Method for Finding the Exact Effective Hamiltonian of Time‐Driven Quantum Systems

• Published in: Annalen der Physik 2019-08, Vol.531 (8), p.n/a
• Main Authors: Sandoval‐Santana, Juan Carlos, Ibarra‐Sierra, Victor Guadalupe, Cardoso, José Luis, Kunold, Alejandro, Roman‐Taboada, Pedro, Naumis, Gerardo
• Format: Article
• Language: English
• Subjects: Algorithms; Cold atoms; effective Hamiltonian; Evolution; evolution operator; Harmonic oscillators; Lie algebra; Lie groups; modulated optical lattice; Optical lattices; Paul trap; System effectiveness; Time dependence
• ISSN: 0003-3804; 1521-3889
• DOI: 10.1002/andp.201900035

Time‐driven quantum systems are important in many different fields of physics as cold atoms, solid state, optics, etc. Many of their properties are encoded in the time evolution operator or the effective Hamiltonian. Finding these operators usually requires very complex calculations that often involve some approximations. To perform this task, a systematic scheme that can be cast in the form of a symbolic computational algorithm is presented. It is suitable for periodic and non‐periodic potentials and, for convoluted systems, can also be adapted to yield numerical solutions. The method exploits the structure of the associated Lie group and a decomposition of the evolution operator on each group generator. To illustrate the use of the method, five examples are provided: harmonic oscillator with time‐dependent frequency (Paul trap), modulated optical lattice, time‐driven quantum oscillator, a step‐wise driving of a free particle, and the non‐periodic Caldirola‐Kanai Hamiltonian. To the extent of the authors' knowledge, whereas the exact form of Paul trap's evolution operator is well known, its effective Hamiltonian was until now unknown. The remaining four examples accurately reproduce previous results. An algebraic method is developed to find the effective Hamiltonian and the evolution operator for time‐dependent quantum systems. This method exploits the structure of the associated Lie group and a decomposition of the evolution operator on each group generator. To illustrate the use of the method, five examples and four symbolic computational algorithms are provided.