Time Evolution in Quantum Systems and Stochastics

We inquire into the time evolution of quantum systems associated with pseudo-or quasi-Hermitian Hamiltonians. We obtain, in the pseudo-Hermitian case, a generalized Liouville-von Neumann equation for closed systems. We show that quantum systems with quasi-Hermitian Hamiltonians admit the proper interpretation in terms of open quantum system and derive a generalized Lindblad-Kossakowski equation. Finally, we extend such formalism to the study of decaying systems.

Accurate simulation of the time evolution of a quantum system under the influence of an environment is critical to making accurate predictions in chemistry, condensed-matter physics, and materials sciences. Whereas there has been a recent surge in interest in quantum algorithms for the prediction of nonunitary time evolution in quantum systems, few studies offer a direct quantum analog to the Lindblad equation. Here, we present a quantum algorithm—utilizing a decomposition of nonunitary operators approach—that models dynamic processes via the unraveled Lindblad equation. This algorithm is employed to probe both a two-level system in an amplitude damping channel as well as the transverse field Ising model in a variety of parameter regimes; the resulting population dynamics demonstrate excellent agreement with classical simulation, showing the promise of predicting population dynamics utilizing quantum devices for a variety of important systems in molecular energy transport, quantum optics, and other open quantum systems.

Time evolution of quantum systems has been shown to be one of the most difficult components of a typical undergraduate quantum mechanics course. In this work, we examine the current literature, and then take a closer look at the process that students use to determine how the quantum state of a spin-1/2 particle evolves with time. We divide the process of writing a time-dependent state into five elements and use these to both directly probe student understanding and guide our coding of student responses. We focus on three elements of this process, including knowledge of the Hamiltonian, the energy eigenstates and eigenvalues, and what basis should be used when writing the state as a function of time using the phase . Analysis of four exam questions given at three institutions suggests that knowledge of the energy eigenbasis and its importance for time evolution may be a weak point in student understanding.

Dans ce papier, nous montrons un principe de comparaison trajectoriel pour l’equation de la chaleur stochastique, fractionnaire, nonlineaire sur $\mathbb{R}$ avec une donnee initiale a valeur mesure. Nous donnons des estimations quantitatives de la proximite a zero d’une solution. Ces resultats etendent le principe de comparaison de Mueller pour l’equation de la chaleur stochastique et permettent de considerer des donnees initiales plus generales telles que des mesures de Dirac et des mesures a queue plus lourde qu’une croissance exponentielle lineaire en ${\pm}\infty$. Ces resultats generalisent un travail recent par Moreno Flores (Ann. Probab. 42 (2014) 1635–1643), qui a prouve la stricte positivite de l’equation de la chaleur stochastique partant d’un Dirac. Comme application, nous etablissons la complete intermittence pour l’equation. Dans une etape intermediaire, nous prouvons la regularite Holder de solutions partant d’une donnee initiale a valeur mesure ce qui generalise, dans un certain sens, un travail recent de Chen and Dalang (Stoch. Partial Differ. Equ. Anal. Comput. 2 (2014) 316–352).

For each quantum system described by an operator algebra [Formula: see text] of physical quantities, we provide a (generalized) state space, notwithstanding the Kochen-Specker theorem. This quantum state space is the spectral presheaf [Formula: see text]. We formulate the time evolution of quantum systems in terms of Hamiltonian flows on this generalized space and explain how the structure of the spectral presheaf [Formula: see text] geometrically mirrors the double role played by self-adjoint operators in quantum theory, as quantum random variables and as generators of time evolution.

We study the time evolution of quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators. The invariant Hermitian operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. With the help of the invariant Hermitian operator we obtain not only the exact solutions of the Schr\"odinger equation but also the time-evolution operator. The adiabatic and nonadiabatic Berry phases are calculated with the exact solutions. \textcopyright{} 1996 The American Physical Society.

We establish a version of the Feynman–Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the Feynman–Kac formula is a weak solution of the stochastic heat equation. From the Feynman–Kac formula, we establish the smoothness of the density of the solution and the Hölder regularity in the space and time variables. We also derive a Feynman–Kac formula for the stochastic heat equation in the Skorokhod sense and we obtain the Wiener chaos expansion of the solution.

The application of the Langevin equation method to the study of fluctuations in the space- and velocity-dependent neutron density as well as in the detector outputs in nuclear reactors is presented. In this case, the Langevin equation is the stochastic linear neutron transport equation with a space- and velocity-dependent random neutron source, often referred to as the noise equivalent source (NES). The power spectral densities (PSDs) of the NESs in the transport equation, as well as in the accompanying detection rate equations, are obtained, and the cross- and auto-power spectral densities of the outputs of pairs of detectors are explicitly calculated. The transport-level expression for the R([omega]) ratio measured in the [sup 252]Cf source-driven noise analysis method is also derived. Finally, the implementation of the Langevin equation approach at different levels of approximation is discussed, and the stochastic one-speed transport and one-group P[sub 1] equations are derived by first integrating the stochastic transport equation over speed and then eliminating the angular dependence by a spherical harmonics expansion. By taking the large transport rate limit in the P[sub 1] description, the stochastic diffusion equation is obtained as well as the PSD of the NES in it. This procedure also leads directlymore » to the stochastic Fick's law.« less

In this article, we study a class of stochastic partial differential equations with fractional differential operators subject to some time-independent multiplicative Gaussian noise. We derive sharp conditions, under which a unique global $$L^p(\Omega )$$ -solution exists for all $$p\ge 2$$ . In this case, we derive exact moment asymptotics following the same strategy as that in a recent work by Balan et al. (Inst Henri Poincaré Probab Stat. To appear, 2021). In the case when there exists only a local solution, we determine the precise deterministic time, $$T_2$$ , before which a unique $$L^2(\Omega )$$ -solution exists, but after which the series corresponding to the $$L^2(\Omega )$$ moment of the solution blows up. By properly choosing the parameters, results in this paper interpolate the known results for both stochastic heat and wave equations.

In this article, we study the following stochastic heat equation where is the generator of a Lévy process X in B is a fractional-colored Gaussian noise with Hurst index in the time variable and spatial covariance function f which is the Fourier transform of a tempered measure After establishing the existence of solution for the stochastic heat equation, we study the regularity of the solution in both time and space variables. Under mild conditions, we establish the exact uniform modulus of continuity and a Chung-type law of iterated logarithm for the sample function These results, to our knowledge, are new even for the classical stochastic heat equation (where ) with space-time white noise and they strengthen the corresponding results of Balan and Tudor (2008) and Tudor and Xiao (2017) where partial regularity results were obtained.

Digital quantum simulators provide a diversified tool for solving the evolution of quantum systems with complicated Hamiltonians and hold great potential for a wide range of applications. Although much attention is paid to the unitary evolution of closed quantum systems, dissipation and noise are vital in understanding the dynamics of practical quantum systems. In this work, we experimentally demonstrate a digital simulation of an open quantum system in a controllable Markovian environment with the assistance of a single ancillary qubit. By Trotterizing the quantum Liouvillians, the continuous evolution of an open quantum system is effectively realized, and its application in error mitigation is demonstrated by adjusting the simulated noise intensities. High-order Trotter for open quantum dynamics is also experimentally investigated and shows higher accuracy. Our results represent a significant step toward hardware-efficient simulation of open quantum systems and error mitigation in quantum algorithms in noisy intermediate-scale quantum systems.

The Arnol’d-Thorn classification of caustics for the Burgers equation suggests that there should be an analogous one for the wavefronts of the corresponding heat equation. We present a general theorem for Hamiltonian systems char acterizing how the level surfaces of Hamilton’s principal function meet the caustic surface in both the deterministic and stochastic cases. Such a characterization allows one to give a fairly detailed description of the behaviour of the solution of the heat equation in the vicinity of the wavefront and caustic. It allows one to propose some reasons for the “blow-up” of the Burgers velocity field on the caustic. In the case of small noise the shapes of the random wavefront and random caustic may easily be obtained, and to first order the caustic is merely displaced. In the stochastic case we have the possibility of “rapid” changes in the caustic-wavefront intersection. This will engender stochastic turbulence in the Burgers velocity field and, due to its stochasticity, may be of an intermittent nature. There is no analogue of this in the deterministic case. Throughout our studies much use has been made of computer algebra packages in building an understanding of the archetypal cases. Numerical simulations and numerical solutions of the partial differential equations involved have been immensely useful in clarifying conjectures and determining apt characterizations.

The Arnol'd-Thom classification of caustics (shockwaves) for Burgers equation suggests a similar one for the wavefronts of the corresponding Heat equation. We give here a general theorem for Hamiltonian systems characterising how the level surfaces of Hamilton's principal function (wavefronts) meet the caustic surface in both the deterministic and stochastic cases. We further show how these results can be applied to the stochastic Burgers equation. The generic example of a caustic, appearing in the two-dimensional case, is the semicubical parabolic cusp with the corresponding zero level surface being a combination of a generalised hypocycloid and a line pair. Our results explain in terms of classical mechanics the properties of the caustic and wavefront for this archetypal example and characterise the caustic-wavefront intersection for the general stochastic case. We discuss the application of these results to turbulence for the Burgers velocity field.

The general area of stochastic PDEs is interesting to mathematicians because it contains an enormous number of challenging open problems. There is also a great deal of interest in this topic because it has deep applications in disciplines that range from applied mathematics, statistical mechanics, and theoretical physics, to theoretical neuroscience, theory of complex chemical reactions [including polymer science], fluid dynamics, and mathematical finance. The stochastic PDEs that are studied in this book are similar to the familiar PDE for heat in a thin rod, but with the additional restriction that the external forcing density is a two-parameter stochastic process, or what is more commonly the case, the forcing is a ``random noise,'' also known as a ``generalized random field.'' At several points in the lectures, there are examples that highlight the phenomenon that stochastic PDEs are not a subset of PDEs. In fact, the introduction of noise in some partial differential equations can bring about not a small perturbation, but truly fundamental changes to the system that the underlying PDE is attempting to describe. The topics covered include a brief introduction to the stochastic heat equation, structure theory for the linear stochastic heat equation, and an in-depth look at intermittency properties of the solution to semilinear stochastic heat equations. Specific topics include stochastic integrals a la Norbert Wiener, an infinite-dimensional Ito-type stochastic integral, an example of a parabolic Anderson model, and intermittency fronts. There are many possible approaches to stochastic PDEs. The selection of topics and techniques presented here are informed by the guiding example of the stochastic heat equation.

We consider the one-dimensional stochastic heat and wave equations driven by Gaussian noises with constant initial conditions.We study the spatial average of the solutions on an interval of length R and show that the family of laws of the spatial average satisfies the large deviation principle as R goes to infinity.We also present the large deviation principle in the space of continuous functions.We prove these results using the tools of Malliavin calculus to evaluate the covariance of nonlinear functionals of the solution.