Dynamic localization in optical and Zeeman lattices in the presence of spin-orbit coupling

The dynamic localization of a two-level atom in a periodic potential under the action of spin-orbit coupling and a weak harmonically varying linear force is studied. We consider optical and Zeeman potentials that are either in phase or out of phase in two spinor components, respectively. The expectation value for the position of the atom after one oscillation period of the linear force is recovered in authentic resonances or in pseudoresonances. The frequencies of the linear force corresponding to authentic resonances are determined by the band structure of the periodic potential and are affected by the spin-orbit coupling. The width or dispersion of the wave packet in authentic resonances is usually minimal. The frequencies corresponding to pseudoresonances do not depend on the type of potential and on the strength of the spin-orbit coupling, while the evolution of excitations at the corresponding frequencies is usually accompanied by significant dispersion. Pseudoresonances are determined by the initial phase of the linear force and by the quasimomentum of the wave packet. Due to the spinor nature of the system, the motion of the atom is accompanied by periodic, but not harmonic, spin oscillations. Under the action of spin-orbit coupling the oscillations of the wave packet can be nearly completely suppressed in optical lattices. Dynamic localization in Zeeman lattices is characterized by doubling of the resonant oscillation periods due to band crossing at the boundary of the Brillouin zone. We also show that higher harmonics in the Fourier expansion of the energy band lead to effective dispersion, which can be strong enough to prevent dynamic localization of the Bloch wave packet.

Figure 1

Transformation of the (a) first and (b) second bands in the spectrum for an OL with growing $\gamma$. In (a) $\gamma =0$ for curve 1, $\gamma =1.17$ for curve 2, and $\gamma =2.15$ for curve 3. In (b) $\gamma =0$ for curve 1, $\gamma =0.82$ for curve 2, and $\gamma =1.84$ for curve 3. The vertical shift of the bands with increasing $\gamma$ were eliminated by subtracting ${\mu }_n(k=1)$ from the ${\mu }_n\left(k\right)$ dependence. (c) First two bands in the spectrum for a ZL at $\gamma =2$.

Figure 2

(a) Width of the first (curve 1) and second (curve 2) bands in an OL vs $\gamma$. (b) Width of the combined band in a ZL vs $\gamma$.

Figure 3

Resonant curves showing (a), (c) position of the integral center of the wave packet and (b) its width at $t=6\pi /\omega$ vs modulation frequency $\omega$ in the OL. (a), (b) correspond to the first-band excitation at $\gamma =0.5$ (curve 1), $\gamma =1.17$ (curve 2), and $\gamma =2.15$ (curve 3). (c) corresponds to the second-band excitation at $\gamma =0$ (curve 1), $\gamma =0.82$ (curve 2), and $\gamma =1.84$ (curve 3). Red dots show resonances where there is no drift and broadening of the wave packet, while black dots show resonances where wave packet broadens, but does not drift. Green dots mark frequencies at which largest displacement or broadening occurs between two first groups of resonances. In all cases $d(t=0)=7\pi$. Notice the logarithmic scale in the horizontal axis.

Figure 4

Evolution dynamics for the first-band excitations in an OL at (a) $\omega =0.0248, \gamma =0.5$, (b) $\omega =0.0187, \gamma =0.5$, (c) $\omega =0.0135, \gamma =0.5$, and (d) $\omega =0.0135, \gamma =1.17$. Only $|{\psi }_1|$ component is shown. The second row shows the corresponding evolution of the spin components. For illustrative purposes the magnitude of the $S_1$ component was divided by five. In all cases $d(t=0)=7\pi$. In (a), (b), and (d) the evolution time corresponds to four complete oscillation periods, while in (c) it corresponds to three oscillation periods.

Figure 5

Maximal width of the wave packet $d^{\max }$ and its maximal displacement $x_c^{\max }$ between the two highest authentic resonances at $t=6\pi /\omega$ vs $\gamma$ for the (a) first-band and (b) second-band excitations in the OL. In all cases $d(t=0)=7\pi$. (c) Width of the wave packet at $t=6\pi /\omega$ in the first (curve 1) and second (curve 2) resonances vs $\gamma$ for the first-band excitation in the ZL. In all cases $d(t=0)=7\pi$.

Figure 6

Resonant curves showing position of the integral center of broad wave packet from the first band versus modulation frequency in the OL at $\gamma =0.5$ (curve 1) and $\gamma =2.15$ (curve 2). In (a) $\beta =1/11\pi$ and lattice depth is 4, while in (b) $\beta =1/17\pi$ and lattice depth is 8. Red dots–authentic resonances, black dots–pseudoresonances.

Figure 7

Resonant curves showing (a) position of the average center of the wave packet and (b) its width at $t=6\pi /\omega$ vs modulation frequency $\omega$ at $\gamma =2$ in the ZL. First-band (curve 1) and second-band (curve 2) excitations are shown in (a), while in (b) only first-band excitation is shown. Dots indicate resonances for the first-band excitation, which differ from resonances for second-band excitation. In all cases $d(t=0)=7\pi$. The largest resonance 0.04822 (see Table 2) is not shown.

Figure 8

Evolution dynamics for the first-band excitations at (a) $\omega =0.0153$, (b) $\omega =0.0097$, (c) $\omega =0.0070$, and of the second-band excitation at (d) $\omega =0.0097$ in the ZL at $\gamma =2$. Only ${\psi }_1$ component is shown. Second row shows corresponding expectation values of the spin components. In all cases $d(t=0)=7\pi$. In (b)–(d) evolution time corresponds to two complete oscillation periods, while in (a) it corresponds to eight oscillation periods.