Lee-Huang-Yang dynamics emergent from a direct Wigner representation

Abstract

We demonstrate how the beyond-mean-field Lee-Huang-Yang (LHY) corrections and its related physics can be naturally incorporated into the representation of an ultracold Bose gas using the truncated Wigner approach without invoking effective energy terms or local density assumptions. By generating a Bogoliubov ground-state representation with appropriately tailored bare interaction strength $g_0$ and condensate density $n_0$, the expected initial energy and densities are obtained while retaining access to quantum effects beyond the reach of the extended Gross-Pitaevskii equation (EGPE) formulation. This approach enables the study of correlations, coherence decay, single realisations, and the onset of quantum fluctuation effects with growing interaction strength. Numerical demonstrations for a weakly interacting single-component Bose gas show that observables deviate significantly from both the plain GPE and the EGPE incorporating LHY corrections. In regimes of strong interaction, many of the interference effects predicted by the GPE and EGPE suppressed, and the EGPE offers no improvement over the plain GPE compared to the full Wigner model. In the weakly interaction limit, the EGPE appears accurate but resolving its deviation from mean-field results requires extensive ensemble averaging.

Figures (20)

• Figure 1: Kinetic energy dependence on the scaled cutoff $k_c=k_{\rm max}\xi_0$ in 1d.
• Figure 2: Predicted values of $g^{(2)}(0)$ in 1d for $g_0=0.2, n_0=5$. Predictions use (\ref{['deltandisc']}), (\ref{['G2_1']}), (\ref{['mdef']}), and (\ref{['g20def']}). (a) Dependence on $k_L=\Delta k/2=k_{\rm min}\xi_0/2$ (horizontal axis) and on $k_c=k_{\rm max}\xi_0$ (curves with values $k_c=1, 2, 5, 10, 20, 30, 50$, from top to bottom). (b) Dependence on $k_c$ for two values of $k_L=\Delta k/2$, namely $0.005$ (green) and $0.05$ (blue). Integral estimates using (\ref{['Nkmkm']}), (\ref{['G2_1']}), (\ref{['mkmkm']}) and (\ref{['g20def']}) are shown with solid lines. Also shown are numerical results from a TWA representation of the Bogoliubov state with $P=10^7$ trajectories and $\Delta k=0.1$ ($g_0=0.2, n_0=5$; blue circles). The cyan dashed line shows the discrete on-lattice Bogoliubov calculation using (\ref{['deltandisc']}) and (\ref{['mdef']}).
• Figure 3: Dependence of calculated and mean-field energy densities on $k_c$ for uniform systems in 3d (1st column), 2d (2nd column), and 1d (3rd column) when the bare interaction strength $g_0$ is kept constant. Top row (a-c): total energy $E=E_{\rm kin}+E_{\rm int}$ calculated microscopically from all modes using (\ref{['sumkekin']}) and (\ref{['Eint']}) in blue, and the dressed mean-field estimate using full density $E_{\rm mf}=E_{\rm mf}^{\rm bare}+E_{\rm mf}^{\rm dress} =(g_0+g_1)n^2V/2$ in red ($g_1$ is used only in 3d). Also shown are $E_{\rm mf}^{\rm bare}=g_0n^2/2$ (black dashed) and the condensate-only bare mean-field energy $E_{\rm mf}^0=g_0n_0^2/2$ (magenta dashed). Bottom row (d-f): the difference $E_{LHY}=E-E_{\rm mf}$, as per (\ref{['Elhy-def']}), with expected LHY correction (\ref{['LHY']}) shown dashed. Three variants using $g_0n_0$ (black), $g_0n$ (green), and $gn$ (orange, 3d) are displayed. Here, $g_0=0.05$ in 3d and 2d, $g_0=0.2$ in 1d, $n_0=1/g_0$, and the box dimensions are $L=20\pi$ in each dimension.
• Figure 4: An illustration of the contributions to ground-state energy, $E=E_{\rm kin}+E_{\rm int}$, in a numerical implementation. The $g_1$ contribution, $E_{\rm mf}^{\rm dress}=g_1n^2V/2$, is included only in 3d.
• Figure 5: Predicted values of the energy shift relative to the mean-field value $E_{\rm mf}$ for a uniform 1d system. (a): $E_{LHY}^{\rm reg}$ (solid) and the full shift $E_{LHY} =E_{LHY}^{\rm reg} + E_{LHY}^{\rm extra}$ (dashed, colour-matched) as functions of $k_L=\pi/L=k_{\rm min}\xi_0/2$ (horizontal axis) and $k_c=k_{\rm max}\xi_0$ (curves with values $k_c=1, 2, 5, 10, 20, 30, 50$, from top to bottom), for $g_0=0.2, n_0=5$. Also shown as a black dashed line is the standard LHY correction $E_{LHY}^{\infty}(g_0,n_0)$ from (\ref{['LHY1d']}). Data here were calculated on a discrete lattice with $M=4,\dots,2^{12}$$k$-space modes, from top to bottom. (b): Dependence on $k_c$ for two values of $k_L=\Delta k/2 = 0.005$ (green) and $0.05$ (blue), for $g_0=0.2, n_0=5, L=20\pi$. The solid lines use the large box estimates (\ref{['Nkmkm']}), (\ref{['Hke']}), (\ref{['G2_1']}), (\ref{['mkmkm']}), and (\ref{['Eint']}), while the cyan dashed line shows the full lattice-based prediction using (\ref{['deltandisc']}), (\ref{['sumkekin']}), and (\ref{['mdef']}). Also shown are values from a numerical realisation of the TWA ensemble with $P=10^7$ trajectories and $k_L=0.05$ (blue circles), the standard LHY correction (\ref{['LHY1d']}) $E_{LHY}^{\infty}(g_0,n_0)/L = -2/3\pi$ from [Rakshit19b] (black dashed), and the "regular" part of the LHY energy $E_{LHY}^{\rm reg}$ from (\ref{['E0reg']}) (blue dotted). Fig. \ref{['fig:g2o-kmkm']} shows the corresponding $g^{(2)}(0)$ values for the same parameters.