Stationary two-atom entanglement induced by nonclassical two

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INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS
J. Opt. B: Quantum Semiclass. Opt. 6 (2004) S610–S617
PII: S1464-4266(04)72641-5
Stationary two-atom entanglement
induced by nonclassical two-photon
correlations
R Tanaś1 and Z Ficek2
1
Nonlinear Optics Division, Institute of Physics, Adam Mickiewicz University,
Poznań, Poland
2
Department of Physics, School of Physical Sciences, The University of Queensland,
Brisbane, QLD 4072, Australia
E-mail: [email protected]
Received 26 September 2003, accepted for publication 6 January 2004
Published 28 May 2004
Online at stacks.iop.org/JOptB/6/S610
DOI: 10.1088/1464-4266/6/6/022
Abstract
A system of two two-level atoms interacting with a squeezed vacuum field
can exhibit stationary entanglement associated with nonclassical two-photon
correlations characteristic of the squeezed vacuum field. The amount of
entanglement present in the system is quantified by the well known measure
of entanglement called concurrence. We find analytical formulae describing
the concurrence for two identical and nonidentical atoms and show that it is
possible to obtain a large degree of steady-state entanglement in the system.
Necessary conditions for the entanglement are nonclassical two-photon
correlations and nonzero collective decay. It is shown that nonidentical
atoms are a better source of stationary entanglement than identical atoms.
We discuss the optimal physical conditions for creating entanglement in the
system; in particular, it is shown that there is an optimal and rather small
value of the mean photon number required for creating entanglement.
Keywords: entanglement, squeezing, superpositions
1. Introduction
Entanglement between separate quantum systems is one of
the key problems in quantum mechanics. A number of
interesting concepts and methods for creating entanglement
have been proposed involving trapped and cooled ions or
neutral atoms [1–8]. Of particular interest is generation of
entangled states in two-atom systems, since they can represent
two qubits, the building blocks of the quantum gates that
are essential to implement quantum protocols in quantum
information processing. It has been shown that entangled states
in a two-atom system can be created by a continuous driving of
the atoms with a coherent or chaotic thermal field [5, 9–12], or
by a pulse excitation followed by a continuous observation of
radiative decay [13–15]. Moreover, the effect of spontaneous
emission on initially prepared entangled state has also been
discussed [16–19]. These studies, however, have been limited
to the small sample (Dicke) model [20] or the situation
1464-4266/04/060610+08$30.00 © 2004 IOP Publishing Ltd
involving noninteracting atoms strongly coupled to a cavity
mode. The difficulty of the Dicke model is that it does not
include the dipole–dipole interaction among the atoms and
does not correspond to realistic experimental situations of
atoms located (trapped) at different positions. In fact, the
model corresponds to a very specific geometrical configuration
of the atoms confined to a volume much smaller compared with
the atomic resonant wavelength (the small-sample model). The
present atom trapping and cooling techniques can trap two
atoms at distances of order of a resonant wavelength [21–23],
which makes questionable the applicability of the Dicke model
to physical systems.
Recently, we have shown [24] that spontaneous emission
from two spatially separated atoms can lead to a transient
entanglement of initially unentangled atoms. This result
contrasts with the Dicke model where spontaneous emission
cannot produce entanglement from initially unentangled
atoms [10, 18]. We have also found [25] analytical results for
Printed in the UK
S610
Stationary two-atom entanglement
two measures of entanglement and the relation between them
for the two-atom system radiating by spontaneous emission for
quite broad range of initial conditions.
In this paper we study the creation of a stationary
entanglement in a system of two identical as well as
nonidentical two-level atoms separated by an arbitrary distance
r12 and interacting with a squeezed vacuum. The squeezed
vacuum appears here as a source of nonclassical twophoton coherences, essential for the creation of the stationary
entanglement. We use the master equation to describe the
evolution of the system and find the steady-state solutions
for the atomic variables. We present analytical results for
concurrence which is well known and calculable measure of
entanglement. We find a surprising result that nonidentical
atoms with significantly different transition frequencies can
exhibit a larger entanglement than identical atoms. Under
some conditions, the nonidentical atoms can be maximally
entangled with the value of the concurrence equal to unity.
2. Master equation
We consider a system of two two-level atoms at fixed positions
r1 and r2 and coupled to the radiation field, whose the modes
are in a squeezed vacuum state. Each atom has energy levels
|gi and |ei (i = 1, 2) such that E ei − E gi = h̄ωi , transition
dipole moment µi , which we assume equal for both atoms.
We analyse separately the dynamics of identical and
nonidentical atoms. In the case of nonidentical atoms, we
assume different transition frequencies ω1 and ω2 such that
= (ω2 − ω1 )/2 ω0 = (ω1 + ω2 )/2, so that the rotatingwave approximation can be applied to calculate the dynamics
of the system.
The system can be described by the reduced density
operator ρ which, in a Schrödinger picture, satisfies the master
equation [26]
2
∂ρ
1 i j (1 + N ) ρ Si+ S −j + Si+ S −j ρ − 2S −j ρ Si+
=−
∂t
2 i, j =1
−
+
+
1
2
1
2
1
2
−i
2
i j N ρ Si− S +j + Si− S +j ρ − 2S +j ρ Si−
+
i j M ρ Si+ S +j + Si+ S +j ρ − 2S +j ρ Si e−2iωs t
i, j =1
2
i j M ∗ ρ Si− S −j + Si− S −j ρ − 2S −j ρ Si− e2iωs t
i, j =1
2
2
ωi Siz , ρ − i
i j Si+ S −j , ρ .
i=1
i= j
(1)
Here, Si+ and Si− are the raising and lowering operators,
respectively, of the i th atom, N and M = |M| exp(iφs )
characterize squeezing such that |M|2 N (N + 1), where
the equality holds for a minimum-uncertainty squeezed state,
φs is the squeezing phase and ωs is the carrier frequency of the
squeezed vacuum.
The parameters i j , which appear in equation (1), are
spontaneous emission rates, such that
ii ≡ =
ω03 |µ|2
,
3πε0 h̄c3
(i = 1, 2)
|1 = |g1 ⊗ |g2 = |g,
|2 = |e1 ⊗ |e2 = |e,
|3 = |g1 ⊗ |e2 ,
(5)
|4 = |e1 ⊗ |g2 ,
i, j =1
2
is the spontaneous emission rate of the i th atom, assumed to
be equal for both atoms, and
sin(k0 r12 )
3√ 12 = 21 =
1 − (µ̂ · r̂12 )2
2
k0 r12
cos(k0 r12 ) sin(k0 r12 )
+ [1 − 3(µ̂ · r̂12 )2 ]
−
,
(3)
(k0 r12 )2
(k0 r12 )3
are collective spontaneous emission rates arising from the
coupling between the atoms through the vacuum field [36–38],
and
cos(k0 r12 )
3
12 = − 1 − (µ̂ · r̂12 )2
4
k0r12
2 sin(k0 r12 ) cos(k0 r12 )
(4)
+ 1 − 3 µ̂ · r̂i j
+
(k0 r12 )2
(k0 r12 )3
represents the vacuum induced coherent (dipole–dipole)
interaction between the atoms.
In expressions (3) and (4), µ̂ and r̂12 are unit vectors
along the atomic transition dipole moments and the vector
r12 = r2 − r1 , respectively, and k0 = ω0 /c. Later on, we will
assume that the atomic dipole moments µ are perpendicular to
the vector r12 joining the two atoms.
The collective parameters 12 and 12 , which arise from
the mutual interaction between the atoms, significantly modify
the master equation of a two-atom system. The parameter
12 introduces a coupling between the atoms through the
vacuum field such that the spontaneous emission from one of
the atoms influences the spontaneous emission from the other.
The dipole–dipole interaction term 12 introduces a coherent
coupling between the atoms. Owing to the dipole–dipole
interaction, the population is coherently transferred back and
forth from one atom to the other.
The two-atom system can be described in the basis of
product states of the individual atoms
(2)
where |gi and |ei (for i = 1, 2) are the ground and excited
states of the individual atoms. In this basis, the two-atom
system behaves as a single four-level system whose density
matrix can be written as a 4 × 4 matrix. Due to the presence
of the dipole–dipole interaction 12 it is often convenient to
introduce collective atomic states that are eigenstates of the
system of two identical atoms including the dipole–dipole
interaction [20]. They are symmetric and antisymmetric
superpositions of the product atomic states |3 and |4, given
by
1
|s = √ (|3 + |4),
2
(6)
1
|a = − √ (|3 − |4),
2
and the states |g and |e remain unchanged.
On introducing the collective states (6) and using the
master equation (1) we are able to write down equations of
motion for the matrix elements of the density matrix for two
atoms in a squeezed vacuum. We treat separately two cases of
identical and nonidentical atoms.
S611
R Tanaś and Z Ficek
2.1. Identical atoms
For identical atoms separated by an arbitrary distance r12
and interacting with a squeezed vacuum field of the carrier
frequency ωs = ω0 , we obtain the following set of coupled
equations of motion [26]:
ρ̇ee = −2 (N + 1) ρee + N ( + 12 ) ρss
+ ( − 12 ) ρaa + 12 |M|ρu ,
ρ̇ss = ( + 12 ) N − (3N + 1) ρss
− Nρaa + ρee − |M|ρu ,
(7)
ρ̇aa = ( − 12 ) N − (3N + 1) ρaa
− Nρss + ρee + |M|ρu ,
ρ̇u = 212 |M| − (2N + 1) ρu − 2|M|
× ( + 212 ) ρss − ( − 212 ) ρaa ,
where ρu = ρeg exp(−iφs ) + ρge exp(iφs ). It is seen from
equation (7) that the evolution of the populations depends on
the two-photon coherences ρeg and ρge , which can modify
population distribution between the collective states. The
coherences can also create superposition (entangled) states
involving only the ground |g and the upper |e states. The
evolution of the populations depends on 12 , but is completely
independent of the dipole–dipole interaction 12 .
The steady-state solutions of equations (7) depend on
whether 12 = or 12 = . For two atoms separated by
an arbitrary distance r12 , 12 = , and then the steady-state
solutions of equations (7) are
2
N 2 (2N + 1)2 − 4|M|2 + |M|2 γ12
,
ρee =
2
(2N + 1)4 − 4|M|2 (2N + 1)2 − γ12
N (N + 1) (2N + 1)2 − 4|M|2 + |M|2 γ12 (γ12 − 2)
ρss =
,
2
(2N + 1)4 − 4|M|2 (2N + 1)2 − γ12
N (N + 1) (2N + 1)2 − 4|M|2 + |M|2 γ12 (γ12 + 2)
ρaa =
,
2
(2N + 1)4 − 4|M|2 (2N + 1)2 − γ12
2 (2N + 1) |M|γ12
,
ρu =
2
(2N + 1)4 − 4|M|2 (2N + 1)2 − γ12
(8)
where γ12 = 12 / is the dimensionless collective damping
parameter. This result shows that all the collective states
are populated in the steady state even for small interatomic
separations (γ12 ≈ 1). For large interatomic separations
γ12 ≈ 0, and then the symmetric and antisymmetric states
are equally populated. When the interatomic separation
decreases, the population of the state |a increases, whereas
the population of the state |s decreases and ρss = 0 for
very small interatomic separations. This effect results from
the enhanced ( + 12 ) damping rate of the symmetric state,
as seen from equation (7). The two-photon coherences,
represented by ρu , affect the population distribution only
when both |M| and γ12 are nonzero. The coherences are
crucial for getting entanglement in the system. Of particular
interest are population√distributions for maximally squeezed
fields with |M| =
N (N + 1). In this case, the factor
(2N + 1)2 − 4|M|2 = 1, and then the solutions (8) take a
S612
very simple form
ρee =
2
N 2 + N (N + 1)γ12
,
2
1 + 4N (N + 1)(1 + γ12
)
ρss =
N (N + 1)(1 − γ12 )2
,
2
1 + 4N (N + 1)(1 + γ12
)
N (N + 1)(1 + γ12 )2
ρaa =
,
2
1 + 4N (N + 1)(1 + γ12
)
√
2 N (N + 1)(2N + 1)γ12
.
ρu =
2
1 + 4N (N + 1)(1 + γ12
)
(9)
The solutions (8) and (9) will be used for calculation of the
degree of entanglement present in the system. For further
reference it is important to note that the sum of the populations
2
ρss +ρaa tends to 0.5 as the quantity N (N +1)(1+γ12
) becomes
much greater than one, which means that for large values of
the mean number of photons N one-half of the population goes
eventually to the states |s and |a.
2.2. Nonidentical atoms
The population distribution is quite different when the atoms
are nonidentical with = (ω2 − ω1 )/2 = 0. As before
for the identical atoms, we use the master equation (1)
and find four coupled differential equations for the density
matrix elements with time-dependent coefficients oscillating
at frequencies exp(±it) and exp[±2i(ωs − ω0 )t + φs ]. If we
tune the squeezed vacuum field to the middle of the frequency
difference between the atomic frequencies, i.e., ωs = (ω1 +
ω2 )/2, the terms proportional to exp[±2i(ωs − ω0 )t + φs ]
become stationary in time. None of the other time-dependent
components is resonant with the frequency of the squeezed
vacuum field. Consequently, for , the time-dependent
components oscillate rapidly in time and average to zero over
long times. Therefore, we can make a secular approximation
in which we ignore the rapidly oscillating terms and obtain the
following equations of motion:
ρ̇ee = −2 (N + 1) ρee + N (ρss + ρaa ) + 12 |M|ρu ,
ρ̇ss = [N − (3N + 1) ρss − Nρaa + ρee ] − 12 |M|ρu ,
ρ̇aa = [N − (3N + 1) ρaa − Nρss + ρee ] − 12 |M|ρu ,
ρ̇u = 212 |M| − (2N + 1) ρu − 412 |M| (ρss + ρaa ) .
(10)
The steady-state solutions of equations (10) are
1 (2N − 1)
1
,
ρee =
+
2
4
2N + 1
(2N + 1)2 − 4|M|2 γ12
1
1
(11)
,
ρss = ρaa =
1− 2
4
(2N + 1)2 − 4|M|2 γ12
ρu =
2|M|γ12
.
2
(2N + 1) (2N + 1)2 − 4|M|2 γ12
Equations (11) are quite different from equations (8) and
show that in the case of nonidentical atoms the symmetric
and antisymmetric states are equally populated. This fact
will be crucial in the entanglement creation in the system
Stationary two-atom entanglement
and results from the equal damping rates of the symmetric
and antisymmetric states, as seen from equation
√ (10). For
maximally squeezed vacuum with |M| = N (N + 1) the
solutions (11) simplify to
1 2N − 1
1
,
ρee =
+
2
4 2N + 1 1 + 4N (N + 1)(1 − γ12
)
2
N (N + 1)(1 − γ12
)
,
2
1 + 4N (N + 1)(1 − γ12
)
√
2 N (N + 1)γ12
ρu =
.
2
(2N + 1)[1 + 4N (N + 1)(1 − γ12
)]
ρss = ρaa =
(12)
From equations (12) it is evident that for the Dicke model,
for which γ12 = 1, the populations of the symmetric and
asymmetric states are both zero and ρu tends to unity for
large N . In real situations the separation of the atoms is
nonzero, and we always have γ12 < 1, which means that for
2
) 1 the populations ρss and ρaa both
N (N + 1)(1 − γ12
approach the value 0.25, i.e., ρss + ρaa ≈ 0.5. That is, for
sufficiently large N one-half of the population is transferred
to the block spanned by the symmetric and antisymmetric
states, similarly to the identical atoms. There is, however,
one essential difference between the identical and nonidentical
atoms, which is the scale for the saturation—much slower for
nonidentical atoms. For γ12 very close to unity, very large
2
) 1, and
intensities N are required to have N (N + 1)(1 − γ12
consequently ρss + ρaa ≈ 0.5. Thus, for small N and γ12 = 1,
the populations ρss and ρaa are very small, ρss = ρaa ≈ 0.
This fact will have an important effect on the entanglement
creation in the system of nonidentical atoms.
σx is the Pauli matrix, and ρ is the density matrix representing
the quantum state. The range of concurrence is from zero to
unity. For unentangled atoms C = 0 whereas C = 1 for the
maximally entangled atoms.
In the basis (5) of the product atomic states the density
matrix for two atoms in the squeezed vacuum has in the steady
state the following block form:


0
ρ11 ρ12 0
ρ22 0
0 
ρ
(16)
ρ =  21

0
0 ρ33 ρ34
0
0 ρ43 ρ44
with the condition Tr ρ = 1. The matrix ρ̃, required for
calculation of the concurrence, has the form


0
ρ22 ρ12 0
ρ11 0
0 
ρ
(17)
ρ̃ =  21

0
0 ρ44 ρ34
0
0 ρ43 ρ33
and the square roots of the eigenvalues of the matrix R given
by (14) are the following:
√
√
{ λi } =
ρ11 ρ22 − |ρ12 |, ρ11 ρ22 + |ρ12 |,
√
√
ρ33 ρ44 − |ρ34 |, ρ33 ρ44 + |ρ34 | .
(18)
Depending on the particular values of the matrix elements
there are two possibilities for the largest eigenvalue, either
the second term or the fourth term in (18). The concurrence is
thus given by
C = max{0, C1 , C2 },
(19)
with
C2 = 2(|ρ34 | −
3. Steady-state entanglement
To assess how much entanglement is stored in a given
quantum system it is essential to have appropriate measures
of entanglement. A number of measures have been proposed,
which include entanglement of formation [27], entanglement
of distillation [28], relative entropy of entanglement [29] and
negativity [30–33]. For pure states, the Bell states represent
maximally entangled states, but for mixed states represented
by a density matrix there are some difficulties with ordering the
states according to various entanglement measures; different
entanglement measures can give different orderings of pairs
of mixed states and there is a problem in the definition of the
maximally entangled mixed state [34, 35].
Here we use the concurrence to describe the amount of
entanglement created in a two-atom system by the interaction
with the squeezed vacuum. The concurrence introduced by
Wootters [27] is defined as
C = max 0, λ1 − λ2 − λ3 − λ4 ,
(13)
√
√
ρ33 ρ44 ),
ρ11 ρ22 ),
(20)
and we have two alternative expressions for the concurrence
depending on which of them is positive.
In terms of the collective atomic states |g, |e, |s and |a,
the expressions for the concurrence (20) take the form
C1 = 2|ρge | − (ρss + ρss )2 − (ρsa + ρas )2 ,
(21)
√
C2 = |ρss − ρaa + ρsa − ρas | − 2 ρgg ρee .
Having the concurrence expressed in terms of the density
matrix elements, we can apply the steady-state solutions (8)
or (11) and obtain analytical results for the stationary
concurrence. We will discuss the results separately for
identical and nonidentical atoms.
3.1. Identical atoms
In the case of identical atoms the steady-state solutions (8) are
valid, and we find
C1 = |ρu | − (ρss + ρaa )
where {λi } are the the eigenvalues of the matrix
R = ρ ρ̃
C1 = 2(|ρ12 | −
(14)
=2
2
(2N + 1)|M|γ12 − |M|2 γ12
− N(N + 1) (2N + 1)2 − 4|M|2
.
2
(2N + 1)4 − 4|M|2 (2N + 1)2 − γ12
(22)
with ρ̃ given by
ρ̃ = σx ⊗ σx ρ ∗ σx ⊗ σx ;
(15)
It turns out that C2 is always negative, so the only contribution
to the concurrence comes from C1 , and then C = max(0, C1 ).
S613
R Tanaś and Z Ficek
It is clear from (22) that the perfect entanglement in the
system, i.e., the value of concurrence equal to unity, can be
achieved when |ρu | = 1 and ρss + ρaa = 0. This is generally
impossible for identical atoms because there is always some
population stored in the states |s and |a. However, it is
possible to obtain some degree of entanglement in the system
for appropriately chosen values of r12 , N and |M|. The
concurrence, measuring the degree of entanglement, depends
on the interatomic distance through the collective damping
parameter γ12 , and the degree of the two-photon coherences
|M|. It is easy to show from equation (22) that there is
no entanglement possible for |M| N , i.e., for classically
correlated fields. For a √
quantum squeezed field with maximum
correlations |M| =
N (N + 1), the concurrence can be
written as
√
2
(2N + 1)γ12 − N (N + 1)(1 + γ12
)
.
C1 = 2 N (N + 1)
2
1 + 4N (N + 1)(1 + γ12
)
(23)
In this case, C1 can be positive. To show this, we plot in figure 1
the concurrence for two identical atoms in the maximally
squeezed vacuum as a function of the interatomic distance
r12 and the mean number of photons N . It is evident from
figure 1 that there is a range of values of r12 /λ for which
C is positive. The maximum of concurrence is obtained for
r12 ≈ 0 when the atoms are very close to each other. The values
of concurrence decrease as the interatomic distance increases
and reduces to zero at r12 ≈ λ/2, but we can observe revival of
concurrence for longer interatomic distances, although the next
maximum is much weaker. It is interesting that the maximum
of concurrence appears for not very high values of the mean
number of photons N < 0.1. It is easy to check, from (23), that
for γ12 = 1 the number of photons Nmax
for which C1 reaches
√
its maximum is given by Nmax =
(1 + 2)/2 − 1 /2 ≈
0.049. The fact that the maximum of concurrence appears for
moderate values of the mean photon numbers can be important
from the experimental point of view as the present sources
of squeezed fields can produce quantum squeezed fields of
intensities N < 1.
The steady-state entanglement and its presence for only
quantum squeezed fields of small intensities N is associated
with nonclassical two-photon correlations characteristic of the
squeezed vacuum field. To show this, we introduce a parameter
|M|
,
N
TC =
(24)
which characterizes two-photon correlations normalized to the
intensity of the squeezed field. For classical fields, |M| N ,
and then√TC < 1 for all N . For a quantum squeezed field with
|M| = N (N + 1), the parameter becomes
TC =
1+
1
,
N
(25)
which is always greater than one. The result is a strong twophoton correlation, which is greatest for N < 1. Thus, the
nonclassical two-photon correlations are significant for N < 1
and lead to a large entanglement in the system.
S614
Figure 1. Concurrence C for two identical atoms as a function of the
interatomic
√ distance r12 /λ and the mean number of photons N for
|M| = N(N + 1).
3.2. Nonidentical atoms
In the case of nonidentical atoms with = 0, the
steady-state values for the density matrix elements are
given in equation (11). As above for identical atoms, the
concurrence (19) can be expressed by formula (20) which,
with solutions (11), leads to
2|M||γ12 |
C1 = |ρu | − (ρss + ρaa ) =
2
(2N + 1) (2N + 1)2 − 4|M|2 γ12
1
1
−
.
(26)
1−
2
2
(2N + 1)2 − 4|M|2 γ12
Similarly to the case of identical atoms, C2 is always negative.
Moreover, C1 is always negative for |M| N independent of
γ12 . Thus, entanglement is possible only for quantum
√ squeezed
fields, which for the maximum correlations |M| = N (N + 1)
gives
√
2
γ12 − (2N + 1) N (N + 1)(1 − γ12
)
.
C1 = 2 N (N + 1)
2
(2N + 1)[1 + 4N (N + 1)(1 − γ12 )]
(27)
Equation (27) is significantly different from that for identical
atoms, equation (23). For example, if the atoms are close
together, γ12 ≈ 1, and then equation (27) reduces to
√
2 N (N + 1)
C1 =
.
(28)
2N + 1
In this limit the concurrence is always positive, increases with
N and approaches unity at a large N . This is in contrast to the
case of identical atoms where values of concurrence are below
0.25 even for γ12 = 1 and approach zero for large N . This
behaviour can be easily explained by the fact that in the case
of nonidentical atoms and γ12 = 1 the population stored in the
symmetric and antisymmetric states, ρss + ρaa , is equal to zero.
At the same time, ρu tends to unity as N increases, giving the
maximum concurrence C1 = 1.
Stationary two-atom entanglement
1
Concurrence
0.8
0.6
0.4
0.2
0
0
In real situations, we have γ12 < 1, and for large N the
concurrence C1 , given by (27), goes to zero, similarly to the
case of identical atoms. The maximum of C1 for nonidentical
atoms, nonetheless, is much more pronounced than that for
identical atoms. The real scale of large number of photons is
2
) 1 rather than by
in this case given by N (N + 1)(1 − γ12
2
) 1 as is the case for identical atoms.
N (N + 1)(1 + γ12
The dependence of the concurrence C = max(0, C1 ), with
C1 given by formula (27), on the interatomic distance r12 and
the mean number of photons of the squeezed field N is shown
in figure 2. The dependence on the interatomic distance is
similar to that seen for identical atoms with the revival of
concurrence for r12 ≈ 3λ/4 and not too large N . For γ12 = 1
the dependence on N is also similar to that for identical atoms,
except that the maximum values of concurrence for given N are
much higher than for identical atoms. Comparing the solutions
for identical and nonidentical atoms indicates that one reason
for higher values of concurrence for nonidentical as compared
to identical atoms interacting with the squeezed vacuum is the
fact that for nonidentical atoms less population remains in the
lower block of the density matrix (16) represented by states
|3 and |4 (or |a and |s) as the atoms become different, i.e.,
= (ω2 − ω1 )/2 becomes large. To confirm this fact we
plot in figure 3 the concurrence as well as the populations,
ρss + ρaa and ρgg + ρee , which are stored in the two blocks of
the density matrix (16), as a function of . Since for identical
atoms, according to (9), a considerable amount of population
remains in the antisymmetric state in contrast to solutions (12)
for nonidentical atoms, it is clear from figure 3 that as the
transition frequencies of the two atoms become more and more
different the population of the antisymmetric state goes down,
reducing the total population ρss + ρaa of the lower block and
increasing the total population ρgg + ρee of the upper block
of (16), which means higher values of concurrence.
Another physical explanation of the origin of the better
entanglement for nonidentical atoms is provided by the
observation that the stationary state of nonidentical atoms, for
small N for which concurrence is maximal, is close to a pure
state, whilst the stationary state of identical atoms is already far
from a pure state. This is illustrated in figure 4, where we plot
2
2
2
2
+ρee
+ρss
+ρaa
+|ρu |2 /2
the purity measure P = Tr(ρ 2 ) = ρgg
10
20
∆/Γ
30
40
50
Figure 3. Concurrence C (solid curve), populations ρss + ρaa
(dashed curve) and populations ρgg + ρee (dashed–dotted curve) for
two atoms
√ as a function of for r12 /λ = 0.05 and N = 0.1 for
|M| = N(N + 1).
1
0.8
Purity
Figure 2. Concurrence C for two nonidentical atoms as a function of
the interatomic
√ distance r12 /λ and the mean number of photons N
for |M| = N(N + 1).
0.6
0.4
0
0.5
1
1.5
2
N
Figure 4. Purity P = Tr(ρ 2 ) as a function
√ of the mean number of
photons N for r12 /λ = 0.05 and |M| = N(N + 1): identical
atoms (solid curve); nonidentical atoms (dashed curve).
as a function of N for the steady state of two identical as well
as nonidentical atoms. It is seen that in both cases the purity
decreases as the number of photons increases, but in the case
of identical atoms the purity goes down much faster.
It should be emphasized, however, that the main source
of entanglement in the system is the nonclassical two-photon
correlations that create two-photon coherences between the
states |g and |e. The two-photon coherences are nonzero
only when the squeezing parameter |M| is nonzero. In fact,
to have entanglement in the system the squeezed field must
represent quantum correlations with |M| > N . There is,
moreover, one more necessary condition to have nonzero ρu ,
which is a nonzero value of the collective damping parameter
γ12 . The two-photon coherences cause the system to decay into
entangled states involving the ground state |g and the upper
state |e without any involvement of the entangled states |s and
|a. Unfortunately, the spontaneous emission from the state |e
redistributes some of the atomic population over the states |s
and |a, limiting in this way the degree of entanglement.
S615
R Tanaś and Z Ficek
1
Fidelity
0.9
0.8
0.7
0.6
0.5
0
0.5
1
1.5
2
N
Figure 5. Fidelity F+ as a function
√ of the mean number of photons
N for r12 /λ = 0.05 and |M| = N(N + 1): identical atoms (solid
curve); nonidentical atoms (dashed curve).
Since the two-photon coherences create superposition
(entangled) states involving the states |g and |e, one can ask
how close the entangled stationary state of the system is to one
of the maximally entangled Bell states
1
|+ = √ (|g + |e) ,
2
1
|− = − √ (|g − |e) .
2
To answer this question we calculate the fidelities
F+ = + |ρ|+ = 12 ρgg + ρee + ρu ,
F− = − |ρ|− = 12 ρgg + ρee − ρu .
(29)
(30)
The fidelities depend on whether ρu is positive or negative.
For small interatomic distances γ12 is positive, and then the
coherence ρu is positive. Hence, the fidelity F+ becomes large
while the fidelity F− is small. Thus, we can conclude that
the stationary state of the system is close to the maximally
entangled Bell state |+ . In figure 5 we plot the fidelity
F+ as a function of N for identical as well as nonidentical
atoms. Comparing the dependence on N of the fidelity F+ and
concurrence C gives us clear evidence that the entanglement
in the system can be related to the Bell state |+ . The state of
the system is of course mixed, but it is closer to the pure Bell
state |+ for nonidentical atoms than for identical atoms. The
entanglement created by the two-photon correlations present
in the squeezed light is limited by the population stored in
the other states. As the number of photons increases, for
γ12 < 1, more and more population goes to the states |s and
|a, and eventually entanglement disappears for both identical
and nonidentical atoms. There are optimal values of the mean
number of photons for which the highest possible stationary
entanglement can be obtained.
4. Conclusions
In this paper, we have studied analytically the entanglement
creation in a system of two atoms interacting with a squeezed
S616
vacuum field. We have demonstrated that nonclassical twophoton correlations characteristic of the squeezed field can
create a large steady-state entanglement in the system.
In our approach we have used a master equation to describe
a system of two two-level atoms subjected to a squeezed
vacuum field. The two atoms are coupled to each other via the
vacuum field which leads to collective damping and collective
dipole–dipole type interaction between the atoms. We have
assumed the two atoms to be separated by a distance r12 , so
the collective parameters depend explicitly on this distance.
Steady-state solutions for the atomic density matrix have been
found for two cases: (i) identical atoms; (ii) nonidentical
atoms.
We have derived analytical expressions for concurrence
which is used to quantify the amount of entanglement created
in the system. Our results show that the necessary condition
for entanglement is nonclassical two-photon correlations of
the squeezed field. The entanglement also depends on the
interatomic separation and the mean number of photons of the
squeezed vacuum. The necessary condition for entanglement
is quantum correlations of the squeezed field. There is no
entanglement for a classically correlated field. We have
found that the degree of entanglement created in the system
is a result of competition between the coherent process of
transferring two-photon coherences from the squeezed vacuum
to the atomic system and the incoherent process of spontaneous
emission redistributing atomic population over the states not
involved in the former process. In particular, we have shown
that there is an optimum value of the mean number of photons
for which the concurrence takes its maximum, and this happens
for a small number of photons. This is important from the
point of view of practical applications. Moreover, we have
also found that the degree of entanglement obtainable in this
way is much higher when the two atoms are not identical. We
have discussed in detail physical reasons for such behaviour of
the two-atom system.
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