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Journal of the Korean Physical Society

Analytical Solutions of the Second-Order Correlation Function in Resonance Fluorescence for Two-level Systems

Jaewan Kim, Heung‐Ryoul Noh

Published Mar 1, 2020• 1 citation
Publisher

Abstract

We present analytical solutions for the second-order correlation function [ g (2 )( τ )] for light scattered from a two-level atom driven by laser light with arbitrary detuning and the Rabi frequency. By solving the optical Bloch equations for a two-level atom, we obtain analytical solutions for the populations that are subsequently used for calculating g (2 )( τ ). We also obtain g (2)( τ ) for special cases, such as zero detuning or weak laser intensity, directly from the general analytical solutions for g (2 )( τ ).

DOI
10.3938/jkps.76.463

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We present analytical solutions for the second-order correlation function [ g (2 )( τ )] for light scattered from a two-level atom driven by laser light with arbitrary detuning and the Rabi frequency. By solving the optical Bloch equations for a two-level atom, we obtain analytical solutions for the populations that are subsequently used for calculating g (2 )( τ ). We also obtain g (2)( τ ) for special cases, such as zero detuning or weak laser intensity, directly from the general analytical solutions for g (2 )( τ ).
We present a comprehensive analytical study of the optical Bloch equations for a closed two-level atom. We obtain a map in phase space of the Rabi frequency and the laser frequency detuning, discriminating the characteristics of solutions to the optical Bloch equations separated by a boundary in phase space. Outside the boundary, oscillatory behavior is shown, while simple decaying behavior is shown inside the boundary. The behavior of the solutions is also found to be dependent on the relative magnitude of the decay rates of the optical coherence and the excited state population.
The spectrum of resonance fluorescence and the second-order correlation function are calculated for a system of two identical two-level cooperative systems interacting via the dipole-dipole interaction. The results are valid for arbitrary incident-field strengths and for atomic separations much less than the resonant wavelength. The intensity of the sidebands at the Rabi frequency is diminished by the dipole interaction, whereas that of the sidebands at twice the Rabi frequency is enhanced. Both the cooperative behavior and the dipole-dipole interaction act to diminish the photon antibunching, as shown by the second-order correlation function.
The spectral distribution of light scattered by a two-level atom in a laser beam with phase modulation has been calculated. The spectrum has been obtained via numerical solution of the optical Bloch equations using the algorithm of Wilkens and Rza\ifmmode \mbox{\c{}}\else \c{}\fi{}\ifmmode \dot{z}\else \.{z}\fi{}ewski [Phys. Rev. A 40, 3164 (1989)]. The modulation produces additional sidebands in both coherent and incoherent parts of the spectrum; the location of the added components depends on the modulation depth. In the case of two-frequency phase modulation with incommensurate frequencies, wild, ``chaotic'' behavior of the spectrum has been observed, similar to the case of bichromatic amplitude modulation. The intensity distribution is practically insensitive to the chosen constant value of the phase of modulation.
We generalize our earlier formalism to treat the effects of both the finite-bandwidth excitations and finite arbitrary detunings, on the fluorescent spectra and second-order intensity correlation functions in optical double resonance. It is assumed that the bandwidth arises from the phase and/or amplitude fluctuations in the fields driving a three-level atom in cascade configuration. Under the conditions that one or both of these fields are intense, secular approximation and the theory of multiplicative stochastic processes are invoked to derive a Markovian master equation for the atomic-density operator averaged over both phase and amplitude. The quantum regression theorem is used to derive analytical expressions for the fluorescent spectra and the second-order intensity correlation functions which explicitly display the effects due to detunings and fluctuations.
Resonance fluorescence from a single two-level atom driven by a beam of squeezed light is studied in the weak-field limit. We consider the situation where the atom is coupled to the ordinary vacuum and only a few field modes corresponding to the driving field are squeezed. The field produced by the degenerate optical parametric oscillator is used as the driving field. Heisenberg equations of motion are solved in the steady-state and analytic expressions for the fluorescent-light intensity and the spectrum of fluorescent light are derived. We also consider photon statistics of fluorescent light. In particular, squeezing, antibunching, and sub-Poissonian statistics of fluorescent photons are discussed, and analytic expressions for the quadrature variance and the two-time intensity correlation function are presented. Contrary to the case of coherent excitation, the second-order intensity correlation function does not factorize. This and other differences are discussed, and curves are presented to illustrate the behavior of various quantities. We also present results for thermal excitation of the atom.
Resonance fluorescence from a single two-level atom driven by a beam of squeezed light is studied in the weak field limit. We consider the situation where the atom is coupled to the ordinary vacuum and only a few field modes corresponding to the driving field are squeezed. The field produced by the degenerate optical parametric oscillator is used as the driving field. This driving field can be described in terms of two real independent Gaussian processes. Heisenberg equations of motion are solved in the steady-state, and analytic expressions for the fluorescent light intensity and the spectrum of fluorescent light are derived. We also consider photon statistics of fluorescent light. In particular, squeezing, antibunching, and sub-Poissonian statistics of fluorescent photons are discussed and analytic expressions for the quadrature variance and the two-time intensity correlation function are presented. Contrary to the case of coherent excitation the second-order intensity correlation function does not factorize.