Quenching dynamics of vortex in spin-orbit coupled Bose-Einstein condensates

TL;DR

This work investigates vortices in a spin-orbit coupled Bose-Einstein condensate (SOC BEC) under a synthetic magnetic field generated by a position-dependent detuning $\delta(y)=\eta k_r y$. Ground-state vortex lattices with $1$–$6$ vortices are obtained by solving the coupled Gross-Pitaevskii equations as the detuning gradient $\eta$ is varied, and non-equilibrium dynamics are explored after quenches in $\eta$. A key finding is that quenching $\eta$ below its initial value yields long-lived, coherent rotation up to $1000$ ms, with twin vortices showing either scissors-like oscillations or unidirectional rotation depending on $\eta$, and increasing $\eta$ beyond the initial gradient nucleates additional vortices. The dynamics are quantified by fits linking rotation metrics such as the maximum rotation angle $\theta_{\max}$ and rotation period $T$ to $\eta$, including forms $\theta_{\max}=0.34e^{-156(\eta-0.02)}-0.34$ for $(0.00915<\eta\le 0.02)$ and $T=e^{680\eta}+136$ for $(0\le \eta<0.00915)$, suggesting practical applications in gradient magnetometry with sensitivities around $10^{-8}$ Tesla/cm near a dynamical critical point and potential uses of rotating twin vortices for quantum information processing and memory.

Abstract

We investigate the ground states and rich dynamics of vortices in spin-orbit coupled Bose-Einstein condensates (BEC) subject to position-dependent detuning. Such a detuning plays the role of an effective rotational frequency, causing the generation of a synthetic magnetic field. Through scanning the detuning gradient, we numerically obtain static vortex lattice structures containing 1 to 6 vortices using the coupled Gross-Pitaevskii equations. When quenching detuning gradient below its initial value, the vortex lattices exhibit interesting periodic rotation motion, and their dynamical stability can persist for up to 1000ms. In particular, depending on the detuning gradient, the twin vortices exhibit either a scissors-like rotational oscillation or a clockwise periodic rotation, reflecting the response to the magnetic field gradient experienced by the condensates. We fit the numerical results to quantitatively analyze the relation between rotation dynamics and magnetic field gradients. When quenching the detuning gradient beyond its initial value, additional vortices appear. Our findings may motivate further experimental studies of vortex dynamics in synthetic magnetic fields and offer insights for engineering a BEC-based magnetic field gradiometer.

Figures (10)

• Figure 1: The ground states of the vortex lattices. The first row shows the total density distribution of condensates containing 1 to 6 vortices from left to right with $\eta_{0} =0.01999,0.02,0.02005,0.02015,0.0202,0.0203E_{r}$. The second and third rows show the corresponding phase and total velocity field, respectively. The results are obtained from GP simulations with atom number $N=5\times 10^{5}$, Raman coupling $\Omega =10E_{r}$, and isotropic trapping frequencies $(\omega _{x} ,\omega _{y})=2\pi \times (50,50)$ Hz. The green and black circles in two-vortex state mark the vortex position on the $x$ negative semi-axis and positive semi-axis at the initial time, respectively.
• Figure 2: The ground state of each component containing two vortices. (a) and (b) show the density profile of component $n_{1}$ and $n_{2}$, respectively. The corresponding momentum distribution is shown in (c) and (d), where $n_{1} (k)=\mathcal{F}(n_{1})$, $n_{2} (k)=\mathcal{F}(n_{2})$. The system parameters are the same as Fig. \ref{['fig:groundstate']}.
• Figure 3: The velocity field distribution near the vortex core. (a) and (b) show the canonical contribution and the spin contribution to the total velocity field of single-vortex state with $\eta_{0} =0.01999E_{r}$, respectively. (c) and (d) correspond to two-vortex state with $\eta_{0} =0.02E_{r}$. The size of the arrows reflects the magnitude of the velocity field, where long arrows represent large velocities and short ones represent small velocities.
• Figure 4: The dynamical evolution of a single vortex with quenching $\eta =0E_{r}$. (a) shows the total density distribution at different quench times, with the corresponding velocity field (arrows) near the center of vortex core shown in (b). The color map of the background in (b) illustrates the phase distributions of order parameters. (c) and (d) show time evolution of the scissors mode $\left \langle xy \right \rangle$ and quadrupole mode $\left \langle x^{2}-y^{2} \right \rangle$ obtained from GP equations numerical simulations.
• Figure 5: The dynamics of two-vortex state with quenching $\eta=0.015E_{r}$. (a) shows the total density profile (left column) and the corresponding phase and velocity field (right column). The green and black circles correspond to the initial vortex positions marked in Fig. \ref{['fig:groundstate']}. The rotation angle of two vortices $\theta(t)$ is defined as the deviation from initial position, with measurements restricted to the second and third quadrants of the Cartesian coordinate system. The rotation is positive in the second quadrant and negative in third quadrant, ranging from $(-\frac{\pi }{2},\frac{\pi }{2} )$. At $t=0$, $\theta(t)=0$. (b) and (c) display the evolution of vortex positions along $x$- and $y$-directions, respectively, where the green symbols (blue solid lines) and black symbols (red solid lines) correspond to the vortex marked by the green and black circles in (a). (d) and (e) show the rotation angle and motion trajectories. In all panels, the solid lines are the fitted results, while the symbols represent results from GP simulations.