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.
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.
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.
Ultracold atoms in optical lattices constitute a versatile many-body platform with highly tunable parameters, allowing us to emulate a multitude of complex quantum systems, in the laboratory, even those eluding analytical treatment. In particular, the nonequilibrium dynamics of strongly correlated many-body systems represents one of the most challenging problems of modern quantum physics, with applications ranging from thermalization dynamics and transport properties to the management of correlations and the control of the dynamics. Understanding nonequilibrium phenomena of strongly correlated systems is a formidable task. A very promising route to gain insight into such systems is to examine few-body setups which contain only a few relevant degrees of freedom, yet incorporating the quantum correlations between the particles.
The present dissertation contributes to the understanding of the nonequilibrium dynamics of strongly-correlated quantum many-body systems by exploring systems of few-bosons − of one or two species − trapped in optical lattices. The systems are driven out-of-equilibrium either by performing a quench of a Hamiltonian parameter or by considering a time-periodic modulation of the external confinement. In the course of several consecutive works, we showcase different ways to couple the nonequilibrium modes, while unveiling their correlated nature and microscopic origin. To simulate the nonequilibrium dynamics, a sophisticated, highly flexible ab-initio method for numerically solving the time-dependent many-body Schrodinger equation is utilized, namely the Multi-Layer Multi-Configuration Time-Dependent Hartree Method for Atomic Mixtures (ML-MCTDHX).
Within the first part we study in six consecutive works the correlated nonequilibrium dynamics of few-boson systems in one-dimensional finite lattices. Starting from weak interactions, it is shown that a sudden increase of the interaction strength generates a global density-wave tunneling dynamics as well as intrawell breathing and cradle-like excited-band processes. The cradle process is a dipole-like process generated by the quench-induced over-barrier transport and it is one of the central results of the present thesis. Remarkably enough, the interaction quenches couple the density-wave and cradle modes, inducing resonance phenomena between the inter and intrawell dynamics. We further show that the cradle mode is inherently related to the initial delocalization and, following a quench from strong-to-weak interactions, can be excited only for incommensurate setups with filling larger than unity. Alternatively, a sudden ramping down of the lattice depth which favors the spatial delocalization is employed to access the cradle mode for setups with filling smaller than unity. Following a multiple interaction quench protocol, we observe the rise of several lowest-band tunneling modes as well as the cradle and the breathing mode. Besides the cradle mode, all other excited modes are highly tunable possessing different frequencies during and in between the quenches. In the excitation dynamics a monotonic behavior with increasing quench amplitude and a non-linear dependence on the duration of the application of the quenched interaction strength is revealed. Additionally, a periodic population transfer between momenta for quenches of increasing interaction is observed, with a power-law frequency dependence on the quench amplitude. Linear interaction quenches from a superfluid to a Mott-insulator state excite various inter- and intraband tunneling modes. The competition between the quench rate and the interparticle repulsion leads to a resonant dynamical response, at moderate ramp times, being related to avoided-crossings in the many-body eigenspectrum.
The resultant higher-band excitation dynamics is shown to obey an exponential decay possessing two distinct time scales with varying ramp time. Inspecting the crossover from shallow to deep lattices we find that for a diabatic quench the excited-band fraction decreases, while approaching the adiabatic limit it exhibits a nonlinear behavior for increasing height of the potential barrier. Quenching from strong-to-weak interactions leads to a melting of the Mott-insulator and negligible higher-band excitations.
Performing quenches either on the wavevector or the phase of a spatially dependent interaction profile triggers various tunneling channels and a rich excitation dynamics which is amplified for increasing inhomogeneity amplitude. Most importantly, the phase quench is shown to induce a directional transport enabling us to discern, otherwise, energetically degenerate tunneling pathways.
Finally, a periodic population transfer between distinct momenta for quenches of increasing wavevector and a directed occupation of higher momenta following a phase quench is observed. Employing a quench of an additional harmonic trap from strong-to-weak confinement, we find that the competition between the initial localization and the repulsive interaction leads to a resonant response of the system related to avoided-crossings in the many-body eigenspectrum with varying final trap frequency. Furthermore, we show that these avoided-crossings can be utilized to prepare the system in a desired state.
The second part comprises two efforts and is devoted to the study of the nonequilibrium dynamics of finite ultracold bosonic ensembles in periodically driven one-dimensional optical lattices. For a shaken lattice, a wide range of driving frequencies is covered and a resonant behavior of the intrawell dynamics is revealed and found to be related to a rich intraband excitation spectrum.
Moreover, it is shown that for increasing repulsion a strong suppression of the interwell tunneling and an enhanced excitation dynamics occurs. For a vibrating lattice, an additional interaction quench gives rise to admixtures of different excitations in the outer wells, an enhanced breathing in the center and an amplification of the emerging tunneling dynamics. The occurence of multiple resonances between the inter- and intrawell dynamics at different quench amplitudes is revealed, with the position of the resonances being tunable via the driving frequency and thus allowing for further control of the mode coupling in optical lattices.
In the third and final part of this thesis we unravel the quench dynamics of a Bose-Bose and a Bose-Fermi mixture. Utilizing an interspecies interaction quench we cross the miscibility-immiscibility threshold in a harmonically confined Bose-Bose mixture. We show that increasing the interspecies repulsion coefficient results in a filamentation of the density of each species, with the spontaneously generated filaments being strongly correlated and exhibiting domain-wall structures. Strikingly, by following the reverse quench protocol, i.e., upon decreasing the interspecies interaction strength, the formation of multiple dark-antidark solitary waves is observed. These solitary wave structures are found to decay into the many-body environment, soon after their generation in sharp contrast to the predictions of the mean-field approximation. To relate our findings with possible experimental realizations, we simulate, for the first time for binary mixtures, single-shot images showcasing that the growth rate of the variance of a sample of single-shots probes the degree of entanglement inherent in the system. As a next step we investigate the expansion dynamics of a mass balanced Bose-Fermi mixture confined in a one-dimensional optical lattice upon quenching an imposed harmonic trap from strong-to-weak confinement. Tuning the interspecies interaction strength we realize the immiscible and miscible correlated ground state phases. We further show that the system's dynamical response crucially depends on the initial phase and consists of an expansion of each cloud and an interwell tunneling dynamics. Varying the quench amplitude and referring to a fixed phase a multitude of response regimes is unveiled, being richer within the immiscible phase, which are described by distinct expansion strengths and tunneling channels. Finally, in the expansion dynamics a two-body anti-correlated behavior between the predominantly occupied wells is unveiled.
Correlated electron systems represent a wide class of materials which at equilibrium display fascinating properties. Several recent experimental breakthroughs in the field of femtosecond spectroscopy and cold atomic gases allow nowadays to investigate the real time dynamics of these many-body quantum systems. Since strongly correlated systems usually escape single particle approaches, the theoretical study of their dynamics constitutes a formidable problem which necessitates the development of novel techniques. In this Thesis we investigate the out-of-equilibrium physics of simple paradigmatic models that are believed to capture some essential physics of interacting fermions by means of the time dependent extension of the Gutzwiller Variational Approach. After an introductory Chapter on the recent results in this field, in Chapter 2 we present the Gutzwiller Approach in-and-out of equilibrium. In Chapter 3 we investigate the dynamics for the single band Hubbard model after a linear ramp of the Coulomb interaction. We will show that a dynamical transition appears for any duration of the ramp; this dynamical point is adiabatically connected to the zero temperature Metal-to-Insulator transition. We will then consider the role of quantum fluctuations beyond mean field. In Chapter 4 we consider the dynamics of an initial antiferromagnetic state under a quench of the interaction in the single band fermionic Hubbard model. We will show that non-thermal ordered states survive more than expected and that two different nonequilibrium antiferromagnets can be distinguished. Finally in Chapter 5 we will consider a two-band Hubbard model which we believe captures the main physics of the paradigmatic compound vanadium sesquioxide, V2 O3 . After an investigation of the equilibrium properties for this model, we will provide evidences that non-thermal metallic phases can emerge upon an excitation of a Mott insulator.
The study of the interplay between interaction-induced correlations and nonequilibrium initial conditions in many-body systems has recently attracted a lot of attention. New experimental techniques provide high control over many-body systems and systematic access to their nonequilibrium regime. In many systems the nonequilibrium properties of a Fermi liquid can be relevant. A first approach to their understanding is the main content of this thesis. At the beginning, nonequilibrium phenomena, Landau's theory of a Fermi liquid, the Hubbard model, time evolution in quantum mechanics, related experiments, and the flow equation method are reviewed. The key observation of this thesis, namely a characteristic mismatch of expectation values in equilibrium and nonequilibrium, is first illustrated for the squeezed oscillator. For this one-particle model Hamiltonian the perturbative approach can be compared with the exact solution and it is shown that the mismatch holds even beyond perturbation theory. Afterwards these observations are generalized to a larger class of one-particle models. Then the nonequilibrium behavior of a Fermi liquid is examined by analyzing the Fermi liquid phase of the Hubbard model in more than one dimension. After a sudden switch-on of a weak two-particle interaction to the noninteracting Fermi gas the relaxation of the many-body system is observed. For this purpose, the flow equation transformation is implemented for the Hubbard Hamiltonian. Then the discussion of the momentum distribution function and of the kinetic energy displays a three-step relaxation behavior of the Fermi liquid from the initial perturbation until thermalization is reached. Firstly, the sudden switch inserts excitation energy into the system which drives the following dynamics. Then a rapid initial build-up of correlations is caused by initial dephasing and leads to the establishment of a quasiparticle picture. By that the system enters into a quasi-stationary state which can be long-lasting for weak interaction. This state shows prethermalization of the kinetic energy which already has relaxed to its final values. However, the momentum distribution still resembles a zero temperature Fermi liquid. Its later relaxation on a second time scale is caused by a residual two-particle interaction which allows for scattering processes and can be described by a quantum Boltzmann equation. The physical origin of this delayed relaxation can be traced back to an interplay of translational invariance of the Hubbard model and the Pauli principle for fermions; together they restrict the phase space for scattering events. Comparing with similar work I conjecture on the generic nature of the findings made for the quenched Fermi liquid in other many-body systems. Finally I point out to the potential relevance of the delayed relaxation for the observation of further nonequilibrium phenomena, for instance in BCS systems. In order to extend the study of sudden switching to arbitrary switching processes the calculation is repeated using the Keldysh perturbation theory. First evaluations for a linear increase of the interaction strength are given. Extensions of this work are suggested and may motivate further research.
The present Thesis aims to deepen the understanding of the physics of strongly-correlated systems along diverse directions with a common denominator, namely a focus on the physical role played by the collective modes shaping quantum many-body correlations. This investigation has been mostly led through the heuristic application of the Quantum Gutzwiller Approach, a recently developed theory of the quantised excitations above the Gutzwiller approximation for strongly-interacting systems on a lattice. In the first place, the intertwining between quantum correlations and collective effects is explored in the context of Bose-Hubbard models, intended both as simple yet highly non-trivial realisations of complex many-body scenarios, and as controllable environments for the quantum simulation of impurity problems. In particular, we are able to provide a comprehensive and semi-analytical description of quantum fluctuations across the critical regimes induced by the proximity of a Mott transition (Mottness), ranging from accurate predictions for the superfluid density and pair correlations across quantum critical regimes to the Andreev-Bashkin effect in a two-component bosonic mixture. As case studies for the effectiveness and flexibility of our approach, we give a detailed account of the rich behaviours of a quantum impurity, either fixed or itinerant, immersed in a Bose-Hubbard environment, showing that both the decoherence dynamics of spinful particle and the resulting Bose polaron in the second case are extremely sensitive probes of the type of critical fluctuations experienced by the surrounding system. Additionally, we extend our analysis of quantum fluctuations to the Fermi-Hubbard model, where we uncover the interplay between low-energy quasiparticles and bosonic elementary excitations in determining two-particle correlations. A more complex physical scenario addressed in this Thesis is represented by driven-dissipative interacting systems, which find a state-of-the-art technological implementation in circuit QED platforms, among others. Inspired by the experimental interest in these systems as a potential toolbox for the generation of exotic quantum states of light, we consider a strongly-correlated fluid of photons on a lattice stabilised via incoherent drive and dissipation, and examine its collective fluctuations across the out-of-equilibrium insulator-to-superfluid transition of the system. Specifically, establishing a conceptual parallelism with our approach to fluctuations at equilibrium, we develop a linear-response theory of the excitations on top of the Gutzwiller stationary state of the Lindblad dynamics, and characterise their physical role in quantum observables and response functions. Taking advantage of our formalism, we describe also dynamical instabilities and novel physical regimes arising in the ultrastrong coupling limit of the superfluid phase. Among the most remarkable results, we bring to light the paradoxical nature of the Mott-like insulating phase of the system, whose peculiar one-body dynamical fluctuations are shown to echo the behaviour of a lasing state. Encompassing diverse modern domains of many-body physics, our investigation goes in the direction of solidifying the interpretation of collective modes as fundamental actors in correlation-induced phenomena under a unified physical picture.
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○