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arXiv:1807.10885 · Journal of Optics

Introduction to the absolute brightness and number statistics in spontaneous parametric down-conversion

James Schneeloch, Samuel H. Knarr, Daniela F. Bogorin, Mackenzie L Levangie, Christopher C. Tison, Rebecca Frank, Gregory A. Howland, Michael L. Fanto, Paul M. Alsing

Published Feb 8, 2019• 88 citations
arXivHTMLPDF

Abstract

Abstract As a tutorial, we examine the absolute brightness and number statistics of photon pairs generated in spontaneous parametric down-conversion (SPDC) from first principles. In doing so, we demonstrate how the diverse implementations of SPDC can be understood through a single common framework, and use this to derive straightforward formulas for the biphoton generation rate (pairs per second) in a variety of different circumstances. In particular, we consider the common cases of both collimated and focused Gaussian pump beams in a bulk nonlinear crystal, as well as in nonlinear waveguides and micro-ring resonators. Furthermore, we examine the number statistics of down-converted light using a non-perturbative approximation (the multi-mode squeezed vacuum), to provide quantitative formulas for the relative likelihood of multi-pair production events, and explore how the quantum state of the pump affects the subsequent statistics of the down-converted light. Following this, we consider the limits of the undepleted pump approximation, and conclude by performing experiments to test the effectiveness of our theoretical predictions for the biphoton generation rate in a variety of different sources.

DOI
10.1088/2040-8986/ab05a8

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Comments on: "Introduction to the Absolute Brightness and Number Statistics in Spontaneous Parametric Down-Conversion" (2019 J. Opt. 21, 043501)

James Schneeloch

These comments contain two primary improvements to the aforementioned article: Where previously mentioned in passing in section 3.4 of our article, we now provide, using the same basic steps as in Bennink's paper, [Phys. Rev. A 81, 053805 (2010)] (with some expanded discussion), a full derivation for the absolute generation rate of photon pairs from Spontaneous-Parametric Down-Conversion (SPDC) when using a focused gaussian pump beam as input, and collecting the output biphotons in a pair of focused gaussian collection modes. We do this to both show consistency between our formalism and his treatment, and to be able to use it to predict the absolute brightness of photon pair generation in these sources. In doing so, our formula differs from that in [Phys. Rev. A 81, 053805 (2010)] by only a constant factor based on the ratio of products of indices of refraction. Second, we correct an abuse of notation in our main work in which the quantization dimensions of the cavity modes of the electromagnetic field are conflated with the dimensions of the nonlinear medium. While this does not meaningfully affect the final formulas we obtain for the generation rates in the paper (adding a dimensionless correction factor of order unity), we use the gaussian-mode SPDC derivation here to show how the factors of the quantization dimensions disappear when converting from a discrete-momentum basis of rectangular cavity modes to a continuous frequency basis of gaussian spatial modes. In doing this, we find that applying this correction factor to our previous theoretical predictions for the photon pair generation rates brings them closer to the obtained experimental values.

arXiv:2503.02062•Mar 3, 2025•0 citations

Introduction to the transverse spatial correlations in spontaneous parametric down-conversion through the biphoton birth zone

James Schneeloch, John C Howell

Abstract As a tutorial to the spatial aspects of spontaneous parametric downconversion (SPDC), we present a detailed first-principles derivation of the transverse correlation width of photon pairs in degenerate collinear SPDC. This width defines the size of a biphoton birth zone, the region where the signal and idler photons are likely to be found when conditioning on the position of the destroyed pump photon. Along the way, we discuss the quantum-optical calculation of the amplitude for the SPDC process, as well as its simplified form for nearly collinear degenerate phase matching. Following this, we show how this biphoton amplitude can be approximated with a double-Gaussian wavefunction, and give a brief discussion of the measurement statistics (and subsequent convenience) of such double-Gaussian wavefunctions. Next, we use this approximation to get a simplified estimation of the transverse correlation width, and compare it to more accurate calculations as well as experimental results. We then conclude with a discussion of the concept of a biphoton birth zone, using it to develop intuition for the tradeoff between the first-order spatial coherence and bipohoton correlations in SPDC.

arXiv:1502.06996•Journal of Optics•Apr 4, 2016•142 citations

Introduction to the Transverse Spatial Correlations in Spontaneous Parametric Down-Conversion through the Biphoton Birth Zone

James Schneeloch, John C. Howell

As a tutorial to the spatial aspects of Spontaneous Parametric Downconversion (SPDC), we present a detailed first-principles derivation of the transverse correlation width of photon pairs in degenerate collinear SPDC. This width defines the size of a biphoton birth zone, the region where the signal and idler photons are likely to be found when conditioning on the position of the destroyed pump photon. Along the way, we discuss the quantum-optical calculation of the amplitude for the SPDC process, as well as its simplified form for nearly collinear degenerate phase matching. Following this, we show how this biphoton amplitude can be approximated with a Double-Gaussian wavefunction, and give a brief discussion of the measurement statistics (and subsequent convenience) of such Double-Gaussian wavefunctions. Next, we use this approximation to get a simplified estimation of the transverse correlation width, and compare it to more accurate calculations as well as experimental results. We then conclude with a discussion of the concept of a biphoton birth zone, using it to develop intuition for the tradeoff between the first-order spatial coherence and bipohoton correlations in SPDC.

•Jan 16, 2016•111 citations

Modelling parametric down-conversion yielding spectrally pure photon pairs

Fabian Laudenbach, Hannes Hübel, Michael Hentschel +2 more

Pair creation by spontaneous parametric down-conversion (SPDC) has become a reliable source for single-photon states, used in many kinds of quantum information experiments and applications. In order to be spectrally pure, the two photons within a generated pair should be as frequency-uncorrelated as possible. For this purpose most experiments use narrow bandpass filters, having to put up with a drastic decrease in count rates. This article elaborates (theoretically and by numerical evaluation) the alternative method to engineer a setup such that the SPDC-generated quantum states are intrinsically pure. Using pulsed pump lasers and periodically poled crystals this approach makes bandpass filtering obsolete and allows for significantly higher output intensities and therefore count rates in the detectors. After numerically scanning all common wavelength regimes, polarisation configurations and three different non-linear crystals, we present a broad variety of setups which allow for an implementation of this method.

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Maximizing the validity of the Gaussian approximation for the biphoton state from parametric down-conversion

Baghdasar Baghdasaryan, Fabian Steinlechner, S. Fritzsche

Spontaneous parametric down-conversion (SPDC) is widely used in quantum applications based on photonic entanglement. The efficiency of photon pair generation is often characterized by means of a $\mathrm{sinc}(L\mathrm{\ensuremath{\Delta}}k/2)$ function, where $L$ is the length of the nonlinear medium and $\mathrm{\ensuremath{\Delta}}k$ is the phase mismatch between the pump and down-converted fields. In theoretical investigations, the $\mathrm{sinc}$ behavior of the phase mismatch has often been approximated by a Gaussian function $exp(\ensuremath{-}\ensuremath{\alpha}{x}^{2})$ in order to derive analytical expressions for the SPDC process. Different values have been chosen in the literature for the optimization factor $\ensuremath{\alpha}$, for instance, by comparing the widths of $\mathrm{sinc}$ and Gaussian functions or the momentum of down-converted photons. As a consequence, different values of $\ensuremath{\alpha}$ provide different theoretical predictions for the same setup. Therefore an informed and unique choice of this parameter is necessary. In this paper, we present a choice of $\ensuremath{\alpha}$ which maximizes the validity of the Gaussian approximation. Moreover, we also discuss the so-called super-Gaussian and cosine-Gaussian approximations as practical alternatives with improved predictive power for experiments.

arXiv:2210.02340•Physical Review A•Dec 20, 2022•8 citations

Spectral and statistical properties of high-gain parametric down-conversion

Kirill Yu. Spasibko

Parametric down-conversion (PDC) is mostly known in the low-gain (spontaneous) regime, in which the correlated photon pairs are produced. Spontaneous PDC (SPDC) plays a very important role for quantum optics as a variety of quantum states is produced via SPDC. In the high-gain case PDC leads to generation of bright states having up to hundreds mW mean power. With such states almost any nonlinear optical interaction or light-matter interaction becomes more efficient. Even being macroscopically bright, the produced states maintain nonclassical properties as, for example, the fluctuations of electric field quadratures are squeezed below the shot-noise level. The high-gain PDC could be used not only in the same applications as SPDC, it also can provide new ones. Apart from that the high-gain PDC has many remarkable spectral and statistical properties, which are in the focus of this work. The description starts from the PDC generation in normal and anomalous group velocity dispersion ranges. The spectrum and mode content of high-gain PDC is considered as well as their change with the parametric gain are demonstrated. Then, there are the interference effects emerging from the PDC correlations presented, namely the macroscopic analogue of the Hong-Ou-Mandel interference. In addition, it is shown how spatial and temporal walk-off matching could be used for the generation of giant narrowband twin beams. Finally, the statistical properties of high-gain PDC are reviewed as well as their use for multiphoton effects is demonstrated. Photon-number fluctuations of PDC are studied via normalized correlation functions and probability distributions. These fluctuations enhance the generation efficiency for multiphoton effects by orders of magnitude and lead to tremendously fluctuating light described by heavy-tailed photon-number probability distributions.

arXiv:2007.12999•Jul 25, 2020•4 citations
Abstract As a tutorial, we examine the absolute brightness and number statistics of photon pairs generated in spontaneous parametric down-conversion (SPDC) from first principles. In doing so, we demonstrate how the diverse implementations of SPDC can be understood through a single common framework, and use this to derive straightforward formulas for the biphoton generation rate (pairs per second) in a variety of different circumstances. In particular, we consider the common cases of both collimated and focused Gaussian pump beams in a bulk nonlinear crystal, as well as in nonlinear waveguides and micro-ring resonators. Furthermore, we examine the number statistics of down-converted light using a non-perturbative approximation (the multi-mode squeezed vacuum), to provide quantitative formulas for the relative likelihood of multi-pair production events, and explore how the quantum state of the pump affects the subsequent statistics of the down-converted light. Following this, we consider the limits of the undepleted pump approximation, and conclude by performing experiments to test the effectiveness of our theoretical predictions for the biphoton generation rate in a variety of different sources.
These comments contain two primary improvements to the aforementioned article: Where previously mentioned in passing in section 3.4 of our article, we now provide, using the same basic steps as in Bennink's paper, [Phys. Rev. A 81, 053805 (2010)] (with some expanded discussion), a full derivation for the absolute generation rate of photon pairs from Spontaneous-Parametric Down-Conversion (SPDC) when using a focused gaussian pump beam as input, and collecting the output biphotons in a pair of focused gaussian collection modes. We do this to both show consistency between our formalism and his treatment, and to be able to use it to predict the absolute brightness of photon pair generation in these sources. In doing so, our formula differs from that in [Phys. Rev. A 81, 053805 (2010)] by only a constant factor based on the ratio of products of indices of refraction. Second, we correct an abuse of notation in our main work in which the quantization dimensions of the cavity modes of the electromagnetic field are conflated with the dimensions of the nonlinear medium. While this does not meaningfully affect the final formulas we obtain for the generation rates in the paper (adding a dimensionless correction factor of order unity), we use the gaussian-mode SPDC derivation here to show how the factors of the quantization dimensions disappear when converting from a discrete-momentum basis of rectangular cavity modes to a continuous frequency basis of gaussian spatial modes. In doing this, we find that applying this correction factor to our previous theoretical predictions for the photon pair generation rates brings them closer to the obtained experimental values.
Abstract As a tutorial to the spatial aspects of spontaneous parametric downconversion (SPDC), we present a detailed first-principles derivation of the transverse correlation width of photon pairs in degenerate collinear SPDC. This width defines the size of a biphoton birth zone, the region where the signal and idler photons are likely to be found when conditioning on the position of the destroyed pump photon. Along the way, we discuss the quantum-optical calculation of the amplitude for the SPDC process, as well as its simplified form for nearly collinear degenerate phase matching. Following this, we show how this biphoton amplitude can be approximated with a double-Gaussian wavefunction, and give a brief discussion of the measurement statistics (and subsequent convenience) of such double-Gaussian wavefunctions. Next, we use this approximation to get a simplified estimation of the transverse correlation width, and compare it to more accurate calculations as well as experimental results. We then conclude with a discussion of the concept of a biphoton birth zone, using it to develop intuition for the tradeoff between the first-order spatial coherence and bipohoton correlations in SPDC.
As a tutorial to the spatial aspects of Spontaneous Parametric Downconversion (SPDC), we present a detailed first-principles derivation of the transverse correlation width of photon pairs in degenerate collinear SPDC. This width defines the size of a biphoton birth zone, the region where the signal and idler photons are likely to be found when conditioning on the position of the destroyed pump photon. Along the way, we discuss the quantum-optical calculation of the amplitude for the SPDC process, as well as its simplified form for nearly collinear degenerate phase matching. Following this, we show how this biphoton amplitude can be approximated with a Double-Gaussian wavefunction, and give a brief discussion of the measurement statistics (and subsequent convenience) of such Double-Gaussian wavefunctions. Next, we use this approximation to get a simplified estimation of the transverse correlation width, and compare it to more accurate calculations as well as experimental results. We then conclude with a discussion of the concept of a biphoton birth zone, using it to develop intuition for the tradeoff between the first-order spatial coherence and bipohoton correlations in SPDC.
Pair creation by spontaneous parametric down-conversion (SPDC) has become a reliable source for single-photon states, used in many kinds of quantum information experiments and applications. In order to be spectrally pure, the two photons within a generated pair should be as frequency-uncorrelated as possible. For this purpose most experiments use narrow bandpass filters, having to put up with a drastic decrease in count rates. This article elaborates (theoretically and by numerical evaluation) the alternative method to engineer a setup such that the SPDC-generated quantum states are intrinsically pure. Using pulsed pump lasers and periodically poled crystals this approach makes bandpass filtering obsolete and allows for significantly higher output intensities and therefore count rates in the detectors. After numerically scanning all common wavelength regimes, polarisation configurations and three different non-linear crystals, we present a broad variety of setups which allow for an implementation of this method.
Spontaneous parametric down-conversion (SPDC) is widely used in quantum applications based on photonic entanglement. The efficiency of photon pair generation is often characterized by means of a $\mathrm{sinc}(L\mathrm{\ensuremath{\Delta}}k/2)$ function, where LLL is the length of the nonlinear medium and $\mathrm{\ensuremath{\Delta}}k$ is the phase mismatch between the pump and down-converted fields. In theoretical investigations, the sinc\mathrm{sinc}sinc behavior of the phase mismatch has often been approximated by a Gaussian function $exp(\ensuremath{-}\ensuremath{\alpha}{x}^{2})$ in order to derive analytical expressions for the SPDC process. Different values have been chosen in the literature for the optimization factor $\ensuremath{\alpha}$, for instance, by comparing the widths of sinc\mathrm{sinc}sinc and Gaussian functions or the momentum of down-converted photons. As a consequence, different values of $\ensuremath{\alpha}$ provide different theoretical predictions for the same setup. Therefore an informed and unique choice of this parameter is necessary. In this paper, we present a choice of $\ensuremath{\alpha}$ which maximizes the validity of the Gaussian approximation. Moreover, we also discuss the so-called super-Gaussian and cosine-Gaussian approximations as practical alternatives with improved predictive power for experiments.
Parametric down-conversion (PDC) is mostly known in the low-gain (spontaneous) regime, in which the correlated photon pairs are produced. Spontaneous PDC (SPDC) plays a very important role for quantum optics as a variety of quantum states is produced via SPDC. In the high-gain case PDC leads to generation of bright states having up to hundreds mW mean power. With such states almost any nonlinear optical interaction or light-matter interaction becomes more efficient. Even being macroscopically bright, the produced states maintain nonclassical properties as, for example, the fluctuations of electric field quadratures are squeezed below the shot-noise level. The high-gain PDC could be used not only in the same applications as SPDC, it also can provide new ones. Apart from that the high-gain PDC has many remarkable spectral and statistical properties, which are in the focus of this work. The description starts from the PDC generation in normal and anomalous group velocity dispersion ranges. The spectrum and mode content of high-gain PDC is considered as well as their change with the parametric gain are demonstrated. Then, there are the interference effects emerging from the PDC correlations presented, namely the macroscopic analogue of the Hong-Ou-Mandel interference. In addition, it is shown how spatial and temporal walk-off matching could be used for the generation of giant narrowband twin beams. Finally, the statistical properties of high-gain PDC are reviewed as well as their use for multiphoton effects is demonstrated. Photon-number fluctuations of PDC are studied via normalized correlation functions and probability distributions. These fluctuations enhance the generation efficiency for multiphoton effects by orders of magnitude and lead to tremendously fluctuating light described by heavy-tailed photon-number probability distributions.