Floquet-Markov description of the parametrically driven, dissipative harmonic quantum oscillator

A periodically driven quantum system, when coupled to a heat bath, relaxes to a non-equilibrium asymptotic state. In the general situation, the retrieval of this asymptotic state presents a rather non-trivial task. It was recently shown that in the limit of an infinitesimal coupling, using so-called rotating wave approximation (RWA), and under strict conditions imposed on the time-dependent system Hamiltonian, the asymptotic state can attain the Gibbs form. A Floquet-Gibbs st...

We study transitions between the Floquet states of a periodically driven oscillator caused by the coupling of the oscillator to a thermal reservoir. The analysis refers to the oscillator that is driven close to triple its eigenfrequency and displays resonant period tripling. The interstate transitions result in a random ``walk'' over the states. We find the transition rates and show that the walk is nonlocal in the state space: the stationary distribution over the states is f...

We construct a quantum Markovian Master equation for a driven system coupled to a thermal bath. The derivation utilizes an explicit solution of the propagator of the driven system. This enables the validity of the Master equation to be extended beyond the adiabatic limit. The Non-Adiabatic Master Equation (NAME) is derived employing the weak system-bath coupling limit. The NAME is valid when a separation of timescales exists between the bath dynamics and the external driving....

We investigate the asymptotic state of a periodically driven many-body quantum system which is weakly coupled to an environment. The combined action of the modulations and the environment steers the system towards a state being characterized by a time-periodic density operator. To resolve this asymptotic non-equilibrium state at stroboscopic instants of time, we introduce the dissipative Floquet map, evaluate the stroboscopic density operator as its eigen-element and elucidat...

We provide a comprehensive study of the energy transfer phenomenon -- populating a given energy level -- in 3- and 4-level quantum systems coupled to two thermal baths. In particular, we examine the effects of an external periodic driving and the coherence induced by the baths on the efficiency of the energy transfer. We consider the Floquet-Lindblad and the Floquet-Redfield scenarios, which both are in the Born-Markov, weak-coupling regime but differ in the treatment of the ...

We reformulate the Floquet theory for periodically driven quantum systems following a perfect analogy with the proof of Bloch theorem. We observe that the current standard method for calculating the Floquet eigenstates by the quasi-energy alone is incomplete and unstable, and pinpoint an overlooked quantum number, the average energy. This new quantum number resolves many shortcomings of the Floquet method stemming from the quasi-energy degeneracy issues, particularly in the c...

We show theoretically how a driven harmonic oscillator can be used as a quantum simulator for non-Markovian damped harmonic oscillator. In the general framework, the results demonstrate the possibility to use a closed system as a simulator for open quantum systems. The quantum simulator is based on sets of controlled drives of the closed harmonic oscillator with appropriately tailored electric field pulses. The non-Markovian dynamics of the damped harmonic oscillator is obtai...

For a closed system with periodic driving, Floquet theorem tells that the time evolution operator can be written as $ U(t,0)\equiv P(t)e^{\frac{-i}{\hbar}H_F t}$ with $P(t+T)=P(t)$, and $H_F$ is Hermitian and time-independent called Floquet Hamiltonian. In this work, we extend the Floquet theorem from closed systems to open systems described by a Lindblad master equation that is periodic in time. Lindbladian expansion in powers of $\frac 1 \omega$ is derived, where $\omega$ i...

We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper-Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency...

Quantum adiabaticity is defined as the evolution of a quantum system close to an instantaneous eigenstate of a time-dependent Hamiltonian without transition. Using Floquet formalism, we prove a rigorous sufficient condition for quantum adiabaticity in periodically driven systems, valid for arbitrarily long period. Unlike traditional conditions, the Floquet condition is tight, does not require additional constraints, and predicts that adiabaticity may exist at high frequencies...