Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order $$ N^{\kappa -1}$$ N κ - 1 , for some $$\kappa >0$$ κ > 0 . For the range $$ \kappa \in [0, \frac{1}{43})$$ κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $$ \kappa \in [0, \frac{1}{20})$$ κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).

arXiv:2401.00784Annales Henri PoincaréJul 1, 20245 citations

An Introduction to Quantum Filtering

John Edward Gough

The following notes are based on lectures delivered at the research school Modeling and Control of Open Quantum Systems (Modélisation et contrôle des systèmes quantiques ouverts) at CIRM, Marseille, 16-20 April, 2018, as part of the Trimester \textit{Measurement and Control of Quantum Systems: Theory and Experiments} organized at Institut Henri Poincaré, Paris, France. The aim is to introduce quantum filtering to an audience with a background in either quantum theory or classical filtering.

arXiv:1804.09086Apr 24, 20186 citations

Isis

Henri Poincare and the Quantum Theory

Russell K. McCormmach

Published Apr 1, 1967• 26 citations

Publisher

DOI: 10.1086/350182

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Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order $N^{\kappa -1}$ N κ - 1 , for some $\kappa >0$ κ > 0 . For the range $\kappa \in [0, \frac{1}{43})$ κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $\kappa \in [0, \frac{1}{20})$ κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order $$ N^{\kappa -1}$$ N κ - 1 , for some $$\kappa >0$$ κ > 0 . For the range $$ \kappa \in [0, \frac{1}{43})$$ κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for $$ \kappa \in [0, \frac{1}{20})$$ κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).

arXiv:2401.00784Annales Henri PoincaréJul 1, 20245 citations

An Introduction to Quantum Filtering

John Edward Gough

arXiv:1804.09086Apr 24, 20186 citations

Abstract We consider dilute Bose gases on the three-dimensional unit torus that interact through a pair potential with scattering length of order

N^{\kappa -1}

N κ - 1 , for some

\kappa >0

κ > 0 . For the range

\kappa \in [0, \frac{1}{43})

κ ∈ [ 0 , 1 43 ) , Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) proves complete BEC of low energy states into the zero momentum mode based on a unitary renormalization through operator exponentials that are quartic in creation and annihilation operators. In this paper, we give a new and self-contained proof of BEC of the ground state for

\kappa \in [0, \frac{1}{20})

κ ∈ [ 0 , 1 20 ) by combining some of the key ideas of Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021) with the novel diagonalization approach introduced recently in Brooks (Diagonalizing Bose Gases in the Gross–Pitaevskii Regime and Beyond, arXiv:2310.11347 ), which is based on the Schur complement formula. In particular, our proof avoids the use of operator exponentials and is significantly simpler than Adhikari et al. (Ann Henri Poincaré 22:1163–1233, 2021).