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'''Quantum stochastic calculus''' is a generalization of [[stochastic calculus]] to [[Commutative property|noncommuting]] variables.<ref name="hudson">{{cite journal
| last = Hudson
| first = R. L.
| last2 = Parthasarathy
| first2 = K. R.
| author2-link = K. R. Parthasarathy (probabilist)
| title = Quantum Ito's Formula and Stochastic Evolutions
| journal = Communications in Mathematical Physics
| volume = 93
| issue = 3
| pages = 301–323
| date = 1984-09-01
| url = http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103941122
| doi = 10.1007/BF01258530
}}</ref> The tools provided by quantum stochastic calculus are of great use for modeling the random evolution of systems undergoing [[Measurement in quantum mechanics|measurement]], as in quantum trajectories.<ref name="control">{{cite book
| last = Wiseman
| first = Howard M.
| author-link = Howard M. Wiseman
| last2 = Milburn
| first2 = Gerard J.
| title = Quantum Measurement and Control
| place = New York
| publisher = Cambridge University Press
| date = 2010
| isbn = 978-0-521-80442-4
}}</ref>{{rp|148}} Just as the [[Lindblad equation|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 equation|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 [[Thermal reservoir|heat bath]]. It is appropriate in many circumstances to model the heat bath as an assembly of [[Harmonic oscillator (quantum)|harmonic oscillators]]. One type of interaction between the system and the bath can be modeled (after making a canonical transformation) by the following [[Hamiltonian (quantum mechanics)|Hamiltonian]]:<ref name="noise">{{cite book
| last = Gardiner
| first = C. W.
| last2 = Zoller
| first2 = P.
| author2-link = Peter Zoller
| title = Quantum Noise
| place = Berlin Heidelberg
| publisher = Springer-Verlag
| series = Springer Series in Synergetics
| date = 2010
| edition = 3rd
| isbn = 978-3-642-06094-6
}}</ref>{{rp|42, 45}}

:<math>H=H_\mathrm{sys}(\mathbf{Z})+\frac{1}{2}\sum_n\left((p_n-\kappa_nX)^2+\omega_n^2q_n^2\right)\,,</math>

where <math>H_\mathrm{sys}</math> is the system Hamiltonian, <math>\mathbf{Z}</math> is a vector containing the system variables corresponding to a finite number of degrees of freedom, <math>n</math> is an index for the different bath modes, <math>\omega_n</math> is the frequency of a particular mode, <math>p_n</math> and <math>q_n</math> are bath operators for a particular mode, <math>X</math> is a system operator, and <math>\kappa_n</math> quantifies the coupling between the system and a particular bath mode.

In this scenario the equation of motion for an arbitrary system operator <math>Y</math> is called the ''quantum Langevin equation'' and may be written as:<ref name="noise" />{{rp|46–47}}

{{Equation box 1 |indent =: |title= |equation = <math>\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}^tf(t-t_0)\dot{X}(t^\prime)\mathrm{d}t^\prime-f(t-t_0)X(t_0)\right\}\right]\,,</math> |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}}

where <math>[\cdot,\cdot]</math> and <math>\{\cdot,\cdot\}</math> denote the [[Commutator#Ring theory|commutator]] and [[Commutator#Anticommutator|anticommutator]] (respectively), the memory function <math>f</math> is defined as:

:<math>f(t)\equiv\sum_n\kappa_n^2\cos(\omega_nt)\,,</math>

and the time dependent noise operator <math>\xi</math> is defined as:

:<math>\xi(t)\equiv i\sum_n\kappa_n\sqrt{\frac{\hbar\omega_n}{2}}\left(-a_n(t_0)e^{-i\omega_n(t-t_0)}+a^\dagger_n(t_0)e^{i\omega_n(t-t_0)}\right)\,,</math>

where the bath annihilation operator <math>a_n</math> is defined as:

:<math>a_n\equiv\frac{\omega_nq_n+ip_n}{\sqrt{2\hbar\omega_n}}\,.</math>

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]] formalism. In such a case the interaction may be modeled by the Hamiltonian <math>H=H_\mathrm{sys}+H_B+H_\mathrm{int}</math> where:<ref name="gardiner" />{{rp|3762}}

:<math>H_B=\hbar\int_{-\infty}^\infty\mathrm{d}\omega\,\omega b^\dagger(\omega)b(\omega)\,,</math>

and

:<math>H_\mathrm{int}=i\hbar\int_{-\infty}^\infty\mathrm{d}\omega\,\kappa(\omega)\left(b^\dagger(\omega)c-c^\dagger b(\omega)\right)\,,</math>

where <math>b(\omega)</math> are [[Creation and annihilation operators|annihilation operators]] for the bath with the commutation relation <math>[b(\omega),b^\dagger(\omega^\prime)]=\delta(\omega-\omega^\prime)</math>, <math>c</math> is an operator on the system, <math>\kappa(\omega)</math> quantifies the strength of the coupling of the bath modes to the system, and <math>H_\mathrm{sys}</math> describes the free system evolution.<ref name="noise" />{{rp|148}} This model uses the [[rotating wave approximation]] and extends the lower limit of <math>\omega</math> to <math>-\infty</math> 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:<ref name="gardiner" />{{rp|3763}}

:<math>\kappa(\omega)=\sqrt{\frac{\gamma}{2\pi}}\,.</math>

Systems coupled to a bath of harmonic oscillators can be thought of as being driven by a noise input and radiating a noise output.<ref name="noise" />{{rp|43}} The input noise operator at time <math>t</math> is defined by:<ref name="noise" />{{rp|150}}<ref name="gardiner" />{{rp|3763}}

:<math>b_\mathrm{in}(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty\mathrm{d}\omega\,e^{-i\omega(t-t_0)}b_0(\omega)\,,</math>

where <math>b_0(\omega)=\left.b(\omega)\right\vert_{t=t_0}</math>, since this operator is expressed in the [[Heisenberg picture]]. Satisfaction of the commutation relation <math>[b_\mathrm{in}(t),b_\mathrm{in}^\dagger(t^\prime)]=\delta(t-t^\prime)</math> allows the model to have a strict correspondence with a [[Markov property|Markovian]] master equation.<ref name="control" />{{rp|142}}

In the white noise setting described so far, the quantum Langevin equation for an arbitrary system operator <math>a</math> takes a simpler form:<ref name="gardiner" />{{rp|3763}}

{{Equation box 1 |indent =: |title= |equation = <math>\dot{a}=-\frac{i}{\hbar}[a,H_\mathrm{sys}]-[a,c^\dagger]\left(\frac{\gamma}{2}c+\sqrt{\gamma}b_\mathrm{in}(t)\right)+\left(\frac{\gamma}{2}c^\dagger+\sqrt{\gamma}b^\dagger_\mathrm{in}(t)\right)[a,c]\,.</math> |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA |ref=WN1}}

For the case most closely corresponding to classical white noise, the input to the system is described by a [[density operator]] giving the following [[Expectation value (quantum mechanics)|expectation value]]:<ref name="noise" />{{rp|154}}

{{NumBlk|:|<math>\langle b^\dagger_\mathrm{in}(t)b_\mathrm{in}(t^\prime)\rangle_{\rho_\mathrm{in}}=N\delta(t-t^\prime)\,.</math>|{{EquationRef|WN2}}}}

=== Quantum Wiener process ===

In order to define quantum stochastic integration, it is important to define a quantum [[Wiener process]]:<ref name="noise" />{{rp|155}}<ref name="gardiner">{{cite journal
| last1 = Gardiner
| first1 = C. W.
| last2 = Collett
| first2 = M. J.
| title = Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation
| journal = Physical Review A
| volume = 31
| issue = 6
| pages = 3761–3774
| date = June 1985
| doi = 10.1103/PhysRevA.31.3761
| pmid = 9895956
}}</ref>{{rp|3765}}

:<math>B(t,t_0)=\int_{t_0}^tb_\mathrm{in}(t^\prime)\mathrm{d}t^\prime\,.</math>

This definition gives the quantum Wiener process the commutation relation <math>[B(t,t_0),B^\dagger(t,t_0)]=t-t_0</math>. The property of the bath annihilation operators in ({{EquationNote|WN2}}) implies that the quantum Wiener process has an expectation value of:

:<math>\langle B^\dagger(t,t_0)B(t,t_0)\rangle_{\rho(t,t_0)}=N(t-t_0)\,.</math>

The quantum Wiener processes are also specified such that their [[Quasiprobability distribution|quasiprobability distributions]] are [[Gaussian]] by defining the density operator:

:<math>\rho(t,t_0)=(1-e^{-\kappa})\exp\left[-\frac{\kappa B^\dagger(t,t_0)B(t,t_0)}{t-t_0}\right]\,,</math>

where <math>N=1/(e^\kappa-1)</math>.<ref name="gardiner" />{{rp|3765}}

== Quantum stochastic integration ==

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

=== Quantum Itō integral ===

The quantum [[Itō calculus#Integration with respect to Brownian motion|Itō integral]] of a system operator <math>g(t)</math> is given by:<ref name="noise" />{{rp|155}}

:<math>(\mathbf{I})\int_{t_0}^tg(t^\prime)\mathrm{d}B(t^\prime)=\lim_{n\to\infty}\sum_{i=1}^ng(t_i)\left(B(t_{i+1},t_0)-B(t_i,t_0)\right)\,,</math>

where the bold ('''I''') preceeding the integral stands for Itō. One of the characteristics of defining the integral in this way is that the increments <math>\mathrm{d}B</math> and <math>\mathrm{d}B^\dagger</math> commute with the system operator.

=== Itō quantum stochastic differential equation ===

In order to define the Itō {{abbr|QSDE|Quantum Stochastic Differential Equation}}, it is necessary to know something about the bath statistics.<ref name="noise" />{{rp|159}} In the context of the white noise formalism described earlier, the Itō {{abbr|QSDE|Quantum Stochastic Differential Equation}} can be defined as:<ref name="noise" />{{rp|156}}

:<math>(\mathbf{I})\,\mathrm{d}a=-\frac{i}{\hbar}[a,H_\mathrm{sys}]\mathrm{d}t+\gamma\left((N+1)\mathcal{D}[c^\dagger]a+N\mathcal{D}[c]a\right)\mathrm{d}t-\sqrt{\gamma}\left([a,c^\dagger]\mathrm{d}B(t)-\mathrm{d}B^\dagger(t)[a,c]\right)\,,</math>

where the equation has been simplified using the [[Lindblad superoperator]]:<ref name="control" />{{rp|105}}

:<math>\mathcal{D}[A]a\equiv AaA^\dagger-\frac{1}{2}\left(A^\dagger Aa+aA^\dagger A\right)\,.</math>

This differential equation is interpreted as defining the system operator <math>a</math> as the quantum Itō integral of the right hand side, and is equivalent to the Langevin equation ({{EquationNote|WN1}}).<ref name="gardiner" />{{rp|3765}}

=== Quantum Stratonovich integral ===

The quantum [[Stratonovich integral]] of a system operator <math>g(t)</math> is given by:<ref name="noise" />{{rp|157}}

:<math>(\mathbf{S})\int_{t_0}^tg(t^\prime)\mathrm{d}B(t^\prime)=\lim_{n\to\infty}\sum_{i=1}^n\frac{g(t_i)+g(t_{i+1})}{2}\left(B(t_{i+1},t_0)-B(t_i,t_0)\right)\,,</math>

where the bold ('''S''') preceeding the integral stands for Stratonovich. Unlike the Itō formulation, the increments in the Stratonovich integral do not commute with the system operator, and it can be shown that:<ref name="noise" />

:<math>(\mathbf{S})\int_{t_0}^tg(t^\prime)\mathrm{d}B(t^\prime)-(\mathbf{S})\int_{t_0}^t\mathrm{d}B(t^\prime)g(t^\prime)=\frac{\sqrt{\gamma}}{2}\int_{t_0}^t\mathrm{d}t^\prime\,[g(t^\prime),c(t^\prime)]\,.</math>

=== Stratonovich quantum stochastic differential equation ===

The Stratonovich {{abbr|QSDE|Quantum Stochastic Differential Equation}} can be defined as:<ref name="noise" />{{rp|158}}

:<math>(\mathbf{S})\,\mathrm{d}a=-\frac{i}{\hbar}[a,H_\mathrm{sys}]\mathrm{d}t-\frac{\gamma}{2}\left([a,c^\dagger]c-c^\dagger[a,c]\right)\mathrm{d}t-\sqrt{\gamma}\left([a,c^\dagger]\mathrm{d}B(t)-\mathrm{d}B^\dagger(t)[a,c]\right)\,.</math>

This differential equation is interpreted as defining the system operator <math>a</math> as the quantum Stratonovich integral of the right hand side, and is in the same form as the Langevin equation ({{EquationNote|WN1}}).<ref name="gardiner" />{{rp|3766–3767}}

=== Relation between Itō and Stratonovich integrals ===

The two definitions of quantum stochastic integrals relate to one another in the following way, assuming a bath with <math>N</math> defined as before:<ref name="noise" />

:<math>(\mathbf{S})\int_{t_0}^tg(t^\prime)\mathrm{d}B(t^\prime)=(\mathbf{I})\int_{t_0}^tg(t^\prime)\mathrm{d}B(t^\prime)+\frac{1}{2}\sqrt{\gamma}N\int_{t_0}^t\mathrm{d}t^\prime\,[g(t^\prime),c(t^\prime)]\,.</math>

=== Calculus rules ===

Just as with classical stochastic calculus, the appropriate product rule can be derived for Itō and Stratonovich integration, respectively:<ref name="noise" />{{rp|156, 159}}

:<math>(\mathbf{I})\,\mathrm{d}(ab)=a\,\mathrm{d}b+b\,\mathrm{d}a+\mathrm{d}a\,\mathrm{d}b\,,</math>

:<math>(\mathbf{S})\,\mathrm{d}(ab)=a\,\mathrm{d}b+\mathrm{d}a\,b\,.</math>

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 Itō and Stratonovitch integration in order to prove that the Stratonovitch form preserves the rules of noncommuting calculus.<ref name="noise" />{{rp|155}}

== Quantum trajectories ==

Quantum trajectories can generally be thought of as the path through [[Hilbert space#Quantum mechanics|Hilbert space]] that the state of a quantum system traverses over time. In a stochastic setting, these trajectories are often [[Conditioning (probability)|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 {{abbr|QSDE|Quantum Stochastic Differential Equation}}. In the case where the conditioned system state is always [[Pure state|pure]], the unraveling could be in the form of a stochastic [[Schrödinger equation]] (SSE). If the state may become mixed, then it is necessary to use a stochastic master equation (SME).<ref name="control" />{{rp|148}}

=== Example unravelings ===

[[File:Master equation unravelings.svg|thumb|Plot of the evolution of the z-component of the [[Bloch vector]] of a two-level atom coupled to the electromagnetic field undergoing damped [[Rabi oscillation|Rabi oscillations]]. The top plot shows the quantum trajectory for the atom for photon-counting measurements performed on the electromagnetic field, the middle plot shows the same for homodyne detection, and the bottom plot compares the previous two measurement choices (each averaged over 32 trajectories) with the unconditioned evolution given by the master equation.]]

Consider the following Lindblad master equation for a system interacting with a vacuum bath:<ref name="control" />{{rp|145}}

:<math>\dot{\rho}=\mathcal{D}[c]\rho-i[H_\mathrm{sys},\rho]\,.</math>

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 {{abbr|SME|Stochastic Master Equation}} describes the evolution of the system conditioned on the results of a continuous [[Photon counting|photon-counting]] measurement performed on the bath:

:<math>\mathrm{d}\rho_I(t)=\left(\mathrm{d}N(t)\mathcal{G}[c]-\mathrm{d}t\mathcal{H}[iH_\mathrm{sys}+\frac{1}{2}c^\dagger c]\right)\rho_I(t)\,,</math>

where

:<math>\begin{array}{rcl}
\mathcal{G}[r]\rho & \equiv & \frac{r\rho r^\dagger}{\operatorname{Tr}[r\rho r^\dagger]}-\rho \\
\mathcal{H}[r]\rho & \equiv & r\rho+\rho r^\dagger-\operatorname{Tr}[r\rho+\rho r^\dagger]\rho
\end{array}
</math>

are nonlinear superoperators and <math>N(t)</math> is the photocount, indicating how many photons have been detected at time <math>t</math> and giving the following jump probability:<ref name="control" />{{rp|152, 155}}

:<math>\operatorname{E}[\mathrm{d}N(t)]=\mathrm{d}t\operatorname{Tr}[c^\dagger c\rho_I(t)]\,,</math>

where <math>\operatorname{E}[\cdot]</math> denotes the expected value. Another type of measurement that could be made on the bath is [[homodyne detection]], which results in quantum trajectories given by the following {{abbr|SME|Stochastic Master Equation}}:

:<math>\mathrm{d}\rho_J(t)=-i[H_\mathrm{sys},\rho_J(t)]\mathrm{d}t+\mathrm{d}t\mathcal{D}[c]\rho_J(t)+\mathrm{d}W(t)\mathcal{H}[c]\rho_J(t)\,,</math>

where <math>\mathrm{d}W(t)</math> is a Wiener increment satisfying:<ref name="control" />{{rp|161}}

:<math>\begin{array}{rcl}
\mathrm{d}W(t)^2 & = & \mathrm{d}t \\
\operatorname{E}[\mathrm{d}W(t)] & = & 0\,.
\end{array}
</math>

Although these two {{abbr|SME|Stochastic Master Equation}}s look wildly different, calculating their expected evolution shows that they are both indeed unravelings of the same Lindlad master equation:

:<math>\operatorname{E}[\mathrm{d}\rho_I(t)]=\operatorname{E}[\mathrm{d}\rho_J(t)]=\dot{\rho}\mathrm{d}t\,.</math>

=== Computational considerations ===

One important application of quanum trajectories is reducing the computational resources required to simulate a master equation. For a Hilbert space of dimension <var>d</var>, the amount of real numbers required to store the density matrix is of order <var>d</var>2, and the time required to compute the master equation evolution is of order <var>d</var>4. Storing the state vector for a {{abbr|SSE|Stochastic Schrödinger Equation}}, on the other hand, only requires an amount of real numbers of order <var>d</var>, and the time to compute trajectory evolution is only of order <var>d</var>2. The master equation evolution can then be approximated by averaging over many individual trajectories simulated using the {{abbr|SSE|Stochastic Schrödinger Equation}}, a technique sometimes referred to as the ''[[Monte Carlo wave function method|Monte Carlo wave-function approach]]''.<ref name="MCWF">{{cite journal | url=http://link.aps.org/doi/10.1103/PhysRevLett.68.580 | title=Wave-function approach to dissipative processes in quantum optics | author=Dalibard, Jean | authorlink=Jean Dalibard | journal=Phys. Rev. Lett. |date=Feb 1992 | volume=68 | issue=5 | pages=580–583 | doi=10.1103/PhysRevLett.68.580 | author2=Castin, Yvan | author3=Mølmer, Klaus | publisher=American Physical Society | pmid=10045937}}</ref> Although the number of calculated trajectories <var>n</var> must be very large in order to accurately approximate the master equation, good results can be obtained for trajectory counts much less than <var>d</var>2</var>. 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.<ref name="control" />{{rp|153}}

== References ==

{{reflist}}

[[Category:Quantum mechanics]]
[[Category:Quantum optics]]
[[Category:Stochastic calculus]]

Pillai's arithmetical function - Revision history

en>Sdipu at 01:15, 6 January 2014