General solution of the time evolution of two interacting harmonic oscillators

authored by: David Edward Bruschi, G. S. Paraoanu, Ivette Fuentes, Frank K. Wilhelm, Andreas W. Schell
Abstract: We study the time evolution of an ideal system composed of two harmonic oscillators coupled through a quadratic Hamiltonian with arbitrary interaction strength. We solve its dynamics analytically by employing tools from symplectic geometry. In particular, we use this result to completely characterize the dynamics of the two oscillators interacting in the ultrastrong-coupling regime with additional single-mode squeezing on both oscillators, as well as higher-order terms. Furthermore, we compute quantities of interest, such as the average number of excitations and the correlations that are established between the two subsystems due to the evolution. We find that this model predicts a second-order phase transition and we compute the critical exponents and the critical value. We also provide an exact decoupling of the time evolution in terms of simple quantum optical operations, which can be used for practical implementations and studies. Finally, we show how our techniques can be extended to include more oscillators and higher-order interactions.
Organisation(s): Institute of Solid State Physics
QuantumFrontiers
External Organisation(s): Saarland University
Brno University of Technology
Aalto University
University of Nottingham
University of Southampton
National Metrology Institute of Germany (PTB)
Type: Article
Journal: Physical Review A
Volume: 103
ISSN: 2469-9926
Publication date: 11.02.2021
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Atomic and Molecular Physics, and Optics
Electronic version(s): https://doi.org/10.1103/PhysRevA.103.023707 (Access: Closed)

BibTeX

@article{62bbe5fcfbf340d3ad0d5d752d11720a,
title = "General solution of the time evolution of two interacting harmonic oscillators",
abstract = "We study the time evolution of an ideal system composed of two harmonic oscillators coupled through a quadratic Hamiltonian with arbitrary interaction strength. We solve its dynamics analytically by employing tools from symplectic geometry. In particular, we use this result to completely characterize the dynamics of the two oscillators interacting in the ultrastrong-coupling regime with additional single-mode squeezing on both oscillators, as well as higher-order terms. Furthermore, we compute quantities of interest, such as the average number of excitations and the correlations that are established between the two subsystems due to the evolution. We find that this model predicts a second-order phase transition and we compute the critical exponents and the critical value. We also provide an exact decoupling of the time evolution in terms of simple quantum optical operations, which can be used for practical implementations and studies. Finally, we show how our techniques can be extended to include more oscillators and higher-order interactions.",
author = "Bruschi, {David Edward} and Paraoanu, {G. S.} and Ivette Fuentes and Wilhelm, {Frank K.} and Schell, {Andreas W.}",
note = "Funding Information: We thank Pablo Tieben and Anna Okopi{\'n}ska for useful comments and suggestions. G.S.P. acknowledges support from Foundational Questions Institute, from the European Commission project Quantum readout techniques and technologies (Grant No. 862644, FET Open, Horizon 2020), and from the Academy of Finland through Project No. 328193 and through the “Finnish Center of Excellence in Quantum Technology Quantum Technology Finland” Project No. 312296. D.E.B. acknowledges the Central European Institute of Technology Nano RI for partial support. A.W.S. was partly funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC-2123 QuantumFrontiers 390837967.",
year = "2021",
month = feb,
day = "11",
doi = "10.1103/PhysRevA.103.023707",
language = "English",
volume = "103",
journal = "Physical Review A",
issn = "2469-9926",
publisher = "American Physical Society",
number = "2",
}