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Ground-state properties of a droplet of ultracold atomic fermions

Nicola Manini, Luca Salasnich, Roberto Meloni

Published Jan 1, 2009• 0 citations
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Abstract

We analyse the ground-state density profiles of a cloud of ultracold fermions in a harmonic, cilindrically symmetric potential. We study in detail the BCS-BEC crossover as the fermion-fermion interaction strength varies. We evaluate the ground-state wavefunction in a mean field regime, expanding the wavefunction on a truncated harmonic-oscillator basis of eigenfunctions. We use LAPACK routines to solve iteratively the nonlinear matrix problem; we study the convergence properties of this method with respect to the basis size. Advisor: Dr. Nicola Manini Co-Advisor: Dr. Luca Salasnich

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We explore the ground states and dynamics of ultracold atomic droplets in the crossover region from three to two dimensions by solving the two-dimensional and the quasi two-dimensional extended Gross-Pitaevskii equations numerically and with a variational approach. By systematically comparing the droplet properties, we determine the validity regions of the pure two-dimensional description, and therefore the dominance of the logarithmic nonlinear coupling, as a function of the sign of the averaged mean-field interactions and the size of the transverse confinement. One of our main findings is that droplets become substantially extended upon transitioning from negative-to-positive averaged mean-field interactions. This is accompanied by a significant reduction of their binding energies which are approximately inversely proportional to the square of their size. To explore fundamental dynamical properties in the crossover region, we study interaction quenches and show that the droplets perform a periodic breathing motion for modest quench strengths, while larger quench amplitudes lead to continuous expansion exhibiting density ring structures. We also showcase that it is possible to form complex bulk and surface density patterns in anisotropic geometries following the quench. Since we are working with realistic parameters, our results can directly facilitate future experimental realizations.

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The ground-state properties of a two-component Fermi gas with attractive short-range interactions are calculated using the fixed-node diffusion Monte Carlo method. The interaction strength is varied over a wide range by tuning the value a of the s-wave scattering length of the two-body potential. We calculate the energy per particle, the one- and two-body density matrix as a function of the interaction strength. Results for the momentum distribution of the atoms, as obtained from the Fourier transform of the one-body density matrix, are reported as a function of the interaction strength. Off-diagonal long-range order in the system is investigated through the asymptotic behavior of the two-body density matrix. The condensate fraction of pairs is calculated in the unitary limit and on both sides of the BCS-BEC crossover.

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We investigate density fluctuations in a coherent ensemble of interacting fermionic atoms. Adapting the concept of full counting statistics, well known from quantum optics and mesoscopic electron transport, we study second-order as well as higher-order correlators of density fluctuations. Using the mean-field BCS state to describe the whole interval between the BCS limit and the Bose-Einstein condensate (BEC) limit, we obtain an exact expression for the cumulant-generating function of the density fluctuations of an atomic cloud. In the two-dimensional case, we obtain a closed analytical expression. Poissonian fluctuations of a molecular condensate on the BEC side are strongly suppressed on the BCS side. The size of the fluctuations in the BCS limit is a direct measure of the pairing potential. We also discuss the BEC-BCS crossover of the third cumulant and the temperature dependence of the second cumulant.

arXiv:cond-mat/0412269•Physical Review A•Jun 12, 2007•22 citations
We analyse the ground-state density profiles of a cloud of ultracold fermions in a harmonic, cilindrically symmetric potential. We study in detail the BCS-BEC crossover as the fermion-fermion interaction strength varies. We evaluate the ground-state wavefunction in a mean field regime, expanding the wavefunction on a truncated harmonic-oscillator basis of eigenfunctions. We use LAPACK routines to solve iteratively the nonlinear matrix problem; we study the convergence properties of this method with respect to the basis size. Advisor: Dr. Nicola Manini Co-Advisor: Dr. Luca Salasnich
We study the mean-field dynamic of a interacting boson-fermion mixture in an harmonic trap, by means of a Gaussian variational ansatz. We find concentric stationary solutions in the repulsive regime and in the regime of weak boson-fermion attraction. Above a threshold value of the repulsion parameter, a stable symmetry-broken stationary solution bifurcates, characterized by the displacement of the center of mass of the bosonic and fermionic clouds in opposite directions away from the trap center. We observe linear and nonlinear oscillations of the width and center of mass of the boson and fermion distributions whenever we let the system evolve starting from a non stationary initial condition. Advisor: Dr. Nicola Manini Co-Advisor: Dr. Luca Salasnich
We explore the ground states and dynamics of ultracold atomic droplets in the crossover region from three to two dimensions by solving the two-dimensional and the quasi two-dimensional extended Gross-Pitaevskii equations numerically and with a variational approach. By systematically comparing the droplet properties, we determine the validity regions of the pure two-dimensional description, and therefore the dominance of the logarithmic nonlinear coupling, as a function of the sign of the averaged mean-field interactions and the size of the transverse confinement. One of our main findings is that droplets become substantially extended upon transitioning from negative-to-positive averaged mean-field interactions. This is accompanied by a significant reduction of their binding energies which are approximately inversely proportional to the square of their size. To explore fundamental dynamical properties in the crossover region, we study interaction quenches and show that the droplets perform a periodic breathing motion for modest quench strengths, while larger quench amplitudes lead to continuous expansion exhibiting density ring structures. We also showcase that it is possible to form complex bulk and surface density patterns in anisotropic geometries following the quench. Since we are working with realistic parameters, our results can directly facilitate future experimental realizations.
In this thesis, following L. Salasnich and F. Toigo [Phys. Rev. A 78, 053626 (2008)] we use an extended Thomas-Fermi (TF) density functional to describe a gas of interacting fermions composed of an even number N of particle in the unitary regime confined by a spherically symmetric harmonic trap. The great advantage in taking such a functional, relative to functionals based on single-fermion wave functions, is to depend only on the density number n and on its derivatives, so that the numerical load is independent of the number of particles. By minimizing the ETF functional with respect to the density we obtain a time-independent non-linear Schrodinger equation which we solve using the imaginary time propagation method, by integrating numerically the corresponding time-dependent Schrodinger equation. We determine the two characteristic parameters of the functionals ξ and λ by fitting the Monte Carlo total energy data obtained by D. Blume et al. [Phys. Rev. Lett. 99, 233201 (2007)]. We obtain the values ξ = 0.469 and λ = 0.082 leading to a χ2 = 42. These values are to be compared with ξ = 0.455 and λ = 0.13, leading to a χ2 = 282 as proposed by Salasnich and Toigo. We also study the dependence of the ground-state density profile on the values of the two parameters, finding that ξ enhances the width of the profile, while increasing λ smooths out the outer boundary of the fermionic cloud. Advisor: Dr. Nicola Manini Co-Advisor: Dr. Luca Salasnich
In this Thesis we will theoretically address some issues concerning the Physics of ultracold Fermi gases, all of them with experimental relevance. The field of ultracold gases, and more recently of ultracold Fermi gases, is gathering a lot of experimental and theoretical interest for two main reasons: the great control on the relevant parameters of the problem and the relative simplicity of the minimal theory which is able to correctly describe the system. Building upon this solid background, ultracold gases provided an ideal laboratory for testing more refined theories and also for addressing fundamental issues of quantum mechanics as well as for simulating more complex physical systems such as those encountered in condensed matter. Even if the effective hamiltonian of ultracold gases can be simple, due to the diluteness of the system and the low temperature, which imply low energy physics, the solution of the quantum mechanic equations governing the state of these systems is not always simple. For the static properties usually mean-field solutions exist or perturbative expansions can be produced in some regimes. However Quantum Monte Carlo (QMC) techniques provide more accurate results especially in the strongly interacting regimes. For confined systems it is possible to use QMC only for a few particles, so that, for large number of particles, a fruitful combined use of Density Functional Theory (DFT) and QMC is necessary. The study of the dynamics of ultracold gases has received little attention with QMC techniques, due to the intrinsic computational difficulty of the many-body problem, so that general hydrodynamic equations are often used for studying the propagation of smoothly varying perturbations. In this Thesis we use QMC techniques for studying the problem of ferromagnetism in repulsive or effectively repulsive ultracold gases without a lattice and the problem of the Bardeen-Cooper-Schrieffer to Bose-Einstein-Condensation (BCS-BEC) crossover in two dimensions. We use DFT in the Local Density Approximation (LDA) for calculating the density profiles of ultracold Fermi gases in harmonic magneto-optical traps, starting from QMC equations of state. We study the propagation of first and second sound in ultracold Fermi gases in cylindrical geometry, using the hydrodynamic equations of superfluids.
The ground-state properties of a two-component Fermi gas with attractive short-range interactions are calculated using the fixed-node diffusion Monte Carlo method. The interaction strength is varied over a wide range by tuning the value a of the s-wave scattering length of the two-body potential. We calculate the energy per particle, the one- and two-body density matrix as a function of the interaction strength. Results for the momentum distribution of the atoms, as obtained from the Fourier transform of the one-body density matrix, are reported as a function of the interaction strength. Off-diagonal long-range order in the system is investigated through the asymptotic behavior of the two-body density matrix. The condensate fraction of pairs is calculated in the unitary limit and on both sides of the BCS-BEC crossover.
We investigate density fluctuations in a coherent ensemble of interacting fermionic atoms. Adapting the concept of full counting statistics, well known from quantum optics and mesoscopic electron transport, we study second-order as well as higher-order correlators of density fluctuations. Using the mean-field BCS state to describe the whole interval between the BCS limit and the Bose-Einstein condensate (BEC) limit, we obtain an exact expression for the cumulant-generating function of the density fluctuations of an atomic cloud. In the two-dimensional case, we obtain a closed analytical expression. Poissonian fluctuations of a molecular condensate on the BEC side are strongly suppressed on the BCS side. The size of the fluctuations in the BCS limit is a direct measure of the pairing potential. We also discuss the BEC-BCS crossover of the third cumulant and the temperature dependence of the second cumulant.