Comparison between two models of absorption of matter waves by a thin time-dependent barrier

We report a quantitative, analytical, and numerical comparison between two models of the interaction of a nonrelativistic quantum particle with a thin time-dependent absorbing barrier. The first model represents the barrier by a set of time-dependent discontinuous matching conditions, which are closely related to Kottler boundary conditions used in stationary-wave optics as a mathematical basis for Kirchhoff diffraction theory. The second model mimics the absorbing barrier with an off-diagonal $\delta$ potential with a time-dependent amplitude. We show that the two models of absorption agree in their predictions in a semiclassical regime, the regime readily accessible in modern experiments with ultracold atoms.

Figure 1

Maximal deviations of the fidelity $M$ (red squares) and probability ratio $R$ (blue circles) from 1 for four different alkali-metal atoms. (a) Data represent the case of a time-independent barrier with $\mathrm{\Omega }\left(\tau \right)={\mathrm{\Omega }}_0$; the maximal deviations are computed with respect to ${\mathrm{\Omega }}_0$. (b) Data correspond to $\mathrm{\Omega }\left(\tau \right)={\mathrm{\Omega }}_1/\tau$; the maximal deviations are computed with respect to ${\mathrm{\Omega }}_1$. See the text for all parameter values.

Figure 2

Aperture function $\chi \left(\tau \right)$ and the corresponding atom-laser interaction strength $\mathrm{\Omega }\left(\tau \right)$ in three different scenarios: (a) $\chi \left(\tau \right)=min\{{\chi }_{>}\left(\tau \right),1\}$, with ${\chi }_{>}\left(\tau \right)\equiv exp\left[\gamma (\tau -3t_c/2)\right]$ and $\gamma =100{\mathrm{s}}^{-1}$; (b) $\chi \left(\tau \right)=min\{{\chi }_{<}\left(\tau \right),1\}$, with ${\chi }_{<}\left(\tau \right)\equiv exp\left[\gamma (\tau -t_c/2)\right]$ and $\gamma =-100{\mathrm{s}}^{-1}$; (c) $\chi \left(\tau \right)=min\{{\chi }_{\wedge }\left(\tau \right),1\}$, with ${\chi }_{\wedge }\left(\tau \right)\equiv cosh\left[\gamma (\tau -t_c)\right]/cosh(\gamma t_c/2)$ and $\gamma =225{\mathrm{s}}^{-1}$; (d) $\mathrm{\Omega }\left(\tau \right)={\mathrm{\Omega }}_{>}\left(\tau \right)\equiv v_0\sqrt{1/{\chi }_{>}\left(\tau \right)-1}$ for $\tau <3t_c/2$ and $\mathrm{\Omega }\left(\tau \right)=0$ for $\tau \ge 3t_c/2$; (e) $\mathrm{\Omega }\left(\tau \right)=0$ for $\tau \le t_c/2$ and $\mathrm{\Omega }\left(\tau \right)={\mathrm{\Omega }}_{<}\left(\tau \right)\equiv v_0\sqrt{1/{\chi }_{<}\left(\tau \right)-1}$ for $\tau >t_c/2$; and (f) $\mathrm{\Omega }\left(\tau \right)={\mathrm{\Omega }}_{\wedge }\left(\tau \right)\equiv v_0\sqrt{1/{\chi }_{\wedge }\left(\tau \right)-1}$ for $|\tau -t_c|<t_c/2$ and $\mathrm{\Omega }\left(\tau \right)=0$ for $|\tau -t_c|\ge t_c/2$.

Figure 3

Probability densities $|{\mathrm{\Psi }}_{\mathrm{AFM}}{|}^2$ (solid curves) and $|{\mathrm{\Psi }}_{\mathrm{DPM}}{|}^2$ (dash-dotted curves) for (a) ${}^7\text{Li}$, (b) ${}^{23}\text{Na}$, (c) ${}^{41}\text{K}$, and (d) ${}^{87}\text{Rb}$ as functions of the position $x$. Blue curves correspond to the barrier specified in Figs. 2 and 2. Red curves correspond to the barrier specified in Figs. 2 and 2. The dotted green curve represents the scaled probability density of the corresponding free-particle wave packet.

Figure 4

Probability densities $|{\mathrm{\Psi }}_{\mathrm{AFM}}{|}^2$ (solid red curve) and $|{\mathrm{\Psi }}_{\mathrm{DPM}}{|}^2$ (dash-dotted blue curve) for (a) ${}^7\text{Li}$, (b) ${}^{23}\text{Na}$, (c) ${}^{41}\text{K}$, and (d) ${}^{87}\text{Rb}$ as functions of the position $x$. The curves correspond to the barrier specified in Figs. 2 and 2. The dotted green curve represents the scaled probability density of the corresponding free-particle wave packet.

Figure 5

Probability densities $|{\mathrm{\Psi }}_{\mathrm{AFM}}{(x,t)|}^2$ (solid red curve) and $|{\mathrm{\Psi }}_{\mathrm{DPM}}{(x,t)|}^2$ (dash-dotted blue curve) for (a) ${}^7\text{Li}$, (b) ${}^{23}\text{Na}$, (c) ${}^{41}\text{K}$, and (d) ${}^{87}\text{Rb}$ evaluated for the case of the Moshinsky shutter [Eq. (66)]. The dotted green curve represents the probability density of the corresponding free-particle wave packet.