Session P27: Non-Equilibrium Physics in AMO Systems I: Quenches and Thermalization

Recurrences in an isolated quantum many-body system

The evolution of an isolated quantum system is unitary. If the system becomes large and its constituents interact one is not able to follow the evolution of the complex many body eigenstates. A system out of equilibrium it will ‘relax’ for practical observables [1]. The steady state reached can be described by a Generalized Gibbs Ensemble [2]. In our present study [3] we ask the question if, under which conditions the initial coherence can be revived.

In general the complexity of interacting many-body systems prevent the observation of recurrences of quantum states for all but the smallest systems, even if completely isolated from the environement. For large systems one cannot access the full complexity of the quantum states and the requirements to observe a recurrence in experiments reduces to being close to the initial state with respect to the employed observable. Selecting an observable connected to the collective excitations in one-dimensional super-fluids, we demonstrate recurrences of coherence and long range order in an interacting quantum many-body system containing thousands of particles. This opens up a new window into the dynamics of large quantum systems even after they reached a transient thermal-like state.

In the context of Quantum Thermodynamics and building Quantum Machines our experiments offer a direct way for designing a reservoir, that can either look Markovian and classical, and at later time strongly non-Markovian, remembering all the quantum correlations written into it by the last operation of the quantum machine.

[1] M. Gring et al., Science, 337, 1318 (2012); T. Langen et al., Nature Physics, 9, 640–643 (2013).
[2] T. Langen et al., Science 348, 207-211 (2015).
[3] B. Rauer et al. arXiv:1705.08231   

Prethermalization: Quantum and dynamical universality in quench dynamics

A long-lived prethermal state may emerge upon a sudden quench of a quantum system. In this talk, I consider a quantum quench of an initial critical state, and show that the resulting prethermal state exhibits a genuinely quantum and dynamical universal behavior. Specifically, I consider a scenario where the “speed of light” characterizing the propagation of local perturbations is suddenly quenched at criticality. I also briefly discuss how the system approaches the prethermal state in a universal way described by a new exponent that characterizes a kind of quantum aging.   

Why Quasi-equilibrium, Quasi-stationary and Pre-thermalization phases are related - Relaxation in strongly nonlinear systems

In 2004, we introduced the concept of the quasi-equilibrium state in strongly nonlinear classical many particle systems. Later in the same year, Berges et al introduced a similar state and named it pre-thermalization. In recent experiments, this state, also called the quasi-stationary state, has been reported. We will discuss the properties of the quasi-equilibrium phase for the Fermi-Pasta-Ulam-Tsingou and Hertz systems and how equipartitioning can set in from the quasi-equilibrium state.   

Probing quantum phase transitions via quench dynamics in integrable and nearly-integrable systems

The study of quantum phase transitions requires the preparation of a many-body system near its ground state, a task challenging for many experimental systems. The measurement of quench dynamics, on the other hand, is a routine practice now in most cold atomic platforms. Here we show that quintessential ingredients of quantum phase transitions can be probed directly with quench dynamics in many integrable and near-integrable systems. As a paradigmatic example, we study a global quench in a transverse-field Ising model with nearest- and next-nearest-neighbor interactions. When the model is integrable, we find non-analytic behavior of a dynamical order parameter, which reflects the corresponding ground state quantum phase transition. This behavior persists at finite times in the presence of a weak integrability breaking perturbation. This non-analyticity can be readily observed in experimental platforms such as trapped ions or Rydberg atoms.   

Approximate Counter-diabatic Driving Protocols for Non-integrable Quantum Systems

Noise and decoherence caused by the environment are two major challenges in applying adiabatic protocols to quantum technologies. Counter-diabatic (CD) driving protocols, which are also known as "shortcuts-to-adiabaticity," provide powerful alternatives for controlling a quantum system. These protocols allow one to change Hamiltonian parameters rapidly while still mimicking adiabatic dynamics. They have been shown to work well for a wide variety of systems, but it is exponentially hard to find exact CD protocols for non-integrable quantum many-body systems. We study a method to develop approximate CD protocols which avoids exponential sensitivity to perturbations of the Hamiltonian. Our finite-size scaling of CD Hamiltonians reveals remarkable differences between integrable and non-integrable quantum systems. We identify numerically different scaling regimes and show how they arise from the eigenstate thermalization hypothesis.   

Quantum Joule expansion for the one-dimensional Bose Hubbard model

We study the quantum Joule expansion for a Bose Hubbard model in a one dimensional chain initially at unity filling. Particles are first confined to the left half of the system by an infinite barrier in the middle. At t = 0, the barrier is removed and particles begin to expand to the other half of the system. We consider both pure and thermal initial states. We observe revivals in small systems and these are quickly pushed to long times as the size of the system is increased. As an example, we study a system with 10 sites by exact diagonalization. Particle density and entanglement entropy equilibrate quickly during the expansion for comparable values for the hopping amplitude J and on-site repulsion U. We argue that the diagonal ensemble of the density matrix is similar to the canonical ensemble with an effective temperature. This effective temperature can be negative if the energy of the initial system is large enough. The finite size effects of this effective temperature are also studied by finite temperature density matrix renormalization group.   

Non-adiabatic preparation of critical ground states using superluminal quenches

We show that a space- and time-dependent quench protocol allows faster preparation of the ground state of critical theories than adiabatic protocols. Specifically, assuming the system initially resides in the ground state of a corresponding massive model, we show that a superluminally-moving `front' that locally quenches the mass, leaves behind it (in space) a state arbitrarily close to the ground state of the gapless model. The protocol takes time O(L) to produce the ground state of a system of size L^d (in d spatial dimensions), while a fully adiabatic protocol requires time O(L²) to produce a state with exponential accuracy in L. We present exact results for such quenches in free theories of bosons and fermions, and discuss implications for systems with interactions and ultra-violet features. Finally, we discuss results for such (and related) quenches in arbitrary conformal field theories.   

A Toy Model for Thermalization in Systems with Many-body Localization Phase Transition

The transition from ergodic to many-body localized (MBL) phase remains a challenge from both analytical and numerical perspective. The system thermalization involves an exponential number of degrees of freedom encoded in many-body quantum correlations. This leads to the enormous computational complexity of ergodic phase description and limits the ability of numerical simulations. To address this challenge, we propose a phenomenological model for thermalization of generic many-body systems with local many-body interactions. The method allows reducing the number of degrees of freedom from exponential to polynomial in system size. We construct an effective Hamiltonian and show numerically that the model exhibits a transition from Wigner-Dyson to Poisson level statistics with an increase of the disorder strength. Also, we propose a method to derive the growth the bipartite quantum correlations in the model after a quench. Finally, we apply the results to MBL phase transition problem of interacting spinless fermions in the one-dimensional lattice.   

Microscopic justification of the eigenstate thermalization hypothesis (ETH)

The ETH explains how isolated quantum many-body systems thermalize by proposing that each energy eigenstate is already thermal [1]. ETH has been shown to be essential to the understanding of quantum chaos and implies various important thermodynamic relations [2].
While by now there are many numerical verifications of ETH, only few analytical arguments for its validity have been found. One such analytical argument is based on a semiclassical limit [3]. Another important argument was given by Deutsch who showed how a small interaction which is modelled by a random matrix leads to ETH [4].
Our work adopts this idea and analyzes whether a generic quantum system can be treated as a random matrix. To do this we employ continuous unitary transformations to map an initial Hamiltonian to an effective one. This effective Hamiltonian turns out to have a simple banded form and can be compared to a random matrix. By studying its statistical properties we are able to use the analytical flow equation approach to close the gap in Deutsch's reasoning. Our results depict a first step towards a microscopic justification of ETH.

[1] M. Rigol et al., Nature 452 (2008)
[2] L. D'Alessio et al., Adv. in Phys. Vol. 65 (2016)
[3] M. Srednicki, Phys. Rev. E 50 (1994)
[4] J.M. Deutsch, Phys. Rev. A 43 (1991)   

Non-ergodic Quantum Dynamics in Highly Excited States of a Kinematically Constrained Rydberg Chain

Motivated by recent experimental advances in Rydberg atom quantum simulators, we study the possibility of quantum coherent many-body dynamics far from equilibrium. We identify a class of models which contain an extensive number of highly-excited energy eigenstates with low entanglement entropy, which are responsible for the persistent oscillations observed in recent experiments. We develop a novel non-perturbative forward-scattering method which quantitatively describes the structure of these eigenstates and predicts the frequency of the oscillations. Remarkably the existence of these states does not rely on Many Body Localisation and occurs despite almost all of the other eigenstates being thermal. Our study opens the way towards understanding the unusual case of quantum coherent states in otherwise thermalising many-body systems.   

Quantum Quench in a Model Ising Magnet

The Kibble-Zurek (KZ) mechanism predicts the defect density as a function of time after a quench into an ordered state. Extending this formalism from classical to quantum phase transitions (QPTs) entails a quantum tuning parameter that can be varied rapidly compared to the system’s equilibration times. To this end, we study the model transverse Ising magnet LiHoF₄, which undergoes a ferromagnet-paramagnet QPT as a function of transverse magnetic field. Pumping with S-band microwaves can saturate the nuclear spin levels, suppressing an effective rescaling of the electronic spin at temperatures below 0.4 K. This should move the phase boundary through a fixed transverse field at frequencies comparable to the 100 kHz single-spin attempt frequency. Using a compact high-Q resonator, we report on initial measurements of the ordering behavior following a quantum quench in the linear and nonlinear response regimes.   

The level statistics of random spin ½ XY chain model

Different level spacing distributions such as Poisson distribution and Gaussian orthogonal ensemble(GOE) have been used to characterize many-body localization in random quantum systems. For simple quantum many-body systems, the energy level spacing can be obtained by exact diagonalization studies[1, 2]. However, because of heavy computation cost, not all eigenstates can be taken into account in exact diagonalization studies of many-body quantum systems with even moderate sizes. In this talk, we investigate a spin 1/2 random xy chain model where all eigenstates can be obtained up to rather large system size. we find that for high enough disorder, the energy level distribution approaches a Poisson distribution, in agreement with previous results. However, there exists a gapped region where neither Poisson nor GOE are found in weak disorder.
[1] Arijeet Pal and David A. Huse, Phys. Rev. B 82, 174411 (2010).
[2] Rahul Nandkishore and David A. Huse, Annual Review of Condensed Matter Physics 6, 15–38 (2015)   

Probing quantum dynamics using out-of-time-ordered measurements

Recently, out of time-ordered correlators (OTOCs), i.e., correlators of the form 〈W(t)V(0)W(t)V(0)〉 where W and V are local operators, have been used extensively to study quantum dynamics in various quantum statistical phases, such as chaotic, many-body-localized and delocalized integrable systems. These measures capture subtle quantum correlations that are missed by ordinary time-ordered or retarded correlators. However, measuring OTOCs experimentally requires creating two identical copies of the system, applying the out-of-time-ordered operator on one of them, and taking the overlap of the resultant state with the second copy. This procedure becomes prohibitively difficult for any reasonably large system. In this work, we propose an alternate quantity that we dub the out-of-time-ordered measurement (OTOM) for probing quantum dynamics. The OTOM is closely related to the OTOC and hence, inherits many of its properties. However, it can be measured experimentally with a single copy of the system. Thus, it overcomes the experimental challenge of scalability that the OTOC faces, and makes it possible to study correlations in quantum dynamics that are out of reach experimentally using OTOCs.