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Exponential unitary integrators for nonseparable quantum Hamiltonians

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Abstract

Quantum Hamiltonians containing nonseparable products of non-commuting operators, such as \(\hat{{\varvec{x}}}^m \hat{{\varvec{p}}}^n\), are problematic for numerical studies using split-operator techniques since such products cannot be represented as a sum of separable terms, such as \(T(\hat{{\varvec{p}}}) + V(\hat{{\varvec{x}}})\). 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 \(\hat{{\varvec{x}}}^2 \hat{{\varvec{p}}}^2+\hat{{\varvec{p}}}^2 \hat{{\varvec{x}}}^2\). The general applicability of Chin’s approach to any Hamiltonian of polynomial form is proven.

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The codes developed for the current study are available at [13].

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Acknowledgements

We thank both reviewers for their many thoughtful suggestions. D.I.B. was supported by by the W. M. Keck Foundation and Army Research Office (ARO) (Grant W911NF-19-1-0377; program manager Dr. James Joseph). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of ARO or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes thank both reviewers for their many thoughtful suggestions notwithstanding any copyright notation herein.

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Authors and Affiliations

  1. Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, AL10 9AB, UK

    Maximilian Ćirić & Ole Steuernagel

  2. Tulane University, New Orleans, LA 70118, USA

    Denys I. Bondar

Authors
  1. Maximilian Ćirić
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  2. Denys I. Bondar
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  3. Ole Steuernagel
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Correspondence to Denys I. Bondar.

Additional information

Contribution to the Focus Point on “Mathematics and Physics at the Quantum-Classical Interface” edited by D.I. Bondar, I. Joseph, G. Marmo, C. Tronci.

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Ćirić, M., Bondar, D.I. & Steuernagel, O. Exponential unitary integrators for nonseparable quantum Hamiltonians. Eur. Phys. J. Plus 138, 238 (2023). https://doi.org/10.1140/epjp/s13360-023-03819-3

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