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arXiv:2209.03504 · Physical Review A

Quantum-based solution of time-dependent complex Riccati equations

D. Martínez-Tibaduiza, Carlos González-Arciniegas, Carlos Farina, Alan Duriez, Antonio Zelaquett Khoury

Published Apr 14, 2023• 3 citations
arXivHTMLPDF

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 $\mathrm{su}(1,1)$, $\mathrm{su}(2)$, and $\mathrm{so}(2,1)$ 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 $\mathrm{su}(2)$. 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.

DOI
10.1103/physreva.107.042211

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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)\mathrm{su}(1,1)su(1,1), su(2)\mathrm{su}(2)su(2), and so(2,1)\mathrm{so}(2,1)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)\mathrm{su}(2)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.
The time-dependent Schrödinger equation for quadratic Hamiltonians has Gaussian wave packets as exact solutions. For the parametric oscillator with frequency ω(t), the width of these wave packets must be time-dependent. This time-dependence can be determined via the solution of a complex nonlinear Riccati equation or an equivalent real nonlinear Ermakov equation. All quantum dynamical properties of the system can easily be constructed from these solutions, e.g., uncertainties of position and momentum, their correlations, tunnelling currents, ground state energies, etc. In addition, the link to the corresponding classical dynamics is supplied via linearization of the Riccati equation to a complex Newtonian equation, actually representing two equations of the same kind: one for the real and one for the imaginary part. If the solution of one part is known, the missing (linear independent) solution of the other can be obtained via a conservation law for the motion in the complex plane. Knowing these two solutions, the solution of the Ermakov equation can be determined immediately plus the explicit expressions for all the quantum dynamical properties mentioned above. Furthermore, these two solutions also provide the time-dependent Green function for the systems propagation for any given initial condition, not only for Gaussians. It is also possible to obtain the corresponding Wigner function and generalized creation and annihilation operators in this way. In searching for problems with analytical solutions, the crucial point is to identify systems that have either a wave packet solution with time-dependent width (in closed form) that obeys a Riccati equation, or a classical parametric oscillator with frequency ω(t) that still enables an analytical solution of the Newtonian equation. Comparing the wave packets with the solution of the diffusion equation – also a Gaussian with time-dependent width – leads to a real Riccati equation. A first attempt at "complexifying" this equation turns out to be problematic. Nevertheless, it provides a clue to time-dependent frequencies inversely proportional to time that lead to Newtonian equations with analytical solutions. Following a detailed analysis of the corresponding quantum dynamics and energetics, possible extensions of the method are indicated.
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The evolution operator U(t) for a time-independent parity-time-symmetric systems is well studied in the literature.However, for the non-Hermitian time-dependent systems, a closed form expression for the evolution operator is not available.In this paper, we make use of a procedure, originally developed by A.R.P. Rau [Phys.Rev.Lett, 81, 4785-4789 (1998)], in the context of deriving the solution of Liuville-Bloch equations in the product form of exponential operators when time-dependent external fields are present, for the evaluation of U(t) in the interaction picture wherein the corresponding Hamiltonian is time-dependent and in general non-Hermitian.This amounts to a transformation of the whole scheme in terms of addressing a nonlinear Riccati equation the existence of whose solutions depends on the fulfillment of a certain accompanying integrability condition.
Time evolution operator for a {h(1)⊕h(1)}⊎u(2)\left\{ h(1) \oplus h(1) \right\} \uplus u(2){h(1)⊕h(1)}⊎u(2) time-dependent quantum Hamiltonian; a self-consistent resolution method based on Feynman's disentangling rules
In this article the time evolution operator of two interacting quantum oscillators, whose Hamiltonian is an element of the complex {h(1)⊕h(1)}⊎u(2)\left\{ h(1) \oplus h(1) \right\} \uplus u(2){h(1)⊕h(1)}⊎u(2) algebra, is analyzed using the Feynman time ordering operator techniques. This method is consistently used to generate the conditions and to formally find explicit disentangled expressions for such operator. In this way, it is shown that all the problem reduces to solve a complex Riccati-type differential equation. Some closed solutions to this differential equation are found and then concrete disentangling expressions for the time-ordered evolution operator are given. Finally, the time evolution of the coherent states linked to the isotropic 2D quantum oscillator are analyzed under alternative time-independent an time-dependent Hamiltonian systems.
The time-evolution of the maximum and the width of exact analytic wave packet (WP) solutions of the time-dependent Schrödinger equation (SE) represents the particle and wave aspects, respectively, of the quantum system.The dynamics of the maximum, located at the mean value of position, is governed by the Newtonian equation of the corresponding classical problem.The width, which is directly proportional to the position uncertainty, obeys a complex nonlinear Riccati equation which can be transformed into a real nonlinear Ermakov equation.The coupled pair of these equations yields a dynamical invariant which plays a key role in our investigation.It can be expressed in terms of a complex variable that linearizes the Riccati equation.This variable also provides the time-dependent parameters that characterize the Green's function, or Feynman kernel, of the corresponding problem.From there, also the relation between the classical and quantum dynamics of the systems can be obtained.Furthermore, the close connection between the Ermakov invariant and the Wigner function will be shown.Factorization of the dynamical invariant allows for comparison with creation/annihilation operators and supersymmetry where the partner potentials fulfil (real) Riccati equations.This provides the link to a nonlinear formulation of timeindependent quantum mechanics in terms of an Ermakov equation for the amplitude of the stationary state wave functions combined with a conservation law.Comparison with SUSY and the time-dependent problems concludes our analysis.