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Physical Review A

State evolution in the two-photon atom-field interaction with large initial fields

Ho Trung Dung, Nguyen Dinh Huyen

Published Jan 1, 1994• 22 citations
Publisher

Abstract

The state evolution in the two-photon atom-field interaction with large initial fields is studied. It is shown that under certain conditions, for the initial coherent field, at one-quarter and three-quarters of the revival time the atom is converged into unique pure states, and at the revival time both the atom and field return to their initial states. The possibility for the field macroscopic superposition states to be generated is discussed. The effects of the cavity damping, photon statistics, and dynamic Stark shifts are investigated. Our results give some further insight into the validity of the effective two-photon Hamiltonian which neglects the Stark shifts.

DOI
10.1103/physreva.49.473

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The state evolution in the two-photon atom-field interaction with large initial fields is studied. It is shown that under certain conditions, for the initial coherent field, at one-quarter and three-quarters of the revival time the atom is converged into unique pure states, and at the revival time both the atom and field return to their initial states. The possibility for the field macroscopic superposition states to be generated is discussed. The effects of the cavity damping, photon statistics, and dynamic Stark shifts are investigated. Our results give some further insight into the validity of the effective two-photon Hamiltonian which neglects the Stark shifts.
We show that the resonant interaction between a two-level atom and a quantized field mode via two-photon transitions leads to extreme quantum entanglement between the atom and quantum field. Nevertheless during the time evolution there are moments at which the atom-field system becomes asymptotically disentangled. We investigate statistical properties of the pure-field states generated at such times. We show that at the quarter of the revival time the field is produced in the pure superposition state (Schrödinger cat state) composed of two coherent states with the same amplitude but which are out of phase by 90° (we obtain approximate analytical solution for this superposition state). We show that the interference between component states leads to non-classical oscillations in the photon number distribution. At the revival time the field is again in the pure state (we present an approximate analytical solution for the corresponding state vector). This pure state is not a superposition state. Nevertheless it exhibits a certain degree of sub-Poissonian photon statistics. We briefly discuss the influence of the Stark shift on the production of superposition states.
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.
The problem of a single atom making two-photon transitions in the presence of a cavity field that is initially in the general squeezed state or squeezed vacuum state is studied. Closed expressions for the atomic inversion, the second-order coherence function, and the quadrature variances are derived under the assumption of a large initial average photon number and a squeezed field input. The effects of Stark shift on these quantities are analyzed and discussed. We see collapses and revivals of atomic inversion and the second-order coherence function with a general squeezed input, but none are apparent when the input is in the squeezed vacuum state. Perodic revivals of squeezing are seen.
The dynamics of an exact two-photon Hamiltonian is used to study the time evolution of an initially disentangled pure state of the atom-field system as it goes through cycles of entanglement separated by instances of disentanglement. For specific initial states of the electromagnetic field, the output state is a pure quantum superposition of a squeezed vacuum state and an orthogonal, odd-photon-number state. The odd-photon-number state, which is not a squeezed state, exhibits both nonclassical sub-Poissonian and classical super-Poissonian photon statistics. In the latter case the quantum superposition resembles a macroscopic superposition state. Conditions are obtained on the atom-cavity interaction time for such states to represent the steady states in the injection in a high-QQQ cavity of a monoenergetic, low-density beam of three-level atoms in a coherent state.
Atomic-dipole squeezing in the presence of the Stark shift in two-photon Jaynes-Cummings model for an atom initially prepared in a coherent superposition of states, interacting with the coherent- or squeezed-vacuum-state field, is studied. Closed expressions for dispersive and absorptive components of the dipole for the coherent-field input under the large-n\ifmmode\bar\else\textasciimacron\fi{} approximation are derived. It is shown that atomic-dipole squeezing is sensitive to both the phase of the initial field and the relative phase between the two atomic levels. Our results show that, for certain choices of these phases, permanent dipole squeezing can be obtained.