Phase sensitivity of a Mach-Zehnder interferometer with single-intensity and difference-intensity detection
aa r X i v : . [ qu a n t - ph ] N ov Phase sensitivity of a Mach-Zehnder interferometer with single-intensity anddifference-intensity detection
Stefan Ataman
Extreme Light Infrastructure - Nuclear Physics (ELI-NP)30 Reactorului Street, 077125 Bucharest-M˘agurele, Romania ∗ Anca Preda
Faculty of Physics, University of Bucharest, 077125 Bucharest-M˘agurele, Romania † Radu Ionicioiu
Horia Hulubei National Institute of Physics and Nuclear Engineering, 077125 Bucharest–M˘agurele, Romania ‡ (Dated: November 7, 2018)Interferometry is a widely-used technique for precision measurements in both classical and quan-tum contexts. One way to increase the precision of phase measurements, for example in a Mach-Zehnder interferometer (MZI), is to use high-intensity lasers. In this paper we study the phasesensitivity of a MZI in two detection setups (difference intensity detection and single-mode intensitydetection) and for three input scenarios (coherent, double coherent and coherent plus squeezed va-cuum). For the coherent and double coherent input, both detection setups can reach the quantumCram´er-Rao bound, although at different values of the optimal phase shift. The double coherentinput scenario has the unique advantage of changing the optimal phase shift by varying the inputpower ratio. I. INTRODUCTION
Precision measurements are one of key elements inboth science and technology. Indeed, many importantdiscoveries have been made due to the improvement ofmeasurement techniques. More sensitive instruments,like microscopes and telescopes, were paramount in dis-covering new phenomena and in verifying or falsifyingtheoretical predictions. Thus, improving the measure-ment sensitivity is a crucial factor driving the advance-ment of science and technology alike.A very sensitive, hence widely used measurement tech-nique is interferometry, with the Mach-Zehnder interfe-rometer (MZI) as a standard tool. Thus, understanding,controlling and improving the limits of phase sensitivityof an MZI is an active field of research, both theoreticallyand experimentally [1–5].Classically, the sensitivity ∆ ϕ of a measurement isbounded by the standard quantum limit (SQL), alsoknown as the shot-noise limit [4, 6, 7]. This is givenby ∆ ϕ SQL ∼ / p h N i , where h N i is the average numberof photons used to probe the system.It was soon realized that squeezed states of light [8–10] can improve the phase sensitivity of an interferometer[11, 12]. Indeed, this technique has been tested and willbe used at the LIGO detector in the future [3, 13]. In aseminal article Caves [2] has showed that squeezed lightcan improve the phase sensitivity of an interferometerbelow the shot-noise limit. Experimental demonstration ∗ [email protected] † [email protected] ‡ [email protected] with a MZI [14] soon followed, proving the usability ofthe concept in practical measurements. Over the nextdecades both theoretical and experimental studies haveshowed how to improve the sensitivity of a MZI fed byboth a coherent and a squeezed vacuum input [15–18].In a quantum context, however, the phase sensitiv-ity is bounded by the Heisenberg limit [4, 11, 19–21],∆ ϕ HL ∼ / h N i , and this limit is fundamental [22]. Theso-called NOON states [11, 19, 21] saturate this limit,while separable states obey the SQL [21].The Heisenberg limit can be achieved in a MZI by in-jecting a coherent state in one port and squeezed vacuuminto the other [23], if roughly half of the input power goesinto squeezing. This result was confirmed by Lang andCaves [24, 25] who, moreover, showed the input state tobe optimal for the class of coherent ⊗ squeezed vacuumtype of states.Other scenarios considering active SU(1,1) type inter-ferometers were studied in [26, 27]. The authors showeda Heisenberg sensitivity limit achievable in a MZI withsqueezed coherent light in both inputs, if the squeezingpower is roughly 1 / ϕ , as shown by Pezz´eand Smerzi [28]. Moreover, this can be also achieved forthe coherent ⊗ squeezed vacuum input [23].There are several detection methods used to measurethe output of a MZI [29], however in this paper we shallfocus only on two. In the difference intensity detection scheme, as the name suggests, we have two detectors (onefor each output of the MZI) and we measure the differ-ence of the two photo-currents. In the single-mode inten-sity detection scheme we measure only one photo-countof the two. For low-power setups the difference intensitydetection scheme is experimentally preferred. Here weshow that for high input power, the single-mode detec-tion scheme is superior to the difference intensity detec-tion scheme.We also consider the double coherent input case in thispaper. To our best knowledge, this scenario was onlydiscussed by Shin et al. [30]. Moreover, we show thatthis scenario can have a practical interest under certaincircumstances.Although Heisenberg limited metrology has been aconstant theoretical and experimental challenge, thisfavourable scenario happens for NOON states [11], wherethe current record in the number of photons remains verylow [31, 32] or at extremely low laser powers coupled withthe highest squeezing factors achievable today.In this paper we are not interested in pursuing theHeisenberg limit at all costs. Instead we focus on scenar-ios where the squeezing factor is a limited resource, butthe intensity of the coherent source is not constrained[3, 33]. This setup is better suited to present-day exper-iments.The paper is structured as follows. In Section II we in-troduce our parameter estimation method, experimentalsetup, field operator transformations and output opera-tor calculations. We also review the Cram´er-Rao boundand the Fisher information approach. In Section III weconsider a coherent ⊗ vacuum input scenario and evalu-ate its phase sensitivity, comparing both output detectionscenarios with the quantum Cram´er-Rao bound. In Sec-tions IV and V we consider a coherent ⊗ coherent and, re-spectively, coherent ⊗ squeezed vacuum input scenarios.We evaluate their respective phase sensitivities, comparethe output detection scenarios and assess them in respectwith the quantum Cram´er-Rao bound. All three scenar-ios are thoroughly discussed and finally, conclusions aredrawn in Section VI. II. MZI SETUP: DETECTION SENSITIVITIESA. Parameter estimation: a short introduction
We now briefly overview the problem of parameter es-timation in quantum mechanics. An experimentally ac-cessible Hermitian operator ˆ O depends on the parameter ϕ – in our case this is the phase shift in a Mach-Zehnderinterferometer; by itself ϕ may or may not be an observ-able. The average of the operator is h ˆ O ( ϕ ) i = h ψ | ˆ O ( ϕ ) | ψ i (1) FIG. 1. The physical intuition behind equation (3). Thesensitivity ∆ ϕ depends on both the displacement of the aver-age h ˆ O i (due to a change of the parameter ϕ ) and the stan-dard deviation ∆ ˆ O . Here we implicitely assume ∆ ˆ O ( ϕ ) =∆ ˆ O ( ϕ + ∆ ϕ ). where | ψ i is the wave-function of the system. A smallvariation δϕ of the parameter ϕ induces a change h ˆ O ( ϕ + δϕ ) i ≈ h ˆ O ( ϕ ) i + ∂ h ˆ O ( ϕ ) i ∂ϕ δϕ (2)The difference h ˆ O ( ϕ + δϕ ) i − h ˆ O ( ϕ ) i is experimentallydetectable if h ˆ O ( ϕ + δϕ ) i − h ˆ O ( ϕ ) i ≥ ∆ ˆ O ( ϕ ) (3)where ∆ ˆ O ( ϕ ) := [ h ˆ O i − h ˆ O i ] / is the standard devia-tion of ˆ O . One can intuitively understand this conditiongraphically, see Fig. 1. The value of δϕ that saturatesthe inequality (3) is called sensitivity and is denoted by∆ ϕ : ∆ ϕ = ∆ ˆ O (cid:12)(cid:12) ∂∂ϕ h ˆ O i (cid:12)(cid:12) (4)This equation will be pivotal in the following sections. B. Transformations of the field operators
Consider a Mach-Zehnder interferometer composed oftwo mirrors M , and two balanced beam splitters BS , ;the transmission (reflection) coefficient of BS , is T =1 / √ R = i/ √ (cid:26) ˆ a † = − sin (cid:0) ϕ (cid:1) ˆ a † + cos (cid:0) ϕ (cid:1) ˆ a † ˆ a † = cos (cid:0) ϕ (cid:1) ˆ a † + sin (cid:0) ϕ (cid:1) ˆ a † (5)where we ignored global phases. We assume the outputports 4 and 5 are connected to ideal detectors.Usually the input state | ψ in i is given and we calcu-late either the output photo-currents or the differencebetween the output photo-currents. BS M M BS ϕ D D FIG. 2. The two detection schemes for the Mach-Zehnderinterferometer we analyse here. The input state | ψ in i is uni-tarily transformed to the output | ψ out i . The parameter wewant to estimate via a suitable observable is the phase differ-ence ϕ between the two arms of the MZI. In the following we denote by ϕ the total phase shiftinside the interferometer. The total phase has two parts:(i) the unknown (e.g. sensor-generated) phase shift ϕ s ,which is the quantity we want to measure, and (ii) theexperimentally-controllable part ϕ exp : ϕ = ϕ s + ϕ exp (6)We assume that | ϕ s | ≪ | ϕ | so that in order to have thebest performance, the experimenter must adjust ϕ exp asclose as possible to the optimal phase shift, ϕ opt . C. Output observables
Each detection scheme has associated an observablecharacterising the measurement setup. We will discusstwo measurement strategies: (i) difference intensity de-tection and (ii) single-mode intensity detection scheme.For Mach-Zehnder interferometers, a well-known ap-proach of calculating the phase sensitivity is Schwinger’sscheme based on angular momentum operators [4, 34].Although this method gives faster results for a differ-ence intensity detector setup, it is not well-suited forthe single-mode intensity detection scheme we investi-gate here. Alternatively one can use a Wigner-functionbased method [29]. In this paper we use a “brute-force”calculation based on the field operator transformations(5).
1. Difference intensity detection scheme
In the first detection strategy we calculate the differ-ence between the output photo-currents (i.e., detectors D and D , see Fig. 2). Thus, the observable conveyinginformation about the phase ϕ isˆ N d ( ϕ ) = ˆ a † ˆ a − ˆ a † ˆ a (7) Using the field operator transformations eqs. (5) we have h ˆ N d i = cos ϕ (cid:16) h ˆ a † ˆ a i − h ˆ a † ˆ a i (cid:17) − sin ϕ (cid:16) h ˆ a ˆ a † i + h ˆ a † ˆ a i (cid:17) (8)where the expectation values are calculated w.r.t. theinput state | ψ in i . To estimate the phase sensitivity ineq. (4) we need the absolute value of the derivative (cid:12)(cid:12)(cid:12)(cid:12) ∂ h ˆ N d i ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) = | sin ϕ ( h ˆ a † ˆ a i − h ˆ a † ˆ a i ) − cos ϕ ( h ˆ a ˆ a † i + h ˆ a † ˆ a i ) | (9)In the following sections we will calculate thisfor various input states. The standard deviation∆ ˆ N d = [ h ˆ N d i − h ˆ N d i ] / follows from eq. (8) and Ap-pendix B.
2. Single-mode intensity detection scheme
We now consider the single-mode intensity detectionscheme, i.e., we have only one detector coupled at theoutput port 4, see Fig. 2. Thus the operator of interestis ˆ N = ˆ a † ˆ a . From eq. (5) we have h ˆ N i = sin (cid:16) ϕ (cid:17) h ˆ a † ˆ a i + cos (cid:16) ϕ (cid:17) h ˆ a † ˆ a i− sin ϕ h ˆ a ˆ a † i − sin ϕ h ˆ a † ˆ a i (10)and the absolute value of its derivative w.r.t. ϕ is (cid:12)(cid:12)(cid:12)(cid:12) ∂ h ˆ N i ∂ϕ (cid:12)(cid:12)(cid:12)(cid:12) = 12 (cid:12)(cid:12)(cid:12)(cid:12) sin ϕ (cid:16) h ˆ a † ˆ a i − h ˆ a † ˆ a i (cid:17) − cos ϕ (cid:16) h ˆ a ˆ a † i + h ˆ a † ˆ a i (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) (11)As before, the standard deviation ∆ ˆ N follows fromeq. (10) and Appendix B. D. Parameter estimation via Fisher information
The Fisher information is a very elegant approach offinding the best-case solution of parameter estimation[35]. The lower bound for the estimation of a parameter ϕ is given by the Cram´er-Rao bound (CRB) [4, 27, 36]∆ ϕ ≥ p F ( ϕ ) (12)where F ( ϕ ) is the Fisher information. The Fisher in-formation F ( ϕ ) is maximised by the quantum Fisher in-formation (QFI) [35] F ( ϕ ) ≤ H ( ϕ ). This leads to thequantum Cram´er-Rao bound (QCRB)∆ ϕ ≥ p H ( ϕ ) (13)Here H ( ϕ ) = Tr h ˆ ρ ϕ ˆ L ϕ i and ˆ ρ ϕ = | ψ ϕ ih ψ ϕ | is thedensity matrix of our system (see Fig. 3); ˆ L ϕ is BS M ϕ M ea s u r e m en t FIG. 3. In the Fisher information approach the Mach-Zehnderinterferometer is considered up to the phase shift operation(in our case ˆ U ( ϕ ) = e − iϕ ˆ a † ˆ a ) and the detection scheme com-pletely disregarded; we have | ψ ϕ i = ˆ U ( ϕ ) ˆ U BS | ψ in i , whereˆ U BS is the unitary corresponding to BS . the symmetric logarithmic derivative defined as [4, 35,36] ˆ L ϕ ˆ ρ ϕ + ˆ ρ ϕ ˆ L ϕ = 2 ∂ ˆ ρ ϕ /∂ϕ . Moreover, if the sys-tem is in a pure state the quantum Fisher informa-tion is H ( ϕ ) = 4 (cid:0) h ∂ ϕ ψ ϕ | ∂ ϕ ψ ϕ i − |h ∂ ϕ ψ ϕ | ψ ϕ i| (cid:1) , where | ∂ ϕ ψ ϕ i = ∂ | ψ ϕ i /∂ϕ [26, 27, 36].Importantly, calculating the Fisher information for agiven scenario is not always straightforward, and more-over it can lead to different results [37]. Indeed, an ex-ternal phase reference is needed w.r.t. which are definedthe two phase-shifts, each in one arm of the MZI. Forthis reason, a two parameter estimation problem involv-ing a Fisher matrix is used [24], see Appendix A. Whenan external phase reference is not available, one has topay particular attention on what is actually measurablegiven the experimental setup [38].We stress that in the evaluation of QCRB the detec-tion scheme is disregarded, see Fig. 3. The QCRB willalways be a theoretical, best case scenario, which over-looks practical implementations of the detection stage.In the following, for each case discussed in SectionsIII, IV and V we will compare the practically achievableresults with the QCRB from eq. (13). III. SINGLE COHERENT INPUT
In this section we consider the input port 1 in a coher-ent state | α i while input port 0 is kept “dark” (i.e., inthe vacuum state). The input state is | ψ in i = | α i = ˆ D ( α ) | i (14)where ˆ D ( α ) = e α ˆ a † − α ∗ ˆ a is the displacement operator[6, 7, 10]. A. Difference intensity detection scheme
The observable we measure is the difference in thephoto-currents at the outputs 4 and 5, namely the av-erage value of ˆ N d , eq. (7). For the input state (14) we FIG. 4. Phase sensitivity for the single-mode (solid blue line)and difference (dashed red line) intensity detection setupscompared to the quantum Cram´er-Rao bound (thick dashedline) for a single coherent input with | α | = 10 . Both con-figurations reach the Cram´er-Rao bound at their respectiveoptimal phase shifts. find h ˆ N d i = cos ϕ | α | and, using equation (B1), the out-put variance is found to be ∆ ˆ N d = | α | . Consequently,the phase sensitivity of a Mach-Zehnder interferometerdriven by a single coherent source is∆ ϕ diff = 1 | sin ϕ || α | = 1 | sin ϕ | p h N i (15)where the average number of photons is h N i = | α | andthis is the well-known shot noise limit or standard quan-tum limit [4, 28]. B. Single-mode intensity detection scheme
In a single-mode intensity detection setup the averageof the output observable ˆ N gives h ˆ N i = cos (cid:16) ϕ (cid:17) | α | (16)The variance of ˆ N follows from eqs. (B2) and (16), giving∆ ˆ N = cos ( ϕ/ | α | . Thus, the phase sensitivity inthe single-mode intensity detection case is∆ ϕ sing = 1 | sin (cid:0) ϕ (cid:1) || α | = 1 | sin (cid:0) ϕ (cid:1) | p h N i (17) C. Discussion: the quantum Cram´er-Rao bound
For a single input coherent state, the QCRB in equa-tion (13) is [4, 37]∆ ϕ QCRB ≥ | α | = 1 p h N i (18)Both detection schemes reach this limit, but at differentvalues of the total internal phase shift, as depicted inFig. 4.In the differential detection scheme, the optimal sen-sitivity is reached for | sin ϕ | = 1, i.e., ϕ optdiff = π/ kπ , k ∈ Z , see eq. (15). This implies equal output power atthe two outputs (4 and 5). There is no “dark” port inthe case of the difference intensity detection. This canbe a major drawback if one uses a high input power inorder to lower the sensitivity.For single-mode intensity detection, the phase sensi-tivity (17) reaches the QCRB at ϕ optsing = π + 2 kπ , k ∈ Z ,see Fig. 4. This means that the output 4 is a “dark” port.This is a clear advantage for high input power since wecan use extremely sensitive PIN photo-diodes [39]. IV. DOUBLE COHERENT INPUT
An interesting situation arises if we apply a coherentsource in each input port of the interferometer: | ψ in i = | α β i = ˆ D ( α ) ˆ D ( β ) | i (19)where the displacement operator at input port 0 isˆ D ( β ) = e β ˆ a † − β ∗ ˆ a . Here α = | α | e iθ α , β = | β | e iθ β and∆ θ = θ α − θ β is the phase difference between the twoinput lasers. A. Differential detection scheme
Using the input state given in equation (19), the aver-age value of the operator ˆ N d is h ˆ N d i = cos ϕ (cid:0) | α | − | β | (cid:1) − ϕ | αβ | cos ∆ θ (20)After a straightforward computation, the variance is∆ ˆ N d = | α | + | β | = | α | (cid:0) ̟ (cid:1) (21)where ̟ := | β | / | α | . The phase sensitivity for a doublecoherent input is∆ ϕ diff = √ ̟ | α | (cid:12)(cid:12) sin ϕ (1 − ̟ ) + 2 cos ϕ̟ cos ∆ θ (cid:12)(cid:12) (22)We will discuss this result in Section IV C. B. Single-mode intensity detection scheme
In the single-mode intensity detection setup, the aver-age of our output observable is h ˆ N i = | α | (cid:16) sin (cid:16) ϕ (cid:17) ̟ + cos (cid:16) ϕ (cid:17) − sin ϕ̟ cos ∆ θ (cid:17) (23) The variance ∆ ˆ N can be computed as before; alter-natively, we notice that at the output port 4 we havea coherent state, therefore the variance is equal to itsaverage value, ∆ ˆ N = h ˆ N i (24)Thus, the phase sensitivity of a Mach-Zehnder with twoinput coherent sources and a single-mode intensity detec-tion scheme is∆ ϕ sing = q sin (cid:0) ϕ (cid:1) ̟ + cos (cid:0) ϕ (cid:1) − sin ϕ̟ cos ∆ θ | α | (cid:12)(cid:12)(cid:12) sin ϕ (1 − ̟ ) + cos ϕ̟ cos ∆ θ (cid:12)(cid:12)(cid:12) (25) C. Discussion: the quantum Cram´er-Rao bound
For the double coherent input, the quantum Cram´er-Rao bound is (see Appendix A)∆ ϕ QCRB ≥ | α | q ̟ − ̟ ̟ sin ∆ θ (26)Therefore the best sensitivity is achieved when thetwo input lasers are in phase, ∆ θ = 0, resulting in∆ ϕ QCRB = 1 / ( | α |√ ̟ ).In the case of differential detection, one can show thatan optimum phase shift exists, ϕ optdiff = ± arctan (cid:18) | − ̟ | ̟ | cos ∆ θ | (cid:19) + kπ (27)with k ∈ Z and ϕ optdiff brings the sensitivity form equation(22) to the QCRB.For the single-mode intensity detection scheme, if thetwo input lasers are in phase (∆ θ = 0), the optimumphase shift is ϕ optsing = ± (cid:18) ̟ (cid:19) + 2 kπ (28)with k ∈ Z and substituting this value into equation (25)gives the QCRB from equation (26).For comparison, the sensitivity of homodyne detectionwith ∆ θ = 0 is∆ ϕ H ≥ | α sin ϕ + β cos ϕ | (29)The phase sensitivity of a MZI with a double-coherentinput is shown in Fig. 5, for both single-mode and differ-ence intensity detection schemes. As already discussed,we can reach the QCRB in both scenarios.Compared to the single-coherent input, the double-coherent case has an important advantage: we can tunethe value of ϕ optsing at which the sensitivity reaches theQCRB. Experimentally, this can be achieved by vary-ing the power ratio of the two input coherent sources. FIG. 5. Phase sensitivity for the single-mode intensity (solidblue line) and difference intensity (dashed red line) detectionsetups versus the phase shift ϕ . Both detection schemes reachthe quantum Cram´er-Rao bound (thick dashed line). We usedthe parameters | α | = 10 , ̟ = 0 . θ = 0. This avoids the use of piezos or other mechanical-basedmethods to induce phase shifts. As a consequence, ourproposal reduces mechanical vibrations, noise or mis-alignments.In the high-power regime this ability is practically use-ful for a single-mode intensity detection scenario. Indeed,at the optimal phase shift the output 4 is a dark port, i.e., h ˆ N (cid:16) ϕ optsing (cid:17) i →
0, which is exactly the desired situationw.r.t. the photo-detectors in the high-power regime.
V. COHERENT PLUS SQUEEZED VACUUMINPUT
The paradigmatic input state which beats the SQL isthe coherent ⊗ squeezed vacuum | ψ in i = | α r i = D ( α ) S ( r ) | i (30)The squeezed vacuum state is obtained by applying thesqueezing operator S ( ξ ) = e [( ξ ∗ ) ˆ a − ξ (ˆ a † ) ] / [6, 8, 10]with ξ = re iθ . For simplicity, in the following we take θ = 0, hence ξ = r ∈ R + . This input state is of con-siderable practical interest as it was shown to beat theSQL [2, 11, 12, 24, 25], a prediction amply confirmed byexperiments [3, 13, 14, 40]. A. Difference intensity detection scheme
With the coherent ⊗ squeezed vacuum input (30) theaverage of ˆ N d in eq.(7) is h ˆ N d i = cos ϕ (cid:0) | α | − sinh r (cid:1) (31)The variance ∆ ˆ N d can be computed using equations (7)and (B1) with the input state given in (30) and yields∆ ˆ N d = cos ϕ (cid:18) sinh r | α | (cid:19) + sin ϕ (cid:0) sinh r + | α | e − r (cid:1) + sin ϕ | α | sinh 2 r (1 − cos (2 θ α )) (32)For the difference intensity detection scheme, the bestachievable phase sensitivity of a MZI with coherent ⊗ squeezed vacuum is∆ ϕ diff = r(cid:16) | α | + sinh r (cid:17) cot ϕ + sinh r + | α | e − r + | α | sinh 2 r (1 − cos (2 θ α )) | sinh r − | α | | (33)The last term in the numerator of equation (33) is the input noise enhancement due to the misalignment of thecoherent input with respect to the squeezed vacuum (whose phase we considered to be zero, for simplicity). Thesensitivity is minimized if the phase of the coherent light is θ α = 0 (hence α ∈ R ):∆ ϕ diff = r(cid:16) α + sinh r (cid:17) cot ϕ + sinh r + α e − r | α − sinh r | (34)expression that can be found in the literature [4, 23]. B. Single-mode intensity detection scheme
For the input state (30) we have h ˆ N i = sin (cid:16) ϕ (cid:17) sinh r + cos (cid:16) ϕ (cid:17) | α | (35)and the variance is ∆ ˆ N = sin (cid:16) ϕ (cid:17) sinh r (cid:16) ϕ (cid:17) cos (cid:16) ϕ (cid:17) sinh r + cos (cid:16) ϕ (cid:17) | α | + sin (cid:16) ϕ (cid:17) cos (cid:16) ϕ (cid:17) | α | e − r + sin ϕ r | α | (1 − cos 2 θ α ) (36)In the single-mode intensity detection setup, the best achievable sensitivity of a MZI fed by coherent ⊗ squeezedvacuum is ∆ ϕ sing = r tan ( ϕ ) sinh r + sinh r + | α | tan ( ϕ ) + | α | e − r + sinh 2 r | α | (1 − cos 2 θ α ) (cid:12)(cid:12)(cid:12) | α | − sinh r (cid:12)(cid:12)(cid:12) (37)The last term of the square root is again the contribu-tion of the misalignment of the coherent input from port1 with the squeezed vacuum from port 0. The sensi-tivity is maximized for cos 2 θ α = 1, thus θ α = 0 andhence α ∈ R . Therefore, we have now the best achiev-able sensitivity for the squeezed ⊗ coherent input and asingle-mode intensity detection scheme [33]∆ ϕ sing = r tan ( ϕ ) sinh r + sinh r + α tan ( ϕ ) + α e − r (cid:12)(cid:12) α − sinh r (cid:12)(cid:12) (38) C. Discussion: the quantum Cram´er-Rao bound
The quantum Cram´er-Rao bound for the coherent ⊗ squeezed vacuum input is [23–25, 37]∆ ϕ QCRB ≥ q | α | e r + sinh r (39)and is independent of the phase shift ϕ of the MZI, similarto the coherent input case.For comparison, we briefly mention the sensitivity ofthe homodyne detection scheme [29, 41]∆ ϕ H ≥ e − r | α | (40)a result we will use later.For the differential detection scheme, the optimalsensitivity in eq. (34) is reached for cos ϕ = 0, i.e., ϕ optdiff = π/ kπ , k ∈ Z and we find the best achievablesensitivity ∆ ϕ diff = p sinh r + α e − r | α − sinh r | (41)a result also found in the literature [4, 23, 29].For single-mode intensity detection, the optimal sensi-tivity from equation (38) is reached when ϕ optsing = ± s √ | α | sinh 2 r + 2 kπ (42) FIG. 6. Phase sensitivity for the single-mode (solid blue line)and difference (dashed red line) intensity detection setupscompared to the quantum Cram´er-Rao bound (thick dashedline) versus the phase shift ϕ . Here | α | = 10 and r = 2 . with k ∈ Z ; substituting this value in equation (38) givesthe best achievable sensitivity in the case of a single-modeintensity scheme, namely∆ ϕ sing = p sinh r + √ α sinh 2 r + α e − r (cid:12)(cid:12) α − sinh r (cid:12)(cid:12) (43)This result is identical to the one reported in reference[29], equation (10).In Figs. 6 and 7 we plot the best achievable phase sen-sitivity in the single-mode and difference intensity detec-tion schemes together with the Cram´er-Rao bound fromequation (39) for coherent ⊗ squeezed vacuum input ver-sus the phase shift of the MZI. One notes that both de-tection schemes have an optimum, however none reachesthe QCRB. (Although in Fig. 7 it seems that the redcurve corresponding to the difference intensity detectionscenario reaches the QCRB, it actually stays above it.)While the optimum working point for the difference in-tensity detection scheme is constant, in the transitionfrom the low- (Fig. 6) to the high-power regime (Fig. 7) FIG. 7. Phase sensitivity for the single-mode (solid blue line)and difference (dashed red line) intensity detection setupscompared to the quantum Cram´er-Rao bound (thick dashedline) versus the phase shift ϕ . We use | α | = 10 and r = 2 . | α | , in the low-intensity regime.We take the squeezing factor r = 2 . ϕ opt for each detection scenario. Both detectiontechniques are sub-optimal w.r.t. the QCRB, yielding poorperformance especially when | α | ≈ sinh r . the optimum working point ϕ opt shifts, see eq. (43).In Fig. 8 we show both ∆ ϕ diff and ∆ ϕ sing in the low | α | regime. For | α | ≈ sinh r both detection schemesgive poor results while the QCRB reaches the Heisenberglimit ∆ ϕ QCRB ∼ / h N i , where h N i = | α | + sinh r .This behaviour has been explained by Pezz´e and Smerzi[23] and was attributed to the limited information gainedby these phase estimation techniques, notably due to theignorance of the fluctuation in the number of particles.Ideally one would like to enhance the squeezing factor FIG. 9. Phase sensitivity for the single-mode (solid blue line),difference (dashed red line) intensity detection schemes andthe quantum Cram´er-Rao bound (thick dashed line) versusthe coherent input amplitude | α | , in the high-intensity regime.We consider a squeezing factor r = 2 . | α | grows, bothdetection schemes approach the Cram´er-Rao bound. r as much as possible. However, this is experimentallychallenging [16, 17, 40]. The maximum reported squeez-ing was 15 dB corresponding to r ≈ . r constant and small compared to the amplitudeof the coherent state, implying | α | ≫ | α | ≫ sinh r . In-deed, for | α | ≫ sinh r both detection schemes equal thesensitivity of the homodyne detection in eq. (40). TheQCRB in eq. (39) can be approximated by ∆ ϕ ≈ e − r / | α | .Thus, using squeezing in port 0 brings a factor of e − r over the SQL, therefore the coherent ⊗ squeezed vacuumtechnique remains interesting even for large | α | .In Fig. 9 we plot both both ∆ ϕ diff and ∆ ϕ sing in thehigh | α | regime. We conclude that if | α | ≫ | α | ≫ sinh r ,both detection schemes have a similar sensitivity, closeto the QCRB. This agrees with the results of ref. [29].As already mentioned, the optimum phase shift insidethe Mach-Zehnder interferometer for a difference inten-sity detection scheme is constant, ϕ optdiff = π/ kπ . Inthis case each output port receives roughly half of the(large) input power – this regime is clearly not desirablefor the detectors.For a single-mode intensity detection scheme ϕ optsing isgiven by eq. (42). Moreover, in this scenario the port4 is almost “dark”, and consequently we can use ex-tremely sensitive PIN photodiodes. Almost all power ex-its through the output 5 and can be discarded or used fora feedback loop to stabilise the input laser. This is thecrucial difference between the two schemes in the highintensity regime, similarly to the single and double co-herent input cases (see Sections III and IV).In this paper we did not consider losses or decoherence.The impact of losses on various scenarios has been dis-cussed extensively in the literature [29, 42–44]. In the fol-lowing we briefly discuss their effect in the high-intensityregime. Following [43], in the case of a coherent inputwe can replace α → α √ − σ resulting in a quantumCram´er-Rao bound ∆ ϕ QCRB = 1 / | α |√ − σ . The effectof small losses ( σ ≪
1) is marginal for a coherent source.In the case of coherent ⊗ squeezed vacuum input we have[43] the Cram´er-Rao bound∆ ϕ loss QCRB ≈ p σ + (1 − σ ) e − r q (1 − σ ) | α | + σ (1 − σ ) sinh r (44)The effect of losses is obvious for high squeezing factors,the numerator of equation (44) being reduced to √ σ , thuslosing the exponential factor brought from the squeezingof the input vacuum. Nonetheless, for the high intensityregime discussed in this paper we have | α | / sinh r ≫ σ ≪ (1 − σ ) e − r .For simplicity, we did not consider the 1 / √ m scalingfor all phase sensitivities throughout this paper, where m is the number of repeated experiments.We summarized our results in Table I. VI. CONCLUSIONS
The sensitivity of a Mach-Zehnder interferometer de-pends on both the input state and the detection setup.To achieve the best sensitivity we need to find the opti-mum working point(s) of the interferometer.For single coherent and double-coherent input, bothdetection setups achieve the QCRB, although at differ-ent values of ϕ . The double-coherent input allows us toexperimentally tune the point of maximum sensitivity byadjusting the relative intensity of the two coherent states.This is an advantage over other methods involving me-chanically adjusted setups.In the high intensity regime all three input states (co-herent, double coherent and coherent plus squeezed va-cuum) give similar phase sensitivity, at or close to theQCRB. The optimum working point for the single-modeintensity detection has an almost “dark” output port.This ensures one can use highly-efficient PIN photodi-odes and thus avoid potential problems of over-heatingor blinding the photo-detectors. We expect that our re-sults will lead to more sensitive detection systems forinterferometry in the high-power regime. ACKNOWLEDGMENTS
S.A. acknowledges that this work has been supportedby the Extreme Light Infrastructure Nuclear Physics(ELI-NP) Phase II, a project co-financed by the Roma-nian Government and the European Union through theEuropean Regional Development Fund and the Compet-itiveness Operational Programme (1/07.07.2016, COP,ID 1334). R.I. acknowledges support from a grantof the Romanian Ministry of Research and Innova-tion, PCCDI-UEFISCDI, project number PN-III-P1-1.2-PCCDI-2017-0338/79PCCDI/2018, within PNCDI IIIand PN 18090101/2018.
Appendix A: Quantum Cram´er-Rao bound for adouble-coherent input
Following reference [24] we consider the general casewhere each arm of the MZI contains a phase-shift ( ϕ and, respectively, ϕ ). The estimation is treated as ageneral two parameter problem. We define the 2 × F = (cid:20) F ++ F + − F − + F −− (cid:21) (A1)where F ij = 4 Re ( h ∂ i ψ | ∂ j ψ i − h ∂ i ψ | ψ ih ψ | ∂ j ψ i ) (A2)with i, j = ± and ϕ ± = ϕ ± ϕ . From this matrix wecan easily compute the QCRB: h ∆ ϕ i ∆ ϕ j i ≥ ( F − ) ij (A3)The state | ψ i in equation (A2) is: | ψ i = e − i ϕ +2 ( a † a + a † a ) e − i ϕ − ( a † a − a † a ) | ψ i (A4)where | ψ i = U BS | α β i is the state after the first beamsplitter and U BS = e − iπ/ a † ˆ a +ˆ a † ˆ a ) is the unitary trans-formation of BS .The elements of F are: F ++ = | α | + | β | , F + − = F − + = − | αβ | sin ∆ θ and F −− = | α | + | β | .We are interested in the phase difference between thetwo arms, i.e., h (∆ ϕ − ) i ≥ ( F − ) −− , for which we obtainthe following QCRB:∆ ϕ QCRB ≥ q | α | + | β | − | αβ | sin ∆ θ | α | + | β | (A5)which is equivalent to eq. (26) with ̟ = | β | / | α | .0 Input Quantum Difference intensity detection Single-mode intensity detectionstate Cram´er-Rao Optimum Best achievable Optimum Best achievablebound phase shift phase sensitivity phase shift phase sensitivity | ψ in i ∆ ϕ QCRB ϕ optdiff ∆ ϕ diff (cid:0) ϕ optdiff (cid:1) ϕ optsing ∆ ϕ sing (cid:0) ϕ optsing (cid:1) | α i | α | π | α | π | α | | α β i √ ̟ | α | q ( ̟ ) − ̟ sin ∆ θ ± arctan (cid:16) − ̟ ̟ (cid:17) ∆ ϕ QCRB ± (cid:0) ̟ (cid:1) ∆ ϕ QCRB | α ξ i p | α | e r + sinh r π √ sinh r + α e − r | α − sinh r | ± (cid:18)q √ | α | sinh r (cid:19) √ sinh r + √ α sinh 2 r + α e − r (cid:12)(cid:12) α − sinh r (cid:12)(cid:12) TABLE I. The phase sensitivity of an MZI for the input states discussed in the paper. The optimum phase shift has a periodof π (for the difference intensity detection), and respectively, 2 π (for the single-mode intensity detection). Appendix B: Calculation of the output variance
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