Quantum dissipation
Encyclopedia
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.
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, 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.
,
The first two terms correspond to the Hamiltonian of a quantum particle of mass
and momentum 
, in a potential 
at position 
. The third term describes the bath as a sum of infinite harmonic oscillators with masses 
and momentum 
, at positions 
. 
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. 
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.
To provide a good description of the dissipation mechanism, a relevant quantity is the bath spectral function, defined as follows:
The bath spectral function provides a constraint in the choice of the coefficients
. When this function has the form 
, the corresponding classical kind of dissipation can be shown to be Ohmic. A more generic form is
. In this case, if 
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. 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 studying the effective dynamics of the quantum system. As a second step, the limit
must be taken to recover classical mechanics
. To proceed with those technical steps mathematically, the path integral
description of quantum mechanics
is usually employed. The resulting classical equations of motion are:
where:
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:
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
.
allows the problem to be described in terms of
spin
operators. The resulting Hamiltonian is also referred in the literature as the Spin-Boson model, reading:
,
where
are proportional to the Pauli matrices
, and 
is the probability of hopping between the two possible positions. Notice that in this model the counter-term is no longer needed, as the coupling to 
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. The transition is of Kosterlitz-Thouless kind, as can be seen by deriving the Renormalization group
flow equations for the hopping term.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
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
Dissipation
In physics, dissipation embodies the concept of a dynamical system where important mechanical models, such as waves or oscillations, lose energy over time, typically from friction or turbulence. The lost energy converts into heat, which raises the temperature of the system. Such systems are called...
from the framework of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. It shares many features with the subjects of quantum decoherence
Quantum decoherence
In quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...
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 mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
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, and hence, the description of the bath. To include an irreversible flow of energy (i.e., to avoid Poincaré recurrences
Poincaré recurrence theorem
In mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which...
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
Universality (dynamical systems)
In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together...
, 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.
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 . 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
and momentum , in a potential at position . The third term describes the bath as a sum of infinite harmonic oscillators with masses and momentum , at positions . 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. 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 invariantTranslational symmetry
In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...
, in the sense that the coupling is different wherever the quantum particle is located. This gives rise to an unphysical renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
of the potential, which can be shown to be suppressed by including the counter-term.
To provide a good description of the dissipation mechanism, a relevant quantity is the bath spectral function, defined as follows:
The bath spectral function provides a constraint in the choice of the coefficients
. When this function has the form , the corresponding classical kind of dissipation can be shown to be Ohmic. A more generic form is. In this case, if 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. 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 studying the effective dynamics of the quantum system. As a second step, the limit
must be taken to recover classical mechanicsClassical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
. To proceed with those technical steps mathematically, the path integral
Path integral
Path integral may refer to:* Line integral, the integral of a function along a curve* Functional integration, the integral of a functional over a space of curves...
description of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
is usually employed. The resulting classical equations of motion are:
where:
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:
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
Quantum decoherence
In quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...
.
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 spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
allows the problem to be described in terms of
spinSpin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
operators. The resulting Hamiltonian is also referred in the literature as the Spin-Boson model, reading:
,where
are proportional to the Pauli matricesPauli matrices
The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...
, and is the probability of hopping between the two possible positions. Notice that in this model the counter-term is no longer needed, as the coupling to 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
Qubit
In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....
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. The transition is of Kosterlitz-Thouless kind, as can be seen by deriving the Renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
flow equations for the hopping term.
See also
- Dissipation model for a particle in a ring
- Dissipation model for extended environmentDissipation model for extended environmentA unified model for Diffusion Localization and Dissipation , optionally termed Diffusion with Local Dissipation, has been introduced for the study of Quantal Brownian Motion in dynamical disorder....
- Dissipation model with chaotic environment
- Random matrix theory modelling of dissipation
- Open quantum systemOpen quantum systemIn physics, an open quantum system is a quantum system which is found to be in interaction with an external quantum system, the environment...
- Lindblad equationLindblad equationIn quantum mechanics, the Kossakowski–Lindblad equation or master equation in the Lindblad form is the most general type of markovian and time-homogeneous master equation describing non-unitary evolution of the density matrix \rho that is trace preserving and completely positive for any initial...
- Quantum decoherenceQuantum decoherenceIn quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...
- DephasingDephasingDephasing is a name for the mechanism that recovers classical behavior from a quantum system. It is an important effect in condensed matter physics, particularly in the study of mesoscopic devices...