Nonlinear transport of Bose-Einstein condensates through mesoscopic waveguides

We study the coherent flow of interacting Bose-condensed atoms in mesoscopic waveguide geometries. Analytical and numerical methods, based on the mean-field description of the condensate, are developed to study both stationary as well as time-dependent propagation processes. We apply these methods to the propagation of a condensate through an atomic quantum dot in a waveguide, discuss the nonlinear transmission spectrum and show that resonant transport is generally suppressed due to an interaction-induced bistability phenomenon. Finally, we establish a link between the nonlinear features of the transmission spectrum and the self-consistent quasibound states of the quantum dot.

Figure 1

(Color online) Transverse energy $ϵ\left(a_{s}n\right)$ (in units of $\hslash \omega$) as a function of $a_{s}n$. The numerical result (solid line) coincides for large values $a_{s}n$ very well with the Thomas-Fermi result. The interpolation result agrees excellently with the numerical result for small values of $a_{s}n$ and converges toward the Thomas-Fermi result for large $a_{s}n$. The upper inset magnifies the domain of large $a_{s}n$. The lower inset zooms into the region of small $a_{s}n$. The straight dashed line displays the perturbative result (10).

Figure 2

(Color online) Schematic behavior of the function $W\left(n\right)$, here displayed for the low-density regime $\mathcal{E}\left(n\right)=g{n}^2∕2$, in the left-hand panel. The right-hand panel shows typical density profiles of the condensate [lengths are given in units of the healing length $\xi \left(n_2\right)=\hslash ∕\sqrt{2m{n}_{2}g}$]. For given values $\mu$, $j_t$, and $g$, a beam of uniform density has either the density $n_1$ (supersonic solution, horizontal line in the right-hand panel) or the density $n_2$ (subsonic solution). At a given classical energy $E$ (with $E_{\text{min}}<E<E_{\text{max}}$, to assure bounded density oscillations), $n_{-}$ and $n^{+}$ are the minimum and maximum values of the cnoidal density oscillations, displayed in the right-hand panel. Energy values $E$ close to (but lower than) $E_{\text{max}}$ correspond to gray solitons.

Figure 3

(Color online) Adiabatic transition of the interaction parameter $g$ for a proper definition of transmission coefficients: The upper part of the figure displays the adiabatic variation of the position-dependent parameter $g\left(x\right)$ from $g=0$ up to a maximal value $g$. The gray-shaded transition region between $x_1$ and $x_2$ in which $g$ varies with position is assumed to be much larger than the typical periodicity of the condensate density oscillations. The lower part shows the density of a stationary scattering state in the presence of a potential barrier, which is constant in the downstream region and displays oscillations in the upstream region (the position of the barrier potential is marked by the vertical line). The nonlinear cnoidal oscillation of $n$ between the barrier and $x_2$ is adiabatically conveyed, in the transition region between $x_1$ and $x_2$, into a sinusoidal oscillation in the interaction-free domain on the left-hand side of $x_1$. There, the wave function can be linearly decomposed into an incident and a reflected component.

Figure 4

(Color online) Comparison between the approximate and exact transmission values, calculated with Eqs. (42, 34), respectively, for a moderately interacting condensate that propagates through a potential barrier, with the dimensionless parameters $\mu =3$, $g=1∕2$, $j_t=1$, and with variable barrier height. The transmission is plotted as a function of the classical energy transfer $\Delta E$, which is a measure for the backreflection in the upstream region. For increasing backreflections, the approximate result (34) overestimates the “true” value of the transmission coefficient given by Eq. (42).

Figure 5

(Color online) A reservoir of Bose-condensed matter with a given chemical potential $\mu$ is locally coupled at the position $x_0$ to a waveguide with a scattering potential. The reservoir emits a plane matter wave in both directions into the guide. Hence, a coherent beam, with current $j_i$, propagates toward the barriers of the potential where the condensate is partially reflected, with the current $j_r$, and partially transmitted, with the current $j_t$.

Figure 6

Illustration of the relation (50) between the source amplitude $S_0$ and the densities $n_{1,2}$ of a homogeneous flow. The lower branch corresponds to the supersonic solutions with density $n_1$ and the upper branch to the subsonic solutions with density $n_2$. These two branches merge at the maximal value $S_{\text{max}}=\hslash \mu ∕\sqrt{2gm}$ of the source amplitude, where a homogeneous solution with density $n_s=\mu ∕\left(2g\right)$ is found on the saddle point of the classical potential $W\left(n\right)$. The insets illustrate the shape of $W\left(n\right)$ for the different types of solutions.

Figure 7

Time evolution of the condensate density during the adiabatic increase of the source amplitude (the source is located at the vertical dashed lines). The panels (a)–(c) show three snapshots of the condensate in a waveguide without scattering potential: at (a) $t=0.1\Delta T$, (b) $t=\Delta T$, and (c) $t=10\Delta T$. The bottom part of the figure shows the real and imaginary part of the wave function (in units of $\sqrt{n_1}$) whose density is displayed in panel (b). The panels (d)–(f) illustrate the scattering of the matter waves at a repulsive barrier potential (gray-shaded region). Panel (f) clearly shows that a stationary scattering state is populated in the long-time limit $t\gg \Delta T$.

Figure 8

(Color online) Evolution of the condensate density [given in units of the density $n_s=\mu ∕\left(2g\right)$ of the saddle-point solution, see Fig. 6] as a function of the source amplitude $S$ for three different values of the time scale $\Delta T$ in which $S$ is ramped to its maximal value $S_0$ (dashed and dashed-dotted curves). For an increasing ratio $\Delta T∕\tau$ with $\tau \equiv \hslash ∕\mu$, the curves converge toward the supersonic branch of Eq. (50) (solid line). For $S\to S_0$ the supersonic scattering state with constant density $n=n_1$ is reached.

Figure 9

(Color online) Transmission spectrum of the condensate flow through a quantum point contact for different values of the incident current $j_i$ ($\mu$ in units of $\hslash \omega$). The solid lines are found by evaluating the stationary Gross-Pitaevskii equation, the values denoted by crosses are obtained by integrating the time-dependent Gross-Pitaevskii equation in the presence of the source term (the dashed line displays the result for a noninteracting condensate). The inset shows the $j_t-j_i$ current characteristics for a condensate flow without interactions ($g=0$, dashed line) and in presence of interactions ($g=0.034\hslash \omega$, solid line), at $\mu =3\hslash \omega$.

Figure 10

(Color online) Transmission spectra of the double barrier potential at $g=0$ (upper panel), $g=0.002\hslash \omega a_{\perp }$ (middle panel), and $g=0.01\hslash \omega a_{\perp }$ (lower panel). The (green) solid lines show the transmissions of all scattering states, calculated by the “stationary” method based on Eq. (17), that exist at the incident current $j_i=1.0\omega$ of the matter-wave beam. The dashed lines display the spectra obtained from the time-dependent integration approach. The inset shows the longitudinal atom densities (in units of $a_{\perp }^{-1}$) of the first resonant state and the coexisting low-transmission state for $g=0.002\hslash \omega a_{\perp }$ (the position $x$ is given in units of $a_{\perp }$). The gray-shaded curves indicate the positions of the two barriers. The (red) dots (marked by the arrows) designate the positions of the resonant state and the low-transmission state in the transmission spectrum. The chemical potential $\mu$ is given in units of $\hslash \omega$.

Figure 11

(Color online) Current characteristics for the double barrier potential in the vicinity of the onset of the multivalued subzone of the spectrum (for $g=0.002\hslash \omega a_{\perp }$), calculated for different chemical potentials (given in units of $\hslash \omega$). Below the critical $\mu =0.419\hslash \omega$, the $j_i-j_t$ characteristics (dashed-dotted line) intersect only once the horizontal line indicating the fixed incident current $j_i=1.0\omega$. Above this critical value, three intersection points are found (dashed line). At $\mu =0.419\hslash \omega$, the current characteristics (solid line) exhibits a tangent to the horizontal line. The intersection points marked with filled (green) points correspond to scattering states that are populated during a time-dependent propagation process.

Figure 12

Chemical potential ${\mu }_0$ and decay rate ${\gamma }_0$ of the quasibound state within the double barrier potential, calculated as a function of $N_{b}g$ with $N_b$ the population of the quasibound state and $g$ the effective one-dimensional interaction strength. In practice, ${\mu }_0$ and ${\gamma }_0$ were computed at 30 equidistant values of $N_{b}g$ within $0\le N_{b}g\le 1.5$, and cubic interpolation was employed to obtain intermediate values of ${\mu }_0$ and ${\gamma }_0$ for the self-consistent solution of Eq. (81). ${\mu }_0$, $\hslash {\gamma }_0$, and $g∕\sigma$ are given in “natural” energy units of $\hslash \omega$.

Figure 13

(Color online) Transmission spectra of the double barrier potential at $g=0$ (upper panel), $g=0.002\hslash \omega a_{\perp }$ (middle panel), and $g=0.01\hslash \omega a_{\perp }$ (lower panel). The solid line shows the transmissions of all scattering states, calculated by the “stationary” method based on Eq. (17), that exist at the incident current $j_i=1\omega$ of the matter-wave beam. The dashed line is obtained from self-consistent solutions of Eq. (81) at $j_i=1\omega$, which are inserted in the expression (80) for the nonlinear transmission coefficient. The good agreement confirms the one-to-one correspondence between quasibound states of the atomic quantum dot and resonance peaks in the transmission spectrum ($\mu$ in units of $\hslash \omega$)

Figure 14

(Color online) Real and imaginary parts (black solid lines) of the steady-state plane-wave solution (in units of $\sqrt{n_1}$) obtained by integrating the time-dependent Gross-Pitaevskii equation with the numerical source term (A12) using the time step $\Delta t=\hslash ∕\left(50\mu \right)$ and the grid spacing $\Delta x=\lambda ∕30$ with $\lambda =2\pi ∕k$ the wavelength of the condensate. An excellent agreement with the exact analytical result (A13) (red dashed lines) is found. The source is located at $x=0$ (lengths are given in units of $\lambda$).

Figure 15

(Color online) The positive branch of the dispersion relation of a plane wave (black line) is approximated by a linear function (straight blue line). The parameters ${\alpha }_1,{\alpha }_2$ are chosen such that the wave numbers of the plane waves to be absorbed lie within the momentum interval $\Delta k$.

Figure 16

Sketch of the right-hand lattice boundary. The additional point at position $\tilde{x}$ allows for a proper implementation of the absorbing boundary conditions in a grid representation of the wave function.