arXiv:physics/9702018

Mean Field Approach to Quantum Duffing Oscillator

Sang Pyo Kim

Published Feb 19, 1997• 1 citation

Abstract

We propose a mean-field approach to the Duffing oscillator to construct perturbatively the bounded operators that generate the quantum states, and to define a non-Gaussian density operator.

DOI: 10.48550/arxiv.physics/9702018

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arXiv:physics/9702018

Mean Field Approach to Quantum Duffing Oscillator

Sang Pyo Kim

Published Feb 19, 1997• 1 citation

arXiv HTML PDF

Abstract

We propose a mean-field approach to the Duffing oscillator to construct perturbatively the bounded operators that generate the quantum states, and to define a non-Gaussian density operator.

DOI: 10.48550/arxiv.physics/9702018

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Bulletin of the American Physical SocietyMar 21, 20130 citations

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Henstock--Kurzweil Gauge Integral in the Non--Gaussian Regime: A Machine--Verified Construction

Yuri N. Berdinsky

This preprint develops a rigorous, machine-checked construction of non-Gaussian functional integrals using the Henstock–Kurzweil (HK) gauge integral and Chernoff product approximations. The central object is a finite family of bosonic modes with an action containing a quartic self-interaction λ∑iφi4λ∑iφi4. We show that the one-mode non-Gaussian integral is finite, positive, monotone and infinitely differentiable in the coupling λλ on [0,∞)[0,∞), with derivatives given by convergent integrals rather than by the divergent perturbative series. The MM-mode influence functional factorises into one-mode integrals and is bounded by its Gaussian value. A Chernoff/Lie–Trotter splitting handles non-commutativity of the free and non-Gaussian generators. All statements are formalised in Lean 4 / Mathlib and are free of sorry, using only the standard axioms propext, Classical.choice, Quot.sound. The accompanying Lean file HkNonGaussian.lean is included. Illustrative applications are worked out at the level of explicit formulas: the Duffing oscillator, local volatility (CEV) in finance, Wilson–Cowan neural fields, and non-Gaussian quantum reservoirs. This work is a direct continuation of the author’s programme on machine-verified path integrals without Wick rotation.This preprint develops a rigorous, machine-checked construction of non-Gaussian functional integrals using the Henstock–Kurzweil (HK) gauge integral and Chernoff product approximations. The central object is a finite family of bosonic modes with an action containing a quartic self-interaction λ∑iφi4λ∑iφi4. We show that the one-mode non-Gaussian integral is finite, positive, monotone and infinitely differentiable in the coupling λλ on [0,∞)[0,∞), with derivatives given by convergent integrals rather than by the divergent perturbative series. The MM-mode influence functional factorises into one-mode integrals and is bounded by its Gaussian value. A Chernoff/Lie–Trotter splitting handles non-commutativity of the free and non-Gaussian generators. All statements are formalised in Lean 4 / Mathlib and are free of sorry, using only the standard axioms propext, Classical.choice, Quot.sound. The accompanying Lean file HkNonGaussian.lean is included. Illustrative applications are worked out at the level of explicit formulas: the Duffing oscillator, local volatility (CEV) in finance, Wilson–Cowan neural fields, and non-Gaussian quantum reservoirs. This work is a direct continuation of the author’s programme on machine-verified path integrals without Wick rotation.

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Non-Gaussian entanglement is a promising resource in various quantum tasks. A recently defined class identifies entanglement that cannot be generated by applying Gaussian operations to separable inputs. To further explore the entanglement in this context, we introduce a quantitative witness $E_{\rm NG}$ in bipartite bosonic systems, which satisfies $E_{\rm NG}=1$ for all Gaussian-entanglable states, while $E_{\rm NG}>1$ certifies non-Gaussian entanglement. Its ceiling $d=\lceil E_{\rm NG}\rceil$ provides a lower bound on the Schmidt number irreducible by Gaussian transformations, thereby defining a natural hierarchy of non-Gaussian entanglement. For pure states, the condition is sharp and the hierarchy reflects the complexity of state learning. We benchmark the framework with some paradigmatic non-Gaussian states, such as NOON states and squeezed Kerr states, and analyze its robustness against loss. Moreover, we construct an experimentally economical NOON-type witness requiring only four density-matrix element measurements. These results establish an operationally meaningful and experimentally accessible framework for identifying non-Gaussian entanglement resources in continuous-variable quantum platforms.Non-Gaussian entanglement is a promising resource in various quantum tasks. A recently defined class identifies entanglement that cannot be generated by applying Gaussian operations to separable inputs. To further explore the entanglement in this context, we introduce a quantitative witness $E_{\rm NG}$ in bipartite bosonic systems, which satisfies $E_{\rm NG}=1$ for all Gaussian-entanglable states, while $E_{\rm NG}>1$ certifies non-Gaussian entanglement. Its ceiling $d=\lceil E_{\rm NG}\rceil$ provides a lower bound on the Schmidt number irreducible by Gaussian transformations, thereby defining a natural hierarchy of non-Gaussian entanglement. For pure states, the condition is sharp and the hierarchy reflects the complexity of state learning. We benchmark the framework with some paradigmatic non-Gaussian states, such as NOON states and squeezed Kerr states, and analyze its robustness against loss. Moreover, we construct an experimentally economical NOON-type witness requiring only four density-matrix element measurements. These results establish an operationally meaningful and experimentally accessible framework for identifying non-Gaussian entanglement resources in continuous-variable quantum platforms.

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Atomic quadrature squeezing and quantum state transfer in a hybrid atom–optomechanical cavity with two Duffing mechanical oscillators

Farzaneh Momeni, M. H. Naderi

In this paper, we investigate theoretically the quantum state transfer in a laser-driven hybrid optomechanical cavity with two Duffing-like anharmonic movable end mirrors containing an ensemble of identical two-level trapped atoms. The quantum state transfer from the Bogoliubov modes of the two anharmonic oscillators to the atomic mode results in the atomic quadrature squeezing beyond the standard quantum limit of 3 dB, which can be controlled by both the optomechanical and atom-field coupling strengths. Interestingly, the generated atomic squeezing can be made robust against the noise sources by means of the Duffing anharmonicity. Moreover, the results reveal that the presence of the Duffing anharmonicity provides the possibility of transferring strongly squeezed states between the two mechanical oscillators in a short operating time and with high fidelity.In this paper, we investigate theoretically the quantum state transfer in a laser-driven hybrid optomechanical cavity with two Duffing-like anharmonic movable end mirrors containing an ensemble of identical two-level trapped atoms. The quantum state transfer from the Bogoliubov modes of the two anharmonic oscillators to the atomic mode results in the atomic quadrature squeezing beyond the standard quantum limit of 3 dB, which can be controlled by both the optomechanical and atom-field coupling strengths. Interestingly, the generated atomic squeezing can be made robust against the noise sources by means of the Duffing anharmonicity. Moreover, the results reveal that the presence of the Duffing anharmonicity provides the possibility of transferring strongly squeezed states between the two mechanical oscillators in a short operating time and with high fidelity.

arXiv:1811.01761Journal of the Optical Society of America BFeb 27, 20198 citations

E_{\rm NG}

in bipartite bosonic systems, which satisfies

E_{\rm NG}=1

for all Gaussian-entanglable states, while

E_{\rm NG}>1

certifies non-Gaussian entanglement. Its ceiling

d=\lceil E_{\rm NG}\rceil

provides a lower bound on the Schmidt number irreducible by Gaussian transformations, thereby defining a natural hierarchy of non-Gaussian entanglement. For pure states, the condition is sharp and the hierarchy reflects the complexity of state learning. We benchmark the framework with some paradigmatic non-Gaussian states, such as NOON states and squeezed Kerr states, and analyze its robustness against loss. Moreover, we construct an experimentally economical NOON-type witness requiring only four density-matrix element measurements. These results establish an operationally meaningful and experimentally accessible framework for identifying non-Gaussian entanglement resources in continuous-variable quantum platforms.