Geometric Integrators for Schrödinger Equations

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control.Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716TESISPremios Extraordinarios de tesis doctorale

Sixth-order schemes for laser-matter interaction in the Schrödinger equation

Control of quantum systems via lasers has numerous applications that require fast and accurate numerical solution of the Schr\"odinger equation. In this paper we present three strategies for extending any sixth-order scheme for Schr\"odinger equation with time-independent potential to a sixth-order method for Schr\"odinger equation with laser potential. As demonstrated via numerical examples, these schemes prove effective in the atomic regime as well as the semiclassical regime, and are a particularly appealing alternative to time-ordered exponential splittings when the laser potential is highly oscillatory or known only at specific points in time (on an equispaced grid, for instance). These schemes are derived by exploiting the linear in space form of the time dependent potential under the dipole approximation (whereby commutators in the Magnus expansion reduce to a simpler form), separating the time step of numerical propagation from the issue of adequate time-resolution of the laser field by keeping integrals intact in the Magnus expansion, and eliminating terms with unfavourable structure via carefully designed splittings.Comment: 33 pages, 7 figure

Efficient Magnus-type integrators for solar energy conversion in Hubbard models

Strongly interacting electrons in solids are generically described by Hubbardtype models, and the impact of solar light can be modeled by an additional time-dependence. This yields a finite dimensional system of ordinary differential equations (ODE)s of Schr\"odinger type, which can be solved numerically by exponential time integrators of Magnus type. The efficiency may be enhanced by combining these with operator splittings. We will discuss several different approaches of employing exponential-based methods in conjunction with an adaptive Lanczos method for the evaluation of matrix exponentials and compare their accuracy and efficiency. For each integrator, we use defect-based local error estimators to enable adaptive time-stepping. This serves to reliably control the approximation error and reduce the computational effor

Magnus-based geometric integrators for dynamical systems with time-dependent potentials

[ES] Esta tesis trata sobre la integración numérica de sistemas hamiltonianos con potenciales explícitamente dependientes del tiempo. Los problemas de este tipo son comunes en la física matemática, porque provienen de la mecánica cuántica, clásica y celestial. La meta de la tesis es construir integradores para unos problemas relevantes no autónomos: la ecuación de Schrödinger, que es el fundamento de la mecánica cuántica; las ecuaciones de Hill y de onda, que describen sistemas oscilatorios; el problema de Kepler con la masa variante en el tiempo. El Capítulo 1 describe la motivación y los objetivos de la obra en el contexto histórico de la integración numérica. En el Capítulo 2 se introducen los conceptos esenciales y unas herramientas fundamentales utilizadas a lo largo de la tesis. El diseño de los integradores propuestos se basa en los métodos de composición y escisión y en el desarrollo de Magnus. En el Capítulo 3 se describe el primero. Su idea principal consta de una recombinación de unos integradores sencillos para obtener la solución del problema. El concepto importante de las condiciones de orden se describe en ese capítulo. En el Capítulo 4 se hace un resumen de las álgebras de Lie y del desarrollo de Magnus que son las herramientas algebraicas que permiten expresar la solución de ecuaciones diferenciales dependientes del tiempo. La ecuación lineal de Schrödinger con potencial dependiente del tiempo está examinada en el Capítulo 5. Dado su estructura particular, nuevos métodos casi sin conmutadores, basados en el desarrollo de Magnus, son construidos. Su eficiencia es demostrada en unos experimentos numéricos con el modelo de Walker-Preston de una molécula dentro de un campo electromagnético. En el Capítulo 6, se diseñan los métodos de Magnus-escisión para las ecuaciones de onda y de Hill. Su eficiencia está demostrada en los experimentos numéricos con varios sistemas oscilatorios: con la ecuación de Mathieu, la ec. de Hill matricial, las ecuaciones de onda y de Klein-Gordon-Fock. El Capítulo 7 explica cómo el enfoque algebraico y el desarrollo de Magnus pueden generalizarse a los problemas no lineales. El ejemplo utilizado es el problema de Kepler con masa decreciente. El Capítulo 8 concluye la tesis, reseña los resultados y traza las posibles direcciones de la investigación futura.[CA] Aquesta tesi tracta de la integració numèrica de sistemes hamiltonians amb potencials explícitament dependents del temps. Els problemes d'aquest tipus són comuns en la física matemàtica, perquè provenen de la mecànica quàntica, clàssica i celest. L'objectiu de la tesi és construir integradors per a uns problemes rellevants no autònoms: l'equació de Schrödinger, que és el fonament de la mecànica quàntica; les equacions de Hill i d'ona, que descriuen sistemes oscil·latoris; el problema de Kepler amb la massa variant en el temps. El Capítol 1 descriu la motivació i els objectius de l'obra en el context històric de la integració numèrica. En Capítol 2 s'introdueixen els conceptes essencials i unes ferramentes fonamentals utilitzades al llarg de la tesi. El disseny dels integradors proposats es basa en els mètodes de composició i escissió i en el desenvolupament de Magnus. En el Capítol 3, es descriu el primer. La seua idea principal consta d'una recombinació d'uns integradors senzills per a obtenir la solució del problema. El concepte important de les condicions d'orde es descriu en eixe capítol. El Capítol 4 fa un resum de les àlgebres de Lie i del desenvolupament de Magnus que són les ferramentes algebraiques que permeten expressar la solució d'equacions diferencials dependents del temps. L'equació lineal de Schrödinger amb potencial dependent del temps està examinada en el Capítol 5. Donat la seua estructura particular, nous mètodes quasi sense commutadors, basats en el desenvolupament de Magnus, són construïts. La seua eficiència és demostrada en uns experiments numèrics amb el model de Walker-Preston d'una molècula dins d'un camp electromagnètic. En el Capítol 6 es dissenyen els mètodes de Magnus-escissió per a les equacions d'onda i de Hill. El seu rendiment està demostrat en els experiments numèrics amb diversos sistemes oscil·latoris: amb l'equació de Mathieu, l'ec. de Hill matricial, les equacions d'onda i de Klein-Gordon-Fock. El Capítol 7 explica com l'enfocament algebraic i el desenvolupament de Magnus poden generalitzar-se als problemes no lineals. L'exemple utilitzat és el problema de Kepler amb massa decreixent. El Capítol 8 conclou la tesi, ressenya els resultats i traça les possibles direccions de la investigació futura.[EN] The present thesis addresses the numerical integration of Hamiltonian systems with explicitly time-dependent potentials. These problems are common in mathematical physics because they come from quantum, classical and celestial mechanics. The goal of the thesis is to construct integrators for several import ant non-autonomous problems: the Schrödinger equation, which is the cornerstone of quantum mechanics; the Hill and the wave equations, that describe oscillating systems; the Kepler problem with time-variant mass. Chapter 1 describes the motivation and the aims of the work in the historical context of numerical integration. In Chapter 2 essential concepts and some fundamental tools used throughout the thesis are introduced. The design of the proposed integrators is based on the composition and splitting methods and the Magnus expansion. In Chapter 3, the former is described. Their main idea is to recombine some simpler integrators to obtain the solution. The salient concept of order conditions is described in that chapter. Chapter 4 summarises Lie algebras and the Magnus expansion ¿ algebraic tools that help to express the solution of time-dependent differential equations. The linear Schrödinger equation with time-dependent potential is considered in Chapter 5. Given its particular structure, new, Magnus-based quasi-commutator-free integrators are build. Their efficiency is shown in numerical experiments with the Walker-Preston model of a molecule in an electromagnetic field. In Chapter 6, Magnus-splitting methods for the wave and the Hill equations are designed. Their performance is demonstrated in numerical experiments with various oscillatory systems: the Mathieu equation, the matrix Hill eq., the wave and the Klein-Gordon-Fock eq. Chapter 7 shows how the algebraic approach and the Magnus expansion can be generalised to non-linear problems. The example used is the Kepler problem with decreasing mass. The thesis is concluded by Chapter 8, in which the results are reviewed and possible directions of future work are outlined.Kopylov, N. (2019). Magnus-based geometric integrators for dynamical systems with time-dependent potentials [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/118798TESI

Exponential propagators for the Schrödinger equation with a time-dependent potential

[EN] We consider the numerical integration of the Schrodinger equation with a time-dependent Hamiltonian given as the sum of the kinetic energy and a time-dependent potential. Commutator-free (CF) propagators are exponential propagators that have shown to be highly efficient for general time-dependent Hamiltonians. We propose new CF propagators that are tailored for Hamiltonians of the said structure, showing a considerably improved performance. We obtain new fourth-and sixth-order CF propagators as well as a novel sixth-order propagator that incorporates a double commutator that only depends on coordinates, so this term can be considered as cost-free. The algorithms require the computation of the action of exponentials on a vector similar to the well-known exponential midpoint propagator, and this is carried out using the Lanczos method. We illustrate the performance of the new methods on several numerical examples. Published by AIP Publishing.We wish to acknowledge Fernando Casas for his help in the construction of the methods Upsilon3[6]. The authors acknowledge Ministerio de Economia y Competitividad (Spain) for financial support through Project No. MTM2016-77660-P (AEI/FEDER, UE). Additionally, Kopylov has been partly supported by Grant No. GRISOLIA/2015/A/137 from the Generalitat Valenciana.Bader, P.; Kopylov, N.; Blanes Zamora, S. (2018). Exponential propagators for the Schrödinger equation with a time-dependent potential. The Journal of Chemical Physics. 149(24):1-7. https://doi.org/10.1063/1.5036838S1714924Bader, P., Iserles, A., Kropielnicka, K., & Singh, P. (2014). Effective Approximation for the Semiclassical Schrödinger Equation. Foundations of Computational Mathematics, 14(4), 689-720. doi:10.1007/s10208-013-9182-8Bader, P., Iserles, A., Kropielnicka, K., & Singh, P. (2016). Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential. 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Classical Hamiltonian structures in wave packet dynamics. The Journal of Chemical Physics, 100(7), 5011-5022. doi:10.1063/1.467219McLachlan, R. I., & Quispel, G. R. W. (2002). Splitting methods. Acta Numerica, 11, 341-434. doi:10.1017/s0962492902000053Neuhauser, C., & Thalhammer, M. (2009). On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential. BIT Numerical Mathematics, 49(1), 199-215. doi:10.1007/s10543-009-0215-2Thalhammer, M. (2008). High-Order Exponential Operator Splitting Methods for Time-Dependent Schrödinger Equations. SIAM Journal on Numerical Analysis, 46(4), 2022-2038. doi:10.1137/060674636Thalhammer, M. (2012). Convergence Analysis of High-Order Time-Splitting Pseudospectral Methods for Nonlinear Schrödinger Equations. SIAM Journal on Numerical Analysis, 50(6), 3231-3258. doi:10.1137/120866373Gray, S. K., & Manolopoulos, D. E. (1996). 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The Magnus expansion and some of its applications. Physics Reports, 470(5-6), 151-238. doi:10.1016/j.physrep.2008.11.001Blanes, S., & Moan, P. C. (2006). Fourth- and sixth-order commutator-free Magnus integrators for linear and non-linear dynamical systems. Applied Numerical Mathematics, 56(12), 1519-1537. doi:10.1016/j.apnum.2005.11.004Thalhammer, M. (2006). A fourth-order commutator-free exponential integrator for nonautonomous differential equations. SIAM Journal on Numerical Analysis, 44(2), 851-864. doi:10.1137/05063042Alvermann, A., & Fehske, H. (2011). High-order commutator-free exponential time-propagation of driven quantum systems. Journal of Computational Physics, 230(15), 5930-5956. doi:10.1016/j.jcp.2011.04.006Auer, N., Einkemmer, L., Kandolf, P., & Ostermann, A. (2018). Magnus integrators on multicore CPUs and GPUs. Computer Physics Communications, 228, 115-122. doi:10.1016/j.cpc.2018.02.019Munthe–Kaas, H., & Owren, B. (1999). Computations in a free Lie algebra. 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In-medium similarity renormalization group with three-body operators

Over the last decade the in-medium similarity renormalization group (IMSRG) approach has proven to be one of the most powerful and versatile ab initio many-body methods for studying medium-mass nuclei. So far, the IMSRG was limited to the approximation in which only up to two-body operators are incorporated in the renormalization group flow, referred to as the IMSRG(2). In this work, we extend the IMSRG(2) approach to fully include three-body operators yielding the IMSRG(3) approximation. We use a perturbative scaling analysis to estimate the importance of individual terms in this approximation and introduce truncations that aim to approximate the IMSRG(3) at a lower computational cost. The IMSRG(3) is systematically benchmarked for different nuclear Hamiltonians for ${}^{4}\text{He}$ and ${}^{16}\text{O}$ in small model spaces. The IMSRG(3) systematically improves over the IMSRG(2) relative to exact results. Approximate IMSRG(3) truncations constructed based on computational cost are able to reproduce much of the systematic improvement offered by the full IMSRG(3). We also find that the approximate IMSRG(3) truncations behave consistently with expectations from our perturbative analysis, indicating that this strategy may also be used to systematically approximate the IMSRG(3).Comment: 22 pages, 10 figures, updated versio

Particle and energy transport in strongly driven one-dimensional quantum systems

This Dissertation concerns the transport properties of a strongly–correlated one–dimensional system of spinless fermions, driven by an external electric field which induces the flow of charges and energy through the system. Since the system does not exchange information with the environment, the evolution can be accurately followed to arbitrarily long times by solving numerically the time–dependent Schrödinger equation, going beyond Kubo’s linear response theory. The thermoelectric response of the system is here characterized, using the ratio of the induced energy and particle currents, in the nonequilibrium state under the steady applied electric field. Even though the equilibrium response can be reached for vanishingly small driving, strong fields produce quantum–mechanical Bloch oscillations in the currents, which disrupt the proportionality of the currents. The effects of the driving on the local state of the ring are analyzed via the reduced density matrix of small subsystems. The local entropy density can be defined and shown to be consistent with the laws of thermodynamics for quasistationary evolution. Even integrable systems are shown to thermalize under driving, with heat being produced via the Joule effect by the flow of currents. The spectrum of the reduced density matrix is shown to be distributed according the Gaussian unitary ensemble predicted by random–matrix theory, both during driving and a subsequent relaxation. The first fully–quantum model of a thermoelectric couple is realized by connecting two correlated quantum wires. The field is shown to produce heating and cooling at the junctions according to the Peltier effect, by mapping the changes in the local entropy density. In the quasiequilibrium regime, a local temperature can be defined, at the same time verifying that the subsystems are in a Gibbs thermal state. The gradient of temperatures, established by the external field, is shown to counterbalance the flow of energy in the system, terminating the operation of the thermocouple. Strong applied fields lead to new nonequilibrium phenomena. At the junctions, observable Bloch oscillations of the density of charge and energy develop at the junctions. Moreover, in a thermocouple built out of Mott insulators, a sufficiently strong field leads to a dynamical transition reversing the sign of the charge carriers and the Peltier effect

Floquet engineering in periodically driven closed quantum systems: from dynamical localisation to ultracold topological matter

This dissertation presents a self-contained study of periodically-driven quantum systems. Following a brief introduction to Floquet theory, we introduce the inverse-frequency expansion, variants of which include the Floquet-Magnus, van Vleck, and Brillouin-Wigner expansions. We reveal that the convergence properties of these expansions depend strongly on the rotating frame chosen, and relate the former to the existence of Floquet resonances in the quasienergy spectrum. The theoretical design and experimental realisation (`engineering') of novel Floquet Hamiltonians is discussed introducing three universal high-frequency limits for systems comprising single-particle and many-body linear and nonlinear models. The celebrated Schrieffer-Wolff transformation for strongly-correlated quantum systems is generalised to periodically-driven systems, and a systematic approach to calculate higher-order corrections to the Rotating Wave Approximation is presented. Next, we develop Floquet adiabatic perturbation theory from first principles, and discuss extensively the adiabatic state preparation and the corresponding leading-order non-adiabatic corrections. Special emphasis is thereby put on geometrical and topological objects, such as the Floquet Berry curvature and the Floquet Chern number obtained within linear response in the presence of the drive. Last, pre-thermalisation and thermalisation in closed, clean periodically-driven quantum systems are studied in detail, with the focus put on the crucial role of Floquet many-body resonances for energy absorption

Ab initio calculations of nuclei using chiral interactions with realistic saturation properties

Ab initio calculations of nuclei from the valley of stability to the drip lines are a prime challenge in low-energy nuclear theory. The interactions in atomic nuclei, being composed of protons and neutrons, are governed by strong interactions. The fundamental theory of strong interactions is quantum chromodynamics (QCD). Due to the non-perturbative nature of QCD at low energies a direct calculation of nuclear forces from the underlying theory is presently not possible. However, chiral effective field theory (EFT) connects the symmetries of QCD to nuclear forces, enabling a systematic derivation of nuclear interactions, naturally including many-nucleon forces and uncertainty estimates. Chiral EFT interactions are generally softer than phenomenological interactions, but their low- and high-momentum components can still be coupled strongly. Using renormalization group (RG) methods, e.g., the similarity renormalization group, this coupling can be removed by a unitary transformation, resulting in even softer interactions. In addition to advances on nuclear forces and RG methods, several ab initio approaches have been developed in recent years to calculate medium-mass nuclei in a systematically improvable way. We employ some of these advanced many-body approaches in our calculation of nuclei, starting from a set of chiral two- and three-nucleon interactions that, when used in perturbative calculations of symmetric nuclear matter, reproduce empirical saturation properties within theoretical uncertainties. We study ground- and excited-state energies of doubly open-shell nuclei from oxygen to calcium using valence-space interactions derived using many-body perturbation theory. Given the prominent role of the calcium isotopic chain, we perform coupled-cluster calculations to investigate stable and short-lived neutron-rich calcium isotopes. The ab initio calculations reveal that the size of the neutron skin of $^{48}$Ca is much smaller than results from density functional theory. In addition, the very steep increase in charge radii up to $^{52}$Ca measured recently questions the neutron shell closure at $N=32$ and provides an intriguing benchmark for our coupled-cluster calculations. We extend our study to ground states of closed-shell nuclei from $^4$He to $^{78}$Ni using the in-medium similarity renormalization group (IM-SRG). The experimental binding-energy and charge-radius systematics is well described, encouraging the decoupling of valence-space interactions with the IM-SRG to study also open-shell nuclei. The results for ground- and excited-state energies as well as for charge radii of open-shell nuclei achieve a similar level of agreement found in the closed-shell calculations, enabling broad predictions for future experiments up to mass number $\sim80$