Exponential unitary integrators for nonseparable quantum Hamiltonians

Abstract

Quantum Hamiltonians containing nonseparable products of non-commuting operators, such as ${\hat{\mathbf{x}}}^m{\hat{\mathbf{p}}}^n$, are problematic for numerical studies using split-operator techniques since such products cannot be represented as a sum of separable terms, such as $T(\hat{\mathbf{p}})+V(\hat{\mathbf{x}})$. In the case of classical physics, Chin [Phys. Rev. E $80$, 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 ${\hat{\mathbf{x}}}^2{\hat{\mathbf{p}}}^2+{\hat{\mathbf{p}}}^2{\hat{\mathbf{x}}}^2$. The general applicability of Chin's approach to any Hamiltonian of polynomial form is proven.

Fixed typos and extended discussion (6 pages, 1 figure)

References in corpus (5)

• A Straightforward Introduction to Continuous Quantum Measurement
• Explicit symplectic approximation of nonseparable Hamiltonians: algorithm and long time performance
• Anharmonic quantum mechanical systems do not feature phase space trajectories
• Quantum Kerr oscillators' evolution in phase space: Wigner current, symmetries, shear suppression and special states
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Cited by in corpus (2)

• The wave operator representation of quantum and classical dynamics
• Wigner's Phase Space Current for Variable Beam Splitters -Seeing Beam Splitters in a New Light-