Experimental observation of Loschmidt time reversal of a quantum chaotic system

We have performed an experiment to demonstrate the approximate time reversal of a “chaotic” time evolution of atomic de Broglie waves. We use ultracold atoms from a Bose-Einstein condensate in a quantum $\delta$-kicked rotor experiment, and show that an initial state can be approximately re-created even after a period of “chaotic” evolution (a number of kicks). As this mechanism only works for a very narrow range of momenta, the net effect is a narrowing of the momentum distribution after the kick sequence.

Figure 1

The simulated momentum distribution after the kick sequence (solid line) and the initial wave packet (dashed line) are shown. The parameters are $N=10$, ${\varphi }_d~2.5$, and $\epsilon =2$. The inset shows a magnified view of the area around $p=0$.

Figure 2

(Top) Absorption images of the momentum distribution before and after each kick (1–10). (Bottom) The momentum distribution of the atoms at $N$ kicks (bottom left) for the experiment (solid line) and the simulation (dashed line). The central part of the same curve is also shown (bottom right), with the initial state represented by the dotted red line. The parameters are $N=10$, ${\varphi }_d\approx 2$, and $\epsilon =1$.

Figure 3

The normalized height of the zero-momentum peak $P$(0), as obtained from the numerical simulation (circles) and from the experiment (stars). Parameters are $\epsilon =1$ and (a) ${\varphi }_d~2$ and (b) ${\varphi }_d~3$.

Figure 4

Width of the zero momentum peak from experiment (diamonds) and simulation (circles) for a range of $\epsilon$ values. Also shown (inset) is a theoretical simulation of the central part of the momentum distribution for the same range of $\epsilon$ values with no convolution applied. Other parameters are $N=10$ and ${\varphi }_d~2$.