Tachyonic and parametric instabilities in an extended bosonic Josephson junction

TL;DR

This work analyzes the decay of phase coherence in two tunnel-coupled one-dimensional Bose-Einstein condensates, focusing on the pi-trapped state. By combining linear stability analysis with Truncated Wigner simulations, it identifies tachyonic and parametric instabilities that drive rapid pair production into finite-momentum modes and subsequently trigger nonlinear secondary growth. The study provides explicit instability conditions, resonant momenta, and a clear physical picture of energy transfer via correlated quasiparticle pairs, along with experimentally realistic parameter regimes. It further outlines measurable signatures, such as momentum-resolved correlations g^{(2)}(k,-k) and interferometric observables, to observe these nonequilibrium phenomena in ultracold-atom setups, thereby linking extended Josephson junction dynamics to fundamental instability mechanisms with potential cosmological analogs.

Abstract

We study the dynamics and decay of quantum phase coherence for Bose-Einstein condensates in tunnel-coupled quantum wires. The two elongated Bose-Einstein condensates exhibit a wide variety of dynamic phenomena where quantum fluctuations can lead to a rapid loss of phase coherence. We investigate the phenomenon of self-trapping in the relative population imbalance of the two condensates, particularly $π$-trapped oscillations that occur when also the relative phase is trapped. Though this state appears stable in mean-field descriptions, the $π$-trapped state becomes dynamically unstable due to quantum fluctuations. Nonequilibrium instabilities result in the generation of pairs excited from the condensate to higher momentum modes. We identify tachyonic instabilities, which are associated with imaginary parts of the dispersion relation, and parametric resonance instabilities that are triggered by oscillations of the relative phase and populations. At early times, we compute the instability chart of the characteristic modes through a linearized analysis and identify the underlying physical process. At later times, the primary instabilities trigger secondary instabilities due to the build-up of non-linearities. We perform numerical simulations in the Truncated Wigner approximation in order to observe the dynamics also in this non-linear regime. Furthermore, we discuss realistic parameters for experimental realizations of the $π$-trapped state in ultracold atom setups.

Figures (13)

• Figure 1: Sketch of the experimental setup under study, consisting of two elongated condensates (left $1$ and right $2$). The BECs are characterized by an atomic density $\rho_{1, 2}$ and a phase $\phi_{1, 2}$. The BECs are coupled by a single-particle tunnel interaction $J$.
• Figure 2: Parameter space $(\Lambda, z_0)$ showing the dynamical regimes delineated by $\Lambda_b$ [Eq. \ref{['eq:lower_bound']}] and $\Lambda_u$ [Eq. \ref{['eq:upper_bound']}], for an initial phase difference of $\phi_0 = \pi$. The horizontal lines represent $\Lambda = \{1.00, 3.55, 7.10\}$.
• Figure 3: The dynamics of the mean fields $\bar{\phi}$ and $\bar{z}$ corresponding to Fig. \ref{['fig:dynamical_regimes']}. For $\Lambda < \Lambda_b(z_0)$, the system exhibits $\pi_0$-oscillations, characterized by $\langle \bar{\phi} \rangle_t =\pi, \langle \bar{z} \rangle_t =0$ as visible in the left and center panels. For $\Lambda_b(z_0) < \Lambda < \Lambda_u(z_0)$, the system displays $\pi$-oscillations, where $\langle \bar{\phi} \rangle_t =\pi, \langle \bar{z} \rangle_t \neq 0$ as visible in the center and right panels (blue lines). For $\Lambda > \Lambda_u(z_0)$, the system exhibits MQST self-trapped modes, characterized by $\langle \bar{\phi} \rangle_t =0, \langle \bar{z} \rangle_t \neq 0$ and it is shown in the right plot (green lines). In all plots, the gradient bar indicates the initial condition for $z_0$.
• Figure 4: Top: Squared dispersion relation $\omega^2_\pm$ [Eq. \ref{['eq:omega^2']}] as a function of momentum $\tilde{k}$ for $\Lambda= 3.55$. The real parts $\operatorname{Re}{(\omega^2_\pm)}$ and imaginary parts $\operatorname{Im}{(\omega^2_\pm)}$ of the squared dispersion relation $\omega^2_\pm$ are represented by solid and dashed lines, respectively. Bottom: Growth rates $\gamma_+$ (dashed line) and $\gamma_-$ (thin solid line) corresponding to the imaginary part of $\omega_+$ and $\omega_-$. The growth rate $\gamma_{lin}$ (thick solid line) is obtained from solving Eq. \ref{['eq:linearized_eq']}.
• Figure 5: Physical processes related to the primary instability of the coupled BEC system with components $\Psi_1$ and $\Psi_2$, indicated by the upper and lower elongated shapes, respectively: $a)$ excitation of a pair of particles from the first condensate, where one particle further tunnels to the second condensate; $b)$: excitation of a pair from the first condensate that tunnels to the second one; $c)$ and $d)$: excitation of a pair of particles from each of the two condensates.