A Lower Bound on the Critical Momentum of an Impurity in a Bose–Einstein Condensate

Benjamin Hinrichs; Jonas Lampart

doi:10.5802/crmath.652

Article de recherche - Physique mathématique

A Lower Bound on the Critical Momentum of an Impurity in a Bose–Einstein Condensate
[Borne inférieure à la vitesse critique d’une impureté dans un condensat de Bose–Einstein]

Benjamin Hinrichs ¹ ; Jonas Lampart ²

¹ Universität Paderborn, Institut für Mathematik, Institut für Photonische Quantensysteme, Warburger Str. 100, 33098 Paderborn, Germany
² CNRS & Laboratoire Interdisciplinaire Carnot de Bourgogne (UMR 6303), Université de Bourgogne Franche-Comté, 9 Av. A. Savary, 21078 Dijon Cedex, France

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1399-1411.

Résumé
Abstract

Dans le modèle Bogoliubov–Fröhlich, nous prouvons qu’une impureté immergée dans un condensat de Bose–Einstein forme une quasi-particule stable lorsque la quantité de mouvement totale est inférieure à sa masse multipliée par la vitesse du son. Le système présente donc un comportement superfluide, car cette quasi-particule ne subit pas de frottement. Nous ne supposons aucune régularisation infrarouge ou ultraviolette du modèle, qui contient des excitations sans masse et des interactions ponctuelles.

In the Bogoliubov–Fröhlich model, we prove that an impurity immersed in a Bose–Einstein condensate forms a stable quasi-particle when the total momentum is less than its mass times the speed of sound. The system thus exhibits superfluid behavior, as this quasi-particle does not experience friction. We do not assume any infrared or ultraviolet regularization of the model, which contains massless excitations and point-like interactions.

Reçu le : 2023-12-12
Accepté le : 2024-05-27
Publié le : 2024-11-14

DOI : 10.5802/crmath.652

Classification : 81V73, 81Q10, 47A10
Keywords: Polaron, energy-momentum spectrum, Cherenkov transition, renormalization
Mot clés : Polaron, spectre énergie-impulsion, transition Cherenkov, renormalisation

Affiliations des auteurs :

Benjamin Hinrichs ¹ ; Jonas Lampart ²

¹ Universität Paderborn, Institut für Mathematik, Institut für Photonische Quantensysteme, Warburger Str. 100, 33098 Paderborn, Germany
² CNRS & Laboratoire Interdisciplinaire Carnot de Bourgogne (UMR 6303), Université de Bourgogne Franche-Comté, 9 Av. A. Savary, 21078 Dijon Cedex, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2024__362_G11_1399_0,
     author = {Benjamin Hinrichs and Jonas Lampart},
     title = {A {Lower} {Bound} on the {Critical} {Momentum} of an {Impurity} in a {Bose{\textendash}Einstein} {Condensate}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1399--1411},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.652},
     language = {en},
}

TY  - JOUR
AU  - Benjamin Hinrichs
AU  - Jonas Lampart
TI  - A Lower Bound on the Critical Momentum of an Impurity in a Bose–Einstein Condensate
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 1399
EP  - 1411
VL  - 362
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.652
LA  - en
ID  - CRMATH_2024__362_G11_1399_0
ER  -

%0 Journal Article
%A Benjamin Hinrichs
%A Jonas Lampart
%T A Lower Bound on the Critical Momentum of an Impurity in a Bose–Einstein Condensate
%J Comptes Rendus. Mathématique
%D 2024
%P 1399-1411
%V 362
%I Académie des sciences, Paris
%R 10.5802/crmath.652
%G en
%F CRMATH_2024__362_G11_1399_0

Benjamin Hinrichs; Jonas Lampart. A Lower Bound on the Critical Momentum of an Impurity in a Bose–Einstein Condensate. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1399-1411. doi : 10.5802/crmath.652. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.652/

Bibliographie
Cité par

[1] Laurent Bruneau; Stephan De Bièvre A Hamiltonian model for linear friction in a homogeneous medium, Commun. Math. Phys., Volume 229 (2002) no. 3, pp. 511-542 | DOI | MR | Zbl

[2] Volker Bach; Jürg Fröhlich; Israel Michael Sigal Quantum electrodynamics of confined nonrelativistic particles, Adv. Math., Volume 137 (1998) no. 2, pp. 299-395 | DOI | MR | Zbl

[3] Volker Betz; Fumio Hiroshima; József Lőrinczi; Robert A. Minlos; Herbert Spohn Ground state properties of the Nelson Hamiltonian: a Gibbs measure-based approach, Rev. Math. Phys., Volume 14 (2002) no. 2, pp. 173-198 | arXiv | DOI | MR | Zbl

[4] Laurent Bruneau The ground state problem for a quantum Hamiltonian model describing friction, Can. J. Math., Volume 59 (2007) no. 5, pp. 897-916 | DOI | MR | Zbl

[5] Rasmus Søgaard Christensen; Jesper Levinsen; Georg M. Bruun Quasiparticle Properties of a Mobile Impurity in a Bose–Einstein Condensate, Phys. Rev. Lett., Volume 115 (2015), 160401, 5 pages | arXiv | DOI

[6] Thomas Norman Dam Absence of ground states in the translation invariant massless Nelson model, Ann. Henri Poincaré, Volume 21 (2020) no. 8, pp. 2655-2679 | arXiv | DOI | MR | Zbl

[7] Stephan De Bièvre; Jérémy Faupin; Baptiste Schubnel Spectral analysis of a model for quantum friction, Rev. Math. Phys., Volume 29 (2017) no. 6, 1750019, 49 pages | arXiv | DOI | MR | Zbl

[8] Dirk-André Deckert; Jürg Fröhlich; Peter Pickl; Alessandro Pizzo Effective dynamics of a tracer particle interacting with an ideal Bose gas, Commun. Math. Phys., Volume 328 (2014) no. 2, pp. 597-624 | arXiv | DOI | MR | Zbl

[9] J. Dereziński; C. Gérard Asymptotic completeness in quantum field theory. Massive Pauli–Fierz Hamiltonians, Rev. Math. Phys., Volume 11 (1999) no. 4, pp. 383-450 | DOI | MR | Zbl

[10] Thomas Norman Dam; Benjamin Hinrichs Absence of ground states in the renormalized massless translation-invariant Nelson model, Rev. Math. Phys., Volume 34 (2022) no. 10, 2250033, 42 pages | DOI | MR | Zbl

[11] Wojciech Dybalski; Alessandro Pizzo Coulomb scattering in the massless Nelson model I. Foundations of two-electron scattering, J. Stat. Phys., Volume 154 (2014) no. 1-2, pp. 543-587 | arXiv | DOI | MR | Zbl

[12] Wojciech Dybalski; Herbert Spohn Effective mass of the polaron – revisited, Ann. Henri Poincaré, Volume 21 (2020) no. 5, pp. 1573-1594 | arXiv | DOI | MR | Zbl

[13] Daniel Egli; Jürg Fröhlich; Zhou Gang; Arick Shao; Israel Michael Sigal Hamiltonian dynamics of a particle interacting with a wave field, Commun. Partial Differ. Equations, Volume 38 (2013) no. 12, pp. 2155-2198 | arXiv | DOI | MR | Zbl

[14] Daniel Egli; Zhou Gang Some Hamiltonian models of friction II, J. Math. Phys., Volume 53 (2012) no. 10, 103707, 35 pages | arXiv | DOI | MR

[15] Jürg Fröhlich; Zhou Gang Ballistic motion of a tracer particle coupled to a Bose gas, Adv. Math., Volume 259 (2014), pp. 252-268 | arXiv | DOI | MR | Zbl

[16] Jürg Fröhlich; Zhou Gang Emission of Cherenkov radiation as a mechanism for Hamiltonian friction, Adv. Math., Volume 264 (2014), pp. 183-235 | arXiv | DOI | MR | Zbl

[17] Jürg Fröhlich; Zhou Gang; Avy Soffer Some Hamiltonian models of friction, J. Math. Phys., Volume 52 (2011) no. 8, 083508, 13 pages | arXiv | DOI | MR | Zbl

[18] Jürg Fröhlich; Zhou Gang; Avy Soffer Friction in a model of Hamiltonian dynamics, Commun. Math. Phys., Volume 315 (2012) no. 2, pp. 401-444 | arXiv | DOI | MR | Zbl

[19] Jürg Fröhlich On the infrared problem in a model of scalar electrons and massless, scalar bosons, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A, Volume 19 (1973), pp. 1-103 | MR | Zbl

[20] Fabian Grusdt; Eugene Demler New theoretical approaches to Bose polarons, Quantum Matter at Ultralow Temperatures (Proceedings of the international school of physics “Enrico Fermi”), Volume 191 (2016), pp. 325-411 | arXiv | DOI

[21] Marcel Griesemer; Elliott H. Lieb; Michael Loss Ground states in non-relativistic quantum electrodynamics, Invent. Math., Volume 145 (2001) no. 3, pp. 557-595 | arXiv | DOI | MR | Zbl

[22] Leonard Gross Existence and uniqueness of physical ground states, J. Funct. Anal., Volume 10 (1972), pp. 52-109 | DOI | MR | Zbl

[23] Fabian Grusdt; Richard Schmidt; Yulia Shchadilova; Eugene Demler Strong-coupling Bose polarons in a Bose-Einstein condensate, Phys. Rev. A, Volume 96 (2017) no. 1, 013607, 25 pages | arXiv | DOI

[24] C. Gérard On the existence of ground states for massless Pauli–Fierz Hamiltonians, Ann. Henri Poincaré, Volume 1 (2000) no. 3, pp. 443-459 | DOI | MR | Zbl

[25] David Hasler; I. Herbst Absence of ground states for a class of translation invariant models of non-relativistic QED, Commun. Math. Phys., Volume 279 (2008) no. 3, pp. 769-787 | arXiv | DOI | MR | Zbl

[26] David Hasler; Benjamin Hinrichs; Oliver Siebert On existence of ground states in the spin boson model, Commun. Math. Phys., Volume 388 (2021) no. 1, pp. 419-433 | arXiv | DOI | MR | Zbl

[27] David Hasler; Benjamin Hinrichs; Oliver Siebert Non-Fock ground states in the translation-invariant Nelson model revisited non-perturbatively, J. Funct. Anal., Volume 286 (2024) no. 7, 110319, 44 pages | DOI | MR | Zbl

[28] Benjamin Hinrichs Existence of Ground States for Infrared-Critical Models of Quantum Field Theory, Ph. D. Thesis, Friedrich Schiller University Jena (2022) | DOI

[29] Fumio Hiroshima; Oliver Matte Ground states and associated path measures in the renormalized Nelson model, Rev. Math. Phys., Volume 34 (2022) no. 2, 2250002, 84 pages | DOI | Zbl

[30] David Hasler; Oliver Siebert Ground states for translationally invariant Pauli-Fierz models at zero momentum, J. Funct. Anal., Volume 284 (2023) no. 1, 109725, 28 pages | arXiv | DOI | MR | Zbl

[31] Jonas Lampart A nonrelativistic quantum field theory with point interactions in three dimensions, Ann. Henri Poincaré, Volume 20 (2019) no. 11, pp. 3509-3541 | arXiv | DOI | MR | Zbl

[32] Jonas Lampart The renormalized Bogoliubov–Fröhlich Hamiltonian, J. Math. Phys., Volume 61 (2020) no. 10, 101902, 12 pages | arXiv | DOI | MR | Zbl

[33] Jonas Lampart The resolvent of the Nelson Hamiltonian improves positivity, Math. Phys. Anal. Geom., Volume 24 (2021) no. 1, 2, 17 pages | arXiv | DOI | MR | Zbl

[34] Jonas Lampart Hamiltonians for polaron models with subcritical ultraviolet singularities, Ann. Henri Poincaré, Volume 24 (2023) no. 8, pp. 2687-2728 | arXiv | DOI | MR | Zbl

[35] Michael Loss; Tadahiro Miyao; Herbert Spohn Lowest energy states in nonrelativistic QED: atoms and ions in motion, J. Funct. Anal., Volume 243 (2007) no. 2, pp. 353-393 | arXiv | DOI | MR | Zbl

[36] Jonas Lampart; Peter Pickl Dynamics of a tracer particle interacting with excitations of a Bose–Einstein condensate, Ann. Henri Poincaré, Volume 23 (2022) no. 8, pp. 2855-2876 | arXiv | DOI | MR | Zbl

[37] Jonas Lampart; Julian Schmidt On Nelson-type Hamiltonians and abstract boundary conditions, Commun. Math. Phys., Volume 367 (2019) no. 2, pp. 629-663 | arXiv | DOI | MR | Zbl

[38] Jonas Lampart; Arnaud Triay The excitation spectrum of a dilute Bose gas with an impurity (2024) (https://arxiv.org/abs/2401.14911)

[39] Tristan Léger Scattering for a particle interacting with a Bose gas, Commun. Partial Differ. Equations, Volume 45 (2020) no. 10, pp. 1381-1413 | arXiv | DOI | MR | Zbl

[40] Oliver Matte Continuity properties of the semi-group and its integral kernel in non-relativistic QED, Rev. Math. Phys., Volume 28 (2016) no. 5, 1650011, 90 pages | arXiv | DOI | MR | Zbl

[41] Krzysztof Myśliwy; Robert Seiringer Microscopic derivation of the Fröhlich Hamiltonian for the Bose polaron in the mean-field limit, Ann. Henri Poincaré, Volume 21 (2020) no. 12, pp. 4003-4025 | arXiv | DOI | MR | Zbl

[42] Jacob Schach Mø ller The translation invariant massive Nelson model. I. The bottom of the spectrum, Ann. Henri Poincaré, Volume 6 (2005) no. 6, pp. 1091-1135 | DOI | MR | Zbl

[43] Edward Nelson Interaction of nonrelativistic particles with a quantized scalar field, J. Math. Phys., Volume 5 (1964), pp. 1190-1197 | DOI | MR | Zbl

[44] Alessandro Pizzo One-particle (improper) states in Nelson’s massless model, Ann. Henri Poincaré, Volume 4 (2003) no. 3, pp. 439-486 | DOI | MR | Zbl

[45] Michael Reed; Barry Simon Methods of modern mathematical physics. I. Functional analysis, Academic Press Inc., 1972, xvii+325 pages | MR

[46] Julian Schmidt On a direct description of pseudorelativistic Nelson Hamiltonians, J. Math. Phys., Volume 60 (2019) no. 10, 102303, 21 pages | arXiv | DOI | MR | Zbl

[47] Julian Schmidt The massless Nelson Hamiltonian and its domain, Mathematical challenges of zero-range physics – models, methods, rigorous results, open problems (Springer INdAM Series), Volume 42, Springer, 2021, pp. 57-80 | arXiv | DOI | MR

[48] Robert Seiringer The polaron at strong coupling, Rev. Math. Phys., Volume 33 (2021) no. 1, 2060012, 21 pages | arXiv | DOI | MR | Zbl

[49] Herbert Spohn The polaron at large total momentum, J. Phys. A. Math. Gen., Volume 21 (1988) no. 5, pp. 1199-1211 | DOI | MR

[50] Herbert Spohn Ground state of a quantum particle coupled to a scalar Bose field, Lett. Math. Phys., Volume 44 (1998) no. 1, pp. 9-16 | DOI | MR | Zbl

[51] Kushal Seetharam; Yulia Shchadilova; Fabian Grusdt; Mikhail Zvonarev; Eugene Demler Dynamical Quantum Cherenkov Transition of Fast Impurities in Quantum Liquids, Phys. Rev. Lett., Volume 127 (2021), 185302, 6 pages | arXiv | DOI

[52] Kushal Seetharam; Yulia Shchadilova; Fabian Grusdt; Mikhail Zvonarev; Eugene Demler Quantum Cherenkov transition of finite momentum Bose polarons (2021) (https://arxiv.org/abs/2109.12260)

[53] César R. de Oliveira Intermediate spectral theory and quantum dynamics, Progress in Mathematical Physics, 54, Birkhäuser, 2009, xvi+410 pages | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique