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

Soon Hoe Lim, Jan Wehr, Aniello Lampo, Miguel Ángel García-March, Maciej Lewenstein

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


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= m0ϵ, the reduced Planck constant to equal ħ= ϵ and the cutoff frequency to equal Λ= EΛ/ ϵ, where m0 and EΛ are positive constants, so that the particle’s de Broglie wavelength and the largest energy scale of the bath are fixed as ϵ→ 0. We study the limit as ϵ→ 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.

Original languageEnglish (US)
Pages (from-to)351-377
Number of pages27
JournalJournal of Statistical Physics
Issue number2
StatePublished - Jan 1 2018


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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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