ОПТИКА И СПЕКТРОСКОПИЯ, 2010, том 108, № 3, с. 384-392
АТОМНО-ФОТОННЫЕ ВЗАИМОДЕЙСТВИЯ
УДК 535.14
SPONTANEOUSLY CREATED ENTANGLEMENT BETWEEN TWO ATOMS
© 2010 Z. Ficek***
* Department of Physics, School of Physical Sciences, The University of Queensland Brisbane, 4072 Australia **The National Centre for Mathematics and Physics, KACST, P.O. Box 6068, Riyadh, 11442 Saudi Arabia
E-mail:z ficek@kacst.edu.sa
Received August 3, 2009
Abstract—Spontaneous emission as a potential tool for creation of entanglement between two atoms is investigated. We assume that the atoms are coupled to the same environment and study entanglement engineering between the atoms and its transfer between different states. The role of the atomic coherence induced by spontaneous emission will be explored which, in contrast to what is generally believed, can create entanglement between initially unentangled atoms. We quantify entanglement by the concurrence and find that it exhibits threshold properties that can lead to interesting noncontinuous phenomena of sudden birth and sudden death of entanglement. In addition, we consider the mechanism involved in creation of entanglement between distant atoms coupled to a single-mode cavity field. We include a possible variation of the coupling constants between the atoms and the cavity mode with location of the atoms in a standing-wave cavity mode. Effectively, we engineer two coupled atoms whose the dynamics are analogous to that of interacting and collectively damped two nonidentical atoms. We illustrate the interesting result that spatial variations of the coupling constants can lead to a stationary entanglement between the atoms. We explain this effect in terms of the trapping phenomenon of atomic population in a non-decaying entangled state.
INTRODUCTION
The role of a noisy environment in entanglement creation and entanglement processing is one of the fundamental problems in quantum optics and quantum information [1—7]. It is well known from the early days of quantum optics that the interaction of a system with a noisy environment leads to spontaneous emission that is usually recognized as an irreversible loss of coherence and information encoded in the internal states of the system and thus is regarded as the main obstacle in practical implementation of coherent effects and entanglement [8].
In this paper, we investigate the role of spontaneous emission in entanglement creation and its evolution in the system of two identical atoms coupled to the same vacuum environment. We explore some surprising results that contrary to our intuition that spontaneous emission should have a destructive effect on coherence, we find some certain circumstances where this irreversible process can in fact induce coherence leading to entanglement between separated systems. We also show how spontaneous emission reveals a competition between the Bell states of a two-atom system that leads to interesting "sudden" features in the temporal evolution of entanglement [9—17]. The coherence induced by spontaneous emission is significant only if the distance between the atoms is smaller or comparable to the resonant wavelength of the atomic transition. In the following, we also demonstrate how one can create coherence and resulting from that entan-
glement between distant independent atoms coupled to a single mode cavity field [18—20]. We will demonstrate that a non-radiating (trapping) state can be created in the bad cavity limit which be accessible by spontaneous emission only if the atoms are in non-equivalent positions inside the cavity mode.
TWO INTERACTING ATOMS
We consider two identical two-level atoms with the ground state g) , the upper state \et) , transition frequencies ®0 and located at positions r1 and r2. We assume an allowed dipole transition between the atomic states that is described by three spin variables, the raising S+ , lowering S- and the energy difference Sz operators. We also allow the atoms to be coupled to the same environment whose the modes are in a vacuum state. The dynamics of the atoms are determined by the density operator p which satisfies the following master equation [8]
2 2 |e = - i ®c X [ sZ, p] - iX s+ s-, p] -
'=1 f*j (1)
2
- 1- X Yj([pS+, Sj] + [S+, S-p]),
i,j = 1
where yH = y are the spontaneous decay rates of the atoms caused by their direct coupling to the vacuum
Fig. 1. Two-atom system represented in the basis of collective states by an equivalent single four-level system composed of the ground state jg), two intermediate states js) and ja) separated in energy by and the upper state je).
field. The parameters Q and Yy (i ^ j) depend on the distance between the atoms and describe the dipoledipole interaction and the collective damping mediated by the vacuum field. The collective parameters are defined, respectively, by
Qw = - y
i] 4
cos kr^ sin krii cos krf]\
krn
+
(krti) (kri])
3 ) '
and
Yi] = 2 Yl
3 ( sin kr. cos kr. sin kr¡]\
kr
3),
(2)
(3)
(kri]) (kri]) where k = œ0/c, and rj = |i) — r; | is the distance between the atoms. Here, we assume, with no loss of generality, that the atomic dipole moments are parallel to each other and are polarized in the direction perpendicular to the interatomic axis. Clearly, the vacuum field plays the role of a transport system which allows for coherent and incoherent exchanges of excitation between the atoms. As we shall see, the collective parameters are essential for creation of entanglement between the atoms.
The master equation (1) is written in the basis of the atomic Hilbert space. We introduce instead the collective states, so called Dicke states of the two-atom system, defined as [21]
|e) = lei) ® |*2>,
g) = gl) ® |g2>, S) = (|gi) ® !e2> + h)®|g2>) /72, a> = (gi) ® |e2>(-|ei) ® |g2>))/72.
(4)
In this basis, the two-atom system behaves as a single four-level system with the ground state g), two intermediate states, the symmetric state |s) and the antisymmetric state |a) , and the upper state |e), as illustrated in Fig. 1. Note that the intermediate states are maximally entangled states, created by the dipole-dipole interaction between the atoms. These states can be independently populated by spontaneous emission from the upper state |e) . There is no any correlation between these states as far as the atoms are identical.
In the following we study dynamical properties of this collective system with a particular attention how one could create an entanglement between the atoms via the dissipative process of spontaneous emission. To do this, we first briefly explain what we mean by entanglement and how it could be created by directly or non-directly interacting atoms.
ENTANGLEMENT CRITERIA
The usual way to identify entanglement between two atoms (qubits) in a mixed state is to examine the concurrence, an entanglement measure that relates entangled properties to the coherence properties of the atoms [22]. For a system described by the density matrix p, the concurrence % is defined as
%(t) = max(0,^(t) - X2(t) - (t) - (t)), (5)
where {^,(t)} are the square roots of the eigenvalues of the non-Hermitian matrix p(t) p (t) with
P ( t) = Vy
Vy P
*( t )Vy ® Vy
(6)
and vy is the Pauli matrix. The range of concurrence is from 0 to 1. For unentangled (separated) atoms %(t) = = 0, whereas %(t) = 1 for the maximally entangled atoms.
In order to compute %(t) one has to know the density operator of a given system. We write the density operator in a matrix form using the collective state basis. In general, the matrix is composed of sixteen nonzero density matrix elements. However, in the case of a dissipative evolution of the system without any external coherent excitations, the density matrix takes a simple form
(
P( t) =
Pgg( t ) 0 0
Peg( t )
0
Pss(t )
P as(t) 0
0
Psa( t )
Paa(t) 0
Pge( t ) 0 0
Pee( t )
(7)
in which we put the one-photon coherences, except Pas(t) and Psa(t), equal to zero. We also include the two-photon coherences Pge(t) and Peg(t) that can be present in the system.
For a system described by the density matrix (7), the concurrence has a simple analytical form
%(t) = max{0, %(t), %(t)}, (8)
with
= 2 | Pge(t)| - (P„( t) + Paa(t)) ,
%2(t) = J(P„(t) - Paa(t))2 - (Psa(t) - Pas(t))2
(9)
(10)
- 2Vpgg(0p ee (t).
From this it is clear that the concurrence %(t) can always be regarded as being made up of the sum of two contributions of the weights ^(t) and ^2(t) associated with two different classes of entangled states that can be generated in a two atom system. From the form of the entanglement weights it is obvious that ^(t) provides a measure of an entanglement produced by a two-photon coherence between the ground |g) and the upper |e) states of the system, whereas %2(t) provides a measure of an entanglement produced by a distribution of the population between the symmetric and antisymmetric states. Inspection of Eq. (8) shows that the necessary condition for %i(t) to be positive is that the two-photon coherence p^ is different from zero. The sufficient condition is that the coherence overweights the sum of the population of the one-photon states. On the other hand, the necessary condition for %2(t) to be positive is that the symmetric and antisymmetric states are not equally populated, and the sufficiency for entanglement creation is provided by the requirement that the population difference over-
weights the square root of the product of the populations of the ground and the upper states.
Thus, we see that are evident thresholds for the coherence at which the system becomes entangled. The criteria show that the thresholds for entanglement depend on the distribution of the population between the entangled and separable states. Notice that the threshold for entanglement creation through %2(t) depends on the population of the upper state |e). Thus, no threshold features can be observed in entanglement creation by spontaneous emission for qubits initially prepared in a s
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