Recently universal dynamic scaling is observed in several systems, which exhibit a spatiotemporal self-similar scaling behavior, analogous to the spatial scaling near phase transition. The latter one arises from the emergent continuous scaling symmetry. Motivated by this, we investigate the possible relation between the scaling dynamics and the continuous scaling symmetry in this paper. We derive a theorem that the scaling invariance of the quenched Hamiltonian and the initial density matrix can lead to the universal dynamic scaling. It is further demonstrated both in a two-body system analytically and in a many-body system numerically. For the latter one, we calculate the dynamics of quantum gases quenched from the zero interaction to a finite interaction via the non-equilibrium high-temperature virial expansion. A dynamic scaling of the momentum distribution appears in certain momentum-time windows at unitarity as well as in the weak interacting limit. Remarkably, this universal scaling dynamics persists approximately with smaller scaling exponents even if the scaling symmetry is fairly broken. Our findings may offer a new perspective to interpret the related experiments. We also study the Contact dynamics in the BEC–BCS crossover. Surprisingly, the half-way time displays a maximum near unitarity while some damping oscillations occur on the BEC side due to the dimer state, which can be used to detect possible two-body bound states in experiments.
Recently universal dynamic scaling is observed in several systems, which exhibit a spatiotemporal self-similar scaling behavior, analogous to the spatial scaling near phase transition. The latter one arises from the emergent continuous scaling symmetry. Motivated by this, we investigate the possible relation between the scaling dynamics and the continuous scaling symmetry in this paper. We derive a theorem that the scaling invariance of the quenched Hamiltonian and the initial density matrix can lead to the universal dynamic scaling. It is further demonstrated both in a two-body system analytically and in a many-body system numerically. For the latter one, we calculate the dynamics of quantum gases quenched from the zero interaction to a finite interaction via the non-equilibrium high-temperature virial expansion. A dynamic scaling of the momentum distribution appears in certain momentum-time windows at unitarity as well as in the weak interacting limit. Remarkably, this universal scaling dynamics persists approximately with smaller scaling exponents even if the scaling symmetry is fairly broken. Our findings may offer a new perspective to interpret the related experiments. We also study the Contact dynamics in the BEC–BCS crossover. Surprisingly, the half-way time displays a maximum near unitarity while some damping oscillations occur on the BEC side due to the dimer state, which can be used to detect possible two-body bound states in experiments.
The study of the quantum dynamics of ultra-cold atomic gases has become a forefront of atomic research. Experiments studying dynamics have become routine in laboratories, and a plethora of phenomena have been studied. Theoretically, however, the situation is often intractable unless one resorts to numerical or semiclassical calculations. In this thesis we apply the symmetry associated with scale invariance to study the dynamics of atomic gases, and discuss the implications of this symmetry on the full quantum dynamics. In particular we study the time evolution of an expanding two-dimensional Bose gas with attractive contact interactions, and the three-dimensional Fermi gas at unitarity. To do this we employ a quantum variational approach and exact symmetry arguments. It is shown that the time evolution due to a scale invariant Hamiltonian produces an emergent conformal symmetry. This emergent conformal symmetry has implications on the time evolution of an expanding quantum gas. In addition, we examine the effects of broken scale symmetry on the expansion dynamics. To do this, we develop a non-perturbative formalism that classifies the possible dynamics that can occur. This formalism is then applied to two systems, an ensemble of two-body systems, and for the compressional and elliptic flow of a unitary Fermi gas, both in three spatial dimensions.
Understanding nonequilibrium dynamics of strongly interacting quantum systems represents one of the most challenging problems in many-body physics. Here, quantum thermalization dynamics are explored in real-time in an ultracold Fermi gas suddenly quenched to the BEC-BCS crossover. When quenched to unitarity, it is observed that the cloud size remains unchanged in the early evolution while the momentum distribution emerges two prethermal states with a lifetime difference of two orders of magnitude in the early and intermediate stage before very slowly evolving to the final stationary state. It is revealed that a crossover momentum, at which the momentum distribution remains nearly unchanged, is determined by the thermal wavelength at high temperatures and the Fermi momentum while at low temperatures. It is identified that the universal prethermal dynamics scaling where momentum distributions with different temperatures collapse onto one curve. When quenched to the BEC side, the thermalization rapidly relaxes into a prethermal state and exhibits the low energy oscillation related to the molecular bound states. This work provides benchmarks for the study of quantum thermalization in strongly interacting fermionic many-body systems.
Motivated by an unexpected experimental observation from the Cambridge group, [Eigen et al., Nature 563, 221 (2018)], we study the evolution of the momentum distribution of a degenerate Bose gas quenched from the weakly interacting regime to the unitary regime. For the two-body problem, we establish a relation that connects the momentum distribution at a long time to a subleading term in the initial wave function. For the many-body problem, we employ the time-dependent Bogoliubov variational wave function and find that, in certain momentum regimes, the momentum distribution at long times displays the same exponential behavior found by the experiment. Moreover, we find that this behavior is universal and is independent of the short-range details of the interaction potential. Consistent with the relation found in the two-body problem, we also numerically show that this exponential form is hidden in the same subleading term of the Bogoliubov wave function in the initial stages. Conceptually, our results show that, for quench to the universal regime and coherent quantum dynamics afterward, the universal longtime behavior is hidden in the initial state.
Scale invariance is conventionally discussed in the context of critical phenomena near a continuous phase transition. In atomic gases, a new class of scale invariant systems emerges, which we call “intrinsic scale invariance.” In these systems, the scaling symmetry does not rely on a phase transition, but comes from the balance of kinetic and interaction energy at different length scales. Three examples are discussed here: three-dimensional unitary Fermi gas, two-dimensional dilute Bose gas, and unitary Bose gas. The first two are invariant under continuous scaling transformation, and thus they share the same universal thermodynamics. Unitary Bose gas is a special case with expected discrete scaling symmetry because of Efimov three-body physics.We show that the discrete scale invariance can be captured by a complex scaling dimension and suggest a universal form for the observables. Introduction The achievement of atomic Bose-Einstein condensation (BEC) in 1995 initiated a new adventure to explore quantum many-body phenomena in the gas phase, a journey that has yielded much excitement and is still progressing fast. Because of their diluteness, ultracold atomic gases can be precisely modeled and characterized with high accuracy. In addition, new opportunities have emerged in the past decade to prepare cold atoms in the strong coupling regime. By loading the samples into optical lattices [1] or tuning them near a Feshbach resonance [2], for instance, cold atoms offer a unique platform to investigate strongly correlated quantum phenomena. Examples include the superfluid-Mott insulator transition [3] (see also Chapter 13), BEC-BCS (Bardeen-Cooper-Schrieffer) crossover [4, 5, 6, 7] (see also related overview in Chapter 12), Efimov trimer states [8], and Berezinskii- Kosterlitz-Thouless transition in two dimensions [9] (see also Chapter 10). One key motivation in these explorations is to uncover new, universal laws that underpin the behavior of strong-coupled few- and many-body quantum systems. In this chapter, we discuss “scale invariance” as a generic attribute in various cold atom systems, including all of the aforementioned examples. While scale invariance is conventionally discussed in the context of critical phenomena, the cold atom systems that interest us belong to a new class, which we call “intrinsic scale invariance.” In the following, we first overview the main features of different types of scale invariance and discuss related examples in cold atom systems.
Recent advances in the tunability of ultracold atomic gases have created opportunities for studying interesting quantum many-body systems. Fano-Feshbach resonances, in particular, allow experimenters to freely adjust the scattering of atoms by controlling an external magnetic field. By rapidly changing this field near a resonance, it is possible to drive systems out of equilibrium towards novel quantum states where correlations between atoms change dynamically. In this thesis, we take a wave-function-based approach to theoretically examine the response of several interesting systems to suddenly-switched, or "quenched", interactions.\nWe first calculate the time evolution of a Bose-Einstein condensate that is quenched to the unitarity regime, where the scattering length 'a' diverges. Working within the time-dependent variational formalism, we find that the condensate does not deplete as quickly as the usual Bogoliubov theory would suggest. We also make a quantitative prediction for the dynamics of short-range pair correlations, encoded in Tan's contact. We then consider the dynamics of these correlations for quenches to small 'a', and we find that bound states can cause high-contrast oscillations of the contact. These dynamics can be modelled quantitatively at short times by using a properly-chosen two-body model. Finally, we characterize the nonlocal correlation waves that are generated by an interaction quench in arbitrary dimensionality. Our analysis demonstrates that the large-momentum limit of the post-quench momentum distribution can sometimes include contributions from both the short range and the long range, depending on the quench protocol.
High temperature virial expansion is a powerful tool in equilibrium statistical mechanics. In this Letter we generalize the high temperature virial expansion approach to treat far-from-equilibrium quench dynamics. As an application of our framework, we study the dynamics of a Bose gas quenched from noninteracting to unitarity, and we compare our theoretical results with unexplained experimental results by the Cambridge group [Eigen et al., Nature 563, 221 (2018)]. We show that, during the quench dynamics, the momentum distribution decreases for the low-momentum part with k<k^{*}, and increases for high-momentum part with k>k^{*}, where k^{*} is a characteristic momentum scale separating the low- and the high-momentum regimes. We determine the universal value of k^{*}λ that agrees perfectly with the experiment, with λ being the thermal de Broglie wavelength. We also find a jump of the halfway relaxation time across k^{*}λ and the nonmonotonic behavior of energy distribution, both of which agree with the experiment. Finally, we address the issue whether the longtime steady state thermalizes or not, and we find that this state reaches a partial thermalization, namely, it thermalizes for the low-energy part with kλ≲1 but does not thermalize for the very high momentum tail with kλ≫1. Our framework can also be applied to quench dynamics in other systems.