冉 冊 冉 冊 兺 兺 兺

Dark periods and revivals of entanglement in a two-qubit system

Z. Ficek1and R. Tanaś2

1_{Department of Physics, School of Physical Sciences, The University of Queensland, Brisbane 4072, Australia} 2

Nonlinear Optics Division, Institute of Physics, Adam Mickiewicz University, Poznań, Poland 共Received 8 April 2006; published 30 August 2006兲

In a recent paper Yu and Eberly 关Phys. Rev. Lett. 93, 140404 共2004兲兴 have shown that two initially entangled and afterward not interacting qubits can become completely disentangled in a finite time. We study transient entanglement between two qubits coupled collectively to a multimode vacuum field, assuming that the two-qubit system is initially prepared in an entangled state produced by the two-photon coherences, and find the unusual feature that the irreversible spontaneous decay can lead to a revival of the entanglement that has already been destroyed. The results show that this feature is independent of the coherent dipole-dipole interaction between the atoms but it depends critically on whether or not collective damping is present. DOI:10.1103/PhysRevA.74.024304 PACS number共s兲: 03.67.Mn, 42.50.Fx, 42.50.Lc

The problem of controlling the evolution of entanglement between atoms共or qubits兲 that interact with the environment has received a great deal of attention in recent years关1–4兴. The environment may be treated as a reservoir and it is well known that the interaction of an excited atom with the res-ervoir leads to spontaneous emission which is one of the major sources of decoherence. In light of the experimental investigations, the spontaneous emission leads to irreversible loss of information encoded in the internal states of the sys-tem and thus is regarded as the main obstacle in practical implementations of entanglement.

This justifies the interest in finding systems where spon-taneous emission is insignificant. However, in many treat-ments of the entanglement creation and entanglement dy-namics, the coupling of atoms to the environment is simply ignored or limited to the interaction of the atoms with a single-mode cavity关5–7兴.

It is well known that under certain circumstances a group of atoms can act collectively so that the radiation field emit-ted by an atom of the group may influence the dynamics of the other atoms关8–11兴. The resulting dynamics and the spon-taneous emission from the atoms may be considerably modi-fied. It was recently suggested that two suitably prepared atoms can be entangled through the mutual coupling to the vacuum field关1,2,12–14兴. Stationary two-atom entanglement is possible when the atoms are damped to the squeezed vacuum 关15兴. It has also been predicted that two initially entangled and afterward not interacting atoms can become completely disentangled in a time much shorter than the de-coherence time of spontaneous emission. This feature has been studied by Yu and Eberly关16兴 and Jakóbczyk and Jam-róz关17兴, who termed it the “sudden death” of entanglement, and elucidated many new characteristics of entanglement evolution in systems of two atoms. Their analysis, however, concentrated exclusively on systems of independent atoms.

In this paper, we consider a situation where the atoms are coupled to the multimode vacuum field and demonstrate the occurrence of multiple dark periods and revivals of entangle-ment induced by the irreversible spontaneous decay. We fully incorporate collective interaction between the atoms and study in detail the dependence of the revival time on the initial state of the system and on the separation between the atoms. We emphasize that the revival of entanglement in a

pure spontaneous emission process contrasts with the situa-tion of the coherent exchange of entanglement between at-oms and a cavity mode关5,6兴.

We consider two identical two-level atoms共qubits兲 having lower levels兩gi典 and upper levels 兩ei典 共i=1,2兲 separated by energyប␻0, where␻0is the transition frequency. The atoms are coupled to a multimode radiation field whose modes are initially in the vacuum state兩兵0其典. The atoms radiate sponta-neously and their radiation fields exert a strong dynamical influence on one another through the vacuum field modes. The time evolution of the system is studied using the Lehmberg–Agarwal关9–11兴 master equation, which reads

⳵␳ ⳵t = − i␻0

兺

_i=1 2 关Siz ,␳兴 − i

兺

i⫽j 2 ⍀ij关Si+_S j −_{,␳兴 −}1 2_i,j=1

兺

2 ␥ij共关␳Si+,S−j兴 +关Si+,Sj −_{␳兴兲, 共1兲}

where Si+共Si−兲 are the dipole raising 共lowering兲 operators and

Si z

is the energy operator of the ith atom; ␥ii⬅␥ are the spontaneous decay rates of the atoms caused by their direct coupling to the vacuum field. The parameters ␥ij and ⍀ij共i⫽ j兲 depend on the distance between the atoms and describe the collective damping and the dipole-dipole inter-action defined, respectively, by

␥ij=3 2␥

冉

sin krij krij +cos krij 共krij兲2 − sin krij 共krij兲3

冊

共2兲 and ⍀ij=3 4␥

冉

− cos krij krij +sin krij 共krij兲2 + cos krij 共krij兲3

冊

, 共3兲 where k =␻0/ c, and rij=兩rជj− rជi兩 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. The effect of the collective parameters on the time evo-lution of the entanglement in the system is the main concern of this paper.

It will prove convenient to work in the basis of four col-lective states, so-called Dicke states, defined as关8兴

PHYSICAL REVIEW A 74, 024304共2006兲

兩e典 = 兩e1典丢兩e2典,

兩g典 = 兩g1典丢兩g2典,

兩s典 = 共兩g1典丢兩e2典 + 兩e1典丢兩g2典兲/

冑

2,

兩a典 = 共兩g1典丢兩e2典 − 兩e1典丢兩g2典兲/

冑

2. 共4兲 In this basis, the two-atom system behaves as a single four-level system with the ground state 兩g典, two intermediate states兩s典 and 兩a典, and the upper state 兩e典.

In order to determine the amount of entanglement be-tween the atoms and the entanglement dynamics, we use the concurrence, which is a widely accepted measure of en-tanglement. The concurrence introduced by Wootters关18兴 is defined as

C共t兲 = max共0,

冑

␭1−

冑

␭2−

冑

␭3−

冑

␭4兲, 共5兲 where兵␭i其 are the eigenvalues of the matrix

R =␳␳˜ with˜ =␳ ␴y丢␴y␳*_{␴y丢␴y, 共6兲} and␴yis the Pauli matrix. The range of concurrence is from 0 to 1. For unentangled atoms C共t兲=0 whereas C共t兲=1 for maximally entangled atoms.

The density matrix, which is needed to calculateC共t兲, is readily evaluated from the master equation共1兲. Following Yu and Eberly, we choose the atoms to be at the initial time 共t=0兲 prepared in an entangled state of the form

兩⌿0典 =

冑

p兩e典 +

冑

1 − p兩g典, 共7兲

where 0ⱕpⱕ1. The state 兩⌿0典 is a linear superposition of only those states of the system in which both or neither of the atoms is excited. As discussed in Refs. 关16,17兴, in the absence of the coupling between the qubits, the initial en-tangled state of the form 共7兲 disentangles in a finite time. They termed this feature the sudden death of entanglement. In what follows, we examine the time evolution of the entanglement of two atoms coupled to the multimode vacuum field. If the atoms are initially prepared in the state 共7兲, it is not difficult to verify that the initial one-photon coherences are zero, i.e., ␳es共0兲=␳ea共0兲=␳sg共0兲=␳ag共0兲 =␳as共0兲=0. This implies that, for all times, the density ma-trix of the system represented in the collective basis共4兲, is given in the block-diagonal form

␳共t兲 =

冢

␳ee共t兲 ␳eg共t兲 0 0 ␳eg*_{共t兲 ␳gg共t兲 0 0} 0 0 ␳ss共t兲 0 0 0 0 ␳aa共t兲

冣

, 共8兲

with the density matrix elements evolving as ␳ee共t兲 = pe−2␥t_, ␳eg共t兲 =

冑

p共1 − p兲e−␥t, ␳ss共t兲 = pe−␥t␥+␥12 ␥−␥12共e −␥12t_{− e}−␥t兲, ␳aa共t兲 = pe−␥t␥−␥12 ␥+␥12共e ␥12t_{− e}−␥t兲, 共9兲 subject to conservation of probability ␳gg共t兲=1−␳ss共t兲 −␳aa共t兲−␳ee共t兲.

Note that the evolution of the system depends crucially on the initial conditions, and for the present initial conditions the evolution of the density matrix elements is independent of the dipole-dipole interaction between the atoms, but is profoundly affected by the collective damping ␥12. If the initial conditions are chosen differently, such that there is a nonzero coherence ␳as which oscillates with the frequency 2⍀12, the dipole-dipole interaction manifests its presence in oscillatory behavior of the concurrence关14兴.

Given the density matrix Eq.共8兲, we can now calculate the concurrenceC共t兲 and examine the transient dynamics of the entanglement. First, we find that the square roots of the eigenvalues of the matrix R are

冑

␭1,2共t兲 = 兩␳ge共t兲兩 ± 关␳ss共t兲 +␳aa共t兲兴,

冑

␭3,4共t兲 = 关␳ss共t兲 −␳aa共t兲兴 ±

冑

␳gg共t兲␳ee共t兲, 共10兲 from which it is easily verified that for a particular value of the matrix elements there are two possibilities for the largest eigenvalue, either

冑

␭1共t兲 or

冑

␭3共t兲. The concurrence is thus given by

C共t兲 = max兵0,C1共t兲,C2共t兲其, 共11兲 with

C1共t兲 = 2兩␳ge共t兲兩 − 关␳ss共t兲 +␳aa共t兲兴,

C2共t兲 = 兩␳ss共t兲 −␳aa共t兲兩 − 2

冑

␳gg共t兲␳ee共t兲. 共12兲 From this it is clear that the concurrenceC共t兲 can always be regarded as being made up of the sum of nonnegative con-tributions of the weightsC1共t兲 and C2共t兲 associated with two different classes of entangled states that can be generated in a two-qubit system. From the form of the entanglement weights it is obvious that C1共t兲 provides a measure of an entanglement produced by linear superpositions involving the ground兩g典 and the upper 兩e典 states of the system, whereas

C2共t兲 provides a measure of an entanglement produced by a distribution of the population between the symmetric and antisymmetric states. Inspection of Eq. 共12兲 shows that for

C1共t兲 to be positive it is necessary that the two-photon coher-ence ␳eg is different from zero, whereas the necessary con-dition forC2共t兲 to be positive is that the symmetric and anti-symmetric states are not equally populated.

We consider first the effect of the collective damping on the sudden death of an initial entanglement determined by the state 共7兲. The entanglement weights C1共t兲 and C2共t兲, which are needed to construct C共t兲, are readily calculated from Eqs.共7兲 and 共12兲. We see that the system initially pre-pared in the state共7兲 can be entangled according to the cri-terion C₁, and the degree to which the system is initially entangled isC1共0兲=2

冑

p共1−p兲.

If the atoms radiate independently, ␥12= 0, and then we

BRIEF REPORTS PHYSICAL REVIEW A 74, 024304共2006兲

find from Eq.共9兲 that ␳ss共t兲=␳aa共t兲. It is clear by inspection of Eq.共12兲 that in this case C2共t兲 is always negative, so we immediately conclude that no entanglement is possible ac-cording to the criterionC2, and the atoms can be entangled only according to the criterion C₁. The initial entanglement decreases in time because of the spontaneous emission and disappears at the time

td= 1 ␥ ln

冉

p +

冑

p共1 − p兲

2p − 1

冊

, 共13兲

from which we see that the time it takes for the system to disentangle is a sensitive function of the initial atomic con-ditions. We note from Eq.共13兲 that the sudden death of the entanglement of independent atoms is possible only for p ⬎1/2. Since␳ee共0兲=p, we must conclude that entanglement sudden death is ruled out for a system that is initially not inverted.

For a collective system, when the atoms are close to each other,␥12⫽0, and then the sudden death appears in less re-stricted ranges of the parameter p. This is shown in Fig. 1, where we plot the death time as a function of p for several separations between the atoms. We see that the range of p over which the sudden death occurs increases with decreas-ing r12, and for small separations the sudden death occurs over the entire range of p.

The most interesting consequence of the collective damp-ing is the possibility of entanglement revival. We now use Eqs.共9兲 and 共12兲 to discuss the ability of the system to revive entanglement in the simple process of spontaneous emission. Figure 2 shows the deviation of the time evolution of the concurrence for two interacting atoms from that of indepen-dent atoms. In both cases, the initial entanglement falls as the transient evolution is damped by the spontaneous emission. For independent atoms we observe the collapse of the en-tanglement without any revivals. However, for interacting atoms, the system collapses over a short time and remains disentangled until a time tr⬇1.7/␥ at which, somewhat counterintuitively, the entanglement revives. This revival then decays to zero, but after some period of time a new

revival begins. Thus, we see two time intervals共dark peri-ods兲 at which the entanglement vanishes and two time inter-vals at which the entanglement revives. To estimate the death and revival times, we use Eqs.共12兲 and 共9兲, and find that for ␥12⬇␥, the entanglement weightC1共t兲 vanishes at times sat-isfying the relation

␥t exp共−␥t兲 =

冑

1 − p

p , 共14兲

which for p⬎0.88 has two nondegenerate solutions td and

tr⬎td. The time td gives the collapse time of the entangle-ment beyond which the entangleentangle-ment disappears. The death zone of the entanglement continues until the time trat which the entanglement revives. Thus, for the parameters of Fig.2, the entanglement collapses at td= 0.6/␥ and revives at the time tr= 1.7/␥.

The origin of the dark periods and the revivals of the entanglement can be understood in terms of the populations of the collective states and the rates with which the popula-tions and the two-photon coherence decay. One can note from Eq.共9兲 that for short times␳aa共t兲⬇0, but␳ss共t兲 is large. Thus, the entanglement behavior can be analyzed almost en-tirely in terms of the population of the symmetric state and the coherence␳eg共t兲.

Figure3shows the time evolution ofC共t兲, the population ␳ss共t兲, and the coherence ␳eg共t兲. As can be seen from the graphs, the entanglement vanishes at the time at which the population of the symmetric state is maximal and remains zero until the time tr at which ␳ss共t兲 becomes smaller than ␳eg共t兲. We may conclude that the first dark period arises due to the significant accumulation of the population in the sym-metric state. The impurity of the state of the two-atom sys-tem is rapidly growing and entanglement disappears.

The reason for the occurrence of the first revival seen in Fig.2 is that the two-photon coherence ␳eg共t兲 decays more slowly than the population of the symmetric state. Once ␳ss共t兲 falls below 2兩␳eg共t兲兩, entanglement emerges again. Thus, the coherence can become dominant again and en-tanglement regenerated over some period of time during the

0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 td p

FIG. 1. The death time of the entanglement prepared according to the criterionC1and plotted as a function of p for different

sepa-rations between the atoms: r₁₂=␭ 共solid line兲, ␭/3 共dashed line兲, ␭/6 共dash-dotted line兲, and r12=␭/20 共dotted line兲.

0 0.02 0.04 0.06 0.08 0.1 0 2 4 6 8 10 C( t) γt

FIG. 2. Transient evolution of the concurrenceC共t兲 for the initial state兩⌿₀典 with p=0.9. The solid line represents C共t兲 for the collec-tive system with the interatomic separation r12=␭/20. The dashed

line showsC共t兲 for independent atoms,␥₁₂= 0.

BRIEF REPORTS PHYSICAL REVIEW A 74, 024304共2006兲

decay process. This is the same coherence that produced the initial entanglement. Therefore, we may call the first revival an “echo” of the initial entanglement that has been unmarked by destroying the population of the symmetric state. It is interesting to note that the entanglement revival appears only for large values of p, and is most pronounced for p⬎0.88. This is not surprising because for p⬎1/2 the system is ini-tially inverted, which increases the probability of spontane-ous emission.

We have seen that the short-time behavior of the entangle-ment is determined by the population of the symmetric state of the system. A different situation occurs at long times. As is

seen from Fig. 2, the entanglement revives again at longer times and decays asymptotically to zero as t→⬁. The second revival has completely different origin from the first one. At long times both␳ss共t兲 and ␳eg共t兲 are almost zero. However, the population␳aa共t兲 is sufficiently large as it accumulates on the time scale t = 1 /共␥−␥12兲 which is very long when ␥12 ⬇␥. A careful examination of Eq.共9兲 shows that C1共t兲⬍0 at long times, so that the long-time entanglement is determined solely by the weight C₂, which is negative for short times, and it becomes positive after a finite time tr

2共second revival time兲 given approximately by the formula

tr₂⬇ 1 ␥12ln

冉

1

冑

p 4␥ ␥−␥12

冊

. 共15兲

It follows from the above analysis and Fig. 2 that the en-tanglement prepared according to the criterionC1 is a rather short-lived affair compared with the long-lived entanglement prepared with the criterion C2. Asymptotically, the concur-rence is equal to the population␳aa共t兲.

We conclude with an example of possible experimental observation of the features predicted in this paper. In prin-ciple this system may be realized in a scheme similar to that used by Osnaghi et al.关6兴 to observe entanglement between two atoms. The scheme involves two Rydberg atoms travers-ing a semiconductor microwave cavity of the resonant wave-length ⬃0.6 cm. At such long wavelengths, the interatomic separations smaller than the resonant wavelength that we have assumed here could be realized without much difficulty. This work was supported in part by the Australian Re-search Council and the Polish Ministry of Education and Science Grant No. 1 P03B 064 28.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

γt

FIG. 3. Origin of the first dark period and revival of the en-tanglement of a collective system. The time evolution of the coher-ence 2兩␳eg共t兲兩 共dashed line兲 is compared with the evolution of the

population␳ss共t兲 共dash-dotted line兲 for the same parameters as in

Fig.2. The solid line is the time evolution of the concurrenceC共t兲 =C1共t兲.

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