Experimental Observation of Thermalization with Noncommuting Charges

Quantum simulators have recently enabled experimental observations of the internal thermalization of quantum many-body systems. Often, the global energy and particle number are conserved and the system is prepared with a well-defined particle number—in a microcanonical subspace. However, quantum evolution can also conserve quantities, or charges, that fail to commute with each other. Noncommuting charges have recently emerged as a subfield at the intersection of quantum thermodynamics and quantum information. Until now, this subfield has remained theoretical. We initiate the experimental testing of its predictions, with a trapped-ion simulator. We prepare 6–21 spins in an approximate microcanonical subspace, a generalization of the microcanonical subspace for accommodating noncommuting charges, which cannot necessarily have well-defined nontrivial values simultaneously. We simulate a Heisenberg evolution using laser-induced entangling interactions and collective spin rotations. The noncommuting charges are the three spin components. We find that small subsystems equilibrate to near a recently predicted non-Abelian thermal state. This work bridges quantum many-body simulators to the quantum thermodynamics of noncommuting charges, the predictions of which can now be tested.

Popular Summary

A hot cup of coffee exchanges heat with its environment until cooling to the environment's temperature. This process is called thermalization and marks the arrow of time. How quantum systems thermalize remains somewhat mysterious. A quantum system can exchange with its environment not only heat but also quantities that participate in uncertainty relations—that cannot be measured simultaneously. How does such a quantum system look upon "cooling"? Theorists posited a prediction by 2015, but experimental evidence has been lacking. This paper presents the first experimental evidence of the predicted state, observed in a 21-ion system.

Two energy levels of each ion served as an effective spin-1/2 particle. We trapped the ions using oscillating electromagnetic fields and manipulated the ions with laser beams. Two of the ions formed the system of interest, and the rest, an effective environment. We initialized the system of interest out of equilibrium with the environment. Then, the system interacted with its environment via an effective long-range Heisenberg interaction. The interaction globally preserved three quantities that failed to commute with each other—the total spin's x-, y-, and z-components—while transporting the quantities locally. Measuring the system's longtime state, we found that it lies close to the theoretical prediction, termed “the non-Abelian thermal state.”

Our work opens a growing subfield—the quantum thermodynamics of noncommuting conserved quantities—to experimental tests. Furthermore, the interactions that we engineered can now be applied to explore the Heisenberg model's many-body physics.

Figure 1

The experimental setup and protocol. (a) A linear ion crystal of $N\le$ 21 qubits is trapped in a linear Paul trap. A small system exchanges charges (local instances of quantities that are conserved globally) with the surrounding environment: the energy, $E$, and all the components of angular momentum. (b) We Trotter approximate the Heisenberg evolution by evolving the state under each of the three terms of the Hamiltonian [Eq. (3)] consecutively, in short time steps. We experimentally realize two terms directly and generate the third term using resonant $\pi /2$ pulses ($R_x$ and $R_y$). This pulse sequence further protects the state against dephasing noise (ii). The Trotter sequence contains building blocks $E_{\pm }$. Alternating between them reduces pulse-length errors (iii). For further details, see Appendix pp1. (c) The observed evolution of the 12-qubit initial state, $|y+,x+,z+{⟩}^{\otimes 4}$, under the Trotter-approximated Heisenberg model (wherein $J_0=356\text{rad/s}$ and $\alpha =0.70$). To characterize the dynamics fully, we derive the spin-excitation hopping rate in Appendix pp2.

Figure 2

The average distance from the system-of-interest state to the thermal prediction versus time. The ion chain consists of $N=21$ qubits. Each nearest-neighbor pair forms a small system of interest. We measure the distance of the state to the NATS (blue points), using the relative entropy [8], and average over the pairs in the chain. The markers show experimental data, while the lines are calculated numerically from Eq. (3). Each data point is formed from 250 repetitions. The error bars are estimated by bootstrapping [66]. The entropies are measured in units of nats (are to base $e$). We also compare the NATS prediction with two competitor thermal predictions, following Refs. [21, 47, 58]: canonical and grand canonical states. At all times, the NATS predicts the state unambiguously more accurately than the competitors do.

Figure 3

The average distance from the long-time system-of-interest state to the thermal prediction versus total number of qubits. The markers show experimental data, while the lines are calculated numerically, using unitary dynamics, from Eq. (3). The NATS predicts the final state the most accurately at all system sizes. Depolarizing noise appears to explain the experiment-theory discrepancies. The $N=6$ point is an outlier due to the small size of the global system.

Figure 4

Relative entropies from simulations with and without depolarizing noise versus time. We simulate 12 qubits subject to Trotterized Heisenberg evolution alone (filled markers) or with depolarization (empty markers). The blue circles show relative-entropy distances to the NATS and the orange squares show distances to the grand canonical state.

Figure 5

The relative entropy to the NATS for each qubit pair, as a function of time. The spatiotemporal fluctuations show that different qubit pairs thermalize to different extents. The chain consists of $N=21$ ions.

Figure 6

Dynamical decoupling. The simulation is performed with 12 ions, a power-law approximation to the coupling, $J_0=510\mathrm{rad}/\mathrm{s}$, $\alpha =1.02$, and 10 ms of evolution. The fidelity compares the simulated Trotter-approximated state with the exact ideal state: ${\left(\mathrm{Tr}\sqrt{\sqrt{{\rho }_{\mathrm{exact}}}{\rho }_{\mathrm{Trotter}}\sqrt{{\rho }_{\mathrm{exact}}}}\right)}^2$. (a) The introduction of $\pi /2$ rotations into the Trotter sequence guards against detuning errors. The right-hand side (rhs) of Eq. (H5) defines the sequence $U^{(0)}$, the rhs of Eq. (H22) defines $U^{(1)}$, and the rhs of Eq. (H23) defines $U^{(1\mathrm{alt})}$. (b) Alternating the direction of the rotations guards against systematic rotation errors. (c) The response to oscillations of a time-varying magnetic field $\mathbf{B}=B_{\mathrm{amp}}\cos (2\pi ft)\hat{z}$, wherein $B_{\mathrm{amp}}\ge 0$ (15 ms evolution). The dynamically decoupled Trotter sequence $U^{(1\mathrm{alt})}$ allows the fidelity to drop. The drops occur when the frequency of the field, $f$, is an integer multiple of $f_1=\frac{1}{2}(4T_{\mathrm{tot}}/N_{\mathrm{Trotter}}{)}^{-1}=300\mathrm{Hz}$. We can understand this behavior most simply when $J_0=0$ (top curves): the qubits do not interact, so each qubit remains in a superposition, the relative phase of which changes undesirably under $\mathbf{B}$.

Figure 7

The measured fidelity of the state resulting from the Trotter approximation. The fidelity of the experimentally observed state to the ideal state is averaged over ion pairs and plotted against time. The error bars indicate the standard deviation over the pairs. The smaller subplots show the fidelities, at the start and end of the evolution, to the observed states of individual ion pairs. The two types of markers represent two cases that we compare. First, we realize a 12-qubit system using a chain of only 12 ions (filled blue markers). The filled gray dots show the corresponding theoretical prediction. Second, we realize a 12-qubit system using a 21-ion chain, by hiding the extraneous ions from the interactions (empty orange markers). The open gray dots show the theoretical prediction for this case. The measurements are carried out (a) for the Trotter-approximated Heisenberg Hamiltonian and (b) for the Trotter-approximated $XY$ Hamiltonian. The fidelity drops and revives at early times. This behavior results from the failure of the Trotter steps to conserve the charges of the exact Hamiltonian, leading to periodic errors.