Constructing Arbitrary Coherent Rearrangements in Optical Lattices

Abstract

Coherent control of motional degrees of freedom of ultracold atoms in optical lattices offers a promising route towards programmable quantum dynamics with massive particles. We propose and analyze a scheme for implementing coherent rearrangement of ultracold atoms, corresponding to arbitrary unitary transformations on single-particle motional states. Exploiting an analogy between dynamics in optical superlattices and discrete linear optics, we employ the Clements scheme to systematically construct any global $N$-dimensional single-particle unitary from tunneling and phase shifts in arrays of double wells. Tunneling is controlled globally, while local operations are achieved through site-resolved potential shifts. We numerically investigate the susceptibility of the scheme to intensity noise and addressing crosstalk. We identify key subroutines enabled by this unitary construction, including the Discrete Fourier Transform and the implementation of non-native Hamiltonians. Extending the scheme to two dimensions enables all-to-all atomic rearrangement with a circuit depth that scales sublinearly with the atom number, providing a high-density and highly scalable approach to atom rearrangement.

Figures (7)

• Figure 1: Parallels of Quantum Optics and Optical Lattices. (a) An arbitrary global unitary $U_\textrm{global}$ is systematically constructed from local two-mode unitary gates $U_\textrm{local}$. (b) An optical interferometer constructed using the Clements scheme realizes arbitrary $N$-mode transformations using brick-wall lattices of generalized beam splitters. The inset shows a single two-mode operation composed of two 50:50 beam splitters and phase shifters applying phases $\gamma$ and $\delta$. (c) Atomic dynamics between discrete modes in optical superlattices can be constructed by analogy to optical interferometers. An $\hat{X}$ rotation (beam splitter) is performed by enabling tunneling, while a $\hat{Z}$ rotation (phase shifter) is performed by applying a local energy offset $\mu_i$. The brick-wall structure is realized by alternating the superlattice dimerization between dimer I and dimer II, which is achieved by adjusting the superlattice phase. In each time step, a two-mode transformation $T(\theta_i, \phi_i)$ is implemented in parallel on all dimers of the current dimerization through a sequence of native $\hat{X}$ and $\hat{Z}$ gates.
• Figure 2: Clements Scheme.(a) A Givens rotation multiplied from the right (left) mixes two columns (rows) to eliminate a specific target element of an arbitrary unitary $U$. A multiplication from the right (left) is colored in blue (red). (b) The Clements scheme transforms a unitary into a diagonal matrix by eliminating the sub diagonal elements in a "snake-like" pattern, alternating between left and right multiplied Givens rotations.(c) The physical brick-wall circuit that implements the decomposition. Each rectangular gate corresponds to a single Givens rotation $T_{n,m}(\theta,\phi)$ used to nullify one element in panel (b). The different sections correspond to the subdiagonals of $U$.
• Figure 3: Decomposition Details of a DFT.(a) A single particle, initially localized at a specific site (a position eigenstate), is coherently transformed into a delocalized state with a periodic phase structure across the entire array (a momentum eigenstate). (b) The circuit decomposition of the DFT unitary for $N=8$ (left) and $N=13$ (right) systems. Each colored rectangle represents a two-mode gate. The upper rectangle indicates the gate's phase $\phi$, while the lower rectangle represents its transmission probability, $T = \cos^2(\theta)$. A translational pattern emerges for odd $N$. (c) Histogram of the required $\hat{Z}$ gate rotation angles for an $N=30$ DFT, showing that angles are distributed across the full $[0, 2\pi]$ range.
• Figure 4: Simulating Non-Native Hamiltonian Dynamics. The framework can simulate dynamics not natively generated by the hardware by decomposing the time-evolution operator $\hat{U}(\tau)=e^{-i\hat{H}\tau/\hbar}$ into a local gate sequence. We show the time evolution of a particle initially localized in the center of a 16-site array (left panels) and the corresponding circuit decomposition of the unitary operator for a given time $\tau$ (right panels). (a) Evolution under a fast scrambling Hamiltonian with non-local, tree-like interactions as described in bentsen2019treelike. The particle's probability rapidly spreads across the system. (b) Evolution under a Hamiltonian with nearest-neighbor (NN) and next-nearest-neighbor (NNN) hopping ($\frac{t_{nnn}}{t_{nn}}=2$) beyond the tight-binding approximation.
• Figure 5: Atom Rearrangement in 1D and 2D(a) A 1D permutation is decomposed into a network of two-site gates that either swap modes (red) or apply the Identity (gray). A $\widehat{\text{SWAP}}$ gate is realized by setting $\{\theta, \phi\} = \{\frac{3\pi}{2},\pi\}$ while the Identity $\hat{\mathbb{I}}$ is realized through $\{\theta, \phi\} = \{0,0\}$ in Eq. \ref{['eq:Z3X2']}. (b) The "Horizontal-Vertical-Horizontal" (HVH) scheme is used for conflict-free 2D atom rearrangement. The three steps are: (1) a horizontal sort moves atoms into a temporary buffer region (orange); (2) a vertical sort places atoms in their correct target rows; and (3) a final horizontal placement moves them into the desired target columns. (c) We show the mean buffer size $L_{\text{buffer}}$ required to generate a conflict-free rearrangement using the HVH scheme for an $L\times L$ fully filled lattice. Each data point is the average over 100,000 random target configurations, with error bars indicating the standard deviation. The points are connected as a guide to the eye. The worst possible buffer size needed is $L-1$ although this is rarely the case.