Heterosymmetric states of rotating quantum droplets under confinement

Abstract

We investigate the rotational response of a confined, two-dimensional quantum droplet, which emerges in an attractive binary Bose mixture that is stabilized against collapse by beyond-mean-field effects. We consider both a harmonic and an anharmonic form for the external confining potential. We go beyond the widely employed ``phase-locked" single-order-parameter model, maintaining two separate order parameters for the two components, and calculating the lowest-energy state for various values of the angular momentum. For a population-balanced quantum droplet and sufficiently tight confinement, we find that near certain half-integer values of the angular momentum the droplet is excited in a ``heterosymmetric" manner, with the two components carrying different vorticities. This mode is naturally missed by the single-order-parameter model. We additionally investigate the effects of a small population imbalance in the droplet. Apart from an energy increase associated with the population difference, the imbalance also lifts the double degeneracy of the heterosymmetric states, which characterizes the $\mathbb{Z}_2$-symmetric balanced droplet. The heterosymmetric mode is found to be favored by the energy term which captures the beyond-mean-field effects in the mixture.

Figures (14)

• Figure 1: The critical value of the total number of atoms $N$ for which the droplet carries its angular momentum via surface waves instead of center-of-mass motion, as a function of the population imbalance ratio $\delta N/N$. Here $\omega=0.05$ and $D=25$. This plot may also be viewed as the low-angular-momentum phase diagram (produced for $\ell=0.2$) for an imbalanced droplet, involving center-of-mass excitation below the curve, and surface-wave excitation above it.
• Figure 2: Solid line, with data points: the critical value of the trapping frequency $\omega$ for which a heterosymmetric state, with a vortex only in the $\downarrow$ component, appears as the yrast state, as a function of the total number of atoms $N$. Here $L = N_\mathrm{\downarrow}$, $\delta N/N = 0$, and $D=25$. Dashed line: the semianalytic result we have derived for the crossover.
• Figure 3: (a)–(c) The densities (left column, in units of $\Psi_0^2$) and the phases (right column) of the droplet order parameters, in the yrast state, for $N = 200$, $\delta N = 0$, $\omega = 0.05$, $D = 25$, and (a) $\ell = 0.4$, (b) $\ell = 0.47$, and (c) $\ell = 0.5$. The unit of length is $x_0$.
• Figure 3: (Cont.) (d) and (e) Same as (a)–(c) but for $\ell = 0.57$ and $0.6$, respectively. (f) The corresponding dispersion relation in the rotating frame, with $\Omega = 0.03$. The unit of energy is $E_0$ and the unit of angular momentum is $\hbar$.
• Figure 4: The angular momenta of component $\uparrow$ (orange line) and component $\downarrow$ (black line) in the yrast state, as a function of the total angular momentum per particle $\ell$. Here $N = 200$, $\delta N = 0$, $\omega = 0.05$, and $D = 25$. The upper dotted horizontal line corresponds to $L = N_\uparrow = N_\downarrow = N/2$. The unit of angular momentum is $\hbar$.