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Effective Evolution Equations from Quantum Dynamics
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Quantum state diffusion QSD is one of these unraveling techniques. QSD reproduces the master equation in the mean. And this is what is meant by an unraveling of the master equation. Expectation values for operators obey a similar relationship:. The use of QSD as a practical algorithm to solve master equations has been widely investigated [ 16 , 17 ].
This includes calculations of output spectra in quantum optics [ 18 ]. As a practical method of computation, QSD gains over the direct solution of the master equation, because of a basis of N states, QSD needs a computer store with N elements, and the time of computation is also proportional to N.
For the direct solution these are proportional to N 2. In this paper, we take advantage of the system simulation method to simulate the evolutionary behavior of the open quantum systems and thus calculate and analyze various physical properties of the ensemble of open quantum systems.
However, noting that we cannot usually get the analytical solution of 2 , an alternative way is to find the numerical solution of the system evolution and investigate various control algorithms and control strategies based on the simulation method, which is a powerful tool built on the systems science, system identification, control theory, and computer technology for the analysis and synthesis of complex systems, especially large-scale systems [ 19 ].
In the system simulation, we should pay attention to the problem that the physical description of the stochastic process is relied on the master equation in the Lindblad operator form. If we want to get the numerical solution, the Lindblad operator must be explicitly quantified. Fortunately, existing results have summarized various forms of decoherence Lindblad operator in open quantum systems for reference; a general form of 2 can be written as [ 20 ].
According to the above system model, in the given interval [0, t f ], a sample of realization can be generated by the following algorithm [ 20 ]. According to the above framework, an iterative algorithm is described as follows:. We consider the process describing the direct photodetection of a driven two-level system, and the piecewise deterministic process is given by the following equation [ 21 , 22 ]:.
From a sample of realizations this probability is estimated by determining the average. According [ 21 , 22 ], the analytical solution of the process is. Hence, it is possible to compare the numerical results with the analytical results. In order to discuss the performance of the algorithm, we introduced the classic Runge-Kutta iterative algorithm for generating the sample of realizations as follows [ 23 ]:.
As can be seen from Table 1 , when the simulation step size is increased, there is little change in the errors in the proposed algorithm, while the errors using the Runge-Kutta method increase linearly.
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That is, in a certain step length range, the proposed iterative algorithm can generate more accurate approximation numerical solution than the Runge-Kutta algorithm in comparison to the analytical solution. At the same time, we can draw a conclusion from Table 2 : when obtaining a more accurate numerical solution using the proposed algorithm, the computational time is on the same order of magnitude as the Runge-Kutta method consumes.
And it is not difficult to find that, when the step length is gradually reduced, accompanied by improving the accuracy, the computational time of the Runge-Kutta algorithm grows faster. Taking the above two advantages comparing to the classical Runge-Kutta algorithm, the proposed algorithm can generate a more accurate sample of realizations while paying lower computational costs, which reflects the superiority and practicality of the proposed algorithm.
We denote X t k as the true analytical solution and X k as the numerical approximation. Before our discussion, several conventions should be made as follows. The single-step error of a certain numerical scheme may then be expressed through the difference. In actual applications, one tends to care about this weaker form of convergence especially when considering the approximation of functionals of the stochastic variable. When investigating the stochastic differential equations and numerical simulation solution process, if the issues are related to the numerical simulation of the stochastic process, the evaluation criteria of the solutions convergence are usually defined as the numerical approximation curve.
It must be sufficiently close to the real trace of the evolution. That is to say, the higher the convergence order is, the closer the distribution of the numerical solution is with the analytical solution of the distribution. Thus, it is really a higher-order strategy in the weaker convergence sense.
The decoherence of open quantum systems usually makes the system evolve from the initial pure state to mixed states in some cases, may also be mixed state into a pure state. Being a powerful tool for investigating the open quantum systems, the quantum master equation can give a quantitative description of the transition, dissipation, and decoherence caused by the interaction between the closed system and the environment. Taking this as the starting point of our research, in order to obtain the evolution of the open quantum systems according to its dynamic characteristics, we used the system simulation method to get the numerical solution to the reduced density operator of a typical open quantum system.
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And its effectiveness and superiority were verified in comparison with the classical algorithm. Further research includes the control scheme [ 25 , 26 ] for quantum manipulation based on the characteristics of quantum dynamics. National Center for Biotechnology Information , U. Journal List ScientificWorldJournal v. Published online May Author information Article notes Copyright and License information Disclaimer. Received Apr 11; Accepted Apr This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract The quantum stochastic differential equation derived from the Lindblad form quantum master equation is investigated. Open in a separate window. Figure 1. Conclusions The decoherence of open quantum systems usually makes the system evolve from the initial pure state to mixed states in some cases, may also be mixed state into a pure state. References 1. Nelson E.
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