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arXiv:0706.0032 · Physical Review A

Bifurcations and bistability in cavity-assisted photoassociation of Bose-Einstein-condensed molecules

M. Jääskeläinen, Jae‐Yoon Jeong, Christopher P. Search

Published Dec 18, 2007• 4 citations
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Abstract

We study the photoassociation of Bose-Einstein condensed atoms into molecules using an optical cavity field. The driven cavity field introduces a dynamical degree of freedom into the photoassociation process, whose role in determining the stationary behavior has not previously been considered. The semiclassical stationary solutions for the atom and molecules as well as the intracavity field are found and their stability and scaling properties are determined in terms of experimentally controllable parameters including driving amplitude of the cavity and the nonlinear interactions between atoms and molecules. For weak cavity driving, we find a bifurcation in the atom and molecule number occurs that signals a transition from a stable steady state to nonlinear Rabi oscillations. For a strongly driven cavity, there exists bistability in the atom and molecule number.

DOI
10.1103/physreva.76.063615

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We explore the quantum dynamics of photoassociation of Bose-Einstein condensed atoms into molecules using an optical cavity field. Inside of an optical resonator, photoassociation of quantum degenerate atoms involves the interaction of three coupled quantum fields for the atoms, molecules, and the photons. The feedback created by a high-QQQ optical cavity causes the cavity field to become a dynamical quantity whose behavior is linked in a nonlinear manner to the atoms inside and where vacuum fluctuations have a more important role than in free space. We develop and compare several methods for calculating the dynamics of the atom-molecule conversion process with a coherently driven cavity field. We first present an alternate operator representation for the Hamiltonian from which we derive an improved form of mean field theory and an approximate solution of the Heisenberg-Langevin (HL) equations that properly accounts for quantum noise in the cavity field. It is shown that our improved mean field theory corrects several deficiencies in traditional mean field theory based on expectation values of annihilation and creation operators. Also, we show by direct comparison to numerical solutions of the density matrix equations that our approximate quantum solution of HL equations gives an accurate description of weakly or undriven cavities where mean field theories break down.
In quantum optics, the most interesting processes occur due to nonlinear interactions between the electromagnetic field and various media. The parametric processes which produce nonclassical states of the electromagnetic field have analogs in the processes of photoassociation and dissociation of atomic and molecular Bose-Einstein condensates. Just as the optical processes need a full quantum theoretical treatment in some regimes, we show that photoassociation and dissociation require full quantum treatments to be confident of the theoretical predictions we can make. If the processes take place inside an optical cavity, we show that we may also need to consider the quantum nature of the intracavity electromagnetic field. As the process of intracavity photoassociation that we analyze has no exact mapping onto a Fokker-Planck equation, we show how quantum-field-theoretical techniques can be used to write stochastic difference equations in a generalized positive-P representation, which may be solved numerically.
We experimentally demonstrate an optical bistability between two hyperfine ground states of trapped, cold atoms, using a single mode of an optical resonator in the collective strong coupling regime. Whereas in the familiar case, the bistable region is created through atomic saturation, we report an effect between states of high quantum purity, which is essential for future information storage. The source of nonlinearity is a cavity-assisted pumping between ground states of the atoms and the stability depends on the intensity of two driving lasers. We interpret the phenomenon in terms of the recent paradigm of first-order, driven-dissipative phase transitions, where the transmitted and driving fields are understood as the order and control parameters, respectively. A semiclassical mean-field theory is invoked to describe the nontrivial two-dimensional phase diagram arising from the competition of the two drive. The saturation-induced bistability is recovered for infinite drive in one of the controls. The order of the transition is confirmed experimentally by hysteresis in the order parameter when either of the two control parameters is swept repeatedly across the bistability region.
We demonstrate that a photon blockade effect exists in the intracavity coherent photoassociation of an atomic Bose-Einstein condensate and that the dynamics of the coupled atomic and molecular condensates can only be successfully described by a quantum treatment of all the interacting fields. We show that the usual mean-field calculational approaches give answers that are qualitatively wrong, even for the mean fields. The quantization of the fields gives a degree of freedom that is not present in analogous nonlinear optical processes. The difference between the semiclassical and quantum predictions can actually increase as the three fields increase in size so that there is no obvious classical limit for this process.
We study the optical bistability and dynamics in a hybrid optomechanical system, where the cavity mode is coupled to the mechanical resonator via radiation pressure and to a two-level atom via the Jaynes–Cummings coupling. The nonlinearity due to optomechanical coupling and the addition of an atom leads to bistable behaviors of the steady-state intracavity photon number and atomic population inversion. We show that the optical bistability can be effectively controlled by the Rabi frequency of the atomic driving field, the cavity–atom coupling strength, and the amplitude and frequency of the cavity pump field. Moreover, the effects of different initial conditions of the cavity field and the amplitude of the cavity pump field on the dynamics of the system also have been discussed.
The interplay of quantum fluctuations with nonlinear dynamics is a central topic in the study of open quantum systems, connected to fundamental issues (such as decoherence and the quantum-classical transition) and practical applications (such as coherent information processing and the development of mesoscopic sensors and amplifiers). With this context in mind, we here present a computational study of some elementary bifurcations that occur in a driven and damped cavity quantum electrodynamics (cavity QED) model at low intracavity photon number. In particular, we utilize the single-atom cavity QED master equation and associated stochastic Schr\"odinger equations to characterize the equilibrium distribution and dynamical behavior of the quantized intracavity optical field in parameter regimes near points in the semiclassical (mean-field, Maxwell-Bloch) bifurcation set. Our numerical results show that the semiclassical limit sets are qualitatively preserved in the quantum stationary states, although quantum fluctuations apparently induce phase diffusion within periodic orbits and stochastic transitions between attractors. We restrict our attention to an experimentally realistic parameter regime.