Signatures of classical structures in the leading eigenstates of quantum dissipative systems

By analyzing a paradigmatic example of the theory of dissipative systems—the classical and quantum dissipative standard map—we are able to explain the main features of the decay to the quantum equilibrium state. The classical isoperiodic stable structures typically present in the parameter space of these kinds of systems play a fundamental role. In fact, we have found that the period of stable structures that are near in this space determines the phase of the leading eigenstates of the corresponding quantum superoperator. Moreover, the eigenvectors show a strong localization on the corresponding periodic orbits (limit cycles). We show that this sort of scarring phenomenon (an established property of Hamiltonian and projectively open systems) is present in the dissipative case and it is of extreme simplicity.

Figure 1

Top panel (a) shows the participation ratio $\eta$ vs parameters $k$ and $\gamma$ (for details see main text). Bottom panels (b) and (c) illustrate probability in momentum $P\left(p\right)$ as a function of $k$ for $\gamma =0.33$ corresponding to the ranges shown in green lines in panel (a).

Figure 2

Participation ratio $\eta$ vs parameters $k$ and $\gamma$ (see main text). Detail of Fig. 1 for $0.32\le \gamma \le 0.34$. In the left panel $4\le k\le 6$, while in the right panel $6.7\le k\le 6.9$.

Figure 3

100 largest eigenvalues of the quantum superoperator $e^{\mathrm{\Lambda }}$ for (a) $k=4.0$, (b) $k=5.13$, (c) $k=5.5$, (d) $k=6.0$. In all cases $\gamma =0.33$, and ${\hslash }_{\mathrm{eff}}=0.042$.

Figure 4

Husimi representation of the invariant state (left panel) and leading eigenstate (right panel) for (a) $k=4.0$, (b) $k=5.13$, (c) $k=5.5$, (d) $k=6.0$. In all cases $\gamma =0.33$, and ${\hslash }_{\mathrm{eff}}=0.042$. In the right panels of (a) and (b) the periodic points of the ISSs are marked with crosses.

Figure 5

Largest eigenvalues of the quantum superoperator $e^{\mathrm{\Lambda }}$ (with ${\hslash }_{\mathrm{eff}}=0.042$) and of the Perron-Frobenius superoperator (with $N=300$) for (a) $k=6.86$, (b) $k=6.8$. In all cases $\gamma =0.33$. Black dots correspond to the quantum model while (red) gray squares to the classical one.

Figure 6

Husimi representation of the invariant state (left panel) and leading eigenstate (right panel) for (a) $k=6.86$, (b) $k=6.8$. In all cases $\gamma =0.33$, and ${\hslash }_{\mathrm{eff}}=0.042$. In the right panels the periodic points of the ISSs are marked with crosses.