Quantum-classical correspondence

The classical calculation of Section 2.1 and the quantum LRT calculation of Section 2.2 gave the same answers for heating rate. This is remarkable because the energy evolution (spreading profile) was different: the classical spreading is gradual (for $t > \tau_{{\mbox{\tiny cl}}}$), whereas the quantum spreading involves jumps in units of $\hbar \omega$ (for driving frequency $\omega$). It is also remarkable because the quantum result is therefore independent of $\hbar$. However the heating rates agree because the expressions for spreading rate (second moment of spreading profile) were the same. The only difference is the replacement of the classical definition of $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ by the corresponding quantum-mechanical definition $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$. If these two agree, then there is quantum-classical correspondence (QCC) as far as dissipation (friction $\mu$) is concerned [47,46]. Cohen calls this `restricted correspondence' because of the different spreading profiles. Here I present numerical evidence supporting this claim of QCC in a real system (a 2D cavity). Hence in the following chapters I will move freely between the quantum and classical pictures. In particular, the term `band profile' will then refer to both quantum and classical $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ auto-correlation spectra.

First I briefly mention QCC outside the LRT regime (whose limits were outlined in the previous section). For extremely small $V$ in the quantum-adiabatic regime, there is no QCC expected because the level spacing distribution $P(s)$ for small $s\ll \Delta$ dominates the heating rate [200]. This is a purely quantum effect (determined by certain quantum symmetries [35,146]), and has no reason to agree with any classical quantity. At the other extreme, as $\hbar\rightarrow0$ the maximum $V$ where LRT is valid also vanishes. Therefore if one is to have QCC in the semiclassical limit (a fundamental requirement of quantum mechanics being that it reduces to classical mechanics in this limit), some new mechanism is required. As explained by Cohen[47,46], in this limit QCC is achieved through semiclassical agreement of the spreading profiles (`detailed correspondence').

Figure 2.5: Agreement between quantum and classical band profile in the billiard of Fig. 2.6, for three example deformations: DI (dilation), W2 (periodic oscillation around the perimeter), P (wide `piston' existing only on the top edge). In each case classical is shown as a thick line (RMS estimation error of 3%), and quantum a thin line (RMS error of 10%, increasing at higher $\omega$); the agreement is excellent. Note that the $y$-axis has been displaced to clearly show the differing $\omega\rightarrow0$ behavior. The singular peak at $\omega = \pi$ is due to the `bouncing ball' orbit. The units are such that $m=v=1$. The quantum calculation was performed at wavenumber $k \approx 400$ using 451 adjacent eigenstates.

Figure 2.6: The example (undeformed) 2D billiard used for comparison of quantum and classical band profiles: the Bunimovich quarter-stadium. I typically choose $a/R = 1$ since this maximizes the average Lyapunov exponent (a measure of chaoticity). The overall size is set by $R=1$.