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arXiv:1508.00582 · Physical review. B./Physical review. B

Flight of a heavy particle nonlinearly coupled to a quantum bath

Mohammad F. Maghrebi, Matthias Krüger, Mehran Kardar A

Published Jan 28, 2016• 9 citations
arXivHTMLPDF

Abstract

Fluctuation and dissipation are byproducts of coupling to the ``environment.'' The Caldeira-Leggett model, a successful paradigm of quantum Brownian motion, views the environment as a collection of harmonic oscillators linearly coupled to the system. However, symmetry considerations may forbid a linear coupling, e.g., for a neutral particle in quantum electrodynamics. We argue that the absence of linear couplings can lead to a fundamentally different behavior. Specifically, we consider a heavy particle quadratically coupled to quantum fluctuations of the bath. In one dimension the particle undergoes anomalous diffusion, unfolding as a power-law distribution in space, reminiscent of L\'evy flights. We suggest condensed matter analogs where similar effects may arise.

DOI
10.1103/physrevb.93.014309

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Fluctuation and dissipation are byproducts of coupling to the ``environment.'' The Caldeira-Leggett model, a successful paradigm of quantum Brownian motion, views the environment as a collection of harmonic oscillators linearly coupled to the system. However, symmetry considerations may forbid a linear coupling, e.g., for a neutral particle in quantum electrodynamics. We argue that the absence of linear couplings can lead to a fundamentally different behavior. Specifically, we consider a heavy particle quadratically coupled to quantum fluctuations of the bath. In one dimension the particle undergoes anomalous diffusion, unfolding as a power-law distribution in space, reminiscent of L'evy flights. We suggest condensed matter analogs where similar effects may arise.
We study a classical system-bath model in which a system particle is linearly coupled to a bath of harmonic oscillators. In a system subject to white and correlated Gaussian noises, we derive an expression for the joint probability density, and the mean squared values of the system-bath particles are calculated. For white noise, the mean squared values of the system-bath particles exhibit an anomalous time dependence, which is different from that of normal diffusion. In particular, for a correlated Gaussian noise, the mean squared displacement and mean squared velocity of the bath particle show superspreading growths of t5t^5t5 and t3t^3t3 in t≪τt{\ll}τt≪τ, respectively. When τ=0τ=0τ=0, the mean squared velocity of a quantum particle under random noise is proportional to ttt, while the mean squared velocity of a bath particle in the presence of correlated Gaussian noise increases in proportion to t2t^2t2. This anomalous transport phenomenon results from the mixed derivative structure of the master equation, which couples with the transport coordinates in the diffusion dynamics of the relative coordinates. This result shows that the removal of dissipation in the Caldeira-Leggett framework leads to fundamentally different transport mechanisms characterized by non-diffusive quantum diffusion.
The Caldeira-Leggett model of quantum Brownian motion is generalized using a generic velocity-dependent coupling. That leads to the description of a set of models able to capture Markovian and non-Markovian versions of Brownian and L\'evy statistics, depending on the functional form of the coupling and on the spectral function of the reservoir. One specific coupling force is found that establishes a connection with L\'evy statistics of cold atoms in Sisyphus laser cooling. In the low-velocity limit, this also gives rise to additional inertia of the Brownian particle, reproducing the Abraham-Lorentz equation from first principles for a superohmic bath. Through path-integral quantization in Euclidean time, the environment is integrated out, leaving a set of nonlocal effective actions. These results further serve as starting points for several numerical calculations, particularly decoherence properties of nonohmic baths.
We study the Caldeira-Leggett model where a quantum Brownian particle interacts with an environment or a bath consisting of a collection of harmonic oscillators in the path-integral formalism. Compared to the contours that the paths take in the conventional Schwinger-Keldysh formalism, the paths in our study are deformed in the complex time plane as suggested by the recent study by C. Aron, G. Biroli, and L. F. Cugliandolo [SciPost Phys. 4, 008 (2018)10.21468/SciPostPhys.4.1.008]. This is done to investigate the connection between the symmetry properties in the Schwinger-Keldysh action and the equilibrium or nonequilibrium nature of the dynamics in an open quantum system. We derive the influence functional explicitly in this setting, which captures the effect of the coupling to the bath. We show that in equilibrium the action and the influence functional are invariant under a set of transformations of path-integral variables. The fluctuation-dissipation relation is obtained as a consequence of this symmetry. When the system is driven by an external time-dependent protocol, the symmetry is broken. From the terms that break the symmetry, we derive a quantum Jarzynski-like equality for a quantum mechanical worklike quantity given as a function of fluctuating quantum trajectory. In the classical limit, the transformations becomes those used in the functional integral formalism of the classical stochastic thermodynamics to derive the classical fluctuation theorem.
We model a quantum system coupled to an environment of damped harmonic oscillators by following the approach of Caldeira-Leggett and adopting the Caldirola-Kanai Lagrangian for the bath oscillators. In deriving the master equation of the quantum system of interest (a particle in a general potential), we show that the potential is modified non-trivially by a new inverted harmonic oscillator term, induced by the damping of the bath oscillators. We analyze numerically the case of a particle in a double-well potential, and find that this modification changes both the rate of decoherence at short times and the well-transfer probability at longer times. We also identify a simple rescaling condition that keeps the potential fixed despite changes in the environmental damping. Here, the increase of environmental damping leads to a slowing of decoherence.
Quantum diffusion is a major topic in condensed-matter physics, and the Caldeira-Leggett model has been one of the most successful approaches to study this phenomenon. Here, we generalize this model by coupling the bath to the system through a Liouville fractional derivative. The Liouville fractional Langevin equation is then derived in the classical regime, without imposing a non-Ohmic macroscopic spectral function for the bath. By investigating the short- and long-time behavior of the mean squared displacement, we show that this model is able to describe a large variety of anomalous diffusion. Indeed, we find ballistic, sub-ballistic, and super-ballistic behavior for short times, whereas for long times, we find saturation and sub- and super-diffusion.
A binary collision model for phenomena of quantum dissipation is developed. Unlike the harmonic oscillator model, widely used for over thirty years, this model assumes nonlinear coupling between system and environment, and is applicable to both bosonic and fermionic baths. The system interacts with an ideal bath through binary collisions only. Solutions for the classical and quantum-mechanical problems in the case of free Brownian motion are presented, and the quantum-classical correspondence for nonequilibrium processes is established. It is shown that in the Brownian motion limit the two models lead to identical dynamical behavior, provided the coupling coefficients in the harmonic oscillator Hamiltonian are temperature dependent. For cases of bath particles of finite mass and number the two models lead to different results. Linear response theory for the model is developed, and the results are compared with those for the harmonic oscillator model. At the end, possible applications of the model are suggested.