Canonical Phase Measurements in the Presence of Photon Loss

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by  Aravind Chiruvelli

Canonical Phase Measurements in the Presence of Photon Loss
Aravind Chiruvelli and Hwang Lee
Hearne Institute for Theoretical Physics, Department of Physics and Astronomy
Louisiana State University, Baton Rouge, LA 70803, USA
(Dated: January 7, 2009)
We analyze the optimal state under the canonical phase measurement, as given by Berry and
Wiseman [Phys. Rev. Lett 85, 5098, (2000)], in the presence of photon loss. The model of photon
loss is a generic fictitious beam splitter, and we present the full density matrix calculations, which
are more direct and do not involve any approximations. We find for a given amount of loss the
upper bound for the input photon number that yields a sub-shot noise estimate.
PACSnumbers: 42.50.St,42.50.Ar,42.50.Dv,42.50.-p
9 I. INTRODUCTION Indefiningthis,theSchwinger’srepresentationisused,
0 and for completeness we wish to outline the notation.
20 Canonical phase measurement in quantum mechanics The three angular momentum components Jˆx,Jˆy and Jˆz
isasignificantproblem,forthemainreasonthatphaseis areveryeffectiveinanalyzingtwo-port,lossless,interfer-
n aquantitythatisconjugatetothenumber,N,ofphotons ometers[5,6,18]. Forthetwomodes,aˆandˆboftheMZI
a
J inaparticularelectromagneticmode[1,2,3]. Duetothis (Fig. 1), these two mode operators are,
conjugate nature, the phase estimate ∆ϕ is ultimately
7 limited by the number N of the photons as ∆ϕ = 1/N, Jˆx =(aˆ†ˆb+aˆˆb†)/2, Jˆy =(aˆ†ˆb−aˆˆb†)/2i,

] which is conventionally referred as the Heisenberg limit. Jˆ =(aˆ†aˆ−ˆb†ˆb)/2, Jˆ2 =Jˆ2+Jˆ2+Jˆ2.
h z x y z
Intheusualclassicalsetting,suchasinterferometrywith
p
- lasers, for a given num√ber of input resources, N, phase In the context of the MZI, Jˆx implements the opera-
nt estimate scales as 1/ N, which is usually referred as tion of a 50-50 beam splitter as ei(π/2)Jˆx and Jˆz defines
shot-noise limit.
a the photon number difference in two modes. The simul-
u Accuratephaseestimationhasmanypracticalapplica- taneous eigenvector of Jˆ2 and Jˆ, |j,m(cid:105) represents the
q tionssuchasmetrology,imagingandsensing[4]. Achiev- z z
joint input state |j+m(cid:105) and |j−m(cid:105) in the Fock-
[ ing Heisenberg limit in practice is not a trivial problem ain bin
state basis, and the total input number of photons is
2 and there have been numerous proposals to achieve this N = 2j. The simultaneous eigenvector of Jˆ2 and Jˆ,
y
v limit[5,6,7,8,9,10,11,12,13,14,15]. Canonicalphase which is |j,n(cid:105) , represents the joint state within the in-
0 measurementhasbeenfirstdealtwithHelstrom[16]and terferometer. yThe beamsplitter transformation in this
8 Shapiro[17],andlaterSandersandMilburn[18,19]used representation performs a rotation about the Jˆ . The
7 it to obtain a phase estimate in a Mach-Zehender inter- x
phase states discussed above, are defined in terms of
2
ferometer (MZI) as shown in Fig.1 (excluding the loss
. states within the interferometer in Fig. 1 and thus the
1 part). The phase estimate thus obtained is independent
output modes or detectors are irrelevant for the present
1 of the system phase, unlike in other methods [5, 6, 7]
purpose. Thustheprobabilitydistributionforthesystem
8 where the ultimate limit is achieved for a particular sys-
phase φ is obtained as
0
tem phase. Also, the measurement specified by Sanders
:
v and Milburn is not particular to a specific input state. P(ϕ)dϕ=(cid:104)ψ|Fˆ(ϕ)|ψ(cid:105)dϕ=Tr[ρFˆ(ϕ)]dϕ. (2)
i Motivated by this work, Berry and Wiseman [8] an-
X
alytically derived an input state, called as the optimal Note that ϕ is the estimate of the system phase φ. The
ar state, subject to the canonical measurement. They also optimal state, to be specified below, is derived condi-
suggested an adaptive method of approximately imple- tioned upon minimizing the Holevo variance calculated
menting the canonical phase measurement [8, 20], which from the above probability distribution. The Holevo
has been used in the recent experimental realization of phase variance is defined as [22]
theHeisenberglimitedphasemeasurementbyHigginset
(∆ϕ)2 ≡−1+|(cid:104)eiϕ(cid:105)|−2, (3)
al. [21]. The canonical measurement, written as a Pos-
itive Operator Valued Measure(POVM) by Sanders and where |(cid:104)eiϕ(cid:105)| = (cid:82)2πdϕP(ϕ)eiϕ−ϕ¯, is also called the
Milburn [18] is, 0
sharpness. Here ϕ¯ is a mean phase and we take it to
Fˆ(ϕ)dϕ= 2j+1|jϕ(cid:105)(cid:104)jϕ|dϕ, (1) be zero. The Holevo variance for the optimal state is
2π given by Ref. [8, 20],
in terms of the phase states
π π2
(∆ϕ)2 =tan2( )≈ , (4)
1 (cid:88)j N +2 N2
|jϕ(cid:105)= √ eiαϕ|jα(cid:105) .
2j+1 y
thus giving rise to a phase estimate that scales as the
α=−j
2
In this paper we study the performance of optimal
state andthe optimal POVMin the presenceof the pho-
tonlossassociatedwithpropagation. Weusethegeneric
beam splitter model for photon loss [23], as shown in
Fig. 1. The input mode cˆfor this fictitious beam splitter
is a vacuum mode, and the output mode cˆ(cid:48) is then to
be traced out. This typically implies that the photons
that are lost in mode cˆ(cid:48), due to the nonzero reflection
coefficient r, correspond to the photon loss.
InthefollowingtwoSectionswedescribeoptimalstate
inpresenceofphotonlossandcarryouttheexplicitden-
sity matrix calculation. In Section IV, we quantitatively
describe the effect of photon loss on the canonical phase
estimation. Section V concludes with numerical results
and discussions.
FIG. 1: Schematic of Mach-Zehender Interferometer (MZI) II. OPTIMAL STATE IN PRESENCE OF
withphotonlossatthephaseshift. Thestate|ψ(cid:105) represents PHOTON LOSS
in
thejointinputstateataˆ andˆb . Photonsinthelowerarm
in in
first encounter the fictitious beam splitter (BS), for which
We now proceed to develop the mathematical frame-
vacuum enters through the other input, and depending the
work to study optimal-state canonical interferometry
transmission coefficient, some of them are scattered into the
with photon loss. The beam splitter representing loss,
mode cˆ(cid:48), which are ignored (traced out), and the remaining
with arbitrary transmission and reflection, can be char-
pass through the phase shifter.
acterizedbyanangleθ,suchthattransmissionandreflec-
tion coefficientsare τ =cos2(θ/2) and r =1−cos2(θ/2),
respectively. Therefore, the loss is simply the reflection
Heisenberglimit. Besidestheoptimalstate,otherpromi-
coefficient,
nentstatesthatachievetheHeisenberglimitaresocalled
the NOON states [5, 13] and the dual-fock state [7, 10].
L=r =1−cos2(θ/2). (7)
Formally the NOON state is,
The action of such an arbitrary beam splitter on an ar-
|N(cid:105) |0(cid:105) +|0(cid:105) |N(cid:105)
|ψ(cid:105) = aˆ ˆb√ aˆ ˆb. (5) bitrary joint input state |j,a(cid:105) in Schwinger notation, is
N 2
simply given as [6],
NotethattheabovestateisnotaninputstateattheMZI
j
showninFig.1,butastatewithintheinterferometerand eiθJˆx|j,a(cid:105)= (cid:88) eiθ(b−a)dj (θ)|j,b(cid:105), (8)
the subscripts aˆ andˆb denotes the internal modes of the a,b
b=−j
MZI. The dual-fock input state is given as,
wheredj (θ)istheusualrotationalmatrixelementgiven
|ψ(cid:105) =|N(cid:105) |N(cid:105) (6) a,b
D aˆin ˆbin as:
(cid:115)
The detection scheme for both, the NOON and the
(j−a)!(j+a)!
dual-fock state that results in the Heisenberg limited dj (θ)=(−1)a−b2−a P(a−b,a+b)(cosθ)
a,b j−b)!(j+b)! j−a
phase estimate is the parity detection [10, 24], first pro-
posed by Bollinger et al. [13] in the context of frequency ×(1−cosθ)a−2b(1+cosθ)a+2b, (9)
metrology with trapped ions. The shot-noise and sub
shot-noise limit with matter wave interferometry which where P(α,β)(x) is the Jacobi polynomial [28]. Also
n
deals with Bose-Einstein condensates at the input of the
it is worth noting that when converted to Fock-basis,
MZI is studied in Refs. [27].
a two mode joint state in Schwinger notation, |j,a(cid:105) is
Itisnaturaltoquestiontheperformanceofsuchstates
|j+a(cid:105)|j−a(cid:105).
or the detection schemes in a more realistic conditions
The optimal state was originally derived by Berry and
suchasphotonlossassociatedwiththepropagation. The
Wiseman [8] conditioned on minimizing the phase vari-
analysis of the NOON states under propagation loss was
ance with the canonical probability distribution given in
carried out independently by Gilbert et al. [25] and by
Eq. (2). Formally the optimal state is,
Rubin and Kaushik [26], where they used pure state for-
malism. In Ref. [25, 26] for NOON state, the minimum j (cid:20) (cid:21)
(cid:88) 1 (µ+j+1)π
number of photons required to achieve a minimum de- |ψ(cid:105) = √ sin |jµ(cid:105) . (10)
opt j+1 2j+2 y
tectable phase in presence of loss is also given.
µ=−j
3
Recall that the simultaneous eigenstate of Jˆ2 and Jˆ de- The fictitious beam-splitter transforms modes aˆ and cˆ
y
note the state within the interferometer, and thus the to modes aˆ(cid:48) and cˆ(cid:48) respectively. Making use of the
abovestateisafterthefirstbeamsplitter, while|j+µ(cid:105) Schwinger representation for modes aˆ and cˆ, the input
a
and |j−µ(cid:105) represents the fock state corresponding to |j+µ(cid:105) |0(cid:105) for the fictitious beam splitter can be writ-
b a c
modesaˆ andˆbrespectively. Rewritingtheabovestatein ten as a joint state: (cid:12)(cid:12)j+µ,j+µ(cid:11) . Letting k =(j+µ)/2
2 2 a,c
a more explicit form as the product of the states at the and using Eq. (8) we have the output as,
two arms of the interferometer as,
k
|ψ(cid:105) = (cid:88)j ψ |j+µ(cid:105) |j−µ(cid:105) , (11) eiθJˆx|k,k(cid:105)a,c = (cid:88) eiπ2(m−k)dkmk(θ)|k,m(cid:105)a(cid:48)c(cid:48)
opt µ a b m=−k
µ=−j
k
where = (cid:88) eiπ2(m−k)dkmk(θ)|k+m(cid:105)a(cid:48)|k−m(cid:105)c(cid:48). (13)
m=−k
(cid:20) (cid:21)
1 (µ+j+1)π
ψ = √ sin .
µ j+1 2j+2 Therefore the pure state of the inner modes aˆ(cid:48), ˆb of the
interferometer, and mode cˆ(cid:48) of the lost photons is given
With the loss in mode aˆ, which is represented by the fic-
as,
titious beam splitter, and |0(cid:105) is the state entering the
c
other input port, the combined input state for the ficti-
j k
btieoucsonbseiadmeresdpliist,teris: |j+µ(cid:105)a|0(cid:105)c,andthusthestateto |ψ(cid:105)= (cid:88) (cid:88) ψµeiπ2(m−k)dkmk(θ)
µ=−jm=−k
(cid:88)j ×|k+m(cid:105)a(cid:48)|k−m(cid:105)c(cid:48)|j−µ(cid:105)b, (14)
|ψ(cid:105)=|ψ(cid:105) ⊗|0(cid:105) = ψ |j+µ(cid:105) |j−µ(cid:105) |0(cid:105) .
opt c µ a b c
where k = (j +µ)/2 and dk (θ) is the usual rotational
µ=−j mk
(12) matrix element, as defined earlier.
III. DENSITY MATRIX DESCRIPTION
The state specified in Eq. (14) is a pure state and cannot be used further, so we need to calculate the reduced
densitymatrix,bytracingout,modecˆ(cid:48),aswehavenomoreaccesstothelostphotons. Thusfirstweneedtocalculate
the total density matrix, representing the pure state of Eq. (14),
ρ=|ψ(cid:105)(cid:104)ψ|
j k k(cid:48)
= (cid:88) (cid:88) (cid:88) ψµψνdkm,k(θ)dkn(cid:48),k(cid:48)(θ)eiπ2(m−k)e−iπ2(n−k(cid:48))|k+m(cid:105)a(cid:48)|k−m(cid:105)c(cid:48)|j−µ(cid:105)ba(cid:48)(cid:104)k(cid:48)+n|c(cid:48)(cid:104)k(cid:48)−n|b(cid:104)j−ν|, (15)
µ,ν=−jm=−kn=−k(cid:48)
where k(cid:48) = (j + ν)/2. The total density matrix given in Eq. (15) explicitly represents the state within the
interferometer for a given loss, characterized by the angle θ, and is useful in analyzing lossy interferometers such
as the present one.
Asthemodecˆ(cid:48) istobeignored,weneedthereduceddensitymatrixbytracingoutthatmodefromthetotaldensity
matrix given in Eq. (15). Thus we have,
ρ(cid:48) =Tr [|ψ(cid:105)(cid:104)ψ|]
c(cid:48)
j k k(cid:48)
= (cid:88) (cid:88) (cid:88) ψµψνdkm,k(θ)dkn(cid:48),k(cid:48)(θ))eiπ2(m−k)e−iπ2(n−k(cid:48))|k+m(cid:105)a(cid:48)|j−µ(cid:105)ba(cid:48)(cid:104)k(cid:48)−n|b(cid:104)j−ν|[c(cid:48)(cid:104)k−m|k(cid:48)−n(cid:105)c(cid:48)]. (16)
µ,ν=−jm=−kn=−k(cid:48)
Noting that (cid:104)k−m|k(cid:48)−n(cid:105) =δ , which eliminates the two exponential terms in Eq. (16). This leads to:
c(cid:48) c(cid:48) k−m,k(cid:48)−n
j k k(cid:48)
ρ(cid:48) = (cid:88) (cid:88) (cid:88) ψ ψ dk (θ)dk(cid:48) (θ)δ |k+m(cid:105) |j−µ(cid:105) (cid:104)k(cid:48)−n| (cid:104)j−ν|. (17)
µ ν m,k n,k(cid:48) k−m,k(cid:48)−n a(cid:48) ba(cid:48) b
µ,ν=−jm=−kn=−k(cid:48)
We can now use Eq. (17) in Eq. (2) to obtain the probability distribution and thus the minimum detectable phase as
a function of θ-which characterizes the photon loss-and the input photon number 2j.
4
FIG. 2: Log-Log graph of minimum detectable phase versus the input photon number N =2j for two different values of loss.
√
(a) L = 10−3 and (b) L = 0.3. The dotted line in both figures is the shot noise limit 1/ 2j and the dashed line in (a) is
the Heisenberg limit given be Eq. (4), for comparison.The solid lines are numerically evaluated using Eq. (3) and Eq. (24).
Depending on the loss, the minimum detectable phase starts to diverge at certain photon numbers, which we shall call the
optimal number, N .
opt
IV. PHASE ESTIMATE IN PRESENCE OF Recalling k(cid:48) = (j +ν)/2, k = (j +µ)/2, and because
PHOTON LOSS β = µ and α = ν as a consequence of the last two inner
productsintheaboveequation,wegetfromthefirsttwo
To get an estimate using Holevo phase variance, we innerproductsn=k(cid:48) andm=k respectively. Hencethe
needtocalculatetheprobabilitydistribution,P(ϕ)using probability distribution explicitly given as,
Eq.(2). We thus, in presence of photon loss have,
P(ϕ)=Tr[ρ(cid:48)Fˆ(ϕ)]. (18)
j
Note that ϕ is an estimate of the system phase φ. P(ϕ)= 21π (cid:88) ψµψνei(ν−µ)ϕdkk,k(θ)dkk(cid:48)(cid:48),k(cid:48)(θ). (22)
The POVM, Fˆ(ϕ)dϕ [Eq. (1)] is given in terms of the µ,ν=−j
Jˆ eigen states. Noting that the Jˆ eigen states are the
y y
states of the modes within the interferometer, we can With this probability distribution we can calculate any
rewrite the POVM as, moments of the phase estimate as a function of loss, θ.
Here we calculate the Holevo phase variance rather than
j
Fˆ(ϕ)= 1 (cid:88) ei(α−β)ϕ|j,α(cid:105) (cid:104)j,β|. (19) the standard variance, because the optimal state is de-
2π y rivedbyminimizingtheHolevophasevariance[8]. Using
α,β=−j Eq. (22) we now calculate sharpness |(cid:104)eiϕ(cid:105)| is given as,
Sincethejointstate|j,α(cid:105) iswrittenas|j+α(cid:105) |j−β(cid:105) ,
y a(cid:48) b
we have, (cid:90) 2π
|(cid:104)eiϕ(cid:105)|= P(ϕ)eiϕdϕ
j
Fˆ(ϕ)= 1 (cid:88) ei(α−β)ϕ|j+α(cid:105) |j−α(cid:105) 0
2π a(cid:48) b j
α,β=−j = (cid:88) δ ψ ψ dk (θ)dk(cid:48) (θ), (23)
ν,µ−1 µ ν k,k k(cid:48),k(cid:48)
⊗ (cid:104)j+β| (cid:104)j−β| (20)
a(cid:48) b µ,ν=−j
After the fictitious beamsplitter which represents pho-
ton loss associated with propagation, the inner modes of where δ = 1 (cid:82)2πe(ν−µ+1)ϕdϕ. Also using
the MZI are aˆ(cid:48) and ˆb, and so the Jˆ eigen states are the the rotaνt,iµo−n1al matr2ixπ e0lement in Eq. (9), we have
y
product states of these modes. Thus using Eq. (20) in dk (θ)dk(cid:48) (θ) = (cid:2)cos2 θ(cid:3)(k+k(cid:48)) and invoking the defi-
Eq. (18) along with Eq. (17) for the reduced density ma- k,k k(cid:48),k(cid:48) 2
nition of k and k(cid:48) in Eq. (23), we obtain
trix,aftercarryingoutthetraceoperationforthemodes
aˆ(cid:48) andˆb,wehavetheprobabilitydistributioninpresence
of loss as,
(cid:88)j (cid:20) θ (cid:21)j+µ−12
j k k(cid:48) |(cid:104)eiϕ(cid:105)|= ψ ψ cos2( ) , (24)
P(ϕ)= 1 (cid:88) (cid:88) (cid:88) [ψ ψ dk (θ)dk(cid:48) (θ) µ µ−1 2
2π µ ν m,k n,k(cid:48) µ=−j
µ,ν=−jm=−kn=−k(cid:48)
α,β=−j
fortheexpressionoftheso-calledsharpnessasafunction
×ei(α−β)ϕ(cid:104)k(cid:48)−n|j+α(cid:105)(cid:104)j+β|k+m(cid:105) of loss. We use Eq. (24) in Eq. (3) to obtain the phase
×(cid:104)j−β|j−µ(cid:105)(cid:104)j−ν|j−α(cid:105)](21) estimate as a function of loss.
5
and it clearly decreases as the loss L increases.
Further,ouranalysisshowsforsmallloss,about0.05%
loss,whichisequivalentto99.5%oftransmission,wecan
stillhavetheinputnumberofphotons,whichgivesasub-
shot noise limited phase estimate.
Ontheotherhand,theupperboundforacertainnum-
ber of input photons can be found where ∆ϕ meets the
shot-noise limit in Fig. 2(a). That is to say, for a small
amount of loss, we can still operate the interferometer,
with the input number of photons less than the upper
bound and achieve a sub-shot noise limited phase esti-
mate.
FIG. 3: Log-Log plot of the optimal number, N as a func-
opt To summarize, we analyzed the canonical phase mea-
tion of loss for various values of loss L. The sub-shot noise
surement in presence of photon loss. Our formalism is
rangeindicatestherangeofloss,soastoobtainthesub-shot
based on the density matrix, which describes the mixed
noise phase estimate
states, which naturally arise due to the presence of loss.
Ouranalysisshowsthattheminimumdetectablephaseis
V. NUMERICAL RESULTS AND DISCUSSION notmonotonicallydecreasingfunctionandwouldtendto
increase at certain photon numbers, depending the loss
present. Nevertheless, we can for small loss have con-
Nowtheminimumdetectablephaseshiftcanbefound
siderably high number of input photons for the optimal
with the uncertainty in phase estimate by plugging
state, that achieve sub-shot noise level phase estimates.
Eq. (24) in Eq. (3). In Fig. 2 we numerically plot the
We also have numerically plotted the optimal number,
minimum detectable phase ∆ϕ as a function of 2j =N,
where the minimum detectable phase shift starts to di-
the number of input photons, for two different values of
verge, as a function of loss, L, which would determine
loss, L = 10−3 and L = 0.3. For comparison we also
the optimal number of the input photon number for the
plot the Heisenberg limit for the optimal state given in
optimal state, in presence of given amount of loss.
Eq. (4). We can immediately observe that the minimum
detectable phase is not always a continuously decreasing
function,dependingontheloss,theminimumdetectable
phasestartstodivergeatcertainphotonnumbers,which Acknowledgments
we shall call the optimal number, N .
opt
As we do not have a closed form expression for the We acknowledge financial support from the US De-
Holevovariance,asafunctionoflossLandinputphoton fense Advanced Research Projects Agency, the US Intel-
number N, we have not found an analytical form for the ligenceAdvancedResearchProjectsActivity,andtheUS
optimalnumber, N asfunctionofloss, L[seeEq.(7)]. Army Research Office. We would like to acknowledge J.
opt
So we numerically plot N as a function of L in Fig. 3 P. Dowling for helpful discussions.
opt
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