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Journal of Modern Optics

Fractional revivals in optical parametric interactions

Iyyanki V. Jyotsna, G. S. Agarwal

Published Feb 1, 1997• 10 citations
Publisher

Abstract

A study is presented of the long-time dynamical behaviour of the degenerate parametric oscillator in the semiclassical and quantum regimes for various initial conditions. It is shown that the eigenstates of the trilinear Hamiltonian lead to the collapse and revival phenomena as well as the novel phenomena of fractional revivals in the photon statistics of the field modes. It is also demonstrated that fractional revivals become more prominent if the pump mode possesses sub-Poissonian statistics.

DOI
10.1080/09500349708241872

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A study is presented of the long-time dynamical behaviour of the degenerate parametric oscillator in the semiclassical and quantum regimes for various initial conditions. It is shown that the eigenstates of the trilinear Hamiltonian lead to the collapse and revival phenomena as well as the novel phenomena of fractional revivals in the photon statistics of the field modes. It is also demonstrated that fractional revivals become more prominent if the pump mode possesses sub-Poissonian statistics.
We consider a trilinear Hamiltonian in boson operators describing various physical processes such as frequency conversion, Raman or Brillouin scattering, or the interaction of N two-level atoms with a single-mode radiation field. Due to the fact that two independent integrals of motion can be found, the solution of the dynamics of the system is reduced to the diagonalization of a finite matrix (as was already shown by Walls and Barakat [Phys. Rev. A 1, 446 (1970)]). Performing a numerical diagonalization, we analyze the statistical properties of the field modes (sub-Poissonian statistics, anticorrelation, squeezing). We also pay attention to the appearance of collapses and revivals in the mean photon number of the modes. The relation of this model to the model of two coupled modes with an intensity-dependent coupling constant is pointed out.
A fully quantum treatment of the non-degenerate optical parametric oscillator in the extreme quantum limit of small photon numbers and very high nonlinear coupling strength is presented. When the nonlinear coupling constant becomes comparable with the cavity decay rate, the sharp threshold that is usually encountered disappears, a situation reminiscent of microlasers. Furthermore the output light exhibits unusual statistical properties such as strong super-Poissonian behaviour of the pump mode for (very) large coupling constants.
We analyze the quantum dynamics of the fractional-time Jaynes-Cummings model using a recent unitary framework for the fractional-time Schrödinger equation. We examine how the fractional derivative order ααα influences non-classical features under different initial conditions. For an initial Fock state, fractional evolution introduces transient dynamics and heightened sensitivity to coupling strength. Through an inverse problem approach, we interpret these effects as arising from an effective time-dependent coupling with a strong initial pulse. For an initial coherent state, the fractional order tunes the system between dynamical regimes, with a transition at $α= 0.50 $ where standard collapse-and-revival is replaced by stable, periodic evolution. This regime enhances non-classical field properties, including stronger sub-Poissonian statistics, periodic quadrature squeezing, and the formation of Schrödinger cat states.
We present a theoretical method to study a trapped gas of bosonic two-level atoms interacting with a single mode of a microwave cavity. This interaction is described by a trilinear Hamiltonian that is formally completely equivalent to the one describing parametric down-conversion in quantum optics. A system of differential equations describing the evolution, including the long-time behavior, of not only the mean value but also the variance of the number of excited atoms is derived and solved analytically. For different initial states the mean number of excited atoms exhibits periodically reappearing dips, with an accompanying peak in the variance, or fractional collapses and revivals. Closed expressions for the period and the revival time are obtained.
Abstract In this paper the interaction of a three-level atom in V -configuration with a two-mode quantized field in cavity optomechanics is studied. To achieve the purpose, we first deduce the effective Hamiltonian and evaluate the explicit time-dependent form of the state vector of the whole system by choosing special initial conditions for atom, field and the oscillatory mirror. Interestingly, we can obtain the time evolution of atomic linear entropy, population inversion, quantum statistics and squeezing, both analytically and numerically. The results show that the entanglement between the atom and the subsystem of field and mirror, and all above-mentioned physical quantities can be appropriately controlled by the initial atom-field state condition, the parameters of cavity optomechanics as well as atom-field coupling strengths. In particular, the appearance of collapse-revival phenomenon in the entanglement and quantum photon statistics, also the full sub-Poissonian statistics in the two modes of field as well as in the mechanical mode of optomechanical system are noticeable features of the work.
It is shown that if the initial photon distribution is narrow (sub-Poissonian), long-time behavior of the atomic inversion is dominated by very spectacular and almost-periodical superstructures. These superstructures generalize phenomena described previously (fractional revivals). They result from beating of not-nearest-neighbor eigenstates of the system, and we show which beats lead to particular types of revivals. In addition, the cavity field develops macroscopically distinguishable states (optical Schr\"odinger-cat states). These stages emerge when the fractional revivals with a frequency of 2/tR{\mathit{t}}_{\mathit{R}}tR​, 3tR{\mathit{t}}_{\mathit{R}}tR​, and 4/tR{\mathit{t}}_{\mathit{R}}tR​, where tR{\mathit{t}}_{\mathit{R}}tR​ is the single-revival period, appear for the second, fourth, sixth, etc., time in the superstructure pattern.