Multi-outcome homodyne detection in a coherent-state light interferometer
aa r X i v : . [ qu a n t - ph ] A p r Multi-outcome homodyne detection in a coherent-state light interferometer
J. Z. Wang, Z. Q. Yang, A. X. Chen, ∗ W. Yang, † and G. R. Jin ‡ Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China Beijing Computational Science Research Center, Beijing 100084, China (Dated: April 2, 2019)The Cram´er-Rao bound plays a central role in both classical and quantum parameter estimation, but findingthe observable and the resulting inversion estimator that saturates this bound remains an open issue for generalmulti-outcome measurements. Here we consider multi-outcome homodyne detection in coherent-light Mach-Zehnder interferometer and construct a family of inversion estimators that almost saturate the Cram´er-Rao boundover the whole range of phase interval. This provides a clue on constructing optimal inversion estimators forphase estimation and other parameter estimation in any multi-outcome measurement.
I. INTRODUCTION
Interference of light fields is important in astronomy [1, 2],spectroscopy [3, 4], and various fields of quantum technol-ogy [5]. For instance, in optical lithography [6], the light-intensity measurement gives rise to an oscillatory interfero-metric signal ∝ sin ( φ/
2) or cos ( φ/ λ/
2, determined by the wavelength λ . Thisis often referred to the classical resolution limit of interfer-ometer, or the Rayleigh resolution criterion in optical imag-ing [6]. To beat this classical limit, it has been proposed to use N -photon entangled states ( | N , i + | , N i ) / √ λ/ (2 N ), i.e., N -times improvementbeyond the classical resolution limit. However, the N -photonentangled states are di ffi cult to prepare and are subject to pho-ton losses [11–13]. In the absence of quantum entanglement,the super-resolution can be also attainable from di ff erent typesof measurement schemes, such as coincidence photon count-ing [14], parity detection [15, 16], and homodyne detectionwith post-processing [17].To realize a high-precision measurement of an unknownphase φ , an optimal measurement scheme with a proper choiceof data processing is important to improve both the resolu-tion and the phase sensitivity [5, 18]. Given a phase-encodedstate and a properly chosen observable ˆ Π , the ultimate phaseestimation precision is determined by the Cram´er-Rao lowerbound (CRB) [19–22], i.e., δφ ≥ δφ CRB = / p F ( φ ), where F ( φ ) is the classical Fisher information (CFI), dependent onthe measurement probabilities. To saturate the CRB, it re-quires complicated data-processing techniques such as max-imal likelihood estimation or Bayesian estimation [23, 24].However, they lack physical transparency and require a lot ofcomputational resources. By contrast, the simplest data pro-cessing is to equate the theoretical expectation value h ˆ Π i φ = f ( φ ) with the experimentally measured value Π exp , whichgives a simple inversion estimator φ inv ≡ f − ( Π exp ) to theunknown phase φ , with a precision determined by the error- ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] propagation formula: δφ = ∆ ˆ Π (cid:12)(cid:12)(cid:12) ∂ h ˆ Π i φ /∂φ (cid:12)(cid:12)(cid:12) . (1)Here ∆ ˆ Π ≡ ( h ˆ Π i φ − h ˆ Π i φ ) / is the root-mean-square fluc-tuation of the observable ˆ Π . Due to its physical transparencyand computational simplicity, the inversion estimator has beenwidely used in various phase estimation schemes. More-over, for binary-outcome measurements (such as parity detec-tion [25–31], single-photon detection [16], and on-o ff mea-surement), the inversion estimator asymptotically approachesthe maximum likelihood estimator [32–34] and hence satu-rates the CRB within the whole phase interval, e.g., φ ∈ [ − π, π ]. However, for general multi-outcome measurements,the performance of the inversion estimator depends stronglyon the chosen observable ˆ Π and usually cannot saturate theCRB [17]. Therefore, an important problem in high-precisionphase estimation is to find an optimal inversion estimator thatsaturates the CRB. At present, this problem remains open.In this paper, we investigate the performance of the in-version estimator in multi-outcome measurements and reportsome interesting findings. We begin with a binary observ-able, i.e., a multi-outcome measurement with only two di ff er-ent eigenvalues. Such a binary observable plays a key role inachieving deterministic super-resolution in a Mach-Zehnderinterferometer fed by a coherent-state light in a recent experi-ment [17]. We find that using a binary observable to constructthe inversion estimator is equivalent to binarizing the originalmulti-outcome measurement into an e ff ective binary-outcomeone. As a result, the precision of the inversion estimator satu-rates the CRB determined by the CFI of this binary-outcomemeasurement and is independent of the choices of the eigen-values. We show that the observable adopted in Ref. [17]belongs to this class. Next, we consider the homodyne de-tection of Ref. [17] as a paradigmatic example to study thedependence of the precision of the inversion estimator on theobservable. Surprisingly, we find that when the neighboringeigenvalues of the observable have alternating signs, the re-sulting inversion estimator is nearly optimal, i.e., its preci-sion almost saturates the CRB. This may provide a clue onconstructing optimal inversion estimators for phase estimationand other parameter estimation (e.g., optical angular displace-ments [35–39]). II. QUANTUM PHASE ESTIMATION WITH AMULTI-OUTCOME MEASUREMENT
For a ( N + Π = N X k = µ k ˆ Π k , (2)where µ k ( ˆ Π k ) is the eigenvalue (projector) associated with the k th outcome. The output signal is the average of the observ-able ˆ Π with respect to the phase-encoded state ˆ ρ ( φ ): h ˆ Π i φ = N X k = µ k P ( k | φ ) ≈ N X k = µ k N k N , (3)where P ( k | φ ) = Tr[ ˆ ρ ( φ ) ˆ Π k ] is the conditional probabilityof the k th outcome, which can be measured by the occur-rence frequency N k / N . Specially, one can perform N inde-pendent measurements at each phase shift φ ∈ [ − π, π ] andrecord the occurrence numbers {N k } . For large enough N , P ( k | φ ) ≈ N k / N . Numerical simulation of a special kind ofmulti-outcome measurement will be shown at the end of thiswork.We begin with a binary observable – the multi-outcomemeasurement with only two di ff erent eigenvalues. Withoutlosing generality, we take µ N = µ N − = · · · = µ , then we canuse ˆ Π k ˆ Π k ′ = δ kk ′ ˆ Π k to obtain h ˆ Π i φ = µ P (0 | φ ) + µ P ( ∅| φ ) , h ˆ Π i φ = µ P (0 | φ ) + µ P ( ∅| φ ) , where P ( ∅| φ ) ≡ P Nk = P ( k | φ ) and P (0 | φ ) = − P ( ∅| φ ). There-fore, the phase sensitivity of the inversion estimator based onthe binary observable ˆ Π is independent of the eigenvalues µ and µ : ( δφ ) = P ( ∅| φ ) P (0 | φ ) (cid:2) P ′ ( ∅| φ ) (cid:3) , (4)where P ′ ≡ ∂ P /∂φ . Interestingly, if we binarize the N + ∅ ”, i.e., we regard the out-comes 1 , , · · · , N as a single outcome “ ∅ ”, then P ( ∅| φ ) is justthe conditional probability for “ ∅ ” and the CFI of this e ff ectivebinary measurement coincides with 1 / ( δφ ) . In other words,the phase sensitivity δφ of the inversion estimator based onthe binary observable ˆ Π always saturates the CRB of the ef-fective binary-outcome measurement [32]. Since this e ff ec-tive binary-outcome measurement is obtained from the orig-inal multi-outcome measurement by coarse-graining, its CFIis smaller than the CFI of the original one: F ( φ ) ≡ N X k = (cid:2) P ′ ( k | φ ) (cid:3) P ( k | φ ) . (5)As a result, δφ cannot saturate the CRB of the original multi-outcome measurement δφ CRB ≡ / p F ( φ ) [19–22]. There-fore, it is important to find an optimal choice of eigenvalues { µ k } such that δφ = δφ CRB . The above results are applicable to an arbitrary measure-ment scheme and arbitrary input state. In the following, weconsider the homodyne detection at one port of Mach-Zehnderinterferometer with a coherent-state input [17] as a paradig-matic example to illustrate these results. Interestingly, we finda nearly optimal observable that almost saturates the CRB forall φ ∈ [ − π, π ]. FIG. 1: (a) Homodyne detection at one port of the coherent-stateinterferometer, equivalent to measuring quadrature operator ˆ p withrespect to the output state. (b) Conditional probability P ( p | φ ) againstthe phase shift φ and the measured quadrature p , given by Eq. (11),and the post-processing method by separating the measured data intoseveral bins [17], where the bin’s center is kb (for k = ±
1, ... ± k f )and the width is 2 a , equivalent to a multi-outcome measurement. (c)and (d): Output signal h ˆ Π i φ and phase sensitivity δφ for k f = δφ min . (e) and (f): Density plots of the ratios π/ and / √ ¯ n δφ min asfunctions of the average photon number ¯ n ( = α ) and the bin size a .Dashed lines: contours of the two ratios. Solid lines and below: aregion that the visibility of the signal ≥ As depicted by Fig. 1(a), a coherent state | α i and a vacuumstate | i are injected from the input ports. The output stateis given by | ψ out ( φ ) i = ˆ U ( φ ) | ψ in i , where ˆ U ( φ ) is an unitaryoperatorˆ U ( φ ) = exp (cid:18) − i π J y (cid:19) exp (cid:16) − i φ ˆ a † ˆ a (cid:17) exp (cid:18) − i π J y (cid:19) , (6)which represents a sequence actions of the 50:50 beamsplitterat the output port [40], the phase accumulation at one of thetwo paths, and the 50:50 beamsplitter at the input port. Forbrevity, we have introduced Schwinger’s representation of theangular momentum ˆ J = (ˆ a † , ˆ b † ) ˆ σ (cid:16) ˆ a ˆ b (cid:17) , where ˆ a † (ˆ a ) and ˆ b † (ˆ b )denote the creation (annihilation) operators of the two fieldmodes and ˆ σ = ( ˆ σ x , ˆ σ y , ˆ σ z ) the Pauli matrix.In general, a homodyne detection at one of two output portsgives the measured quadrature p ∈ ( −∞ , ∞ ), with the condi-tional probability P ( p | φ ) = Z ∞−∞ dx Z ∞−∞ dX Z ∞−∞ dPW out ( α, β ; φ ) , (7)where α = x + ip ( x , y ∈ R ) and β = X + iP ( X , P ∈ R ), and W out is the Wigner function of the output state [41, 42], W out ( α, β ; φ ) = W in ( ˜ α φ , ˜ β φ ) , (8)with ( ˜ α φ = α e i φ − + β e i φ + , ˜ β φ = − α e i φ + − β e i φ − . (9)Note that Eqs. (8) and (9) are valid for the two-path interfer-ometer described by ˆ U ( φ ), independent from the input stateand the measurement scheme.For the coherent-state input | ψ in i = | α i ⊗ | i , the Wignerfunction is given by [40] W in ( α, β ) = π ! e − | α − α | e − | β | . (10)Replacing ( α, β ) by ( ˜ α φ , ˜ β φ ), one can obtain the Wigner func-tion of the output state, as shown by Eq. (8). For α = √ ¯ n ∈ R ,we obtain the probability to detecting an outcome p , P ( p | φ ) = r π exp " − (cid:18) p + α φ (cid:19) , (11)in agreement with our previous result [32]. In Fig. 1(b), weshow density plot of P ( p | φ ) against the phase shift φ and themeasured quadrature p . The green dashed line is given by theequation p = − α sin( φ ) /
2, indicating the peak of P ( p | φ ). Thecommonly used observable in a traditional homodyne mea-surement is given by ˆ Π = R ∞−∞ p | p ih p | d p , where ˆ p | p i = p | p i and ˆ p ≡ (ˆ a − ˆ a † ) / (2 i ). The output signal is the average of thisobservable h ˆ Π i φ = R ∞−∞ d ppP ( p | φ ) ∝ √ ¯ n sin φ , which exhibitsthe full width at half maximum (FWHM) 2 π/
3, and hence theRayleigh limit in fringe resolution. The resolution can be im-proved by choosing a suitable observable [17].
III. BINARY-OUTCOME HOMODYNE DETECTION
We first consider the binary-outcome homodyne detection,where the measured data has been divided into two bins [17]: p ∈ [ − a , a ] as an outcome, denoted by “ + ”, and p < [ − a , a ] asan another outcome “ − ”, with the bin size 2 a . Using Eq. (11),it is easy to obtain the conditional probabilities of the out-comes “ ± ”, namely P ( + | φ ) = Z + a − a d pP ( p | φ ) =
12 Erf (cid:2) g − ( φ ) , g + ( φ ) (cid:3) , (12)and hence P ( −| φ ) = − P ( + | φ ). Here, Erf (cid:2) x , y (cid:3) = erf( y ) − erf( x ) denotes a generalized error function, and g ± ( φ ) = √ (cid:18) α φ ± a (cid:19) . (13)Obviously, this is a binary-outcome measurement with the ob-servable ˆ Π = µ + ˆ Π + + µ − ˆ Π − , where ˆ Π + = R + a − a | p ih p | d p andˆ Π − = ˆ1 − ˆ Π + . The signal is the average of ˆ Π D ˆ Π E φ = µ + P ( + | φ ) + µ − P ( −| φ ) , (14)where h ˆ Π ± i φ = P ( ±| φ ). According to Eq. (4), we obtain thephase sensitivity of the inversion estimator: δφ = p P ( + | φ ) P ( −| φ ) | P ′ ( + | φ ) | , (15)which is independent of the eigenvalues µ ± of the binary ob-servable ˆ Π . The CFI of this binary-outcome measurement isgiven by [32–34] F ( φ ) = X k = ± (cid:2) P ′ ( k | φ ) (cid:3) P ( k | φ ) = ( δφ ) − , (16)where, in the last step, we have used the relation P ( + | φ ) + P ( −| φ ) =
1. Therefore, the sensitivity δφ of the inversionestimator based on the binary observable ˆ Π always saturatesthe CRB of the binary-outcome measurement [32–34].In Figs. 1(c) and 1(d), we take µ + = / erf( √ a ) and µ − = φ , where erf( √ a ) is a normalization factor [17]. Thevertical lines determine the resolution and the best sensitiv-ity δφ min = / p F ( φ min ). From Figs. 1(e) and 1(f), one canfind that a higher resolution with the FWHM ∼ π/ √ ¯ n , canbe obtained as the bin size a →
0. However, the best sensi-tivity occurs as a ≥ /
2. Therefore, as a trade-o ff , one cansimply take a = ( ∆ ˆ p ) | α i = / √ ¯ n . Numer-ically, it has been shown that the best sensitivity can reach δφ min ∼ . / √ ¯ n [17].We now investigate the visibility of the interferometric sig-nal and its relationship with a and ¯ n that have not been ad-dressed by Ref. [17]. From Fig.1(c), one can note that thevisibility can be determined by V = D ˆ Π E φ = − D ˆ Π E φ = π/ D ˆ Π E φ = + D ˆ Π E φ = π/ , (17)where φ = ± π/ a and ¯ n fora given V . The solid line of Fig. 1(e) corresponds to V = . V > .
9. Our numericalresults show that the visibility is larger than 90% only whenthe average number of photons is not too small (at least ¯ n > . IV. MULTI-OUTCOME HOMODYNE DETECTION
To proceed, let us consider the multi-outcome case by sep-arating the measured quadrature into several bins [17], i.e.,treating p ∈ [ b k − a , b k + a ] as an outcome k , where b k is cen-ter of each bin. When p does not lie within any bin, then weidentify p as outcome “ − ”. The occurrence probability of the k -th outcome is P ( k | φ ) = Z b k + ab k − a d pP ( p | φ ) =
12 Erf h g − ( φ ) + √ b k , g + ( φ ) + √ b k i , (18)where g ± ( φ ) is defined in Eq. (13). The occurrence probabilityfor the outcome “ − ” is simply given by P ( −| φ ) = − P k P ( k | φ ).According to Distante et al. [17], one can take b k = kb withthe integers k = ± · · · , ± k f and a factor b to be deter-mined, so the total number of the outcomes is 2( k f + b > a , the overlap of the conditional probabilities be-tween neighbour outcomes is vanishing, and k f b ∼ α / n = α and b ( > a ), we take k f = [ α / (2 b )] such that the outcomes for | k | > k f give almostvanishing contribution, where [ x ] is the integer closest to x .In Fig. 2, we take α = √ a = /
2, and b = . φ . Here, thetotal number of the outcomes 2( k f + =
6, since k f =
2. Thevisibility of P (0 | φ ) is similar to the binary-outcome problemand is larger than 90% as long as ¯ n = α > .
8. Numericalsimulations of the occurrence probabilities are shown usingrandom numbers ranged from 0 to 1 (see below).The observable corresponding to this multi-outcome homo-dyne detection can be written as ˆ
Π = µ − ˆ Π − + P k µ k ˆ Π k , whereˆ Π k ≡ R b k + ab k − a | p ih p | d p and ˆ Π − ≡ ˆ1 − P k ˆ Π k , with the eigenvalues { µ k } and µ − . The signal is the average of ˆ Π : D ˆ Π E φ = k f X k = − k f µ k P ( k | φ ) + µ − P ( −| φ ) , (19)Previously, Distante et al. [17] have derived the signal and thesensitivity by taking µ k = / erf( √ a ) and µ − =
0. For ar-bitrary µ k = µ + and µ − =
0, it is easy to reduce the signalas h ˆ Π i φ = µ + − ( µ + − µ − ) P ( −| φ ). Furthermore, the sensitiv-ity is found independent on the values of µ ± ; see Eq. (4). InFigs. 3(a) and 3(b), we simply take µ k = µ + = µ − = h ˆ Π i φ = − P ( −| φ ) and the sensitivity against FIG. 2: Conditional probabilities P ( k | φ ) for k = ±
1, ..., ± k f ,and P ( −| φ ) = − P k P ( k | φ ). The parameters: ¯ n = a = / b = .
8, and hence the total number of the outcomes 2( k f + = k f = P ( ± | φ ) for the blue solid lines,and P ( ± | φ ) for the red dashed lines. Numerical simulations: av-eraged occurrence frequency N k / N (the solid circles) and its stan-dard derivation (the bars) of each outcome after M =
10 replicas of N =
200 independent measurements. φ . Similar to Ref. [17], the signal exhibits a multi-fold oscil-latory pattern, with the peaks located at φ k = arcsin b k α ! , (20)and also π − φ k . If the bin’s center b k = kb and | k | < k f , itis easy to obtain φ k ≈ kb /α and hence the first dark pointof the signal φ dark ≈ φ ± / ≈ ± b /α ; see the vertical lines ofFig. 3(a).When all { µ k } are the same, we obtain a binary observableˆ Π and the sensitivity is similar to Eq. (4), δφ = p P ( + | φ ) P ( −| φ ) | P ′ ( −| φ ) | ≥ p F ( φ ) , (21)where P ( + | φ ) ≡ P k P ( k | φ ) = − P ( −| φ ), and F ( φ ) is the CFIof the multi-outcome homodyne measurement, F ( φ ) = k f X k = − k f (cid:2) P ′ ( k | φ ) (cid:3) P ( k | φ ) + (cid:2) P ′ ( −| φ ) (cid:3) P ( −| φ ) . (22)In Fig. 3(b), we show the sensitivity δφ and its ultimate lowerbound δφ CRB = / p F ( φ ) against φ . One can see that the sen-sitivity δφ diverges at certain values of φ (e.g., φ dark ≈ ± b /α ),but δφ CRB does not (see the red dashed line). The singularityof δφ means that complete no phase information can be in-ferred. Usually, it takes place when the slope of signal is van-ishing, i.e., P ′ ( −| φ ) =
0. On the other hand, δφ CRB divergeswhen F ( φ ) =
0, i.e., P ′ ( −| φ ) = P ′ ( k | φ ) = φ . As depicted by Figs. 3(a) and 3(b), the sensitivity δφ di-verges at the extreme values of the signal; While for δφ CRB ,however, the divergences only occur at the peaks of the sig-nal. The reason why the sensitivity shows a series of extra divergences at the minima of the output signal could be un-derstood by the fact that the signal is a sum of highly sharpphase distribution as Fig. 2, weighted by positive eigenvalues µ k = +
1. It is therefore important to investigate the depen-dence of the signal and the sensitivity on di ff erent choices ofthe eigenvalues, which has not been addressed in Ref. [17]. FIG. 3: Output signal h ˆ Π i φ and phase sensitivity δφ for µ − = ff erent choices of { µ k } . The parameters: ¯ n = a = / b = .
8, and k f =
2. (a) and (b): µ k = k ’s. (c) and (d): { µ − , µ − , µ , µ , µ } = {− . , . , . , − . , . } . (e) and(f): { µ − , µ − , µ , µ , µ } = { , − , , − , } . The red dashed lines: theCRB 1 / p F ( φ ). The vertical lines: locations of the first dark points φ dark ≈ ± b / √ ¯ n . The horizontal lines in right panel: the best sensitiv-ity δφ min ≈ . / √ ¯ n . For a general multi-outcome measurement, di ff erentchoices of { µ k } correspond to di ff erent observables ˆ Π = P k µ k ˆ Π k . Therefore, both the signal and the sensitivity de-pend on the eigenvalues { µ k } . To see it clearly, we take ran-dom numbers of { µ k } , ranged from − +
1. As shown bythe blue solid lines of Figs. 3(c) and 3(d), one can see thatboth the signal and the sensitivity are di ff erent with that ofFigs. 3(a) and 3(b). However, near φ =
0, the sensitivityin Fig. 3(d) coincides with that of Fig. 3(b), indicating thatEq. (21) still works to predict the best sensitivity. Remark-ably, one can also see that the sensitivity does not diverge atthe locations of the arrows. The singularity of δφ can be fur-ther suppressed using alternating signs of { µ k } for neighbouroutcomes (i.e., µ k ± = − µ k ). As shown in Figs. 3(e) and 3(f),one can see δφ ≈ δφ CRB = / p F ( φ ) within the whole phaseinterval. In other words, the inversion estimator associatedwith the so-chosen observable almost saturates the CRB.We now adopt Monte Carlo method to simulate the abovemulti-outcome measurement [24, 34]. Specially, we first gen-erate N random numbers { ξ , ξ , ..., ξ N } , according to the oc-currence probabilities { P ( k | φ ) } at a given φ , where ξ i (for i = · · · , N ) can be regarded as the outcome k = − k f , pro-vided 0 ≤ ξ i ≤ P ( − k f | φ ). It can be regarded as the outcome k = − k f +
1, provided P ( − k f | φ ) < ξ i ≤ P ( − k f | φ ) + P ( − k f + | φ ),and so on. If ξ i obeys P ( − k f | φ ) + P ( − k f + | φ ) + · · · + P ( k f | φ ) <ξ i ≤
1, then we treat it as the outcome “ − ”. In this way, we ob-tain the occurrence numbers of all the outcomes {N k } . Next,we repeat the above process for any value of φ ∈ ( − π, π ) andobtain the occurrence frequencies {N k / N} . As depicted byFig. 2, we show the averaged N k / N (the solid circles) andits standard deviation (the bars) after M =
10 replicas of theabove simulations. With large enough N ( = n = α = b = .
2, which gives k f = { µ k } and vanishing µ − .In real experiments, the dependence of P ( k | φ ) on φ is ob-tained from replicas of N independent measurements at eachgiven phase shift. This comprises a calibration of the inter-ferometer. With all known occurrence probabilities and thesignal, one can infer unknown value of φ via the phase es-timation. As the simplest protocol, we adopt the inversionphase estimator φ inv = g − ( P k µ k N k / N ), where g − is the in-verse function of the average signal g ( φ ) = h ˆ Π i φ and N k / N is the occurrence frequency of the k th outcome in a single N independent measurements. After M replicas, one can obtainthe estimators { φ (1)inv , φ (2)inv , ..., φ ( M )inv } . The mean value of the es-timators ¯ φ inv = h φ ( i )inv i s and its standard deviation are shownin the inset of Fig. 4(b), where the statistical average is de-fined as h ( · · · ) i s ≡ P Mi = ( · · · ) / M . The standard deviation (thebars) is larger than ( ¯ φ inv − φ ) indicates that the inversion es-timator is unbiased; see the inset of Fig. 4(b). For an e ff ec-tive single-shot measurement, the phase uncertainty is definedby σ = √N q h ( φ ( i )inv − φ ) i s , which almost follows the lowerbound of phase sensitivity δφ CRB = / p F ( φ ); see the solidcircles of Fig. 4(b). FIG. 4: Output signal h ˆ Π i φ and phase sensitivity δφ for alternatingsigns of { µ k } and the parameters: ¯ n = a = / b = .
2, and k f = φ inv obtained with N =
200 and M = δφ min ≈ . / √ ¯ n . Inset: Di ff erence between theaverage value of the inversion estimator ¯ φ inv = P Mi = φ ( i )inv / M and thetrue value of phase shift φ . The bars are the standard deviations ofthe estimators { φ (1)inv , φ (2)inv , ..., φ ( M )inv } . Finally, it should be mentioned that we have discussed theachievable sensitivity close to the shot-noise limit with thecoherent-state input. However, it is possible to surpass thisclassical limit once the interferometer is fed by nonclassicalstates of light. Recently, Sch¨afermeier et al [43] have demon-strated that both the resolution and the sensitivity can surpasstheir classical limits using the binary-outcome homodyne de-tection, where a coherent state and a squeezed vacuum areused as the input. For any binary-outcome measurement, wehave shown that the phase estimator by inverting the averagesignal is good enough to saturate the CRB [32–34]. For amulti-outcome detection, the inversion estimator is less opti-mal due to the divergence of the phase sensitivity [23, 24]. We show here that the singularity can be suppressed when the sig-nal is a sum of positive P ( k | φ ), weighted by alternating signsof eigenvalues. V. CONCLUSION
In summary, we have considered quantum phase estima-tion with multi-outcome homodyne detection in the coherent-state light Mach-Zehnder interferometer. Compared with theultimate phase sensitivity determined by the classical Fisherinformation, we show that (i) the phase sensitivity shows aseries of extra divergences at the minima of the output sig-nal; (ii) these extra divergences can be removed by using ob-servables whose eigenvalues associated with neighboring out-comes have alternating signs. This result provides a familyof nearly optimal inversion estimators that almost saturate theCram´er-Rao bound over the whole range of phases. We fur-ther perform numerical simulations using such observablesand demonstrate that phase uncertainty of the inversion esti-mator almost follows the Cram´er-Rao bound. Our method forremoving extra divergences of the phase sensitivity may alsobe applicable to other kinds of multi-outcome measurements.
Funding
Science Foundation of Zhejiang Sci-Tech University(18062145-Y); National Natural Science Foundation of China(NSFC) (91636108, 11775190, and 11774021); the NSFCprogram for “Scientific Research Center” (U1530401).
Acknowledgments
We thank Professor C. P. Sun for helpful discussions. [1] M. Born and E. Wolf,
Principle of Optics (Cambridge Univer-sity, 1999)[2] C. M. Caves, “Quantum-mechanical noise in an interferome-ter,” Phys. Rev. D , 1693 (1981).[3] D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen,“Squeezed atomic states and projection noise in spectroscopy,”Phys. Rev. A. , 67 (1994).[4] D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini,W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “To-ward Heisenberg-Limited Spectroscopy with Multiparticle En-tangled States,” Science , 1476-1478 (2004).[5] J. P. Dowling, “Quantum optical metrology-the lowdown onhigh-N00N states,” Contemp. Phys. , 125-143 (2008).[6] A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P.Williams, and J. P. Dowling, “Quantum Interferometric OpticalLithography: Exploiting Entanglement to Beat the Di ff ractionLimit,” Phys. Rev. Lett. , 2733 (2000).[7] M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature , 161-164 (2004).[8] P. Walther, J. W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni,and A. Zeilinger, “de Broglie wavelength of a non-local four-photon state,” Nature , 158-161 (2004).[9] Y. A. Chen, X. H. Bao, Z. S. Yuan, S. Chen, B. Zhao, and J. W.Pan, “Heralded Generation of an Atomic NOON State,” Phys.Rev. Lett. , 043601 (2010).[10] I. Afek, O. Ambar, and Y. Silberberg, “High-NOON States byMixing Quantum and Classical Ligh,” Science , 879-881(2010).[11] D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino,M. W. Mitchell, and S. Pirandola, “Quantum-enhanced mea-surements without entanglement,” Rev. Mod. Phys. , 035006(2018).[12] U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lun-deen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Opti-mal Quantum Phase Estimation,” Phys. Rev. Lett. , 040403(2009).[13] Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presenceof photon loss,” Phys. Rev. A , 043832 (2013).[14] K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J.Pryde, J. L. O’Brien, and A. G. White, “Time-Reversal andSuper-Resolving Phase Measurements,” Phys. Rev. Lett. ,223601 (2007).[15] Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, andJ. P. Dowling, “Super-resolution at the shot-noise limit with co-herent states and photon-number-resolving detectors,” J. Opt.Soc. Am. B , A170-A174 (2010).[16] L. Cohen, D. Istrati, L. Dovrat, and H. S. Eisenberg, “Super-resolved phase measurements at the shot noise limit by paritymeasurement,” Opt. Express , 11945-11953 (2014).[17] E. Distante, M. Jeˇzek, and U. L. Andersen, “Deterministic Su-perresolution with Coherent States at the Shot Noise Limit,”Phys. Rev. Lett. , 033603 (2013).[18] P. R. Bevington, Data Reduction and Error Analysis for thePhysical Sciences (McGraw-Hill, 1969).[19] C. W. Helstrom,
Quantum Detection and Estimation Theory (Academic, 1976).[20] S. L. Braunstein and C. M. Caves, “Statistical distance and thegeometry of quantum states,” Phys. Rev. Lett. , 3439 (1994).[21] S. L. Braunstein, C. M. Caves, and G. J. Milburn, “GeneralizedUncertainty Relations: Theory, Examples, and Lorentz Invari-ance,” Ann. Phys. (N.Y.) , 135-173 (1996).[22] M. G. A. Paris, “Quantum estimation for quantum technology,”Int. J. Quantum. Inform. , 125-137 (2009).[23] B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1)interferometers,” Phys. Rev. A , 4033 (1986).[24] L. Pezz´e, A. Smerzi, G. Khoury, J. F. Hodelin, and D.Bouwmeester, “Phase Detection at the Quantum Limit withMultiphoton Mach-Zehnder Interferometry,” Phys. Rev. Lett. , 223602 (2007).[25] J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen,“Optimal frequency measurements with maximally correlatedstates,” Phys. Rev. A , R4649 (1996).[26] C. C. Gerry, “Heisenberg-limit interferometry with four-wavemixers operating in a nonlinear regime,” Phys. Rev. A ,043811 (2000).[27] C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear in-terferometer as a resource for maximally entangled photonicstates: Application to interferometry,” Phys. Rev. A , 013804(2002).[28] C.C. Gerry and J. Mimih, “The parity operator in quantum op-tical metrology,” Contemp. Phys. , 497-511 (2010).[29] P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick,S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrologywith Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. , 103602 (2010).[30] A. Chiruvelli and H. Lee, “Parity measurements in quantumoptical metrology,” J. Mod. Opt. , 945-953 (2011).[31] K. P. Seshadreesan, S. Kim, J. P. Dowling, and H. Lee, “Phaseestimation at the quantum Cram-Rao bound via parity detec-tion,” Phys. Rev. A , 043833 (2013).[32] X. M. Feng, G. R. Jin, and W. Yang, “Quantum interferometrywith binary-outcome measurements in the presence of phasedi ff usion,” Phys. Rev. A. , 013807 (2014).[33] L. Ghirardi, I. Siloi, P. Bordone, F. Troiani, and M. G. A. Paris,“Quantum metrology at level anticrossing,” Phys. Rev. A ,012120 (2018).[34] G. R. Jin, W. Yang, and C. P. Sun, “Quantum-enhanced mi-croscopy with binary-outcome photon counting,” Phys. Rev. A , 013835 (2017).[35] J. D. Zhang, Z. J. Zhang, L. Z. Cen, J. Y. Hu, and Y. Zhao, “De-terministic super-resolved estimation towards angular displace-ments based upon a Sagnac interferometer and parity measure-ment,” arXiv:1809.04830 (2018).[36] J. D. Zhang, Z. J. Zhang, L. Z. Cen, S. Li, Y. Zhao, and F. Wang,“Super-resolution and super-sensitivity of entangled squeezedvacuum state using optimal detection strategy,” Chin. Phys. B , 094204 (2017).[37] Z. J. Zhang, T. Y. Qiao, L. Z. Cen, J. D. Zhang, F. Wang, and Y.Zhao, “Optimal quantum detection strategy for super-resolvingangular-rotation measurement,” Appl. Phys. B , 148 (2017).[38] Q. Wang, L. L. Hao, H. X. Tang, Y. Zhang, C. H. Yang, X.Yang, L. Xu, and Y. Zhao, “Super-resolving quantum LiDARwith even coherent states sources in the presence of loss andnoise,” Phys. Lett. A , 3717 (2016).[39] Q. Wang, L. L. Hao, Y. Zhang, L. Xu, C. H. Yang, X. Yang, andY. Zhao, “Super-resolving quantum lidar: entangled coherent-state sources with binary-outcome photon counting measure-ment su ffi ce to beat the shot-noise limit,” Opt. Express ,5045-5056 (2016).[40] C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).[41] K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling,“Parity detection achieves the Heisenberg limit in interferome-try with coherent mixed with squeezed vacuum light,” New J.Phys. , 083026 (2011).[42] Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhancedinterferometry using squeezed thermal states and even or oddstates,” Phys. Rev. A , 053822 (2014).[43] C. Schafermeier, M. Jezex, L. S. Madsen, T. Gehring, and U.L. Andersen, “Deterministic phase measurements exhibitingsuper-sensitivity and super-resolution,” Optica5