Work statistics in the periodically driven quartic oscillator: Classical versus quantum dynamics

In the thermodynamics of nanoscopic systems, the relation between classical and quantum mechanical description is of particular importance. To scrutinize this correspondence we study an anharmonic oscillator driven by a periodic external force with slowly varying amplitude both classically and within the framework of quantum mechanics. The energy change of the oscillator induced by the driving is closely related to the probability distribution of work for the system. With the amplitude λ(t) of the drive increasing from zero to a maximum λ_{max} and then going back to zero again, the initial and final Hamiltonian coincide. The main quantity of interest is then the probability density P(E_{f}|E_{i}) for transitions from initial energy E_{i} to final energy E_{f}. In the classical case nondiagonal transitions with E_{f}≠E_{i} mainly arise due to the mechanism of separatrix crossing. We show that approximate analytical results within the pendulum approximation are in accordance with numerical simulations. In the quantum case numerically exact results are complemented with analytical arguments employing Floquet theory. For both the classical and quantum case we provide an intuitive explanation for the periodic variation of P(E_{f}|E_{i}) with the maximal amplitude λ_{max} of the driving.