The Wigner function negative value domains and energy function poles of the polynomial oscillator

Perepelkin, E. E.; Садовников, Б. И.; Иноземцева, Н. Г.; Burlakov, E. V.; Afonin, P. V.

doi:10.1016/j.physa.2022.127339

Cited by 4 publications

(4 citation statements)

References 16 publications

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“…Описанная схема детектирования, используется не только для CV QKD, но и для проведения, так называемой квантовой томографии [41], восстановлении квантового состояния по данным гомодинирования. Так, например, квадратурные компоненты являются аргументами функции Вигнера W(q,p) [42][43][44][45] -важного объекта квантовой оптики, представляющего собой функцию квази-распределения. Типичная схема реализации протокола CV QKD с дискретной модуляцией, выглядит следующим образом [11]:…”

Section: описание метода балансного гомодинного детектированияunclassified

Key Features of Continuous-Variable Quantum Key Distribution Protocols

Бурлаков,

Коробов

2024

RENSIT

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В работе рассмотрены ключевые особенности протоколов квантового распределения ключей на непрерывных переменных. Обоснована мотивация изучения и разработки методов квантовой криптографии. Выделены основные моменты, присущие протоколам на непрерывных переменных, и проведена их классификация по различным признакам, особенностям и вариантам реализации. Подробно рассмотрен пример протокола с дискретной модуляцией и регистрацией сигнала по средствам балансного гомодинного детектирования. Приведена классификация атак на квантовый протокол. Дана общая характеристика методам постобработки. Обозначена роль шумов, и рассмотрены подходы к оценке уровня секретности, которые применимы для протоколов на непрерывных переменных. В заключении приведены основные выводы касательно текущего статуса данной проблемы, а также определены аспекты, которые требуют дальнейшего изучения.

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Section: описание метода балансного гомодинного детектированияunclassified

Key Features of Continuous-Variable Quantum Key Distribution Protocols

Бурлаков,

Коробов

2024

RENSIT

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“…The Wigner function has phase regions of negative and positive values (see figure 2), at the boundary of these regions, summand O µ has poles (2.12). The presence of the poles leads to infinite energy barriers [18][19][20][21], near which particles can «reflect» sharply changing their trajectory.…”

Section: 18)mentioning

confidence: 99%

Is the Moyal equation for the Wigner function a quantum analogue of the Liouville equation?

Perepelkin,

Sadovnikov,

Inozemtseva

et al. 2023

J. Stat. Mech.

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The Moyal equation describes the evolution of the Wigner function of a quantum system in the phase space. The right-hand side of the equation contains an infinite series with coefficients proportional to powers of the Planck constant. There is an interpretation of the Moyal equation as a quantum analogue of the classical Liouville equation. Indeed, if one uses the notion of the classical passage to the limit as the Planck constant tends to zero, then formally the right-hand side of the Moyal equation tends to zero. As a result, the Moyal equation becomes the classical Liouville equation for the distribution function. In this paper, we show that the right side of the Moyal equation does not explicitly depend on the Planck constant, and all terms of the series can make a significant contribution. The transition between the classical and quantum descriptions is related not to the Planck constant, but to the spatial scale. For a model quantum system with a potential in the form of a «quadratic funnel», an exact 3D solution of the Schrödinger equation is found and the corresponding Wigner function is constructed in the paper. Using trajectory analysis in the phase space, based on the representation of the right-hand side of the Moyal equation, it is shown that on the spatial microscale there is an infinite number of «trajectories» of the particle motion (thereby the concept of a trajectory is indefinite), and when passing to the macroscale, all «trajectories» concentrate around the classical trajectory.

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“…    á ñ ( ) (i.13), which is velocity dependent. The velocity dependence of approximation (i.13) can persist for the classical limit 0   [33]. Therefore, the use of approximation (i.12) may result in incorrect results of plasma simulation, which require modifying the numerical method [36,37].…”

Section: The Maxwell Equationsmentioning

confidence: 99%

“…The second Vlasov equation (i.3) with approximation (i.12) became very popular in astrophysics, statistical, solid state and plasma physics. In papers [33] the approximation v 1,2   á ñ was shown to be extended in terms of the Wigner function formalism for quantum systems in the phase space. Using the dynamic Vlasov-Moyal approximation for the kinematical average value…”

Section: Introductionmentioning

confidence: 99%

The Wigner-Vlasov formalism for time-dependent quantum oscillator

Perepelkin,

Sadovnikov,

Inozemtseva

et al. 2023

Phys. Scr.

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This paper presents a comprehensive investigation of the problem of a harmonic oscillator with time-depending frequencies in the framework of the Vlasov theory and the Wigner function apparatus for quantum systems in the phase space. A new method is proposed to find an exact solution of this problem using a relation of the Vlasov equation chain with the Schrödinger equation and with the Moyal equation for the Wigner function. A method of averaging the energy function over the Wigner function in the phase space can be used to obtain time-dependent energy spectrum for a quantum system. The Vlasov equation solution can be represented in the form of characteristics satisfying the Hill equation. A particular case of the Hill equation, namely the Mathieu equation with unstable solutions, has been considered in details. An analysis of the dynamics of an unstable quantum system shows that the phase space square bounded with the Wigner function level line conserves in time, but the phase space square bounded with the energy function line increases. In this case the Vlasov equation characteristic is situated on the crosspoint of the Wigner function level line and the energy function line. This crosspoint moves in time with a trajectory that represents the unstable system dynamics. Each such trajectory has its own energy, and averaging these energies by the Wigner function results in time-dependent discreet energy spectrum for the whole system. An explicit expression has been obtained for the Wigner function of the 4th rank in the generalized phase space {r,p,p',p''}.

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The Wigner function negative value domains and energy function poles of the polynomial oscillator

Cited by 4 publications

References 16 publications

Key Features of Continuous-Variable Quantum Key Distribution Protocols

Key Features of Continuous-Variable Quantum Key Distribution Protocols

Is the Moyal equation for the Wigner function a quantum analogue of the Liouville equation?

The Wigner-Vlasov formalism for time-dependent quantum oscillator

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