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arXiv:1411.6420 · Physical Review A

Hilbert transform: Applications to atomic spectra

Kate A. Whittaker, James Keaveney, Ifan G. Hughes, Charles S. Adams

Published Mar 25, 2015• 46 citations
arXivHTMLPDF

Abstract

In many areas of physics, the Kramers-Kronig relations are used to extract information about the real part of the optical response of a medium from its imaginary counterpart. In this paper we discuss an alternative but mathematically equivalent approach based on the Hilbert transform. We apply the Hilbert transform to transmission spectra to find the group and refractive indices of a Cs vapor and thereby demonstrate how the Hilbert transform allows indirect measurement of the refractive index, group index, and group delay while avoiding the use of complicated experimental setups.

DOI
10.1103/physreva.91.032513

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In many areas of physics, the Kramers-Kronig relations are used to extract information about the real part of the optical response of a medium from its imaginary counterpart. In this paper we discuss an alternative but mathematically equivalent approach based on the Hilbert transform. We apply the Hilbert transform to transmission spectra to find the group and refractive indices of a Cs vapor and thereby demonstrate how the Hilbert transform allows indirect measurement of the refractive index, group index, and group delay while avoiding the use of complicated experimental setups.
Abstract The Kramers–Kronig relations are a pivotal foundation of linear optics and atomic physics, embedding a physical connection between the real and imaginary components of any causal response function. A mathematically equivalent, but simpler, approach instead utilises the Hilbert transform. In a previous study, the Hilbert transform was applied to absorption spectra in order to infer the sole refractive index of an atomic medium in the absence of an external magnetic field. The presence of a magnetic field causes the medium to become birefringent and dichroic, and therefore it is instead characterised by two refractive indices. In this study, we apply the same Hilbert transform technique to independently measure both refractive indices of a birefringent atomic medium, leading to an indirect measurement of atomic magneto-optical rotation. Key to this measurement is the insight that inputting specific light polarisations into an atomic medium induces absorption associated with only one of the refractive indices. We show this is true in two configurations, commonly referred to in literature as the Faraday and Voigt geometries, which differ by the magnetic field orientation with respect to the light wavevector. For both cases, we measure the two refractive indices independently for a Rb thermal vapour in a 0.6 T magnetic field, finding excellent agreement with theory. This study further emphasises the application of the Hilbert transform to the field of quantum and atomic optics in the linear regime.
The refractive index of a system is often viewed as the collective response of a medium to an electromagnetic field. Nevertheless, even if targeted at a single atom, the incoming light undergoes the dispersion. By studying the propagation of a single photon interacting with a two-level atom, we examine the dispersion behavior, calculate the phase and group velocity of the photon wave packet, and further analyze the dispersion experienced by the photon. These findings are critical for advancing long-distance quantum communications.
This thesis presents the study of thermal alkali-metal vapours confined in a layer with a sub-micron thickness. This confinement enables the study of high density media without the loss of signal present in usually thermal vapours, but also has additional effects on spectra acquired from the system. Such effects include the suppression of the Doppler broadening and the interaction of the atoms with nearby surfaces. Herein, we present a study of this atom-surface interaction in both Rubidium and Caesium atoms, demonstrating that the interaction follows a power law of 1/rα1/r^\alpha1/rα, where α=3.02±0.06\alpha = 3.02 \pm 0.06α=3.02±0.06. We also study Rabi oscillations at high densities, driving GHz Rabi oscillations in a Rb vapour at densities up to 101710^{17}1017~cm−3^{-3}−3. We find that the results do not have sufficient agreement with an optical Bloch simulation, but Maxwell Bloch simulations indicate the possible presence of simultons; simultaneously propagating solitons. Such phenomena have not yet been observed out of crystalline media. We also present a study of causality relations in atomic media. We first discuss the equivalency of the Hilbert transform to the Kramers-Kronig relations, well known in signal processing, but rarely applied in atomic physics. We then demonstrate that the Hilbert transform can be applied to atomic transmission spectra to quickly generate refractive index spectra. The final section of this thesis fully details the successful design and fabrication of vapour cells with a thickness of 500-1500~nm. The cells are assembled using a combination of traditional scientific glass blowing techniques and thermally annealed optically contacted plates. We fully outline the production process, and then present evidence of their successful functionality and longevity.
In the context of quantum mechanics we employ the technique of integration within an ordered product of operators to recast the classical wavelet transform to a squeezing-displacing transform between the mother wavelet vector and the state vector to be transformed. In this way we propose the wavelet-transform spectrum for quantum optical states. For some typical states we obtain numerical results which imply that the spectrum can be used to recognize a variety of quantum optical states, and the inverse wavelet transform has the possibility to play a role in quantum state engineering.
The real and imaginary parts of the complex index of refraction in the R-line region of ruby have been measured near liquid-nitrogen temperature with sufficient resolution that the R-line doublet structure was well resolved. A modification of the usual Babinet compensator technique was used for the refractive-index measurements, which were found to correlate with the absorption measurements as predicted by the Kramers-Kronig relations. An attempt at direct quantitative verification of the dispersion relations for an optically pumped sample is also described.