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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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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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
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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DOI: https://doi.org/10.1007/s11128-015-1196-5