Three-dimensional Bose-Fermi droplets at nonzero temperatures

Abstract

Using numerical methods, we study the formation of self-bound quantum Bose-Fermi droplets at nonzero temperatures. We describe an attractive atomic Bose-Fermi mixture using quantum hydrodynamics enriched by beyond-mean-field corrections and thermal fluctuations, together with a simplified self-consistent Hartree-Fock model. With these models, we determine that low-temperature droplets with finite lifetimes can exist in free space when the attraction between bosons and fermions is sufficiently strong. Additionally, Bose-Fermi droplets at nonzero temperatures can exist in a box potential in equilibrium with bosonic and fermionic vapor. We discuss the properties of Bose-Fermi droplets at nonzero temperatures in terms of the initial condensate fraction, total atom number, and interspecies attraction strength.

Figures (6)

• Figure 1: Column densities of a condensate and thermal fraction and of fermionic component for $a_{BF}/a_B=-3$ and the trapping frequencies $2\pi \times 250\,$Hz and $2\pi \times 4000\,$Hz for bosons and fermions, respectively. The number of condensate, thermal, and fermionic atoms is $N_0=5600$, $N_{th}=8400$, and $N_F=860$. The temperature of the sample is $T=286\,$nK which corresponds to the condensate fraction $n_0=0.4$. Solid lines show the results obtained using the hydrodynamic approach while the dashed lines comes from the simplified self-consistent Hartree-Fock treatment.
• Figure 2: Condensate fraction as a function of temperature. The top panel shows four samples distinguished by the number of atoms (given in the legend) and for $a_{BF}/a_B=-3$. The bottom panel shows four samples distinguished by the mutual scattering length (given in the legend) and for $N_B=1460$ and $N_F=100$. These curves were obtained by solving the self-consistent Hartree-Fock equations as described in Appendix \ref{['second']}.
• Figure 3: Column densities for the bosonic component are shown for $a_{BF}/a_B=-3$ and for different amounts of energy pumped into the system. The number of bosonic atoms is $N_B=14000$, and the condensate fraction is indicated in the legend. These results were obtained using the hydrodynamic approach.
• Figure 4: (a) Condensate fraction as a function of time for the absorbing (solid line) and periodic (dotted line) boundary conditions. (b) Number of bosons (solid line) and fermions (dotted line) as a function of time for absorbing boundary conditions. The initial number of atoms is $N_B=14000$ and $N_F=860$, and $a_{BF}/a_B=-3$. The temperatures are $268\,$nK (black lines) and $355\,$nK (red lines), which corresponds to condensate fractions of $0.4$ and $0.03$, respectively.
• Figure 5: Phase diagram showing the region in which Bose-Fermi droplets exist, as a function of total number of bosons and the initial (i.e., before opening the trap) condensate fraction, for $a_{BF}/a_B=-3$. The number of fermions is $N_F=N_B/14.6$. Too high temperature (too low condensate fraction) destroys the droplet after the trap is removed. The data $(T,N_0)$ displayed in the diagram show the highest temperature at which the droplet is not destroyed and the corresponding number of condensed atoms for an initial total number of bosonic atoms $N_B$, as shown on the $x$-axis.