Exponential unitary integrators for nonseparable quantum Hamiltonians

TL;DR

This paper extends a classical method to quantum systems for numerically evolving Hamiltonians with nonseparable operator products, enabling more accurate simulations of complex quantum dynamics.

Contribution

We generalize Chin's classical approach to quantum Hamiltonians, allowing for the numerical evolution of nonseparable polynomial operator products.

Findings

Successfully evolved a Kerr-type oscillator with nonseparable Hamiltonian.

Proved the general applicability of Chin's approach to polynomial Hamiltonians.

Demonstrated improved numerical techniques for nonseparable quantum operators.

Abstract

Quantum Hamiltonians containing nonseparable products of non-commuting operators, such as $\hat{\bf x}^m \hat{\bf p}^n$, are problematic for numerical studies using split-operator techniques since such products cannot be represented as a sum of separable terms, such as $T(\hat{\bf p}) + V(\hat{\bf x})$. In the case of classical physics, Chin [Phys. Rev. E $\bf 80$, 037701 (2009)] developed a procedure to approximately represent nonseparable terms in terms of separable ones. We extend Chin's idea to quantum systems. We demonstrate our findings by numerically evolving the Wigner distribution of a Kerr-type oscillator whose Hamiltonian contains the nonseparable term $\hat{\bf x}^2 \hat{\bf p}^2 + \hat{\bf p}^2 \hat{\bf x}^2$. The general applicability of Chin's approach to any Hamiltonian of polynomial form is proven.