Tachyonic and parametric instabilities in an extended bosonic Josephson junction

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.

💡 Research Summary

This paper presents a comprehensive theoretical investigation of the decay of quantum phase coherence in an extended bosonic Josephson junction (eBJJ) formed by two tunnel‑coupled one‑dimensional Bose‑Einstein condensates (BECs). The authors focus on the π‑trapped state, a dynamical regime where the relative phase between the condensates is locked near π while the population imbalance remains confined. In a mean‑field description this state appears stable, but the inclusion of quantum fluctuations reveals two distinct early‑time instability mechanisms: tachyonic (imaginary‑frequency) and parametric‑resonance instabilities.

The model starts from a microscopic Hamiltonian containing kinetic energy, contact interaction (strength g) and tunneling (amplitude J) for the two fields Ψ₁ and Ψ₂. By employing the Madelung representation the authors rewrite the dynamics in terms of densities ρⱼ and phases φⱼ, and introduce the relative imbalance z = (ρ₁−ρ₂)/(ρ₁+ρ₂) and relative phase φ = φ₁−φ₂. In the spatially uniform limit the equations reduce to the well‑known two‑mode Josephson equations, characterized by the dimensionless interaction‑to‑tunneling ratio Λ = μ/ħJ. For an initial phase φ₀ = π the phase diagram in the (Λ, z₀) plane exhibits three regimes: pure phase trapping (π₀‑oscillations), combined phase‑and‑density trapping (π‑oscillations) and macroscopic quantum self‑trapping (MQST). The π‑oscillation region is bounded by Λ_b(z₀) and Λ_u(z₀), given analytically in Eqs. (9) and (10).

To assess the stability of the π‑trapped trajectory when spatial degrees of freedom are restored, the authors linearize the Gross‑Pitaevskii equations around the time‑dependent mean‑field solution. Fourier transforming the fluctuations yields a dispersion relation ω(k) for each longitudinal mode k. When ω acquires an imaginary part, the corresponding mode grows exponentially. In the limit of weak tunneling (large Λ) the dispersion takes the form ω²(k) ≈ (ħk²/2m)² − Δ², where Δ² > 0 produces a purely imaginary ω, i.e. a tachyonic instability. The growth rate γ(k) = |Im ω(k)| defines an instability band in momentum space that can be computed analytically.

Because the π‑trapped state is not strictly static—its mean‑field trajectory exhibits small oscillations of z(t) and φ(t)—the linearized equations acquire a periodic coefficient. This maps onto a Mathieu‑type equation, leading to parametric resonances when the driving frequency Ω satisfies ω₀ ≈ 2Ω. Consequently, additional instability tongues appear at specific k values. The authors provide a detailed “instability chart” that distinguishes tachyonic from parametric bands, together with analytical expressions for the corresponding growth rates.

Beyond the early linear regime, the authors employ the Truncated Wigner Approximation (TWA) to simulate the full nonlinear dynamics. In TWA the initial quantum noise is sampled as random phase fluctuations, and the subsequent evolution follows the classical Gross‑Pitaevskii equations for many stochastic realizations. The simulations confirm that the primary tachyonic and parametric modes grow as predicted, and then act as seeds for secondary (non‑linear) instabilities. These secondary processes transfer energy to higher‑momentum modes, leading to a rapid broadening of the momentum distribution and a pronounced decay of the relative phase coherence. The authors also explore finite‑temperature effects, showing that thermal noise accelerates the onset of instability but does not qualitatively change the growth rates.

The final sections discuss experimental feasibility. Using realistic parameters for ⁸⁷Rb (scattering length ≈ 5 nm, tunneling J ≈ 2π·10 Hz, interaction g ≈ 2π·0.5 Hz·µm) and a trap length of ≈ 200 µm, the π‑trapped regime can be prepared by imprinting a π phase difference and adjusting the initial imbalance. Quantum‑gas microscopy provides spatially resolved density and phase measurements, while time‑of‑flight imaging can resolve momentum‑space correlations. The authors highlight twin‑beam correlations g^{(2)}(k, −k) as a clear signature of pair production associated with both tachyonic and parametric instabilities. They also outline protocols for state preparation, detection of imbalance dynamics, and measurement of correlation functions, demonstrating that the predicted phenomena are within reach of current ultracold‑atom technology.

In summary, the paper establishes that the π‑trapped state of an extended bosonic Josephson junction, while appearing stable at the mean‑field level, is intrinsically vulnerable to quantum‑fluctuation‑driven tachyonic and parametric instabilities. These instabilities initiate exponential growth of specific longitudinal modes, which subsequently trigger a cascade of secondary nonlinear processes, ultimately destroying phase coherence. The combined analytical and numerical treatment, together with concrete experimental proposals, provides a solid foundation for future investigations of non‑equilibrium dynamics and instability‑driven relaxation in low‑dimensional superfluid systems.