Dissipation model for extended environment

(a) The Brownian particle in the Caldeira-Leggett model experiences a fluctuating homogeneous field of force. (b) In case of the DLD model the fluctuating field is farther characterized by a finite correlation distance. The background image is a "snapshot" of the fluctuating environment. Namely, the gray levels correspond to the "height" of an instantaneous potential which is experienced by the Brownian particle.

A unified model for Diffusion Localization and Dissipation (DLD), optionally termed Diffusion with Local Dissipation, has been introduced for the study of Quantal Brownian Motion (QBM) in dynamical disorder. ^{[1] [2]} It can be regarded as a generalization of the familiar Caldeira-Leggett model.

${\displaystyle {\mathcal {H}}={\frac {p^{2}}{2m}}+V(x)+{\mathcal {H}}_{int}+{\mathcal {H}}_{bath}}$

${\displaystyle {\mathcal {H}}_{bath}=\sum _{\alpha }\left({\frac {P_{\alpha }^{2}}{2m_{\alpha }}}+{\frac {1}{2}}m\omega _{\alpha }^{2}Q_{\alpha }^{2}\right)}$

${\displaystyle {\mathcal {H}}_{int}=-\sum _{\alpha }c_{\alpha }Q_{\alpha }u(x-x_{\alpha })}$

where ${\displaystyle Q_{\alpha }}$ denotes the dynamical coordinate of the ${\displaystyle \alpha }$ scatterer or bath mode. ${\displaystyle u(x-x_{\alpha })}$ is the interaction potential, and ${\displaystyle c_{\alpha }}$ are coupling constants. The spectral characterization of the bath is analogous to that of the Caldeira-Leggett model:

${\displaystyle {\frac {\pi }{2}}\sum _{\alpha }{\frac {c_{\alpha }^{2}}{m_{\alpha }\omega _{\alpha }}}\delta (\omega -\omega _{\alpha })\ \delta (x-x_{\alpha })\ =\ J(\omega )}$

i.e. the oscillators that appear in the Hamiltonian are distributed uniformly over space, and in each location have the same spectral distribution ${\displaystyle J(\omega )}$. Optionally the environment is characterized by the power spectrum of the fluctuations ${\displaystyle {\tilde {S}}(q,\omega )}$, which is determined by ${\displaystyle J(\omega )}$ and by the assumed interaction ${\displaystyle u(r)}$. See examples.

The model can be used to describes the dynamics of a Brownian particle in an Ohmic environment whose fluctuations are uncorrelated in space. ^{[3] [4]} This should be contrasted with the Zwanzig-Caldeira-Leggett model, where the induced fluctuating force is assumed to be uniform in space (see figure).

At high temperatures the propagator possesses a Markovian property and one can write down an equivalent Master equation. Unlike the case of the Zwanzig-Caldeira-Leggett model, genuine quantum mechanical effects manifest themselves due to the disordered nature of the environment.

Using the Wigner picture of the dynamics one can distinguish between two different mechanisms for destruction of coherence: scattering and smearing. The analysis of dephasing can be extended to the low temperature regime by using a semiclassical strategy. In this context the dephasing rate SP formula can be derived. ^{[5] [6]} Various results can be derived for ballistic, chaotic, diffusive, and both ergodic and non-ergodic motion.

See also

• Quantum dissipation
• dephasing
• The dephasing rate SP formula

References

1. Cohen, Doron (1997-02-01). "Unified model for the study of diffusion localization and dissipation". Physical Review E. 55 (2): 1422–1441. arXiv: chao-dyn/9611013 . Bibcode:1997PhRvE..55.1422C. doi:10.1103/physreve.55.1422. ISSN   1063-651X. S2CID   51749412.
2. Cohen, Doron (1997-04-14). "Quantum Dissipation versus Classical Dissipation for Generalized Brownian Motion". Physical Review Letters. 78 (15): 2878–2881. arXiv: chao-dyn/9704016 . Bibcode:1997PhRvL..78.2878C. doi:10.1103/physrevlett.78.2878. ISSN   0031-9007. S2CID   51786519.
3. Cohen, Doron (1998-10-09). "Quantal Brownian motion - dephasing and dissipation". Journal of Physics A: Mathematical and General. IOP Publishing. 31 (40): 8199–8220. arXiv: cond-mat/9805023 . Bibcode:1998JPhA...31.8199C. doi:10.1088/0305-4470/31/40/013. ISSN   0305-4470. S2CID   11609074.
4. Driven chaotic mesoscopic systems,dissipation and decoherence, in Proceedings of the 38th Karpacz Winter School of Theoretical Physics, Edited by P. Garbaczewski and R. Olkiewicz (Springer, 2002). https://arxiv.org/abs/quant-ph/0403061
5. Cohen, Doron; Imry, Yoseph (1999-05-01). "Dephasing at low temperatures". Physical Review B. American Physical Society (APS). 59 (17): 11143–11146. arXiv: cond-mat/9807038 . Bibcode:1999PhRvB..5911143C. doi:10.1103/physrevb.59.11143. ISSN   0163-1829. S2CID   51856292.
6. Cohen, Doron; von Delft, Jan; Marquardt, Florian; Imry, Yoseph (2009-12-08). "Dephasing rate formula in the many-body context". Physical Review B. 80 (24): 245410. arXiv: 0909.1441 . Bibcode:2009PhRvB..80x5410C. doi:10.1103/physrevb.80.245410. ISSN   1098-0121. S2CID   51754321.