Observation of string breaking on a (2 + 1)D Rydberg quantum simulator

TL;DR

The paper demonstrates the experimental observation of string breaking in a (2+1)D U(1) lattice gauge theory implemented on a programmable Rydberg-atom array. By encoding gauge fields on links as spin-1/2 and matter as hardcore bosons, and enforcing Gauss's law with blockade constraints, they realize a confining regime with a tunable string tension arising from long-range interactions. They map out the equilibrium phase diagram distinguishing unbroken and broken string sectors and observe real-time string breaking dynamics, including resonant quenches that enhance broken-string formation. This work provides a scalable platform for studying confinement and string-breaking phenomena beyond 1+1D and paves the way for analog meson scattering and non-Abelian LGT simulations.

Abstract

Lattice gauge theories (LGTs) describe a broad range of phenomena in condensed matter and particle physics. A prominent example is confinement, responsible for bounding quarks inside hadrons such as protons or neutrons. When quark-antiquark pairs are separated, the energy stored in the string of gluon fields connecting them grows linearly with their distance, until there is enough energy to create new pairs from the vacuum and break the string. While such phenomena are ubiquitous in LGTs, simulating the resulting dynamics is a challenging task. Here, we report the observation of string breaking in synthetic quantum matter using a programmable quantum simulator based on neutral atom arrays. We show that a (2+1)D LGT with dynamical matter can be efficiently implemented when the atoms are placed on a Kagome geometry, with a local U(1) symmetry emerging from the Rydberg blockade, while long-range Rydberg interactions naturally give rise to a linear confining potential for a pair of charges, allowing us to tune both their masses as well as the string tension. We experimentally map out the corresponding phase diagram by adiabatically preparing the ground state of the atom array in the presence of defects, and observe substructure of the confined phase, distinguishing regions dominated by fluctuating strings or by broken string configurations. Finally, by harnessing local control over the atomic detuning, we quench string states and observe string breaking dynamics exhibiting a many-body resonance phenomenon. Our work paves a way to explore phenomena in high-energy physics using programmable quantum simulators.

Figures (11)

• Figure 1: Emergent confinement and string breaking on a ($2+1$)D Rydberg atom array: (a) U(1) gauge theories satisfy local Gauss's law constraints, where electric charges $\pm Q$ are connected by strings of electric field $\vec{E}$. In a confined phase, the potential energy $U(d)$ between charges increases linearly with their distance $d$, until the strings break by creating particle pairs with masses $2m$. (b) Equivalence between atom configurations satisfying the Rydberg blockade constraint (R) and the corresponding gauge-invariant states (G), shown for the A (upper) and B (lower) sites in the unit cell of (c). For the vacuum $S^z_\ell = -1/2$ (black lines) $\forall \ell$, while defects correspond to $Q_x=\pm 1$ charges (blue/red circles) and $S^z_\ell = +1/2$ strings (blue lines). (c) Neutral atoms trapped in optical tweezers are arranged on the links of a hexagonal lattice, with a two-site unit cell (green) and lattice spacing $a$. The figure depicts a Rydberg ordered configuration / LGT vacuum, with a line defect / string connecting two static charges, forced by removing atoms from the array (dashed circles). Rydberg atoms in the string interact with their second neighbors, giving rise to a confining potential with string tension $\sigma\approx V(\sqrt{3}a) - V(2a)$. (d) String breaking in the Rydberg system, where particle-pairs are created by flipping the state of one atom. (e) State preparation protocol involving local detuning, where we show experimental snapshots of the atom array in the initial disorder phase, and after deterministically preparing a broken string state (top) or one of the 6 degenerate unbroken string states (bottom). Dark spots correspond to atoms detected in the $\ket{g}$ state. (f) and (g) show the real-space configurations of these two prepared states, obtained by collecting these snapshots (results for the remaining 5 string states for this placement of charges are shown in Extended Data Fig.\ref{['fig:figm1']}). We depict the expectation value of the Rydberg densities (left), and the corresponding LGT observables (right). (h) Phase diagram calculated theoretically for the Rydberg Hamiltonian \ref{['eq:Rydberg_H']} as a function of $R_{\rm b}$ and $\delta / \Omega$, for the geometry displayed in (c). We plot the difference $p_{\rm b} - p_{\rm s}$ between the probability of the broken $p_{\rm b}$ and the unbroken string configurations $p_{\rm s}$ in the ground state, showing two distinct regions within the ordered phase found in Samajdar_2021, marked here by black dots (see Methods).
• Figure 1: Experimental Hamiltonian evolution protocols (a) Quasi-adiabatic state preparation used to obtain the ground state of the Rydberg Hamiltonian \ref{['eq:Rydberg_H']}, where the local detuning (dashed tuquoise line) remains at zero throughout the sweep. (b) Quasi-adiabatic state preparation with applied local detuning $\delta_0(t)$ (dashed tuquoise line), which strongly shifts the atoms off resonance, ensuring they remain in their ground states throughout the global detuning sweep. By applying the appropriate local detuning pattern (e.g. the one shown in (d)), one can selectively prepare either one of the string states or the broken string state. (c) Quasi-adiabatic state preparation followed by a quench in local detuning $\delta_0$ (dashed tuquoise line) that tensions the initially prepared string, such that the energies of the broken and unbroken string configurations become comparable. (d) For the string breaking studies with $d=2$ charge separation, the local detuning pattern applied is shown in open tuquoise circles over the atom geometry. This same detuning pattern can be used to initially prepare the broken string state via the protocol described in (b).
• Figure 2: String breaking in equilibrium (a) Product state configurations corresponding to a broken string (0) as well as to unbroken strings of the same (minimal) length (1--6) connecting two charges displaced in both directions, $d_0 = d_1 = 2$, for a system with $L_0 = L_1 = 5$. (b) Quasi-adiabatic state preparation protocol (details in Methods), where the final value of $\delta$ is varied to scan along the x-axis in the phase diagram. (c) Values of $p_{\rm b}$ and $p_{\rm s}$ as a function of $R_{\rm b}$ and fixed $\delta / \Omega = 3.67$ [dashed line in (e) and (f)] obtained experimentally by quasi-adiabatically preparing the ground state of the Rydberg atom array. Here $p_{\rm s}$ grows from zero to a finite value in the string region, while it decreases again in the broken string region. At the same time, $p_{\rm p}$ acquires a finite value in the latter, indicating string breaking, consistent with the theoretical prediction computed numerically (d). The coexistence of broken and unbroken configurations in the experiment is likely due to imperfect adiabatic state preparation, since each of them is a low-energy excited state in the region dominated by the other one. (e) and (f) show the values of $p_{\rm s}$ and $p_{\rm b}$ obtained experimentally as a function of $R_{\rm b}$ and $\delta / \Omega$, respectively. (g) and (h) show the distribution of probabilities $p_i$, where $i=0,...,6$ correspond to the classical configurations shown in (a), for $R_{\rm b} = 1.2$ and $\delta / \Omega = 2.3$ (white circle) and $R_{\rm b} = 1.7$ and $\delta / \Omega = 3.2$ (black circle), respectively, both for theory and experiment. (i) and (j) show the corresponding real-space configurations obtained in the experiment.
• Figure 2: Experimentally prepared (2+1)D strings: (a) -- (f) Classical string states (1) -- (6) prepared in the Rydberg atom array using the quasi-adiabatic state preparation protocol assisted by local detuning patterns. The real-space average Rydberg occupation results are represented on the left, while the extracted corresponding LGT observables are represented on the right.
• Figure 3: String breaking dynamics: (a) Different string configurations involved in a collective process required to transition from the unbroken to the fully broken string, where we consider a system with $L_0 = 5$ and $L_1 = 3$, and static charges separated by a distance $d = 2$. The arrows indicate the direct transitions driven by $\Omega$. (b) Experimental protocol employed to adiabatically prepare a string state by ramping up the detuning globally, followed by a quench in the local detuning $\delta_0$. The initial state is prepared at $R_{\rm b} = 1.2$ and $\delta / \Omega = 2.3$, for the geometry in (a), for which, under ideal preparation, the probability of the unbroken string [s in (a)] is $p_{\rm s}(t=0)\approx 0.8$. (c) -- (e) show the time-evolved probabilities obtained experimentally for the unbroken $p_{\rm s}$, broken string $p_{\rm b}$, [b in (a)] and fully charged configurations $p_{\rm c}$ [c in (a)] for different values of $\delta_0 / \Omega$. If we do not quench the system, $p_{\rm b}$ and $p_{\rm c}$ do not grow in time, while they show a fast growth followed by damped oscillations if we quench to $\delta_0 / \Omega = 1.0$ and $\delta_0 / \Omega = 3.0$, respectively. (f) Real-space configuration at time $t = 0$ and $t = 0.4$$\mu$s obtained experimentally for a quench to $\delta_0 / \Omega = 1$, showing the initially unbroken and the broken strings, respectively.