The dephasing rate formula (1997-1999, 2007-2008)

What is the dephasing rate for a particle in a dissipative environment? As discussed in [1,2] the general case requires to take into account not only the temporal correlations but also the spatial correlations of the fluctuations. These can be characterized by the spectral form factor \blue\tilde{S}(q,\omega), while the motion of the particle is characterized by its power spectrum \blue\tilde{P}(q,\omega). Consequently at finite temperature the expression for the dephasing rate takes the form [3]:

\blue\Gamma_{\varphi} \ \ = \ \  \int d{q} \int \frac{d\omega}{2\pi} \,\tilde{S}({q},\omega) \, \tilde{P}(-{q},-\omega)

It has been further argued in [4] that due to inherent limitations of the semiclassical (stationary phase) approximation the physically correct procedure is to use the  non-symmetrized quantum versions of  \blue\tilde{S}(q,\omega) and \blue\tilde{P}(q,\omega). The argument was based on the analogy of the the above expression with Fermi-golden-rule calculation of the transitions which are induced by the system-environment interaction. In order to derive the dephasing rate formula from first principles a purity based definition of the dephasing factor has been adopted in [5,6]. The purity \blue P(t)=e^{-F(t)} describes how a pure state becomes mixed due to the entanglement of the system with the environment. Using perturbation theory one recovers at finite temperatures at the long time limit \blue F(t)=\Gamma_{\varphi}t where the decay constant is given by the dephasing rate formula with non symmetrized spectral functions as expected. We further discuss in this work the possibility to get power law decay of \blue P(t) at the limit of zero temperature.

The dephasing rate formula has been further applied in [5,6] for the purpose of analyzing the dephasing of a particle in a ring under the influence of a dirty metal environment. The proper way to incorporate Pauli blocking within this calculation is being discussed with the Munich group (not yet published).

[1] D. Cohen, Phys. Rev. E 55, 1422-1441 (1997). [arXiv] [pdf]
[2] D. Cohen, Phys. Rev. Lett. 78, 2878 (1997). [arXiv] [pdf]
[3] D. Cohen, J. Phys. A 31, 8199-8220 (1998). [arXiv] [pdf]
[4] D. Cohen and Y. Imry, Phys. Rev. B 59, 11143 (1999). [arXiv] [pdf]
[5]  D. Cohen and B. Horovitz, J. Phys. A 40, 12281 (2007). [arXiv] [pdf]
[6] D. Cohen and B. Horovitz, Europhysics Letters 81, 30001 (2008). [arXiv] [pdf]