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, , (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 Fick’s 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 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:

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:
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 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.

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.