Steady-state entanglement of two atoms created by classical driving field
Özgür Çakir, Alexander A. Klyachko, and Alexander S. ShumovskyFaculty of Science, Bilkent University, Bilkent, Ankara 06800, Turkey 共Received 12 June 2004; published 10 March 2005兲
The stabilization of entanglement caused by action of a classical driving field in the system of two-level atoms with the dipole interaction accompanied by spontaneous emission is discussed. An exact solution shows that the maximum amount of concurrence that can be achieved in the Lamb-Dicke limit is 0.43. Dependence of entanglement on interatomic distance and the classical driving field, beyond the Lamb-Dicke limit, is examined numerically.
DOI: 10.1103/PhysRevA.71.034303 PACS number共s兲: 03.67.Mn, 03.65.Ud, 32.80.⫺t, 42.50.Ct
The main aim of this paper is to show that classical driv-ing field can be used to stabilize entanglement in atomic systems.
It is well known that two-level atoms can be successfully used to model entangled states of qubits as well as realiza-tion of different quantum communicarealiza-tion protocols. In fact, two-level atoms have been used for decades as the main tool for testing fundamentals of quantum mechanics 共see Refs.
关1,2兴 and references therein兲.
The practical applications require the robust entangled states. This notion includes long enough lifetime of the states and high amount of entanglement 共as close to perfect en-tanglement as possible兲. However, in many cases entangle-ment of two-level atoms is not stable enough. In the case of atoms trapped in high-quality cavities, absence of stability is caused mainly by Rabi oscillations. In free space, entangle-ment related to exited atomic states decays because of the spontaneous emission processes.
To stabilize atomic entanglement, an engineered environ-ment can be utilized. For example, it was shown in Refs.
关3–5兴 that the presence of a squeezed vacuum field can
sta-bilize entanglement of a pair of two-level atoms with dipole-dipole interaction. The use of a bad cavity as a stabilizing environment was considered in Ref. 关6兴. Stabilization in a bad cavity with optical white-noise field was discussed in Ref.关7兴. A scheme of stabilization based on the use of three-level ⌳-type atoms in two-mode cavities with leakage and absorption was proposed in Ref. 关8兴 and then discussed in Ref.关9兴.
In this note, we show that a reasonable amount of steady-state entanglement can be achieved in a system of two-level atoms in the weak-coupling regime共high losses兲, in particu-lar for free space, in the presence of a classical driving field. The collective effects, i.e., dipole-dipole interaction and col-lective spontaneous emission, are the mechanisms respon-sible for generation of entanglement. However, in the ab-sence of a special environment that compensates the losses of energy caused by spontaneous emission, the entanglement is a transient one. We show that instead of a more sophisti-cated squeezed vacuum field the simple classical driving field can be succesfully used for this aim. The classical driv-ing field alone acts only locally on the atoms, so that it can-not create specific quantum correlations between the atoms peculiar for the entangled state. However, it continuously provides atomic excitations that are responsible for survival
of the collective effects thus enabling a steady-state entangle-ment.
The system of two identical two-level atoms in free space is governed by the master equation关10兴
˙ = − i关H,兴 +1 2i,j=1
兺
2 ⌫ij共2− i + j −+i−j−+i−j兲, 共1兲 H =兺
i=1 2冋
2z i + E共+ieikជ·rជi−it+ − i e−ikជ·rជi+it兲册
+⍀共+1−2++2−1兲, 共2兲 where the atomic dipoles are aligned in the same direction along the interatomic axis and driven by a linearly polarized classical field, with dipole coupling constant E. Here +i =共−i兲†=兩e典i具g兩i andz
i=兩e典i具e兩i−兩g典i具g兩i with 兩e典i,兩g典i
denot-ing the excited and the ground states of the ith atom,⌫ii=⌫ is the single atom decay rate,⌫=3兩ជ兩2/ 3ប⑀
0c3, andជis the
atomic dipole moment. The collective decay rates are
⌫12=⌫21= − 3⌫
冋
cos kr共kr兲2 −
sin kr
共kr兲3
册
, 共3兲and the coupling constant for dipole-dipole interaction has the form ⍀ = −3 2⌫
冉
sin kr 共kr兲2 + cos kr 共kr兲3冊
. 共4兲We are going to consider the case when the classical field is in phase at the atomic locations, namely kជ· rជ12= 0. If the
density matrix is initially block diagonal,
=
冋
T 0 0 S册
→T=冤
11 12 13 21 22 23 31 32 33冥
, S=44, 共5兲in the total angular momentum basis, consisting of the four states 兵兩ee典, 兩s典=共兩eg典+兩ge典兲/
冑
2, 兩a典=共兩eg典−兩ge典兲/冑
2, 兩gg典其, then it will always preserve the block diagonal form. HereTis defined in the triplet part of the Hilbert space spanned by the symmetric vectors in the above basis, while S
corre-sponds to singlet subspace with antisymmetric base vector PHYSICAL REVIEW A 71, 034303共2005兲
兩a典. This fact directly follows from the equations of motion forTandS, ˙T= − i共HTT−THT†兲 +⌫ + ⌫12 2 J−SJ+ +共⌫ − ⌫12兲S兩gg典具gg兩, ˙S= −共⌫ − ⌫12兲共S−具ee兩T兩ee典兲. 共6兲 Here HT denotes the non-Hermitian Hamiltonian,
corre-sponding to the interaction picture, which can be represented in the triplet part of the basis as follows:
HT=
冤
− i⌫
冑
2E 0冑
2E ⍀ −2i共⌫ + ⌫共12兲兲冑
2E0
冑
2E 0冥
. 共7兲
From Eq.共6兲 it is clearly seen that if ⌫=⌫12, the population of the antisymmetric state will remain constant, i.e.,
equa-tions of motion for T and S will decouple. In this case, there are two independent steady-state solutions. Otherwise there will be only one solution.
It is evident from Eq.共7兲 that in the absence of the clas-sical driving field, all states except兩gg典 are damped, so that the steady-state entanglement at E = 0 is impossible, and the system evolves towards the unentangled ground state兩gg典.
Because we are interested in the robust entanglement, let us consider the steady-state solutions of the master equation
共6兲 forT. Consider first the Lamb-Dicke limit of short
in-teratomic separation. Then, it follows from the definition of the decay rate共3兲 that
⌫共12兲⬇ ⌫.
In this case, assuming that the atoms are initially prepared in their ground states, the steady-state density matrix will be determined in the triplet sector as follows:
T=
1 N
冤
64E4 − 16iE3
冑
2 8E2共2i⍀ − 1兲16iE3
冑
2 8E2共1 + 8E2兲 − 2E冑
2共2⍀ + i + 8iE2兲 − 8E2共2i⍀ + 1兲 − 2E冑
2共2⍀ − i − 8iE2兲 4共⍀2+ 2E2+ 16E4兲 + 1冥
. 共8兲
Here N is the normalization factor and⍀ and E are replaced by the dimensionless parameters⍀/⌫ and E/⌫, respectively. To determine the settings, leading to the maximum pos-sible amount of entanglement in the system under consider-ation, we choose ⍀=E2, where is a dimensionless
constant to be determined upon the maximization of concur-rence. This factor in the Lamb-Dicke limit can be repre-sented as follows:
= 3 4␣关共kr兲
3Qn¯V兴−1, 共9兲
where ␣= 1 / 137 is the fine-structure constant, Q denotes atomic quality factor 共Q=0T, and T is the lifetime of the
excited atomic state兲, n¯ is the mean number of photons per unit volume in classical driving field, and V denotes the vol-ume of interaction between atom and field, so that n¯V gives the mean number of photons interacting with atom during the time T.
The concurrence共measure of entanglement in the case of two-qubit system兲 is defined as follows 关11兴:
C = max共1−2−3−4,0兲, 共10兲 where denotes the spectrum of matrix R=共
冑
¯冑
兲1/2and¯denotes the complex conjugation of Eq.共8兲 in the so-called “magic basis” 关11兴. The maximum entangled state provides C = 1, while the unentangled states give C = 0.
One can see from Eq.共4兲 that at fixedand in the Lamb-Dicke limit kជ0· rជⰆ1, both dimensionless parameters
⍀/⌫,E/⌫Ⰷ1. In this case, the density matrix 共8兲 takes the
form T⬇ 1 2+ 48
冢
16 0 4i 0 16 0 − 4i 0 16 +2冣
. 共11兲To our surprise, the concurrence共10兲 in this limit turns out to be rational function of,
C共兲 =8− 16
2+ 48, 艌 2,
extended by zero at艋2. Thus entanglement is impossible if 艋2. The maximum value of the concurrence
Cmax=
2
冑
13 + 1⬇ 0.43 is attained atmax= 2 + 2
冑
13⬇ 9.21.The corresponding amount of entanglement关11兴 is
Emax= H
冉
1 −
冑
1 − Cmax 22
冊
⬇ 0.285 ebit.Taking into account the form of the dimensionless parameter given by Eq.共9兲, we can examine the dimensionless
inter-BRIEF REPORTS PHYSICAL REVIEW A 71, 034303共2005兲
atomic distance kជ0· rជ, corresponding to the maximum
en-tanglement provided bymax= 9.21, as a function of the
num-ber of photons n¯V, which should obey the condition n¯VⰇ1 in the case of classical driving field. It is seen that in the case of a mean number of photons n¯V⬃10, the interatomic dis-tance should be of the order of 10−2 共where is the
wave-length兲 to achieve the maximum possible amount of en-tanglement. An increase of the mean number of photons in the driving field, considered as a coherent state 兩␣典 with
兩␣兩2Ⰷ1, decreases the interatomic distance, which is
re-quired to have a maximum amount of entanglement. So far we have discussed the Lamb-Dicke limit. The re-sults of numerical calculations beyond the Lamb-Dicke limit for different values of the classical driving field are shown in Fig. 1. Both cooperations, the dipole coupling and collective decay, are oscillating functions of distance关Eqs. 共3兲 and 共4兲兴, and even when one of them becomes zero, the other can still give rise to entanglement共see Fig. 2兲. The deviation from the Lamb-Dicke limit decreases the cooperation effects, thus de-creases steady-state entanglement.
Summarizing, we have examined the system of two iden-tical two-level atoms, interacting with each other by means of vacuum induced dipole forces and collective decay. The dissipation of energy in the system is provided by the spon-taneous decay of the excited atomic states. The compensa-tion of losses is provided by a classical driving field.
It is shown that in the absence of the classical driving field, the system evolves towards an unentangled state共both atomic dipoles are in the ground state兲. The presence of the classical driving field stabilizes the entanglement.
In the Lamb-Dicke limit of a pointlike system, we ob-tained an exact solution for the steady-state density matrix, that manifests a high amount of entanglement 共the
concur-rence Cmax= 0.43 and the entanglement Emax= 0.285 ebit兲.
This amount is much higher than in a number of recent pro-posals. In particular, it is higher than that in the case when the squeezed vacuum is used for stabilization of entangle-ment instead of the classical driving field关3兴.
Outside the Lamb-Dicke limit, i.e., when ⌫12⬍⌫, both the triplet and the singlet sectors of the density matrix共5兲 are populated, and this leads to a decrease in the amount of entanglement.
In free space small, interatomic distances are required for strong atomic cooperation. However, atoms can exhibit col-lective effects in cavities, or in the vicinity of dielectric bod-ies 关12兴 even when they are spatially well separated. The prescribed scheme of steady-state entanglement generation can as well be applied to these cases.
In the above consideration, we always assumed that atoms are identical. It seems interesting to extend our consideration to the case of nonidentical atoms. In view of the result of Ref. 关3兴, we can expect that this may lead to a significant increase of entanglement.
We also restricted our consideration to the case of polar-ization of the classical driving field parallel to the inter-atomic axis. The alternative choice of the polarization per-pendicular to the interatomic axis can lead to a strong change of picture as well. First of all, the change of polarization changes the form of the coupling constant 共4兲. Then, it causes the consideration of the different values of the classi-cal driving field in the atomic locations.
The detailed analysis of the above-mentioned two exten-sions of the model deserves special consideration.
FIG. 1. Numerical dependence of concurrence on the inter-atomic distance and classical driving field. The dimensionless quan-tities r / and E/⌫ are used here. is the wavelength corresponding to atomic transition.
FIG. 2. The dipole interaction constant ⍀ 关Eq. 共4兲兴 共dashed curve兲 and collective decay rate ⌫12 关Eq. 共3兲兴 共solid curve兲 as a
function of interatomic separation r. Here r is given in terms of wavelength corresponding to atomic transition.
BRIEF REPORTS PHYSICAL REVIEW A 71, 034303共2005兲
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