# LPTMS Internship Proposal: Searching for topological physics in dissipative Ytterbium gases

Dissipation is ubiquitous in experiments on quantum matter and it typically reduces the timescales

over which pristine quantum phenomena can be investigated or lowers the quality of the

measurements. It’s an “enemy” that has to be fought harshly and roughly. In this internship we

will change the paradigm and consider dissipation as a resource. Dissipation can induce genuine

and interesting quantum effects (see for instance Ref. 1) and we are interesting in proposing

realistic experiments that can reveal them.

We will focus on the experiments on ultracold ytterbium gases that are currently realized in

several laboratories around the world, among which those at Collège de France in Paris (see

Ref. 2). The goal of this internship is to characterize theoretically the interplay between (i) the

dissipative mechanisms that distinguish these atoms and (ii) the unavoidable presence of atomatom

interactions (Ref. 3 presents some first data obtained in Hamburg, Germany). We will

inspect whether the dissipation-induced topological properties presented in the model of

reference 4, where interactions are neglected, can be observed in Ytterbium gases, where

interactions cannot be neglected. The main investigation tool will be advanced numerical

algorithms based on matrix-product states, that allow for the study of dissipative many-body

systems (see Ref. 5 for an article where such methods have been used to characterize dissipative

topological models).

References:

1. F. Verstraete, M. W. Wolf and J. I. Cirac, Nature Physics 5, 633 (2009).

2. R Bouganne et al., New J. Phys. 19, 113006 (2017).

3. K. Sponselee et al., arXiv:1805.11853 (2018).

4. M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 102, 065703 (2009).

5. F. Iemini, D. Rossini, R. Fazio, S. Diehl and L. Mazza, Phys. Rev. B 93, 115113 (2016)

Director:

Leonardo MAZZA

leonardo.mazza@u-psud.fr

https://sites.google.com/site/leonardmaz/

# LPTMS PhD Proposal: Algebraic area enumeration of two-dimensional closed random lattice walks

### Responsable: OUVRY Stéphane + 33 (0)1 69 15 36 30

Thesis proposal :

The quantum Hofstadter problem [1] (a charge particle hopping on a square lattice coupled to a perpendicular magnetic field) is fascinating for several reasons : its spectrum is a rare example of a fractal emerging from quantum mechanics, its transport properties can shed an interesting light on the quantum Hall effect,...

The Hofstadter model happens to be related to closed random walks on a square lattice,more precisely there is a mapping between the n-th moment of the Hofstadter Hamiltonian and the generating function for the enumeration of close lattice walks of length n enclosing a given algebraic area.

Recently [2] a formula for the algebraic area enumeration has been obtained starting from the so called Kreft coefficients [3] which encode the Schrodinger equation for the Hofstadter model.

Several key observations made in [2] and on which the enumeration relies happen to be incompletely understood and not yet seated on solid mathematical grounds.

Also the final enumeration formula has a complexity which increases exponentially with n making it difficult to be used for walks with a large number of steps.

The thesis will focus on a better understanding and improving of [2] namely :

i) derive some or all of the observations in [2] which lead to the enumeration formula

ii) interpret in terms of 1d random walks characteristics (if possible) the building blocks of the enumeration formula and use this interpretation to push further

its understanding

iii) also if possible simplify and reduce the formula to make it more tractable for the algebraic aera enumeration of walks with a large number of steps Last but not the least, coming back to the Hofstadter model via the mapping discussed above,can the enumeration formula gives new insights on the Hofstadter spectrum in the irrational limit where the flux per plaquette becomes an irrational number?

Last but not the least, coming back to the Hofstadter model via the mapping discussed above, can the enumeration formula gives new insights on the Hofstadter spectrum in the irrational limit where the flux per plaquette becomes an irrational number?

[1] D.R. Hofstadter, Phys. Rev. B 14 (1976) 2239.

[2] S. Ouvry and S. Wu, arXiv: 1810.04098 « The algebraic area of closed lattice random walks »

[3] C. Kreft, « Explicit Computation of the Discriminant for the Harper Equation with Rational Flux », SFB 288 Preprint No. 89 (1993)

# LPTMS Internship Proposal: Pairing and topological phases in cold atoms with long-range interaction

Correlated quantum systems in low dimensions show fascinating properties that distinguish them from their three dimensional counterparts as a consequence of the enhancement of quantum fluctuations. Interacting fermions and bosons in one-dimension (1D) can exhibit many exotic phases of matter. Although short-range interacting particles in 1D are rather well understood, much less is known for long-range interacting systems.Seminal efforts are underway in the control of artificial quantum systems to simulate arbitrary model Hamiltonians which are now barely accessible to classical computation methods. Ultra-cold dipolar or Rydberg atoms can realize Bose or Fermi gases with long-range interactions.

Internship director surnames:Leonardo MAZZA, Guillaume ROUX and Pascal SIMON

E-mails:leonardo.mazza@u-psud.fr, guillaume.roux@u-psud.fr,pascal.simon@u-psud.fr

Web page:http://lptms.u-psud.fr/and https://www.lps.u-psud.fr

Internship location:Orsay(labs will be neighbor in January)

Formulaire_Proposition_Stage_These_SimonRouxMazza