Dissipation model for extended environment

{{Use American English|date = April 2019}}

{{Short description|Mathematical model}}

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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.{{cite journal | last=Cohen | first=Doron | title=Unified model for the study of diffusion localization and dissipation | journal=Physical Review E | volume=55 | issue=2 | date=1997-02-01 | issn=1063-651X | doi=10.1103/physreve.55.1422 | pages=1422–1441| arxiv=chao-dyn/9611013 | bibcode=1997PhRvE..55.1422C | s2cid=51749412 }}{{cite journal | last=Cohen | first=Doron | title=Quantum Dissipation versus Classical Dissipation for Generalized Brownian Motion | journal=Physical Review Letters | volume=78 | issue=15 | date=1997-04-14 | issn=0031-9007 | doi=10.1103/physrevlett.78.2878 | pages=2878–2881| arxiv=chao-dyn/9704016 | bibcode=1997PhRvL..78.2878C | s2cid=51786519 }} It can be regarded as a generalization of the familiar Caldeira-Leggett model.

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

$\backslash mathcal\{H\}\_\{bath\}=\backslash sum\_\{\backslash alpha\}\backslash left(\backslash frac\{P\_\{\backslash alpha\}^2\}\{2m\_\{\backslash alpha\}\}\; +\backslash frac\{1\}\{2\}\; m\; \backslash omega\_\{\backslash alpha\}^2\; Q\_\{\backslash alpha\}^2\backslash right)$

$\backslash mathcal\{H\}\_\{int\}\; =\; -\; \backslash sum\_\{\backslash alpha\}\; c\_\{\backslash alpha\}\; Q\_\{\backslash alpha\}\; u(x-x\_\{\backslash alpha\})$

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

$\backslash frac\{\backslash pi\}\{2\}\; \backslash sum\_\{\backslash alpha\}\; \backslash frac\{c^2\_\{\backslash alpha\}\}\{m\_\{\backslash alpha\}\backslash omega\_\{\backslash alpha\}\}\; \backslash delta(\backslash omega-\backslash omega\_\{\backslash alpha\})\; \backslash \; \backslash delta(x-x\_\{\backslash alpha\})\; \backslash \; =\; \backslash \; J(\backslash omega)$

i.e. the oscillators that appear in the Hamiltonian are distributed uniformly over space, and in each location have the same spectral distribution $J(\backslash omega)$. Optionally the environment is characterized by the power spectrum of the fluctuations $\backslash tilde\{S\}(q,\backslash omega)$, which is determined by $J(\backslash omega)$ and by the assumed interaction $u(r)$. 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.{{cite journal | last=Cohen | first=Doron | title=Quantal Brownian motion - dephasing and dissipation | journal=Journal of Physics A: Mathematical and General | publisher=IOP Publishing | volume=31 | issue=40 | date=1998-10-09 | issn=0305-4470 | doi=10.1088/0305-4470/31/40/013 | pages=8199–8220| arxiv=cond-mat/9805023 | bibcode=1998JPhA...31.8199C | s2cid=11609074 }}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 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.{{cite journal | last1=Cohen | first1=Doron | last2=Imry | first2=Yoseph | title=Dephasing at low temperatures | journal=Physical Review B | publisher=American Physical Society (APS) | volume=59 | issue=17 | date=1999-05-01 | issn=0163-1829 | doi=10.1103/physrevb.59.11143 | pages=11143–11146| arxiv=cond-mat/9807038 | bibcode=1999PhRvB..5911143C | s2cid=51856292 }}{{cite journal | last1=Cohen | first1=Doron | last2=von Delft | first2=Jan | last3=Marquardt | first3=Florian | last4=Imry | first4=Yoseph | title=Dephasing rate formula in the many-body context | journal=Physical Review B | volume=80 | issue=24 | date=2009-12-08 | issn=1098-0121 | doi=10.1103/physrevb.80.245410 | page=245410| arxiv=0909.1441 | bibcode=2009PhRvB..80x5410C | s2cid=51754321 }} Various results can be derived for ballistic, chaotic, diffusive, and both ergodic and non-ergodic motion.