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arXiv:cond-mat/0608233 · Physical Review Letters

Critical Rotational Frequency for Superfluid Fermionic Gases across a Feshbach Resonance

Hui Zhai, Tin-Lun Ho

Published Nov 3, 2006• 25 citations
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Abstract

We present a method to determine the critical rotational frequencies for superfluidity of both uniform and trapped Fermi gases across a wide Feshbach resonance. It is found that as one approaches the resonance from the BCS side, beyond a critical scattering length, pairing is so robust that superfluidity cannot be destroyed by rotation. Moreover, the critical frequency has a sequence of jumps revealing the appearance of Landau levels, which are particularly prominent for systems up to a few thousand particles. For rotational frequency below an ultimate critical frequency, defined to be the lowest frequency at which the center of the cloud goes normal, a trapped gas has a superfluid core surrounded by a normal gas, as seen in recent experiments (C. H. Schunck, cond-mat/0607298).

DOI
10.1103/physrevlett.97.180414

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We present a method to determine the critical rotational frequencies for superfluidity of both uniform and trapped Fermi gases across a wide Feshbach resonance. It is found that as one approaches the resonance from the BCS side, beyond a critical scattering length, pairing is so robust that superfluidity cannot be destroyed by rotation. Moreover, the critical frequency has a sequence of jumps revealing the appearance of Landau levels, which are particularly prominent for systems up to a few thousand particles. For rotational frequency below an ultimate critical frequency, defined to be the lowest frequency at which the center of the cloud goes normal, a trapped gas has a superfluid core surrounded by a normal gas, as seen in recent experiments (C. H. Schunck, cond-mat/0607298).
We study a rotating atomic Fermi gas near a narrow sss-wave Feshbach resonance in a uniaxial trap with frequencies ${\ensuremath{\Omega}}_{\ensuremath{\perp}}$, ${\ensuremath{\Omega}}_{z}$. We predict the upper-critical angular velocity, ${\ensuremath{\omega}}_{c2}(\ensuremath{\delta},T)$, as a function of temperature TTT and detuning $\ensuremath{\delta}$ across the BEC-BCS crossover. The suppression of superfluidity at ${\ensuremath{\omega}}_{c2}$ is distinct in the BCS and BEC regimes, with the former controlled by depairing and the latter by the dilution of bosonic molecules. At low TTT and ${\ensuremath{\Omega}}_{z}\ensuremath{\ll}{\ensuremath{\Omega}}_{\ensuremath{\perp}}$, in the BCS and crossover regimes of $0\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}{\ensuremath{\delta}}_{c}$, ${\ensuremath{\omega}}_{c2}$ is implicitly given by $\ensuremath{\hbar}\sqrt{{\ensuremath{\omega}}_{c2}^{2}+{\ensuremath{\Omega}}_{\ensuremath{\perp}}^{2}}\ensuremath{\approx}2\ensuremath{\Delta}\sqrt{\ensuremath{\hbar}{\ensuremath{\Omega}}_{\ensuremath{\perp}}/{ϵ}_{F}}$, vanishing as ${\ensuremath{\omega}}_{c2}\ensuremath{\sim}{\ensuremath{\Omega}}_{\ensuremath{\perp}}(1\ensuremath{-}\ensuremath{\delta}/{\ensuremath{\delta}}_{c}{)}^{1/2}$ near ${\ensuremath{\delta}}_{c}\ensuremath{\approx}2{ϵ}_{F}+\frac{\ensuremath{\gamma}}{2}{ϵ}_{F}\mathrm{ln}({ϵ}_{F}/\ensuremath{\hbar}{\ensuremath{\Omega}}_{\ensuremath{\perp}})$ (with $\ensuremath{\Delta}$ the BCS gap and $\ensuremath{\gamma}$ the resonance width), and extending the bulk result $\ensuremath{\hbar}{\ensuremath{\omega}}_{c2}\ensuremath{\approx}2{\ensuremath{\Delta}}^{2}/{ϵ}_{F}$ to a trap. In the BEC regime of $\ensuremath{\delta}<0$ we find ${\ensuremath{\omega}}_{c2}\ensuremath{\rightarrow}{\ensuremath{\Omega}}_{\ensuremath{\perp}}^{\ensuremath{-}}$, where molecular superfluidity is destroyed only by large quantum fluctuations associated with comparable boson and vortex densities.
We have analyzed a single vortex at T=0 in a 3D superfluid atomic Fermi gas across a Feshbach resonance. On the BCS side, the order parameter varies on two scales: k(F)(-1)and the coherence length xi, while only variation on the scale of xi is seen away from the BCS limit. The circulating current has a peak value jmax which is a nonmonotonic function of 1/k(F)a(s) implying a maximum critical velocity approximately v(F) at unitarity. The number of fermionic bound states in the core decreases as we move from the BCS to the BEC regime. Remarkably, a bound state branch persists even on the BEC side reflecting the composite nature of bosonic molecules.
We address the phase of a highly polarized Fermi gases across a narrow Feshbach resonance starting from the problem of a single down-spin fermion immersed in a Fermi sea of up spins. Both polaron and pairing states are considered using the variational wave function approach, and we find that the polaron-to-pairing transition will take place on the BCS side of the resonance, strongly in contrast to a wide resonance where the transition is located at the BEC side. For the pairing phase, we find the critical strength of the repulsive interaction between pairs above which the mixture of pairs and fermions will not phase separate. Therefore, nearby a narrow resonance, it is quite likely that magnetism can coexist with sss-wave BCS superfluidity at large Zeeman fields, which is a remarkable property absent in conventional BCS superconductors (or fermion-pair superfluids).
We consider a dilute atomic gas of two species of fermions with unequal concentrations under a Feshbach resonance. We find that the system can have distinct properties due to the unbound fermions. The uniform state is stable only when either (a) beyond a critical coupling strength, where it is a gapless superfluid, or (b) when the coupling strength is sufficiently weak, where it is a normal Fermi gas mixture. Phase transition(s) must therefore occur when the resonance is crossed, in contrast to the equal population case where a smooth crossover takes place.
A two-component Fermi gas with attractive sss-wave interactions forms a superfluid at low temperatures. When this gas is confined in a rotating trap, fermions can unpair at the edges of the gas and vortices can arise beyond certain critical rotation frequencies. We compute these critical rotation frequencies and construct the phase diagram in the plane of scattering length and rotation frequency for different total numbers of particles. We work at zero temperature and consider a cylindrically symmetric harmonic trapping potential. The calculations are performed in the Hartree-Fock-Bogoliubov approximation which implies that our results are quantitatively reliable for weak interactions.
We investigate the dynamical properties of a superfluid gas of trapped fermionic atoms in the BCS phase. As a simple example we consider the reaction of the gas to a slow rotation of the trap. It is shown that the currents generated by the rotation can be understood within a two-fluid model similar to the one used in the theory of superconductors, but with a position-dependent ratio of normal and superfluid densities. The rather general result of this paper is that already at very low temperatures, far below the critical one, an important normal-fluid component appears in the outer regions of the gas. This renders the experimental observation of superfluidity effects more difficult and indicates that reliable theoretical predictions concerning other dynamical properties, like the frequencies of collective modes, can only be made by taking into account temperature effects.