Spin Faraday pattern formation in a circular spin-orbit coupled Bose-Einstein condensate with stripe phase

TL;DR

This work analyzes spin Faraday pattern formation in a pancake-shaped, stripe-phase spin-orbit-coupled BEC under periodic modulation of interatomic interactions. By contrasting in-phase and out-of-phase modulation, the study uncovers $L$-fold Faraday patterns, with $L=6$ acting as a critical value under out-of-phase excitation and higher-order $L$-fold patterns ($L=6$–$9$) accessible with in-phase modulation without external noise. The patterns exhibit supersolid-like characteristics, and their symmetry, radial nodes, and radii can be tuned by the modulation frequency, offering a versatile platform for probing supersolidity and nonlinear excitations in SOC systems. These results extend Faraday-wave physics into spin-orbit-coupled quantum fluids and provide new control handles on pattern formation in driven BECs.

Abstract

We investigate the spin Faraday pattern formation in a periodically driven, pancake-shaped spin-orbit-coupled (SOC) Bose-Einstein condensate (BEC) prepared with stripe phase. By modulating atomic interactions using in-phase and out-of-phase protocols, we observe collective excitation modes with distinct rotational symmetries (L-fold). Crucially, at the critical modulation frequency, out-of-phase modulation destabilizes the L = 6 pattern, whereas in-phase modulation not only preserves high symmetry but also excites higher-order modes. Unlike conventional binary BECs, Faraday patterns emerge here without initial noise due to SOC-induced symmetry breaking, with all patterns exhibiting supersolid characteristics. Furthermore, we demonstrate control over pattern symmetry, radial nodes, and pattern radius by tuning the modulation frequency, providing a new approach for manipulating quantum fluid dynamics. This work establishes a platform for exploring supersolidity and nonlinear excitations in SOC systems with stripe phase.

Figures (8)

• Figure 1: (a) Schematic of the harmonically trapped BEC with the atomic interaction periodically modulated. Here, 1 and 2 represent pseudo spin-up (yellow balls) and spin-down (blue balls). The scattering length $a_{11}$ and $a_{22}$ can be periodically modulated, while $a_{12}$ is a constant. (b) Dimensionless dispersion relation of stripe phase with $\Omega=0.1E_r,\delta=0$. The two minima correspond to the spin states $\left |\uparrow \right \rangle$ and $\left |\downarrow \right \rangle$. (c)(d) The momentum distributions of the two spin states in the x-direction are opposite.
• Figure 2: The time evolution of the system from 0 to 220 ms. The spin Faraday wave is excited with $in$-$phase$ modulation when $\Omega$ = 0.1 $E_r$, $f\equiv \omega _m/2\pi$ = 600 Hz, $A$ = 8$a_0$. (a)(b) The evolution of the density distribution. Periodic variations in the spin density distribution emerge around 174 $\sim$ 190 ms. (c) The evolution of energy over time.
• Figure 3: The energy evolution of different systems when $\omega _m/2\pi$ = 600 Hz, $A$ = 8$a_0$. Blue: $in$-$phase$ modulation without noise (SOC BEC); Yellow: $in$-$phase$ modulation with noise (normal binary BEC); Red: $in$-$phase$ modulation without noise (normal binary BEC).
• Figure 4: The density distribution patterns in the row of $n_1$ represent spin-up component, and those in the row of $n_2$ represent spin-down component. Different modulation methods exhibit distinct density distribution patterns and rotational symmetry. (a) $out$-$of$-$phase$ modulation is employed with the corresponding modulation frequencies (from left to right) $\omega _m/2\pi$ = {50, 120, 255, 400, 520, 560, 600} Hz at $t$ = {16, 553, 284, 390, 414, 370, 304} ms, where the modulation amplitude is $A$ = 8$a_0$. (b) $in$-$phase$ modulation is employed with the corresponding modulation frequencies (from left to right) $\omega _m/2\pi$ = {180, 300,330,600,780,835,900} Hz at $t$ = {93, 164, 119, 166, 132, 168, 164} ms, where the modulation amplitude is $A$ = 8$a_0$.
• Figure 5: A comparison of the $out$-$of$-$phase$ and $in$-$phase$ modulation with L = 6 and $\omega _m/2\pi$ = 600 Hz. (a)(b) Total density and spin density in the x-direction, $n_t(x)=\int (n_1+n_2)dy$, $n_s(x)=\int (n_1-n_2)dy$. (c)(d) Fourier transform of the $n_t$ and $n_s$.