Functions For Quantum Systems Research Articles

Measurement-Based Control for Minimizing Energy Functions in Quantum Systems

In variational quantum algorithms (VQAs), the most common objective is to find the minimum energy eigenstate of a given energy Hamiltonian. In this paper, we consider the general problem of finding a sufficient control Hamiltonian structure that, under a given feedback control law, ensures convergence to the minimum energy eigenstate of a given energy function. By including quantum non-demolition (QND) measurements in the loop, convergence to a pure state can be ensured from an arbitrary mixed initial state. Based on existing results on strict control Lyapunov functions, we formulate a semidefinite optimization problem, whose solution defines a non-unique control Hamiltonian, which is sufficient to ensure almost sure convergence to the minimum energy eigenstate under the given feedback law and the action of QND measurements. A numerical example is provided to showcase the proposed methodology.

Small mode volume topological photonic states in one-dimensional lattices with dipole-quadrupole interactions

We study the topological photonic states in one-dimensional lattices analog to the Su-Schrieffer-Heeger model beyond the dipole approximation. The electromagnetic resonances of the lattices supported by near-field interactions between the plasmonic nanoparticles are studied analytically with coupled dipole-quadrupole method. The topological phase transition in the bipartite lattices is determined by the change of Zak phase. Our results reveal the contribution of quadrupole moments to the near-field interactions and the band topology. It is found that the topological edge states in nontrivial lattices have both dipolar and quadrupolar nature. The quadrupolar edge states are not only orthogonal to the dipolar edge states, but also spatially localized at different sublattices. Furthermore, the quadrupolar topological edge states, which coexist at the same energy with the quadrupolar flat band have shorter localization length and hence smaller mode volume than the conventional dipolar edge states. The findings deepen our understanding in topological systems that involve higher-order multipoles, or in analogy to the wave functions in quantum systems with higher-orbital angular momentum, and may be useful in designing topological systems for confining light robustly and enhancing light-matter interactions.

Scale invariant distribution functions in quantum systems with few degrees of freedom

Scale invariance usually occurs in extended systems where correlation functions decay algebraically in space and/or time. Here we introduce a new type of scale invariance, occurring in the distribution functions of physical observables. At equilibrium these functions decay over a typical scale set by the temperature, but they can become scale invariant in a sudden quantum quench. We exemplify this effect through the analysis of linear and non-linear quantum oscillators. We find that their distribution functions generically diverge logarithmically close to the stable points of the classical dynamics. Our study opens the possibility to address integrability and its breaking in distribution functions, with immediate applications to matter-wave interferometers.

Thermal out-of-time-order correlators, KMS relations, and spectral functions

We describe general features of thermal correlation functions in quantum systems, with specific focus on the fluctuation-dissipation type relations implied by the KMS condition. These end up relating correlation functions with different time ordering and thus should naturally be viewed in the larger context of out-of-time-ordered (OTO) observables. In particular, eschewing the standard formulation of KMS relations where thermal periodicity is combined with time-reversal to stay within the purview of Schwinger-Keldysh functional integrals, we show that there is a natural way to phrase them directly in terms of OTO correlators. We use these observations to construct a natural causal basis for thermal n-point functions in terms of fully nested commutators. We provide several general results which can be inferred from cyclic orbits of permutations, and exemplify the abstract results using a quantum oscillator as an explicit example.

Quantum Monte Carlo estimation of complex-time correlations for the study of the ground-state dynamic structure function.

We present a method based on the path integral Monte Carlo formalism for the calculation of ground-state time correlation functions in quantum systems. The key point of the method is the consideration of time as a complex variable whose phase δ acts as an adjustable parameter. By using high-order approximations for the quantum propagator, it is possible to obtain Monte Carlo data all the way from purely imaginary time to δ values near the limit of real time. As a consequence, it is possible to infer accurately the spectral functions using simple inversion algorithms. We test this approach in the calculation of the dynamic structure function S(q, ω) of two one-dimensional model systems, harmonic and quartic oscillators, for which S(q, ω) can be exactly calculated. We notice a clear improvement in the calculation of the dynamic response with respect to the common approach based on the inverse Laplace transform of the imaginary-time correlation function.

Preface: Theory of Distribution Functions for Quantum Systems

Preface: Theory of Distribution Functions for Quantum Systems

Path Integrals for Electronic Densities, Reactivity Indices, and Localization Functions in Quantum Systems

The density matrix theory, the ancestor of density functional theory, provides the immediate framework for Path Integral (PI) development, allowing the canonical density be extended for the many-electronic systems through the density functional closure relationship. Yet, the use of path integral formalism for electronic density prescription presents several advantages: assures the inner quantum mechanical description of the system by parameterized paths; averages the quantum fluctuations; behaves as the propagator for time-space evolution of quantum information; resembles Schrödinger equation; allows quantum statistical description of the system through partition function computing. In this framework, four levels of path integral formalism were presented: the Feynman quantum mechanical, the semiclassical, the Feynman-Kleinert effective classical, and the Fokker-Planck non-equilibrium ones. In each case the density matrix or/and the canonical density were rigorously defined and presented. The practical specializations for quantum free and harmonic motions, for statistical high and low temperature limits, the smearing justification for the Bohr’s quantum stability postulate with the paradigmatic Hydrogen atomic excursion, along the quantum chemical calculation of semiclassical electronegativity and hardness, of chemical action and Mulliken electronegativity, as well as by the Markovian generalizations of Becke-Edgecombe electronic focalization functions – all advocate for the reliability of assuming PI formalism of quantum mechanics as a versatile one, suited for analytically and/or computationally modeling of a variety of fundamental physical and chemical reactivity concepts characterizing the (density driving) many-electronic systems.

Knitting algorithm for calculating Green functions in quantum systems

We propose a fast and versatile algorithm to calculate local and transport properties such as conductance, shot noise, local density of state, and local currents in mesoscopic quantum systems. Within the nonequilibrium Green function formalism, we generalize the recursive Green function technique to tackle multiterminal devices with arbitrary geometries. We apply our method to analyze two recent experiments: an electronic Mach--Zehnder interferometer in a two-dimensional gas and a Hall bar made of graphene nanoribbons in a quantum Hall regime. In the latter case, we find that the Landau edge state pinned to the Dirac point gets diluted upon increasing carrier density.

DENSITY-MATRIX RENORMALISATION GROUP FOR DYNAMIC CORRELATION FUNCTIONS

The calculation of dynamic correlation functions in quantum systems is formulated as a variational problem. For low-dimensional many-body systems this variational problem can be solved numerically using the density-matrix renormalisation group (DMRG). This dynamic DMRG method is demonstrated on the linear optical conductivity in the Mott insulating phase of the one-dimensional extended Hubbard model at half filling. The full optical spectrum of this model can be calculated almost exactly for chains with more than 100 sites, which is large enough to investigate the spectral properties in the thermodynamic limit. The accuracy of the method is illustrated by comparison with analytical results in the field-theoretical regime and in the strong-coupling limit.

On the Bayesian approach to calculating time correlation functions in quantum systems; reaction dynamics and spectroscopy

We assess the current status, advantages and limitations of the numerical analytic continuation approach to computing time correlation functions in large many-body quantum systems characteristic of condensed phase chemical processes. We determine the quantum correlation function as a function of complex time, and use its analytic properties to select a suitable contour in the complex time plane along which the function can be evaluated efficiently by stochastic simulation methods. The simulation data are then used to obtain the values of the correlation function along the real-time axis through a maximum entropy numerical analytic continuation procedure. This approach is used to compute the dynamical properties of several condensed phase processes including vibrational relaxation lineshapes and canonical reaction rates. We discuss how to improve the accuracy of the numerical analytic continuation methods.

Methods for simulating time correlation functions in quantum systems

A review of the basic strategies that we have developed for the simulation of dynamic correlation functions in quantum systems is provided. Three methods are considered: (a) the analytic continuation of imaginary (thermal) time correlation functions to real times; (b) direct evaluation of thermally symmetrized version of the time correlation functions; (c) direct simulation of the power spectrum. All of these methods are based on the path integral representation of time corellation functions. The generalization of these methods for the computation of electronic spectrum, which involves the consideration of two potential energy surfaces, is also presented. We also discuss the approach based on numerical integration of an effective time-dependent Schrödinger equation for the simulation of electronic absorption and emission spectra for mixed quantum classical systems.

Time correlation functions in quantum systems

First Page

Dynamic correlation functions in quantum systems: A Monte Carlo algorithm

We propose a new algorithm to compute dynamic correlation functions in quantum systems at zero temperature using Monte Carlo techniques. The method, which avoids inverse-Laplace transforms, is based on a representation of the Dirac $\ensuremath{\delta}$ function as the limit of a Gaussian. We illustrate the method on the one-dimensional harmonic oscillator.

Quasiprobabilities and correlation functions in quantum systems

A general survey is given of the different methods that exist for the description of time correlations in quantum systems. A complete equivalence between all these methods is established on the basis of a consequent rule for the transcription of operator products into phase space. Particular attention is given to possible ambiguities in the definition of quantum mechanical Markov processes. The classical limit is discussed in detail.