Manipulating matter waves in an optical superlattice

We investigate the potential for controlling a noninteracting Bose-Einstein condensate loaded into a one-dimensional optical superlattice. Our control strategy combines Bloch oscillations, originating from accelerating the lattice, with time-dependent control of the superlattice parameters. We investigate two experimentally viable scenarios, very low and very high potential depths, in order to gain a better understanding of matter wave control available within the system. Multiple lattice parameters and a versatile energy band structure allow us to obtain a wide range of control over energy band populations. Finally, we consider several examples of quantum state preparation in the superlattice structure that may be difficult to achieve in a regular lattice.

Figure 1

The top curve shows the modulus square of the initial wave function (1) localized into the lowest energy band with an initial Gaussian quasimomentum distribution with standard deviation $\sigma =2k_r/5$. The bottom curve is the potential (5) plotted on the same scale. Here the superlattice parameters are $A_1=3E_R$, $A_2=2E_R$, and $\varphi =0$.

Figure 2

(a) Superlattice plotted in real space. (b) Lowest four and (c) lowest five energy bands plotted in momentum space. Here $A_1=3E_R$, $A_2=2E_R$, and $\varphi =0$. (e)–(g) have the same potential depths but $\varphi =\pi /8$. (d) and (h) display the band structure for the simple lattice when $V_0=2E_R$ and $V_0=20E_R$, respectively.

Figure 3

Time evolution of the wave function ${\left|\psi (x,t)\right|}^2$ for one Bloch period when $A_1=3E_R$, $A_2=2E_R$, and $\varphi =0$ with $F=1\times {10}^{-4}{E}_R/4d$ (left) and $F=2\times {10}^{-4}{E}_R/4d$ (right). The initial quasimomentum distribution, localized in the lowest energy band, is described by a Gaussian function centered on $k=0$ with standard deviation $\sigma =2k_r/5$.

Figure 4

Shown on top is an example of transition widths: the lowest four energy bands in the superlattice for $A_1=A_2=2E_R$ and $\varphi =0$. The widths of the transitions (where interband transitions can occur) are depicted by the black boxes. On the bottom left is a diagram showing the range of parameters of $A_1$ and $A_2$ for which the transitions $1\to 2$ and $2\to 3$ do or do not overlap for $\varphi =0$ (red) and $\varphi =\pi /8$ (orange). The bottom right is a similar diagram for the transitions $2\to 3$ and $3\to 4$. Here $F=0.05E_R/4d$.

Figure 5

Example of population transition probabilities at each avoided crossing when $A_1=2E_R$ and $\varphi =\pi /8$. Shown are $T_{12}$ (red solid curve with circles) and $T_{34}$ (green dashed curve with squares) measured at $k=\pi /4d$ in the Brillouin zone and $T_{23}$ (blue dotted curve with diamonds) measured at $k=0$. The curves represent the theoretical predictions for these transitions from the Landau-Zener formula; the markers represent our simulations.

Figure 6

Gap between each band (${\mathrm{\Delta }}_{\alpha \beta }$) when $A_1=8E_R$ and $A_2=5E_R$, corresponding to a band structure with very flat bands, with changing $\varphi$. The lines shown are ${\mathrm{\Delta }}_{12}$ (blue solid line), ${\mathrm{\Delta }}_{23}$ (orange dotted line), ${\mathrm{\Delta }}_{34}$ (green dashed line), and ${\mathrm{\Delta }}_{45}$ (red dot-dashed line).

Figure 7

Tunneling between energy bands for a very deep potential. Each image has $A_1=8E_R$ and $A_2=5E_R$. The top left shows the population exchange between the first and second energy bands for $\varphi =\pi /8$. The top right shows the population transfer between the second and third bands when $\varphi =0$. The bottom left shows the transfer between the third and fourth bands with $\varphi =\pi /8$. The bottom right shows particles loaded into the second energy band and no two bands are coupled, with $\varphi =\pi /20$. Even after ten Bloch periods we retain $99.89\%$ in the second energy band. Small population transfer to the lowest energy band does occur as the bands are not mathematically flat.

Figure 8

Shown on top is the relevant band structure for the simulation in Sec. 4 lasting $1.6{\tau }_B^{\mathrm{super}}$ when $A_1=0.5=2A_2$ and $\varphi =\pi /8$. The BEC starts at $k=0$ in the lowest energy band. The bottom shows the band population probabilities plotted against time. The wave packet starts in the first energy band (blue solid line) before making complete transitions into the second band (yellow dotted line), the third band (green dashed line), and finally in the fourth energy band (red dot-dashed line). The evolution has been extended beyond the final time of $1.6{\tau }_B^{\mathrm{super}}$ to provide clarity on the line styles.

Figure 9

Shown on top is the relevant band structure for the simulation in Sec. 4 lasting ${\tau }_B^{\mathrm{super}}$ when $A_1=A_2=0.5E_R$ and $\varphi =\pi /8$. The bottom shows the band population probabilities plotted against time. The particles are initially populating the lowest energy band (blue solid line) transferring $100\%$ into the second band (yellow dotted line) and transferring $50\%$ into the third band (green dashed line). This plot moves slightly beyond ${\tau }_B^{\mathrm{super}}$ to illustrate the $50/50$ superposition.

Figure 10

Values of superlattice parameters we use for the process in Sec. 4d: $A_1$ (red squares) and $A_2$ (blue triangles) over time. The potential depth changes are performed at $t/{\tau }_B^{\mathrm{super}}=2.6$ and 3.2. The inset shows the relative phase $\varphi$ versus time. The changes are performed at $t/{\tau }_B^{\mathrm{super}}=2$, 2.6, 3.2, and 4.8.

Figure 11

Population probabilities of each energy band plotted against time in the simulation in Sec. 4. The wave packet begins in a state evenly spread across the lowest four energy bands and we dynamically manipulate its movement through the superlattice to regain $96\%$ into the lowest energy band (blue solid line). Time evolution of the second energy band population is shown in the yellow dotted line, the third energy band in the green dashed line, and the fourth in the red dot-dashed line.