Time evolution of entangled biatomic states in a cavity

Abstract

We study the time evolution of entangled states of a pair of identical atoms, considered in the harmonic approximation, coupled to an environment represented by an infinite set of free oscillators, with the whole system confined within a spherical cavity of radius R. Taking the center-of-mass and the relative-position coordinates, and using the dressed-state approach, we present the time evolution of some quantities measuring the entanglement, for both limits of a very large and a small cavity; the chosen examples are simple and illustrate these very distinct behaviors. PACS number(s): 37.30.+i; 03.67.Mn; 42.50.Pq; 03.65.Ud IntroductionInteracting or noninteracting parts of a quantum system can share entangled states that hold quantum correlations [1,2]. An recent review covering all relevant aspects of the subject is in [3]. Entanglement is a purely quantum phenomenon due to the attribution of physical meaning to superposed states, a concept with no correspondence in classical physics. Entanglement means that individual parts of a quantum system are not independent of each other, even if they do not interact, and their quantum properties are described by their common wavefunction. In particular, entanglement properties of bipartite systems have been largely investigated along recent years. Within this context, we consider here a simple biatomic system, in which each atom is modeled by a harmonic oscillator. In this case, studies have been performed for a noninteracting bipartite system, with different approaches, as for instance in Refs. [4,5] and when an interaction between the oscillators is taken into account [6][7][8][9][10][11][12][13][14].In this Brief Report, we study the time evolution of a superposition of two biatomic states of identical atoms which interact indirectly via the coupling with the harmonic modes of a field force representing the environment. We consider the two atoms in the harmonic approximation and assume that the whole system resides inside a spherical cavity of radius R. Our basic objects will be dressed states, corresponding to the atoms dressed by the field. The biatomic system will be consistently described by the pair consisting of the "center-of-mass" and the "relative-position" oscillators, a procedure already employed in the literature, as for instance in Refs. [6,7]. In our case, these oscillators will be appropriately dressed by the field. We will consider the entangled state formed by the superposition of two kinds of states: one state in which the "center-of-mass" oscillator is in its first excited level and the "relative-position" oscillator is in the ground state; this state is superposed with another state in which the oscillators have their role reversed. We will be concerned by the system at zero temperature, i.e. all the field modes are in their ground states. Actually, it

Entanglement of a two-atom system driven by the quantum vacuum in arbitrary cavity size

“…We call atention to the fact that in Ref. [36] the authors considered the same model here. However the authors considered the time evolution of a superposition of dressed states related to the center of mass and relative coordinates.…”

“…where η = √ 2g∆ω k . Finally, the infinite sum that appears in (36) or (41) can be done numerically, by noting that for large Ω r , the coefficient (t r 0 ) 2 approaches zero as 1/(Ω r ) 2 .…”

Uncertainty principle in a cavity at finite temperature

“…A detailed exposition of our formalism and of its meaning, for both zero-and finite temperature can be found in Refs. [6][7][8][9][10].…”

“…At finite temperature it is obtained in [8], for a small cavity, that the occupation number of a simple atom in a heated environment has an oscillatory behavior with time and that its mean value increases with increasing temperature. In [9,10] the behavior of an entangled bipartite system at zero and finite temperature is investigated; taking two measures of entanglement, it results an oscillatory behavior for a small cavity (entanglement is preserved at all times), while it disappears as t → ∞ for free space.…”

Spontaneous Decay of a Dressed Harmonic Oscillator Inside a Spherical Cavity

“…[19] as a method to account, in a nonperturbative way, for the oscillator radiation process in free space. In subsequent works, the concept was used to study the spontaneous emission of atoms inside small cavities [20], the quantum Brownian motion [21,22], the thermalization process [23][24][25], the time evolution of bipartite systems [26][27][28], the entanglement of biatomic systems [29][30][31], and other related issues [32][33][34][35][36]. For a clear explanation, see reference [33].…”