Finite-temperature study of Bose-Fermi superfluid mixtures

Ultracold atom experiments offer the unique opportunity to study mixing of different types of superfluid states. Our interest is in superfluid mixtures comprising of particles with different statistics—Bose and Fermi. Such scenarios occur naturally, for example, in dense quantum chromodynamics (QCD) matter. Interestingly, cold atomic experiments are performed in traps with finite spatial extent, thus critically destabilizing the occurrence of various homogeneous phases. Critical to this analysis is the understanding that the trapped system can undergo phase separation, resulting in a unique situation where phase transition in either species (bosons or fermions) can overlap with the phase separation between possible phases. In the present work, we illustrate how this intriguing interplay manifests in an interacting two-species atomic mixture—one bosonic and another fermionic with two spin components—within a realistic trap configuration. We further show that such interplay of transitions can render the nature of the ground state to be highly sensitive to the experimental parameters and the dimensionality of the system.

Figure 1

(a) Phase diagram of the Bose-Fermi superfluid mixture at $T=0$ [21]. Solid (green) curve is the dynamical stability contour while the dashed (blue) contour $C$ denotes points in phase space with a fixed value of ${\lambda }_f$ such that $T_c=0.11\hspace{0.5em}{T}_f$. The filled (red) circle represents the critical point for this specific experimental realization, at which the homogeneous mixture enters the dynamically and mechanically stable region. (b) Plot of the free energy densities of pure fermions (solid) and the homogeneous mixture (dashed) against ${\mu }_b[C]$. Free energy of homogeneous mixture is lower than that of pure fermions only beyond the critical point represented by the filled (red) circle. Here, free energy density is a dimensionless quantity [22].

Figure 2

Schematic depicting possible first-order transitions occurring in Bose-Fermi mixtures across $T_{c,\mathrm{mix}}$. Horizontal axis denotes the $R^5$ phase space of experimental parameters defined by $\{{\lambda }_b,{\lambda }_{\mathit{bf}},{\mu }_f,n_b,\Delta \}$.

Figure 3

Phase diagram of the Bose-Fermi superfluid mixture at temperatures $T_c>T$. Solid (green) curves show dynamical stability criteria and the (blue) surface represents points of fixed ${\lambda }_f$, such that $T_c$$=$ 0.11 $T_f$ [20, 21]. Dashed (blue) curves $C_i$ are contours of fixed temperature $T_i$ along this surface and (red) circles indicate critical points at which the homogeneous mixture enters the region of both dynamical and mechanical stability.

Figure 4

Subplots top, middle, and bottom show boson density profiles (slice along $y=0$ plane) computed within the LDA for temperatures $T=$0.13, 0.072, 0.055 $T_f$, respectively, with $T_c=0.11\hspace{0.5em}{T}_f$. ${\mu }_b$ is adjusted to ensure number conservation ($~$$42\hspace{0.5em}000$ atoms). $X$ and $Z$ are in $\mu$m. The color bar shows density variations in scale of $n_b{a}_b^3$ (10${}^{-6}$). Chosen trap parameters $V_0$ (several mW), $\sigma =12$ $\mu$m, and ${\omega }_z$/${\omega }_{\perp }=0.08$.

Figure 5

(a) Phase diagram of the Bose-Fermi superfluid mixture in 1D at $T=0$ [for the same parameters used in Fig. 1a]. Solid (green) curve is the dynamical stability contour, while the dashed (blue) contour $C$ denotes phase-space points with a fixed value of ${\lambda }_{1\mathrm{D}\hspace{0.5em}f}$. The dotted (black) curve represents the region within which the homogeneous mixture is mechanically unstable. The filled circle (red) represents the critical point for this specific experimental realization at which the homogeneous mixture enters the mechanically unstable region. (b) Plot of the free energy densities of pure bosons (dotted), of pure fermions (solid), and the homogeneous mixture (dashed) against ${\mu }_b[C]$. Free energy density of the homogeneous mixture becomes higher than that of pure bosons at the critical point represented by the filled (red) circle, resulting in the phase separation of pure bosons out of the mixture. Here free energy density is a dimensionless quantity [22].