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We develop a Magnus formalism for periodically driven systems which provides an expansion both in the driving term and the inverse driving frequency, applicable to isolated and dissipative systems. We derive explicit formulas for a driving term with a cosine dependence on time, up to fourth order. We apply these to the steady state of a classical parametric oscillator coupled to a thermal bath, which we solve numerically for comparison. Beyond dynamical stabilization at second order, we find that the higher orders further renormalize the oscillator frequency, and additionally create a weakly renormalized effective temperature. The renormalized oscillator frequency is quantitatively accurate almost up to the parametric instability, as we confirm numerically. Additionally, a cut-off dependent term is generated, which indicates the break-down of the hierarchy of time scales of the system, as a precursor to the instability. Finally, we apply this formalism to a parametrically driven chain, as an example for the control of the dispersion of a many-body system.
To investigate a system coupled to a harmonic oscillator bath, we propose a new approach based on a phonon number representation of the bath. Compared to the method of the hierarchical equations of motion, the new approach is computationally much les
In this paper we examine some foundational issues of a class of quantum engines where the system consists of a single quantum parametric oscillator, operating in an Otto cycle consisting of 4 stages of two alternating phases: the isentropic phase is
We propose a new concept for the dynamics of a quantum bath, the Chebyshev space, and a new method based on this concept, the Chebyshev space method. The Chebyshev space is an abstract vector space that exactly represents the fermionic or bosonic bat
The theory of probability shows that, as the fraction $X_n/Yto 0$, the conditional probability for $X_n$, given $X_n+Y in h_{delta}:=[h, h+delta]$, has a limit law $f_{X_n}(x)e^{-psi_n(h_delta)x}$, where $psi_n(h_delta) $ equals to $[partial ln P(Y i
The rectification of unbiased fluctuations, also known as the ratchet effect, is normally obtained under statistical non-equilibrium conditions. Here we propose a new ratchet mechanism where a thermal bath solicits the random rotation of an asymmetri