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.
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.
A new field of research, which gained interest in the past years, is the field of non- equilibrium physics of strongly interacting fermionic systems. In this thesis we present a novel apparatus to study an ultracold strongly interacting superfluid Fermi gas driven out of equilibrium. In more detail, we study the excitation of a collective mode, the Higgs mode, and the dynamics occurring after a rapid quench of the interaction strength. Ultracold gases are ideal candidates to explore the physics of strongly interacting Fermi gases due to their purity and the possibility to continuously change many different system parameters, such as the interaction strength. The novel apparatus enabling the experiments outlined below is built from scratch. The setup is described and characterized in detail. We produce a fermionic superfluid of 4 × 10^6 6Li atoms at a temperature of T/T_F = 0.07 ± 0.02, where T_F is the Fermi temperature. This is achieved within 23 s by a combination of laser cooling, radio frequency evaporation of 23Na and simultaneous sympathetic cooling of 6Li in an optical plugged magnetic trap and subsequent evaporative cooling of 6Li in a dipole trap. In this thesis we provide the first experimental evidence of the Higgs mode in a strongly interacting Fermi gas in the crossover from a BCS superfluid to a molecular Bose-Einstein condensate. We develop a novel excitation method, which directly couples to the amplitude of the order parameter. This is achieved by continuously changing the population of one spin component by driving a radio frequency transition to a previously unoccupied third hyperfine state. This effectively modulates the interaction strength and the amplitude of the order parameter. We spectroscopically observe a resonance behavior at twice the gap frequency. For strong coupling, the peak width broadens and eventually the Higgs mode disappears when the Cooper pairs turn into tightly bound dimers signaling the instability of the Higgs mode. It has been suggested that the Higgs mode frequency is a precise measure of the superconducting gap in the BEC-BCS crossover, where the exact value of the gap is yet unknown and numerical calculations are challenging. Hence, our novel method provides a unique technique to determine the superconducting gap. Moreover, we perform rapid variations of the interaction parameter and vary both the initial interaction strength and the amplitude of the quench. The rapid quenches are performed by rapidly transferring one spin component of a two-component spin mixture into a third state with a different interaction strength by using a radio frequency transition. Using this novel method, we can perform the quenches in half the Fermi time, which is experimentally very close to a sudden change and to date faster than any other research group performing interaction quenches with ultra cold gases. In the experiment, we observe a fast relaxation to a zero order parameter for large quenches, whereas for small quenches we observe, after a sudden drop, a revival of the order parameter and equilibration to a long-term superfluid steady state. Our measurement provides the first evidence of the collapse and subsequent revival of order in a strongly interacting fermionic system.
The study of ultracold trapped fermionic gases is a rapidly exploding subject [1-10] which is defining new directions in condensed matter and atomic physics. It has also captured the attention of physicists who study color superconducting quark matter as well as nuclear matter [11-13]. Indeed, it is hard, in recent times, to find a subfield of physics which appeals this broadly to the research community. As we come to understand the gases, experimentalists will move to address fermionic atoms in optical lattices; this will provide important insight to condensed matter physicists as analogue systems for “solving ” intractable many body problems. What makes these gases (and lattices) so important is their remarkable tunability and controllability. Using a Feshbach resonance, one can tune the attractive two-body interaction from weak to strong, and thereby make a smooth crossover from a BCS superfluid to a Bose-Einstein condensation (BEC) [14, 15]. Not only does this allow high transition temperatures Tc (relative to the Fermi energy EF) but it may also provide insights into the high temperature cuprate superconductivity [16-19]. Furthermore, one can tune the population of the two spin states, essentially at will, in this way, allowing exploration [20-22] of exotic polarized phases such as the FFLO [23-25] superfluids, of interest to condensed matter, nuclear and particle physicists. One will be able to tune the lattice parameters such as bandwidth, on-site repulsion, even random disorder, etc and thereby study the famous, and as yet, unsolved Hubbard model Hamiltonian. This paper will concentrate on those issues relating to the effects which are of current interest to experimentalists. In particular, we will study BCS-BEC crossover in atomic Fermi gases, looking at a wide range of different experiments. Our group has been principally interested in the effects of finite temperature [26, 17, 18] and in this review, we will discuss how temperature T enters into the standard crossover theory, and how temperature can be measured, and how temc○
This thesis presents experiments probing physics in the crossover between Bose-Einstein condensation (BEC) and BCS superconductivity using an ultracold gas of atomic fermions. Scattering resonances in these ultracold gases (known as Feshbach resonances) provide the unique ability to tune the fermion-fermion interactions. The work presented here pioneered the use of fermionic Feshbach resonances as a highly controllable and tunable system ideal for studying the cusp of the BCS-BEC crossover problem. Here pairs of fermionic atoms have some properties of diatomic molecules and some properties of Cooper pairs. I present studies of a normal Fermi gas at a Feshbach resonance and the work required to cool the gas to temperatures where superfluidity in the crossover is predicted. These studies culminated in our observation of a phase transition at the cusp of the BCS-BEC crossover through condensation of fermionic atom pairs. I also discuss subsequent work that confirmed the crossover nature of the pairs in these condensates.
This thesis summarizes our study on the properties of ultracold Fermions near a Feshbach resonance. The Feshbach resonance allows for a continuous tuning of the interaction strength between the atoms. With this tunability, many properties predicted for the BCS-BEC crossover region in solids can be shown to exist in this ultracold Fermi gas. With the highly controllable parameters and the clean environment in the experiments, the study of the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein Condensation (BEC) crossover in ultracold Fermions provides valuable insights to the strongly interacting crossover region, which has never been achieved before in solids. We first examine the properties of 6 Li and 40 K gases. We find that in the unitary region the properties of the system do not depend on the atom species: outside the unitary region, the properties of the system do depend on the scattering properties of the different atom species. We study the dynamics of the ultracold Fermions in the crossover region in response to the variations in the external magnetic field. We find that the response depends on the time scale and the amplitude of the variations; which is in qualitative agreement with the experimental observations. We derive a set of dynamic mean-field equations for a Fermi gas in an arbitrary potential trap near a wide Feshbach resonance. The equations are more general than the local density approximation (LDA), and less demanding computationally as the Bogoliubov-de-Gennes (BdG) equation. We then analyze the solubility of the equations, as well as solve a simplified version of the equations for illustration and for self-consistency check on the assumptions made during the derivation. We systematically study the possible phases for a trapped Fermi gas with population imbalance near a Feshbach resonance. Our results agree qualitatively well with the recent experimental results, and predict the existence of two exotic non-BCS superfluid phases for the strongly interacting gas with population unbalance: the Breached Pair (BP) phase and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. Finally, to guide the experimental efforts in the search of new quantum phases, we propose a detection scheme for the possible exotic phases.
Strongly correlated quantum systems display a myriad of complex, fascinating behaviour. The competition between charge, spin, orbital, or lattice degrees of freedom and their respective energy scales gives rise to rich and complex emergent phases. More recently, the non-equilibrium properties of correlated quantum many-body systems have come into the focus of active research, with remarkable advances. While the advent of tensor network based simulations on the theoretical side has allowed access to the full time-evolution of the quantum state as it traverses the Hilbert space, the experimental progress in the ultracold atom community has allowed physicists to enter the paradigm of analog quantum simulation. Their tunability and isolation from the environment makes them ideal platforms to emulate and study open problems in condensed matter physics and go beyond the simulational capabilities of state of the art classical simulations. In this thesis, we explore the dynamics of correlated Fermi gases. We develop and present different facets of applicability of radiofrequency modulation techniques to drive the system out of equilibrium. We begin by considering a homogeneous, three-dimensional Fermi gas in the BCS-BEC crossover and develop an off-resonant radiofrequency transfer scheme to excite the Higgs mode of the superfluid. In a subsequent study we investigate the tunability of the coherence between Cooper pairs by tuning the duration of interaction ramps of the internal interaction strength. These works highlight the possibility to externally tune and stabilise complex quantum many-body states away from equilibrium. Using quasi-exact matrix product state simulations, we study the response of an attractively interacting, one-dimensional Fermi-Hubbard model to weak radiofrequency perturbations. We reveal the emergence of two distinct dynamical regimes in the time evolution and are able to relate it back to the underlying excitation spectrum of the system. Finally, we explore the possibility to perform quantum quenches through radiofrequency π-pulses, achieving nearly complete population transfer between different interacting fermionic states. We reveal non-trivial dynamical effects in the pair correlation of the final state, as well as the excitation of a collective trap mode of the system.
We study ultracold gases of alkali-metal atoms in the quantum degenerate regime. The interatomic interactions in these type of systems can be tuned using resonances induced by magnetic or electric fields. The tunability of the interactions, together with the possibility of confining the atoms with several kinds of external potentials, allows for a completely novel approach to study basic problems in many-body physics, and moreover, allows to enter regimes which have never been accessible in condensed matter or nuclear physics. For example, this has led to the experimental demonstration of an intimate relation between two types of superfluidity: the crossover from Bose-Einstein condensation of tightly bound molecules to the superfluid behavior related to weakly bound BCS-like pairs. Another experimental landmark was reached when the existence of universal Efimov three-body bound states was proven in experiments with ultracold bosonic cesium atoms. In this thesis, we study several aspects of these strongly interacting and ultracold atomic gases. We develop an analytical model that encapsulates all of the relevant scattering physics in atomic systems where open-channel shape resonances and closedchannel Feshbach resonances give rise to complicated and non-trivial scattering properties. This model provides lots of physical insight and is shown to describe important quantities, such as the molecular energies and scattering phase shifts, with a high level of accuracy. The model is compared to full numerical coupled-channel calculations in two atomic systems: rubidium and lithium. We study the BCS-BEC crossover using a many-body description of the ultracold gas that includes the non-trivial energy dependence of the scattering model. We show that it gives rise to superfluid behavior associated with the formation of BCS-like pairs while the low-energy interactions are repulsive in character. The energy dependence of the interactions is crucial, as it gives rise to attractive interactions at the Fermi energy, necessary for the formation of Cooper pairs. We demonstrate new ways of controlling the interatomic interactions using a combination of magnetic and electric fields. This leads to experimental control of, for instance, the three-body parameter in the context of Efimov physics and of non-universal behavior in the BCS-BEC crossover in fermionic gases. Using a four-body method based on first principles, we solve the molecule-molecule scattering problem to calculate several important properties of bosonic and fermionic Summary 145 molecules that consist of light and heavy atoms. These type of molecules are of current experimental interest, and we predict several exciting relations between three- and four-body observables in these type of systems.