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

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

James Schneeloch Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA Center for Coherence and Quantum Optics, University of Rochester, Rochester, New York 14627, USA Air Force Research Laboratory, Information Directorate, Rome, New York, 13441, USA    John C. Howell Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA Center for Coherence and Quantum Optics, University of Rochester, Rochester, New York 14627, USA
July 15, 2019

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.

03.67.Mn, 03.67.-a, 03.65-w, 42.50.Xa

I Introduction

In continuous-variable quantum information, there are many experiments using entangled photon pairs generated by spontaneous parametric downconversion (SPDC) 111There are many dozens (if not hundreds) of experimental papers either using or exploring the spatial entanglement between photon pairs from SPDC, but some papers representative of the scope of research are: Reid et al. (2009); Brougham and Barnett (2012); Tasca et al. (2011); Leach et al. (2012); Schneeloch et al. (2013); Edgar et al. (2012); Moreau et al. (2012); Howell et al. (2004); Walborn et al. (2011); Howland and Howell (2013); Ali Khan and Howell (2006); Walborn et al. (2010); Bennink et al. (2004); Barreiro et al. (2008); Mazzarella et al. (2013). In short, SPDC is a -nonlinear optical process occurring in birefringent crystals 222In order to produce SPDC, one does not necessarily need a birefringent crystal, but this is a popular way to ensure a constant phase relationship (also known as phase matching) between the pump photon, and the signal/idler photon pair. where high energy “pump” photons are converted into pairs of low energy “signal” and “idler” photons. In particular, the pump field interacts coherently with the electromagnetic quantum vacuum via a nonlinear medium in such a way that as an individual event, a pump photon is destroyed, and two daughter photons (signal and idler) are created (this event happening many times). As this process is a parametric process (i.e., one by definition in which the initial and final states of the crystal are the same), the total energy and total momentum of the field must each be conserved. Because of this, the energies and momenta of the daughter photons are highly correlated, and their joint quantum state is highly entangled. These highly entangled photon pairs may be used for any number of purposes, ranging from fundamental tests of quantum mechanics, to almost any application requiring (two-party) quantum entanglement.

In this tutorial, we discuss a particularly convenient and common variety of SPDC used in quantum optics experiments. In particular, we consider illuminating a nonlinear crystal with a collimated continuous-wave pump beam, and filtering the downcoverted light to collect only those photon pairs with frequencies nearly equal to each other (each being about half of the pump frequency). This degenerate collinear SPDC process is amenable to many approximations, especially considering that most optical experiments are done in the paraxial regime, where all measurements are taken relatively close to the optic axis (allowing many small-angle approximations). With this sort of experimental setup in mind, we discuss the theoretical treatment of such entangled photon pairs (from first principles) in sufficient detail so as to inform the understanding and curiosity of anyone seeking to discuss or undertake such experiments 333For more extensive treatments of spontaneous parametric downconversion, we recommend the Ph.D. theses of Lijun Wang Wang (1992), Warren Grice Grice (1997), and Paul Kwiat Kwiat (1993), as well as the Physics Reports article by S.P. Walborn et al.Walborn et al. (2010).. Indeed as we discuss all the necessary concepts preceding each approximation, much of this discussion will be useful in understanding non-collinear and non-degenerate SPDC as well.

The rest of this paper is laid out as follows. In Section 2, we discuss the derivation of the quantum biphoton field state in SPDC, as discussed in Hong and Mandel (1985), and Mandel and Wolf (1995). In addition to this, we point out what factors contribute not only to the shape of the biphoton wavefunction (defined later), but also to the magnitude of the amplitude for the biphoton generation to take place. This is important, as it determines the overall likelihood of downconversion events, and gives important details to look for in new materials in the hopes of creating brighter sources of entangled photon pairs. In Section 3, we simplify the biphoton wavefunction for the case of degenerate, collinear SPDC in the paraxial regime using the results in Monken et al. (1998). We also use geometrical arguments to explain the approximations allowed in the paraxial regime. In Section 4, we show how to further approximate this approximate biphoton wavefunction as a Double-Gaussian (as seen in Law and Eberly (2004) and Fedorov et al. (2009)), as the multivariate Gaussian density is well studied, and is easier to work with. In doing so, we give derivations of common statistical parameters of the Double-Gaussian wavefunction, showing its convenience in multiple applications. In Section 5, we provide a calculation of the transverse correlation width, defined as the standard deviation of the transverse distance between the signal and idler photons’ positions at the time of their creation. In Section 6, we explore the utility of the transverse correlation width, and introduce the concepts of the biphoton birth zone, and of the birth zone number as a measure of biphoton correlation. We conclude by using the birth zone number to gain a qualitative understanding of the tradeoff between the first-order spatial coherence and the measurable correlations between photon pairs in the downconverted fields.

Ii Foundation: The Quantum - Optical Calculation of the Biphoton state in SPDC

The procedure to quantize the electromagnetic field as it is used in quantum optics Mandel and Wolf (1995); Loudon (2000) (as opposed to quantum field theory), is to: decompose the electromagnetic field into a sum over (cavity) modes; find Hamilton’s equations of motion for each field mode; and assign to the classically conjugate variables (generalized coordinates and momenta), quantum-mechanically conjugate obervables, whose commutator is . From these field observables, one can obtain a Hamiltonian operator describing the evolution of the quantum electromagnetic field, and in so doing, describe the evolution of any quantum-optical system.

SPDC is a -nonlinear process. To describe it Mandel and Wolf (1995), we begin with the classical Hamiltonian of the electromagnetic field;


where . Since the electric field amplitude of the incident light on a nonlinear medium is usually substantially smaller than the electric field strength binding the atoms in a material together, we can express the polarization field as a power series in the electric field strength Boyd (2007), so that


Since the nonlinear interaction beyond second order is considered here to not appreciably affect the polarization, the classical Hamiltonian for the electromagnetic field can be broken up into two terms, one linear, and one nonlinear;




Next, since the nonlinear susceptibility depends on pump, signal, and idler frequencies444At this point we would like to point out that we use the Einstein summation convention for ., each of which are determined by their respective wave numbers, the nonlinear Hamiltonian is better broken down into its frequency components:


where subscripts and , are understood to refer to signal and idler modes, respectively.

To condense this paper, we note that when the field quantization is carried out, our electric field functions are replaced by the field observables , which separate into a sum of positive and negative frequency contributions , and , where


and is the hermitian conjugate of . Here, is an index indicating component of polarization, is a unit polarization vector, and is the photon annihilation operator at time . In addition, is the quantization volume 555The quantization volume is the volume of the hypothetical cavity containing the modes of the electromagnetic field. For simplicity, the cavity is taken to be rectangular, so the sum over modes is straightforward using boundary conditions in Cartesian coordinates. To get an accurate representation of the electromagnetic field in free space, we may take the quantization volume to be arbitrarily large., which in the standard quantization procedure, would be the volume of a cavity which can be taken to approach infinity for the free-space case.

With the electric field observables thus defined, we can obtain the quantum Hamiltonian of the electromagnetic field:


where the first term is the linear contribution to the Hamiltonian. Though this Hamiltonian looks relatively simple, the field operator , so that the integral in (II) actually contains eight terms. These terms correspond to all different processes (e.g., sum-frequency generation, difference-frequency generation, optical rectification, etc.), each of which has its own probability amplitude of occurring. However, given that we have a single input field (i.e., the pump field), and start with no photons in either of the signal and idler fields, the only energy-conserving contributions to the Hamiltonian (i.e., the only significant contributions 666The reason the non-energy-conserving terms in (II) can be neglected is due to the rotating wave approximation. In calculating the amplitude for the downconversion process, and converting to the interaction picture, all other contributions to this amplitude will have complex exponentials oscillating much faster than . Since each of these contributions (oscillating at frequency ) when integrated give amplitudes proportional to , and the propagation time through the crystal is fixed, these Sinc functions become negligibly small for large . Since is small for nearly degenerate SPDC, the energy-conserving contribution dominates over the non-conserving contributions.) are transitions (forward and backward) where pump photons are annihilated, and signal-idler photon pairs are created.

Our first approximation (beyond what was done to get (II) to begin with) is that the pump beam is bright enough to be treated classically, and that the pump intensity is not significantly diminished due to downconversion events. This “undepleted pump” approximation, along with keeping only the energy-conserving terms, gives us the simplified Hamiltonian:


which we then expand in the modes of the signal, and idler fields;


Note that here and throughout this paper, stands for hermitian conjugate.

Next, we assume the pump is sufficiently narrowband, so that we can, to a good approximation, separate out the time dependence of the pump field as a complex exponential of frequency . In addition, we assume the pump field to be sufficiently well-collimated so that, to a good approximation, we can also separate out the longitudinal dependence of the pump field 777For a reference that examines in detail how the downconverted light is affected by the pump spatial profile, we recommend the reference Pittman et al. (1996a). For a reference that treats SPDC with short pump pulses (as opposed to continuous-wave), see Keller and Rubin (1997).. At this point, we define the transverse momenta , , and ,as the projections of the pump wave vector , the signal wave vector , and the idler wave vector , onto the plane transverse to the optic axis respectively. We also define , , and , as the longitudinal components of the corresponding wave vectors. With this in mind, we express the pump field as an integral over plane waves:


By separating out the transverse components of the wave vectors, we make the Hamiltonian easier to simplify in later steps. As one additional simplification, we define the pump polarization vector , so that . With the transverse components separated out, and the narrowband pump approximation made, the Hamiltonian takes the form:


where we define , and .

In most experimental setups (including the one we consider here), the nonlinear crystal is a simple rectangular prism, centered at , and with side lengths , , and . Here, we assume the crystal is isotropic, so that does not depend on . To simplify the subsequent calculations, we assume the crystal to be embedded in a linear optical medium of the same index of refraction to avoid dealing with multiple reflections. Alternatively, we could assume the crystal has an anti-reflective coating to the same effect. We can then carry out the integral over the spatial coordinates (from to in each direction) (such that ), to get the Hamiltonian:


Note that the Sinc function, , is defined here as .

To obtain the state of the downconverted fields, one can readily use first-order time-dependent perturbation theory. To see why this is, we can compare the nonlinear classical Hamiltonian to the linear Hamiltonian using typical experimental parameters of a pump field intensity of , and signal/idler intensities of about . As such, the nonlinear contribution to the total Hamiltonian is indeed very small relative to the linear part, and the consequent results we obtain from these first order calculations should be quite accurate. Though this is the treatment we discuss, alternative higher order and non-perturbative derivations of the quantum state of down-converted light are also useful in examining the photon number statistics of down converted light, particularly when a sufficiently intense pump beam makes it significantly probable that multiple pairs will be generated at once through the simultaneous absorption of multiple pump photons. Indeed, the general two photon state is described as a multimode squeezed vacuum state, whose photon number statistics have been shown experimentally (and theoretically) to be such that the number of pairs created in a given time interval is approximately Poisson-distributed Avenhaus et al. (2008); Christ et al. (2011).

Using first-order time-dependent perturbation theory, the state of the signal and idler fields in the interaction picture can be computed as follows:


Note that in the interaction picture, operators evolve according to the unperturbed Hamiltonian, so that . Here, the initial state of the signal and idler fields is given to be the vacuum state , which means that the hermitian conjugate (with its lowering operators) will not contribute to the state of the downconverted photon fields.

Before we calculate the state of the downconverted photon fields (up to a normalization factor), we make use of some simplifying assumptions. First, we assume that the polarizations of the downconverted photons are fixed, so that we can neglect the sums over and , effectively making the sum over one value. With this, the sum over the components of the nonlinear susceptibility is proportional to the value , which is the effective experimentally determined coefficient for the nonlinear interaction. Second, we assume that the nonlinear crystal is much larger than the optical wavelengths considered here, so that the sums over and can be replaced by integrals in the following way:


With these simplifications, we can express the nonlinear Hamiltonian in such a way that both the sums are replaced by integrals, while still accurately reflecting the relative likelihood of downconversion events;


where , and , and is a constant. Note that here, and , to condense notation.

With one additional assumption, that the slowly-varying pump amplitude (excluding ) is essentially constant over the time light takes to propagate through the crystal, the integral over this nonlinear Hamiltonian becomes an integral of a constant times . With this integral, we get our first look at the state of the downconverted field exiting the crystal:


Here, is the time it takes light to travel through the crystal; is the intensity of the pump beam; is the (Fock) vacuum state with zero photons in the signal mode and zero photons in the idler mode, and;

is, up to a normalization constant, the biphoton wavefunction in momentum space (where is the normalized pump amplitude spectrum). To see how this works, we note that the biphoton probability amplitude can be expressed as . When we normalize this probability amplitude, by integrating its magnitude square over all values of , and , and setting this integral equal to unity, the resulting normalized probability amplitude has the necessary properties (for our purposes) of a biphoton wavefunction 888Though it is debatable whether it is correct to speak of a biphoton wavefunction since expectation values are in fact carried out with , and does not evolve according to the Schrödinger equation ( does, though), is a square-integrable function in a two- particle joint Hilbert space that accurately describes the relative measurement statistics of the biphotons.. With approximations to be made in the next section, only will govern the transverse momentum probability distribution of the biphoton field999Note that the biphoton wavefunction (II) is expressed as an integral over the rectangular crystal shape. For those interested in a derivation of the integral giving the biphoton wavefunction for a generalized crystal shape, see Saldanha and Monken (2013).

The factors preceding the biphoton wavefunction are still important to understand because (with some algebra) they contribute to the rate of downconversion events . In particular 101010Note that this downconversion rate comes from the approximation of near-perfect energy conservation: i.e., . Helt et al. (2012):


where is the pump power (in Watts). We also note from equation (16) that the rate of downconversion events is also proportional to , though this factor is essentially unity for the nearly degenerate frequencies of the signal and idler photons considered here. This proportionality also follows from more rigorous calculations of the rate of downconversion events Hong and Mandel (1985); Kleinman (1968), though only in the approximation where the minuscule signal/idler fields don’t appreciably contribute to the likelihood of downconversion events. In those more rigorous calculations, the conversion efficiency (biphotons made per incident pump photon) is of the order , which again shows just how weak these signal/idler fields are relative to the pump field 111111Including the collection/coupling efficiencies in many quantum-optical experiments, the measured conversion efficiency is closer to ., and why first-order perturbation theory is sufficient to get a reasonably accurate representation of the state of the downconverted fields. We also note that although beam size doesn’t affect the global rate of downconversion events, it does affect the fraction of those downconversion events that are likely to be counted by a detector near the optic axis 121212The rate of downconversion events yielding biphotons propagating close to the optic axis increases with a smaller beam size, but only to a point. For a good summary, see Ling et al. (2008). For a more detailed discussion on how focusing affects the fraction of downconverted light propagating near the optic axis, see Ljunggren and Tengner (2005). For a more rigorous discussion of how the rate of downconversion events (i.e., the signal/idler power) changes with the crystal length, see Loudon (2000).. Even so, these factors are useful to know when selecting a crystal as a source of entangled photon pairs. For example, with a constant power pump beam, a longer crystal will be a brighter source of photon pairs. However, there is a tradeoff; the degree of correlation between the signal and idler photons decreases with increasing crystal length (as we shall show).

Iii Approximation for degenerate collinear SPDC

To obtain a relatively simple expression for the biphoton field in SPDC, we have made multiple (though reasonable) simplifying assumptions. We have assumed that the pump is narrowband and collimated so that it is nearly monochromatic, while also having a momentum spectrum whose longitudinal components dominate over its transverse components. We next assumed that the pump is bright enough to be treated classically, but not so bright that the perturbation series approximation to the nonlinear polarization breaks down. In addition, we assumed that we need not consider multiple reflections, and that the crystal is large compared to an optical wavelength so that sums over spatial modes may be replaced by integrals. We have also assumed that the pump is bright enough that it is not attenuated appreciably due to downconversion events.

Now, we consider the experimental case where we place frequency filters over photon detectors, so that we may only examine downconversion events which are degenerate (where ),and perfectly energy-conserving (). In this case, along with all the previous assumptions made, we define a new constant of normalization (absorbing factors outside the integrals), and obtain the following simplified expression for the state of the downconverted field as seen in Monken et al. (1998);


Here, the biphoton wavefunction is as defined previously (II). Next, we use the fact that the transverse dimensions of the crystal are much larger than the pump wavelength to carry out the integral over the transverse pump momentum.


The significant contributions of the Sinc functions to the integral will come from when, for example, , where is the ratio of the width of the crystal to the pump wavelength . Where the crystal is much wider than a pump wavelength, is large, and we see the Sinc function will only contribute significantly when is only a very small fraction of . Thus, with a renormalization, the sincs act like delta functions, setting , and giving us the biphoton wavefunction:


where is a normalization constant.

Figure 1: Diagram of the relationship between the pump, signal, and idler momenta, in standard (not necessarily collinear) phase matching.

Since most experiments are done in the paraxial regime, we use such approximations to get the Sinc-Gaussian biphoton wavefunction, ubiquitous in the literature. With the previous assumptions already made, we point out that in degenerate, collinear SPDC, , which we redefine as ,, and , to simplify notation. In addition, since the transverse pump momentum is essentially equal to the sum of the transverse signal and idler momenta, the three vectors can be readily drawn on a plane, as seen in Fig. 1.

Let be the angle between the pump momentum and the signal or idler momentum vectors and . This angle is small enough that we may use the small-angle approximation to find an expression for in terms of easier-to-measure quantities. Using the conservation of each component of the total momentum, we get the following equations:


Using the small-angle approximation, and substituting one equation into the other, we find:


Finally, when we assume the transverse pump momentum profile is a Gaussian,


with being the pump radius in position space 131313The pump radius in position space is defined as the standard deviation of . This is justified by noting that the pump radius in momentum space is explicitly given by the standard deviation of , and using the properties of Fourier transformed Gaussian wavefunctions., we renormalize, and find the biphoton wavefunction to be:


where the minus sign in the argument of the sinc function is eliminated since the sinc function is an even function. Interestingly, we can use the radius to the first zero of the Sinc function along with (22) to derive a simple formula for the half-angle divergence of the degenerate collinear SPDC light:


In experiments where no filtering takes place to isolate the degenerate portion of the SPDC light, this angle will be larger since the non-degenerate frequencies of SPDC light have a wider, ring-shaped distribution.

To obtain a transverse correlation width from this biphoton wavefunction, we need to transform it to position space. Fortunately, this biphoton wavefunction is approximately 141414For small values of and , . For typical experimental parameters, the argument of the Sinc function is of the order , even for transverse momenta as large as the pump momentum. With the paraxial approximation, the transverse momenta are much smaller than the pump momentum, and so the arguments of the Sinc functions are very small indeed. separable (subject to our paraxial approximation) into horizontal and vertical wavefunctions (i.e., into a product of functions, one dependent on only x-coordinates, and the other dependent on only y-coordinates). In addition, we can find an orthogonal set of coordinates in terms of sums and differences of momenta that allows us to transform this wavefunction by transforming the Sinc function and Gaussian individually. While transforming the Gaussian is extremely straightforward, transforming the concurrent Sinc-based function is more challenging, owing to that it is a Sinc function of the square of a momentum coordinate, and is not in most dictionaries of transforms.

Iv The Double-Gaussian Approximation

In what follows here, we approximate the Sinc-Gaussian biphoton momentum-space wavefunction (26) as a Double-Gaussian function (as seen in Law and Eberly (2004) andFedorov et al. (2009)), by matching the second order moments in the sums and differences of the transverse momenta. Transforming this approximate wavefunction to position space, and computing the correlation width gives us an estimate of the true correlation width seen experimentally that we later compare with more exact calculations and experimental data. In addition, we take a moment to explore the conveniences that come with the Double-Gaussian wavefunction.

In this analysis, we consider only the horizontal components of the transverse momenta, since the statistics are identical (with our approximations) in both transverse dimensions. The transverse pump profile is already assumed to be a Gaussian. Our first step is to transform to a rotated set of coordinates to separate the Sinc function from the Gaussian.



With these rotated coordinates, the (horizontal) biphoton wavefunction becomes:


Taking the modulus-squared and integrating over , we isolate the probability density for :


where , for convenience. is an even function, so its first-order moment, the expectation . The second-order moment is nonvanishing, with a value . With this second order moment, we can fit to a Gaussian by matching these moments. In doing so, is approximately a Gaussian with width .

To see how good this Gaussian approximation of is, we show in Fig. 2, both the Sinc-based probability density (in momentum space) and the approximate Gaussian density with matched moments. The overall scale of the central peak is captured but the shape is significantly different. However, a Gaussian probability density for the position difference density, , appropriately scaled is a good approximation for values near the central peak (though the oscillatory behavior of the wings is still not captured). In Fig. 3, we plot various choices of an approximate Gaussian density for our transformed Sinc-based position difference density function . We find that simply setting the central maxima of both densities equal to each other works very well, as discussed in the next section.

Figure 2: Plot comparing estimates of the momentum difference probability density . The solid (blue) curve with wavy side-bands gives our Sinc-based probability density estimate (30), where we set for convenience. The dashed (magenta) Gaussian curve gives our Gaussian-based probability density estimate with matching means and variances.

By approximating the Sinc-Gaussian wavefunction as a Double-Gaussian wavefunction, the inverse Fourier transform to position space becomes very straightforward. We note that and form an orthogonal pair of coordinates, as and do. Because of this, the inverse Fourier transform is separable 151515The Fourier transform convention we use is the unitary convention: , and . Since the Fourier transform is invariant under rotations (i.e., since the argument in the exponential can be thought of as an inner product between two vectors), we get identical formulas for the Fourier transform in rotated coordinates. In particular, we find ., and we find:


where , , and where


Figure 3: Plot comparing different estimates of . The solid blue wavy curve is our most accurate estimate from the transformed Sinc-based distribution (V). The tall dashed (magenta) curve is the Gaussian distribution obtained from matching momentum means and variances, while the shallow dashed (red) curve is the Gaussian distribution obtained by matching position means and variances. The solid (green) curve gives us a refined Gaussian approximation, by setting the central maximums equal to one another. The flat (gold) line, gives the height of the half maximum of the Sinc-based probability density (V) (blue curve). We see that the widths of half maximum are nearly identical (off by less than ) for the Sinc-based and refined Gaussian distributions Again, we set for convenience.

iv.1 Usefulness of the Double-Gaussian approximation

Here, we digress to discuss the usefulness of the Double-Gaussian approximation. To begin, we express and in terms of and , and take the magnitude-squared of to get a Double-Gaussian probability density :


Figure 4: Plot of the Double-Gaussian probability density for unit and units. The horizontal dotted line indicates a particular value of , so that the half-width of the Gaussian along that dotted line is the conditioned half-width . The equivalent Sinc-Gaussian distribution will have subtle side bands (at about the maximum intensity) parallel to the long axis of this Double-Gaussian as in Fig. 3.
Name Value
marginal means
conditioned mean
marginal variance
conditioned variance
Pearson r value
joint entropy
marginal entropy ;
mutual information ;
Probability notation
Table 1: Statistics of the Double Gaussian

Here, we define the transverse correlation width as the standard deviation of the distance between and (i.e., ). This is not defined as a half-width, since it represents the full width (at of the maximum) of the signal and idler photons’ position distributions conditioned on the location of the prior pump photon (see Section VI). For the Double-Gaussian density, the transverse correlation width, , is .

Alternatively, the Double-Gaussian density can be put into the standard form of a bi-variate Gaussian density function;




This Double-Gaussian probability density has a number of useful properties. First, it is separable into single Gaussians in rotated coordinates, making many integrals straightforward to do analytically. Second, the marginal and conditional probability densities of the Double Gaussian density function are also Gaussian density functions. Because of this, many statistics of the Double-Gaussian density have particularly simple forms. For examples, consider the statistics in Table 1.

In addition, the Double-Gaussian is uniquely defined by its marginal and conditioned means and variances. As seen in Fig. 4, these values give a straightforward characterization of the overall shape of the Double-Gaussian distribution.

iv.1.1 Fourier-Transform Limited properties of the Double-Gaussian

Gaussian wavefunctions are minimum uncertainty wavefunctions in that they are Fourier-transform limited; Heisenberg’s uncertainty relation;


is satisfied with equality. The Double-Gaussian wavefunction (33) factors into a product of two Gaussians (one in and the other in ), and so the standard deviations of these rotated coordinates also saturate the Heisenberg relation:


Remarkably, the simple expressions for the statistics of the double-Gaussian distribution (see Table 1) show that these are not the only pairs that are related this way. Since conditioning measurements on a single ensemble of events doesn’t change the fact that those measurements must satisfy an uncertainty relation, we find:


is still a valid uncertainty relation. In addition, since conditioning on average reduces the variance 161616That conditioning on average reduces the variance can be seen from the law of total variance. Given two random variables and , the variance of is equal to the mean over of the conditioned variance plus the variance over of the conditioned mean . Both of these terms are non negative, so the mean conditioned variance never exceeds the unconditioned variance., we arrive at the relations:


These relations are also useful for understanding how narrowband frequency filters undermine the resolution of temporal correlations as discussed in Appendix C. For the Double-Gaussian state, these relations are saturated as well. From these properties, we may find many other useful identities for the double-Gaussian including:


where and is the Pearson correlation coefficient for the position and momentum statistics of the Double-Gaussian, respectively.

iv.1.2 Propagating the Double-Gaussian field

One especially useful aspect of the Double-Gaussian wavefunction, is that it is simple to propagate (in the paraxial regime). Given the transverse momentum amplitude profile of a nearly monochromatic optical field in one transverse plane, we can find the transverse momentum profile at another optical plane by multiplying it by the paraxial free-space transfer function 171717The free space transfer function comes about due to the momentum decomposition of an optical field being a sum (or integral) over plane waves. For each plane wave defined by , , and , we add a phase corresponding to the plane wave translating a total forward distance . The particular form of the free space transfer function used here is due to the small angle- or paraxial approximation. For a good reference on this topic, see Goodman et al. (1968).:


For an entangled pair of optical fields at half the pump frequency, the full transfer function becomes:


Since a global constant phase can come outside the Fourier transform integral, and the relative phases and amplitudes in position space will be independent of this factor, we can remove it from the transfer function, and express the remaining transfer function simply as a product of a horizontal and vertical transfer function: