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Physical Review A

Pancharatnam phase for the Jaynes-Cummings model modified by a sudden perturbation of the two-level system

Istok P. Mendaš

Published Apr 15, 2003• 3 citations
Publisher

Abstract

The geometrical (total minus dynamical) phase of generally nonadiabatic and noncyclic evolution of initial dressed states, and also arbitrary entangled states, of the two-level system coupled to a quantum oscillator is calculated in closed form. A sudden perturbation of the two-level system leads to a discontinuous phase jump at the instant when the system makes a transition from the initial state. A possible experimental verification in the field of quantum optics is proposed.

DOI
10.1103/physreva.67.044101

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The geometrical (total minus dynamical) phase of generally nonadiabatic and noncyclic evolution of initial dressed states, and also arbitrary entangled states, of the two-level system coupled to a quantum oscillator is calculated in closed form. A sudden perturbation of the two-level system leads to a discontinuous phase jump at the instant when the system makes a transition from the initial state. A possible experimental verification in the field of quantum optics is proposed.
The coherent Jaynes-Cummings (JC) dynamics modified by a quasi-instantaneous perturbation of the two-level system is discussed, and the corresponding evolution operator is determined with the help of the anticommutator analogue of the Baker-Hausdorff lemma. The evolution operator leads to a coherent superposition of two JC evolutions, one with the original values of the coupling constant and detuning, and the other with the modified, effective values of these parameters followed (in the resonant case) by a phase pulse. This leads to various possibilities for controlling and modifying the final state of the harmonic oscillator depending on the choice of the initial state and the perturbing phase and on the instant the perturbation acts.
The Pancharatnam phase for ordinary and generalized squeezed states of the harmonic oscillator is discussed. In particular, the general expressions for the total and dynamical phases are obtained yielding, from their difference, the geometric phase for a nonadiabatic and noncyclic evolution from an initial generalized squeezed state at time t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH to the final state at a time t\ensuremath{''}. The special case of a two-photon interaction process, such as parametric amplification, is subsequently discussed. In particular, the conditions leading to cyclic evolution are examined and the corresponding Aharonov-Anandan phase is found from the general expression for the Pancharatnam phase.
We study the quasienergy spectrum and the nonadiabatic time dynamics of the open quantum system, which incorporates the two-level atom dipole coupled to both the quantized mode and the polychromatic classical field. Both the numeric calculations and the original perturbation technique are applied to track the quasilevel positions against the strength of the classical field. The first-order perturbation theory predicts the singular behavior of the system including the infinitely large squeezing of the quantized mode quadratures at certain values of the field strength. The singularity is removed here by allowing for the nonadiabatic transitions among the quasienergy states and the higher perturbations as well.
The formalism of path integral is used to treat the influence of Stark shift on the atomic population inversion (API) and Pancharatnam phase in the Jaynes–Cummings model with a nonlinear Kerr cavity. The propagators are given explicitly as perturbation series. In the case of a quantized wave interacting with a two-level system, these are summed up exactly, and the corresponding Pancharatnam phase as well as atomic population inversion, and energy spectrum with corresponding wave functions are deduced.
The Pancharatnam phase of deformed Jaynes-Cummings(J-C) model was obtained analytically.The evolution of Pancharatnam phase for deformed Jaynes-Cummings model,was also discussed.The difference between the Pancharatnam phase evolution for deformed J-C model and that of conventional J-C model has been investigated.The results show that the Pancharatnam phase reveals some information of population reversion of an atom and the quantum nature of the cavity field.And the larger is the average photon number,the larger is the difference of Pancharatnam phase evolution between deformed J-C model and conventional J-C model.This exhibits that the evolution of Pancharatnam phase is closely relative to the quantum algebra structure of the system.
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.