Faraday patterns, spin textures, spin-spin correlations and competing instabilities in a driven spin-1 antiferromagnetic Bose-Einstein condensate

TL;DR

This work addresses how periodic modulation of the $s$-wave scattering lengths $a_0$ and $a_2$ in a driven spin-1 AFM Bose-Einstein condensate induces transient Faraday patterns and spin textures in both quasi-1D and quasi-2D geometries. By analyzing the Bogoliubov spectrum with three branches, employing Mathieu stability for time-periodic interactions, and simulating with the GPE plus truncated-Wigner noise, the authors map driving frequency relative to the gapped spin mode gap $\\Delta$ to distinct pattern outcomes: below-gap drives excite two gapless modes producing density and spin patterns with dimension-dependent spin-spin correlations (Gaussian in 1D, Bessel in 2D), while above-gap drives trigger population transfer to $m=0$ and random or anomalous spin textures. Modulating only $a_0$ and modulating $a_2$ yield qualitatively different regimes, and simultaneous modulation can generate competing instabilities, leading to complex, highly structured population transfer as a function of the quadratic Zeeman field $q$. The results reveal rich, dimensionally tuned spinor pattern formation in driven AFM spin-1 BECs with potential for engineered spin textures and controlled correlations. $ $Delta$ and the Bogoliubov branches play central roles in predicting the observed phenomena.

Abstract

We study the formation of transient Faraday patterns and spin textures in driven quasi-one-dimensional and quasi-two-dimensional spin-1 Bose-Einstein condensates under the periodic modulation of $s$-wave scattering lengths $a_0$ and $a_2$, starting from the anti-ferromagnetic phase. This phase is characterized by a Bogoliubov spectrum consisting of three modes: one mode is gapped, while the other two are gapless. When $a_0$ is modulated and half of the modulation frequency lies below the gapped mode, density and spin Faraday patterns emerge. In that case, in quasi-one-dimension, the spin texture is characterized by periodic domains of opposite $z$-polarizations. When driven above the gap, the spin texture is characterized by random orientations of spin vectors along the condensate axis. Qualitatively new features appear in the driven quasi-two-dimensional condensate. For instance, when driven above the gap, the spin textures are characterized by anomalous vortices and antivortices that do not exhibit phase winding in individual magnetic components. Below the gap, the spin texture exhibits irregular ferromagnetic patches with opposite polarizations. The spatial spin-spin correlations in quasi-one-dimension exhibit a Gaussian envelope, whereas they possess a Bessel function dependence in quasi-two-dimension. Under the $a_2$-modulation, the density patterns dominate irrespective of the driving frequency, unless the spin-dependent interaction strength is sufficiently smaller than that of the spin-independent interaction. The intriguing scenario of competing instability can emerge when both scattering lengths are simultaneously modulated. Finally, we show that the competing instabilities result in a complex relationship between the population transfer and the strength of the quadratic Zeeman field, while keeping all other parameters constant.

Figures (12)

• Figure 1: (color online). Bogoliubov spectrum of a quasi-1D spin-1 BEC with the spin order parameter $(1, 0, 1)/\sqrt{2}$ for (a) $q/\hbar\omega_\rho=-0.2$, $c_0n_H^{1D}/2\pi l_\rho^2\hbar\omega_\rho=0.4$, $c_1n_H^{1D}//2\pi l_\rho^2\hbar\omega_\rho=0.2$ and (b) $q/\hbar\omega_\rho=-0.2$, $c_0n_H^{1D}/2\pi l_\rho^2\hbar\omega_\rho=0.2$, $c_1n_H^{1D}/2\pi l_\rho^2\hbar\omega_\rho=0.4$, where $l_\rho=\sqrt{\hbar/M\omega_\rho}$. $k$ is the quasi-particle momentum along the axial $z$ direction.
• Figure 2: (color online). Results for $\omega_0<\Delta$. The (a) density ($n$) and (b) spin ($F_z$) Faraday patterns in a Q1D spin-1 BEC starting from an AFM phase for $\omega_0=0.2\omega_\rho$, $q/\hbar\omega_\rho=-0.2$, $\bar{c}_{0}n_H^{1D}/2\pi l_\rho^2\hbar\omega_\rho=\bar{c}_{1}n_H^{1D}/2\pi l_\rho^2\hbar\omega_\rho=0.1$, and $\alpha_0=0.45$. The value of gap is $\Delta=0.283\omega_\rho$. (c) The spin texture at $\omega_\rho t=600$, showing periodic domains of spins pointing up and down along the $z$-axis. (d) The dynamics of condensate density in the momentum space, $\tilde{n}(k)$ reveals the unstable momentum $k_u^{(+)}$ and the peak at $k=0$ corresponds to the homogeneous density.
• Figure 3: (color online). Q1D spin-spin correlations of for $\omega_0<\Delta$: (a) $C_x$ and $C_y$, and (b) $C_z$, for the same parameters as in Fig. \ref{['fig:2']}. $C_z$ exhibits a Gaussian envelope. The time average is taken from $t_i=300\omega_\rho^{-1}$ to $t_f=650\omega_\rho^{-1}$. We further take an average over 40 realizations of TWA noises. The dashed line shows the Gaussian fit for the envelope. The central peak, $C_\alpha(z=0)=1$ is removed for clarity.
• Figure 4: (color online). Results for $\omega_0>\Delta$. (a) The dynamics of condensate density in the momentum space, $\tilde{n}(k)$ reveals the unstable momentum $k_u^{(+)}$ and $k_u^{(0)}$. The peak at $k=0$ corresponds to the homogeneous density. (b) The spin dynamics exhibiting population transfer from $m=\pm 1$ components to $m=0$ component due to the unstable $\epsilon_{0}(k)$ mode, where $N_m(t)=\int dz|\psi_m(z, t)|^2dz$ and $N=\sum_mN_m$. The dashed line is $\exp[\sigma^{(0)}t]$, which captures the initial exponential population growth in $m=0$ accurately and also indicates the linear regime of the dynamics. (c) shows the dynamics of $F_x$ and (d) the snap shot of random spin texture at $\omega_\rho t=500$. The values of system parameters are $\omega_0=0.4\omega_\rho$, $q/\hbar\omega_\rho=-0.2$, $\bar{c}_{0}n_H^{1D}/2\pi l_\rho^2\hbar\omega_\rho=\bar{c}_{1}n_H^{1D}/2\pi l_\rho^2\hbar\omega_\rho=0.1$, and $\alpha_0=0.45$. The value of gap is $\Delta=0.283\omega_\rho$.
• Figure 5: (color online). Spin-spin correlations for $\omega_0>\Delta$: (a) $C_x$ and $C_y$, and (b) $C_z$, for the same parameters as in Fig. \ref{['fig:4']}. All three correlations exhibit a Gaussian envelope. The time average is taken from $t_i=200\omega_\rho^{-1}$ to $t_f=600\omega_\rho^{-1}$. We further take an average over 40 realizations of TWA noises. The dashed line shows the Gaussian fit for the envelope and the central peak, $C_\alpha(z=0)=1$ is removed for clarity.