Wave chaos as signature for depletion of a Bose-Einstein condensate

We study the expansion of repulsively interacting Bose-Einstein condensates (BECs) in shallow one-dimensional potentials. We show for these systems that the onset of wave chaos in the Gross-Pitaevskii equation (GPE), that is, the onset of exponential separation in Hilbert space of two nearby condensate wave functions, can be used as an indication for the onset of depletion of the BEC and the occupation of excited modes within a many-body description. Comparison between the multiconfigurational time-dependent Hartree for bosons method and the GPE reveals a close correspondence between the many-body effect of depletion and the mean-field effect of wave chaos for a wide range of single-particle external potentials. In the regime of wave chaos the GPE fails to account for the fine-scale quantum fluctuations because many-body effects beyond the validity of the GPE are non-negligible. Surprisingly, despite the failure of the GPE to account for the depletion, coarse-grained expectation values of the single-particle density such as the overall width of the atomic cloud agree very well with the many-body simulations. The time-dependent depletion of the condensate could be investigated experimentally, for example, via decay of coherence of the expanding atom cloud.

Figure 1

The initial ($t<0$) harmonic trapping potential with longitudinal frequency ${\omega }_0$ (black line). At $t=0$ the longitudinal harmonic trapping potential is switched off (the radial trapping potential with frequency ${\omega }_r$ remains switched on). Simultaneously, either a periodic (dashed red line) or a disorder (blue solid line) potential is switched on. The system expands for $t\ge 0$ in the 1D external potential. The length scale $l_0$ corresponds to $4.6\mu$m.

Figure 2

The initial state of the BEC in the harmonic trap with ${\omega }_0=2\pi \times 5.4$ Hz and $l_0\approx 4.6\mu$m. The interaction strength is given by the nonlinearity $g_0$ [Eq. (8)]. Results for GPE (black line) and MCTDHB with $N=1000$, $M=3$ (red short dashed line) are indistinguishable within the graphical resolution; blue dashed line, MCTDHB with $N=100$, $M=3$. The two local maxima for $N=100$ are due to depletion of the condensate in the initial state and indicate deviations from the Thomas-Fermi limit.

Figure 3

(a) The density of an expanding cold atom cloud in a periodic potential within the MCDTHB for $N={10}^5$ at $t=3t_0$. The inverted parabola in the blue dashed line indicates the shape of an expanding cloud in the absence of the potential. The parameters of the periodic potential [Eq. (3)] are $l=0.54811l_0$ and $V_{\mathrm{A}}=0.2e$, where $e$ is the energy per particle on the mean-field level. (b) The corresponding Fourier spectrum again compared to its form for free expansion (blue long dashed parabolic curve). The vertical red short dashed lines correspond to $\pm k_L=\pm 4.434l_0^{-1}$, the momenta assiociated with the Landau velocity (see text).

Figure 4

Time dependence of the distance function $d^{\left(2\right)}\left(t\right)$ within the GPE for a periodic potential with period $l=0.54811l_0$ and amplitude $V_{\mathrm{A}}=0.2e$ (black solid line) as well as for free space expansion (blue dashed line) after release from the harmonic trap. The initial wave functions ${\psi }_1(x,0)$ and ${\psi }_2(x,0)$ correspond to weakly perturbed ground-state wave functions given in Fig. 2 with $d^{\left(2\right)}\left(0\right)\approx {10}^{-7}$ (for details, see Ref. 13). The nonlinearity is $g_0\approx 390e_0{l}_0$ [Eq. (8)]. $d^{\left(2\right)}\left(t\right)$ for the periodic potential is fitted to an exponential function (red short dashed line). The time $t_e$ is determined from the crossing point between the exponential and the free space expansion. The saturation of $d^{\left(2\right)}\left(t\right)$ for $t\gtrsim 15t_0$ near unity indicates approximate orthogonality of ${\psi }_1(x,t)$ and ${\psi }_2(x,t)$.

Figure 5

(a) The potential $V\left(x\right)$ in units of the energy per particle $e$ for varying amplitude $V_{\mathrm{A}}$; the period $l$ is $l=0.54811l_0$. (b) The state entropy $S_N\left(t\right)$ within MCTDHB and (c) the $d^{\left(2\right)}\left(t\right)$ function of two close wave functions within the GPE expanding in the periodic potentials of (a). In (b) and (c) $V_{\mathrm{A}}$ in units of the energy per particle $e$ is indicated next to the curves. The black dashed line in (a), (b), and (c) refers to vanishing potential. For the many-body system the particle number is $N={10}^4$, the orbital number is $M=2$. The onset of exponential growth $t_e$ as well as the onset of depletion $t_d$ for $V_{\mathrm{A}}=0.2e$ are marked by arrows. $t_d$ corresponds to a time of $\approx$60 ms. Note that $S_N$ is strictly zero within the GPE.

Figure 6

The depletion rate $\eta$ as a function of $V_{\mathrm{A}}$ as compared to the Lyapunov exponent $\lambda$. $\eta$ and $\lambda$ are calculated from $S_N\left(t\right)$ and $d^{\left(2\right)}\left(t\right)$, respectively. The MCDTHB calculation is for $N={10}^4$ particles. All other parameters as in Fig. 5.

Figure 7

(a) The periodic and a sample realization of the disorder potential with amplitude $V_{\mathrm{A}}=0.2e$. The period of the periodic potential is $l=0.54811l_0\approx 5.8\xi$. The correlation length of the disorder potential is $\sigma =0.7\xi$. (b) Onset of depletion within the MCTDHB for different particle and orbital numbers for propagation in the periodic and disorder potential of (a). The numbers next to the curves indicate the particle number $N$. (c) Onset of chaos within the GPE. In (a), (b), and (c) the thick light gray line corresponds to the disorder potential. Within the GPE $d^{\left(2\right)}$ has been determined by averaging over 90 realizations of the disorder potential. Within the MCTDHB 44 realizations have been used for $S_N\left(t\right)$. The onset of depletion $t_d$ and exponential divergence $t_e$ are marked by arrows.

Figure 8

Particle density at $t=4t_0$ within MCTDHB for $N={10}^4$ (green thick line) and GPE (red dashed line). The condensate density is given by $n_1^{\mathrm{NO}}\left(t\right){\left|{\Phi }_1^{\mathrm{NO}}(x,t)\right|}^2$ [blue (gray) line]; the density of excited atoms is determined by $n_2^{\mathrm{NO}}\left(t\right){\left|{\Phi }_2^{\mathrm{NO}}(x,t)\right|}^2$ (black line).

Figure 9

Two-particle normalized correlation function $g^{\left(2\right)}(x_1,x_2,x_1,x_2;t=4t_0)$ for $N={10}^4$ bosons as well as the density of the second natural orbital $|{\phi }_2^{\mathrm{NO}}(x,t=4t_0){|}^2$ (upper frame) obtained from MCTDHB. The color code in the online version is chosen to highlight deviations from full coherence (white): red corresponds to correlations ($g^{\left(2\right)}>1$) and blue to anticorrelations ($g^{\left(2\right)}<1$). Note that $g^{\left(2\right)}(x_1,x_2,x_1,x_2;t)$ is a real function [see Eq. (14)] and is equal to unity within the GPE. For a movie of the time dependence of $g^{\left(2\right)}(x_1,x_2,x_1,x_2;t)$ as well as $n_2^{\mathrm{NO}}\left(t\right){\left|{\phi }_2^{\mathrm{NO}}(x,t)\right|}^2$, see Ref. 41.

Figure 10

(a) The width $\Delta x$ as a function of time as predicted by the GPE (solid black line) and the MCTDHB for $N={10}^3$ (red squares), $N={10}^4$ (green circles), and $N={10}^5$ (blue triangles). (b) Average of the square of momentum $\langle k^2\rangle$ (or mean kinetic energy) as a function of time, symbols as in (a).