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arXiv:1202.0584 · Physical Review A

Scale-invariant Fermi gas in a time-dependent harmonic potential

Sergej Moroz

Published Jul 6, 2012• 27 citations
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Abstract

A scale-invariant two-component Fermi gas in a time-dependent isotropic harmonic potential is investigated. The exact time evolution of the density distribution in position space in any spatial dimension is obtained. Two experimentally relevant examples, an abrupt change and a periodic modulation of the trapping frequency, are solved. Small deviations from scale invariance are addressed within first-order perturbation theory. The consequences for experiments with ultracold quantum gases, such as the excitation of a tower of undamped breathing modes and an alternative for measuring the Tan contact, are discussed.

DOI
10.1103/physreva.86.011601

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A scale-invariant two-component Fermi gas in a time-dependent isotropic harmonic potential is investigated. The exact time evolution of the density distribution in position space in any spatial dimension is obtained. Two experimentally relevant examples, an abrupt change and a periodic modulation of the trapping frequency, are solved. Small deviations from scale invariance are addressed within first-order perturbation theory. The consequences for experiments with ultracold quantum gases, such as the excitation of a tower of undamped breathing modes and an alternative for measuring the Tan contact, are discussed.
A unitary Fermi gas in an isotropic harmonic trap is predicted to show scale and conformal symmetry that have important consequences in its thermodynamic and dynamical properties. By experimentally realizing a unitary Fermi gas in an isotropic harmonic trap, we demonstrate its universal expansion dynamics along each direction and at different temperatures. We show that as a consequence of SO(2,1) symmetry, the measured release energy is equal to that of the trapping energy. We further observe the breathing mode with an oscillation frequency twice the trapping frequency and a small damping rate, providing the evidence of SO(2,1) symmetry. In addition, away from resonance when scale invariance is broken, we determine the effective exponent γ that relates the chemical potential and average density along the BEC-BCS crossover, which qualitatively agrees with the mean field predictions. This Letter opens the possibility of studying nonequilibrium dynamics in a conformal invariant system in the future.
The frequency of the breathing mode of a two-dimensional Fermi gas with zero-range interactions in a harmonic confinement is fixed by the scale invariance of the Hamiltonian. Scale invariance is broken in the quantized theory by introducing the two-dimensional scattering length as a regulator. This is an example of a quantum anomaly in the field of ultracold atoms and leads to a shift of the frequency of the collective breathing mode of the cloud. In this work, we study this anomalous frequency shift for a two-component Fermi gas in the strongly interacting regime. We measure significant upwards shifts away from the scale-invariant result that show a strong interaction dependence. This observation implies that scale invariance is broken anomalously in the strongly interacting two-dimensional Fermi gas.
Recent experiments on a 2D Fermi gas find an undamped breathing mode at twice the trap frequency over a wide range of parameters. To understand this seemingly scale-invariant behavior in a system with a scale, we derive two exact results valid across the entire Bardeen-Cooper-Schrieffer--Bose-Einstein condensation (BCS--BEC) crossover at all temperatures. First, we relate the shift of the mode frequency from its scale-invariant value to ${\ensuremath{\gamma}}_{d}\ensuremath{\equiv}(1+2/d)P\ensuremath{-}\ensuremath{\rho}(\ensuremath{\partial}P/\ensuremath{\partial}\ensuremath{\rho}{)}_{s}$ in ddd dimensions. Next, we relate ${\ensuremath{\gamma}}_{d}$ to dissipation via a new low-energy bulk viscosity sum rule. We argue that 2D is special, with its logarithmic dependence of the interaction on density, and thus ${\ensuremath{\gamma}}_{2}$ is small in both the BCS--BEC regimes, even though $P\ensuremath{-}2\ensuremath{\epsilon}/d$, sensitive to the dimer binding energy that breaks scale invariance, is not.
Single-component quantum gas confined in a harmonic potential, but otherwise isolated, is considered. From the invariance of the system of the gas under a displacement-type transformation, it is shown that the center of mass oscillates along a classical trajectory of a harmonic oscillator. It is also shown that this harmonic motion of the center has, in fact, been implied by Kohn's theorem. If there is no interaction between the atoms of the gas, the system in a time-independent isotropic potential of frequency ${\ensuremath{\nu}}_{c}$ is invariant under a squeeze-type unitary transformation, which gives collective radial breathing motion of frequency $2{\ensuremath{\nu}}_{c}$ to the gas. The amplitudes of the oscillating and breathing motions from the exact invariances could be arbitrarily large. For a Fermi system, appearance of $2{\ensuremath{\nu}}_{c}$ mode of the large breathing motion indicates that there is no interaction between the atoms, except for a possible long-range interaction through the inverse-square-type potential.
We investigate collective excitations of a harmonically trapped two-dimensional Fermi gas from the collisionless (zero sound) to the hydrodynamic (first sound) regime. The breathing mode, which is sensitive to the equation of state, is observed with an undamped amplitude at a frequency 2 times the dipole mode frequency for a large range of interaction strengths and different temperatures. This provides evidence for a dynamical SO(2,1) scaling symmetry of the two-dimensional Fermi gas. Moreover, we investigate the quadrupole mode to measure the shear viscosity of the two-dimensional gas and study its temperature dependence.