Feedback-stabilization of an arbitrary pure state of a two-level atom

TLDR: In this article, the authors reanalyze Hofmann, Mahler, and Hess' proposal using the technique of stochastic master equations and show that any point on the upper or lower-half, but not the equator, of the sphere may be stabilized.

Abstract: Unit-efficiency homodyne detection of the resonance fluorescence of a two-level atom collapses the quantum state of the atom to a stochastically moving point on the Bloch sphere. Recently, Hofmann, Mahler, and Hess [Phys. Rev. A 57, 4877 (1998)] showed that by making part of the coherent driving proportional to the homodyne photocurrent one can stabilize the state to any point on the bottom-half of the sphere. Here we reanalyze their proposal using the technique of stochastic master equations, allowing their results to be generalized in two ways. First, we show that any point on the upper- or lower-half, but not the equator, of the sphere may be stabilized. Second, we consider nonunit-efficiency detection, and quantify the effectiveness of the feedback by calculating the maximal purity obtainable in any particular direction in Bloch space.

Figures

FIG. 4. Locus of the solutions to the Bloch equations with o timal feedback for different values of detector efficiencyh. From the outside in, we haveh51, 0.8, 0.6, 0.4, 0.2, 0.

FIG. 1. Diagram of the experimental apparatus. The laser b is split to produce both the local oscillatorb and the fielda0, which is modulated using the currentI (t). The modulated beam, with amplitude proportional toa1lI (t), drives an atom at the center o the parabolic mirror. The fluorescence thus collected is subjec homodyne detection using the local oscillator, and gives rise to homodyne photocurrentI (t).

FIG. 2. Locus of the solutions to the Bloch equations. The lipse in the lower-half plane is the locus for the equations w driving only. The full circle~minus the points on the equator! is the locus for the equations with optimal driving and feedback, as fined in Sec. III.

FIG. 6. Typical quantum trajectories for optimal feedback w h51, shown byxc ~pluses! andzc ~dots! as functions of time.~a! A single trajectory foru5p/6 and~b! two trajectories foru5p/2.

Full text

Feedback-stabilization of an arbitrary pure state of a two-level atom
Jin Wang
1
and H. M. Wiseman
2
1
Centre for Laser Science, Department of Physics, The University of Queensland, Brisbane, Queensland 4072, Australia
2
School of Science, Griffith University, Brisbane, Queensland 4111, Australia
共Received 24 July 2000; published 16 November 2001兲
Unit-efficiency homodyne detection of the resonance fluorescence of a two-level atom collapses the quantum
state of the atom to a stochastically moving point on the Bloch sphere. Recently, Hofmann, Mahler, and Hess
关Phys. Rev. A 57, 4877 共1998兲兴 showed that by making part of the coherent driving proportional to the
homodyne photocurrent one can stabilize the state to any point on the bottom-half of the sphere. Here we
reanalyze their proposal using the technique of stochastic master equations, allowing their results to be gen-
eralized in two ways. First, we show that any point on the upper- or lower-half, but not the equator, of the
sphere may be stabilized. Second, we consider nonunit-efficiency detection, and quantify the effectiveness of
the feedback by calculating the maximal purity obtainable in any particular direction in Bloch space.
DOI: 10.1103/PhysRevA.64.063810 PACS number共s兲: 42.50.Lc, 42.50.Ct, 03.65.Ta
I. INTRODUCTION
Although classical models of feedback schemes have
been used for a long time to control dynamical noise, an
analogous quantum theory of feedback has been developed
only in the last 15 years 关1–7兴. Recently there has been con-
siderable interest in quantum feedback as a way to fight de-
coherence in isolated quantum systems, using the approach
of Refs. 关4,5兴. The central idea is to use a continuous mea-
surement record, whose existence is due to the coupling of
the system to a bath, to control the dynamics of the system so
as to counteract the noise introduced by that bath and possi-
bly other baths. For example, it has been suggested as a way
to create optical squeezed states 关8兴, to create micromaser
number states 关9兴, to correct errors in quantum bits 关10兴, and
to protect optical and microwave Schro
¨
dinger cat states
against dissipation 关11–13兴.
Decoherence in quantum systems can be loosely defined
as loss of purity. Therefore the ultimate success in using
feedback to fight decoherence would be to create an arbitrary
stable pure state in the presence of dissipation. This goal was
realized 共better even that they realized兲 by Hoffman, Mahler,
and Hess 共HMH兲关14,15兴 for a very simple system: a reso-
nantly driven two-level atom. They showed that by using the
photocurrent derived from unit-efficiency homodyne detec-
tion of the atom’s fluorescence to control part of the driving
field of the atom, it is possible to exactly cancel the noise
introduced by the electromagnetic vacuum field when the
atom is in a particular pure state. By choosing the driving
strength and feedback strength appropriately, any pure state
on the Bloch sphere may be picked out, although HMH
claimed that only pure states on the lower-half of the sphere
would be stable under their scheme.
HMH chose to describe detection and feedback in their
system in a way different from 共but equivalent to兲 the stan-
dard approach in Refs. 关4,5兴. In this paper we reformulate
their theory using the latter approach. This has the advantage
of enabling a number of generalizations of their results. First,
we revisit the question of stability and find that, contrary to
the claims of HMH, the states in the upper-half of the Bloch
sphere can be stabilized as well as those in the lower-half
共this is what was better than they realized兲. The only states
that cannot be stabilized, in the sense that an arbitrary initial
state would not always end up in the desired state, are those
on the equator of the Bloch sphere; that is, those that are
equal superpositions of excited and ground states.
Our second generalization is to consider how effective
feedback is with
␩
⬍ 1; i.e., with nonunit-efficiency detec-
tion. In this case it is not possible to stabilize the atom at any
pure state, except the ground state, which is trivially stable
by setting the driving and feedback to zero. Instead, we aim
to produce a steady state that is as close as possible to a
given pure state. For the two-level atom, this is equivalent to
trying to create a state that is as pure as possible in a par-
ticular direction in Bloch space. Not surprisingly 共given the
above result兲, we find that states near the equator cannot be
well protected against decoherence. We also find an echo of
the distinction HMH found between the upper- and lower-
halves of the Bloch sphere, in that states in the upper-half
sphere are affected much more by loss of detection efficiency
that those in the lower-half.
The paper is organized as follows. In Sec. II we present
the model of a driven two-level atom, including the stochas-
tic Schro
¨
dinger equation for unit-efficiency homodyne detec-
tion. In Sec. III we use this equation to derive the driving and
feedback required to stabilize the atom in an arbitrary pure
state. These results agree with those of HMH. However, our
stability analysis disagrees substantially with theirs. In Sec.
IV we present entirely new analytical results relating to the
effect of nonunit-efficiency detection. In Sec. V we give nu-
merical simulations of the stochastic evolution equations, il-
lustrating the issues discussed in the preceding two sections.
In Sec. VI we summarize and interpret our results, explain
their significance, and discuss the possibility of future work.
II. HOMODYNE DETECTION
A. Master equation
Consider an atom, with two relevant levels
兵
兩
g
典
,
兩
e
典
其
and
lowering operator
␴
⫽
兩
g
典具
e
兩
. Let the decay rate be
␥
, and let
it be driven by a resonant classical driving field with Rabi
frequency 2
␣
. This is as shown in Fig. 1, where for the
PHYSICAL REVIEW A, VOLUME 64, 063810
1050-2947/2001/64共6兲/063810共9兲/$20.00 ©2001 The American Physical Society64 063810-1

moment we are omitting feedback by setting ␭⫽ 0. This sys-
tem is well approximated by the master equation
␳
˙
⫽
␥
D
关
␴
兴
␳
⫺ i
␣
关
␴
y
,
␳
兴
, 共2.1兲
where the Lindblad 关16兴 superoperator is defined as usual
D
关
A
兴
B⬅ABA
†
⫺
兵
A
†
A,B
其
/2. In this master equation we
have chosen to define the
␴
x
⫽
␴
⫹
␴
†
and
␴
y
⫽ i
␴
⫺ i
␴
†
quadratures of the atomic dipole relative to the driving field.
The effect of driving is to rotate the atom in Bloch space
around the y axis. The state of the atom in Bloch space is
described by the three-vector (x,y,z). It is related to the state
matrix
␳
by
␳
⫽
1
2
共
I⫹ x
␴
x
⫹ y
␴
y
⫹ z
␴
z
兲
. 共2.2兲
It is easy to show that the stationary solution of the master
equation 共2.1兲 is
x
ss
⫽
⫺ 4
␣␥
␥
2
⫹ 8
␣
2
, 共2.3兲
y
ss
⫽ 0, 共2.4兲
z
ss
⫽
⫺
␥
2
␥
2
⫹ 8
␣
2
. 共2.5兲
For
␥
fixed, this is a family of solutions parametrized by the
driving strength
␣
苸 (⫺ ⬁,⬁). All members of the family are
in the x-z plane on the Bloch sphere. Thus, for this purpose,
we can reparametrize the relevant states using r and
␪
by
x⫽ r sin
␪
, 共2.6兲
z⫽ r cos
␪
, 共2.7兲
where
␪
苸
关
⫺
␲
,
␲
兴
. Since
Tr
关
␳
2
兴
⫽
1
2
共
1⫹ x
2
⫹ y
2
⫹ z
2
兲
共2.8兲
is a measure of the purity of the Bloch sphere, r
⫽
冑
x
2
⫹ z
2
, the distance from the center of the sphere, is also
a measure of purity. Pure states correspond to r⫽ 1 and
maximally mixed states to r⫽ 0.
The locus of solutions in this plane 共an ellipse兲 is shown
in Fig. 2. Since z
ss
⬍ 0, all solutions are in the lower-half of
the Bloch sphere. That is, we are restricted to
兩
␪
兩
⬎
␲
/2.
Also, it is evident that the smaller
兩
␪
兩
is 共i.e., the more ex-
cited the atom is兲 the smaller r is 共i.e., the less pure the atom
is兲.At
兩
␪
兩
⫽
␲
, the stationary state is pure, but this is not
surprising as it is simply the ground state of the atom with no
driving. As
兩
␪
兩
→
␲
/2 we have r→ 0. This can only be ap-
proached asymptotically as
兩
␣
兩
→⬁. In summary, the station-
ary states we can reach by driving the atom are limited, and
generally far from pure.
B. Homodyne detection
Now consider subjecting the atom to homodyne detection.
As shown in Fig. 1, we assume that all of the fluorescence of
the atom is collected and turned into a beam 共represented in
Fig. 1 by placing the atom at the focus of a mirror兲. Ignoring
the vacuum fluctuations in the field, the annihilation operator
for this beam is
冑
␥
␴
, normalized so that the mean intensity
␥
具
␴
†
␴
典
is equal to the number of photons per unit time in
the beam. This beam then enters one port of a 50:50 beam
splitter, while a strong local oscillator
␤
enters the other. To
FIG. 1. Diagram of the experimental apparatus. The laser beam
is split to produce both the local oscillator
␤
and the field
␣
0
, which
is modulated using the current I(t). The modulated beam, with
amplitude proportional to
␣
⫹ ␭I(t), drives an atom at the center of
the parabolic mirror. The fluorescence thus collected is subject to
homodyne detection using the local oscillator, and gives rise to the
homodyne photocurrent I(t).
FIG. 2. Locus of the solutions to the Bloch equations. The el-
lipse in the lower-half plane is the locus for the equations with
driving only. The full circle 共minus the points on the equator兲 is the
locus for the equations with optimal driving and feedback, as de-
fined in Sec. III.
JIN WANG AND H. M. WISEMAN PHYSICAL REVIEW A 64 063810
063810-2

ensure that this local oscillator has a fixed phase relationship
with the driving laser used in the measurement, it would be
natural to utilize the same coherent light field source in the
driving process and as the local oscillator in the homodyne
detection. This is as shown in Fig. 1.
Again ignoring vacuum fluctuations, the two field opera-
tors exiting the beam splitter, b
1
and b
2
, are
b
k
⫽
关
冑
␥
␴
⫺
共
⫺ 1
兲
k
␤
兴
/
冑
2. 共2.9兲
When these two fields are detected, the two photocurrents
produced have means
I
¯
k
⫽
具
兩
␤
兩
2
⫺
共
⫺ 1
兲
k
共
冑
␥␤
␴
†
⫹
冑
␥
␴
␤
*
兲
⫹
␥
␴
†
␴
典
/2. 共2.10兲
The middle two terms represent the interference between the
system and the local oscillator.
Equation 共2.10兲 gives only the mean photocurrent. In an
individual run of the experiment for a system, what is re-
corded is not the mean photocurrent, but the instantaneous
photocurrent. This photocurrent will vary stochastically from
one run to the next, because of the irreducible randomness in
the quantum measurement process. This randomness is not
just noise, however. It is correlated with the evolution of the
system and thus tells the experimenter something about the
state of the system. In fact, if the detection efficiency is per-
fect, the system is collapsed into a pure state, rather than the
mixed state, which is the solution of the master equation. The
stochastic evolution of the state of the system conditioned on
the measurement record is called a ‘‘quantum trajectory’’
关17兴. Of course, the master equation is still obeyed on aver-
age, so the set of possible quantum trajectories is called an
unraveling of the master equation 关17兴. It is the conditioning
of the system state on the photocurrent record that allows
feedback of the photocurrent to control the system state. The
application of an appropriate feedback loop to this continu-
ous measurement process 共to be considered in Sec. III兲 real-
izes an effective ‘‘reservoir engineering’’ to control the sys-
tem at the quantum level.
The ideal limit of homodyne detection is when the local
oscillator amplitude goes to infinity, which in practical terms
means
兩
␤
兩
2
Ⰷ
␥
. In this limit, the rate of the photodetections
goes to infinity and thus it should be possible to change the
point process of photocounts into a continuous photocurrent
with white noise. Also, the only relevant quantity is the dif-
ference between the two photocurrents. Suitably normalized,
this is 关17,18兴
I
共
t
兲
⫽
I
1
共
t
兲
⫺ I
2
共
t
兲
兩
␤
兩
⫽
冑
␥
具
e
⫺ i⌽
␴
†
⫹ e
i⌽
␴
典
c
共
t
兲
⫹
␰
共
t
兲
. 共2.11兲
A number of aspects of Eq. 共2.11兲 need to be explained. First,
⌽⫽ arg
␤
, the phase of the local oscillator 共defined relative
to the driving field兲. Second, the subscript c means condi-
tioned and refers to the fact that if one is making a homo-
dyne measurement then this yields information about the
system. Hence, any system averages will be conditioned on
the previous photocurrent record. Third, the final term
␰
(t)
represents Gaussian white noise, so that
␰
共
t
兲
dt⫽ dW
共
t
兲
, 共2.12兲
an infinitesimal Wiener increment defined by 关19兴
dW
共
t
兲
2
⫽ dt, 共2.13兲
E
关
dW
共
t
兲
兴
⫽ 0. 共2.14兲
Since the stationary solution of the master equation confines
the state to the x-z plane, it makes sense to follow HMH by
setting ⌽⫽ 0. In that case,
I
共
t
兲
⫽
冑
␥
具
␴
x
典
c
共
t
兲
⫹
␰
共
t
兲
. 共2.15兲
That is, the deterministic part of the homodyne photocurrent
is proportional to x
c
⫽
具
␴
x
典
c
. This should be useful for con-
trolling the dynamics of the state in the x-z plane by feed-
back, as we will consider in Sec. III. Of course, all that really
matters here is the relationship between the driving phase
and the local oscillator phase, not the absolute phase of ei-
ther.
The conditioning process referred to above can be made
explicit by calculating how the system state changes in re-
sponse to the measured photocurrent. Assuming that the state
at some point in time is pure 共which will tend to happen
because of the conditioning anyway兲, its future evolution can
be described by the stochastic Schro
¨
dinger equation 共SSE兲
关17,18兴
d
兩
␺
c
共
t
兲
典
⫽ A
ˆ
c
共
t
兲
兩
␺
c
共
t
兲
典
dt⫹ B
ˆ
c
共
t
兲
兩
␺
c
共
t
兲
典
dW
共
t
兲
. 共2.16兲
This is an Ito
ˆ
stochastic equation 关19兴 with a drift term and a
diffusion term. The operator for the drift term is
A
ˆ
c
共
t
兲
⫽
␥
2
关
⫺
␴
†
␴
⫹
具
␴
x
典
c
共
t
兲
␴
⫺
具
␴
x
典
c
2
共
t
兲
/4
兴
⫺ i
␣
␴
y
, 共2.17兲
while that for the diffusion is
B
ˆ
c
共
t
兲
⫽
冑
␥
关
␴
⫺
具
␴
x
典
c
共
t
兲
/2
兴
. 共2.18兲
Both of these operators are conditioned in that they depend
on the system average
具
␴
x
典
c
共
t
兲
⫽
具
␺
c
共
t
兲
兩
␴
x
兩
␺
c
共
t
兲
典
. 共2.19兲
As stated above, on average the system still obeys the master
equation 共2.1兲. This is easiest to see from the stochastic mas-
ter equation 共SME兲, which allows for impure initial condi-
tions. The SME can be derived from the SSE by constructing
d
共
兩
␺
c
典具
␺
c
兩
兲
⫽
共
d
兩
␺
c
典
)
具
␺
c
兩
⫹
兩
␺
c
典
共
d
具
␺
c
兩
兲
⫹
共
d
兩
␺
c
典
兲
共
d
具
␺
c
兩
兲
,
共2.20兲
using the Ito
ˆ
rule 共2.13兲, and then identifying
兩
␺
c
典具
␺
c
兩
with
␳
c
. The result is
d
␳
c
⫽ dt
␥
D
关
␴
兴
␳
c
⫺ idt
␣
关
␴
y
,
␳
c
兴
⫹ dW
共
t
兲
冑
␥
H
关
␴
兴
␳
c
,
共2.21兲
where H
关
A
兴
B⬅AB⫹ BA
†
⫺ Tr
关
AB⫹ BA
†
兴
. Although this
has been derived assuming pure initial conditions, it is valid
FEEDBACK-STABILIZATION OF AN ARBITRARY PURE... PHYSICAL REVIEW A 64 063810
063810-3

for any initial conditions 关18兴. This is also an Ito
ˆ
equation,
which means the evolution for the ensemble average state
matrix
␳
共
t
兲
⫽ E
关
␳
c
共
t
兲
兴
共2.22兲
is found simply by averaging over the photocurrent noise
term by using Eq. 共2.14兲. This procedure yields the original
Master equation 共2.1兲 again. The term ‘‘quantum trajectory’’
can be applied to any stochastic conditioned evolution of the
system, be it described by a SSE or SME.
III. FEEDBACK WITH UNIT-EFFICIENCY DETECTION
A. SSE including feedback
We now include feedback onto the amplitude of the driv-
ing on the atom, proportional to the homodyne photocurrent,
as done by HMH. This is as shown in Fig. 1, where the
driving field passes through an electro-optic amplitude
modulator controlled by the photocurrent, yielding a field
proportional to
␣
⫹ ␭I(t). This means that the feedback can
be described by the Hamiltonian
H
fb
⫽ ␭
␴
y
I
共
t
兲
. 共3.1兲
In this paper we are assuming instantaneous feedback, where
the time delay in the feedback loop is negligible.
Since the homodyne photocurrent 共2.11兲 is defined in
terms of system averages and the noise dW(t), the SSE in-
cluding feedback can still be written as an equation of the
form 共2.16兲. The effect of the feedback Hamiltonian can be
shown 关4,8兴 to change the drift and diffusion operators to
A
ˆ
c
共
t
兲
⫽
␥
2
关
⫺
␴
†
␴
⫹
具
␴
x
典
c
共
t
兲
␴
⫺
具
␴
x
典
c
2
共
t
兲
/4
兴
⫺ i
␣
␴
y
⫹
␭
2
冑
␥
关
⫺ i
具
␴
x
典
c
共
t
兲
␴
y
⫺ 2
␴
†
␴
兴
⫺ ␭
2
/2, 共3.2兲
B
ˆ
c
共
t
兲
⫽
冑
␥
关
␴
⫺
具
␴
x
典
c
共
t
兲
/2
兴
⫺ i␭
␴
y
. 共3.3兲
Say we wish to stabilize the pure state with Bloch angle
␪
,as
defined in Eqs. 共2.6兲 and 共2.7兲, with r⫽ 1 of course. In terms
of the ground and excited states, this state is
兩
␪
典
⫽ cos
␪
2
兩
e
典
⫹ sin
␪
2
兩
g
典
. 共3.4兲
Now for this state to be stabilized we must have
关
A
ˆ
c
共
t
兲
dt⫹ B
ˆ
c
共
t
兲
dW
共
t
兲
兴
兩
␪
典
⬀
兩
␪
典
. 共3.5兲
We cannot say the left-hand side should equal zero because a
change in the overall phase still leaves the physical state
unchanged. However, we can work with this equation, and
simplify it by dropping all terms proportional to the identity
operator in A
ˆ
c
(t) and B
ˆ
c
(t). We can also demand that it be
satisfied for the deterministic and noise terms separately, be-
cause dW(t) can take any value. This gives the two equa-
tions
共
冑
␥
␴
⫺ i␭
␴
y
兲
兩
␪
典
⬀
兩
␪
典
, 共3.6兲
关
␥
共
⫺
␴
†
␴
⫹ sin
␪
␴
兲
⫺ i2
␣
␴
y
⫹ ␭
冑
␥
共
⫺ i sin
␪
␴
y
⫺ 2
␴
†
␴
兲
兴
兩
␪
典
⬀
兩
␪
典
, 共3.7兲
where we have put
具
␴
x
典
c
(t) equal to sin
␪
, its value for the
state
兩
␪
典
.
Solving the first equation easily yields the condition
␭⫽⫺
冑
␥
2
共
1⫹ cos
␪
兲
. 共3.8兲
This is equivalent to the feedback condition derived by
HMH, stated as Eq. 共35兲 of Ref. 关15兴. Substituting this into
the second equation gives, after some trigonometric manipu-
lation, the second condition
␣
⫽
␥
4
sin
␪
cos
␪
. 共3.9兲
Again, this agrees with the driving strength of HMH, given
as Eq. 共44兲 of Ref. 关15兴. It is worth emphasizing that the
derivation given here is entirely different in detail from that
of HMH, and so is an independent verification of their result.
These functions are plotted in Fig. 3. Note that there are two
points with the same values of both ␭ and
␣
,at
␪
⫽⫾
␲
/2.
B. Stability
The preceding derivation seems to show that any pure
state can be stabilized by a suitable choice of driving and
feedback. Indeed our derivation proves that that if one pre-
pares a state in exactly the pure state one desires, then the
feedback scheme of HMH, which we have analyzed, will
keep the system in that state. However, to discuss stability
we need to know what will happen for states that are not
FIG. 3. Plot of the optimal driving (
␣
, solid兲 and feedback (␭,
dashed兲 required to produce a pure state with Bloch angle
␪
. For
this plot we have set
␥
⫽ 1 so that
␣
and ␭ are dimensionless. The
purity (r
2
, starred兲 is one except for
␪
⫽⫾
␲
/2, where the feedback
is not stable.
JIN WANG AND H. M. WISEMAN PHYSICAL REVIEW A 64 063810
063810-4

initially in the desired state. To deal with this it is much more
convenient to use the SME rather than the SSE, as will be-
come apparent.
The SME can be constructed from the SSE in the same
way as before. The result is 关4,8兴
d
␳
c
⫽ dt
␥
D
关
␴
兴
␳
c
⫺ idt
␣
关
␴
y
,
␳
c
兴
⫺ idt␭
关
␴
y
,
␴␳
c
⫹
␳
c
␴
†
兴
⫹ dt
共
␭
2
/
␥
兲
D
关
␴
y
兴
␳
c
⫹ dW
共
t
兲
H
关
冑
␥
␴
⫺ i␭
␴
y
兴
␳
c
.
共3.10兲
Also as before, this is an Ito
ˆ
stochastic equation, which
means that the ensemble average can be found simply by
dropping the stochastic terms. This time, the result is not the
original master equation, but rather the feedback-modified
master equation
␳
˙
⫽⫺i
关
␣
␴
y
,
␳
兴
⫹ D
关
冑
␥
␴
⫺ i␭
␴
y
兴
␳
⬅L
␳
. 共3.11兲
Here we have put the Liouvillian superoperator L in a mani-
festly Lindblad form.
Now we have shown already that the pure state
␳
⫽
兩
␪
典具
␪
兩
must be a solution of this master equation, for the
appropriate choices of ␭关Eq. 共3.8兲兴 and
␣
关Eq. 共3.9兲兴. But
for it to be a stable solution we require all of the eigenvalues
of the resulting L
␪
to have a negative real part 共except for the
one eigenvalue that is always zero, as required for L
␪
to be
normpreserving兲. It is not difficult to find these eigenvalues,
and in terms of
␪
they are
⫺
␥
/2,⫺
␥
/2,⫺
␥
cos
2
␪
. 共3.12兲
Evidently the state
兩
␪
典
will be stable for all
␪
except
␪
⫽
⫾
␲
/2. That is, all states are stable except those on the equa-
tor. This is contrary to the conclusion of HMH 关15兴, based
on a linearized stability analysis, that ‘‘long-term stability
of . . . inverted states 共i.e., states in the upper-half-plane兲
cannot be achieved.’’ We emphasize that our stability analy-
sis contains no approximations.
In hindsight, the lack of stability for pure states on the
equator could have been predicted from expressions 共3.8兲
and 共3.9兲. As discussed above and shown in Fig. 3, the values
for driving and feedback for
␪
⫽
␲
/2 are the same as those
for
␪
⫽⫺
␲
/2. This means that both
␳
⫽
兩
␲
/2
典具
␲
/2
兩
and
␳
⫽
兩
⫺
␲
/2
典具
⫺
␲
/2
兩
are solutions of L
␪
␳
⫽ 0 for
␪
⫽
␲
/2 or
⫺
␲
/2. By linearity, any mixture of
兩
␲
/2
典具
␲
/2
兩
and
兩
⫺
␲
/2
典具
⫺
␲
/2
兩
will be a solution also. Hence any deviation
away from one of these pure states will not necessarily be
suppressed, and the system lacks stability. With random ex-
ternal perturbations, the system will eventually reach an
equal mixture of
兩
␲
/2
典具
␲
/2
兩
and
兩
⫺
␲
/2
典具
⫺
␲
/2
兩
, which is a
state with r⫽ 0 共minimum purity兲. This is why we have plot-
ted a value of r⫽ 0 in Fig. 3 for
兩
␪
兩
⫽
␲
/2. We also plot r as
a function of
␪
in Bloch space in Fig 2, giving the locus of
states that can be stabilized by feedback. This can compared
to the locus of the mixed states that are accessible by driving
alone. We will return to the stability issue in the context of
stochastic dynamics in Sec. V.
IV. FEEDBACK WITH NONUNIT-EFFICIENCY
DETECTION
We have seen that the stochastic master equation is a very
useful representation of a quantum trajectory, as it allows the
unconditioned 共deterministic兲 master equation to be derived
easily, and this latter equation is all that is required for a
completely rigorous stability analysis. The SME is also su-
perior to the SSE in that it allows inefficient detection to be
described. In a real experiment this has to be taken into ac-
count. The effect of nonunit
␩
on feedback in the present
system is of interest both in itself, and because of the ex-
treme nonlinearity of the system dynamics as compared to
other quantum optical feedback systems such as considered
in Ref. 关8兴.
As explained in Ref. 关18兴, the homodyne photocurrent
from a detection scheme with efficiency
␩
is
I
共
t
兲
⫽
冑
␥
具
␴
x
典
c
共
t
兲
⫹
␰
共
t
兲
/
冑
␩
. 共4.1兲
Here we have used a normalization such that the determin-
istic part does not depend on
␩
. The effect of decreased
efficiency is increased noise. This means that we can retain
the same feedback Hamiltonian as above 关Eq. 共3.1兲兴, without
changing the significance of the feedback parameter ␭. The
SME with
␩
⬍ 1, including feedback, is 关8兴
d
␳
c
⫽ dt
␥
D
关
␴
兴
␳
c
⫺ idt
␣
关
␴
y
,
␳
c
兴
⫺ idt␭
关
␴
y
,
␴␳
c
⫹
␳
c
␴
†
兴
⫹ dt
共
␭
2
/
␥
␩
兲
D
关
␴
y
兴
␳
c
⫹ dW
共
t
兲
H
⫻
关
冑
␥
␩
␴
⫺ i␭
␴
y
/
冑
␩
兴
␳
c
. 共4.2兲
The no-feedback SME, analogous to Eq. 共2.21兲, can be ob-
tained simply by setting ␭⫽ 0, and was derived in Ref. 关18兴.
Once again, it is easiest for the moment to just consider
the ensemble-average evolution by averaging dW to zero.
The Lindblad form of the resulting master equation is
␳
˙
⫽⫺i
关
␣
␴
y
,
␳
兴
⫹ D
关
冑
␥
␴
⫺ i␭
␴
y
兴
␳
⫹
共
␭
2
/
␩
兲
D
关
␴
y
兴
␳
.
共4.3兲
We do not know a priori what values of ␭ and
␣
to choose
to give the best results with inefficient detection, as the SSE
analysis in Sec. III A obviously does not apply. Hence we
simply solve for the stationary matrix in terms of
␣
and ␭.
Using the Bloch representation we find
x
ss
⫽⫺4
␣
␩
2
共
␥
⫹ 2
冑
␥
␭
兲
/D, 共4.4兲
y
ss
⫽ 0, 共4.5兲
z
ss
⫽⫺
冑
␥
␩
共
冑
␥
⫹ 2␭
兲
共
␥
␩
⫹ 4
冑
␥
␩
␭⫹ 4␭
2
兲
/D, 共4.6兲
where
D⫽
␥
2
␩
2
⫹ 6
␥
3/2
␩
2
␭⫹ 2
␥
␩
共
3⫹ 4
␩
兲
␭
2
⫹ 16
冑
␥
␩
␭
3
⫹ 8
共
␣
2
␩
2
⫹ ␭
4
兲
. 共4.7兲
The question now arises, what do we mean by ‘‘best results’’
for the feedback system? We cannot hope anymore to pro-
FEEDBACK-STABILIZATION OF AN ARBITRARY PURE... PHYSICAL REVIEW A 64 063810
063810-5

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Feedback-stabilization of an arbitrary pure state of a two-level atom" ?

Here the authors reanalyze their proposal using the technique of stochastic master equations, allowing their results to be generalized in two ways. First, the authors show that any point on the upperor lower-half, but not the equator, of the sphere may be stabilized. Second, the authors consider nonunit-efficiency detection, and quantify the effectiveness of the feedback by calculating the maximal purity obtainable in any particular direction in Bloch space. 

Q2. What are the future works mentioned in the paper "Feedback-stabilization of an arbitrary pure state of a two-level atom" ?

This is an issue the authors plan to explore in future work. 

Q3. What is the annihilation operator for the Bloch sphere?

Ignoring the vacuum fluctuations in the field, the annihilation operator for this beam is Ags , normalized so that the mean intensity g^s†s& is equal to the number of photons per unit time in the beam. 

Q4. What is the effect of a small perturbation away from one fixed pure state?

A small perturbation away from one fixed pure state leads to a proportionally small fraction of the ensemble ending up in the diametrically opposite pure state. 

Q5. how does the system state vary in the long-time limit?

That is, the system state continues to vary stochastically in the long-time limit, but is constrained so that the time-averaged state is equal to the solution of the deterministic master equation. 

Q6. What is the effect of nonunit h on feedback in the present system?

The effect of nonunit h on feedback in the present system is of interest both in itself, and because of the extreme nonlinearity of the system dynamics as compared to other quantum optical feedback systems such as considered in Ref. @8#. 

Q7. What is the way to determine the phase relationship of the local oscillator?

To0-2ensure that this local oscillator has a fixed phase relationship with the driving laser used in the measurement, it would be natural to utilize the same coherent light field source in the driving process and as the local oscillator in the homodyne detection. 

Q8. What is the deterministic part of the homodyne photocurrent?

In that case,I~ t !5Ag^sx&c~ t !1j~ t !. ~2.15!That is, the deterministic part of the homodyne photocurrent is proportional to xc5^sx&c . 

Q9. What does the purity of the states in the upper half of the Bloch sphere mean?

The authors find that the purity ~which measures this closeness! of states thus produced decays to zero as h decreases to zero, for states in the upper half of the Bloch sphere. 

Q10. Why does the photocurrent vary from one run to the next?

This photocurrent will vary stochastically from one run to the next, because of the irreducible randomness in the quantum measurement process. 

Q11. how many times do the authors minimize the noise terms in the SBE Eq.?

5US 2Aghxss2 1~Agh12l/Ah!zss1Agh2Aghxssy ss 2~Agh12l/Ah!xss2Aghxsszss D U 2. ~5.8!That is, the authors minimize the noise terms in the SBE Eq. ~5.2!. 

Q12. How can one obtain an asymmetry between the upper and lower halves of the Bloch?

It is nevertheless possible to obtain an asymmetry between the upper and lower halves of the Bloch sphere, reminiscent of the conclusion of HMH, if one considers detection efficiencies less than one. 

Q13. What is the convenient way to treat the stochastic dynamics in general?

So far the authors have considered the stochastic conditioned dynamics for the system state in order to derive the parameters l and a such that for h51 those dynamics are banished in the steady state.