Thermalization of isolated and open quantum systems has been studied extensively. However, being the subject of investigation by different scientific communities and being analysed using different mathematical tools, the connection between the isolated (IQS) and open (OQS) approaches to thermalization has remained opaque. Here we demonstrate that the fundamental difference between the two paradigms is the order in which the long time and the thermodynamic limits are taken. This difference implies that they describe physics on widely different time and length scales. Our analysis is carried out numerically for the case of a double quantum dot (DQD) coupled to a fermionic lead, also known as the interacting resonant level model in quantum impurity physics. We show how both OQS and IQS thermalization can be explored in this model on equal footing, allowing a fair comparison between the two. We find that while the quadratically coupled (free) DQD experiences no isolated thermalization, it of course does experience open thermalization. For the non-linearly interacting DQD coupled to a fermionic lead, the many-body interaction in the DQD breaks the integrability of the whole system. We find that this system shows strong evidence of both OQS and IQS thermalization in the same dynamics, but at widely different time scales, consistent with reversing the order of the long time and the thermodynamic limits.
Thermalization of isolated and open quantum systems has been studied extensively. However, being the subject of investigation by different scientific communities and being analysed using different mathematical tools, the connection between the isolated (IQS) and open (OQS) approaches to thermalization has remained opaque. Here we demonstrate that the fundamental difference between the two paradigms is the order in which the long time and the thermodynamic limits are taken. This difference implies that they describe physics on widely different time and length scales. Our analysis is carried out numerically for the case of a double quantum dot (DQD) coupled to a fermionic lead, also known as the interacting resonant level model in quantum impurity physics. We show how both OQS and IQS thermalization can be explored in this model on equal footing, allowing a fair comparison between the two. We find that while the quadratically coupled (free) DQD experiences no isolated thermalization, it of course does experience open thermalization. For the non-linearly interacting DQD coupled to a fermionic lead, the many-body interaction in the DQD breaks the integrability of the whole system. We find that this system shows strong evidence of both OQS and IQS thermalization in the same dynamics, but at widely different time scales, consistent with reversing the order of the long time and the thermodynamic limits.
In this Thesis we discussed several issues related to the dynamics of isolated quantum many-body systems. The brief overview of Chapter 1 highlighted a series of open fundamental theoretical questions which are crucial for the investigation of nonequilibrium phenomena. Among them, we mainly focused on the emergence of thermal behaviors and on the characterization of dynamical phase transitions and of the associated universal dynamics. To this end, we analyzed some paradigmatic models, such as the Fermi-Hubbard model, the quantum Ising chain, and the O(N) model. -Chapter 2: Absence of thermalization in a Fermi liquid. In this Chapter we report two results that question the common expectation of thermalization in a Landau-Fermi liquid after an interaction quench. We first consider the perturbative expansion in the interaction strength of the long-wavelength structure factor S(q) and show that it does not satisfy the hypothesis that steady-state averages correspond to thermal ones. In particular, S(q) has an analytical component ∼ const. + O(q^2) for q → 0, compatible with thermalization, but it retains also a nonanalytic term ∼ |q| characteristic of a Fermi liquid at zero-temperature. In real space, this nonanalyticity turns in a power-law decay of the density-density correlations, in contrast with the exponential decay associated to thermalization. We next analyze the low-density case, where one can obtain results valid at any order in interaction but at leading in the density, and find that in the steady state the momentum distribution jump at the Fermi surface is strictly finite, though smaller than at equilibrium. -Chapter 3: Nonadiabatic stationary behavior in a driven Ising chain. In this Chapter we discuss the emergence of a nonadiabatic behavior in the dynamics of the order parameter in a low-dimensional system subject to a linear ramp of one of its parameters within a gapped phase, which should be the most favorable situation for an adiabatic evolution. We study in details the dynamics of the spontaneous magnetization m_x(t) in a quantum Ising chain after a linear variation in time of the transverse field within the ordered phase. In particular, focusing on the value of the order parameter at the end of the ramp m_x(τ), we show that the smaller the switching rate of the transverse field is the closer m_x(τ) gets to its ground state value. Nonetheless, this apparently small correction to adiabaticity eventually leads to the disruption of the order exponentially fast in the subsequent time evolution, no matter how slowly the ramp is performed. -Chapter 4: Aging and coarsening in isolated quantum systems after a quench. This Chapter analyzes the nonequilibrium dynamics of an isolated quantum system after a sudden quench to the dynamical critical point, where the emergence of scaling and universal exponents is expected, due to the absence of time and length scales. We explore these features for a bosonic interacting scalar field theory with O(N) symmetry in the large-N limit, where the model is exactly solvable and the exponents and scaling forms of the relevant two-time correlation functions can be analytically calculated. Moreover, we provide evidence of the emergence of a dynamical scaling behavior also for quenches below the dynamical critical point, associated with coarsening. We find that the latter case is characterized by the same scaling functions as those describing the critical case, yet with different exponents. -Chapter 5: Dynamical transitions and statistics of excitations. In this Chapter we study the dynamics of the O(N) model (introduced in the previous Chapter) resulting from a different protocol: a linear ramp of one of its parameters. We find that the presence of a dynamical phase transition, as well as its critical properties, are robust against the change of the protocol. We show that a characterization based on the critical dimensions and exponents would suggest that the dynamical phase transition is analogous to the equilibrium thermal one. However, its nonequilibrium nature becomes evident analyzing the statistics of excitations produced in the ramp process. In particular, the critical properties are encoded in the fluctuations in the number of excitations, which display qualitatively different behaviors depending on the ramp being performed above, at, or below the dynamical critical point. These behaviors bear no dependence on the duration of the protocol.
Open quantum systems have drawn attention over decades due to its applicability in the foundation of theoretical physics, e.g. statistical mechanics, quantum mechanics and condensed matter physics. The dynamics of open quantum systems has been described as separate entities from their surrounding environment that consists of a very large number of modes, somehow coupled to the mode of the system. Even though the exact solution of the dynamical behavior of the system is impossible to calculate, we obtain a tentative solution using the crucial Markov approximation. The input-output formalism of the quantum Langevin equation (QLE) has been considered as a useful tool which provides a semi-classical description of the dynamics of the system, whereas the master equation provides a complete picture of the dynamics of the system expressed in terms of density matrix. While studying the dynamics of nonlinear system/environment coupling using QLE, we see, for a small value of external field, that the steady state system field does not change much from the steady state field obtained in the absence of nonlinear dissipation. However, in a case where a stronger external field is applied, we see that the deviation becomes substantial from the solution of linear system. We also see that the nonlinear coupling introduces significant difference in the cavity fluctuation spectrum. The description, therefore, provides a potential explanation of parametric effects in terms of nonlinear dissipation phenomena associated with the nonlinear coupling. Even though the theories developed in the context of open quantum systems have proven to be powerful tools, they do not provide a satisfactory platform to be implemented on non-linear Hamiltonians. We often approximate it by linearizing over nonlinear steady state field amplitude, and therefore, the interesting effects are often overlooked. The limitation of the analytics provokes us to simulate open quantum dynamics numerically. The numerical method consists of transformation of the environmental degrees of freedom to a one-dimensional many-body chain, and the computational technique includes numerical diagonalization and renormalization process. The time-adaptive density matrix renormalisation group (t-DMRG) is known as one of the most powerful techniques for the simulation of strongly-correlated many-body quantum systems. In this thesis, along with the theoretical modeling, we implement DMRG numerical scheme for the simulation of canonical S/B model by mapping it to one-dimensional harmonic chain with nearest neighbor interactions, and use the method to investigate the dynamics of the free dissipative system. The thermalization of open quantum systems is also studied by generating minimally entangled typical thermal states (METTS) through imaginary time evolution, and real-time evolving an empty system in the presence of the thermal bath. Further, we simulate coherently driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and t-DMRG algorithm, and demonstrate how the numerical results are analogous to classical bistability. By comparing with analytics, we see that the DMRG numerics is analogous to the quantum-mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation. The comparison between two different numerical techniques shows that the semi-classical Euler's method determines the dynamics of the system field of one among two coherent branches, whereas DMRG numerics gives the superposition of both of them. Hence, DMRG-determined time dynamics undergoes generating non-classical states. Our approach of dealing with nonlinearity represents an important contribution in the developments of technique to study the dynamical and steady-state behavior of open quantum systems, which is a fundamental aspect of quantum physics.
The last decade has witnessed remarkable progress in our understanding of thermalization in isolated quantum systems. Combining the eigenstate thermalization hypothesis with quantum measurement theory, we extend the framework of quantum thermalization to open many-body systems. A generic many-body system subject to continuous observation is shown to thermalize at a single trajectory level. We show that the nonunitary nature of quantum measurement causes several unique thermalization mechanisms that are unseen in isolated systems. We present numerical evidence for our findings by applying our theory to specific models that can be experimentally realized in atom-cavity systems and with quantum gas microscopy. Our theory provides a general method to determine an effective temperature of quantum many-body systems subject to the Lindblad master equation and thus should be applicable to noisy dynamics or dissipative systems coupled to nonthermal Markovian environments as well as continuously monitored systems. Our work provides yet another insight into why thermodynamics emerges so universally.
Abstract A nearly-integrable isolated quantum many-body system reaches a quasi-stationary prethermal state before a late thermalization. Here, we revisit a particular example in the settings of an open quantum system (OQS). We consider a collection of non-interacting atoms coupled to a spatially correlated bosonic bath characterized by a bath correlation length. Our result implies that the integrability of the system depends on such a correlation length. If this length is much larger than the distance between the atoms, such a system behaves as a nearly-integrable OQS. We study the properties of the emerging prethermal state for this case, i.e. the state’s lifetime, the extensive number of existing quasi-conserved quantities, the emergence of the generalized Gibbs state, and the scaling of von Neumann entropy, etc. We find that for the prethermal state, the maximum growth of entropy is logarithmic with the number of atoms, whereas such growth is linear for the final steady state, which is the Gibbs state in this case. Finally, we discuss how such prethermal states can have significant applications in quantum entanglement storage devices.
Abstract The issue of thermalization in open quantum systems is explored from the perspective of fermion models with quadratic couplings and linear baths. Both the thermodynamic state and the stationary solution of the Lindblad equation are rendered as a matrix-product sequence following a reformulation in terms of underlying algebras, allowing to characterize a family of stationary solutions and determine the cases where they correspond to thermal states. This characterization provides insight into the operational mechanisms that lead the system to thermalization and their interplay with mechanisms that tend to drive it out of thermal equilibrium.
Quantum thermalization describes how interacting quantum systems relax toward thermal equilibrium, a central problem in modern physics. Yet most experimental information on many-body systems comes from short-time transition spectroscopy, typically interpreted within Kubo's linear-response framework. These perspectives-long-time equilibration versus short-time response-seem fundamentally disconnected. Here we establish an exact link between them: the \emph{Kubo-Thermalization correspondence}, which connects long-time thermalized magnetization under weak driving to short-time linear-response spectra for a spin coupled to a thermal bath. The correspondence holds even when the steady state differs substantially from the initial state and when each regime is individually difficult to describe theoretically. We experimentally confirm the correspondence using effective spin-1/2 impurities realized with ultracold fermions in two internal states coupled to a Fermi sea. Our results provide a rare exact statement about quantum thermalization and offer a novel route to infer thermalization dynamics from equilibrium response measurements in strongly interacting quantum systems, independent of microscopic details of the system--bath coupling.