Recovering properties of correlation functions is typically challenging. On the one hand, experimentally, it requires measurements with a temporal resolution finer than the system's dynamics. On the other hand, analytical or numerical analysis requires solving the system evolution. Here, we use recent results of quantum metrology with continuous measurements to derive general lower bounds on the relaxation and second-order correlation times that are both easy to calculate and measure. These bounds are based solely on steady-state expectation values and their derivatives with respect to a system parameter, and can be readily extended to the autocorrelation of arbitrary observables. We validate our method on two examples of critical quantum systems: a critical driven-dissipative resonator, where the bound matches analytical results for the dynamics, and the infinite-range Ising model, where only the steady state is solvable, and thus the bound provides information beyond the reach of existing analytical approaches. Our results can be applied to the experimental characterization of ultrafast systems and to the theoretical analysis of many-body models whose dynamics are hard to compute.
Recovering properties of correlation functions is typically challenging. On the one hand, experimentally, it requires measurements with a temporal resolution finer than the system's dynamics. On the other hand, analytical or numerical analysis requires solving the system evolution. Here, we use recent results of quantum metrology with continuous measurements to derive general lower bounds on the relaxation and second-order correlation times that are both easy to calculate and measure. These bounds are based solely on steady-state expectation values and their derivatives with respect to a system parameter, and can be readily extended to the autocorrelation of arbitrary observables. We validate our method on two examples of critical quantum systems: a critical driven-dissipative resonator, where the bound matches analytical results for the dynamics, and the infinite-range Ising model, where only the steady state is solvable, and thus the bound provides information beyond the reach of existing analytical approaches. Our results can be applied to the experimental characterization of ultrafast systems and to the theoretical analysis of many-body models whose dynamics are hard to compute.
We give rigorous analytical results on the temporal behavior of two-point correlation functions-also known as dynamical response functions or Green's functions-in closed many-body quantum systems. We show that in a large class of translation-invariant models the correlation functions factorize at late times ⟨A(t)B⟩_{β}→⟨A⟩_{β}⟨B⟩_{β}, thus proving that dissipation emerges out of the unitary dynamics of the system. We also show that for systems with a generic spectrum the fluctuations around this late-time value are bounded by the purity of the thermal ensemble, which generally decays exponentially with system size. For autocorrelation functions we provide an upper bound on the timescale at which they reach the factorized late time value. Remarkably, this bound is only a function of local expectation values and does not increase with system size. We give numerical examples that show that this bound is a good estimate in nonintegrable models, and argue that the timescale that appears can be understood in terms of an emergent fluctuation-dissipation theorem. Our study extends to further classes of two point functions such as the symmetrized ones and the Kubo function that appears in linear response theory, for which we give analogous results.
Estimating the steady-state properties of open many-body quantum systems is a fundamental challenge in quantum science and technologies. In this work, we present a scalable approach based on semidefinite programming to derive certified bounds on the expectation value of an arbitrary observable in the steady state of Lindbladian dynamics. We illustrate our method on a series of many-body systems, including paradigmatic spin-1/2 chains and two-dimensional ladders, considering both equilibrium and nonequilibrium steady states. We benchmark our method with state-of-the-art tensor-network approaches that, unlike our method, are only able to provide estimates, with no guarantee, on steady-state quantities. For the tested models, only modest computational effort is needed to obtain certified nontrivial bounds for system sizes intractable by exact methods. Our method introduces the first general numerical tool for bounding steady-state properties of open quantum dynamics, opening an avenue in the understanding of stable configurations in many-body systems.
Ising models, and the physical systems described by them, play a central role in generating entangled states for use in quantum metrology and quantum information.In particular, ultracold atomic gases, trapped ion systems, and Rydberg atoms realize long-ranged Ising models, which even in the absence of a transverse field can give rise to highly non-classical dynamics and long-range quantum correlations.In the first part of this paper, we present a detailed theoretical framework for studying the dynamics of such systems driven (at time t = 0) into arbitrary unentangled non-equilibrium states, thus greatly extending and unifying the work of Foss-Feig et al (2013 Phys.Rev. A 87 042101).Specifically, we derive exact expressions for closed-time-path ordered correlation functions, and use these to study experimentally relevant observables, e.g.Bloch vector and spin-squeezing dynamics.In the second part, these correlation functions are then used to derive closed-form expressions for the dynamics of arbitrary spin-spin correlation functions in the presence 5
We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. In particular, dynamic correlation functions appear in the fluctuation-dissipation theorem, where the response of a many-body system to an external perturbation is given in terms of the relaxation function of the unperturbed system, provided the disturbance is small. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numericallly exact diagonalizaton method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations. The method of recurrence relations relies on the solution to the coefficients of a continued fraction for the Laplace transformed relaxation function. The calculation of those coefficients becomes very involved and, only a few cases offer exact solution. We shall concentrate our efforts on the cases where extrapolation schemes must be used to obtain solutions for long times (or low frequency) regimes. We also cover numerical work based on the exact diagonalization of finite sized systems. The numerical work provides some thermodynamically exact results and identifies some difficulties intrinsic to the method of recurrence relations.
The exact classical simulation of non-integrable quantum systems rapidly becomes intractable beyond small system sizes, yet its role is central to predicting many-body dynamics. Approximate methods alleviate this cost but typically lack rigorous guarantees on their deviation from the exact result. For the infinite-temperature correlation function, we derive a rigorous error bound for approximating the full dynamics within a computable finite-sized region. Applied to autocorrelations, this finite-region approach is especially accurate across one-dimensional systems and beyond, and can further provide rigorous error bounds for other approximate methods, demonstrated here for matrix-product-operator simulations.
The study of quantum dynamics is an increasingly important aspect of atomic, molecular and optical physics, as well as condensed-matter physics and other disciplines. It is essential in order to understand non-equilibrium phenomena which do not fit within the framework of equilibrium thermodynamics, such as many-body localization, dynamical quantum phase transitions, and non-equilibrium steady states in driven-dissipative systems. However, the exact calculation of dynamical observables is extremely difficult in general for several reasons, which makes it challenging for theoretical predictions and analyses to keep pace with rapidly advancing experiments. In this thesis I use and develop several computational methods to investigate the dynamics of quantum many-body systems, with the two goals of better understanding the evolution of local observables and correlation functions in spin and Fermion systems as well as creating and testing more effective numerical methods for solving quantum dynamics problems. I explore quantum dynamics by using Hamiltonian parameter quenches in several commonly-used models in AMO physics and elsewhere: the Fermi-Hubbard model, the XXZ model, and the transverse-Ising model. The first represents a system of short-range interacting Fermions, while the latter two represent spin systems with different spin-spin couplings and magnetic field terms. All of these models can be realized in current experiments. First, I derive an exact solution for an extreme interaction quench in the Fermi-Hubbard model, from a strongly-interacting initial state to a noninteracting final Hamiltonian. From this result I show that there are nontrivial transient connected correlations, despite the fact that the system has very high temperature and a noninteracting Hamiltonian. Those features would suppress such correlations in equilibrium, but the dynamical correlations are still significant. Then, I develop and examine two numerical methods for solving dynamics of the spin expectation values and correlation functions in the transverse-Ising and XXZ models. Both methods rely on approximating a thermodynamically large system with a much smaller and more computationally tractable cluster of lattice sites, or a series of clusters. The first method is a dynamical numerical linked cluster expansion (NLCE). NLCEs have already been used successfully on systems in thermodynamic equilibrium, and I extended the technique to directly compute the dynamics of an observable after a quantum quench. I review the general NLCE method, describe my dynamical variant of it, and compute the quench dynamics of spin and correlation observables in the transverse-Ising and XXZ models. The second method, which I term the adaptive-boundary cluster method, relies on recently refined bounds on information propagation in quantum systems to choose a cluster of lattice sites that minimizes the finite size error bound of a dynamical computation. I briefly review these bounds, and describe a new algorithm which uses them to construct a cluster whose boundary is chosen according to a related error estimate. Such an adaptive-boundary cluster has the property that there exists a computable upper bound on the finite size error of observables measured at the cluster center, and this error bound is nearly minimal when compared to that of other clusters of the same size. I then compute the quench dynamics of spin and correlation observables in the transverse-Ising model using the adaptive boundary clusters. I apply this method to both uniform and strongly disordered systems. To evaluate the effectiveness of both dynamical NLCE and adaptive-boundary clusters, I compare the results obtained using them to results from a standard and general purpose numerical method: exact solvers applied a rectangular cluster with periodic boundary conditions. Both of the new methods usually converge faster and are more accurate than the standard method for comparable cluster sizes. This thesis illustrates that choosing numerical methods which take advantage of the geometric and entanglement structure of the initial state and Hamiltonian can substantially enhance our ability to study quantum dynamics, by reducing the computational cost of numerically exact solutions. I conclude with a discussion of various avenues for future research to build on this work.