Suppression and Revival of Weak Localization through Control of Time-Reversal Symmetry

We report on the observation of suppression and revival of coherent backscattering of ultracold atoms launched in an optical disorder in a quasi-2D geometry and submitted to a short dephasing pulse, as proposed by Micklitz, Müller, and Altland [Phys. Rev. B 91, 064203 (2015)]. This observation demonstrates a novel and general method to study weak localization by manipulating time reversal symmetry in disordered systems. In future experiments, this scheme could be extended to investigate higher order localization processes at the heart of Anderson (strong) localization.

Figure 1

Principle of CBS suppression and revival following a dephasing kick. (a) CBS results from the constructive interference between the scattering amplitudes of direct (1, solid line) and reciprocal (2, dashed) multiple scattering paths, for a plane matter wave launched with a wave vector ${\mathbf{k}}_{\mathrm{i}}$ and detected in the backward direction, at ${\mathbf{k}}_{\mathrm{f}}=-{\mathbf{k}}_{\mathrm{i}}$. The time diagram shows the switching on at 0, and off after time $t$ of the disordered potential (amplitude $V_R$). (b) CBS suppression: A pulsed perturbation at time $T\ne t/2$ entails a phase difference [${\varphi }_{\text{kick}}({\mathbf{r}}_1)\ne {\varphi }_{\text{kick}}({\mathbf{r}}_2)$] between the amplitudes associated with paths 1 and 2, where ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ are the positions on each path at time $T$. The constructive interference is destroyed. (c) CBS revival: For the special case $T=t/2$, one has ${\mathbf{r}}_1={\mathbf{r}}_2$ and the constructive interference between the direct and reciprocal paths is restored. A CBS revival is then expected at $t=2T$.

Figure 2

Generalization of the revival mechanism for strong momentum changes. All multiple scattering paths that participate to the momentum distribution at ${\mathbf{k}}_{\mathrm{f}}=-{\mathbf{k}}_{\mathrm{i}}$ must conserve the modulus of the momentum at the kick: $|\mathbf{k}(T_{+})|=|\mathbf{k}(T_{-})+\mathrm{\Delta }\mathbf{k}|=|\mathbf{k}(T_{-})|$, where $T_{-}$ and $T_{+}$ are the times just before and after the kick (see inset). For the observation time $t=2T$, each direct path (solid line) satisfying that condition has a time-reversed counterpart (dashed line) satisfying the same condition, and for which the momentum change happens at the same position. The constructive interference between these reciprocal paths leads to the CBS revival at ${\mathbf{k}}_{\mathrm{f}}=-{\mathbf{k}}_{\mathrm{i}}$, ideally with a contrast of unity.

Figure 3

Experimental setup. (a) and (b) A cloud of ultracold atoms (orange disk) is launched upwards, along the $z$ axis, with an initial momentum $\hslash {\mathbf{k}}_{\mathrm{i}}$. The disordered potential is created by an anisotropic laser speckle, elongated along the $x$ axis, obtained by passing a laser beam through a scattering plate (shaded blue). The atoms are suspended against gravity by magnetic levitation provided by the pair of large coils (yellow). The small red coil is used to create a magnetic gradient pulse resulting in a downward momentum change $\hslash \mathrm{\Delta }\mathbf{k}$, along the $z$ axis, to the atoms. After a time of flight of 150 ms, fluorescence imaging recorded by an EMCCD camera along $x$ gives the transverse momentum distribution (in the $y-z$ plane, see text). (c) 3D false color representation of the disordered potential.

Figure 4

Observation of the CBS revival. The images represent the 2D k-vector distribution in the $y-z$ plane, $n(\mathbf{k},t)$. Each image results from an average over 30 experimental runs and the color scale is kept unchanged, except for the first two images in (a). (a) Initial evolution of $n(\mathbf{k},t)$ for atoms launched at $t=0$ in the disorder with momentum $\hslash {\mathbf{k}}_{\mathrm{i}}$. The initial peak at ${\mathbf{k}}_{\mathrm{i}}$ decays while the atoms’ $\mathbf{k}$-vectors are redistributed on a ring of radius $|k_{\mathrm{i}}|$, and the CBS peak grows at $-{\mathbf{k}}_{\mathrm{i}}$. (b) and (c) Evolution for $t>T$, without and with the dephasing kick at $T=1\mathrm{ms}$. The dashed circles are centered on the origin, and have a radius $|k_{\mathrm{i}}|$. The amplitude of the kick $\mathrm{\Delta }k$ is measured from the first image in (c). The rectangular boxes below each image show the extracted coherent fraction ${\mathcal{C}}_{\mathrm{coh}}(\mathbf{k},t)$ around $-{\mathbf{k}}_{\mathrm{i}}$ (see text). (d) and (e) Coherent fractions around $-{\mathbf{k}}_{\mathrm{i}}$, as in (b) and (c), but renormalized by the reference CBS peak at the same time (see text).

Figure 5

Amplitude of the CBS revivals ${\gamma }_{\mathrm{rev}}(t)$ for various choices of the dephasing times $T$. The data points triangle, circle, square, diamond correspond to $T=0.7$, 1.0, 1.3, and 1.6 ms, respectively, the kick’s strength $\mathrm{\Delta }k$ being fixed (same as in Fig. 4). The dotted vertical lines indicate the times $2T$ when the revivals are expected. Solid lines are Gaussian fits with zero offset. The observed revival times, determined by the fits, are, respectively, (in ms): $1.3\pm 0.08$, $1.99\pm 0.08$, $2.65\pm 0.05$, and $3.18\pm 0.09$, in good agreement with the predicted values. Uncertainties correspond to the 95% confidence intervals.