Classical and quantum coupled oscillators: symplectic structure

Magalhães, A.R. Bosco de; Fonseca, C.H. d'Àvila; Nemes, M. C.

doi:10.1088/0031-8949/74/4/011

Cited by 12 publications

(7 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Such a behavior of spectral anticrossing is a typical sign of strong coupling. The observed spectral behavior can easily be explained and fitted (blue curve) by a model assuming strong coupling between two harmonic oscillators: , the cavity field (measured gray Lorentzian line shape functions for the on-axis cavity mode) and the longitudinal plasmon mode of the GNR represented by the brown Lorentzian curve with the fixed spectral maximum and spectral width as fitting parameters.…”

mentioning

confidence: 99%

“…Both can be regarded as damped Lorentzian oscillators scriptL i with intrinsic eigenfrequency ω i and damping constant γ i where i = R , G symbolizing the resonator ( R ) and the GNR ( G ). The respective differential equations (see Supporting Information) for such coupled systems can be solved for the coupled resonance frequencies ω ± by introducing a complex eigenfrequency for i : ω̃ i 2 = ω i 2 − i ω i γ i with solutions , ω̃ ± 2 = 1 2 [ ω̃ R 2 + ω̃ G 2 ± ( ω̃ R 2 − ω̃ G 2 ) 2 + 4 g 2 ] where the coupling g is given in terms of the coupling frequencies Ω i by g = normalΩ G normalΩ R which are only equal for identical oscillators. As can be seen from eq the coupled system shows a high frequency scriptL + and a low frequency mode …”

mentioning

confidence: 99%

See 1 more Smart Citation

Strong and Coherent Coupling of a Plasmonic Nanoparticle to a Subwavelength Fabry–Pérot Resonator

et al. 2015

View full text Add to dashboard Cite

A major aim in experimental nano- and quantum optics is observing and controlling the interaction between light and matter on a microscopic scale. Coupling molecules or atoms to optical microresonators is a prominent method to alter their optical properties such as luminescence spectra or lifetimes. Until today strong coupling of optical resonators to such objects has only been observed with atom-like systems in high quality resonators. We demonstrate first experiments revealing strong coupling between individual plasmonic gold nanorods (GNR) and a tunable low quality resonator by observing cavity-length-dependent nonlinear dephasing and spectral shifts indicating spectral anticrossing of the luminescent coupled system. These phenomena and experimental results can be described by a model of two coupled oscillators representing the plasmon resonance of the GNR and the optical fields of the resonator. The presented reproducible and accurately tunable resonator allows us to precisely control the optical properties of individual particles.

show abstract

mentioning

confidence: 99%

mentioning

confidence: 99%

Strong and Coherent Coupling of a Plasmonic Nanoparticle to a Subwavelength Fabry–Pérot Resonator

et al. 2015

View full text Add to dashboard Cite

show abstract

“…Clearly, this corresponds to coupled harmonic oscillators. As demonstrated in Appendix A, this Hamiltonian can be solved exactly using the symplectic transformation [58,59], which decouples the coupled harmonic oscillators into the following form:…”

Section: Beyond the Mean-field Approachmentioning

confidence: 99%

Quantum Phase Transitions in a Generalized Dicke Model

Liu,

Duan

2023

Entropy

View full text Add to dashboard Cite

We investigate a generalized Dicke model by introducing two interacting spin ensembles coupled with a single-mode bosonic field. Apart from the normal to superradiant phase transition induced by the strong spin–boson coupling, interactions between the two spin ensembles enrich the phase diagram by introducing ferromagnetic, antiferromagnetic and paramagnetic phases. The mean-field approach reveals a phase diagram comprising three phases: paramagnetic–normal phase, ferromagnetic–superradiant phase, and antiferromagnetic–normal phase. Ferromagnetic spin–spin interaction can significantly reduce the required spin–boson coupling strength to observe the superradiant phase, where the macroscopic excitation of the bosonic field occurs. Conversely, antiferromagnetic spin–spin interaction can strongly suppress the superradiant phase. To examine higher-order quantum effects beyond the mean-field contribution, we utilize the Holstein–Primakoff transformation, which converts the generalized Dicke model into three coupled harmonic oscillators in the thermodynamic limit. Near the critical point, we observe the close of the energy gap between the ground and the first excited states, the divergence of entanglement entropy and quantum fluctuation in certain quadrature. These observations further confirm the quantum phase transition and offer additional insights into critical behaviors.

show abstract

“…In this section, we establish a connection between Lyapunov exponents and the rate of information loss, in this model, which we generalize to periodic quadratic Anosov systems in the following section and exemplify with a physically more sound system of coupled parametric oscillators. The solution of the Heisenberg equations of motion for the vector ẑ = ( x1 , p1 , x2 , p2 ) T is as follows [22]:…”

Section: Inverted Harmonic Environment (Ihe)mentioning

confidence: 99%

An analytical relation between entropy production and quantum Lyapunov exponents for Gaussian bipartite systems

Romero

Parreira

Souza

et al. 2008

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

Abstract. We study and compare the information loss of a large class of Gaussian bipartite systems. It includes the usual Caldeira-Leggett type model as well as Anosov models (parametric oscillators, the inverted oscillator environment, etc), which exhibit instability, one of the most important characteristics of chaotic systems. We establish a rigorous connection between the quantum Lyapunov exponents and coherence loss and show that in the case of unstable environments, coherence loss is completely determined by the upper quantum Lyapunov exponent, a behavior which is more universal than that of the Caldeira-Leggett type model.

show abstract

Classical and quantum coupled oscillators: symplectic structure

Abstract: We consider a set of N linearly coupled harmonic oscillators and show that the diagonalization of this problem can be put in geometrical terms. The matrix techniques developed here allowed for solutions in both the classical and quantum regimes.

Cited by 12 publications

References 19 publications

Strong and Coherent Coupling of a Plasmonic Nanoparticle to a Subwavelength Fabry–Pérot Resonator

Strong and Coherent Coupling of a Plasmonic Nanoparticle to a Subwavelength Fabry–Pérot Resonator

Quantum Phase Transitions in a Generalized Dicke Model

An analytical relation between entropy production and quantum Lyapunov exponents for Gaussian bipartite systems

Contact Info

Product

Resources

About