Dissipative parametric resonance in a modulated 1D Bose gas

TL;DR

The paper demonstrates that dissipative parametric resonance in a modulated 1D Bose gas can be described by a dissipative extension of the Busch-2014 model, with the decay rate matching Micheli-2022's three-wave mixing calculation. Through exact analytic solutions at resonance and fully nonlinear Truncated Wigner simulations, it shows how growth and saturation compete with dissipation and how nonseparability evolves under these conditions. It also systematically reanalyzes Micheli-2022 data with an improved template, clarifying early-time behavior and finite-size effects, and confirms the existence of two growth regimes controlled by the dimensionless ratio α = Γ/G. The work highlights the quantum-origin seeds of entanglement via vacuum fluctuations, identifies distinct growth and separability thresholds, and discusses decoherence mechanisms beyond the current model, informing future experiments and theory in analogue gravity and driven quantum gases.

Abstract

We synthesize results of previous works to give a coherent and self-consistent account of parametric resonance in a modulated quasi-1D Bose gas in the presence of a dissipative mechanism. The resonant behaviour is shown to be largely in line with the predictions of a phenomenological model published in 2014, while the associated dissipation rate is consistent with that derived in 2022 from three-wave mixing processes between the produced phonons and thermal fluctuations.

Figures (11)

• Figure 1: Diagrammatic representation of the Beliaev (left) and Landau (right) three-body phonon scattering processes. Note that the arrows do not represent charge but momentum direction, whose magnitude is given above the edges. The arrows corresponding to phonons in the thermal part of the spectrum ($q \approx 0$) are shown in red.
• Figure 2: Evolution of the number of excitations $n_k$ in the resonant mode with positive wavenumber $k \xi$ for different values of $\rho_{0} \xi$ as given in Tab. \ref{['tab:fig5_parameters']}. The other parameters are fixed to the values given in Tab. \ref{['tab:fixed_param']}. The colored dots with error bars correspond to the results of TWA simulations. The values of $\alpha_k$ reported are the best-fit values to template (\ref{['eq:n_c_initial_thermal']}) with details of the procedure in Sec. \ref{['subsec:fit']}. The full (resp. dashed) curves represent the evolution predicted by template (\ref{['eq:n_c_initial_thermal']}) for best-fit values (resp. predicted values given in Tab. \ref{['tab:fig5_parameters']}) of $\alpha_k$.
• Figure 3: Evolution of the nonseparability parameter $\Delta_k$ of the resonant modes during modulation as a function of adimensionalised time $G_k t$. Green dots show results of TWA simulations for different gas density $\rho_0 \xi$, so different expected $\alpha_k$ computed from Eq. \ref{['eq:Gamma_formula']}. Upper and lower panels correspond respectively to resonant modes $k \xi = \pm 1.0$ and $k \xi = \pm 3.1$. Red and blue shaded regions correspond to $\Delta_k > 0$, separable states, and $-0.5 < \Delta_k < 0$, entangled states. The region $\Delta_k < -0.5$ is left blank as it should be excluded for physical states. Finite statistics might still lead to points in the region. Relevant parameters are listed in the figure and in Tabs. \ref{['tab:fixed_param']}-\ref{['tab:fig5_parameters']}. The error bars correspond to one standard deviation on each side of the mean value, see App. \ref{['app:error']} for more details. Dashed lines show predictions \ref{['eq:n_c_initial_thermal']} using the value of $\alpha_k$ quoted in the figure.
• Figure 4: Best-fit values for the quasiparticle dissipation rate $\Gamma_k$ extracted from TWA simulations data for two different pairs of resonant modes $k \xi = \pm 1.0$ for the upper panel and $k \xi = \pm 3.1$ for the lower panel. These values are compared to the prediction \ref{['eq:Gamma_formula']}. The figure is an updated version of Fig. 5 in Micheli-2022. The value of the reduced $\chi^2$ for the fits performed jointly over $n_k$ (for the mode with positive wavenumber) and $|c_k|$ are reported in Tab. \ref{['tab:fig5_parameters']}.
• Figure 5: Evolution of the pair correlation $|c_k|$ of the resonant modes during modulation as a function of adimensionalised time $G_k t$. Coloured dots show the results of TWA simulations for different modulation amplitudes $a$, so different expected $\alpha_k$ computed from Eq. \ref{['eq:Gamma_formula']}. The amplitude decreases from blue to red. Relevant parameters are listed in Tabs. \ref{['tab:omegamod3_fixed_parameters']}-\ref{['tab:omegamod3_varyingA_parameters']}. The error bars correspond to one standard deviation on each side of the mean value, see App. \ref{['app:error']} for more details. Dashed lines are predictions \ref{['eq:n_c_initial_thermal']} using the value of $\alpha_k$ quoted in the figure.