Measuring out-of-time-order correlations and multiple quantum spectra in a trapped ion quantum magnet

TL;DR

The study establishes and experimentally implements a time-reversal protocol to measure out-of-time-order correlators in a large trapped-ion quantum magnet, revealing a spectrum of multi-spin coherences via the Fourier components of a fidelity signal $F_\phi(\tau)$. By linking $F_\phi(\tau)$ to the MQC spectrum $\sum_m I_m e^{-im\phi}$, the authors quantify scrambling and the growth of $m$-body correlations in an all-to-all Ising model with $N>100$ ions, and they benchmark the system against a full Lindblad master equation. The work demonstrates the buildup of up to eight-body coherences, provides a framework to study many-body localization, phase transitions, and holographic duality, and delivers a robust methodology for diagnosing decoherence and validating quantum simulators in large spin networks. The combination of precise spin-motion control, time-reversal dynamics, and symmetry-aware simulations enables high-fidelity probing of complex quantum information flow in many-body systems.

Abstract

Controllable arrays of ions and ultra-cold atoms can simulate complex many-body phenomena and may provide insights into unsolved problems in modern science. To this end, experimentally feasible protocols for quantifying the buildup of quantum correlations and coherence are needed, as performing full state tomography does not scale favorably with the number of particles. Here we develop and experimentally demonstrate such a protocol, which uses time reversal of the many-body dynamics to measure out-of-time-order correlation functions (OTOCs) in a long-range Ising spin quantum simulator with more than 100 ions in a Penning trap. By measuring a family of OTOCs as a function of a tunable parameter we obtain fine-grained information about the state of the system encoded in the multiple quantum coherence spectrum, extract the quantum state purity, and demonstrate the buildup of up to 8-body correlations. Future applications of this protocol could enable studies of many-body localization, quantum phase transitions, and tests of the holographic duality between quantum and gravitational systems.

Figures (9)

• Figure 1: Illustration of the many-body echo scheme.a, Experimental sequence. The global $-\pi/2$ rotation $\hat{R}_y$ about the $y$-axis prepares an initial state with all spins pointing along the $x$-axis, and enables a measurement in this same basis. The generalized Bloch spheres illustrate the evolution of the state (Husimi distribution). In the case of $\phi=0$ (blue) the spins return to the initial state, while for $\phi=\pi/2$ (green) the overlap of the final state $\hat{\rho}_f$ with the initial state is small. b, Fidelity signal for an idealized case with $N=6$ spins and different evolution times $\tau$ given in c. c, The Fourier transforms of the fidelity signals of b. The Fourier amplitudes are identical to the MQCs $I_m$, which quantify the coherence of the state $\hat{\rho}(\tau)$. The small squares on the right show the absolute values of the density matrix elements of $\hat{\rho}(\tau)$ in the basis of symmetric Dicke states. Thus, $I_m$ is the sum of the squares of all matrix elements at a distance $m$ from the diagonal. The times are given in units of the time to reach the Schrödinger cat state $t_{\text{cat}}=\pi\hbar N/(4J)$. d, Simulated dynamics of the Fourier amplitudes of fidelity, $I_m$, and magnetization, $A_m$, for purely coherent evolution of $48$ ions, illustrating complementary probes of the flow of quantum information. The vanishing odd Fourier components are not shown.
• Figure 2: Phonon-mediated, reversible spin-spin coupling in a Penning trap.a, (left) Illustration of Penning trap cross-section. Ions (blue circles) are confined axially to a single 2D plane (shown in b) with static electric fields from potentials on the electrodes (gold). Rotation of the ions in the axial magnetic field $\vec{B}$ produces radial confinement from the Lorentz force. A pair of detuned ODF beams (green) interfere and form a traveling wave optical lattice, producing spin-dependent COM mode excitations that couple the spins to the axial phonon mode. Shown here are two of $(2N+1)$ excitations: all ions in $\left| \uparrow \right\rangle$ (purple) and all in $\left| \downarrow \right\rangle$ (orange). (right) The phonon wave packets experience equal and opposite displacement in the axial potential $V_z$. Spin-dependent motion, along with the Coulomb interaction, generates the spin-spin coupling. b, Rotating frame image of 2D array of $^9$Be$^+$ ions, integration time 2.1 s. c, Residual spin-phonon coupling for drive frequencies away from the decoupling points $\pm \delta$ appears as a decrease in the magnetization measured after the experimental sequence from Fig. \ref{['fig:seq']}, with $\phi = \pi$, and without inverting $\hat{H}_{\text{zz}}$. Here $\tau = 200\,\mu$s. Note that decoupling points appear at $\pm \delta$ with $+\delta$ giving an anti-ferromagnetic interaction, and $-\delta$ giving a ferromagnetic interaction used for the time reversal of the $\hat{H}_{\text{zz}}$ dynamics.
• Figure 3: Measured fidelity and coherence spectrum of $N=48$ ions.a,b, Dependence of the fidelity $\mathcal{F}_\phi(\tau)$ on the rotation angle $\phi$. The experimental sequence in b includes an additional $\pi$ pulse in the middle of each evolution period $\tau$. The dashed lines are simulations including off-resonant light scattering as the only source of decoherence, with $\Gamma=62\,$s$^{-1}$. The solid lines include effects of COM mode and magnetic field fluctuations, with COM mode frequency fluctuations $\Delta_{\text{COM}}/\omega_z=8.0\times10^{-5}$ RMS, and magnetic field noise $\Delta_B/B = 0.32 \times 10^{-9}\,$ RMS (Methods). Note that for each interaction time $\tau$ the detuning is chosen so that $\delta = 2\pi/\tau$ (a) or $\delta = 4\pi/\tau$ (b). In each case, the spin-spin coupling also varies as $J/\hbar = \Omega_0^2/(2\delta)$ where $\Omega_0 = 7850$ s$^{-1}$. c, Fourier amplitudes of fidelity (b) as a function of time. Solid lines are simulations including all known decoherence processes. $I_2$ and $I_4$ clearly show the buildup of higher order MQCs. Odd coherences and coherences $m\geq 6$ are zero within the statistical error. For $I_0$, decoherence induced decay (dashed) and approximate analytic curve (dotted, see text) are shown. The data points at $\tau=0.3$ and $0.9$ (not shown in b) have been added. The longest measured evolution time of $\tau=1\,$ms corresponds to $6.5\%$ of $t_{\text{cat}}$ (cf. Fig. \ref{['fig:seq']}d). All error bars denote the statistical error of $1$ standard deviation (SD) of the mean.
• Figure 4: Probing scrambling through magnetization dynamics.a, Dependence of the normalized component $F_\phi(\tau)=(2/N)\langle \hat{S}_x\rangle$ of the total spin on the rotation angle $\phi$, measured in an array of $N=111(2)$ ions. Lines are the solutions of the full master equation with (solid) and without (dashed) magnetic field noise, where $\Delta_B/B = 0.32 \times 10^{-9}\,$ RMS. The effect of COM mode fluctuations is negligible here. Error bars denote the statistical error of $1$ SD of the mean. b, Fourier amplitudes $A_m$ as a function of time. In the theory plot, the case without magnetic field noise (dashed lines in a) was used. The interaction parameter varies as $J/\hbar = \Omega_0^2/(2\delta)$ where $\Omega_0 = 7450$ s$^{-1}$ and $\Gamma=91\,$s$^{-1}$. The longest measured evolution time of $\tau=1.2\,$ms corresponds to $7.3\%$ of $t_{\text{cat}}$. c, Ideal case for $N=111$, neglecting all decoherence effects. This corresponds to the lower panel of Fig. \ref{['fig:seq']}d. The box in the left panel shows the experimentally accessed region which is magnified in the right panel.
• Figure 5: Extraction of fidelity.a, Reference histogram taken with all ions optically pumped into the "dark" state $\left| \downarrow \right\rangle$. b, Example of a photon count histogram obtained with the full MQC sequence.