Journal of Statistical Physics

, Volume 170, Issue 2, pp 351–377 | Cite as

On the Small Mass Limit of Quantum Brownian Motion with Inhomogeneous Damping and Diffusion

  • Soon Hoe LimEmail author
  • Jan Wehr
  • Aniello Lampo
  • Miguel Ángel García-March
  • Maciej Lewenstein


We study the small mass limit (or: the Smoluchowski–Kramers limit) of a class of quantum Brownian motions with inhomogeneous damping and diffusion. For Ohmic bath spectral density with a Lorentz–Drude cutoff, we derive the Heisenberg–Langevin equations for the particle’s observables using a quantum stochastic calculus approach. We set the mass of the particle to equal \(m = m_{0} \epsilon \), the reduced Planck constant to equal \(\hbar = \epsilon \) and the cutoff frequency to equal \(\varLambda = E_{\varLambda }/\epsilon \), where \(m_0\) and \(E_{\varLambda }\) are positive constants, so that the particle’s de Broglie wavelength and the largest energy scale of the bath are fixed as \(\epsilon \rightarrow 0\). We study the limit as \(\epsilon \rightarrow 0\) of the rescaled model and derive a limiting equation for the (slow) particle’s position variable. We find that the limiting equation contains several drift correction terms, the quantum noise-induced drifts, including terms of purely quantum nature, with no classical counterparts.


Quantum Brownian motion Heisenberg–Langevin equation Small mass limit Smoluchowski–Kramers limit Noise-induced drifts Quantum stochastic calculus 



S. Lim and J. Wehr were partially supported by NSF Grant DMS 1615045. This work has been funded by a scholarship from the Programa Másters d’Excel-léncia of the Fundació Catalunya-La Pedrera, ERC Advanced Grant OSYRIS (ERC-2013-AdG Grant 339106), EU IP SIQS (FP7-ICT-2011- 9600645), EU PRO QUIC (H2020-FETProAct-2014 641122), EU STREP EQuaM (FP7/2007-2013, No. 323714), Fundació Cellex, the Spanish MINECO (SEVERO OCHOA GRANT SEV-2015-0522, FISICATEAMO FIS2016-79508-P), and Generalitat de Catalunya (SGR 874 and CERCA/Program).


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Soon Hoe Lim
    • 1
    Email author
  • Jan Wehr
    • 1
  • Aniello Lampo
    • 2
  • Miguel Ángel García-March
    • 2
  • Maciej Lewenstein
    • 2
    • 3
  1. 1.Department of Mathematics and Program in Applied MathematicsUniversity of ArizonaTucsonUSA
  2. 2.ICFO-Institut de Ciències FotòniquesThe Barcelona Institute of Science and TechnologyCastelldefels (Barcelona)Spain
  3. 3.ICREABarcelonaSpain

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