Quantum dissipation

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.[2] It describes a quantum particle in one dimension coupled to a bath. The Hamiltonian reads:

${\displaystyle H={\frac {P^{2}}{2M}}+V(X)+\sum _{i}\left({\frac {p_{i}^{2}}{2m_{i}}}+{\frac {1}{2}}m_{i}\omega _{i}^{2}q_{i}^{2}\right)-X\sum _{i}{C_{i}q_{i}}+X^{2}\sum _{i}{\frac {C_{i}^{2}}{2m_{i}\omega _{i}^{2}}}}$,

The first two terms correspond to the Hamiltonian of a quantum particle of mass ${\displaystyle M}$ and momentum ${\displaystyle P}$, in a potential ${\displaystyle V}$ at position ${\displaystyle X}$. The third term describes the bath as an infinite sum of harmonic oscillators with masses ${\displaystyle m_{i}}$ and momentum ${\displaystyle p_{i}}$, at positions ${\displaystyle q_{i}}$. ${\displaystyle \omega _{i}}$ 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. ${\displaystyle C_{i}}$ 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 employing real potentials.[3]

To provide a good description of the dissipation mechanism, a relevant quantity is the bath spectral function, defined as follows:

${\displaystyle J(\omega )={\frac {\pi }{2}}\sum _{i}{\frac {C_{i}^{2}}{m_{i}\omega _{i}}}\delta (\omega -\omega _{i})}$

The bath spectral function provides a constraint in the choice of the coefficients ${\displaystyle C_{i}}$. When this function has the form ${\displaystyle J(\omega )=\eta \omega }$, the corresponding classical kind of dissipation can be shown to be Ohmic. A more generic form is ${\displaystyle J(\omega )\propto \omega ^{s}}$. In this case, if ${\displaystyle s>1}$ the dissipation is called "super-ohmic", while if ${\displaystyle s<1}$ 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 ${\displaystyle \hbar \rightarrow 0}$ 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:

${\displaystyle M{\frac {d^{2}}{dt^{2}}}X(t)=-{\frac {\partial V(X)}{\partial X}}-\int _{0}^{T}dt'\alpha (t-t')(X(t)-X(t'))}$

where:

${\displaystyle \alpha (t-t')={\frac {1}{2\pi }}\int _{0}^{\infty }J(\omega )e^{-\omega |t-t'|}d\omega }$

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:

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

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 is a particular realization of the Caldeira–Leggett model that 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 rather a continuous degree of freedom. This reduced Hilbert space allows the problem to be described in terms of ½-spin operators. This is sometimes referred in the literature as the spin-boson model, and it is closely related to the Jaynes–Cummings model.

The Hamiltonian for the dissipative two-level system reads:

${\displaystyle H=\Delta {\frac {\sigma _{x}}{2}}+\sum _{i}\left({\frac {p_{i}^{2}}{2m_{i}}}+{\frac {1}{2}}m_{i}\omega _{i}^{2}q_{i}^{2}\right)+{\frac {\sigma _{z}}{2}}\sum _{i}{C_{i}q_{i}}}$,

where ${\displaystyle \sigma _{x}}$ and ${\displaystyle \sigma _{z}}$ are the Pauli matrices and ${\displaystyle \Delta }$ is the amplitude of hopping between the two possible positions. Notice that in this model the counter-term is no longer needed, as the coupling to ${\displaystyle S_{z}}$ 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 system 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.

A different approach to describe energy dissipation is to consider time dependent Hamiltonians. Against a common misunderstanding, the resulting unitary dynamics can describe energy dissipation. However, the quantum mechanical state of the system stays pure, thus such an approach can not describe dephasing. Dephasing leads to quantum decoherence or information dissipation and is often important when describing open quantum systems. However, this approach is typically used e.g. in the description of optical experiments. There a light pulse (described by a time dependent semi-classical Hamiltonian) can change the energy in the system by stimulated absorption or emission.