Phase domain walls in coherently driven Bose-Einstein condensates

TL;DR

This work analyzes a coherently driven two-component Bose fluid, such as a polariton condensate, under a resonant drive that explicitly breaks U(1) symmetry. Using mean-field, dissipative dynamics, it identifies two vacuum states and classifies domain walls by the total phase variation $q \in \{-1,0,1\}$, showing that walls can form spontaneously in 1D and 2D and can host bound HQV molecules. It reveals two distinct topological wall types: $q=0$ walls, analogous to magnetic solitons with potential spin polarization depending on motion, and $q=\pm1$ walls with broken spatiotemporal symmetry that move along preferred directions and couple to HQVs into composite solitons; these walls induce long-range ordering through vortex-wall interactions. The results, supported by analytical and numerical modeling, predict observable signatures in exciton-polariton microcavities and advance understanding of how driven-dissipative spinor condensates relate to freely evolving BECs, including Kibble-Zurek-like symmetry breaking and self-organization into complex topological patterns.

Abstract

We consider coherent states of weakly interacting bosons under the conditions of external resonant excitation, with a focus on a two-dimensional polariton fluid driven by a plane electromagnetic wave near the ground state. The coherent driving breaks the U(1) symmetry explicitly, which prevents the occurrence of quantum vortices in a uniform scalar condensate. Surprisingly, a spinor (two-component) system of the same kind admits topological excitations, such as domain walls of relative phase or confined half-vortex molecules, typical of a freely evolving spinor Bose system. Opposite-phase domains arise from the spontaneous breakdown of the spin symmetry $(\mathbb{Z}_2)$. Domain walls form with time even when the initial state of the system is uniform or completely disordered; they fall into different topological types distinguished by the total phase variation in the transverse direction. One type of domain walls is similar to ``magnetic'' solitons in Bose-Einstein condensates and exhibits nonzero spin polarization whose sign depends on the direction of motion. Domain walls of the second type, by contrast, behave like monopoles with broken spatiotemporal symmetry and tend to move in certain preferred directions. The interaction of vortices and domain walls results in a long-range ordering of the system.

Figures (11)

• Figure 1: Vacuum states depending on $f^2$. (a) Density of the full number of particles according to \ref{['eq:vac:asymm:f']} for $f < f_2$ and \ref{['eq:vac:symm']} for $f \ge f_2$. The dashed curves show unstable solutions. (b) Cosine of the relative phase of $\psi_\pm$ according to \ref{['eq:vac:asymm:phase-diff']} (exactly) or \ref{['eq:asymm:phase']} (approximately). The arrows in (a) and (b) indicate individual phases $\alpha_\pm$. (c) Energy gap between the vacuum state and elementary excitations according to \ref{['eq:exc:asymm']} for $f < f_2$ and \ref{['eq:exc:symm']} for $f \ge f_2$ (exactly) or \ref{['eq:exc:asymm:gap']} for $f < f_2$ (approximately). Parameters: $D = \Omega$, $\gamma / \Omega = 5 \times 10^{-3}$.
• Figure 2: Spectra of elementary excitations depending on wave number $k$ at $f = f_1$ or $f = f_2$ (a) and $f = \sqrt{2} f_2$ (b) according to \ref{['eq:exc:asymm']} (exactly) or \ref{['eq:exc:top']} (approximately). The dashed curve shows the bare dispersion law; the central dot is the driven mode. The gapless spectrum in (a) obeys the Bogoliubov formula \ref{['eq:exc:bgl']}. The arrows indicate polarization directions of excitations, horizontal $(\leftrightarrow)$ or vertical $(\updownarrow)$. Parameters as in Fig. \ref{['fig:vac']}; $\xi = \hbar / \sqrt{2 m \mu}$.
• Figure 3: (a)--(f) Three solutions to boundary-value problem \ref{['eq:soliton']}--\ref{['eq:bc2']} for $v = 0$ and $f^2 / f_2^2 = 0.1$. The left and right sides represent $\phi_\pm = \arg \psi_\pm$ and $n_\pm = |\psi_\pm|^2$, respectively; solid and dashed curves correspond to individual components ($+$ and $-$). (g) Quantity $K$\ref{['eq:K']} across the $q = \pm 1$ domain walls. Parameters: $D = \Omega$, $\gamma / \Omega = 10^{-6}$.
• Figure 4: Peak values of $n_\pm$ (a) and spatial widths $\sigma_\pm$ (b) of motionless domain walls depending on $f^2$. The dashed vertical line indicates the limiting value of $f^2$ at which the $q = \pm 1$ domain walls cease to exist. The solution with $q = -1$ (not shown) is obtained by exchanging '$+$' and '$-$' for $q = 1$. Parameters as in Fig. \ref{['fig:bvp:types']}.
• Figure 5: Energy \ref{['eq:energy']} (a, b) and peak values of $n_\pm$ (c, d) as functions of $v$ for the domain walls with $q = 0$ (a, c) and $+1$ (b, d) at $f^2 / f_2^2 = 0.025$. The solution with $q = -1$ (not shown) is obtained by reversing time $(v \mapsto -v)$ and exchanging '$+$' and '$-$' for $q = 1$. Parameters as in Fig. \ref{['fig:bvp:types']}; $H_\mathrm{vac}$ is the vacuum energy; $v_s = \sqrt{\mu / m}$.