Constrained Dynamics via the Zeno Effect in Quantum Simulation: Implementing Non-Abelian Lattice Gauge Theories with Cold Atoms

We show how engineered classical noise can be used to generate constrained Hamiltonian dynamics in atomic quantum simulators of many-body systems, taking advantage of the continuous Zeno effect. After discussing the general theoretical framework, we focus on applications in the context of lattice gauge theories, where imposing exotic, quasilocal constraints is usually challenging. We demonstrate the effectiveness of the scheme for both Abelian and non-Abelian gauge theories, and discuss how engineering dissipative constraints substitutes complicated, nonlocal interaction patterns by global coupling to laser fields.

Figure 1

Dissipative protection of gauge invariance in implementations of a lattice gauge theory (LGT). (a) The dynamics $H_0$ happens within the physically relevant subspace $H_{\mathcal{P}}$, defined by $G_x^a|\psi ⟩=0$, but gauge-variant perturbations $H_1$ may drive the system into the unphysical subspace $H_{\mathcal{Q}}$ (where $G_x^a|\psi ⟩\ne 0$). (b) LGTs consist of fermions ${\psi }_x$ living on sites, coupled to gauge fields ${\mathcal{U}}_{x,x+1}$ living on links. The dynamics can be constrained to the physical subspace by coupling independent noise sources linearly to each generator. The multisite structure of the generators implies that the noise has to be correlated quasilocally in space.

Figure 2

(a) System dynamics of a U(1) LGT in the ideal case ($H_0$, top) and under the effect of undesired single particle tunneling ($H_1$, bottom); (b) on-site Hilbert spaces. (c)–(f) Dissipative protection of quench dynamics in a $U(1)$ QLM with $N_s=4$ sites connected by $N_l=3$ links (open boundary conditions). (c) Population of gauge-invariant subspace. (d), (f) Average violation of gauge constraint, quantified by $g^2=\sum _x{G}_x^2/N_s$. (e) Average electric field $E=\sum _x{E}_{x,x+1}/N_l$. In panels (c)–(e), blue curves indicate the ideal dynamics. Thin red curves show the detrimental influence of gauge-variant fermion tunneling ($\lambda /J=0.25$). The arrows (from red to blue) show how increasing $\kappa$ restores the ideal dynamics (green dashed curves; $\kappa /J=1$, $2.5$, 5, 10, 20, 40, 80). Panel (f) shows the scaling with $\kappa$ of $⟨g^2⟩$ at a fixed time; the black line is a guide to the eye indicating a scaling $\propto 1/\kappa$. All results are obtained from the full master equation with Hamiltonian $H_0+H_1$ (see text), starting from the eigenstate of $H_0$ for $m\to \infty$.

Figure 3

U(2) LGT. (a) Basic dynamics of Eq. (5). The matter/gauge-field coupling corresponds to a simultaneous color-conserving tunneling of one fermion at site $x$ to the link $x$, $x+1$ (red) and a rishon at $x$, $x+1$ to the site $x+1$ (blue, dashed). Matter and rishon sites are denoted by squares and circles, respectively; the blocks $x$ and $x+1$ are indicated by continuous and dashed contours. (b) An example for a gauge-variant process: correlated tunneling similar to panel (a), but accompanied by a change of color. (c) Numerical analysis of two building blocks, evolving under $H_0$ plus various error terms (calculations for the full master equation (2); for details see the Supplemental Material [33]). Thin red line: $\kappa =0$, green dashed lines: $\kappa /J=5$, 10, 20, 40, 80, 160 (increasing along the arrow).