Density-and-phase domain walls in a condensate with dynamical gauge potentials

TL;DR

The paper addresses how density-dependent gauge potentials in a harmonically trapped BEC can generate ground-state domain walls with localized synthetic electromagnetic fields, separating regions of high and low density near a critical density $\rho_c$. By combining a hydrodynamic framework with a Gross-Pitaevskii energy functional, it clarifies the distinct roles of the vector potential $\mathbfcal{A}(\rho)$ and the scalar potential $\mathcal{B}(\rho)$ and the resulting fields $\mathbf{E}$ and $\mathbf{B}$, showing that the wall concentrates the EM fields. One-dimensional numerical results reveal that stronger repulsive interactions stabilize a sharper wall and that an electrostatic field localizes at the wall, with a peak field scaling as $E_{\max}\sim \hbar^2 k_0^3 N l / d$. The wall can undergo a discontinuous first-order transition into a flatter density state, ending at a critical point as the field weakens, providing a controllable analog of false-vacuum decay in a cold-atom platform and a route to explore dynamical gauge-field phenomena in quantum gases.

Abstract

We show how one can generate domain walls that separate high- and low-density regions with opposite momenta in the ground state of a harmonically trapped Bose-Einstein condensate using a density-dependent gauge potential. Within a Gross-Pitaevskii framework, we elucidate the distinct roles of vector and scalar potentials and how they lead to synthetic electromagnetic fields that are localized at the domain wall. In particular, the kinetic energy cost of a steep density gradient is compensated by an electrostatic field that pushes particles away from a special value of density. We show numerically in one dimension that such a domain wall is more prominent for repulsive contact interactions, and becomes metastable at strong electric fields through a first-order phase transition that ends at a critical point as the field is reduced. Our findings build on recent experimental developments and may be realized with cold atoms in a shaken optical lattice, providing insights into collective phenomena arising from dynamical gauge fields.

Figures (5)

• Figure 1: (a,b) Ground-state density profiles for $x>0$ of a 1D BEC with $N$ bosons in a harmonic trap of length $d$ with interaction strength $g$ in the presence of a density-dependent gauge potential given by Eq. \ref{['eq:tanh']} with $N l / d = 30$ for (a) $\tilde{g} = 0$ and (b) $\tilde{g} = 40$, where $\tilde{g} \coloneqq (2 N d / \hbar^2) g$ and $k_0 \coloneqq p_0/\hbar$. The solid and dashed horizontal lines correspond to $\rho_c$ and $\rho_c \pm 1/l$, respectively. As the gauge potential is increased a steep slope emerges where $\rho = \rho_c$, becoming more prominent for stronger repulsive interactions. (c) Reversal of the phase gradient (dark blue) and a synthetic, localized electrostatic field [Eq. \ref{['eq:EMfields']}] (red) for the $k_0 d = 3$ curve in (b), where $E_0 \coloneqq \hbar^2 k_0^3 N l / d$. The vertical and horizontal lines show where $\rho = \rho_c$ and $\varphi^{\prime} = 0$, respectively.
• Figure 2: (a) Discontinuous transition in the ground-state density for $\tilde{g} = 40$, $N l / d = 30$, and $k_0 d = 5$. As $\rho_c$ crosses above the transition point (horizontal line), it becomes energetically favorable to annihilate the domain structure (red) and create a flatter profile (blue) below $\rho_c$. (b) Phase diagram for $\tilde{g} = 10$ and $k_0 d = 2$, where $\rho_0$ is the peak density for $k_0=0$. The phase boundary (black curve) ends at a critical point for small $l$ where the electric field is weak. The color tracks the potential energy difference, $\mathcal{E}_{\text{pot}} / \mathcal{E}_{0,\text{pot}} - 1$, where $\mathcal{E}_{0,\text{pot}}$ is the potential energy for $k_0 = 0$, showing one approaches the unperturbed ground state for $l \to 0$ and $l \to \infty$.
• Figure 3: (a) One-parameter family of curves that yield a piecewise linear variation of the gauge potential for $\eta \to \infty$. (b) Resulting ground-state density profiles for $\tilde{g} = 40$, $N l / d = 30$, $k_0 d = 3$, and $\rho_c d / N = 0.2$ (solid horizontal line), corresponding to the red curve in Fig. \ref{['fig:profiles']}(b), with the same color convention as in (a). As $\mathbfcal{A}(\rho)$ becomes sharper the domain wall gets more pronounced between $\rho_c \pm 1/l$ (dashed horizontal lines).
• Figure 4: Slope at the domain wall location $x_c$, where $\rho(x_c) \coloneqq \rho_c$, for $Nl/d=30$, corresponding to Figs. \ref{['fig:profiles']}(a) (blue) and \ref{['fig:profiles']}(b) (orange). In both cases $\rho^{\prime}(x_c)$ approaches $-2k_0 \rho_c$ at large $k_0$, in accordance with our estimate from Eq. \ref{['eq:domainwallenergy']}.
• Figure 5: Variation of the (a) peak density and (b) potential energy, measured relative to the unperturbed ground state, across the phase diagram for $\tilde{g}=10$ and $k_0 d = 2$ shown in Fig. \ref{['fig:phasetran']}(b). The observables exhibit an infinite slope at the critical point $l_c$ (orange), a jump across a first-order transition for $l > l_c$ (blue), and a smooth crossover for $l < l_c$ (green).