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Journal of the Optical Society of America B

Quantum theory of nondegenerate multiwave mixing: effect of pump phase fluctuations

A. H. Toor, Muhammad Suhail Zubairy

Published Jun 1, 1997• 1 citation
Publisher

Abstract

We extend the quantum theory of nondegenerate four-wave mixing by including the effects of the finite bandwidth of the driving-pump field. The interaction of a beam of two-level atoms with the two opposite driving-pump fields that have finite bandwidth inside a bimodal cavity is considered. The master equation for the cavity-field modes, averaged over the stochastic process, is derived. We use our theory to study the effects of phase fluctuations, associated with the driving-pump field, on the generation of two-mode squeezing inside the cavity. The steady-state squeezing is achieved with the same driving-pump field as a local oscillator in the balanced homodyne-detection system. Our results show that, in spite of instantaneous phase locking between the cavity field and the local oscillator, the time-delay effects associated with the exponential decay of atomic coherence relate the steady-state squeezing to the diffusion constant of the driving-pump field.

DOI
10.1364/josab.14.001280

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We extend the quantum theory of nondegenerate four-wave mixing by including the effects of the finite bandwidth of the driving-pump field. The interaction of a beam of two-level atoms with the two opposite driving-pump fields that have finite bandwidth inside a bimodal cavity is considered. The master equation for the cavity-field modes, averaged over the stochastic process, is derived. We use our theory to study the effects of phase fluctuations, associated with the driving-pump field, on the generation of two-mode squeezing inside the cavity. The steady-state squeezing is achieved with the same driving-pump field as a local oscillator in the balanced homodyne-detection system. Our results show that, in spite of instantaneous phase locking between the cavity field and the local oscillator, the time-delay effects associated with the exponential decay of atomic coherence relate the steady-state squeezing to the diffusion constant of the driving-pump field.
We apply our quantum theory of nondegenerate multiwave mixing [ Phys. Rev. A37, 2017 ( 1988)] to squeezed-state generation experiments with two-level atoms. Our main interest is to predict the amount of squeezing achievable with a Doppler-broadened two-level medium. We are particularly interested in the single-beam configuration, in which all four interacting beams are spatially degenerate. We analytically solve the coupled-mode quantum Langevin equations for nondegenerate four-wave mixing. The solutions are used to compute the amount of squeezing. In the computation the effects of pump–probe phase mismatch, collisions, Doppler broadening, and Gaussian-intensity variation are comprehensively taken into account for the first time to our knowledge. Simple rules of thumb as to where one can see squeezing in both degenerate- and nondegenerate-frequency cases are derived by examining the limit of a short medium. We then present the case of an infinitely long medium, in which maximum squeezing is achieved when there is no pump–probe phase mismatch. With the inclusion of pump–probe phase mismatch, however, the maximum amount of squeezing is obtained with a finite-length medium instead. This prompts us to investigate in detail the finite-length medium case. Our results show that the effects of Doppler broadening and Gaussian-intensity variation can be largely circumvented by detuning the pump frequency more than three Doppler half-widths from resonance and that good broadband squeezing can be achieved even with a Doppler-broadened medium that has a moderate amount of collision broadening. Under these circumstances it is found that the effect of pump self-focusing or defocusing will be the major factor that limits the amount of achievable squeezing. In particular, the spatially varying nonlinear refractive indices seen by the pump and the probe modes are quite different, which causes the former to become spatially mismatched with the latter in the region in which strong squeezing is otherwise expected.
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.
In this first paper of a series on the quantum theory of nondegenerate multiwave mixing applicable to traveling-wave interaction geometries, we describe our problem formulation. We consider the explicit dynamics of a subset of the field quantization modes interacting with a system of stationary two-level atoms contained in a volume much smaller than the field quantization volume. Because we make the realistic assumption of leaving the remaining infinite set of field modes as a common thermal-field reservoir, the resulting Langevin equations contain extra decay terms due to collective spontaneous emission or super-radiance. We show that all but one of these super-radiance terms are negligible in the following two limits: (a) when the number of atoms in a diffraction volume is small; (b) when the atoms are pumped far off resonance. There is, however, an anomalous decay term which does not appear in a classical model of super-radiance based upon coherently phased atomic dipoles. The magnitude of this anomalous term is neither dependent upon the number of atoms nor on the pump-frequency detuning and may not be negligible at a low pump intensity. Neglecting the super-radiance terms, we then present a general Fourier-expansion solution technique for obtaining the atomic polarization in the presence of any number of field modes. The expansion is shown to be convergent in a commonly occurring situation in which all the strong pump modes are frequency degenerate and the remaining nondegenerate modes are all weak compared to the atomic saturation intensity. In subsequent papers of this series, we will present methods to treat, with some rigor, the spatial propagation of an interacting multimode quantum field and apply these methods to traveling-wave squeezed-state generation experiments.
A quantum theory of nondegenerate multiwave mixing in a two-level atomic medium is formulated, with emphasis on traveling-wave interaction geometries. Two equivalent alternative methods are developed to treat, with some rigor, the spatial propagation of the interacting multimode traveling-wave field. The theory is applied to the case of a single beam propagating through the medium to obtain the spectrum of squeezing. The effects of spontaneous emission, propagation loss, atomic collisions, and pump-probe phase mismatch are comprehensively taken into account for the first time.
The fluctuations in the phase of the pump in a degenerate optical parametric oscillator are modeled by a classical phase-diffusion process, and their influence on squeezing is studied. To this end, appropriate quadrature variables, relative to the instantaneous phase of the pump, are introduced. The squeezing is calculated both inside and outside the cavity. In both cases it is found to be degraded by the phase fluctuations, which may be understood in terms of the time lag in the response of the system to an instantaneous change in the pump phase. This effect might account for a small part of the residual noise present in current experiments. Correlation functions for the field of the oscillator for finite pump linewidth are derived and discussed.