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Physical Review A

Systematic method for deriving effective Hamiltonians

S. Swain

Published Apr 1, 1994• 15 citations
Publisher

Abstract

A systematic procedure for deriving effective Hamiltonians to any order is presented, which is applicable to any time-independent Hamiltonian. The method is based on a continued-fraction approach and avoids the singularities which may occur with perturbation theory. It is illustrated by deriving the effective Hamiltonian for the one-photon, dressed-state laser to second order.

DOI
10.1103/physreva.49.2816

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A systematic procedure for deriving effective Hamiltonians to any order is presented, which is applicable to any time-independent Hamiltonian. The method is based on a continued-fraction approach and avoids the singularities which may occur with perturbation theory. It is illustrated by deriving the effective Hamiltonian for the one-photon, dressed-state laser to second order.
Although the fundamental principles of time-independent perturbation theory are well-established and widely applied, the specific treatment of degeneracy lifted to the second order remains largely unexplored. This work fills the gap by presenting a systematic derivation of time-independent perturbation theory that accounts for the added complexity introduced by non-trivial degeneracies. We provide a general procedure and corresponding formulae for calculating state and energy corrections to any order, with degeneracy lifted to the second order. Additionally, we apply these formulae to solve a prototype problem of the dressed quark system. This work not only advances the theoretical framework but also provides practical tools for tackling complex problems in quantum physics.
James' effective Hamiltonian method has been extensively adopted to investigate largely detuned interacting quantum systems. This method only corresponds to the second-order perturbation theory and cannot be exploited to treat problems which should be solved by using the third- or higher-order perturbation theory. In this paper, we generalize James' effective Hamiltonian method to the higher-order case. Using the method developed here, we reexamine two recently published examples [L. Garziano et al., Phys. Rev. Lett. 117, 043601 (2016); Ken K. W. Ma and C. K. Law, Phys. Rev. A 92, 023842 (2015)]; our results turn out to be the same as the original ones derived from the third-order perturbation theory and adiabatic elimination method, respectively. For some specific problems, this method can simplify the calculating procedure and the resultant effective Hamiltonian is more general.
In this paper, we present a systematic approach to building useful time-dependent effective Hamiltonians in molecular quantum electrodynamics. The method is based on considering part of the system as an open quantum system and choosing a convenient unitary transformation based on the evolution operator. We illustrate our formalism by obtaining four Hamiltonians, each suitable to a different class of applications. We show that we may treat several effects of molecular quantum electrodynamics with a direct first-order perturbation theory. In addition, our effective Hamiltonians shed light on interesting physical aspects that are not explicit when employing more standard approaches. As applications, we discuss three examples: two-photon spontaneous emission, resonance energy transfer, and dispersion interactions.
We show how to derive fixed-point Hamiltonians in quantum mechanics from a proposed renormalization group invariance approach that relies in a subtraction procedure at a given energy scale. The scheme is valid for arbitrary interactions that originally contain point-like singularities (as the Dirac-delta and/or its derivatives). One example of diagonalization of a fixed-point Hamiltonian illustrates the procedure for an interaction that contains regular and singular parts. In another example, the fixed point Hamiltonian is derived for the case of higher order singularities in the interaction.
We discuss a general and systematic method for obtaining effective Hamiltonians that describe different nonlinear optical processes. The method exploits the existence of a nonlinear deformation of the usual su(2) algebra that arises as the dynamical symmetry of the original model. When some physical parameter, dictated by the process under consideration, becomes small, we immediately get a diagonal effective Hamiltonian that correctly represents the dynamics for arbitrary states and long times. We extend the technique to su(3) and su(N), finding the corresponding effective Hamiltonians when some resonance conditions are fulfilled.
A general and systematic method for obtaining eiective Hamiltonians that describe diierent nonlinear optical processes is discussed. The method exploits the existence of a nonlinear deformation of the usual su(2) algebra that arises as the dynamical symmetry of the original model. When some physical parameter, dictated by the process under consideration, becomes small, a diagonal eiective Hamiltonian is obtained immediately, that correctly represents the dynamics for arbitrary states and long times. The technique is extended to su(3) and su(N), finding the corresponding eiective Hamiltonians when some resonance conditions are fulfilled.