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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Cooperative Transients In Inter-Atomic Correlation In The Presence Of An Externally Applied Coherent Field: Relation To Intrinsic Mirrorless Optical Bistability

Charles M. Bowden, Chi C. Sung

Published Aug 2, 1982• 3 citations
Publisher

Abstract

Our model, presented earlier, which predicts the circumstances under which intrinsic mirrorless optical bistability (OB) can occur due to atomic pair correlation in a small volume, is outlined and the results presented. These results, which Predict a first-order phase transition in steady state for an externally-driven collection of a large number of atoms far removed from thermodynamic equilibriun, forms the motivation for a detailed microscopic examination of the dynamical behavior of atonic pair correlation in the presence of externally-applied coherent radiation. 4 model is presented and results are discussed for the transient dynamic evolution of two, two-level atoms separated from each other by a distance r in the presence of an externally-applied coherent radiation field. The results predict collective radiation reaction, frequency shifts, relaxation in terms of the atomic seoaration r (assumed much larger than single atom dimensions), the externally-applied field intensity and spacial uniformity of the field with respect to the inter-atomic volume.

DOI
10.1117/12.933114

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Physical Review A•Jun 1, 1979•125 citations
Our model, presented earlier, which predicts the circumstances under which intrinsic mirrorless optical bistability (OB) can occur due to atomic pair correlation in a small volume, is outlined and the results presented. These results, which Predict a first-order phase transition in steady state for an externally-driven collection of a large number of atoms far removed from thermodynamic equilibriun, forms the motivation for a detailed microscopic examination of the dynamical behavior of atonic pair correlation in the presence of externally-applied coherent radiation. 4 model is presented and results are discussed for the transient dynamic evolution of two, two-level atoms separated from each other by a distance r in the presence of an externally-applied coherent radiation field. The results predict collective radiation reaction, frequency shifts, relaxation in terms of the atomic seoaration r (assumed much larger than single atom dimensions), the externally-applied field intensity and spacial uniformity of the field with respect to the inter-atomic volume.
Dynamical transient response and longitudinal spatial evolution of the first-order phase transition are presented for intrinsic optical bistability of a system of interacting spatially distributed, coherently driven two-level atoms. It is shown that a discontinuity in the nonlinear dielectric function is established internal to the medium due to transient response to a coherent, externally applied, time-dependent driving field, following the attainment of a critical value for the input intensity. If the externally applied field intensity reaches the steady state, beyond the critical value, the discontinuity of the nonlinear dielectric function in the medium reaches a stable stationary condition and forms a spatial boundary between states of high and low polarization in the medium. Thus it is shown that intrinsic optical bistability corresponds to a spatial as well as a temporal first-order phase transition in light-matter interactions, far from thermodynamic equilibrium, where the spatial boundary between different phases is established by transient dynamical response to the incident field.
A general formulation, developed earlier, is applied to the problem of optical bistability in a sample with dimensions that are small compared with the wavelength of light. A self-consistent model of radiative relaxation is introduced in which atom–atom correlations are taken into account. This is based on the screened interaction approximation of the many-body theory. The degree of correlation is internally determined by the dynamics of the system. The validity of the S-conservation approximation is investigated; it is concluded that this approximation may be valid only in the weak-field limit, since the local-field fluctuations become appreciable as the pump-field strength is increased. As a result, the system undergoes a first-order phase transition, in contrast to the second-order transition predicted by the S-conserving model. The method is quite general, and it can be applied to other damping mechanisms. As an illustration, a simple collision model is also considered. In the limit of a zero external field and thermodynamic equilibrium, our results reduce to the results of existing theories; the system exhibits a second-order phase transition.
We use a fully quantum, many-body treatment to show mirrorless optical bistability in terms of the spacial properties of interacting two-level systems. The bistability effects are related to the coherent dipole-dipole interactions which introduce an inversion-dependent renormalization of the frequency (or relaxation). Conditions of stability and instability are discussed for a thin sample of two-level atoms with a width smaller than a wavelength. For a long sample of two-level atoms, we find that the two phases of high transmissivity and low transmissivity may coexist spacially in the material.
We use a quantum-electrodynamical, many-body treatment to show mirrorless optical bistability in terms of the spatial properties of coherent dipole-dipole interactions among interacting two-level atoms. The general theory is applied to two special cases: (1) a thin sample of two-level atoms, with a width smaller than a resonance wavelength and (2) a long sample of two-level atoms, with dimensions very large relative to a resonance wavelength. While for the thin sample we are able to use a mean-field approximation with validity, for the long sample we are compelled to take into account retardation and propagation. In both cases bistability is found to be related to a renormalization of the frequency (or relaxation rate) that is inversion dependent. For the long sample the frequency renormalization is significant for high atomic densities and for large oscillator strengths.
The authors consider a system comprised of a collection of identical two-level atoms interacting with the electromagnetic field in the dipole approximation and with an externally applied steady-state coherent driving field. The atoms are considered to interact with each other only via the electromagnetic field and are assumed to be contained within a volume much smaller than a resonance wavelength. The system is treated as a quantum-statistical ensemble in the rotating frame, rotating at the carrier frequency of the externally applied coherent field. The Hamiltonian in this frame is explicitly time independent and the exact analog of a spin temperature is defined. The thermodynamic-Green's-function method is used to determine the thermodynamic equilibrium properties of the system in the rotating frame. This results in a nonlinear relation between the applied field and the field in the volume containing the atoms. The expression is a nonlinear function of the effective or "spin" temperature as well, and has the form of a first-order phase transition with the macroscopic atomic polarization as order parameter. In the low-"spin"-temperature limit, and for perfect tuning, the expression reduces to the exact form of the nonlinear relation for optical bistability derived by Bonifacio and Lugiato and relating the incident and transmitted fields for atoms in a ring cavity in the mean-field approximation. The results contain absorptive as well as dispersive contributions. These results predict essentially different behavior as a function of off tuning compared with behavior predicted by others using statistical steady-state models far from thermodynamic equilibrium. In the limit of zero value for the applied field amplitude, the results reduce to the conditions for the well-known second-order superradiant" phase transition. The condition for the existence of the second-order phase transition is shown to depend on the cavity tuning and the photon escape rate. The authors also show why, in light of the experiments in optical bistability, this second-order phase is difficult to observe.