Resonant trapping in the transport of a matter-wave soliton through a quantum well

We theoretically investigate the scattering of bright solitons in a Bose-Einstein condensate on narrow attractive potential wells. Reflection, transmission, and trapping of an incident soliton are predicted to occur with remarkably abrupt transitions upon varying the potential depth. Numerical simulations of the nonlinear Schrödinger equation are complemented by a variational collective coordinate approach. The mechanism for nonlinear trapping is found to rely both on resonant interaction between the soliton and bound states in the potential well and on the radiation of small-amplitude waves. These results suggest that solitons can be used to probe bound states that are not accessible through scattering with single atoms.

Figure 1

A soliton being scattered on a rectangular well with width $2a$ and depth $V_0$.

Figure 2

Reflection ($R$), transmission ($T$), and trapping ($L$ for localized) as a function of $V_0$ and a fixed velocity $v=0.3$ by solving Eq. (2). The lower panel zooms into one of the structures shown in the upper panel. The well width $2a=1$ is kept constant and we use $A=1$ and $g=-1$. The same parameters are used throughout the paper unless explicitly mentioned otherwise.

Figure 3

The upper panel shows a snapshot of the condensate density at $t=77$ for $V_0=5.2$, where trapping is maximized in the second $\mathit{RL}$ window. One can see the trapped mode in the first excited bound state (see text) and the radiation that stabilizes the trapped soliton. The lower panel shows partial trapping with a reflected soliton at $t=65$.

Figure 4

Time and spatial dependence of the condensate density $|\psi (x,t)|{}^2$ in grayscale (normalized to a maximum amplitude of 1) for four different $V_0$ as in Fig. 2 but with $l_{\mathrm{box}}=40$. (a) The case of full reflection, (b) the case of full transmission, (c) a fully trapped soliton, and (d) the case of partial trapping and reflection (the additional reflection toward the end comes from the hard wall boundary conditions).

Figure 5

$L(V_0)$ (solid line) from the time-independent solutions and $N_{L,\mathrm{rel}}(V_0)$ (dashed line) from the time-independent calculations for a fixed chemical potential $\mu =-\frac{1}{2}$. Both quantities show similar behavior, even for different potential widths and different bound states.

Figure 6

Comparison of $T$ for solitons ($g=-1$, solid line) with the analytical solution $T_{\mathrm{lin}}(V_0)$ [Eq. (8)] for linear waves ($g=0$, dashed line). From top to bottom we increased $v_{\mathrm{initial}}$ of the incoming soliton. For increasing velocities, the kinetic term in the Gross-Pitaevskii equation becomes dominant and therefore both curves approach each other.

Figure 7

The phase difference $\Delta \Phi$ for different potential depths and for the time when the incoming soliton reaches the quantum well at $t\approx 30$ from solutions of Eqs. (15). The solid line and the dotted line clearly show a phase difference close to 0 where transmission or trapping, respectively, occurs. An example for reflection with a turning point at $t\approx 29$ is given by the dashed line where $\Delta \Phi \approx \pm \pi$ (for initial conditions see text).

Figure 8

Results for soliton scattering on a quantum well from the collective coordinate Eqs. (15). From top to bottom: reflection $R$ and transmission $T$ versus $V_0$, the total trapping $L_{\mathrm{total}}$, fraction $L_t$ in the trapping mode, and fraction $L_s$ in the soliton mode.

Figure 9

Condensate density as a function of time by solving Eqs. (15), analogous to Fig. 4. For $V_0=0.2$, panel (a) shows the typical situation of a fully transmitted soliton. Partial trapping at $V_0=0.7$ is shown in panel (b). Panels (c) and (d) present the cases for $V_0=0.38$ and $V_0=0.78$. The soliton is being trapped for (c) due to the population of the trapped mode and sloshes around the well for (d) while populating the soliton mode only. The last plot (e) for $V_0=1.75$ shows the whole soliton being completely reflected.

Figure 10

Transmission $T_{\mathrm{rel}}(V_0)=T(V_0)/T(4.85)$ and velocity contribution to the kinetic energy, $E_{\mathcal{kin},\mathcal{rel}}^v(V_0)=E_{\mathrm{kin}}^v(V_0)/E_{\mathrm{kin}}^v(4.85)$, relative to the values at $V_0=4.85$ where transmission is maximal. As the transmitted fraction of the incoming soliton decreases for deeper wells, so does its velocity $v_t$.

Figure 11

Time dynamics of different kinetic and interaction energies for the transmitted part of the soliton for $V_0=4.9$ (solid line) and $V_0=5.1$ (dashed line) well after the collision ($t\approx 45$). The soliton undergoes breathing oscillations after the scattering process that show up as oscillations in the energy. For deeper wells, $E_{\mathrm{kin}}^v$ becomes smaller (i.e., the velocity of the transmitted soliton $v_t$ is smaller).

Figure 12

Logarithmic density plot of the condensate for different times at $V_0=4.9$. A reflected component (radiation) is clearly visible.

Figure 13

Logarithmic density plot of the condensate for different times at $V_0=5.1$. Compared to Fig. 12, the reflected part is clearly larger.

Figure 14

Time dynamics of different kinetic and interaction energies for the reflected part of the soliton for $V_0=4.9$ and $V_0=5.1$. The ratio $|E_{\mathrm{int}}/E_{\mathrm{kin}}^d|\ll 1$, unlike what one would expect from a soliton. Because the interaction energy is small, the reflected part is mainly radiation. This is only valid in the transmission window; the reflected fraction is a soliton again for the case of trapping.

Figure 15

Plot showing the maximum amount of atoms $L_{\max }$ that can be trapped around the second transmission resonance under variation of the well depth $V_0$.

Figure 16

Density plot for (a) $V_0=5.182$ and (b) $V_0=5.183$ ($l_{\mathrm{box}}=40$). In part (a), the soliton decelerates and remains at the edges of the well before it continues to move to the right-hand side. In (b) the soliton slows down and remains at the edges of the well before it decides to move back to get trapped by the well.

Figure 17

Time delay of the soliton due to the existence of the potential well (solid line). These are results for the simulations in Sec. 3. Near the transition point from full reflection to full transmission (left vertical dashed line), the time delay increases very fast and becomes very large. This means the soliton remains in the vicinity of the trap for a very long time scale. Within the transmission region it decreases again until the trapping mechanism kicks in (right vertical dashed line). This picture for the nonlinear regime differs from the analytically calculated time delay for the linear case (dashed-dotted line) not only in the position and value of its maximum but also for the nonexistent negative time delay (right of the dotted vertical line).

Figure 18

Density plot for $V_0=4.842$ ($l_{\mathrm{box}}=40$). The soliton decelerates and more than 99% of the initial soliton is being trapped for a long time period (see also Fig. 17). This is connected to temporal trapping of a linear wave packet.

Figure 19

The scaled number of trapped atoms $N_L$ vs $\mu$ from time-dependent simulations of Eq. (2) is compared with $N_S(\mu )$ for stationary solutions from Eq. (30). Stationary solutions were also found with our numerical code for checking numerical accuracy. Results taken over a range of initial velocities $v_i$ show a consistent picture. The most significant deviations occur at the onset of trapping around the location of the linear bound state at $E_b=-0.5$. This feature is most clearly distinguished for the smallest velocities.

Figure 20

Scaled particle number of the trapped component $N_L$ vs $\mu$ after soliton-defect scattering as in Fig. 19 for different values of the defect strength $V_0$. Short vertical lines indicate the energy of the bound state $E_b$ for each of the values of $V_0$. Square-root fits to the data (as explained in the text) provide estimates $\beta$ for $E_b$ from the scattering data. Values found for $\beta$ are $-$0.120, $-$0.505, $-$1.12, $-$1.97, $-$3.08, and $-$4.43, which correspond to the analytical values for $E_b$ given by $-$0.125, $-$0.5, $-$1.125, $-$2, $-$3.125, and $-$4.5, respectively.