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Quantum analysis of Young\'s interference experiment for electromagnetic fields

Anjan Shrestha

Published Nov 26, 2014• 0 citations

Abstract

In this work, we formulate a quantum-mechanical description of interference of electromagnetic fields in Young's interference experiment, thereby taking into account the polarization properties of the field and describing them in terms of quantum analogs of classical Stokes parameters.Commencing with the classical theory of interference of scalar fields, we proceed to a relatively advanced approach to electromagnetic interference, bringing into the equation cross-spectral density tensor, polarization matrix and Stokes parameters to analyze the polarization properties.Subsequently, the same phenomenon is analyzed in the domain of quantum optics, thereby expressing the fields as operators and observables as the expectation values.Firstly, an outline of the scalar approach of the operators in the interference experiment is presented to establish the foundation to base the electromagnetic approach on, followed by a full description of quantum analog of electromagnetic interference in Young's experiment.In particular, Stokes parameters are adopted to calculate the polarization effects in quantum theory. PrefaceThis report attempts to condense and present the theories related to the interference in the canonical Young's double pinhole experiment, prior to heading to investigate the phenomenon with the quantum analysis of electromagnetic fields.My interest in the realm of quantum mechanics has encouraged me to work on this thesis, wherefrom I have learnt a lot during these times.I am really grateful to my supervisors, Prof. Ari T. Friberg and Prof. Jani Tervo for guiding me and believing in me.I will always be indebted for their constant support and confidence on me.I would also like to offer my special thanks to the Faculty of Forestry and Sciences for providing the financial

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In this work, we formulate a quantum-mechanical description of interference of electromagnetic fields in Young's interference experiment, thereby taking into account the polarization properties of the field and describing them in terms of quantum analogs of classical Stokes parameters.Commencing with the classical theory of interference of scalar fields, we proceed to a relatively advanced approach to electromagnetic interference, bringing into the equation cross-spectral density tensor, polarization matrix and Stokes parameters to analyze the polarization properties.Subsequently, the same phenomenon is analyzed in the domain of quantum optics, thereby expressing the fields as operators and observables as the expectation values.Firstly, an outline of the scalar approach of the operators in the interference experiment is presented to establish the foundation to base the electromagnetic approach on, followed by a full description of quantum analog of electromagnetic interference in Young's experiment.In particular, Stokes parameters are adopted to calculate the polarization effects in quantum theory. PrefaceThis report attempts to condense and present the theories related to the interference in the canonical Young's double pinhole experiment, prior to heading to investigate the phenomenon with the quantum analysis of electromagnetic fields.My interest in the realm of quantum mechanics has encouraged me to work on this thesis, wherefrom I have learnt a lot during these times.I am really grateful to my supervisors, Prof. Ari T. Friberg and Prof. Jani Tervo for guiding me and believing in me.I will always be indebted for their constant support and confidence on me.I would also like to offer my special thanks to the Faculty of Forestry and Sciences for providing the financial

An electromagnetic Young's interference experiment is considered with arbitrary, spectrally partially coherent and partially polarized electric fields at the pinholes. Any correlations that exist at the openings lead to modulations of the Stokes parameters on the observation screen. The electromagnetic degree of coherence can be viewed, in direct analogy to the usual scalar case, as a measure of the modulation contrasts of not just the optical intensity but also the polarization state. The electromagnetic degree of coherence can be obtained experimentally via four visibility measurements using appropriate wave plates. If the field at the pinholes is fully polarized, a periodic variation of the polarization state appears on the observation screen, leading to a Pancharatnam–Berry phase. It is shown that this geometric phase has a straightforward but elegant relationship to the Stokes parameters at the pinholes.

Young's double slit experiment is formulated in the framework of canonical quantum field theory in view of the modern quantum optics. We adopt quantum scalar fields instead of quantum electromagnetic fields ignoring the vector freedom in gauge theory. The double slit state is introduced in Fock space corresponding to experimental setup. As observables, expectation values of energy density and positive frequency part of current with respect to the double slit state are calculated which give the interference term. Classical wave states are realized by coherent double slit states in Fock space which connect quantum particle states with classical wave states systematically. In case of incoherent sources, the interference term vanishes by averaging random phase angles as expected.

The mathematical representation of each of the four interference laws of Fresnel and Arago is presented. Each of the laws is characterized by a unique Stokes vector. These vectors are arrived at by a matrix analysis of the, interference experiments of Fresnel and Arago in which only polarizers are used. The historical motivation that led Stokes to introduce the parameters that now bear his name is noted. The similarity in the approach taken by Stokes to represent the fundamental properties of the classical optical field in terms of observables is compared with Heisenberg's approach to the formulation of quantum mechanics. The mathematical statements that characterize unpolarized light are developed. These statements are then used to determine the Stokes vectors that describe each of the four interference laws of Fresnel and Arago. In a final section of the paper the mathematical apparatus developed here is used to describe Young's interference experiment with unpolarized light.

Any polarization state of light occupies intrinsic quantum uncertainties due to the noncommutative behavior of the Stokes operators, which are manifested as unavoidable quantum fluctuations in the corresponding classical Stokes parameters.In this thesis, we explore such quantum polarization uncertainties exhibited by vectorial light fields in Young's seminal double-slit interference experiment.As a main result, by adopting a proper Stokes operator representation of double-slit interference, we establish general quantum polarization uncertainty relations for the Stokes parameters in the observation plane.The relations show how the lower bounds of the polarization uncertainties are periodically varying across the observation screen and how they are affected by the polarization and coherence properties of the light at the two slits.We also present various noncommutative, interlinked connections between coherence and polarization in the slit plane which have not been considered earlier in the literature.Our work sets the fundamental precision limits imposed by quantum physics for characterizing any polarization state in double-slit interference, which could find use in photonic applications exploiting polarimetric and interferometric techniques for high-resolution quantum-enhanced sensing.

The principle of complementarity relationship introduced by Niels Bohr in 1927 states that particle and wave behavior are two mutually exclusive properties for a quantum system.This relationship has taken the attention of researchers for decades due to its conceptual importance.This thesis contains a theoretical research work conducted on quantum complementarity of vectorial light fields and in the doublepinhole interference.Here we have adopted the classical Stokes parameters within a quantum framework and calculated the Stokes parameters in the observation plane in double-pinhole interference.Based on these calculations, we further evaluated visibility and distinguishability parameters for an arbitrary pure single-photon state and for a specific single-photon pure state, where the photon is x polarized at the first pinhole and y polarized at the second pinhole.We have then examined and interpreted the wave-particle duality for the mentioned two states.Finally, we performed mixed state calculation where fully unpolarized photon is incident on the two pinholes, but with the photon at one pinhole rotated by an angle of π/2.From the obtained results we concluded that a priori which path information (WPI) regarding the complementarity relations for the single photon pure vectorial quantum state couples not to the intensity visibility but to the total visibility.We further found that the vectorial quantum state gives rise to polarization modulation.From the results obtained for the mixed state calculations, we discovered that the last Stokes parameter has a non-zero value and it contributes to the polarization modulation.Thereby we further recommended to repeat the calculation for different mixed states other than what we have considered here to extend the exploration about the nature of polarization modulation.