Elsevier

Annals of Physics

Volume 325, Issue 4, April 2010, Pages 816-834
Annals of Physics

Dynamics of entanglement in two-qubit open system interacting with a squeezed thermal bath via dissipative interaction

https://doi.org/10.1016/j.aop.2010.01.003Get rights and content

Abstract

We study the dynamics of entanglement in a two-qubit system interacting with a squeezed thermal bath via a dissipative system–reservoir interaction with the system and reservoir assumed to be in a separable initial state. The resulting entanglement is studied by making use of concurrence as well as a recently introduced measure of mixed state entanglement via a probability density function which gives a statistical and geometrical characterization of entanglement by exploring the entanglement content in the various subspaces spanning the two-qubit Hilbert space. We also make an application of the two-qubit dissipative dynamics to a simplified model of quantum repeaters.

Introduction

Open quantum systems take into account the effect of the environment (reservoir or bath) on the dynamical evolution of the system of interest thereby providing a natural route for discussing damping and dephasing. One of the first testing grounds for open system ideas was in quantum optics [1]. Its application to other areas gained momentum from the works of Caldeira and Leggett [2], and Zurek [3], among others. The total Hamiltonian is H=HS+HR+HSR, where S stands for the system, R for the reservoir and SR for the system–reservoir interaction. Depending upon the system–reservoir (SR) interaction, open systems can be broadly classified into two categories, viz., quantum non-demolition (QND) or dissipative. A particular type of quantum non-demolition (QND) SR interaction is given by a class of energy-preserving measurements in which dephasing occurs without damping the system, i.e., where [HS,HSR]=0 while the dissipative systems correspond to the case where [HS,HSR]0 resulting in decoherence along with dissipation [4].

A prototype of dissipative open quantum systems, having many applications, is the quantum Brownian motion of harmonic oscillators. This model was studied by Caldeira and Leggett [2] for the case where the system and its environment were initially separable. The above treatment of the quantum Brownian motion was generalized to the physically reasonable initial condition of a mixed state of the system and its environment by Hakim and Ambegaokar [5], Smith and Caldeira [6], Grabert et al. [7], and for the case of a system in a Stern–Gerlach potential [8], and also for the quantum Brownian motion with non-linear system-environment couplings [9], among others.

The interest in the relevance of open system ideas to quantum information has increased in recent times because of the impressive progress made on the experimental front in the manipulation of quantum states of matter towards quantum information processing and quantum communication. Myatt et al. [10] and Turchette et al. [11] have performed a series of experiments in which they induced decoherence and decay by coupling the atom (their system-S) to engineered reservoirs, in which the coupling to, and the state of, the environment are controllable.

Quantum entanglement is the inherent property of a system to exhibit correlations, the physical basis being the non-local nature of quantum mechanics [12], and hence is a property that is exclusively quantum in nature. Entanglement plays a central role in quantum information theory [13] as in interesting non-classical applications such as quantum computation [14] and quantum error correction [15]. A number of methods have been proposed for creating entanglement involving trapped atoms [16], [17], [18].

An important issue is to study how quantum entanglement is affected by noise [19], [20], [21], [22], [23], which can be thought of as a manifestation of an open system effect [24]. In [25] entanglement of a two-mode squeezed state in a phase-sensitive Gaussian environment was studied and the criteria for the necessary and sufficient condition for separability of Gaussian continuous-variable states [26] was employed as a measure of entanglement. In [27] the entanglement between charge qubits induced by a common dissipative environment was analyzed using concurrence as the measure. Another interesting feature that has been studied is the phenomena of entanglement birth and death [28]. The interesting effect that irreversible spontaneous decay, due to interaction with a vacuum bath, can have on the revival of entanglement between two qubits was studied in [29]. Some recent experimental investigations on the influence of decoherence on the dynamics of entanglement have been made in [30], [31]. In a related work [32], this issue was taken up with the noise coming from the effect of the environment modelled by a QND SR interaction. Here we complement this program by studying the effect of noise, modelled by a dissipative SR interaction with the reservoir in an initial squeezed-thermal state [4], [33], on the entanglement evolution between two spatially separated (and initially uncorrelated) qubits, brought out by interaction with the bath. This would be of relevance to evaluate the performance of two-qubit gates in practical quantum information processing systems.

Since we are dealing here with a two qubit system which very rapidly evolves into a mixed state, a study of entanglement would necessarily involve a measure of entanglement for mixed states. Entanglement of a bipartite system [34] in a pure state is unambigious and well defined. However, mixed state entanglement (MSE) is not so well defined. Thus, although a number of criteria such as entanglement of formation [35], [36], [37] and separability [38] exist, there is a realization [35] that a single quantity is inadequate to describe MSE. This was the principal motivation for the development of a new prescription of MSE [39] in which it is characterized not as a function, but as a probability density function (PDF). The known prescriptions such as concurrence and negativity emerge as particular elements in the set of parameters that characterize the probability density function. We will make use of concurrence as well as this measure in our study of entanglement in the two-qubit system.

The plan of the paper is as follows. In Section 2, the master equation describing the dynamical evolution of the two-qubit system interacting with a squeezed thermal bath, via a dissipative SR interaction, is given which is then used in Section 3, to study in detail the dynamics of the system interacting with a vacuum bath with zero bath squeezing in Section 3.1 and with a general squeezed thermal bath in Section 3.2. In Section 4, we make a brief application of the model to practical quantum communication, in the form of a quantum repeater [40], [41]. In Section 5, we recapitulate for consistency, the recently developed entanglement measure of MSE [39]. Section 6 deals with the entanglement analysis of the two-qubit open system using concurrence as well as the PDF as a measure of entanglement. We dwell on the scenarios where the two qubits effectively interact via localized SR interactions, called the independent decoherence model, as also when they interact collectively with the bath, called the collective decoherence model. The usefulness of the PDF measure of entanglement is that it allows us to demonstrate the existence of noise regimes where even though entanglement vanishes, the state is still available for applications like NMR quantum computation, because of the presence of a pseudo-pure component. In Section 7, we make our conclusions.

Section snippets

Two-qubit dissipative interaction with a squeezed thermal bath

We consider the Hamiltonian, describing the dissipative interaction of N qubits (two-level atomic system) with the bath (modelled as a 3-D electromagnetic field (EMF)) via the dipole interaction as [42]H=HS+HR+HSR=n=1NωnSnz+ksωk(bksbks+1/2)-iksn=1N[μn.gks(rn)(Sn++Sn-)bks-h.c.].Here μn are the transition dipole moments, dependent on the different atomic positions rn andSn+=|engn|,Sn-=|gnen|,are the dipole raising and lowering operators satisfying the usual commutation

Dynamics of the two-qubit dissipative interaction with a vacuum and squeezed thermal bath

Here we present the solutions of the density matrix equation (5) for the case of a two-qubit system interacting with a (A) vacuum bath and (B) squeezed thermal bath. These results will be of use in the investigation of the dynamics of entanglement subsequently.

An application to quantum communication: quantum repeaters

The technique of entanglement purification [40] can be adapted for quantum communication over long distances, the key idea behind a quantum repeater[41]. The efficiency of quantum communication over long distances is reduced due to the effect of noise, which can be considered as a natural open system effect. For distances much longer than the coherence length of a noisy quantum channel, the fidelity of transmission is usually so low that standard purification methods are not applicable. In a

Characterization of mixed state entanglement through a probability density function

Here we briefly recapitulate the characterization of mixed state entanglement (MSE) through a PDF as developed in [39]. As pointed out in the Introduction, the above criterion was evolved from the motivation that for the characterization of MSE, a single parameter is inadequate. The basic idea is to express the PDF of entanglement of a given system density matrix (in this case, a two-qubit) in terms of a weighted sum over the PDF’s of projection operators spanning the full Hilbert space of the

Entanglement analysis

In this section, we will study the development of entanglement in the two qubit system, both for the independent as well as the collective decoherence model interacting with a squeezed thermal bath. A well known measure of MSE is the concurrence [36] defined asC=max0,λ1-λ2-λ3-λ4,where λi are the eigenvalues of the matrixR=ρρ˜,with ρ˜=σyσyρσyσy and σy is the usual Pauli matrix. C is zero for unentangled states and one for maximally entangled states. The reduced dynamics of the two-qubit

Conclusions

Here we have analyzed the dynamics of entanglement in a two-qubit system interacting with its environment, taken to be in a general squeezed thermal state, via a dissipative SR interaction. The analysis of the mixed state entanglement has been made using a measure involving a probability density function (PDF). This paper has a two-fold purpose: (a) To study two-qubit open system dynamics, in detail, from a quantum information perspective and to study the impact of various parameters such as

Acknowledgments

We wish to thank Shanthanu Bhardwaj for numerical help.

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