Quadratic open quantum harmonic oscillator

Abstract

We study the quantum open system evolution described by a Gorini–Kossakowski–Sudarshan–Lindblad generator with creation and annihilation operators arising in Fock representations of the \(\mathfrak {sl}_2\) Lie algebra. We show that any initial density matrix evolves to a fully supported density matrix and converges towards a unique equilibrium state. We show that the convergence is exponentially fast and we exactly compute the rate for a wide range of parameters. We also discuss the connection with the two-photon absorption and emission process.

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Acknowledgements

A. Dhahri acknowledges support by Fondo professori stranieri DHG9VARI01 Politecnico di Milano. The research by H. J. Yoo was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03936006).

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Correspondence to Ameur Dhahri.

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Dhahri, A., Fagnola, F. & Yoo, H.J. Quadratic open quantum harmonic oscillator. Lett Math Phys 110, 1759–1782 (2020). https://doi.org/10.1007/s11005-020-01274-0

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Keywords

  • Quantum harmonic oscillator
  • Quantum Markov semigroup
  • Fock representations of the \(\mathfrak {sl}_2\) algebra
  • Spectral gap

Mathematics Subject Classification

  • 46L55
  • 82C10
  • 60J27