Splitting and connecting singlets in atomic quantum circuits

Abstract

Gate operations composed in quantum circuits form the basis for digital quantum simulation and quantum processing. While two-qubit gates generally operate on nearest neighbours, many circuits require nonlocal connectivity and necessitate some form of quantum information transport. Yet, connecting distant nodes of a quantum processor still remains challenging, particularly for neutral atoms in optical lattices. Here, we create singlet pairs of two magnetic states of fermionic potassium-40 atoms in an optical lattice and use a bi-directional topological Thouless pump to transport, coherently split, and separate the pairs, as well as to demonstrate interaction between them via tuneable $($swap$)^α$-gate operations. We achieve pumping with a single-shift fidelity of 99.78(3)% over 50 lattice sites and split the pairs within a decoherence-free subspace. Gates are implemented by superexchange interaction, allowing us to produce interwoven atomic singlets. For read-out, we apply a magnetic field gradient, resulting in single- and multi-frequency singlet-triplet oscillations. Our work shows avenues to create complex patterns of entanglement and new approaches to quantum processing, sensing, and atom interferometry.

Figures (10)

• Figure 1: Quantum circuits based on topological pumping for splitting and connecting atomic singlet pairs in an optical superlattice. (a) Schematic of the experimental architecture, showing an exemplary circuit of depth five consisting of several parallel ${\text{\sc swap}}$ and $({\text{\sc swap}})^2$ gates. Left inset: input state preparation of an atomic singlet $\ket{s}$ occupying two separate bands. Right inset: illustration of one operation cycle of the Thouless pump. Atoms in the ground (right-moving, Chern number $C = +1$) and first excited (left-moving, $C = -1$) bands each feature quantised and state-independent displacement in opposite directions. As a result, singlet pairs are separated by many lattice sites. (b) Circuit representation of the process displayed in panel (a) whose input consists of randomly positioned singlets in the lattice (filled dots and magenta lines).
• Figure 2: Shuttle and gate operations enabled by topological pumping and controlled superexchange interactions. (a) Illustration of shuttle operations for noninteracting atomic singlets via topological pumping. (b) In-situ absorption images of the atomic cloud during the shuttle operation for different numbers of operation cycles $N_{\text{cyc}}$. (c) Measured double occupancy as function of $N_{\text{cyc}}$, including one pump reversal halfway. The low and high double occupancies correspond to the dimerised ($\Delta=0$) and staggered configurations ($\Delta=\Delta_0$), respectively. The solid lines denote the respective exponential fits of the data in the two configurations, yielding an operation fidelity of $F=0.9978(3)$. (d) Visualisation of the configurable gate operations achieved by controlling the superexchange interaction $J_{\text{ex}}$ when two atoms meet during pumping. The example shows a swap gate. (e) Two-particle Bloch sphere displaying the Hamiltonians $H_{\text{ex}}$ and $H_{\text{STO}}$. The remaining equatorial states are $\ket{i_{-}}=\left(\ket{\downarrow,\uparrow}-i\ket{\uparrow,\downarrow}\right)/\sqrt{2}$ and $\ket{i_{+}}=\left(\ket{\downarrow,\uparrow}+i\ket{\uparrow,\downarrow}\right)/\sqrt{2}$. The initialisation of the product state $\ket{\uparrow,\downarrow}$ is indicated by light dashed lines. The time evolution of the state under the superexchange Hamiltonian $H_{\text{ex}}$ corresponds to a rotation around the $z$-axis by an angle $\varphi$ (dashed purple line). Gate operations are controlled by adjusting $\varphi$, which is determined by measuring the projection of the final state onto the $y$-axis of the Bloch sphere (Appendix). We realise the gates as a function of the pump period $T$ in units of the average tunnelling over one period, $t_{\text{hop}}$ [(f)], and as a function of the optical lattice depth $V_{\text{X}}$ [(g)]. The upper axis in (g) corresponds to the strength of $J_{\text{ex}}$ in the dimerised configuration. The insets indicate the parameters used for the implementation of the respective gates. The error bars in (c), (f), and (g) correspond to the standard error for at least 9, 14, and 12 repetitions, respectively. The lines in (f) and (g) are damped sinusoidal fits with respect to $T$ ($J_{\text{ex}}$).
• Figure 3: Determining the separation between two entangled atoms with singlet-triplet oscillations (STOs). (a) All- swap circuit of depth three ($N_{\text{cyc}}=3$), resulting in a separation of $s=7$ lattice sites. The magenta trace within the circuit illustrates the path of an exemplary atomic singlet. (b) Schematic of singlet-triplet oscillations of atoms split by multiple lattice sites. A magnetic gradient $\Delta B$ lifts the degeneracy between $\ket{\downarrow,\uparrow}$ and $\ket{\uparrow,\downarrow}$ states, leading to an oscillation between the singlet and triplet states. The oscillation frequency is proportional to the separation $s=2N_{\text{cyc}}+1$. (c) Measurement of the STO frequency for different numbers of operation cycles $N_{\text{cyc}}=0,2,3,$ and $5$ (from top to bottom) with swap gates, corresponding to separations up to $11$ sites. The orange lines denote sinusoidal fits to the data and the error bars are the standard error for 3-6 repetitions. (d) STO frequencies as a function of the number of operation cycles $N_{\text{cyc}}$. The coloured arrows point to the respective time traces in (c). Time traces for $N_{\text{cyc}}=1,7,$ and $9$ are plotted in Fig. \ref{['fig:S6']}. The solid black line is a proportional fit to the data, used to determine the STO base frequency $f_{1}$. The error bars, representing the uncertainty of the sinusoidal fit, are smaller than the data points. (e) Distribution of frequency components, corresponding to different values of atom separation $s$. These distributions are evaluated by applying a twelve-frequency sinusoidal fit to the four time traces in (c) with frequencies $s \times f_{1}$ up to $s=12$. The amplitudes $A_{s}$ of the respective components are plotted as function of $s$, where error bars denote the amplitude uncertainty of the fit.
• Figure 4: Quantum states generated by configurable gate sequences exhibiting multi-frequency singlet-triplet oscillations. (a-c) Singlet fraction as a function of STO time $(\tau_{\text{STO}})$ after a total of five operation cycles, each containing one [(a)], two [(b)], or all [(c)] ( swap)$^2$ gates, while all remaining gates are swap gates. The order of the respective gates is visualised in the insets. The orange lines represent twelve-frequency sinusoidal fits of the data with a base frequency of $f'_{1}=218(1)Hz$. The error bars denote the standard error for at least 3 [(a, b)] or 6 [(c)] experimental repetitions, respectively. (d-f), Amplitudes $A_s$ of the corresponding fits, directly revealing atomic singlets separated by different distances. The error bars correspond to the uncertainty of the fit. (g-i), Circuit representations of the gate sequences implemented in (a-c).
• Figure A1: Optical lattice potential and resulting Hamiltonian parameters. (a) The superposition of a short, static (shown as red) and a long, moving (shown as yellow) lattice creates the bipartite superlattice potential (black) with tunnellings $t_x$, $t_x'$, and site-offset $\Delta$. The lattice is formed of a single laser wavelength $\lambda = 1064nm$ and the colours merely illustrate the two main contributions to the pumping lattice. Double occupancies are suppressed due to a large repulsive on-site energy $U>2\Delta_0$. (b) Dependence of tunnelling $t_x$ and site-offset $\Delta$ variations within one pump cycle for two different lattice depths $V_\mathrm{X}=\{5.7, 7.7\} E_{\text{rec}}$. The tunnelling $t_x'$ is omitted for simplicity, as $t_x' = t_x(\tau + T/2)$. (c) Calculated superexchange energy $J_\mathrm{ex}$ as a function of time within one pump cycle for two different lattice depths $V_{\text{X}}$. The superexchange energy $J_\mathrm{ex}$ is negligible in the staggered lattice configuration ($\pm 0.25T$) due to large $\Delta$ and small $t_x$. It takes on larger values when entering and exiting the balanced double-well configuration ($\Delta=0$). (d) Maximum tunnelling $t_x$, site-offset energy $\Delta$, interaction strength $U$, and resulting maximum of $J_\mathrm{ex}$ as a function of lattice depth $V_{\mathrm{X}}$.