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Physical Review A

Evolution of wave functions in the two-photon Jaynes-Cummings model: The generation of superpositions of coherent states

Tahira Nasreen, Khalid Zaheer

Published Jan 1, 1994• 22 citations
Publisher

Abstract

The pure-state evolution of the atomic as well as the field states in the resonant two-photon Jaynes-Cummings model with the field initially in the coherent state is studied. Analytic expressions for various quantities involved are obtained under the large-n\ifmmode\bar\else\textasciimacron\fi{} approximation. It is shown that the atom as well as the field are approximately in the pure state for gt=k\ensuremath{\pi}/4, where k is an integer and in a completely pure state for gt=k\ensuremath{\pi}. The field states generated at the times gt=k\ensuremath{\pi}/4 are shown to be a quantum superposition of the coherent states for a particular choice of the relative phase. The interference between these states leads to a spiked photon-number distribution but the field is still Poissonian.

DOI
10.1103/physreva.49.616

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arXiv:quant-ph/0106031•Journal of Modern Optics•Jan 1, 2002•9 citations
The pure-state evolution of the atomic as well as the field states in the resonant two-photon Jaynes-Cummings model with the field initially in the coherent state is studied. Analytic expressions for various quantities involved are obtained under the large-n\ifmmode\bar\else\textasciimacron\fi{} approximation. It is shown that the atom as well as the field are approximately in the pure state for gt=k\ensuremath{\pi}/4, where k is an integer and in a completely pure state for gt=k\ensuremath{\pi}. The field states generated at the times gt=k\ensuremath{\pi}/4 are shown to be a quantum superposition of the coherent states for a particular choice of the relative phase. The interference between these states leads to a spiked photon-number distribution but the field is still Poissonian.
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.
We study the evolution of the wavefunction in the intensity dependent Jaynes—Cummings model. It is shown that apart from the periodic revivals indicated by the dynamics of inversion, half-time revivals of the wavefunction take place in the limit of [nbar] ≫ 1. The decoupled atomic state is a superposition of the upper and lower levels and is independent of the initial atomic condition. The decoupled field state is an even or odd coherent state with a spiked distribution function, depending on the initial atomic state.
The evolution of the atomic state in the resonant Jaynes-Cummings model (a two-level atom interacting with a single mode of the quantized radiation field) with the field initially in a coherent state is considered. It is shown that the atom is to a good approximation in a pure state in the middle of what has been traditionally called the ``collapse region.'' This pure state exhibits no Rabi oscillations and is reached independently of the initial state of the atom. For most initial states a total or partial ``collapse of the wave function'' takes place early during the interaction, at the conventional collapse time, following which the state vector is recreated, over a longer time scale.