Long-time coherence in echo spectroscopy with $\pi /2$-$\pi$-$\pi /2$ pulse sequence

Motivated by atomic optics experiments, we investigate a class of fidelity functions describing the reconstruction of quantum states by time-reversal operations as $M_{\mathrm{Da}}(t)=|\langle \psi |e^{i{H}_{2}t/2}{e}^{i{H}_{1}t/2}{e}^{-i{H}_{2}t/2}{e}^{-i{H}_{1}t/2}|\psi \rangle |{}^2$. We show that the decay of $M_{\mathrm{Da}}$ is quartic in time at short times and that it freezes well above the ergodic value at long times, when $H_2-H_1$ is not too large. The long-time saturation value of $M_{\mathrm{Da}}$ contains easily extractable information on the strength of decoherence in these systems.

Figure 1

Trajectories of the unperturbed (${\gamma }_1$) and perturbed (${\gamma }_2$, ${\gamma }_2^{\prime }$, and ${\gamma }_2^{″}$) systems together with the associated action differences ($\Delta S$, $\Delta S^{\prime }$, and $\Delta S^{″}$).

Figure 2

Average echo decay for the kicked rotator model with $K_1=57$, $N=8192$, $K_2-K_1=5\times {10}^{-5}$, $1.2\times {10}^{-4}$, $2.1\times {10}^{-4}$, and $5\times {10}^{-4}$ ($M_{\mathrm{L}}$, red solid lines from top to bottom), and $K_2-K_1=5\times {10}^{-5}$, $9\times {10}^{-5}$, $1.2\times {10}^{-4}$, $1.6\times {10}^{-4}$, $2.1\times {10}^{-4}$, $3.1\times {10}^{-4}$, and $5\times {10}^{-4}$ ($M_{\mathrm{Da}}$, black dashed lines, from top to bottom). Curves are averages over 500 initial states.

Figure 3

Long-time saturation value of $M_{\mathrm{Da}}$ for $K_1=57$ and $N=256$ (black up-pointing triangles), 512 (red circles), 1024 (green squares), 2048 (blue down-pointing triangles), 4096 (violet diamonds), and 8192 (cyan left-pointing triangles). (Main panel) Rescaled data confirming the analytical prediction of Eq. (20). The straight black line indicates a slope of $\propto$$1/x^{3.8}$. (Inset) Raw data as function of the difference of dimensionless kicking strengths $\delta \mathrm{K̃}={\mathrm{K̃}}_2-{\mathrm{K̃}}_1$, with ${\mathrm{K̃}}_{1(2)}=K_{1(2)}/\tau$.