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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).
Linear Response
Consider the absorption of energy by system subjected to a perturbation with
spectral intensity
.
Linear response theory gives rate of absorption
For some response function
The response is a linear functional of the intensity, satisfying:
in the form:
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.
A possible experiment
Response of a light-sensitive resistor:
Both ‘red’ and ‘green’ photons are required to allow percolation of electrons:
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
Time-dependent perturbation theory
Expand state in terms of eigenfunctions :
Find equation of motion for expansion coefficients
Integrate equation of motion with initial condition
probabilities
: find
The transition rates are:
:
and determine
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
Level-number diffusion
On long timescales the master equation describes
diffusion of occupation probability between levels.
Consider a coarse-grained probability
a continuity equation.
.This must satisfy
The probability current is expected to satisfy Fick’s law:
Probability obeys diffusion equation:
Note that when
there can be ‘bottlenecks’ due to weak transitions.
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
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 Kirchoff’s 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:
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:
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).
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:
Linear Response Theory – Prediction
:
Linear response:
This is evaluated using the two-level correlation function:
The result (expressed as diffusion constant) is
For small
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.
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.