Semilinear Response

1
Semilinear Response

• Michael Wilkinson (Open University), Bernhard
  Mehlig (Gothenburg University), Doron Cohen (Ben
  Gurion University)

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
Semilinear Response Theory,M. Wilkinson, B.
Mehlig and D. Cohen,Europhys. Lett, 75,709-15,
(2006).
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).
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).
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
3
Semilinear Response
This is a newly discovered phenomenon it is
possible for the response to a small perturbation
to satisfy simple linearity
while it does not satisfy the criterion to be a
linear functional
We describe one example, pertaining to a quantum
system driven by a weak perturbation which
contains predominantly low frequency components,
satisfying
In this case we find
The result is applicable to the absorption of
far-infrared/microwave radiation by small
metallic particles.
4
A possible experiment
Response of a light-sensitive resistor
Both red and green photons are required to
allow percolation of electrons
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
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
7
Master Equation and Linear Response
The probability for the nth eigenstate to be
occupied satisfies a master equation (or rate
equation)
The energy of the system is
The rate of absorption of energy is
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
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.
The probability current is expected to satisfy
Ficks law
Probability obeys diffusion equation
Note that when there can be
bottlenecks due to weak transitions.
9
Energy Diffusion and Dissipation
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
to evaluate time derivative
When p(E,t) decreases rapidly at the Fermi
energy, we have
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
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
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).
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
Required average is
For and find
14
Linear Response Theory Prediction
Linear response

This is evaluated using the two-level correlation
function
The result (expressed as diffusion constant) is
For small
15
Numerical demonstration

• Simulations of master equation, using energy
  levels from a GOE matrix, dimension N4000.

Rate of absorption initially agrees with LR
theory, then crosses over to SLR theory.
Exact diffusion constant D compared with LR and
SLR approximations.
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.