Basis choice, perturbation theory and the FGR

Dissipation in a quantum system will be the result of `driving' a
parameter on which the Hamiltonian depends.
The parameter is externally controllable with any (real)
function of time ;
it does not have any dynamics of its own (it is not a degree of freedom).
We start by linearizing about ,

The Schrodinger-picture wavefunction is represented as

where the time-independent states are eigenstates of the unperturbed Hamiltonian: . The basis vectors are fixed but include phase evolution under (following [128,160]), thus the coefficients are equivalent to an `interaction picture' representation. The full time-evolution

gives a first-order differential equation for the coefficients

(2.37) |

I take the initial condition as

that is, a single initial pure eigenstate , well above the ground state. The generalization to any incoherent ensemble of initial eigenstates (

which is called first-order perturbation theory (FOPT). Clearly this is a valid approximation only when remains close to unity and for all . If this is true, the response of any expectation value is linear in : this is what is meant by the word `linear' in LRT (rather that the linearization (2.34)).

I will now specialize to the case of periodic driving at amplitude ,

(2.42) |

where sinc as usual means . After many periods ( ) the sinc functions become localized delta-like functions of width ,

and the interference terms become irrelevant as these functions become separated. In order that these delta-like functions do not become narrow enough to resolve discrete energy levels, we are limited by the Heisenberg time, . Substituting the delta functions (from which I now drop the width subscript ), the probabilities in states can be seen to grow linearly, with transition rates

recognizable as the usual Fermi Golden Rule (FGR) in the presence of both positive and negative frequency driving.

In textbook treatments of the FGR [128],
the squared matrix element
is assumed
to be a smooth function of , enabling it to be taken as constant
over the width of the delta-function--
this is often true in integrable systems.
In contrast, in
a chaotic system
it takes essentially
random, uncorrelated values at each and (this assumption
underlies any Random Matrix Theory[35,146]
description of chaos).
The *average* transition rate
is given by the local mean value near the location in the matrix
(see Fig. 2.3).
This will next be formalised using the `band profile' of the matrix.