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arXiv:1205.4936 · Physical Review A

Effect of retardation on the dynamics of entanglement between atoms

Qurrat-ul-Ain Gulfam, Z. Ficek, Jörg Evers

Published Aug 22, 2012• 11 citations
arXivHTMLPDF

Abstract

The role of retardation in the entanglement dynamics of two distant atoms interacting with a multimode field of a ring cavity is discussed. The retardation is associated with a finite time required for light to travel between the atoms located at a finite distance and between the atoms and the cavity boundaries. We explore features in the concurrence indicative of retardation and show how these features evolve depending on the initial state of the system, distance between the atoms, and the number of modes to which the atoms are coupled. In particular, we consider the short-time and the long-time dynamics for both the multi- and subwavelength distances between the atoms. It is found that the retardation effects can qualitatively modify the entanglement dynamics of the atoms not only at multi- but also at subwavelength distances. We follow the temporal evolution of the concurrence and find that at short times of the evolution the retardation induces periodic sudden changes of entanglement. To analyze where the entanglement lies in the space spanned by the state vectors of the system, we introduce the collective Dicke states of the atomic system that explicitly account for the sudden changes as a periodic excitation of the atomic system to the maximally entangled symmetric state. At long times, the retardation gives rise to periodic beats in the concurrence that resemble the phenomenon of collapses and revivals in the Jaynes-Cummings model. In addition, we identify parameter values and initial conditions at which the atoms remain separable or are entangled without retardation during the entire evolution time, but exhibit the phenomena of sudden birth and sudden death of entanglement when the retardation is included.

DOI
10.1103/physreva.86.022325

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The role of retardation in the entanglement dynamics of two distant atoms interacting with a multimode field of a ring cavity is discussed. The retardation is associated with a finite time required for light to travel between the atoms located at a finite distance and between the atoms and the cavity boundaries. We explore features in the concurrence indicative of retardation and show how these features evolve depending on the initial state of the system, distance between the atoms, and the number of modes to which the atoms are coupled. In particular, we consider the short-time and the long-time dynamics for both the multi- and subwavelength distances between the atoms. It is found that the retardation effects can qualitatively modify the entanglement dynamics of the atoms not only at multi- but also at subwavelength distances. We follow the temporal evolution of the concurrence and find that at short times of the evolution the retardation induces periodic sudden changes of entanglement. To analyze where the entanglement lies in the space spanned by the state vectors of the system, we introduce the collective Dicke states of the atomic system that explicitly account for the sudden changes as a periodic excitation of the atomic system to the maximally entangled symmetric state. At long times, the retardation gives rise to periodic beats in the concurrence that resemble the phenomenon of collapses and revivals in the Jaynes-Cummings model. In addition, we identify parameter values and initial conditions at which the atoms remain separable or are entangled without retardation during the entire evolution time, but exhibit the phenomena of sudden birth and sudden death of entanglement when the retardation is included.
The effect of retardation in the atom-field interaction on the dynamics of entanglement in a double Jaynes-Cummings system is investigated. We consider large cavities in which a finite time necessary for light to travel between the atoms and the cavity mirrors may result in retardation effects. Our results demonstrate the qualitatively new behaviour observable in the time evolution of entanglement when the retardation effects are included. Solutions for single and double excitation in the system are presented. We follow the temporal evolution of an initial entanglement and find that the evolution is affected drastically by the retardation effects. In particular, the harmonic oscillations of the atomic populations and the concurrence, characteristic of single-mode Jaynes-Cummings systems, are suppressed when the retardation effects are included. The process of revival of the entanglement degrades with an increasing number of the cavity modes to which the atoms are coupled. It is also found that the effect of the retardation on the doubly excited states is more drastic than on the single excitation states that at relatively short times, the retardation leads to a complete distortion of entanglement carried by a doubly excited state.
We investigate the entanglement dynamics via the concurrence of two distant atoms interacting off-resonantly with two cavity fields in Fock states, respectively. We find that the evolution of entanglement has sudden death and sudden birth phenomena, that with the increase of photon numbers in the two cavities, the alternate frequency of sudden death and sudden birth turns fast, and that the amplitude of concurrence oscillates regularly with oscillation frequency becoming slow when the cavity fields have the same photon numbers. While, the maximum of concurrence declines and the amplitude of concurrence oscillates irregularly when the two cavity fields have different photon numbers. In addition, we find the length of death time is dependent on the initial entanglement.
We consider the time evolution of quantum entanglement in the context of interactions between atoms and the electromagnetic field. We explore the influence of interatomic separation and the degree to which this can change the qualitative character of those dynamics, including entanglement generation, protection, and sudden death (SD). We find that the qualitative features of entanglement dynamics can be changed entirely in few-mode models when atomic spacing is varied, allowing for particular choices of configuration that are favorable for maintaining entanglement. We also examine the inaccuracies introduced by the use of common approximations: We characterize unexpected errors that result from using perturbative master equations as well as those that result from using the rotating-wave approximation (RWA). We find that in dissipative systems the errors introduced by these approximations can lead to an incorrect picture of late-time dynamics. Standard perturbative master equations using the RWA are constrained to predict that late-time SD occurs to only some initial states at zero temperature, but this is merely an artifact of those approximation and generally not correct. The same master equations predict that at finite temperature all states are separable asymptotically at late times and must undergo SD. In fact a proper accounting of environmentally-induced corrections to the steady state of the system shows that for low temperatures it is possible to have asymptotic entanglement in some cases. We derive a master equation for two atoms interacting with the free field without using the RWA and solve it to obtain the dynamics, including the effects of distance. From these dynamics we find that, in fact, all initial states of atoms separated by any positive distance undergo SD even at zero temperature, though there are sub-radiant states that can be quite long-lived for closely spaced atoms.
An investigation is reported of the collective effects and the dynamics of atom–atom entanglement in a system of two distant two-level atoms which are coupled via an optical element. In the system under consideration, the two atoms, which are trapped in the foci of a lens, are coupled to a common environment being in the vacuum state and they emit photons spontaneously. A fraction of the emitted photons from each atom is thus focused on the position of the other atom. The presence of optical element between two distant atoms leads to the occurrence of delayed collective effects, such as delayed dipole–dipole interaction and delayed collective spontaneous emission, which play the crucial role in the dynamical behaviour of the entanglement. We discuss the phenomena of entanglement sudden birth, entanglement sudden death, and revival of entanglement for both cases of initial one-photon and initial two-photon unentangled atomic states. We show that the evolution of the entanglement is sensitive not only to the interatomic distance but also to the initial state of the system as well as to the properties of the optical element.
Extending the double Jaynes–Cummings model to a more complicated case where the mode–mode competition is considered, we investigate the entanglement character of two isolated atoms by means of concurrence, and discuss the dependence of atom–atom entanglement on the different initial state and the relative coupling strength between the atom and the corresponding cavity field. The results show that the amplitude and the period of the atom–atom entanglement evolution can be controlled by the choice of initial state and relative coupling strength, respectively. We find that the phenomenon of entanglement sudden death (ESD) is sensitive to the initial conditions. The length of the time interval for zero entanglement depends not only on the initial degree of entanglement between two atoms but also on the relative coupling strength of atom–field interaction. The ESD effect can be weakened by enhancing the mode–mode competition between the three- and single-photon processes.