Correspondence behavior of classical and quantum dissipative directed transport via thermal noise

We systematically study several classical-quantum correspondence properties of the dissipative modified kicked rotator, a paradigmatic ratchet model. We explore the behavior of the asymptotic currents for finite ${\hslash }_{\mathrm{eff}}$ values in a wide range of the parameter space. We find that the correspondence between the classical currents with thermal noise providing fluctuations of size ${\hslash }_{\mathrm{eff}}$ and the quantum ones without it is very good in general with the exception of specific regions. We systematically consider the spectra of the corresponding classical Perron-Frobenius operators and quantum superoperators. By means of an average distance between the classical and quantum sets of eigenvalues we find that the correspondence is unexpectedly quite uniform. This apparent contradiction is solved with the help of the Weyl-Wigner distributions of the equilibrium eigenvectors, which reveal the key role of quantum effects by showing surviving coherences in the asymptotic states.

Figure 1

(a) We display the distance between the classical and quantum asymptotic currents given by $|J_{\mathrm{c}}^{\mathrm{th}}-J_{\mathrm{q}}|$. We have considered this in order to better reflect the discrepancy in the approximation. (b) We show the sign difference $[\mathrm{sgn}\left(J_{\mathrm{c}}^{\mathrm{th}}\right)-\mathrm{sgn}\left(J_{\mathrm{q}}\right)]/2$.

Figure 2

Average distance between classical and quantum eigenvalues $\lambda$ in complex plane vs $k$ for values of $\gamma =0.4$, 0.45, 0.5, 0.55, and 0.6 from (e) to (a), correspondingly. Green, red, and violet solid lines show the average distance for eigenvalues with moduli larger than $0.25,0.35$ and $0.4$, respectively (values are shown in left axis). Dashed black lines represent classical currents (with values shown in right axis) with dotted blue lines showing zero current.

Figure 3

Black circles and red squares show the eigenvalues $\lambda$ of the classical and quantum systems, respectively, displayed in the complex plane. The green lines represent the distances between them, they are only computed when classical and quantum eigenvalues are larger than $0.35$ (in modulus). We have taken $\gamma =0.6$ and $k$ between 5 and 10 (with steps $\mathrm{\Delta }k=0.05$). The highlighted area (with a dashed violet line) of (a) is shown in (b).

Figure 4

Three examples of distributions of classical (left column) and quantum (right column) equilibrium states. Panels (a) and (b) correspond to $k=5$ and $\gamma =0.5$, panels (c) and (d) to $k=5.55$ and $\gamma =0.55$, and panels (e) and (f) to $k=9.25$ and $\gamma =0.55$. Negative values are highlighted in blue (gray) while red (darker shades of gray) stand for the positive ones. The scale is fixed in each panel, taking into account the maximum of the corresponding distribution.

Figure 5

$P\left(p\right)$ probability distributions as a function of $p$ for the three cases shown in Fig. 4, in the same order. Blue (black) lines correspond to the classical distributions with thermal noise, green (gray) lines to the ones without thermal noise, and cyan (light gray) lines to the quantum distributions.