Temperature driven false vacuum decay in coherently coupled Bose superfluids

TL;DR

This work investigates finite-temperature false vacuum decay in a two-dimensional, coherently coupled Bose-Bose mixture using the Stochastic Gross-Pitaevskii equation to prepare thermal false-vacuum states and monitor decay through global magnetization. The authors show that the decay rate follows a thermal instanton form, $\Gamma \propto A e^{-\beta E_c}$, with an effective critical bubble energy $E_c$ extracted from two complementary analyses, and they observe that the relative phase $\varphi$ dynamically participates in the decay process. The results demonstrate that SGPE is a robust framework for capturing both magnetization and phase dynamics in this platform, linking ultracold-atom experiments to field-theoretic instanton physics. They also highlight the need for a dedicated instanton theory for complex scalar fields that jointly involve $Z$ and $\varphi$, and discuss future directions including finite-size scaling and detailed comparisons with higher-dimensional and experimental implementations.

Abstract

The relaxation of a quantum field from a metastable state (false vacuum) to a stable one (true vacuum), also known as false vacuum decay, is a fundamental problem in quantum field theory and cosmology. We study this phenomenon using a two-dimensional interacting and coherently coupled Bose-Bose mixture, a platform that has already been employed experimentally to investigate false vacuum decay in one dimension. In such a mixture, it is possible to define an effective magnetization that acts as a quantum field variable. Using the Stochastic Gross-Pitaevskii equation (SGPE), we prepare thermal equilibrium states in the false vacuum and extract decay rates from the magnetization dynamics. The decay rates show an exponential dependence on temperature, in line with the thermal theory of instantons. Since the SGPE is based on complex scalar fields, it also allows us to explore the behavior of the phase, which turns out to become dynamic during decay. Our results confirm the SGPE as an effective tool for studying coupled magnetization and phase dynamics and the associated instanton physics in ultracold quantum gases.

Figures (6)

• Figure 1: Schematic illustration of false vacuum decay through the energy landscape. Here $\varepsilon$ is the energy and $Z$ is the magnetization. The metastable false vacuum corresponds to the fully polarized ${Z}=+1$ state on the right (blue plane). It decays by nucleating bubbles (i.e., regions of opposite polarization) which then evolve towards the true vacuum $Z=-1$ state on the left (red plane) through thermalization processes.
• Figure 2: Panels Ⓐ and Ⓑ show the energy landscapes, in arbitrary units, described by Eq. (\ref{['eq:energy_density']}). In Ⓐ, for an initial positive detuning $\delta_i = 0.5 \Omega$, the system is prepared at equilibrium near the global minimum ($Z \approx 1$) at temperature $T = 5.5T_s$. For $\tau_r< t < 0$, with $\tau_r=-1.6$ s, the detuning is linearly varied from the initial $\delta_i$ to a final negative value $\delta_f = -1.0 \Omega$, hence transferring the system into a metastable state, as in Ⓑ. The system subsequently decays into the true vacuum. Panels (a)–(i) show typical snapshots of the local magnetization $z=(n_1-n_2)/(n_1+n_2)$ in the $x$–$y$ plane at three different times during the decay process, namely $12$, $16$ and $21$ ms. Each square box has length $25~\mu \rm{m}$ and periodic boundary conditions. Each row corresponds to a distinct stochastic noise realization. The figure shows bubbles of condensate $2$ (red) forming and growing in condensate $1$ (blue).
• Figure 3: Representative trajectories of the rescaled magnetization ${\cal Z}(t)$ obtained by solving the PGPE from $20$ different stochastic realizations and two different temperatures. Here the detuning is $\delta_f = -\Omega$. The false vacuum decay occurs when ${\cal Z}(t)$ exhibits a sharp decrease, corresponding to the growth of bubbles with $Z \approx -1$ (true vacuum) in a medium with $Z \approx 1$ (false vacuum). The variation in decay times reflects the intrinsic stochasticity of the process. At $T = 7T_s$, the trajectories exhibit a narrower spread and decay rapidly, whereas at $T = 5.5T_s$ the decay occurs slowly with greater variability.
• Figure 4: Logarithm of survival probability vs. time, at various temperatures and for $\delta_f = -\Omega$. In (a) the survival probability is defined as the ensemble average of the rescaled magnetization, while in (b) is obtained by counting the number $P(t)$ of trajectories which are not yet decayed at time $t$. All curves clearly exhibit exponential behavior, with both $\langle {\cal Z}(t) \rangle$ and $P(t)$ being proportional to $e^{-\Gamma t}$. The black dashed lines correspond to the fitting functions used to extract the decay rate $\Gamma$. For each temperature, the time interval used for the fit starts approximately when the first growing bubble appears and ends when the true vacuum occupies about half of the box.
• Figure 5: Temperature dependence of the decay rate $\Gamma$, extracted for $\delta_f=-\Omega$ and $\delta_f=-0.95\Omega$. The solid and empty markers represent the decay rate obtained from Fig. \ref{['fig:lnft_time']}(a) and (b), respectively. The vertical error bars denote the statistical uncertainties in the estimated decay rates, obtained via the bootstrapping procedure Billam_19Bootstrapping. The numerical data exhibit excellent agreement with the instanton prediction, $\Gamma \propto e^{-\beta E_c}$. For $\delta_f=-0.95\Omega$, the higher potential barrier leads to a reduced decay rate at any given temperature. The straight lines are the fitted curves to the data represented through the markers.