Controlled decoherence in a quantum Lévy kicked rotator

We develop a theory describing the dynamics of quantum kicked rotators (modeling cold atoms in a pulsed optical field) which are subjected to combined amplitude and timing noise generated by a renewal process (acting as an engineered reservoir). For waiting-time distributions of variable exponent (Lévy noise), we demonstrate the existence of a regime of nonexponential loss of phase coherence. In this regime, the momentum dynamics is subdiffusive, which also manifests itself in a non-Gaussian limiting distribution and a fractional power-law decay of the inverse participation ratio. The purity initially decays with a stretched exponential which is followed by two regimes of power-law decay with different exponents. The averaged logarithm of the fidelity probes the sprinkling distribution of the renewal process. These analytical results are confirmed by numerical computations on quantum kicked rotators subjected to noise events generated by a Yule-Simon distribution.

Figure 1

(Color online) (a) Schematic illustration of the kicking sequence in the Lévy kicked rotator. The period of the kicks is set to unity. The timing of the noise is generated in a renewal process, which selects the integer-valued waiting times ${\tau }_n$ from a distribution $\omega \left(\tau \right)$. (b) Waiting-time distributions of the Yule-Simon form (24) for different values of the parameter $\alpha$. The distributions display power-law tails, specified in Eq. (25), and hence generate Lévy noise of variable exponent. The lines are a guide to the eye.

Figure 2

(Color online) Time dependence of the variance of momentum $\text{var}{p}_0\left(t\right)$ for a noiseless kicked rotator with regular kicking strength $K=7.5$. The dashed curve is the theoretical prediction (5) of dynamical-localization theory [13]. A single-parameter fit delivers the value $D^{\ast }={\hslash }^2{t}^{\ast }=45.28$, which is used throughout the rest of the present work. The inset zooms in onto the region where dynamical localization is established.

Figure 3

(Color online) Averaged distribution of momenta $P\left(p;t\right)$ at selected times $t$ for kicked rotators which are subjected to Lévy noise generated by a Yule-Simon distribution. The average is over ${10}^3$ realizations. The smooth curves are fits to double-sided exponential and Gaussian distributions. The unit of momentum is the localization length $p^{\ast }=D^{\ast }/\hslash$. (a) In the upper panels $\alpha =2.0$, $\kappa =1/30000$, corresponding to a decoherence time $t_c=12.3t^{\ast }$ (where $t^{\ast }$ is the localization time). At $t=15t^{\ast }\approx 1.2t_c$ the distribution function fluctuates around a double-sided exponential function. At $t=30t^{\ast }\approx 2.4t_c$ the distribution function is still exponential, but the fluctuations are suppressed. At $t=120t^{\ast }\approx 9.8t_c$ the distribution function is well described by a Gaussian. (b) In the lower panels $\alpha =0.5$, $\kappa =1/3$, corresponding to stronger but nonstationary, increasingly rare noise events. In this case, a decoherence time cannot be defined and the distributions function remains exponential for all times $t>t^{\ast }$. Note that the peak at $p=0$ arises from coherent backscattering.

Figure 4

(Color online) Same as the left-hand panels in Fig. 3, but for $\kappa =1/300$ $\left(t_c=0.12t^{\ast }\right)$, so that $t=15t^{\ast }⪢t_c$.

Figure 5

(Color online) Time dependence of the momentum spreading $\text{var}p\left(t\right)$ for kicked rotators which are subjected to Lévy noise generated by a Yule-Simon distribution with noise exponent $\alpha =2.0$ [panel (a)] and $\alpha =0.5$ [panel (b)]. The results of numerical computations (solid curves) are compared to the theoretical prediction from Eq. (49) (dashed curves). The straight solid line labeled “classical” is the classical diffusion.

Figure 6

(Color online) Time dependence of the inverse participation ratio IPR for kicked rotators which are subjected to Lévy noise generated by a Yule-Simon distribution with $\kappa =1/300$. The main panels are single logarithmic while the insets show the same data in a double-logarithmic presentation. (a) Noise with exponent $\alpha =2.0$, corresponding to a decoherence time $t_c=0.12t^{\ast }$ (where $t^{\ast }$ is the localization time) so that the momentum distribution $P\left(p;t\right)$ is always well approximated by a Gaussian. The dashed line is the quasiclassical prediction (58). (b) Noise with exponent $\alpha =0.5$, where for $t>t^{\ast }$ the momentum distribution $P\left(p;t\right)$ is always well approximated by a double-sided exponential function. The dashed line is the quasiclassical prediction (59).

Figure 7

(Color online) Time dependence of the purity $\text{tr}{\rho }^2$ of Lévy kicked rotators for different noise exponents $\alpha$ and fixed noise strength $\kappa =1/300$. The results of numerical simulations (solid curves) are compared to the theoretical predictions from Eq. (66) (dashed curves)

Figure 8

(Color online) Averaged logarithm of the fidelity of Lévy kicked rotators for different noise exponents $\alpha$ and fixed noise strength $\kappa =1/300$. The results of numerical simulations (solid curves) are compared to the theoretical predictions from Eq. (68) (dashed curves), which does not account for the saturation background.