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Journal of Physics A Mathematical and Theoretical

Hill’s determinant approach to single-mode spin-boson model

Qing-Wei Wang, Yu-Liang Liu

Published Oct 11, 2013• 4 citations
Publisher

Abstract

An efficient and stable numerical scheme based on Hill’s determinant is proposed for computing the whole spectrum of the single-mode spin-boson model. Its shown that for the low-lying levels a simple version of Hill’s determinant method works well, while for high levels a normalization of the determinant is necessary and this normalization results in an entire function whose zeros correspond to the energy levels. It is very convenient to use this entire function to compute all the energy levels numerically. Some high levels of the model are computed and compared with Irish’s general rotating-wave approximation. A scaling law in the large photon number limit is numerically tested and several statistical properties of the spectrum are investigated.

DOI
10.1088/1751-8113/46/43/435303

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An efficient and stable numerical scheme based on Hill’s determinant is proposed for computing the whole spectrum of the single-mode spin-boson model. Its shown that for the low-lying levels a simple version of Hill’s determinant method works well, while for high levels a normalization of the determinant is necessary and this normalization results in an entire function whose zeros correspond to the energy levels. It is very convenient to use this entire function to compute all the energy levels numerically. Some high levels of the model are computed and compared with Irish’s general rotating-wave approximation. A scaling law in the large photon number limit is numerically tested and several statistical properties of the spectrum are investigated.
The single-mode spin-boson model exhibits behavior not included in the rotating-wave approximation (RWA) in the ultra and deep-strong coupling regimes, where counter-rotating contributions become important. We introduce a symmetric rotating-wave approximation that treats rotating and counter-rotating terms equally, preserves the invariances of the Hamiltonian with respect to its parameters, and reproduces several qualitative features of the spin-boson spectrum not present in the original rotating-wave approximation both off-resonance and at deep-strong coupling. The symmetric rotating-wave approximation allows for the treatment of certain ultra- and deep-strong coupling regimes with similar accuracy and mathematical simplicity as does the RWA in the weak-coupling regime. Additionally, we symmetrize the generalized form of the rotating-wave approximation to obtain the same qualitative correspondence with the addition of improved quantitative agreement with the exact numerical results. The method is readily extended to higher accuracy if needed. Finally, we introduce the two-photon parity operator for the two-photon Rabi Hamiltonian and obtain its generalized symmetric rotating-wave approximation. The existence of this operator reveals a parity symmetry similar to that in the Rabi Hamiltonian as well as another symmetry that is unique to the two-photon case, providing insight into the mathematical structure of the two-photon spectrum, significantly simplifying the numerics, and revealing some interesting dynamical properties.
A system of N two-level atoms interacting with a resonant single-mode quantum field (the Dicke model) is described using the operator method for solving the Schrödinger equation. The spectrum of states is calculated without recourse to the rotating-wave approximation and the assumption regarding smallness of the linear sizes of the system as compared to the radiation wavelength. Analytical approximate expressions are obtained for the energy spectrum. These expressions approximate the energy spectrum over the entire range of Hamiltonian parameters in the normal and collective states of the system and make it possible to calculate the thermodynamic characteristics.
The single-mode spin-boson model (SMSBM) has extensive application in different subfields of physics. In the absence of rotating-wave approximation (RWA), we try to solve the SMSBM analytically. We argue that the analytical expression obtained might be the most exact approximation to the solution of the system in our case under the assumption of the Abel-Ruffini theorem, which works for a wide range of the parameters such as coupling strength and detuning and would be practical for currently available experiments.
A diagonalization scheme for the Rabi Hamiltonian, which describes a qubit interacting with a single-mode radiation field via a dipole interaction, is proposed. It is shown that the Rabi Hamiltonian can be solved almost exactly using a progressive scheme that involves a finite set of one variable polynomial equations. The scheme is especially efficient for the lower part of the spectrum. Some low-lying energy levels of the model with several sets of parameters are calculated and compared to those provided by the recently proposed generalized rotating-wave approximation and a full matrix diagonalization.