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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Quantum optical coherence: From linear to nonlinear interferometers

Kai-Hong Luo, Matteo Santandrea, Michael Stefszky, Jan Sperling, Marcello Massaro, Alessandro Ferreri, Polina R. Sharapova, Harald Herrmann, Christine Silberhorn

Published Jun 21, 2021• 4 citations
Publisher

Abstract

Interference between two waves is fundamentally driven by the coherence, an intrinsic property of waves. The classical coherence of two fields then defines the degree of first-order coherence, as quantified by their cross-correlation function. Here, we study the coherence properties of various interferometer configurations, spanning from linear over semi-nonlinear to fully nonlinear settings [1] – [3] .

DOI
10.1109/cleo/europe-eqec52157.2021.9542465

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Quantum optical coherence: From linear to nonlinear interferometers

Kai-Hong Luo, Matteo Santandrea, Michael Stefszky +6 more

Interferometers provide a highly sensitive means to investigate and exploit the coherence properties of light in metrology applications. However, interferometers come in various forms and exploit different properties of the optical states within. In this paper, we introduce a classification scheme that characterizes any interferometer based on the number of involved nonlinear elements by studying their influence on single-photon and photon-pair states. Several examples of specific interferometers from these more general classes are discussed, and the theory describing the expected first-order and second-order coherence measurements for single-photon and single-photon-pair input states is summarized and compared. These theoretical predictions are then tested in an innovative experimental setup that is easily able to switch between implementing an interferometer consisting of only one or two nonlinear elements. The resulting singles and coincidence rates are measured in both configurations and the results are seen to fit well with the presented theory. The measured results of coherence are tied back to the presented classification scheme, revealing that our experimental design can be useful in gaining insight into the properties of the various interferometeric setups containing different degrees of nonlinearity.

arXiv:2104.02641•Physical Review A•Oct 21, 2021•21 citations

The quantum theory of optical coherence

Roy J. Glauber

The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The $n\mathrm{th}$ order function expresses the correlation of values of the fields at $2n$ different points of space and time. Certain values of these functions are measurable by means of $n$-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.

•Jan 1, 1986•0 citations

The Quantum Theory of Optical Coherence

Roy J. Glauber

The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The $n\mathrm{th}$ order function expresses the correlation of values of the fields at $2n$ different points of space and time. Certain values of these functions are measurable by means of $n$-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.

Physical Review•Jun 15, 1963•4,074 citations

An Introduction to Quantum Optics: Photon and Biphoton Physics

Yanhua H. Shih

Electromagnetic Wave Theory and Measurement of Light Electromagnetic Wave Theory of Light Classical Superposition Measurement of Light Intensity of Light: Expectation and Fluctuation Measurement of Intensity: Ensemble Average and Time Average Coherence Property of Light-The State of the Radiation Coherence Property of Light Temporal Coherence Spatial Coherence Diffraction and Propagation Diffraction Field Propagation Optical Imaging A Classic Imaging System Fourier Transform via a Lens First-Order Coherence of Light First-Order Temporal Coherence First-Order Spatial Coherence Second-Order Coherence of Light Second-Order Coherence of Coherent Light Second-Order Correlation of Chaotic-Thermal Radiation and the HBT Interferometer The Physical Cause of the HBT Phenomenon Near-Field Second-Order Spatial Coherence of Thermal Light Nth-Order Coherence of Light Nth-Order Near-Field Spatial Coherence of Thermal Light Homodyne Detection and Heterodyne Detection of Light Optical Homodyne and Heterodyne Detection Balanced Homodyne and Heterodyne Detection Balanced Homodyne Detection of Independent and Coupled Thermal Fields Quantum Theory of Light: Field Quantization and Measurement The Experimental Foundation-Part I: Blackbody Radiation The Experimental Foundation-Part II: Photoelectric Effect The Light Quantum and the Field Quantization Photon Number State of Radiation Field Coherent State of Radiation Field Density Operator and Density Matrix Composite System and Two-Photon State of Radiation Field A Simple Model of Incoherent and Coherent Radiation Source Pure State and Mixed State Product State, Entangled State, and Mixed State of Photon Pairs Time-Dependent Perturbation Theory Measurement of Light: Photon Counting Measurement of Light: Joint Detection of Photons Field Propagation in Space-Time Quantum Theory of Optical Coherence Quantum Degree of First-Order Coherence Photon and Effective Wavefunction Measurement of the First-Order Coherence or Correlation Quantum Degree of Second-Order Coherence Two-Photon Interference vs. Statistical Correlation of Intensity Fluctuations Second-Order Spatial Correlation of Thermal Light Photon Counting and Measurement of G(2) Quantum Entanglement EPR Experiment and EPR State Product State, Entangled State, and Classically Correlated State Entangled States in Spin Variables Entangled Biphoton State EPR Correlation of Entangled Biphoton System Subsystem in an Entangled Two-Photon State Biphoton in Dispersive Media Quantum Imaging Biphoton Imaging Ghost Imaging Ghost Imaging and Uncertainty Relation Thermal Light Ghost Imaging Classical Simulation of Ghost Imaging Turbulence-Free Ghost Imaging Two-Photon Interferometry I: Biphoton Interference Is Two-Photon Interference the Interference of Two Photons? Two-Photon Interference with Orthogonal Polarization Franson Interferometer Two-Photon Ghost Interference Delayed Choice Quantum Eraser Two-Photon Interferometry II: Quantum Interference of Chaotic Light Two-Photon Young's Interference Two-Photon Anticorrelation with Incoherent Chaotic Light Two-Photon Interference with Incoherent Orthogonal Polarized Chaotic Light Bell's Theorem and Bell's Inequality Measurement Hidden Variable Theory and Quantum Calculation for the Measurement of Spin 1/2 Bohm State Bell's Theorem and Bell's Inequality Bell States Bell State Preparation

•Jan 24, 2011•40 citations

First-order optical coherence of photonic-dimer coherent states

Qihang Liu, Jung‐Tsung Shen

Photon-photon correlations assume a pivotal significance in optical coherence. Recently, a new, to the best of our knowledge, type of quantum photonic states, the coherent state of photonic dimers, has been introduced, wherein the fundamental building blocks are two-photon bound states, instead of individual photons as in conventional lasers. In this Letter, we investigate the first-order coherence properties of the photonic-dimer coherent states, as well as the interference patterns in a double-slit interferometer setup, and compare with the coherence properties of other optical light sources, e.g., the conventional laser and the thermal light.

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Coherence in Classical Electromagnetism and Quantum Optics

Hanne-Torill Mevik

This thesis is a study of coherence theory in light in classical\nelectromagnetism and quantum optics. %The coherence is quantified\nSpecifically two quantities are studied: The degree of first-order\ntemporal coherence, which quantifies the field-field coherence, and\nthe degree of second-order coherence, quantifying the intensity-intensity\ncoherence. In the first part of the thesis these concepts are applied to classical\nelectric fields; to both the ideal plane wave and to chaotic light.\nWe then study how they can be measured using two interferometer\ntechnologies from optical astronomy, specifically with the\nMichelson stellar interferometer and the intensity interferometer.\n\nIn the second part we define the\nquantum degrees of first- and second-order coherence. These are calculated\nfor light in a quantum coherent state, in a Fock state and for light\nin a mixed thermal state. The results for the coherent state and\nthe thermal state are found to be analogous to\nthose obtained for the ideal plane wave and chaotic light, respectively,\nfrom the classical coherence theory seen in the first part.\n\nWe proceed to investigate the properties of the three-level laser with\nthe aim of showing that far above threshold it develops similar photon\nstatistics and values for the degrees of first- and second-order coherence,\nto light in a coherent state. The mechanism of phase-drift in the laser\nis also looked into. Subsequently the Mølmer-model is discussed, where it is\ndemonstrated that the coherent state is not a necessary construct, but\nmerely a convenient one, in describing phenomena in quantum optics.

•Jan 1, 2009•2 citations
Interference between two waves is fundamentally driven by the coherence, an intrinsic property of waves. The classical coherence of two fields then defines the degree of first-order coherence, as quantified by their cross-correlation function. Here, we study the coherence properties of various interferometer configurations, spanning from linear over semi-nonlinear to fully nonlinear settings [1] – [3] .
Interferometers provide a highly sensitive means to investigate and exploit the coherence properties of light in metrology applications. However, interferometers come in various forms and exploit different properties of the optical states within. In this paper, we introduce a classification scheme that characterizes any interferometer based on the number of involved nonlinear elements by studying their influence on single-photon and photon-pair states. Several examples of specific interferometers from these more general classes are discussed, and the theory describing the expected first-order and second-order coherence measurements for single-photon and single-photon-pair input states is summarized and compared. These theoretical predictions are then tested in an innovative experimental setup that is easily able to switch between implementing an interferometer consisting of only one or two nonlinear elements. The resulting singles and coincidence rates are measured in both configurations and the results are seen to fit well with the presented theory. The measured results of coherence are tied back to the presented classification scheme, revealing that our experimental design can be useful in gaining insight into the properties of the various interferometeric setups containing different degrees of nonlinearity.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nthn\mathrm{th}nth order function expresses the correlation of values of the fields at 2n2n2n different points of space and time. Certain values of these functions are measurable by means of nnn-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
The concept of coherence which has conventionally been used in optics is found to be inadequate to the needs of recently opened areas of experiment. To provide a fuller discussion of coherence, a succession of correlation functions for the complex field strengths is defined. The nthn\mathrm{th}nth order function expresses the correlation of values of the fields at 2n2n2n different points of space and time. Certain values of these functions are measurable by means of nnn-fold delayed coincidence detection of photons. A fully coherent field is defined as one whose correlation functions satisfy an infinite succession of stated conditions. Various orders of incomplete coherence are distinguished, according to the number of coherence conditions actually satisfied. It is noted that the fields historically described as coherent in optics have only first-order coherence. On the other hand, the existence, in principle, of fields coherent to all orders is shown both in quantum theory and classical theory. The methods used in these discussions apply to fields of arbitrary time dependence. It is shown, as a result, that coherence does not require monochromaticity. Coherent fields can be generated with arbitrary spectra.
Electromagnetic Wave Theory and Measurement of Light Electromagnetic Wave Theory of Light Classical Superposition Measurement of Light Intensity of Light: Expectation and Fluctuation Measurement of Intensity: Ensemble Average and Time Average Coherence Property of Light-The State of the Radiation Coherence Property of Light Temporal Coherence Spatial Coherence Diffraction and Propagation Diffraction Field Propagation Optical Imaging A Classic Imaging System Fourier Transform via a Lens First-Order Coherence of Light First-Order Temporal Coherence First-Order Spatial Coherence Second-Order Coherence of Light Second-Order Coherence of Coherent Light Second-Order Correlation of Chaotic-Thermal Radiation and the HBT Interferometer The Physical Cause of the HBT Phenomenon Near-Field Second-Order Spatial Coherence of Thermal Light Nth-Order Coherence of Light Nth-Order Near-Field Spatial Coherence of Thermal Light Homodyne Detection and Heterodyne Detection of Light Optical Homodyne and Heterodyne Detection Balanced Homodyne and Heterodyne Detection Balanced Homodyne Detection of Independent and Coupled Thermal Fields Quantum Theory of Light: Field Quantization and Measurement The Experimental Foundation-Part I: Blackbody Radiation The Experimental Foundation-Part II: Photoelectric Effect The Light Quantum and the Field Quantization Photon Number State of Radiation Field Coherent State of Radiation Field Density Operator and Density Matrix Composite System and Two-Photon State of Radiation Field A Simple Model of Incoherent and Coherent Radiation Source Pure State and Mixed State Product State, Entangled State, and Mixed State of Photon Pairs Time-Dependent Perturbation Theory Measurement of Light: Photon Counting Measurement of Light: Joint Detection of Photons Field Propagation in Space-Time Quantum Theory of Optical Coherence Quantum Degree of First-Order Coherence Photon and Effective Wavefunction Measurement of the First-Order Coherence or Correlation Quantum Degree of Second-Order Coherence Two-Photon Interference vs. Statistical Correlation of Intensity Fluctuations Second-Order Spatial Correlation of Thermal Light Photon Counting and Measurement of G(2) Quantum Entanglement EPR Experiment and EPR State Product State, Entangled State, and Classically Correlated State Entangled States in Spin Variables Entangled Biphoton State EPR Correlation of Entangled Biphoton System Subsystem in an Entangled Two-Photon State Biphoton in Dispersive Media Quantum Imaging Biphoton Imaging Ghost Imaging Ghost Imaging and Uncertainty Relation Thermal Light Ghost Imaging Classical Simulation of Ghost Imaging Turbulence-Free Ghost Imaging Two-Photon Interferometry I: Biphoton Interference Is Two-Photon Interference the Interference of Two Photons? Two-Photon Interference with Orthogonal Polarization Franson Interferometer Two-Photon Ghost Interference Delayed Choice Quantum Eraser Two-Photon Interferometry II: Quantum Interference of Chaotic Light Two-Photon Young's Interference Two-Photon Anticorrelation with Incoherent Chaotic Light Two-Photon Interference with Incoherent Orthogonal Polarized Chaotic Light Bell's Theorem and Bell's Inequality Measurement Hidden Variable Theory and Quantum Calculation for the Measurement of Spin 1/2 Bohm State Bell's Theorem and Bell's Inequality Bell States Bell State Preparation
Photon-photon correlations assume a pivotal significance in optical coherence. Recently, a new, to the best of our knowledge, type of quantum photonic states, the coherent state of photonic dimers, has been introduced, wherein the fundamental building blocks are two-photon bound states, instead of individual photons as in conventional lasers. In this Letter, we investigate the first-order coherence properties of the photonic-dimer coherent states, as well as the interference patterns in a double-slit interferometer setup, and compare with the coherence properties of other optical light sources, e.g., the conventional laser and the thermal light.
This thesis is a study of coherence theory in light in classical\nelectromagnetism and quantum optics. %The coherence is quantified\nSpecifically two quantities are studied: The degree of first-order\ntemporal coherence, which quantifies the field-field coherence, and\nthe degree of second-order coherence, quantifying the intensity-intensity\ncoherence. In the first part of the thesis these concepts are applied to classical\nelectric fields; to both the ideal plane wave and to chaotic light.\nWe then study how they can be measured using two interferometer\ntechnologies from optical astronomy, specifically with the\nMichelson stellar interferometer and the intensity interferometer.\n\nIn the second part we define the\nquantum degrees of first- and second-order coherence. These are calculated\nfor light in a quantum coherent state, in a Fock state and for light\nin a mixed thermal state. The results for the coherent state and\nthe thermal state are found to be analogous to\nthose obtained for the ideal plane wave and chaotic light, respectively,\nfrom the classical coherence theory seen in the first part.\n\nWe proceed to investigate the properties of the three-level laser with\nthe aim of showing that far above threshold it develops similar photon\nstatistics and values for the degrees of first- and second-order coherence,\nto light in a coherent state. The mechanism of phase-drift in the laser\nis also looked into. Subsequently the Mølmer-model is discussed, where it is\ndemonstrated that the coherent state is not a necessary construct, but\nmerely a convenient one, in describing phenomena in quantum optics.