Cent. Eur. J. Phys.

DOI: 10.2478/s11534-013-0326-x

VERS ITA

Central European Journal of Physics

Coherence-controlled stationary entanglement between two atoms embedded in a bad cavity injected with squeezed vacuum

Research Article

Xiang-Ping Liao1*, Jian-Shu Fang1, Mao-Fa Fang2

1 College of Science, Hunan University of Technology, Zhuzhou, Hunan 412007, China

2 Department of Physics and Information Science, Hunan Normal University, Changsha, 410081, Hunan, China

Received 07 November 2012; accepted 04 November 2013

Abstract: We investigate the entanglement between two atoms in an overdamped cavity injected with squeezed

vacuum when these two atoms are initially prepared in coherent states. It is shown that the stationary entanglement exhibits a strong dependence on the initial state of the two atoms when the spontaneous emission rate of each atom is equal to the collective spontaneous emission rate, corresponding to the case where the two atoms are close together. It is found that the stationary entanglement of two atoms increases with decreasing effective atomic cooperativity parameter. The squeezed vacuum can enhance the entanglement of two atoms when the atoms are initially in coherent states. Valuably, this provides us with a feasible way to manipulate and control the entanglement, by changing the relative phases and the amplitudes of the polarized atoms and by varying the effective atomic cooperativity parameter of the system, even though the cavity is a bad one. When the spontaneous emission rate of each atom is not equal to the collective spontaneous emission rate, the steady-state entanglement of two atoms always maintains the same value, as the amplitudes of the polarized atoms varies. Moreover, the larger the degree of two-photon correlation, the stronger the steady-state entanglement between the atoms.

PACS C200B): 03.65.Ud; 03.67.Mn; 03.65.Yz

Keywords: quantum entanglement • the squeezed vacuum • coherent states • quantum control

© Versitasp. zo.o.

1. Introduction

Entanglement Is a kind of quantum correlation that has played a central role in quantum information. It has been

^E-mail: lLaoxp1@126.com

found to be an indispensable resource In various quantum Information processes [1-5], but the inevitability of the interaction between the system of interest and its environment may cause decoherence and disentanglement. Many authors have shown that the collective interaction with a common thermal environment can cause the entanglement of qubits [6-9].

The study of the controlled entanglement dynamics is of

Springer

considerable Importance to the prospects of maintaining quantum information [10-16]. Model systems that theoretically exhibit the rebirth of entanglement have been proposed and discussed in several cases [17-19]. Clark and Parkins [20] have put forward a scheme to control-lably entangle the internal states of two atoms trapped in a high-finesse optical cavity by using quantum-reservoir engineering. Duan and Kimble have proposed an efficient scheme to engineer multi-atom entanglement by detecting cavity decay through single-photon detectors for generating multipartite entanglement [21]. In Ref. [22, 23], it has been shown that white noise may play a positive role in generating the controllable entanglement in some specific conditions. Malinovsy and Sola [24] have proposed a method of controlling entanglement in a two-qubit system by changing a relative phase of the pulses. In a recent paper, Yu and Eberly [25] have shown that two entangled qubits can become completely disentangled in a finite time under the influence of pure vacuum noise. In Ref. [26], Ficek and Tanas have investigated the concept of time-delayed creation of entanglement by the dissipa-tive process of spontaneous emission. They have found a threshold effect for the creation of entanglement, whereby the initially unentangled qubits can be entangled after a finite time despite the fact that the coherence between the qubits exists for all times. In Ref. [27], the author has studied the entanglement and the nonlocality of two qubits interacting with a thermal reservoir. The results show that the common thermal resevoir can enhance the entanglement of two qubits when two qubits are initially in coherent states.

In this paper, we investigate the entanglement between two atoms in an overdamped cavity injected with squeezed vacuum when these two atoms are initially prepared in coherent states. It is shown that the stationary entanglement exhibits a strong dependence on the initial state of the two atoms when the spontaneous emission rate of each atom is equal to the collective spontaneous emission rate. Conversely, when the spontaneous emission rate of each atom is not equal to the collective spontaneous emission rate, the steady-state entanglement of two atoms always maintain the same value as the amplitudes of the polarized atoms varies. Moreover, the larger the degree of two-photon correlation, the stronger the steady-state entanglement between the atoms.

2. The master equation

We study the dynamics of entanglement between two two-level atoms embedded in an overdamped cavity injected with a broadband squeezed vacuum. We assume that two

identical atoms are located in a single-mode cavity. The Hamiltonian of this system is

H = ca+a + oSz + g(a+S_ + aS+), (1)

where a and a+ are annihilation and creation operators for the cavity field, and Sz and S± are the collective pseudospin operators, which are defined as Sz = T.j=1S{z') and

S± = ^j=1S±'. The squeezing of all modes seen by the atom is difficult to realize experimentally. Instead the cavity environment, where only those modes centred around the privileged cavity mode need be squeezed, provides a much more realistic scenario for experimental investigation. The simplest situation to examine is the bad cavity limit. Moreover, it is experimentally easy to realize a cavity with the squeezed vacuum input on the one side. Thus the broadband squeezed vacuum is injected into the cavity via its lossy mirror (the other mirror is assumed to be perfect). Taking the spontaneous emission into account, the time evolution of the system of atom-field interaction is given by the following master equation [28-30]

ip = _i-[H, p] + LaP + Lop, (2)

LaP = y(2S_pS+ _ S+S_p _ pS+S_)

+ (Y12 _ y)(2S_»pS+2) + 2S_2»pS+1) _ S+S_P _ S+)S_)p _ pS(:]S_ _ pS+S-. (3)

Lop = k(N + 1)(2apa+ _ a+ap _ pa+a) + kN(2a+pa _ aa+p _ paa+) + kMeie(2a+pa+ _ a+2p _ pa+2) + kMe_ie(2apa _ a2p _ pa2), (4)

where y is the spontaneous emission rate of each atom, and y12 is the collective spontaneous emission rate stemming from the coupling between the atoms through the vacuum field, which is dependent on the separation of the atoms. If the atomic separation is much larger than the resonant wavelength, then y12 K 0; if it is much smaller than the resonant wavelength, then y12 K y. The parameter k denotes the cavity decay constant. The parameter N is the mean photon number of the broadband squeezed vacuum field. M measures the strength of two-photon correlations. They obey the relation M = + 1),

(0 < n < 1). 9 is the phase of the squeezed vacuum. We term n the degree of two-photon correlation. The injected field is an ideal squeezed vacuum when n = 1 or

a nonideal one when n == 1; if n = 0 then this implies no squeezing and our cavity field is then equivalently damped by a chaotic field.

Here, we are interested in the bad-cavity limit;that is, k » g » y, but with Ci = g2/ky finite. Ci is the effective cooperativity parameter of a single atom familiar from optical bistability. To ensure the validity of the broadband squeezing assumption, the bandwidth of squeezing must also be large compared to k. In the following, it will be convenient for us to use the basis of the collective states

[31-33] B = {|e), |s),\a), |g}}, where |e) =

\s) = -^(|ei)|g2) + |gi)k2)),

\a) = -1=(|ei)|g2)-|gi)|e2)), g = |gi)|g2). (5)

The most important property of the collective states is that the symmetric state |s) and antisymmetric state |a) are maximally entangled states.

We take the initial coherent state of the two atoms as ip(0) = cos(a^e-^gj) + sin(a)e'pe2), which can be generated by controlling the relative phase of the external fields [34]. Here p is a relative phase. Using the BornMarkoff approximation and tracing over the field state [3537], and using the atomic basis as B = {|e), |s), |a), |g)}, we can write down the time evolution equations of the density matrix elements for the atoms

= [-+ 1) - 4y]pee + 4|2Npss - 2|2nVN(N + 1) exp(-i9)pe, - 2f2iVN(N + 1) exp(i8)pge,

Pss = [4j2(N + 1)+2(y + Yi2]Pee - [^(2N + 1) + 2(y + Yn)]pss + ^nVN(N + 1)exp(-i9)peg

nVN(N + 1) exp(iQ)pge + 4kg2Npgg,

Paa = 2(y - Yn)pee - 2(y - Yn)paa,

Peg = -2k^nVN(N + 1) exp(iQ)pee + 4g2nVN(N + 1) exp(i6)pss + [-2^(2N + 1) - 2Y]peg

- lVNW+1) exp(i9)pgg,

Pge = - 2k^ nVN (N + 1) exp(-i6)pee + ^ nV N (N + 1) exp(-i6)pss + [-^(2N + 1) - 2y ]pge

- nVNW+1) exp(-i9)pgg,

Pgg = [4g-(N + 1)+2(Y + Yn)]pss +2(y - Y12)paa - ^ nVWftTV) exp(-i6)peg

- 2g2 nVNN + 1) exp(i9)pge - Npgg,

. 2(2N + 1^

Pas = [-2Y--pas,

. 2 2(2N + 1)

Psa = [-2Y--]psa,

with the condition pgg + pee + pss + paa = i. It is evident that the other matrix elements retain their initial zero values, and only the set of eight equations (6) can have non-zero solutions when the atoms are initially in

the above coherent state. It is difficult to obtain analytical solutions to the Eqs. (6). We use fourth-order Runge-Kutta method to solve these equations with the relevant initial conditions. In order to calculate the concurrence,

we next transform them into the original basis |e-|)® \e2), |e-i)®\g2), |gi)®|e2), g^®^). In section 3, we will explore quantum entanglement between two two-level atoms embedded in a bad cavity injected with a squeezed vacuum.

3. Entanglement between two atoms

In order to quantify the degree of entanglement, we choose the Wootters concurrence C [38, 39], defined as

C = max(0, Vi _ V2 _ V3 _ V^), (7)

where A1, ...,A4 are the eigenvalues of the non-Hermition matrix p = p(ay < ay)p*(ay < ay) . p is the density matrix which represents the quantum state. The matrix elements are taken with respect to the 'standard' eigenbasis |e1)< e), |ei) < |g2), ^i )< e), ^i) < g). The concurrence varies from C = 0 for unentangled atoms to C = i for the maximally entangled atoms.

First, we consider the case of y = yi2 = 0.i, corresponding to the case where the two atoms are close together; i.e., the atomic separation is much smaller than the resonant wavelength.

In Fig. 1, we plot the time evolution of the entanglement between two atoms for y = yi2 = 0.i, n = i,-k2 = i0 and 9 = n, with (a) N = 0.05, (b)N = 0.5, (c)N = 2, when two atoms are initially in different states with the same phase p = 2n/3. From bottom to top, the lines correspond to a = n/2, a = n/2.08, a = n/2.i363, a = n/2.48i3, a = n/3 and a = n/4. This figure shows the dynamical entanglement as a varies and phase remains constant at p = 2n/3. It can be seen that the stationary entanglement of two atoms increases as the amplitude of the polarized atoms cos(a) increases (ie. a decreases). It reaches its maximum when a = n/4. We also see from Fig 1(a), (b) and (c) that the stationary entanglement decreases with increasing average photon number N. However, these steady entangled states are very robust at high temperature (see Fig. 1(c)).

Figure 2 displays the time evolution of the entanglement between two atoms for N = 0.00i, y = yi2 = 0.i, n = 0.i, gj = 3 and 9 = n, for different phases p (from bottom to top, the lines correspond to p = 0, p = n/3, p = n/2, p = 2n/3, p = 5n/6 and p = n) and for given a: (a) a = ir/2.294i, (b) a = n/i0, (c) a = n/4. This figure displays the dynamical entanglement as the relative phase p varies and a remains unchanged, and depicts how the entanglement of the two atoms depends on the relative phase p. It can be seen from this figure that

§ 0.6

0 5 10 15 20 25 30

¡5 0.6

0 5 10 15 20 25 30

Figure 1. The entanglement of two atoms versus t for y = yi2 = 0.i with (a)N = 0.05, (b)N = 0.5, (c)N = 2 when two qubits are initially in different states with the same phase p = 2n/3. From bottom to top, the lines correspond to a =

n/2, a = ttI2.08, a = nl2.i363, a = nl2.48i3, a = n/3, a = n/4.

the stationary entanglement increases with the increasing of the relative phase p. This is particularly valuable in that it provides us with a feasible way to manipulate and control the entanglement by changing the relative phases. In particular, when p = n, the entanglement of all the states (0 < a < n/4) is larger than their initial

Figure 2. The entanglement of two atoms versus t for y = yi2 = 0.i, N = 0.00i, for different phases p (from bottom to top, the lines correspond to p = 0, p = n/3, p = n/2, p = 2n/3, p = 5n/6, p = n) and given a:(a)a = nl2.294i, (b)a = nli0, (c)a = n/4.

values; this indicates the possibility of obtaining steady entangled states with a larger amount of entanglement originating from entangled states with a smaller amount of entanglement. It should be noted that the initial state with a = n/4, p = n always maintains maximal entanglement, regardless of time and temperature, as shown in Fig. 2(c). The two atoms in this initial state do not seem

to interact with the squeezed vacuum all the time. This result can be explained as follows: in an interaction picture with respect to H = ca+a + wSz, after some lengthy algebra and tracing over the field state, from the master equation (2) we find the following master equation for the atoms pa :

^ = - S-Pa + Pa S+ S- - 2S-pa S+ )

+ LaPa,

LaPa = Y(2S-PaS+ - S+S-Pa - PaS+S-)

+ (yi2 - y)(2S-)PaS+ + 2S-)PaS+) - S _ „ ç(1)<;(2) _ „ ç(2)S(1)

+ - Pa Pa + - Pa

vS+ + v *S-vS+ + vS-,

VNTÏ ,

y/N exp(i6).

One can see that the (Bell) state ^_) = ^(|i0) _ |0i ))(corresponding to the initial condition a = n/4, p = n) is in fact a "dark'state of the system, i.e. S_|^_) = S+|^_) = 0, implying that this state is not influenced by coupling to the squeezed vacuum reservoir (hence the straight line in Fig. 2(c)). However, when p = 0, Fig. 2(a), (b) and (c) show that entanglement can fall abruptly to zero before entanglement recovers to a stationary state value. The time at which the entanglement falls to zero is dependent upon the degree of entanglement of the initial state. The bigger the initial degree of entanglement, the later the entanglement vanishes. This implies that two-atom entanglement may terminate abruptly in a finite time under the influence of the squeezed vacuum. This phenomenon is referred to as "sudden death' of entanglement [25] and it has elucidated a number of new characteristics of entanglement evolution in systems of two qubits.

Figure 3 displays the entanglement of two atoms versus t for N = 0.05, y = Yi2 = 0.i, n = i, a = n/i0 and 9 = n, for different phases p (from bottom to top, the lines correspond to p = 0, p = n/3, p = n/2, p = 2n/3, p = 5n/6, p = n) and given k: (a)k = 4, (b) f2 = i0, (c)k = i00, (d)g; = i000. This figure displays the dynamical entanglement as g2 (cf. the effective atomic cooperativity parameter Ci = g2/kY) varies. It is shown that the stationary entanglement of two atoms increases with increasing g

(c) (d)

Figure 3. The entanglement of two atoms versus t for y = Yi2 = 0.i, N = 0.05, a = n/i0, for different phases p (from bottom to top, the lines correspond to p = 0, p = n/3, p = n/2, p = 2n/3, p = 5n/6, p = n) and given \: (a)k = 4, (b)k = i0, (c)k = i00,(d)k = i000.

(ie. decreasing Ci = g2/kY). From figure 3(d), we can see that, when the parameter C is very small, which corresponds to the field's inside cavity being more chaotic, the stationary entanglement of two atoms is very large. This is possible, since dissipation plays a crucial role in the generation of the stationary entanglement. From figure 3(a), we have found an interesting phenomenon: when p = 0 and k is not large, the entanglement can fall abruptly to zero twice before entanglement recovers to a stationary state value. We see two time intervals (dark periods) at which the entanglement vanishes and two time intervals at which the entanglement revives. And, with the increase of p, though the phenomenon of "sudden death'of entanglement does not occur, the rate of evolution of entanglement can suddenly change twice. Meanwhile, with the increase of gj, the entanglement can fall abruptly to zero only once before entanglement recovers to a stationary state value when p = 0 (see Figure 3(b)-(d)). Furthermore, the bigger the parameter k, the shorter the state will stay in the disentangled separable state. So, we can steer the evolution of entanglement between two atoms by varying the

effective atomic cooperativity parameter Ci of the system. Next, we discuss the situation of y = Yi2, which means that the separation between two atoms is not very small. In Fig. 4, we plot the entanglement of two atoms versus t for y = 0.i, Yi2 = 0.06, N = 0.05,k = i0 and 6 = n, with (a) n = i, (b)n = 0.7, (c)n = 0.2 when two atoms are initially in different states with the same phase p = 2n/3. From bottom to top, the lines correspond to a = n/2, a = jt/2.08, a = n/2.i363, a = n/2.48i3, a = n/3 and a = n/4. When the degree of two-photon correlation n = i, the injected field in the cavity is an ideal squeezed vacuum, corresponding to the reservoir being in an ideal or minimum uncertainty squeezed state. When the degree of two-photon correlation n = i, the injected field in the cavity is not ideal, which means that some of the photon pairs in the squeezed field are not correlated due to the cavity effect. From this figure, which displays the dynamical entanglement as a varies and phase remains constant at p = 2n/3, it is discovered that the steady-state entanglement of two atoms always remains constant as the amplitudes of the polarized atoms a vary. This result can

0 20 40 60 80 100

0 20 40 60 80 100

0 20 40 60 80 100

Figure 4. The entanglement of two atoms versus t for y = 0.i, yi2 = 0.06 with (a)n = i, (b)n = 0.7, (c)n = 0.2 when two qubits are initially in different states with the same phase p = 2n/3. From bottom to top, the lines correspond to a =

n/2, a = n/2.08, a = n/2.i363, a = n/2A8i3, a = n/3, a = n/4.

be explained as follows: when y = Yi2, Eqs.(6) imply that paa = pee. Therefore, regardless of whether the asymmetric state |a) is initially populated, in the long-time limit, due to the interaction of the nonclassical field, the asymmetric state will be equally as populated as the upper lever. Given a long time, the values of pas and psa tend

to zero, and the symmetric state |s), peg and pge tend to certain values, regardless of the initial states of the atoms. In addition, from Fig 3.(a), (b) and (c), we can see that the larger the degree of two-photon correlation n, the stronger the steady-state entanglement between the atoms. Thus, the nonclassical two-photon correlations of the injected squeezed vacuum are significant for the stationary entanglement in the system.

4. Conclusion

In this paper, we have investigated the entanglement between two atoms in an overdamped cavity injected with squeezed vacuum when these two atoms are initially prepared in coherent states. It is shown that the stationary entanglement exhibits a strong dependence on the initial state of the two atoms when y = Yi2, corresponding to the case where the two atoms are close together. It is found that the stationary entanglement of two atoms increases with decreasing effective atomic cooperativity parameter. The squeezed vacuum can enhance the entanglement of two atoms when two atoms are initially in coherent states. Valuably, this provides us with a feasible way to manipulate and control the entanglement by changing the relative phases and the amplitudes of the polarized atoms, and by varying the the effective atomic cooperativity parameter of the system even though the cavity is a bad one. When y = Yi2, the steady-state entanglement of two atoms always remains constant as the amplitudes of the polarized atoms a vary. Moreover, the larger the degree of two-photon correlation n, the stronger the steady-state entanglement between the atoms. Thus, the nonclassical two-photon correlations are significant for the entanglement in the system.

Acknowledgement

This work is supported by the National Natural Science Foundation of China (Grant No. 11074072 and No.61174075), Hunan Provincial Natural Science Foundation of china (Grant No. 10JJ3088 and No.11JJ2038) and by the Major Program for the Research Foundation of Education Bureau of Hunan Province of China (Grant No. 10A026).

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