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Quantum Optics Journal of the European Optical Society Part B

Quantum noise reduction in a one-photon optical bistability and a laser with injected signal

L. A. Lugiato, E. M. Pessina

Published Dec 1, 1993• 1 citation
Publisher

Abstract

We perform a systematic comparison between the spectrum of the intensity fluctuations and the spectrum of squeezing optimized with respect to the phase of the local oscillator, for the case of a one-photon single-mode optical bistability. In the bad-cavity limit the quantum noise reduction in the intensity fluctuations becomes comparable to the optimum level of squeezing. We consider also the case of a one-photon, single-mode laser with injected signal and show that under appropriate conditions the system can exhibit a meaningful level of squeezing.

DOI
10.1088/0954-8998/5/6/003

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We perform a systematic comparison between the spectrum of the intensity fluctuations and the spectrum of squeezing optimized with respect to the phase of the local oscillator, for the case of a one-photon single-mode optical bistability. In the bad-cavity limit the quantum noise reduction in the intensity fluctuations becomes comparable to the optimum level of squeezing. We consider also the case of a one-photon, single-mode laser with injected signal and show that under appropriate conditions the system can exhibit a meaningful level of squeezing.
It is shown that a laser with a two-photon saturable absorber exhibits a phase instability similar to that of two-photon optical bistability. The authors calculate the spectrum of the intensity fluctuations in the output field under conditions of exact resonance between the amplifier and the absorber. In the bad-cavity limit the relevant part of the spectrum assumes a simple Lorentzian form; as the phase-instability threshold is approached, the reduction of quantum noise becomes perfect for small frequencies.
A two-photon laser is modeled as NNN two-level atoms interacting via a two-photon transition with a single resonant-cavity field mode. The quadratic form of this interaction makes such a system a candidate to exhibit reduced quantum fluctuations (squeezing) in one quadrature of the output field. A Fokker-Planck equation containing all leading quantum noise terms is derived following the method of Haken. A linearized fluctuation analysis reveals that there is no squeezing in a two-photon laser with injected signal. A small amount of squeezing may be present in two-photon optical bistability. Our calculations confirm the results of Lugiato and Strini, who adopted a factorization of moment equations.
We study the spectrum of squeezing in the light transmitted by a system of two-level atoms in an optical cavity driven by a coherent field, as recently observed by Kimble and collaborators. Our analysis does not introduce any restriction on the detuning parameters or on the relative order of magnitude of the atomic and cavity relaxation rates. Using the dressed-mode formalism, we derive explicit analytic expressions for the spectrum of squeezing and for the second-order intensity correlation function. We describe how the spectrum depends on the various parameters of the system; this study provides general indications on how to optimize the conditions to observe squeezing in optical bistability. A special attention is devoted to the changes of the spectrum under variation of the detuning parameters or on the ratio of cavity to atomic damping rate. In all cases, the changes are continuous and smooth, with quantitative variations in the level of squeezing. This indicates the existence of a single avenue to squeezing in this system. The bad-cavity direction is identified as especially convenient to obtain a significant amount of squeezing.
We present an exact nonlinear derivation of the spectrum of fluctuations in the difference of output intensities for the signal and idler beams generated by intracavity two-photon gain, when the cavity decay rates are the same for the two modes, and taking into account the atomic fluctuations. For the field inside the cavity, we show that the variance in the difference of intensities may be up to 50% below the shot-noise value. We also consider the case of different decay rates, and show that the noise in the output-intensity difference vanishes at zero frequency, for single-ported cavities, and as long as the only source of field decay is the transmission through the mirror. Within this general approach, we show that for the two-photon laser there is a strong correlation between signal and idler output beams when the relay-level detuning is larger than the power broadening. We calculate, in lowest order, the correction to the noise spectrum of the difference of intensities due to the population of the relay level.
In Chapter 1 we consider the question of squeezing in a two-photon laser. A two-photon laser is modelled as N two-level atoms interacting via a two-photon transition with a single resonant cavity field mode. A Fokker-Planck equation containing all leading noise terms is derived. A linearised fluctuation analysis reveals that there is no squeezing in a two-photon laser with injected signal. A small amount of squeezing is present in two-photon optical bistability. In Chapter 2 a microscopic model of degenerate four wave mixing including the quantisation of the medium is presented. Time-dependent solutions show ideal squeezing to be obtained in the limits of large detuning, low saturation and zero loss. The effect of nonideal atomic parameters is analysed. Equations describing the spatial propagation of the field through the medium are also derived. In Chapters 3 and 4 we are concerned with quantum effects, in particular violation of Bell's Inequalities, present in optical systems. An optical field produced by an intracavity four wave mixing is shown to exhibit the nonclassical features of photon antibunching, squeezing and violation of Cauchy-Schwarz and Bell's Inequalities. These intrinsic quantum mechanical effects are shown to be associated with the nonexistence of a positive normalizable Glauber-Sudarshan P-function.
We develop a systematic theory of quantum fluctuations in the driven optical parametric oscillator, including the region near threshold. This allows us to treat the limits imposed by nonlinearities to quantum squeezing and noise reduction in this nonequilibrium quantum phase transition. In particular, we compute the squeezing spectrum near threshold and calculate the optimum value. We find that the optimal noise reduction occurs at different driving fields, depending on the ratio of damping rates. The largest spectral noise reductions are predicted to occur with a very high-QQQ second-harmonic cavity. Our analytic results agree well with stochastic numerical simulations. We also compare the results obtained in the positive-PPP representation, as a fully quantum-mechanical calculation, with the truncated Wigner phase-space equation, also known as the semiclassical theory.