Effects of interactions and noise on tunneling of Bose-Einstein condensates through a potential barrier

We investigate theoretically the tunneling of a dilute Bose-Einstein condensate through a potential barrier. This scenario is closely related to recent experimental studies of condensates trapped in one-dimensional optical lattices. We derive analytical results for the tunneling rate of the condensate with emphasis on the effects of atom-atom interactions. Furthermore, we consider the effect of fluctuating barrier height to the tunneling rate. We have computed the tunneling rate as a function of the characteristic frequency of the noise. The result is seen to be closely related to the excitation spectrum of the condensate. These observations should be experimentally verifiable.

Figure 1

A schematic diagram of the tunneling problem. The external potential $V_{\mathrm{ext}}\left(x\right)$ is represented by the solid curve, and the square of the wave function $\mid \psi \left(x\right){\mid }^2$ by the dashed curve. The chemical potential $\mu$ is denoted by the horizontal dotted line and the corresponding classical turning points $a$ and $b$ by the vertical dashed lines. The region II in the vicinity of the left-hand classical turning point where the order parameter is approximately of the Airy form is bounded by the vertical solid lines. The size of this region has been exaggerated for clarity.

Figure 2

Approximations used in the derivation of the analytic expression for the decay rate $\Gamma$. The Thomas-Fermi (TF) form for the ground state is given by the dotted line, the Airy function is given by the dashed line, and the actual stationary tunneling state wave function is given by the solid line. The points $a-\xi$ and $a+\xi$ are denoted by the solid vertical lines, and the left-hand classical turning point $a$ is denoted by the vertical dashed line. The TF approximation is used in region I, $x\le a-\xi$, the Airy solution in region II, $a-\xi \le x\le a+\xi$, and the semiclassical approximation in region III, $a+\xi \le x\le b$.

Figure 3

Comparison of decay rates $\Gamma$ obtained from simulation (solid line) and by using the analytic expression of Eq. (10) (dashed line) for two values of the barrier height parameter, $s=50$ (a) and $s=100$ (b). The expression fails to produce the correct decay rate for small values of $\tilde{g}$ and for values of the chemical potential close to the barrier height.

Figure 4

Decay rate $\Gamma$ as a function of the RTN correlation time ${\tau }_c$ for four values of the effective interaction strength $\tilde{g}∕(E_r∕q)=0$, $6.91$, $8.98$, and $10.4$.

Figure 5

Decay rate $\Gamma$ as a function of $\pi ∕\Omega$, where $\Omega$ is the frequency of the harmonic drive, for four values of the effective interaction strength $\tilde{g}∕(E_r∕q)=0$, $6.91$, $8.98$, and $10.4$. The crosses denote the frequency of the dipole mode for each interaction strength.

Figure 6

Time-averaged population $⟨\mid p_q\left(t\right){\mid }^2{⟩}_t$ of the dipole (solid), quadrupole (dashed), octupole (dash-dot), and hexadecapole (dotted) modes as a function of the correlation time ${\tau }_c$ of the RTN signal. The effective interaction was set to the value $\tilde{g}=3.46E_r∕q$.

Figure 7

Time-averaged populations $⟨\mid p_q\left(t\right){\mid }^2{⟩}_t$ of the dipole (solid), quadrupole (dashed), octupole (dash-dot), and hexadecapole (dotted) modes as functions of $\pi ∕\Omega$, where $\Omega$ is the frequency of the harmonic drive. The effective interaction was set to the value $\tilde{g}=3.46E_r∕q$.