Coarsening of binary Bose superfluids: an effective theory

TL;DR

This work addresses phase-ordering dynamics in binary Bose superfluids after a quench into the immiscible regime. It develops an effective, conservative equation of motion for the density-imbalance order parameter $\phi$ derived from a microscopic Bose mixture Hamiltonian, generalizing the Cahn–Hilliard framework to quantum fluids. The main findings show that domain growth scales as $L(t) \sim t^{2/3}$ due to a competition between interspecies interactions and quantum pressure, with Porod's law in the structure factor and a calculable interfacial tension; hydrodynamic flows are not the primary driver in weak segregation. This framework provides a unified description of coarsening in quantum fluids, aligning with ab initio simulations and offering avenues to explore 1D dynamics, unbalanced mixtures, and fluctuation effects in ultracold gases.

Abstract

We derive an effective equation of motion for binary Bose mixtures, which generalizes the Cahn-Hilliard description of classical binary fluids to superfluid systems. Within this approach, based on a microscopic Hamiltonian formulation, we show that the domain growth law $L(t)\sim t^{2/3}$ observed in superfluid mixtures is not driven by hydrodynamic flows, but arises from the competition between interactions and quantum pressure. The effective theory allows us to derive key properties of superfluid coarsening, including domain growth and Porod's laws. This provides a new theoretical framework for understanding phase separation in superfluid mixtures.

Figures (7)

• Figure 1: Density plot of the interfacial potential $u^2/(1-\phi^2)-\phi^2/\xi_s^2$ appearing in the effective Hamiltonian density (\ref{['eq:effectiveH']}), here shown for $\xi_s=1$. The potential has a saddle point at $(u,\phi)=(0,0)$, and two global minima at $(u,\phi)=(0,\pm1)$ that correspond to the formation of domains.
• Figure 2: Density plots of the density imbalance $\phi(x,y,t)$ at different times [from upper left to lower right: $t/t_\text{NL}=0, 10, 20, 60, 120, 180$]. The initial condition for $\phi(x,y,t)$ is a uniformly distributed random field of zero mean and correlation function $\langle\phi({\boldsymbol{r}},0)\phi({\boldsymbol{r}}',0)\rangle=\epsilon^2\exp[-({\boldsymbol{r}}-{\boldsymbol{r}}')^2/4\sigma^2]$, where we here choose $\sigma=2\xi_s$ and $\epsilon=0.01$. Domains start to form around $t/t_\text{NL}\simeq 10$ and then grow in time.
• Figure 3: Variance of the density imbalance as a function of time $t/t_\text{NL}$. The dashed curve is the result of the effective EOM (\ref{['eq:effective_nodim']}), and the dotted curve the analytical prediction (\ref{['eq:short-timeapprox']}) for short times. Solid colored curves show results from ab initio simulations for different values of the segregation parameter $g_{12}/g$, based on coupled nonlinear Shrödinger equations. Here $\sigma/\xi_s=2$.
• Figure 4: Correlation function $g(r,t)= \langle\phi({\boldsymbol{r}},t)\phi(0,t)\rangle$ at different times, numerically computed from from Eq. (\ref{['eq:effective_nodim']}). In (a), $g_1$ is shown as a function of $r/\xi_s$, and in (b) as a function of the rescaled position $r/L(t)$, where $L(t)$ is the first zero of $g$. The inset (c) shows the rescaled structure factor $S(q,t)/L^2(t)$ [with $S$ defined in Eq. (\ref{['eq:Skdef']})], as a function of the rescaled momentum $qL(t)$. The dashed line highlights the Porod's scaling $\sim 1/q^3$ expected for $1/L\ll q\ll1/\xi_s$. Here $\sigma/\xi_s=2$, and the correlation function includes an angular average.
• Figure 5: Mean domain size $L(t)$ versus time, numerically computed from the first zero of the correlation function $g(r,t)$, and for several values of $\sigma/\xi_s$ (with $\sigma$ the correlation length of the initial noise). At long times, all curves scale as $t^{2/3}$, with a prefactor independent of $\sigma/\xi_s$.