Quantum stochastic calculus

Quantum stochastic calculus is a generalization of stochastic calculus 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.

An important physical scenario in which a quantum stochastic calculus is needed is the case of a system interacting with a heat bath. It is appropriate in many circumstances to model the heat bath as an assembly of harmonic oscillators. One type of interaction between the system and the bath can be modeled (after making a canonical transformation) by the following Hamiltonian:

where ${\displaystyle H_{\mathrm {sys} }}$ is the system Hamiltonian, ${\displaystyle \mathbf {Z} }$ is a vector containing the system variables corresponding to a finite number of degrees of freedom, ${\displaystyle n}$ is an index for the different bath modes, ${\displaystyle \omega _{n}}$ is the frequency of a particular mode, ${\displaystyle p_{n}}$ and ${\displaystyle q_{n}}$ are bath operators for a particular mode, ${\displaystyle X}$ is a system operator, and ${\displaystyle \kappa _{n}}$ quantifies the coupling between the system and a particular bath mode.

In this scenario the equation of motion for an arbitrary system operator ${\displaystyle Y}$ is called the quantum Langevin equation and may be written as:

${\displaystyle {\dot {Y}}(t)={\frac {i}{\hbar }}[H_{\mathrm {sys} },Y(t)]-{\frac {i}{2\hbar }}\left[X,\left\{Y(t),\xi (t)-\int _{t_{0}}^{t}f(t-t_{0}){\dot {X}}(t^{\prime })\mathrm {d} t^{\prime }-f(t-t_{0})X(t_{0})\right\}\right]\,,}$

where ${\displaystyle [\cdot ,\cdot ]}$ and ${\displaystyle \{\cdot ,\cdot \}}$ denote the commutator and anticommutator (respectively), the memory function ${\displaystyle f}$ is defined as:

and the time dependent noise operator ${\displaystyle \xi }$ is defined as: