Anomalous diffusion in a dynamical optical lattice

Motivated by experimental progress in strongly coupled atom-photon systems in optical cavities, we study theoretically the quantum dynamics of atoms coupled to a one-dimensional dynamical optical lattice. The dynamical lattice is chosen to have a period that is incommensurate with that of an underlying static lattice, leading to a dynamical version of the Aubry-André model which can cause localization of single-particle wave functions. We show that atomic wave packets in this dynamical lattice generically spread via anomalous diffusion, which can be tuned between superdiffusive and subdiffusive regimes. This anomalous diffusion arises from an interplay between Anderson localization and quantum fluctuations of the cavity field.

Figure 1

Schematic of the experimental setup. Atoms in an optical lattice and a standing-wave cavity are driven by a transverse laser. The frequency ${\omega }_{\mathrm{p}}$ of the pump laser is far detuned from the atomic transition line but close to the cavity-mode frequency ${\omega }_{\mathrm{c}}$.

Figure 2

(a) Phase diagram of the mean-field steady state. (b1) and (b2) Second-order phase transition along line $B$ in (a). (b1) Mean cavity field $\alpha$. (b2) Inverse partition ratio in real-space $p$. (c1) and (c2) First-order phase transition along line $C$ in (a). Here $L=201,\beta =(\sqrt{5}-1)/2,\varphi =\pi /2,\mathrm{\Delta }/J=1$, and $\kappa /J=1.2$. The square dots $p_i(i=1,...,5)$ represent the pumping strengths $\lambda /J=1.5,2.5,3.5,4.5,5.5$, respectively, and $U/J=3$.

Figure 3

(a1) and (a2) Time evolution of wave-packet width $\sigma \left(t\right)$ and phonon number $n_{\mathrm{p}}\left(t\right)$ from the mean-field dynamics. The dashed line is a guide for a ballistic ($\gamma =1$) or a saturated ($\gamma =0$) behavior. (b1) and (b2) Dissipationless nonequilibrium dynamics. (c1) and (c2) Dynamics in the large dissipation limit. The dashed line is a guide for diffusion $\gamma =1/2$. Here $\beta =(\sqrt{5}-1)/2,\varphi =\pi /2,\mathrm{\Delta }/J=1,\kappa /J=1.2$, and $U=0$.

Figure 4

(a) Wave-packet width and photon number evolution from the exact quantum trajectory method. The parameters are given by the point $p_4$ in Fig. 2. (b) Evolution of the wave-packet width. The dashed lines are the results of fitting to $\sigma =a{t}^{\gamma }$, and the corresponding numbers are the exponents. The two solid lines represent the ballistic $\gamma =1$ and diffusive $\gamma =1/2$ cases. (c) The exponent $\gamma$ crosses over from superdiffusion to subdiffusion with increasing pump strength. [The parameters of $p_i$ can be found in Fig. 2.] For comparison, the solid line is the mean-field result showing a transition from ballistic transport $\gamma =1$ to localization $\gamma =0$, whereas the dashed line marks the diffusive value $\gamma =1/2$.

Figure 5

Individual quantum trajectories of the stochastic evolution in: (a) the superdiffusive regime and (b) the subdiffusive regime. The solid line is the wave-packet width, and the dashed line is the photon number [multiplied by 20 in (a)]. Here $\beta =(\sqrt{5}-1)/2,\varphi =\pi /2,\mathrm{\Delta }/J=1,\kappa /J=1.2$, and $U=0$.