CaldeiraLeggett

The Caldeira–Leggett model is a theoretical framework in quantum mechanics used to describe dissipation and decoherence of a quantum system due to its interaction with an environment. It was introduced in the early 1980s by Antonio O. Caldeira and Anthony J. Leggett. The model treats the environment as a large collection of independent harmonic oscillators that couple linearly to the coordinate of the system of interest. This setup provides a tractable way to study how coupling to an environment affects quantum dynamics, including loss of coherence and dissipative energy exchange.

In the standard formulation, the total Hamiltonian consists of three parts: the system Hamiltonian H_S, the bath Hamiltonian H_B that describes the oscillators, and the system–bath interaction H_SB. The bath is characterized by a spectral density J(ω), which encodes how strongly environmental modes of frequency ω couple to the system. A common choice is Ohmic dissipation, where J(ω) ∝ η ω at low frequencies with a high-frequency cutoff. A counterterm is often included to cancel renormalization effects on the system’s potential, ensuring a well-defined dynamics.

Analytically, the model is frequently treated with path-integral methods, yielding the Feynman–Vernon influence functional that effectively incorporates dissipation and memory effects into the system’s dynamics. In certain limits, this leads to a quantum Brownian motion description and, for specific regimes, to master equations such as the Caldeira–Leggett equation.

Applications span quantum tunneling in dissipative environments, decoherence studies in macroscopic quantum systems, and modeling of superconducting circuits, quantum dots, and nanoelectromechanical devices. The model provides a foundational link between open quantum systems theory and practical descriptions of dissipation in condensed matter and quantum information contexts.