Creation of entangled states of distant atoms by interference
C. Cabrillo,
1
J. I. Cirac,
2
P. Garcı
´
a-Ferna
´
ndez,
1
and P. Zoller
2
1
Instituto de Estructura de la Materia, Serrano 123, E-28006 Madrid, Spain
2
Institut fu
¨
r Theoretische Physik, Universita
¨
t Innsbruck, A-6020 Innsbruck, Austria
~Received 28 September 1998!
We propose a scheme to create distant entangled atomic states. It is based on driving two ~or more! atoms
with a weak laser pulse, so that the probability that two atoms are excited is negligible. If the subsequent
spontaneous emission is detected, the entangled state is created. We have developed a model to analyze the
fidelity of the resulting state as a function of the dimensions and location of the detector, and the motional
properties of the atoms. @S1050-2947~99!04502-3#
PACS number~s!: 03.65.Bz, 42.50.Vk
I. INTRODUCTION
The preparation of entangled atomic states is one of the
goals of atomic physics and quantum optics. These states are
a key ingredient for studying some fundamental issues of
quantum mechanics @1#, as well as for certain applications
related to quantum information @2#. Methods proposed so far
to ‘‘engineer’’ entanglement between atoms in the laboratory
are based on achieving and controlling an effective interac-
tion between the atoms that are to be entangled. Typically,
these interactions are mediated by the electromagnetic field.
For example, in cavity QED, two atoms can be entangled if
they both interact with the same cavity mode @3#. This cou-
pling of the two atoms to the field mode can be simultaneous
or sequential ~that is, one atom interacts first with the cavity
mode, and then the other one!. With trapped ions, entangled
states can be produced by using the Coulomb repulsion be-
tween the ions, together with some laser couplings @4#. With
these methods, it is always necessary that the atoms inter-
change some particles ~photons! or that they are very close to
each other.
In this paper we propose a scheme to prepare entangled
atomic states using a different approach. In particular, the
entangled state is not produced by an effective interaction
between the atoms, but rather by an interference effect and
state projection accompanying a measurement. Imagine that
we have two atoms A and B, situated in distant locations,
both in an excited state
u
0
&
. These atoms may decay to the
state
u
1
&
due to spontaneous emission, producing one pho-
ton. A detector is placed at half the way between the atoms.
After some time, if the detector clicks and we cannot distin-
guish from where the detected photon came, we will have
produced an entangled state
u
C
&
5
1
A
2
~
u
0
&
A
u
1
&
B
1 e
i
f
u
1
&
A
u
0
&
B
), ~1!
where
f
is a fixed phase. Entanglement is then achieved as a
consequence of two facts: first, the impossibility to deter-
mine from the detection event which atom emitted the pho-
ton, second, the projection postulate in quantum mechanics,
which indicates that after the detection the state of the atoms
is projected onto the one which is compatible with the out-
come of the measurement. The first effect is precisely the
one that would give rise to interference fringes at the detector
position if one would repeat several times the experiment, as
it has been shown by the NIST group at Boulder @5,6#. The
second effect has been used, for example, in the preparation
of nonclassical states of a cavity mode @7#. Using this
method to prepare entangled states, the atoms do not need to
interact, and no interchange of particles ~photons! is re-
quired. In fact, the entanglement can be produced ~in prin-
ciple! in a time which is half the distance between the atoms
divided by the speed of light.
In practice, the method described above might not be very
useful. First, it is very unlikely that the photon emitted by
one of the atoms is detected. Second, and more important,
even if one photon is detected, the second atom will eventu-
ally decay to the ground state thus yielding the state
u
1
&
A
u
1
&
B
, which is not entangled. Here we will analyze in
some detail how an experiment can be performed in a real-
istic setup. The idea is to use two three-level atoms with a
Lambda configuration ~see Fig. 1!. The states
u
0
&
and
u
1
&
are
the two ground states, so that once the state ~1! is prepared, it
will stay. Both atoms are initially prepared in the state
u
0
&
.
The excitation is achieved by using a very short laser pulse,
which ~with a small probability! excites one of the two atoms
to level
u
2
&
. If following the excitation a spontaneously emit-
ted photon is detected, an entangled state of the two atoms
FIG. 1. Sketch of the experimental setup as well as of the inter-
nal level structure of the atoms corresponding to the proposed ex-
periment.
PHYSICAL REVIEW A FEBRUARY 1999VOLUME 59, NUMBER 2
PRA 59
1050-2947/99/59~2!/1025~9!/$15.00 1025 ©1999 The American Physical Society
will be produced. The method presented here seems particu-
larly timely, in view of the spectacular experimental progress
reported by the NIST group of observation of interference
fringes of the light emitted by two independent atoms @5#.In
fact, the same experimental setup could be used to prepare
atomic entangled states using our proposal.
In order to estimate the conditions that must be fulfilled to
create an entangled state, we have developed a theoretical
model describing the whole process of laser excitation of the
two atoms, spontaneous emission of a photon, and detection.
The idea is to represent the detector as a collection of atoms,
and then to use master equation methods to describe the
projection occurring when a detection event is recorded. In
this way, the electromagnetic field does not appear explicitly
in the formulas, making the calculations simpler. We empha-
size that the model is equivalent to the one in which the
whole state of the electromagnetic field is taken into account
at all times, and the measurement projects its state along with
the state of the atoms. This model can be easily generalized
to other situations in which there are more atoms present,
yielding entangled states of more than two atoms.
The paper is organized as follows. In Sec. II we explain
qualitatively the details of our proposal and discuss the main
results, and some of the practical problems. In Sec. III we
present the theoretical model. In Sec. IV we obtain an ana-
lytical formula for the fidelity of the final state as a function
of the physical parameters involved in the problem. Finally,
in Sec. IV we discuss the results and point out some possible
generalizations.
II. QUALITATIVE DESCRIPTION
Let us consider two atoms A and B separated by a dis-
tance 2d. Each of the atoms has an internal structure which
can be described in terms of a three-level Lambda system
~see Fig. 1!. It consists of two ground levels
u
0
&
and
u
1
&
, and
an excited state
u
2
&
. A photodetector is located at a distance
D from the segment connecting atoms A and B ~see Fig. 1!.
The detector is sensitive to photons of wavelength l
1
~and/or
polarization! corresponding to the transition
u
2
&
→
u
1
&
, which
is characterized by a spontaneous emission rate G
1
. It is not,
however, sensitive to the ones corresponding to the other
transition.
Both atoms are initially prepared in the state
u
0
&
. Then,
they are driven by a very short laser pulse on resonance with
the transition
u
2
&
↔
u
0
&
. As a consequence, sometimes one of
the atoms ~or both! will spontaneously emit a photon of
wavelength l
1
, which might be recorded at the photodetec-
tor. Most of the times, no photon will be detected after a
waiting time t@ G
1
. In such a case, the atoms are pumped
back to the original state
u
0
&
, and the experiment is repeated
until the detector clicks. Once this occurs, the state of both
atoms will be described by a density operator
r
A,B
. The goal
is to obtain a state as close as possible to the maximally
entangled state ~1! where
f
is a phase that does not change
from experiment to experiment. That is, we wish to obtain a
fidelity
F5
^
C
u
r
A,B
u
C
&
, ~2!
close to 1.
The physical idea is that the laser pulse prepares a super-
position state of the two atoms, which apart from the state
u
0
&
A
u
0
&
B
also contains a coherent superposition of the states
u
0
&
A
u
2
&
B
and
u
2
&
A
u
0
&
B
. Detection of a photon implies that a
transition
u
2
&
→
u
1
&
has taken place in one of the atoms, pro-
ducing a photon of wavelength l
1
that is detected. The term
u
0
&
A
u
0
&
B
will thus be projected out from the atomic state,
since it is incompatible with that event ~the state
u
1
&
of one
of the atoms must be present in the atomic state!. Moreover,
given the fact that the detector cannot distinguish among
photons emitted by different atoms, the superposition of the
states
u
0
&
A
u
2
&
B
and
u
2
&
A
u
0
&
B
will be transformed into a su-
perposition of the states
u
0
&
A
u
1
&
B
and
u
1
&
A
u
0
&
B
, i.e., it will
be close to the entangled state ~1!.
In order to obtain an entangled state close to the ideal Bell
state ~1!, several conditions have to be satisfied. ~i! First, the
laser pulse has to be such that the probability of exciting both
atoms to the state
u
2
&
A
u
2
&
B
has to be much smaller than the
probability of exciting the relevant coherent superposition.
Otherwise, it may happen that although we detect a photon
emitted by one of the atoms, the other atom also emits a
photon albeit in another direction which is not detected; this
would spoil the fidelity F since the final state of this process
would be
u
1
&
A
u
1
&
B
. In order to avoid this problem one must
use a sufficiently weak or short laser pulse. In that case, the
probability of exciting two atoms
e
2
is of the order of the
square of the probability of exciting only one atom .2
e
.By
choosing
e
! 1 one avoids the two-atom excitation. Notice,
however, that the laser beam cannot be too weak since it
would take a very long time to detect one spontaneously
emitted photon, given that the detection probability is pro-
portional to
e
. ~ii! Second, the detector has to be sufficiently
small. At each point of the detector the phase
f
will have a
different value spoiling the fidelity since a detection does not
specify the exact location of the event, and therefore the
exact phase is unknown. Thus, the detector has to be such
that at all points the phase is practically the same. In order to
estimate the required size of the detector surface one can use
the analogy between the situation considered here and the
double slit experiment: the distance traveled by a photon
coming from one atom or the other will be somewhat differ-
ent at different positions, and therefore the accumulated
phase depends on the position in which it is detected. The
phase will be essentially constant over regions where the
corresponding interference fringes have a constant visibility.
Thus, the length L
x
of the detector along the XZ plane has to
be much smaller than the interfringe distance L
x
! l
1
D/d.
However, the detection probability is proportional to the size
of the detector and therefore we cannot take L
x
arbitrarily
small. ~iii! Furthermore, the dynamics of the atoms during
the absorption emission cycle will also affect the final fidel-
ity. In fact, every absorption or emission of photons by an
atom is always accompanied by a recoil, which changes the
atomic motional state. This leaves a trace of which atom has
emitted the photon, thus also destroying the entanglement. In
order to avoid this problem, one has to find a way ‘‘not to
leave information about the motional states behind.’’ This
can be done, for example, by using trapped particles and
operating in the Lamb-Dicke limit, where the recoil energy
does not suffice to change the atomic motional state ~similar
to the Mo
¨
sbauer effect!. However, the extent to which this
1026 PRA 59CABRILLO, CIRAC, GARCI
´
A-FERNA
´
NDEZ, AND ZOLLER
effect can be reduced will also depend on the temperature of
the atoms in the trap, as well as on the propagation directions
of the laser beams.
In the following sections we will solve in detail a theoret-
ical model to answer all of these questions. Our result is a
simple formula for the fidelity in which these effects are
clearly separated. We consider a situation where the atoms
are trapped in identical isotropic harmonic potentials, char-
acterized by a frequency
n
and initial temperature T.We
obtain
F5
cos
2
~
u
las
!
2
~
11 F
geo
F
dyn
!
, ~3!
where
u
las
is the pulse area ~Rabi frequency times time!, and
F
dyn
and F
geo
represent a dynamical and a geometrical factor,
respectively. More specifically,
F
geo
5 sinc
F
dL
x
2l
1
A
d
2
1D
2
G
, ~4!
where sinc(x)5 sin(x)/x. We also have
F
dyn
5
E
0
`
d
t
e
2
t
exp
H
2 2
h
2
coth
S
\
n
2k
B
T
D
3
F
12 cos
~
x
!
cos
S
n
t
G
D
G
J
. ~5!
Here,
h
5 2
p
a
tp
/l
1
is the so-called Lamb-Dicke parameter,
with a
tp
5
A
\/2m
n
the size of the harmonic trapping poten-
tial ground state, G is the total spontaneous emission rate
from level
u
2
&
, and
x
is the angle between the propagation
direction of the laser acting on an atom and the line that
connects the atom with the center of the detector ~we take
this angle to be the same for atoms A and B).
The first factor in Eq. ~3! accounts for the effects due to
the laser excitation. That is, when
u
las
increases, the fidelity
decreases due to the fact that both atoms may be simulta-
neously excited. The geometrical factor is related to the size
of the detector with respect to the interfringe distance. For
small detectors compared with such a distance, this factor
approaches 1. Finally, the dynamical factor shows that the
fidelity increases for small Lamb-Dicke parameters and low
temperatures, and depends on the ratio
n
/G as well as the
direction of the lasers. The highest fidelity occurs for
cos(
x
).1 and
h
2
coth(\
n
/2k
B
T)! (G/
n
)
2
. The first condi-
tion means that the laser direction and the direction of the
photon emitted and recorded at the detector has to be prac-
tically the same. In that case the recoil given by the laser is
compensated by the recoil experienced by the atom in the
spontaneous emission process that is monitored at the pho-
todetector, and therefore no trace of which atom has emitted
is left behind. Under such circumstances a
n
! G~weak con-
finement! is needed so that the atom does not have time to
oscillate in the trap before the spontaneous emission takes
place—this would destroy the compensation of the recoils
between the absorption-emission process. In these limits we
can approximate
F
geo
.12
1
6
F
dL
x
2l
1
A
d
2
1D
2
G
2
, ~6a!
F
dyn
.12 2
h
2
coth
S
\
n
2k
B
T
DS
n
G
D
2
. ~6b!
On the other hand, under conditions of strong confinement
(G!
n
) although it is not possible to compensate for the
harmful effect of the recoil by choosing the laser propagation
direction, the dynamical factor can be very close to one in
the Lamb-Dicke limit (
h
! 1). In particular, for
h
2
coth(\
n
/2k
B
T)! 1 we have
F
dyn
.12 2
h
2
coth
S
\
n
2k
B
T
D
. ~7!
III. MODEL
A. Master equation for the atoms and photodetector
We consider two identical atoms A and B, centered at
positions r
0
A
and r
0
B
, separated by a distance 2d5
u
r
0
A
2 r
0
B
u
.
Each of the atoms has an internal structure which can be
described in terms of a three-level Lambda system ~see Fig.
1!. It consists of two ground levels
u
0
&
and
u
1
&
, and an ex-
cited state
u
2
&
. Spontaneous emission from level
u
2
&
to both
ground levels is possible, and is characterized by the rates
G
0,1
and wave vectors k
0,1
(V), where V represents a direc-
tion and G5 G
0
1 G
1
the total decay width of the excited
state.
A detector of surface dimensions S5 L
x
L
y
and efficiency
h
D
is situated in the XY plane, at a distance D from the
segment connecting atoms A and B. The center of the detec-
tor r
0
and the center of atoms A and B define the XZ plane.
We will describe the detector as a collection of independent
point atoms located at position r, with r varying along the
detector surface @8#. These atoms have two internal discrete
levels
u
g
&
and
u
e
&
, which are resonant with the wavelength
l
1
5 2
p
/k
1
. The level
u
e
&
is monitored for population at
time intervals
d
t which we will take to be sufficiently small
so that the atomic dynamics can be neglected during that
time. The level
u
e
&
has a width
g
: for sufficiently large val-
ues of
g
our model corresponds to a broadband detector,
whereas for small values it corresponds to a narrowband de-
tector. The results will be independent of the specific value
of
g
. We will concentrate on a given atom C of the detector
coupled to the quantized electromagnetic field, which in turn
is coupled to atoms A and B. We will calculate the state in
which those atoms are left when the atom C is found in the
state
u
e
&
, and we will add incoherently the contributions cor-
responding to different detection times and different posi-
tions r. In such a way we will be finally able to derive an
expression for the density operator of atoms A and B condi-
tioned to the observation of a click of the detector.
PRA 59 1027CREATION OF ENTANGLED STATES OF DISTANT . . .
Using standard methods of quantum optics, one can trace
out the electromagnetic field and obtain a master equation for
the atoms A, B, and C
d
dt
r
5
F
L
C
1
(
a
5A,B
~
L
a
1S
a
,C
1J
a
,C
!
G
r
, ~8!
where L
a
denotes the Liouvillian superoperator describing
the evolution of atom
a
alone, and
S
a
,C
r
52i
g
˜
2
G
~
u
r
a
2r
u
!
~
s
eg
C
^
s
12
a
1
s
ge
C
^
s
21
a
!
r
1 H. c.,
~9a!
J
a
,C
r
5
g
˜
E
dV
4
p
e
2 ik
~
V
!
•r
a
s
12
a
rs
eg
C
e
ik
~
V
!
•r
1H. c.,
~9b!
with
s
ij
a
5
u
i
&
a
^
j
u
~superscripts indicate the atom, whereas
subscripts indicate the states!. Here and in the following we
will use the symbol
^ ~tensor product! whenever we feel
that it clarifies the corresponding expression. The vectors r
A
and r
B
are the position operators of the atoms A and B, while
the vector r is treated as a c number. The presence of the
factor G(r)52exp(ik
1
u
r
u
)/(k
1
u
r
u
) is due to the dipole-dipole
interaction ~real part! and reabsorption ~imaginary part! be-
tween atoms A,B, and C,
g
˜
giving the typical strength of this
interaction. These two terms give rise to the excitation of
atom C via a photon absorption from atoms A and/or B,
which leads to a detection event. We have assumed k(d
2
1 D
2
)
1/2
@ 1, so that only the far–field part contributes to the
dipole-dipole interaction.
The Liouvillian action on atom C ~detector! is given by
L
C
r
52
g
2
~
s
ee
C
r
1
rs
ee
C
!
1
g
s
ge
C
rs
eg
C
. ~10!
In the absence of laser excitation, we have (
a
5 A,B)
L
a
5
1
i\
@
H
tp
a
,
r
#
2
G
2
~
s
22
a
r
1
rs
22
a
!
1 G
0
E
dV
4
p
N
0
~
V
!
e
2 ik
0
~
V
!
•r
a
s
02
a
rs
20
a
e
ik
0
~
V
!
•r
a
1 G
1
E
dV
4
p
N
1
~
V
!
e
2 ik
1
~
V
!
•r
a
s
12
a
rs
21
a
e
ik
1
~
V
!
•r
a
.
~11!
Here, H
tp
is the Hamiltonian describing the motion of an
atom in an isotropic harmonic potential of frequency
n
, and
N
0
and N
1
describe the dipole emission pattern correspond-
ing to transitions
u
2
&
→
u
0
&
and
u
2
&
→
u
1
&
, respectively.
The master equation ~8! can be solved formally as
r
~
t
!
5 e
~
L
A
1 L
B
1 L
C
!
~
t2 t
0
!
r
~
t
0
!
1
E
t
0
t
d
t
e
~
L
A
1 L
B
1 L
C
!
~
t2
t
!
3
@
S
A,C
1 J
A,C
1 S
B,C
1 J
B,C
#
r
~
t
!
. ~12!
This integral equation can be iterated to obtain a formal ex-
pansion in terms of S and J. Since each of these terms scales
as 1/k(d
2
1 D
2
)
1/2
! 1, we can stop at the first nonvanishing
order of the equation. The even terms of the expansion cor-
respond to physical processes in which excitations ~photons!
are interchanged between atoms A and C ~or B and C). We
have not included in Eq. ~8! the ~dipole-dipole! interactions
between atoms A and B which would give rise to processes
describing photon exchange, because they correspond to a
very small correction of the order of 1/kd!1 to the final
result. Note that we should only consider the case in which
atom C is detected in
u
e
&
, which can only occur if a photon
coming from A or B is absorbed; that is, the first nonvanish-
ing process in our expansion will correspond to the emission
of a photon from atom A or B subsequently absorbed by
atom C. This will give a contribution of the order 1/k
2
(d
2
1 D
2
). Processes in which more than one photon are inter-
changed between atoms A ~or B) and C, or in which ~apart
from the photon absorbed by C) other photons are inter-
changed between atoms A and B would give higher order
contributions, at least of the order of 1/k
4
(d
2
1 D
2
)
2
or
1/k
4
(d
2
1 D
2
)d
2
, respectively.
B. Initial state of atoms A and B: Laser interaction
So far, we have ignored the initial state of atoms A and B.
Let us assume that they are driven by a very short laser pulse
of duration t
las
! G
2 1
,
n
2 1
. The state of atom
a
after the
interaction is
r
˜
a
~
t
0
!
5 e
2 ih
las
a
r
a
~
t
0
!
e
ih
las
a
, ~13!
where
r
a
(t
0
)5
s
00
a
^
r
tp
a
(t
0
), with
r
tp
a
~
t
0
!
}exp
~
2 H
tp
a
/k
B
T
!
~14!
being the initial motional state corresponding to a thermal
distribution at temperature T in the trapping potential, and
e
2 ih
las
a
acts in the subspace span
$
u
0
&
a
,
u
2
&
a
%
as
e
2 ih
las
a
5 cos
~
u
las
!
2 isin
~
u
las
!
@
s
20
a
e
ik
a
•r
a
1 H. c.
#
. ~15!
Here,
u
las
is the rotation angle due to the laser interaction and
k
a
the laser wave vector acting on atom
a
.
According to these equations, the effect of the laser on
each of the atoms is twofold: on one hand, it excites a su-
perposition of the internal states
u
0
&
and
u
2
&
, on the other
hand, it gives a kick to the atom. The coefficient of the su-
perposition
u
las
can be easily varied by changing the laser
intensity or duration.
1028 PRA 59CABRILLO, CIRAC, GARCI
´
A-FERNA
´
NDEZ, AND ZOLLER
C. Detection
We will use the following model for the detection @9#.
The initial state of the atom detector is
u
g
&
. The evolution
time is divided in time steps t
1
,t
2
,...,t
n
,...,ofduration
d
t! G
2 1
,
n
2 1
. After each time interval
d
t, the internal state
of atom C is measured and the state of the whole system is
projected onto
u
g
&
or
u
e
&
depending on the outcome. Let us
consider the case in which the detection at time t
1
,t
2
,...,t
n
has yielded the outcome
u
g
&
, and the detection at time t
n1 1
has yielded
u
e
&
. To lowest order in our expansion, the un-
normalized state of atoms A and B at time t→ ` once we
have made the corresponding projections will be
r
n
5 K lim
t→ `
e
~
L
A
1 L
B
!
~
t2 t
n
!
R
~
t
n
!
, ~16!
where K is a constant that only depends on
g
,
g
˜
, and
d
t, and
R
~
t
!
5 G
~
r
A
2 r
!
s
12
A
r
A
~
t
!
s
21
A
G
~
r
A
2 r
!
†
^
r
B
~
t
!
1 G
~
r
A
2 r
!
s
12
A
r
A
~
t
!
^
r
B
~
t
!
s
21
B
G
~
r
A
2 r
!
†
1 same with A↔B, ~17!
with
r
a
(t)5 e
L
a
t
r
a
(0). This expression along with other
intermediate results are calculated in the Appendix.
Since we do not know a priori at which time the detection
will take place, we have to perform the sum over all the
operators
r
(t
n
). This sum can be transformed into an inte-
gral given the fact that
d
t is smaller than any dynamical
parameter corresponding to the evolution of atoms A and B.
Moreover, we also have to integrate to all positions r corre-
sponding to the detector; that is, to all positions of atom C.
By doing so, we are adding incoherently all the contributions
coming from detections at different points of the detector.
Finally, we have to trace over the motional states of atoms A
and B. The result, properly normalized, will give the aver-
aged density operator provided the detector has performed a
click ~i.e., detected one photon!.
IV. RESULTS
A. Density operator and fidelity
As it is shown in the Appendix, the reduced density op-
erator describing the internal state of atoms A and B in the
case of detection can be written as the sum of two contribu-
tions
r
AB
5
R
1
1R
2
tr
~
R
1
1 R
2
!
, ~18!
where
R
1
5 cos
2
~
u
las
!
sin
2
~
u
las
!
@
M
A,A
u
1,0
&^
1,0
u
1 M
B,B
u
0,1
&^
0,1
u
1 M
A,B
u
1,0
&^
0,1
u
1 M
B,A
u
0,1
&^
1,0
u
#
, ~19a!
R
2
5 sin
4
~
u
las
!
F
G
0
G
M
A,A
u
1,0
&^
1,0
u
1
G
0
G
M
B,B
u
0,1
&^
0,1
u
1
G
1
G
~
M
A,A
1 M
B,B
!
u
1,1
&^
1,1
u
G
. ~19b!
Here, we have defined
M
a
,
b
5
E
S
dr
E
0
`
dtGe
2Gt
tr
tp
$
G
~
r
a
~
t
!
2 r
!
e
ik
a
•r
a
~
0
!
r
tp
A
~
0
!
3
r
tp
B
~
0
!
e
2 ik
b
•r
b
~
0
!
G
~
r
b
~
t
!
2 r
!
†
%
, ~20!
where the first integral is extended to the detector surface,
the trace is taken over the motional states of both atoms, and
r
tp
A,B
(0) denote the initial motional states ~14!. The time-
dependent operators r
a
(t)5 exp(iH
tp
a
t)r
a
exp(2iH
tp
a
t) are de-
fined in the interaction picture with respect to the harmonic
potential.
The interpretation of Eq. ~18! is very simple. The term R
1
comes from processes in which only one atom is excited by
the laser pulses and the subsequent photon emission is cap-
tured at the detector. This can be easily understood if one
writes such a term as
R
1
5
E
S
dr
E
0
`
dtGe
2Gt
tr
tp
$
u
c
~
t
!
&^
c
~
t
!
u
%
, ~21!
with
u
c
~
t
!
&
A,B
5 G
~
r
A
~
t
!
2 r
!
e
ik
A
•r
A
~
0
!
u
1,0
&
A,B
1 G
~
r
B
~
t
!
2 r
!
e
ik
B
•r
B
~
0
!
u
0,1
&
A,B
. ~22!
The state
u
c
(t)
&
is the superposition of two states. The first
one comes from the process in which at time zero the laser
excites atom A, including the corresponding recoil; then, at
time t the atom emits a photon which is detected by the
atomic detector at position r. The factor G
@
r
A
(t)2 r
#
in-
cludes the phase acquired during the propagation from the
position of atom A to the detector as well as the attenuation
of the probability of reaching the detector which is inversely
proportional to the distance traveled ~a solid angle factor!.
The second term has the same contribution but for the pro-
cess in which atom B is excited. Since we do not take into
account the exact time at which the photon is detected, we
have to multiply
u
c
(t)
&^
c
(t)
u
by the probability density that
the photon is emitted at time t, proportional to e
2 Gt
, and
integrate over time. On the other hand, since we do not know
the point at the detector where the photon arrives, we have
also to integrate the resulting expression over the detector
surface, resulting in Eq. ~21!. Notice that retardation effects
are not included in our formulation. They can be simply
incorporated to this formula by changing t→t2
u
r
A,B
(t)
2 r
u
/c. Since here r
A,B
and r vary over very small distances
~size of the atomic wave packets and detector size, respec-
tively!, the result will not be affected by retardation effects.
On the other hand, expanding the term R
2
in a similar way as
Eq. ~21! one can readily see that it comes from the process in
which both atoms are excited by laser pulses, one photon
emission is detected and the other not. The terms propor-
tional to G
0
correspond to the case in which the undetected
photon is emitted in the transition
u
2
&
→
u
0
&
, whereas the
ones with G
1
correspond to the
u
2
&
→
u
1
&
transition.
With these expressions, we can easily calculate the fidel-
ity ~2! as
PRA 59 1029CREATION OF ENTANGLED STATES OF DISTANT . . .