Stability and dynamics across magnetic phases of vortex-bright type excitations in spinor Bose-Einstein condensates

The static properties, i.e., existence and stability, as well as the quench-induced dynamics of vortex-bright type excitations in two-dimensional harmonically confined spin-1 Bose-Einstein condensates are investigated. Linearly stable vortex-bright-vortex and bright-vortex-bright solutions arise in both antiferromagnetic and ferromagnetic spinor gases upon quadratic Zeeman energy shift variations. Their deformations across the relevant transitions are exposed and discussed in detail, evincing also that emergent instabilities can lead to pattern formation. Spatial elongations, precessional motion, and spiraling of the nonlinear excitations when exposed to finite temperatures and upon crossing the distinct phase boundaries, via quenching of the quadratic Zeeman coefficient, are unveiled. Spin-mixing processes triggered by the quench lead, among others, to changes in the waveform of the ensuing configurations. Our findings reveal an interplay between pattern formation and spin-mixing processes accessible in contemporary cold atom experiments.

Figure 1

(a) Schematic illustration of the phase diagram containing the distinct VBV and BVB stationary solutions in the $(c_1,q)$ plane as well as their corresponding deformations under QZ energy shift variations. (b) Intervals of existence of nonlinear excitations of the VB type for AF $c_1>0$ (top rows) and FM $c_1<0$ (bottom rows) interactions corresponding, respectively, to a spin-1 BEC consisting of ${}^{23}\mathrm{Na}$ and ${}^{87}\mathrm{Rb}$ atoms. From left to right, each column depicts the occupation of a single (1C), two (2C), and all three (3C) $m_F$ components. A spin component is treated as unpopulated when its occupation, $n_{m_F}$, is less than $1/N$. Here, the total number of particles $N={10}^4$, while the in- and out-of-plane trapping frequencies are $\omega =2\pi \times 20$ Hz and ${\omega }_z=2\pi \times 400$ Hz, respectively. The values of $q$ are given in units of $\hslash \omega$.

Figure 2

Polarization, $P$, with respect to the QZ coefficient $q$ for VBV (orange solid line) and BVB (dashed purple line) equilibrium states existing in AF ($c_1>0$) spin-1 BECs. Differently colored opaque and transparent markers indicate the value of $q$ for which the distinct solutions are provided. Insets (a1)–(a6) [(b1)–(b6)] illustrate representative density profiles, $|{\mathrm{\Psi }}_{m_F}{(x,y)|}^2$, of a VBV [BVB] 3C configuration (top row) and its corresponding deformations towards a 2C (middle row) and a 1C (bottom row) stationary state. The components that are not depicted possess zero population. For both types of solutions singly quantized vortices are considered for the relevant in each case $m_F$ component (see legends). The QZ coefficient is expressed in units of $\hslash \omega$.

Figure 3

BdG spectra of 3C (a1, b1) VBV and (a2, b2) BVB stationary states upon a $q$ variation for $c_1>0$. In both cases the anomalous modes (AMs) are depicted by light blue circles while the background ones are depicted by black circles. Notice the two AMs present for VBV structures when compared to the single pair occurring for BVB configurations. (c1–g1) [(c2–e2)] 2D contour plots of the difference $\mathrm{\Delta }{\mathrm{\Psi }}_{m_F}\equiv |{\tilde{\mathrm{\Psi }}}_{m_F}{(x,y)|}^2-{\left|{\mathrm{\Psi }}_{m_F}(x,y)\right|}^2$, demonstrating the effect that the perturbation has on a VBV [BVB] solution for $q=-0.6$ [$q=0.4$]. The AF BEC consists of $N={10}^4$ sodium atoms confined in a quasi-2D harmonic trap. Note that length and density are given in units of $\mu \mathrm{m}$ and ${\mu \mathrm{m}}^{-2}$, respectively; the excitation frequencies $\mathrm{\Omega }$ are measured in units of $\omega$ and $q$ in units of $\hslash \omega$.

Figure 4

Polarization, $P$, as a function of the QZ coefficient $q$ for the BVB and VBV equilibrium states occurring in the distinct phases of a spin-1 FM ($c_1<0$) condensate. Insets on the left-hand [right-hand] side of ($\mathrm{a}1$)–($\mathrm{a}6$) [($\mathrm{b}1$)–($\mathrm{b}6$)] showcase representative examples of the $m_F$ densities, $|{\mathrm{\Psi }}_{m_F}{(x,y)|}^2$, of a 3C VBV [BVB] configuration together with its relevant 1C and 2C deformed structures (see legends), as $q$ is varied. The corresponding QZ values in each of the aforementioned cases are also indicated for the individual stationary solutions, with opaque and transparent markers in the polarization curves pointing explicitly at each specific value. Our results are presented in dimensional units, i.e., length is measured in $\mu \mathrm{m}$, density in ${\mu \mathrm{m}}^{-2}$, and $q$ in $\hslash \omega$.

Figure 5

Same as Fig. 3 but for $c_1<0$. Light blue circles denote the AMs present in the spectra and black circles are used for the background modes. Contrary to AF interactions, FM VBV entities feature three anomalous mode pairs but a single pair is present for BVB configurations. (c1–g1) [(c2–e2)] $\mathrm{\Delta }{\mathrm{\Psi }}_{m_F}\equiv |{\tilde{\mathrm{\Psi }}}_{m_F}{(x,y)|}^2-{\left|{\mathrm{\Psi }}_{m_F}(x,y)\right|}^2$ quantifies the difference between a perturbed and an equilibrium VBV [BVB] solution for $q=-1.2$ [$q=0.2$]. The FM BEC consists of $N={10}^4$ rubidium atoms confined in a quasi-2D harmonic trap, while length, density, $\mathrm{\Omega }$, and $q$ are given in units of $\mu \mathrm{m}, {\mu \mathrm{m}}^{-2}, \omega$, and $\hslash \omega$.

Figure 6

($\mathrm{a}1$–$\mathrm{c}5$) [($\mathrm{d}1$–$\mathrm{f}5$)] Snapshots of the $m_F$-component density, $|{\tilde{\mathrm{\Psi }}}_{m_F}{(x,y)|}^2$, of a perturbed FM BVB [VBV] solution for $q=0.2$ [$q=-1.0$]. The distinct spin components are illustrated, respectively, from top to bottom (see the legends) while each column corresponds to a fixed time instant ranging from $t=0\mathrm{s}$ to $0.9\mathrm{s}$. For both the BVB and the VBV structure the prevailing feature is their regular and irregular precessional motion, being activated upon adding the eigenvector associated with the single and the ${\text{AM}}_3$ negative-energy mode, respectively (see the text). All quantities shown are given in dimensionful units.

Figure 7

Instantaneous density profiles of a BVB solution illustrating the $m_F=-1$ (top), $m_F=0$ (middle), and $m_F=+1$ (bottom) components, upon considering quenches that $(\mathrm{a}1$–$\mathrm{c}4)$ either cross the phase boundary separating the PO and the AF phase of an AF spin-1 BEC or $(\mathrm{d}1$–$\mathrm{f}4)$ enter deeper in the PO regime for a FM spinor gas, respectively (see legends). For AF interactions, the structural deformation of the BVB entity corresponding to a precession and simultaneous spatial elongation of all components is monitored for times up to $t=1.1\mathrm{s}$. The vortical pattern at $m_F=0$ acquires a dipolar spatial form. In contrast, the precessional motion of the BVB configuration along with a simultaneous population transfer from the $m_F=\pm 1$ states to the $m_F=0$ component dominates the evolution for FM interactions. In both cases the damping parameter is $\gamma =0.0023$ while each system contains $N={10}^4$ atoms. Note that time, length, and density are measured in units of $\mathrm{s}, \mu \mathrm{m}$, and ${\mu \mathrm{m}}^{-2}$, respectively, while QZ is measured in $\hslash \omega$.

Figure 8

$(\mathrm{aj}$–$\mathrm{cj})$ [$(\mathrm{dj}$–$\mathrm{fj})$] With $j=1,2$, temporal evolution of the populations, $n_{m_F}\left(t\right)$, of the different spin components considering quenches of the QZ coefficient at finite temperatures both within the same magnetic phase and upon crossing distinct phases of an AF [FM] spinor gas (see legends). Spin mixing is triggered in all cases being more suppressed for VBV excitations as compared to the BVB ones and for both types of entities when the relevant transition entails quenches within the same phase. In all cases, the initial-state configuration, having a prequench value $q\equiv q_i$, refers to the underlying in each phase 3C stationary state transitioning either from $3\mathrm{C}\to 2\mathrm{C}$ or from $3\mathrm{C}\to 1\mathrm{C}$ states being characterized by different postquench QZ coefficients $q\equiv q_f$ (see legends). All quenches are considered for fixed $\gamma$ that is either $\gamma =0.0023$ or $\gamma =0.01$ (see legends). The AF [FM] condensate $c_1>0$ [$c_1<0$] consists of $N={10}^4 {}^{23}\mathrm{Na}$ [${}^{87}\mathrm{Rb}]$ atoms. Note also that time is provided in dimensional units being of the order of few seconds and $q$ is given in units of $\hslash \omega$.

Figure 9

Polarization, $P$, in terms of the QZ coefficient $q$ for (a) a VBV and (b) a BVB configuration upon also varying the vortex charge $S$ (see legend). An increasing $S$ prolongs the region of existence of the 3C state with respect to $q$. Note that QZ is provided in units of $\hslash \omega$. $(\mathrm{c}1–\mathrm{c}3)$ [$(\mathrm{d}1–\mathrm{d}3)$] Density contours of a stationary VBV state of charge $S=2$ [$S=3$] for $q=-1.0$ [$q=-2.2$], i.e., within the EA phase. The FM spin-1 BEC mixture contains $N={10}^4 {}^{87}\mathrm{Rb}$ atoms.

Figure 10

$\left(\mathrm{a}1\right)$ [$\left(\mathrm{a}2\right)$] BdG spectra of doubly quantized, $S=2$, VBV [BVB] stationary states for $q=-1.0$ [$q=0.6$] and for FM interactions ($c_1<0$). Notice the absence of imaginary eigenfrequencies for both entities that demonstrates their spectral stability. Remarkably seven $\text{AM}i$ (with $i=1,2,...,7$) pairs, being marked by light blue circles, are present for the VBV configuration in contrast to the three found for a BVB solution. $(\mathrm{b}1–{\mathrm{j}}_1)$ [$(\mathrm{b}2–{\mathrm{j}}_2)$] 2D contour plots measuring the density difference $\mathrm{\Delta }{\mathrm{\Psi }}_{m_F}(x,y)\equiv |{\tilde{\mathrm{\Psi }}}_{m_F}{(x,y)|}^2-{\left|{\mathrm{\Psi }}_{m_F}(x,y)\right|}^2$ for a VBV [BVB] solution and for $q=-1.0$ [$q=0.6$] (see legends). (${\mathrm{k}}_1$–${\mathrm{m}}_3$) Snapshots of the density, $|{\tilde{\mathrm{\Psi }}}_{m_F}{(x,y)|}^2$, of an $S=2$ BVB solution for $q=0.6$ perturbed via the eigenvector associated with ${\text{AM}}_1$. The distinct spin components are shown, respectively, for $t=0$, 0.5, and $1.0\mathrm{s}$ (see legends). All densities are illustrated in dimensional units. The BdG frequencies $\mathrm{\Omega }$ are in units of $\omega$ and $q$ is provided in terms of $\hslash \omega$.