Relaxation of a periodically-driven coherent many body system
Recent experimental techniques have allowed to realize, in the so-called cold atomic systems, the quantum-coherent dynamics of many particles undergoing an arbitrary perturbation. Among the many application, one which has attracted a lot of interest is the possibility to study the evolutionof a non-equilibrium state of the Hamiltonian: the so-called quantum-quench protocols. When the spectrum of the Hamiltonian obeys some quite general conditions, local observables relax to a steady state described by the so-called “diagonal ensemble density matrix”, strictly depending on the eigenstates of the Hamiltonian. A question dating back to the origins of quantum mechanics is when the steady state is a thermal equilibrium one. It results, indeed, that this happens when the eigenstates are random states obeying the so-called Eigenstate Thermalization condition, but there is no general theory telling us precisely when this happens. This is quite different from classical thermalization induced by ergodic microscopic dynamics.
After having reviewed these results, clarifying the meaning of the words “integrable” and “ergodic”, I will discuss my recent work aimed to generalize them to the case of periodically driven systems. Noteworthy, even in this case, under conditions very similar to those of the quantum
quench, local observables relax to a steady condition described by the so-called “Floquet-diagonal density matrix”, very similar to its quenched counterpart. Here the observables in the steady condition are time-periodic and the energy-eigenstates are replaced by the Floquet states, which are the eigenstates of the stroboscopic dynamics. I will show numerical results on the Quantum Ising chain, which never thermalizes being integrable, and on its fully-connected counterpart, the Lipkin model. This quantum system has a well defined classical limit and it thermalizes to T = ∞ whenever the classical dynamics is ergodic. This thermalization is induced by the Floquet states obeying ETH and being extended in the Hilbert space. I will conclude the seminar moving from the discussion of local observables, with a well-defined classical limit, to truly quantum non-local objects like the fidelity.
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