We derive the optimal N-photon two-mode input state for obtaining an estimate
\phi of the phase difference between two arms of an interferometer. For an
optimal measurement [B. C. Sanders and G. J. Milburn, Phys. Rev. Lett. 75, 2944
(1995)], it yields a variance (\Delta \phi)^2 \simeq \pi^2/N^2, compared to
O(N^{-1}) or O(N^{-1/2}) for states considered by previous authors. Such a
measurement cannot be realized by counting photons in the interferometer
outputs. However, we introduce an adaptive measurement scheme that can be thus
realized, and show that it yields a variance in \phi very close to that from an
optimal measurement.Comment: 4 pages, 4 figures, journal versio

A. S. Holevo

B. C. Sanders

B. Yurke

C. M. Caves

D. W. Berry

G. S. Summy

H. M. Wiseman

M. J. Holland

Z. Hradil

English

arXiv

We derive the optimal N-photon two-mode input state for obtaining an estimate f of the phase difference between two arms of an interferometer. For an optimal measurement [B. C. Sanders and G. J. Milburn, Phys. Rev. Lett. 75, 2944 (1995)], it yields a variance (?f)2?p2/N2, compared to O(N-1) or O(N-1/2) for states considered by previous authors. Such a measurement cannot be realized by counting photons in the interferometer outputs. However, we introduce an adaptive measurement scheme that can be thus realized, and show that it yields a variance in f very close to that from an optimal measurementNo Full Tex

Berry, DW.

Wiseman, Howard

Griffith Research Online

Optimal States and Almost Optimal Adaptive Measurements for Quantum Interferometry

We derive the optimal N-photon two-mode input state for obtaining an estimate φ of the phase difference between two arms of an interferometer. For an optimal measurement [B. C. Sanders and G. J. Milburn, Phys. Rev. Lett. 75, 2944 (1995)], it yields a variance (Δφ)≈π/N2, compared to O(N) or O(N) for states considered by previous authors. Such a measurement cannot be realized by counting photons in the interferometer outputs. However, we introduce an adaptive measurement scheme that can be thus realized, and show that it yields a variance in φ very close to that from an optimal measurement

Berry, D. W.

Wiseman, H. M.

University of Queensland eSpace

VOLUME 85, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 11 DECEMBER 2000
5Optimal States and Almost Optimal Adaptive Measurements for Quantum Interferometry
D. W. Berry1 and H. M. Wiseman2
1Department of Physics, The University of Queensland, St. Lucia 4072, Australia
2School of Science, Griffith University, Nathan, Brisbane, Queensland 4111, Australia
(Received 19 June 2000)
We derive the optimal N-photon two-mode input state for obtaining an estimate f of the phase dif-
ference between two arms of an interferometer. For an optimal measurement [B. C. Sanders and G. J.
Milburn, Phys. Rev. Lett. 75, 2944 (1995)], it yields a variance Df2  p2N2, compared to ON21
or ON212 for states considered by previous authors. Such a measurement cannot be realized by
counting photons in the interferometer outputs. However, we introduce an adaptive measurement scheme
that can be thus realized, and show that it yields a variance in f very close to that from an optimal
measurement.
PACS numbers: 42.50.Dv, 03.67.–a, 07.60.Ly, 42.50.LcInterferometry is the basis of many high-precision mea-
surements. The ultimate limit to the precision is due to
quantum effects. This limit is most easily explored for a
Mach-Zehnder interferometer (see Fig. 1, where F should
be ignored for the moment). The outputs of this device can
be measured to yield an estimate f of the phase difference
w between the two arms of the interferometer. It is well
known that this can achieve the standard quantum limit
for phase sensitivity of Df2  N21 when an N-photon
number state enters one input port. Several authors [1–5]
have proposed ways of reducing the phase variance to the
Heisenberg limit of N22. Here N is the fixed total num-
ber of photons in the inputs [6].
Most of these proposals [1–3] are limited in that they
require that the phase difference w between the two arms
be zero or very small in order to obtain the N22 scaling.
Sanders and Milburn [4,5] considered an ideal or canonical
measurement, for which the N22 scaling is independent of
w. Unfortunately they do not explain how this measure-
ment can be performed, and it can be shown [7] that it can-
not be realized by counting photons in the outputs of the
interferometer. In this Letter we show that there is an ex-
perimentally realizable measurement scheme using photo-
detectors and feedback which is almost as good as the
canonical measurement.
Before introducing our adaptive scheme, we find the op-
timal input states for the canonical interferometric mea-
surements. These will then be used as the input states for
our adaptive scheme, to demonstrate a Df2 scaling al-
most as good as N22. The optimal input states are inter-
esting in themselves, in that they differ significantly from
the input states considered in Refs. [1–5]. In particular,
our rigorous analysis shows that those nonoptimal states,
in fact, exhibit a worse scaling than the standard quantum
limit of N21.
The canonical measurement.—Using the same notation
as Sanders and Milburn [4], we designate the two annihi-
lation operators for the two input modes as aˆ and bˆ, and
we use the Schwinger representation
Jˆx  aˆybˆ 1 aˆbˆy2, Jˆy  aˆybˆ 2 aˆbˆy2i , (1)098 0031-90070085(24)5098(4)$15.00Jˆz  aˆyaˆ 2 bˆybˆ2, Jˆ2  Jˆ2x 1 Jˆ
2
y 1 Jˆ
2
z . (2)
We use the notation j jmz for the common eigenstate of
Jˆz and Jˆ2 with eigenvalues m and j j 1 1, respectively.
This state corresponds to Fock states with j 1 m and
j 2 m photons entering ports a and b, respectively.
From Ref. [5], the optimal probability operator measure
(POM) for phase measurements is the canonical one,
Eˆfdf 
2j 1 1
2p
j jf  jfjdf , (3)
where the phase states j jf are defined in terms of the Jˆy
eigenstates by j jf  2j 1 1212Pjm2j eimfj jmy .
In terms of the Jˆy eigenstates, the POM is
Eˆfdf 
1
2p
jX
m,n2j
eim2nfj jmy jnjdf . (4)
The POM defines the probability distribution for f, the
best estimate for the interferometer phase w, by
Pfdf  cjEˆfjcdf . (5)
FIG. 1. The Mach-Zehnder interferometer, with the addition
of a controllable phase F in one arm. The unknown phase to
be estimated is w. Both beam splitters (BS) are 50:50.© 2000 The American Physical Society
VOLUME 85, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 11 DECEMBER 2000Here jc is the two-mode interferometer input state having
N  2j photons.
The optimal input state.—A short examination reveals
that the canonical POM (4) has the same matrix elements
as the POM for ideal measurements on a single mode with
an upper limit of N on the photon number. The optimal
state in this case has been considered before [8,9]. Here
we follow the procedure of Ref. [9], which minimizes the
Holevo phase variance [10]
Df2  Vf  S22f 2 1 , (6)
where Sf [ 	0, 1
 is the sharpness of the phase distribu-
tion, defined as
Sf  jeifj 
Z 2p
0
dfPfeif2f¯, (7)
where the “mean phase” f¯ is defined so that Sf is positive.
The Holevo variance is the natural quantifier for dispersion
in a cyclic variable. If the variance is small, then it is easy
to show that
Vf 
Z 2p
0
4 sin2
µ
f 2 f¯
2
∂
Pfdf , (8)
from which the equivalence to the usual definition of vari-
ance for well-localized distributions is readily apparent.
Using the Holevo variance enables a simple analytic
solution. The minimum variance is
tan2
µ
p
N 1 2
∂

p2
N 1 22
1 ON24 , (9)
and the optimal state (chosen here to have a mean relative
phase of zero) is
jcopt  1pj 1 1
jX
m2j
sin
∑
m 1 j 1 1p
2j 1 2
∏
j jmy .
(10)
To obtain the state in terms of the eigenstates of Jˆz , we
use y jm j jnz  eip2 n2mIjmnp2, where [4]
Ijmnp2  2
2m
∑
 j 2 m!
 j 2 n!
 j 1 m!
 j 1 n!
∏12
3 P
m2n,m1n
j2m 0 ,
for m 2 n . 21,m 1 n . 21 , (11)
where Pa,bn x are the Jacobi polynomials, and the other
matrix elements are obtained using the symmetry relations
I
j
mnu  21m2nIjnmu  Ij2n,2mu.
An example of the optimal state for 40 photons is plotted
in Fig. 2. This state contains contributions from all the
Jˆz eigenstates, but the only significant contributions are
from 9 or 10 states near m  0. The distribution near the
center is fairly independent of photon number N  2j.
To demonstrate this, the distribution near the center for
1200 photons is also shown in Fig. 2.
In order to compare this state with j j0z , where equal
photon numbers are fed into both input ports (as considered
in Refs. [3–5]), the exact phase variance for this case was−20 −10 0 10 20
0
0.1
0.2
0.3
0.4
0.5
0.6
 µ
 
z 
<
j,µ
|ψ  
o
pt
>
FIG. 2. The coefficients z jm jcopt for the state optimized
for minimum phase variance under ideal measurements. All
coefficients for a photon number of 2j  40 are shown as the
continuous line, and those near m  0 for a photon number of
2j  1200 as crosses.
calculated for a range of photon numbers up to 25 600.
Since the phase must be measured modulo p for this state,
rather than using the Holevo phase variance, we used the
following measure for the dispersion:
Df2  V2f4  je2ifj22 2 14 , (12)
where the expectation value is again determined using (5).
The phase variances for this state and the optimal state are
shown in Fig. 3. The exact Holevo phase variance of the
state where all the photons are incident on one port, j jjz ,
is also shown for comparison.
As can be seen, the phase variance for j j0z scales down
with photon number much more slowly than the phase
variance for optimal states [11], and even more slowly than
the variance for j jjz , which scales as N21. In fact, this
101 102 103 104
10−8
10−6
10−4
10−2
100
 
(∆
φ)2
 2j
FIG. 3. Variances in the phase estimate versus input photon
number 2j. The lines are exact results for canonical measure-
ments on optimal states jcopt (continuous line), on states with
all photons incident on one input port j jjz (dashed line), and
on states with equal photon numbers incident on both input ports
j j0z (dotted line). The crosses are the numerical results for the
adaptive phase measurement scheme on jcopt, and the plusses
are those on a j j0z input state. The circles are numerical re-
sults for a nonadaptive phase measurement scheme on jcopt.
All variances for the j j0z state are for phase modulo p.5099
VOLUME 85, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 11 DECEMBER 2000figure shows that the phase variance for j j0z scales as
N212, which agrees with what can be calculated from the
asymptotic formula for Pf given in Ref. [5]. This is
a radically different result from the N22 scaling found in
Refs. [3–5]. The state j j0z 1 j j1z
p
2, considered in
Ref. [2], is even worse than the state j j0z .
The reason for this discrepancy is that the results found
in Refs. [3–5] are all based on the width of the central peak
in the distribution, but the main contribution to the vari-
ance is from the tails of the distribution. This can be seen
from the probability distribution multiplied by sin2f, since
Eqs. (8) and (12) imply that Df2 
R
sin2f 2 w 3
Pfdf. We plot this in Fig. 4 for N  80 and w  0.
In practice this means that the error in the phase will be
small most of the time, but there will be a significant num-
ber of results with a large error.
Adaptive measurements.—Although the quantum inter-
ferometry problem is now formally solved, this is of little
practical use because (even if the optimal input states could
be produced) it is not known how to implement the canoni-
cal measurement scheme. In particular, it is impossible to
implement it by counting photons in the two output ports of
the interferometer [7], as an experimenter would expect to
do. Nevertheless, as we will show, it is possible to closely
approximate the canonical measurement by counting out-
put photons if one makes the measurement adaptive. The
situation is as in Fig. 1. The unknown phase we wish to
measure, w, is in one arm of the interferometer, and we
introduce a known phase shift, F, into the other arm of
the interferometer. After each photodetection we adjust
this introduced phase shift in order to minimize the ex-
pected uncertainty of our best phase estimate after the next
photodetection.
The annihilation operators cˆ1 and cˆ0 for the two output
modes shown in Fig. 1 are related to the inputs by
cˆuw,F  aˆ sin
w 2 F 1 up
2
1 bˆ cos
w 2 F 1 up
2
.
(13)
Before the mth photon has been detected, the phaseF will
be fixed to the value Fm by an adaptive algorithm to be
−1.5 −1 −0.5 0 0.5 1 1.50
0.01
0.02
0.03
0.04
 
si
n2
 
φ P
(φ)
 φ
FIG. 4. The canonical phase probability distribution for j j0z
multiplied by sin2f for 2j  80 photons.5100specified later. Clearly the feedback loop which adjusts
Fm must act much faster than the average time between
photon arrivals. It is the ability to change Fm during the
passage of a single (two-mode) pulse that makes photon
counting measurements more general than a measurement
of the output Jˆz considered in Refs. [2,3].
An adjustable second phase F is of use even without
feedback [12]. By setting
Fm  F0 1
mp
N
, (14)
where F0 is chosen randomly, an estimate f of w can be
made with an accuracy independent of w. However, as
we will show, this phase estimate has a variance scaling as
ON21 rather than ON22.
Let us denote the result u from the mth detection as
um (which is 0 or 1 according to whether the photon is
detected in mode c0 or c1), and the measurement record
up to and including the mth detection as the binary string
nm  um · · · u2u1. The input state after m detections will
be a function of the measurement record and w, and we
denote it as jc˜nm,w. It is determined by the initial
condition jc˜n0,w  jc and the recurrence relation
jc˜umnm21,w  cˆum w,Fm
jc˜nm21,wp
N 1 1 2 m
. (15)
These states are unnormalized, and their norm represents
the probability for the record nm, given w,
Pnm jw  c˜nm,w j c˜nm,w . (16)
Thus the probability of obtaining the result um at the mth
measurement, given the previous results nm21, is
Pum jw, nm21  c˜umnm21,w j c˜umnm21,wc˜nm21,w j c˜nm21,w .
(17)
Also, the posterior probability distribution for w is
Pw j nm  Nmnm c˜nm,w j c˜nm,w , (18)
whereNmnm is a normalization factor. To obtain this we
have used Bayes’ theorem assuming a flat prior distribution
for w (that is, an initially unknown phase). A Bayesian
approach to interferometry has been considered previously,
and even realized experimentally [12]. However, this was
done only with nonadaptive measurements and with all
particles incident on one input port.
With this background, we can now specify the adaptive
algorithm for Fm. The sharpness of the distribution after
the mth detection is given by
Swumnm21 
ÇZ 2p
0
Pw jumnm21eiw dw
Ç
. (19)
We choose the feedback phase before the mth detection,
Fm, to maximize the sharpness. Since we do not know um
beforehand, we weight the sharpnesses for the two alterna-
tive results by their probability of occurring based on the
previous measurement record. Therefore the expression
VOLUME 85, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 11 DECEMBER 2000we maximize is
MFm jnm 
X
um0,1
Pum j nm21Swumnm21 . (20)
Using Eqs. (17), (18), and (19), and ignoring the constant
Nmnm, the maximand can be rewritten as
X
um0,1
ÇZ 2p
0
c˜umnm21,w j c˜umnm21,weiw dw
Ç
.
(21)
The controlled phaseFm appears implicitly in (21) through
cˆum w,Fm in (13), which appears in the recurrence rela-
tion (15). The maximizing solution Fm can be found ana-
lytically, but we will not exhibit it here.
The final part of the adaptive scheme is choosing the
phase estimate f of w from the complete data set nN . To
maximize Re	eif2w
 (that is, to minimize the deviation
of f from w), we choose f to be the mean of the posterior
distribution Pw j nN , which from Eq. (18) is
f  arg
Z 2p
0
c˜nN ,w j c˜nN ,weiw dw . (22)
We can determine the approximate phase variance
Df2 under this adaptive scheme using a stochastic
method. The phase w is taken to be zero, which leads
to no loss of generality since the initial controlled phase
F1 is chosen randomly. The measurement results are
determined randomly with probabilities determined using
w  0, and the final estimate f determined as above.
From Eq. (6), an ensemble fmMm1 of M final estimates
allows the phase variance to be approximated by Df2 
21 1 jM21PMm1 eifm j22. It is also possible to deter-
mine the phase variance exactly by systematically going
through all the possible measurement records and aver-
aging over F1. This method is feasible only for photon
numbers up to 20 or 30, however.
The results of using this adaptive phase measurement
scheme on the optimal input states determined above are
shown in Fig. 3. The phase variance is very close to the
phase variance for ideal measurements, with scaling very
close toN22. The phase variances do differ relatively more
from the ideal values for larger photon numbers, however,
indicating a scaling slightly worse than N22. It is possible
that the actual scaling is logNN2, as is the case for op-
timal single-mode adaptive phase measurements based on
dyne detection [13,14]. For comparison, we also show the
variance from the nonadaptive phase measurement definedby Eq. (14). As is apparent, this has a variance scaling as
N21. Finally, the results for j j0z input states with this
adaptive phase measurement scheme (modified to estimate
w modulo p , as we must for these states) are also shown.
The variances are again close to those for ideal measure-
ments, scaling as N212.
To conclude, we have shown that although the j j0z
input states considered by previous authors do not give a
phase variance that scales as 1N2 under ideal measure-
ments, it is straightforward to derive optimal input states
that do give this scaling. These states require significant
contributions from only 9 or 10 Jˆz eigenstates (photon
eigenstates for the two input modes). We have also demon-
strated a practical measurement scheme (i.e., one based on
photodetection of the output modes) to approximate ideal
measurements. This scheme, which uses feedback, allows
phase measurements with a variance scaling close to 1N2
with optimized input states.
[1] C. M. Caves, Phys. Rev. D 23, 1693 (1981).
[2] B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A
33, 4033 (1986).
[3] M. J. Holland and K. Burnett, Phys. Rev. Lett. 71, 1355
(1993).
[4] B. C. Sanders and G. J. Milburn, Phys. Rev. Lett. 75, 2944
(1995).
[5] B. C. Sanders, G. J. Milburn, and Z. Zhang, J. Mod. Opt.
44, 1309 (1997).
[6] Except for Ref. [1], the above proposals have involved
input states where the total number of photons is fixed.
For simplicity, we follow this convention.
[7] We have found that for N  3 and N  4 our feedback
scheme as applied to optimum input states is globally
optimum, and also that the phase variance is above the
canonical phase variance. This demonstrates that canonical
measurements cannot be realized by counting photons in
the interferometer outputs, even if feedback is allowed.
[8] G. S. Summy and D. T. Pegg, Opt. Commun. 77, 75 (1990).
[9] H. M. Wiseman and R. B. Killip, Phys. Rev. A 56, 944
(1997).
[10] A. S. Holevo, Springer Lecture Notes Math. 1055, 153
(1984).
[11] The phase variance starts out less than that for the optimum
state, but this is merely due to using a slightly different
measure of the variance.
[12] Z. Hradil et al., Phys. Rev. Lett. 76, 4295 (1996).
[13] H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169
(1998).
[14] D. W. Berry and H. M. Wiseman, Phys. Rev. A (to be
published).5101


Optimal states and almost optimal adaptive measurements for quantum interferometry

Crossref

Physical Review Letters

Optimal states and almost optimal adaptive measurements for quantum
  interferometry

Optimal states and almost optimal adaptive measurements for quantum interferometry

Abstract

Similar works

Full text

Available Versions

Griffith Research Online

University of Queensland eSpace

Crossref