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Related Experiment Video

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Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators

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Published on: May 30, 2014

Nonperturbative Quantum Physics from Low-Order Perturbation Theory.

Héctor Mera¹, Thomas G Pedersen², Branislav K Nikolić¹

¹Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716-2570, USA.

Physical Review Letters

|November 10, 2015

Summary

This study shows how to accurately approximate complex quantum system energies using analytic continuation. This method, utilizing divergent perturbation series, offers superior precision over traditional techniques for nonperturbative quantum phenomena.

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Area of Science:

Quantum Mechanics
Atomic Physics
Theoretical Chemistry

Background:

Perturbation theory in quantum mechanics often leads to divergent series for complex systems.
The Stark effect in hydrogen and anharmonic oscillators exhibit nonperturbative behavior due to tunneling ionization.
Approximating complex eigenvalues (real and imaginary parts) is crucial for understanding these systems.

Purpose of the Study:

To demonstrate a novel method for approximating complex eigenvalues of quantum systems.
To show that low-order coefficients of divergent perturbation series can yield accurate results.
To introduce analytic continuation functions tailored with Bender-Wu dispersion relations for improved accuracy.

Main Methods:

Utilizing low-order coefficients from divergent perturbation series.
Employing analytic continuation functions, specifically Gauss hypergeometric functions.
Tailoring the singularity structure of these functions using Bender-Wu dispersion relations.
Comparing the performance against Padé and Borel-Padé approximants.

Main Results:

Achieved excellent approximations for both real and imaginary parts of perturbed ground-state eigenenergies.
Gauss hypergeometric functions, informed by fourth-order perturbation theory, provided highly accurate complex eigenvalues.
The proposed method significantly outperformed standard Padé and Borel-Padé approaches, even for large coupling constants.

Conclusions:

Analytic continuation functions with tailored singularity structures offer a powerful approach to handle divergent perturbation series in quantum mechanics.
This method provides a robust and accurate way to calculate complex eigenvalues in nonperturbative quantum systems.
The Gauss hypergeometric functions present a Borel-consistent and superior alternative to existing approximation techniques.