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Journal of Modern Optics

On the preparation of pure states in resonant microcavities

Per Kristian Rekdal, Bo-Sture Skagerstam, P. L. Knight

Published Dec 30, 2003• 2 citations
Publisher

Abstract

We consider the time evolution of the radiation field (R) and a two-level atom (A) in a resonant microcavity in terms of the Jaynes-Cummings model with an initial general pure quantum state for the radiation field. It is then shown, using the Cauchy-Schwarz inequality and also a Poisson resummation technique, that perfect coherence of the atom can in general never be achieved. The atom and the radiation field are, however, to a good approximation in a pure state ∥ψ〉A⊗R=∥ψ〉A×∥ψ〉R in the middle of what has been traditionally called the ‘collapse region’, independent of the initial state of the atoms, provided that the initial pure state of the radiation field has a photon number probability distribution which is sufficiently peaked and phase differences that do not vary significantly around this peak. An approximate analytic expression for the quantity Tr[ρ2 A(t)], where ρA (t) is the reduced density matrix for the atom, is derived. We also show that under quite general circumstances an initial entangled pure state will be disentangled to the pure state ∥ψ〉PSA⊗R.

DOI
10.1080/0950034031000121731

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We consider the time evolution of the radiation field (R) and a two-level atom (A) in a resonant microcavity in terms of the Jaynes-Cummings model with an initial general pure quantum state for the radiation field. It is then shown, using the Cauchy-Schwarz inequality and also a Poisson resummation technique, that perfect coherence of the atom can in general never be achieved. The atom and the radiation field are, however, to a good approximation in a pure state ∥ψ〉A⊗R=∥ψ〉A×∥ψ〉R in the middle of what has been traditionally called the ‘collapse region’, independent of the initial state of the atoms, provided that the initial pure state of the radiation field has a photon number probability distribution which is sufficiently peaked and phase differences that do not vary significantly around this peak. An approximate analytic expression for the quantity Tr[ρ2 A(t)], where ρA (t) is the reduced density matrix for the atom, is derived. We also show that under quite general circumstances an initial entangled pure state will be disentangled to the pure state ∥ψ〉PSA⊗R.
We consider the time evolution of the radiation field (R) and a two-level atom (A) in a resonant microcavity in terms of the Jaynes-Cummings model with an initial general pure quantum state for the radiation field. It is then shown, using the Cauchy-Schwarz inequality and also a Poisson resummation technique, that perfect coherence of the atom can in general never be achieved. The atom and the radiation field are, however, to a good approximation in a pure state ∥ψ〉A⊗R=∥ψ〉A×∥ψ〉R in the middle of what has been traditionally called the ‘collapse region’, independent of the initial state of the atoms, provided that the initial pure state of the radiation field has a photon number probability distribution which is sufficiently peaked and phase differences that do not vary significantly around this peak. An approximate analytic expression for the quantity Tr[ρ2 A(t)], where ρA (t) is the reduced density matrix for the atom, is derived. We also show that under quite general circumstances an initial entangled pure state will be disentangled to the pure state ∥ψ〉PSA⊗R.
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.
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 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 two-level atoms with degenerate Raman two-photon transitions interacting with one-mode coherent or thermal radiation field in cavity is presented. Asymptotic solution for system state vector is obtained in the approximation of large initial coherent fields. The atom-field 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 is shown. Conditions and times of disentanglement are derived. The atom-atom entanglement is investigated with using negativity. The possibility of sudden death and birth of atom-atom entanglement is predicted for a coherent field with large mean photon numbers.
An exact solution of the problem of two two-level atoms with degenerate two-photon transitions interacting with a one-mode coherent radiation field 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 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 the model considered is shown. The conditions and times of disentanglement are derived.