James E. Williams

Longitudinal Spin Waves in a Dilute Bose Gas

We present a kinetic theory for a dilute noncondensed Bose gas of two-level atoms that predicts the transient spin segregation observed in a recent experiment. The underlying mechanism driving spin currents in the gas is due to a mean-field effect arising from the quantum interference between the direct and exchange scattering of atoms in different spin states. We numerically solve the spin Boltzmann equation, using a one-dimensional model, and find excellent agreement with experimental data.

Linear spin waves in a trapped Bose gas

An ultracold Bose gas of two-level atoms can be thought of as a spin-½ Bose gas. It supports spin-wave collective modes due to the exchange mean field. Such collective spin oscillations have been observed in recent experiments at JILA with 8 7 Rb atoms confined in a harmonic trap. We present a theory of the spin-wave collective modes based on the moment method for trapped gases. In the collisionless and hydrodynamic limits, we derive analytic expressions for the frequencies and damping rates of modes with dipole and quadrupole symmetry. We find that the frequency for a given mode is given by a temperature-independent function of the peak density n, and falls off as 1/n. We also find that, to a very good approximation, excitations in the radial and axial directions are decoupled. We compare our model to the numerical integration of a one-dimensional version of the kinetic equation and find very good qualitative agreement. The damping rates, however, show the largest deviation for intermediate densities, where one expects Landau damping-which is unaccounted for in our moment approach-to play a significant role.

Phase diagrams for an ideal gas mixture of fermionic atoms and bosonic molecules

We calculate the phase diagrams for a harmonically trapped ideal gas mixture of fermionic atoms and bosonic molecules in chemical and thermal equilibrium, where the internal energy of the molecules can be adjusted relative to that of the atoms by use of a tunable Feshbach resonance. We plot the molecule fraction and the fraction of Bose-condensed molecules as functions of the temperature and internal molecular energy. We show the paths traversed in the phase diagrams when the molecular energy is varied either suddenly or adiabatically. Our model calculation helps to interpret the adiabatic phase diagrams obtained in recent experiments on the Bose–Einstein condensation to Bardeen–Cooper–Schrieffer crossover, in which the condensate fraction is plotted as a function of the initial temperature of the Fermi gas measured before a sweep of the magnetic field through the resonance region.

Kinetic Theory of a Spin-1/2 Bose-Condensed Gas

We derive a kinetic theory for a spin-1/2 Bose-condensed gas of two-level atoms at finite temperatures. The condensate dynamics is described by a generalized Gross–Pitaevskii equation for the two-component spinor order parameter, which includes the interaction with the uncondensed fraction. The noncondensate atoms are described by a quantum kinetic equation, which is a generalization of the spin kinetic equation for spin-polarized quantum gases to include couplings to the condensate degree of freedom. The kinetic equation is used to derive hydrodynamic equations for the noncondensate spin density. The condensate and noncondensate spins are coupled directly through the exchange mean field. Collisions between the condensate and noncondensate atoms give rise to an additional contribution to the spin diffusion relaxation rate. In addition, they give rise to mutual relaxation of the condensate and noncondensate due to lack of local equilibrium between the two components.

Mesoscopic effects in quantum phases of ultracold quantum gases in optical lattices

We present a wide array of quantum measures on numerical solutions of one-dimensional Bose- and Fermi-Hubbard Hamiltonians for finite-size systems with open boundary conditions. Finite-size effects are highly relevant to ultracold quantum gases in optical lattices, where an external trap creates smaller effective regions in the form of the celebrated wedding cake structure and the local density approximation is often not applicable. Specifically, for the Bose-Hubbard Hamiltonian we calculate number, quantum depletion, local von Neumann entropy, generalized entanglement or Q measure, fidelity, and fidelity susceptibility; for the Fermi-Hubbard Hamiltonian we also calculate the pairing correlations, magnetization, charge-density correlations, and antiferromagnetic structure factor. Our numerical method is imaginary time propagation via time-evolving block decimation. As part of our study we provide a careful comparison of canonical versus grand canonical ensembles and Gutzwiller versus entangled simulations. The most striking effect of finite size occurs for bosons: we observe a strong blurring of the tips of the Mott lobes accompanied by higher depletion, and show how the location of the first Mott lobe tip approaches the thermodynamic value as a function of system size.

Theory of Feshbach molecule formation in a dilute gas during a magnetic field ramp

Starting with coupled atom?molecule Boltzmann equations, we develop a simplified model to understand molecule formation observed in recent experiments. Our theory predicts several key features: (i) the effective adiabatic rate constant is proportional to density; (ii) in an adiabatic ramp, the dependence of molecular fraction on magnetic field resembles an error function whose width and centroid are related to the temperature; and (iii) the molecular production efficiency is a universal function of the initial phase space density, the specific form of which we derive for a classical gas. Our predictions show qualitative agreement with the data from Hodby et al (2005 Phys. Rev. Lett. 94 120402) without the use of adjustable parameters.

Atom?molecule equilibration in a degenerate Fermi gas with resonant interactions

We present a nonequilibrium kinetic theory describing atom–molecule population dynamics in a two-component Fermi gas with a Feshbach resonance. Key collision integrals emerge that govern the relaxation of the atom–molecule mixture to chemical and thermal equilibrium. Our focus is on the pseudogap regime where molecules form above the superfluid transition temperature. In this regime, we formulate a simple model for the atom–molecule population dynamics. The model predicts the saturation of molecule formation that has been observed in recent experiments, and indicates that a dramatic enhancement of the atom–molecule conversion efficiency occurs at low temperatures.

Ultracold Fermion Cooling Cycle Using Heteronuclear Feshbach Resonances

We consider an ideal gas of Bose and Fermi atoms in a harmonic trap, with a Feshbach resonance in the interspecies atomic scattering that can lead to the formation of fermionic molecules. We map out the phase diagram for this three-component mixture in chemical and thermal equilibrium. Considering adiabatic association and dissociation of the molecules, we identify a possible cooling cycle, which in ideal circumstances can yield an exponential increase of the phase-space density.

Adiabatic Phase Diagram of an Ultracold Atomic Fermi Gas with a Feshbach Resonance

We determine the adiabatic phase diagram of a resonantly-coupled system of Fermi atoms and Bose molecules confined in a harmonic trap by using the local density approximation. The adiabatic phase diagram shows the fermionic condensate fraction composed of condensed molecules and Cooper paired atoms. The key idea of our work is conservation of entropy through the adiabatic process, extending the study of Williams et al. [New J. Phys. 6 (2004) 123] for an ideal gas mixture to include the resonant interaction in a mean-field theory. We also calculate the molecular conversion efficiency as a function of initial temperature. Our work helps to understand recent experiments on the BCS–BEC crossover, in terms of the initial temperature measured before a sweep of the magnetic field.

Adiabatic Phase Diagram of a Degenerate Fermi Gas with a Feshbach Resonance

AbstractnWe determine the adiabatic phase diagram of a resonantly-coupled system of Fermi atoms and Bose molecules confined in a harmonic trap by using the local density approximation. The key idea of our work is conservation of entropy through the adiabatic process. We also calculate the molecular conversion efficiency as a function of the initial temperature. Our work helps to understand recent experiments on the BCS-BEC crossover, in terms of the initial temperature measured before a sweep of the magnetic field.n