Particle-hole origin of thermal beating in dipole-compression modes of a 1D Bose gas

Abstract

Using generalized hydrodynamics, we study the thermal behavior of dipole-compression collective oscillations in a harmonically trapped one-dimensional (1D) Bose gas across the crossover from weak to strong repulsive contact interactions. A key scale controlling this behavior is the temperature of the hole-induced anomaly, associated with the thermal population of hole excitations. In contrast to classical hydrodynamics, which predicts a single oscillation mode, we find a beating signal composed of two frequencies. As the temperature increases, both frequencies evolve from the low-temperature phononic hydrodynamic regime toward the collisionless limit around the anomaly temperature, without saturating at the values expected in the high-temperature collisional hydrodynamic regime. The lower frequency originates from hole excitations and is associated to low-energy oscillations, while the higher frequency emerges from particle excitations and corresponds to the dipole-compression mode. The thermal evolution of the relative excitation strengths of the two frequencies reflects the changing population imbalance between particle and hole spectral states across the anomaly. Our results reveal direct connections between excitations, thermodynamics, correlations, dynamics, and interparticle collisions, and may prove relevant to other atomic, nuclear, solid-state, electronic, and spin systems exhibiting similar anomalies or thermal second-order phase transitions.

Figures (4)

• Figure 1: Diagram of regimes of the harmonically trapped 1D Bose gas in terms of the interaction strength $\gamma_0$ and temperature $\mathcal{T}$\ref{['Eq:gamma and T parameters']}. Regimes are Kerr2024: I. Bogoliubov (quantum), $\mathcal{T}/2 \ll \gamma_0^{-1}, \gamma_0 \ll 1$; II. Bogoliubov (thermal), $2 \gamma_0^{-1} \ll \mathcal{T} \ll 2\gamma_0^{-3/2}$; III. Nearly ideal Bose gas (degenerate), $2\gamma_0^{-3/2}\ll \mathcal{T}\ll \gamma_0^{-2}$; IV. Nearly ideal Bose gas (non-degenerate), $\mathcal{T} \gg \text{max}\{1,\gamma_0^{-2}\}$; V. Strong interactions (high $T$), $\pi^2/\left( \gamma_0 + 2\right)^2 \ll \mathcal{T} \ll 1$, $\gamma_0 \gg 1$; VI. Strong interactions (low $T$), $\mathcal{T} \ll \pi^2/\left( \gamma_0 + 2\right)^2$, $\gamma_0 \gg 1$. Red thick solid line denotes the anomaly temperature $\mathcal{T}_A$ vs. $\gamma_0$, where the red square markers represent the values of $\mathcal{T}_A$ estimated from the position of the peak in the specific heat DeRosi2022. Datasets (a)-(e) (blue points), are used for the study of dipole compression modes in regimes I-VI, and are calculated at fixed $\mathcal{T}$.
• Figure 2: Frequencies $\omega/\omega_x$ of the dipole compression modes vs. the interaction strength $\gamma_0$, Eq. \ref{['Eq:gamma and T parameters']}, for datasets (a)-(e), shown in the panels, which span the regimes I-VI of the 1D Bose gas of Fig. \ref{['Fig:Regimes_Diagram']}. Additional parameter values of our simulations are listed in Appendix B, Table \ref{['Tab:Parameters']}. The two dominant frequency components, $\omega_1$ (lower) and $\omega_2$ (higher), are extracted from generalized hydrodynamics (GHD) calculations of the Fourier transform of the skewness (Appendix D). The sizes of the respective yellow and blue markers are proportional to the excitation strengths $K_1$ and $K_2$. Red triangles indicate the predictions from Ref. Hu2014 obtained using a classical hydrodynamic (CHD) variational ansatz. Dotted horizontal lines at $\omega/\omega_x=\{1,\sqrt{6},\sqrt{7},3\}$ mark the analytic limits of Table \ref{['Tab:DC freq']}. Vertical solid red lines denote the interaction strength at the hole-induced anomaly temperature $\gamma_0^A \!\equiv \!\gamma_0\!\left(\mathcal{T}\!=\! \mathcal{T}_A\right)$, whose values are: (a) $\gamma_0^A\!=\!2.86 \times 10^2$; (b) $6.57$; (c) $1.51$; (d) $1.23 \times 10^{-2}$.
• Figure 3: Total atom number $N$ of a harmonically confined 1D Bose gas vs. the interaction strength $\gamma_0$, Eq. \ref{['Eq:gamma and T parameters']}, defined at the trap centre. Each dataset (a)-(e) (see also Fig. \ref{['Fig:Regimes_Diagram']}) is shown for a fixed value of both temperature $\mathcal{T}$, Eq. \ref{['Eq:gamma and T parameters']}, and the 1D scattering length $\tilde{a}_{\mathrm{1D}}$, Eq. \ref{['Eq:a_1D']}, as in Table \ref{['Tab:Parameters']}.
• Figure 4: (1): Example of the skewness, Eq. \ref{['Eq:Skew']}, vs. the dimensionless time $t \omega_x$, calculated for dataset (c) (see Table \ref{['Tab:Parameters']}) at $\gamma_0\!=\!5.035$, corresponding to $N\!=\!1412$ in Fig. \ref{['Fig:N_gamma0']}. (2): Fourier transform of the skewness in (1), $\rm {FT}\left[ \rm{Skew} \right]$, Eq. \ref{['Eq:FT Skew']}, vs. frequency $\omega/\omega_x$. The yellow and blue markers at the two peaks correspond to the DC oscillation frequencies $\omega_1$ and $\omega_2$.