Quantum phase space with a basis of Wannier functions

Fang, Yuan; Wu, Fan; Wu, Biao

doi:10.1088/1742-5468/aaac54

Cited by 11 publications

(9 citation statements)

References 17 publications

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“…In 1929, von Neumann proposed to construct a set of orthonormal basis by assigning each Planck cell a localized wave function [30,31]. This idea has been further developed with the help of the Wannier functions [21,27,29].…”

Section: (A)(b)mentioning

confidence: 99%

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Microscope for Quantum Dynamics with Planck Cell Resolution

Wang,

Feng,

2021

Preprint

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We introduce the out-of-time-correlation(OTOC) with the Planck cell resolution. The dependence of this OTOC on the initial state makes it function like a microscope, allowing us to investigate the fine structure of quantum dynamics beyond the thermal state. We find an explicit relation of this OTOC to the spreading of the wave function in the Hilbert space, unifying two branches of the study of quantum chaos: state evolution and operator dynamics. By analyzing it in the vicinity of the classical limit, we clarify the dependence of the OTOC's exponential growth on the classical Lyapunov exponent.

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Section: (A)(b)mentioning

confidence: 99%

“…These states |x form a set of complete orthonormal basis[appendix. A], and they are localized in both position and momentum space [27][28][29]. Operators Q = x |x Q x x| and P = x |x P x x| are so-called macroscopic position and momentum operators [21,[29][30][31].…”

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confidence: 99%

Microscope for Quantum Dynamics with Planck Cell Resolution

Wang,

Feng,

2021

Preprint

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“…As shown in Fig. 2, the classical phase space is divided into Planck cells and each Planck cell is assigned a Wanner function |w j [14][15][16]. These orthonormal Wanner functions |w jxjp form FIG.…”

Section: Examples Of Physical Distancementioning

confidence: 99%

Genuine Quantum Chaos and Physical Distance Between Quantum States

Wang,

2020

Preprint

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We propose a physical distance between two quantum states. Qualitatively different from existing distances for quantum states, for example, the Fubini-Study distance, the physical distance between two orthogonal quantum states can be very small. This allows us to discuss the dynamical divergence of two quantum states that are initially very close by physical distance and define quantum Lyapunov exponent. We are also able to use physical distance to define quantum chaos measure, which can lead to quantum analogue of the classical Poincaré section. Three different systems, kicked rotor, threesite Bose-Hubbard model, and spin-1/2 XXZ model, are used to illustrate the physical distance.

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“…We emphasize that this basis can be constructed as long as one has the action-angle pairs (p, x), where x has periodic boundary condition. If the natural coordinate of the classical system is not the angle variable, one can also numerically obtain an orthonormal and complete basis of Wannier functions efficiently [23].…”

Section: B Construction Of Quantum Phase Spacementioning

confidence: 99%

“…In this work we propose a different approach to capture the division of eigenstates, and thus the quantum KAM effect. In our approach, we divide the phase phase into Planck cells and assign a Wannier function to each Planck cell [22][23][24]. These Wannier functions form an orthonormal and complete basis and they allow us to project a wave function unitarily to phase space .…”

Section: Introductionmentioning

confidence: 99%

Wannier basis method for the Kolmogorov-Arnold-Moser effect in quantum mechanics

Yin

Chen

2019

Phys. Rev. E

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The effect of Kolmogorov-Arnold-Moser (KAM) theorem in quantum systems is manifested in dividing eigenstates into regular and irregular states. We propose an effective method based on Wannier basis in phase space to illustrate this division of eigenstates. The quantum kicked-rotor model is used to illustrated this approach, which allows us to define the area and effective dimension of an eigenstate and the length of a Planck cell. The area and effective dimension can be used to distinguish quantitatively regular and irregular eigenstates. The length of a Planck cell measures how many Planck cells the system will traverse if it starts at the given Planck cell. Moreover, with this Wannier approach, we are able to clarify the distinction between KAM effect and Anderson localization.

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Quantum phase space with a basis of Wannier functions

Cited by 11 publications

References 17 publications

Microscope for Quantum Dynamics with Planck Cell Resolution

Microscope for Quantum Dynamics with Planck Cell Resolution

Genuine Quantum Chaos and Physical Distance Between Quantum States

Wannier basis method for the Kolmogorov-Arnold-Moser effect in quantum mechanics

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