Controlled wave-packet manipulation with driven optical lattices

Motivated by recent experimental progress achieved with ultracold atoms in kilohertz-driven optical lattices, we provide a theoretical discussion of mechanisms governing the response of a particle in a cosine lattice potential to strong forcing pulses with smooth envelope. Such pulses effectuate adiabatic motion of a wave packet's momentum distribution on quasienergy surfaces created by spatiotemporal Bloch waves. Deviations from adiabaticity can then be deliberately exploited for exerting coherent control and for reaching target states which may not be accessible by other means. As one particular example, we consider an analog of the $\pi$ pulses known from optical resonance. We also suggest adapting further techniques previously developed for controlling atomic and molecular dynamics by laser pulses to the coherent control of matter waves in shaken optical lattices.

Figure 1

Response of the initial wave packet (20) with momentum distribution (21) in an optical lattice with depth $V_0/E_{\mathrm{r}}=5.7$ to a short pulse (19) with nonresonant scaled frequency $\hslash \omega /E_{\mathrm{r}}=1.640$, maximum scaled amplitude $K_0^{\max }=0.8$, and pulse length $T_{\mathrm{P}}/T=10$. (a) shows the density $|g_1^{\mathrm{B}}{(k,t)|}^2$ in the crystal-momentum representation. For comparison, the white-dashed line is the first moment $k_{\mathrm{c}}\left(t\right)$, as predicted by the acceleration theorem (12). (b) depicts the Floquet density $|g_1{(k,t)|}^2$, obtained by expanding the same wave packet with respect to the instantaneous spatiotemporal Bloch waves.

Figure 2

Final escape probabilities from the lowest Bloch band of an optical lattice with depth $V_0/E_{\mathrm{r}}=5.7$, calculated for the same initial wave packet as considered in Fig. 1, after pulses with squared-sine envelope (22) and length $T_{\mathrm{P}}/T=50$. Observe the window of almost adiabatic response appearing between the two-photon and the single-photon resonances.

Figure 3

Final momentum distributions in the lowest (a) and in the first excited (b) Bloch bands after the initial state has been exposed to pulses with squared-sine envelope (22) and length $T_{\mathrm{P}}/T=50$, with varying maximum scaled amplitudes $0⩽K_0^{\max }⩽0.8$. Here the scaled driving frequency is $\hslash \omega /E_{\mathrm{r}}=4.690$, implying $\hslash \omega =E_2\left(0\right)-E_1\left(0\right)$, so that both bands are exactly resonant at $k/k_{\mathrm{L}}=0$. (c) compares the final distributions in the first (dotted) and in the second (dashed) bands to the initial distribution (full line), for $K_0^{\max }=0.186$. This particular situation corresponds to a “$\pi$ pulse.”

Figure 4

(a) Quasienergy surfaces underlying the excitation pattern observed in Fig. 3. The upper surface originates from the unperturbed Bloch band $n=1$, the lower one from the band $n=2$. Because of the resonant frequency, both surfaces are degenerate at $k/k_{\mathrm{L}}=0$ for vanishing instantaneous amplitude $K_0$. The quasienergy lines at $k/k_{\mathrm{L}}=0$ are emphasized for better visibility. (b) Section through the surfaces at $k/k_{\mathrm{L}}=0$, showing the removal of the initial degeneracy.

Figure 5

Floquet representation of the evolution of a wave packet initially prepared in the lowest Bloch band, under the action of a resonant $\pi$ pulse with $K_0^{\max }=0.186$. This figure shows how the final bimodal distribution $|g_1(k,T_{\mathrm{P}}){|}^2$ depicted in Fig. 3c appears after an initial reduction of the original density, corresponding to the partial occupation of the other resonantly coupled quasienergy surface; and after a period of almost adiabatic motion during the middle of the pulse.

Figure 6

Final momentum distributions in the lowest (a) and in the first excited (b) Bloch bands after the initial state has been exposed to pulses with squared-sine envelope (22) and length $T_{\mathrm{P}}/T=50$, with varying maximum scaled amplitudes $0.7⩽K_0^{\max }⩽1.3$. Here the nonresonant scaled driving frequency is $\hslash \omega /E_{\mathrm{r}}=1.640$, as in Fig. 1. (c) compares the final distributions in the first (dotted) and in the second (dashed) bands to the initial distribution (full line) for $K_0^{\max }=1.3$.

Figure 7

(a) Quasienergy surfaces underlying the excitation pattern observed in Fig. 6. The almost flat surface originates from the comparatively narrow Bloch band $n=1$, the one with larger curvature from the band $n=2$. Both surfaces undergo an avoided crossing along a parabola with apex at $k/k_{\mathrm{L}}=0$ and $K_0\approx 0.9$. The quasienergy lines at $k/k_{\mathrm{L}}=0$ are emphasized for better visibility. (b) Section through the surfaces at $k/k_{\mathrm{L}}=0$, showing the narrow avoided crossing.

Figure 8

Floquet representation of the evolution of a wave packet initially prepared in the lowest Bloch band, under the action of a nonresonant pulse with $K_0^{\max }=1.3$. After an initial period of almost adiabatic motion, the center of the distribution undergoes partial Landau-Zener transitions to the anticrossing surface depicted in Fig. 7, and later returns to the initial surface, subject to Stückelberg oscillations.

Figure 9

Contour plot of the real-space density $|{\psi }_1{(x,t)|}^2$ associated with the lowest Bloch energy band (a), and of the density $|{\psi }_2{(x,t)|}^2$ associated with the first excited band (b), as resulting from a pulse with $\hslash \omega /E_{\mathrm{r}}=1.640$, $K_0^{\max }=1.5$, and $T/T_{\mathrm{P}}=30$. The phase $\varphi$ in Eq. (27) has been set to zero. (c) shows the gradual loss of population from the lowest band.

Figure 10

As Fig. 9, but for $\varphi =\pi$.