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Journal of Modern Optics

Cooperativity and Entanglement of Atom-field States

Igor K. Kudryavtsev, Astrid Lambrecht, H. M. Moya-Cessa, P. L. Knight

Published Aug 1, 1993• 47 citations
Publisher

Abstract

The Jaynes-Cummings model of a single two-level atom interacting with a quantized single-mode coherent field generates at the half-revival time a dynamically disentangled atom-field state. At such times, the field is in asymptotically pure Schrödinger cat state, a macroscopic superposition of distinct field eigenmodes. In this paper we address the problem of field purity when a second atom is allowed to interact with the cavity mode and becomes entangled with the first atom via their mutual cavity field with which they interact. We employ the collective Dicke states to describe the cooperative effects on the entanglement and show that the second atom spoils the purity of the field state except for special cases of the atom-field coupling or of initial conditions.

DOI
10.1080/09500349314551631

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The Jaynes-Cummings model of a single two-level atom interacting with a quantized single-mode coherent field generates at the half-revival time a dynamically disentangled atom-field state. At such times, the field is in asymptotically pure Schrödinger cat state, a macroscopic superposition of distinct field eigenmodes. In this paper we address the problem of field purity when a second atom is allowed to interact with the cavity mode and becomes entangled with the first atom via their mutual cavity field with which they interact. We employ the collective Dicke states to describe the cooperative effects on the entanglement and show that the second atom spoils the purity of the field state except for special cases of the atom-field coupling or of initial conditions.
The Jaynes-Cummings model of optical resonance describes the simplest fully quantized interaction between two quantum systems of different nature: a two-level atom (fermionic system) and a quantized field mode (bosonic system). This interaction leads to extreme quantum entanglement of the atom and field. However, the model also predicts that, at precisely half of the revival time, the atom and field become asymptotically disentangled. This disentanglement becomes more exact as the coherent-state amplitude increases. In this paper we investigate the nature of the pure-field-state superposition generated at such times. We show that this superposition is of distinguishable states of the field with the same amplitude but opposite phase. Interference between these components leads to nonclassical oscillations in photon-number distributions and squeezing in quadratures of the field. The Schr\"odinger-cat states of the field are highly transient, and depend very sensitively on the interaction time, the initial intensity of the field, and the atom-field detuning.
An exact solution of the problem of two-atom one- and two-mode Jaynes-Cummings model with intensity-dependent coupling is presented. Asymptotic solutions for system state vectors are obtained in the approximation of large initial coherent fields. The atom-field entanglement is investigated on the basis of the reduced atomic entropy dynamics. The possibility of the system being initially in a pure disentangled state to revive into this state during the evolution process for both models is shown. Conditions and times of disentanglement are derived.
An exact solution of the problem of two-atom one- and two-mode Jaynes-Cummings model with intensity- dependent coupling is presented. Asymptotic solutions for system state vectors are obtained in the approximation of large initial coherent fields. The atom-field entanglement is investigated on the basis of the reduced atomic entropy dynamics. The possibility of the system being initially in a pure disentangled state to revive into this state during the evolution process for both models is shown. Conditions and times of disentanglement are derived.
For large initial fields, the evolution of the Jaynes-Cummings model is shown to simplify considerably. For two special initial atomic states (the eigenstates of the semiclassical problem) one finds a disentangled, quasipure-state evolution for both the field and atom. For an arbitrary initial atomic state the evolution is a linear super-imposition of these two privileged alternative paths. New results include the fact that the state of the atom at half-revival time is pure and independent of the initial state (nonunitary evolution), and that the state of the field at that time is a coherent super-imposition of macroscopically distinct states (Schrodinger cat).
An asymptotic result is derived for the Jaynes-Cummings model of a two-level atom interacting with a quantized single-mode field, which is valid when the field is initially in a coherent state with a large average photon number. It is shown that for certain initial atomic states the joint atom-field wave function factors into an atomic and a field part throughout the interaction, so that each system remains separately in a pure state. The atomic part of the wave function displays a crossing of trajectories in the atom Hilbert space that leads to a unique state for the atom, independent of its initial state, at a specific time t0{\mathit{t}}_{0}t0​ (equal to half the revival time). The field part of the wave function resembles a crescent squeezed state. The well-known collapses and revivals are investigated from this perspective. The collapse appears to be associated with a ``measurement'' of the initial state of the atom with the field as the measuring apparatus. The measurement is not complete for finite average photon number: the system is instead left in a coherent superposition of macroscopically distinct states. At the half-revival time t0{\mathit{t}}_{0}t0​ this superposition of states is entirely in the field part of the state vector, so that the (pure) state of the field at that time is of the form sometimes referred to as a ``Schr\"odinger cat.'' The revivals of the population inversion are seen to be entirely due to the fact that the linear superposition of the two macroscopically distinct field states is coherent (i.e., a pure state), as opposed to an incoherent mixture.
In this paper we consider a system of identical three two-level atoms interacting at resonance with a single-mode of the quantized field in a lossless cavity. The initial cavity field is prepared in the coherent state while the atoms are taken initially to be either in the uppermost excited state "∣eee>|eee>∣eee>" or The GHZ\textmd{GHZ}GHZ-state or the W\textmd{W}W-state. For this system we investigate different kinds of atomic inversion and entanglement, which arise between the different parts of the system due to the interaction. Also the relationship, between entanglement and some other nonclassical effects in the statistical properties, such as collapses and revivals in the atomic inversion where superharmonic effects appear, is discussed. The QQQ-functions for different cases are discussed. Most remarkably it is found that the GHZ\textmd{GHZ}GHZ-state is more robust against energy losses, showing almost coherent trapping and Schrödinger-cat states can not be produced from such state. Also the entanglement of GHZ\textmd{GHZ}GHZ-state is more robust than the W\textmd{W}W-state. Another interesting feature found is that the state which has no pairwise entanglement initially will have a much improvement of such pairwise entanglement through the evolution. Sudden death and sudden revival of atoms-pairwise entanglement are produced with the W\textmd{W}W-state.