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.

Abstract (VO)
Résumé

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.

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.

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

Zbl

DOI : 10.5802/crmath.652

Classification : 81V73, 81Q10, 47A10
Keywords: Polaron, energy-momentum spectrum, Cherenkov transition, renormalization
Mots-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},
     zbl = {07945483},
     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 | Zbl

[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 | Numdam | 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 | MR | 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