Transport of ultracold atoms in superpositions of $S$- and $D$-band states in a moving optical lattice

Ultracold atoms in a moving optical lattice with high controllability are a feasible platform to research the transport phenomenon. Here, we study the transport process of ultracold atoms at the $D$ band in a one-dimensional optical lattice and manipulate the transport of superposition states with different superposition weights of $S$-band and $D$-band atoms. In the experiment, we first load ultracold atoms into an optical lattice using the shortcut method and then accelerate the optical lattice by scanning the phase of lattice beams. The atomic transport at the $D$ band and $S$ band is demonstrated, respectively. The group velocity of atoms at the $D$ band is opposite to that at the $S$ band. By preparing superposition states with different superposition weights of the $D$-band and $S$-band atoms, we realize the manipulation of atomic group velocity from positive to negative, and observe the quantum interference between atoms at different bands. The influence of the lattice depth and acceleration on the transport process is also studied. Moreover, the multiorbital simulations are coincident with the experimental results. Our paper sheds light on the transport process of ultracold atoms at higher bands in optical lattices and provides a useful method to manipulate the transport of atomic superposition states.

Figure 1

Transport process of atoms in a moving optical lattice. (a1) and (a2) show the ultracold atoms in a moving optical lattice in the laboratory frame and moving frame, respectively. In (a1), the solid blue line represents the optical lattice, which is moving with acceleration $a_L$. The dashed blue line shows the motion of the optical lattice. $d_L$ is the lattice constant, and $V_0$ is the lattice depth. The arrow $x$ denotes the position coordinate. In (a2), $F=-m{a}_L$ is the inertia force, and $v_g^{\prime }$ is the group velocity of atoms in the moving frame. (b). The transport process of $D$-band atoms in the moving frame. The orange, yellow, blue, and purple lines (from bottom to top) represent the dispersion relation of $S$, $P$, $D$, and $F$ bands in quasimomentum space, respectively. The lattice depth is $10E_r$. The dashed arrows denote Bloch oscillation (BO) and Landau-Zener tunnel (LZT). The red ellipses denote the atoms performing Bloch oscillation at a single band. The blue ellipses denote the atoms which perform Landau-Zener tunneling to other bands or scattering to another edge of the first Brillouin zone (BZ). The dashed vertical lines show the edges of the first Brillouin zone. (c). The group velocity of atoms in the laboratory frame and moving frame. The dashed-dot orange line and solid blue line represent $v_g^{\prime }$ of atoms at $S$ band and $D$ band in the moving frame, and the dashed line denotes the velocity $-v_L=q/m$ of optical lattice. The distance between $-v_L$ and $v_g^{\prime }$ of the $S$ band and $D$ band shows the group velocity $v_g$ in the laboratory frame.

Figure 2

The experimental sequence and typical images. (a) The experimental sequence. The horizontal axis denotes the experimental time, and there are several stages of the experiment. The first stage is the preparation of the BEC. In the second stage, the shortcut method is used to prepare the atomic states in the optical lattice. $t_i^{\mathrm{on}\left(\mathrm{off}\right)}$ shows the pulse sequence of the shortcut method. In the third stage, the optical lattice is accelerated by changing the phase of lattice beam. After the time of flight 32 ms, the absorption imaging is taken to detect the atoms. $t_1\text{–}{t}_3$ are three typical moments: the BEC, the atoms loaded into the optical lattice, and the atoms after accelerating. (b) shows the typical images corresponding to the moments $t_1\text{–}{t}_3$. The above curves show the fitting of images. The horizontal axis represents the momentum $p$.

Figure 3

The transport process of atoms at the $S$ band and $D$ band. (a1) and (a2) are the absorption images of transport process of $S$-band and $D$-band atoms, respectively. The horizontal axis represents momentum $p$, and the vertical axis is time $t$. The color denotes the normalized density of atoms. (b) The blue (lower) data and orange (upper) data show the evolution of group velocity $v_g$ of the $D$-band atoms and $S$-band atoms. The circles are the experimental data, and the dashed lines are the simulations. The error bar denotes the standard error of five measurements. The parameters in the figure: $V_0=10E_r$, $a_L=15\mathrm{m}/{\mathrm{s}}^2$.

Figure 4

The group velocity $v_g$ for the different superposition weights of $D$ band $W_d$ over time. The top, middle, and bottom figures correspond to the case of $W_d=0.2$, 0.4, and 0.7, respectively. The dashed lines denote the theoretical superposition group-velocity curves, calculated by the multiorbital simulation method. The gray dotted lines denote the theoretical classical group velocity curves, calculated by weighted averaging of the $S$-band and $D$-band group velocity. The error bar denotes the standard error of five measurements. The parameters in the figure: $V_0=10E_r$, $a_L=15\mathrm{m}/{\mathrm{s}}^2$.

Figure 5

Transport process of $D$-band atoms with the different lattice depths $V_0$. (a) The green squares, red circles, and yellow diamonds represent the group velocity of $D$-band atoms in the Lab frame with $V_0=5,10,15E_r$, respectively. The dashed lines are the simulations. The dashed-dot line denotes half of one Bloch oscillation period $T_B/2$. The acceleration is $15\mathrm{m}/{\mathrm{s}}^2$. (b) The theoretical group-velocity $v_g^{\prime }$ of the $D$ band in the moving frame with different lattice depths. The dotted green, solid red, and dashed-dot yellow lines denote the $v_g^{\prime }$ with $V_0=5,10,15E_r$, respectively. The dashed line denotes the velocity $-v_L=q/m$ of the optical lattice. In (c), the blue (lower) points and the orange (upper) points represent transport distance $S_a$ for atoms at the $D$ band and $S$ band with $V_0=5,8,10,12,15E_r$, respectively. The dashed lines are the simulations. $d_L=\lambda /2$ is the lattice constant. The error bar denotes the standard error of five measurements.

Figure 6

Atomic transport distance $S_a$ with the different lattice depths $V_0$ and acceleration $a_L$. (a) The green squares, red circles, and yellow diamonds show transport distance $S_a$ with $V_0=5,10,15E_r$, $a_L=15\mathrm{m}/{\mathrm{s}}^2$, respectively. The purple triangles show transport distance $S_a$ with $V_0=10E_r$, $a_L=30\mathrm{m}/{\mathrm{s}}^2$. (b) shows the group velocity over time of $D$-band atoms in the accelerated optical lattice when $a_L=120\mathrm{m}/{\mathrm{s}}^2$ when $V_0=10E_r$. The dashed-dot line denotes half of one Bloch oscillation period $T_B/2$. The dashed lines are the simulations. The error bar denotes the standard error of five measurements.