Quantum stochastic calculus is a generalization ofstochastic calculus tononcommuting variables.^[1] The tools provided by quantum stochastic calculus are of great use for modeling the random evolution of systems undergoingmeasurement, as in quantum trajectories.^[2]: 148 Just as theLindblad master equation provides a quantum generalization to theFokker–Planck equation, quantum stochastic calculus allows for the derivation of quantum stochastic differential equations (QSDE) that are analogous to classicalLangevin equations.

For the remainder of this articlestochastic calculus will be referred to asclassical 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 aheat bath. It is appropriate in many circumstances to model the heat bath as an assembly ofharmonic oscillators. One type of interaction between the system and the bath can be modeled (after making a canonical transformation) by the followingHamiltonian:^{[3]: 42, 45}

${\displaystyle H=H_{\mathrm{s}\mathrm{y}\mathrm{s}}(\mathbf{Z})+\frac{1}{2}\sum _n\left((p_n-{\kappa }_{n}X{)}^2+{\omega }_n^2{q}_n^2\right),}$

where${\displaystyle H_{\mathrm{s}\mathrm{y}\mathrm{s}}}$ 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 thequantum Langevin equation and may be written as:^{[3]: 46–47}

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

${\displaystyle f(t)\equiv \sum _n{\kappa }_n^2\cos ({\omega }_{n}t),}$

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

${\displaystyle \xi (t)\equiv i\sum _n{\kappa }_n\sqrt{\frac{\hslash {\omega }_n}{2}}\left(-a_n(t_0)e^{-i{\omega }_n(t-t_0)}+a_n^{†}(t_0)e^{i{\omega }_n(t-t_0)}\right),}$

where the bath annihilation operator${\displaystyle a_n}$ is defined as:

${\displaystyle a_n\equiv \frac{{\omega }_n{q}_n+i{p}_n}{\sqrt{2\hslash {\omega }_n}}.}$

Oftentimes this equation is more general than is needed, and further approximations are made to simplify the equation.

For many purposes it is convenient to make approximations about the nature of the heat bath in order to achieve awhite noise formalism. In such a case the interaction may be modeled by the Hamiltonian${\displaystyle H=H_{\mathrm{s}\mathrm{y}\mathrm{s}}+H_B+H_{\mathrm{i}\mathrm{n}\mathrm{t}}}$ where:^[4]: 3762

${\displaystyle H_B=\hslash {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{d}\omega \omega b^{†}(\omega )b(\omega ),}$

and

${\displaystyle H_{\mathrm{i}\mathrm{n}\mathrm{t}}=i\hslash {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{d}\omega \kappa (\omega )\left(b^{†}(\omega )c-c^{†}b(\omega )\right),}$

where${\displaystyle b(\omega )}$ areannihilation operators for the bath with the commutation relation${\displaystyle [b(\omega ),b^{†}({\omega }^{\mathrm{\prime }})]=\delta (\omega -{\omega }^{\mathrm{\prime }})}$,${\displaystyle c}$ is an operator on the system,${\displaystyle \kappa (\omega )}$ quantifies the strength of the coupling of the bath modes to the system, and${\displaystyle H_{\mathrm{s}\mathrm{y}\mathrm{s}}}$ describes the free system evolution.^[3]: 148 This model uses therotating wave approximation and extends the lower limit of${\displaystyle \omega }$ to${\displaystyle -\mathrm{\infty }}$ 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:^[4]: 3763

${\displaystyle \kappa (\omega )=\sqrt{\frac{\gamma }{2\pi }}.}$

Systems coupled to a bath of harmonic oscillators can be thought of as being driven by a noise input and radiating a noise output.^[3]: 43 The input noise operator at time${\displaystyle t}$ is defined by:^{[3]: 150 [4]: 3763}

${\displaystyle b_{\mathrm{i}\mathrm{n}}(t)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{d}\omega e^{-i\omega (t-t_0)}{b}_0(\omega ),}$

where${\displaystyle b_0(\omega )={b(\omega )|}_{t=t_0}}$, since this operator is expressed in theHeisenberg picture. Satisfaction of the commutation relation${\displaystyle [b_{\mathrm{i}\mathrm{n}}(t),b_{\mathrm{i}\mathrm{n}}^{†}(t^{\mathrm{\prime }})]=\delta (t-t^{\mathrm{\prime }})}$ allows the model to have a strict correspondence with aMarkovian master equation.^[2]: 142

In the white noise setting described so far, the quantum Langevin equation for an arbitrary system operator${\displaystyle a}$ takes a simpler form:^[4]: 3763

${\displaystyle \dot{a}=-\frac{i}{\hslash }[a,H_{\mathrm{s}\mathrm{y}\mathrm{s}}]-[a,c^{†}]\left(\frac{\gamma }{2}c+\sqrt{\gamma }{b}_{\mathrm{i}\mathrm{n}}(t)\right)+\left(\frac{\gamma }{2}{c}^{†}+\sqrt{\gamma }{b}_{\mathrm{i}\mathrm{n}}^{†}(t)\right)[a,c].}$

(WN1)

For the case most closely corresponding to classical white noise, the input to the system is described by adensity operator giving the followingexpectation value:^[3]: 154

${\displaystyle ⟨b_{\mathrm{i}\mathrm{n}}^{†}(t)b_{\mathrm{i}\mathrm{n}}(t^{\mathrm{\prime }}){⟩}_{{\rho }_{\mathrm{i}\mathrm{n}}}=N\delta (t-t^{\mathrm{\prime }}).}$

WN2

In order to define quantum stochastic integration, it is important to define a quantumWiener process:^{[3]: 155 [4]: 3765}

${\displaystyle B(t,t_0)={\int }_{t_0}^t{b}_{\mathrm{i}\mathrm{n}}(t^{\mathrm{\prime }})\mathrm{d}{t}^{\mathrm{\prime }}.}$

This definition gives the quantum Wiener process the commutation relation${\displaystyle [B(t,t_0),B^{†}(t,t_0)]=t-t_0}$. The property of the bath annihilation operators in (WN2) implies that the quantum Wiener process has an expectation value of:

${\displaystyle ⟨B^{†}(t,t_0)B(t,t_0){⟩}_{\rho (t,t_0)}=N(t-t_0).}$

The quantum Wiener processes are also specified such that theirquasiprobability distributions areGaussian by defining the density operator:

${\displaystyle \rho (t,t_0)=(1-e^{-\kappa })\exp \left[-\frac{\kappa B^{†}(t,t_0)B(t,t_0)}{t-t_0}\right],}$

where${\displaystyle N=1/(e^{\kappa }-1)}$.^[4]: 3765

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

The quantumItô integral of a system operator${\displaystyle g(t)}$ is given by:^[3]: 155

${\displaystyle (\mathbf{I}){\int }_{t_0}^{t}g(t^{\mathrm{\prime }})\mathrm{d}B(t^{\mathrm{\prime }})=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^{n}g(t_i)\left(B(t_{i+1},t_0)-B(t_i,t_0)\right),}$

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${\displaystyle \mathrm{d}B}$ and${\displaystyle \mathrm{d}{B}^{†}}$ commute with the system operator.

In order to define the ItôQSDE, it is necessary to know something about the bath statistics.[3]^: 159 In the context of the white noise formalism described earlier, the ItôQSDE can be defined as:[3]^: 156

${\displaystyle (\mathbf{I})\mathrm{d}a=-\frac{i}{\hslash }[a,H_{\mathrm{s}\mathrm{y}\mathrm{s}}]\mathrm{d}t+\gamma \left((N+1)\mathcal{D}[c^{†}]a+N\mathcal{D}[c]a\right)\mathrm{d}t-\sqrt{\gamma }\left([a,c^{†}]\mathrm{d}B(t)-\mathrm{d}{B}^{†}(t)[a,c]\right),}$

where the equation has been simplified using theLindblad superoperator:^[2]: 105

${\displaystyle \mathcal{D}[A]a\equiv Aa{A}^{†}-\frac{1}{2}\left(A^{†}Aa+a{A}^{†}A\right).}$

This differential equation is interpreted as defining the system operator${\displaystyle a}$ as the quantum Itô integral of the right hand side, and is equivalent to the Langevin equation (WN1).^[4]: 3765

The quantumStratonovich integral of a system operator${\displaystyle g(t)}$ is given by:^[3]: 157

${\displaystyle (\mathbf{S}){\int }_{t_0}^{t}g(t^{\mathrm{\prime }})\mathrm{d}B(t^{\mathrm{\prime }})=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^n\frac{g(t_i)+g(t_{i+1})}{2}\left(B(t_{i+1},t_0)-B(t_i,t_0)\right),}$

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:^[3]

${\displaystyle (\mathbf{S}){\int }_{t_0}^{t}g(t^{\mathrm{\prime }})\mathrm{d}B(t^{\mathrm{\prime }})-(\mathbf{S}){\int }_{t_0}^t\mathrm{d}B(t^{\mathrm{\prime }})g(t^{\mathrm{\prime }})=\frac{\sqrt{\gamma }}{2}{\int }_{t_0}^t\mathrm{d}{t}^{\mathrm{\prime }}[g(t^{\mathrm{\prime }}),c(t^{\mathrm{\prime }})].}$

The StratonovichQSDE can be defined as:[3]^: 158

${\displaystyle (\mathbf{S})\mathrm{d}a=-\frac{i}{\hslash }[a,H_{\mathrm{s}\mathrm{y}\mathrm{s}}]\mathrm{d}t-\frac{\gamma }{2}\left([a,c^{†}]c-c^{†}[a,c]\right)\mathrm{d}t-\sqrt{\gamma }\left([a,c^{†}]\mathrm{d}B(t)-\mathrm{d}{B}^{†}(t)[a,c]\right).}$

This differential equation is interpreted as defining the system operator${\displaystyle a}$ as the quantum Stratonovich integral of the right hand side, and is in the same form as the Langevin equation (WN1).^{[4]: 3766–3767}

The two definitions of quantum stochastic integrals relate to one another in the following way, assuming a bath with${\displaystyle N}$ defined as before:^[3]

${\displaystyle (\mathbf{S}){\int }_{t_0}^{t}g(t^{\mathrm{\prime }})\mathrm{d}B(t^{\mathrm{\prime }})=(\mathbf{I}){\int }_{t_0}^{t}g(t^{\mathrm{\prime }})\mathrm{d}B(t^{\mathrm{\prime }})+\frac{1}{2}\sqrt{\gamma }N{\int }_{t_0}^t\mathrm{d}{t}^{\mathrm{\prime }}[g(t^{\mathrm{\prime }}),c(t^{\mathrm{\prime }})].}$

Just as with classical stochastic calculus, the appropriate product rule can be derived for Itô and Stratonovich integration, respectively:^{[3]: 156, 159}

${\displaystyle (\mathbf{I})\mathrm{d}(ab)=a\mathrm{d}b+b\mathrm{d}a+\mathrm{d}a\mathrm{d}b,}$

${\displaystyle (\mathbf{S})\mathrm{d}(ab)=a\mathrm{d}b+\mathrm{d}ab.}$

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.^[3]: 155

Quantum trajectories can generally be thought of as the path throughHilbert space that the state of a quantum system traverses over time. In a stochastic setting, these trajectories are oftenconditioned 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 tounravel the Lindblad equation by choosing a consistentQSDE. In the case where the conditioned system state is alwayspure, the unraveling could be in the form of a stochasticSchrödinger equation (SSE). If the state may become mixed, then it is necessary to use a stochastic master equation (SME).^[2]: 148

Consider the following Lindblad master equation for a system interacting with a vacuum bath:^[2]: 145

${\displaystyle \dot{\rho }=\mathcal{D}[c]\rho -i[H_{\mathrm{s}\mathrm{y}\mathrm{s}},\rho ].}$

This describes the evolution of the system state averaged over the outcomes of any particular measurement that might be made on the bath. The followingSME describes the evolution of the system conditioned on the results of a continuousphoton-counting measurement performed on the bath:

${\displaystyle \mathrm{d}{\rho }_I(t)=\left(\mathrm{d}N(t)\mathcal{G}[c]-\mathrm{d}t\mathcal{H}[i{H}_{\mathrm{s}\mathrm{y}\mathrm{s}}+\frac{1}{2}{c}^{†}c]\right){\rho }_I(t),}$

where

are nonlinear superoperators and${\displaystyle N(t)}$ is the photocount, indicating how many photons have been detected at time${\displaystyle t}$ and giving the following jump probability:^{[2]: 152, 155}

${\displaystyle \mathrm{E}[\mathrm{d}N(t)]=\mathrm{d}t\mathrm{Tr}[c^{†}c{\rho }_I(t)],}$

where${\displaystyle \mathrm{E}[\cdot ]}$ denotes the expected value. Another type of measurement that could be made on the bath ishomodyne detection, which results in quantum trajectories given by the followingSME:

${\displaystyle \mathrm{d}{\rho }_J(t)=-i[H_{\mathrm{s}\mathrm{y}\mathrm{s}},{\rho }_J(t)]\mathrm{d}t+\mathrm{d}t\mathcal{D}[c]{\rho }_J(t)+\mathrm{d}W(t)\mathcal{H}[c]{\rho }_J(t),}$

where${\displaystyle \mathrm{d}W(t)}$ is a Wiener increment satisfying:[2]^: 161

Although these twoSMEs look wildly different, calculating their expected evolution shows that they are both indeed unravelings of the same Lindlad master equation:

${\displaystyle \mathrm{E}[\mathrm{d}{\rho }_I(t)]=\mathrm{E}[\mathrm{d}{\rho }_J(t)]=\dot{\rho }\mathrm{d}t.}$

One important application of quantum trajectories is reducing the computational resources required to simulate a master equation. For a Hilbert space of dimensiond, the amount of real numbers required to store the density matrix is of orderd², and the time required to compute the master equation evolution is of orderd⁴. Storing the state vector for aSSE, on the other hand, only requires an amount of real numbers of orderd, and the time to compute trajectory evolution is only of orderd². The master equation evolution can then be approximated by averaging over many individual trajectories simulated using theSSE, a technique sometimes referred to as theMonte Carlo wave-function approach.^[5] Although the number of calculated trajectoriesn must be very large in order to accurately approximate the master equation, good results can be obtained for trajectory counts much less thand². 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.^[2]: 153

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