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arXiv:quant-ph/0106031 · Journal of Modern Optics

Kerr cat states from the four-photon Jaynes-Cummings model

HONGCHEN FU, A. I. Solomon

Published Jan 1, 2002• 9 citations
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Abstract

We investigate the dynamics of a four-photon Jaynes—Cummings model for large photon number. It is shown that at certain times the cavity field is in a pure state which is a superposition of two Kerr states, analogous to the Schrödinger cat state (superposition of two coherent states) which occurs in the one and two photon cases.
DOI
10.1080/09500340110072936

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We demonstrate that the generalization of the two-level Jaynes-Cummings model (JCM) to an N-level atom leads to the creation of up to N macroscopically distinct field states. These field states are Schmidt-orthogonalized superpositions of Fock states. They correspond to macroscopic states of the field, attainable with large mean photon numbers. Unlike the situation with a two-level atom and a coherent-state field, which evolves into a macroscopic coherent superposition state (a Schrodinger cat), we find that when the additional levels participate strongly in the excitation (e.g all transitions are resonant with equal dipole moments) then the system does not evolve into a pure state. We will present some examples of special cases, giving insight into the behavior of three-level atoms and the two-level two-photon JCM.
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.
We demonstrate that the generalization of the two-level Jaynes-Cummings model to an N-level atom leads to the creation of up to N macroscopically distinct field states. These field states are Schmidt-orthogonalized superpositions of Fock states. They correspond to macroscopic states of the field, attainable with large mean photon numbers. Unlike the situation with a two-level atom and a coherent-state field, which evolves into a macroscopic coherent superposition state (a Schr\"odinger cat), we find that when the additional levels participate strongly in the excitation (e.g., all transitions are resonant with equal dipole moments) then the system does not evolve into a pure state.
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.
The dynamics of the Q-function for a two-level system coupled to one-cavity mode (Jaynes-Cummings model) in the presence of a Kerr-like medium is investigated. Starting the field mode in a coherent state and the atom in its upper state, the Q-function splits into distinct distribution functions as in the case of the Jaynes-Cummings model. In addition to this splitting the Kerr medium introduces an additional rotation and broadening of the counter-rotating contributions to the Q-function. It is further shown that the discrete superpositions of coherent states produced by the Kerr medium split into additional peaks is due to atomic coupling.