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arXiv:1305.2604 · Journal of Physics B Atomic Molecular and Optical Physics

Bound entanglement in the Jaynes–Cummings model

Nicolás Quesada, Anna Sanpera

Published Nov 7, 2013• 10 citations
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Abstract

We study in detail entanglement properties of the Jaynes-Cummings model assuming a two-level atom (qubit) interacting with the first $N$ levels of an electromagnetic field mode (qudit) in a cavity. In the Jaynes-Cummings model, the number operator is the conserved quantity that allows for the exact diagonalization of the Hamiltonian and thus we study states that commute with this conserved quantity and whose structure is preserved under the Jaynes-Cummings dynamics. Contrary to the common belief, we show that there are bound entangled states that satisfy the symmetries imposed by the conservation of the number of excitations when $N>3$. Furthermore we show that \emph{the Jaynes-Cummings interaction can be used to generate bound-entanglement} between the atom and the mode.

DOI
10.1088/0953-4075/46/22/224002

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Dynamics of entanglement due to intensity-dependent interaction between a two-level atom and a single-mode electromagnetic field in a Kerr medium is studied. The form of the interaction is such that the Hamiltonian evolution is exactly solvable. The Hamiltonian is shown to be a deformed Jaynes-Cummings model admitting a closed, symmetric algebra. Dynamics of population inversion and atom-field entanglement are studied taking the initial state of the field to be either a coherent state or a squeezed vacuum. Analysis is extended to the case of a two-mode cavity field interacting with a two-level atom. For the two-mode case, the initial field is a pair coherent state or a two-mode squeezed vacuum. Effects due to nonlinearity, intensity-dependent interaction and detuning on the dynamics are discussed and compared with those of the single-mode case.
The proposed schemes in this paper involve the interaction of a two-level atom with single- or two-mode quantized cavity fields (for different purposes) in the presence of a classical field. Indeed, following the path of Solano et al. in [Phys. Rev. Lett.90, 027903 (2003)], the behavior of the entire atom-field system may be described by the Jaynes–Cummings (JC)- and anti-Jaynes–Cummings (anti-JC)-like models. It is illustrated that, under specific conditions, the effective Hamiltonian of the system can be switched from a JC- to an anti-JC-like model. During the process, the two-level atom in the cavity is alternately affected by the above two effective interactions. Ultimately, after the occurrence of the desired interactions in appropriate setups, the cavity field will arrive at a specific superposition of number states, a fixed number state, and in particular, two-mode binomial field states. Moreover, the entanglement property of the two-mode binomial state is investigated by evaluating the entropy criterion. While there exist various proposals for preparation of number states and their superpositions in the literature, our scheme has the advantage that it is independent of the detection of the atomic state after the interaction occurs.
We study the entanglement dynamics in the system of coupled quantum fields. We prove that if the coupling is linear, that is if the total Hamiltonian is a quadratic form of field operators, entanglement can only be transferred between the fields. We show that entanglement is produced in the model of the two-mode self-interacting boson field with the characteristic Gaussian decay of coherence in the limit of high number of particles. The interesting feature of this system is that the particles in different modes become entangled even if there is no direct interaction between the modes. We apply these results for analysis of the entanglement dynamics in the two-mode Jaynes-Cummings model in the limit of large number of photons. While the photon-atom interaction is assumed to conserve helicity the photons with different polarizations still get entangled due to an effective interaction mediated by the atom with the characteristic entanglement time linearly increasing with the number of photons.
We examine the dynamics of bipartite quantum entanglement between a two-level atom and the electromagnetic field. We begin with the Jaynes-Cummings model with a single field mode and examine in detail the exact time evolution of entanglement, including cases where the atomic state is initially mixed and the atomic transition is detuned from resonance. We then explore the effects of other nearby modes by calculating the exact time evolution of entanglement in more complex systems with two, three, and five field modes. For these cases we can obtain exact solutions which include the strong-coupling regime. Finally, we consider the entanglement of a two-level atom with the infinite collection of modes present in the intracavity field of a Fabre-Perot cavity. In contrast to the usual treatment of atom-field interactions with a continuum of modes using the Born-Markov approximation, our treatment in all cases describes the full non-Markovian dynamics of the atomic subsystem under general couplings. Only when an analytic expression for the infinite mode case is desired do we need to make a weak coupling assumption which at late times approximates Markovian dynamics.
The dynamics of the much studied Jaynes–Cummings model, of a single two-level atom interacting with a single quantized field mode, is simple to solve for because the number of quanta in the system is strictly conserved. Including the effects of cavity damping makes the model significantly more difficult to treat. Here we present a simple derivation of the evolution of the atomic inversion in the presence of cavity damping. This will enable us to explain the physical meaning of the solution, first derived more than 20 years ago, and also to show that it is valid for much stronger damping than was appreciated at that time.