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Quantum-based solution of time-dependent complex Riccati equations

D. Martínez-Tibaduiza1,*, C. González-Arciniegas2, C. Farina3, A. Cavalcanti-Duriez1, and A. Z. Khoury1

  • 1Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea, Gragoata, 24210-346 Niteroi, RJ, Brazil
  • 2Department of Physics, University of Virginia, Charlottesville, Virginia 22904-4714, USA
  • 3Instituto de Física, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, RJ, Brazil

  • *Corresponding author: danielmartinezt@gmail.com
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Phys. Rev. A 107, 042211 – Published 14 April, 2023

DOI: https://doi.org/10.1103/PhysRevA.107.042211

Abstract

Using the Wei-Norman theory, we obtain a time-dependent complex Riccati equation (TDCRE) as the solution of the time evolution operator (TEO) of quantum systems described by time-dependent (TD) Hamiltonians that are linear combinations of the generators of the su(1,1), su(2), and so(2,1) Lie algebras. Using a recently developed solution for the time evolution of these quantum systems, we solve the TDCRE recursively as generalized continued fractions, which are optimal for numerical implementations, and establish the necessary and sufficient conditions for the unitarity of the TEO in the factorized representation. The inherited symmetries of quantum systems can be recognized by a simple inspection of the TDCRE, allowing effective quantum Hamiltonians to be associated with it, as we show for the Bloch-Riccati equation whose Hamiltonian corresponds to that of a generic TD system of the Lie algebra su(2). As an application, but also as a consistency test, we compare our solution with the analytic one for the Bloch-Riccati equation considering the Rabi frequency driven by a complex hyperbolic secant pulse generating spin inversion, showing an excellent agreement.

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Outline

Information

Phys. Rev. A 107, 042211– Published 14 April, 2023
  • Received 16 January 2023
  • Accepted 30 March 2023
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