Observation of ballistic plasma and memory in high-energy gauge theory dynamics

Abstract

Gauge theories describe the fundamental forces of nature. However, high-energy dynamics, such as the formation of quark-gluon plasmas, is notoriously difficult to model with classical methods. Quantum simulation offers a promising alternative in this regime, yet experiments have mainly probed low energies. Here, we observe the formation of a ballistic plasma and long-time memory effects in high-energy gauge theory dynamics on a high-precision quantum simulator. Both observations are unexpected, as the initial state - fully filled with particle-antiparticle pairs - was thought to rapidly thermalize. Instead, we find correlations spreading ballistically to long distances and a memory of charge clusters. Our observations cannot be explained by many-body scars, but are captured by a new theory of plasma oscillations between electric field and current operators, persisting all the way to the continuum limit of the (1+1)D Schwinger model, of which we simulate a lattice version. Adapting techniques from quantum optics, we visualize plasma oscillations as rotations of Wigner distributions, leading to a novel set of predictions which we test in experiment and numerics. The new framework encompasses both our scenario and scars, which show up as coherent states of the plasma. The experimental surprises we observe in the high-energy dynamics of a simple gauge theory point to the potential of high-precision quantum simulations of gauge theories for general scientific discovery.

Figures (16)

• Figure 1: Experimental observations.a. Correspondence between the Rydberg atom array and a lattice gauge theory. Atomic states map onto configurations of electric fields and charges. The Rydberg blockaded spin configurations map onto the matter-gauge configurations that satisfy Gauss' Law and the Kogut-Susskind fermion staggering. b. We perform an out-of-equilibrium quench experiment on a Rydberg atom array, equivalent to a one-dimensional quantum link model. The initial state $|0\rangle^{\otimes N}$ corresponds to state fully filled with particles and anti-particles. Experimental measurements yield atomic fluorescence images, with bright and dark (dashed circles) indicating $|0\rangle/|1\rangle$, which translate to field and charge configurations. c. Connected electric-field correlations $\langle E_j E_{{j+d}}\rangle_c = (-1)^d(\langle Z_j Z_{{j+d}}\rangle-\langle Z_j \rangle \langle Z_{{j+d}}\rangle)$ reveal correlations propagating ballistically to large distances and short-range correlations which persist to long times, both defying conventional expectations of thermalization. d. Consistent with ballistic transport, the average distance of the correlations (Methods) grows with a velocity that quantitatively agrees with analytical predictions (dashed line) as well as exact simulations of small system sizes (gray). e. Late-time correlations deviate from the infinite-temperature value (black dashed) at short distances, in agreement with numerical simulations (grey), which reveal further oscillations over time.
• Figure 1: Ballistic correlations.a. Cross sections of experimental electric field correlators at three points in time: the propagating correlations are visible, along with remaining athermal correlations at long times. Numerical simulations corroborate both observations, but also reveal high-frequency oscillations which are not seen in experimental data due to the interval of time samples. Plotting the numerical simulations only at the experimental times aliases this high-frequency oscillations, in agreement with the experimental data. b. To estimate the effective distance and amplitude of the ballistic propagation, we first subtract a background $C^{EE}(d,\infty)$ of approximately static correlations at small distances ($d\leq 5$), estimated from the average of the experimental correlators over the last five available time-points (from 11.2---14.3 cycles). This background subtraction removes the long-lived non-thermal correlations, and only the ballistic correlations remain in $\Delta C^{EE}(d,t)$ (Methods). We plot the squared correlations $\Delta C^{EE}(d,t)^2$ at two points in time to illustrate the propagation and decay of the correlations: we use this quantity to define the correlation distance in Fig. \ref{['fig:experimental_data']} (Methods). We also see that the experimental amplitudes decay exponentially over time as $\exp(-2\bar{\gamma}t)$ (dashed line), where $\bar{\gamma}\equiv \mathbb{E}_q~\gamma(q)$ is an average decay rate, as well as numerical simulation of small system sizes ($N=16,20,24$, gray filled lines). $\gamma(q)$ is estimated by fitting the numerically-obtained iDSF to a Lorentzian function (bottom).
• Figure 2: Plasma band structure.a. Ballistic correlations are due to plasma oscillations on an infinite temperature background. Plasma oscillations are induced by pair production and backreaction processes which couple electric field and charge current. b. Plasma oscillations are evident in the narrow band $\omega_0(q)$ of the infinite-temperature dynamical structure factor (iDSF) of the electric field operator (computed at $N=24$, but whose features are independent of system size), well approximated by a mean-field theory (blue line, Methods). c. The lattice Schwinger model [Eq. \ref{['eq:lattice_Schwinger_Hamiltonian']}] provides a class of models containing plasma oscillations, spanning the large $\mathcal{J}$ PXP limit to the $\mathcal{J}\rightarrow 0$ field-theoretic limit. This is evident in the ballistic propagation of correlators (top row) and the band structure of the iDSF (bottom) of the fermion occupation operator. d. Our phenomenology is not due to exceptional eigenstates (so-called scars) and requires all eigenstates to participate. Top: The iDSF is the sum of a large number of terms $|\langle \mathcal{E}_i |E(q)|\mathcal{E}_j\rangle|^2$ for eigenstate pairs with energy difference $\omega = \mathcal{E}_i - \mathcal{E}_j$. Summing over all average energies $(\mathcal{E}_i+\mathcal{E}_j)/2$ gives the iDSF (data for $N=20$ system plotted for momentum $q=0$). Bottom: the iDSF is delocalized among generic eigenstate pairs, as confirmed by the operator participation ratio (Methods) which asymptotically saturates the maximum expected growth rate of $\varphi^{2N}$ (dashed), where $\varphi \approx 1.618$ is the golden ratio.
• Figure 2: Entanglement dynamics.a. A spacetime plot of the negativity $\mathcal{N}(\rho_{AB})$ for the reduced state $\rho_{AB}$ associated with disjoint subsystems $A$ and $B$ (details in SI SI) reveals the ballistic propagation of entanglement, schematically illustrated in the inset. $A$ and $B$ are contiguous three-site subsystems separated by $d_{AB}$ sites. At a given time, the negativity takes substantial positive values only at a certain $d_{AB}$ which grows linearly with time (in addition to local entanglement that remains at $d_{AB} = 1$). A spacetime plot of the quantum mutual information $I(A:B)$ of $\rho_{AB}$ also shows ballistic spreading of correlations. Unlike the negativity, these correlations may be classical in nature and are more pronounced. Bottom row: Averaging the negativity over all bitstring initial states reveals that entanglement transport is a generic feature, independent of initial state. In contrast, the $|\mathbb{Z}_2\rangle$ initial state shows limited entanglement transport. All numerical simulations are for $N=24$ systems with periodic boundary conditions. b. Plots of the negativity and mutual information for subsystems of size $|A|=|B|=1,2$ show similar but weaker behavior. c. The MFIM (top) does not show such entanglement growth, while the TFIM (bottom) shows repeated entanglement propagation, possibly due to its underlying free-fermion description. Numerical simulations for $N=20$ systems with parameters described in the caption of Ext. Dat. Fig. \ref{['fig:MFIM_TFIM_XXZ_structure_factors']}, and with $|A|=|B|=3$.
• Figure 3: Wigner distributions.a. The many-body Wigner distribution reveals semi-classical structure in a phase space of electric field and current. The evolved $|0\rangle^{\otimes N}$ state is akin to a squeezed state: its Wigner distribution is always centered about the origin, with a stretched axis (initially along the current direction [dashed ovals]) that rotates over time. This holds in all momentum sectors, illustrated for $q=0$ (top) and $q=0.2\pi$ (bottom). b. Experimental data of the field mean and variance for the evolved $|0\rangle^{\otimes N}$ state confirms the phase-space oscillations. While $\langle E(q)\rangle$ (orange triangles) is always zero, $\text{var}[E(q)]$ oscillates over time for all momenta $q$. c. Top: The spacetime Fourier transform of the variance oscillations (normalized for each $q$, Methods) reflect their momentum-dependent frequencies and experimentally reconstructs the iDSF band-structure $2\omega_0(q)$ (blue dashed line). Bottom: The experimental full-counting-statistics of electric field broadens before sharpening again, in accordance with the Wigner distribution dynamics. d. In contrast, the $|\mathbb{Z}_2\rangle$ state is analogous to a coherent state in the $q=0$ mode. Top: Its Wigner distribution is displaced in phase space and moves along the circular trajectory traced by its expectation values (dotted line). Accordingly, both the mean and variance of the uniform electric field $E(0)$ oscillate. The $|\mathbb{Z}_2\rangle$ state is not displaced for $q\neq 0$ modes and like $|0\rangle^{\otimes N}$, behaves as a squeezed state. Accordingly, numerical simulations reveal variance oscillations for $E(q\neq0)$.