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

Abstract

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 $\hslash =\epsilon$ and the cutoff frequency to equal $\Lambda =E_{\Lambda }/\epsilon$, where $m_0$ and $E_{\Lambda }$ are positive constants, so that the particle's de Broglie wavelength and the largest energy scale of the bath are fixed as $\epsilon \to 0$. We study the limit as $\epsilon \to 0$ 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.

29 pages

References in corpus (12)

• Quantum Feedback Networks: Hamiltonian Formulation
• Multiple-scale stochastic processes: decimation, averaging and beyond
• Bose polaron as an instance of quantum Brownian motion
• Quantum state-dependent diffusion and multiplicative noise: a microscopic approach
• The Langevin equation from Markovian Quantum Central Limits
• From non equilibrium quantum Brownian motion to impurity dynamics in 1D quantum liquids
• Quantum Brownian Motion with Inhomogeneous Damping and Diffusion
• Quantum Smoluchowski equation for driven systems
• Anomalous diffusion in quantum Brownian motion with colored noise
• Singular perturbation of quantum stochastic differential equations with coupling through an oscillator mode
• Adiabatic elimination in quantum stochastic models
• Zeno Dynamics for Open Quantum Systems

Cited by in corpus (6)

• Using polarons for sub-nK quantum non-demolition thermometry in a Bose-Einstein condensate
• Bose polaron as an instance of quantum Brownian motion
• Non-Markovian polaron dynamics in a trapped Bose-Einstein condensate
• Quantum dynamics of Bose-polaron in a $d$-dimensional Bose Einstein condensate
• Quantum partition of energy for a free Brownian particle: Impact of dissipation
• Quantum pure noise-induced transitions: A truly nonclassical limit cycle sensitive to number parity