Exponential unitary integrators for nonseparable quantum Hamiltonians

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