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Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions

A basis- and integral-free representation of time-dependent perturbation theory via the Omega matrix calculus

Antônio Francisco Neto

Published Jul 12, 2023• 2 citations
Publisher

Abstract

We obtain a basis- and integral-free representation of the n -term in time-dependent perturbation theory of the quantum time-evolution operator. The main technical tool to construct the aforementioned representation comprises the Omega matrix calculus; that is, an extension of MacMahon’s partition analysis to the realm of matrix calculus. In particular, we show that if we specialize our formulation to the time-independent and finite-dimensional case, we recover Putzer (1966) original formulation to compute the matrix exponential avoiding the Jordan canonical form. Furthermore, if we use an explicit basis for the time-independent part of the generator of evolution, then our work implies a representation of the perturbation expansion similar to an approach discussed in the recent work of Kalev and Hen (2021) using divided differences. Finally, we obtain closed form expressions which generalize and unify all the perturbation calculations considered in Kalev and Hen (2021) and advance some considerations regarding degeneracy associated with the unperturbed Hamiltonian in the context of adiabatic switching.

DOI
10.4171/aihpd/173

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We obtain a basis- and integral-free representation of the n -term in time-dependent perturbation theory of the quantum time-evolution operator. The main technical tool to construct the aforementioned representation comprises the Omega matrix calculus; that is, an extension of MacMahon’s partition analysis to the realm of matrix calculus. In particular, we show that if we specialize our formulation to the time-independent and finite-dimensional case, we recover Putzer (1966) original formulation to compute the matrix exponential avoiding the Jordan canonical form. Furthermore, if we use an explicit basis for the time-independent part of the generator of evolution, then our work implies a representation of the perturbation expansion similar to an approach discussed in the recent work of Kalev and Hen (2021) using divided differences. Finally, we obtain closed form expressions which generalize and unify all the perturbation calculations considered in Kalev and Hen (2021) and advance some considerations regarding degeneracy associated with the unperturbed Hamiltonian in the context of adiabatic switching.
The computational complexity of time-dependent perturbation theory is well known to be largely combinatorial whatever the chosen expansion method and family of parameters (combinatorial sequences, Goldstone and other Feynman-type diagrams, etc.). We show that a very efficient perturbative expansion, both for theoretical and numerical purposes, can be obtained through an original parametrization by trees and generalized iterated integrals. We emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics. These tools are applied to the adiabatic approximation and the effective Hamiltonian. We prove perturbatively and nonperturbatively the convergence of Morita’s generalization of the Gell–Mann and Low wave function. We show that summing all the terms associated with the same tree leads to an utter simplification where the sum is simpler than any of its terms. Finally, we recover the Rayleigh-Schrödinger time-independent equation for the wave operator and we give an explicit nonrecursive expression for the term corresponding to an arbitrary tree.
The computational complexity of time-dependent perturbation theory is well-known to be largely combinatorial whatever the chosen expansion method and family of parameters (combinatorial sequences, Goldstone and other Feynman-type diagrams...). We show that a very efficient perturbative expansion, both for theoretical and numerical purposes, can be obtained through an original parametrization by trees and generalized iterated integrals. We emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics. These tools are applied to the adiabatic approximation and the effective Hamiltonian. We prove perturbatively and non-perturbatively the convergence of Morita's generalization of the Gell-Mann and Low wavefunction. We show that summing all the terms associated to the same tree leads to an utter simplification where the sum is simpler than any of its terms. Finally, we recover the time-independent equation for the wave operator and we give an explicit non-recursive expression for the term corresponding to an arbitrary tree.
Starting from our idea of combining the Feynman path integral spirit and the Dyson series kernel, we find an explicit and general form of time evolution operator that is a ccc-number function and a power series of perturbation including all order approximations in the unperturbed Hamiltonian representation. Based on it, we obtain an exact solution of the Schrödinger equation in general quantum systems independent of time. Comparison of our exact solution with the existed perturbation theory makes some features and significance of our exact solution clear. The conclusions expressly indicate that our exact solution is obviously consistent with the usual time-independent perturbation theory at any order approximation, it explicitly calculates out the expanding coefficients of the unperturbed state in the non-perturbation method, and it fully solves the recurrence equation of the expansion coefficients of final state in the unperturbed Hamiltonian representation from a view of time-dependent perturbation theory. At the same time, the exact solution of the von Neumann equation is also given. Our results can be thought of as theoretical developments of quantum dynamics, and are helpful for understanding the dynamical behavior and related subjects of general quantum systems in both theory and application. Our exact solution, together with its sequence studies on perturbation theory [An Min Wang, quant-ph/0611217] and open system dynamics [An Min Wang, quant-ph/0601051] can be used to establish the foundation of theoretical formulism of quantum mechanics in general quantum systems. Further applications of our exact solution to quantum theory can be expected.
The Dyson series is an infinite sum of multi-dimensional time-ordered integrals, which serves as a formal representation of the quantum time-evolution operator in the interaction-picture. Using the mathematical tool of divided differences, we introduce an alternative representation for the series that is entirely free from both time ordering and integrals. In this new formalism, the Dyson expansion is given as a sum of efficiently-computable divided differences of the exponential function, considerably simplifying the calculation of the Dyson expansion terms, while also allowing for time-dependent perturbation calculations to be performed directly in the Schrödinger-picture. We showcase the utility of this novel representation by studying a number of use cases. We also discuss several immediate applications.
We present a time-dependent extension of logarithmic perturbation theory for nonrelativistic quantum dynamics governed by the Schrödinger equation, in which the logarithm of the wave function is expanded in powers of a coupling constant. The resulting hierarchy of equations defining the perturbative corrections is governed by a gauge-rotated Hamiltonian of the unperturbed system and leads to closed-integral expressions for the time-dependent corrections based on Duhamel's formula. This closed-integral structure of corrections is a hallmark of time-independent logarithmic perturbation theory and is preserved in the present extension. This structure, in particular, provides a computable expression for the instantaneous energy shift. Furthermore, dynamic energy shifts arise naturally within this framework in the form of time-averaged expectation values of pseudopotentials and can be related, for example, to AC Stark shifts and electric polarizabilities. As an illustration, we apply the method to the harmonic oscillator and the hydrogen atom, both driven by a time-dependent laser field. The harmonic oscillator provides a proof of principle for which the exact solution is recovered, while the hydrogen atom illustrates the method applied to atomic systems. Supported by numerical simulations, we demonstrate the applicability to obtain relevant physical observables with high accuracy. The present approach offers a promising alternative for analytical studies of time-dependent multi-photon processes in the perturbative regime.