LPTMS Internship and PhD proposal: Correlations and temperature regimes in the one-dimensional Bose-Hubbard model

Contact : Guillaume Roux & Leonardo Mazza (guillaume.roux@universite-paris-saclay.fr, leonardo.mazza@universite-paris-saclay.fr)

One-dimensional bosonic gases, despite being described by a simple Hamiltonian, display rich physics due to the fact that quantum and thermal fluctuations are particularly strong in such low-dimensional system and compete with strong interactions [1]. Furthermore, thanks to the existence of powerful one-dimensional analytical and numerical techniques and to many experimental realizations, they have become a playground for the study of quantum fluids, in particular out-of-equilibrium. While thermal regimes are well understood for bosons in the continuum (Lieb-Liniger model) [2] and are relevant to recent experiments [3], we would like to systematically understand them for bosons on the lattice (Bose-Hubbard model). This is motivated by recent experiments [4] and in particular by the development of a new setup at Institut d’Optique (IOGS), which offers the perspective of an active collaboration with the group of Marc Cheneau.

Using exact diagonalization, matrix-product state techniques and quantum Monte-Carlo, we wish to explore the different regimes of the Bose-Hubbard model from zero to high temperatures, and from weak to strong interactions. We aim at devising simple phenomenological laws by connecting them with the behavior of elementary quasiparticle excitations in the system [5]. This would prepare for a PhD extending and confronting these results to two-dimensions and out-of-equilibrium situations relevant for the IOGS experiment.

[1] M. A. Cazalilla, et al, Rev. Mod. Phys. 83 , 1405 (2011)
[2] D. S. Petrov, et al, Phys. Rev. Lett. 85 , 3745 (2000)
[3] Bess Fang, et al, Phys. Rev. Lett. 116 , 050402 (2016)
[4] Chiara D’Errico, et al, Phys. Rev. Lett. 113 , 095301 (2014)
[5] Guillaume Roux, et al, (2013) New J. Phys. 15 055003

key-words: cold atoms gases, numerical techniques, Luttinger liquids, Bose-Hubbard model, temperature effect, spectral functions

LPTMS Internship: Conformal field theory: Integrable structures and statistical applications

Contact : Raoul Santachiara (raoul.santachiara@universite-paris-saclay.fr)

LPTMS Internship and PhD proposal: Modes non-linéaires de spin dans les condensats de Bose-Einstein à deux composantes

Contact : Nicolas Pavloff (nicolas.pavloff@universite-paris-saclay.fr)

De nombreux systèmes au comportement fortement quantique hébergent des phénomènes non-linéaires. On a ainsi pu observer des solitons, des vortex, des ondes de choc et des murs de domaine dans des condensats de vapeurs atomiques ultra-froides, dans des systèmes de polaritons condensés en micro-cavités, dans l'hélium superfluide, dans des supra-conducteurs... Cette double caractéristique, quantique et non-linéaire, fait de ces systèmes des modèles permettant de mettre en évidence des effets tels le rayonnement de Hawking et/ou l'effet Casimir dynamique. Dans de telles études, les fluctuations quantiques de l'état fondamental du système (le "vide" de la théorie) peuvent être non triviales grâce à la possible non homogénéité du vide, qui est elle même sous-tendue par la non-linéarité de l'onde décrivant le système.

Le sujet de ce stage (qui sera normalement suivi par une thèse) porte sur la description, dans une approche de champ moyen, de la dynamique non-linéaire d'un condensat de Bose-Einstein à deux composantes. Le diagramme de phase de ces systèmes est divisé entre un domaine de paramètres pour lequel le mélange homogène est stable, et un autre dans lequel ce mélange est instable (les deux composantes subissent alors une séparation de phase). Au voisinage de la transition entre ces deux comportements, les modes de densité relative des deux composants peuvent être décrits par l'équation de Landau-Lifshitz qui est usuellement utilisée pour décrire la dynamique non-linéaire de l'aimantation dans un système ferro-magnétique.

Le but de ce stage sera de se familiariser avec la phénoménologie du système (dans le domaine des condensats à deux composantes, mais aussi des systèmes ferromagnétiques) et de classifier les différentes excitations non-linéaires de spin dans l'équation de Landau-Lifshitz. Si le temps le permet, on s'intéressera également à la superfluidité de spin, aux oscillations de Bloch non-linéaires et à l'implémentation d'une configuration imitant un trou noir gravitationnel (un "trou à onde de spin").

Ces sujets seront abordés dans la thèse qui devrait naturellement suivre ce stage. Il s'agit donc d'une thèse théorique durant laquelle on s'attachera à décrire des systèmes susceptibles d'être étudiés expérimentalement. La thèse aura une composante numérique dont la proportion pourra varier en fonction de la pertinence d'une description précise (au vu des possibles réalisations expérimentales). Le cadre théorique général est celui des équations aux dérivées partielles non-linéaires (décrivant la dynamique quasi-classique du système) et de la théorie quantique des champs (pour l'étude des fluctuations quantiques).

LPTMS / PMMH ESPCI Internship and PhD Proposal: Stress reversal by a strong nonlinearity: an elastic sheet toy model

Living cells move thanks to nanometer-size molecular motors whose forces are transmitted up to the scale of the cell by a fiber network known as the cytoskeleton. On much larger length scales, individual cells generate forces that are similarly transmitted to the tissue level through the fibrous extracellular matrix. While the biology of these processes is rather well characterized, the simple problem of force transmission through these highly nonlinear elastic media is far from trivial, and leads to a conversion of local extensile forces to contractile stresses, with crucial biological implications.
To better understand this surprising physical behavior, we will set up a model force transmission experiment where the role of the nonlinear elastic medium will be played by a thin plastic sheet floating on water. By locally exerting extensile forces at the center of the sheet by inflating a balloon, we will directly observe how the forces are rectified through the wrinkling of the sheet. The goal is to help explain why the cytoskeleton is always contractile despite containing a significant number of extensile motors, and to inspire the design of counter-intuitive materials that contract when they should extend.

Expected skills: The student will have a taste for experimental physics. He/She will set up and run a model experiment, and participate in the theoretical analysis of the measurements.

Contact: Martin Lenz, Etienne Reyssat, José Bico, Benoît Roman / @: prenom.nom@espci.fr

Internship location: barre Cassan A, campus Jussieu, 75005 Paris


LPTMS Internship and PhD Proposal: Frustrated self-assembly with multiple particle types

Self-organization is key to the function of living cells – but sometimes goes wrong! In Alzheimer’s and many other diseases, normally soluble proteins thus clump up into pathological fiber-like aggregates. While biologists typically explain this on the grounds of detailed molecular interactions, we have started proving that such fibers are actually expected from very general physical principles. We thus show that geometrical frustration builds up when mismatched objects self-assemble, and leads to non-trivial aggregate morphologies, including fibers.
While we have shown that collections of identical particles form aggregates of various dimensionalities, realistic biological examples often involve multiple proteins. We will thus investigate how collections of several types of different particles typically interact and interfere. Our study will first consist in developing multi-geometries variants of the lattice-based numerical model presented in the illustration. We will then ask whether species with different geometries tend to phase separate, or conversely whether the mutiplicity of interactions they offer eases geometrical frustration and favors co-assembly. We will also wonder how this combinatorics affects the dimensionality of the aggregates, and whether we can identify generic features of the particles that distinguish between the two scenarios. We will then conduct off-lattice simulations to assess the robustness of these scenarios. Finally, we will attempt to construct a mean-field theory describing the co-assembly of a large variety of particles (> 10 or so) thus revealing the interplay between frustration and combinatorial freedom in self-assembly.
Beyond protein aggregation, this project opens investigations into a new class of “disordered” systems where the disorder is carried by each identical particle, as opposed to sprinkled throughout the system. This will help define the much-debated notion of frustration in dilute systems. This project will be conducted in collaboration with Pierre Ronceray (Turing Center for Living Systems, Marseille), who will co-direct a possible PhD project.

Expected skills:
A taste for statistical mechanics and numerical simulations connected to analytical aspects.
PMMH at ESPCI & Sorbonne U. and/or LPTMS at U. Paris-Saclay (Orsay)
martin.lenz@espci.fr or martin.lenz@u-psud.fr


LPTMS Internship and PhD Proposal: Self-assembly in space and time

Video: Self-assembly in space and time - Martin Lenz - LPTMS

Recent experimental developments have made assembling machines at the nanometer scales that mimic or even attempt to surpass the functions of biological objects an increasingly reasonable goal (as recognized in 2016). Despite remarkable progress in manufacturing individual nanometer-sized objects with controlled shapes however (see an example in the illustration), assembling many of them into larger structures remains an open challenge and an active field of research.
In this project we will undertake an additional challenge, namely to self-assemble such objects not only in space, but also in time. Specifically, we will explore the design principles for DNA origami particles produced by our collaborator Seth Fraden (Brandeis University, USA) to assemble over a given sequence over time, which will allow for an actin-like treadmilling (coordinated polymerization from one end, depolymerization from the other) of a polymer-like structure under e.g., temperature cycling. Suchmechanisms could be key in controlling the motor action of prospective molecular machines.
In a second stage (e.g., during a PhD), the intern may develop simulations tools to optimize particle shapesfor self-assembly of printed particles produced at PMMH in collaboration with Julien Heuvingh and Olivia du Roure.

Expected skills:  A taste for statistical mechanics, numerical simulations and working with experimentalists.

PMMH at ESPCI & Sorbonne U. and/or LPTMS at U. Paris-Saclay (Orsay)
martin.lenz@espci.fr or martin.lenz@u-psud.fr