Quantum dissipation

Quantum Dissipation is the branch of physics that studies the quantum analogous 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.

Models of Quantum Dissipation

The main problem to address to study 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 on 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, so it does the description of the bath. In order to include an irreversible flow of energy (i.e., to avoid Poincaré recurrences in which the energy eventually flows back to the system), it is required that the bath contains 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 [ R. P. Feynman and F. L. Vernon, Ann. Phys. (N. Y.) 24, 118 (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.

The Caldeira-Leggett model

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 [ A. Caldeira and A. J. Leggett, "Influence of dissipation on quantum tunneling in macroscopic systems", Phys. Rev. Lett., vol. 46, p. 211, 1981 ] . It describes a quantum particle in one-dimension coupled to a bath. The Hamiltonian reads:

$H=frac\{P^2\}\{2\; M\}\; +\; V(X)\; +\; sum\_ileft(frac\{p\_i^2\}\{2\; m\_i\}+frac\{1\}\{2\}m\_i\; omega\_i^2\; q\_i^2\; ight)\; +\; X\; sum\_i\{C\_i\; q\_i\}\; +\; X^2\; sum\_i\; frac\{C\_i^2\}\{2m\_i\; omega\_i^2\}$,

The first two terms correspond to the Hamiltonian of a quantum particle of mass $M$ in a potential $V(X)$, being $P$ the momentum of the particle, and $X$ its position. The third term describes the bath as a sum of infinite harmonic oscillators with masses $m\_i$ and momentum $p\_i$, at positions $q\_i$. $omega\_i$ are the frequencies of the harmonic oscillators. Next term describes the way system and bath are coupled. In Caldeira - Leggett model the bath is coupled to the position of the particle. $C\_i$ 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 the 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 suppresed by including the counter-term.

In order to provide a good description of the dissipation mechanism, a very relevant quantity is the bath spectral function:

$J(omega)\; =\; frac\{pi\}\{2\}\; sum\_i\; frac\{C\_i^2\}\{m\_i\; omega\_i^2\}\; delta(omega\; -\; omega\_i)$

The bath spectral function provides a constraint in the choice of the coefficients $C\_i$. When this function has the form $J(omega)\; =\; eta\; omega$, the corresponding classical kind of dissipation can be shown to be Ohmic. A more generic form is$J(omega)\; propto\; omega^s$. In this case, if $s\; >\; 1$ the dissipation is called "super-ohmic", while if is sub-ohmic. An example of a super-ohmic bath is the electro-magnetic field under certain circumstances.

As mentioned, the main idea in the field of quantum dissipation is to explain the way classical dissipation can be described from a quantum mechanics point of view. In order to get the classical limit of the Caldeira - Leggett model the bath must be integrated out (or traced out), which can be understood as taking the average over all the possible realizations of the bath and study the effective dynamics of the quantum system. As a second step, the limit $hbar\; ightarrow\; 0$ must be taken to recover classical mechanics. In order to proceed with those technical steps mathematically, the path integral description of quantum mechanics is usually employed. The resulting classical equations of motion are:

$M\; frac\{d^2\}\{dt^2\}\; X(t)\; =\; -\; frac\{partial\; V(X)\}\{partial\; X\}\; -\; int\_0^\{T\}\; d\; t\text{'}\; alpha\; (t\; -\; t\text{'})\; (X(t)\; -\; X(t\text{'}))$ where:

$alpha\; (t\; -\; t\text{'})\; =\; frac\{1\}\{2pi\}\; int\_0^\{infty\}\; J(omega)\; e^\{-omega|t-t\text{'}\; d\; omega$

is a kernel which characterizes the effective force that affects the motion of the particle in the presence of dissipation. For so-called Markovian baths, which do not keep memory of the interaction with the system, and for Ohmic dissipation, the equations of motion simplify to the classical equations of motion of a particle with friction:

$M\; frac\{d^2\}\{dt^2\}\; X(t)\; =\; -\; frac\{partial\; V(X)\}\{partial\; X\}\; -\; eta\; frac\{d\; X(t)\}\{d\; t\}$

Hence, one can see how Caldeira-Leggett model fulfills the goal of getting classical dissipation from the quantum mechanics framework. The Caldeira-Leggett model has been used to study quantum dissipation problems since its introduction in 1981, being extensively used as well in the field of quantum decoherence.

The dissipative two-level system

This particular realization of the Caldeira - Leggett model deserves special attention due to its interest in the field of Quantum Computation. The aim of the model is to study the effects of dissipation in the dynamics of a particle that can hop between two different positions. This reduced Hilbert space allows the problem to be described in terms of $1/2$ spin operators. The resulting Hamiltonian is also referred in the literature as the Spin-Boson model, reading:

$H=\; Delta\; S\_x\; +\; sum\_ileft(frac\{p\_i^2\}\{2\; m\_i\}+frac\{1\}\{2\}m\_i\; omega\_i^2\; q\_i^2\; ight)\; +\; S\_z\; sum\_i\{C\_i\; q\_i\}$,

where $S\_i\; =\; frac\{sigma\_i\}\{2\},\; i\; =\; x,y,z$ are proportional to the Pauli matrices $sigma\_i$, and $Delta$ is the probability of hopping between the two posible positions. Notice that in this model the counter-term is no longer needed, as the coupling to $S\_z$ gives already homogeneous dissipation.

The model has many applications. In quantum dissipation it is used as a simple model to study the dynamics of a dissipative particle confined in a double-well potential. In the context of Quantum Computation it represents a qubit coupled to an environment, which can produce decoherence. In the study of amorphous solids it provides the basis of the standard theory to describe their thermodynamic properties.

The dissipative two-level systems represents also a paradigm in the study of quantum phase transitions. For a critical value of the coupling to the bath it shows a phase transition from a regime in which the particle is delocalized among the two positions to another in which it is localized in only one of them.

Notes

References

*U. Weiss, "Quantum Dissipative Systems" (1992), World Scientific
*A. Leggett, S. Chakravarty, A. Dorsey, M. Fisher, A. Garg, and W. Zwerger, "Dynamics of the dissipative two-state system", Rev. Mod. Phys., vol. 59, p. 1, 1987.