Chiral Quantum Droplet in a Spin-Orbit Coupled Bose Gas

TL;DR

The paper addresses how spin-orbit coupling (SOC) and broken Galilean invariance affect quantum droplets in a two-component Bose gas. It develops a moving-frame theoretical framework incorporating SOC and Raman coupling, computing mean-field and Lee-Huang-Yang (LHY) corrections to obtain a velocity-dependent effective interaction $g_{\rm eff}$ that can become negative, enabling self-binding. A chiral quantum droplet emerges only when the system moves toward a specific direction, with $p_{0x}$ and the spin polarization $S$ adjusting dynamically to velocity; the authors map a four-phase diagram in the $\delta$-$v$ plane for realistic $^{39}$K mixtures, including gas, droplet, and their coexistence, and identify a first-order gas–droplet transition. The work provides a mechanism to engineer topological chiral states in driven quantum gases and outlines experimental pathways for observing chiral droplets and related phases under SOC and motion.

Abstract

We report the formation of chiral quantum droplet in a spin-orbit coupled Bose gas, where the system turns to a self-bound droplet when moving towards a particular direction and remains gaseous otherwise. The chirality arises from the breaking of Galilean invariance by spin-orbit coupling, which enables the system to dynamically adjust its condensation momentum and spin polarization in response to its velocity. As a result, only towards a specific moving direction and beyond a critical velocity, the acquired spin polarization can trigger collective interactions sufficient for self-binding and drive a first-order transition from gas to droplet. We have mapped out a phase diagram of droplet, gas and their coexistence for realistic spin-orbit coupled 39K mixtures with tunable moving velocity and magnetic detuning. Our results have revealed the emergence of chirality in spin-orbit coupled quantum gases, which shed light on general chiral phenomena in moving systems with broken Galilean invariance.

Figures (3)

• Figure 1: (Color online.) (a)Single-particle dispersion along $k_x$ ($k_y=k_z=0$) for static system with SOC. According colors denote spin polarizations $S$. Here we take $\Omega=5E_q$ and $\delta=0.07E_q$. (b) Total and mean-field effective interactions as functions of $k_x$. Here we take the same $\Omega$ and $\delta$ as in (a), and the tunable scattering length as $a_{\downarrow\downarrow}=65a_B$. The units of energy, momentum, and effective interaction are $E_q$, $q$ and $4\pi a_B/m$, respectively.
• Figure 2: (Color online.) (a) Condensation momentum ($p_{0x}$) as a function of moving velocity $v$ in low-density limit. Solid and Dashed lines respectively show the true $p_{0x}$ and the linear dependence $p_{0x}=mv$. Their deviation comes from the fact that the system breaks Galilean invariance under SOC. (b) Mean-field ($g^{\rm mf}_{\rm eff}$) and total ($g_{\rm eff}$) effective interactions as functions of moving velocity ($v$) in low-density limit. The single-particle and interaction parameters are the same as in Fig.\ref{['fig_spectrum']}. The units of momentum, velocity and effective interaction are $q$, $q/m$ and $4\pi a_B/m$, respectively.
• Figure 3: (Color online.) Ground-state phase diagram in the parameter plane of magnetic detuning ($\delta$, in unit of $E_q$) and moving velocity ($v$, in unit of $q/m$). Four phases are found: (a) a single gaseous ground state, (b) a gaseous ground state and a droplet metastable state; (c) a droplet ground state and a metastable gaseous state; (d) a single droplet ground state. Their typical $\epsilon/n\sim n$ are plotted in (a,b,c,d) accordingly, at a fixed $\delta=-0.07E_q$ and different $mv/q=0 (a), -0.02 (b), -0.04(c), -0.08(d)$. Here the units of $\epsilon/n$ and $n$ are $E_q$ and $q^2/(4\pi a_B)$, respectively. The other parameters are the same as in Fig.\ref{['fig_spectrum']}. Different colors in the diagram are just to distinguish different phases.