Lattice of double wells for manipulating pairs of cold atoms

We describe the design and implementation of a two-dimensional optical lattice of double wells suitable for isolating and manipulating an array of individual pairs of atoms in an optical lattice. Atoms in the square lattice can be placed in a double well with any of their four nearest neighbors. The properties of the double well (the barrier height and relative energy offset of the paired sites) can be dynamically controlled. The topology of the lattice is phase stable against phase noise imparted by vibrational noise on mirrors. We demonstrate the dynamic control of the lattice by showing the coherent splitting of atoms from single wells into double wells and observing the resulting double-slit atom diffraction pattern. This lattice can be used to test controlled neutral atom motion among lattice sites and should allow for testing controlled two-qubit gates.

Figure 1

2D lattices with four beams. (a) Lattices formed by interfering two independent standing waves must be actively stabilized to be topologically phase stable against phase noise caused by vibration of mirrors. (b) Lattices formed from a folded retroreflected beam have intrinsic topological phase stability.

Figure 2

Calculated intensities for in-plane lattice (a) and the out-of-plane lattice (b). Cross sections taken on the white dashed line are shown below their respective plot; a cross is used to denote the origin in each plot. The in-plane lattice has the familiar ${\cos }^2$ profile typical of $\lambda ∕2$ lattices, while the out-of-plane lattice has a ${\cos }^4$ profile and periodicity of $\lambda$. The flat portion of the (b) cross section shows the intersection of two nodal lines.

Figure 3

Adjustment of the phases $\delta \theta$ and $\delta \varphi$ allow for nearest-neighbor pairing with all four nearest neighbors. “+” marks the location of a lattice site located at the origin which can be paired with any of its four nearest neighbors (shown with $엯$) depending upon the choice of phase: (a) $\delta \theta =\pi ∕2$, $\delta \varphi =-\pi ∕2$, (b) $\delta \theta =-\pi ∕2$, $\delta \varphi =\pi ∕2$, (c) $\delta \theta =-\pi ∕2$, $\delta \varphi =0$, (d) $\delta \theta =\pi ∕2$, $\delta \varphi =0$.

Figure 4

Cross sections of example double well potentials. Solid line represents the double well potential; dotted line shows the placement and amplitude of the out-of-plane lattice. (a) The barrier height, labeled above by the quantity $\gamma$, of the double well can be adjusted by placing the out-of-plane lattice “in the barrier” and adjusting the ratio of $I_{xy}∕I_z$. (b) The “tilt” of the double well (the relative offset between adjacent sites) can be changed by adjusting $\delta \theta$ and $\delta \varphi$.

Figure 5

(Color online) Lattice imperfections causing (a) modulation of the lattice depth, $\mathrm{\Delta }U$, between neighboring sites and (b) state dependent modulation of the barrier height by a polarization lattice. In (b) the solid line is the cross section of the intensity lattice; the dashed line is the cross section of the state dependent lattice resulting from unbalanced beam intensities. Atoms in the ground state of each well are shown schematically.

Figure 6

(Color online) Schematic of the experimental implementation of the 2D double well lattice made from a single folded, retroreflected beam. Mirrors $M1$ and $M2$, lenses $L1$ and $L2$, and EOM$\theta$ are mounted on a fixed plate.

Figure 7

Experimental images of atom diffraction from a $3\text{−}\mathrm{\mu }\mathrm{s}$ pulse of (a) the in-plane lattice and (b) the out-of-plane lattice after $13\mathrm{ms}$ time of flight.

Figure 8

(a) Schematic of the momentum components that contribute to $G=(N_{1k}-N_{2k})∕(N_{1k}+N_{2k})$. $N_{2k}$ is the sum of atoms in the momentum components designated with filled circles, and $N_{1k}$ is the sum of atoms in momentum components designated with open circles. (b) Calibration of $\mathrm{EOM}\beta$: $\beta \approx 52\mathrm{mrad}$ (open circles), $\beta \approx 34\mathrm{mrad}$ (triangles), and $\beta \approx 0\mathrm{mrad}$ (filled circles).

Figure 9

Time dependence of the value $G$ characterizing the diffraction patterns for atoms loaded into lattices with a small offset energy $\mathrm{\Delta }U$ caused by $ϵ\ne 0$. Open circles are data points; solid lines are a fit to the data using a exponentially decaying sinusoid. The frequency of the oscillations given from the fit is inset in each image. From this frequency we determine $ϵ$: (a) $\nu =3900\mathrm{Hz}$ corresponds to $ϵ\simeq 7\mathrm{mrad}$, and (b) $\nu =775\mathrm{Hz}$ corresponds to $ϵ\simeq 1.4\mathrm{mrad}$. The data in (a) were taken after the initial pentaprism alignment; the data in (b) were taken after several iterations of measuring the frequency and then realigning the beams to further improve the angle. Schematics of the diffraction patterns corresponding to different values of $G$ at different times are shown in the insets. The initial phase of $G$ in (a) and (b) is arbitrary; it depends only on how much phase has been wound up during the loading time.

Figure 10

Experimental images after $13\mathrm{ms}$ TOF of atoms filling the first Brillouin zone for (a) the in-plane lattice and (b) the out-of-plane lattice. The shapes of the BZs reflect the momentum components in each lattice.

Figure 11

(a) Single-slit diffraction pattern resulting from a loss of phase coherence among out-of-plane lattice sites. (b) Double-slit interference pattern caused by coherence between atoms within a particular double well but not among the ensemble of double wells.