Quantum theory of optical temporal phase and
instantaneous frequency. II. Continuous-time
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Tsang, Mankei , Jeffrey H. Shapiro, and Seth Lloyd. “Quantum
theory of optical temporal phase and instantaneous frequency. II.
Continuous-time limit and state-variable approach to phaselocked loop design.” Physical Review A 79.5 (2009): 053843. (C)
2010 The American Physical Society.
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http://dx.doi.org/10.1103/PhysRevA.79.053843
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American Physical Society
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PHYSICAL REVIEW A 79, 053843 共2009兲
Quantum theory of optical temporal phase and instantaneous frequency. II. Continuous-time
limit and state-variable approach to phase-locked loop design
Mankei Tsang,1,* Jeffrey H. Shapiro,1 and Seth Lloyd1,2
1
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
共Received 17 February 2009; published 21 May 2009兲
2
We consider the continuous-time version of our recently proposed quantum theory of optical temporal phase
and instantaneous frequency 关M. Tsang et al., Phys. Rev. A 78, 053820 共2008兲兴. Using a state-variable
approach to estimation, we design homodyne phase-locked loops that can measure the temporal phase with
quantum-limited accuracy. We show that postprocessing can further improve the estimation performance if
delay is allowed in the estimation. We also investigate the fundamental uncertainties in the simultaneous
estimation of harmonic-oscillator position and momentum via continuous optical phase measurements from the
classical estimation theory perspective. In the case of delayed estimation, we find that the inferred uncertainty
product can drop below that allowed by the Heisenberg uncertainty relation. Although this result seems
counterintuitive, we argue that it does not violate any basic principle of quantum mechanics.
DOI: 10.1103/PhysRevA.79.053843
PACS number共s兲: 42.50.Ct, 42.50.Dv, 03.65.Ta
I. INTRODUCTION
Optical phase measurements at the fundamental quantum
limit of accuracy are an important goal in science and engineering and crucial for future metrology, sensing, and communication applications. While the single-mode case has
been extensively studied, less attention has been given to the
quantum measurements of a temporally varying phase. Theoretically, the temporal-phase positive operator-valued measure 共POVM兲 describes the optimal quantum measurements
关1兴, but it is difficult to perform such measurements in practice. Adaptive homodyne detection 关2,3兴 is a much more feasible approach, and Berry and Wiseman 关4兴 proposed the use
of a homodyne phase-locked loop to estimate the phase when
the mean phase is a classical Wiener random process 关5兴. On
the other hand, we have recently shown in Ref. 关1兴 how
homodyne phase-locked loops can be designed using classical estimation theory to perform quantum-limited temporalphase measurements when the mean phase is any stationary
Gaussian random process.
The main purpose of this paper is to unify and generalize
the two distinct approaches undertaken by Berry and Wiseman 关4兴 and ourselves under the common framework of classical estimation theory. In Sec. II, we first extend our
discrete-time theory proposed in Ref. 关1兴 to the continuoustime domain. In Sec. III A, we generalize the results of Berry
and Wiseman 关4兴 to a much wider class of random processes
using the Kalman-Bucy filtering theory 关6,7兴. The KalmanBucy approach guarantees the real-time estimation efficiency
provided that the phase-locked loop operates in the linear
regime. Our approach also significantly simplifies the design
of phase-locked loops, compared to the more computationally expensive Bayesian state estimation approach suggested
by Pope et al. 关5兴. In Sec. III B, we show that the Wiener
filtering technique used in our previous paper 关1兴 is equivalent to Kalman-Bucy filtering at steady state. In Sec. IV, we
*mankei@mit.edu
1050-2947/2009/79共5兲/053843共10兲
point out that results of Berry and Wiseman are not optimal
if delay is permitted in the phase estimation process, and
postprocessing can further improve the phase estimation performance beyond that offered by Kalman-Bucy or Wiener
filtering. We illustrate these concepts by considering the specific cases of the mean phase being an Ornstein-Uhlenbeck
random process as well as the Wiener process studied by
Berry and Wiseman 关4兴. Apart from the theoretical importance of our results in the context of quantum estimation and
control theory, they should also be of immediate interest to
experimentalists and engineers who wish to achieve
quantum-limited temporal-phase measurements, as we expect our proposals to be realizable using current technology.
In Sec. V, we investigate the fundamental problem of simultaneous harmonic-oscillator position and momentum estimation at the quantum limit by continuous optical phase
measurements. The problem can be cast directly in the
framework of classical estimation theory for Gaussian states.
The use of Kalman-Bucy filtering for real-time position and
momentum estimation has been proposed by Belavkin and
Staszewski 关8兴 and Doherty et al. 关9兴, who have shown that
the quantum state of the harmonic-oscillator conditioned
upon the real-time measurement record is a pure-Gaussian
state. Here we show that the inferred position and momentum estimation errors according to classical estimation
theory can be further reduced below the Heisenberg uncertainty product if delay is allowed in the estimation. While
counterintuitive, we explain in Sec. V C why this result does
not violate the basic principles of quantum mechanics.
II. PHASE IN THE CONTINUOUS TIME DOMAIN
For completeness, we first review the continuous-time
limit of our discrete-time theory of temporal phase 关1兴, as
previously described in Ref. 关10兴. Consider the optical envelope annihilation and creation operators Â共t兲 and †共t兲, respectively, in the slowly varying envelope regime, with the
time-domain commutation relation
053843-1
©2009 The American Physical Society
PHYSICAL REVIEW A 79, 053843 共2009兲
TSANG, SHAPIRO, AND LLOYD
兩␾共t兲典 ⬅
冋
兺 exp i 兺j dn共␶ j兲␾共␶ j兲
dn共t兲
=
冋冕
兺 exp
dn共t兲
⬁
i
册
兩dn共t兲典
册
dtI共t兲␾共t兲 兩dn共t兲典.
−⬁
共2.9兲
In terms of the temporal-phase states, a temporal-phase
POVM can be defined as
FIG. 1. 共Color online兲 A realization of the continuous-time
discrete-photon-number random process.
关Â共t兲,†共t⬘兲兴 = ␦共t − t⬘兲.
共2.1兲
⌸̂关␾共t兲兴 ⬅ 兩␾共t兲典具␾共t兲兩,
which is the continuous limit of the one defined in Ref. 关1兴
and can be normalized using a path integral with the paths
restricted to a range of 2␲,
冕
Let dn共t兲 be a continuous-time discrete-photon-number random process, a realization of which is depicted in Fig. 1, and
␶ j be the times at which dn共␶ j兲 is nonzero.
A Fock state with a definite dn共t兲 can be defined as
兩dn共t兲典 ⬅
再兿
j
1
冑dn共␶ j兲!
冎
关†共␶ j兲冑dt兴dn共␶ j兲 兩0典,
共2.2兲
I共t兲 ⬅
共2.3兲
dn共t兲
= 兺 dn共␶ j兲␦共t − ␶ j兲.
dt
j
共2.4兲
The Fock states form a complete orthogonal basis of the
continuous-time Hilbert space,
兺 兩dn共t兲典具dn共t兲兩 = 1̂,
dn共t兲
P关dn共t兲兴 = Tr兵␳ˆ 兩dn共t兲典具dn共t兲兩其,
兺 P关dn共t兲兴 = 1.
D␾共t兲 ⬅ lim 兿
␦t→dt j
d␾共t j兲
,
2␲
p关␾共t兲兴 = Tr兵␳ˆ ⌸̂关␾共t兲兴其,
冕
D␾共t兲p关␾共t兲兴 = 1.
共2.12兲
It is difficult to analytically calculate p关␾共t兲兴 for most quantum states of interest, so perturbative or numerical methods
should be sought.
For the design of homodyne phase-locked loops, the
Wigner distribution is of more interest. For a Gaussian state
with uncorrelated quadratures, it can be written as
冋
W关␰1共t兲, ␰2共t兲兴 ⬀ exp −
1
兺
2 j=1,2
冕
where ␰ j共t兲 are quadrature processes,
␰1共t兲 ⬅ A共t兲e−i␪共t兲 + Aⴱ共t兲ei␪共t兲 − 具A共t兲e−i␪共t兲 + Aⴱ共t兲ei␪共t兲典,
共2.14兲
共2.6兲
冋
N̄
兩A共t兲典 = exp − +
2
N̄ ⬅
冕
⬁
dt兩A共t兲兩2,
冕
⬁
␰2共t兲 ⬅ − i关A共t兲e−i␪共t兲 − Aⴱ共t兲ei␪共t兲兴
册
− 具− i关A共t兲e−i␪共t兲 − Aⴱ共t兲ei␪共t兲兴典,
dtA共t兲Â 共t兲 兩0典,
†
−⬁
Â共t兲兩A共t兲典 = A共t兲兩A共t兲典,
共2.7兲
冕
where A共t兲 is the mean field. The photon-number probability
density is then
P关dn共t兲兴 = lim e
␦t→dt
兿j
关兩A共t j兲兩2␦t兴dn共t j兲
t j ⬅ t0 + j␦t,
dn共t j兲!
共2.15兲
A共t兲 is the complex field variable in phase space, ␪ is an
arbitrary phase, and K−1
j 共t , ␶兲 is defined in terms of the covariance functions K j共t , ␶兲 as
−⬁
¯
−N
册
dtd␶␰ j共t兲K−1
j 共t, ␶兲␰ j共␶兲 ,
共2.13兲
dn共t兲
For example, a coherent state is defined as
共2.11兲
The temporal-phase probability density is thus given by
共2.5兲
where the sum is over all realizations of dn共t兲. For a quantum
state ␳ˆ , the photon-number probability distribution is
D␾共t兲⌸̂关␾共t兲兴 = 1̂,
␾0共t兲 ⱕ ␾共t兲 ⬍ ␾0共t兲 + 2␲ .
which is an eigenstate of the photon-number flux operator
†共t兲Â共t兲,
†共t兲Â共t兲兩dn共t兲典 = I共t兲兩dn共t兲典,
共2.10兲
duK j共t,u兲K−1
j 共u, ␶兲 = ␦共t − ␶兲,
K j共t, ␶兲 ⬅ 具␰ j共t兲␰ j共␶兲典.
共2.16兲
共2.17兲
The covariance functions must satisfy the uncertainty
relation
,
冕
共2.8兲
which describes a Poisson process, as is well known 关11兴.
A temporal-phase state can be defined as the functional
Fourier transform of the Fock states,
duK1共t,u兲K2共u, ␶兲 ⱖ ␦共t − ␶兲,
共2.18兲
which becomes an equality if and only if the state is pure. In
particular, the covariance functions for a coherent state are
053843-2
PHYSICAL REVIEW A 79, 053843 共2009兲
QUANTUM THEORY OF… . II. CONTINUOUS-TIME…
具z共t兲z共␶兲典 = Z共t兲␦共t − ␶兲,
FIG. 2. 共Color online兲 A homodyne phase-locked loop. LO
denotes local oscillator.
K1共t, ␶兲 = K2共t, ␶兲 = ␦共t − ␶兲.
共2.19兲
III. PHASE-LOCKED LOOP DESIGN
A. Kalman-Bucy filtering
Consider the homodyne phase-locked loop illustrated in
Fig. 2. The output of the homodyne detection can be written
as
¯ 共t兲 − ␾⬘共t兲兴 + z共t兲,
␩共t兲 = sin关␾
共3.1兲
¯ 共t兲 is the mean phase of the optical field, which
where ␾
contains the message to be estimated, ␾⬘共t兲 is the localoscillator phase, and z共t兲 is the quantum noise. For a phasesqueezed state with squeezed quadrature ␰2共t兲 and antisqueezed quadrature ␰1共t兲, z共t兲 can be written as
z共t兲 ⬅
1
¯ 共t兲 − ␾⬘共t兲兴 + ␰2共t兲cos关␾
¯ 共t兲 − ␾⬘共t兲兴其,
兵␰1共t兲sin关␾
2兩A兩
冤冥
x1共t兲
x2共t兲
,
x共t兲 ⬅
]
¯ 共t兲 − ␾⬘共t兲 + z共t兲,
␩共t兲 ⬇ ␾
共3.3兲
with the mean phase proportional to the first one,
C共t兲 ⬅ 关␤,0, . . . ,0兴.
共3.4兲
In the Kalman-Bucy formalism, x共t兲 is modeled as zeromean random processes that satisfy a set of linear differential
equations,
dx共t兲
= A共t兲x共t兲 + B共t兲u共t兲,
dt
共3.5兲
where A共t兲 and B共t兲 are n ⫻ n and n ⫻ m matrices, respectively, and u共t兲 is a vector of m zero-mean white Gaussian
inputs with autocorrelation
具u共t兲 丢 u共␶兲典 = U␦共t − ␶兲.
¯ 共t兲 − ␾⬘共t兲兴2典 Ⰶ 1.
具关␾
共3.9兲
The threshold constraint ensures that the phase-locked loop
is phase locked.
If the canonical measurements characterized by the
temporal-phase POVM can be performed, we can instead
modulate the phase of the incoming field by −␾⬘共t兲 and perform the canonical measurements, producing an output
¯ 共t兲 − ␾⬘共t兲 + z共t兲…,
␩c共t兲 = f„␾
共3.10兲
where f共␾兲 must be a periodic function, such as a sawtooth
function,
共3.11兲
¯ 共t兲
and z共t兲 is the quantum phase noise and independent of ␾
¯
and ␾⬘共t兲 for any quantum state. Because ␾共t兲 may exceed
the 2␲ range, it is still necessary to use the phase-locked
loop to perform phase unwrapping. The following analysis
can be applied to canonical temporal-phase measurements
and arbitrary quantum states if ␩c共t兲 is linearized as
¯ 共t兲 − ␾⬘共t兲 + z共t兲
␩c共t兲 ⬇ ␾
共3.12兲
and z共t兲 is approximated as a white Gaussian noise. The
same threshold constraint given by Eq. 共3.9兲 ensures that the
periodic nature of ␩c共t兲 can be neglected and the linearization is valid.
The linearization allows us to use Kalman-Bucy filtering
to produce the real-time minimum-mean-square-error estimates of x共t兲 关6,7兴, which we denote as x⬘共t兲,
dx⬘
= Ax⬘ + ⌫␩ .
dt
共3.6兲
We focus on coherent states, so that z共t兲 can be modeled as
an independent white Gaussian noise according to its Wigner
distribution,
共3.8兲
when the following condition, called the threshold constraint
in classical estimation theory 关1,6兴, is satisfied:
f共␾兲 = 关共␾ − ␲兲mod 2␲兴 − ␲ ,
xn共t兲
¯ 共t兲 = C共t兲x共t兲,
␾
1
ប␻0
, 共3.7兲
=
4兩A兩2 4P
where ␻0 is the optical carrier frequency and P is the average optical power. The additive white Gaussian noise allows
us to apply classical estimation techniques directly. Coherent
states should also be of more immediate interest to experimentalists and engineers, as they are easier to generate and
more robust to loss compared to nonclassical states. For a
¯ 共t兲
phase-squeezed state, the statistics of z共t兲 depend on ␾
− ␾⬘共t兲, but one may still wish to approximate z共t兲 as an
independent Gaussian noise by neglecting the antisqueezed
quadrature ␰1共t兲 in order to take advantage of classical estimation techniques 关1兴.
The purpose of the phase-locked loop is to make ␾⬘共t兲 the
¯ 共t兲, using the measurement record of
optimal estimate of ␾
␩共␶兲 in the period t0 ⱕ ␶ ⱕ t, such that we can linearize Eq.
共3.1兲,
共3.2兲
¯ 兲 is the mean field. For generality,
where A ⬅ 具Â典 = 兩A兩exp共i␾
we let the message be a vector of n random processes,
Z共t兲 ⬅
共3.13兲
This is called the Kalman-Bucy estimator equation. ␩ is
called the innovation, defined in terms of a general vectoral
observation process y共t兲 as
053843-3
PHYSICAL REVIEW A 79, 053843 共2009兲
TSANG, SHAPIRO, AND LLOYD
⌫共t兲 =
4␤P
ប␻0
⌺共t兲.
共3.22兲
The variance equation can be solved analytically,
␮−
⌺共t兲 = ⌺ss
FIG. 3. 共Color online兲 A phase-locked loop that implements
Kalman-Bucy filtering for angle demodulation.
␮⬅
␥/k + 1
exp关− 2␥共t − t0兲兴
␥/k − 1
,
␮ + exp关− 2␥共t − t0兲兴
␥/k + 1 + ⌳⌺共t0兲
,
␥/k − 1 − ⌳⌺共t0兲
␩共t兲 ⬅ y共t兲 − C共t兲x⬘共t兲,
y共t兲 ⬅ C共t兲x共t兲 + z共t兲,
共3.14兲
where z共t兲 is a vectoral Gaussian white noise with mean
具z共t兲典 = 0 and covariance 具z共t兲 丢 z共␶兲典 ⬅ Z共t兲␦共t − ␶兲. For
phase-locked loops, the homodyne output ␩共t兲 can be used
directly as the innovation, so z共t兲 = z共t兲 and Z共t兲 = Z共t兲. ⌫ is
called the gain, given by
冤冥
⌺11
4␤P ⌺21
,
⌫ = ⌺CTZ−1 =
ប␻0 ]
⌺n1
⌺共t兲 → ⌺ss ⬅
1
⌳
⌺ss ⬇
␬⌳
+1
k
冉 冊
␬
k⌳
共3.17兲
共3.18兲
共3.20兲
⌺共t兲 = ⌺ss
t − t0 Ⰷ
−1 ,
␬⌳
Ⰷ 1,
k
1/2
,
1
,
␥
共3.25兲
共3.26兲
共3.27兲
共3.28兲
␮ − exp关− 2␥共t − t0兲兴
,
␮ + exp关− 2␥共t − t0兲兴
⌫共t兲 =
4P
ប␻0
⌺共t兲,
共3.29兲
⌺ss =
1
2 冑N
,
␮⬅
␥ ⬅ 2␬冑N,
1 + 2冑N⌺共t0兲
1 − 2冑N⌺共t0兲
N⬅
P
ប ␻ 0␬
,
共3.30兲
.
At steady state,
⌺共t兲 → ⌺ss =
2
and the gain is
1/2
we can either follow the same procedure as before to derive
the Kalman-Bucy filter or take the results for the OrnsteinUhlenbeck process to the limit k → 0. Either way, assuming
␤ = 1 for simplicity, we find
The variance equation becomes
␬ ⬅ B 2U
共3.24兲
,
dx
= Bu,
dt
Apart from phase estimation, Kalman-Bucy filtering can also
be used to simultaneously estimate other parameters that depend linearly on the phase. The instantaneous frequency, for
instance, can be estimated by defining x2 ⬀ dx1 / dt. The
phase-locked-loop implementation of Kalman-Bucy filtering
for general angle demodulation is depicted in Fig. 3.
For example, consider the message as an OrnsteinUhlenbeck process,
4␤ P 2
d⌺
= − 2k⌺ −
⌺ + ␬,
dt
ប␻0
,
When the message is a Wiener random process,
⌺共t0兲 = 具x共t0兲 丢 x共t0兲典. 共3.19兲
dx
= − kx + Bu.
dt
1/2
␬
⌳ Ⰷ ␤4 .
k
共3.16兲
and the initial conditions are
x⬘共t0兲 = 具x共t0兲典 = 0,
冊
and the threshold constraint is
Equations 共3.13兲–共3.17兲 are much simpler to solve than the
conditional probability density equation suggested by Pope
et al. 关5兴 for phase estimation. The threshold constraint becomes
␤2⌺11 Ⰶ 1,
ប ␻ 0k
␬⌳
+1
k
冋冉 冊 册
which satisfies the variance equation,
d⌺
= A⌺ + ⌺AT − ⌺CTZ−1C⌺ + BUBT .
dt
4 ␤ 2P
冉
where the subscript ss denotes the steady state,
共3.15兲
and ⌺ is the estimation covariance matrix, defined as
⌺共t兲 ⬅ 具关x共t兲 − x⬘共t兲兴 丢 关x共t兲 − x⬘共t兲兴典,
⌳⬅
␥⬅k
共3.23兲
共3.21兲
1
,
2 冑N
⌫共t兲 → 2␬冑N,
t − t0 Ⰷ
1
.
␥
共3.31兲
The threshold constraint is
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PHYSICAL REVIEW A 79, 053843 共2009兲
QUANTUM THEORY OF… . II. CONTINUOUS-TIME…
4P
4N =
ប ␻ 0␬
Ⰷ 1.
共3.32兲
These results for the Wiener process agree with the result of
Berry and Wiseman 关4兴.
B. Wiener filtering
In addition to the Kalman-Bucy state-variable approach,
Wiener’s frequency-domain approach can also be used to
design the phase-locked loop 关1,6,12兴. Defining
¯ 共t兲 + z共t兲,
y共t兲 ⬅ ␾
共3.33兲
FIG. 4. 共Color online兲 A phase-locked loop implementation of
Wiener filtering when the mean phase is an Ornstein-Uhlenbeck
process.
it can be shown that Kalman-Bucy filtering is equivalent to
the integral equation 关6,7兴
x⬘共t兲 =
冕
Sxy共␻兲 ⬅
冕
⬁
dtKxy共t兲exp共− i␻t兲 =
−⬁
d␶H共t, ␶兲y共␶兲,
共3.34兲
the Wiener filter in the frequency domain is
冋 册
t
d␶H共t, ␶兲Ky共␶, ␴兲,
共3.35兲
t0
Kxy共t, ␴兲 ⬅ 具x共t兲y共␴兲典,
Ky共␶, ␴兲 = 具y共␶兲y共␴兲典.
共3.36兲
If x共t兲 and y共t兲 are stationary and we let t0 → −⬁, Eq. 共3.35兲
becomes the Wiener-Hopf equation,
Kxy共t − ␴兲 =
冕
t
d␶H共t − ␶兲Ky共␶ − ␴兲,
共3.37兲
Kx共t, ␴兲 ⬅ 具x共t兲x共␴兲典 = Kx共t − ␴兲,
S x共 ␻ 兲 ⬅
冕
⬁
dtKx共t兲exp共− i␻t兲 =
−⬁
共3.38兲
␬
.
2
␻ + k2
冕
S y共␻兲 ⬅
冕
冉 冊
4P
d␻ Sxy共␻兲
exp共i␻t兲 =
ⴱ
2 ␲ H +共 ␻ 兲
ប␻0
−⬁
H共␻兲 =
⌫ss
,
i␻ + ␥
x⬘共t兲 =
共3.39兲
ប␻0
4P
冕
⬇
1/2
␥−k
.
␤
共3.45兲
t
d␶L共t − ␶兲␩共␶兲
共3.46兲
冕
t
d␶L共t − ␶兲关y共␶兲 − ␤x⬘共␶兲兴,
−⬁
␤␬
ប␻0
.
2
2 +
4P
␻ +k
冉 冊
⌫ss ⬅
−⬁
共3.47兲
L共␻兲
= H共␻兲,
1 + ␤L共␻兲
To solve for H共t − ␶兲, we rewrite Sy共␻兲 as
H +共 ␻ 兲 =
␤␬
关U共− t兲e␥t + U共t兲e−kt兴,
␥+k
To implement the Wiener filter in the phase-locked loop
shown in Fig. 4, the loop filter that relates the homodyne
output ␩共t兲 to the estimate x⬘共t兲 is
共3.40兲
Sy共␻兲 = H+共␻兲H+ⴱ 共␻兲,
1/2
where U共t兲 is the Heaviside step function. The realizable part
is then obtained by multiplying Eq. 共3.44兲 by U共t兲 and performing the Fourier transform. After some algebra, we obtain
2
dtKy共t兲exp共− i␻t兲 =
共3.43兲
共3.44兲
The power spectral density for y共t兲 is then
⬁
,
+
where the subscript + denotes the realizable part. To calculate the realizable part, first perform the inverse Fourier
transform,
−⬁
which can be solved by a well-known frequency-domain
technique 关1,6,12兴. For example, if x共t兲 is an OrnsteinUhlenbeck process, its power spectral density in the limit of
t0 → −⬁ is
1
Sxy共␻兲
H+共␻兲 H+ⴱ 共␻兲
H共␻兲 =
where H共t , ␶兲 is called the optimum realizable filter and satisfies the integral equation
冕
共3.42兲
t
t0
Kxy共t, ␴兲 =
␤␬
,
␻2 + k2
L共␻兲 =
H共␻兲
⌫ss
=
.
1 − ␤H共␻兲 i␻ + k
共3.48兲
i␻ + ␥
,
i␻ + k
共3.41兲
where ␥ is given in Eq. 共3.24兲 and H+共␻兲 and 1 / H+共␻兲 are
causal filters. Defining
The resulting phase-locked-loop structure is equivalent to
that obtained by Kalman-Bucy filtering at steady state, as
both approaches implement the optimum realizable filter.
The mean-square error of Wiener filtering is given by the
well-known expression 关1,6,12兴
053843-5
PHYSICAL REVIEW A 79, 053843 共2009兲
TSANG, SHAPIRO, AND LLOYD
⌺=
=
ប␻0
4 ␤ 2P
1
⌳
冕
⬁
−⬁
冋
4␤2PSx共␻兲
d␻
ln 1 +
2␲
ប␻0
冋冉 冊 册
␬⌳
+1
k
册
1/2
共3.49兲
−1 ,
which obviously must be the same as the steady-state error
⌺ss obtained by Kalman-Bucy filtering. The interested reader
is referred to Ref. 关6兴 for an excellent treatment of Wiener
filters.
The advantage of Kalman-Bucy filtering over Wiener filtering is that the former can also deal with a wide class of
nonstationary random processes that can be described by a
system of linear equations 关Eq. 共3.5兲兴 whereas Wiener filtering works only for stationary processes. In the special case of
a Wiener process, however, we can first design a Wiener
filter for an Ornstein-Uhlenbeck process and take the limit
k → 0. The result for ␤ = 1 is
L共␻兲 →
⌫ss
,
i␻
⌫ss → 2␬冑N,
⌺→
1
,
2 冑N
共3.50兲
side of Eq. 共4.3兲 to zero and using the steady-state ⌺ss as ⌺.
Again using the Ornstein-Uhlenbeck process as an example, the steady-state smoothing error, also called the
“irreducible” error 关6,12兴, is given by
⌸ss =
Both Kalman-Bucy filtering and Wiener filtering provide
real-time estimates of x共t兲 based on the measurement record
up to time ␶ = t. If we allow delay in the estimation, we can
use the additional information from more advanced measurements to improve upon the estimation. In the following we
consider the optimal estimation of x共t兲 given the full measurement record in the interval t0 ⱕ t ⱕ T, also called smoothing in classical estimation theory 关6,7兴.
共4.5兲
This result is identical to that derived in 关1兴 using a
frequency-domain approach. In the limit of ⌳ Ⰷ k / ␬,
⌸ss →
冉 冊
1 ␬
2 k⌳
1/2
1
⬇ ⌺ss ,
2
共4.6兲
which is smaller than the error from Kalman-Bucy or Wiener
filtering by approximately a factor of 2. For the Wiener process, the smoothing error is
⌸ss =
1
1
= ⌺ss ,
4 冑N 2
共4.7兲
which is smaller than the filtering error by exactly a factor of
2.
which is again the same as the steady-state Kalman-Bucy
filter.
IV. SMOOTHING
␬
.
2k共␬⌳/k + 1兲1/2
B. Two-filter smoothing
An equivalent but more intuitive form of the optimal
smoother was discovered by Mayne 关14兴 and Fraser and Potter 关15兴, who treated the smoother as a combination of two
filters, one running forward in time to produce a prediction
x⬘共t兲 via Kalman-Bucy filtering using the past measurement
record, as specified in Eqs. 共3.13兲–共3.17兲, and one running
backward in time to produce a retrodiction x⬙共t兲 using the
advanced measurement record,
dx⬙
= Ax⬙ − ⌼␩ ,
dt
共4.8兲
d⌶
= A⌶ + ⌶AT + ⌶CTZ−1C⌶ − BUBT ,
dt
共4.9兲
⌼ = ⌶CTZ−1 ,
共4.10兲
A. State-variable approach
Given the output x⬘ of the homodyne phase-locked loop
designed by Kalman-Bucy filtering and the associated covariance matrix ⌺, the optimal smoothing estimates of x共t兲,
which we define as x̃共t兲, can be calculated using a statevariable approach, first suggested by Bryson and Frazier
关7,13兴. x̃共t兲 and the smoothing covariance matrix,
⌸共t兲 ⬅ 具关x共t兲 − x̃共t兲兴 丢 关x共t兲 − x̃共t兲兴典,
⌶−1共T兲x⬙共T兲 = 0,
共4.1兲
can be obtained by solving the following equations backward
in time:
dx̃
= Ax̃ + BUBT⌺−1共x̃ − x⬘兲,
dt
with final conditions
共4.3兲
⌸共T兲 = ⌺共T兲.
共4.12兲
⌸ = 共⌺−1 + ⌶−1兲−1 .
共4.13兲
−1
−1 −1
⌸ss = 共⌺ss
+ ⌶ss
兲 .
共4.4兲
In the t0 Ⰶ t Ⰶ T limit, we can calculate the steady-state
smoothing covariance matrix ⌸ss by setting the right-hand
x̃ = ⌸共⌺−1x⬘ + ⌶−1x⬙兲,
The steady-state smoothing covariance matrix ⌸ss can be
calculated by combining the steady-state predictive and retrodictive covariance matrices,
with the final conditions
x̃共T兲 = x⬘共T兲,
共4.11兲
The smoothing estimates and covariance matrix, taking into
account both the prediction and the retrodiction, are given by
共4.2兲
d⌸
= 共A + BUBT⌺−1兲⌸ + ⌸共A + BUBT⌺−1兲T − BUBT ,
dt
⌶−1共T兲 = 0.
共4.14兲
C. Frequency-domain approach
For stationary Gaussian random processes and in the limit
of t0 Ⰶ t Ⰶ T, a frequency-domain approach can also be used
053843-6
PHYSICAL REVIEW A 79, 053843 共2009兲
QUANTUM THEORY OF… . II. CONTINUOUS-TIME…
FIG. 5. 共Color online兲 A homodyne phase-locked loop with a
postloop filter F共␻兲 that realizes optimal smoothing.
FIG. 6. 共Color online兲 Position and momentum estimation by
optical phase measurements.
to obtain the optimal smoother 关1,6,12兴. The optimal smoothing estimates can be written in terms of y共t兲 as 关6,7,12兴
x̃共t兲 =
冕
⌸=
T
d␶G共t, ␶兲y共␶兲,
冕 冋
⬁
−⬁
共4.15兲
册
兩Sxy共␻兲兩2
d␻
␬
S x共 ␻ 兲 −
=
,
2␲
S y共␻兲
2k共␬⌳/k + 1兲1/2
共4.22兲
t0
where G共t , ␶兲 obeys
冕
Kxy共t, ␴兲 =
T
d␶G共t, ␶兲Ky共␶, ␴兲.
共4.16兲
which is the same as the smoothing error derived by the
state-variable approach in Eq. 共4.5兲 as expected. The interested reader is again referred to Ref. 关6兴 for an excellent
treatment of optimal frequency-domain filters and smoothers.
t0
For Kxy共t , ␴兲 = Kxy共t − ␴兲, G共t , ␶兲 = G共t − ␶兲, and Ky共␶ , ␴兲
= Ky共␶ − ␴兲 and in the limit of t0 → −⬁ and T → ⬁, we can
solve Eq. 共4.16兲 by Fourier transform,
G共␻兲 =
Sxy共␻兲
.
S y共␻兲
共4.17兲
G共␻兲 is called the optimum unrealizable filter 关6兴. For an
Ornstein-Uhlenbeck process, G共␻兲 is
G共␻兲 =
4␤P
␬
.
ប␻0 ␻ + ␥2
共4.18兲
2
To implement this filter, one can use the homodyne phaselocked loop designed by Wiener filtering and a postloop filter
given by
k+␥
G共␻兲
=
.
F共␻兲 =
H共␻兲 − i␻ + ␥
V. QUANTUM POSITION AND MOMENTUM
ESTIMATION BY OPTICAL PHASE MEASUREMENTS
A. Quantum Kalman-Bucy filtering
So far we have assumed that the mean phase of the optical
field contains classical random processes to be estimated in
the presence of quantum optical noise. In this section we
investigate the estimation of inherently quantum processes
carried by the optical phase. Specifically, we revisit the classic problem of quantum-limited mirror position and momentum estimation by optical phase measurements. First we review the problem of optimal real-time estimation by
Kalman-Bucy filtering, previously studied by Belavkin and
Staszewski 关8兴 and Doherty et al. 关9兴.
We model the mirror as a harmonic oscillator, as depicted
in Fig. 6,
f共t兲 ⬅
冕
−⬁
=
再
d␻
F共␻兲exp共i␻t兲
2␲
共k + ␥兲exp共␥t兲, t ⱕ 0
0,
t ⬎ 0,
共5.1兲
2Mប␻0 cos ␪
dp̂
2
x̂ +
= − m␻m
Î,
dt
c
共5.2兲
共4.19兲
The postloop filter impulse response is
⬁
dx̂ p̂
= ,
dt m
共4.20兲
冎
共4.21兲
which is anticausal, so one must introduce a time delay td
Ⰷ 1 / ␥ for F to be approximated by a causal filter. The optimal smoother designed by the frequency-domain approach is
depicted in Fig. 5.
The variance of the optimal frequency-domain smoother
is 关1,6,12兴
where x̂共t兲 and p̂共t兲 are quantum position and momentum
operators, m is the harmonic-oscillator mass, ␻m is the mechanical harmonic-oscillator frequency, the last term of Eq.
共5.2兲 is the radiation pressure term, M is the number of times
the optical beam hits the mirror, ␪ is the angle at which the
optical beam hits the mirror, and Î共t兲 is the optical flux operator, consisting of a mean and a quantum noise term,
Î共t兲 =
P
ប␻0
+ ⌬Î共t兲.
共5.3兲
The constant force term can be eliminated by redefining the
position of the harmonic oscillator, so we shall neglect the
constant radiation pressure term from now on. ⌬Î共t兲 is ap-
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PHYSICAL REVIEW A 79, 053843 共2009兲
TSANG, SHAPIRO, AND LLOYD
proximately a white Gaussian noise term for a high-power
optical coherent state,
具⌬Î共t兲⌬Î共␶兲典 ⬇
P
ប␻0
␦共t − ␶兲.
共5.4兲
The Gaussian approximation neglects the discreteness of
photon number and is valid if the number of photons within
the relaxation time of the filter impulse response is much
larger than 1. We can then write the quantum system model
as
冋册 冋
0
1/m
d x̂
=
2
0
dt p̂
− m␻m
册冋 册 冋 册
x̂
p̂
+
0
1
具p̂共t0兲典 = 0,
共5.6兲
,
␤ ⬅ 2Mk0 cos ␪ .
冋册
x̂
p̂
具ẑ共t兲ẑ共␶兲典 = Z␦共t − ␶兲,
共5.8兲
共5.9兲
+ ẑ,
ប␻0
.
4P
共5.10兲
Our linearized model is consistent with the general model of
continuous quantum nondemolition 共QND兲 measurements
关8,9,16兴.
To apply Kalman-Bucy filtering to the estimation of mirror position and momentum, let us define
⌬x̂ ⬅ x̂ − x⬘,
具⌬x̂2典
1
具⌬x̂⌬p̂ + ⌬p̂⌬x̂典
2
⌬p̂ ⬅ p̂ − p⬘ ,
1
具⌬x̂⌬p̂ + ⌬p̂⌬x̂典
2
具⌬p̂2典
ប␻0
共5.16兲
.
共5.17兲
ប1
关共1 + Q2兲1/2 − 1兴,
2Q
共5.18兲
共⌺22兲ss =
បm␻m 冑2
关共1 + Q2兲1/2 − 1兴1/2共1 + Q2兲1/2 ,
2 Q
共5.19兲
where
Q⬅
Z⬅
4 ␤ 2P
ប 冑2
关共1 + Q2兲1/2 − 1兴1/2 ,
2m␻m Q
共⌺12兲ss = 共⌺21兲ss =
ŷ = 关␤ 0 兴
冤
共5.15兲
共5.7兲
The mirror position is observed via optical phase measurements using a phase-locked loop. In the linearized regime,
we can define the quantum observation process as
⌺=
d⌺22
2
共⌺12 + ⌺21兲 − V⌺12⌺21 + U,
= − m␻m
dt
共⌺11兲ss =
2
␻0
共5.14兲
The steady state is given by the condition d⌺ / dt = 0. After
some algebra,
具û共t兲û共␶兲典 = U␦共t − ␶兲,
U⬅
d⌺12 1
2
⌺11 − V⌺11⌺12 ,
= ⌺22 − m␻m
dt
m
V⬅
and radiation pressure acting as the quantum Langevin noise,
ប␤ P
共5.13兲
共5.5兲
û,
with initial conditions,
具x̂共t0兲典 = 0,
d⌺11 1
2
,
= 共⌺21 + ⌺12兲 − V⌺11
dt
m
共5.11兲
冥
冑UV
2
m␻m
=
2 ␤ 2P
2
m ␻ 0␻ m
is a dimensionless parameter that characterizes the strength
of the measurements. The position uncertainty 具⌬x̂2典ss
= 共⌺11兲ss is squeezed due to the continuous QND measurements, while the momentum uncertainty 具⌬p̂2典ss = 共⌺22兲ss is
antisqueezed due to the radiation pressure. These results
have also been derived by various groups of people 关8,9兴,
although here we have shown how one can realistically
implement the optical measurements of a mechanical oscillator.
The Kalman-Bucy gain is
⌫=
.
冋 册
4␤P ⌺11
.
ប␻0 ⌺21
The estimator equation becomes
冋册冋
共5.12兲
In the linearized model, the Wigner distribution remains
Gaussian and non-negative provided that the initial Wigner
distribution is Gaussian, so it can be regarded as a classical
phase-space probability distribution, x̂共t兲, p̂共t兲, ⌬Î共t兲, and
ŷ共t兲, can be regarded as classical random processes with statistics governed by the Wigner distribution, and we can apply
classical estimation theory directly. The off-diagonal components of the variance matrix are written in terms of symmetrized operators to ensure that they are Hermitian and also
obey Wigner-distribution statistics.
The Kalman-Bucy variance equations hence become
共5.20兲
− V⌺11
1/m
d x⬘
=
2
p
dt ⬘
− V⌺21 − m␻m 0
共5.21兲
册冋 册 冋 册
x⬘
V ⌺11
+
y,
p⬘
␤ ⌺21
共5.22兲
where y is the measurement record of ŷ. The filter relaxation
time is on the order of
tf ⬃
1
1
=
,
V共⌺11兲ss 冑2␻m关共1 + Q2兲1/2 − 1兴1/2
共5.23兲
which decreases for increasing Q, so the steady state can be
reached faster for a larger Q. For Q → 0, t f → ⬁, and a steady
state does not exist. The photon number within the filter
053843-8
PHYSICAL REVIEW A 79, 053843 共2009兲
QUANTUM THEORY OF… . II. CONTINUOUS-TIME…
共⌶11兲ss = 共⌺11兲ss,
relaxation time is much larger than 1, and the assumption of
white Gaussian radiation pressure noise is valid when
Pt f
ប␻0
⬃
m␻m
Q
Ⰷ 1.
2 关共1 + Q2兲1/2 − 1兴1/2
冑
2 2␤ ប
共5.24兲
On the other hand, the threshold constraint, which ensures
that the linearized analysis of the phase-locked loop is valid,
is
␤2ប 关共1 + Q2兲1/2 − 1兴1/2
Ⰶ 1.
␤2共⌺11兲ss =
冑2m␻m
Q
共⌶22兲ss = 共⌺22兲ss ,
2
兲ss =
det共⌶ss兲 = 共⌶11⌶22 − ⌶12
After some algebra,
共5.25兲
共⌸11兲ss =
冋
and satisfies the Heisenberg uncertainty principle for all Q,
as one would expect. Furthermore, the covariances satisfy
the following relation for pure-Gaussian states 关8,9兴:
册
共⌸12兲ss = 0,
共⌸22兲ss =
共5.35兲
បm␻m
关共1 + iQ兲1/2 + 共1 − iQ兲1/2兴.
8
共5.36兲
共⌸11兲ss共⌸22兲ss =
冋
册
1
ប2
1+
,
32
共1 + Q2兲1/2
共5.37兲
which is smaller than the Heisenberg uncertainty product
ប2 / 4 by four to eight times.
共5.27兲
indicating that the harmonic oscillator conditioned upon the
real-time measurement record becomes a pure-Gaussian state
at steady state.
B. Smoothing errors
From the classical estimation theory perspective, we
should be able to improve upon Kalman-Bucy filtering if we
allow delay in the estimation and apply smoothing. Here we
calculate the steady-state smoothing errors using the twofilter approach described in Sec. IV B. The steady-state
smoothing covariance matrix is
−1
−1 −1
+ ⌶ss
兲 ,
⌸ss = 共⌺ss
共5.33兲
1
1
ប
+
, 共5.34兲
8m␻m 共1 + iQ兲1/2 共1 − iQ兲1/2
共5.26兲
ប2
,
4
ប2
.
4
These results can be confirmed using the frequency-domain
approach outlined in Sec. IV C. The position-momentum uncertainty product becomes
ប2 2
关1 + Q2 − 共1 + Q2兲1/2兴
4 Q2
2
兲ss =
det共⌺ss兲 = 共⌺11⌺22 − ⌺12
共5.32兲
and also satisfy the pure-Gaussian-state relation
This condition, apart from a factor of 4, is the same as the
large-photon-number assumption given by Eq. 共5.24兲 and ensures that the linearized system and measurement model is
self-consistent.
The mirror position-momentum uncertainty product at
steady state is
具⌬x̂2典ss具⌬p̂2典ss = 共⌺11⌺22兲ss =
共⌶12兲ss = − 共⌺12兲ss ,
共5.28兲
where ⌺ss is the steady-state forward-filter covariance matrix, already solved and given by Eqs. 共5.17兲–共5.19兲. The
backward-filter covariances obey the following equations:
d⌶11 1
2
,
= 共⌶21 + ⌶12兲 + V⌶11
dt
m
共5.29兲
d⌶12 1
2
= ⌶22 − m␻m
⌶11 + V⌶11⌶12 ,
dt
m
共5.30兲
d⌶22
2
共⌶12 + ⌶21兲 + V⌶12⌶21 − U.
= − m␻m
dt
共5.31兲
The steady-state values for the backward filter turn out to be
almost identical to the ones for the forward filter,
C. Discussion
While counterintuitive, the sub-Heisenberg uncertainties
given by Eqs. 共5.34兲–共5.37兲 do not violate any basic law of
quantum mechanics. The reason is that we only estimate the
position and momentum of the mirror some time in the past
as if they were classical random processes with Wignerdistribution statistics, but it is impossible to verify our estimates by comparing them against the mirror in the past,
which has since been irreversibly perturbed by the unknown
radiation pressure noise. In classical estimation, x共t兲 and p共t兲
are classical random processes unknown to the observer but
can in principle be perfectly measured or simply decided at
will by another party, so it is possible to compare one’s delayed estimates against the perfect versions and verify the
smoothing errors. In the quantum regime, however, one cannot measure the mirror in the past more accurately without
disturbing it further, and the only way for us to obtain perfect
information about the mirror position or momentum is to
perform a strong projective measurement. Unlike the
Kalman-Bucy estimates, which predict the mirror position
and momentum at present and can be verified by performing
a projective measurement at present, it is obviously impossible to go back to the past and perform a strong projective
measurement on the mirror to verify our delayed estimates
without changing our model of the problem.
It is also impossible to perfectly reverse the dynamics of
the mirror in time and recreate the past quantum state without introducing additional noise because quantum-limited
optical phase measurements prevent us from obtaining any
information about the optical power fluctuations, and the dy-
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TSANG, SHAPIRO, AND LLOYD
namics of the mirror subject to the unknown radiation pressure noise is irreversible. Thus, even though classical estimation theory indicates that we can achieve more accurate
estimates than the Heisenberg uncertainty principle would
allow, quantum mechanics seem to forbid one from experimentally verifying the violation. In this sense the apparent
paradox is analogous to the Einstein-Podolsky-Rosen paradox 关17兴 and may yet have implications for the interpretation
of quantum mechanics.
In practice, while one may argue from a frequentist point
of view that delayed estimation of quantum processes is
meaningless if it cannot be verified, smoothing should still
be able to improve the estimation of a classical random process in a quantum system, such as a classical force acting on
a quantum harmonic oscillator 关18兴.
VI. CONCLUSION
modyne measurement scheme for nonclassical states remains
an open problem. Along this direction Berry and Wiseman
关4兴 recently suggested the use of Bayesian estimation for
narrow band squeezed states when the mean phase is a
Wiener process 关19兴. Generalization of their scheme to more
general random processes is challenging but may be facilitated by classical nonlinear estimation techniques 关6,7兴.
When we apply the same classical techniques to the
quantum-limited estimation of harmonic-oscillator position
and momentum, we find that the two conjugate variables can
be simultaneously estimated with inferred accuracies beyond
the Heisenberg uncertainty relation if smoothing is performed. Although quantum mechanics seems to forbid one
from verifying the delayed estimates by destroying the evidence, this result remains counterintuitive and may have implications for the interpretation of quantum mechanics. In the
general context of quantum trajectory theory 关20兴, quantum
smoothing deserves further investigation and should be useful for quantum sensing and communication applications
关18,21兴.
In conclusion, we have used classical estimation theory to
design homodyne phase-locked loop for quantum optical
phase estimation and shown that the estimation performance
can be improved when delay is permitted and smoothing is
applied. We have focused on coherent states, as it can be
regarded as a classical field with additive phase-insensitive
noise upon homodyne detection, and classical estimation
techniques can be applied directly. The optimal adaptive ho-
Discussion with Howard Wiseman is gratefully acknowledged. This work was financially supported by the W. M.
Keck Foundation Center for Extreme Quantum Information
Theory.
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ACKNOWLEDGMENTS
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