Quantifying Tripartite Spatial and Energy-Time Entanglement in Nonlinear Optics

In this work, we provide a means to quantify genuine tripartite entanglement in arbitrary (pure and mixed) continuous-variable states as measured by the Tripartite Entanglement of formation -- a resource-based measure quantifying genuine multi-partite entanglement in units of elementary Greenberger-Horne-Zeilinger (GHZ) states called gebits. Furthermore, we predict its effectiveness in quantifying the tripartite spatial and energy-time entanglement in photon triplets generated in cascaded spontaneous parametric down-conversion (SPDC), and find that ordinary nonlinear optics can be a substantial resource of tripartite entanglement.


I. INTRODUCTION
As quantum networking and computing platforms grow more sophisticated, it is more important than ever to develop means of efficiently characterizing the quantum resources present. To that end, many advances have been made over the last few years in quantifying the entanglement present between two groups of increasingly high-dimensional systems [1,2]. However, when it comes to quantifying multi-partite entanglement in highdimensional systems, the field remains relatively underdeveloped with notable exceptions [3,4] using generalizations of the three-tangle [5], a monotone based on the residual entanglement [6], which identifies most but not all tripartite entangled states. Recently, resourcebased entanglement measures that both faithfully identify all multi-partite entangled states, and are additive over copies of the state to be measured have been developed [7,8], but the fundamental challenge of efficiently quantifying genuine multi-partite entanglement in highdimensional systems remains to be answered [9].
As experimental sources of entanglement have a finite preparation uncertainty, strategies toward quantifying multi-partite entanglement must be applicable to mixed as well as pure states. To that end, there do exist multiple witnesses of genuine continuous-variable tripartite entanglement that apply to arbitrary quantum states [10][11][12][13][14], which have been used successfully in experiment [15]. In general, witnesses of genuine tripartite entanglement date back as early as 1987 [16], but quantifying more than a nonzero amount of tripartite entanglement present has remained elusive.
Within the last year, the challenge of quantifying genuine tripartite entanglement in mixed states has been answered in part in [17], where correlations between observables of qudits were used to place a lower bound on the tripartite entanglement of formation E 3F -a measure of genuine tripartite entanglement that compares the * james.schneeloch.1@us.af.mil arbitrary state being measured to a comparable number of three-qubit GHZ states, known as three-party gebits. This strategy to quantify genuine tripartite entanglement was explicitly dependent on the dimension d of the quantum systems, so that adapting it for continuous-variable degrees of freedom remained an open challenge [18].
In this work, we present a strategy to quantify genuine tripartite entanglement of arbitrary (pure and mixedstate) continuous-variable systems using the correlations naturally present in many of these systems. In particular, we examine the tri-partite spatial and energy-time entanglement present in cascaded χ (2) spontaneous parametric down-conversion (SPDC), in which one pump photon is split into two daughter photons, followed by one daughter photon down-converting into two granddaughter photons. The spatial correlations in this system are qualitatively identical to those in χ (3) SPDC (a single-step photon triplet generation process in nonlinear optics), and generation rates are comparable with one another [19][20][21][22], but we focus on cascaded SPDC, as this process is more well-studied [23].

II. FOUNDATIONS AND MOTIVATION: QUANTIFYING GENUINE TRIPARTITE CONTINUOUS-VARIABLE ENTANGLEMENT
Entanglement is defined with respect to separability. Any quantum stateρ of three parties A, B, and C, that factors out into a product of states for each party, or any mixture of such factorable states is defined to be separable:ρ All other states are entangled. With more than two parties, there are multiple forms of separability, defining multiple forms of entanglement. For example, states of the form A ⊗ BC: are known as biseparable because they can be expressed as mixtures of states that factor out as a product of two terms -in this case, one for A and another for the joint state of BC. To demonstrate tri-partite entanglement, the state must at least be in no way biseparable, but there is more to it. Proving ABC is genuinely tripartite entangled requires showing not just that the state is outside all three classes of biseparable states (i.e., A ⊗ BC, B ⊗ AC and C ⊗ AB), but that the state cannot be made out of any arbitrary mixture of states coming from one or more of these classes. This distinction is important because it is possible to combine biseparable states from multiple classes to obtain mixed states that are outside all of these sets. Such fully inseparable states are not genuinely tripartite entangled. The measure of genuine tripartite entanglement we will be using in this paper is the tripartite entanglement of formation E 3F , which for parties A, B, and C, is given by: where the first minimum is taken over all pure state decompositions ofρ ABC and the second minimum is of the entanglement entropy over all bipartitions of each constituent pure state in the decomposition. This measure, first discussed in [7], is a generalization of the regular entanglement of formation, and is: (1) greater than zero if and only if the state is genuinely tripartite entangled; (2) invariant under local unitary transformations; (3) monotonically decreasing under local operations and classical communication (LOCC), and (4) at least additive over tensor products for pure states [24], so that m copies of a given entangled state will have m times the value of E 3F that one copy does. This facilitates side-by side comparisons of multiple low-dimensional entangled states with fewer high-dimensional entangled states. For pure tripartite states, E 3F is simply equal to the minimal entropy between subsystems A, B, and C. In [17], we were able to lower-bound E 3F using correlations between observables of d-dimensional systems, but in this article, we show how one can also do this for continuous-variable (high-dimensional) systems.
To witness genuine tripartite entanglement in both pure and mixed states, one can start with a convex witness of genuine tripartite entanglement for pure states. Here, the convex witness is any convex function f of the quantum state |ψ ABC such that for some value η, f > η witnesses genuine tripartite entanglement. The convexity allows us to immediately apply these witnesses to mixed states because the average value of a convex function cannot increase under mixing.
Once a convex witness of genuine tripartite entanglement for pure states is found, it is readily adapted to fully general (i.e., mixed) states. Given f is convex, ifρ has a pure state decomposition: the witness will obey the inequality: Since f > η witnesses genuine tripartite entanglement in a pure state, it must also follow that for any mixture of pure statesρ, that f (ρ) > η witnesses genuine tripartite entanglement as well. This forces at least one element in any pure state decomposition ofρ to be genuinely tripartite entangled, which is sufficient to prove genuine tripartite entanglement in the mixed state case. This general strategy of constructing convex witnesses for pure states to adapt them for mixed states has been used to great success to construct multi-partite entanglement witnesses from uncertainty relations in [11].
To quantify genuine tripartite entanglement, we use convex entanglement witnesses that bound the quantum conditional entropy. In particular, one can show from what we found in [25], that for Fourier-conjugate position x and momentum k = p/ : is the continuous Shannon entropy [26] of the joint probability density of x A , x B , and x C . In addition, S(A|BC) = S(ABC) − S(BC) is the quantum conditional entropy where for example, S(ABC) is the von Neumann entropy of density matrixρ ABC . The left hand sides of (6) witness entanglement in their respective bipartitions when they are greater than zero. All logarithms are taken to be base two, since we measure entropy in bits. For a more detailed discussion behind the derivation of these relations, see Appendix A.
With the preceding three relations (6), we find functions of x and k that bound the left hand side of all three at once. For momentum k, we have for the entropy of a linear combination of the three momenta: where (β A , β B , β C ) are real-valued coefficients and |·| denotes absolute value. Similarly for the linear combination of positions, we have the relation: See proof in Appendix B. Together, these allow us to consolidate these separate bounds (6) into one: To obtain a bound for the tripartite entanglement of formation for pure states, we use the relation for pure states that −S(A|BC) = S(A), and re-arrange the previous inequality to obtain: where |η||β| = min i {|η i ||β i |}.This relation is true for every element in the pure state decomposition ofρ ABC , and therefore for any mixture. Then, because this relation must hold, even if we choose the pure state decomposition that minimizes the left hand side of this relation, we have our bound for the tripartite entanglement of formation E 3F : With this bound, we can place a conservative lower limit to E 3F on continuous-variable systems where direct calculation is generally intractable even with full knowledge of the state. Moreover, this relation can be adapted into one using standard deviations σ or variances σ 2 instead of entropies (as was accomplished previously for bipartite entanglement in [25]), because the Gaussian distribution is the maximum entropy distribution for a fixed variance: Resource measures of multi-partite entanglement are still relatively underdeveloped, but we expect that tools such as these will spur new growth in the field.

Generality of application
While the entanglement bound (11) covers arbitrary linear combinations of positions or momenta, we will focus for the rest of the paper on the relation adapted for correlations seen in simple nonlinear-optical sources of spatially entangled photon triplets. In particular, we consider the case of a pump photon being converted into an FIG. 1: Basic diagram of degenerate-cascaded SPDC using 517nm pump light to produce triplets at 1550nm.
First, light at λ p is split into wavelengths λ 1 and λ 2 . The light at λ 2 is then split into light at λ 3 and λ 4 .
entangled photon triplet through nonlinear-optical processes. Conservation of momentum implies that the uncertainty in the sum of the triplet's momenta k A +k B +k C is bounded by the uncertainty in the pump momentum, which may be made arbitrarily narrow. If we consider that these photon triplets may arise from a common birthplace, then we may expect the mean squared distance between one of the photons x A , and their common centroid (x A +x B +x C )/3 to be small as well, so that the uncertainty in the linear combination x A − (x B + x C )/2 could also be arbitrarily narrow. In cascaded SPDC, we expect the uncertainty in x A − (x B + x C )/2 to be narrow as well. In this case, there are two birthplaces; one between x A and the centroid (x B +x C )/2 for the first downconversion event, and one between x B and x C for the second event. The principal distance here is between x A and the centroid (x A + (x B + x C )/2)/2 which gives, up to a constant factor the quantity x A − (x B + x C )/2 just as before. For the rest of the paper, we will be considering the form of our relation where (η A , η B , η C ) = (1, −1/2, −1/2) and (β A , β B , β C ) = (1, 1, 1). Although we derived our tripartite entanglement bound (11) using Fourier-conjugate position x and momentum k = p/ , any pair of Fourier-conjugate variables will apply, including time t and (angular) frequency ω, or functions of conjugate field quadratures in quantum optics as studied in [27,28].
Indeed, in [15], Shalm et al demonstrated tri-partite energy-time entanglement between photon triplets generated in cascaded SPDC, subject to the assumption that σ(ω 1 + ω 2 + ω 3 ) = σ(ω p ). Their source generates photon triplet detection events at a rate of approximately seven per hour of the course of three days, but using their maximum time uncertainty σ(t 2 − t 1 ) = 3.7 × 10 −10 s as an approximation toward the uncertainty σ(t A − t B +t C 2 ), and their pump uncertainty σ(ω p ) = 3.77 × 10 7 /s, one could verify as much as 3.72 gebits of tripartite energy-time entanglement, which is already more entanglement than an 11-qubit or 2200-dimensional state can support.

III. EFFECTIVENESS IN CASCADED SPDC
Having developed our quantitative bound for tripartite entanglement, we now test its effectiveness for a realis-Approved for Public Release; Distribution Unlimited: PA #: AFRL-2022-1492 tic source of tripartite entanglement. In this work, we consider a source of spatially entangled photon triplets generated in degenerate cascaded spontaneous parametric down-conversion (See Fig. 1 for basic diagram). The source would be a pump laser at frequency ω p interacting with two χ (2) -nonlinear crystals each of length L z . The first crystal would be phase-matched to produce signal/idler photon pairs at frequencies 2ω p /3 and ω p /3, respectively. The signal photons would then be directed toward the second crystal chosen to be phase matched for degenerate SPDC taking idler photons at 2ω p /3 and producing photon pairs at ω p /3.
As shown in Appendix C, the transverse spatial amplitude of the photon triplets generated in this process is of the form: where here: In these expressions: α qp is the transverse momentum amplitude of the pump field where q p is set equal to ( q 1 + q 3 + q 4 ) by transverse momentum conservation; q 1 is the projection onto the transverse plane of the momentum k 1 of the lower energy idler photon exiting the first crystal; q 3 and q 4 are the corresponding transverseprojected momenta of the photons created in the second crystal; kp = k p + k Λ1 + k Λ2 , where k Λ1 and k Λ2 are the poling momenta of the first and second crystals; and N is a normalization constant. If no periodic poling or quasiphase matching is employed to achieve these processes, then (k Λ1 = k Λ2 = 0). To simplify notation, we will let k 1 refer to the first transverse component of k 1 , and define k 3 and k 4 similarly. Since for small arguments of the Sinc function, sinc(x 2 + y 2 ) ≈ sinc(x 2 )sinc(y 2 ), we have as our model for the triphoton wavefunction (for one spatial component): This function is symmetric under permutations of k 1 , k 3 and k 4 . Moreover, the argument of the sinc function is a quadratic form, so we can simplify it dramatically by transforming to a new basis of coordinates. Taking this, together with the Gaussian approximation of the sinc function in [29], we obtain a simple triple-Gaussian wavefunction for the photon triplets: techniques. The upper dashed yellow curve gives the exact value for E 3F for the triple-Gaussian wavefunction. In the limit of high correlation, the difference between these two curves raplidly approaches a constant of 2 log 2 (e) − 1 or about 1.88 gebits.
such that a = 3Lz 4kp , and: and σ p is the ordinary pump beam radius (i.e., one quarter of the 1/e 2 beam diameter). Remarkably, our tripartite entanglement bound can also be expressed in terms of these rotated coordinates: which is further simplified by virtue of its being a Gaussian distribution to: with variances: and we obtain the final result: A. Predicted tripartite entanglement for reasonable experimental parameters Let us consider two crystals of periodically poled lithium niobate PPLN whose poling is chosen to be phase Approved for Public Release; Distribution Unlimited: PA #: AFRL-2022-1492 matched to the appropriate down-conversion process. At the pump wavelength of 516.67nm, the index of refraction n p is approximately 2.240, and the poling periods will have to be about 8.84µm and 18.99µm for each crystal, respectively. From this, the effective pump momentum kp is approximately 2.60 × 10 7 /m. Bulk crystals come in a variety of lengths, but let us assume L z = 3mm. The only remaining parameter to fix is the Gaussian beam radius σ p .
In Fig. 2, we have plotted our lower bound for E 3F as a function of σ p and find that this source has potentially a substantial amount of entanglement. In particular, we find for these experimental parameters that a modest beam radius of 1 mm will generate in excess of five gebits of tripartite entanglement in each transverse dimension, giving us in excess of 10 gebits in total. As a basis of comparison, ten gebits of tripartite entanglement is the maximum amount of tripartite entanglement that a 30-qubit or billion-dimensional state can support! In order to gauge the effectiveness of our technique at quantifying tripartite entanglement, we need to compare the entanglement we can quantify to the total entanglement present in the triple-Gaussian state. Luckily, the triple-Gaussian wavefunction is simple enough to find its Schmidt decomposition using properties of the double-Gaussian wavefunction [30] (see Appendix D for details). For both the double-Gaussian wavefunction of two parties, and the triple-Gaussian wavefunction of three parties, the reduced density operator ρ A has an identical form. The marginal eigenvalues of ρ A are then determined, and from them, the von Neumann entropy S(A) as well. Finally, because the triple-Gaussian wavefunction corresponds to a pure state, and because it is symmetric under permutations of parties, the exact tripartite entanglement of formation is given as E 3F (ABC) = S(A), the von Neumann entropy of system A. In Fig. 2, we plot both the exact value for E 3F for the triple-Gaussian wavefunction along with our lower bound to it (23) based on measured correlations (11), and find the gap between the witnessed and total tripartite entanglement rapidly approaches a constant of about 1.88 gebits in the limit of high correlation.
As the pump beam radius σ p grows wider (alternatively σ xu ), the corresponding uncertainty in the transverse momentum σ ku grows smaller while the other principal variances σ kv and σ kw remain constant. This results in the momentum distribution becoming more correlated to a flat plane, and the position distribution more correlated to a single line. That both the position and momentum distributions for three particles would be strongly correlated to single lines is actually forbidden by the uncertainty principle [17], a surprising result since this restriction on correlations does not exist between two particles. Even so, the tripartite correlations discussed in this proposed experiment are optimal in that they approach a continuous-variable analogue of the GHZ state.

B. Considered Experimental Setup
In [2], we showed how one can use a simple experimental setup along with techniques in information theory to adaptively sample the transverse position and transverse momentum correlations in such a way that a very small number of measurements (relative to the state space) can faithfully extract those correlations without overestimating the entanglement present. In this subsection, we discuss how to do the same for the tripartite entanglement generated in our considered setup, referring to Fig. 3 for details.
To begin with, we consider a 517 nm pump laser incident on a nonlinear crystal (labeled NLC 1) quasi-phasematched for type-I collinear nondegenerate SPDC with signal light centered at λ 1 = 775 nm and idler light centered at λ 2 = 1550 nm, respectively. After passing through a pump removal filter, the light would be split by a dichroic beamsplitter (labeled DBS) that transmits 775nm light and reflects 1550nm light. The length of the reflected 1550nm path following the DBS would terminate with a digital micromirror device (DMD) array whose pixels are computer-controlled to reflect the downconverted light toward or away from a photon-counting detector. This arm would be fitted with optics to either image the plane of NLC 1, or its Fourier transform, giving us access to either the position or momentum statistics of that idler photon, respectively. The transmitted arm following the DBS would terminate with the second crystal (labeled NLC 2), and be fitted with 4F imaging optics to preserve the amplitude and phase of the signal photon.
Next, the signal light at λ 2 = 775nm would pass through the second nonlinear crystal phase-matched and periodically poled for type-0 degenerate collinear SPDC from 775 nm converting into photon pairs centered at λ 3 = λ 4 = 1550 nm. Following NLC2, the residual 775 nm light would be filtered out, and the 1550 nm photon pairs would be split by a 50/50 beamsplitter. Each arm of this wing of the experiment would terminate in a DMD array with optics to image either the field at the plane at NLC 2 onto them, or to image its Fourier transform, thus allowing us access to the position and momentum statistics of this pair as well.
By connecting all three photon detectors to a multichannel photon correlator (PC), we can record the triplet photon coincidences that occur. By correlating the triplet coincidence count rate to the settings on each DMD array, we can build up a tri-partite joint probability distribution for the positions or the momenta of the photon triplets.
If we were to build up the joint position and momentum statistics one pixel triplet at a time, acquiring the entire distribution at decent resolution would rapidly become intractable. Indeed, current triplet generation rates are improving for both χ (2) and χ (3) processes [21,22,31], but still are well below one triplet per second per mW of pump power. However, using multi-resolution sampling techniques employed in [2] can solve this scaling issue as more efficient sources of entangled photon triplets continue to be developed. By sampling first at the lowest possible resolution (i.e., 2 × 2 × 2), and next subsampling only in regions with a significant triplet coincidence rate, and iteratively subsampling in the brightest areas of those sub-regions, one can obtain a coarse-grained approximation to these distributions that will never overestimate the entanglement present, and requires a minuscule fraction of the total number of measurements that would be required otherwise. Indeed, in [2], these multiresolution techniques improved the required acquisition time by at least a factor of 10 7 , and this advantage will only be more dramatic in the tripartite case, due to the higher dimensional space in which these sparse correlations reside.

IV. DISCUSSION: RESOURCE MEASURE CHALLENGES AND FUTURE APPLICATIONS
The major issue with generating tripartite spatial entanglement via cascaded SPDC is dealing with a low generation rate. Even with an optimistic free-space generation rate of 10 8 photon pairs per second per mW of pump power, this implies about 1 in 2.9 × 10 7 pump photons get converted in each crystal, so that only about 1 in 1.5 × 10 15 pump photons end up yielding triplets. For 1 mW of pump power at 517 nm, this would imply a total generation rate of about 2.6 triplets per second. Incorporating reasonable sources of loss reduces this rate by an order of magnitude, which puts it on par with the recent demonstration of a triplet generation rate of 12.4 triplets per minute [31]. Even with the highly efficient structured sensing approach discussed here and in [2], the total acquisition time required would make acquiring these spatial correlations at high resolution and statistical significance beyond current capabilities. However, with the recent advent of spatially-resolving photon detectors (e.g., SPAD arrays), demonstrating tripartite spatial entanglement, even at these low intensities becomes possible. With SPAD arrays one can acquire the maximum information from every photon triplet since the position that they strike each detector would be immediately stored, and a usable tripartite position probability distribution can be acquired with a comparably small number of photons.

V. CONCLUSION
The utilization and characterization of multi-partite entanglement is rapidly developing, even while fundamental questions remain to be answered. Here we have presented the first (to our knowledge) technique to quantify genuine tripartite entanglement in continuousvariable systems without the restriction to pure states, enabling us to employ these techniques experimentally. Moreover, we explore a natural source of these tripartite correlations in cascaded spontaneous parametric downconversion, and find that for reasonable experimental pa-rameters, there is already more tripartite entanglement present than 2-3 dozen qubits can support. On top of this, we were able to gauge the effectiveness of our technique because the high symmetry of the triple-Gaussian wavefunction allowed an explicit calculation of its tripartite entanglement of formation. With current sources of high-dimensional tripartite entanglement owing their strength to the correlations arising from conserved quantities in their interaction, measurement techniques capitalizing on those correlations are highly efficient. Moreover, using entropy-based tools will allow us to efficiently acquire these correlations at variable resolution without ever over-estimating the entanglement present, as was accomplished for two-party entanglement in [2].

ACKNOWLEDGMENTS
We gratefully acknowledge support from the Air Force Office of Scientific Research LRIR 18RICOR028, as well as insightful discussions with Dr. A. Matthew Smith, Dr. H Shelton Jacinto, and an insightful anonymous referee who improved the generality of our results.
The views expressed are those of the authors and do not reflect the official guidance or position of the United States Government, the Department of Defense or of the United States Air Force. The appearance of external hyperlinks does not constitute endorsement by the United States Department of Defense (DoD) of the linked websites, or of the information, products, or services contained therein. The DoD does not exercise any editorial, security, or other control over the information you may find at these locations. uncertainty relation In [25], we proved the relation starting from the entropic uncertainty principle in the presence of quantum memory [32], where for a pair of N -dimensional observablesQ andR, we have: where Ω QR is an uncertainty bound which approaches N in the limit thatQ andR are mututally unbiased.
Here, H(Q A |B) is a quantum conditional entropy S(A|B) where observableQ A has been measured. This measurement acts as a sum of projectors |q Ai q Ai | ⊗ I B , leaving system A in a mixed state of eigenstates ofQ A , but with B unperturbed. From the fact that quantum conditional entropy can be expressed as relative entropy, we can use the monotonicity of relative entropy to say that any subsequent measurement of system B cannot decrease the left hand side of (A2), which gives for arbitrary observablesV B andŴ B , the relation: Here, there is no assumption that the dimension of system A be the same as system B or that observablesV B andŴ B are in any way related toQ A andR A . To obtain the relation (A1) from (A3), we selectedQ A andR A to be a pair of observables related by a quantum Fourier transform, took a pair of continuum limits, and noted that the continuous entropy of the discrete approximation of a random variable is an upper bound to the true continuous entropy of the variable itself. For a full discussion, see [25].
To prove the tripartite version of (A1): h(x A |x B , x C ) + h(k A |k B , k C ) ≥ log(2π) + S(A|BC) (A4) we start by taking system B in (A1) to be the joint system BC and let the observablesV B andŴ B be joint measurements of local observables (V B ,V C ) and (Ŵ B ,Ŵ C ), of system BC. From this, one can take precisely the same steps to derive (A4) as were used to derive (A1) in [25]. In Section II, we gave the following relations for continuous entropy: and for position, the relation: To prove these relations, we will use the following five properties of continuous entropy: (a) The scaling law for continuous entropy: which also implies: (b) the continuous entropy is constant under reflection: To prove the position and momentum relations requires the same sequence of properties, so we prove the position relation here.
To obtain the bound for h(x A |x B , x C ), we use property (c): and then property (d): and finally, we use property (a): so that the bound becomes: Using the same sequence of properties, we can also derive: which proves the general position relation (B2).
Appendix C: Derivation of triphoton wavefunction for cascaded SPDC In this section, we show how to derive the triphoton spatial wavefunction in cascaded SPDC. To begin, we have the Hamiltonian for the SPDC process: To describe the cascaded SPDC process, we use firstorder time-dependent perturbation theory to describe the evolution of the field after interacting with each crystal. After the first interaction (k p → k 1 + k 2 ), and after the second interaction, (k 2 → k 3 + k 4 ). The approximate state of the down-converted field after these interactions is: To first non-trivial order in the generation of photon triplets, the state of the photon triplets is described by: where the simplification is carried out by the identity: In [29], the spatially varying components of G kp,k1,k2 is given by: where χ (2) ef f ( r) is the spatially varying second order nonlinear susceptibility taken to be a constant inside the nonlinear medium unless performing quasi phase matching by periodic poling in which case it flips sign with the flipping poling. Then, for a rectangular crystal of length L z (and other dimensions L x and L y ), this integrates to: where N is a normalization constant; k Λi is the i-th component of the poling momentum 2π/Λ; and Λ is the poling period. In the bulk crystal case without periodic poling, k Λ = 0. With the approximate expression for G kp,k1,k2 , and assuming the pump is bright enough to replace its annihilation operator with the corresponding coherent state amplitude, we can express the state of the cascaded downconverted light in terms of a triphoton amplitude: With the approximate expression for G kp,k1,k2 , we approximate the sums in the triphoton amplitude as integrals and note the following simplifications. We assume the pump is sufficiently narrowband in frequency that its longitudinal momentum takes on one value in this sum. Next, we take the small-angle/paraxial approximation so that the pump amplitude α kp factors into the product of a longitudinal amplitude α pz and a transverse amplitude α qp . Together, this gives: where N is a normalization constant, and q p is the projection of k p onto the transverse (xy) plane. Here, we are assuming that cascaded SPDC is achieved either simultaneously in the same crystal using two different poling momenta Λ 1 and Λ 2 , or using a sequence of two identical crystals both of length L z for simplicity. This integral has the form of a convolution, and can be solved to give: Through the rest of this derivation N will be a normalization constant absorbing factors not dependent on transverse momentum including the longitudinal pump amplitude. Next, we assume the transverse crystal dimensions L x and L y are large enough to wholly encompass the beam without clipping any side, which in turn is much larger than the pump wavelength. The transverse sinc functions can only contribute significantly for values less than the order of 2π/L x or 2π/L y , which is multiple orders of magnitude smaller than the pump momentum 2π/λ p . Because of this, they can be treated as delta functions when integrating over the transverse components of the pump momentum. In addition, this enforces transverse momentum conservation.
For simplification, we will assume the periodic poling (when necessary) only exists in the longitudinal direction so that for i = (x, y), k Λ1i = k Λ2i = 0. At this point, we have a three-dimensional triphoton aplitude, describing both longitudinal and transverse components of the photons' momenta. To isolate the transverse spatial component of the triphoton amplitude, we express the longitudinal momentum components in the sinc function terms of the respective momentum magnitudes and the transverse spatial components through the Pythagorean formula: The approximation comes from the small-angle approximation, in which the magnitude | q| is considered small relative to | k|. Initially, this complicates the sinc function: where λ n = 4r (r + 1) 2 r − 1 r + 1 2n ; (D3) r is the ratio of σ(x A + x B )/σ(x A − x B ) which simplifies to σ + /σ − , and φ n (x) is the nth-order Hermite-Gaussian wavefunction as seen in the solutions to the quantum harmonic oscillator. For the double-Gaussian wavefunction, θ n (x) = φ n (x), though for arbitrary wavefunctions, they can be different. If we take the marginal density operator of the double-Gaussian state, we find that: What is important to note are two points. First is that the Schmidt eigenvalues λ n are geometrically distributed, and second, that the ratio between successive Schmidt coefficients √ λ n is given by: 1. The reduced density operator for the triple-Gaussian wavefunction The Triple-Gaussian state |ψ ABC is given by: where ψ(x A , x B , x C ) is the triple-Gaussian wavefunction.
where σ w is set equal to σ v for rotational symmetry around the x u -axis.
The density matrixρ ABC corresponding to this state is |ψ ABC ψ| ABC : Next, we obtainρ A by tracing over B and C so that we may later obtain the marginal eigenvalues: Now, for the triple-Gaussian wavefunction, η(x A , x A ) has a double-Gaussian form: Since this function is up to a normalization constant, equal to a double-Gaussian wavefunction, we can define its correlation ratio R as σ(x A + x A )/σ(x A − x A ), and obtain the result: