The excitation spectrum of the Bose gas in the Gross–Pitaevskii regime

Boccato, Chiara

doi:10.1142/s0129055x20600065

Cited by 8 publications

(2 citation statements)

References 33 publications

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“…There are many other papers that tackle the problems of quantum fluctuations around the mean field limit and the excitation spectrum of the Bose gas in the mathematics literature; see, e.g., [7,6,10,47,48,46,49,50,41,42,20,56,14]. A comprehensive review of some of the challenges in analyzing many-body excitations in the periodic box is given by Seiringer [57].…”

Section: On Related Past Workmentioning

confidence: 99%

Many-body excitations in trapped Bose gas: A non-Hermitian view

Grillakis,

Margetis,

Sorokanich

2021

Preprint

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We provide the analysis of a physically motivated model for a trapped dilute Bose gas with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states by accounting for the scattering of atoms in pairs from the macroscopic state (condensate). We formally construct a many-body Hamiltonian, H app , that is quadratic in the Boson field operators for noncondensate atoms. This H app conserves the total number of atoms. Inspired by Wu (J. Math. Phys., 2:105-123, 1961), we apply a non-unitary transformation to H app . Key in this non-Hermitian view is the pair-excitation kernel, which in operator form obeys a Riccati equation. In the stationary case, we develop an existence theory for solutions to this operator equation by a variational approach. We connect this theory to the one-particle excitation wave functions heuristically derived by Fetter (Ann. Phys., 70:67-101, 1972). These functions solve an eigenvalue problem for a J-self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter's excitation spectrum. Our approach leads to a description of the excited eigenstates of the reduced Hamiltonian in the N-particle sector of Fock space.

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Section: On Related Past Workmentioning

confidence: 99%

Many-body excitations in trapped Bose gas: A non-Hermitian view

Grillakis,

Margetis,

Sorokanich

2021

Preprint

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“…It is important to note that the BEC density introduces a large parameter in our theory, which is unrelated to the expected number of fluctuation particles. Works analyzing the HFB dynamics include [3,20,21,36,37,38,41,42,43,46,47,51,71,72,73,75,76,111,112,113,137,138,146,166,167,168,170,171,172,173,178] and references therein. See also [142,175] for more details.…”

Section: Introductionmentioning

confidence: 99%

On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate

Chen¹,

Hott²

2021

Preprint

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In this paper, we study the quantum fluctuation dynamics in a Bose gas on a torus Λ Ă R 3 that exhibits Bose-Einstein condensation, beyond the leading order Hartree-Fock-Bogoliubov (HFB) fluctuations. Given a mean-field Hamiltonian and Bose-Einstein condensate (BEC) with density N , we extract a quantum Boltzmann type dynamics from a second-order Duhamel expansion upon subtracting both the HFB dynamics and the BEC dynamics. Using a Fock-space approach, we provide explicit error bounds. Given an approximately quasi-free initial state, we determine the time evolution of the centered correlation functions xay, xaay ´xay 2 , xa `ay ´|xay| 2 for mesoscopic time scales. For large but finite N , we consider both the case of fixed system size |Λ| " 1, and the case |Λ| " N 6 53 ´. In the case |Λ| " 1, we show that the Boltzmann collision operator contains subleading terms that can become dominant, depending on time-dependent coefficients assuming particular values in Q; this phenomenon is reminiscent of the Talbot effect. For the case´, we prove that the collision operator is well approximated by the expression predicted in the literature. THOMAS CHEN AND MICHAEL HOTT 7.1. Discrete case 78 7.2. Approximation by continuous limit 81 Appendix A. Introductory observations 83 Appendix B. Proof of convergence to mean field equations 85 B.1. Error bounds due to HFB evolution 89 B.2. Various error bounds 90 References 95

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Correlation Corrections as a Perturbation to the Quasi-free Approximation in Many-Body Quantum Systems

Benedikter

Boccato

2022

Perturbation Theory

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The excitation spectrum of the Bose gas in the Gross–Pitaevskii regime

Cited by 8 publications

References 33 publications

Many-body excitations in trapped Bose gas: A non-Hermitian view

Many-body excitations in trapped Bose gas: A non-Hermitian view

On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate

Correlation Corrections as a Perturbation to the Quasi-free Approximation in Many-Body Quantum Systems

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