Ultra-cold Fermi gases display diverse quantum mechanical properties, including the transition from a fermionic superfluid BCS state to a bosonic superfluid BEC state, which can be probed experimentally with high precision. However, the theoretical description of these properties is challenging due to the onset of strong pairing correlations and the non-perturbative nature of the interaction among the constituent particles. This work introduces a novel Pfaffian-Jastrow neural-network quantum state that includes backflow transformation based on message-passing architecture to efficiently encode pairing, and other quantum mechanical correlations. Our approach offers substantial improvements over comparable ansätze constructed within the Slater-Jastrow framework and outperforms state-of-the-art diffusion Monte Carlo methods, as indicated by our lower ground-state energies. We observe the emergence of strong pairing correlations through the opposite-spin pair distribution functions. Moreover, we demonstrate that transfer learning stabilizes and accelerates the training of the neural-network wave function, enabling the exploration of the BCS-BEC crossover region near unitarity. Our findings suggest that neural-network quantum states provide a promising strategy for studying ultra-cold Fermi gases.
Ultra-cold Fermi gases display diverse quantum mechanical properties, including the transition from a fermionic superfluid BCS state to a bosonic superfluid BEC state, which can be probed experimentally with high precision. However, the theoretical description of these properties is challenging due to the onset of strong pairing correlations and the non-perturbative nature of the interaction among the constituent particles. This work introduces a novel Pfaffian-Jastrow neural-network quantum state that includes backflow transformation based on message-passing architecture to efficiently encode pairing, and other quantum mechanical correlations. Our approach offers substantial improvements over comparable ansätze constructed within the Slater-Jastrow framework and outperforms state-of-the-art diffusion Monte Carlo methods, as indicated by our lower ground-state energies. We observe the emergence of strong pairing correlations through the opposite-spin pair distribution functions. Moreover, we demonstrate that transfer learning stabilizes and accelerates the training of the neural-network wave function, enabling the exploration of the BCS-BEC crossover region near unitarity. Our findings suggest that neural-network quantum states provide a promising strategy for studying ultra-cold Fermi gases.
Abstract Ultra-cold Fermi gases exhibit a rich array of quantum mechanical properties, including the transition from a fermionic superfluid Bardeen-Cooper-Schrieffer (BCS) state to a bosonic superfluid Bose-Einstein condensate (BEC). While these properties can be precisely probed experimentally, accurately describing them poses significant theoretical challenges due to strong pairing correlations and the non-perturbative nature of particle interactions. In this work, we introduce a Pfaffian-Jastrow neural-network quantum state featuring a message-passing architecture to efficiently capture pairing and backflow correlations. We benchmark our approach on existing Slater-Jastrow frameworks and state-of-the-art diffusion Monte Carlo methods, demonstrating a performance advantage and the scalability of our scheme. We show that transfer learning stabilizes the training process in the presence of strong, short-ranged interactions, and allows for an effective exploration of the BCS-BEC crossover region. Our findings highlight the potential of neural-network quantum states as a promising strategy for investigating ultra-cold Fermi gases.
The search for exotic phases of matter in quantum systems remains a central goal in condensed matter physics. Recent progress in machine learning has driven the development of powerful new computational methods in quantum chemistry and physics. In particular, the application of neural networks to solve the Schrödinger equation within the variational Monte Carlo (VMC) framework has led to the development of the successful neural network VMC (NNVMC) method. This thesis extends and explores the application of NNVMC to the study of strongly interacting Fermi gases. We first investigated the unitary Fermi gas (UFG)---a system with strong, short-range attractions known to possess a superfluid ground state that is difficult to describe quantitatively. By applying the NNVMC method with the FermiNet Ansatz, we first identified its key limitations before proposing a simple modification. This change led to a substantial improvement in accuracy over the original FermiNet in the UFG, outperforming results from the state-of-the-art simulations while being consistent with experiments. Next, we explored the phases of the 2D spin-imbalanced Fermi gas. The combined power of NNVMC and our modified FermiNet enabled us not only to observe the long-predicted Fulde-Ferrell-Larkin-Ovchinnikov phase but also to discover a previously unknown phase, characterised by the crystallisation of tightly bound Cooper pairs within a uniform sea of unpaired fermions. Lastly, we studied an anisotropic Fermi gas featuring spin-dependent splitting in momentum space --- a prime signature of the recently discovered form of collinear magnetism, termed altermagnetism. We mapped its 2D phase diagram, providing unbiased estimations for the locations of various phases beyond the mean-field approximation. In conclusion, our work has demonstrated that the NNVMC method is a powerful and versatile tool for investigating strongly correlated systems, confirming its potential to uncover novel phenomena in the vast landscape of quantum systems.
This thesis covers two fields of research carried out in the context of ultracold Fermi gases: Non-equilibrium physics in the BEC-BCS crossover and the use of machine learning methods in physics. \n\t \nThe BEC-BCS crossover is of particular interest for the investigation of low-temperature superconductivity and related many-body phenomena. We realise the BEC-BCS crossover experimentally by cooling a mixture of two hyperfine states of fermionic lithium atoms far below degeneracy. To tune the interaction between the constituents, we make use of a broad magnetic Feshbach resonance between the states. We achieve temperatures far in the degenerate regime that can be tuned by a trap release heating procedure. Being able to tune temperature and interaction allows us to access the whole BEC-BCS crossover regime and investigate related phenomena. \n\t \nOne key quantity across the crossover is the critical temperature of the thermal to superfluid phase transition that signals condensation of pairs. The critical temperature can be calculated in the limits of weak interactions, however, for strong interactions theoretical treatment is difficult. In this thesis, we use neural networks to explore the critical temperature in the whole crossover regime with unprecedented accuracy. We show that a supervisedly trained neural network can not only detect the critical temperature, but also distinguish between Cooper pairs and Feshbach molecules, which was not possible before. Moreover, we show that an unsupervisedly trained autoencoder network is able to determine the critical temperature without any external input other than time-of-flight pictures. \n\t \nAnother important quantity in the crossover regime is the superfluid gap parameter. We measure the gap by exciting the Higgs mode through radio-frequency modulation of the atom population. In the BCS limit, the frequency of the Higgs mode is given by twice the gap frequency. By observing the frequency dependent response of the system we can therefore measure the gap parameter. Towards strong interactions, the observed feature broadens, signalling an instability of the mode due to the gradual violation of the required particle-hole symmetry. \n\t \nExploring the non-equilibrium behaviour of a Fermi gas in the BEC-BCS crossover might hold the key to understand the formation dynamics of pairs and how to induce superconductivity in a system that does not exhibit superconductivity in its equilibrium state. In this work, we present two ways to expose the system to a sudden change of interaction, resulting in a non-equilibrium response. Firstly, we use a fast radio-frequency transfer of one of the hyperfine states into a different, third state. We observe dynamics on several timescales in the momentum distribution and condensate fraction. Secondly, we develop a new coil system that is able to change the magnetic field at the atom position by tens of Gauss within a few microseconds. We detail the construction, testing and alignment process and present investigations into the long-time response of the system. \n\t \nMachine learning procedures can not only be used for data analysis, but also to improve upon the control over experimental systems. In this work we present a novel approach to improve the detection and correction of experimental errors exponentially. We use two machine learning approaches, principal component analysis and small, fully connected neural networks to create an empirical model of the experimental setup. Using this model allows us to reduce the required amount of data for compensation of noise sources exponentially. We demonstrate both approaches in the context of micromotion compensation on a single trapped ytterbium ion in a radio-frequency Paul trap.
Understanding superfluidity remains a major goal of condensed matter physics. Here, we tackle this challenge utilizing the recently developed fermionic neural network (FermiNet) wave function [D. Pfau , .] for variational Monte Carlo calculations. We study the unitary Fermi gas, a system with strong, short-range, two-body interactions known to possess a superfluid ground state but difficult to describe quantitatively. We demonstrate key limitations of the FermiNet in studying the unitary Fermi gas and propose a simple modification based on the idea of an antisymmetric geminal power singlet (AGPs) wave function. The new AGPs FermiNet outperforms the original FermiNet significantly in paired systems, giving results which are more accurate than fixed-node diffusion Monte Carlo and are consistent with experiment. We prove mathematically that the new , which differs from the original only by the method of antisymmetrization, is a strict generalization of the original FermiNet architecture, despite the use of fewer parameters. Our approach shares several advantages with the original FermiNet: The use of a neural network removes the need for an underlying basis set; sand the flexibility of the network yields extremely accurate results within a variational quantum Monte Carlo framework that provides access to unbiased estimates of arbitrary ground-state expectation values. We discuss how the method can be extended to study other superfluid. Published by the American Physical Society 2024
Developing accurate numerical methods for strongly interacting fermions is crucial for improving our understanding of various quantum many-body phenomena, especially unconventional superconductivity. Recently, neural quantum states have emerged as a promising approach for studying correlated fermions, highlighted by the hidden fermion and backflow methods, which use neural networks to model corrections to fermionic quasiparticle orbitals. In this work, we expand these ideas to the space of Pfaffians, a wave-function that naturally expresses superconducting pairings, and propose the hidden fermion Pfaffian state (HFPS), which flexibly represents both unpaired and superconducting phases and scales to large systems with favorable asymptotic complexity. In our numerical experiments, HFPS provides state-of-the-art variational accuracy in different regimes of both the attractive and repulsive Hubbard models. We show that the HFPS is able to capture general phases with and without pairing, and therefore may be a useful tool for modeling phases with unconventional superconductivity.
We report results of quantum Monte Carlo calculations of the ground state of dilute Fermi gases with attractive short-range two-body interactions. The strength of the interaction is varied to study different pairing regimes which are characterized by the product of the s-wave scattering length and the Fermi wave vector, akF. We report results for the ground-state energy, the pairing gap $\ensuremath{\Delta}$, and the quasiparticle spectrum. In the weak-coupling regime, $1∕a{k}_{F}<\ensuremath{-}1$, we obtain Bardeen-Cooper-Schrieffer (BCS) superfluid and the energy gap $\ensuremath{\Delta}$ is much smaller than the Fermi gas energy EFG. When a>0, the interaction is strong enough to form bound molecules with energy Emol. For $1∕a{k}_{F}\ensuremath{\gtrsim}0.5$, we find that weakly interacting composite bosons are formed in the superfluid gas with $\ensuremath{\Delta}$ and gas energy per particle approaching $\ensuremath{\mid}{E}_{\mathrm{mol}}\ensuremath{\mid}∕2$. In this region, we seem to have Bose-Einstein condensation (BEC) of molecules. The behavior of the energy and the gap in the BCS-to-BEC transition region, $\ensuremath{-}0.5<1∕a{k}_{F}<0.5$, is discussed.