Journal of Optics B: Quantum and Semiclassical Optics
Persistent entanglement in threelevel atomic
systems
To cite this article: M Ali Can et al 2004 J. Opt. B: Quantum Semiclass. Opt. 6 S13
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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS
J. Opt. B: Quantum Semiclass. Opt. 6 (2004) S13–S17 PII: S14644266(04)664901
Persistent entanglement in threelevel
atomic systems
MAliCan,
¨
Ozg
¨
ur ¸Cakir, Alexander Klyachko and
Alexander Shumovsky
Faculty of Science, Bilkent University, Bilkent, Ankara, 06533, Turkey
Email: cakir@fen.bilkent.edu.tr
Received 24 June 2003, accepted for publication 1 August 2003
Published 5 March 2004
Online at stacks.iop.org/JOptB/6/S13 (
DOI: 10.1088/14644266/6/3/003)
Abstract
We discuss the evolution towards persistent entangled state in an
atom–photon system. A maximally entangled state can be stabilized at a
local minimum of the system by draining some energy, thus obtaining a
persistent entangled state. This scheme can be realized in threelevel, type
atomic systems since the third level is a metastable state. In particular, we
compare dynamical description based on the exact and effective models.
Some experimental realizations are discussed.
Keywords: entanglement, Jaynes–Cummings model, Markov approximation,
threelevel atom
This paper reports some new results relating to the
entanglement in atomic systems. It builds upon our earlier
investigations [1–3].
The generation and manipulation of entangled states in
atom–photon systems has recently attracted a great deal of
interest in the context of quantum information processing and
quantum computing [4–10]. Most studies on entanglement in
atomic systems have used the twolevel atoms, interacting with
photons via dipole transitions (e.g., see [11–13] and references
therein). In this case, the lifetime of maximum entangled state
(MES) in atomic subsystem is chieﬂy determined by the life
of excited atomic state, that is by the natural line breadth.
Usually, this time is quite short[14, 15]. At the same time, the
quantum information processing needs more or less durable
entanglement.
According to the result obtained in [1], the maximum
entanglement in a system corresponds to the maximum total
local variance, describing the quantum ﬂuctuations of all
measurements. Thus, to achieve a longlived maximum
entanglement, we should ﬁrst prepare a state with maximum
quantum ﬂuctuations and then stabilize it by draining energy
right up to a (local) minimum, conserving at the same time
thelevel of quantum ﬂuctuations. This can be done via an
interaction with a proper environment.
It was noticed in [1, 2] that the lifetime of atomic
entanglement can be improved by entangling atoms with
respect to the two atomic states, between which the dipole
transition is forbidden. Besides that, since the system is
1
2
3
Figure 1. Scheme of three level type atomic conﬁguration.
initially disentangled, to achieve the persistent entanglement
an irreversible evolution of the system should be realized.
An important example is provided by the socalled type
threelevel structure [16, 17], which is illustrated in ﬁgure 1.
Here the levels 1 and 2 are coupled by dipole transition as well
as the levels 2 and 3. Then, because of the selection rules with
respect to parity, dipole transition between the levels 3 and 1
is forbidden [14]. The level 3 can be populated by a sort of
Raman process, when atom ﬁrst absorbs a pumping photon
by 1 → 2transition and then emits Stokes photon by 2 → 3
transition. If Stokes photon is then discarded, the atom will
stay in the state 3 for a long time determined by either multipole
or nonradiative processes.
Elimination of Stokes photon can be caused by a number
of physical processes, involving dissipative environment. For
example, in the case of atom in a cavity, Stokes photon
can be absorbed by the cavity walls. Another possibility is
14644266/04/030013+05$30.00 © 2004 IOP Publishing Ltd Printed in the UK S13
MACanet al
provided by a twomode cavity that is resonant with respect
to the pumping mode and transparent for Stokes photons.
In the former case, Stokes photons interact with continuum
of ‘phonon’ modes, describing the standard Heisenberg–
Langevin mechanism of cavity losses [18, 19]. In the latter
case, atoms interact with continuum of modes, corresponding
to the natural line width for the level 2.
Following [1–3], we can assume now that there are two
identical threelevel atoms in a cavity. The atoms are supposed
to be prepared initially both in the ground state 1, while the
cavity contains a single pump photon. Then, the irreversible
evolution caused by either of above mentioned processes will
lead to the following atomic state
1
√
2
(3, 1 + 1, 3) (1)
that deﬁnitely manifests maximum entanglement. The process
of evolution towards the state (1) can be described through the
use of two models. Within the ﬁrst model, the interaction part
of the Hamiltonian has the form
H
int
=
f
[λ
P
R
21
( f )a
P
+ λ
S
R
23
( f )a
S
]+H.c., (2)
where a
P
, a
S
are the annihilation operators of pumping and
Stokes photons, R
ij
=i j is the atomic transition operator,
f marks the atom, and λ
P
, λ
S
are the coupling constants.
Instead of the onephoton threelevel process described by
theHamiltonian (2), an effective model of twophoton process
in twolevel atom can also be considered [20, 21]. In this case,
it is assumed that the cavity is tuned consistent with twophoton
energy conservation, i.e.
E
3
− E
1
≈ ω
P
− ω
S
,
where E
i
denotes the energy of atomic level i.Inthiscase,
only one detuning parameter
= E
2
− E
1
− ω
P
= E
2
− E
3
− ω
S
can be taken into account. Under the condition E
3
− E
1
the level 2 can be adiabatically eliminated [20], so that the
dynamics of the system can be described by the effective
interaction Hamiltonian of the form
H
int
=
f
λR
31
a
+
S
a
P
+H.c., (3)
where λ = 2λ
S
λ
P
/ is a certain effective coupling constant.
The effective scheme of transitions is shown in ﬁgure 2.
The two main objectives of this paper are on the one hand
to show that both models describe the deterministic evolution
towards the state (1), and on the other hand to ascertain the
difference in the evolution process caused by the speciﬁc
structure of the models.
Assume that the two identical atoms ( f = 1, 2) are
located in a cavity that has quite high quality with respect to
pumping mode ω
P
,while absorb easily Stokes photons with
ω
S
.Weassume that interatomic distance is much less than
the wavelength of pumping and Stokes ﬁelds, so that the intra
cavity registration of Stokes photon cannot be used to identify
1
2
3
Figure 2. Effective scheme in the existence of strong detuning.
which atom is the source of this radiation. Consider ﬁrst the
complete model with the Hamiltonian
H = H
0
+ H
loss
+ H
int
,
H
0
= ω
P
a
+
P
a
P
+ ω
S
a
+
S
a
S
+
f
(ω
21
R
22
( f ) + ω
31
R
33
( f )),
H
loss
=
q
η
q
(b
+
q
a
S
+ a
+
S
b
q
) +
q
q
b
+
q
b
q
,
(4)
where H
int
coincides with (2), ω
21
= E
2
−E
1
, ω
31
= E
3
−E
1
,
and b
+
q
and b
q
arethe Bose operators of ‘phonons’ in the cavity
walls [18, 19].
With the total Hamiltonian (4) in hand, we can now write
master equation for the reduced density matrix for the atom
ﬁeld system that is obtained through the use of the socalled
dressedatom approximation (e.g., see [22–25]):
˙ρ =−i[H
0
+ H
int
,ρ]+κ{2a
S
ρa
+
S
− a
+
S
a
S
ρ − ρa
+
S
a
S
}. (5)
Here ρ denotes the density matrix and κ is a physical quantity
such that κ
−1
determines the lifetime of Stokes photon in the
cavity and Q = ω
23
/κ is the cavity quality factor with respect
to Stokes photons.
Assume that the system is initially prepared in the state
ψ
1
=1, 1⊗1
P
, 0
S
, (6)
so that both atoms are in the ground state 1 and cavity contains
single pumping photon. Then, the evolution described by (5)
takes place in the Hilbert space H spanned by the states (6) and
ψ
2
=
1
√
2
(1, 2 + 2, 1) ⊗0
P
, 0
S
ψ
3
=
1
√
2
(1, 3 + 3, 1) ⊗0
P
, 1
S
ψ
4
=
1
√
2
(1, 3 + 3, 1) ⊗0
P
, 0
S
.
(7)
Here ψ
2
gives an intermediate entangled atomic state with
short lifetime determined by the dipole decay either to the
initial state ψ
1
(6) or to another entangled state ψ
3
.Inturn,
absorption of Stokes photon by cavity walls lead to irreducible
evolution from ψ
3
to ψ
4
that corresponds to the persistent
atomic entangled state (1). Then, the density matrix in H takes
the form
ρ(t) =
j,
ρ
j
(t)ψ
j
ψ
, j, = 1, 2, 3, 4. (8)
Use of (8) allows equation (5) to be cast into the form of
sixteen differential equations that can be analysed numerically
S14
Persistent entanglement in threelevel atomic systems
Probability of Persistent Entanglement
0.0
0.2
0.4
0.6
0.8
1.0
0.0 5.0 10.0
κt
Figure 3. Evolution to the persistent entangled state in complete
model (dotted curve) and in effective model (solid curve) for
κ = 0.1λ
P
,
P
=
S
= 10λ
P
, λ
S
= λ
P
.
at different values of parameters. The results are shown in
ﬁgure 3. It is seen that always
ρ
j
(t) →
1, at j = = 4
0, otherwise
at t κ
−1
.Thus, the system described by the Hamiltonian (4)
and prepared initially in the state (6) evolves towards the
persistent entangled atomic state (1) without failure.
It is seen in ﬁgure 3 that evolution ρ
44
→ 1isnot described
by a smooth curve but has a stairslike structure that becomes
more visible with increase of κ.Thisspeciﬁc behaviour is
caused by the competition between the processes on transitions
1 ↔ 2and2↔ 3.
Consider now dynamics of the system within the
framework of effective model with adiabatically eliminated
excited level 2. Then, in the Hamiltonian (4), the interaction
Hamiltonian H
int
(3) should be substituted instead of H
int
(2)
and
H
0
→ H
0
= ω
P
a
+
P
a
P
+ ω
S
a
+
S
a
S
+
f
ω
31
R
33
( f ).
The master equation (5) can be used again with the same form
of the Liouville term but with the change
H
0
+ H
int
→ H
0
+ H
int
.
Since the level 2 is omitted, the state ψ
2
in (7) should be
discarded. Thus, the density matrix involves the states ψ
1
,
ψ
3
and ψ
4
and consists of only nine elements instead of
sixteen in previous case that enables us to fairly simplify the
numerical analysis.
It can be seen again that the system evolves towards the
persistent entangled state (1). At the same time, the absence
of the contribution coming from the transition 1 ↔ 2leads to
amorerapidevolution at the same κ (ﬁgures 3, 4).
Consider now the case of twomode cavity, such that
Stokes photons can leave it freely, while pumpingmode
excitations exist for averylong time. Let us again begin
with the onephoton threelevel model. Then, the interaction
Probability of Persistent Entanglement
0.0
0.2
0.4
0.6
0.8
1.0
0.0 100.0 200.0
κt
Figure 4. Evolution to the persistent entangled state in complete
model (dotted curve) and in effective model (solid curve) for
κ = λ
S
= λ
P
,
P
=
S
= 10λ
P
.
Hamiltonian (2) should be changed by
H
int
=
f
λ
P
R
21
( f )a
P
+
k
λ
Sk
R
23
( f )a
Sk
+H.c., (9)
where summation over wavenumber k corresponds to the
natural line breadth of the level 2 with respect to the transition
2 ↔ 3. In turn, the unperturbed partofthe Hamiltonian should
be chosen as follows
H
0
= ω
P
a
+
P
a
P
+
k
ω
Sk
a
+
Sk
a
Sk
+
f
[ω
21
R
22
( f ) + ω
31
R
33
( f )]. (10)
The Hilbert space of the system with the Hamiltonian
H
= H
0
+ H
int
is spanned by the vectors ψ
1
(6), ψ
2
in (7), and
ψ
3k
=
1
√
2
(1, 3 + 3, 1) ⊗0
P
, 1
Sk
, (11)
forming an inﬁnite dimensional subspace.
The timedependent wavefunction of the system (10) has
the form
(t)=C
1
(t)ψ
1
+ C
2
(t)ψ
2
+
k
C
3k
(t)ψ
3k
(12)
with the initial conditions
C
(0) =
1, if = 1
0, otherwise.
(13)
Then, the Schr
¨
odinger equation
i
∂
∂t
(t)=H
(t)
with the Hamiltonian H
leads to the following set of linear
differential equations
i
˙
C
1
= ω
P
C
1
+ λ
P
√
2C
2
i
˙
C
2
= ω
21
C
2
+ λ
P
√
2C
1
+
k
λ
Sk
C
3k
i
˙
C
3k
= (ω
31
+ ω
Sk
)C
3k
+ λ
Sk
C
2
.
(14)
S15
MACanet al
To ﬁnd solution of (14), let us express C
3k
in terms of C
2
:
C
3k
(t) =−iλ
Sk
t
0
C
2
(τ )e
i(ω
31
+ω
Sk
)(τ −t )
dτ,
convert summation over k into an integration over continuum
of Stokes modes corresponding to the natural line breadth of
the level 2, and use Markovianapproximation [19]. Then, the
irreversible dynamics of the system is described by the two
equations
˙
C
1
=−iω
P
C
1
− iλ
P
√
2C
2
˙
C
2
=−iω
21
C
2
− iλ
P
√
2C
1
− C
2
,
(15)
where = πρ(ω
23
)λ
Sk

2
(k = ω
23
/c) is thespontaneous
decay rate for the transition 2 → 3andρ(ω
23
) denotes the
density of states of Stokes photons.
To within the second order in λ
P
/Z ,whereZ = −i
P
and
P
= ω
P
−ω
12
is the detuning for the pumping mode, we
get
C
1
(t) ≈ e
−iω
P
t
−
2λ
2
P
Z
2
e
−Zt
+
Z
2
− 2λ
2
P
Z
2
e
−
2λ
2
P
t
Z
, (16)
C
2
≈−
λ
P
√
2
iZ
[e
−t
− e
2λ
2
P
+iZ
P
Z
t
]e
−iω
12
t
. (17)
The above approximations are reasonable because
λ
P
,λ
Sk
.
In view of equation (12), the probability to achieve the
states,corresponding to the persistent entanglement (1), is
P(t) =
k
C
3k
(t)
2
= 1 −C
1
(t)
2
−C
2
(t)
2
. (18)
Since equations (16) and (17) describe the damped oscillations,
we get
lim
t→∞
P(t) = 1.
The detailed evolution of the probability (18) is shown in
ﬁgure 5. It is seen that the typical time required for persistent
entanglement is
τ ∼
Z 
2
λ
2
P
. (19)
Since Z
2
=
2
+
2
P
,theincrease of detuning for the
pumping mode leads to a deceleration of evolution towards
the persistent atomic entanglement.
Dynamics of the system described by the effective
Hamiltonian in the cavity transparent for Stokes photons was
examined in [2]. Comparing the above results with those
of [2], one can conclude that the effective model gives only
rough picture of purely exponential evolution toward the
state (1), completely deleting from the consideration the Rabi
oscillations between the levels 1 and 2. Besides that, the role
that detuning for the pumping mode
P
plays in acceleration
and deceleration of evolution is lost within the framework of
effective model.
Summarizing, we should stress that we have studied the
quantum irreversible dynamics of a system of two three
level type atoms, interacting with two modes of quantized
electromagnetic ﬁeld in a cavity under the assumption that
Stokes photon either leave cavity freely or is strongly damped.
1
2
3
4
0.0
0.2
0.4
0.6
0.8
1.0
λt
Σ
k
C
k

2
0.0 5000.0 10000.0 15000.0
Figure 5. Time evolution of probability (18) to have the persistent
entanglement at λ
P
= 0.001 for (1)
P
= 0; (2)
P
= ;
(3)
P
= 2;(4)
P
= 4.
It is shown that in both cases system evolves towards the
persistent atomic entangled state (1). Such an evolution takes
place if the atoms are initially prepared in the ground state,
while cavity contains single photon of the pumping mode. It
is easily seen that another realistic choice of the initial state,
when one of the atoms is excited and cavity ﬁeld is in the
vacuum state, does not lead to evolution towards the persistent
entangled state (1).
It is also shown that the effective model with adiabatically
removed atomic state 2 predicts correct asymptotic behaviour,
while is incapable of description of details of evolution.
Concerning the experimental realization, let us point
out that the transitions 1 ↔ 2and2 ↔ 3should have
quite different frequencies. Only in this case it seems to be
possible to design a multimode cavity with high quality with
respect to ω
12
,permitting either leakage or strong absorption
of Stokes photons. An important example is provided by
the 3S ↔ 4P and 4P ↔ 4S transitions in sodium atom
and in other alkaline atoms (see [26]). These atoms are
widely used in quantum optics, in particular in investigation
of Bose–Einstein condensation [27]. The multimode cavities
are also usedinquantum optics for years (e.g., see [28] and
references therein). In particular, the cavities with required
properties may be assembled using distributed Bragg reﬂectors
(DBR) and double DBR structures to single out wavelengths
corresponding to ω
12
and ω
23
.
The initial state can be prepared in the same way as in
[29–31]. First, a singlephoton excitation corresponding to
the pumping mode is prepared in the cavity. Then, the atoms
can propagate through the cavity, using either the same opening
or two different but closed openings. The velocity of atoms
should be chosen in a proper way, so that the time they spend in
the cavity obeys the condition τ Z
2
/(λ
2
P
), 1/κ (19). All
measurements aimed at the detection of atomic entanglement
can be performed outside the cavity.
Thus, the realization of persistent entanglement in three
level type atoms seems to be feasible with present
experimental technique. It should be emphasized that
the singleatom Ramantype process has been observed
recently [32]. At the same time, recent progress in
investigation of atomic entanglement in the presence of
S16