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Physical Review A

Treatment of the spectrum of squeezing based on the modes of the universe. I. Theory and a physical picture

Julio R. Gea-Banacloche, Ning Lü, Leno M. Pedrotti, Sudhakar Prasad, Marlan O. Scully, Krzysztof Wódkiewicz

Published Jan 1, 1990• 92 citations
Publisher

Abstract

For a leaky cavity driven by an active medium, we clarify the connection between the concepts of single-quasimode squeezing (for the intracavity field) and spectral squeezing (for the input and output fields). Our analysis is based on a rigorous construction of the Fox-Li quasimodes of an empty cavity by a superposition of the true modes of a perfect, large (infinitely large in the limit) cavity which fully subsumes the leaky cavity. To illustrate the physical importance of spectral squeezing (as defined here and as used by other authors), we consider a novel model of a microscopic system that is sensitive to the generalized quadratures of the field. We obtain a simple picture of the reasons why the quantum-noise reduction is in general different inside and outside the cavity. An important special case of our picture is that if the intracavity field is perfectly squeezed, then the output field shows no squeezing at all. Our considerations apply to a wide class of systems, examples of which are discussed in the companion paper.

DOI
10.1103/physreva.41.369

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For a leaky cavity driven by an active medium, we clarify the connection between the concepts of single-quasimode squeezing (for the intracavity field) and spectral squeezing (for the input and output fields). Our analysis is based on a rigorous construction of the Fox-Li quasimodes of an empty cavity by a superposition of the true modes of a perfect, large (infinitely large in the limit) cavity which fully subsumes the leaky cavity. To illustrate the physical importance of spectral squeezing (as defined here and as used by other authors), we consider a novel model of a microscopic system that is sensitive to the generalized quadratures of the field. We obtain a simple picture of the reasons why the quantum-noise reduction is in general different inside and outside the cavity. An important special case of our picture is that if the intracavity field is perfectly squeezed, then the output field shows no squeezing at all. Our considerations apply to a wide class of systems, examples of which are discussed in the companion paper.
In the preceding paper we developed a formalism based on the true modes of the universe containing a leaky cavity to analyze the relationship of quantum noise inside that cavity to that outside. We illustrate this formalism here by studying several active systems. We analyze both an ordinary laser and a laser operating in the phase-locked regime. We calculate the extracavity quadrature variances for the phase-locked laser and find that the spectrum of noise reduction in the squeezed quadrature is Lorentzian. Our formalism is also applied to a two-photon correlated-spontaneous-emission laser. We find the familiar result that with 50% squeezing in the phase quadrature of the field inside the cavity, one has nearly 100% phase squeezing outside the cavity. However, under different conditions the phase quadrature of the intracavity field is nearly perfectly squeezed, but the extracavity field shows almost no squeezing in its phase quadrature. We also analyze the effect of finiteness of measurement time on the extracavity quadrature variances.
The recent realizations of squeezed states and phase-noise quenching via correlated spontaneous emission have sharpened the question of the relationship of quantum noise inside and outside a transmitting cavity. Although many authors1 have succeeded in demonstrating that the degree of quantum noise reduction in the transmitted field can be quite different from (and even much larger than) that in the intracavity field, a rigorous yet simple physical picture of the reasons for this curious fact has not emerged.
The authors present a quantum analysis of four-wave mixing in a non-resonant nonlinear medium inside a cavity which is driven by two driving fields. In this process a nonlinear medium couples two pump modes with a signal mode of the frequency equal to the half-sum of the two pump field frequencies. Three types of stable steady-state solutions of the deterministic problem, describing three regimes of oscillation, are found: one below threshold and two different above-threshold regimes. The spectra of fluctuations of the quadrature-phase amplitudes for all three modes are calculated. It is found that in the above-threshold regime each of three modes may display a squeezed noise reduction. Below threshold a squeezing is realized for the central mode.
Master equation methods provide useful tools for analyzing the statistics of quantum optical systems. They focus attention on a simple system of interest (a single cavity mode, for example), accounting for its interaction with its environment in parameters describing decay rates and environmental fluctuations. Intracavity systems for generating squeezed light are prime candidates for analysis by these methods. It has been shown, however, that it is important in such analysis to distinguish between the statistics of the intracavity field and the statistics of the field generated outside the cavity by transmission through the cavity mirrors.1,2 It is desirable to perform calculations with the single mode inside the cavity. Then the statistics of the multimode field outside the cavity must be related to those of the internal mode. The situation is analogous to that in resonance fluorescence, where the statistics of the fluorescent field can be calculated in terms of the source dipole operator. In this paper the field outside an optical cavity is expressed in terms of the intracavity oscillator as a source. The measurement of squeezing in the external field is then related to the correlation properties of the cavity source operator. Measurement by photon counting of homodyned light is related to the spectrum of squeezing calculated by Walls and co-workers.
The theory of a two-mode nondegenerate parametric amplifier in a cavity is reformulated in terms of quadrature-phase-amplitude variables. The corrrespondence with a genuine classical stochastic linear process is found (non-negative-definite diffusion matrices) for the case of a cavity device immersed in thermal or ordinary (nonsqueezed) vacuum sources. A special kind of squeezing, i.e., quadrature squeezing [B. L. Schumaker, Phys. Rep. 135, 318 (1986)], is found to be characteristic of the internal steady state for the case of a cavity model subjected to an additional phase-sensitive noise source coupled to one of the two internal modes. Finally, the usual squeezing spectrum of the output field is calculated in both cases by means of an input-output formalism based upon a symmetric ordering scheme of noncommuting operators, well adapted for the quadrature-phase description of the fields involved in the interaction.
Previous theories of laser radiation have described the electromagnetic field in terms of a discrete set of quasimodes (Fox and Li "modes") of the laser cavity. Each quasimode was assumed to have a finite quality factor Q. In steady state (when gain due to lasing atoms cancels the losses) and in the absence of noise sources, the semiclassical theory predicts a $\ensuremath{\delta}$-function line shape. In the present semiclassical analysis (fluctuations are ignored), this theory is generalized for a maser with a cavity having a semitransparent wall as one of the mirrors so that there are now many modes of the universe corresponding to each Fox-Li-type quasimode. It is demonstrated that the normal modes of the universe associated with a single Fox-Li "mode" may, under proper conditions, lock together and the $\ensuremath{\delta}$-function laser line shape may be regained. Specifically, we investigate the conditions under which the coupled multimode equations associated with laser oscillation can be reduced to an equation of the form found in the usual single-quasimode theory.