Fermi–Bose Correspondence and Bose–Einstein Condensation in the Two-Dimensional Ideal Gas

The ideal uniform two-dimensional (2D) Fermi and Bose gases are considered both in the thermodynamic limit and the finite case. We derive May's Theorem, viz. the correspondence between the internal energies of the Fermi and Bose gases in the thermodynamic limit. This results in both gases having the same heat capacity. However, as we shall show, the thermodynamic limit is never truly reached in two dimensions and so it is essential to consider finite-size effects. We show in an elementary manner that for the finite 2D Bose gas, a pseudo-Bose–Einstein condensate forms at low temperatures, incompatible with May's Theorem. The two gases now have different heat capacities, dependent on the system size and tending to the same expression in the thermodynamic limit.     

Bose and Fermi Gases with Lennard?CJones Interactions

We study a model for cold Bose and Fermi gases based on the Lennard?CJones interaction, using the optimized (Fermi-)hypernetted-chain ((F)HNC-EL) method. For comparison, we also have carried out path integral ground state Monte Carlo (PIGSMC) simulations in the Bose case. By varying the density and the coupling strength for the Lennard?CJones potential, we cover the whole range of dilute, weakly interacting gases up to the dense, strongly interacting case of liquid ³He and ⁴He. Below about 20?percent helium equilibrium density, the simplest version of the (F)HNC-EL theory is accurate within better than 1?percent.     

Anisotropic Spin Diffusion in Trapped Boltzmann Gases

No Heading Recent experiments in a mixture of two hyperfine states of trapped Bose gases show behavior analogous to a spin-1/2 system, including transverse spin waves and other familiar Leggett-Rice-type effects. We have derived the kinetic equations applicable to these systems, including the spin dependence of interparticle interactions in the collision integral, and have solved for spin-wave frequencies and longitudinal and transverse diffusion constants in the Boltzmann limit. We find that, while the transverse and longitudinal collision times for trapped Fermi gases are identical, the Bose gas shows diffusion anisotropy. Moreover, the lack of spin isotropy in the interactions leads to the non-conservation of transverse spin, which in turn has novel effects on the hydrodynamic modes.PACS numbers: 03.75.Mn,05.30Jp,05.60.Gg,51.10.+y.67.20.+k.     

Padé approximants applied to the equation of state and heat capacity of quantum ideal gases

Padé approximants have been applied to the equation of state and heat capacity of the quantum ideal gases. For the Bose gas the agreement is almost perfect. For the Fermi gas, the maximum relative error is 0.03% for the former and 0.4% for the latter.     

Quasi 1 and 2d Dilute Bose Gas in Magnetic Traps: Existence of Off-Diagonal Order and Anomalous Quantum Fluctuations

Current magnetic traps can be made so anisotropic that dilute Bose gases confined in these traps will occupy the lowest quantum state in the tightly confining direction, while still in the Thomas-Fermi limit in the loosely confining direction. As a result, the trapped Bose gas behaves like a quasi one or two dimensional systems. Unlike the homogeneous case, quantum phase fluctuations do not destroy macroscopic off-diagonal order of trapped Bose gases in d2 because they are suppressed by the the trapping potential. In the dilute limit, quantum fluctuations increase, remain constant, and decrease with size for 3, 2, 1 d respectively. These behaviors are due to the combination of a finite gap and the universal spectrum of the collective mode.     

Zero-temperature attenuation and transverse spin dynamics in fermi liquids. II. Dilute fermi systems

This is the second in a series of papers on a consistent microscopic theory of transverse dynamics in spin-polarized or binary Fermi liquids. We demonstrate when and how the exact theory of Ref. 1 reduces to the conventional theory of highly polarized degenerate low-density Fermi liquids and gases. In the lowest approximations, i.e. for an ideal polarized Fermi gas and in the first (Born) order, our theory assumes the standard form. In the next order in density and/or interaction, the main equations still have a fairly conventional form, though they already contain the peculiar zero-temperature attenuation which is missing in the standard theory. This attenuation can be incorporated into the standard Fermi liquid formalism by adding an imaginary part to a mixed spin component of the Landau interaction function. The source of this imaginary contribution atT = 0 is a pole in the integral expression for the Landau interaction function (the situation is very similar to the case of collisionless Landau damping). In the next order, the standard theory fails completely, and even the form of the equations of transverse dynamics becomes very unconventional. We calculated explicitly the parameters of transverse spin dynamics and the spectrum of spin waves, including the zero-temperature attenuation, and, as a by-product, the polarization dependencies of thermodynamic parameters. The calculation includes a possible non-locality of the interaction. An application of the results to³He-⁴He mixtures covers the non-locality in the direct interaction channel as well as the non-locality and retardation associated with a phonon-mediated part of particles' interaction.     

Spin-polarized3He Fermi systems

Transport phenomena in spin-polarized³He systems are studied (normal Fermi liquid³He↑,³He↑-He II solutions, gas³He↑,³He↑-⁴He gaseous mixtures). The transport coefficients, including the spin diffusion, spin thermodiffusion, and spin bulk viscosity coefficients, are calculated for spinpolarized Fermi liquids and gaseous mixtures. The analogy of the “spin rotation effect” in polarized nondegenerate gases with similar phenomena in degenerate Fermi systems and with collisionless spin oscillations is discussed.     

Shape Resonance at a Lifshitz Transition for High Temperature Superconductivity in Multiband Superconductors

We discuss the shape resonance in the superconducting gaps of a two-band superconductor by tuning the chemical potential at a Lifshitz transition for Fermi surface neck collapsing and for spot appearing. The high temperature superconducting scenario for complex matter shows the coexistence of a first BCS condensate made of Cooper pairs in the first band and a second boson-like condensate made of bosons like bipolarons, in the second band where the chemical potential is tuned near a Lifshitz transition. The interband coupling controls the shape resonance in the pair exchange between the two condensates. We discuss the particular BCS–Bose crossover that occurs at the shape resonance tuning the Lifshitz parameter (the energy difference between the chemical potential and the Lifshitz topological transition) like tuning the external magnetic field for the Feshbach resonances in ultracold gases. This superconducting phase provides a particular case of topological superconductivity with multiple condensates of different winding numbers.     

Slip in Quantum Fluids

In this review we describe theoretical and experimental investigations of general slip phenomena in context with the flow of the quantum liquids³He,⁴He and their mixtures at low temperatures. The phenomenon of slip is related to a boundary effect. It occurs when sufficiently dilute gases flow along the wall of an experimental cell. A fluid is said to exhibit slip when the fluid velocity at the wall is not equal to the wall’s velocity. Such a situation occurs whenever the wall reflects the fluid particles in a specular-like manner, and/or if the fluid is describable in terms of a dilute ordinary gas (classical fluid) or a dilute gas of thermal excitations (quantum fluid). The slip effect in quantum fluids is discussed theoretically on the basis of generalized Landau-Boltzmann transport equations and generalized to apply to a regime of ballistic motion of the quasiparticles in the fluid. The central result is that the transport coefficient of bulk shear viscosity, which typically enters in the Poiseuille flow resistance and the transverse acoustic impedance, has to be replaced by geometry dependent effective viscosity, which depends on the details of the interaction of the fluid particles with the cell walls. The theoretical results are compared with various experimental data obtained in different geometries and for both Bose and Fermi quantum fluids. Good agreement between experiment and theory is found particularly in the case of pure normal and superfluid³He, with discrepancies probably arising because of deficiencies in characterization of the experimental surfaces.     

Fermionic Superfluidity: From High T c Superconductors to Ultracold Fermi Gases

We present a pairing fluctuation theory which self-consistently incorporates finite momentum pair excitations in the context of BCS—Bose–Einstein condensation (BEC) crossover, and we apply this theory to high T _c superconductors and ultracold Fermi gases. There are strong similarities between Fermi gases in the unitary regime and high T _c superconductors. Here, we address key issues of common interest, especially the pseudogap. In the Fermi gases, we summarize recent experiments including various phase diagrams (with and without population imbalance), as well as evidence for a pseudogap in thermodynamic and other experiments. This work was supported by NSF PHY-0555325 and NSF-MRSEC Grant No. DMR-0213745.     

Bose condensation of the localized pairs in two-band superconductors

Superconductor with two singularities inherent in high-T_c materials, low carrier density and overlapping of the energy bands on the Fermi surface, is considered. Provided T=0, the order parameter ‡_n and chemical potential Μ are determined in the mean-field approximation for the state with Bose condensation of the localized pairs (Μ < 0). The equation for the energy of the bound state ε_b is also obtained and the relationship ε_b = 2Μ is established. The method of functional integration concerning the two-band model is developed. On the basis of this method, the crossover between the Fermi picture and the Bose one of the elementary excitations is demonstrated for the system in the presence of the two-particle bound state. The expression for the temperature of Bose condensationT _k is obtained and the contribution of the residual interaction between bosons for the systems with arbitrary dimension is determined.     

Boundary conditions for normal-superfluid counterflow and internal Kapitza resistance in superfluids

We calculate the spatial variation of the temperature and the normal fluid velocity of superfluid Fermi and Bose liquids between parallel plates in the presence of a stationary heat flow normal to the plates. The system is modeled by a quasiparticle kinetic equation and a diffuse scattering boundary condition at the walls. We derive integral equations for the hydrodynamic fields, which are solved numerically for arbitrary ratio of the mean free path to the plate separationL. For small /L we recover the hydrodynamic boundary conditions proposed recently by Grabinski and Liu and are able to determine the three surface Onsager coefficients. In the Kundsen regime (large /L), we find the thermal boundary resistance to increase exponentially with decreasing temperature.     

Quartet Structures of Particles and Quasiparticles in the BCS Model

We study the possibility of multiple paring for more than two particles or two quasiparticles in terms of the BCS model. We consider the multiple pairs of particles in terms of the BCS Hamiltonian for two different ground states: in a quiescent Fermi sea model and in the BCS ground state. In case of quasiparticles, we consider the multiple quasiparticle pairs in terms of the BCS ground state only. Although there is no interaction between Cooper’s type pairs in terms of the BCS Hamiltonian, yet we have shown that four particles or two/four quasiparticles can be paired and form a bound state in the singlet state with just twice the bound state energy of a single Cooper’s pair. In the case of a pair of quasiparticles, the bound state exists as a result of the large quasiparticle density of states and the residual interaction between two quasiparticles which is described by the off-diagonal terms in the BCS Hamiltonian written in terms of quasiparticles. In the case of four particles, the bound state exists only as a result of the Pauli principle and the sharp Fermi edge. In the BCS model, a quartet of bound fermions indeed represents a boson, and the many-particle system of the composite of bosons can undergo the conventional Bose condensation of a boson gas. The temperature of the Bose condensation corresponds to the conventional temperature of Bose condensation for non-interacting bosons where each boson has quadruple mass (4m) and the boson density is one-quarter (n/4) of fermions density. A similar conclusion remains valid in the case of the particle–hole resonance in a quiescent Fermi sea. We have shown that in the particle–hole channel there exists the multiple particle–hole resonance for four particles and four holes in a quiescent Fermi sea model similar to the case of two particles and two holes resonance. We have shown that there is no particle–hole resonance in the case of the BCS ground state because there is no hole-type of excitations in the BCS ground state. Nevertheless we have shown that in the BCS ground state quasiparticle pairs can form bound pairs similar to the Cooper’s pair of particles due to the off-diagonal terms in the BCS Hamiltonian. The bound pairs of quasiparticles exist as a result of extremely large quasiparticle density of states. We also discuss the formation of a quartet of quasiparticles and the Bose condensation of the multiple pairs of quasiparticles.     

Two-Fluid Hydrodynamics in Trapped Bose Gases and in Superfluid Helium

A review is given of recent theoretical work on the superfluid dynamics of trapped Bose gases at finite temperatures, where there is a significant fraction of non-condensate atoms. One can now reach large enough densities and collision cross-sections needed to probe the collective modes in the collision-dominated hydrodynamic region where the gas exhibits characteristic superfluid behavior involving the relative motions of the condensate and non-condensate components. The precise analogue of the Landau-Khalatnikov two-fluid hydrodynamic equations was recently derived from trapped Bose gases, starting from a generalized Gross-Pitaevskii equation for the condensate macroscopic wavefunction and a kinetic equation for the non-condensate atoms.     

Flow and Critical Velocity of an Imbalanced Fermi Gas through an Optical Potential

Optical lattices offer the possibility to investigate the superfluid properties of both Bose condensates and Fermionic superfluid gases. When a population imbalance is present in a Fermi mixture, this leads to frustration of the pairing, and the superfluid properties will be affected. Here, we investigate how imbalance will influence the flow of a Fermi superfluid through an optical lattice. The flow through the lattice is analysed by taking into account coupling between neighboring layers of the optical lattice up to second order in the interlayer tunneling amplitude for single atoms. We find that the critical velocity of flow through the lattice decreases monotonically to zero as the imbalance is increased to 100%. Closed-form analytical expressions are given for the tunneling contribution to the action and for the critical velocity as a function of the binding energy of pairs in the (quasi) two-dimensional Fermi superfluid and as a function of the imbalance. These results are obtained in the mean-field approximation which is known to provide only a qualitative picture near unitarity. Nevertheless this mean-field result should provide a useful benchmark for theories that take into account fluctuations beyond the saddle-point.      

Boundary slip in spin-polarized quantum systems

We describe the effects of boundary slip in spin-polarized quantum liquids and gases. The slip coefficients in boundary conditions form a 3 × 3 matrix. The off-diagonal coefficients are expressed via each other with the help of the Onsager relations. We calculate accurate lower and upper bounds of all slip coefficients for polarized degenerate Fermi liquids and for dilute gases at arbitrary temperatures. The calculations are based on the transport equation for spin-polarized systems with diffuse boundary conditions. The results for gases are especially simple in the limiting cases of low-temperature degenerate systems or in the high-temperature classical Boltzmann regime. All slip coefficients are proportional to the mean free path and increase with increasing spin polarization. As a by-product the theory describes the slip effects in binary mixtures of classical gases or Fermi liquids when the role of spin polarization is played by the concentration of the mixture.     

Thermodynamics of a Trapped Bose-Condensed Gas

We investigate the thermodynamic behaviour of a Bose gas interacting with repulsive forces and confined in a harmonic anisotropic trap. We develop the formalism of mean field theory for non uniform systems at finite temperature, based on the generalization of Bogoliubov theory for uniform gases. By employing the WKB semiclassical approximation for the excited states we derive systematic results for the temperature dependence of various thermodynamic quantities: condensate fraction, density profiles, thermal energy, specific heat and moment of inertia. Our analysis points out important differences with respect to the thermodynamic behaviour of uniform Bose gases. This is mainly the consequence of a major role played by single particle states at the boundary of the condensate. We find that the thermal depletion of the condensate is strongly enhanced by the presence of repulsive interactions and that the critical temperature is decreased with respect to the predictions of the non-interacting model. Our work points out an important scaling behaviour exhibited by the system in large N limit. Scaling permits to express all the relevant thermodynamic quantities in terms of only two parameters: the reduced temperature t = T/T _c ⁰ and the ratio between the T = 0 value of the chemical potential and the critical temperature T _c ⁰ for Bose-Einstein condensation. Comparisons with first experimental results and ab-initio calculations are presented.     

Ultracold Fermi Gases : The BEC-BCS Crossover

We give an introduction to the present state of research in the field of ultracold atomic Fermi gases. Of particular interest is the search for (BCS) superfluidity in these systems. The existence of Feshbach resonance have made these systems especially flexible, allowing to control at will the strength and the sign of the effective interaction. In particular it has allowed an experimental realization of the Bose-Einstein Condensation (BEC)-BCS crossover allowing to display on one side a Bose condensate of molecules and on the other side a BCS superfluid produced by Cooper pairs. Then we focus more specifically to the shift of the molecular threshold, produced by Pauli exclusion in these degenerate gas. We discuss also briefly collective modes corresponding to oscillations of the gas in the harmonic trap in which it is confined.      

Collective Excitations of a One-Dimensional Fermi Gas Under Harmonic Confinement

We study the spectrum of collective excitations of a spin-polarized Fermi gas confined in a one-dimensional harmonic trap at zero temperature. In the collisionless regime we evaluate exactly the dynamic structure factor, while in the collisional regime we solve analytically the linearized equations of hydrodynamics in the Thomas-Fermi approximation. We also verify the validity of the Thomas-Fermi theory by solving numerically a time-dependent nonlinear Schroedinger equation with a fifth-order interaction term. We find that in both the collisionless and the collisional regime the excitation frequencies of the Fermi gas are multiples of the trap frequency, analogously to the case of the one-dimensional homogeneous Fermi fluid where the velocities of zero and first sound coincide. Due to boson-fermion dynamical mapping our results for the spectrum apply as well to a one-dimensional Bose gas with hard-core point-like interactions (Tonks gas).     

Dynamics of correlations in atomic Bose-Einstein condensates

Abstract

The Gross-Pitaevskii equation has been extremely successful in the theory of weakly interacting Bose-Einstein condensates. However, present-day experiments reach beyond the regime of its validity due to the significant role of correlations. We review a method for tackling the dynamics of correlations in Bose condensed gases, in terms of non-commutative cumulants. This new approach has a wide applicability in the areas of current interest, e.g. the production of molecules and the manipulation of interactions in condensates. It also offers an interesting perspective on the classical-field methods for partly condensed Bose gases.