Exotic domain walls in Bose-Einstein condensates with double-well dispersion

We study the domain walls which form when Bose condensates acquire a double-well dispersion. Experiments have observed such domain walls in condensates driven across a ${\mathbb{Z}}_2$ symmetry-breaking phase transition in a shaken optical lattice. We derive a generic model to describe the dispersion and to compute the wave functions and energies of the domain walls. We find two distinct regimes which demand different physical pictures. In the weak-coupling regime, where interactions are weak compared to the kinetic-energy barrier, “density-wave domain walls” form that support an extended density wave and a series of phase steps. These features can be understood as the quantum interference between domains with distinct momenta. In the strong-coupling regime where interaction dominates, the system forms “phase domain walls” which have the minimum width allowed by the uncertainty principle and suppressed density modulation. Analytic results for the domain-wall wave functions are obtained in the two regimes. The energy of domain walls behaves similarly to that of topological defects in paradigmatic field theories.

Figure 1

Emergence of double-well dispersion for atoms in a shaken optical lattice. (a) Ground-band structures for atoms in a shaken lattice of spacing $a$ calculated by diagonalizing Eq. (1) have a single-well structure for small shaking amplitudes (bottom curve), become quartic at $k=0$ at the critical shaking amplitude (second-lowest curve), and develop a double-well structure for large shaking amplitudes (top three curves). The double-well dispersion is characterized by the kinetic-energy barrier $\varepsilon$, shown in terms of the ground-band tunneling energy $t_0$, and the wave numbers at the minima $k=\pm k^{*}$. (b) An averaged image of Bose condensates with a single domain wall. The color represents atoms condensed to wave number $k=k^{*}$ (red, left half) and $k=-k^{*}$ (cyan, right half).

Figure 2

Wave functions for domain walls at different coupling strengths. The (a) density $\rho$ and (b) phase $\varphi$ of the condensate wave functions $\psi =\sqrt{\rho }{e}^{i\varphi }$ with a domain wall at $x=0$ vary dramatically with the coupling strength $\mu /\varepsilon$. Away from the domain wall, the asymptotic density and phase are ${\rho }_{\infty }=\mu /g$ and $\varphi =\pm k^{*}x$, respectively. In the weak-coupling regime $\mu /\varepsilon \ll 1$, the period of the density oscillations and the step size of the phase jumps are $\pi /k^{*}$ and $\pi$, respectively. Red dashed curves show the analytic wave functions given in Eq. (10) in the weak-coupling regime and Eq. (12) in the strong-coupling regime. (c) The width of the domain wall $L$. The solid curves indicate the asymptotic behaviors $L\propto {(\mu /\epsilon )}^{-1/2}/k^{*}$ for weak coupling and $L\approx 0.81\pi /k^{*}$ for strong coupling.

Figure 3

Illustrations of the domain wall in the strong- and weak-coupling regimes. Strong-coupling regime: (a) the calculated density profile at $\mu /\varepsilon =100$ shows a smooth transition from $k^{*}$ (gray) to $-k^{*}$ (white) across the domain wall; (b) the wave function plotted in the complex plane stays near the surface of a cylinder and merely switches helicity at the domain wall. Weak-coupling regime: (c) the wave function calculated with $\mu /\varepsilon =0.01$ is well described by two domains with distinct wave numbers $k=k^{*}$ (gray) and $k=-k^{*}$ (white), which overlap near $x=0$; (d) the wave function in the complex plane takes on a highly elliptical trajectory near the domain wall at $x=0$.

Figure 4

Energy of a domain wall. Energies evaluated based on numerical calculations (symbols) are compared with asymptotic curves (gray lines) based on Eq. (20) in the weak-coupling regime and Eq. (21) in the strong-coupling regime. In dimensionless units the excitation energy $E_{\mathrm{g}}$ (open squares), kinetic energy $\mathrm{\Delta }{E}_{\mathrm{k}}$ (solid squares), interaction energy $\mathrm{\Delta }{E}_{\mathrm{i}}$ (solid circles), and $\mu \mathrm{\Delta }N$ (open circles) all depend on the ratio $\mu /\varepsilon$. The gray shaded region indicates the regime that has been explored by previous experiments [22, 23]. Note that $\mathrm{\Delta }{E}_{\mathrm{i}}$ and $\mathrm{\Delta }N$ are negative.