Periodic Steady Regime and Interference in a Periodically Driven Quantum System

We study the coherent dynamics of a quantum many-body system subject to a time-periodic driving. We argue that in many cases, destructive interference in time makes most of the quantum averages time periodic, after an initial transient. We discuss in detail the case of a quantum Ising chain periodically driven across the critical point, finding that, as a result of quantum coherence, the system never reaches an infinite temperature state. Floquet resonance effects are moreover observed in the frequency dependence of the various observables, which display a sequence of well-defined peaks or dips. Extensions to nonintegrable systems are discussed.

Figure 1

(Upper inset) Sketch of the dynamics within each $k$ subspace: At each avoided crossing the system can be either reflected by the gap or transmitted with amplitude $r_k$ and $t_k$. (Main figure) Evolution of the average energy density $e(t)=⟨\Psi (t)|\widehat{H}(t)|\Psi (t)⟩/L$ versus $t$ with a driving field $h(t)=1+\cos ({\omega }_{0}t)$, for $\hslash {\omega }_0/J=2$. The lower and upper curves are the instantaneous ground state and maximum energy density, versus $t$. Notice the initial transient (left panel, obtained from numerical solution with $L=2000$), and the final periodic behaviour of $e(t)$ (right panel, obtained from $e^{\mathrm{per}}(t)$, again with $L=2000$).

Figure 2

(a) The density of defects for a driving field $h(t)=1+\cos ({\omega }_{0}t)$ at the end of half-periods, i.e., at times $t=t_{2n+1}=(2n+1)\pi /{\omega }_0$ when the transverse field vanishes; $n=0$ is the result of a single LZ crossing. (b) The total work (per spin) done on the system $w=e(t_{2n}\to \infty )-e(0)$ versus the frequency ${\omega }_0$ of the transverse field.

Figure 3

(Left panel) The Floquet spectrum for an Ising chain with a Gaussian inhomogeneity with $l=20$, $h_G=2.8$, ${\omega }_0=10$, $A=1$ and different values of $L$. Notice a finite $L$-independent number of discrete quasienergies. (Upper right panel) The average transverse magnetization $m(t)$ probed at $t=n\tau$, showing fluctuations that decrease for increasing $L$. (Lower right panel) The transverse magnetization at the center of the inhomogeneity, $m_{j_c}(t=n\tau )$, whose fluctuations persist for all $L$.