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
), whereas the quantum
spreading
involves jumps in units of (for driving frequency ).
It is also remarkable because the quantum result is therefore independent
of .
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
by the corresponding
quantum-mechanical definition
.
If these two agree, then there is *quantum-classical correspondence* (QCC)
as far as dissipation (friction ) 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
auto-correlation spectra.

First I briefly mention QCC outside the LRT regime (whose limits were outlined in the previous section). For extremely small in the quantum-adiabatic regime, there is no QCC expected because the level spacing distribution for small 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 the maximum 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').

- Semiclassical connection between quantum and classical band profiles
- Numerical comparison of band profiles
- Quantum structure beyond the band profile
- Effects below the quantum level spacing