Transport of ultracold Bose gases beyond the Gross-Pitaevskii description

We explore atom-laser-like transport processes of ultracold Bose-condensed atomic vapors in mesoscopic waveguide structures beyond the Gross-Pitaevskii mean-field theory. Based on a microscopic description of the transport process in the presence of a coherent source that models the outcoupling from a reservoir of perfectly Bose-Einstein condensed atoms, we derive a system of coupled quantum evolution equations that describe the dynamics of a dilute condensed Bose gas in the framework of the Hartree-Fock–Bogoliubov approximation. We apply this method to study the transport of dilute Bose gases through an atomic quantum dot and through waveguides with disorder. Our numerical simulations reveal that the onset of an explicitly time-dependent flow corresponds to the appearance of strong depletion of the condensate on the microscopic level and leads to a loss of global phase coherence.

Figure 1

Possible waveguide geometry for a guided atom-laser experiment with Bose-Einstein condensates in the presence of an optical scattering potential. The lower panel shows the effective potentials along the longitudinal coordinate $x$ for the reservoir ($V_r$, green parabola) and the waveguide atoms ($V_g$, black horizontal curve). A radio-frequency field with suitable frequency ${\omega }_{\mathrm{rf}}$ (indicated by the vertical red line) can be used to resonantly couple atoms from the trapped condensate (whose chemical potential is denoted by the horizontal blue line) into the waveguide.

Figure 2

Time-dependent sweep across a resonance of the double barrier potential [Eq. (29)] (which is displayed in the lower panel of Fig. 3). The upper panel shows the nonlinear transmission spectrum at the incident current $j_i=1.6{\omega }_{\perp }$ obtained from the integration of the Gross-Pitaevskii equation [Eq. (30)] with the interaction strength $g\simeq 0.034\hslash {\omega }_{\perp }{a}_{\perp }$. Dashed lines mark the resonance branches that are not directly accessible by means of a straightforward injection of matter waves at fixed chemical potential. The dotted line represents the linear transmission spectrum at $g=0$. The middle and lower panels show the time-dependent transmission that is obtained by sweeping the chemical potential according to Eq. (31) during the propagation process. This sweep allows one to populate the upper branch of the resonance peak that corresponds to the fifth excited quasibound state within the cavity. Apart from a slight shift of the distorted resonance peak (due to a renormalization of the effective interaction strength), the calculation based on the dynamical Hartree-Fock–Bogoliubov (HFB) equations [Eqs. (19, 20, 21)] (lower panel) reproduces quite well the Gross-Pitaevskii (GP) calculation (middle panel).

Figure 3

Density of the condensate wave function (upper panel) and of the noncondensed atoms (middle panel; also shown as red dashed line in the upper panel) near a resonance of the double barrier potential [Eq. (29)]. The plot represents a snapshot of the time-dependent sweep across the resonance shown in Fig. 2 at the evolution time at which the chemical potential equals $\mu =1.1\hslash {\omega }_{\perp }$, corresponding to a transmission of $T\simeq 0.85$. The lower panel shows the double barrier potential.

Figure 4

Density of the condensate wave function (upper panel) and of the noncondensed atoms (middle panel; also shown as red dashed line in the upper panel) during the time-dependent propagation process of the condensate through a disorder potential, at the evolution time $t=1000/{\omega }_{\perp }$ (corresponding to $160$ ms). The incident current $j_i=1.5{\omega }_{\perp }$ was chosen such that a Gross-Pitaevskii calculation of the transport process yields a quasistationary scattering state. During the propagation process, noncondensed atoms are found to slowly accumulate preferably around the minima of the disorder potential, which is shown in the lower panel. The total depletion, however, is still relatively low at $t=1000/{\omega }_{\perp }$.

Figure 5

Same as Fig. 4 for a higher incident current $j_i=4.775{\omega }_{\perp }$ for which the Gross-Pitaevskii calculation yields permanently time-dependent scattering. The figure shows a snapshot at the evolution time $t=500/{\omega }_{\perp }$. Here the density of noncondensed atoms $n_{\mathrm{nc}}$ is of the same order as the condensate density, which indicates strong depletion and the destruction of the condensate on the microscopic level.

Figure 6

Total current of the condensate and the noncondensed atoms (narrow blue and wide red lines, respectively) as a function of the propagation time, for the incident currents $j_i=1.5{\omega }_{\perp }$ (upper panel) and $j_i=4.775{\omega }_{\perp }$ (lower panel), in the presence of the disorder potential that is shown in Fig. 4. The plots include the time interval of the initial ramping process of the source amplitude, which lasts until $t\simeq 300/{\omega }_{\perp }$. In the case of a rather low incident current (upper panel) for which a Gross-Pitaevskii calculation would yield quasistationary transport, depletion remains rather low on finite time scales. At higher incident currents for which permanently time-dependent scattering manifests on the Gross-Pitaevskii level, we find strongly enhanced depletion and a current of noncondensed atoms that is comparable to the current of the condensate fraction. This indicates, on a microscopic level, that the condensate becomes destroyed during this time-dependent scattering process.

Figure 7

Relative current of noncondensed atoms (normalized by the total current) as a function of time for 17 different disorder potentials, at the incident current $j_i=2.387{\omega }_{\perp }$. The disorder potentials were randomly generated and selected such that a stable, time-independent current is observed on the level of the Gross-Pitaevskii equation. With one exception (brown dotted line) stable flow is also found with our Hartree-Fock–Bogoliubov approach, and with two more exceptions [violet (top) and light-blue dashed lines] depletion in the transmitted current remains below 20% within the considered time interval.

Figure 8

Total atomic current across the disorder potential as a function of time, calculated for the critical incident current $j_i=2.387{\omega }_{\perp }$ at which time-dependent scattering begins to appear on the level of the Gross-Pitaevkii equation (upper panel). A simulation of this scattering process with the kinetic Hartree-Fock–Bogoliubov equations (lower panel), however, yields quasistationary behavior on a time scale of the order of $1000{\omega }_{\perp }^{-1}$, with a reasonably low amount of depletion (between 15% and 20% of the total density), while dynamical instability and strong depletion sets in at ${\omega }_{\perp }t\simeq 1500$.