a squeezing-enhanced version of Raman spectroscopy that overcomes the shot-noise limit of sensitivity withenhancement of the Raman signal and inherent background suppression, while remaining fully compatible with standard Ramanspectroscopy methods. By incorporating the Raman sample between two phase-sensitive parametric ampli fi ers that squeeze thelight along orthogonal quadrature axes, the typical intensity measurement of the Raman response is converted into a quantum-limited, super-sensitive estimation of phase. The resonant Raman response in the sample induces a phase shift to signal-idlerfrequency-pairs within the fi ngerprint spectrum of the molecule, resulting in ampli fi cation of the resonant Raman signal by thesqueezing factor of the parametric ampli fi ers, whereas the non-resonant background is annihilated by destructive interference.Seeding the interferometer with classical coherent light stimulates the Raman signal further without increasing the background,effectively forming squeezing-enhanced versions of CARS and SRS, where the quantum enhancement is achieved on top of theclassical stimulation. npj Quantum Information (2019) 5:81 ; https://doi.org/10.1038/s41534-019-0197-0 INTRODUCTION
Quantum-enhanced measurements utilize the unique correlationproperties of non-classical light for highly sensitive detection.Common examples include NOON and squeezing-based inter-ferometers that employ entangled quantum states to achieve sub-shot-noise phase sensitivity. This enhancement can be useful formeasurements of extremely weak signals, with a crowningexample being the detection of gravitational waves. A major fi eld that can greatly bene fi t from sub-shot-noise detection isRaman spectroscopy, which is widely used for chemical sensing, – due to its ability to identify the molecular contents of a samplebased on its Raman fi ngerprint spectrum. Raman spectroscopy is,therefore, an ideal contrasting method for chemically-resolvedmicroscopy with no prior preparation or fl uorescent tagging ofthe target molecule required. However, the major challenge forRaman sensing is the relative weakness of the Raman response,which is orders of magnitude weaker than fl uorescence, and mayoften be obscured by the shot-noise of other stimulated light-matter interactions.In coherent anti-stokes Raman spectroscopy (CARS), a Ramansample is excited by a strong pump wave (frequency ω p ) and aStokes wave (idler, frequency ω i ) that interact within the sample togenerate an anti-Stokes (signal) wave at frequency ω s = ω p − ω i via four-wave mixing (FWM). When the frequency differencebetween the pump and Stokes fi eld matches a molecularvibration/rotation in the sample, the generated anti-Stokes fi eldis resonantly enhanced, indicating that the Raman frequency shiftof the signal (with respect to the pump) acts as a molecular fi ngerprint. However, since FWM is a parametric process, non-resonant FWM can also occur via virtual levels, resulting in a non-resonant background that is not chemically speci fi c. In dilutedsamples, where the target molecule is surrounded by largequantities of background molecules (e.g., a protein dissolved inwater within a biological cell), the non-resonant background fromthe environment (water) can become a major limiting factor to the sensitivity of measurement, since it dominates over and obscuresthe weak resonant Raman signal from the target molecule(protein). The fundamental limit to the sensitivity of standardCARS is, therefore, the noise associated with the non-resonantbackground, indicating that suppression of this background is amajor goal for CARS spectroscopy, and several methods havebeen proposed in past research to address it: Pulse shaping wasapplied to reduce the peak power of the exciting pulses (andhence the non-resonant background), epi-CARS that detects onlythe back-scattered Raman signal (which is primarily resonant),and polarization CARS that rejects the non-resonant signal basedon polarization. All these methods rely on some speci fi cproperty of the sample/light to suppress the background, andeven with ideal suppression, all classical detection methods areultimately limited by shot-noise. In stimulated Raman spectro-scopy (SRS), only the resonant Raman response is observed,through sensitive measurement of the weak nonlinear gain thatthe Stokes fi eld experiences in the presence of a strong pump.While SRS is free of the non-resonant background, its sensitivity isfundamentally limited by the shot-noise of the coherentStokes seed.Here, we propose and present the theoretical analysis of asample-independent con fi guration for measurement of theresonant Raman response beyond the shot-noise limit, byrecasting the typical measurement of the Raman intensity into aquantum-enhanced estimation of the nonlinear Raman phase. Inour proposal, we place the Raman sample between two phase-sensitive optical parametric ampli fi ers (OPAs) — forming a non-linear version of the well-known SU(1,1) interferometer (Fig. 1a),which employs the squeezing of the OPAs to measure aninduced optical phase shift beyond the shot-noise limit (see a briefoverview of the SU(1,1) interferometer in the Methods andcomprehensive reviews in ref. ). By probing the Ramaninteraction in the sample with broadband two-mode squeezedlight (see Fig. 1b), generated via FWM in the OPAs, the resonant Department of Physics and BINA Center for Nanotechnology, Bar-Ilan University, Ramat Gan 5290002, Israel. *email: [email protected]
Published in partnership with The University of New South Wales () :,; ignal of the sample is enhanced by the squeezing ratio of theOPAs, while the non-resonant background is completely elimi-nated by destructive interference due to the inherent π /2difference in the phase response between the resonant andnon-resonant interactions. This con fi guration effectively forms aquantum-enhanced version of interferometric CARS. Our theoretical analysis of this squeezing-enhanced Ramanspectroscopy method is organized as follows: Section “ Squeezing-enhancement of spontaneous Raman spectroscopy ” highlightsthe squeezing enhancement of the resonant Raman signal byanalysis of a resonant Raman sample placed inside a lossless SU(1,1) interferometer. Section “ Complete suppression of the non-resonant background ” analyses the additional effect of a non-resonant interaction in the Raman sample, by distinguishingbetween the non-resonant background and the resonant signal,which yields complete elimination of the background due to theinherent ± π /2 phase difference between the resonant and non-resonant interactions. Section “ Coherent seeding of the inter-ferometer: squeezing-enhanced CARS and SRS ” introduces coher-ent seeding of the interferometer by classical fi elds, showing thatthe squeezing enhancement of section “ Squeezing-enhancementof spontaneous Raman spectroscopy ” can be added on top of theclassical stimulation of the Raman interaction. This squeezing-enhancement can be applied to any classical Raman method, suchas CARS and SRS. Finally, section “ Detection sensitivity in thepresence of loss ” incorporates loss into the analysis, both internaland external to the interferometer, which is a critical considerationfor experimental realizations of any squeezing application,showing that the scheme maintains its sub-shot-noise sensitivityeven with practical levels of loss, as long as the phase of OPA2 isappropriately tuned. RESULTS
Squeezing-enhancement of spontaneous Raman spectroscopyThe Raman sample, which can be considered as a weak parametricampli fi er via Raman-based FWM, is placed inside a nonlinearinterferometer that is composed of two external OPAs (see Fig. 2afor illustration), where each OPA ampli fi es one quadrature (andattenuates the other quadrature) of the two-mode signal-idler fi eld. Using the fi eld operators of the signal and idler ^ a s ; i , we cande fi ne the quadratures of the two-mode fi eld as: ^ X ¼ ^ a s þ ^ a y i and ^ Y ¼ i ð ^ a y s (cid:2) ^ a i Þ (See Methods and ref. for additional informa-tion). The two-mode quadratures represent the cosine and sineoscillation components of the combined signal-idler fi eld andform a pair of conjugate variables, which obey the standardcommutation relation ^ X ; ^ Y (cid:1) (cid:3) ¼ i . As previously stated, an OPAampli fi es one quadrature ^ X ! ^ Xe G and attenuates the other ^ Y ! ^ Ye (cid:2) G , and the quadratures evolve as shown in Fig. 2 for eachampli fi er. The two OPAs are arranged in a “ crossed ” con fi guration,where the quadrature attenuation axis of OPA2 matches theampli fi cation axis OPA1 and vice-versa, such that OPA2 exactlyreverses the squeezing of OPA1 (equal gain, see Fig. 2b). Theorientation of the ampli fi cation/attenuation axes of the OPAs (andthe Raman sample) is determined by the relative phase ϕ r = ϕ p − ( ϕ i + ϕ s ) between the pump and signal-idler fi elds. This relativephase is a tunable parameter that can be controlled by adjustingthe phase of either the pump, signal, or idler (for example, byadjusting the path of the beams). The squeezed signal-idler pairsgenerated in OPA1 (with a relative phase ϕ r =
0) interactparametrically with the Raman sample at an intermediate relativephase ϕ r = π /2. In practice, this relative phase could be achievedby changing the optical path of the beams, but if OPA1 and OPA2generate non-resonant light (for example, by using PhotonicCrystal Fibers ), then the resonant Raman ampli fi cation of thesample will occur at ϕ r = π /2 with no additional phase adjust-ments required (see section “ Complete suppression of the non-resonant background ” hereon for further discussions about theresonant and non-resonant interactions). This relative phase istranslated to an ampli fi cation axis of 45° with respect to OPA1,which in-turn rotates the squeezing ellipse. OPA2 then ampli fi es atan opposite relative phase ϕ r = π , which without a Raman samplewill completely reconvert the signal-idler pairs back to pump light(leaving a vacuum output). In the presence of a Raman sample,the cancellation is incomplete, elevating the signal intensity at theoutput above zero (from its initial vacuum state).Let us fi rst calculate the light intensity (number of photons) atthe output of this squeezing-enhanced Raman con fi guration.When passing through the ampli fi ers (and the sample), we can usethe input-output relations of the fi eld operators at each Fig. 1
Squeezing-enhanced Raman detection in an SU(1,1) interferometer. a The standard SU(1,1) interferometric detection of a linear phase:Two phase-sensitive OPAs of equal gain and opposite pump phases are arranged in series (balanced con fi guration). OPA1 ampli fi es signal-idler pairs at the input (vacuum), while OPA2 is shifted in pump phase to exactly reverse the ampli fi cation of OPA1 and return the light back toits original input state. When a linear phase shift θ is introduced, the cross-cancellation of the ampli fi ers is no longer complete, and additionallight is detected at the output. This light directly corresponds to the induced phase shift, which is detected with sub-shot-noise sensitivity dueto the squeezing of the OPAs. b Our scheme for squeezing-enhanced Raman detection: The Raman sample is introduced into a balanced SU(1,1) interferometer. The sample acts as a phase-sensitive parametric ampli fi er that induces a phase shift into the interferometer, and thereforegenerates some light at the (previously) dark output, as explained in the text Y. Michael et al. npj Quantum Information (2019) 81 Published in partnership with The University of New South Wales () :,; mpli fi er: ^ a ð Þ s ; i ¼ A ^ a ð Þ s ; i þ B ^ a yð Þ i ; s ; (1) ^ a ð Þ s ; i ¼ C ^ a ð Þ s ; i e i π = þ D ^ a yð Þ i ; s e (cid:2) i π = ; (2) ^ a ð Þ s ; i ¼ A ^ a ð Þ s ; i e i π = þ B ^ a yð Þ i ; s e (cid:2) i π = ; (3)where A and B represent the operation of the OPAs: A = cosh( G )and B = sinh( G ), with G the gain of the ampli fi ers, and C and D areassociated with the Raman sample, C = cosh( g r ) and D = sinh( g r )with g r the gain of the Raman sample (assumed for now to bepurely resonant and narrowband). The relative phase between thepump, signal and idler varies throughout the different parametricampli fi ers, and is adjusted such that the two external OPAs areorthogonal to each other, with the Raman sample set to anintermediate phase of ϕ r = π /2. Thus, we apply a π /4 phase toboth the signal and idler fi elds (accumulating to ϕ r = π /2) twice: fi rst after OPA1, which rotates the quadrature ampli fi cation axis ofthe Raman sample to 45°, and a second time after the sample,which sets OPA2 orthogonal to OPA1. The total accumulatedrelative phase for the FWM light before OPA2 is, therefore, ϕ r = π ,indicating that in the absence of a Raman sample ( C = D = ^ N ð Þ s D E ¼ j D j ðj A j þ j B j þ j A jj B jÞ¼ sinh ð g r Þ cosh ð G Þ : (4)Equation (4) shows the signal at the output of the interferom-eter, and highlights the increase in the overall signal due to thestimulation of the Raman sample by the two-mode squeezed lightgenerated in OPA1, whose noise is then eliminated by OPA2. We Fig. 2
Overview of the signal-idler two-mode quadratures within the squeezing-enhanced Raman con fi guration. a Conceptual illustration ofthe scheme, where photonic crystal fi bers represent the OPAs. b , c Show the two-mode quadrature dynamics of the light throughout thedifferent ampli fi ers for resonant and non-resonant interactions respectively. The top line of graphs in b , c portrays the quadrature map of thetwo-mode light at various locations along the interferometer, and the bottom part shows the ampli fi cation/attenuation axes of each ampli fi er(indicated by arrows pointing outwards/inwards). OPA1 and OPA2 are set in a crossed (orthogonal) con fi guration, and are assumed to be idealsqueezers. In the resonant case b the Raman sample is set to a relative phase of ϕ r = π /2 (45° ampli fi cation axis), which rotates the squeezing-ellipse of the two-mode light, resulting in non-vacuum output. The transparent ellipses/circles illustrate the state of the light in the absence ofa sample, where the output returns exactly to the input state (vacuum). c For a non-resonant Raman sample, the relative phase is ϕ r = fi cation axis of the non-resonant sample is aligned with that of OPA1, and the two-mode ellipse is not rotated, butrather further squeezed in the same direction. The non-resonant gain g nr unbalances the interferometer by effectively increasing/decreasingthe gain of OPA1 G , which can be negated by tuning the gain of OPA2 to G = G + g nr Y. Michael et al. Published in partnership with The University of New South Wales npj Quantum Information (2019) 81 an compare the result of Eq. (4) to that of spontaneous Ramanemission: ^ N ð Þ s D E ¼ sinh ð g r Þ , which corresponds to setting theOPA gain to zero G =
0. Thus, Eq. (4) illustrates the couplingbetween the parameters of the sample ( C , D , g r ) and theparameters of the crossed ampli fi ers ( A , B , G ), creating an effectivesqueezing-enhanced signal, but with no added background (thenon-resonant background can be eliminated by tuning the gain ofthe external OPAs, as we explain in the next section). Thus,although the sample is stimulated by the FWM light of OPA1, theinterferometer conceals this stimulation completely, acting as aneffective black-box, which to an external observer appears just likea spontaneous Raman scatterer, but with a higher effective(squeezing-enhanced) signal. An analogy from the fi eld ofinterferometry can be given here: In standard Mach-Zender SU(2) interferometers, the sensitivity of the interferometer is limitedby shot-noise, but can be improved by seeding the unused port ofthe interferometer with squeezed light, resulting in “ squeezing-enhanced ” estimation of phase. In the scheme proposed here, weplace the Raman sample between two-mode quadrature-squeez-ing OPAs: OPA1 generates squeezed light that stimulates theRaman interaction in the sample, while OPA2 attempts tounsqueeze the light back into vacuum. The overall result is a “ squeezing-enhanced ” detection of the Raman signal. As will beexplained in section “ Coherent seeding of the interferometer:squeezing-enhanced CARS and SRS ” , this effective Raman black-box remains applicable also for stimulated Raman techniques,such as CARS and SRS, by seeding the interferometer with acoherent idler (rather than vacuum).It should also be noted that due to dispersion, the Ramansample will also induce a linear phase shift which is not Raman-speci fi c. Control over this phase is possible using other dispersivemedia, as shown in ref. effectively negating it.Complete suppression of the non-resonant backgroundIn the previous section, we considered only the contribution of theresonant Raman process to the output intensity. The Ramanresponse of the sample is generally complex with respect to thepump drive, and its phase varies spectrally across the resonancefrom ϕ r = ϕ r = π /2 on resonance to ϕ r = π above resonance (not to be confused with the phases ofthe interferometer). The imaginary part of the Raman response isassociated with the Raman absorption/gain, which is maximal onresonance, whereas the real part of the response is associatedwith dispersion, which is nulled on resonance, just like a driventwo-level system. Thus, we should treat the Raman sample as aparametric ampli fi er capable of two separate ampli fi cationssimultaneously: First, the resonant, phase-shifting ampli fi cationat ϕ r = π /2 (discussed thus far) and an additional non-resonant,non-phase-shifting ampli fi cation at ϕ r = ϕ r = π aboveresonance). In the two-mode quadrature representation, thisampli fi cation occurs “ on-axis ” with the ampli fi cation axis of OPA1,and acts as direct extension to the gain of OPA1 (see Fig. 2c) — itampli fi es the same quadrature (the same relative phase) as OPA1,indicating that the non-resonant contribution can be completelynulled by varying the gain of the OPAs. This concept is similar tothe experiments of Lupke and Lee, where the non-resonantbackground of the sample was canceled by placing an additional,non-Raman FWM medium, whose non-resonant backgrounddestructively interfered with the background from the sample.The scheme proposed here bene fi ts from the backgroundcancellation in combination with the inherent squeezing of theexternal OPAs to enhance the Raman signal.To observe this background cancellation mathematically, let us fi rst consider a purely non-resonant sample: after passing throughOPA1 (with gain G ), the sample performs non-resonant ampli fi ca-tion with gain g nr and at a relative phase ϕ r =
0. This phase isidentical to that of OPA1, which effectively extends the same ampli fi cation process, and in the quadrature picture, performs on-axis ampli fi cation (see Fig. 2c). OPA2 is set as before to negate theampli fi cation of OPA1, this time with different gain G ≠ G . Thenumber of photons at the output is then: ^ N ð Þ s D E nr ¼ sinh ð G þ g nr (cid:2) G Þ : (5)Setting the gain of OPA2 to G = G + g nr will null the non-resonant output completely.In practice, real Raman samples will have both resonant Ramangain from the molecule of interest and non-resonant gain frombackground molecules (e.g., solvent). For simplicity, we assumehere that the Raman sample is composed of a target substancethat produces a resonant Raman signal, surrounded by a uniformmedium whose vibrational resonances do not overlap with that ofthe target molecule, such that it produces a non-resonantbackground (in our example, below the resonance). We can thinkof the sample as a mixture of many in fi nitesimal parametricampli fi ers that perform either resonant ampli fi cation at ϕ r = π /2 ornon-resonant ampli fi cation at ϕ r = g nr , g r << G ), it is fair to neglect the cross-interaction between the resonant and non-resonant gains withinthe sample ð sinh ð g r Þ sinh ð g nr ÞÞ compared to the interaction ofthe sample with the external OPAs. Speci fi cally, we can see from Eq.(5) that the two extreme possibilities of ordering the gain in thesample - either setting the non-resonant gain entirely before theresonant gain (adding slightly to the gain of OPA1 G → G + g nr ) orentirely after the resonant gain (decreasing slightly G → G − g nr ),both lead to the same gain balancing outcome — increase G tocompensate for g nr . Thus, the effect of ordering within the samplemust be of higher order, and we may neglect it for weak samples.We, therefore, treat the sample as two separate parametricampli fi ers (resonant and non-resonant) that are placed in series,which yields the following photon number at the output: ^ N ð Þ s D E ¼ sinh ð g r Þ cosh ð G Þ : (6)The result of Eq. (6) remains similar to Eq. (4) even in thepresence of the non-resonant background, which is now fullysuppressed, indicating that the interferometric scheme stillbehaves as a spontaneous squeezing-enhanced Raman black-box. Based on this result, we can assume in the upcoming sectionsthat the non-resonant contribution is canceled by gain balancing.Additional information about the unbalanced interferometer isprovided in the Methods.Coherent seeding of the interferometer: squeezing-enhancedCARS and SRSLet us now examine the CARS response of our squeezing-enhancedRaman con fi guration from section “ Squeezing-enhancement ofspontaneous Raman spectroscopy ” by subjecting it to a strongcoherent idler input. Note fi rst that by simple extension of thetreatment in section “ Complete suppression of the non-resonantbackground ” , we may assume that the non-resonant background issuppressed by gain balancing of the external OPAs, and consideronly the resonant response (the stimulation affects both the OPAsand the sample in the same manner, leaving the gain balancingunchanged). This assumption holds true regardless of the inputstate of the signal or idler. The output photon number of the signal ^ N ð Þ s D E for the seeded con fi guration |0 s , α i 〉 (input of vacuum signal,strong coherent state idler) is then (see Methods for derivation) ^ N ð Þ s D E ¼ ð þ j α i j Þ cosh ð G Þ sinh ð g r Þ ; (7)which is directly equivalent to the expression of standard CARSwith the additional enhancement of the Raman signal due to the Y. Michael et al. npj Quantum Information (2019) 81 Published in partnership with The University of New South Wales queezing inside the interferometer, and with full inherentsuppression of the non-resonant background.To consider our scheme for SRS, let us examine the increase inintensity (photon number) of the idler beam: ^ N ð Þ i D E (cid:2) j α i j ¼ ð þ j α i j Þð þ cosh ð G ÞÞ sinh ð g r Þ¼ ð þ j α i j Þ cosh ð G Þ sinh ð g r Þ : (8)In contrast to CARS, SRS requires lock-in detection to separatethe Raman signal from the coherent idler input, but in principal,both methods bene fi t from the squeezing-enhancement of thesignal due to the OPAs.Consequently, the ‘ Raman black-box ’ concept, which claims thatthe interferometer can behave exactly like a normal Ramansample but with an enhanced signal due to the squeezing,extends also to stimulated interactions. The squeezing effect onthe sample appears only internally between the crossed OPAs,and an external observer will not be able to differentiate the idealinterferometer con fi guration from a simple, high-signal resonantRaman sample (unless he can ‘ look inside the box ’ ).Detection sensitivity in the presence of lossLet us now evaluate the sensitivity of measurement for oursuggested scheme under practical conditions by calculating theminimum detectable resonant Raman gain g min (cid:4) (cid:5) of the sample inthe presence of photon-loss. This sensitivity can be calculated byerror-propagation analysis: g min ¼ ^ N s (cid:6) (cid:7) (cid:2) ^ N s (cid:6) (cid:7) ddg ^ N s (cid:6) (cid:7) j g ¼ (cid:8)(cid:8)(cid:8) (cid:8)(cid:8)(cid:8) ; (9)which states that the variation of the signal must be comparableto the noise of the output intensity.For the ideal seeded con fi guration, where no losses are present,the sensitivity is (see Methods for derivation): g min ¼ ð G Þð þ j α i j Þ ; (10)which is similar to the minimum detectable phase of the standardSU(1,1) interferometer, and indicates the detection of a singleoutput photon during the fi nite measurement time.Both internal loss (between the two OPAs) and external/detection loss (after the interferometer) affect the sensitivity ofmeasurement, although in a different manner. External loss takesplace after the nonlinear interference and does not affect thesqueezing, thereby reducing the measured signal by a loss factor | r ext | , identical to the effect of losses on classical light. Internalloss, on the other hand, hinders the quantum correlationsbetween the signal and idler, effectively diminishing the squeez-ing, which in-turn degrades the contrast of the nonlinearinterference and elevates the dark fringe level (along with itsassociated noise), resulting in a lower detection sensitivity.Let us calculate the average photon number and the noiseassociated with the dark fringe (background) due to internal lossesfor exactly crossed OPAs. We use the standard modeling of loss asa beam-splitter (BS) placed inside the interferometer, wherevacuum may enter through the unused port of the BS. We applythe BS loss to the ideal squeezing-enhanced Raman scheme,seeded with a strong (classical) coherent state | α i 〉 for the idler,and obtain for the resonant signal: ^ N ð Þ s D E ¼ j t j ð þ j α i j Þ sinh ð g r Þ cosh ð G Þ þ j r j sinh ð G Þ ; (11)where r represents the loss and t the transmission inside theinterferometer (| r | + | t | = fi cient | t | , and the loss term (right, proportionalto | r | ) which corresponds to ampli fi cation of the vacuum byOPA2, and causes direct elevation of the dark fringe level. Sincethis term does not depend on the Raman sample, it limits thesensitivity of the measurement. The noise associated with thebackground due to loss is: σ loss ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ N s (cid:6) (cid:7) loss (cid:2) ^ N s (cid:6) (cid:7) loss q ¼ j r j sinh ð G Þ : (12)Equation (12) represents the background noise of the darkfringe, which limits the ability to detect the small Raman signal.Note that plugging Eq. (12) and the derivative of Eq. (11) intoEq. (9) results in a diverging expression for the sensitivity g min forexactly crossed ampli fi ers ( ϕ r = π ). Thus, the optimal workingpoint of the interferometer (de fi ned as the relative phase betweenthe OPAs where the sensitivity is optimal) may vary with theinternal loss. This behavior is shown in Fig. 3a, which displays g min as a function of the phase of OPA2 ( ϕ , where ϕ r = π + ϕ ) forvarious loss values. For no loss, the optimal working point (de fi nedas the value of ϕ which bestows a minimum on g min ), is ϕ opt = ϕ opt increases due to thedark fringe noise. However, the squeezing enhancement stillimproves the minimum detectable gain below the shot-noise limiteven in the presence of considerable losses, as long as the phaseof OPA2 is tuned appropriately. Figure 3b introduces coherentseeding of the idler port for various intensities (number ofphotons) with constant 20% loss. The stimulation partiallycompensates for the effect of loss, improving the sensitivity plotin two ways: First, It pushes the optimal phase ϕ opt towards theideal case ϕ opt =
0, and also reduces the minimum detectable gain g min . This improvement occurs because the seeding alsostimulates the external OPAs to generate more signal-idler photonpairs, thereby increasing the total number of correlated photonsthat probe the sample.Note that in our analysis, we mostly focused on theinterferometer itself and on the way it interacts with the Ramansample, using the most straightforward detection — direct intensitymeasurements at the signal port. It is also important to considerother detection methods, such as: parity detection which isideal for loss-free schemes, homodyne measurement whichachieves higher sensitivity for the seeded interferometer, para-metric homodyne which is optically broadband and robust todetection losses, or a truncated interferometer which may bepreferred if the parametric ampli fi ers are lossy. In the Methods, wealso discuss measurement at the idler port using lock-in detection(squeezing-enhanced version of SRS). The calculations can also becarried out using Fisher information analysis, which extractsthe phase from positive-operator valued measure, rather thanthrough the error-propagation method we discussed here. DISCUSSION
We presented a new method for Raman spectroscopy, whichutilizes the squeezed light inside a nonlinear interferometer toenhance the resonant Raman signal and to suppress the non-resonant background in the most general Raman sample. Thiscon fi guration can be considered as a “ Raman black box ” with asqueezing-enhanced signal that is compatible with any standardRaman technique — spontaneous or stimulated, CARS or SRS.Being a “ black box ” , it is applicable in combination with any otherclassical interference scheme that may be used to enhance thedetection or to fi lter out noise as is the case in many variations ofRaman spectroscopy. The generality of this squeezing-enhancement along with its resilience to loss and to experimentalerrors, deem it highly applicable for any fi eld where spectroscopicdetection of trace-chemicals is needed. Y. Michael et al. Published in partnership with The University of New South Wales npj Quantum Information (2019) 81
ETHODS
In the following section, we fi rst provide a short overview of the two-modequadratures discussed in the text. We then review the SU(1,1) inter-ferometer, and derive its squeezing-enhancement of the measurement ofa linear optical phase. We then derive in more detail the major quantumexpressions presented in the article: the average photon number at thesignal output and its fl uctuations (noise), the minimum detectable gainwith coherent seeding of the interferometer, and with internal loss. Toanalyze the robustness of the proposed Raman interferometer toexperimental imperfections, we provide additional results for non-idealcon fi gurations, such as unbalanced interferometric detection or inaccurateorientation (phase) of the Raman sample. Two-mode quadratures
The basic description of the signal-idler oscillation consists of two fi eldswith frequencies ω s , i where Ω ¼ ω s þ ω i is the center frequency (and, in FWM, the frequency of the pump) and ω ¼ ω s (cid:2) ω i is the frequency separation. Westart from the electric fi eld operator of a two-mode (signal and idler)oscillation: E ð t Þ ¼ a s e (cid:2) i ω s t þ a i e (cid:2) i ω i t þ c : c ; (13)where a s , i are the annihilation operators of the signal/idler modes atfrequencies ω s , i . The standard procedure is to rewrite the electric fi eld insingle-mode quadrature components using the standard de fi nition ofsingle-mode quadrature operators for the signal/idler modes X s ; i ¼ a s ; i þ a y s ; i and Y s ; i ¼ i ð a s ; i (cid:2) a y s ; i Þ , which de fi nes the quadratures of each mode,yielding E ð t Þ ¼ X s cos ð ω s t Þ þ Y s sin ð ω s t Þþ X i cos ð ω s t Þ þ Y i sin ð ω i t Þ ; (14)Since the signal and idler are generated symmetrically around thedegenerate carrier frequency Ω (the pump frequency in FWM), theirfrequencies can be written as ω s = Ω + ω and ω i = Ω − ω , which allows torearrange the electric fi eld of Eq. (14), using simple trigonometricmanipulations: E ð t Þ ¼ ½ð X s þ X i Þ cos ð ω t Þ þ ð Y s (cid:2) Y i Þ sin ð ω t Þ(cid:3) cos ð Ω t Þþ ½ð Y s þ Y i Þ cos ð ω t Þ þ ð X i (cid:2) X s Þ sin ð ω t Þ(cid:3) sin ð Ω t Þ (15)We can now identify: X M ð t Þ (cid:4) ð X s þ X i Þ cos ð ω t Þ þ ð Y s (cid:2) Y i Þ sin ð ω t Þ ; (16) Y M ð t Þ (cid:4) ð Y s þ Y i Þ cos ð ω t Þ þ ð X i (cid:2) X s Þ sin ð ω t Þ ; (17)as the cosine and sine quadrature components of the combined two-modeelectric fi eld, but now with respect to the common local oscillator at thecarrier frequency Ω , just like the standard case of a single-mode fi eld.Speci fi cally, when the intensity difference ( X i − X s ) and phase sum ( Y s + Y i )are squeezed, this is equivalent squeezing of one two-mode quadrature,just like single-mode squeezing with the same squeezing ellipse. The onlydifference compared to the standard degenerate squeezing is that thetwo-mode quadratures in this de fi nition are explicitly time-dependent.We can now re-insert the single-mode quadratures X s ; i ¼ a s ; i þ a y s ; i and Y s ; i ¼ i ð a s ; i (cid:2) a y s ; i Þ into Eqs (16) and (17) to obtain: X M ð t Þ ¼ ð a s þ a y i Þ e (cid:2) i ω t þ c : c ; (18) Y M ð t Þ ¼ i ð a y s (cid:2) a i Þ e (cid:2) i ω t þ c : c ; (19)which directly leads us to the de fi nition of X ¼ a s þ a y i and Y ¼ i ð a y s (cid:2) a i Þ .This de fi nition is elegant because it describes within one uni fi edframework both two-mode squeezing and single-mode squeezing andtransfers intact the intuitive picture of squeezing of the quadrature mapfrom the single mode onto the two-mode case. Mathematically, the two-mode quadratures obey the standard commutation relation ^ X ; ^ Y (cid:1) (cid:3) ¼ i just like single-mode quadratures, and evolve in a parametric ampli fi er justlike the standard squeezing operation, with ^ X out ! ^ X in e G , ^ Y out ! ^ Y in e (cid:2) G (where G is the squeezing ratio). Although the two-mode quadratures arenon-Hermitian X ≠ X y , the two-mode quadrature is a measurable quantity,since it commutes with its Hermitian conjugate X ; X y (cid:1) (cid:3) ¼ f X g ¼ X þ X y ¼ ð X s þ X i Þ and Im f X g ¼ i ð X y (cid:2) X Þ ¼ ð Y s (cid:2) Y i Þ areHermitian and can be measured simultaneously to provide the completequadrature information. A more expanded study about this two-modeterminology along with its detection is presented in. Review of the SU(1,1) interferometer
Interferometric measurements allow for highly sensitive detection of anyphysical phenomenon that induces an optical phase shift. The phasesensitivity of an interferometric scheme depends on both the illuminationsource and the con fi guration of the interferometer: Standard SU(2)interferometers, such as the Michelson or Mach-Zehnder, achieve a phasesensitivity ð θ min Þ of 1/ N — the shot-noise limit, when fed with coherent light( N — the average number of photons that traversed the interferometerduring the detection time). SU(2) interferometers can surpass this limitwhen the unused port of the interferometer is fed with squeezed light. TheSU(1,1) nonlinear interferometer is a fundamentally different type ofinterferometric detector, where nonlinear gain media (OPAs) replace thebeam splitters, and squeezed light is generated within the interferometeritself without the need to feed it externally. Additionally, the SU(1,1)interferometer can be robust to detection losses. Fig. 3
Sensitivity of the squeezing-enhanced Raman schemecompared to the shot-noise limit, for two different signal-idlerinputs: a vacuum, and b seeded by a coherent idler, where the gainof the OPAs is fi xed at a moderate level of G = fi gurationsof seeding and internal loss as a function of the phase of OPA2 (at ϕ = a Shows the sensitivity of the unseeded schemerelative to spontaneous Raman spectroscopy (0 dB) for variousvalues of internal loss, indicating that sub-shot-noise sensitivity canbe obtained with practical levels of internal loss (up to 30%). b shows the sensitivity of a Raman interferometer with fi xed internalloss (20%) when seeded with various intensities of a coherent idler(| α i | =
0, 10, 100, 1000). Note that all sensitivities are shown relativeto the shot-noise limit at their speci fi c seeding level (1/| α i | ). Clearlythe seeded con fi guration maintains substantial reduction below theshot-noise limit with practical levels of internal loss (and showing animprovement over the unseeded case) Y. Michael et al. npj Quantum Information (2019) 81 Published in partnership with The University of New South Wales n a parametric process, the direction of energy transfer - from the pumpto the signal and idler or vice-versa — depends on the relative phasebetween the pump and a signal-idler pair, such that either ampli fi cation( ϕ r =
0) or attenuation ( ϕ r = π ) of the signal-idler pair occurs. The two OPAsof an SU(1,1) interferometer are arranged in series with opposite phase,where the attenuation axis of OPA2 matches the ampli fi cation axis OPA1and vice-versa (setting ϕ r = ϕ r = π in OPA2). Thus, if the gainof both OPAs is equal, the output quantum state of the light remainsunchanged from the input (unless the phase of the light is altered betweenthe ampli fi ers). Indeed, when the standard SU(1,1) interferometermeasures a linear phase shift θ between the two OPAs (assuming vacuuminput for now), the number of signal (or idler) photons at the output isgiven by: ^ N s (cid:6) (cid:7) SU ð ; Þ ¼ sinh ð G Þ sin θ (cid:10) (cid:11) ; (20)where G is the gain of the OPAs. The second moment is: ^ N s (cid:6) (cid:7) SU ð ; Þ ¼ sinh ð G Þ sin θ (cid:4) (cid:5) þ sinh ð G Þ sin θ (cid:4) (cid:5) : (21)If no phase shift is present, the output is vacuum (identical to the input),with ^ N s (cid:6) (cid:7) ; ^ N s (cid:6) (cid:7) ¼ (cid:4) (cid:5) , allowing signal detection which is background-free.The phase sensitivity, obtained by error-propagation analysis (see Eq. 28for additional details), is given by: θ min ¼ ^ N s h i (cid:2) ^ N s h i dd θ ^ N s h i j θ ¼ j j ; ¼ ð G Þ (cid:5) N sq ; (22)where N sq is the number of squeezed signal and idler photons generatedinside the interferometer. The result of Eq. (22) shows sub-shot-noisescaling, allowing for super sensitive phase detection (ideally Heisenbergsensitivity of 1 = N sq ). Ideal squeezing-enhanced con fi guration We start by expressing the output fi eld operators (creation andannihilation) of the signal and idler as a function of the input fi eldoperators in a three-stage propagation: through OPA1 (optical parametricampli fi er), through the Raman sample, and last through OPA2. Assumingan undepleted (classical) pump, the signal-idler fi eld operators evolve in anOPA as ^ a ð Þ s ; i ¼ cosh ð G Þ þ i Δ qz G sinh ð G Þ (cid:4) (cid:5) ^ a ð Þ s ; i þ γ I p zG sinh ð G Þ ^ a yð Þ i ; s ; (23)where G is the gain of the OPA, Δ q = Δ k + γ I p is the total phase mismatch,comprised of the bare phase mismatch Δ k = k p − k i − k s and 2 γ I p thenonlinear phase induced by the pump ( I p is the pump intensity and γ is theKerr nonlinear coef fi cient), z is the length of the medium, and ^ a ð Þ s ; i , ^ a yð Þ s ; i arethe fi eld operators before the OPA. The gain is given by G ¼ z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ I p (cid:2) Δ q q .For simplicity, we assume perfect phase matching, Δ q =
0. This assumptionis not critical for the calculation, which can be carried out just as well withany phase mismatch, and served only for simpli fi cation of the fi nalexpression. From a practical point of view, OPA1 and OPA2 can be phasematched across a broad spectrum with careful designing of the nonlinearmedium, while phase matching in the Raman medium depends on thenonlinear properties of the sample itself. We may now de fi nethe parameteres A ≡ cosh( G ), B ≡ sinh( G ). Using Eq. (23), we can expressthe fi eld operators at the output of OPA1, the Raman sample and OPA2with the fi eld operators at the input. OPA1 performs ampli fi cation at ϕ r =
0, such that: ^ a ð Þ s ; i ¼ A ^ a ð Þ s ; i þ B ^ a yð Þ i ; s : (24)The Raman sample then follows up on the ampli fi cation of OPA1, but ata relative phase of φ r ¼ π and different gain parameters ( g r , C , D ). Thus, weapply a phase of π /4 to both the signal and the idler as we propagate theoperators through the sample: ^ a ð Þ s ; i ¼ C ^ a ð Þ s ; i e i π = þ D ^ a yð Þ i ; s e (cid:2) i π = : (25) Finally, OPA2 is set orthogonal to OPA1 (a total relative phase of ϕ r = π ),and accordingly we once more apply a π /4 to the signal and idler: ^ a ð Þ s ; i ¼ A ^ a ð Þ s ; i e i π = þ B ^ a yð Þ i ; s e (cid:2) i π = : (26)The photon number at the fi nal output can be calculated using therelation between the operators before and after interferometer, whileapplying 0 s ; i ^ a yð Þ s ; i ^ a ð Þ s ; i (cid:8)(cid:8)(cid:8) (cid:8)(cid:8)(cid:8) s ; i D E ¼ ½ ^ a ð Þ s ; i ; ^ a yð Þ s ; i (cid:3) ¼
1, resulting in: ^ N ð Þ s D E ¼ s ; i h j ^ a yð Þ s ^ a ð Þ s s ; i j i¼ j D j ðj A j þ j B j þ j A jj B jÞ¼ sinh ð g r Þ cosh ð G Þ ; (27)The minimum detectable Raman gain is calculated using an error-propagation analysis, demanding that g min ¼ ^ N s (cid:6) (cid:7) (cid:2) ^ N s (cid:6) (cid:7) ddg r ^ N s (cid:6) (cid:7) j g r ¼ (cid:8)(cid:8)(cid:8) (cid:8)(cid:8)(cid:8) (28)where the numerator represents the background noise and thedenominator represents the relative change of the signal intensity as theRaman gain is varied. For the ideal, lossless case, we obtain: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ N s (cid:6) (cid:7) (cid:2) ^ N s (cid:6) (cid:7) q ¼ j C jj D jðj A j þ j B j Þ ; (29) ddg r ^ N s (cid:6) (cid:7) ¼ j C jj D jðj A j þ j B j Þ ; (30)both terms approach 0 at g r = D = C =
1) but their ratio is fi nite.taking the limit of g r →
0, we obtain: g min ¼ ð G Þ : (31) Seeded con fi guration The calculation shown in Eq. (31) can be generalized for the seededcon fi guration. In the main text, we present the result for a coherent idlerseed | α i 〉 used as the input to OPA1. The calculation for this case isconceptually the same as the unseeded case, but applied to a differentinput. The average of the photon number requires no additionalassumptions, and is given by: ^ N ð Þ s D E ¼ s ; α i j ^ a yð Þ s ^ a ð Þ s j s ; α i D E ¼ ð þ j α i j Þ sinh ð g r Þ cosh ð G Þ ; (32)which forms a squeezing-enhanced signal for CARS. Alternatively, theaverage photon number at the seeded (idler) port is given by: ^ N ð Þ i D E ¼ j α i j (cid:2) þ ðj α i j þ Þ h (cid:6)ð ð G Þ cosh ð g r Þ (cid:2) cosh ð G ÞÞ (cid:3) ; (33)where Eq. (33) shows the ampli fi cation of the idler. In SRS, the netampli fi cation of the idler is measured via lock-in detection. The expressionfor SRS is then: ^ N ð Þ i D E (cid:2) j α i j ¼ ð þ j α i j Þ sinh ð g r Þð þ cosh ð G ÞÞ¼ ð þ j α i j Þ cosh ð G Þ sinh ð g r Þ : (34)Following the same steps of Eq. (31), the minimum detectable gain forthe seeded case is: g min ¼ ð G Þð þ j α i j Þ : (35) Robustness to loss, unbalanced detection and phase inaccuracy
Loss between the two OPAs can be treated as a beam-splitter, which mixesthe fi eld operators ^ a ð Þ s ; i with an additional vacuum mode ^ b ð Þ s ; i , indicatingthat the operators at the entrance of OPA2 are ^ a ð Þ s ; i ¼ t ^ a ð Þ s ; i ± r ^ b ð Þ s ; i , where t , r are the amplitude transmission and re fl ection coef fi cients of the assumedbeam-splitter. The number of photons is given by: ^ N ð Þ s D E ¼ j t j sinh ð g r Þ cosh ð G Þ þ j r j sinh ð G Þ ; (36)Y. Michael et al. Published in partnership with The University of New South Wales npj Quantum Information (2019) 81 nd the background noise is: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ N s (cid:6) (cid:7) loss (cid:2) ^ N s (cid:6) (cid:7) loss q ¼ j r j sinh ð G Þ : (37)Plugging Eq. (37) into Eq. (28) results in a diverging limit at g r →
0. Thus,the relative phase before OPA2 must also be taken into considerationwhen calculating the sensitivity for the squeezing-enhanced Ramanscheme with internal loss. Analytical expressions are no longer simpleand intuitive enough to be provided in this paper, so it is best to refer tothe numerical plots of Fig. 3 provided in the main text.The scheme is also effective in non-symmetric conditions, such asunbalanced interferometric detection ( G ≠ G ): This can occur becauseof different pump intensities before the different OPAs, or due tounsuppressed non-resonant background. This behavior is shown in Fig. 4afor the lossless case, where the sensitivity remains identical to thebalanced case, but the optimal working point of the interferometer shiftsin phase to compensate for the added noise at ϕ =
0. If internal loss isintroduced, major unbalancing of the OPAs can somewhat compensate forthe loss and improve the sensitivity, as shown in Fig. 4b, and also discussedin.
The unbalanced interferometer boasts other potential advantages,such as high-bandwidth two-mode homodyne detection. This is achievedby setting G > G , where the main idea is that for a suf fi ciently strongdifference in gain, the light output of OPA2 is proportional to thequadrature it ampli fi es, using the pump as the local oscillator. It is also important to consider the ampli fi cation angle of the Ramansample, which ideally would be at the mid-point between the ampli fi ers,as previously discussed in Fig. 2. We now discuss the non-ideal case,where the relative phase of the light at the Raman sample deviates from ϕ r = π /2 by some adverse phase θ . This phase arises, for example, frominaccurate compensation of linear dispersion inside the interferometer. InFig. 5, we show the sensitivity for different values of the phase deviation,where for relatively small phase deviations of the Raman sample, thesensitivity is nearly unaffected by this error. Note that we have assumedhere that the sample has a pure resonant response, and in general, this behavior depends on the exact ratio between the resonant and non-resonant gains. DATA AVAILABILITY
All the data and calculations that support the fi ndings of this study are available fromthe corresponding author upon reasonable request. CODE AVAILABILITY
The code used to calculate the results given in this paper will be made available fromthe corresponding author upon reasonable request.
Received: 14 April 2019; Accepted: 9 September 2019;
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This research was supported by grant No.44/14 from the Israel Science Foundation(FIRST program for high-risk high-gain research). We thank Dr. Emanuele Dalla Torreand Prof. Richard Berkovits for their thoughtful comments. We are grateful to Jose LuisGomez-Munoz and Francisco Delgado for their “ Quantum notation ” add-on to theMathematica software, which we have used for the quantum calculations in this work. AUTHOR CONTRIBUTIONS
Y.M. proposed the main concept and carried out the calculations, L.B. and M.R.participated in the analysis of results and continuous discussions, Y.M. and A.P.developed the theoretical framework. All authors contributed to writing of the paper.
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The authors declare no competing interests.
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