Therefore, for simplicity, we assume the system and the environment are separate at \(t=0\), and the environment is in the vacuum state, \(|\Psi (0)\rangle = |\psi _0\rangle \otimes |0\rangle\). Consequently, the operator \(\mathcal {M}(R(t)\bar{O}^\dagger )\) can be explicitly expressed as \(\frac{\Gamma pop over here {M}(R(t))L^\dagger\). In order to apply it, we need to slightly adjust the form of \(\tilde{b_k}(t)\) towhere the term \(e^{i\omega _k (t+s)}\) is zero, due to the rotating wave approximation. And we define the two termswhich can be simulated independently with the same set of stochastic noise \(z_t^*\). In quantitative finance, the theory is known as Ito Calculus.

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Many stochastic processes are based on functions which are continuous, but nowhere differentiable. Most often, the reduced density matrix of the central system is the key quantity to describe the interested central system, and is defined as the partial trace of the total system-reservoir density matrix over the degrees of freedom of the reservoir, \(\rho _{S}(t) = \mathrm {Tr}_R[\rho _{tot}(t)]\). (32),In the first term of the above expression, it should be noted that \(\tilde{b_k}(t) = U(t)b_kU^\dagger (t)\) cannot apply the analytical solution of \(b_k(t)\) in Eq. Firstly, let us consider the observable \(b_k(t)= U^\dagger (t)b_k U(t)\) in the Heisenberg picture and its evolution equation acquiresInserting the formal solution of \(b_k(t)\),into the Eq. A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process.

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org/licenses/by/4. We can then finally use a no-arbitrage argument to price a European call option via the derived Black-Scholes equation. Z. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. Our approach will only employ one stochastic noise \(z_t^*\) to simulate the TTCF, which can be easily extended to a complicate-structured quantum model.

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 2c, the \(t’\) is changed to 10. The Fourier spectral functions of the two TTCFs are shown in Fig. It is also the notation used in publications on numerical methods for solving stochastic differential equations. A weak solution consists of a probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space. This method allows us to turn the hard open quantum system problems into a numerical ensemble average problem over all possible trajectories. The following is a typical existence and uniqueness theorem for Itô SDEs taking values in n-dimensional Euclidean space Rn and driven by an m-dimensional Brownian motion B; the proof may be found in Øksendal (2003, §5.

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Without loss of generality, the initial read review of the system is chosen to be \(|\psi _0\rangle = (|0\rangle + |1\rangle )/\sqrt{2}\). In Fig. For simplicity, we study a zero-temperature environment and focus on the quantum fluctuations only. Note that our method works for arbitrary spectral functions and the corresponding correlation functions of the environment, including Ohmic, super- and sub-Ohmic, etc.

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Moreover, our method can be extended to study the multiple time correlation functions. Therefore, the following is the most general class of SDEs:
where

x

X

{\displaystyle x\in X}

is the position in the system in its phase (or state) space,

X

{\displaystyle X}

, assumed to be a differentiable manifold, the

F

T
X

{\displaystyle F\in TX}

is a flow vector field representing deterministic law of evolution, and

g

T
X

this article {\displaystyle g_{\alpha }\in TX}

is a set of vector fields that define the coupling of the system to Gaussian white noise,

{\displaystyle \xi ^{\alpha }}

. .

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