Observation of a Dynamical Quantum Phase Transition by a Superconducting Qubit Simulation

A dynamical quantum phase transition can occur during time evolution of sudden quenched quantum systems across a phase transition. It corresponds to the nonanalytic behavior at a critical time of the rate function of the quantum-state return amplitude, analogous to nonanalyticity of the free-energy density at the critical temperature in macroscopic systems. A variety of many-body systems can be represented in momentum space as a spin-1/2 state evolving on the Bloch sphere, where each momentum mode is decoupled and thus can be simulated independently by a single qubit. Here, we report the observation of a dynamical quantum phase transition in a superconducting qubit simulation of the quantum-quench dynamics of many-body systems. We take the Ising model with a transverse field as an example for demonstration. In our experiment, the spin state, which is initially polarized longitudinally, evolves based on a Hamiltonian with adjustable parameters depending on the momentum and strength of the transverse magnetic field. The time-evolving quantum state is read out by state tomography. Evidence of dynamical quantum phase transitions, such as paths of time-evolution states on the Bloch sphere, nonanalytic behavior of the dynamical free energy, and the emergence of Skyrmion lattice in momentum-time space, is observed. The experimental data agrees well with theoretical and numerical calculations. The experiment demonstrates explicitly the topological invariant, both topologically trivial and nontrivial, for dynamical quantum phase transitions. Our results show that the quantum phase transitions of this class of many-body system can be simulated successfully with a single qubit by varying certain control parameters over the corresponding momentum range.

Figure 1

Photography of qubit chip and external circuitries. (a) Microscopic photography of our Xmon qubit chip. The red part is the Xmon qubit. Its frequency can be adjusted by applying dc current through its $Z$ control line. The transmission line coupled to the readout cavity is to measure the qubit state. Basic information of the qubit is listed in the table. The experiment data of the energy relaxation time $T_1$, dephasing time $T_{2\ast }$ and spin-echo dephasing time $T_{2se}$ are also shown. (b) Sketch of our experiment circuit setup, the blue part is for $Z$ bias, the green part is for $XY$ control, and the brown part is for readout. (c) Qubit parameters are presented in the table.

Figure 2

Evolution of states on the Bloch sphere. The state evolves depending on a fixed momentum. Data of the evolving path are presented on the Bloch sphere. Here $g_i=0.2$ is fixed, cases with $g_f=1.5$ are presented in (a)–(g) in the upper panel, cases of $g_f=0.5$ are presented in (h)–(n) in the lower panel. The momenta are chosen to be $k=0$, $0.2\pi$, $0.4\pi$, $0.6\pi$, $0.8\pi$, $\pi$, presented, respectively, on up-down pairs of subfigures: (a),(h); (b),(i); (c),(j); (d),(k); (e),(l); (f),(m). Data for two different cases are summarized together in (g) for $g_f=1.5$, and (n) for $g_f=1.5$, respectively. We can find that the whole Bloch sphere in (g) is covered, in contrast in (n), only a partial region of the Bloch sphere is covered. We emphasize that the initial state is always prepared on the equator of the Bloch sphere in the $X$ axis.

Figure 3

Dynamical free energy. (a) Dynamical free energies for different $g_f$ are presented, while $g_i=0.2$ is fixed. We take $70$ time points for each period. The error bar represents the deviation of the average value of 5000 single-shot measurements from the fitting value for the evolution path of 70 points on the Bloch sphere, see the Appendix for details. (b) The dynamical free energies near the critical time ${\tau }_c$ for a different number of momenta implemented in experiment are given, corresponding to different sizes. The exact results for $N\to \mathrm{\infty }$ are presented as the solid line. Here we take $g_i=0.2$ and $g_f=1.5$.

Figure 4

Skyrmion and DQPT. (a),(b) Dynamical free energies for $g_f=1.5$ and $g_f=0.5$, both with $g_i=0$, respectively. The number of momenta is $N=30$. For each momentum mode $k$, states at $2\times 70=140$ time points are read out. The error bar is the deviation of the average value from the fitting value, see the Appendix for details. (c),(d) Expectation values of the initial spin operator about the evolving state $|\varphi (k,t)>$. The Skyrmion is shown obviously in (c), while no Skyrmion appears in (d).

Figure 5

Equivalent circuits of (a) the isolated transmon circuit and (b) its Xmon variety with control. (a) Isolated transmon circuit, (b) Xmon circuit with control.

Figure 6

Experiment control sequence. The initial state is prepared at the state-initialization period by control quantity $A_0\cos (\omega t+{\phi }_0)$. The state is prepared as $|{\varphi }_i⟩$. Then for a quantum quench, by controlling ${\phi }_k$ depending on momentum $k$, we adjust the direction of the rotation axis, shown in Eq. (B7).