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Physical Review A

Spatial and temporal evolution of the first-order phase transition in intrinsic optical bistability

Ramarao Inguva, Charles M. Bowden

Published Feb 1, 1990• 44 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.41.1670

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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 system of coherently driven spatially distributed two-level atoms, coupled with each other via dipole–dipole interaction, exhibits intrinsic optical bistability (IOB) at solid-state densities,1 i.e., optical bistability without optical feedback. The origin of the IOB was shown to be due to inversion-dependent renormalization of the single-atom resonance frequency caused by dipole–dipole interaction among the atoms located mainly within an atomic resonance wavelength.2 For normally incident plane waves of intensity beyond threshold, a spatially dependent discontinuity in the nonlinear refractive index was shown to exist inside the medium,1 corresponding to spatial separation of the high and low absorption phases of the IOB. We present numerical results for the time-dependent solutions of the equations of motion and dynamics of the field propagation.
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.
Longitudinal spatial effects in the nonlinear medium are analyzed in a model for mirrorless optical bistability due to induced absorption that treats heat conduction, electromagnetic field propagation, and temperature-dependent absorption. We consider two specific cases: the temperature-induced shift of the absorption peak of bound I2 excitons in CdS single crystals and the temperature-induced shrinkage of the band gap in GaAs/GaAlAs quantum-well material. For both cases it is shown that the inhomogeneous spatial distribution of the nonlinear absorption profile along the direction of propagation of the electric field undergoes a first-order-like phase transition commensurate with the transition of the output intensity from one value to another for fixed input intensity. Stability conditions are presented for the steady states of the output intensity as a function of the input intensity for the cases treated. There is a multiplicity of bistability curves, and the stability analysis produced the expected branches of stable solutions.
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.
Using the nonlinear oscillator model as a prototype medium exhibiting intrinsic optical bistability, we investigate the inhomogeneous absorption of the electromagnetic field. The forward- and backward-field amplitudes and diffraction effects are retained in the mathematical description. Analytic results are given in the limit of plane-wave propagation under steady-state conditions. The transmitted and reflected intensity exhibit a structure that is determined by the spatial inhomogeneity of the absorption in the longitudinal direction. The transmitted intensity has a structure that is dependent on the length of the medium. The reflected intensity has an interference structure from light reflected at the front surface and the internal boundary separating a high-polarization from a low-polarization branch. A degenerate-four-wave-mixing experiment is predicted to be a very sensitive probe of the internal boundary and the interference between the forward and backward field. The phase-conjugate signal develops large oscillations as the input field is varied. Numerical results for diffraction effects are also given, and we find that the plane-wave results for the center of the beam remain reliable down to Fresnel numbers of order unity and in media that are smaller than the linear absorption length.
Optical bistability is a first-order nonequilibrium phase transition, which is characterized by the spatial or temporal coexistence of two phases. In systems with resonators this phase coexistence can be realized in the form of transverse patterns of high and low intensity values, whereas in a resonatorless optical bistability the phase coexistence becomes manifest in a longitudinal intensity variation. In systems in which strong diffusion of the elementary excitations suppresses spatial phase coexistence, random fluctuations establish at least in principal a temporal phase coexistence by stochastic switching between the two bistable states.