Quench induced collective excitations: from breathing to acoustic modes

TL;DR

This work investigates quench-induced collective excitations in a harmonically trapped two-dimensional Bose-Einstein condensate using Gross-Pitaevskii simulations and analytical frameworks. It reveals a dual regime: at low momentum the spectrum is a hybrid of hydrodynamic modes and conformal-symmetry modes due to scale-invariance breaking from the trap and finite cut-offs, while at high momentum the excitations follow a trap-modified Bogoliubov dispersion with an effective chemical potential. The high-k dispersion takes the form $\omega_k=\sqrt{(\tilde{\mu}_{\text{eff}}/m)k^2+(\hbar^2/4m^2)k^4}$ with $\mu_{\text{eff}}=2\mu/3$ and $\tilde{\mu}_{\text{eff}}=g_x\mu_{\text{eff}}$, and the post-quench structure factor $S_q(\mathbf{k},t)$ reveals oscillations at twice the mode frequency. Lifetimes scale as $T_s \propto r_0/v_k$, reflecting the confinement-induced decay of well-defined momentum eigenmodes. Overall, the results reconcile theory with experiments and provide a practical spectroscopy framework for non-equilibrium many-body states in trapped 2D BECs.

Abstract

In trapped Bose-Einstein condensates, interaction quenches which are abrupt changes of the interaction strength typically implemented via Feshbach tuning, are a practical and widely used protocol to address far-from-equilibrium collective modes. Using both numerical Gross Pitaevskii and analytical schemes we study these interaction-quench-induced collective modes in a harmonically trapped two-dimensional Bose--Einstein condensate contrasting the behavior found at low and high energies. In the low-lying regime, we characterize realistic circumstances in which there is a breakdown of the expected scale invariance so that the collective excitations follow hydrodynamic theory instead of the predictions given by SO(2,1) conformal symmetry. In the high energy regime, we focus on important trap effects associated with acoustic oscillations which have been of interest experimentally. This comprehensive analysis of the collective excitations in trapped two-dimensional Bose-Einstein condensates is experimentally accessible. Through their frequencies and damping, this reflects an important built-in spectroscopy of such many-body states.

Figures (6)

• Figure 1: (a) Scaling of the collective-mode frequencies of a trapped condensate with trap frequency $\omega_0$. The panel plots the extracted mode frequencies $\omega_n$ ($n=1,2,3,4$ and $l=0$), obtained from the spectral peaks of $\rho(r,\omega_n)$ at $r=10$ grid points, as $\omega_0$ is varied from $60 ~\mathrm{rad/s}$ to $125~\mathrm{rad/s}$. A linear regression yields $\omega_n/\omega_0 = 1.98,\,3.49,\,4.94$ and $6.38$, in close agreement with the theoretical predictions $2.00,\,3.46,\,4.90,$ and $6.32$ from Eq. \ref{['Sfrequency']}. (b) Spectral distribution for $\omega_0 = 125~\mathrm{rad/s}$, where the lowest four dominant peaks (purple arrows) mark the collective modes, and additional minor peaks (orange arrows) are associated with conformal-symmetry modes which are suppressed and deformed at this $r$.
• Figure 2: (a) Dependence of the spectral response $\rho(r,\omega_n)$ on quench strength at a fixed short observation length scale $r$. The two panels show results for $g_x = 1.0$ and $g_x = 1.6$. As the quench strength increases, the conformal mode contribution (orange arrows) becomes increasingly dominant relative to the hydrodynamical modes (purple arrows). (b) Dependence on observational length scale for the same strong quench ($g_x = 1.6$). The two panels correspond to different $r$ values. At larger lengths $r$, the conformal modes (orange arrow) become more pronounced compared with the hydrodynamical modes (purple arrows) predicted by Eq. \ref{['Sfrequency']}.
• Figure 3: (a) At a fixed trapping frequency $\omega_0=60~\mathrm{rad/s}$, the predictions from Bogoliubov theory with $\mu_{\text{eff}}$ (blue line) agree with the numerical data (red dots) within a reasonable error from $0.41\%$ to $2.95\%$. By contrast, the standard Bogoliubov theory (green line), which assumes a homogeneous density distribution, fails to reproduce these oscillation frequencies. For small wavenumbers ($k<4k_0$), the excitation frequencies saturate to a constant value independent of $k$, indicating a crossover to low-lying collective modes. (b) When excitations with a given $\mathbf{k} = (0, 2.6)\mu\mathrm{m}^{-1}$ are measured under different trapping frequencies $\omega_0$ (ranging from $60~\mathrm{rad/s}$ to $125~\mathrm{rad/s}$), their excitation frequencies remain unchanged once the modes enter the large-$k$ regime. Meanwhile, the peak amplitudes $\mathrm{Re}\left[\psi(\mathbf{k},\omega_k)\right]$ decrease as the trapping frequency increases.
• Figure 4: (a) An example of the oscillation of $\operatorname{Re}[\psi(\mathbf{k},t)]$ over time at $(k_x,k_y) = (0,\,2.6)\,\mu\mathrm{m}^{-1}$, from which the excitation frequency is extracted. The unit for each computational time step is $16~\mu\mathrm{s}$. (b) The corresponding Fourier transform of the signal shown in (a).
• Figure 5: This figure shows good agreement between the modified Bogoliubov theory given in Eq. \ref{['dispersionquench']} (solid lines) and the numerical data (hollow symbols) of the excitation frequencies extracted from $S_q(\mathbf{k},t)$ for various $k$ (from $9\,k_0$ to $17\,k_0$). We show both quench up (red) at $g_0 = 3$ with $g_x = 2$ and quench down (blue) at $g_0 = 13$ and $g_x = 0.25$ cases. The Bogoliubov theory without trap effects (dashed lines) fails to capture these oscillation frequencies as expected. We extracted the corresponding frequencies from the first five oscillations of structure factors.