Effect of phase-damped cavity on dynamics of tangles of a nondegenerate two-photon JC model

doi:10.1016/j.optcom.2008.06.076

Optics Communications

Volume 281, Issue 20, 15 October 2008, Pages 5189-5193

https://doi.org/10.1016/j.optcom.2008.06.076 Get rights and content

Abstract

A two-mode cavity field coupled to a two-level atom and damped by the environment through a phase-damped process is considered. For a chosen initial state, the effects of phase damping on the purity loss of the global system and different bipartite partitions of the system (atom–two modes, mode–(atom–mode)) through the tangles are considered. In particular, the effect of phase damping on the amount of entanglement between atom and field is evaluated by the negativity.

Introduction

Environment-induced decoherence has been widely accepted as the mechanism responsible for the destruction of quantum superpositions. The environment which is represented by a thermal reservoir always exists and affects the system considered. Because of the dissipations and fluctuations originating from the coupling to the environment, the primary system collapses from a correlated state into a statistical mixture [1] irreversibly (which is usually called decoherence or purity loss). However, for the dissipative variants of the Jaynes–Cummings (JC) model, the theoretical efforts has been stimulated by experimental progress in investigation of the interaction of a single atom with electromagnetic field inside a cavity [2]. The experiments with highly excited Rydberg atoms allowed some of the predictions of the extended version of JC model to be proved. Besides the experimental drive, there also exists a theoretical motivation [3], [4], [5], [6] to include relevant damping mechanism to JC model because it makes its dynamics more interesting.

Recently entanglement in many-body physical systems has been vastly studied and it has become clear that the entanglement plays an important role in the understanding of critical and thermodynamical properties of quantum many-body systems [7]. Unlike classical correlations, quantum entanglement cannot be freely shared among many objects. For the tripartite system, the entanglement distribution amongst subsystems has been studied for the case of three qubits [8], [9], [10], [11], [12]. If the tripartite system consists of a two two-level atoms resonantly coupled to a single mode of the electromagnetic field, the entanglement distribution has been studied in [10]. But, when it consists of a two-level atom coupled to a two-mode quantum field in a cavity with Kerr-like medium via two-photon process it was studied in [11]. The entanglements for the different bipartite partitions of the system, i.e., atom–two modes, mode–mode, mode–(atom+mode), have been studied in terms of the relative entropy. While, if it consists of three-level atom and the two-mode cavity field, the entanglement distribution has been discussed in [12].

It is important to note that, in these studies of the tripartite system, the influence of the environment is not taken into account. But for the bipartite systems composed of two-level subsystems in a dissipative environment, the purity loss (decoherence) (by using the linear entropy), of the global system and of atomic and field subsystems and entanglement of the global atom-field state have been studied [13], [14], [15]. On the other hand, it is known that two-photon processes are very important in atomic systems due to the high degree of correlation between the emitted photons. Hence, one valuable extension of the JC model is the well-known two-photon JC model. With the experimental realization of two-photon micromaser [16], [17] the dissipative two-photon one-mode JC has attracted a great deal of attention [18]. It is worth mentioning that two-mode nonclassical state fields have been studied extensively both theoretically and experimentally [19], [20], [21]. An important feature of the two-mode nonclassical state field is its correlation between the two modes which has been classified as classically correlated (separable) and quantum-mechanically correlated (entangled).

In order to advance one step further in the investigation of two-photon processes, the JC model with two-mode two-photon interaction or nondegenerate two-photon JC model were are proposed. So in this paper, we study the effect of phase-damped cavity (which occurs when there is no energy exchange between the system and environment) on the tangles dynamics, which are used to measure the purity loss, and entanglement in the dispersive regime [22], [23], [24], [25]. This article is organized as follows: in Section 2 we find the time development of the atom-field state for a chosen initial state. In Section 3, the time evolution of the purity loss of global and reduced density operators of different bipartite partitions of the system (atom–two modes, mode–(atom–mode)) are calculated explicitly in order to study their purity loss. We use the negativity as an entanglement measure to study the effect of phase damping on the amount of entanglement between atom and field. Some concluding remarks are given in Section 4.

Section snippets

The nondegenerate two-photon JC model in the dispersive regime

In the two-photon processes, some intermediate states $| f 〉$ of the atom are involved. The transitions from the excited state $| e 〉$ through $| f 〉$ to the ground state $| g 〉$ and from $| g 〉$ through $| f 〉$ to $| e 〉$ with the emission and absorption of two photons at a time are taking place in these processes. However, it has been shown that, when certain conditions are satisfied this intermediate states can be adiabatically eliminated [26]. In that case, the Hamiltonian for two-photon JC model in nondegenerate case

Effect of the phase damping on the evolution of uncorrelated initial state

In order to analyze what happens to the evolution of the mentioned uncorrelated initial state with the phase damping, we study in the following section.

Conclusions

The effect of phase-damped cavity on the purity loss and the amount of entanglement of the global atom-field state in the dispersive regime is studied. The purity loss of the global system and different bipartite partitions of the system (atom–two modes, mode–(atom–mode)) can be quantified by their tangles. The time evolution of the tangle $I_{A (f_{1} f_{2})}$ have no change due to the effect of the phase damping unlike its counterpart $I_{f_{1} f_{2} (A)}$ . The output of the tangle for one-mode and the remainder of

Acknowledgements

The authors wish to record their thanks to the referees for their valuable comments that resulted in improvements of the article in many aspects.

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Effect of phase-damped cavity on dynamics of tangles of a nondegenerate two-photon JC model

Abstract

Introduction

Section snippets

The nondegenerate two-photon JC model in the dispersive regime

Effect of the phase damping on the evolution of uncorrelated initial state

Conclusions

Acknowledgements

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