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Physical Review A

Quantum theory of nondegenerate multiwave mixing. II. Adiabatic elimination via slowly varying amplitude approximation

Seng‐Tiong Ho, Prem Kumar, Jeffrey H. Shapiro

Published Apr 1, 1991• 4 citations
Publisher

Abstract

In this paper, we continue our development of the quantum theory of nondegenerate multiwave mixing in an atomic medium [Phys. Rev. A 37, 2017 (1988)]. In our development, the atomic variables are eliminated using a frequency-domain approach employing a slowly varying amplitude approximation that is more rigorous than the usual adiabatic approximation. We then specialize to the case of four-wave interaction with two strong pump beams and obtain noise correlations of the atomic polarization that are applicable to nondegenerate four-wave mixing. The noise correlations include the effect of atomic collisions that is crucial to the atomic-vapor experiments. After making the usual rotating-wave approximation, the atomic-polarization equations give us a set of temporal coupled-mode equations for the c-number variables corresponding to the annihilation operators. We then further specialize to the single-beam case in which all the relevant modes of interest are collinear and obtain a paired set of coupled-mode equations. In order to apply the theory to experiments employing traveling-wave interaction geometries, in the following paper of this series we will present a formalism to treat the spatial propagation of a quantum field.

DOI
10.1103/physreva.43.3939

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Effects of Injected Atomic Coherence on Multiwave Mixing.

Timothy Carty

Discussion begins with a brief account of atomic level-pumping and reasons why atomic coherence is typically not considered in cw work on optical interactions. This dissertation is divided into four parts: semiclassical treatments of one-photon electric- dipole atom-field single-mode interactions and multimode interactions, and corresponding treatments for the two-photon interaction. We present the effects of injected atomic coherence on the polarization of the medium, the slowly varying envelope wave equation, the single- and multiwave mixing coefficients, and weak field propagation in a homogeneously broadened two-level medium. Spatial and temporal phase matching of the injected coherence to a field mode is crucial throughout, since the field may not be able to remain in phase with the induced and injected polarizations. One-photon injected coherence contributes directly to the polarization at the atomic resonance frequency. The perfectly phase-matched case leads to a linear superposition of an exponentially decaying field (Beer's law) and a constant field driven by the injected coherence. The interaction of an injected coherence with a detuned field produces frequency-symmetric sidebands about the pump field polarization. The sideband spacing equals the atom-field detuning. To probe the injected coherence we inject a weak resonant field. The resulting three-wave mixing leads to multiwave mixing coefficients that are unaffected to first-order in the weak sidemodes, but the injected coherence adds inhomogeneous terms to the coupled-mode equations. For both single- and multimode interactions the injected coherence does not affect the exponential growth/decay of the sidemodes, but it supports a weak field that may propagate if properly phase matched. For two-photon media the injected coherence requires at least one field interaction in order to produce a polarization, which then appears in the single- and multiwave mixing coefficients. The exponential growth/decay rate is modified by the injected coherence. For a centrally-tuned pump the injected coherence contributes the standard multiwave mixing terms as well as additional effects. Four-wave mixing is discussed as a means of relaxing the spatial phase matching constraint on the injected coherence.

•Jan 1, 1989•0 citations
In this paper, we continue our development of the quantum theory of nondegenerate multiwave mixing in an atomic medium [Phys. Rev. A 37, 2017 (1988)]. In our development, the atomic variables are eliminated using a frequency-domain approach employing a slowly varying amplitude approximation that is more rigorous than the usual adiabatic approximation. We then specialize to the case of four-wave interaction with two strong pump beams and obtain noise correlations of the atomic polarization that are applicable to nondegenerate four-wave mixing. The noise correlations include the effect of atomic collisions that is crucial to the atomic-vapor experiments. After making the usual rotating-wave approximation, the atomic-polarization equations give us a set of temporal coupled-mode equations for the c-number variables corresponding to the annihilation operators. We then further specialize to the single-beam case in which all the relevant modes of interest are collinear and obtain a paired set of coupled-mode equations. In order to apply the theory to experiments employing traveling-wave interaction geometries, in the following paper of this series we will present a formalism to treat the spatial propagation of a quantum field.
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.
A quantum theory of degenerate four-wave mixing is presented in which the atomic medium consists of stationary four-level atoms, each with three degenerate excited states, and the interacting light beams are allowed to be in different states of polarization. This vector-field theory differs from the scalar-field theory of Reid and Walls [Phys. Rev. A 31, 1622 (1985)] in that there are new atomic variables in the Langevin equations which are related to the induced coherence between the upper atomic states. It is found, for the assumed set of atomic levels, that this seemingly different mechanism of four-wave-mixing gain does not circumvent the degrading effect of spontaneous emission on squeezing obtainable via degenerate four-wave mixing. The theory is applied to both forward and backward degenerate four-wave mixing with nearly collinear geometry, and specialized to the case in which the polarization states of the two pump modes are mutually orthogonal. It is found that for both forward and backward configurations, the range of pump intensity for which squeezing can be achieved in the vector-field case is larger than that in the scalar-field case, and the maximum amount of squeezing obtainable at a particular pump detuning is comparable in both cases.
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.
Discussion begins with a brief account of atomic level-pumping and reasons why atomic coherence is typically not considered in cw work on optical interactions. This dissertation is divided into four parts: semiclassical treatments of one-photon electric- dipole atom-field single-mode interactions and multimode interactions, and corresponding treatments for the two-photon interaction. We present the effects of injected atomic coherence on the polarization of the medium, the slowly varying envelope wave equation, the single- and multiwave mixing coefficients, and weak field propagation in a homogeneously broadened two-level medium. Spatial and temporal phase matching of the injected coherence to a field mode is crucial throughout, since the field may not be able to remain in phase with the induced and injected polarizations. One-photon injected coherence contributes directly to the polarization at the atomic resonance frequency. The perfectly phase-matched case leads to a linear superposition of an exponentially decaying field (Beer's law) and a constant field driven by the injected coherence. The interaction of an injected coherence with a detuned field produces frequency-symmetric sidebands about the pump field polarization. The sideband spacing equals the atom-field detuning. To probe the injected coherence we inject a weak resonant field. The resulting three-wave mixing leads to multiwave mixing coefficients that are unaffected to first-order in the weak sidemodes, but the injected coherence adds inhomogeneous terms to the coupled-mode equations. For both single- and multimode interactions the injected coherence does not affect the exponential growth/decay of the sidemodes, but it supports a weak field that may propagate if properly phase matched. For two-photon media the injected coherence requires at least one field interaction in order to produce a polarization, which then appears in the single- and multiwave mixing coefficients. The exponential growth/decay rate is modified by the injected coherence. For a centrally-tuned pump the injected coherence contributes the standard multiwave mixing terms as well as additional effects. Four-wave mixing is discussed as a means of relaxing the spatial phase matching constraint on the injected coherence.