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Quantum droplets in a resonant Bose-Fermi mixture

Sam Foster, Olivier Bleu, Jesper Levinsen, Meera M. Parish

TL;DR

The paper introduces a versatile variational ansatz for strongly interacting Bose-Fermi mixtures that unifies the Fermi and Bose polaron limits while capturing many-body correlations around a Fermi sea. By renormalizing BF and BB interactions and evaluating the resulting free energy, the authors demonstrate the emergence of self-bound quantum droplets at unitarity and near-resonant BF attraction, where Fermi pressure balances resonant BF attraction. They show that droplets exist on a first-order phase boundary between a uniform BF mixture and vacuum, with a liquid-gas–like critical point arising upon varying fermion density, and they map the phase behavior across mass ratios around equality. The framework reproduces known perturbative results at weak coupling and provides access to strong-coupling regimes beyond previous second-order theories, offering predictions that are accessible to current ultracold-atom experiments and potentially to other platforms with resonant BF interactions. Overall, the work highlights first-order quantum phase transitions and droplet formation as central features in the phase diagram of resonantly interacting BF mixtures.

Abstract

We study the canonical problem of a Fermi gas interacting with a weakly repulsive Bose-Einstein condensate at zero temperature. To explore the quantum phases across the full range of boson-fermion interactions, we construct a versatile variational ansatz that incorporates pair correlations and correctly captures the different polaron limits. Remarkably, we find that self-bound quantum droplets can exist in the strongly interacting regime, preempting the formation of boson-fermion dimers, when the Fermi pressure is balanced by the resonant boson-fermion attraction. This scenario can be achieved in experimentally available Bose-Fermi mixtures for a range of boson-fermion mass ratios in the vicinity of equal masses. We furthermore show that a larger fermion density instead yields phase separation between a Bose-Fermi mixture and excess fermions, as well as behavior reminiscent of a liquid-gas critical point. Our results suggest that first-order quantum phase transitions play a crucial role in the phase diagram of Bose-Fermi mixtures.

Quantum droplets in a resonant Bose-Fermi mixture

TL;DR

The paper introduces a versatile variational ansatz for strongly interacting Bose-Fermi mixtures that unifies the Fermi and Bose polaron limits while capturing many-body correlations around a Fermi sea. By renormalizing BF and BB interactions and evaluating the resulting free energy, the authors demonstrate the emergence of self-bound quantum droplets at unitarity and near-resonant BF attraction, where Fermi pressure balances resonant BF attraction. They show that droplets exist on a first-order phase boundary between a uniform BF mixture and vacuum, with a liquid-gas–like critical point arising upon varying fermion density, and they map the phase behavior across mass ratios around equality. The framework reproduces known perturbative results at weak coupling and provides access to strong-coupling regimes beyond previous second-order theories, offering predictions that are accessible to current ultracold-atom experiments and potentially to other platforms with resonant BF interactions. Overall, the work highlights first-order quantum phase transitions and droplet formation as central features in the phase diagram of resonantly interacting BF mixtures.

Abstract

We study the canonical problem of a Fermi gas interacting with a weakly repulsive Bose-Einstein condensate at zero temperature. To explore the quantum phases across the full range of boson-fermion interactions, we construct a versatile variational ansatz that incorporates pair correlations and correctly captures the different polaron limits. Remarkably, we find that self-bound quantum droplets can exist in the strongly interacting regime, preempting the formation of boson-fermion dimers, when the Fermi pressure is balanced by the resonant boson-fermion attraction. This scenario can be achieved in experimentally available Bose-Fermi mixtures for a range of boson-fermion mass ratios in the vicinity of equal masses. We furthermore show that a larger fermion density instead yields phase separation between a Bose-Fermi mixture and excess fermions, as well as behavior reminiscent of a liquid-gas critical point. Our results suggest that first-order quantum phase transitions play a crucial role in the phase diagram of Bose-Fermi mixtures.
Paper Structure (8 sections, 59 equations, 4 figures, 1 table)

This paper contains 8 sections, 59 equations, 4 figures, 1 table.

Table of Contents

  1. Model and ansatz
  2. Approximate normalization of the ansatz
  3. Boson-fermion interaction
  4. Boson-boson interaction
  5. Total free energy
  6. Equations of Motion and variational parameters
  7. Fermi polaron
  8. Bose polaron

Figures (4)

  • Figure 1: (a) Phase diagram of a Bose-Fermi mixture as a function of $\mu_\mathrm{B}$ and interaction $a_\mathrm{BF}$ at fixed $\mu_\mathrm{F}=0$. Here we take $m_\mathrm{B}/m_\mathrm{F}=23/40$, and we scale lengths by $a_\mathrm{BB}$ and energies by $\varepsilon_{\mathrm{BB}}= 1/m_\mathrm{B} a_{\mathrm{BB}}^2$. The quantum droplet exists along the first-order phase boundary separating a uniform BF mixture (purple) from the vacuum (white), and it corresponds to a minimum in the free energy which is degenerate with the vacuum (inset). Beyond the purple circles, the droplet is replaced by a second-order phase boundary, either between vacuum and an atomic Bose gas at $\mu_\mathrm{B}=0$, or between vacuum and a Fermi gas of dimers where $\mu_\mathrm{B}$ corresponds to the dimer energy $-1/2m_\mathrm{r} a_\mathrm{BF}^2$ (black line). (b) Boson, condensate, and fermion densities (blue, green, red) in the droplet phase.
  • Figure 2: Densities in the quantum droplet phase at unitarity, $1/a_\mathrm{BF} = 0$. The boson and fermion densities are lowest for mass ratios in the vicinity of equal masses, which can be accessed with experimentally available BF mixtures (indicated on the top axis).
  • Figure 3: (a) Phase diagram of a Bose-Fermi mixture at unitarity for $m_\mathrm{B}/m_\mathrm{F}=23/40$. The red and purple regions denote an ideal Fermi gas and BF mixture, respectively. The purple line corresponds to a first-order phase boundary that evolves from a BF droplet at $\mu_\mathrm{F}=0$ to liquid-gas phase separation (lines I and V) before terminating in a critical end point (square, zoomed-in inset), where two finite-density minima in the free energy merge. The solid black line denotes a second-order phase boundary where $\mu_\mathrm{B}$ corresponds to the polaron energy of a single bosonic impurity supp. Following its intersection (circle) with the purple line, it becomes a spinodal line (dot-dashed) separating regions II and III. (b) The corresponding free energies for the regions labeled by roman numerals in (a). Each panel has a different scale, with the value of the free energy indicated by the color bar.
  • Figure S1: (a) Free energy as a function of the condensate density and the Fermi momentum for parameters $a_\mathrm{BB}/a_\mathrm{BF} = 0$, $\mu_\mathrm{F}=0$, $m_\mathrm{B}/m_\mathrm{F}=23/40$ and $\mu_\mathrm{B}/\epsilon_\mathrm{BB} = -2.077\times 10^{-3}$. Overlayed are the equations of motion $\partial\Omega/\partial n_0=0$ and $\partial\Omega/\partial n_\mathrm{F}=0$ giving the red, solid and green dashed line, respectively. The white dot indicates the droplet phase. (b) The free energy curve as a function of the condensate density plotted along the minimization conditions. The solid (dashed) line corresponds to the solid (dashed) equation of motion in panel (a).