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

Generalized dispersion relations and optical bistability in squeezed vacua

R. Bullough, Hanan Ali Batarfi, Saad S. M. Hassan, Munir Ibrahim, R. Saunders

Published May 6, 1996• 5 citations
Publisher

Abstract

We derive c-number Bloch-Maxwell (B-M) type equations from a quantum theory for 2-level atoms occupying a macroscopic slab-like region V of width L stimulated by a coherent-state electromagnetic field of radial frequency (omega) plus a broad-band correlated squeezed vacuum field with carrier frequency (omega) p. The squeezed vacuum field is in principle of arbitrary 3-dimensional geometry including geometries created by an optical parametric oscillator. The c-number spatially dependent B-M equations are solved self-consistently and 'exactly' in terms of a nonlinear dispersive refractive index m((omega) ) while the slab V acts, also self-consistently, as a natural Fabry-Perot cavity of finite spacing L. Very preliminary results for optical bistability in the squeezed vacuum and more exact results for anomalous dispersion and absorption in the squeezed vacuum in linearized approximation are reported. Finally we compute the effect of the dielectric on the free-field modes of the squeezed vacuum.

DOI
10.1117/12.239848

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Optical bistability is reexamined as a candidate system for the generation of squeezed states of light. Previous analyses have concluded that significant squeezing can be achieved only in a rather extreme range of parameter space relative to current experimental capabilities and only over a restricted range of frequencies near the carrier.1 By contrast we have identified a new regime in which appreciable squeezing should be obtainable for rather modest operating conditions and without excessive demands on the stability of external control parameters. Following the treatment by Carmichael,2 we have conducted a theoretical search for optimal squeezing in optical bistability in the general case without adiabatic elimination of atomic or cavity variables. Our analysis is based on a linearization of the generalized Fokker-Planck equation for a system of N atoms coupled to a single mode of a high finesse cavity. For comparable atomic and cavity decay rates, there is a mode splitting in the eigenvalue structure of the system which is responsible for the vacuum field Rabi splitting. It is near the spectral features associated with these vacuum field Rabi sidebands that we find significant squeezing. The concepts of optimum squeezing as indicated by Mandel's Q parameter and by the spectrum of squeezing are discussed with reference to the various operating regimes in optical bistability.

Annual Meeting Optical Society of America•Jan 1, 1986•2 citations
We derive c-number Bloch-Maxwell (B-M) type equations from a quantum theory for 2-level atoms occupying a macroscopic slab-like region V of width L stimulated by a coherent-state electromagnetic field of radial frequency (omega) plus a broad-band correlated squeezed vacuum field with carrier frequency (omega) p. The squeezed vacuum field is in principle of arbitrary 3-dimensional geometry including geometries created by an optical parametric oscillator. The c-number spatially dependent B-M equations are solved self-consistently and 'exactly' in terms of a nonlinear dispersive refractive index m((omega) ) while the slab V acts, also self-consistently, as a natural Fabry-Perot cavity of finite spacing L. Very preliminary results for optical bistability in the squeezed vacuum and more exact results for anomalous dispersion and absorption in the squeezed vacuum in linearized approximation are reported. Finally we compute the effect of the dielectric on the free-field modes of the squeezed vacuum.
A composite system consisting of a degenerate optical parametric oscillator (DOPO) and N two-level atoms interacting with a broadband squeezed vacuum (SV) centred at frequency ωs and an input monochromatic pumping field with a frequency ωp is analysed. The corresponding explicit analytical steady-state solutions in the central mode ωp = ωs are derived and the result displays optical bistability (OB). In addition, the influence of the broadband SV on the bistable behaviour is analysed in detail.
We review the use of operator reaction field theory in quantum optics by considering the problem of one atom in an isotropic three-dimensional broad-band squeezed vacuum. In these terms we give a quantum electrodynamical analysis of this system exact up to all contributions of order one in the ratio γω 0 -1 , the ratio of the A -coefficient to the atomic resonance frequency. These terms include all of the counter-rotating terms at this order. At zero order, namely in the usual rotating wave approximation, we find the usual Einstein rate equation while we confirm that the fluorescence spectrum consists of two Lorentzian peaks. However, at order one we find an additional oscillating term in the rate equation and additional resonant dispersive terms correct for each of the two Lorentzian peaks in the fluorescence spectrum. Similar results are expected for broad-band correlated squeezed vacua of arbitrary, rather than isotropic, geometry.
Master equation methods are used to investigate the effects of a broad-band squeezed vacuum on a three-level atom of the lambda configuration. The two-mode squeezed vacuum is treated as a Markovian reservoir in a non-stationary phase-dependent state. In addition to the squeezed vacuum the atom is driven by two coherent laser fields each of which, depending on the polarization, can couple to one or both of the atomic transitions. We show that in general the optical Bloch equations for the atomic density matrix elements have oscillatory coefficients, thereby necessitating the use of Floquet methods. For the case of equal laser frequencies, which are also equal to the carrier frequency of the squeezed vacuum, the coefficients of the Bloch equations become time independent and stationary solutions for the populations and coherences are obtained by standard matrix methods. For the ordinary vacuum the usual coherent population trapping effect at two-photon resonance is obtained, with the upper state population being zero. An unsqueezed thermal field partially destroys the trapping effect as the upper state population is no longer zero at two-photon resonance. The squeezed vacuum has the effect of improving the trapping in that the coherence hole becomes more pronounced for some values of the relative phase between the squeezed vacuum and the driving fields. The additional effects of a coherence transfer rate between the two optical coherences, which occurs for special choices of angular momentum quantum numbers are also studied. For the case of equal laser frequencies, the inclusion of this coherence transfer process destroys population trapping and reduces the lambda system to a two-level system. However, for the case of unequal laser frequencies, the coherence transfer process in combination with the squeezed vacuum can restore to some extent the population trapping. We show that other features that do not occur for two-level atoms, such as stationary population inversions between pairs of the atomic levels, also depend on the relative phase and can be enhanced in the squeezed vacuum. In the case of unequal frequencies of the driving fields the population in the upper state depends on the relative phase only when the carrier frequency of the squeezed vacuum is equal to one of the two frequencies of the driving fields. When the carrier frequency of the squeezed vacuum is slightly detuned from both frequencies of the driving fields, the population in the upper state is insensitive to the relative phase but is dependent on the degree of squeezing. For large detunings, the population does not show any dependence on the degree of squeezing and its distribution in function of the two-photon detuning is similar to that in the thermal vacuum field.
We analyze fluctuation effects of a squeezed electromagnetic reservoir (also termed squeezed vacuum, or SV) on the phenomenon of intrinsic optical bistability (IOB) in a dense collection of two-level atoms. A quantum statistical treatment is developed in the limit of a bad cavity and large dephasing in the medium. Under these conditions we obtain a single operator Langevin equation for the atomic inversion by adiabatic elimination of the atomic polarization variables. By using Ito calculus, the corresponding Fokker-Planck equation is derived, the stationary solution of which yields the probability distribution function for the bistable system as a function of the inversion. In a narrow region of transition, the probability distribution is double peaked with the two peaks corresponding to two metastable states in IOB. Quantum fluctuations can induce the system to jump randomly from one metastable state to the other with certain characteristic (passage) times. Squeezed quantum noise (as in the SV) is found to result in drastic alteration of the intrinsic bistable behavior of the system. Interestingly, the probability distribution and the passage times now exhibit critical dependence on the relative phase between the external driving field and the squeezed vacuum.
The steady state behaviour of a bistable system of a homogeneously/inhomogeneously broadened two-level atomic medium in contact with a squeezed vacuum field and placed in a ring cavity driven by a coherent field is analysed within the mean field limit where the transverse effect of the radiation field is taken into account within the single-transverse-mode model. Effects on the input–output relation and the critical value of the cooperative parameter C crit are examined and compared with these results in the plane wave approximation. The 'cooperative switching effect' between the output states of the system due to variation of the cooperative parameter C is also investigated. Linear stability analysis for the model Maxwell–Bloch equations is presented in the homogeneously broadened case.
Optical bistability is reexamined as a candidate system for the generation of squeezed states of light. Previous analyses have concluded that significant squeezing can be achieved only in a rather extreme range of parameter space relative to current experimental capabilities and only over a restricted range of frequencies near the carrier.1 By contrast we have identified a new regime in which appreciable squeezing should be obtainable for rather modest operating conditions and without excessive demands on the stability of external control parameters. Following the treatment by Carmichael,2 we have conducted a theoretical search for optimal squeezing in optical bistability in the general case without adiabatic elimination of atomic or cavity variables. Our analysis is based on a linearization of the generalized Fokker-Planck equation for a system of N atoms coupled to a single mode of a high finesse cavity. For comparable atomic and cavity decay rates, there is a mode splitting in the eigenvalue structure of the system which is responsible for the vacuum field Rabi splitting. It is near the spectral features associated with these vacuum field Rabi sidebands that we find significant squeezing. The concepts of optimum squeezing as indicated by Mandel's Q parameter and by the spectrum of squeezing are discussed with reference to the various operating regimes in optical bistability.