From Quantum Hydrodynamics to Koopman Wavefunctions II

Gay–Balmaz, François; Tronci, Cesare

doi:10.1007/978-3-030-80209-7_35

Cited by 6 publications

(6 citation statements)

References 5 publications

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“…. Then, upon dropping subscripts and changing the notation as q 2 → x and Ψ → Υ, one obtains the following quantum-classical wave equation [10,23,54] i…”

Section: The Quantum-classical Wave Equationmentioning

confidence: 99%

“…Then, one may think of extending the exact factorization method to allow for a more general type of relation that is solved by a wider class of ψ. Following [19,54], this extension may be realized upon resorting to von-Naumann operators. In this case, the quantum-classical wave equation ( 19)…”

Section: Remark 41 (Von Neumann Operators)mentioning

confidence: 99%

“…would return a pure transport equation for ρ c = D, whose sign would be then preserved in time. Recently, similar approaches were used in [22,54] to devise simple closure models based on von Neumann operators in the Koopman setting. Here, we take an alternative route by exploiting the full power of the exact factorization method.…”

Section: Closure Modelmentioning

confidence: 99%

“…Compound systems comprising both quantum and classical degrees of freedom are generally referred to as hybrid quantum-classical systems, while "mixed quantum-classical dynamics" is more common in the chemical physics literature [56]. Following earlier ideas by Sudarshan [14,48], we recently obtained a hybrid quantum-classical model [10,23,54] by combining Koopman's Hilbert-space approach to classical mechanics [38] with van Hove's unitary representations in prequantization [57]. This hybrid theory is written on the tensor-product Hilbert space of Schrödinger and Koopman wavefunctions and, unlike other available models, satisfies all of the following three properties: 1) hybrid dynamics is energy-conserving and Hamiltonian; 2) the quantum density matrix is positive-definite; 3) standard quantum and classical dynamics are recovered in the absence of quantum-classical interaction.…”

Section: Introductionmentioning

confidence: 99%

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Koopman wavefunctions and classical states in hybrid quantum-classical dynamics

Gay-Balmaz,

Tronci

2021

Preprint

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We deal with the reversible dynamics of coupled quantum and classical systems. Based on a recent proposal by the authors, we exploit the theory of hybrid quantum-classical wavefunctions to devise a closure model for the coupled dynamics in which both the quantum density matrix and the classical Liouville distribution retain their initial positive sign. In this way, the evolution allows identifying a classical and a quantum state in interaction at all times. After combining Koopman's Hilbert-space method in classical mechanics with van Hove's unitary representations in prequantum theory, the closure model is made available by the variational structure underlying a suitable wavefunction factorization. Also, we use Poisson reduction by symmetry to show that the hybrid model possesses a noncanonical Poisson structure that does not seem to have appeared before. As an example, this structure is specialized to the case of quantum two-level systems.

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“…. Then, upon dropping subscripts and changing the notation as q 2 → x and Ψ → Υ, one obtains the following quantum-classical wave equation [10,23,54] i…”

Section: The Quantum-classical Wave Equationmentioning

confidence: 99%

Section: Remark 41 (Von Neumann Operators)mentioning

confidence: 99%

Section: Closure Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Koopman wavefunctions and classical states in hybrid quantum-classical dynamics

Gay-Balmaz,

Tronci

2021

Preprint

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“…In terms of the measure-valued density matrix P = ΥΥ † , the hybrid wave equation (7.11) reads i ∂ t P+i div( P X H ) = [ H, P] , where X H = Tr( PX H )/ P and we recall P = Υ 2 = ρ c . Recently, this model was also obtained as a closure model directly from the original theory in Section 4; see [77]. The equations for the classical density ρ c = P and the quantum density matrix ρq = ´ Pd 2 z are…”

Section: Specializations and Further Commentsmentioning

confidence: 99%

Evolution of hybrid quantum-classical wavefunctions

Gay-Balmaz,

Tronci

2021

Preprint

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A gauge-invariant wave equation for the dynamics of hybrid quantum-classical systems is formulated by combining the variational setting of Lagrangian paths in continuum theories with Koopman wavefunctions in classical mechanics. We identify gauge transformations with unobservable phase factors in the classical phase-space and we introduce gauge invariance in the variational principle underlying a hybrid wave equation previously proposed by the authors. While the original construction ensures a positive-definite quantum density matrix, the present model also guarantees the same property for the classical Liouville density. After a suitable wavefunction factorization, gauge invariance is achieved by resorting to the classical Lagrangian paths made available by the Madelung transform of Koopman wavefunctions. Due to the appearance of a phase-space analogue of the Berry connection, the new hybrid wave equation is highly nonlinear and it is proposed here as a platform for further developments in quantum-classical dynamics. Indeed, the associated model is Hamiltonian and appears to be the first to ensure a series of consistency properties beyond positivity of quantum and classical densities. For example, the model possesses a quantum-classical Poincaré integral invariant and its special cases include both the mean-field model and the Ehrenfest model from chemical physics.

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Evolution of hybrid quantum–classical wavefunctions

Gay–Balmaz

Tronci

2022

Physica D: Nonlinear Phenomena

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From Quantum Hydrodynamics to Koopman Wavefunctions II

Cited by 6 publications

References 5 publications

Koopman wavefunctions and classical states in hybrid quantum-classical dynamics

Koopman wavefunctions and classical states in hybrid quantum-classical dynamics

Evolution of hybrid quantum-classical wavefunctions

Evolution of hybrid quantum–classical wavefunctions

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