Dynamic formation of supersolid phase in a mixture of ultracold bosonic and fermionic atoms

Abstract

We numerically study the dynamical properties of a mixture consisting of a dipolar condensate and a degenerate Fermi gas in a quasi-one-dimensional geometry. In particular, we focus on the system's response to a temporal variation in the interaction strength between bosons and fermions. When the interspecies attraction becomes sufficiently strong, we observe a phase transition to a supersolid state. This conclusion is supported by the emergence of an out-of-phase Goldstone mode in the excitation spectrum.

Figures (6)

• Figure 1: (a) Sketch of the physical system under consideration before the quench. (b) A sketch of the system after the quench, when two density peaks indicate the transition to the supersolid phase. (c) Details of the initial trapping disturbance and preparation of the initial state. (d) Details of the quench, including the time dependence of the interspecies attraction.
• Figure 2: Low-energy excitation spectrum of the bosonic component after a quench: $g_{BF}=-1.2 \rightarrow g_{BF}=-3.8$, at constant $g_B=0.01$, within a duration of $500$. Three peaks are clearly visible, representing the in-phase Goldstone mode at $\omega=0.01$ (in units of $\omega_{B\perp}$), the out-of-phase Goldstone mode at $\omega \approx 0.008$, and the Higgs mode at $\omega \approx 0.002$. See movies available at Refs. movies24movies25, showing the dynamics of the Goldstone and Higgs modes.
• Figure 3: Correlation between the imbalance and the displacement for out-of-phase (upper frame) and in-phase (lower frame) Goldstone modes. The modes are excited after quenching: $g_{BF}=-1.2 \rightarrow g_{BF}=-3.8$, with constant $g_B=0.01$, within a duration of $500$. The imbalance for a given mode is defined as the difference in density (obtained from Eq. (\ref{['eq5']})) maxima between the right and left peaks. Displacement is defined as the average of the left and right peaks center of mass (CM) positions: $(z_{CM}^L+z_{CM}^R)/2$. Here, $z_{CM}^L=\int_{-\infty}^0 z\, n_B^{(\omega_j)}(z,t)\, dz$ and $z_{CM}^R=\int_0^{\infty} z\, n_B^{(\omega_j)}(z,t)\, dz$ and, again, the density $n_B^{(\omega_j)}(z,t)$ is calculated from Eq. (\ref{['eq5']}). The red points in each frame indicate the mode parameters (imbalance and displacement) immediately after the quench.
• Figure 4: (a)-(c) Examples of long-wavelength modes found by the quench. The red curve shows the bosonic density immediately after the quench. The blue curves indicate the densities at specific times and reveal the unique properties of each mode. The horizontal and vertical arrows in frames (a) and (b) show the dynamics of the crystalline and superfluid structures, respectively. When the crystalline structure moves to the left, the superfluid moves to the right, raising the right peak (frame (a)). This is a characteristic of the out-of-phase Goldstone mode. For the in-phase Goldstone mode, both the crystal and the superfluid move in the same direction (see frame (b)). In the case of the Higgs mode, the positions of the density peaks remain essentially motionless while their maxima simultaneously increase and decrease over time.
• Figure 5: Low-energy excitation spectrum of the bosonic component after a quench: $g_{BF}=-3.7 \rightarrow g_{BF}=-3.8$, at constant $g_B=0.0055$, within a duration of $500$. Three peaks are clearly visible, representing the in-phase Goldstone mode at $\omega=0.01$, the out-of-phase Goldstone mode at $\omega \approx 0.007$, and the Higgs mode at $\omega \approx 0.001$. See movies available at Refs. movies24movies25, showing the dynamics of Goldstone modes. An additional peak is visible at approximately $\omega \approx 0.012$, representing the in-phase motion of the two density peaks (see the magenta line in Fig. 3 of Ref. Lewkowicz25). This peak is the crystal phonon mode.