In this paper, properties of a homogeneous Bose gas with a Feshbach resonance are studied in the dilute region at zero temperature. The stationary state contains condensations of atoms and molecules. The ratio of the molecule density to the atom density is pna3. There are two types of excitations, molecular excitations and atomic excitations. Atomic excitations are gapless, consistent with the traditional theory of a dilute Bose gas. The molecular excitation energy is finite in the long-wavelength limit as observed in recent experiments on 85Rb. In addition, the decay process of the condensate is studied. The coefficient of the three-body recombination rate is about 140 times larger than that of a Bose gas without a Feshbach resonance, in reasonably good agreement with the experiment on 23Na.

In this paper, collective excitations in a homogeneous fermion-fermion mixture with different Fermi surfaces are studied. In the Fermi liquid phase, the zero-sound velocity is found to be larger than the largest Fermi velocity. With attractive interactions, the superfluid phase appears below a critical temperature, and the phase mode is the low-energy collective excitation. The velocity of the phase mode is significantly different from the zero-sound velocity. The difference between the two velocities may serve as a tool to detect the superfluid phase.

We study the properties of a phase slip in a superfluid Fermi gas near a Feshbach resonance. The phase slip can be generated by the phase imprinting method. Below the superfluid transition temperature, it appears as a dip in the density profile and becomes more pronounced when the temperature is lowered. Therefore the phase slip can provide direct evidence of the superfluid state. The condensation energy of the superfluid state can be extracted from the density profile of the phase slip, due to the unitary properties of the Fermi gas near the resonance. The width of the phase slip is inversely proportional to the square root of the difference between the transition temperature and the temperature. The signature of the phase slip in the density profile becomes more robust across the BCS-Bose-Einstein-condensate crossover.

In this paper, we study the three-body recombination rate of a homogeneous dilute Bose gas with a Feshbach resonance at zero temperature. The ground state and excitations of this system are obtained. The three-body recombination in the ground state is due to the breakup of an atom pair in the quantum depletion and the formation of a molecule by an atom from the broken pair and an atom from the condensate. The rate of this process is in good agreement with the experiment on 23Na in a wide range of magnetic fields.

The existence of a dynamical potential with both local and global meanings in general nonequilibrium processes has been controversial. Following an earlier heuristic argument in a letter by one of us, in the present paper we show rigorously its existence for a generic class of situations in physical and biological sciences. The local dynamical meaning of this potential function is demonstrated via a special stochastic differential equation and its global steadystate meaning via a novel and explicit form of the Fokker-Planck equation. We also give a procedure to obtain the special stochastic differential equation for any given Fokker-Planck equation. No detailed balance condition is required in our demonstration. For the first time we obtain here a formula to describe the noise-induced shift in drift force compared to the steady-state distribution, a phenomenon extensively observed in numerical studies.