Explanation and observability of diffraction in time

Diffraction in time (DIT) is a fundamental phenomenon in quantum dynamics due to time-dependent obstacles and slits. It is formally analogous to diffraction of light, and is expected to play an increasing role in the design of coherent matter wave sources, as in the atom laser, to analyze time-of-flight information and emission from ultrafast pulsed excitations, and in applications of coherent matter waves in integrated atom-optical circuits. We demonstrate that DIT emerges robustly in quantum waves emitted by an exponentially decaying source and provide a simple explanation of the phenomenon, as an interference of two characteristic velocities. This allows for its controllability and optimization.

Figure 1

Unnormalized density versus time at $x=1000$ for a constant or exponentially decaying source.

Figure 2

Normalized probability densities versus time for $k_0=1-0.0015i$ at $x=1000$. Exact (red solid line), approximate [Eq. (7); black dashed line], saddle term (blue circles), pole term (orange triangles), and interference term [Eq. (8); green squares].

Figure 3

Time intervals $T_{n+1}-T_n$ between two consecutive maxima: exact (circles), and approximation from Eq. (9) (boxes). Same parameters as in Fig. 2. The symbol size is to help the eye, and is not related to errors. (Inset) Nonlinear position of the first maximum versus time for $k_{0I}=-0.08$ (solid line) and onset of the pole term $x_c=2(1+k_{0I})t$ (dotted line).

Figure 4

Saddle and pole terms for $k_{0I}=-0.03$ (solid blue line) and $k_{0I}=-0.13$ (long-dashed red line). $x=80$. The inset shows the corresponding exact densities. The shorter lifetime suppresses DIT.

Figure 5

Density $|\mathrm{\psi ̃}(x,t)|{}^2$ for $k_{0I}=-0.003$ showing the transition from pure exponential decay to a DIT pattern.

Figure 6

Stability of DIT for decay from $U\delta (x)$ (Winter’s model). The initial states are the ground states of the two wells of the inset. At $t=0$ the right wall is substituted by the delta. $L=3.14$, $x=157.05$, $U=161.35$, $V=202.72$.