Floquet dynamical quantum phase transitions

Dynamical quantum phase transitions (DQPTs) are manifested by time-domain nonanalytic behaviors of many-body systems. Introducing a quench is so far understood as a typical scenario to induce DQPTs. In this work, we discover a type of DQPTs, termed Floquet DQPTs, as intrinsic features of time-periodic systems. Floquet DQPTs occur within each period of continuous driving, without the need for any quenches. In particular, in a harmonically driven spin chain model, we find analytically the existence of Floquet DQPTs in and only in a parameter regime hosting a certain nontrivial Floquet topological phase. The Floquet DQPTs are further characterized by a dynamical topological invariant defined as the winding number of the Pancharatnam geometric phase versus quasimomentum. These findings are experimentally demonstrated with a single spin in diamond. This work thus opens a door for future studies of DQPTs in connection with topological matter.

Figure 1

Sketch of conventional DQPTs and Floquet DQPTs. (a) Scenario of conventional DQPTs following a quench. $H_i$ and $H_f$ denote the Hamiltonians before and after the quench. (b) Scenario of Floquet DQPTs. $H\left(t\right)$ denotes a periodically and continuously modulated Hamiltonian. The occupied energy or quasienergy bands are embroidered with halos. (c) and (d) show rate functions under the scenario illustrated in (a) and (b), respectively.

Figure 2

Experimental system and method. (a) NV center in diamond. (b) Electronic ground state of a negatively charged NV center. The energy splitting depends on the magnetic field which is parallel to the NV axis in this experiment. The two levels $\left|m_s=0\right.〉$ and $\left|m_s=-1\right.〉$ are encoded as a qubit which is manipulated by microwaves (MW). (c) Pulse sequence for qubit control and measurement. In the pulse sequence illustrated here, the microwave pulse in the measurement section aims for acquiring the return probability. The Bloch vector of the quantum state and the direction of the rotating field are shown below the pulse sequence.

Figure 3

Return probabilities and rate functions. (a), (b) Return probabilities for ${\delta }_2=2\pi \times 5$ MHz and $-2\pi \times 5$ MHz, respectively. Experimental data are shown in the upper panels and theoretical results are presented in the lower panels. All panels in (a) and (b) share the same color bar. The associated rate functions are shown in panels (c) and (d), with black circles and red curves representing experimental data and theoretical values.

Figure 4

Measured geometric phases versus system parameter $k$ and time $t$ for (a) ${\delta }_2=2\pi \times 5$ MHz and (b) ${\delta }_2=-2\pi \times 5$ MHz. The bottom two panels represent theoretical results for the sake of comparison. In left panels, a change in the net jumps of the geometric phase along the $k$ dimension depicts a change in a dynamical topological invariant, thus signifying DQPTs at $t=0.1\mu \mathrm{s},0.3\mu \mathrm{s}$, and $0.5\mu \mathrm{s}$. No such behavior is found in the right panels.

Figure 5

Expectation values of ${\sigma }_x,{\sigma }_y$, and ${\sigma }_z$. (a)–(c) Expectation values of ${\sigma }_x,{\sigma }_y$, and ${\sigma }_z$ as a function of the synthetic quasimomentum $k$ and the scaled time $\tau$ for ${\delta }_2=5$ MHz. (d)–(f) Expectation values of ${\sigma }_x,{\sigma }_y$, and ${\sigma }_z$ as a function of the synthetic quasimomentum $k$ and the scaled time $\tau$ for ${\delta }_2=-5$ MHz. Calculations based on ideal situations are on the left and experimental data are on the right.