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 $\lambda \left(t\right)$ of the drive increasing from zero to a maximum ${\lambda }_{\max }$ and then going back to zero again, the initial and final Hamiltonian coincide. The main quantity of interest is then the probability density $P\left(E_f\right|E_i)$ for transitions from initial energy $E_i$ to final energy $E_f$. In the classical case nondiagonal transitions with $E_f\ne E_i$ 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 $P\left(E_f\right|E_i)$ with the maximal amplitude ${\lambda }_{\max }$ of the driving.

Figure 1

External force (2) over time for ${\lambda }_{\max }=3.5, \omega =7.7$, and $t_f=1002\pi /\omega$.

Figure 2

Histogram of work values as defined in (15) obtained from numerical integrations of the classical equation of motion (10). Parameter values are ${\lambda }_{\max }=3.5, \omega =7.7$, and $t_{\mathrm{f}}=10002\pi /\omega$. The inverse temperature of the bath is $\beta =1/E_{\omega }$, with $E_{\omega }$ defined by (12). The average value resulting from this distribution is indicated by the red line.

Figure 3

The classical transition probability $P_t^C\left(E_f\right|E_i)$ color-coded as function of the initial and final energy. The parameters are the same as in Fig. 2. Outside the transition window $0.5\lesssim E_i/E_{\omega }\lesssim 1.6$, the final energy is almost always identical with the initial one. The green line shows the analytical result from the pendulum approximation.

Figure 4

Poincaré sections showing the dependence of the action-angle variables $(I,\theta )$ of the undriven system under a dynamics with $\lambda =\mathrm{const}$. Left: $\lambda =0$ implying $I\left(\theta \right)=\mathrm{const}$. Shown are lines for six different choices of $E_i$. Right: $\lambda ={\lambda }_{\max }$. The six lines $I\left(\theta \right)$ are now bent and may meet, forming closed loops, such that transitions by separatrix crossing to values $E_f\ne E_i$ may take place.

Figure 5

Poincaré sections for the original system (24) (full lines) and for the pendulum approximation (31) (dashed lines) for $\lambda =1$. Note that the colors do not correspond to the same values of $E_i$ in Fig. 4 since the value of $\lambda$ is different.

Figure 6

Quantum transition probability $P_t^Q\left(E_f\right|E_i)$ color coded as a function of the initial and final energy. The parameters are the same as in Fig. 3, and the time step used in the numerical integration of the Schrödinger equation is $\mathrm{\Delta }t=4\times {10}^{-5}$.

Figure 7

Histogram of work values for the quantum case. The parameters are the same as in Fig. 2, and the time step used in the numerical integration of the Schrödinger equation is $\mathrm{\Delta }t=4\times {10}^{-5}$. The average value of the work resulting from the histogram and indicated by the red line is again positive, as required by the Jarzynski equality (16).

Figure 8

Selection of quasienergies ${\epsilon }_n$ divided by $\omega$ for $n=9,\cdots ,29$ as function of $\lambda$ for the parameter values of Fig. 6. The red lines correspond to the transitions on the secondary diagonal in Fig. 6, the black ones to the transitions between states 3 and 9.

Figure 9

Classical (blue) and quantum (red) transition probability $P\left(E_{16}\right|E_{16})$ as a function of ${\lambda }_{\max }$. Parameter values are the same as in the other figures.

Figure 10

Poincaré sections of classical orbits originating from $E_i=E_{16}$ (lower parts) and $E_i=E_{22}$ (upper parts) just before the two parts separate again in the course of decreasing $\lambda \left(t\right)$ for ${\lambda }_{\max }=2.05$ (left) and ${\lambda }_{\max }=2.085$ (right). For both initial energy values, 2500 classical trajectories were generated (red points), and 500 randomly selected initial conditions with $E_i=E_{16}$ give rise to the black dots. In the left figure almost all of these black points return to their initial energy value; in the right one most end up at $E_f=E_{22}$.