Quantum kicked harmonic oscillator in contact with a heat bath

We consider the quantum harmonic oscillator in contact with a finite-temperature bath, modeled by the Caldeira-Leggett master equation. Applying periodic kicks to the oscillator, we study the system in different dynamical regimes between classical integrability and chaos, on the one hand, and ballistic or diffusive energy absorption, on the other. We then investigate the influence of the heat bath on the oscillator in each case. Phase-space techniques allow us to simulate the evolution of the system efficiently. In this way, we calculate high-resolution Wigner functions at long times, where the system approaches a quasistationary cyclic evolution. Thereby, we perform an accurate study of the thermodynamic properties of a nonintegrable, quantum chaotic system in contact with a heat bath at finite temperature. In particular, we find that the heat transfer between harmonic oscillator and heat bath is governed by Fourier's law.

Figure 1

Wigner function for the quantum KHO for $q=4$, coupled to a heat bath (with $\beta =0.1$ and $D=5$), with an initial coherent state positioned at $z=0,p=0$. Different rows correspond to different choices of $\kappa$ and $\eta$. Resonant case ($\kappa =-0.8,{\eta }^2=\pi$) (a) after the 35th kick, (b) before the 36th kick, and (c) after the 36th kick. (d)–(f) Nonresonant case ($\kappa =-0.8,{\eta }^2=0.7\pi$) at the same instances in time. (g)–(i) Chaotic case ($\kappa =-4.5,{\eta }^2=1$).

Figure 2

Energy of the dimensionless harmonic oscillator $\langle H_0\rangle$ vs the number of kicks for the same three cases shown in Fig. 1. The lowest curve (red line) corresponds to the resonant case with $\kappa =-0.8$ and ${\eta }^2=\pi$; the curve slightly above (green line) corresponds to the nonresonant case with $\kappa =-0.8$ and ${\eta }^2=0.7\pi$. The top curve (blue line) corresponds to the chaotic case with $\kappa =-4.5$ and ${\eta }^2=0.7\pi$.

Figure 3

Difference between the temperature of the system, measured by $D^{\prime }$ from Eq. (16), and the temperature of the heat bath vs the heat current $dQ/d\tau$ from Eq. (19) for $q=4$ and different KHO parameters $\kappa$ and ${\eta }^2$, as well as coupling strengths $\beta$ (see the legend). For each of the four cases, ${\eta }^2/\pi$ varies from 0.2 to 1 in integer steps. Larger values of ${\eta }^2$ mean smaller kick amplitudes, and hence the corresponding points are closer to the origin. The black dashed lines show the theoretical expectation according to Fourier's law, Eq. (19).

Figure 4

Wigner functions right after the 36th kick for different coupling strengths: $\beta =0.001$ (first column), $\beta =0.01$ (second column), and $\beta =0.1$ (third column) for the resonant case (first row) with $q=4,\kappa =-0.8$, and ${\eta }^2=\pi$; the nonresonant case (second row) with $q=4, \kappa =-0.8$, and ${\eta }^2=(1+\sqrt{5})\pi /2$; and the chaotic case (third row) with $\kappa =-4.5,{\eta }^2=1$ but a kick period with $q=6$.

Figure 5

Energy evolution in time in the resonant case, $q=6,\kappa =-0.8,{\eta }^2=2\pi /\sqrt{3}$, for different coupling strengths from $\beta ={10}^{-5}$ to $\beta =0.1$. The increase in energy shown by the different curves is slower the larger $\beta$ is.

Figure 6

Plots of the number of kicks required to reach a mean energy of $\langle H_0\rangle =50$ for the resonant case (with $\kappa =-0.8,q=4$) and different coupling strength values $\beta$; the resonance is in ${\eta }^2/\pi =0.5$.