Motivated by recent experiments on rotating Bose-Einstein condensates, we investigate a rotating, polarized Fermi gas trapped in an anharmonic potential. We apply a semiclassical expansion of the density of states in order to determine how the thermodynamic properties depend on the rotation frequency. The accuracy of the semiclassical approximation is tested and shown to be sufficient for describing typical experiments. At zero temperature, rotating the gas above a given frequency $Ω_{\rm DO}$ leads to a `donut'-shaped cloud which is analogous to the hole found in two-dimensional Bose-Einstein condensates. The free expansion of the gas after suddenly turning off the trap is considered and characterized by the time and rotation frequency dependence of the aspect ratio. Temperature effects are also taken into account and both low- and high-temperature expansions are presented for the relevant thermodynamical quantities. In the high-temperature regime a virial theorem approach is used to study the delicate interplay between rotation and anharmonicity.
We present a theoretical model to describe the dynamics of Bose-Einstein condensates in anharmonic trapping potentials. To first approximation the center-of-mass motion is separated from the internal condensate dynamics and the problem is reduced to the well known scaling solutions for the Thomas-Fermi radii. We discuss the validity of this approach and analyze the model for an anharmonic waveguide geometry which was recently realized in an experiment \cite{Ott2002c}.
We determine the quantitative effect, in the maximum number of particles and other static observables, due to small anharmonic terms added to the confining potential of an atomic condensed system with negative two-body interaction. As an example of how a cubic or quartic anharmonic term can affect the maximum number of particles, we consider the trap parameters and the results given by Roberts et al. [Phys. Rev. Lett. 86, 4211 (2001)]. However, this study can be easily transferred to other trap geometries to estimate anharmonic effects.
There is currently intensive research into creating a large-scale quantum computer with trapped ions. It is well known that for a linear ion crystal in a harmonic potential, the ions near the center are more closely spaced compared to the ions near the ends. This is problematic as the number of ions increases. Here, we consider a linear ion crystal in an anharmonic potential that is purely quartic in position. We find that the ions are more evenly spaced compared to the harmonic case. We develop a variational approach to calculate the properties of the ground state. We also characterize the zi…
We present an overview of the various phase transitions that we anticipate to occur in trapped fermionic alkali gases. We also discuss the prospects of observing these transitions in (doubly) spin-polarized Li-6 and K-40 gases, which are now actively being studied by various experimental groups around the world.
We investigate theoretically the properties of an ideal trapped gas in a time-dependent harmonic potential. Using a scaling formalism, we are able to present simple analytical results for two important classes of experiments: free expansion of the gas upon release of the trap; and the response of the gas to a harmonic modulation of the trapping potential is investigated. We present specific results relevant to current experiments on trapped Fermions.
We present an analytical approximation for nonlinear dynamics of trapped Bose-co ndensed gases. The new approximation is a substantial improvement over the Thomas-Fermi approximation and is shown to be applicable for systems with a rather small number of atoms $N$. The calculated aspect ratios after ballistic expansion are found to be in good agreement with those observed in recent experiments.
We discuss BEC in (quasi)2D trapped gases and find that well below the transition temperature $T_c$ the equilibrium state is a true condensate, whereas at intermediate temperatures $T<T_c$ one has a quasicondensate (condensate with fluctuating phase). The mean-field interaction in a quasi2D gas is sensitive to the frequency $ω_0$ of the (tight) confinement in the "frozen" direction, and one can switch the sign of the interaction by changing $ω_0$. Variation of $ω_0$ can also reduce the rates of inelastic processes, which opens prospects for tunable BEC in trapped quasi2D gases.
We discuss stability and phase coherence of 1D trapped Bose gases and find that inelastic decay processes, such as 3-body recombination, are suppressed in the strongly interacting (Tonks-Girardeau) and intermediate regimes. This is promising for achieving these regimes with a large number of particles. "Fermionization" of the system reduces the phase coherence length, and at T=0 the gas is fully phase coherent only deeply in the weakly interacting (Gross-Pitaevskii) regime.
We investigate the width of the Ginzburg critical region and experimental signatures of critical behavior in strongly interacting trapped Fermi gases close to unitarity, where the s-wave scattering length diverges. Despite the fact that the width of the critical region is of the order unity, evidence of critical behavior in the bulk thermodynamics of trapped gases is strongly suppressed by their inhomogeneity. The specific heat of a harmonically confined gas, for instance, is \textit{linear} in the reduced temperature $t = (T-T_{\mathrm{c}})/T_{\mathrm{c}}$ above $T_{\mathrm{c}}$. We also disc…
In this comment, I discuss a recent path-integral Monte Carlo calculation by Pearson, Pang, and Chen (Phys. Rev. A 58, 4796 (1998)). For bosons with a small hard-core interaction in a harmonic trap, the authors find a critical temperature which does not change with respect to the non-interacting gas. The calculation suffers from a serious discretization error of the many-particle density matrix.
We derive variationally the HFB equations for a trapped self-interacting Bose gas at finite temperature. In the Thomas-Fermi limit, we obtain simple expressions for the condensate, the non condensate and the anomalous densities. Their behavior in terms of the condensate fraction meets qualitatively the experimental data. In particular, the non condensate and the anomalous densities are peaked at the center of the trap and not at the edges as predicted by the self-consistent HFB calculations.
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 …
We apply the closed time-path formalism to evaluate the dynamics of the BCS transition to the superfluid state in trapped atomic $^6$Li. We find that the Fokker-Planck equation for the probability distribution of the order parameter is, sufficiently close to the critical temperature, identical to the equation that describes the switching on of a single-mode laser.
We present the general dielectric formalism for Bose-Einstein condensed systems in external potential at finite temperatures. On the basis of a model arising within this framework as a first approximation in an intermediate temperature region for large condensate we calculate the damping of low-energy excitations in the collisionless regime.
We discuss the regimes of quantum degeneracy in a trapped 1D gas and obtain the diagram of states. Three regimes have been identified: the BEC regimes of a true condensate and quasicondensate, and the regime of a trapped gas of Tonks (gas of impenetrable bosons). The presence of a sharp cross-over to the BEC regime requires extremely small interaction between particles. We discuss how to distinguish between true and quasicondensates in phase coherence experiments.
We study a rotating Bose-Einstein Condensate in a strongly anharmonic trap (flat trap with a finite radius) in the framework of 2D Gross-Pitaevskii theory. We write the coupling constant for the interactions between the gas atoms as $1/ε^2$ and we are interested in the limit $ε\to 0$ (TF limit) with the angular velocity $Ω$ depending on $ε$. We derive rigorously the leading asymptotics of the ground state energy and the density profile when $Ω$ tends to infinity as a power of $1/ε$. If $Ω(ε)=Ω_0/ε$ a ``hole'' (i.e., a region where the density becomes exponentially small as $1/ε\to\infty$) deve…
We study some aspects of equilibrium and off equilibrium quantum dynamics of dilute bosonic gases in the presence of a trapping potential. We consider systems with a fixed number of particles N and study their scaling behavior with increasing the trap size. We focus on one-dimensional (1D) bosonic systems, such as gases described by the Lieb-Liniger model and its Tonks-Girardeau limit of impenetrable bosons, and gases constrained in optical lattices as described by the Bose-Hubbard model. We study their quantum (zero-temperature) behavior at equilibrium and off equilibrium during the unitary t…
The two-fluid and Landau criteria for superfluidity are compared for trapped Bose gases. While the two-fluid criterion predicts translational superfluidity, it is suggested, on the basis of the homogeneous Gross-Pitaevski limit, that a necessary part of Landau's criterion, adequate for non-translationally invariant systems, d oes not hold for trapped Bose gases in the GP limit. As a consequence, if the compressibility is detected to be very large (infinite by experimental standards) the two-fluid criterion is seen to be the relevant one in case the system is a tranlational superfluid, while th…
We derive a model to describe decoherence of atomic clouds in atom-chip traps taking the excited states of the trapping potential into account. We use this model to investigate decoherence for a single trapping well and for a pair of trapping wells that form the two arms of an atom interferometer. Including the discrete spectrum of the trapping potential gives rise to a decoherence mechanism with a decoherence rate $Γ$ that scales like $Γ\sim 1/r_0^4$ with the distance $r_0$ from the trap minimum to the wire.
We exploit and clarify the use of random matrix theory for the calculation of the entanglement entropy of free Fermi gases. We apply this method to obtain analytic predictions for Renyi entanglement entropies of a one-dimensional gas trapped by a harmonic potential in all the relevant scaling regimes. We confirm our findings with accurate numerical calculations obtained by means of an ingenious discretisation of the reduced correlation matrix.
We study the creation of vortex states in a trapped Bose-Einstein condensate by a rotating force. For a harmonic trapping potential the rotating force induces only a circular motion of the whole condensate around the trap center which does not depend on the interatomic interaction. For the creation of a pure vortex state it is necessary to confine the atoms in an anharmonic trapping potential. The efficiency of the creation can be greatly enhanced by a sinusodial variation of the force's angular velocity. We present analytical and numerical calculations for the case of a quartic trapping poten…
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 …
We present a unified overview, from the mean-field to the unitarity regime, of the stability of a trapped Fermi gas with short range attractive interactions. Unlike in a system of bosons, a Fermi gas is always stable in these regimes, no matter how large the particle number. However, when the interparticle spacing becomes comparable to the range of the interatomic interactions, instability is not precluded.
We present a theoretical study of the collective excitations of a trapped imbalanced fermion gas at unitarity, when the system consists of a superfluid core and a normal outer shell. We formulate the relevant boundary conditions and treat the normal shell both hydrodynamically and collisionlessly. For an isotropic trap, we calculate the mode frequencies as a function of trap polarization. Out-of-phase modes with frequencies below the trapping frequency are obtained for the case of a hydrodynamic normal shell. For the collisionless case, we calculate the monopole mode frequencies, and find that…
The problem of particle fluctuations in arbitrary nonuniform systems with Bose-Einstein condensate is considered. This includes the case of trapped Bose atoms. It is shown that the correct description of particle fluctuations for any nonuniform system of interacting atoms always results in thermodynamically normal fluctuations.
We study the properties of a spin-polarized Fermi gas in a harmonic trap, using the semiclassical (Thomas-Fermi) approximation. Universal forms for the spatial and momentum distributions are calculated, and the results compared with the corresponding properties of a dilute Bose gas.
We show that the slow driving of a focused laser beam through the cloud of trapped cold fermions allows for a creation of the collective excitation in the system. The method, proposed originally by us for bosons, seems to be quite feasible experimentally -- it requires only an appropriate change in time of the potential in atomic traps, as realized in laboratories already.
We study population imbalanced Fermi mixtures under quasi-two-dimensional confinement at zero temperature. Using mean-field theory and the local-density approximation, we study the ground state configuration throughout the BEC-BCS crossover. We find the trapped system to be either fully normal or to consist of a superfluid core surrounded by a normal shell, which is itself either fully or partially polarized. Upon changing the trap imbalance, the trap configuration may undergo continuous transitions between the different ground states. Finally, we argue that thermal equilibration throughout th…
We calculate the dynamic single-particle and many-particle correlation functions at non-zero temperature in one-dimensional trapped repulsive Bose gases. The decay for increasing distance between the points of these correlation functions is governed by a scaling exponent that has a universal expression in terms of observed quantities. This expression is valid in the weak-interaction Gross-Pitaevskii as well as in the strong-interaction Girardeau-Tonks limit, but the observed quantities involved depend on the interaction strength. The confining trap introduces a weak center-of-mass dependence i…
