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Physical Review A

Microscopic dynamics of ultracold particles in a ring-cavity optical lattice

Wolfgang Niedenzu, Rainer Schulze, András Vukics, Helmut Ritsch

Published Oct 8, 2010• 26 citations
Publisher

Abstract

The quantum dynamics of particles optically trapped in a symmetrically pumped high-$Q$ 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.

DOI
10.1103/physreva.82.043605

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•Jan 1, 2018•0 citations
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.
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.
Ultracold bosonic gases in optical lattices are strongly correlated quantum systems similar to solids. The strong correlation between the electrons in a solid on the one hand, and the bosonic atoms in optical lattice on the other, exhibit various quantum phenomena like insulation, conductivity, localization of electrons and atoms, respectively. Controlled by the intensity of the lattice laser, the ultracold bosonic gas can be transferred from a regime with superfluid character for shallow lattices into a regime of strong correlations, the Mott insulator. As an additional external parameter besides the lattice depth, one can generate spatial inhomogeneities by superimposing an additional standing wave (so-called two-color superlattices), which gives rise to localization effects or the formation of a Bose-glass phase. In the present work, numerical simulations are employed in order to investigate characteristic signatures of the quantum phases in the low-energy excitation spectrum of one-dimensional systems. We simulate temporal small amplitude modulations of the optical lattice in analogy to experiments, and evaluate the response of the system from the time-evolved initial state. The lattice systems are described in the framework of the Bose-Hubbard model. For the evaluation of the time-evolved state, we employ several numerical methods. We analyze systems of small size (6 particles on 6 sites) using an exact time-evolution by integration of the time-dependent Schrödinger equation. The formulation of an importance truncation scheme enables us to retain only the relevant components of the model space in the strongly correlated regime and, thus, allows for the investigation of systems with 10 particles on 10 sites using exact time-evolution. Based on this method, we present results of the Mott-insulating regime as well as for the Bose-glass phase. Furthermore, we employ particle-hole methods, which allow for the treatment of systems with experimental lattice sizes and particle numbers. Starting from the equation of motion method we adapt the Tamm-Dancoff approximation as well as the random-phase approximation for the occupation number representation of the Bose-Hubbard model. We present results of simulations of up to 50 particles on 50 sites and discuss the impact of the lattice depth on the low-energy excitations (U-resonance). Moreover, the impact of a two-color superlattice and the variation of its amplitude is investigated.
Cavity cooling of an atom works best on a cyclic optical transition in the strong coupling regime near resonance, where small-cavity photon numbers suffice for trapping and cooling. A straightforward application to the cooling of the translational motion of other polarizable particles without sharply defined two-level transitions (such as molecules) fails as optical pumping transfers the particle into uncoupled states. An alternative operation in the far-off-resonant regime generates only very slow cooling due to the reduced field-particle coupling. We suggest one can overcome this by using a strongly driven ring cavity operated in the sideband cooling regime. The dynamics can be mapped onto the optomechanics setup with a movable mirror and allows one to take advantage of a collectively enhanced field-particle coupling by large photon numbers. A linearized analytical treatment confirmed by full numerical quantum simulations predicts fast cooling despite the off-resonant small single-particle--single-photon coupling. Even ground-state translational cooling (in the external potential) can be obtained by tuning the cavity field close to the Anti-stokes sideband for sufficiently high trapping frequency. Numerical simulations show quantum jumps of the particle between the lowest two trapping levels, which can be directly and continuously monitored via scattered light intensity detection.
We formulate a method for studying the quantum field dynamics of ultracold Bose gases confined within optical lattice potentials, within the lowest Bloch-band Bose-Hubbard model. Our formalism extends the two-sites results of Vardi and Anglin [Phys. Rev. Lett. 86, 568 (2001)] to the general case of MMM lattice sites. The methodology is based on mapping the Bose-Hubbard Hamiltonian to an SU(M)\mathrm{SU}(M)SU(M) pseudospin problem and truncating the resulting hierarchy of dynamical equations for correlation functions, up to pair correlations between SU(M)\mathrm{SU}(M)SU(M) generators. Agreement with few-site exact many-particle calculations is consistently better than the corresponding Hartree-Fock-Bogoliubov approximation. Moreover, our approximation compares favorably with a more elaborate two-particle irreducible effective action formalism, at a fraction of the analytic and numerical effort.
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.
Ultracold atoms in optical lattices are a powerful platform for the study of quantum many-body physics. The combination of a high degree of isolation from the environment and external control over all relevant parameters makes these systems ideal candidates for the quantum simulation of fundamental lattice models. However, since the atoms are trapped in standing waves of interfering laser beams, the available trap geometries are constrained to regular lattices and single-site control is limited. In this thesis, an alternative experimental platform is investigated. Here, the combination of a microlens array and a spatial light modulator is used to provide a two-dimensional optical microtrap array for ultracold atoms. This setup allows for versatile trap geometries and comprehensive single-site control. The experimental feasibility of the described platform is investigated in the following way. First, the light field generating the microtrap array is simulated using a detailed model of the optical setup. The computed intensity distribution is proportional to the optical dipole potential for the atoms. Second, the simulation results are used to obtain the Hubbard parameters for multiple alkalies from numerical calculations as well as approximative analytical methods. It is shown that the strongly correlated regimes of the Bose-Hubbard model can be reached at sufficiently large tunneling rates. In addition, the impact of fluctuations in the trap parameters is investigated. Third, two approaches are considered for the preparation of low-entropy many-body states. On the one hand, a loading scheme is investigated which starts from a Bose-Einstein condensate and is used in optical lattice experiments. Here, the depth of the microtrap array is increased adiabatically. On the other hand, an array of isolated traps, which is initialized with one atom per site in the respective motional ground state, is considered as starting point. The itinerant regime of the Hubbard model is accessed by an adiabatic decrease of the trap depth. An analysis of ramp-induced excitations and external heating processes shows the feasibility of both approaches. Demonstrating the potential of the investigated platform, two applications are described. On the one hand, the tunneling dynamics of ultracold atoms between weakly coupled ring lattices is analyzed. Controlled by the interaction strength, multiple phenomena can be observed: Josephson oscillations exhibiting collapse and revival, inter-action-induced self-trapping, and tunneling resonances. On the other hand, the implementation of a scheme for universal quantum computing based on time-continuous quantum walks of interacting particles is proposed. Here, the information is encoded into the position of atomic wave packets moving through a planar graph which is built from optical microtraps and implements a quantum circuit. Details of an experimental implementation are discussed for both applications using the results derived in the preceding parts of this thesis.