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arXiv:0802.2402 · Physical Review A

Cavity nonlinear optics with few photons and ultracold quantum particles

András Vukics, Wolfgang Niedenzu, Helmut Ritsch

Published Jan 29, 2009• 20 citations
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Abstract

The light force on particles trapped in the field of a high-$Q$ cavity mode depends on the quantum state of field and particle. Different photon numbers generate different optical potentials and different motional states induce different field evolution. Even for negligible internal particle excitation, which yields linear polarizability, the quantum character of particle motion generates nonlinear field dynamics. We derive a corresponding effective field Hamiltonian containing all the powers of the photon number operator, which predicts nonlinear phase shifts and squeezing even at the few-photon level. Simulations of the full particle-field dynamics confirm this and show significant particle-field entanglement.

DOI
10.1103/physreva.79.013828

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The light force on particles trapped in the field of a high-QQQ cavity mode depends on the quantum state of field and particle. Different photon numbers generate different optical potentials and different motional states induce different field evolution. Even for negligible internal particle excitation, which yields linear polarizability, the quantum character of particle motion generates nonlinear field dynamics. We derive a corresponding effective field Hamiltonian containing all the powers of the photon number operator, which predicts nonlinear phase shifts and squeezing even at the few-photon level. Simulations of the full particle-field dynamics confirm this and show significant particle-field entanglement.
Polarizable particles trapped in a resonator-sustained optical-lattice potential generate strong position-dependent backaction on the intracavity field. In the quantum regime, particles in different energy bands are connected to different intracavity light intensities and optical-lattice depths. This generates a highly nonlinear coupled particle-field dynamics. For a given pump strength and detuning, a factorizing mean-field approach predicts several self-consistent stationary solutions of strongly distinct photon numbers and motional states. Quantum Monte Carlo wave-function simulations of the master equation confirm these predictions and reveal complex multimodal photon-number and particle-momentum distributions. Using larger nanoparticles in such a setup thus constitutes a well-controllable playground to study nonlinear quantum dynamics and the buildup of macroscopic quantum superpositions at the quantum-classical boundary.
This is a review of state-of-the-art theory and experiment of the motion of cold and ultracold atoms coupled to the radiation field within a high-finesse optical resonator in the dispersive regime of the atom-field interaction with small internal excitation. The optical dipole force on the atoms together with the backaction of atomic motion onto the light field gives rise to a complex nonlinear coupled dynamics. As the resonator constitutes an open driven and damped system, the dynamics is nonconservative and in general enables cooling and confining the motion of polarizable particles. In addition the emitted cavity field allows for real-time monitoring of the particle's position with minimal perturbation up to subwavelength accuracy. For many-body systems, the resonator field mediates controllable long-range atom-atom interactions, which set the stage for collective phenomena. Besides the correlated motion of distant particles, one finds critical behavior and nonequilibrium phase transitions between states of different atomic order in conjunction with superradiant light scattering. Quantum-degenerate gases inside optical resonators can be used to emulate optomechanics as well as novel quantum phases such as supersolids and spin glasses. Nonequilibrium quantum phase transitions as predicted by, e.g., the Dicke Hamiltonian can be controlled and explored in real time via monitoring the cavity field. In combination with optical lattices, the cavity field can be utilized for nondestructive probing Hubbard physics and tailoring long-range interactions for ultracold quantum systems.
The quantum dynamics of particles optically trapped in a symmetrically pumped high-QQQ ring cavity exhibits much richer physics than for a standing-wave resonator. In addition to modifying the lattice depth, light scattering by the particles shifts and reshapes the trapping potential. We calculate the corresponding changes in tunneling amplitudes and damping by an effective bipotential (two-level) model for the particle motion. As a crude truncation of the Bose-Hubbard model, expansion to the lowest band decouples particle and field dynamics. Only including excitations to higher bands can capture this essential additional physics and correctly describe decoherence, damping, and long-range correlations of the particle dynamics. The validity limits of the analytic models are confirmed by quantum Monte Carlo wave-function simulations, which exhibit correlated particle-field quantum jumps as unambiguous quantum signature of the system dynamics.
Two recent experiments have reported the trapping of individual atoms inside optical resonators by the mechanical forces associated with single photons [Hood et al., Science 287, 1447 (2000); Pinkse et al., Nature (London) 404, 365 (2000)]. Here we analyze the trapping dynamics in these settings, focusing on two points of interest. First, we investigate the extent to which light-induced forces in these experiments are distinct from their free-space counterparts, and whether or not there are qualitatively different effects of optical forces at the single-photon level within the setting of cavity QED. Second, we explore the quantitative features of the resulting atomic motion, and how these dynamics are mapped onto experimentally observable variations of the intracavity field. Toward these ends, we present results from extensive numerical simulations of the relevant forces and their fluctuations, as well as a detailed derivation of our numerical simulation method, based on the full quantum-mechanical master equation. Not surprisingly, qualitatively distinct atomic dynamics arise as the coupling and dissipative rates are varied. For the experiment of Hood et al., we show that atomic motion is largely conservative and is predominantly in radial orbits transverse to the cavity axis. A comparison with the free-space theory demonstrates that the fluctuations of the dipole force are suppressed by an order of magnitude. This effect is based upon the Jaynes-Cummings eigenstates of the atom-cavity system and represents distinct physics for optical forces at the single-photon level within the context of cavity QED. By contrast, even in a regime of strong coupling in the experiment of Pinkse et al., there are only small quantitative distinctions between the potentials and heating rates in the free-space theory and the quantum theory, so it is not clear that a description of this experiment as a novel single-quantum trapping effect is necessary. The atomic motion is strongly diffusive, leading to an average localization time comparable to the time for an atom to transit freely through the cavity, and to a reduction in the ability to infer aspects of the atomic motion from the intracavity photon number.
A cold gas of polarizable particles moving in the optical potential of a standing wave high finesse optical resonator acts as a dynamic refractive index. For a sufficiently strong cavity pump the optical forces generated by the intra cavity field perturb the particles phase space distribution, which shifts the optical resonance frequency and induces a nonlinear optical response. By help of the corresponding Vlasov equation we predict that beyond the known phenomenon of optical bi-stability one finds regions in parameter space, where no stable stationary solution exists. The atom field dynamics then exhibits oscillatory solutions converging to stable limit cycles of the system. The linearized analytical predictions agree well with corresponding numerical solutions of the full time dependent equations and first experimental observation in both cases.
Summary form only given.Nonlinear optical effects are normally limited to high-intensity fields containing a large number of photons because the electric field associated with a single photon is usually very weak. Nonlinear phase shifts at the two-photon level previously have been observed using high-Q cavities and atomic beams. The authors recently predicted that nonlocal effects involving pairs of atoms in a medium could enhance the nonlinear interaction between two photons to the point that nonlinear phase shifts could be obtained in an ordinary medium without the use of resonant cavities.