We present an experimental demonstration of the power of real-time feedback
in quantum metrology, confirming a theoretical prediction by Wiseman regarding
the superior performance of an adaptive homodyne technique for single-shot
measurement of optical phase. For phase measurements performed on weak coherent
states with no prior knowledge of the signal phase, we show that the variance
of adaptive homodyne estimation approaches closer to the fundamental quantum
uncertainty limit than any previously demonstrated technique. Our results
underscore the importance of real-time feedback for reaching quantum
performance limits in coherent telecommunication, precision measurement and
information processing.Comment: RevTex4, color PDF figures (separate files), submitted to PR

A. C. Doherty

Andrew C. Doherty

D. W. Berry

Hideo Mabuchi

John K. Au

John K. Stockton

K. Schneider

Michael A. Armen

W. P. Smith

Y. Yamamoto

English

arXiv

We present an experimental demonstration of the power of feedback in quantum metrology, confirming the predicted [H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] superior performance of an adaptive homodyne technique for single-shot measurement of optical phase. For measurements performed on weak coherent states with no prior knowledge of the signal phase, adaptive homodyne estimation approaches closer to the intrinsic quantum uncertainty than any previous technique. Our results underscore the importance of real-time feedback for reaching quantum limits in measurement and control

Armen, Michael A.

Au, John K.

Stockton, John K.

Doherty, Andrew C.

Mabuchi, Hideo

University of Queensland eSpace

VOLUME 89, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 23 SEPTEMBER 2002Adaptive Homodyne Measurement of Optical Phase
Michael A. Armen,* John K. Au, John K. Stockton, Andrew C. Doherty, and Hideo Mabuchi
Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, California 91125
(Received 1 April 2002; published 4 September 2002)133602-1We present an experimental demonstration of the power of feedback in quantum metrology,
confirming the predicted [H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] superior performance of
an adaptive homodyne technique for single-shot measurement of optical phase. For measurements
performed on weak coherent states with no prior knowledge of the signal phase, adaptive homodyne
estimation approaches closer to the intrinsic quantum uncertainty than any previous technique. Our
results underscore the importance of real-time feedback for reaching quantum limits in measurement
and control.
DOI: 10.1103/PhysRevLett.89.133602 PACS numbers: 42.50.Dv, 03.65.Ta, 03.67.–a, 06.90.+vperiments — there has yet to be any demonstration of a
measurement procedure that is capable even in principle
phases must also be measured optimally but where there is
a guarantee that the phases vary only over a range muchQuantum mechanics complicates metrology in two
complementary ways. First it forces us to accept the
existence of intrinsic uncertainty in the value of observ-
ables such as the position, momentum, and phase of an
oscillator. Such quantum uncertainty exists even when the
state of the system being measured has been prepared in a
technically flawless way. Even the vacuum state of a
single-mode optical field, for example, exhibits ‘‘zero-
point’’ fluctuations in the amplitudes of its electric and
magnetic components. The same is true of the optical
coherent states representative of the output of an ideal
laser. These uncertainties limit the sensitivity of ubiqui-
tous measurement techniques such as laser interferome-
try. Squeezed states of light are challenging to produce
but have reduced uncertainties that could be exploited
for improved sensitivity in optical metrology [1]. To
date, however, the achievable performance gains have
not been sufficient to motivate their use in practical
applications.
Quantum mechanics presents a second major obstacle
to precision measurement by making it generally quite
difficult to realize ideal measurement procedures, whose
inherent inaccuracy is small enough to reveal the intrinsic
uncertainty limits associated with quantum states. Let us
refer to such ideal measurements as being uncertainty-
limited (UL). Clearly, the implementation of UL meas-
urement procedures is essential for any application that
seeks to take advantage of exotic quantum states with
reduced intrinsic uncertainties. In connection with
squeezed states of light, measurements of optical quad-
rature amplitudes do constitute an important class of UL
measurements that actually can be implemented in prac-
tice (via homodyne detection). But this is an unusual case
as UL measurement schemes have not previously been
demonstrated even for closely related observables such as
optical phase, despite intense historical interest in their
quantum properties [2,3]. This shortcoming is not merely
one of achievable signal-to-noise ratio in realistic ex-0031-9007=02=89(13)=133602(4)$20.00of achieving UL estimation of true optical phase (as
opposed to phase-quadrature amplitude), for coherent or
any other pure states of an optical field.
In this Letter we present an experimental demonstra-
tion of the surprising efficacy of real-time feedback in the
development of UL measurement procedures. We do this
in the context of measuring the optical phases of weak
pulses of light, following a theoretical proposal by
Wiseman [4]. This metrological task can be motivated
by a coherent optical communication scenario in which
information is encoded in the phase of laser pulses that
must travel long distances between the sender and re-
ceiver. In such a context the receiver is likely to be faced
with decoding information carried by optical wave pack-
ets whose quantum states correspond to coherent states
with low mean photon number (as a result of optical
attenuation). If the sender is encoding information in an
efficient manner, the variation of phase from one optical
wave packet to the next should be uniformly distributed
over the entire interval from zero to 2. Hence, the
receiver would ideally like to implement a single-shot
UL measurement procedure for estimating the phase of
each individual pulse [5]. The variance of such optimal
phase estimates should be limited only by the intrinsic
quantum uncertainty associated with optical coherent
states of the given mean photon number.
There is no known experimental procedure to accom-
plish this goal exactly. Prior to Wiseman’s proposal [4] it
had been widely believed [6] that the best feasible strategy
for single-shot phase decoding should be heterodyne de-
tection, which for coherent signal states can in principle
achieve a measurement variance that is only a factor of 2
greater than the intrinsic uncertainty limit [7]. In what
follows we will thus consider heterodyne phase estima-
tion as the benchmark we need to surpass. It is important
to compare the current measurement scenario with what
arises in applications such as the implementation of opti-
cal frequency standards (optical clocks), where optical 2002 The American Physical Society 133602-1
VOLUME 89, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 23 SEPTEMBER 2002less than =2. In such cases, as mentioned above, essen-
tially UL measurement can be achieved by employing
fixed-quadrature homodyne detection with a local oscil-
lator whose phase is held at an offset of =2 radians from
the expected signal phase. Real-time feedback can thus be
seen as a key ingredient in formulating an UL scheme for
measurement scenarios in which we have no prior knowl-
edge of the signal phase —feedback enables protocols in
which the local oscillator phase is adapted, in real-time,
to the phase of each individual signal pulse [8]. As each
optical pulse has some spatiotemporal extent, the meas-
urement signal generated by the leading edge of a given
pulse can be used to form a preliminary estimate of its
phase, which is used promptly to adjust the local oscil-
lator settings to be optimal for that pulse. (Note that this
essentially amounts to the implementation of a quantum-
noise limited phase-lock loop; the scheme relies on lock
acquisition from the first one or two photons of signal in
each pulse.) Detailed theoretical analyses of such schemes
by Wiseman and co-workers have led to the striking
realization that, despite extremely low signal-to-noise
ratio in the feedback loop, adaptive homodyne measure-
ment can be essentially UL for optical pulses with mean
photon number of order 10 (or greater), and benefits of
real-time adaptation should still be evident for mean
photon numbers 1. We turn now to our experimental
test of these predictions.
The data plotted in Fig. 1(a) demonstrate the superi-
ority of an adaptive homodyne measurement procedure
(‘‘adaptive’’) to the benchmark heterodyne measurement
procedure (‘‘heterodyne’’) for making single-shot esti-
mates of optical phase. As described in detail below, we133602-2perform these measurements on optical pulses of 50 s
duration derived from an intensity-stabilized cw laser. We
have also plotted the theoretical prediction for the var-
iance of ideal heterodyne measurement, both with (thin
solid line) and without (dotted line) correction for a small
amount of excess electronic noise in the balanced photo-
current. The excellent agreement between the heterodyne
data and theory indicates that we have no excess phase
noise (technically ideal preparation of coherent signal
states) and validates electronic calibrations involved in
our data analysis [9]. In the range of10–300 photons per
pulse, the adaptive data lie well below the heterodyne
curve that has been corrected for electronic noise (which
also has a detrimental effect on the adaptive data), and a
few of the data points lie significantly below the absolute
limit. Quantitatively, for  50 photons per pulse the
adaptive data point sits 6.5 standard deviations below
the absolute heterodyne limit (note logarithmic scale).
For intermediate values of the photon number the
adaptive variances should ideally be even lower [10], but
the performance of our experiment is limited by finite
feedback bandwidth. For signals with large mean photon
number, the adaptive scheme is inferior to heterodyne
because of excess technical noise in the feedback loop.
As the intrinsic phase uncertainty of coherent states
becomes large for very low photon numbers, the relative
differences among the expected variances for adaptive,
heterodyne, and ideal estimation become small.
Accordingly, we have been unable to beat the heterodyne
limit for phase estimation variances with adaptive homo-
dyne measurement for mean photon numbers N & 8.
(Note that all theoretical curves in Fig. 1(a) correspondFIG. 1 (color). Experimental results from
the adaptive and heterodyne measurements.
(a) Adaptive (blue circles) and heterodyne
(red crosses) phase estimate variance vs
pulse photon number. The blue dash-dotted
line is a second order curve through the
adaptive data, to guide the eye. The thin
lines are the theoretical curves for hetero-
dyne detection with (solid) and without
(dotted) corrections for detector electronic
noise. The thick solid line denotes the
fundamental quantum uncertainty limit,
given our overall photodetection efficiency.
(b) Phase-estimator distributions for adap-
tive (blue circles) and heterodyne (red
crosses) measurements, for pulses with
mean photon number  2:5. (c) Polar plot
showing the variance of adaptive phase
estimates (blue dots) for different signal
phases (mean photon number  50). The
solid blue line is a linear fit to the data. The
double red lines indicate the 1 scatter for
our heterodyne data, averaged over initial
phase.
133602-2
FIG. 2 (color). (a) Apparatus used to perform both adaptive
homodyne and heterodyne measurements (see text). Solid lines
denote optical paths, and dashed lines denote electrical paths.
(b) Photocurrent It (red above), and feedback signal t
(green below), for three consecutive adaptive homodyne meas-
urements with N  8. The x axis represents time for both
signals. The y axis scale indicates the absolute phase shifts
made by the feedback signal. The photocurrent is plotted on an
arbitrary scale.
VOLUME 89, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 23 SEPTEMBER 2002to asymptotic expressions valid for large mean photon
numbers; corrections are small for N * 10.) However, we
are able to show that the estimator distribution for adap-
tive homodyne remains narrower than that for heterodyne
detection even for pulses with mean photon number down
to N  0:8. In Fig. 1(b) we display the adaptive and
heterodyne phase-estimator distributions for a signal
size corresponding to  2:5 photons. Note that we have
plotted the distributions on a logarithmic scale, and that
the mean phase has been subtracted off so that the dis-
tributions are centered at zero. The horizontal axis can
thus be identified with estimation error. The adaptive
phase distribution has a smaller ‘‘Gaussian width’’ than
heterodyne but exhibits rather high tails, in qualitative
agreement with predictions [11] based on quantum esti-
mation theory. Theoretical analysis suggests that the high
tails are a consequence of changes in the sign of the
photocurrent caused by vacuum fluctuations of the optical
field, which become comparable to the coherent optical
signal for N  1. The feedback algorithm responds to
these photocurrent inversions by locking to incorrect
phase values with an error that tends towards .
Accurately assessing the performance of a single-shot
measurement requires many repetitions of the measure-
ment under controlled conditions. Figure 2(a) shows a
schematic of our apparatus. Light from a single-mode
cw Nd:YAG laser is first stripped of excess intensity noise
by passage through a high-finesse Fabry-Perot cavity (not
pictured) with ringdown time  16 s; the transmitted
beam is shot-noise limited above 50 kHz. This light
enters the Mach-Zehnder interferometer (MZI) at beam
splitter 1 (BS1), creating two beams with well-defined
relative phase. The local oscillator (LO) is generated
using an acousto-optic modulator (AOM) driven by an
rf synthesizer RF1 at 84.6 MHz, yielding 230 W of
frequency-shifted light. The signal beam corresponds to a
frequency sideband created by an electro-optic modulator
(EOM) driven by an rf synthesizer (RF2) that is phase
locked to RF1. The power (5 fW to 5 pW) and pulse length
(50 s) of the signal beam are controlled by changing the
amplitude of RF2 and by switching it on/off. A pair of
photodetectors collect the light emerging from the two
output ports of the final 50/50 beam splitter (BS2); the
difference of their photocurrents [balanced photocurrent
It] provides the basic signal used for either heterodyne
or adaptive phase estimation [9]. At our typical LO
power, the photodetectors supply 6 dB of shot noise
(over electronic noise) in the difference photocurrent
from 1 kHz to 10 MHz. We perform adaptive homodyne
measurement by feedback to the phase of RF2, which sets
the (instantaneous) relative phase between signal and LO
[12]. Our feedback bandwidth 1:5 MHz is limited by
the maximum slew rate of RF2. Real-time electronic
signal processing for the feedback algorithm is per-
formed by a field programmable gate array (FPGA) that
can execute complex computations with very low delay133602-3[13]. Our feedback and phase-estimation procedures cor-
respond to the ‘‘Mark II’’ scheme of Wiseman and co-
workers [11], in which the photocurrent is integrated with
time-dependent gain to determine the instantaneous
feedback signal: t  =2 / Rt0 dv Iv=

v
p
; where t
is scaled such that the pulse has duration 1. The final
phase estimate depends on integrals of the photocurrent
and feedback signal according to the Mark II scheme. For
heterodyne measurements we turn off the feedback to
RF2 and detune it from RF1 by 1.8 MHz; phase estimates
are made by the standard method of I=Q demodulating
the photocurrent beat note. For both types of measure-
ment we store the photocurrent It and feedback signal
t on a computer for postprocessing.
In Fig. 2(b) we show the photocurrent and feedback
signal of three consecutive adaptive phase measurements133602-3
VOLUME 89, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 23 SEPTEMBER 2002for N  8. At the beginning of each measurement, feed-
back of the photocurrent shot noise causes the relative
phase between the signal and LO to vary randomly. As
more of the pulse is detected, the information gained is
used to drive the signal-LO phase towards an optimal
value. Phase estimation variances are established using
ensembles of  150 consecutive single-shot measure-
ments. Although each optical pulse is the subject of an
independent single-shot measurement, we fix the signal
phase over the length of each ensemble in order to accu-
rately determine the estimation variance. It is important
to note that each data point in Fig. 1(a) corresponds to an
average of variance estimates from many ensembles, each
with a different (random) signal phase.
Ideally, the performance of a single-shot phase mea-
surement procedure should be independent of the signal
phase. In our experiment, this implies that the measure-
ment variances should be independent of the initial rela-
tive phase between signal and local oscillator. In Fig. 1(c)
we display a polar plot of the adaptive variance (radial)
versus initial signal phase (azimuth). The data were taken
using signals with mean photon number  50. Each data
point corresponds to the variance of an ensemble of phase
measurements taken at a fixed phase. The double solid
lines indicate the 1 scatter of our heterodyne data. It
is clear from this data that the performance of the adap-
tive homodyne scheme is essentially independent of ini-
tial phase, and is consistently superior to heterodyne
measurement.
The photon number per pulse, N, is determined by
extracting optical amplitude information from the bal-
anced photocurrent in heterodyne mode. The typical
experimental procedure is to fix the signal amplitude,
take an ensemble of heterodyne measurements (which
yields both heterodyne phase estimates and an estimate
of N), and then take an ensemble of adaptive homodyne
measurements. The photon number assigned to the sub-
sequent ensemble of adaptive measurements is 0:95N,
where the relative calibration factor arises from the mea-
sured response of the EOM.
In conclusion, we have presented an experimental dem-
onstration of adaptive homodyne phase measurement.
For pulses with mean photon number 10–300 our mea-
sured variances approach closer to the intrinsic phase-
uncertainty limit of coherent states than any previously
demonstrated technique. These results establish the fea-
sibility of broadband quantum-noise-limited feedback
for adaptive quantum measurement [14], quantum feed-
back control [15], quantum error correction [16], and
studies of conditional quantum dynamics [17,18].133602-4This work was supported by the NSF (PHY-9987541,
EIA-0086038) and ONR (N00014-00-1-0479).*Electronic address: armen@caltech.edu
[1] K. Schneider et al., Opt. Express 2, 59 (1998).
[2] See special issue of Phys. Scr. T48 (1993).
[3] U. Leonhardt et al., Phys. Rev. A 51, 84 (1995).
[4] H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995).
[5] Note that this precludes the use of techniques such as
homodyne tomography, which rely on multiple measure-
ments on an ensemble of identically prepared systems.
[6] C. M. Caves and P. D. Drummond, Rev. Mod. Phys. 66,
481 (1994).
[7] This factor of 2 derives from the fact that heterodyne de-
tection samples both the amplitude and phase-quadrature
amplitudes, which are complementary observables.
[8] Y. Yamamoto et al., in Progress in Optics, edited by
E. Wolf (North-Holland, Amsterdam, 1990), Vol. 28,
p. 150.
[9] Our calibration of the horizontal axis has been adjusted
for overall photodetection efficiency , which has no
bearing on the comparison of adaptive and heterodyne
procedures but does affect our placement of curves on
the graph; had the measurements been of unit efficiency,
the data and theory variances would decrease by a factor
of . We have independently measured   0:56 (detec-
tor quantum efficiency  0:85, homodyne efficiency 
0:66). For phase measurements performed on squeezed
states  would need to be optimized, but its value and
affect on the photon-number calibration are not of direct
concern in our current work involving coherent states.
[10] For pulses with mean photon number > 10, an ideal
adaptive measurement would lie within 1% of the un-
certainty limit. Accounting for technical noise, the var-
iance of the adaptive measurement could at best lie a
factor of 2 below the noise-corrected heterodyne curve.
[11] H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169
(1998).
[12] In principle one can modulate either the signal phase or
the LO phase; we do the former to avoid interference
between the adaptive homodyne feedback loop and an
auxiliary feedback loop used to stabilize the MZI.
[13] J. K. Stockton, M. Armen, and H. Mabuchi, J. Opt. Soc.
Am. B (to be published).
[14] D.W. Berry and H. M. Wiseman, Phys. Rev. A 63, 013813
(2001).
[15] A. C. Doherty et al., Phys. Rev. A 62, 012105 (2000).
[16] C. Ahn, A. C. Doherty, and A. Landahl, Phys. Rev. A, 65,
042301 (2002).
[17] W. P. Smith et al., Phys. Rev. Lett. 89, 133601 (2002).
[18] H. Mabuchi and H. M. Wiseman, Phys. Rev. Lett. 81, 4620
(1998).133602-4


Adaptive homodyne measurement of optical phase

We present an experimental demonstration of the power of feedback in quantum metrology, confirming the predicted [H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] superior performance of an adaptive homodyne technique for single-shot measurement of optical phase. For measurements performed on weak coherent states with no prior knowledge of the signal phase, adaptive homodyne estimation approaches closer to the intrinsic quantum uncertainty than any previous technique. Our results underscore the importance of real-time feedback for reaching quantum limits in measurement and control

Caltech Authors - Main

ar
X
iv
:q
ua
nt
-p
h/
02
04
00
5v
1 
 1
 A
pr
 2
00
2
Adaptive homodyne measurement of optical phase
Michael A. Armen, John K. Au, John K. Stockton, Andrew C. Doherty, and Hideo Mabuchi
Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, CA 91125
(Dated: March 31, 2002)
We present an experimental demonstration of the power of real-time feedback in quantum metrol-
ogy, confirming a theoretical prediction by Wiseman [1] regarding the superior performance of an
adaptive homodyne technique for single-shot measurement of optical phase. For phase measure-
ments performed on weak coherent states with no prior knowledge of the signal phase, we show
that the variance of adaptive homodyne estimation approaches closer to the fundamental quantum
uncertainty limit than any previously demonstrated technique. Our results underscore the impor-
tance of real-time feedback for reaching quantum performance limits in coherent telecommunication,
precision measurement and information processing.
Quantum mechanics complicates metrology in two
complementary ways. First it forces us to accept the
existence of intrinsic uncertainty in the value of observ-
ables such as the position, momentum, and phase of an
oscillator. Such quantum uncertainty exists even when
the state of the system being measured has been prepared
in a technically flawless way. Even the vacuum state of
a single-mode optical field, for example, exhibits ‘zero-
point’ fluctuations in the amplitudes of its electric and
magnetic components. The same is true of the optical
coherent states representative of the output of an ideal
laser. These uncertainties limit the sensitivity of ubiqui-
tous measurement techniques such as laser interferome-
try. Squeezed states of light are challenging to produce
but have reduced uncertainties that can in principal be
exploited for improved sensitivity in optical metrology
[2]. To date, however, the achievable performance gains
have not been sufficient to motivate their use in practical
applications.
Quantum mechanics presents a second major obsta-
cle to precision measurement by making it generally
quite difficult to realize ideal measurement procedures,
whose inherent inaccuracy is small enough to reveal
the intrinsic uncertainty limits associated with quantum
states. Let us refer to such ideal measurements as being
Uncertainty-Limited (UL). Clearly, the implementation
of UL measurement procedures is essential for any ap-
plication that seeks to take advantage of exotic quantum
states with reduced intrinsic uncertainties. In connection
with squeezed states of light, measurements of optical
quadrature amplitudes do constitute an important class
of UL measurements that actually can be implemented
in practice (via homodyne detection). But this is an un-
usual case as UL measurement schemes have not previ-
ously been demonstrated even for closely related observ-
ables such as optical phase, despite intense historical in-
terest in their quantum properties [3, 4]. This shortcom-
ing is not merely one of achievable signal-to-noise ratio
in realistic experiments – there has yet to be any demon-
stration of a measurement procedure that is capable even
in principle of achieving UL estimation of true optical
phase (as opposed to phase-quadrature amplitude), for
coherent or any other pure states of an optical field.
In this article we present an experimental demonstra-
tion of the surprising efficacy of real-time feedback in the
development of UL measurement procedures. We do this
in the context of measuring the optical phases of weak
pulses of light, following a theoretical proposal by Wise-
man [1]. This metrological task can be motivated by a
coherent optical communication scenario in which infor-
mation is encoded in the phase of laser pulses that must
travel long distances between the sender and receiver. In
such a context the receiver is likely to be faced with de-
coding information carried by optical wave-packets whose
quantum states correspond to coherent states with low
mean photon number (as a result of optical attenuation).
If the sender is encoding information in an efficient man-
ner, the variation of phase from one optical wave-packet
to the next should be uniformly distributed over the en-
tire interval from zero to 2π. Hence, the receiver would
ideally like to implement a single-shot UL measurement
procedure for estimating the phase of each individual
pulse [5]. The variance of such optimal phase estimates
should be limited only by the intrinsic quantum uncer-
tainty associated with optical coherent states of the given
mean photon number.
There is no known experimental procedure to accom-
plish this goal exactly. Prior to Wiseman’s proposal [1] it
had been widely believed [6] that the best feasible strat-
egy for single-shot phase decoding should be heterodyne
detection, which for coherent signal states can in princi-
ple achieve a measurement variance that is only a factor
of two greater than the intrinsic uncertainty limit [7]. In
what follows we will thus consider heterodyne phase esti-
mation as the benchmark we need to surpass. It is impor-
tant to compare the current measurement scenario with
what arises in applications such as the implementation of
optical frequency standards (optical clocks), where opti-
cal phases must also be measured optimally but where
there is a guarantee that the phases vary only over a
range much less than π/2. In such cases, as mentioned
above, essentially UL measurement can be achieved by
employing fixed-quadrature homodyne detection with a
local oscillator whose phase is held at an offset of π/2
2
radians from the expected signal phase. Real-time feed-
back can thus be seen as a key ingredient in formulating
an UL scheme for measurement scenarios in which we
have no prior knowledge of the signal phase – feedback
enables protocols in which the local oscillator phase is
adapted, in real time, to the phase of each individual sig-
nal pulse [8]. As each optical pulse has some spatiotem-
poral extent, the measurement signal generated by the
leading edge of a given pulse can be used to form a pre-
liminary estimate of its phase, which is used promptly to
adjust the local oscillator settings to be optimal for that
pulse. (Note that this essentially amounts to the imple-
mentation of a quantum-noise limited phase-lock loop;
the scheme relies on lock acquisition from the first one
or two photons of signal in each pulse.) Detailed the-
oretical analyses of such schemes by Wiseman and co-
workers have led to the striking realization that, despite
extremely low signal-to-noise ratio in the feedback loop,
adaptive homodyne measurement can be essentially UL
for optical pulses with mean photon number of order ten
(or greater), and benefits of real-time adaptation should
still be evident for mean photon numbers ∼ 1. We turn
now to our experimental test of these predictions.
The data plotted in Fig. 1A demonstrate the supe-
riority of an adaptive homodyne measurement proce-
dure (‘adaptive’) to the benchmark heterodyne measure-
ment procedure (‘heterodyne’) for making single-shot es-
timates of optical phase. As described in detail below,
we perform these measurements on optical pulses of 50
µs duration derived from an intensity-stabilized cw laser.
We have also plotted the theoretical prediction for the
variance of ideal heterodyne measurement, both with
(thin solid line) and without (dotted line) correction for
a small amount of excess electronic noise in the balanced
photocurrent. The excellent agreement between the het-
erodyne data and theory indicates that we have no excess
phase noise (technically-ideal preparation of coherent sig-
nal states) and validates electronic calibrations involved
in our data analysis [9]. In the range of ∼ 10 − 300
photons per pulse, most of the adaptive data lies below
the absolute theoretical limit for heterodyne measure-
ment (dotted line), and all of it lies below the curve that
has been corrected for excess electronic noise (which also
has a detrimental effect on the adaptive data). Quan-
titatively, we note that for ≈ 50 photons per pulse the
adaptive data point sits 6.5 standard deviations below
the absolute heterodyne limit (note logarithmic scale).
For intermediate values of the photon number the
adaptive variances should ideally be even lower, but the
performance of our experiment is limited by finite feed-
back bandwidth. For signals with large mean photon
number, the adaptive scheme is inferior to heterodyne
because of excess technical noise in the feedback loop.
As the intrinsic phase uncertainty of coherent states be-
comes large for very low photon numbers, the relative
differences among the expected variances for adaptive,
heterodyne, and ideal estimation become small. Ac-
cordingly, we have been unable to beat the heterodyne
limit for phase-estimation variances with adaptive ho-
modyne measurement for mean photon numbers N <
∼
8.
(Note that all theoretical curves in Fig. 1A correspond to
asymptotic expressions valid for large mean photon num-
bers; corrections are small for N >
∼
10.) However, we are
able to show that the estimator distribution for adaptive
homodyne remains narrower than that for heterodyne de-
tection even for pulses with mean photon number down
to N ≈ 0.8. In Fig. 1B we display the adaptive and het-
erodyne phase-estimator distributions for a signal size
corresponding to ≈ 2.5 photons. Note that we have plot-
ted the distributions on a logarithmic scale, and that the
mean phase has been subtracted off so that the distribu-
tions are centered at zero. The horizontal axis can thus
be identified with estimation error. The adaptive phase
distribution has a smaller ‘Gaussian width’ than the het-
erodyne distribution, but exhibits rather high tails. The
observed shape of the adaptive distribution agrees qual-
itatively with predictions [10] based on quantum esti-
mation theory. Theoretical analysis suggests that the
high tails are a consequence of changes in the sign of the
photocurrent caused by vacuum fluctuations of the opti-
cal field, which become comparable to the coherent op-
tical signal for N ∼ 1. The feedback algorithm responds
to these photocurrent inversions by locking to incorrect
phase values with an error that tends towards π.
Accurately assessing the performance of a single-shot
measurement requires many repetitions of the measure-
ment under controlled conditions. Fig. 2A shows a
schematic of our apparatus. Light from a single-mode
cw Nd:YAG laser is first stripped of excess intensity noise
by passage through a high-finesse Fabry-Perot cavity (not
pictured) with ringdown time ≈ 16 µs; the transmitted
beam is shot-noise limited above ∼ 50 kHz. This light
enters the Mach-Zehnder interferometer (MZI) at beam-
splitter 1 (BS1), creating two beams with well-defined
relative phase. The local oscillator (LO) is generated
using an acousto-optic modulator (AOM) driven by an
RF synthesizer RF1 at 84.6 MHz, yielding ∼ 230 µW of
frequency-shifted light. The signal beam corresponds to
a frequency sideband created by an electro-optic modu-
lator (EOM) driven by an RF synthesizer (RF2) that is
phase locked to RF1. The power (5 fW to 5 pW) and
pulse length (50 µs) of the signal beam are controlled
by changing the amplitude of RF2 and by switching it
on/off. A pair of photodetectors collect the light emerg-
ing from the two output ports of the final 50/50 beam-
splitter (BS2); the difference of their photocurrents (bal-
anced photocurrent) provides the basic signal used for
either heterodyne or adaptive phase estimation. At our
typical LO power, the photodetectors supply 6 dB of shot
noise (over electronic noise) in the difference photocur-
rent from to ∼ 1 kHz to 10 MHz. We perform adaptive
homodyne measurement by feedback to the phase of RF2,
3
which sets the (instantaneous) relative phase between sig-
nal and LO [11]. Our feedback bandwidth ∼ 1.5 MHz is
limited by the maximum slew rate of RF2. Real-time
electronic signal processing for the feedback algorithm is
performed by a Field Programmable Gate Array (FPGA)
that can execute complex computations with very low
latency [12]. Our feedback and phase estimation proce-
dures correspond to the ‘Mark II’ scheme of Wiseman and
co-workers [10], in which the photocurrent is integrated
with time-dependent gain to determine the instantaneous
feedback signal. The final phase estimate depends on
the full history of the photocurrent and feedback signal.
For heterodyne measurements we turn off the feedback to
RF2 and detune it from RF1 by 1.8MHz; phase estimates
are made by the standard method of I/Q demodulating
the photocurrent beat note. For both types of measure-
ment we store the photocurrent I(t) and feedback signal
Φ(t) on a computer for post-processing.
In Fig. 2B we show the photocurrent and feedback sig-
nal of three consecutive adaptive phase measurements for
N ≈ 8. At the beginning of each measurement, feedback
of the photocurrent shot noise causes the relative phase
between the signal and LO to vary randomly. As more
of the pulse is detected, the information gained is used
to drive the signal-LO phase towards an optimal value.
Phase estimation variances are established using ensem-
bles of ≈ 150 consecutive single-shot measurements. Al-
though each optical pulse is the subject of an independent
single-shot measurement, we fix the signal phase over the
length of each ensemble in order to accurately determine
the estimation variance. It is important to note that
each data point in Fig. 1A corresponds to an average
of variance estimates from many ensembles, each with a
different (random) signal phase.
Ideally, the performance of a single-shot phase mea-
surement procedure should be independent of the signal
phase. In our experiment, this implies that the mea-
surement variances should be independent of the initial
relative phase between signal and local oscillator. In
Fig. 1C we display a polar plot of the adaptive variance
(radial) versus initial signal phase (azimuth). The data
were taken using signals with mean photon number ≈ 50.
Each data point corresponds to the variance of an ensem-
ble of phase measurements taken at a fixed phase. The
double solid lines indicate the (one sigma) performance of
our heterodyne measurements. It is clear from this data
that the performance of the adaptive homodyne scheme
is essentially independent of initial phase, and is consis-
tently superior to heterodyne measurement.
The photon number per pulse, N , is determined by ex-
tracting optical amplitude information from the balanced
photocurrent in heterodyne mode. The typical experi-
mental procedure is to fix the signal amplitude, take an
ensemble of heterodyne measurements (which yields both
heterodyne phase estimates and an estimate of N), and
then take an ensemble of adaptive homodyne measure-
ments. The photon number assigned to the subsequent
ensemble of adaptive measurements is 0.95N , where the
relative calibration factor arises from the measured re-
sponse of the EOM.
In conclusion, we have presented an experimental
demonstration of adaptive homodyne phase measure-
ment. For pulses with mean photon number ∼ 10 − 300
our measured variances approach closer to the intrinsic
phase-uncertainty limit of coherent states than any pre-
viously demonstrated technique, including ideal hetero-
dyne detection. These results establish the technical fea-
sibility of broadband quantum-noise-limited feedback for
adaptive quantum measurement [13], quantum feedback
control [14], quantum error correction [15], and studies
of conditional quantum dynamics [16, 17].
This work was supported by the NSF (PHY-9987541,
EIA-0086038) and ONR (N00014-00-1-0479). JKS ac-
knowledges the support of a Hertz Fellowship.
[1] H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995).
[2] K. Schneider et al., Opt. Express 2, 59 (1998).
[3] See special issue of Phys. Scr., T48 (1993).
[4] U. Leonhardt et al., Phys. Rev. A 51, 84 (1995).
[5] Note that this precludes the use of techniques such as
homodyne tomography, which rely on multiple measure-
ments on an ensemble of identically-prepared systems.
[6] C. M. Caves, P. D. Drummond, Rev. Mod. Phys. 66, 481
(1994).
[7] This factor of two derives from the fact that heterodyne
detection samples both the amplitude and phase quadra-
ture amplitudes, which are complementary observables.
[8] Y. Yamamoto et al., in Progress in Optics, ed. E. Wolf
(North-Holland, Amsterdam, 1990), Vol. 28, p. 150.
[9] Our absolute calibration of the horizontal axis has been
adjusted for overall photodetection efficiency η, which
has no bearing on the relative comparison of adaptive
and heterodyne measurement procedures but does affect
our placement of the fundamental uncertainty limit asso-
ciated with coherent states. We have independently mea-
sured η ≈ 0.56, including detector quantum efficiency
and homodyne fringe visibility, and have indicated the
fundamental limit for phase estimation given this value of
η. For phase measurements performed on squeezed states
η would need to be optimized, but its value and the cor-
responding correction to the photon-number calibration
are not of direct concern in our current investigation in-
volving coherent states.
[10] H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169
(1998).
[11] In principle one can modulate either the signal phase or
the LO phase; we do the former to avoid interference
between the adaptive homodyne feedback loop and an
auxiliary feedback loop used to stabilize the MZI.
[12] J. K. Stockton, M. Armen, and H. Mabuchi, in prepara-
tion (quant-ph/0203143 at http://www.arXiv.org).
[13] D. W. Berry, H. M. Wiseman, Phys. Rev. A 6301, 013813
(2001).
[14] A. C. Doherty et al., Phys. Rev. A 6201, 012105 (2000).
4
[15] C. Ahn, A. C. Doherty, and A. Landahl, Phys. Rev. A,
65, 042301 (2002).
[16] W. P. Smith et al., in preparation (quant-ph/0202063 at
http://www.arXiv.org).
[17] H. Mabuchi and H. M. Wiseman, Phys. Rev. Lett. 81,
4620 (1998).
5
FIG. 1: Experimental results from the adaptive and heterodyne measurements. (a) Adaptive (blue circles) and Heterodyne (red
crosses) phase estimate variance vs. pulse photon number. The blue dashed-dotted line is a second order fit to the adaptive data.
The thin lines are the theoretical curves for heterodyne detection with (solid) and without (dotted) corrections for detector
electronic noise. The thick solid line denotes the fundamental quantum uncertainty limit, given our overall photodetection
efficiency. (b) Phase-estimator distributions for adaptive (blue circles) and heterodyne (red crosses) measurements, for pulses
with mean photon number ≈ 2.5. (c) Polar plot showing the variance of adaptive phase estimates (blue dots) for different
signal phases (mean photon number ≈ 50). The solid blue line is a linear fit to the data. The double red lines indicate the one
sigma range for heterodyne measurements averaged over initial phase.
FIG. 2: (a) Apparatus used to perform both adaptive homodyne and heterodyne measurements (see text). Solid lines denote
optical paths, and dashed lines denote electrical paths. (b) Photocurrent I(t) (red above), and feedback signal Φ(t) (green
below), for three consecutive adaptive homodyne measurements with N ≈ 8. The x-axis represents time for both signals. The
y-axis scale indicates the absolute phase shifts made by the feedback signal. The photocurrent is plotted on an arbitrary scale.


Adaptive Homodyne Measurement of Optical Phase

Crossref

https://authors.library.caltech.edu/1826/4/0204005.pdf

Adaptive homodyne measurement of optical phase

Abstract

Similar works

Full text

Available Versions

University of Queensland eSpace

Caltech Authors - Main

Crossref