Universal random matrix behavior of a Fermionic quantum gas
Maxime Dixmerias, Giuseppe Del Vecchio Del Vecchio, Cyprien Daix, Joris Verstraten, Tim de Jongh, Bruno Peaudecerf, Pierre Le Doussal, Grégory Schehr, Tarik Yefsah
Since its initial use by Wigner in describing heavy atom spectra, random matrix theory (RMT) has found applications and revealed universal laws in diverse physical contexts, from quantum field theory to disordered systems and wireless communications. A particularly compelling application arises in the mathematical structure of the many-body wavefunction of non-interacting Fermi gases, which underpins a complex spatial organization driven by Pauli’s exclusion principle. However, experimental validation of the counting statistics predicted in such systems has remained elusive. Here, we probe at the single-atom level ultracold atomic Fermi gases made of two interacting spin states, obtaining direct access to their counting statistics in situ. We show that, while the system is strongly attractive, each spin component is described to a high accuracy by RMT predictions based on Fredholm determinants. Our results constitute an experimental validation of the Fermi-sphere point process through the lens of RMT, and establish its relevance for strongly-interacting systems. Random matrix theory (RMT) is used to describe non-interacting ultracold Fermi gases. Here, the authors directly probe the full counting statistics in a 2D system of 6Li atoms with two interacting spin states, by means of a continuum quantum gas microscopy technique. The measured number and spacing statistics agree well with predictions from RMT, also in the strongly interacting regime.
Since its initial use by Wigner in describing heavy atom spectra, random matrix theory (RMT) has found applications and revealed universal laws in diverse physical contexts, from quantum field theory to disordered systems and wireless communications. A particularly compelling application arises in the mathematical structure of the many-body wavefunction of non-interacting Fermi gases, which underpins a complex spatial organization driven by Pauli’s exclusion principle. However, experimental validation of the counting statistics predicted in such systems has remained elusive. Here, we probe at the single-atom level ultracold atomic Fermi gases made of two interacting spin states, obtaining direct access to their counting statistics in situ. We show that, while the system is strongly attractive, each spin component is described to a high accuracy by RMT predictions based on Fredholm determinants. Our results constitute an experimental validation of the Fermi-sphere point process through the lens of RMT, and establish its relevance for strongly-interacting systems. Random matrix theory (RMT) is used to describe non-interacting ultracold Fermi gases. Here, the authors directly probe the full counting statistics in a 2D system of 6Li atoms with two interacting spin states, by means of a continuum quantum gas microscopy technique. The measured number and spacing statistics agree well with predictions from RMT, also in the strongly interacting regime.
This thesis presents a series of experimental studies that leverage atom-resolved imaging to probe fundamental properties of ultracold Fermi gases in continuous space. We demonstrate the viability of this continuum microscopy by reliably observing the expansion of a Gaussian wave packet and the Fermi hole of a two-dimensional ideal Fermi gas, measuring its second- and third-order correlation functions. Building upon this foundation, we introduce a novel local thermometer based on the relationship between fluctuations and correlations. This method enables accurate global and local temperature measurement without requiring trap calibration. Further investigations reveal sub-extensive fluctuations absent from the well-known fluctuation-dissipation theorem. We describe how these fluctuations are linked to entanglement entropy and observe a violation of the area law. Following this we connect the spatial organization of non-interacting fermions to random matrix theory via the Fredholm determinant. Measurements of the counting statistics and nearest-neighbor spacing are compared with numerical predictions, showing excellent agreement. Finally, we explore the physics of two-dimensional interacting fermions in the BEC-BCS crossover. Our measurements of two- and three-point correlation functions show significant discrepancies with mean-field BCS theory, but are corroborated by exact quantum Monte Carlo calculations. We also perform a precise measurement of Tan's contact. As a whole, this work provides a unique and unprecedented microscopic perspective on the fundamentals of Fermi gases, and establishes quantum gas microscopy as a powerful and versatile tool for exploring quantum gases in the continuum.
Atom-based quantum simulators offer a unique platform that enables the imaging of each particle in a many-body system. Until now, however, this capability has been limited to quantum systems in discretized space such as optical lattices and tweezers, where spatial degrees of freedom are quantized. Here, we introduce a novel method for imaging atomic quantum many-body systems in the continuum, allowing for in situ resolution of every particle. We demonstrate the capabilities of our approach on a two-dimensional atomic Fermi gas. We probe the density correlation functions, resolving their full spatial functional form, and reveal the shape of the Fermi hole arising from Pauli exclusion as a function of temperature. Our method opens the door to probing strongly correlated quantum gases in the continuum with unprecedented spatial resolution, providing in situ access to spatially resolved correlation functions of arbitrarily high order across the entire system.
We use ab initio electronic-structure methods to investigate random-matrix theory (RMT) universality in molecular electronic structure. Using single-reference electronic structure methods, including Hartree-Fock, configuration-interaction singles (CIS), density functional theory, and linear-response time-dependent density-functional theory, we compute single-particle orbital energies and many-electron excitations of several representative molecules (benzene, alanine, 1-phenylethylamine, methyloxirane, and helicene chains). For generic low-symmetry geometries, the unfolded spectra of these ab initio Hamiltonians exhibit Wigner-Dyson level statistics of the Gaussian orthogonal ensemble (GOE). For extended helicene chains we explicitly restrict to bound valence excitations below the ionization threshold and still observe GOE statistics, indicating that the RMT universality is present for physical states of direct relevance to real molecules. We further explore the electric and magnetic field dependence of the molecular electronic spectra. The variance of electric polarizability (level curvature K) is predicted to be non-analytic in the magnetic field which serves as an infrared cutoff, <K^2> proportional to log(1/|B|). We observe a transition to the Gaussian unitary ensemble (GUE) by increasing the magnetic fields, although it occurs only at magnetic fields far beyond experimentally accessible scales. Our results indicate that random matrix universality provides a general framework for organizing ab initio predictions of interacting electron spectra in complex systems.
The quantum many-body problem is at the heart of some of the most formidable open problems in modern physics, such as high-Tc superconductivity or the behaviour of neutron stars. Ultracold atomic systems can now be used to simulate model Hamiltonians of condensed matter or nuclear physics, in very well-controlled environnement. In this thesis, we have developed a general method to probe the thermodynamics of homogeneous quantum systems using trapped atomic gases. These measurements are directly compared to the predictions of theories of the quantum many-body problem. We have applied this technique to the spin-1/2 Fermi gas and the Bose with short-range interactions. Using fermionic 6Li, we explored a part of the wide parameter space by changing the interaction strength, the spin-population imbalance or the temperature of the gas. This system exhibits remarkably rich physics, such as normal/superfluid phase transitions (that can be of thermal or quantum character) or a Fermi liquid-type behaviour of the normal phase. We have also used this method to probe a Bose gas of 7Li atoms close to a Feshbach resonance. We have measured the Equation of State of the Bose gas as a function of interactions at very low temperature. For the first time, we measured quantitatively the Lee-Huang-Yang beyond mean-field correction to the ground-state energy of the system, first predicted in 1957. We compared the experimental results to Quantum Monte-Carlo calculations. We have extended this study using out-of-equilibrium measurements of the Bose gas in the strongly interacting regime, which gives a first hint on the properties of the hypothetical unitary Bose gas.
In this Letter, we explore two-dimensional attractive Fermi gases at the microscopic level by probing spatial charge and spin correlations in situ. Using atom-resolved continuum quantum gas microscopy, we directly observe fermion pairing and study the evolution of two- and three-point correlation functions as interspin attraction is increased. The precision of our measurement allows us to reveal nonlocal anticorrelations in the pair correlation function, fundamentally forbidden by the mean-field result based on BCS theory but whose existence we confirm in exact auxiliary-field quantum Monte Carlo calculations. We demonstrate that the BCS prediction is critically deficient not only in the superfluid crossover regime but also deep in the weakly attractive side. Guided by our measurements, we find a remarkable relation between two- and three-point correlations that establishes the dominant role of pair correlations. Finally, leveraging local single-pair losses, we independently characterize the short-range behavior of pair correlations, via the measurement of Tan's contact, and find excellent agreement with numerical predictions. Our measurements provide a novel microscopic view into strongly correlated two-dimensional Fermi gases in the continuum.
Abstract We review recent advances in the theory of trapped fermions using techniques borrowed from random matrix theory (RMT) and, more generally, from the theory of determinantal point processes. In the presence of a trap, and in the limit of a large number of fermions , the spatial density exhibits an edge, beyond which it vanishes. While the spatial correlations far from the edge, i.e. close to the center of the trap, are well described by standard many-body techniques, such as the local density approximation (LDA), these methods fail to describe the fluctuations close to the edge of the Fermi gas, where the density is very small and the fluctuations are thus enhanced. It turns out that RMT and determinantal point processes offer a powerful toolbox to study these edge properties in great detail. Here we discuss the principal edge universality classes, that have been recently identified using these modern tools. In dimension d = 1 and at zero temperature T = 0, these universality classes are in one-to-one correspondence with the standard universality classes found in the classical unitary random matrix ensembles: soft edge (described by the ‘Airy kernel’) and hard edge (described by the ‘Bessel kernel’) universality classes. We further discuss extensions of these results to higher dimensions and to finite temperature. Finally, we discuss correlations in the phase space, i.e. in the space of positions and momenta, characterized by the so called Wigner function.