Direct Observation of Dynamical Quantum Phase Transitions in an Interacting Many-Body System

The theory of phase transitions represents a central concept for the characterization of equilibrium matter. In this work we study experimentally an extension of this theory to the nonequilibrium dynamical regime termed dynamical quantum phase transitions (DQPTs). We investigate and measure DQPTs in a string of ions simulating interacting transverse-field Ising models. During the nonequilibrium dynamics induced by a quantum quench we show for strings of up to 10 ions the direct detection of DQPTs by revealing nonanalytic behavior in time. Moreover, we provide a link between DQPTs and the dynamics of other quantities such as the magnetization, and we establish a connection between DQPTs and entanglement production.

Figure 1

Schematic comparison of conventional and dynamical quantum phase transitions. (a) Equilibrium temperature—control parameter phase diagram. A quantum phase transition (QPT) is a continuous phase transition occurring at $T=0$ and separating two phases, e.g., a ferromagnet for $g<g_c$ from a paramagnet for $g>g_c$ (black arrows). At $g_c$, physical quantities become nonanalytic upon varying $g$ (yellow arrows), triggered by a change in the spectrum of the system Hamiltonian $H=H_0+H_1(g)$, which becomes gapless as indicated by the schematic energy-level structure. Though only occurring at $T=0$, QPTs control the system’s properties also in the quantum critical region at $T>0$. (b) Dynamics in the energy density—time plane. A DQPT occurs along the $ϵ=0$ axis at $t=t_c$, separating two regimes of, e.g., opposite magnetization (black arrows). The DQPT is not associated with a change in the spectrum but with an incisive redistribution of occupations between the eigenstates of the initial Hamiltonian $H_0$, induced by the perturbation $H_1$. In the present experiment, $H_0$ exhibits two degenerate ground states of opposite magnetization, and the DQPT is caused by a sudden change of the low-energy occupations from one of the two ground states to the other. Though the mean energy density (red line), where many observables acquire their dominant contribution, lies at $ϵ>0$, the nonequilibrium dynamics of observables can still be controlled by the underlying DQPT (white space).

Figure 2

Observation of dynamical quantum phase transition. (a) Measured rate function $\lambda (\tau )$ for three different system sizes at $J/B\approx 0.42$, showing a nonanalytical behavior (with $\tau =tB$ being the dimensionless time). Dots are experimental data with error bars estimated from quantum projection noise; lines are numerical simulations with experimental parameters. In a lighter black color we have included data for the subdominant contributions at $N=6$. Inset: The transition between the normalized ground-state probabilities $P_{\Rightarrow }/P$ (solid lines) and $P_{\Leftarrow }/P$ (dashed lines) becomes sharper for larger $N$. (b) The first-order correction to the critical time ${\tau }_{\mathrm{crit}}$, i.e., the occurrence of the first DQPT, is linear as a function of ${(J/B)}^2$ for small $J/B$, and approximately independent of interaction range. Error bars are $1\sigma$ confidence intervals of the fits on $\log [P_{\Rightarrow ,\Leftarrow }(\tau )]$ from which we extract ${\tau }_{\mathrm{crit}}$ (see Supplemental Material [24]). Inset: DQPT for $(J/B)=0$, 0.392, and 0.734 (light blue, dark blue, and black dots). The grey dashed lines indicate ${\tau }_{\mathrm{crit}}$ for $(J/B)=0$.

Figure 3

Control of the magnetization dynamics by a DQPT. DQPTs, indicated by kinks in $\lambda (\tau )$ (a), control the average magnetization in the $x$ direction, $M_x$ (b). (c) This connection becomes apparent when resolving the magnetization against energy density $\epsilon$, with the nonanalyticity at $\epsilon =0$ radiating out to $\epsilon >0$. For details on the measurement of the energy-resolved magnetization, see Supplemental Material [24]. In (a) and (b), dots indicate experimental data with errors derived from quantum projection noise; solid lines denote numerical simulations ($J/B=0.5$).

Figure 4

Entanglement production. Dynamics of (a) the half-chain entropy $\mathcal{S}$ and (b) spin squeezing ${\xi }_S^2$ for $N=6$ spins at $\alpha \approx 0$. For nonzero interactions, both entanglement quantifiers show a marked increase in the vicinity of the DQPTs, indicated by dashed lines [$J/B=0.223$ in (a) and 0.25 in (b)]. (a) Comparison of the measured half-chain entropy obtained from quantum tomography (circles) with the entropies resulting from solving the Schrödinger equation using our experimental parameters, with the ideal input state $|\Rightarrow ⟩$ (red line) and a slightly depolarized input state (blue line). Entropies obtained from simulating the tomographic reconstruction including projection noise are slightly higher, as indicated for the mixed initial state by the shaded area ($1\sigma$ confidence region). (b) The change in ${\xi }_s^2(t)$ signals qualitatively similar entanglement production (red symbols). For $J/B=0$, no entanglement is created (black symbols).