   Next: Generalized force-force correlation and Up: Review of the linear Previous: Review of the linear

## 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 , (2.34)

The hermitian operator is the generalized force' conjugate to the parameter : it is an actual force if is a displacement, it is a dipole moment if is an applied electric field magnetic field, it is a current around a ring (magnetization) if is magnetic flux enclosed by the ring, and so forth. I shall now use time-dependent perturbation theory, choosing to perform this in a fixed basis of the eigenstates of , which we call the unperturbed' Hamiltonian. This choice of basis is conventional in textbooks [128,99,118] so is ideal for introductory purposes. However this choice does not allow an understanding of the limitations of conventional LRT; to understand breakdown and possible extensions to LRT it is necessary to consider transitions in the so-called adiabatic' basis of the local eigenstates of ([200,46,48]. The adiabatic basis is much more natural for unifying with the classical picture . It also removes certain difficulties, for instance that of the infinite matrix elements which would otherwise occur for any deformation of a hard-walled billiard system. As another example, the case of constant velocity driving cannot be shown to give diffusive spreading in a fixed basis--the adiabatic basis is required (Appendix B of ).

The Schrodinger-picture wavefunction is represented as (2.35)

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 (2.36)

gives a first-order differential equation for the coefficients (2.37)

where and . The formal solution (using notation , and the matrix ) is a time-ordered exponential (Dyson series ) (2.38)

I take the initial condition as (2.39)

that is, a single initial pure eigenstate , well above the ground state. The generalization to any incoherent ensemble of initial eigenstates (e.g. canonical ensemble) can be achieved by an occupation-weighted average of the results over . Formulations of LRT ([122,84,99,118,200], and Ingold in ) using the density matrix are equivalent to this, although the notation may superficially look different. Using (2.39) and keeping only the first-order term in in (2.38) gives (2.40)

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 , (2.41)

Substituting into FOPT and squaring the absolute value gives    (2.42)

where sinc as usual means . After many periods ( ) the sinc functions become localized delta-like functions of width , (2.43)

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   (2.44)  (2.45)

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

In textbook treatments of the FGR , 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.    Next: Generalized force-force correlation and Up: Review of the linear Previous: Review of the linear
Alex Barnett 2001-10-03