Semilinear Response Michael Wilkinson (Open University),
Bernhard Mehlig (Gothenburg University), Doron Cohen (Ben Gurion
University) I also use the energy diffusion theory for dissipation,
introduced in: Statistical Aspects of Dissipation by Landau-Zener
Transitions, M. Wilkinson, J. Phys. A, 21, 4021-37, (1988).
Semilinear Response Theory, M. Wilkinson, B. Mehlig and D. Cohen,
Europhys. Lett, 75,709-15, (2006). A newly discovered variant of
linear response theory, in which quantum perturbation theory is
used to derive a master equation which is solved
non-perturbatively. Reference: The example which is worked out is
the absorption of low-frequency radiation by small metal particles,
a topic which was pioneered by Kubo: A. Kawabata and R. Kubo, J.
Phys. Soc. Japan, 21, 1765, (1966). Slide 2 Linear Response
Consider the absorption of energy by system subjected to a
perturbation with spectral intensity. Linear response theory gives
rate of absorption in the form: For some response function The
response is a linear functional of the intensity, satisfying: Slide
3 Semilinear Response This is a newly discovered phenomenon: it is
possible for the response to a small perturbation to satisfy simple
linearity We describe one example, pertaining to a quantum system
driven by a weak perturbation which contains predominantly low
frequency components, satisfying while it does not satisfy the
criterion to be a linear functional: In this case we find: The
result is applicable to the absorption of far-infrared/microwave
radiation by small metallic particles. Slide 4 A possible
experiment Response of a light-sensitive resistor: Both red and
green photons are required to allow percolation of electrons: Slide
5 Hamiltonian We consider a Hamiltonian of the form: where the
time-dependent fields are random functions: with spectral
intensity: Practical application: could be the single-electron in a
small metallic particle, are the (screened) dipole operators, and
are the components of the electric field with spectral intensity.
For numerical investigations we used Slide 6 Time-dependent
perturbation theory Expand state in terms of eigenfunctions : Find
equation of motion for expansion coefficients : Integrate equation
of motion with initial condition and determine probabilities : find
The transition rates are: Slide 7 Master Equation and Linear
Response The probability for the nth eigenstate to be occupied
satisfies a master equation (or rate equation): Standard linear
response theory is obtained if we treat this perturbatively, using
the initial occupation probabilities: write and expand probability
differences to first order to obtain The rate of absorption of
energy is: The energy of the system is: Slide 8 Level-number
diffusion On long timescales the master equation describes
diffusion of occupation probability between levels. Consider a
coarse-grained probability.This must satisfy a continuity equation.
Note that when there can be bottlenecks due to weak transitions.
The probability current is expected to satisfy Ficks law:
Probability obeys diffusion equation: Slide 9 Energy Diffusion and
Dissipation to evaluate time derivative: Dissipation results from
diffusion of electrons from filled states below Fermi level to
empty states. Energy of system is Approximate sum by integral, and
use diffusion equation When p(E,t) decreases rapidly at the Fermi
energy, we have Slide 10 Random resistor network and level
diffusion To determine the energy level diffusion constant,
consider a steady state probability current J. The steady-state of
the master equation is analogous to Kirchoffs eqaution for a
resistor network (idea of Miller and Abrahams (1960), but applied
in energy rather than space): is solved in steady state by
considering a resistor network: Level number diffusion constant is
the conductivity: Slide 11 Low frequency limit: resistors in series
When the characteristic photon energy is smaller than the mean
level spacing, only nearest neighbour transitions are significant.
In this case the transition network behaves as a series circuit,
for which resistances are added. The conductivity (diffusion
constant) is the harmonic mean of the transition rates: Slide 12
Resistors in parallel When parallel connections dominate, we
estimate: There are ( n-m ) resistors connecting links separated by
( n-m ) links, and the potential difference is proportional to (
n-m ). Slide 13 Estimate of harmonic mean of transition rate The
transition rate must be averaged over distributions of matrix
elements (variance ) and level spacing,. Distributions are Gaussian
and Wigner surmise, respectively: For and find: Required average
is: Slide 14 Linear Response Theory Prediction : This is evaluated
using the two-level correlation function: Linear response: The
result (expressed as diffusion constant) is For small Slide 15
Numerical demonstration Simulations of master equation, using
energy levels from a GOE matrix, dimension N =4000. Rate of
absorption initially agrees with LR theory, then crosses over to
SLR theory. Exact diffusion constant D compared with LR and SLR
approximations. Slide 16 Conclusions The response of a system to a
weak disturbance can be treated by deriving a master equation in
perturbation theory. If the master equation is itself treated
perturbatively, we obtain linear response theory. If we examine the
long-time behaviour using a non-perturbative approach, we may see
semilinear response, in which the response to a sum of two
different probe intensities is greater than the sum of the response
to each intensity applied separately. We have discussed one example
in detail: the absorption of far infrared electromagnetic radiation
by small metal particles. After an initial transient, the
absorption starts to be limited by bottlenecks caused by large
level spacings. Other realisations are possible, and the
experimental signature is very simple.
