Table Of ContentPhase detection at the quantum limit with multi-photon Mach-Zehnder interferometry
L. Pezz´e, A. Smerzi
BEC-CNR-INFM and Dipartimento di Fisica, Universit`a di Trento, I-38050 Povo, Italy
G. Khoury, J. F. Hodelin and D. Bouwmeester
Department of Physics, University of California, Santa Barbara, California 93106, USA
We study a Mach-Zehnder interferometer fed by a coherent state in one input port and vacuum
in the other. We explore a Bayesian phase estimation strategy to demonstrate that it is possible
to achieve the standard quantum limit independently from the true value of the phase shift and
specificassumptionsonthenoiseoftheinterferometer. Wehavebeenabletoimplementtheprotocol
7
usingparalleloperationoftwophoton-number-resolvingdetectorsandmultiphotoncoincidencelogic
0
electronicsattheoutputportsofaweakly-illuminatedMach-Zehnderinterferometer. Thisprotocol
0
2 is unbiased and saturates the Cramer-Rao phase uncertainty bound and, therefore, is an optimal
phaseestimation strategy.
n
a
PACSnumbers: 42.50St,42.50-p
J
6
2 TheMach-Zehnder(MZ)interferometer[1,2]isatruly
ubiquitous device that has been implemented using pho-
2
tons, electrons [3], and atoms [4, 5]. Its applications
v
8 range from micro- to macro-scales, including models of
5 aerodynamics structures, near-field scanning microscopy
1 [6]andthemeasurementofgravityaccelerations[7]. The
1
central goal of interferometry is to estimate phases with
0
thehighestpossibleconfidence[8,9,10]whiletakinginto
7
0 account sources of noise. Recent technological advances
/ make it possible to reduce or compensate the classical
h FIG. 1: (a) Schematic of a Mach-Zehnder interferometer. A
noisetothelevelwhereadifferentandirreduciblesource
p
phase sensitive measurement is provided by the detection of
- of uncertainty becomes dominant: the quantum noise.
t the number of particles Nc and Nd at the two output ports.
n Givenafiniteenergyresource,quantumuncertaintyprin- (b)Pulseheightdistributionforavisiblelightphotoncounter
a ciples and back reactions limit the ultimate precision of (VLPC) used in the experiment. The power incident on the
u a phase measurement. In the standard configuration of detector is 144 fW at a wavelength of 780 nm. The vertical
q
the MZ interferometer, a coherent optical state with an lines show the decision thresholds [23].
:
v average number of photons n¯ = α2 enters input port
i | |
X a and the vacuum enters input port b, as illustrated in
Figure(1). Thegoalistoestimatethevalueofthephase
r eventuallyvanishes. Asaconsequence,thisinterferomet-
a shift θ after measuring a certain number of photons N
c ricprotocoldoesnotallowthe measurementofarbitrary
and N at output ports c and d, which, in the experi-
d phaseshifts. Thiscanbe a seriousdrawbackforapplica-
ment discussed in this Letter, is made possible by two
tions likelasergyroscopes,the synchronizationofclocks,
number-resolving photodetectors.
orthealignmentofreferenceframes. Furthermore,toes-
The conventional phase inference protocol estimates
timatesmallphaseshiftswiththe highestresolution,the
the true value of the phase shift θ as [11, 12, 13]:
interferometer has to be actively stabilized around π/2.
M This generally requires the addition of a feedback loop,
p
Θest =arccos(cid:16) n¯ (cid:17), (1) which can be quite costly in terms of time and energy
resources.
whereM = p (N(k) N(k))/pisthephotonnumber
p Pk=1 c − d It was first noticed by Yurke, McCall and Klauder
difference detected at the output ports, averaged over p
(YMK) in [14] that the estimator (1) does not take into
independent measurements. The phase uncertainty of
account all the available information, and, in particular,
estimator (1) is
the fluctuations in the total number of photons at the
output ports. The possibility to improve Eq.(2) is con-
1
∆Θ= , (2) firmed by the analysis of the Cramer-Rao lower bound
√pn¯sinθ
(CRLB) [15, 16], which provides, given an input state
which follows from a linear error propagation theory. andchoiceofobservables,the lowestuncertaintyallowed
Eq.(2)predictsanoptimalworkingpointataphaseshift byQuantumMechanics. Forageneric,unbiased,estima-
θ = π/2, where the average photon number difference tor, ∆Θ =1/ pF(θ), where F(θ) is the Fisher in-
CRLB p
varies most quickly with phase. As θ approaches 0 or π, formation[17],which,ingeneral,candependonthetrue
theconfidenceofthemeasurementbecomesverylowand valueofthephaseshiftθ andthenumberofindependent
2
0.2 0.4 1
probability distribution 000... 1320 AD 0000000 ....0...3245132 CB (i)(i)φ P ( | {N , N } )pci=1...pd 0000 ....045232468 AC 010 1 . ..088242 BD
0.1 0.1 1 4
0 0 0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
φ / π φ / π φ / π φ / π
FIG. 2: (color online). Phase distribution P(φ|Nc,Nd), for: FIG.3: (color online). Phaseprobabilitydistribution Eq.(6),
A) Nc =0, Nd =0; B) Nc =0, Nd =1; C) Nc =1, Nd =1; obtained after p independent experimental measurements
Dco)lleNccte=din0,thNedca=lib2r.atTiohnepcairrctleosftahreeetxhpeereixmpeenritm,tehnetadladshaetda {Nc(i),Nd(i)}i=1..p: A) p = 1, B) p = 10, C) p = 100, D)
p = 1000. The true value of the phase shift is θ/π = 0.24,
line is the ideal phase distribution Eq.(4). The solid line is a
shown bythe vertical dashed line.
fit of thedata according toEq.(5).
measurements p. A direct calculation of the Fisher in- vides both anestimate onthe phase andthe uncertainty
formation for the coherent vacuum input state, gives in this estimate.
⊗
F(θ)=n¯. Therefore, the Cramer-Rao lower bound is There are several advantages to using a Bayesianpro-
tocol. Notably, it can be applied to any number p of
1
independent measurements, it does not require statisti-
∆Θ = , (3)
CRLB
√pn¯ calconvergenceoraveraging,anditprovidesuncertainty
estimates tailored to the specific measurement results.
which, in contrast with the result of Eq.(2), is indepen-
For instance, with a single measurement, p = 1, it pre-
dent of the true value of the phase shift. The only as- dicts anuncertaintythatscalesas ∆Θ 1 . Since
sumptionhereisthattheobservablemeasuredattheout- ≈ √Nc+Nd
Eq.(4)doesnotdependonn¯,theestimationisinsensitive
putportsisthenumberofparticles. Itiswellknown(see
to fluctuations of the input laser intensity. Most impor-
for instance [18]) that the Maximum Likelihood (ML)
tantly,itsuncertainty,inthelimitp>>1,is∆Θ= 1
estimator, defined as the maximum, Θ , of the Like- √pn¯
ML
which coincides with the CRLB.
lihood function P(N ,N φ) (see below), saturates the
c d
CRLB, but only asympto|tically in the number of mea- To implement the proposed protocol we have real-
surements p. In the current literature there have been ized a polarization Mach-Zehnder interferometer with
alternative suggestions to obtain an unbiased estima- photon-number-resolving coincidence detection. In a re-
tor and a phase independent sensitivity with a Mach- cent paper we reported on the analysis of a coherent
Zehnder interferometer [10, 14, 19, 20, 21]. They will be state using a single photon-number-resolving detector
specifically addressed at the end of this Letter. [23]. We have extended this experimental capability to
Here we develop a protocol based on a Bayesian two simultaneously operating visible light photon coun-
analysis of the measurement results [8, 9, 18, ters(VLPCs)[24],cryogenicphotodetectorsthatprovide
19]. The goal is to determine P(φ N ,N ), acurrentpulseofapproximately40,000electronsperde-
c d
the probability that the phase equals |φ given tected photon. The VLPCs were maintained at 8 K in a
the measured N and N . Bayes’ theorem pro- helium flow cryostat, and their photocurrent was ampli-
c d
videsthis: P(φN ,N )=P(N ,N φ) P(φ)/P(N ,N ), fiedbylow-noise,roomtemperatureamplifiers. Wemea-
c d c d c d
where P(N ,N| φ) is the probabili|ty to detect N and suredadetectionefficiencyof35%andadarkcountrate
c d c
N when the ph|ase is φ [22], P(φ) quantifies our prior of 3 105 for each detector under our operating condi-
d ×
knowledge about the true value of the phase shift, and tions. Customelectronicsprocessedthe amplifiedVLPC
P(N ,N ) is fixed by normalization. Assuming no prior current pulses to perform gated, fast coincidence detec-
c d
knowledge of the phase shift, P(φ) = 1/π. In the ideal tion. Wewerethusabletodetermine,foreachpulse,how
case, the Bayesian phase probability distribution can be many photons were detected at both ports c and d. A
calculated analytically for any value of N and N , Ti:sapphire pulsed laser, attenuated such that n¯ = 1.08
c d
photons,providedtheinputstate. Sinceacoherentstate
φ 2Nc φ 2Nd maintains its form under linear loss [11], the presence of
P(φN ,N )= cos sin , (4)
| c d C(cid:16) 2(cid:17) (cid:16) 2(cid:17) loss after the interferometer is completely equivalent to
a losslessinterferometerfed by a weakerinput state. We
where = Γ(1/2+Nc)Γ(1/2+Nd) is a normalization con- use n¯ to signify the average number of photons in the
C Γ(1+Nc+Nd)
stant. In practice, one must measure P(N ,N φ) and, detected state per pulse, after all losses. We were lim-
c d
|
fromthis,determineP(φN ,N ). Thisdistributionpro- ited by the amplifiers to measuring up to four photons
c d
|
3
0.03 10
0.02
8
Θ ) / πest 0.0 01 ∆Θ 6
− p
θ −0.01 4
(
−0.02 2
−0.03 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
θ / π θ / π
FIG. 4: (color online). Difference between the mean value FIG. 5: (color online). Phase sensitivity as a function of
of the phase estimator, Θ , obtained after 150 replica of
est the true value of the phase shift. Circles are obtained from
h i
p=1000independentmeasurementsandthetruevalueofthe
Bayesian distributions Eq.(6), with p=1000. Theerror bars
phase shift θ. The vertical bars are the mean squarefluctua-
give the fluctuations of √p∆Θ obtained with 150 indepen-
tions σe2st =h(Θest−hΘesti)2i. The result σest ≫(θ−Θest) dent replica of the experiment. The solid black line is the
proves that our protocol provides an unbiased experimental
theoretical prediction Eq.(3). The dashed blue line is the
phase estimation.
CRLB calculated with the experimental distributions. Tri-
angles are the uncertainty obtained with a generalization of
theestimatorEq.(1)takingintoaccounttheexperimentalim-
perfections, while the dotted red line is the phase sensitivity
per pulse Fig. 1(b), but at n¯ = 1.08 the probability of
predicted by Eq.(2).
detecting five or more photons is negligible. The phase
shiftθwaschangedbytiltingabirefringentcrystalinside
the interferometer.
distributions associated with each experimental result:
The first part of the experiment consists of the cali-
brationoftheinterferometer. Atdifferent,known,values p
P φ N(i),N(i) P φN(i),N(i) . (6)
of the phase shift, we measured Nc and Nd for each of (cid:0) |{ c d }i=1...p(cid:1)∝Y fit(cid:0) | c d (cid:1)
200,000 laser pulses. This procedure allows us to deter- i=1
mineexperimentallybothP(N ,N φ)andP(φN ,N ).
c d c d The phase estimator Θ is given by the mean value of
| | est
In Fig.(2) we compare the ideal and the experimental the distribution, Θ = πdφφP φ N(i),N(i) ,
phase distributions. The agreement is quite good, and est R0 (cid:0) |{ 1 2 }i=1...p(cid:1)
and the phase uncertainty ∆Θ is the 68.27% con-
the discrepanciescanbe attributed to imperfect photon-
fidence interval around Θ . An example of
est
number discrimination by the detectors. To fit the data
(i) (i)
P φ N ,N isgiveninFig.(3),forθ/π =0.24
inFig.(2),weintroducetheprobabilityP(Nc,Nd|Nc′,Nd′) an(cid:0)d|f{ordcifferednt}vi=a1lu..e.ps(cid:1)ofp. Sincetheaveragenumberof
tomeasureN andN whenN andN photonswerere-
c d c′ d′ photonsofthecoherentinputstateissmall,forp 1the
ally present. Taking this into account, the experimental ∼
phase uncertainty is of the order of the prior knowledge,
phase probability distribution is
∆Θ π. As p increases,the probability distribution be-
≃
comesGaussianandthesensitivityscalesas∆Θ 1/√p,
∝
Pfit(φ|Nc,Nd)= X P(φ|Nc′,Nd′)P(Nc′,Nd′|Nc,Nd), in agreement with the central limit theorem.
N′,N′ In Fig.(4), we show the difference between the mean
c d
(5) value of the phase estimator Θ , obtained from 150
est
h i
whereP(φ|Nc′,Nd′)aretheidealprobabilitiesEq.(4). The phaseestimations,eachwithp=1000,andthetruevalue
weights P(Nc′,Nd′|Nc,Nd) can be retrieved from a fit of ofthephaseshift. Thebarsarethemeansquarefluctua-
the experimental calibration distributions P(φNc,Nd), tion. The important result is that our protocol provides
|
seeFig.(2). ThequantityP(Nc,Nd Nc,Nd),equaltoone anexperimentallyunbiasedphaseestimationovertheen-
|
in the ideal case, is 0.54 in Fig.(2A) (corresponding to tire phase interval.
the worstcaseamongalldistributions), 0.67in (2B) and The main result of this Letter is presented in Fig.(5).
(2C), and 0.87 in (2D). We show the phase sensitivity for different values of the
After the calibration, we can proceed with the phase shift θ, calculated from the distribution Eq.(6)
Bayesian phase estimation experiment. For a certain with p = 1000 photon-number measurements. The
value of the phase shift, we input one laser pulse and circles are the mean value of ∆Θ, and the bars give
detect the number of photons N and N . We repeat the corresponding mean square fluctuation, obtained
c d
this procedure p times obtaining a sequence of indepen- from 150 independent phase measurements. The dashed
dent results N(i),N(i) . The p photon-number blue line is the CRLB calculated with the experimen-
{ c d }i=1...p tal probability distributions, ∆θ =1/ pF (θ), where
measurements comprise a single phase estimation. The fit p fit
overall phase probability is given by the product of the Ffit(θ) = PN1,N2 Pfit(N11,N2θ)(∂Pfit(N∂θ1,N2|θ))2 [26]. For
|
4
0.1 . θ/π . 0.9, it follows the theoretical prediction put ports of a weakly-illuminated interferometer. Yet,
(solid black line), Eq.(3), where n¯ = 1.08 has been in- the method can be generalizedto the case of high inten-
dependently calculated from the collected data. Around sity laser interferometry and photodiode detectors. In
θ = 0,π, where the photons have higher probability to this case, the limit Eq.(3) becomes harder to achieve be-
exit through the same output port, ∆θ increases as a causeoflargerelectronicnoiseandlowerphotonnumber
fit
consequence of the decreased sensitivity of our detectors resolution, however it should still be possible to demon-
to higher photon number states. Even though the phase strateaphaseindependentsensitivity. Ourresultsareof
sensitivityofourapparatusbecomesworsenearθ =0,π, importance to quantum inference theory and show that
it never diverges. Triangles show the phase uncertainty the MZ interferometer does not require phase-locking in
obtainedwithEq.(1), but takinginto accountthe exper- order to reach an optimal sensitivity.
imental noise. The estimator is obtained by inverting Acknowledgment. This work has been partially sup-
the equation M = n¯cos(a+θ)+b [25]. This strategy ported by the US DoE and NSF Grant No. 0304678.
p
provides an unbiased estimation with a sensitivity close
to the one predicted by Eq.(2) (dotted red line). The
reasonfor the superiorperformance ofthe Bayesianpro-
tocol can be understood by noticing that, in Eq.(2), the
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67,1426(1991);J.Rehacek,etal.,Phys. Rev.A60,473
putdetection)anddiscussedtheoreticallythe MZ.Their
(1991).
analysis wasbased onspecific assumptions about the in- [22] The Quantum Mechanical probability to detect Nc and
terferometricclassicalnoisewhicharenotsatisfiedinthe N particles at the output port c and d, respectively,
d
case discussed in this Letter. Different approaches with is given by P(Nc,Nd φ) = ( Nc Nd )e−iφJˆy ψinp 2,
| | h |h | | i|
adaptive measurements [20] and positive operator value whereJˆy =(aˆ†ˆb ˆb†aˆ)/2i, withaˆ,ˆbbosonicannihilation
−
measurements[10]havebeenalsosuggested. Whilethese operators. For the configuration studied in this Letter
sHteraisteengbieesrgmliigmhitt,bteheiymparoertnaonttnfoecreisnstaerryfeirnomouertrcyasaet. the [23] GP.(NKch,oNudr|yφ,)e=t a|lα.(,N|2PcN)h!e(y−Ns|dα.)|!R2`evsi.nLφ2et´t2.N9c6`,co2s03φ26´021Nd(.2006).
In conclusion, we have presented a Bayesian phase es- [24] G.B.Turner,etal.,ProceedingsoftheWorkshoponScin-
timationprotocolforaMZinterferometerfedbyasingle tillating Fiber Detectors, Notre Dame University, edited
by R. Ruchti(World Scientific, Singapore, 1994).
coherent state. The protocol is unbiased and provides
a phase sensitivity that saturates the ultimate Cramer- [25] AfitofthecalibrationdatagiveshNc−Ndi(θ)=n¯cos(θ+
a)+b, with a=0.024 and b=0.008.
Rao uncertainty bound imposed by quantum fluctua-
[26] ThedistributionsP (N ,N θ)areobtainedfromEq.(5)
fit 1 2
tions. We have been able to implement the protocol with the Bayes theorem. |
with two photon-number-resolving detectors at the out- [27] Manuscript in preparation.
