Evaporative depolarization and spin transport in a unitary trapped Fermi gas

Two-component atomic Fermi gases provide an ideal experimental system in which to investigate fermion pairing and superfluidity in a controllable manner. 1 For example, one can use a magnetically-tunable Feshbach resonance to access the unitary regime, where the scattering length diverges and one has a strongly-interacting fermionic superfluid that is 'universal'. 2 Of particular interest is the case where there is a spin imbalance that frustrates pairing between fermion species, because this is a situation that arises in many fields of physics, ranging from QCD to superconductivity. 3 Here, a central question has been: what is the critical spin polarization δ c at which pairing and superfluidity are destroyed for a unitary trapped Fermi gas at equilibrium? However, current experiments produce different answers. The experimental group at MIT finds that δ c ≃ 77%, 4,5,6,7 while experiments on highly-elongated trapped gases at Rice University 8 suggest that δ c is at least 90%. Moreover, even though the critical polarization is known to be a strong function of temperature, where δ c decreases with increasing temperature, the different δ c 's observed in experiment are unlikely to be caused by differences in temperature since both experimental groups have claimed that their temperatures are low enough to yield δ c 's that are close to the zero-temperature result. Thus, there is a real discrepancy in the measured δ c and a resolution of this problem has potentially important implications for the nature of the paired superfluid phase in a finitesized system. Here we propose that the high δ c observed in the Rice experiment was due to their trapped spinimbalanced gas being out of equilibrium. We show that a combination of the trap geometry and the evaporative cooling scheme implemented in the Rice experiment can induce a spin current along the trap axis, which in turn creates a substantially non-equilibrium polarization pattern that favors a superfluid at the trap center.

To understand how such a spin current can be generated, one must first examine the evaporative cooling process. Here, the temperature, and entropy per atom, of a trapped gas is lowered when the most energetic atoms escape over the “lip” of the trap -the route of escape with the lowest potential barrier to be surmounted. For a partially-polarized Fermi gas at temperature T , the rate The flux of spin-up atoms j0 is exiting the gas over the lip of the trap that is located at the axial center (z = 0). This flux must be drawn from the fully-polarized (FP) normal region. Thus, for a quasi-1D gas (a), a spin current must flow through the intervening partially-polarized (PP) normal region and across the normal/superfluid(SF) interface. In the 3D regime (b), the atoms can evaporate directly from the surrounding fully-polarized layer.

of thermal activation over this barrier is larger for the majority species by a factor of exp [(µ ↑ -µ ↓ )/k B T ], assuming the two species are subject to the same trapping potential, where µ ↑ , µ ↓ are the chemical potentials for the majority and minority species, respectively. Thus, at low T, the flux of evaporating atoms passing over the lip is essentially fully polarized, and we have evaporative depolarization in addition to evaporative cooling (as stated in Ref. 4). The Rice experiments we are considering 8,9 had both a long, thin, high-aspect-ratio optical trap and a nonuniform magnetic field that contributed to the axial confinement of the gas. As a result, the lowest barrier for the atoms to escape from the trap was at the axial center, with the atoms escaping over this “lip” in the radial direction (and downwards, due to gravity; see Fig. 1(a)). Elsewhere in the trap the barrier to escape was significantly higher, so at their lowest temperatures, essentially all of the evaporating atoms escaped radially at the axial center of their high-aspect-ratio trap.

To achieve low temperatures in the Rice experiment, the height of this barrier was lowered by reducing the in-tensity of the optical trap. It was at the lowest temperatures that the unusually large δ c was observed, along with strong deviations from the equilibrium local density approximation (LDA) in the shapes of the regions occupied by the superfluid and partially-polarized normal phases. 8 What we propose happened here is that the evaporation, with the ↑ atoms rapidly escaping radially over the trap lip, greatly depleted any excess unpaired ↑ atoms from the axially central region of the cloud occupied by the paired superfluid phase. This depletion, which is apparent in the in situ density measurements, 8,9 substantially suppressed (µ ↑ -µ ↓ ) in that region (evaporative depolarization). The flux of evaporating ↑ atoms over the lip then had to come from the fully-polarized normal regions at the axial ends of the cloud and be driven through the partially-polarized region and across the normal/superfluid interface by a substantial axial gradient of (µ ↑ -µ ↓

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