Return to the Origin as a Probe of Atomic Phase Coherence

We report on the observation of the coherent enhancement of the return probability [“enhanced return to the origin” (ERO)] in a periodically kicked cold-atom gas. By submitting an atomic wave packet to a pulsed, periodically shifted, laser standing wave, we induce an oscillation of ERO in time that is explained in terms of a periodic, reversible dephasing in the weak-localization interference sequences responsible for ERO. Monitoring the temporal decay of ERO, we exploit its quantum-coherent nature to quantify the decoherence rate of the atomic system.

Figure 1

Experimental observation of enhanced return to the origin. (a) Momentum distribution $\mathrm{\Pi }(p,t)$ at an even ($t=20$, red) and an odd ($t=21$, blue) kick. The distribution around $p=0$ at $t=20$ is enhanced with respect to the distribution at $t=21$, as evidenced by the difference signal. (b) The zero-momentum population ${\mathrm{\Pi }}_0$ vs $t$ shows a clear oscillation between even kicks (red circles) and odd kicks (blue squares). The contrast attenuation is due to decoherence. $K=12$, $\mathit{ҟ}=1.5$, $a=0.04$.

Figure 2

Paths in momentum space at the origin of ERO. A path is formed of free evolution between kicks (horizontal segments) and momentum changes at kicks (vertical segments). Momentum changes at odd kicks are associated with an $a$-dependent phase, see Eq. (3). The green-dotted REV (reversed) paths are the time reversal of the black-solid DIR (direct) ones and should be read from right to left (green time scale on the top). (a) For four kicks, the accumulated phases for DIR and REV paths are identical: ${\mathrm{\Phi }}_{\mathrm{DIR}}={\mathrm{\Phi }}_{\mathrm{REV}}=3a+2a=2a+3a=5a$, allowing constructive interference leading to the ERO peak. (b) For five kicks, the accumulated phases for DIR and REV paths are reversed, ${\mathrm{\Phi }}_{\mathrm{DIR}}=a+a+3a=5a$, ${\mathrm{\Phi }}_{\mathrm{REV}}=-3a-a-a=-5a$, and the ERO peak is suppressed.

Figure 3

Zero-momentum probability density ${\mathrm{\Pi }}_0$ vs $t$. A decoherer beam is applied between the 21st and 23rd pulses (green-shadowed region), quenching the oscillations; $K=12$, $\mathit{ҟ}=1.5$, and $a=0.04$. The decoherer beam induces an average of $N_{\mathrm{sp}}=2$ spontaneous emission events per atom. The inset shows the reduction in the difference signal ${\mathrm{\Delta }}_t$ as a function of $N_{\mathrm{sp}}$; the black line is the expected exponential decay $\exp (-N_{\mathrm{sp}})$ (it is not a fit).

Figure 4

Probing decoherence with ERO. Inset: the decay of the difference signal ${\mathrm{\Delta }}_t$ vs $t$ is fitted by an exponential (black line) from which the decoherence rate $\mathrm{\Gamma }$ is extracted. In the absence of externally applied decoherence, this gives the bare decoherence rate ${\mathrm{\Gamma }}_0$. This procedure is repeated in the presence of the decoherer beam for several values of the imposed decoherence rate ${\mathrm{\Gamma }}_{\mathrm{ext}}$. The fact that the excess rate $\mathrm{\Gamma }-{\mathrm{\Gamma }}_0$ measured using the decay of the ERO signal agrees perfectly with the externally added rate ${\mathrm{\Gamma }}_{\mathrm{ext}}$ shows that ERO is a faithful measure of decoherence.