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Laser Physics

Quantum dynamics of a dipolar Fermi gas in free expansion

Takushi Nishimura, Tomoyuki Maruyama

Published Apr 2, 2010• 0 citations
Publisher

Abstract

We presented our theoretical study on quantum dynamics of a polarized dipolar Fermi gas in free expansion. The dipole-dipolar interparticle interaction induces axisymmetrical deformation of the expanding gas not only in the spatial space but also in the momentum space, so that, in order to obtain proper results in the time-of-flight method for the dipolar Fermi gas, it is necessary to deal with time-evolution of the deformation. To solve the free expansion problem, we develop the Hartree-Fock and Landau-Vlasov approaches and a new time-evolution ansatz for the quantum dynamics. In conclusion, we obtain exact predictions for the minimal quadrupole deformation of the high-temperature Maxwell-Boltzmann and zero-temperature Thomas-Fermi gases in the week-interaction and small-deformation regime, and also reveal a scaling law associated with the Liouville’s theorem in the long-time behaviors of the MB and TF gases.

DOI
10.1134/s1054660x10090148

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We presented our theoretical study on quantum dynamics of a polarized dipolar Fermi gas in free expansion. The dipole-dipolar interparticle interaction induces axisymmetrical deformation of the expanding gas not only in the spatial space but also in the momentum space, so that, in order to obtain proper results in the time-of-flight method for the dipolar Fermi gas, it is necessary to deal with time-evolution of the deformation. To solve the free expansion problem, we develop the Hartree-Fock and Landau-Vlasov approaches and a new time-evolution ansatz for the quantum dynamics. In conclusion, we obtain exact predictions for the minimal quadrupole deformation of the high-temperature Maxwell-Boltzmann and zero-temperature Thomas-Fermi gases in the week-interaction and small-deformation regime, and also reveal a scaling law associated with the Liouville’s theorem in the long-time behaviors of the MB and TF gases.
We theoretically investigate a polarized dipolar Fermi gas in free expansion. The inter-particle dipolar interaction deforms phase-space distribution in trap and also in the expansion. We exactly predict the minimal quadrupole deformation in the expansion for the high-temperature Maxwell-Boltzmann and zero-temperature Thomas-Fermi gases in the Hartree-Fock and Landau-Vlasov approaches. In conclusion, we provide a proper approach to develop the time-of-flight method for the weakly-interacting dipolar Fermi gas and also reveal a scaling law associated with the Liouville's theorem in the long-time behaviors of the both gases.
A recent time-of-flight (TOF) expansion experiment with polarized fermionic erbium atoms measured a Fermi surface deformation from a sphere to an ellipsoid due to dipole-dipole interaction, thus confirming previous theoretical predictions. Here we perform a systematic study of the ground-state properties and TOF dynamics for trapped dipolar Fermi gases from the collisionless to the hydrodynamic regime at zero temperature. To this end, we solve analytically the underlying Boltzmann-Vlasov equation within the relaxation-time approximation in the vicinity of equilibrium by using a suitable rescaling of the equilibrium distribution. The resulting ordinary differential equations for the respective scaling parameters are then solved numerically for experimentally realistic parameters and relaxation times that correspond to the collisionless, collisional, and hydrodynamic regimes. The equations for the collisional regime are first solved in the approximation of a fixed relaxation time, and then this approach is extended to include a self-consistent determination of the relaxation time. The presented analytical and numerical results are relevant for a detailed quantitative understanding of ongoing experiments and the design of future experiments with ultracold fermionic dipolar atoms and molecules. In particular, the obtained results are relevant for systems with strong dipole-dipole interaction, which turn out to affect significantly the aspect ratios during the TOF expansion.
In this paper, we investigate the dynamical properties of a one-component Fermi gas with dipole–dipole interaction between particles. Using a variational function based on the Thomas–Fermi density distribution in phase space representation, the total energy is described as a function of deformation parameters in both real and momentum spaces. Various thermodynamic quantities of a uniform dipolar Fermi gas are derived, and then instability of this system is discussed. For a trapped dipolar Fermi gas, the collective oscillation frequencies are derived with the energy-weighted sum rule method. The frequencies for the monopole, quadrupole, radial and axial modes are calculated, and softening against collapse is shown as the dipolar strength approaches the critical value. Finally, we investigate the expansion dynamics of the Fermi gas and show how the dipolar interaction manifests itself in the shape of an expanded cloud.
By means of the Boltzmann-Vlasov kinetic equation we investigate dynamical properties of a trapped one-component Fermi gas at zero temperature, featuring the anisotropic and long-range dipole-dipole interaction. To this end, we determine an approximate solution by rescaling both space and momentum variables of the equilibrium distribution, thereby obtaining coupled ordinary differential equations for the corresponding scaling parameters. Based on previous results on how the Fermi sphere is deformed in the hydrodynamic regime of a dipolar Fermi gas, we are able to implement the relaxation-time approximation for the collision integral. Then, we proceed by linearizing the equations of motion around the equilibrium in order to study both the frequencies and the damping of the low-lying excitation modes all the way from the collisionless to the hydrodynamic regime. Our theoretical results are expected to be relevant for understanding current experiments with trapped dipolar Fermi gases.
By means of the Boltzmann-Vlasov kinetic equation we investigate dynamical properties of a trapped, one-component Fermi gas at zero temperature, featuring the anisotropic and long-range dipole-dipole interaction. To this end, we determine an approximate solution by rescaling both space and momentum variables of the equilibrium distribution, thereby obtaining coupled ordinary differential equations for the corresponding scaling parameters. Based on previous results on how the Fermi sphere is deformed in the hydrodynamic regime of a dipolar Fermi gas, we are able to implement the relaxation-time approximation for the collision integral. Then, we proceed by linearizing the equations of motion around the equilibrium in order to study both the frequencies and the damping of the low-lying excitation modes all the way from the collisionless to the hydrodynamic regime. Our theoretical results are expected to be relevant for understanding current experiments with trapped dipolar Fermi gases.
The quest for quantum degenerate Fermi gases interacting through the anisotropic and long-range dipole-dipole interaction is an exciting and fast developing branch within the cold-atom research program. Recent experimental progress in trapping, cooling, and controlling polar molecules with large electric dipole moments has, therefore, motivated much theoretical effort. In a recent letter, we have briefly discussed the application of a variational time-dependent Hartree-Fock approach to study theoretically both the static and the dynamic properties of such a system in a cylinder-symmetric harmonic trap. We focused on the hydrodynamic regime, where collisions ensure the equilibrium locally. Here we present a detailed theory extended to encompass the general case of a harmonic trap geometry without any symmetry. After deriving the equations of motion for the gas, we explore their static solutions to investigate key properties like the aspect ratios in both real and momentum space as well as the stability diagram. We find that, despite the lack of symmetry of the trap, the momentum distribution remains cylinder symmetric. The equations of motion are then used to study the low-lying hydrodynamic excitations, where, apart from the quadrupole and monopole modes, the radial quadrupole mode is also investigated. Furthermore, we study the time-of-flight dynamics as it represents an important diagnostic tool for quantum gases. We find that the real-space aspect ratios are inverted during the expansion, while that in momentum space becomes asymptotically unity. In addition, anisotropic features of the dipole-dipole interaction are discussed in detail. These results could be particularly useful for future investigations of strongly dipolar heteronuclear polar molecules deep in the quantum degenerate regime.