Quantum stochastic calculus
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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 ...
: :H=H_\mathrm(\mathbf)+\frac\sum_n\left((p_n-\kappa_nX)^2+\omega_n^2q_n^2\right)\,, where H_\mathrm is the system Hamiltonian, \mathbf is a vector containing the system variables corresponding to a finite number of degrees of freedom, n is an index for the different bath modes, \omega_n is the frequency of a particular mode, p_n and q_n are bath operators for a particular mode, X is a system operator, and \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 Y is called the ''quantum Langevin equation'' and may be written as: where cdot,\cdot/math> and \ denote the commutator and
anticommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
(respectively), the memory function f is defined as: :f(t)\equiv\sum_n\kappa_n^2\cos(\omega_nt)\,, and the time dependent noise operator \xi is defined as: :\xi(t)\equiv i\sum_n\kappa_n\sqrt\left(-a_n(t_0)e^+a^\dagger_n(t_0)e^\right)\,, where the bath annihilation operator a_n is defined as: :a_n\equiv\frac\,. Oftentimes this equation is more general than is needed, and further approximations are made to simplify the equation.


White noise formalism

For many purposes it is convenient to make approximations about the nature of the heat bath in order to achieve a
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines ...
formalism. In such a case the interaction may be modeled by the Hamiltonian H=H_\mathrm+H_B+H_\mathrm where: :H_B=\hbar\int_^\infty\mathrm\omega\,\omega b^\dagger(\omega)b(\omega)\,, and :H_\mathrm=i\hbar\int_^\infty\mathrm\omega\,\kappa(\omega)\left(b^\dagger(\omega)c-c^\dagger b(\omega)\right)\,, where b(\omega) are annihilation operators for the bath with the commutation relation (\omega),b^\dagger(\omega^\prime)\delta(\omega-\omega^\prime), c is an operator on the system, \kappa(\omega) quantifies the strength of the coupling of the bath modes to the system, and H_\mathrm describes the free system evolution. This model uses the
rotating wave approximation The rotating-wave approximation is an approximation used in atom optics and magnetic resonance. In this approximation, terms in a Hamiltonian that oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic rad ...
and extends the lower limit of \omega to -\infty in order to admit a mathematically simple white noise formalism. The coupling strengths are also usually simplified to a constant in what is sometimes called the first Markov approximation: :\kappa(\omega)=\sqrt\,. Systems coupled to a bath of harmonic oscillators can be thought of as being driven by a noise input and radiating a noise output. The input noise operator at time t is defined by: :b_\mathrm(t)=\frac\int_^\infty\mathrm\omega\,e^b_0(\omega)\,, where b_0(\omega)=\left.b(\omega)\right\vert_, since this operator is expressed in the
Heisenberg picture In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but ...
. Satisfaction of the commutation relation _\mathrm(t),b_\mathrm^\dagger(t^\prime)\delta(t-t^\prime) allows the model to have a strict correspondence with a Markovian master equation. In the white noise setting described so far, the quantum Langevin equation for an arbitrary system operator a takes a simpler form: For the case most closely corresponding to classical white noise, the input to the system is described by a
density operator In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, usin ...
giving the following
expectation value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
:


Quantum Wiener process

In order to define quantum stochastic integration, it is important to define a quantum
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
: :B(t,t_0)=\int_^tb_\mathrm(t^\prime)\mathrmt^\prime\,. This definition gives the quantum Wiener process the commutation relation (t,t_0),B^\dagger(t,t_0)t-t_0. The property of the bath annihilation operators in () implies that the quantum Wiener process has an expectation value of: :\langle B^\dagger(t,t_0)B(t,t_0)\rangle_=N(t-t_0)\,. The quantum Wiener processes are also specified such that their
quasiprobability distribution A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Quasiprobabilities share several of general features with ordinary probabilities ...
s are
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
by defining the density operator: :\rho(t,t_0)=(1-e^)\exp\left \frac\right,, where N=1/(e^\kappa-1).


Quantum stochastic integration

The stochastic evolution of system operators can also be defined in terms of the stochastic integration of given equations.


Quantum Itô integral

The quantum Itô integral of a system operator g(t) is given by: :(\mathbf)\int_^tg(t^\prime)\mathrmB(t^\prime)=\lim_\sum_^ng(t_i)\left(B(t_,t_0)-B(t_i,t_0)\right)\,, where the bold (I) preceding the integral stands for Itô. One of the characteristics of defining the integral in this way is that the increments \mathrmB and \mathrmB^\dagger commute with the system operator.


Itô quantum stochastic differential equation

In order to define the Itô , it is necessary to know something about the bath statistics. In the context of the white noise formalism described earlier, the Itô can be defined as: :(\mathbf)\,\mathrma=-\frac ,H_\mathrmmathrmt+\gamma\left((N+1)\mathcal ^\dagger+N\mathcal \right)\mathrmt-\sqrt\left( ,c^\daggermathrmB(t)-\mathrmB^\dagger(t) ,cright)\,, where the equation has been simplified using the
Lindblad superoperator In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation, named after Vittorio Gorini, Andrzej Kossakowski, George Sudarshan and Göran Lindblad), master equation in Lindblad form, quantum Liouvillian, or Li ...
: :\mathcal \equiv AaA^\dagger-\frac\left(A^\dagger Aa+aA^\dagger A\right)\,. This differential equation is interpreted as defining the system operator a as the quantum Itô integral of the right hand side, and is equivalent to the Langevin equation ().


Quantum Stratonovich integral

The quantum
Stratonovich integral In stochastic processes, the Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the Itô integral. Although the Itô integral is the usual choice in a ...
of a system operator g(t) is given by: :(\mathbf)\int_^tg(t^\prime)\mathrmB(t^\prime)=\lim_\sum_^n\frac\left(B(t_,t_0)-B(t_i,t_0)\right)\,, where the bold (S) preceding the integral stands for Stratonovich. Unlike the Itô formulation, the increments in the Stratonovich integral do not commute with the system operator, and it can be shown that: :(\mathbf)\int_^tg(t^\prime)\mathrmB(t^\prime)-(\mathbf)\int_^t\mathrmB(t^\prime)g(t^\prime)=\frac\int_^t\mathrmt^\prime\, (t^\prime),c(t^\prime),.


Stratonovich quantum stochastic differential equation

The Stratonovich can be defined as: :(\mathbf)\,\mathrma=-\frac ,H_\mathrmmathrmt-\frac\left( ,c^\dagger-c^\dagger ,cright)\mathrmt-\sqrt\left( ,c^\daggermathrmB(t)-\mathrmB^\dagger(t) ,cright)\,. This differential equation is interpreted as defining the system operator a as the quantum Stratonovich integral of the right hand side, and is in the same form as the Langevin equation ().


Relation between Itô and Stratonovich integrals

The two definitions of quantum stochastic integrals relate to one another in the following way, assuming a bath with N defined as before: :(\mathbf)\int_^tg(t^\prime)\mathrmB(t^\prime)=(\mathbf)\int_^tg(t^\prime)\mathrmB(t^\prime)+\frac\sqrtN\int_^t\mathrmt^\prime\, (t^\prime),c(t^\prime),.


Calculus rules

Just as with classical stochastic calculus, the appropriate product rule can be derived for Itô and Stratonovich integration, respectively: :(\mathbf)\,\mathrm(ab)=a\,\mathrmb+b\,\mathrma+\mathrma\,\mathrmb\,, :(\mathbf)\,\mathrm(ab)=a\,\mathrmb+\mathrma\,b\,. As is the case in classical stochastic calculus, the Stratonovich form is the one which preserves the ordinary calculus (which in this case is noncommuting). A peculiarity in the quantum generalization is the necessity to define both Itô and Stratonovitch integration in order to prove that the Stratonovitch form preserves the rules of noncommuting calculus.


Quantum trajectories

Quantum trajectories can generally be thought of as the path through Hilbert space that the state of a quantum system traverses over time. In a stochastic setting, these trajectories are often conditioned upon measurement results. The unconditioned Markovian evolution of a quantum system (averaged over all possible measurement outcomes) is given by a Lindblad equation. In order to describe the conditioned evolution in these cases, it is necessary to ''unravel'' the Lindblad equation by choosing a consistent . In the case where the conditioned system state is always
pure Pure may refer to: Computing * A pure function * A pure virtual function * PureSystems, a family of computer systems introduced by IBM in 2012 * Pure Software, a company founded in 1991 by Reed Hastings to support the Purify tool * Pure-FTPd, F ...
, the unraveling could be in the form of a stochastic
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
(SSE). If the state may become mixed, then it is necessary to use a stochastic master equation (SME).


Example unravelings

Consider the following Lindblad master equation for a system interacting with a vacuum bath: :\dot=\mathcal rho-i _\mathrm,\rho,. This describes the evolution of the system state averaged over the outcomes of any particular measurement that might be made on the bath. The following describes the evolution of the system conditioned on the results of a continuous photon-counting measurement performed on the bath: :\mathrm\rho_I(t)=\left(\mathrmN(t)\mathcal \mathrmt\mathcal H_\mathrm+\fracc^\dagger cright)\rho_I(t)\,, where :\begin \mathcal rho & \equiv & \frac-\rho \\ \mathcal rho & \equiv & r\rho+\rho r^\dagger-\operatorname \rho+\rho r^\daggerrho \end are nonlinear superoperators and N(t) is the photocount, indicating how many photons have been detected at time t and giving the following jump probability: :\operatorname mathrmN(t)\mathrmt\operatorname ^\dagger c\rho_I(t),, where \operatorname
cdot CDOT may refer to: *\cdot – the LaTeX input for the dot operator (⋅) *Cdot, a rapper from Sumter, South Carolina *Centre for Development of Telematics, India * Chicago Department of Transportation * Clustered Data ONTAP, an operating system from ...
/math> denotes the expected value. Another type of measurement that could be made on the bath is
homodyne detection In electrical engineering, homodyne detection is a method of extracting information encoded as modulation of the phase and/or frequency of an oscillating signal, by comparing that signal with a standard oscillation that would be identical to the s ...
, which results in quantum trajectories given by the following : :\mathrm\rho_J(t)=-i _\mathrm,\rho_J(t)mathrmt+\mathrmt\mathcal rho_J(t)+\mathrmW(t)\mathcal rho_J(t)\,, where \mathrmW(t) is a Wiener increment satisfying: :\begin \mathrmW(t)^2 & = & \mathrmt \\ \operatorname mathrmW(t)& = & 0\,. \end Although these two s look wildly different, calculating their expected evolution shows that they are both indeed unravelings of the same Lindlad master equation: :\operatorname mathrm\rho_I(t)\operatorname mathrm\rho_J(t)\dot\mathrmt\,.


Computational considerations

One important application of quantum trajectories is reducing the computational resources required to simulate a master equation. For a Hilbert space of dimension d, the amount of real numbers required to store the density matrix is of order d2, and the time required to compute the master equation evolution is of order d4. Storing the state vector for a , on the other hand, only requires an amount of real numbers of order d, and the time to compute trajectory evolution is only of order d2. The master equation evolution can then be approximated by averaging over many individual trajectories simulated using the , a technique sometimes referred to as the '' Monte Carlo wave-function approach''. Although the number of calculated trajectories n must be very large in order to accurately approximate the master equation, good results can be obtained for trajectory counts much less than d2. Not only does this technique yield faster computation time, but it also allows for the simulation of master equations on machines that do not have enough memory to store the entire density matrix.


References

{{reflist Quantum optics Stochastic calculus