Quantum dissipation

Quantum dissipation is the branch of physics that studies the quantum analogues of the process of irreversible loss of energy observed at the classical level. Its main purpose is to derive the laws of classical dissipation from the framework of quantum mechanics. It shares many features with the subjects of quantum decoherence and quantum theory of measurement. Quantum dissipation is the branch of physics that studies the quantum analogues of the process of irreversible loss of energy observed at the classical level. Its main purpose is to derive the laws of classical dissipation from the framework of quantum mechanics. It shares many features with the subjects of quantum decoherence and quantum theory of measurement. The main problem to address dissipation at the quantum level is the way to envisage the mechanism of irreversible loss of energy. Quantum mechanics usually deal with the Hamiltonian formalism, where the total energy of the system is a conserved quantity. So in principle it would not be possible to describe dissipation in this framework. The idea to overcome this issue consists of splitting the total system in two parts: the quantum system where dissipation occurs, and a so-called environment or bath where the energy of the former will flow towards. The way both systems are coupled depends on the details of the microscopic model, and hence, the description of the bath. To include an irreversible flow of energy (i.e., to avoid Poincaré recurrences in which the energy eventually flows back to the system), requires that the bath contain an infinite number of degrees of freedom. Notice that by virtue of the principle of universality, it is expected that the particular description of the bath will not affect the essential features of the dissipative process, as far as the model contains the minimal ingredients to provide the effect. The simplest way to model the bath was proposed by Feynman and Vernon in a seminal paper from 1963. In this description the bath is a sum of an infinite number of harmonic oscillators, that in quantum mechanics represents a set of free bosonic particles. In 1981, Amir Caldeira and Anthony J. Leggett proposed a simple model to study in detail the way dissipation arises from a quantum point of view. It describes a quantum particle in one dimension coupled to a bath. The Hamiltonian reads: The first two terms correspond to the Hamiltonian of a quantum particle of mass M {displaystyle M} and momentum P {displaystyle P} , in a potential V {displaystyle V} at position X {displaystyle X} . The third term describes the bath as an infinite sum of harmonic oscillators with masses m i {displaystyle m_{i}} and momentum p i {displaystyle p_{i}} , at positions q i {displaystyle q_{i}} . ω i {displaystyle omega _{i}} are the frequencies of the harmonic oscillators. The next term describes the way system and bath are coupled. In the Caldeira–Leggett model, the bath is coupled to the position of the particle. C i {displaystyle C_{i}} are coefficients which depend on the details of the coupling. The last term is a counter-term which must be included to ensure that dissipation is homogeneous in all space. As the bath couples to the position, if this term is not included the model is not translationally invariant, in the sense that the coupling is different wherever the quantum particle is located. This gives rise to an unphysical renormalization of the potential, which can be shown to be suppressed by including the counter-term.