Finite-temperature phase diagram of a spin-polarized ultracold Fermi gas in a highly elongated harmonic trap

We investigate the finite-temperature properties of an ultracold atomic Fermi gas with spin population imbalance in a highly elongated harmonic trap. Previous studies at zero temperature showed that the gas stays in an exotic spatially inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid state at the trap center; while moving to the edge, the system changes into either a nonpolarized Bardeen-Cooper-Schrieffer superfluid $(P<P_c)$ or a fully polarized normal gas $(P>P_c)$, depending on the smallness of the spin polarization $P$, relative to a critical value $P_c$. In this work, we show how these two phase-separation phases evolve with increasing temperature, and thereby construct a finite-temperature phase diagram. For typical interactions, we find that the exotic FFLO phase survives below one-tenth of Fermi degeneracy temperature, which seems to be accessible in the current experiment. The density profile, equation of state, and specific heat of the polarized system have been calculated and discussed in detail. Our results are useful for the ongoing experiment at Rice University on the search for FFLO states in quasi-one-dimensional polarized Fermi gases.

Figure 1

(Color online) Finite temperature phase diagram of a 1D trapped spin-polarized Fermi gas at a given coupling constant ${\gamma }_0=1.6$ as a function of the spin polarization $P=(N_{\uparrow }-N_{\downarrow })∕(N_{\uparrow }+N_{\downarrow })$, where $N_{\uparrow }$ and $N_{\downarrow }$ are the number of spin-up and spin-down atoms, respectively. $T_F$ is the Fermi temperature of an ideal, noninteracting Fermi gas in a harmonic trap. The solid line gives the boundary of the second-order phase transition from a superfluid to a normal state. The dashed and dot-dashed lines distinguish the different superfluid phases: a pure BCS phase, a phase separation with a FFLO at center, and a BCS outside (FFLO-BCS), and a phase separation with FFLO and normal components (FFLO-N). Due to the presence of harmonic traps, these boundaries may be better understood as crossover, rather than phase transition lines.

Figure 2

(Color online) Spin-up and spin-down density profiles (thick solid lines), the density difference (thin solid lines), and the order parameter (dot-dashed lines) of a trapped 1D Fermi gas for a spin polarization $P=0.075$, at several temperatures $T=0.01T_F$ (upper panel), $T=0.08T_F$ (middle panel), and $T=0.15T_F$ (bottom panel). For a better illustration, the scale of the density difference has been doubled. At zero temperature, the system stays in a phase separation phase with a FFLO state at center and an outside BCS shell. The FFLO phase at the trap center, signaled by the oscillations in the order parameter, suppresses with increasing temperature, and leaves out a pure BCS state throughout the trap at about $0.10T_F$. At a higher temperature around $0.20T_F$, the system becomes fully normal through a second-order superfluid phase transition.

Figure 3

(Color online) Same plots as in Fig. 2, but for a larger spin polarization $P=0.175$. With this value of spin polarizations, at zero temperature the atoms at the trap edge stay in a normal state. With increasing temperature, the FFLO phase at the trap center vanishes gradually, which eventually leads to a normal phase throughout the trap at $T\simeq 0.15T_F$.

Figure 4

(Color online) (a) Density difference of a spin-polarized Fermi gas at $P=0.175$ and $T=0.01T_F$ for different numbers of total atoms as indicated. (b) The number dependence of the amplitude of the density oscillations $A$ at the trap center. The amplitude $A$ decreases slowly with increasing the number of atoms. (c) The number dependence of the periodicity of the density oscillations $d$ at the trap center.

Figure 5

(Color online) Order parameters of a spin-polarized Fermi gas at $P=0.175$ and $T=0.01T_F$ for different numbers of total atoms: (a) $N=50$, (b) $N=100$, and (c) $N=200$. The two dashed lines indicate the value of the order parameter at the trap center.

Figure 6

(Color online) Temperature dependence of the chemical potential [panel (a)] and of the chemical potential difference [panel (b)] at three spin polarizations: $P=0$ (solid lines), 0.075 (dashed lines), and 0.175 (dash-dotted lines).

Figure 7

(Color online) Temperature dependence of the total entropy per particle [panel (a)] and of the total energy per particle [panel (b)] at three spin polarizations: $P=0$ (solid lines), 0.075 (dashed lines), and 0.175 (dash-dotted lines).

Figure 8

(Color online) Temperature dependence of the specific heat at three spin polarizations: $P=0$ (solid line), 0.075 (dashed line), and 0.175 (dash-dotted line). For $P=0.075$, the dashed arrow indicates the transition from a FFLO-BCS phase separation phase to a pure BCS state, while the solid arrow marks the second-order phase transition from a BCS superfluid to a normal gas.