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arXiv:2004.07008 · New Journal of Physics

Bistability of Bose–Fermi mixtures

Tomasz Karpiuk, Mariusz Gajda, Mirosław Brewczyk

Published Sep 30, 2020• 7 citations
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Abstract

We study formation of a Bose–Fermi droplet in realistic experimental setting. We start with a typical experimental arrangement of two atomic species—bosons and fermions in a harmonic axially symmetric confinement. The variational analysis shows that the system exhibits bistability. For weak repulsion between bosons, one of the equilibrium states, smaller in size, spherically symmetric, and with negative energy, corresponds to quantum droplet; the other with always positive energy represents the elongated droplet-like state immersed in the sea of a fermionic cloud. For stronger repulsion between bosons the bifurcation is seized and only the former state is left. Now it represents an elongated object which, for strong enough boson–fermion attraction, gets negative energy. It becomes an excited Bose–Fermi droplet when the trap is released, what is demonstrated by solving the quantum hydrodynamics equations for the Bose–Fermi system. To depict our ideas we consider the 133 Cs– 6 Li mixture under ideal conditions, i.e. we assume no losses.

DOI
10.1088/1367-2630/abbcef

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We study formation of a Bose–Fermi droplet in realistic experimental setting. We start with a typical experimental arrangement of two atomic species—bosons and fermions in a harmonic axially symmetric confinement. The variational analysis shows that the system exhibits bistability. For weak repulsion between bosons, one of the equilibrium states, smaller in size, spherically symmetric, and with negative energy, corresponds to quantum droplet; the other with always positive energy represents the elongated droplet-like state immersed in the sea of a fermionic cloud. For stronger repulsion between bosons the bifurcation is seized and only the former state is left. Now it represents an elongated object which, for strong enough boson–fermion attraction, gets negative energy. It becomes an excited Bose–Fermi droplet when the trap is released, what is demonstrated by solving the quantum hydrodynamics equations for the Bose–Fermi system. To depict our ideas we consider the 133 Cs– 6 Li mixture under ideal conditions, i.e. we assume no losses.
We study stability of a zero temperature mixture of attractively interacting degenerate bosons and spin-polarized fermions in the absence of confinement. We demonstrate that higher order corrections to the standard mean field energy of the system can lead to a formation of liquid droplets -- self-bound incompressible systems in a three-dimensional space. In addition to the quantum Lee-Huang-Yang correction to the boson-boson interaction energy, a higher order term in the Bose-Fermi coupling constant is crucial for stabilization of the system. We study a stability region of droplets for three different mixtures of species with different mass ratios, namely: 41^{41}41K \,-40^{40}40K, 87^{87}87Rb \,-40^{40}40K, and 133^{133}133Cs \,-6^{6}6Li as a function of the ratio of Bose-Fermi to Bose-Bose scattering length. The stability analysis is supported by a numerical simulations of droplet formation dynamics starting with a trapped system which is subsequently released from the trap. Our results indicate that Bose-Fermi droplets can be realized experimentally.
We consider the formation of bright solitons in a mixture of Bose and Fermi degenerate gases confined in a three-dimensional elongated harmonic trap. The Bose and Fermi atoms are assumed to effectively attract each other whereas bosonic atoms repel each other. Strong enough attraction between bosonic and fermionic components can change the character of the interaction within the bosonic cloud from repulsive to attractive making thus possible the generation of bright solitons in the mixture. On the other hand, such structures might be in danger due to the collapse phenomenon existing in attractive gases. We show, however, that under some conditions (defined by the strength of the Bose-Fermi components attraction) the structures which neither spread nor collapse can be generated. For elongated enough traps the formation of solitons is possible even at the ``natural'' value of the mutual Bose-Fermi ($^{87}\mathrm{Rb}\text{\ensuremath{-}}^{40}\mathrm{K}$ in our case) scattering length.
We theoretically study the static and dynamic behavior of a BEC immersed in a large Fermi sea of ultracold atoms under conditions of tunable interspecies interaction. The degenerate Bose-Fermi mixture is kept in an elongated trap, typical for a single-beam optical dipole trap. We focus on the case of repulsive Bose-Fermi interaction and develop mean-field models to simulate the system over a wide range of repulsion strength. We further get analytical solutions in the regimes of phase separation and weak interaction. We obtain static density profiles and the frequency of the radial breathing mode, which is an elementary dynamic phenomenon of the mixture. Our results unveil the structure of the Bose-Fermi interface and describe the origin of the frequency shift of the breathing mode when the components become phase-separated at strong repulsion. We show that the mediated interaction between bosons induced by the Fermi sea can be understood as an adiabatic second-order mean-field effect, which is valid also beyond the weak-interaction regime for relevant experimental conditions. These results are consistent with our recent observations [Lous et al., Phys. Rev. Lett. 120, 243403 (2018); Huang et al., Phys. Rev. A 99, 041602(R) (2019)] in a mixture of 41K^{41}\mathrm{K}41K and 6Li^{6}\mathrm{Li}6Li.
We study the formation of quantum droplets in the mixture of a single-component Bose–Einstein condensate (BEC), and a two-species Fermi superfluid across a wide Feshbach resonance. With repulsive boson-boson and attractive boson-fermion interactions, we show that quantum droplets can be stabilized by attractive fermion-fermion interactions on the Bardeen–Cooper–Schrieffer (BCS) side of the resonance, and can also exist in the deep BEC regime under weak boson-fermion interactions. We map out the phase diagram for stable droplets with respect to the boson-boson and boson-fermion interactions, and discuss the role of different types of quantum fluctuations in the relevant regions of the BCS-BEC crossover. Our work reveals the impact of fermion pairing on the formation of quantum droplets in Bose–Fermi mixtures, and provides a useful guide for future experiments.
We study the stability of a zero temperature mixture of attractively interacting degenerate bosons and spin-polarized fermions in the absence of confinement. We demonstrate that higher order corrections to the standard mean-field energy can lead to a formation of Bose-Fermi liquid droplets – self-bound systems in three-dimensional space. The stability analysis of the homogeneous case is supported by numerical simulations of finite systems by explicit inclusion of surface effects. We discuss the experimental feasibility of formation of quantum droplets and indicate the main obstacle – inelastic three-body collisions.
By analyzing the breathing mode of a Bose-Einstein condensate repulsively interacting with a polarized fermionic cloud, we further the understanding of a Bose-Fermi mixture recently realized by Lous et al. [Phys. Rev. Lett. 120, 243403 (2018)PRLTAO0031-900710.1103/PhysRevLett.120.243403]. We show that a hydrodynamic description of a domain wall between bosonic and fermionic atoms reproduces the experimental data of Huang et al. [Phys. Rev. A 99, 041602(R) (2019)PLRAAN2469-992610.1103/PhysRevA.99.041602]. Two different types of interaction renormalization are explored, based on lowest-order constrained variational and perturbation techniques. In order to replicate nonmonotonic behavior of the oscillation frequency observed in the experiment, temperature effects have to be included. We find that the frequency down-shift is caused by the fermion-induced compression and rethermalization of the bosonic species as the system is quenched into the strongly interacting regime.