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Realizing the dynamics of a non-Markovian quantum system by Markovian coupled oscillators: a Green’s function-based root locus approach

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Abstract

In this paper, we develop a Green’s function-based root locus approach to realizing a Lorentzian-noise-disturbed non-Markovian quantum system by Markovian coupled oscillators in an extended Hilbert space. By using a Green’s function-based root locus method, we design an ancillary oscillator for Markovian coupled oscillators to be a Lorentzian noise generator. Thus a principal oscillator coupled to the ancillary oscillator via a direct interaction can capture the dynamics of a Lorentzian-noise-disturbed non-Markovian quantum system. By matching the root locus in the frequency domain, conditions for the realization are obtained and a critical transition in the non-Markovian quantum system can also be observed in the Markovian coupled oscillators.

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Acknowledgments

This research was supported under Australian Research Councils Laureate Fellowships funding schemes (Projects FL110100020) and the Chinese Academy of Sciences President’s International Fellowship Initiative (No. 2015DT006).

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Correspondence to Shibei Xue.

Appendix: Calculation of \(G^m(s)\)

Appendix: Calculation of \(G^m(s)\)

Equation (28) can be solved via the Laplace transform method and an expression for \(G^m(s)\) is given as

$$\begin{aligned}&G^m(s)=\mathcal {L}[G^m(t)]=\nonumber \\&\left\{ \begin{array}{cc} \frac{1}{\chi }\left[ \begin{array}{cc} \frac{\left( \xi +\frac{\chi }{2}\right) -i\epsilon }{s-p_+}+\frac{\left( -\xi +\frac{\chi }{2}\right) +i\epsilon }{s-p_-} &{} -\kappa \hbox {e}^{-i\frac{\theta }{2}}\left( \frac{1}{s-p_+}-\frac{1}{s-p_-}\right) \\ \kappa \hbox {e}^{-i\frac{\theta }{2}}\left( \frac{1}{s-p_+}-\frac{1}{s-p_-}\right) &{} \frac{\left( -\xi +\frac{\chi }{2}\right) +i\epsilon }{s-p_+}+\frac{\left( \xi +\frac{\chi }{2}\right) -i\epsilon }{s-p_-} \\ \end{array} \right] \nonumber , &{} ~~p_+\ne p_-\\ \left[ \begin{array}{cc} \frac{\frac{\gamma }{4}-i\frac{\varDelta }{2}}{(s-p)^2}+\frac{1}{s-p} &{} \frac{\kappa }{(s-p)^2} \\ \frac{-\kappa }{(s-p)^2} &{} \frac{-\frac{\gamma }{4}+i\frac{\varDelta }{2}}{(s-p)^2}+\frac{1}{s-p} \\ \end{array} \right] ,&p_\pm =p \end{array}\right. ~ \end{aligned}$$

By using the inverse Laplace transform, an expression of \(G^m(t)\) can be obtained as

with \(\xi =\frac{\gamma }{4}\cos \frac{\theta }{2}-\frac{\varDelta }{2}\sin \frac{\theta }{2}\), \(\epsilon =\frac{\gamma }{4}\sin \frac{\theta }{2}+\frac{\varDelta }{2}\cos \frac{\theta }{2}\), where \(G^m(s)=\mathcal {L}[G^m(t)]\) is the Laplace transform of \(G^m(t)\) and the inverse Laplace transform is denoted as \(\mathcal {L}^{-1}[\cdot ]\).

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Xue, S., Petersen, I.R. Realizing the dynamics of a non-Markovian quantum system by Markovian coupled oscillators: a Green’s function-based root locus approach. Quantum Inf Process 15, 1001–1018 (2016). https://doi.org/10.1007/s11128-015-1196-5

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