1. Non-equilibrium dynamics of strongly interacting Bose gas

Breathing-mode dynamics of the TG gas following a confinement quench. (a,c) Density distribution (b,d) momentum distribution for zero temperature (upper row) and finite temperature (bottom row)  [Physical Review A 95 , 043622 (2017)].

Square of the two body wave function in momentum space for (a) bosons and (b) fermions in a box potential undergoing a quench in its size and at half revival time. 

I am currently working on non-equilibrium properties of strongly interacting Bose gas known as Tonks Girardeau gas. Ultra-cold quantum gases offer a fascinating and remarkably insightful playground for exploring and understanding the fundamentals of out-of- equilibrium behaviour of quantum matter. Experimental systems whose non-equilibrium dynamics can be described by exactly solvable models of quantum many-body theory play a particularly important role. A paradigmatic example in this realm is the Tonks-Girardeau (TG) gas of strongly interacting bosons previously treated only for zero temperature dynamics. While previous work have been focusing on either dynamics at zero temperature or equilibrium at zero and finite temperature, I have proposed an exact study of finite temperature dynamics of TG in both real and momentum space thus encompassing previous studies. I have developed an exact finite temperature dynamical theory of a harmonically trapped TG gas, and applied it to the problem of breathing-mode oscillations after a sudden confinement quench.

In particular, I have identified physical regimes for observing a phenomenon of frequency doubling in the oscillations of the momentum distribution of the gas. The same behaviour is surprisingly well captured by a finite temperature hydrodynamic approach which I have also developed. The method is based on the Fredholm determinant expression of the density matrix at finite temperature. I have shown that it is possible to simplify it as a double sum over single particle eigenfunctions of the relevant problem dressed by the correct fermi weight (see Physical Review A 95 , 043622 (2017)PDF and arXiv :1612.04593PDF for more details).The results provide a straightforward and efficient approach to compute the one particle density matrix of TG at finite temperature and can be easily used to include temperature effects on previous studies done at zero temperature. It also offers possibilities to study various quench protocol which are relevant in current experiments done in cold atomic physics.

2. Quantum spin chains, quantum chaos and random matrices

My thesis was devoted to the study of some aspects of quantum chaos in quantum spins chains. I investigated spectral properties of quantum spins model such as the quantum Ising model in longitudinal and tranverse fields. One very important tool used in quantum chaos is random matrix theory (RMT). It was introduced half a century ago in order to describe statistical properties of energy levels of complex atomic nuclei. Since then, it has proven to be very useful in a great variety of different fields ranging from nuclear physics to wireless communication. In quantum chaos, RMT accurately accounts for the spectral statistics of systems whose classical counterpart is chaotic.

Histogram of the ratio of consecutive level spacings for the quantum Ising model (black) and the zeros of the Riemann zeta function (violet). Full lines correspond to the Wigner-like surmise [Physical Review Letters 110 (8), 084101 (2013)].

While for quantum Hamiltonians which classical counterpart is integrable, the Berry-Tabor conjecture states that their level statistics follows a Poisson law, Bohigas, Giannoni, and Schmit conjectured that the case of quantum Hamiltonians with chaotic classical dynamics must fall into one of the three classical ensembles of RMT. These three ensembles correspond to Hermitian random matrices whose entries are independently distributed, respectively, as real (GOE), complex (GUE), or quaternionic (GSE) random variables.
In Physical Review Letters 110 (8), 084101 (2013), I derived expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices using an argument à la  Wigner. This ratio distribution was introduced as a new measure of chaoticity to study spectral properties of quantum many-body problems but no theoretical prediction was available back then. One main advantage of the ratio distribution is that, as opposed to the standard level spacing distributions, it does not depend on the local density of states. This in turn, greatly simplify statistical study of quantum many body systems where the density of states is generally not known and one has to rely on numerical approximation which introduce errors in the analysis. 

3. Multi-Gaussian approximations for the level density of quantum spin chains

Spectral density for the quantum Ising model with N = 14 spins and for increasing value of transverse field λ. The red circles are numerically calculated densities. The blue lines indicate the multi-Gaussian approximation [Journal of Physics A : Mathematical and Theoretical 47 (33), 335201 (2014)].

The study of spectral statistics led me to investigate the density of states of interacting quantum many body systems. Level densities of quantum models are important and simple quantities which permit to characterise spectral properties of systems with large number of degree of freedom. Due to the central limit theorem, the level density of integrable systems with spectrum obtained by filling of fermionic type excitations tends to the normal law in the thermodynamic limit and in the bulk of the spectrum. This is the case for the integrable Ising model in transverse field. However, in certain limit of coupling with the magnetic field, we observe the appearance of peaks in the level density. In the strict limit of number of spins N → ∞, peaks overlap and disappear (and we recover the gaussian shape) but for values of N accessible in numerical calculations they often strongly influence spectral densities as well as other quantities. In practice, the thermodynamic limit is not reachable and one of the main result I have shown that the knowledge of the two first moments of the Hamiltonian in the degenerated subspace associated with each peak give good approximation of the level density (see Journal of Physics A : Mathematical and Theoretical 47 (33), 335201 (2014)PDF.  Similar to the integrable case, I have also demonstrated the existence of Gaussian and multi-Gaussian regimes for the non integrable Ising model in transverse and longitudinal field.

4.  Multifractality and quantum criticality

Binomial measure for N = 12 steps of the binomial cascade corresponding to the N-th line of the binomial tree. The coefficients of the ground state wave function of the XY model on the critical line λ2+ γ2 = 1 can be mapped to a binomial cascade [Phil. Trans. R. Soc. A 372 (2007), 20120520 (2014)].

Many-body quantum systems are a recurring subject of theoretical physics. Today’s increase in computer power makes it possible to treat problems with a few tens of particles numerically, which opens up new possibilities for their investigation. One other aspects of my thesis was related to the localisation properties of the spins models’ wave functions. I have investigated different one-dimensional quantum spin- 1/2 chain models, and by combining analytical and numerical calculations I have proven that their ground state wave functions in the natural spin basis are multifractals with, in general, nontrivial fractal dimensions. Multifractality is a general notion introduced to characterise quantitatively irregular structures appearing in different problems, such as turbulence, dynamical systems, geophysics… Fractal dimensions give a concise description of eigenfunction moments in the limit of large dimensionality of the Hilbert space and serve as simple indicators of wave function spreading in different scales. In other words, the multifractal dimension is related to the degree of localisation in the Hilbert space. In this work, I have investigated various spins chain model, and by combining numerical and analytical methods I have demonstrated that the multifractality of the ground state wave function is a universal property of all of them. Another important discovery was that quantum criticality can be observed through the multifractal dimensions which means they can be used as a probe of quantum phase transition.

5. Statistical properties of quantum spin chains wave functions

Particular coefficient of the wave functions of the Ising model in transverse field α = 1 and longitudinal field λ = 1 for N = 17 spins in the sector with zero translational momentum [Journal of Physics A : Mathematical and Theoretical 50 385102 (2017)].

Participation ratio for the Ising model for indicated values of α and λ and different N in zero translational momentum sector. From top to bottom N = 17, N = 16, N = 15, and N = 14. Small crosses are numerical results. White lines are the corrected statistical approximation [Journal of Physics A : Mathematical and Theoretical 50 385102 (2017)].

Direct numerical calculation in quantum many-body systems is a challenging problem of modern computational physics. In these systems, the size of the Hilbert space generally grows exponentially with the number of particles, and the development of approximate statistical description remains thus of great importance. In recent years, there has been an increasing interest in this field, mostly due to progress made on the experimental side and the realisation of simple models of quantum many body physics in the laboratory. This in turn has stimulated theoretical studies, and enabled meaningful formulations of questions about fundamental principles of quantum thermodynamics and foundations of quantum statistical physics. In particular, the questions of knowing how/which quantum systems thermalize or what corrections to the thermodynamic limit are appropriate in finite systems are still under investigation.



In this domain, I have constructed and carefully checked a statistical model for eigenfunctions of the quantum Ising model in transverse and longitudinal fields. The investigation of the model is restricted to the chaotic regime when all coupling constants are of the same order. It is attested that for large number of spins, eigenfunction coefficients in the bulk of the spectrum are well approximated by Gaussian functions with zero mean and variance determined analytically from the Hamiltonian. Such asymptotic results are supposed to give a good description of wave functions only in the thermodynamic limit, when the number of spins tends to infinity. For numbers of spins accessible in numerical calculations there exist small but noticeable deviations from asymptotic formulae, and a large part of my work was devoted to calculations of different types of correction from the asymptotic limit. One type of correction is related to higher order moments of the Hamiltonian, and can be taken into account by Gibbs-like formulae. Other corrections are due to symmetry contributions, which manifest as different numbers of non-zero real and complex coefficients in the wave function expansion. The statistical model with these corrections included agreed well with numerical calculations of wave function moments.

6. Painlevé equation and two dimensional diffraction

Two dimensional diffraction pattern generated by an incident plane wave characterised by the wave number k on a perfectly conducting circular obstacle with radius R. The figure represents the amplitude of the total field (incident plus diffracted) in the (x,y) plane and for Rk=9/2. 

It is not well known that the problem of diffraction of an electromagnetic plane wave by a conducting strip or by a slit in a conducting screen has an exact solution. Using contour integral representation, Sommerfeld obtained in 1896 the exact solution of the diffraction by a straight edge. Sieger found the solution in terms of Mathieu functions for the diffraction through a slit by separating the wave equation in elliptic cylinder coordinates and then by flattening the cylinder to form a strip.  In this work, I unveiled an unexpected link between Mathieu functions and Painlevé third transcendent. The result was obtained by solving a two dimensional diffraction problem, namely diffraction by a slit, using two different formalism. The first formalism consisted in solving the Helmholtz equation for the diffracted field in elliptic coordinates and then flattening the parameter of the ellipse to obtain a strip. The solution can be expressed as an infinite serie of Mathieu functions which appears naturally from the elliptic coordinate system.  The diffraction field can also be obtained from an integral equation using the propagator of the Helmholtz equation in two dimension. This technique constitutes the second formalism used and I have shown that the solution in this approach is somehow related to the third Painlevé equation. By relating the two solutions, I was able to write a new expression of the Painlevé third transcendent as a ratio of two series involving Mathieu functions.