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arXiv:2604.26524 · Brazilian Journal of Physics

Over forty years of research towards the understanding of Quantum Brownian Motion -- the contributions of A. O. Caldeira

Marcus V. S. Bonança, Sebastian Deffner, Gert-Ludwig Ingold

quant-phcond-mat.stat-mechSubmitted Apr 29, 2026• Announced Apr 30, 2026• 0 citations

Abstract

This article presents a brief account of Amir O. Caldeira's contributions to the theory of quantum Brownian motion. Motivated by its importance, we outline the description of Brownian motion in the quantum regime following Caldeira's first works. In this context, we particularly highlight the effect of dissipation on the tunneling rate out of a metastable state. We then journey along the alternative ways to approach quantum Brownian motion developed by Caldeira during his career, which go beyond the so-called Caldeira-Leggett model. We conclude by summarizing some of Caldeira's contributions to contemporary fields such as the theory of quantum decoherence and quantum thermodynamics, that were strongly inspired by his eponymous approach to quantum Brownian motion.

Journal reference: Braz. J. Phys. 56, 209 (2026)
DOI: 10.1007/s13538-026-02127-2
Comments: Brief review submitted to the Special Issue "Frontiers in Quantum Mechanics and the 45 Years of the Caldeira-Leggett model" of the Brazilian Journal of Physics to celebrate the career of A. O. Caldeira

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Caldeira and Leggett (CL), in a seminal paper, derived a master equation describing Markovian quantum Brownian motion. Such an equation suffered from not being completely positive, and many efforts have been made to solve this issue. We show that when a careful mathematical analysis is performed, the model considered by CL leads to a nondissipative master equation. We argue that the correct way to understand the master equation derived in the CL regime is to consider it non-Markovian. Moreover, we show that if one wants to provide a microscopic description of quantum Brownian motion with the CL model, one always needs to consider a non-Markovian dynamics. We conclude that dissipation is a genuinely non-Markovian feature.

Caldeira and Leggett (CL) in a seminal paper derived a master equation describing Markovian Quantum Brownian motion. Such an equation suffered of not being completely positive, and many efforts have been made to solve this issue. We show that, when a careful mathematical analysis is performed, the model considered by CL leads to a non dissipative master equation. We argue that the correct way to understand the master equation derived in the CL regime is to consider it non-Markovian. Moreover, we show that if one wants to provide a microscopic description of Quantum Brownian motion one always needs to consider a non-Markovian dynamics. We conclude that dissipation is a genuinely non-Markovian feature.

We investigate the heat statistics in a relaxation process of quantum Brownian motion described by the Caldeira-Leggett model. By employing the normal mode transformation and the phase-space formulation approach, we can analyze the quantum heat distribution within an exactly dynamical framework beyond the traditional paradigm of Born-Markovian and weak-coupling approximations. It is revealed that the exchange fluctuation theorem for quantum heat generally breaks down in the strongly non-Markovian regime. Our results may improve the understanding about the nonequilibrium thermodynamics of open quantum systems when the usual Markovian treatment is no longer appropriate.

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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.

We develop a non-Markovian extension to the caldeira-leggett master equation for quantum Brownian motion by using the timescale of the bath as a small parameter around which to perform a perturbative expansion. This framework allows us to deal directly with the time non-locality of the non-Markovian regime. We derive an exact description of the evolution of the reduced density matrix and obtain a new quantum master equa- tion with non-Markovian corrections added perturbatively to the standard Lindblad form. The resulting simulations show new dynamics and a strong deviation from Markovian behaviour. We observe that quantum coherences survive far longer than in the Marko- vian limit. In particular, when the system is fully non-Markovian, quantum phenomena experience a form of decoherence that behaves functionally like an inverse-power law. These results may have important implications in quantum technologies and biology as they suggest environments can be engineered to support long-lived quantum coherence.