Cavity-loss-induced generation of entangled atoms
M. B. Plenio, S. F. Huelga,
*
A. Beige, and P. L. Knight
Optics Section, Blackett Laboratory, Imperial College London, London SW7 2BZ, England
~Received 14 September 1998!
We discuss the generation of entangled states of two two-level atoms inside an optical resonator. When the
cavity decay is continuously monitored, the absence of photon counts is associated with the presence of an
atomic entangled state. In addition to being conceptually simple, this scheme can be demonstrated with
presently available technology. We describe how such a state is generated through conditional dynamics, using
quantum jump methods, including both cavity damping and spontaneous emission decay, and evaluate the
fidelity and relative entropy of entanglement of the generated state compared with the target entangled state.
@S1050-2947~99!08303-1#
PACS number~s!: 42.50.Lc, 42.50.Dv, 03.67.2a
I. INTRODUCTION
Superposition effects in composite systems are well
known in classical physics. However, when the superposition
principle is combined with a tensor product structure for the
space of states, an entirely quantum-mechanical effect arises:
Quantum states can be entangled @1#. This fact was early
recognized as the characteristic of the quantum formalism
@2#. However, early work concentrated on the implications of
entanglement on the nonlocal structure of quantum theory
@3#, and it was considered by many as a purely philosophical
issue. The reason for the renewed interest in the fundamental
aspects of quantum mechanics is twofold. On the one hand,
it was discovered that Bell’s inequalities do not provide a
good criterion for discriminating between classical and quan-
tum correlations when dealing with mixed states @4#. New
criteria for characterizing the separability of a given quantum
state have been proposed @5#, and measures of entanglement
have been introduced @6,7#. On the other hand, it has been
realized that entangled states allow new practical applica-
tions, ranging from quantum computation @8# and secure
cryptographic schemes @9# to improved optical frequency
standards @10#. The feasibility of some these applications has
been demonstrated in recent experiments @11#. In particular,
recent advances in ion trapping technology @12# and cavity
QED @13# provide suitable scenarios for manipulating small
quantum systems.
In this paper we will discuss a scheme that allows the
generation of a maximally entangled state of two two-level
atoms within a single-mode cavity field. The underlying idea
is conceptually simple, and relies on the concept of condi-
tional dynamics due to continuous observation of the cavity
field. The key to understanding how the entangled state is
generated in this scheme is population trapping @14#. There
are three dressed states of the combined two-atom plus cav-
ity field mode system; one has a zero eigenvalue, which is
therefore stationary, whereas the other two decay in time.
Provided no photon leaks out of the cavity ~which is why
conditional dynamics is necessary!,apure entangled state
between the two atoms results. From the experimental point
of view, this proposal is feasible with presently available
technology.
The paper is organized as follows. In Sec. II we describe
the system of interest. This consists of two trapped atoms
inside an optical resonator. Certain aspects of the dynamics
of this system, when driven by an external field, have been
addressed, for instance, in the context of the two-atom mi-
crolaser @15#. The coherence properties of the fluorescence
from close-lying atoms in an optical cavity have been con-
sidered recently using the quantum jump approach @16#. Our
proposal provides a probabilistic scheme @17# for generating
an entangled state of the two atoms. This will require an
initial preparation, which involves the selective excitation of
one of the atoms and the continuous monitoring of photons
leaking out of the cavity. The time evolution under the con-
dition of no-photon detection is discussed in Sec. III. We
will show that the quantum jump approach provides a suit-
able theoretical framework for analyzing the dynamics in a
simple and intuitive way. The fidelity with respect to a maxi-
mally entangled state and the relative entropy of entangle-
ment of the final atomic state will be evaluated in Sec. IV.
II. DESCRIPTION OF THE PHYSICAL SYSTEM
Our system consists of two two-level ions confined in a
linear trap which has been surrounded by a leaky optical
cavity. We will refer to atom a and atom b when the context
requires us to differentiate them, but otherwise they are sup-
posed to be identical. We denote the atomic ground and ex-
cited states by
u
0
&
i
and
u
1
&
i
, and call 2G (G5 G
a
5 G
b
) the
spontaneous emission rate from the upper level. We assume
that the distance between the atoms is much larger than an
optical wavelength, and that therefore dipole-dipole interac-
tions can be neglected @18#. In addition, this requirement
allows us to assume that each atom can be individually ad-
dressed with laser light. The cavity mode is assumed to be
resonant with the atomic transition frequency, and we will
denote the cavity decay rate by
k
. For the sake of generality
we allow the coupling between each atom and the cavity
*
Permanent address: Departamento de Fı
´
sica, Avda de Calvo
Sotelo n/s, 33007 Oviedo, Spain.
PHYSICAL REVIEW A MARCH 1999VOLUME 59, NUMBER 3
PRA 59
1050-2947/99/59~3!/2468~8!/$15.00 2468 ©1999 The American Physical Society
mode, g
i
, to be different.
1
The relaxation of the ion-cavity
system can take place through two different channels, at rates
k
~cavity decay! and G~spontaneous decay!.
In what follows we will assume that the coupling con-
stants and the decay rates are such that
g
i
,
k
@ G. ~1!
The experimental setup is depicted in Fig. 1. Note the pres-
ence of a single photon detector D in our scheme. This setup
will allow us to monitor the decay of the system through the
fast channel, i.e., photons leaking through the cavity mirrors.
On the other hand, spontaneously emitted photons from the
slow decay channel in the regime of Eq. ~1!, will not be
detected. The initial state of the system is of the form
u
0
&
^
u
0
&
a
^
u
0
&
b
[
u
000
&
, ~2!
where the first index refers to the cavity field state. Now
applying a
p
pulse to atom a, we introduce an excitation into
the system, and the initial conditions for our scheme will be
given by the composite state
u
c
0
&
5
u
0
&
^
u
1
&
a
^
u
0
&
b
[
u
010
&
. ~3!
In the following we will use Eq. ~3! as the basis for all the
following discussions. It is important to emphasize that our
scheme only requires the atoms to be cooled to the Lamb-
Dicke limit, i.e. each atom is localized within one wave-
length of the emitted light. But no further cooling to the
motional ground state is necessary. This notably simplifies
the experimental realizability of the proposal.
Experiments on ions in optical cavities are underway, for
example, in Innsbruck. In these experiments the S
1/2
-D
5/2
transition of calcium ions @lifetime (2G)
2 1
5 1 s] couples to
an optical cavity which has a decay rate
k
between 1 and 10
kHz. The ions are separated by many optical wavelengths,
and can therefore be addressed separately using focused laser
beams @19#.
III. ATOM-CAVITY SYSTEM WITHOUT DECAY
In order to illustrate the main idea underlying this pro-
posal, let us ignore any relaxation process for the moment.
The unitary time evolution of the system will then be gov-
erned by the Hamiltonian
H5
(
i5 a,b
\
v
i
u
1
&
ii
^
1
u
1\
n
b
†
b1i\
3
(
i5a,b
~
g
i
b
u
1
&
ii
^
0
u
2H.c.
!
, ~4!
where b and b
†
denote the annihilation and creation opera-
tors for the single-mode cavity field. The fourth term in this
expression is the familiar Jaynes-Cummings ~JC! interaction
between each atomic system and the cavity mode. Moving to
an interaction picture with respect to the unperturbed Hamil-
tonian,
H
0
5
(
i5 a,b
\
v
i
u
1
&
ii
^
1
u
1\
n
b
†
b, ~5!
and assuming exact resonance between the cavity mode and
the atomic transition,
n
5
v
i
, we find
H
I
5 i\
(
i5 a,b
~
g
i
b
u
1
&
ii
^
0
u
2H.c.
!
~6!
where the coupling constants g
i
have been taken to be real.
In the basis B5 (
u
100
&
,
u
010
&
,
u
001
&
), the interaction picture
Hamiltonian reads
H
I
5
\
i
S
0 g
a
g
b
2 g
a
00
2g
b
00
D
. ~7!
It is easy to check that the eigenvalues associated with this
operator are given by
l
0
5 0, ~8!
l
1,2
56\
A
g
a
2
1g
b
2
, ~9!
with corresponding eigenvectors
u
l
0
&
5
1
A
g
a
2
1 g
b
2
~
g
a
u
001
&
2 g
b
u
010
&
), ~10!
u
l
1,2
&
5
1
A
2
S
u
100
&
6
i
A
g
a
2
1 g
b
2
~
g
b
u
001
&
1 g
a
u
010
&
D
.
Note that when g
a
5 g
b
, the solution
u
l
0
&
is a tensor product
of the cavity field in the vacuum state and the maximally
entangled atomic state
u
f
2
&
5
1
A
2
~
u
01
&
2
u
10
&
). ~11!
To prepare an entangled state of the atoms one now needs a
mechanism that destroys the population of the cavity mode.
1
A symmetric location of the atoms with respect to the center of
the trap suffices to make g
a
5 g
b
. However, experimentally this
may well be hard to achieve.
FIG. 1. Experimental setup. The system consists of two two-
level atoms placed inside a leaky cavity. The decay rate G describes
the spontaneous emission of the atoms, while the rate
k
refers to
photons leaking through the cavity mirrors. The latter can be moni-
tored by the detector D.
PRA 59
2469CAVITY-LOSS-INDUCED GENERATION OF ENTANGLED ATOMS
One possibility is to use a leaking cavity, and to detect all
photons coming through the cavity mirrors. If a photon is
detected, the system is in the ground state
u
000
&
. Then the
experiment has to be repeated. But if not, the system goes
over into a state which cannot decay. Therefore, the atoms
should end up in state
u
l
0
&
, the entangled state, where the
cavity mode is not populated.
Using the quantum jump approach, we will see that the
dynamics under the condition that no photon has been de-
tected outside the cavity is governed by an effective Hamil-
tonian whose solutions keep track of the structure illustrated
above. More precisely, for sufficiently large times the state
of the system will be a tensor product of the cavity field in
the vacuum state and an entangled state of the two atoms.
IV. ATOM-CAVITY SYSTEM INCLUDING DECAY
Let us now consider the experimental situation depicted in
Fig. 1, in which the decay of the cavity field is monitored by
means of the detector D. For the moment we will assume
that the detector has 100% efficiency, but later this constraint
will be relaxed. The time evolution is now governed by the
Hamiltonian
H5
(
i5 a,b
\
v
i
u
1
&
ii
^
1
u
1\
n
b
†
b1
(
kl
\
v
kl
a
kl
†
a
kl
1i\
(
i5a,b
~
g
i
b
u
1
&
ii
^
0
u
2H.c.
!
1 i\
(
i5 a,b
(
kl
~
g
kl
a
kl
u
1
&
ii
^
0
u
e
i
~
v
i
2
v
kl
!
t
2H.c.
!
1 i\
(
kl
~
s
kl
a
kl
b
†
e
i
~
n
2
v
kl
!
t
2 H.c.
!
, ~12!
where a
kl
†
and a
kl
denote the free radiation field creation and annihilation operators of a photon in the mode (k,l). The two
remaining terms including the coupling constants g
kl
and s
kl
describe, respectively, the coupling of the atoms and the cavity
mode to the free radiation field. The initial state of the system,
u
c
0
&
, is given by Eq. ~3!. At a time t, and provided that no
photon leaking through the cavity mirrors has been detected, the state of the system can be described in terms of a density
operator of the form
r
~
t,
c
0
!
5 „P
0
~
t,
c
0
!
u
c
ˆ
coh
~
t
!
&^
c
ˆ
coh
~
t
!
u
1 P
spon
~
t,
c
0
!
u
000
&^
000
u
…/tr
~
!
. ~13!
Here P
0
(t,
c
0
) is the probability for no photon emission, where neither the cavity field nor the atoms have decayed until t, and
u
c
ˆ
coh
(t)
&
denotes the normalized state resulting from the coherent evolution in this case. Later we will also use the notation
u
c
coh
&
for the unnormalized state. The second term of the mixture takes into account that spontaneously emitted photons are
not observed. If an atom emits a spontaneous photon, then the state of the atom-cavity system is reduced to the state
u
000
&
. Our
main task consists of evaluating the explicit form of the state
u
c
ˆ
coh
(t)
&
of P
0
(t,
c
0
), and the probability P
spon
(t,
c
0
) for
spontaneously decay in (0,t). The quantum jump approach ~also called the quantum trajectories method!@20–22#~See Ref.
@23# for a recent review! provides a suitable theoretical framework for this analysis.
A. Derivation of the conditional time evolution
Let us consider an idealized situation where both the photons leaking through the cavity and the spontaneously emitted
photons could be detected. In the derivation of the quantum jump approach, one envisages an equally spaced sequence of
gedanken photon measurements at times t
1
,t
2
,...,t
n21
,t
n
, such that t
i
2 t
i2 1
5 Dt. According to the projection postulate, the
subensemble for which no photon has been detected until time t
n
is described by the ~unnormalized! state vector
u
c
coh
~
t
!
&
5 P
0
U
~
t
n
,t
n2 1
!
P
0
...P
0
U
~
t
1
,t
0
!
u
0
ph
&
u
c
~
t
0
!
&
[
u
0
ph
&
U
cond
~
t
n
,t
0
!
u
c
~
t
0
!
&
, ~14!
where we have defined the projector
P
0
5
u
0
ph
&
I
A
^
0
ph
u
, ~15!
and I
A
denotes the identity over the atomic variables. Therefore, the operator U
cond
(t
n
,t
0
) describes the time evolution of the
system under the condition that no photon has been detected. Using our previous notation, the state of the system at a time t
n
will be given by U
cond
(t
n
,t
0
)
u
c
(t
0
)
&
when the system has not relaxed through either the fast or the slow channel. Taking into
account Eq. ~12! and the form of the projector P
0
, our problem reduces to evaluating expressions of the form
^
0
ph
u
U(t
n
,t
n2 1
)
u
0
ph
&
, which can be done easily using second-order perturbation theory. The calculations can be simplified
moving to an appropriate interaction picture with respect to the unperturbed Hamiltonian
H
0
5
(
i5 a,b
\
v
i
u
1
&
ii
^
1
u
1\
n
b
†
b1
(
kl
\
v
kl
a
kl
†
a
kl
. ~16!
In second-order perturbation theory one obtains
2470 PRA 59M. B. PLENIO, S. F. HUELGA, A. BEIGE, AND P. L. KNIGHT
^
0
ph
u
U
~
t
n
,t
n2 1
!
u
0
ph
&
5 12
1
\
E
t
n21
t
n
dt
8
^
0
ph
u
H
I
~
t
8
!
u
0
ph
&
2
1
\
2
E
t
n21
t
n
dt
8
E
t
n21
t
8
dt
9
^
0
ph
u
H
I
~
t
8
!
H
I
~
t
9
!
u
0
ph
&
, ~17!
where the interaction Hamiltonian reads
H
I
5 H
a-c
1 H
a-f
1 H
c-f
5 i\
(
i5 a,b
~
g
i
b
u
1
&
ii
^
0
u
2H.c.
!
1 i\
(
i5 a,b
(
kl
~
g
kl
a
kl
u
1
&
ii
^
0
u
e
i
~
v
i
2
v
kl
!
t
2H.c.
!
1 i\
(
k,l
~
s
kl
a
kl
b
†
e
i
~
n
2
v
kl
!
t
2 H.c.
!
. ~18!
In first-order perturbation theory, only the JC term contrib-
utes to Eq. ~17! since both
^
0
ph
u
a
kl
u
0
ph
&
and
^
0
ph
u
a
kl
†
u
0
ph
&
are zero. On the other hand, the second-order contribution
from the JC term is quadratic in gDt and can be neglected. A
contribution from the term H
a-f
i
(i5 a,b) appears only in
second-order perturbation theory and can be evaluated using
the usual Markov approximation @24#. Then one finds
2
1
\
2
E
t
n21
t
n
dt
8
E
t
n21
t
8
dt
9
^
0
ph
u
H
a-f
~
t
8
!
H
a-f
~
t
9
!
u
0
ph
&
52G
i
u
1
&
ii
^
1
u
Dt, ~19!
where
G
i
5
e
2
6
p
e
0
\c
3
d
2
v
i
3
. ~20!
Similarly, one can show that the term H
c-f
yields a formally
analogous contribution, now replacing the atomic decay rate
by the cavity decay rate
k
. The form of the conditional
Hamiltonian is now easily inferred, taking into account that
)
i5 1
n
^
0
ph
u
U
~
t
n
,t
n2 1
u
0
ph
&
5 U
cond
~
t
n
,0
!
5 T exp
S
2
i
\
E
0
t
n
dt
8
H
cond
~
t
8
!
D
,
~21!
where T indicates a time ordered expression. We find
H
cond
5
\
i
S
k
g
a
g
b
2 g
a
G 0
2 g
b
0 G
D
[
\
i
M ~22!
in the basis B5 (
u
100
&
,
u
010
&
,
u
001
&
). The corresponding ei-
genvalues of M are given by
l
0
5 G; ~23!
l
1,2
5
~
k
1 G6 iS
!
/2, ~24!
with S5
A
4(g
a
2
1g
b
2
)2(
k
2G)
2
. The eigenvector of the
smallest eigenvalue is the same entangled state as in Eq.
~10!, i.e.,
u
l
0
&
5
1
A
g
a
2
1 g
b
2
~
g
a
u
001
&
2 g
b
u
010
&
). ~25!
M has three normalized eigenvectors
u
l
i
&
, which are in gen-
eral not orthogonal. The reciprocal vectors
^
l
i
u
are defined
by
^
l
i
u
l
j
&
5
d
ij
. Then one can write M5 (
i
l
i
u
l
i
&^
l
i
u
. For
the conditional time evolution operator, one has the represen-
tation
U
cond
~
t,0
!
5 e
2 Mt
5
(
i51
3
e
2l
i
t
u
l
i
&^
l
i
u
. ~26!
Therefore, provided that no photon has been detected during
the time interval
@
0,t
#
and t satisfies
G
2 1
@ t@
k
2 1
, ~27!
the exponentials exp(2l
1/2
t) can be neglected while
exp(2l
0
t) is still close to unity and the system will be in the
state
u
c
ˆ
coh
~
t
!
&
5 U
cond
~
t,0
!
u
c
0
&
5 e
2 l
0
t
u
l
0
&^
l
0
u
c
0
&
/z
uu
z5
u
l
0
&
.
~28!
This state factorizes as a tensor product between the cavity
field in the vacuum state and an entangled state of the two
atoms.
More precisely, the conditional time evolution operator
U
cond
can be calculated as
e
2 Mt
5
~
M2l
1
!
~
M2l
2
!
~
l
0
2l
1
!
~
l
0
2l
2
!
e
2l
0
t
1
~
cyclic permutations
!
,
~29!
which can easily be verified by application to the eigenvec-
tors @25#. Applying this operator to our initial state, Eq. ~3!,
we obtain
PRA 59 2471CAVITY-LOSS-INDUCED GENERATION OF ENTANGLED ATOMS
u
c
ˆ
coh
~
t
!
&
5
1
g
a
2
1 g
b
2
F
g
b
e
2 Gt
S
0
g
b
2 g
a
D
1 g
a
e
2
~
1/2
!
~
k
1 G
!
t
3
H
S
0
g
a
g
b
D
cos
~
St/2
!
1
1
S
S
2 2
~
g
a
2
1 g
b
2
!
g
a
~
k
2 G
!
g
b
~
k
2 G
!
D
3 sin
~
St/2
!
J
G
. ~30!
The probability amplitudes for the three basis states are
plotted in Fig. 2. As expected, on a time scale such that
G
2 1
@ t@
k
2 1
, the contribution from terms multiplied by a
damping factor proportional to the sum
k
1 G becomes neg-
ligible, and the conditional state vector is a two-particle en-
tangled state correlated with the cavity field in the vacuum
state
u
l
0
&
.
B. Calculation of the detection probabilities
After the derivation of the conditional time evolution, we
are now in a position to calculate the probabilities for photon
emissions. We first calculate the probability that there is no
decay at all, neither spontaneous emissions by the atoms nor
photons leaking out of the cavity. Subsequently we will de-
rive the probability for ~a! having a spontaneous decay from
the atoms, and ~b! for having photon emission from the cav-
ity.
The probability to have no photon emission ~neither spon-
taneously emitted nor leaking through the cavity mirrors!
until time t is given by the norm squared of Eq. ~30!, i.e.
P
0
~
t,
c
0
!
5 z
u
U
cond
~
t,0
!
u
c
0
&
u
z
2
. ~31!
This general expression can be simplified considerably for
large times t. The probability to detect no photon until time t
with t@
k
2 1
is equal to
P
0
~
t,
c
0
!
5
g
b
2
g
a
2
1 g
b
2
e
2 2Gt
. ~32!
In our experimental setup ~see Fig. 1!, only photons leak-
ing through the cavity mirrors are monitored and, as we have
pointed out, the state of the system will be the mixture given
by Eq. ~13!. The quantum jump approach @21–23# provides a
transparent way to evaluate the weight of the component
u
000
&
, i.e., the probability for a spontaneous emission from
an atom.
Let us denote by t
8
an intermediate time within the inter-
val
@
0,t
#
. The probability P of having an emission at any
time in that interval will be given by
P5
E
0
t
dt
8
w
1
~
t
8
,
c
0
!
, ~33!
where w
1
(t
8
,
c
0
) denotes the probability density for the first
photon at time t
8
for the given initial state
u
c
0
&
@26,27#.
Since w
1
(t
8
,
c
0
)dt equals P
0
(t
8
,
c
0
)2 P
0
(t
8
1 dt
8
,
c
0
), one
has
w
1
~
t
8
,
c
0
!
52
d
dt
8
P
0
~
t
8
,
c
0
!
5
^
c
0
u
e
2M
†
t
8
~
M1M
†
!
e
2Mt
8
u
c
0
&
. ~34!
Taking into account the explicit form of M in Eq. ~22!,we
find
w
1
~
t
8
,
c
0
!
5 2
k
z
^
100
u
U
cond
~
t
8
,0
!
u
c
0
&
z
2
1 2G„z
^
010
u
U
cond
~
t
8
,0
!
u
c
0
&
z
2
1 z
^
001
u
U
cond
~
t
8
,0
!
u
c
0
&
z
2
…. ~35!
As expected, both relaxation channels contribute separately to the decay rate w
1
. Setting t
8
equal to 0, one finds that the
probability density for a photon leaking through the cavity mirrors is given by the population of the state
u
100
&
multiplied by
the cavity decay rate. Similarly, the probability for spontaneous emission is determined by the population of the states
u
010
&
and
u
001
&
.
In our case we are only interested in the contribution to P in Eq. ~33! coming from spontaneously emitted photons. Using
Eq. ~33!, one finds
P
spon
~
t,
c
0
!
5 2G
E
0
t
dt
8
„z
^
010
u
U
cond
~
t
8
,0
!
u
c
0
&
z
2
1 z
^
001
u
U
cond
~
t
8
,0
!
u
c
0
&
z
2
…. ~36!
FIG. 2. The time dependence of the probability amplitudes for
the basis states
u
100
&
,
u
010
&
, and
u
001
&
under the conditional time
evolution that no photon has been detected at all. We have chosen
g
a
5 g
b
5 g5
k
and G5 10
2 3
g. After a short time the cavity mode is
decayed, and the atoms have reached the pure entangled atomic
state.
2472 PRA 59
M. B. PLENIO, S. F. HUELGA, A. BEIGE, AND P. L. KNIGHT
