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The Journal of Chemical Physics

Higher order adiabatic separation of strongly coupled systems. I. Two-body systems and dressed hydrogen atom

T. T. Nguyen‐Dang, André D. Bandrauk

Published Dec 15, 1986• 8 citations
Publisher

Abstract

The phase corrected adiabatic separation method proposed recently by Marechal [J. Chem. Phys. 83, 247 (1985)] is reexamined. We have been able to show that a state independent differential equation exists, which governs the phase coupling function f of this method, if f further satisfies a set of constraints, Eqs. (7) of the present paper. The method is illustrated by its application to three simple, exactly soluble two-body systems: the free hydrogen atom, the linearly forced hydrogen atom, and the system of two harmonically coupled oscillators. We show that for these systems, the present method, which employs an adiabatic separation procedure, after phase correcting the Hamiltonian, leads to exact results, suggesting its utility for more general problem, involving nonseparable potentials. A nonseparable system of particular interest is the dressed hydrogen atom, which is a representative of all dressed systems, molecular or atomic, with respect to radiative couplings. The method of phase corrected adiabatic separation is shown to give, for field frequencies lying below an IR threshold, a better representation for the exact eigenstates of this dressed systems, as ensuing residual couplings are estimated to be of second order in the original radiative coupling constant.

DOI
10.1063/1.451359

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The phase corrected adiabatic separation method proposed recently by Marechal [J. Chem. Phys. 83, 247 (1985)] is reexamined. We have been able to show that a state independent differential equation exists, which governs the phase coupling function f of this method, if f further satisfies a set of constraints, Eqs. (7) of the present paper. The method is illustrated by its application to three simple, exactly soluble two-body systems: the free hydrogen atom, the linearly forced hydrogen atom, and the system of two harmonically coupled oscillators. We show that for these systems, the present method, which employs an adiabatic separation procedure, after phase correcting the Hamiltonian, leads to exact results, suggesting its utility for more general problem, involving nonseparable potentials. A nonseparable system of particular interest is the dressed hydrogen atom, which is a representative of all dressed systems, molecular or atomic, with respect to radiative couplings. The method of phase corrected adiabatic separation is shown to give, for field frequencies lying below an IR threshold, a better representation for the exact eigenstates of this dressed systems, as ensuing residual couplings are estimated to be of second order in the original radiative coupling constant.
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.
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Numerous examples in condensed matter physics possess an effective Hamiltonian structure involving two-body anharmonic interactions between elementary excitations. The effective second-quantised Hamiltonian used to model these systems is essentially of the same form except for the particulars of the interaction parameters and the statistics of the particles which are interacting. The author investigates such systems in the limit of strongly anharmonic interactions which may possibly lead to symmetry breaking. He introduces quantum field operators and calculates their equations of motion. The interaction coefficients are expanded in a Taylor series about the critical mode to second order to illustrate the method, but he gives the expansion to infinite order for completeness. It is demonstrated that both in first and second order the equations of motion are reduced to a system of highly nonlinear coupled partial differential equations (PDE) for the field operator. He shows that the form of the second-order equation is exact due to renormalisation theory, in the case where the Hamiltonian coefficients represent operators in real space which are isotropic, and incorporates symmetries and topologies which would appear in an expansion to infinite order. This special situation arises in a very wide range of physical examples. These equations can be solved exactly using a particular ansatz in conjunction with recent mathematical developments in the field of non-linear analysis. In particular, the method of symmetry reduction for PDE provides a complete set of symmetry variables for the type of equations used in the study.
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Several techniques to solve a hierarchical set of equations of motion for propagating a reduced density matrix coupled to a thermal bath have been developed in recent years. This is either done using the path integral technique as in the original proposal by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1998)] or by the use of stochastic fields as done by Yan et al. [Chem. Phys. Lett. 395, 216 (2004)]. Based on the latter ansatz a compact derivation of the hierarchy using a decomposition of the spectral density function is given in the present contribution. The method is applied to calculate the time evolution of the reduced density matrix describing the motion in a harmonic, an anharmonic, and two coupled oscillators where each system is coupled to a thermal bath. Calculations to several orders in the system-bath coupling with two different truncations of the hierarchy are performed. The respective density matrices are used to calculate the time evolution of various system properties and the results are compared and discussed with a special focus on the convergence with respect to the truncation scheme applied.
The dressed atom approach provides a tool to investigate the dynamics of an atom-laser system by fully retaining the quantum nature of the coherent mode. In its standard derivation, the internal atom-laser evolution is described within the rotating-wave approximation, which determines a doublet structure of the spectrum and the peculiar fluorescence triplet in the steady state. However, the rotating wave approximation may fail to apply to atomic systems subject to femtosecond light pulses, light-matter systems in the strong-coupling regime or sustaining permanent dipole moments. This work aims to demonstrate how the general features of the steady-state radiative cascade are affected by the interaction of the dressed atom with propagating radiation modes. Rather than focusing on a specific model, we analyze how these features depend on the parameters characterizing the dressed eigenstates in arbitrary atom-laser dynamics, given that a set of general hypotheses is satisfied. Our findings clarify the general conditions in which a description of the radiative cascade in terms of transition between dressed states is self-consistent. We provide a guideline to determine the properties of photon emission for any atom-laser interaction model, which can be particularly relevant when the model should be tailored to enhance a specific line. We apply the general results to the case in which a permanent dipole moment is a source of low-energy emission, whose frequency is of the order of the Rabi coupling.