Solution to the Time-Dependent Coupled Harmonic Oscillators Hamiltonian with Arbitrary Interactions

Urzúa, Alejandro R.; Ramos-Prieto, Irán; Fernández‐Guasti, M.; Moya‐Cessa, H.

doi:10.3390/quantum1010009

Cited by 34 publications

(28 citation statements)

References 42 publications

(60 reference statements)

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“…These results will be used to compute the occupation numbers and used to discuss entanglement through logarithmic negativity. Note that Solution ( 16) can also be obtained by using the method presented in Reference [34].…”

Section: Time-dependent Schrödinger Equationmentioning

confidence: 99%

Instability of Meissner Differential Equation and Its Relation with Photon Excitations and Entanglement in a System of Coupled Quantum Oscillators

et al. 2021

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In this work, we investigate the Schrödinger dynamics of photon excitation numbers and entanglement in a system composed by two non-resonant time-dependent coupled oscillators. By considering π periodically pumped parameters (oscillator frequencies and coupling) and using suitable transformations, we show that the quantum dynamics can be determined by two classical Meissner oscillators. We then study analytically the stability of these differential equations and the dynamics of photon excitations and entanglement in the quantum system numerically. Our analysis shows two interesting results, which can be summarized as follows: (i) Classical instability of classical analog of quantum oscillators and photon excitation numbers (expectations Nj) are strongly correlated, and (ii) photon excitations and entanglement are connected to each other. These results can be used to shed light on the link between quantum systems and their classical counterparts and provide a nice complement to the existing works studying the dynamics of coupled quantum oscillators.

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Section: Time-dependent Schrödinger Equationmentioning

confidence: 99%

Instability of Meissner Differential Equation and Its Relation with Photon Excitations and Entanglement in a System of Coupled Quantum Oscillators

et al. 2021

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“…Recently, the Hamiltonian for two coupled harmonic oscillators has been relevant given the applications in several problems. For example, in [29,30], the general timedependent solutions for this Hamiltonian were found. In [31], the entanglement between modes in this particular system was reported, while in [32], the reflection coefficient in such a type of system was presented.…”

Section: Examplementioning

confidence: 99%

Measurement of the Temperature Using the Tomographic Representation of Thermal States for Quadratic Hamiltonians

López-Saldívar

Man’ko

2021

Entropy

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The Wigner and tomographic representations of thermal Gibbs states for one- and two-mode quantum systems described by a quadratic Hamiltonian are obtained. This is done by using the covariance matrix of the mentioned states. The area of the Wigner function and the width of the tomogram of quantum systems are proposed to define a temperature scale for this type of states. This proposal is then confirmed for the general one-dimensional case and for a system of two coupled harmonic oscillators. The use of these properties as measures for the temperature of quantum systems is mentioned.

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“…(83) This is indeed the Hamiltonian of two-coupled harmonic oscillators, with coupling −s1s2κ 2 c , but with the exotic feature that the kinetic terms can have negative signs. The analysis could then be carried on in the line of Bruschi et al (2019); Urzúa et al (2019); Moya-Cessa & Récamier (2020); Ramos-Prieto et al (2020) for example, to quote only recent studies.…”

Section: Coupled Harmonic Oscillator Formmentioning

confidence: 99%

An MHD spectral theory approach to Jeans’ magnetized gravitational instability

Durrive

Keppens

Langer

2021

Monthly Notices of the Royal Astronomical Society

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In this paper, we revisit the governing equations for linear magnetohydrodynamic (MHD) waves and instabilities existing within a magnetized, plane-parallel, self-gravitating slab. Our approach allows for fully non-uniformly magnetized slabs, which deviate from isothermal conditions, such that the well-known Alfvén and slow continuous spectra enter the description. We generalize modern MHD textbook treatments, by showing how self-gravity enters the MHD wave equation, beyond the frequently adopted Cowling approximation. This clarifies how Jeans’ instability generalizes from hydro to magnetohydrodynamic conditions without assuming the usual Jeans’ swindle approach. Our main contribution lies in reformulating the completely general governing wave equations in a number of mathematically equivalent forms, ranging from a coupled Sturm-Liouville formulation, to a Hamiltonian formulation linked to coupled harmonic oscillators, up to a convenient matrix differential form. The latter allows us to derive analytically the eigenfunctions of a magnetized, self-gravitating thin slab. In addition, as an example we give the exact closed form dispersion relations for the hydrodynamical p- and Jeans-unstable modes, with the latter demonstrating how the Cowling approximation modifies due to a proper treatment of self-gravity. The various reformulations of the MHD wave equation open up new avenues for future MHD spectral studies of instabilities as relevant for cosmic filament formation, which can e.g. use modern formal solution strategies tailored to solve coupled Sturm-Liouville or harmonic oscillator problems.

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Solution to the Time-Dependent Coupled Harmonic Oscillators Hamiltonian with Arbitrary Interactions

Cited by 34 publications

References 42 publications

Instability of Meissner Differential Equation and Its Relation with Photon Excitations and Entanglement in a System of Coupled Quantum Oscillators

Instability of Meissner Differential Equation and Its Relation with Photon Excitations and Entanglement in a System of Coupled Quantum Oscillators

Measurement of the Temperature Using the Tomographic Representation of Thermal States for Quadratic Hamiltonians

An MHD spectral theory approach to Jeans’ magnetized gravitational instability

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