Sergio Ciuchi University of L’Aquila (ITALY) <sergio.ciuchi@aquila.infn.it>

Abstract

In this short course the concept of classical dissipation and quantum decoherence will be introduced through the study of simple models in both classical and quantum regimes. An introduction to stochastic dynamics will also be given. I will made all my lectures on the blackboard. Simple codes written in fortran will be available to the students. Here follows the arguments of my lectures, proposed numerical exercises and some references.

A single particle in a bath of harmonic oscillators

A simple model for an open classical system: a single particle interacting with an ensemble of harmonic oscillators (the bath).

Equation of motion and their formal solution.

The Ohmic density for the oscillators and the thermodynamic limit for the bath. Effective equation of motion for the particle.

Langevin and Fokker-Planck equations

The Langevin equation, velocity correlation function and diffusion coefficient, the Browninan motion.

The Fokker-Planck equation and the equilibrium distribution.

Quantum systems

A generalization to quantum system: the spin-boson problem.

The spin-boson interacting with a classical bath: the Ehrenfest dynamics and why it doesn’t work.

Zero temperature relaxation and decoherence using the resolvent technique.


Numerical examples

Here I provide two codes written in Fortran to test the analytical development exposed in the lectures. First of all you have to download and compile the codes. Both are provided as a tarball files. On a unix/linux/OSX untar the files with the command

tar zxvf tarball.tgz

where tarball is the name of the code. This command will create a directory. To compile and run the program see the files named README in the code directory.

I propose the students, as exercise, to check the output of the codes by choosing the parameters as described below.

This code simulates a classical particle interacting with N harmonic oscillators (the bath). You can download the tarball here

Exercise #1 - Run the program with N = 1024 oscillators with integration step 0.001 and 60000 time steps (nstep = 60000), starting from initial position x0=0 and initial momenta p0=10. Charge the temperature of the bath from T=0 to T=1. Check that the total energy is conserved within numerical errors. Does the momentum of the particle relaxes?

Exercise #2 - Change the time steps (nstep = 1000000). Check that the total energy is conserved within numerical errors. See what’s happen.

Exercise #3 - Choose the appropriate number of time steps and time-average the value of of the particle momentum P, compare the results with the Boltzmann equilibrium value.

This code simulates a quantum two-level system coupled to classical bath of harmonic oscillators using the Ehrenfest dynamics. You can download the tarball here

Exercise #1 - Run the program with N = 1024 oscillators with integration step 0.001 and 60000 time steps (nstep = 60000). Check that the total energy is conserved within numerical errors. Change the temperature of the bath and observe the relaxation of the average spin σz starting from the initial state (1,0)

Exercise #2 - Change the time steps (nstep = 1000000) Check that the total energy is conserved within numerical errors. See what’s happen.

Exercise #3 - Choose the appropriate number of time steps and time-average the value of σz, compare the results with the Boltzmann equilibrium value


Notes

Here you can download the notes of the course. They are handwritten and possibly affected by typos and errors. Please let me know by writing at


References:
  • C. Gardiner Stochastic Methods Spinger.com

  • U. Weiss Quantum dissipative systems Worldscientific.com

  • A. J. Leggett,S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, W. Zwerger, Dynamics of the dissipative two-state system, Rev. Mod. Phys. 59, 1 (1987) Prola

  • H. Kleinert Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets Worldscientific.com

  • A paper where the Ehrenfest evolution is compared with quantum dynamics:

    1. V. Parandekar and J. C. Tully Mixed quantum-classical equilibrium AIP

  • A paper where the resolvent technique has been applied to a decoherence problem:

    1. de Pasquale, G. L. Giorgi, and S. Paganelli Phys. Rev. A 71, 042304 (2005) Double-dot chain as a macroscopic quantum bit Prola