Two-photon dipole-dipole blockade

Abstract

We describe a mechanism for a controlled generation of a pure Bell state with
correlated atoms that involve two or zero excitations. The mechanism inhibits
transitions into singly excited collective states of a two-atom system by
shifting them from their unperturbed energies. The shift is accomplished by the
dipole-dipole interaction between the atoms. The creation of the Bell state is
found to be dependent on the relaxation of the atomic excitation. When the
relaxation is not present or can be ignored, the state of the system evolves
harmonically between a separable to the maximally entangled state. We follow
the temporal evolution of the state and find that the concurrence can be
different from zero only in the presence of the dipole-dipole interaction.
Furthermore, in the limit of a large dipole-dipole interaction, the concurrence
reduces to that predicted for an X-state of the system. A general inequality is
found which shows that the concurrence of an X-state system is a lower bound
for the concurrence of the two-atom system. With the relaxation present, the
general state of the system is a mixed state that under a strong dipole-dipole
interaction reduces the system to an X-state form. We find that mixed states
admit of lower level of entanglement, and the entanglement may occur over a
finite range of time. A simple analytical expression is obtained for the
steady-state concurrence which shows that there is a threshold value for the
dipole-dipole interaction relative to the Rabi frequency of the driving field
above which the atoms can be entangled over the entire time of the evolution.

Full text

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PHYSICAL REVIEW A 84, 013831 (2011)
Generating two-photon entangled states in a driven two-atom system
Khulud Almutairi,1,2Ryszard Tana´ s,3,*and Zbigniew Ficek4,†
1Institute for Quantum Information Science, University of Calgary, Calgary, Alberta, T2N 1N4 Canada
2Department of Physics, King Saud University, Riyadh 11451, Saudi Arabia
3Nonlinear Optics Division, Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna´ n, Poland
4The National Centre for Mathematics and Physics, KACST, P. O. Box 6086, Riyadh 11442, Saudi Arabia
(Received 9 March 2011; published 27 July 2011)
WedescribeamechanismforacontrolledgenerationofapureBellstatewithcorrelatedatomsthatinvolvetwo
or zero excitations. The mechanism inhibits transitions into singly excited collective states of a two-atom system
by shifting them from their unperturbed energies. The shift is accomplished by the dipole-dipole interaction
between the atoms. The creation of the Bell state is found to be dependent on the relaxation of the atomic
excitation. When the relaxation is not present or can be ignored, the state of the system evolves harmonically
between a separable to the maximally entangled state. We follow the temporal evolution of the state and find that
the concurrence can be different from zero only in the presence of the dipole-dipole interaction. Furthermore,
in the limit of a large dipole-dipole interaction, the concurrence reduces to that predicted for an X state of the
system. A general inequality is found which shows that the concurrence of an X-state system is a lower bound
for the concurrence of the two-atom system. With the relaxation present, the general state of the system is a
mixed state that under a strong dipole-dipole interaction reduces the system to an X-state form. We find that
mixed states admit of lower level of entanglement, and the entanglement may occur over a finite range of time.
A simple analytical expression is obtained for the steady-state concurrence which shows that there is a threshold
value for the dipole-dipole interaction relative to the Rabi frequency of the driving field above which the atoms
can be entangled over the entire time of the evolution.
DOI: 10.1103/PhysRevA.84.013831PACS number(s): 42.50.Dv, 03.65.Yz, 03.67.Bg, 42.50.Hz
I. INTRODUCTION
It has been known for many years that two types of maxi-
mally entangled Bell states can be generated in a system com-
posedoftwotwo-levelatoms;theso-calledspinanticorrelated
statesthatarelinearsuperpositionsofsingleexcitationproduct
states and spin-correlated states that are linear superpositions
ofdoubleandzeroexcitationproductstates.Recenttheoretical
andexperimentalworkhasdemonstratedthatsingleexcitation
Bell states can be generated in a system of two Rydberg
atoms by the dipole-dipole blockade mechanism [1]. The
mechanism is often referred to as a Rydberg blockade and
bears on the elimination of the simultaneous excitation of the
atoms by shifting the double-excitation states of the system,
an effect attributed to the long-range dipole-dipole interaction
characteristics of Rydberg atoms [2–10]. Rydberg atoms are
highly excited atoms which have large sizes and therefore can
have huge dipole moments, proportional to n2, where n is the
principal quantum number [11]. Because the dipole moments
of Rydberg atoms are so large, the atoms can strongly interact
with each other even at large distances.
The physical origin of the dipole-dipole blockade is
explained clearly in terms of collective states of the two-atom
system. These states provide a more natural basis for interact-
ing atoms [12–15]. The effects of the dipole-dipole interaction
include, in general, the creation of maximally entangled single
excitationstatesandtheshiftofthestatesfromthesingle-atom
energy [16–18]. When the double-excitation states are shifted
formtheir resonances, the two-photon excitation of the system
*tanas@kielich.amu.edu.pl
†zficek@kacst.edu.sa
by a resonant laser field becomes suppressed. The two-photon
excitation is suppressed without destroying the one-photon
excitation. The blockade effect is a simple process for creation
of single-excitation entangled states, and the preparation of
the entangled states via the dipole blockade has recently been
demonstrated experimentally [19–22]. The presence of only
single excitations of the system is manifested by the photon
antibunching effect, which signifies that at given time only
a single photon is emitted by the system [23–26]. In the
experiment of Ga¨ etan et al. [19], a modification of the Rabi
frequency of the driving laser field by√2 has been observed
that is recognized as a signature of not only the dipole-dipole
blockade effect but also of the creation of a single excitation
entangled state. In an earlier dipole blockade experiment,
Heidemann et al. [20] have observed the characteristic
scaling of the Rabi oscillations between the ground state
and the single excitation multiatom entangled state of a laser
driven mesoscopic ensemble of N ultracold Rydberg atoms.
In the experiments of Zhang et al. [21] and Wilk et al.
[22], an entanglement between two Rydberg atoms has been
demonstrated by measuring the state fidelity of F = 0.71 and
F = 0.75, respectively, that are well above the threshold of
F = 0.5requiredforquantumentanglement.Theexperiments
clearly demonstrate that the dipole-dipole blockade can deter-
ministically generate entangled states in an atomic system.
The notion of blockade of multiphoton excitations is not
restricted to the dipole-dipole blockade but has been extended
to photon blockade for the transport of light through an
optical system [27–29], and Coulomb blockade of resonant
transportofelectronsthroughsmallmetallicorsemiconductor
devices [30]. The photon blockade mechanism, similar to the
dipole-dipole blockade, prevents absorption of more than one
√N
013831-1
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KHULUD ALMUTAIRI, RYSZARD TANA´S, AND ZBIGNIEW FICEKPHYSICAL REVIEW A 84, 013831 (2011)
photonbyanopticalsystem.Inanalogy,theCoulombblockade
preventstransportofmorethanoneelectronthroughametallic
or semiconductor device.
The dipole-dipole blockade mechanism described above
applies to the process of the creation of maximally entangled
singleexcitationBellstatesonly.Itisthepurposeofthispaper,
therefore, to propose a mechanism which might be useful
in achieving entangled two-photon Bell states in a system
composed of two coherently driven two-level atoms. As we
have already mentioned, these states are linear superposition
of product states with double and zero excitations. We shall
demonstrate that the states could also be generated with
the help of the dipole-dipole interaction between the atoms.
Specifically, we analyze the dipole-dipole interaction as a
blockadeeffectforasinglephotonabsorptionofthelaserfield.
We work in the collective basis of the system, and find that
in the limit of a large dipole-dipole interaction, the collective
four-level system reduces to an effective two-level two-photon
system. The state of this reduced system is then obtained
analytically and the nature of the state is fully analyzed. We
show that the creation of a pure two-photon Bell state is
dependent on the relaxation of the atomic excitation. When
the relaxation is not present or can be ignored, the state
of the system evolves harmonically between a separable to
the maximally entangled two-photon Bell states. With the
relaxation present, the general state of the system is a mixed
state that under a strong dipole-dipole interaction reduces the
system to an X-state form. We find that mixed states admit of
lower level of entanglement and show that there is a threshold
value for the dipole-dipole interaction relative to the Rabi
frequency of the driving field above which the atoms can be
entangled over the entire time of the evolution.
The paper is organized as follows. In Sec. II, we give a
qualitative explanation of the mechanism for the creation of
two-photon Bell states in a driven two-atom system with the
help of the dipole-dipole interaction. A detailed calculation
of the evolution of the system isolated from a dissipative
environment is studied in Sec. III. Analytical expressions are
obtained for the probability amplitudes of the collective states
of the system along with the expression for the concurrence.
We show how the shift of the single excitation states may
lead to a maximally entangled state involving only the ground
and double excited states of the system. In Sec. IIIB, we
present the conditions for the system to reduce to a two-level
two-photon system described by the density matrix in an
X-stateform.SectionIVdealswithapracticallymorerealistic
modelinwhichtheatomsinteractwithavacuumreservoir.The
interaction results in the dissipation of the atomic excitation
and coherence, and we examine the effect of the dissipation
on the transient evolution and stationary properties of the
concurrence. The remarkably simple analytical expression is
obtained for the steady-state concurrence. We summarize our
results in the concluding Sec. VI. Finally, in the Appendix,
we present the full set of equations of motion for the density
matrix elements and their steady-state solutions.
II. THE MODEL
We begin with a qualitative explanation of the concept of
the generation of two-photon Bell states in a driven two-atom
system with the help of the dipole-dipole interaction. Let us
consider a system composed of two identical nonoverlapping
atoms, separated by a distance r12= |? r2− ? r1| and coupled to
each other through the dipole-dipole interaction. Each atom
is modelled as a two-level system (qubit) with the ground
state |gi? and the excited state |ei?, (i = 1,2) separated by a
transition frequency ω0. In the absence of any external fields,
e.g., a driving laser field, the Hamiltonian of the system is of
the form
H = ¯ hω0
?Sz
1+ Sz
2
?+ ¯ h
2
?
i?=j=1
?ijS+
iS−
j,
(1)
where S+
|gi??gi|)/2 are, respectively, the raising, lowering, and energy
differenceoperatorsoftheithatomand?ijisthedipole-dipole
potential between the atoms. The potential depends on the
distance between the atoms, mutual orientation of the atomic
transition dipole moments, and the orientation of the dipole
moments in respect to the interatomic axis. For the case
of mutually parallel dipole moments, the potential is given
by [13–15]
?
?sin(krij)
where γ is the spontaneous emission rate of the atomic
excitations, assumed to be the same for both atoms; θ is
the angle between the dipole moments of the atoms and the
direction of the interatomic axis, k = ω0/c; and rij is the
distance between the atoms.
We can write the Hamiltonian (1) in a matrix form using
basis states of two noninteracting atoms that are four product
states
i= |ei??gi|, S−
i= |gi??ei|, and Sz
i= (|ei??ei| −
?ij=3
4γ
−(1 − cos2θ)cos(krij)
(krij)2+cos(krij)
krij
+ (1 − 3cos2θ)
??
×
(krij)3
,
(2)
|1? = |g1? ⊗ |g2?,
|3? = |g1? ⊗ |e2?,
|2? = |e1? ⊗ |e2?,
|4? = |e1? ⊗ |g2?,
(3)
and find
H = ¯ h
⎛
⎝
⎜
−ω0
0
0
0
00
0
0
0
0
ω0
0
0
?12
0
?12
⎞
⎠.
⎟
(4)
Note that in the absence of the dipole-dipole interaction, the
single excitation states |3? and |4? are degenerate in energy
with E3= E4= 0 and are separated from the zero |1? and
double |2? excitation states by ¯ hω0, as illustrated in Fig. 1(a).
In the presence of the dipole-dipole interaction, the
Hamiltonian (4) is not diagonal, and a diagonalization results
in the so-called collective or Dicke states [12–15]
|g? = |g1? ⊗ |g2?,
|s? =
1
√2(|e1? ⊗ |g2? − |g1? ⊗ |e2?),
|e? = |e1? ⊗ |e2?,
1
√2(|e1? ⊗ |g2? + |g1? ⊗ |e2?),
(5)
|a? =
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PHYSICAL REVIEW A 84, 013831 (2011)
|2?
|3?|4?
|1?
ω0
ω0
(a)
|e?
|s?
|a?
|g?
ω0
ω0
Ω12
Ω12
(b)
FIG. 1. Schematic illustration of the idea of the creation of
two-photon entangled states in a driven two-atom system. (a) In
the absence of the dipole-dipole interaction between the atoms,
?12= 0,thedrivinglaserfieldoffrequencyωL= ω0isonresonance
with both one and two-photon transitions of the two-atom system.
(b) In the presence of the dipole-dipole interaction, ?12?= 0, and
thentheintermediatestatesareshiftedfromtheone-photonresonance
leaving only the two-photon transition on resonance with the driving
field frequency.
with the corresponding energies
Eg= −¯ hω0,
We see that the dipole-dipole interaction lifts the degeneracy
of the single excitation product states and leads to two nonde-
generate maximally entangled states |s? and |a? separated in
energy by 2¯ h?12, as illustrated in Fig. 1(b).
We are now in a position to explain qualitatively the idea
of the creation of two-photon entangled states with the help of
the dipole-dipole interaction. The idea is illustrated in Fig. 1,
in which the schematic energy-level diagram of a two-atom
system is shown. Independent of the basis used, a system
composed of two two-level atoms is equivalent to a single
four-level system with a ground state, two intermediate states
separated from the ground state by energy ¯ hω0and an upper
state separated from the ground state by energy 2¯ hω0. Assume
that the two-atom system is driven by an external laser field of
frequency ωLresonant with the atomic transition frequency,
i.e., ωL= ω0. Consider separately two cases, the absence
and the presence of the dipole-dipole interaction between the
atoms. In the absence of the dipole-dipole interaction, the
laser field drives on resonance both the one- and two-photon
transitionsofthesystem,asseenfromFig.1(a).Inthiscase,all
states of the system are populated by the laser field. Since the
one-photoncouplingdominatesoverthetwo-photoncoupling,
the intermediate states |3? and |4? are more populated than
the upper state |2?. The situation changes in the presence
of the dipole-dipole interaction. The interaction shifts the
intermediate states from the one-photon resonance, as shown
in Fig. 1(b). As a consequence, the one-photon transitions
become off resonance to the driving field frequency, but the
two-photon transition remains on resonance. Thus, in the
presence of the dipole-dipole interaction, a resonant laser field
effectively couples only to the two-photon transition of the
two-atom system.
A more quantitative attempt is presented below, where
we show that the shift of the intermediate states by the
dipole-dipole interaction can lead to the population of the
Ee= ¯ hω0,Es= ¯ h?12,Ea= −¯ h?12.
(6)
upper state |e? without populating the intermediate states |s?
and |a?, which then may result in a pure maximally entangled
two-photon state of the system.
III. CREATION OF TWO-PHOTON ENTANGLED STATES
Let us first consider the case when the atoms are isolated
from the environment. In this case, there is no relaxation of
the atomic excitation, but there could still exist a nonzero
dipole-dipole coupling between the atoms. In other words,
there are no losses due to spontaneous emission and therefore
the evolution of the system can be determined by the evolution
ofapurestateofthesystem.Inpracticalterms,itcouldbedone
byplacingtheatomsinsideseparatecavitiesandtoarrangethe
coupling between the cavities through an optical waveguide
[31,32]. Alternatively, one could trap single ions or atoms
inside separate potential wells and use an additional trapped
ion as antennae to enhance the dipole-dipole coupling [33].
Assume that the two-atom system described by the Hamil-
tonian (1) is subjected to a continuous driving by an external
coherentlaserfield.Thestrengthofthedrivingischaracterized
by the Rabi frequency ?0. The driving laser field has a
traveling-wave character and is propagating in the direction
orthogonal to the interatomic axis. The orthogonality of the
driving field and the interatomic axis ensues that both atoms
experience the same laser field amplitude and phase. The
Hamiltonian for this case is given by
H = ¯ hω0
?Sz
1+ Sz
2
?+ ¯ h
1+ S+
2
?
i?=j=1
?ijS+
iS−
j
−1
2¯ h?0[(S+
2)e−iωLt+ (S−
1+ S−
2)eiωLt],
(7)
whereωListhelaserfieldfrequency.Thelaser-atomscoupling
part of the Hamiltonian retains only the terms which play
a dominant role in the rotating-wave approximation (RWA).
Antiresonant terms which would make much smaller contri-
butions have been omitted.
When the relaxation of the atomic excitation is not present
or can be neglected, the time evolution of the system is
governed by the Schr¨ odinger equation
i¯ hd |?s(t)?
dt
= H|?s(t)?,
(8)
where |?s(t)? is the wave function of the driven two-atom
system.
Since the Hamiltonian (7) depends explicitly on time, we
make the unitary transformation
|?(t)? = exp[i (H0/¯ h)t]|?s(t)?,
(9)
where
H0= ¯ hωL
?Sz
1+ Sz
2
?,
(10)
and obtain the Schr¨ odinger equation for the transformed state
i¯ hd |?(t)?
dt
= ˜ H|?(t)?,
(11)
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KHULUD ALMUTAIRI, RYSZARD TANA´S, AND ZBIGNIEW FICEKPHYSICAL REVIEW A 84, 013831 (2011)
with the transformed Hamiltonian˜ H of the form
˜ H = ¯ h??Sz
where ? = ω0− ωLis the detuning of the laser frequency
from the atomic resonance.
ThetransformedHamiltonian(12)doesnotdependontime,
so the Schr¨ odinger equation (11) has the formal solution
1+ Sz
2¯ h?0(S+
2
?+ ¯ h?12(S+
1S−
1+ S−
2+ S+
2),
2S−
1)
−1
1+ S+
2+ S−
(12)
|?(t)? = exp[−i(˜ H/¯ h)t]|?(0)?,
where |?(0)? is the initial state of the system at t = 0.
In order to find the time evolution of the wave function for
a given initial state of the system, we write the Hamiltonian˜ H
in the basis of the collective states (5) and find
⎛
⎜
where˜? = ?0/√2.
Since the two-photon coherence, which is of the main
interest here, attains maximal values for the laser frequency
on resonance with the two-photon transition |g? ↔ |e?, we put
? = 0 in Eq. (14) and readily find the following eigenvalues
(energies) of the Hamiltonian
E1= −¯ h?? −1
E3= 0,
with the corresponding eigenvectors (energy states)
?α+
?α−
|ψ3? =
(13)
˜ H = ¯ h
⎜
⎝
−?
0
−˜?
0
0
?
−˜?
0
−˜?
−˜?
?12
0
0
0
0
−?12
⎞
⎟
⎟
⎠,
(14)
2?12
?,
E4= −¯ h?12,
E2= ¯ h?? +1
2?12
?,
(15)
|ψ1? =
2
(|e? + |g?) +√α−|s?,
(|e? + |g?) +√α+|s?,
|ψ2? = −
2
(16)
1
√2(|e? − |g?),
|ψ4? = |a?,
where
α±=?±
2?,
with
?±= ? ±1
2?12,
(17)
and
? =
?
?2
0+1
4?2
12.
(18)
We see from Eq. (15) that the single excitation antisymmetric
state |a? is the eigenstate of the Hamiltonian with eigenvalue
−¯ h?12, the double excitation antisymmetric state |ψ3? is the
eigenstate with eigenvalue zero, and the remaining states are
superpositions of the single and double excitation states with
nondegenerate eigenvalues.
A. Time evolution of the concurrence
We now turn to the problem of determining the form of
an entangled state that could be created by the dipole-dipole
interaction shift of the single excitation states and the degree
of the resulting entanglement. As a measure of entanglement,
we choose the concurrence that for a pure state |?s(t)? of a
two-atom system is given by
C(t) = |??s(t)|˜?s(t)?|,
(19)
where
|˜?s(t)? = σ(1)
y
⊗ σ(2)
y|?∗
s(t)?,
(20)
and σ(i)
For the system considered here, the state |?s(t)? is of the
form
y(i = 1,2) is the Pauli operator for the ith atom.
|?s(t)? = exp[−i(H0/¯ h)t]|?(t)?,
where |?(t)? is given in Eq. (13).
Given the state of the system, it is straightforward to cal-
culate the concurrence. We have two alternative forms for the
state|?s(t)?,thediagonalstates(16)orthecollectivestates(5).
Of these alternatives, we will prefer the collective states as it
is very often found that they form very convenient basis states
to discuss properties of the concurrence. Therefore, we invert
the transformation (16) to find
(21)
|g? =
1
√2(√α+|ψ1? −√α−|ψ2? − |ψ3?),
1
√2(√α+|ψ1? −√α−|ψ2? + |ψ3?),
|s? =√α−|ψ1? +√α+|ψ2?,
and write the state vector of the system as
|e? =
(22)
|a? = |ψ4?,
|?(t)? = Cg(t)|g? + Ce(t)|e? + Cs(t)|s? + Ca(t)|a?,
where Cn(t) is the probability amplitude of the nth state.
The time evolution of the state vector of the system and so
the concurrence depends, of course, on the initial state of the
system. Consider as the initial state the ground state |g(0)? =
|g0?, corresponding to no excitation of the system at t = 0. In
terms of the eigenstates (16), the initial state is of the form
(23)
|g(0)? =
1
√2(√α+|ψ1(0)? −√α−|ψ2(0)? − |ψ3(0)?).
(24)
From Eqs. (13) and (15), we then find that the time evolution
of the coefficients Cn(t) is of the form
Cg(t) =1
Ce(t) =1
1
2√2
Ca(t) = 0.
Note that the amplitude of the antisymmetric state is equal to
zero for all times. This is due to the fact that for the configura-
tionconsideredhere,thelaserfieldpropagatinginthedirection
perpendicular to the interatomic axis, the laser couples exclu-
sively to the symmetric state leaving the antisymmetric state
completely decoupled from the driven states. For the initial
state |g0?, the antisymmetric state is not populated at t = 0,
and therefore the state will remain unpopulated for all t > 0.
2(α+ei?−t+ α−e−i?+t+ 1),
2(α+ei?−t+ α−e−i?+t− 1),
?0
?(ei?−t− e−i?+t).
(25)
Cs(t) =
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PHYSICAL REVIEW A 84, 013831 (2011)
0
0.2
0.4
0.6
0.8
1
C(t)
00.20.40.60.8
γt
11.21.4
FIG. 2. Time evolution of the concurrence C(t) for ?0= 10γ
and different values of the ratio u = ?0/?12: u =√3/4 (solid line),
u = 1.5 (dashed line), u =?15/4 (dashed-dotted line), and u = 2.5
(dotted line).
Using Eqs. (19) and (20), we find that the solution (25)
yields the following expression for the concurrence
??2Cg(t)Ce(t) − C2
2
?
C(t) =
s(t)??
=1
????−1 +
×(α+e2i?t− α−e−2i?t)
??0
?2
e−i?12t+?12
????.
2?e−i?12t
(26)
It is easily verified that the concurrence vanishes when
?12= 0. Thus, crucial for entanglement is the presence of
the dipole-dipole interaction that shifts of the single excitation
statesfromtheirresonantvalues.Theconcurrencediffersfrom
zero and contains exponentials oscillating in time with three
different frequencies, ?12, 2?−, and 2?+. Since, in general,
the three frequencies are not commensurate, the evolution of
concurrence is not periodic. However, for some values of the
ratio ?0/?12, the evolution can be periodic. This is shown in
Fig. 2, where we plot the concurrence versus time t for several
different values of the ratio ?0/?12. For t = 0 the system
is separable, C(0) = 0, due to our choice of the initial states
of the atoms. Immediately afterward, the concurrence begins
to increase and, depending on the ratio ?0/?12, it oscillates
periodically or nonperiodically in time. For these values of
the ratio ?0/?12at which the oscillations are periodical, the
concurrencereachesthemaximalvalueofC(t) = 1thatisseen
tooccurperiodicallyoveralltimes.Forthecaseofnonperiodic
oscillations, the maxima are smaller than one. Thus, the effect
of the nonperiodicity is clearly do decrease the amount of
entanglement.
Let us examine the concurrence a little more closely.
Consider separately the cases of periodic and nonperiodic
oscillations.AswehavealreadynoticedfromFig.2,anditcan
alsobeseenfromEq.(26),forthecaseofperiodicoscillations,
the concurrence becomes unity at certain times, satisfying the
condition
?12t = π
and
?t = nπ,n = 1,2,....
(27)
which can happen only for the discrete values of the ratio
?
?0
?12
=
n2−1
4,n = 1,2,....
(28)
It is easily verified that at the times satisfying the condition
(27), the probability amplitudes (25) reduce to
Cg(t) =
1
√2e(−1)n−1iπ/4,
Cs(t) = 0,
Ce(t) = −1
√2e(−1)niπ/4,
(29)
n = 1,2,....
This shows that at these particular times, the system is in a
superposition of the ground |g? and the upper |e? states with
no excitation of the symmetric state |s?.
When Eq. (29) is used in Eq. (23), we readily find for the
statevectorofthesystemoftheresultingmaximallyentangled
two-photon Bell state
|?(tn)? = −e(−1)niπ/4
√2
(|e? + (−1)ni|g?),n = 1,2,....
(30)
We now turn to a detailed analysis of the concurrence
for the case of nonperiodic oscillations. In this case, the
competing effects of one- and two-photon transitions modify
the Rabi oscillations and cause increasing distortions of the
concurrence. However, we can reduce the destructive effects
of the competing transitions by taking large values of the
dipole-dipole interaction, ?12? ?. It is easily verified from
Eq. (26) that, in the limit of ?12? ?, the amplitudes of the
termsoscillatingwithfrequencies?12and2?+areverysmall,
and then the concurrence can be approximated by
????−1 + exp
t = nπ?12
2?2
0
the concurrence is close to its maximal value of C(t) = 1.
At these times, the state of the system reduces to the
maximally entangled two-photon Bell state
C(t) ≈1
2
?
−i2?2
0
?12t
?????.
(31)
We see that at times
,n = 1,3,5,...
(32)
|?(t)? = −e−iπ/4
√2
(|e? − i|g?).
(33)
It is seen that the system evolves between the ground |g? and
the doubly excited state |e? and at some discrete time can be
found in the maximally entangled two-photon Bell state. The
maximal entanglement is due to the two-photon coherence.
It is straightforward to show using Eq. (25) that the one-
and two-photon coherences evolve in time as
˜ ρgs(t) = Cg(t)C∗
s(t) =1
4
˜?
?[e−i?−t− ei?+t
+α−(e−2i?t− 1) − α+(e2i?t− 1)]
˜ ρes(t) = Ce(t)C∗
+α−(e−2i?t− 1) − α+(e2i?t− 1)]
˜ ρge(t) = Cg(t)C∗
+1
s(t) =1
4
˜?
?[−e−i?−t+ ei?+t
(34)
e(t) = −1
2
?˜?
?
?2
sin2?t
2i(α−sin?+t − α+sin?−t),
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KHULUD ALMUTAIRI, RYSZARD TANA´S, AND ZBIGNIEW FICEKPHYSICAL REVIEW A 84, 013831 (2011)
where ˜ ρge(t) = ρge(t)exp(−2iω0t) is the slowly varying part
of the coherence.
ItfollowsfromEqs.(28),(32),and(34)thattheone-photon
coherences vanish, ˜ ρgs(t) = ˜ ρse(t) = 0, and the two-photon
coherence | ˜ ρge(t)| becomes maximal, | ˜ ρge(t)| = 1/2, at times
the concurrence is maximal. Thus, we must conclude that the
entanglement is created by the two-photon coherence with no
population of the single excitation state |s?.
B. X state as a lower bound for entanglement
The results for entanglement created by the two-photon
coherence bear interesting relation to an X-state entanglement
[34]. Therefore, it is interesting to contrast the concurrence
considered under the two-photon coherence with the concur-
renceoftheX-statesystem.Sincethedipole-dipoleinteraction
shifts the single excitation states from resonance with the
driving field, the one-photon coherences are all equal to zero
and then the density matrix of the system takes the X-state
form
⎛
⎝
The concurrence of the system determined by the X-state
density matrix is readily found to be
ρ(t) =
⎜
ρgg(t)
0
0
ρeg(t)
00
0
ρge(t)
0
0
ρee(t)
ρaa(t)
0
0
ρss(t)
0
⎞
⎠.
⎟
(35)
Cx(t) = 2|ρge(t)| − ρss(t)
= 2|Cg(t)C∗
e(t)| − |Cs(t)|2,
(36)
where we have taken into account that ρaa(t) = 0.
We wish to compare the concurrence Cx(t) with the
concurrence C(t), Eq. (26), and see whether the concurrence
of the system discussed above is larger or smaller than that
predicted by the X-state system. From Eq. (26), we have
C(t) = 2??Cg(t)Ce(t) − C2
??z1z2− z2
we readily find that C(t) ? Cx(t). This inequality always
holds true, so we may conclude that values of Cx(t), which
determines concurrence for the X-state system, are lower
bounds for the concurrence of the present system. In addition,
if the phases of the complex amplitudes are chosen such
that φg+ φe− 2φs= 0, the inequality becomes equality.
Moreover, for the case of Cs(t) = 0, that occurs at times the
concurrence C(t) reaches the maximum C(t) = 1 value, we
have C(t) = Cx(t). This means that under the ideal two-photon
excitation, the system behaves as an X-state system.
The above considerations are illustrated in Fig. 3, which
shows the concurrences C(t) and Cx(t) as a function of
time for two different values of ?12. For a small ?12, the
time evolution of the concurrences exhibits a modulation of
different amplitudes, and the effect of an increasing ?12is
evident in the decrease of the difference between C(t) and
Cx(t).
s(t)??,
(37)
and then, applying the inequality,
3
??? |z1z∗
2| − |z3|2,
(38)
which holds for arbitrary complex numbers z1,z2, and z3,
0
0.2
0.4
0.6
0.8
1
C(t)
01234567
γt
(a)
0
0.2
0.4
0.6
0.8
1
C(t)
01234567
γt
(b)
FIG. 3. Time development of the concurrences C(t) (solid line)
and Cx(t) (dashed line) for ?0= 10γ and different values of ?12:
(a) ?12= γ and (b) ?12= 100γ. In the case (b), both concurrences
overlap.
IV. MASTER EQUATION
Theabovediscussionofthecreationofthetwo-photonBell
state describes a simplified case, which ignores dissipative
effects of spontaneous emission resulting from the coupling of
the atoms to an external multimode reservoir. We now extend
the model to include the dissipation in the system. In this
case, a general state of the system is a mixed state and the
dynamics of the atoms are then determined by the evolution
of the density operator of the system. The density operator
satisfies the master equation which in the interaction picture
can be written as
˙ ρ = −i??Sz
+1
1+ Sz
2,ρ?− i
2
?
i?=j=1
?ij[S+
iS−
j,ρ]
2i?0
2
?
i=1
[S+
i+ S−
i,ρ]
−
2
?
i,j=1
γij(ρS+
iS−
j+ S+
iS−
jρ − 2S−
jρS+
i).
(39)
The term of the master equation, involving γij, represents the
evolutionofρ duetodissipationintheatomicsystem.Theterm
contains contributions, proportional to γii≡ γ representing
thedampingoftheithatombyspontaneousemission,assumed
to be independent of i, so both atoms are equally damped
by the field. Apart from the contribution of the individual
atoms,thetermcontainscontributions,proportionaltoγij(i ?=
j), that represent the collective damping resulting from the
mutual exchange of spontaneously emitted photons through
the common reservoir [12–15]. The parameter γij(i ?= j)
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GENERATING TWO-PHOTON ENTANGLED STATES IN A ...
PHYSICAL REVIEW A 84, 013831 (2011)
depends on the distance between the atoms and the orientation
oftheatomicdipolemomentsinrespecttotheinteratomicaxis
γij=3
2γ
?
(1 − cos2θ)sin(krij)
krij
(krij)2−sin(krij)
+(1 − 3cos2θ)
?cos(krij)
(krij)3
??
.
(40)
In general, γij involves terms oscillating at frequency krij
multiplied by inverse powers of krijranging from (krij)−1to
(krij)−3. Note that γijis symmetric under the exchange of the
atoms, i.e., γ21= γ12. For small atomic separations, krij? 1,
the collective damping γij approaches γ, whereas for large
separations γij vanishes. The later corresponds to the case
where the atoms are independently damped by the reservoir.
This case is equivalent to the situation where the atoms are
damped by their own independent reservoirs.
We now employ the master equation (39) to find equations
of motion for the matrix elements of the atomic density
operator ρ. We use the collective states basis (5) as the
representation of the density operator. As in the previous
sections, we assume that both atoms experience the same
amplitude and phase of the driving field and we focus on the
two-photonresonance,i.e.,weput? = 0.Insodoing,wefind
that the equations of motion split into two independent sets,
onecomposedofnineinhomogeneousandtheothercomposed
of six homogeneous coupled equations. The sets together with
their steady-state solutions are listed in Appendix A.
A. Effect of spontaneous emission on entanglement
We wish to examine the time development of the concur-
rence in the presence of spontaneous decay of the atomic
excitation and coherence. In this case a general state of the
system is a mixed state described by the density operator ρ.
The concurrence of a mixed state of a two-atom system is
defined as [35]
C = max(0,
?
λ1−
?
λ2−
?
λ3−
?
λ4),
(41)
where λiare the the eigenvalues, in decreasing order, of the
matrix
R = ρ ˜ ρ,
(42)
with
˜ ρ = σy⊗ σyρ∗σy⊗ σy,
(43)
and ρ∗denotes the complex conjugate of ρ.
The concurrence is specified by the density matrix of a
given system and thus can be determined from the knowledge
of the density matrix elements. Assume that, initially, prior
to the laser field being turned on at t = 0, the atoms were
in their ground states, i.e., ρgg(0) = 1 and the other matrix
elements equal to zero. It is easily verified from Eq. (A5)
that, in this case, the coherences between the triplet states and
the antisymmetric state are equal to zero for all times. As a
consequence, the density matrix of the system, written in the
collective basis (|g?,|e?,|s?,|a?), takes the simplified form
⎛
⎝
The time evolution of the density matrix elements is found
by solving the set of nine coupled differential equations (A4).
Theseequationsarecumbersomeforananalyticalsolutionbe-
causeofthecouplingbetweenthepopulationsandcoherences.
Therefore, we use a numerical method to find the evolution of
the density matrix elements from which we then compute the
time evolution of the concurrence.
Note that the density matrix (44) is not diagonal in the
basis of the collective states. This means that, in general,
the collective states are not the eigenstates of the system.
In principle, it is possible to find the diagonal states simply
by direct diagonalization of the matrix (44). However, the
diagonal states obtained are complicated in form and thus
difficult to interpret. An alternative way is to compare the
general state (44) with approximate states in which the system
could be found under the pure two-photon excitation.
Of these alternatives, we consider a state described by an
approximate density operator of the form
ρ(t) =
⎜
ρgg(t)
ρeg(t)
ρsg(t)
0
ρge(t)
ρee(t)
ρse(t)
0
ρgs(t)
ρes(t)
ρss(t)
0
0
0
0
ρaa(t)
⎞
⎠.
⎟
(44)
ρ(t) = (1 − 4ρaa(t))|?(t)???(t)| + ρaa(t)I,
where |?(t)? is the pure state of the system, calculated in Sec.
III,andI isthe4 × 4identitymatrix.Theapproximatedensity
operator represents a state of the system that is an incoherent
superposition of the pure state |?(t)? and the antisymmetric
state |a?. This choice is suggested by an observation from
Eq.(A4)thatthespontaneousdecaycouplestheantisymmetric
state to the triplet states. This means that this coupling may
play an important role in the creation of the two-photon Bell
states.
We also compare the general state (44) with a state
describedbythedensitymatrixoftheX-stateform.Therefore,
we consider the criterion Cxfor entanglement of an X-state
system given by [36]
(45)
Cx(t) = max{0,2|ρge(t)|
−
Since for the general state ρsa(t) = 0, the criterion (46)
simplifies to
?
[ρss(t) + ρaa(t)]2− [2Reρsa(t)]2}.
(46)
Cx(t) = max{0,2|ρge(t)| − [ρss(t) + ρaa(t)]}.
We now present some numerical calculations that illustrate
theeffectofspontaneousemissiononthepuretwo-photonBell
state created with the help of the dipole-dipole interaction.
The time development of the concurrence is calculated for
the actual state described by the density matrix (44) and, for
comparison, for an approximate state described by the density
matrix (45) and also for an X state of the system.
Figure 4 shows the time development of the concurrence
of the system of two atoms located at a small distance,
r12= 0.078λ,andcoupledtoacommonreservoir.Alsoshown
are concurrences of the system decoupled from the reservoir
and that determined by the density matrix (45). It is easy
(47)
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KHULUD ALMUTAIRI, RYSZARD TANA´S, AND ZBIGNIEW FICEKPHYSICAL REVIEW A 84, 013831 (2011)
0
0.2
0.4
0.6
0.8
1
C(t)
00.20.40.60.81
γt
FIG. 4. Time development of the concurrence for three different
states of the system. The solid line shows the concurrence of the
pure state of the system decoupled from the reservoir. The dashed
line is the concurrence of the actual state of the atoms coupled to a
common reservoir, and the dashed-dotted line is the concurrence of
the approximate state of the system, Eq. (45). In each case ?0= 10γ
and ?12= ?0/√3/4. In last two cases, the distance between the
atoms is r12= 0.078λ and the dipole moments are polarized in the
direction perpendicular to the interatomic axis, θ = π/2. This choice
of the parameters has been made to get ?12= 11.48γ = ?0/√3/4.
seen that the concurrences evolve in decidedly different ways.
In particular, the concurrence of the state approximated by
the density matrix (45) does not reproduce the concurrence
of the actual state of the system. It is rather close to the
concurrenceofthepurestateofthesystemofatomsdecoupled
from the reservoir. The concurrence oscillates in time with the
same frequency as that of the isolated system and persists
over many Rabi periods. The amplitude of the oscillations
decreases slowly in time and is damped out in a time of order
(γ − γ12)−1.
The reason for this feature of the approximate state of
the system can be understood by referring to the asymmetry
introduced to the system by the competing effects of spon-
taneous emission from the upper state |e? to the symmetric
and antisymmetric states. It is easily verified from Eq. (A4)
that the antisymmetric state |a? is populated by spontaneous
emission from the state |e? at a subradiant rate γ − γ12,
which for small distances between the atoms is much smaller
that the superradiant rate, γ + γ12, at which the symmetric
state |s? is populated. This means that the redistribution of
the noise in the system is not isotropic. Consequently, the
symmetric and antisymmetric states are affected differently
by spontaneous emission. A considerable part of the vacuum
noise is transferred to the symmetric state with only a small
fraction being transferred to the antisymmetric state. Thus,
the population ρaa(t) is not a good measure of the noise
distribution between the states of the system of two atoms
coupled to a common reservoir.
The actual mixed state of the system is much better
approximated by the X-state density matrix. It is convincingly
seen from Fig. 5, where we graph the concurrence C(t) of the
actual state and the concurrence Cx(t) of the X state of the
system for the same parameters as in Fig. 4. The concurrences
behave in a similar fashion that Cx(t) remains quite close to
C(t) for all times. The entanglement lives over a restricted
time range with no oscillations present even though the Rabi
frequency is high. This is an example of the phenomenon of
0
0.2
0.4
0.6
0.8
C(t)
00.20.40.60.81
γt
FIG. 5. The time evolution of the concurrence of the actual state
of the system C(t) (solid line) and the approximate X state Cx(t)
(dashed line) for the same parameters as in Fig. 4.
suddendeathofentanglement[37–39].NotethatCx(t)doesnot
exceed the actual concurrence C(t), same feature as predicted
above for the pure state; see Sec. IIIB.
Although the density matrix (45) differs considerably from
the density matrix of the actual state of the system when the
atoms interact with a common reservoir, it does approximate
quitewelltheactualstateinthecaseofindependentreservoirs,
i.e., when γ12= 0. We illustrate this feature in Fig. 6,
where we plot the time evolution of the concurrence for
two different arrangements of the coupling of the atoms
to an external reservoir. It is seen from Fig. 6 that the
concurrence of the approximate state (45) coincides quite well
with the concurrence of the actual state of the system. The
reason for this similarity is that, in the limit of γ12= 0, the
symmetric and antisymmetric states are equally populated by
the spontaneous emission from the upper state |e?. This results
in an equal redistribution of the noise between the symmetric
and antisymmetric states. In other words, the antisymmetric
state is now fully participating in the dynamics of the system,
and the actual state of the system of atoms coupled to inde-
pendent reservoirs can be well approximated by the density
matrix (45).
0
0.2
0.4
0.6
0.8
1
C(t)
00.20.40.60.81
γt
FIG. 6. Thetimeevolutionoftheconcurrenceofthedipole-dipole
interacting atoms for two different arrangements of the coupling to
an external reservoir. The solid line is the concurrence of the system
decoupledfromthereservoir.Thedashedlineistheconcurrenceofthe
actual state of the atoms coupled to independent reservoirs (γ12= 0),
andthedashed-dottedlineistheconcurrenceoftheapproximatestate
of the atoms, Eq. (45), and also coupled to independent reservoirs. In
each case, ?0= 10γ and ?12= ?0/√3/4.
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GENERATING TWO-PHOTON ENTANGLED STATES IN A ...
PHYSICAL REVIEW A 84, 013831 (2011)
We close this part of the section by pointing out that
following the results presented in Figs. 4–6 one would
conclude that in the presence of spontaneous emission, the
entanglement created by the two-photon coherence may exist
only over a short time of the evolution of the system.
However, a quick look at the figures reveals that the time
range over which the entanglement exists depends on the
value of the parameters involved. A question then arises,
which of the parameters are crucial for the entanglement to
survive over a long time, presumably until steady state? In
order to examine this point more closely, we consider the
steady-state solution for the density matrix elements from
which we then infer conditions for a nonzero steady-state
entanglement.
B. Steady-state entanglement
We now proceed to use the steady-state solutions (A7) in
order to evaluate the concurrence of the stationary state of the
system. To evaluate the concurrence, defined in Eq. (41), we
must first find the eigenvalues of the matrix R. By taking the
explicit form of the matrix R and making use of the steady-
state solutions for the density matrix elements, Eq. (A7), we
readily find the following expressions for the square roots of
the required eigenvalues
?˜?4+ 4γ2|U12|2± 2γ|U12|
?
where U12= γ12+ i?12describes the strength of the interac-
tion between the atoms.
Although the above formulas look complicated, a straight-
forward but lengthy calculation leads to a remarkably simple
analytical expression for the steady-state concurrence
?
?λ1,2=
λ3=
˜?2
4
˜?4+ γ2?2˜?2+ (γ + γ12)2+ ?2
λ4=1
4
12
?,
(48)
?
˜?4
˜?4+ γ2?2˜?2+(γ +γ12)2+?2
12
?,
C(∞) = max0,
˜?2?γ|U12| −1
2˜?2?
˜?4+ γ2?2˜?2+ (γ + γ12)2+ ?2
12
?
?
.
(49)
Equation (49) is a general formula for the steady-state
concurrence valid for an arbitrary Rabi frequency, an arbitrary
distance between the atoms, and for a common (γ12?= 0)
or separate (γ12= 0) reservoirs. It is seen that there is a
threshold for the Rabi frequency below which the atoms are
entangled in the steady state. The threshold depends only on
therelationbetweentheRabifrequencyandthestrengthofthe
interaction between the atoms. The interaction is determined
by the collective parameters ?12and γ12.
In the special case of independent reservoirs, i.e., when
γ12= 0,theconcurrencereducestotheresultrecentlyobtained
by Li and Paraoanu [40]
?
C(∞) = max0,
˜?2?γ?12−1
2˜?2?
(˜?2+ γ2)2+ γ2?2
12
?
,
(50)
which also exhibits a threshold for the steady-state entangle-
ment. The existence of a threshold was also noted by Macovei
et al. [41], who studied numerically the stationary pairwise
entanglement in a system composed of N atoms confined to a
region much smaller than the resonant wavelength, i.e., in the
limit of γ12= γ and ρs
The steady-state concurrence (49) depends on the value of
the Rabi frequency of the driving field relative to strength
of the interatomic interactions. Obviously, the stationary
entanglement is zero for independent atoms, i.e., γ12= 0 and
?12= 0. However, as soon as γ12?= 0 and/or ?12?= 0, the
atoms can be entangled in the steady state. The sufficient
condition for a steady-state entanglement is to maintain the
strength of the interatomic interactions γ|U12| >˜?2/2. Thus,
by changing the value of˜?, we may dynamically switch on or
off the steady-state entanglement.
Looking at the steady-state solutions for the density matrix
elements, Eq. (A7), we see that in the limit of a strong
dipole-dipoleinteraction,?12? γ,thetwo-photoncoherence
ρgeapproaches a large nonzero value, ρge(∞) ≈ −i/5, with
all other coherences being vanishingly small. Evidently, the
source of the two-photon coherence is in the dipole-dipole
interaction that shifts the single excitation states from the
resonance with the driving laser field. The strong correlations
are reflected in the stationary state of the atoms which, despite
theirinteractionwithadissipativereservoir,decaytoastrongly
correlated state.
The existence of the threshold for the steady-state entan-
glement provides a clear explanation of why for the examples
illustrated in Figs. 4–6 the entanglement was confined to short
times. Simply, the chosen values of the parameters corre-
sponded to the below threshold situations of the interatomic
interaction strength γ|U12| <˜?2/2. If, instead, we choose the
values of the parameters such that γ|U12| >˜?2/2, i.e., above
the threshold, we then should observe an entanglement for all
times t > 0. This is illustrated in Fig. 7, where we plot the
concurrences C(t) and Cx(t) for the case of γ|U12| >˜?2/2.
The concurrence builds up in an oscillatory fashion and then
approaches a nonzero steady-state value C(∞) ≈ 0.2. Clearly,
the entanglement is present for all times t > 0. It is also
seen that the concurrence of the X state admits of lower
level of entanglement than the actual state of the system, i.e.,
Cx(t) < C(t) for all times.
aa= 0.
V. POTENTIAL EXPERIMENTAL SCHEME
Finally, we discuss experimental prospects for the realiza-
tion of the two-photon Bell states in a system of two driven
0
0.2
0.4
0.6
0.8
C(t)
00.511.52
γt
2.533.54
FIG. 7. The time evolution of the concurrences C(t) (solid line)
and Cx(t) (dashed line) for ?0= 10γ, θ = π/2, and r12= 0.06λ
(γ12= 0.97γ, ?12= 26.22γ).
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KHULUD ALMUTAIRI, RYSZARD TANA´S, AND ZBIGNIEW FICEKPHYSICAL REVIEW A 84, 013831 (2011)
atoms. A potential experimental system might be based on
the experimental setup of Ga¨ etan et al. [5,19] and the same
set of parameters employed in the observation of the Rydberg
blockade effect. The only difference would be in the tuning of
the excitation laser field to the energy levels of the system.
Namely, instead of satisfying the condition of zeroing the
detuningbetweenthegroundstate|g?andthesingle-excitation
entangled state|s?,thefrequency oftheexcitationlasershould
be kept on the two-photon resonance between the ground state
|g? and one of the upper states shifted by the dipole-dipole
interaction potential ?E±= ±C3/r3
depending on the geometry of the experiment [5,19]. In this
case,thefrequencyshift?ω±= ?E±/¯ happearsasadetuning
of the laser field from the one-photon resonance |g? → |s?.
In other words, a frequency shift ?ω±> γ would prevent the
excitationofthestate|s?.Apracticalvalueofthedipole-dipole
frequency shift ?ω±≈ 50 MHz and a typical spontaneous
emission rate of highly excited states of
γ/2π ∼ 6 MHz satisfy the requirement of ?ω±> γ. One
could argue that such a scheme differs from that considered
in this paper and illustrated in Fig. 1. However, these two
schemes are mathematically completely equivalent to each
other.ItisseenfromEq.(A4)thatthedipole-dipoleinteraction
appears in the equations of motions for the coherences ρgs
and ρseas a detuning of the laser field from the resonance
frequency of the transition |g? → |s?. The detuning ?12can
be achieved either through a shift of the entangled state |s?
from the atomic resonance by ?12or through a shifting of the
laser field frequency from the atomic resonance by the amount
of ?12.
In closing, we briefly comment about a possible experi-
mental observation of the signature of entanglement created
by the two-photon coherence. We have seen that the entangled
state created by the shift of the single excitation states
involves states with zero and double excitations. These states
are intrinsically connected to correlated states known in the
literature as pairwise atomic squeezed states [42–44]. Similar
states occur at long times for a two-atom system in a squeezed
reservoir. Therefore, a signature of such a state should be
seen in squeezing of the emitted fluorescence field. Hence,
the presence of the entangled states could be detected simply
by observing squeezed fluctuations in one of the quadratures
of the emitted fluorescence field. The fluctuations are directly
measurable in schemes involving homodyne or heterodyne
detection.
Alternatively, one could measure fidelity of the entangled
state created by the two-photon coherence in the presence of
spontaneous emission. It could be done in experiments similar
to that of Refs. [21,22], where the fidelity of entangled single
excitation states was observed. Fidelity of the actual mixed
state of the system, determined by the density matrix (44), is
given by
12, where C3is a constant
87Rb atoms of
F(t) = ??|ρ(t)|?? =1
where |?? is the pure two-photon maximally entangled Bell
state, Eq. (33), created by the two-photon coherence in the
absence of spontaneous emission.
Figure 8 shows a plot of the fidelity versus the normalized
time γt given by this equation for the same parameters as in
2[ρee(t)+ρgg(t)] + Im[ρeg(t)],
(51)
0
0.2
0.4
0.6
0.8
1
F(t)
012
γt
34
FIG. 8. The time variation of the fidelity F(t) for ? = 10γ, θ =
π/2, and r12= 0.06λ (γ12= 0.97γ, ?12= 26.22γ).
Fig. 7. We see that for all times, the fidelity of the actual state
is smaller than 1, but the essential point is that the concurrence
exceeds the value of 0.5 required for quantum entanglement.
The fidelity is greatest in the transient regime, where it
reaches the maximum value F(t) ≈ 0.8 and then decreases
to its steady-state value F(t) ≈ 0.6. Therefore, the entangle-
ment created by the two-photon coherence is not washed
out by spontaneous emission and should be observable in
practice.
VI. CONCLUSIONS
Wehavepresentedamechanismforacontrolledgeneration
of pure or mixed Bell states with correlated atoms that involve
double or zero excitations. The mechanism inhibits excitation
of singly excited collective states of a two-atom system by
shiftingthestatesfromtheone-photonresonance.Inparticular,
we have shown that the shift of the energy levels can lead
the system to evolve into a pure entangled Bell state that
involves two-atom states with double or zero excitations.
The crucial for the occurrence of the entangled state is the
presence of a nonzero two-photon coherence. The degree of
entanglement and the purity of the state depend on the
relaxation of the atomic excitation. In the absence of the
atomicrelaxation,thestateofthesystemevolvesharmonically
between a separable to the maximally entangled Bell state. We
have found that the concurrence can be different from zero
only in the presence of the dipole-dipole interaction. By going
into the limit of a large dipole-dipole interaction, we have
shown that the concurrence reduces to that predicted for an
X state of the system. Furthermore, we have demonstrated that
the concurrence of an X-state system is a lower bound for the
concurrence of the two-atom system. In the presence of the
relaxation, the general state of the system is a mixed state that
under a strong dipole-dipole interaction reduces to an X-state
form. We have found that mixed states admit of lower level of
entanglement, and the entanglement may occur over a finite
rangeoftime.Thetimerangefortheentanglementdependson
the relation between the dipole-dipole interaction and the Rabi
frequencyofthelaserfield.Wehavecalculatedthesteady-state
concurrence and have found there is a threshold value for the
dipole-dipole interaction relative to the Rabi frequency above
which the atoms can be entangled for all times.
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GENERATING TWO-PHOTON ENTANGLED STATES IN A ...
PHYSICAL REVIEW A 84, 013831 (2011)
ACKNOWLEDGMENTS
This work was supported in part by a research grant from
the King Abdulaziz City for Science and Technology.
APPENDIX
The matrix elements of the atomic density operator ρ
written in the basis of the collective states of the system
satisfy two independent sets of differential equations. This
is a consequence of using the collective basis and of the fact
of assuming that both atoms experience the same amplitude
and phase of the driving field. The two sets can be written in
compact matrix forms as
˙? X = M? X +?I,
where the vectors? X and?Y are of the form
? X = (ρee,ρss,ρaa,ρge,ρeg,ρes,ρse,ρgs,ρsg)T,
?Y = (ρae,ρea,ρga,ρag,ρsa,ρas)T,
the vector?I of the inhomogeneous terms is of the form
?I = (0,0,0,0,0,0,0, −i˜?,i˜?)T,
and M and Q are, respectively, 9 × 9 and 6 × 6 matrices of
the coefficients of the differential equations.
The equations of motion for the nine components of the
vector? X are
˙ ρee= −4γρee+ i˜?(ρse− ρes),
˙ ρss= −2(γ + γ12)(ρss− ρee)
+i˜?(ρes− ρse+ ρgs− ρsg),
˙ ρaa= −2(γ − γ12)(ρaa− ρee),
˙ ρge= −2γ ρge+ i˜?(ρse− ρgs),
˙ ρeg= −2γ ρeg− i˜?(ρes− ρsg),
˙ ρes= −(3γ + γ12− i?12)ρes+ i˜?(ρss− ρee− ρeg),
˙ ρse= −(3γ + γ12+ i?12)ρse− i˜?(ρss− ρee− ρge),
˙ ρgs= −i˜? − (γ + γ12− i?12)ρgs+ 2(γ + γ12)ρse
+i˜?(2ρss+ ρaa+ ρee− ρge),
˙ ρsg= i˜? − (γ + γ12+ i?12)ρsg+ 2(γ + γ12)ρes
−i˜?(2ρss+ ρaa+ ρee− ρeg),
and the equations of motion for the six components of the
˙?Y = Q?Y,
(A1)
(A2)
(A3)
(A4)
vector?Y are
˙ ρae= −(3γ − γ12− i?12)ρae− i˜?ρas,
˙ ρea= −(3γ − γ12+ i?12)ρea+ i˜?ρsa,
˙ ρga= −(γ − γ12+ i?12)ρga− 2(γ − γ12)ρae+ i˜?ρsa,
˙ ρag= −(γ − γ12− i?12)ρag− 2(γ − γ12)ρea− i˜?ρas,
˙ ρsa= −2(γ + i?12)ρsa+ i˜?(ρea+ ρga),
˙ ρas= −2(γ − i?12)ρas− i˜?(ρae+ ρag),
where˜? = ?0/√2.
The matrices M and Q are nonsingular, so we can readily
solve the equations (A1) by matrix inversion. A formal
integration gives
(A5)
? X(t) =? X(0)exp(Mt) + [exp(Mt) − 1]?I,
?Y(t) =?Y(0)exp(Qt),
where? X(0) ≡? X(t = 0) and?Y(0) ≡?Y(t = 0) are vectors of
initial values of the matrix elements.
It is easy to see from Eqs. (A4) and (A5) that only the
components of the vector? X can have nonzero steady-state
solutions. All the components of the vector?Y are zero in the
steady state.
Afterstraightforwardbutquitetediouscalculations,wefind
that the steady-state solution for the components of the vector
? X are
aa=1
ss=1
4
ge=?ρs
ρs
se
γ˜?[˜?2+ 2γ(γ + U12)]
(A6)
ρs
ee= ρs
4
˜?4
D,
ρs
(˜?2+ 4γ2)˜?2
D
?∗= −1
es=?ρs
?∗= −i
,
ρs
eg
2
γ (γ + U12)˜?2
D
?∗=i
,
(A7)
2
γ˜?3
D
,
ρs
gs=?ρs
sg
2
D
,
where the superscript s stands for the steady-state value,
D =˜?4+ γ2?2˜?2+ (γ + γ12)2+ ?2
and U12= γ12+ i?12.
12
?,
(A8)
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