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Physical Review Letters

Generation of Pair Coherent States and Squeezing via the Competition of Four-Wave Mixing and Amplified Spontaneous Emission

G. S. Agarwal

Published Aug 18, 1986• 242 citations
Publisher

Abstract

Strong competition between four-wave mixing and amplified spontaneous emission in resonant two-photon excitations is shown to generate radiation fields with strong squeezing and antibunching. The generated fields are in a new type of coherent state which is an eigenstate of the operator corresponding to the simultaneous annihilation of photons in two modes.

DOI
10.1103/physrevlett.57.827

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Strong competition between four-wave mixing and amplified spontaneous emission in resonant two-photon excitations is shown to generate radiation fields with strong squeezing and antibunching. The generated fields are in a new type of coherent state which is an eigenstate of the operator corresponding to the simultaneous annihilation of photons in two modes.
Competition between amplified spontaneous emission (ASE) and the four-wave-mixing (FWM) process has been observed under conditions of two-photon resonant excitation of the sodium 3d level. The nature of the competition is that the FWM process is able to prevent the occurrence of ASE, even though the gain of the ASE process calculated in the absence of competition effects is much larger than that of FWM. The ASE is suppressed because the fields generated by the FWM process create a new excitation pathway connecting the ground and 3d levels, and under quite general conditions this pathway interferes destructively with that due solely to the applied laser field. These effects are modeled theoretically by solving perturbatively the density-matrix equations of the atomic system, thereby determining the population in the upper level and the nonlinear polarization of the medium. The coupling between the various optical fields due to the nonlinear polarization is described by coupled amplitude equations. The solution to these equations predicts that when the wave-vector mismatch is not too large the fields evolve spatially to reach steady-state values, and that the population excited to the 3d level by the total steady-state optical field is much smaller than that due to the incident laser field alone. We have observed experimentally the suppression of ASE by FWM and have observed that this suppression does not occur when the medium is excited with counterpropagating beams that cannot efficiently excite the FWM process. In addition, we have conducted a series of experiments that shows that the degree of suppression of ASE depends on the intensity and focusing characteristics of the incident laser as expected on the basis of our theoretical model.
We present a detailed discussion of a type of four-photon coherent state defined as right eigenstates of the operator â 4 where â is the usual annihilation operator. There are actually four sets of states that need to be considered, namely those containing as the lowest number states ¦0>, ¦1>, ¦2>, or ¦3>. These correspond to the possible unique superpositions of the ordinary coherent states ¦±α> and ¦±iα>. We discuss the nonclassical properties of these states such as photon antibunching and squeezing. The usual second order squeezing does not exist for these states but higher order squeezing and square field amplitude squeezing do exist. Also discussed are the quasiprobability distributions, namely the P-function, the Q-function and the Wigner function. Finally, a method of generating these states based on the competition between a four-photon parametric process and incoherent losses from four-photon absorption is presented.
We formulate a quantum-statistical theory of the generation of the fields in a two-photon medium in which several competing nonlinear processes such as four-wave mixing, two-photon absorption, and amplified spontaneous emission are taking place. We derive the equation for the density matrix of the generated fields. We give numerical results for the time-dependent solutions, which enable us to understand the growth and stabilization of the fields in such a medium. We show the generation of new types of the coherent states of the radiation field. Our analysis shows that the generated fields, both in transient and the steady-state domain, have very striking quantum properties leading to squeezing, sub-Poissonian statistics, and violations of Cauchy-Schwarz inequalities. Wherever possible we compare our results with the experiment of Malcuit et al. and the semiclassical theory of Boyd et al.
Two-photon-resonant excitation of the sodium 3d3d3d level can lead to the generation of new frequencies either by amplified spontaneous emission at the $3d\ensuremath{\rightarrow}3p$ transition frequency or by a resonantly enhanced four-wave mixing process. Competition between these two processes has been observed. The four-wave mixing process can suppress amplified spontaneous emission by preventing the excitation of the 3d3d3d level due to an interference between two different pathways connecting the ground (3s3s3s) and 3d3d3d states.
Competition has been observed between amplified spontaneous emission (ASE) and four-wave mixing (FWM). Previously, competition between an incoherent and coherent process was observed in the form of suppression of multiphoton ionization by third harmonic generation.1,2 In our experiment, an input laser was tuned precisely to the 3s → 3d two-photon allowed transition in atomic sodium vapor and the emission was observed in both the forward and backward directions. In these conditions, either FWM or ASE can occur. Ignoring the possibility of competition between these two processes, the calculated gain for ASE is much larger than that for FWM. However, we have observed experimentally that FWM and not ASE occur. To determine that it is the presence of efficient FWM that suppresses ASE, we have excited the atomic vapor instead with counterpropagating pump waves of different frequency. In this case, FWM, being a phase-matched process, cannot occur and strong ASE is observed. The nature of this competition is explained by solving the coupled Maxwell and density matrix equations for a three-level system. It is found that the suppression is due to the destructive interference between two different pathways of excitation of the 3d level which results in trapping of the population in the ground state.
We examine the nonclassical properties of the pair coherent states defined as the simultaneous eigenstates of the operator that annihilates photons in pairs and of the operator that gives the relative occupation number in the two modes. We show that fields in such states have remarkable quantum features such as sub-Poissonian statistics correlations in the number fluctuations, squeezing, and violations of Cauchy–Schwarz inequalities. We show how such pair coherent states can be generated by the competition of different nonlinear processes in a two-photon medium. The quantam features occur not only in the transient domain but also survive in the steady state because of the balance between nonlinear processes.