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arXiv:1711.01615 · Journal of Physics A Mathematical and Theoretical

Quantum mechanical treatment of two-level atoms coupled to continuum with an ultraviolet cutoff

Fatih Dinç, İlke Ercan

Published Jul 5, 2018• 2 citations
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Abstract

Abstract In this paper, we provide a rigorous quantum mechanical derivation for the coherent photon transport characteristics of a two-level atom coupled to a waveguide without linearizing the coupling coefficient between the light and the atom. We propose a novel single frequency sampling method utilizing a UV-cutoff that allows us to treat the singularities in real space scattering potential despite the non-convergence property. We also study the conditions under which the linearization of the coupling coefficient is an accurate assumption and find the resulting spontaneous emission and transport characteristics taking the radiative and non-radiative decay rates into account. This allows us to confirm and expand on the findings of the existing literature while obtaining the dynamic electronic polarizability for the two-level atom confined to a 1D waveguide while using an interaction Hamiltonian with rotating-wave approximation.

DOI
10.1088/1751-8121/aad165

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Abstract In this paper, we provide a rigorous quantum mechanical derivation for the coherent photon transport characteristics of a two-level atom coupled to a waveguide without linearizing the coupling coefficient between the light and the atom. We propose a novel single frequency sampling method utilizing a UV-cutoff that allows us to treat the singularities in real space scattering potential despite the non-convergence property. We also study the conditions under which the linearization of the coupling coefficient is an accurate assumption and find the resulting spontaneous emission and transport characteristics taking the radiative and non-radiative decay rates into account. This allows us to confirm and expand on the findings of the existing literature while obtaining the dynamic electronic polarizability for the two-level atom confined to a 1D waveguide while using an interaction Hamiltonian with rotating-wave approximation.
In this paper, we provide a rigorous quantum mechanical derivation for the coherent photon transport characteristics of a two-level atom coupled to a waveguide without linearizing the coupling coefficient between the light and the atom. We propose a novel single frequency sampling method that allows us to treat the singularities in real space scattering potential despite the non-convergence property. We also study the conditions under which the linearization of the coupling coefficient is an accurate assumption. This allows us to confirm and expand on the findings of the existing literature while obtaining the dynamic electronic polarizability for the two-level atom confined to a 1-D waveguide while using an interaction Hamiltonian with rotating-wave approximation.
We analyze the single-photon transport in a single-mode waveguide coupled to a whispering-gallery-type resonator interacting with a two-level atom. The single-photon transport properties such as the transmission and reflection amplitudes, as well as the resonator and the atom responses, are solved exactly via a real-space approach. The treatment includes the inter-mode backscattering between the two degenerate whispering gallery modes of the resonator, and the dissipations of the resonator and the atom. We also show that a generalized critical coupling condition, that the single-photon transmission at the output of the waveguide goes to zero on resonance for a matched system, holds for the full coupled waveguide-ring resonator-atom system.
The effect of spontaneous decay on the linear polarizability of an atom is typically included by adding imaginary parts to the frequency denominators that appear in the Kramers-Heisenberg formula. It has been shown for a two-level atom with radiative broadening that these (frequency-dependent) imaginary parts must be included in both the resonant and antiresonant frequency denominators [P. W. Milonni and R. W. Boyd, Phys. Rev. A 69, 023814 (2004)]; however, the expression obtained by Milonni and Boyd for the polarizability does not satisfy the optical theorem, if contributions from non-rotating-wave terms are included. In this paper, we derive a more accurate expression for the polarizability. The calculations are rather complicated and require that we go beyond the standard Weisskopf-Wigner approximation. We present calculations carried out in both the Heisenberg and Schr\"odinger pictures, since they offer complementary methods for understanding the dynamics of the Rayleigh scattering associated with the atomic polarizability. Moreover, it is shown that the shifts associated with the excited state are not the Lamb shifts of an isolated atom, but depend on the dynamics of the atom-field interaction. Our results for the polarizability are consistent with those obtained recently by Loudon and Barnett using a Green's-function approach.
The studies of interactions of electromagnetic fields with two level atom systems is a main theme of quantum optics. Here, we shows how the inherent dissipation of atom influences the interaction between atom and single photon propagating in the waveguide. Based on a fully quantum mechanical theory in real space, we exactly calculated the transportation probability of a single photon for a two-level atom system next to the waveguide. We find that the existence of a dissipated atom can be utilized to act as the light router as well as the beam splitter, if properly adjust relationship of atom dissipation 􀟛􀯤and coupling strengthV between atom and single photon in waveguide.
We theoretically investigate quantum Rabi oscillations in a system consisting of a two-level atom (qubit) strongly coupled to a one-dimensional open waveguide. In contrast to conventional cavity quantum electrodynamics, the qubit interacts with a continuum of propagating modes, which gives rise to fundamentally different dynamical behavior. Within the rotating-wave approximation, we express the multimode Jaynes-Cummings Hamiltonian in terms of collective bosonic operators and show that the system possesses two integrals of motion, enabling an exact diagonalization of the Hamiltonian in the single-excitation subspace. We show that for a strongly interacting qubit-photon system, the dynamics are captured by a reduced two-level model. In this framework, each level is defined as the product of the atomic excited state and a specific field mode, which takes the form of a multiphoton Fock-like state. In this picture, the Rabi oscillations represent a collective phenomenon corresponding to oscillations between multiphoton states differing by a single photon. We then extend our analysis to multiphoton processes in which the initial field is a coherent state with a continuous spectrum. In this case, the Rabi frequency is shown to be sensitive to the spectral profile of the function that generates the coherent state.