Quantum stochastic calculus is a generalization of
stochastic calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
to
noncommuting variables.
The tools provided by quantum stochastic calculus are of great use for modeling the random evolution of systems undergoing
measurement, as in quantum trajectories.
Just as the
Lindblad master equation provides a quantum generalization to the
Fokker–Planck equation, quantum stochastic calculus allows for the derivation of quantum stochastic differential equations (QSDE) that are analogous to classical
Langevin equations.
For the remainder of this article ''stochastic calculus'' will be referred to as ''classical stochastic calculus'', in order to clearly distinguish it from quantum stochastic calculus.
Heat baths
An important physical scenario in which a quantum stochastic calculus is needed is the case of a system interacting with a
heat bath
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
. It is appropriate in many circumstances to model the heat bath as an assembly of
harmonic oscillators
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'':
\vec F = -k \vec x,
where ''k'' is a positive consta ...
. One type of interaction between the system and the bath can be modeled (after making a canonical transformation) by the following
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
:
:
where
is the system Hamiltonian,
is a vector containing the system variables corresponding to a finite number of degrees of freedom,
is an index for the different bath modes,
is the frequency of a particular mode,
and
are bath operators for a particular mode,
is a system operator, and
quantifies the coupling between the system and a particular bath mode.
In this scenario the equation of motion for an arbitrary system operator
is called the ''quantum Langevin equation'' and may be written as:
where