Classical Bound for Mach-Zehnder Super-Resolution
aa r X i v : . [ qu a n t - ph ] F e b Classical Bound for Mach-Zehnder Super-Resolution
I. Afek, O. Ambar, and Y. Silberberg
Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel (Dated: November 3, 2018)The employment of path entangled multiphoton states enables measurement of phase with en-hanced precision. It is common practice to demonstrate the unique properties of such quantumstates by measuring super-resolving oscillations in the coincidence rate of a Mach-Zehnder interfer-ometer. Similar oscillations, however, have also been demonstrated in various configurations usingclassical light only; making it unclear what, if any, are the classical limits of this phenomenon.Here we derive a classical bound for the visibility of super-resolving oscillations in a Mach-Zehnder.This provides an easy to apply, fundamental test of non-classicality. We apply this test to experi-mental multiphoton coincidence measurements obtained using photon number resolving detectors.Mach-Zehnder super-resolution is found to be a highly distinctive quantum effect.
PACS numbers: 42.50.-p,42.50.Xa,03.65.Ud
Inroduction. —Experimental observations which ex-hibit uniquely quantum mechanical behavior lie at theheart of quantum optics. Such observations establishthe way we diffrentiate between the quantum and classi-cal worlds. Notable examples include Bell’s inequalities[1], photon anti-bunching [2] and sub-Poissonian numberstatistics [3]. Negativity of the Wigner function is alsowidely used to verify non-classicality [4]. The derivationof additional tests involving non-classical behavior hasmotivated many studies [5–10].Recently, there has been a surge of interest in pathentangled two-mode states of the form | N :: 0 i a,b ≡ √ ( | N, i a,b + | , N i a,b ) known as NOON states [11]. Insuch states, all N photons acquire phase as a collec-tive entity thus enabling phase super-sensitivity [11] andsub-wavelength lithography [12]. A number of experi-mental implementations of NOON states exist to date[13–16]. In these experiments the high NOON statefidelity was demonstrated by measuring a sinusoidallyvarying N photon coincidence rate with a reduced pe-riod corresponding to a wavelength λ/N . Such behavior,coined ’phase super-resolution’ [11] is generally acceptedas a signature of NOON states. However, it has beenshown that similar measurements exhibiting high visibil-ity super-resolution can be obtained using classical lightonly. This has been demonstrated using multiport lin-ear interferometers with coincidence counting [17] or in ascheme for nonlinear sub-Rayleigh lithography [18, 19].It is therefore natural to ask whether super-resolutioncan be used at all as a direct, non-ambiguous test fornon-classicality. Here we answer this question in the af-firmative. We consider a Mach-Zehnder geometry simi-lar to the one employed in the initial proposal for quan-tum lithography [12] and subsequently used in all NOONstate generation experiments (see Fig. 1). We show thatfor any classical state with a positive-definite well be-haved Glauber-Sudarshan P representation [25], there isa strict upper limit for the visibility of super-resolvingfringes. The bound is derived analytically for arbitrary coincidences with m, n photons in the two output portsrespectively. We apply the bound to experimental mul-tiphoton coincidences obtained using both classical andquantum light. The classical results are found to ap-proach the bound from below. For the quantum case,we employ NOON states obtained in an experiment wehave recently reported [15] and a violation of the classicalbound with states of up to five photons is demonstrated. Derivation. —Consider a balanced Mach-Zehnder In-terferometer (MZI) whose input modes are denoted a, b (see Fig. 1). The coincidence rate with m, n photons indetectors D , D respectively is denoted C m,n ( ϕ ) wherethe total number of photons is N = n + m . When employ-ing a NOON state input, C m,n exhibits perfect sinusoidalfringes of the form C m,n ( ϕ ) ∝ N ϕ − δ ) where δ = 0 , π depending on the values of m, n . This patternexhibits N -fold phase super-resolution with 100% visibil-ity. Our goal is to derive a simple unambiguous criterionbased on the visibility of C m,n for distinguishing betweenquantum and classical input states. To this end, considerthe Fourier series of an arbitrary coincidence function, C m,n ( ϕ ) = ∞ X k =0 A k cos( kϕ − δ k ) . (1)We are interested in the visibility of the N -fold oscilla-tions, | A N /A | , which will be referred to as the N -foldvisibility. For pure sinusoidal oscillations with a constantbackground the N -fold visibility coincides with the con-ventional definition of visibility. Considering the N -foldvisibility allows application of this bound to situations inwhich the measured coincidence has contributions oscil-lating at a number of different frequencies (see [15] forexample). The m, n coincidence rate for a MZI is givenby [24], C m,n ( ϕ ) = Tr h ˆ U ( ϕ ) ˆ ρ a,b ˆ U † ( ϕ ) ˆ π n ⊗ ˆ π m i , (2)where ˆ U ( ϕ ) is a unitary operator describing the MZI[23], ˆ ρ a,b = | ψ i a,b h ψ | is the input state density ma- (cid:1)(cid:0)(cid:2)(cid:3) (cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25) (cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) !" ˆ Q ˆ R STUV WX
FIG. 1: (color online) Experimental setup. (a) Schematic ofthe setup, a Mach-Zehnder interferometer with photon num-ber resolving detectors at the output ports. (b) The actualsetup used for obtaining the classical light results shown inFig. 3. A horizontally (H) polarized pulsed Ti:Sapphire os-cillator with 120 fs pulses @1 MHz repetition rate (reducedfrom 80MHz using a pulse picker) is used as the light source.The MZI is implemented in a collinear inherently phase stabledesign using polarization. The phase, ϕ is applied using a liq-uid crystal phase modulator at 45 deg. ( X, Y polarizations).The photon number resolving detectors are Hamamatsu multipixel photon counters (MPPC) module C10507-11-050U [22]. trix and ˆ π n are the positive operator valued measures(POVM’s) of detectors D [24]. For the present deriva-tion we consider ideal photon number resolving detectorsimplying ˆ π n = ˆ π n = | n ih n | . First, we derive the N -foldvisibility for an input state of the form | ψ i a,b = | α i| i a,b .Here, | α i is an arbitrary non-vacuum coherent state de-fined | α i ≡ P ∞ n =0 e − | α | α n √ n ! | n i . Substitution in Eq. (2)yields, C m,n ( ϕ ) = e −| α | n ! m ! | α | m + n ) | sin( ϕ/ | m | cos( ϕ/ | n = e −| α | n ! m ! (cid:12)(cid:12)(cid:12) α (cid:12)(cid:12)(cid:12) m + n ) | e ı ϕ − e − ı ϕ | m | e ı ϕ + e − ı ϕ | n . (3)We denote the N -fold visibility for each photon numberpair, m, n by Γ m,ncl . Using simple combinatorial consid-erations we obtainΓ m,ncl = 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X r =0 ( − r (cid:18) nr (cid:19)(cid:18) mn + m − r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − . (4)We note that Γ m,ncl does not depend on the specific choiceof α . In the following we prove that Γ m,ncl is, in fact, thelargest N -fold visibility obtainable for an arbitrary classi-cal input state. We now consider an input state consist-ing of two arbitrary coherent states | ψ i a,b = | α i| β i a,b .We denote the N th fourier component of C m,n ( ϕ ) forthis state by A α,β,m,nN . Using some straightforward butsomewhat lengthy algebra it can be shown that for suchan input state the N -fold visibility is always bounded byΓ m,ncl i.e. (cid:12)(cid:12)(cid:12) A α,β,m,nN /A α,β,m,n (cid:12)(cid:12)(cid:12) ≤ Γ m,ncl . (5) V i s i b ili t y [ % ] iii FIG. 2: The classical visibility bound, Γ m,ncl (Eq. (4)) as afunction of N = m + n . For each N , the bound is shown fortwo choices of m, n : (i) m, n = ⌊ N/ ⌋ , ⌈ N/ ⌉ i.e. the detectionevents are divided as equally as possible between D and D (see Fig. 1), (ii) m, n = N, D and zero are detected in D . For a given N , the classicallyobtainable visibility for case (i) is always higher than for case(ii). However, for both cases Γ m,ncl decays exponentially with N . Finally, we prove the bound holds for general clas-sical inputs of the form ˆ ρ cla,b = R P ( α, β ) a,b | α i a h α | ⊗| β i b h β | d αd β with a positive-definite, well behaved P function P ( α, β ) a,b . These are precisely the quantummechanical states which posses classical analogues [25].Formally the N -fold visibility is a functional of P whichwe denote V m,n [ P ( α, β )]. The bound ensues from thefollowing inequalities, V m,n [ P ( α, β )] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R P ( α, β ) A α,β,m,nN d αd β R P ( α, β ) A α,β,m,n d αd β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (6) ≤ R P ( α, β ) (cid:12)(cid:12)(cid:12) A α,β,m,nN A α,β,m,n (cid:12)(cid:12)(cid:12) A α,β,m,n d αd β R P ( α, β ) A α,β,m,n d αd β ≤ R P ( α, β )Γ m,ncl A α,β,m,n d αd β R P ( α, β ) A α,β,m,n d αd β = Γ m,ncl , using the positive definiteness of P and Eq. (5). Notethat A α,β,m,n is positive by definition. This completesthe proof and the main result of this letter.The bound, Γ m,ncl , is plotted in Fig. 2 for a selection of m, n pairs. In addition, the numerical values for N ≤ m,ncl decays expo-nentially with N = m + n . However, for a given N the obtainable visibility is much higher when the pho-tons are distributed equally (or almost equally for odd N ) between the two output modes. Interestingly, ourresult for pairs of the form N, − f o l d C o i n c i den c e r a t e [ a r b . un i t s ] − f o l d − f o l d − f o l d , c o i n c . [ K H z ] , c o i n c . [ K H z ] iiiiiiiiiiiiiiiiiiMZ phase [rad] 2 ππ abcd fe π π MZ phase [rad] π π MZ phase [rad]
FIG. 3: (color online) Experimental coincidence scans usingclassical light. (a-d) Coincidence rates (raw data, normalized)as a function of Mach-Zehnder phase with m ’clicks’ in D and n ’clicks’ in D for various values of m, n (see Fig 1).The total number of photons m + n is denoted by N . Theinput state is | ψ i a,b = | α i| i a,b where | α i is a coherent state.The solid lines are obtained using an analytical model of thedetector POVM’s accounting for cross-talk, dark counts andlosses [22]. Error bars are smaller than the displayed markers.(a) N = 2 (i) m, n = 2 ,
0, (ii) m, n = 1 ,
1. (b) N = 3 (i) m, n = 3 ,
0, (ii) m, n = 2 ,
1. (c) N = 4 (i) m, n = 4 , m, n = 2 ,
2, (iii) m, n = 3 ,
1. (d) N=5 (i) m, n = 5 , m, n = 3 ,
2, (iii) m, n = 4 ,
1. The classical visibilities inFig. 4iii (blue bars) are obtained using these measurements.(e) Three-fold sinusoidal oscillations used for obtaining theexperimental visibility of the m, n = 2 , , π , π and superimposed. The redline is the fitted Fourier series. The obtained 3-fold visibility is45 . m, n = 2 , m, n = 2 , , π , π, π and superimposed. The obtained 4-foldvisibility is 29 . ple modes is straightforward [25] and does not alter thefinal result. We have implicitly assumed the use of broad-band detectors such as those typically employed for singlephoton detection. If one allows use of narrow-band mul-tiphoton detection (such as atomic two photon absorp-tion) and appropriate engineering of the input state thenhigher visibilities may be obtained [26, 27]. The classicalvisibility limits of multiphoton interferences have beenpreviously discussed [28, 29]. However, these analyseswere limited to input states with randomly fluctuatingphases and considered the standard rather than N -fold TABLE I: Classical visibility bound, Γ m,ncl , Eq. (4). Numeri-cal values for N ≤ N m, n m,ncl (%) 100 33.3 50 10 33.3 20 2.85 16.67 7.14 0.79 visibility which in our case can always be 100% (as seenexperimentally in Fig. 3). We point out that phasesensitivity beyond the standard quantum limit (knownas phase super-sensitivity) is a distinct quantum effectand can be exploited for quantum noise reduction. How-ever, it cannot be inferred directly from the visibilityand requires additional knowledge about the efficiency[17, 21], making it much less straightforward as a test fornon-classicality. The sensitivity criterion is also typicallyharder to beat experimentally than the present bound. Experimental results. —We now demonstrate the appli-cability of the bound using experiments done in both theclassical and quantum regimes. In the classical case, thebound is approached from below by using a MZI with acoherent state input of the form | ψ i a,b = | α i| i a,b . TheMZI is implemented in a polarization based geometry asshown in Fig. 1b. We measured experimental multipho-ton coincidences using two multi-pixel photon number re-solving detectors [22]. The coincidence sweeps are shownin Fig. 3a-d. For a given N it is readily observable thatthe features are widest when the photons are all detectedin one output. It is therefore not surprising that the vis-ibility bound is lowest in this case. The N -fold visibil-ity was obtained by fitting the experimental curves to aFourier series of the form in Eq. (1) truncated at k = N .For a given N , the fit was applied to N copies of the datashifted by 0 , π N , . . . π N − N and superimposed (see Fig.3e-f). This eliminates the lower frequencies without af-fecting the N -fold visibility thus improving the accuracyof the numerical fit. Although in this case the slowly os-cillating terms were eliminated in post-processing, it canbe shown that for any value of m, n there exists a classi-cal input state which yields a coincidence rate of the form C m,n ( ϕ ) ∝ m,ncl cos( N ϕ + δ ) i.e. pure N -fold oscil-lations with a constant background. The experimentallyobtained visibilities are shown in Fig. 4iii (blue bars).The measured visibility is lower than the classical boundfor all m, n as predicted. We note that limited overall ef-ficiency doesn’t affect the classically obtainable visibilitysince coherent states are transformed to coherent statesby loss [30]. Additional experimental imperfections suchas cross-talk and dark-counts only serve to reduce the N -fold visibility. Thus, the bound is expected to hold forraw experimental data and doesn’t require compensationfor detector imperfections.Surpassing the bound requires use of non-classicallight. To demonstrate this we use path-entangled mul-tiphoton states with high NOON state fidelity that we V i s i b ili t y [ % ] iiiiii m,n =N = 2 3 4 5Photon numbers3,02,1 FIG. 4: (color online) Comparison between the classicalbound and experimental visibilities. Visibilities as a functionof the number of coincidence ’clicks’ in each of the MZI out-puts. (i) The classical bound, Γ m,ncl , Eq. (4). (ii) Experimen-tal visibilities obtained with NOON states using non-classicallight [15]. All the visibilities violate the classical bound (ex-cept 4 , , ± σ statistical uncertainty.(iii) Experimental visibilities obtained using classical inputs(error bars are negligible). The experimental visibilities wereobtained from the coincidence measurements in Fig. 3. Thesevisibilities are lower than the classical bound as expected, thediscrepancy being mainly due to the cross-talk between de-tector pixels. have recently reported [15]. Briefly, the experimental ge-ometry was similar to the one used in Fig. 1, howeverone of the inputs was non-classical resulting in a highlyentangled state in modes c, d of the MZI. The visibilities,obtained from raw data, are in violation of the classicalbound. This verifies the quantum nature of the resultsand demonstrates the feasibility of surpassing the clas-sical bound for a wide range of photon numbers in arealistic experimental setting, see Fig. 4ii (red bars). Conclusion. —We have derived a simple, practical testfor non-classicality using MZI coincidence measurementsand demonstrated its use. The quantity of interest is thevisibility of N -fold oscillations in the N photon coinci-dence rate. The classical bound on this quantity dropsexponentially to zero with the total photon number N while the quantum limit remains constant. This makes itan excellent indicator of non-classicality. It can be shownthat only the NOON component of a given state in theMZI contributes to the N -fold oscillatory term. Thus,our results reflect the fact that the NOON state overlapresulting from any classical input state becomes exceed-ingly low as N grows. We expect this result to serve asa benchmark in future experimental demonstrations of super-resolution.Itai Afek gratefully acknowledges the support of theIlan Ramon Fellowship. Financial support of this re-search by the German Israel Foundation and the MinervaFoundation is gratefully acknowledged. Correspondenceand requests for materials should be addressed to Afek.I (email: [email protected]). [1] A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. , 460 (1981).[2] H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev.Lett. , 691 (1977).[3] R. Short and L. Mandel, Phys. Rev. Lett. , 384 (1983).[4] G. Nogues et al. , Phys. Rev. A , 054101 (2000).[5] E. Waks, E. Diamanti, B. C. Sanders, S. D. Bartlett, andY. Yamamoto, Phys. Rev. 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