A Floquet-Rydberg quantum simulator for confinement in $\mathbb{Z}_2$ gauge theories

TL;DR

The paper addresses real-time simulation of confinement in a $\mathbb{Z}_2$ lattice gauge theory using Floquet engineering in a dipolar Rydberg array. It develops an analog quantum simulator that maps a driven XY chain to a $\mathbb{Z}_2$ LGT by selective dressing of second-order processes and destructive interference among driving phases. By deriving an effective Hamiltonian $H_{\rm eff}$ that hosts the required three-body gauge terms and detailing a dipolar-Rydberg implementation with a ladder geometry and angular suppression of gauge-breaking couplings, the authors demonstrate confinement dynamics in the neutral sector and verify the dynamics with exact diagonalization and matrix-product-state simulations. They also analyze Floquet errors and show gauge violations stay within experimental thresholds for realistic parameters, and discuss extensions to trapped ions or superconducting qubits to broaden implementation platforms.

Abstract

Recent advances in the field of quantum technologies have opened up the road for the realization of small-scale quantum simulators of lattice gauge theories which, among other goals, aim at improving our understanding on the non-perturbative mechanisms underlying the confinement of quarks. In this work, considering periodically-driven arrays of Rydberg atoms in a tweezer ladder geometry, we devise a scalable Floquet scheme for the quantum simulation of the real-time dynamics in a $\mathbb{Z}_2$ LGT. Resorting to an external magnetic field to tune the angular dependence of the Rydberg dipolar interactions, and by a suitable tuning of the driving parameters, we manage to suppress the main gauge-violating terms, and show that an observation of gauge-invariant confinement dynamics in the Floquet-Rydberg setup is at reach of current experimental techniques. Depending on the lattice size, we present a thorough numerical test of the validity of this scheme using either exact diagonalization or matrix-product-state algorithms for the periodically-modulated real-time dynamics.

Figures (5)

• Figure 1: Rydberg tweezer array: (a) The external magnetic field $\boldsymbol{B}_0$ makes an angle $\theta_m = 54.7^{\rm o}$ with respect to the $\boldsymbol{z}$ axis. The projection of $\boldsymbol{B}_0$ onto the $xy$ plane where the atoms reside, $\boldsymbol{B}_0^\perp$, makes an angle of $45^{\rm o}$ with the $\boldsymbol{x}$ axis. (b) Angular distribution of the XY couplings as a function of the angle $\phi_{ij}$ between the interatomic vector $\boldsymbol{R}_{ij}$ and $\boldsymbol{B}_0^\perp$, which vanish at the critical angles $\phi_{ij} = \pm 45^{\rm o}, \pm \,135^{\rm o}$. (c) Ladder configuration of the Rydberg atoms trapped in optical tweezers. The atoms are arranged on the vertices of isosceles right triangles of sides $d$ and base $b = \sqrt{2} \, d$. The interactions are represented as colored lines according to the color-scheme of panel (b). Dashed lines highlight the critical directions of the vanishing XY terms, which forbid a direct coupling between even-even and odd-odd spins. In this paper, we drive the odd atoms detuning according to $H_\text{drive} = \sum_{i \, \text{odd}} \frac{\eta \omega_d}{2} \cos(\omega_d \, t + \varphi_i) \sigma_i^z$, with $\eta \approx 2.4$ and $\phi_{2i+1} = i \frac{\pi}{2}$.
• Figure 2: Dressing parameter:$\tilde{\chi}_{ik} = -\rm i \, \chi_{ik}$ is depicted as a function of the phase difference $\Delta\varphi_{ik}$ and relative driving strength $\eta$. Here $\nu_n$ denotes the $n$-th zero of $\mathsf{J}_0(x)$. For $\eta = \nu_1$, $\tilde{\chi}_{ik}$ has a maximum (minimum) at $\Delta\varphi_{ik} = 60^{\rm o}$ ($-60^{\rm o}$).
• Figure 3: $\mathbb{Z}_2$ confinement: Two particles are initialized at positions $i=3,9$ and connected by an electric-field string. Here, $h = J_{\rm t}$ and $\mu = 0$. Left and right panels correspond to the contour plots of the charge distribution $\langle \rho_n(t)\rangle=\bra{\psi(t)} a_n^\dagger a_n\ket{\psi(t)}$ and the electric field $\langle\tau_{n+1/2}^x(t)\rangle$, where the evolution of $\ket{\psi(t)}$ under ${H}_{\rm XY}(t)$ with $\omega_d = 30J$ is approximated using MPS with bond dimension $= 20$.
• Figure 4: Gauge-invariant dynamics and gauge violation. The left panel displays the driven dynamics (solid line) and the effective gauge-invariant dynamics (dots) arising from the initial state $\ket{L} = \ket{1,-,0}$, in the $N_a=3, \omega_{\rm d} = 30 J_3/d^3$ case. The driven evolution is in full-agreement with the expected periodic oscillations of $\langle a^\dagger_n a_n (t) \rangle$, $\langle \tau^x (t) \rangle$. The right panel corresponds to the average percent error $\bar{\epsilon}(t,\omega_{\rm d})$ accumulated during the evolution from the state $\ket{\psi_0}=\ket{1_1,-,0,- \dots -,0}$, in the case $N_a=9$. For each value of the driving frequency $\omega_d$, only times up to $3\pi/J_{\rm t}(\omega_{\rm d})$ are explored. The dashed line corresponds to a $10 \%$ threshold of gauge violation.
• Figure S1: Plot of the average gauge-violation $\bar{\epsilon}$ as a function of time and of the allowed maximal bond dimension. The plot refers to the same configuration as for Fig.(2) of the main text. The data is plot at integer multiples of T.