Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential

Bader, Philipp; Iserles, Arieh; Kropielnicka, Karolina; Singh, Pranav

doi:10.1098/rspa.2015.0733

Cited by 26 publications

(44 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many works in quantum mechanics are devoted to finding approximate solutions of Schrödinger's equation. Among them, several methods use Zassenhaus expansion formula [13,15]. As already noticed in the previous sections, this paper is more focused on the wide range of possible applications of the proposed method rather than its accuracy.…”

Section: Applications To Quantum Mechanics: Shallow Potential Well Anmentioning

confidence: 98%

See 1 more Smart Citation

Analytical evaluation of the numerical coefficients in the Zassenhaus product formula and its applications to quantum and statistical mechanics

Bologna

2019

Progress of Theoretical and Experimental Physics

View full text Add to dashboard Cite

This paper studies the exponential of the sum of two non-commuting operators as an infinite product of exponential operators involving repeated commutators of increasing order. It will be shown how to determine two coefficients in front of the nested commutators in the Zassenhaus formula. The knowledge of one coefficient is enough to generate a closed formula that has several applications in solving problems ranging from linear differential equations, quantum mechanics to non-linear differential equations.

show abstract

Section: Applications To Quantum Mechanics: Shallow Potential Well Anmentioning

confidence: 98%

“…In particular in Refs. [13,15] the investigation is performed using the Zassenhaus formula. In the present paper, we will find an approximated formula obtained from the Zassenhaus expansion based on the knowledge of the coefficients in front of the nested commutators.…”

Section: Introductionmentioning

confidence: 99%

Analytical evaluation of the numerical coefficients in the Zassenhaus product formula and its applications to quantum and statistical mechanics

Bologna

2019

Progress of Theoretical and Experimental Physics

View full text Add to dashboard Cite

show abstract

“…Remark 5 Along the lines of (Bader et al 2016, Iserles et al 2018, we also assume that spatial derivatives of the potential, as well as its integrals, are either available or inexpensive to compute.…”

Section: )mentioning

confidence: 99%

Solving Schrödinger equation in semiclassical regime with highly oscillatory time-dependent potentials

Iserles

Kropielnicka

Singh

2019

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

Schrödinger equations with time-dependent potentials are of central importance in quantum physics and theoretical chemistry, where they aid in the simulation and design of systems and processes at atomic and molecular scales. Numerical approximation of these equations is particularly difficult in the semiclassical regime because of the highly oscillatory nature of solution. Highly oscillatory potentials such as lasers compound these difficulties even further. Altogether, these effects render a large number of standard numerical methods less effective in this setting. In this paper we will develop a class of high-order exponential splitting schemes that are able to overcome these challenges by combining the advantages of integral-preserving simplified-commutator Magnus expansions with those of symmetric Zassenhaus splittings. This allows us to use large time steps in our schemes even in the presence of highly oscillatory potentials and solutions.

show abstract

“…Reference solution. In the first example the reference solution is obtained by using a sixth-order Magnus-Lanczos method [5], while in the second example the reference solution is obtained using a sixth-order commutator-free Lanczos based method [2]. In both cases we used 5000 spatial grid points and 10 6 time steps.…”

Section: Numerical Examplesmentioning

confidence: 99%

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

Singh

2019

The Journal of Chemical Physics

Self Cite

View full text Add to dashboard Cite

Control of quantum systems via lasers has numerous applications that require fast and accurate numerical solution of the Schrödinger equation. In this paper we present three strategies for extending any sixth-order scheme for Schrödinger equation with time-independent potential to a sixth-order method for Schrödinger 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.

show abstract

Efficient methods for linear Schrödinger equation in the semiclassical regime with time-dependent potential

Cited by 26 publications

References 21 publications

Analytical evaluation of the numerical coefficients in the Zassenhaus product formula and its applications to quantum and statistical mechanics

Analytical evaluation of the numerical coefficients in the Zassenhaus product formula and its applications to quantum and statistical mechanics

Solving Schrödinger equation in semiclassical regime with highly oscillatory time-dependent potentials

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

Contact Info

Product

Resources

About