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

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

James Schneeloch1, 2, 3, ∗ and John C. Howell1, 2, † 1Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA 2Center for Coherence and Quantum Optics, University of Rochester, Rochester, New York 14627, USA 3Air Force Research Laboratory, Information Directorate, Rome, New York, 13441, USA (Dated: March 1, 2018)

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.

CONTENTS C. Heisenberg limited temporal correlations in SPDC 16

I. Introduction 1 References 17

II. Foundation: The Quantum - Optical Calculation of the Biphoton state in SPDC 2 I. INTRODUCTION III. Approximation for degenerate collinear SPDC 6 In continuous-variable quantum information, there are IV. The Double-Gaussian Approximation 7 many experiments using entangled photon pairs gener- A. Usefulness of the Double-Gaussian approximation 8 ated by spontaneous parametric downconversion (SPDC) 1. Fourier-Transform Limited properties of the 1 (2) Double-Gaussian 9 . In short, SPDC is a χ -nonlinear optical process 2. Propagating the Double-Gaussian ﬁeld 10 occurring in birefringent crystals 2 where high energy “pump” photons are converted into pairs of low energy V. Estimating the Transverse Correlation Width 11 arXiv:1502.06996v9 [quant-ph] 27 Feb 2018 A. Comparison with experimental data 11

VI. The Biphoton birth zone 12 1 There are many dozens (if not hundreds) of experimental pa- VII. Conclusion 15 pers either using or exploring the spatial entanglement between photon pairs from SPDC, but some papers representative of the A. The double-Gaussian field propagated to different scope of research are: (Ali Khan and Howell, 2006; Barreiro et al., distances 15 2008; Bennink et al., 2004; Brougham and Barnett, 2012; Edgar B. Schmidt decomposition and quantum entanglement of the et al., 2012; Howell et al., 2004; Howland and Howell, 2013; Leach Double-Gaussian state 16 et al., 2012; Mazzarella et al., 2013; Moreau et al., 2012; Reid et al., 2009; Schneeloch et al., 2013; Tasca et al., 2011; Walborn et al., 2010, 2011) 2 In order to produce SPDC, one does not necessarily need a birefringent crystal, but this is a popular way to ensure a constant ∗ jfi[email protected] phase relationship (also known as phase matching) between the † [email protected] pump photon, and the signal/idler photon pair. 2

“signal” and “idler” photons. In particular, the pump the case of degenerate, collinear SPDC in the paraxial field interacts coherently with the electromagnetic quan- regime using the results in (Monken et al., 1998). We tum vacuum via a nonlinear medium in such a way that also use geometrical arguments to explain the approx- as an individual event, a pump photon is destroyed, and imations allowed in the paraxial regime. In Section 4, two daughter photons (signal and idler) are created (this we show how to further approximate this approximate event happening many times). As this process is a para- biphoton wavefunction as a Double-Gaussian (as seen in metric process (i.e., one by definition in which the initial (Law and Eberly, 2004) and (Fedorov et al., 2009)), as and final states of the crystal are the same), the total the multivariate Gaussian density is well studied, and is energy and total momentum of the field must each be easier to work with. In doing so, we give derivations of conserved. Because of this, the energies and momenta common statistical parameters of the Double-Gaussian of the daughter photons are highly correlated, and their wavefunction, showing its convenience in multiple appli- joint quantum state is highly entangled. These highly en- cations. In Section 5, we provide a calculation of the tangled photon pairs may be used for any number of pur- transverse correlation width, defined as the standard de- poses, ranging from fundamental tests of quantum me- viation of the transverse distance between the signal and chanics, to almost any application requiring (two-party) idler photons’ positions at the time of their creation. In quantum entanglement. Section 6, we explore the utility of the transverse corre- In this tutorial, we discuss a particularly convenient lation width, and introduce the concepts of the bipho- and common variety of SPDC used in quantum optics ex- ton birth zone, and of the birth zone number as a mea- periments. In particular, we consider illuminating a non- sure of biphoton correlation. We conclude by using the linear crystal with a collimated continuous-wave pump birth zone number to gain a qualitative understanding of beam, and filtering the downcoverted light to collect only the tradeoff between the first-order spatial coherence and those photon pairs with frequencies nearly equal to each the measurable correlations between photon pairs in the other (each being about half of the pump frequency). downconverted fields. This degenerate collinear SPDC process is amenable to many approximations, especially considering that most optical experiments are done in the paraxial regime, II. FOUNDATION: THE QUANTUM - OPTICAL where all measurements are taken relatively close to the CALCULATION OF THE BIPHOTON STATE IN SPDC optic axis (allowing many small-angle approximations). The procedure to quantize the electromagnetic field as With this sort of experimental setup in mind, we dis- it is used in quantum optics (Loudon, 2000; Mandel and cuss the theoretical treatment of such entangled photon Wolf, 1995) (as opposed to quantum field theory), is to: pairs (from first principles) in sufficient detail so as to decompose the electromagnetic field into a sum over (cav- inform the understanding and curiosity of anyone seek- ity) modes; find Hamilton’s equations of motion for each ing to discuss or undertake such experiments 3. Indeed field mode; and assign to the classically conjugate vari- as we discuss all the necessary concepts preceding each ables (generalized coordinates and momenta), quantum- approximation, much of this discussion will be useful in mechanically conjugate obervables, whose commutator is understanding non-collinear and non-degenerate SPDC i . From these field observables, one can obtain a Hamil- as well. ~ tonian operator describing the evolution of the quantum The rest of this paper is laid out as follows. In Section electromagnetic field, and in so doing, describe the evo- 2, we discuss the derivation of the quantum biphoton field lution of any quantum-optical system. state in SPDC, as discussed in (Hong and Mandel, 1985), SPDC is a χ(2)-nonlinear process. To describe it (Man- and (Mandel and Wolf, 1995). In addition to this, we del and Wolf, 1995), we begin with the classical Hamil- point out what factors contribute not only to the shape tonian of the electromagnetic field; of the biphoton wavefunction (defined later), but also to the magnitude of the amplitude for the biphoton genera- 1 Z H = d3r (D~ · E~ + B~ · H~ ), (1) tion to take place. This is important, as it determines the EM 2 overall likelihood of downconversion events, and gives im- ~ ~ ~ portant details to look for in new materials in the hopes where D = 0E + P. Since the electric field amplitude of of creating brighter sources of entangled photon pairs. the incident light on a nonlinear medium is usually sub- In Section 3, we simplify the biphoton wavefunction for stantially smaller than the electric field strength binding the atoms in a material together, we can express the polarization field P~ as a power series in the electric field strength (Boyd, 2007), so that 3 For more extensive treatments of spontaneous parametric down- (1) (2) 2 (3) 3 conversion, we recommend the Ph.D. theses of Lijun Wang P~ = 0 χ E~ + χ (E~ ) + χ (E~ ) + ... . (2) (Wang, 1992), Warren Grice (Grice, 1997), and Paul Kwiat (Kwiat, 1993), as well as the Physics Reports article by S.P. Since the nonlinear interaction beyond second order is Walborn et al.(Walborn et al., 2010). considered here to not appreciably affect the polariza- 3 tion, the classical Hamiltonian for the electromagnetic where the first term is the linear contribution to the field can be broken up into two terms, one linear, and Hamiltonian. Though this Hamiltonian looks relatively one nonlinear; simple, the field operator Eˆ(~r, t) = Eˆ+(~r, t) + Eˆ−(~r, t), so that the integral in (7) actually contains eight terms. HEM = HL + HNL, (3) These terms correspond to all different χ2 processes (e.g., where, sum-frequency generation, difference-frequency genera- Z tion, optical rectification, etc.), each of which has its 1 3 (2) own probability amplitude of occurring. However, given HNL = o d r χ˜ijl Ei(~r, t)Ej(~r, t)El(~r, t). (4) 2 that we have a single input field (i.e., the pump field), 2 and start with no photons in either of the signal and Next, since the nonlinear susceptibilityχ ˜ijl depends on pump, signal, and idler frequencies4, each of which are idler fields, the only energy-conserving contributions to the Hamiltonian (i.e., the only significant contributions determined by their respective wave numbers, the nonlin- 6 ear Hamiltonian is better broken down into its frequency ) are transitions (forward and backward) where pump components: photons are annihilated, and signal-idler photon pairs are created. Z 1 3 X (2) ~ ~ ~ Our first approximation (beyond what was done to HNL = √ o d r χ˜ (ω(kp), ω(k1), ω(k2)) 2( 2π)3 ijl get (7) to begin with) is that the pump beam is bright ~ ~ ~ kp,k1,k2 enough to be treated classically, and that the pump inten- ~ ~ ~ × Ei(ω(kp))Ej(ω(k1))El(ω(k2)) , (5) sity is not significantly diminished due to downconversion events. This “undepleted pump” approximation, along where subscripts 1 and 2, are understood to refer to signal with keeping only the energy-conserving terms, gives us and idler modes, respectively. the simplified Hamiltonian: To condense this paper, we note that when the field quantization is carried out, our electric field functions 1 Z Hˆ = H + d3r χ˜(2)(~r)E (~r, t)Eˆ−(~r, t)Eˆ−(~r, t)+h.c., E(~r, t) are replaced by the field observables Eˆ(~r, t), which L 2 o ijl i j l separate into a sum of positive and negative frequency (8) contributions Eˆ+(~r, t), and Eˆ−(~r, t), where which we then expand in the modes of the signal, and s idler fields; ~ ˆ+ 1 X ~ω(k) i~k·~r ˆ E (~r, t) = 1 i aˆ~k,s(t) ~~k,s e , (6) H = HL V 2 20 ~k,s Z 1 3 −1 X X (2) ~ ~ ~ + o d r χ˜ijl (~r; ω(kp), ω(k1), ω(k2)) ˆ− ˆ+ 2 V and E (~r, t) is the hermitian conjugate of E (~r, t). Here, ~k1,s1 ~k2,s2 s is an index indicating component of polarization, ~ is a s 2 ~ ~ ω(k1)ω(k2) ~ ~ unit polarization vector, anda ˆ~k,s(t) is the photon anni- ~ −i(k1+k2)·~r × 2 e Ei(~r, t) hilation operator at time t. In addition, V is the quan- 40 tization volume 5, which in the standard quantization † † procedure, would be the volume of a cavity which can be × aˆ (t)â (t)(~~ )j(~~ )l + h.c. . (9) ~k1,s1 ~k2,s2 k1,s1 k2,s2 taken to approach infinity for the free-space case. With the electric field observables thus defined, we can Note that here and throughout this paper, h.c. stands for obtain the quantum Hamiltonian of the electromagnetic hermitian conjugate. field: Next, we assume the pump is sufficiently narrowband, X so that we can, to a good approximation, separate out Hˆ = ω(~k)â† aˆ ~ ~k,s ~k,s the time dependence of the pump field as a complex ex- ~ k,s ponential of frequency ωp. In addition, we assume the 1 Z + d3r χ˜(2)Eˆ (~r, t)Eˆ (~r, t)Eˆ (~r, t), (7) 2 o ijl i j l 6 The reason the non-energy-conserving terms in (7) can be ne- glected is due to the rotating wave approximation. In calculating the amplitude for the downconversion process, and converting 4 At this point we would like to point out that we use the Einstein to the interaction picture, all other contributions to this ampli- (2) summation convention forχ ˜ijl Ei(~r, t)Ej (~r, t)El(~r, t). tude will have complex exponentials oscillating much faster than 5 The quantization volume is the volume of the hypothetical cavity ∆ω ≡ ω1 +ω2 −ωp. Since each of these contributions (oscillating containing the modes of the electromagnetic field. For simplicity, at frequency ω) when integrated give amplitudes proportional ωT the cavity is taken to be rectangular, so the sum over modes is to sinc 2 , and the propagation time T through the crystal straightforward using boundary conditions in Cartesian coordi- is fixed, these Sinc functions become negligibly small for large nates. To get an accurate representation of the electromagnetic ω. Since ∆ω is small for nearly degenerate SPDC, the energy- field in free space, we may take the quantization volume to be conserving contribution dominates over the non-conserving con- arbitrarily large. tributions. 4 pump field to be sufficiently well-collimated so that, to (such that d3r = dxdydz), to get the Hamiltonian: a good approximation, we can also separate out the lon- ˆ gitudinal dependence of the pump field 7. At this point, H = HL Z we define the transverse momenta ~q , ~q , and ~q ,as the 1 −LxLyLz X X (2) p 1 2 + d2q χ˜ (ω(k~ ), ω(k~ ), ω(k~ )) projections of the pump wave vector ~k , the signal wave 4π o p V ijl p 1 2 p ~ ~ ~ ~ k1,s1k2,s2 vector k1, and the idler wave vector k2, onto the plane s 2 ~ ~ transverse to the optic axis respectively. We also define ~ ω(k1)ω(k2) × (~~ )i(~~ )j(~~ )l k , k , and k , as the longitudinal components of the kp k1,s1 k2,s2 2 pz 1z 2z 40 corresponding wave vectors. With this in mind, we ex- ∆qxLx ∆qyLy ∆kzLz press the pump field as an integral over plane waves: × sinc sinc sinc e−iωptE˜(~q , t) 2 2 2 p Z † † 1 2 i(~q ·~r) i(k z−ω t) E (~r, t) = d q E˜ (~q , t)e p e zp p . (10) × aˆ~ (t)â~ (t) + h.c. . (12) i 2π p i p k1,s1 k2,s2 Note that the Sinc function, sinc(x), is defined here as By separating out the transverse components of the sin(x)/x. wave vectors, we make the Hamiltonian easier to sim- To obtain the state of the downconverted fields, one plify in later steps. As one additional simplification, can readily use first-order time-dependent perturbation we define the pump polarization vector ~ , so that theory. To see why this is, we can compare the nonlin- ~kp Ei(~qp, t) = E(~qp, t)(~ )i. With the transverse compo- ear classical Hamiltonian to the linear Hamiltonian using ~kp nents separated out, and the narrowband pump approx- typical experimental parameters of a pump field inten- 2 imation made, the Hamiltonian takes the form: sity of 1mW/mm , and signal/idler intensities of about 1pW/mm2. As such, the nonlinear contribution to the total Hamiltonian is indeed very small relative to the lin- Hˆ = H L ear part, and the consequent results we obtain from these Z 1 3 2 −1 X X (2) ~ ~ ~ first order calculations should be quite accurate. Though + o d rd qp χ˜ (~r; ω(kp), ω(k1), ω(k2)) 4π V ijl this is the treatment we discuss, alternative higher order ~ ~ k1,s1 k2,s2 and non-perturbative derivations of the quantum state s 2 ~ ~ ~ ω(k1)ω(k2) of down-converted light are also useful in examining the × (~~ )j(~~ )l(~~ )i k1,s1 k2,s2 kp 2 photon number statistics of down converted light, par- 40 ticularly when a sufficiently intense pump beam makes −i(∆~q)·~r −i∆kz z −iωpt ˜ † † it significantly probable that multiple pairs will be gen- × e e e Ei(~qp, t)â~ (t)â~ (t) + h.c. , k1,s1 k2,s2 erated at once through the simultaneous absorption of (11) multiple pump photons. Indeed, the general two photon state is described as a multimode squeezed vacuum state, where we define ∆~q ≡ ~q1 + ~q2 − ~qp, and ∆kz ≡ k1z + whose photon number statistics have been shown experi- k2z − kpz. mentally (and theoretically) to be such that the number In most experimental setups (including the one we con- of pairs created in a given time interval is approximately sider here), the nonlinear crystal is a simple rectangular Poisson-distributed (Avenhaus et al., 2008; Christ et al., prism, centered at ~r = 0, and with side lengths Lx, Ly, 2011). and Lz. Here, we assume the crystal is isotropic, so that Using first-order time-dependent perturbation theory, (2) the state of the signal and idler fields in the interaction χijl does not depend on ~r. To simplify the subsequent calculations, we assume the crystal to be embedded in a picture can be computed as follows: linear optical medium of the same index of refraction to Z t i 0 0 avoid dealing with multiple reflections. Alternatively, we |Ψ(t)i ≈ 1 − dt HNL(t ) |Ψ(0)i. (13) could assume the crystal has an anti-reflective coating to ~ 0 the same effect. We can then carry out the integral over Note that in the interaction picture, operators evolve ac- L L † the spatial coordinates (from − 2 to 2 in each direction) cording to the unperturbed Hamiltonian, so thata ˆ (t) = aˆ†(0)eiωt. Here, the initial state of the signal and idler fields |Ψ(0)i is given to be the vacuum state |01, 02i, which means that the hermitian conjugate (with its low- ering operators) will not contribute to the state of the 7 For a reference that examines in detail how the downconverted downconverted photon fields. light is affected by the pump spatial profile, we recommend the reference (Pittman et al., 1996a). For a reference that treats Before we calculate the state of the downconverted SPDC with short pump pulses (as opposed to continuous-wave), photon fields (up to a normalization factor), we make see (Keller and Rubin, 1997). use of some simplifying assumptions. First, we assume 5 that the polarizations of the downconverted photons are works, we note that the biphoton probability amplitude ~ ~ fixed, so that we can neglect the sums over s1 and s2, can be expressed as h01, 02|aˆ(k1)â(k2)|ΨSPDC i. When effectively making the sum over one value. With this, we normalize this probability amplitude, by integrating ~ ~ the sum over the components of the nonlinear suscep- its magnitude square over all values of k1, and k2, and 1 (2) setting this integral equal to unity, the resulting normal- tibility is proportional to the value deff ≡ 2 χeff , which is the effective experimentally determined coefficient for ized probability amplitude has the necessary properties the nonlinear interaction. Second, we assume that the (for our purposes) of a biphoton wavefunction 8. With nonlinear crystal is much larger than the optical wave- approximations to be made in the next section, only ~ ~ ~ ~ lengths considered here, so that the sums over k1 and k2 Φ(k1, k2) will govern the transverse momentum proba- can be replaced by integrals in the following way: bility distribution of the biphoton field9. The factors preceding the biphoton wavefunction are Z 1 X 1 X 3 still important to understand because (with some alge- lim = d k1. (14) V →∞ V (2π)3 bra) they contribute to the rate of downconversion events ~k ,s s1 1 1 10 (RSPDC ). In particular : With these simplifications, we can express the nonlinear 2 2 Hamiltonian in such a way that both the sums are re- RSPDC ∝ deff PpLz, (18) placed by integrals, while still accurately reflecting the relative likelihood of downconversion events; where Pp is the pump power (in Watts). We also note from equation (16) that the rate of downconver- ZZ q sion events is also proportional to sinc2(∆ωT/2), though H ≈ C d d3k d3k ω(~k )ω(~k ) NL NL eff 1 2 1 2 this factor is essentially unity for the nearly degenerate Z 3 frequencies of the signal and idler photons considered Y ∆kmLm × d2q sinc E˜(~q , t)ei∆ωt here. This proportionality also follows from more rig- p 2 p m=1 orous calculations of the rate of downconversion events † ~ † ~ × aˆ (k1)â (k2), (15) (Hong and Mandel, 1985; Kleinman, 1968), though only in the approximation where the minuscule signal/idler where m = {1, 2, 3} = {x, y, z}, and ∆ω ≡ ω1 + ω2 − ωp, fields don’t appreciably contribute to the likelihood of and CNL is a constant. Note that here, ∆kx = ∆qx and downconversion events. In those more rigorous calcula- ∆ky = ∆qy, to condense notation. tions, the conversion efficiency (biphotons made per in- With one additional assumption, that the slowly- cident pump photon) is of the order 10−8, which again varying pump amplitude (excluding eiωpt) is essentially shows just how weak these signal/idler fields are relative constant over the time light takes to propagate through to the pump field 11, and why first-order perturbation the crystal, the integral over this nonlinear Hamiltonian theory is sufficient to get a reasonably accurate represen- becomes an integral of a constant times ei∆ωt. With this tation of the state of the downconverted fields. We also integral, we get our first look at the state of the down- note that although beam size doesn’t affect the global converted field exiting the crystal: rate of downconversion events, it does affect the fraction of those downconversion events that are likely to be |ΨSPDC i ≈ C0|01, 02i counted by a detector near the optic axis 12. Even so, q ZZ 2 3 3 ~ ~ + C1deff IpT d k1d k2 Φ(k1, k2) q i∆ωT ∆ωT † † × ω(~k )ω(~k )e 2 sinc aˆ (~k )â (~k )|0 , 0 i. 8 1 2 2 1 2 1 2 Though it is debatable whether it is correct to speak of a biphoton wavefunction since expectation values are in fact carried out ~ ~ (16) with |ΨSPDC i, and Φ(k1, k2) does not evolve according to the ~ ~ Schrödingerequation (|ΨSPDC i does, though), Φ(k1, k2) is a Here, T is the time it takes light to travel through the square-integrable function in a two- particle joint Hilbert space crystal; Ip is the intensity of the pump beam; |01, 02i is that accurately describes the relative measurement statistics of the (Fock) vacuum state with zero photons in the signal the biphotons. 9 Note that the biphoton wavefunction (17) is expressed as an mode and zero photons in the idler mode, and; integral over the rectangular crystal shape. For those interested in a derivation of the integral giving the biphoton wavefunction Z 3 2 Y ∆kmLm for a generalized crystal shape, see (Saldanha and Monken, 2013) Φ(~k , ~k ) ≡ d q sinc ν(~q ), 10 1 2 p 2 p Note that this downconversion rate comes from the approxima- m=1 tion of near-perfect energy conservation: i.e., ∆ωT << 1. (Helt (17) et al., 2012) 11 Including the collection/coupling efficiencies in many quantum- optical experiments, the measured conversion efficiency is closer is, up to a normalization constant, the biphoton wave- to 10−10. function in momentum space (where ν(~qp) is the nor- 12 The rate of downconversion events yielding biphotons propagat- malized pump amplitude spectrum). To see how this ing close to the optic axis increases with a smaller beam size, 6 these factors are useful to know when selecting a crystal k k1 2 as a source of entangled photon pairs. For example, with ∆kz a constant power pump beam, a longer crystal will be θ a brighter source of photon pairs. However, there is a kp tradeoff; the degree of correlation between the signal and idler photons decreases with increasing crystal length (as FIG. 1: Diagram of the relationship between the pump, we shall show). signal, and idler momenta, in standard (not necessarily collinear) phase matching.

III. APPROXIMATION FOR DEGENERATE COLLINEAR pump momentum. SPDC ~ ~ ∆kzLz Φ(k1, k2) = sinc To obtain a relatively simple expression for the bipho- 2 Z ton field in SPDC, we have made multiple (though rea- ∆kxLx ∆kyLy × d2q sinc sinc ν(k , k ), sonable) simplifying assumptions. We have assumed that p 2 2 px py the pump is narrowband and collimated so that it is (20) nearly monochromatic, while also having a momentum The significant contributions of the Sinc functions to the spectrum whose longitudinal components dominate over integral will come from when, for example, ∆k < πkp , its transverse components. We next assumed that the x 2Ω where Ω is the ratio of the width of the crystal L to pump is bright enough to be treated classically, but not x the pump wavelength λ . Where the crystal is much so bright that the perturbation series approximation to p wider than a pump wavelength, Ω is large, and we see the nonlinear polarization breaks down. In addition, we the Sinc function will only contribute significantly when assumed that we need not consider multiple reflections, ∆k is only a very small fraction of k . Thus, with a and that the crystal is large compared to an optical wave- x p renormalization, the sincs act like delta functions, setting length so that sums over spatial modes may be replaced ~q = ~q + ~q , and giving us the biphoton wavefunction: by integrals. We have also assumed that the pump is p 1 2 bright enough that it is not attenuated appreciably due ∆k L Φ(~k , ~k ) = N sinc z z ν(~q + ~q ), (21) to downconversion events. 1 2 2 1 2 Now, we consider the experimental case where we place where N is a normalization constant. frequency filters over photon detectors, so that we may Since most experiments are done in the paraxial only examine downconversion events which are degen- regime, we use such approximations to get the Sinc- erate (where ω = ω ),and perfectly energy-conserving 1 2 Gaussian biphoton wavefunction, ubiquitous in the lit- (∆ω = 0). In this case, along with all the previous as- erature. With the previous assumptions already made, sumptions made, we define a new constant of normal- we point out that in degenerate, collinear SPDC, |~k | = ization C˜ (absorbing factors outside the integrals), and 1 1 |~k | = |~k |/2, which we redefine as k ,k , and k /2, to obtain the following simplified expression for the state of 2 p 1 2 p simplify notation. In addition, since the transverse pump the downconverted field as seen in (Monken et al., 1998); momentum is essentially equal to the sum of the transverse signal and idler momenta, the three vectors can be |ΨSPDC i ≈ C0|01, 02i (19) readily drawn on a plane, as seen in Fig. 1. ZZ ~ +C˜ d3k d3k Φ(~k , ~k )â†(~k )â†(~k )|0 , 0 i. Let θ be the angle between the pump momentum kp 1 1 2 1 2 1 2 1 2 ~ ~ and the signal or idler momentum vectors k1 and k2. This angle θ is small enough that we may use the small- ~ ~ Here, the biphoton wavefunction Φ(k1, k2) is as defined angle approximation to find an expression for ∆kz in previously (17). Next, we use the fact that the transverse terms of easier-to-measure quantities. Using the conser- dimensions of the crystal are much larger than the pump vation of each component of the total momentum, we get wavelength to carry out the integral over the transverse the following equations:

kp = (k1 + k2) cos(θ) − ∆kz, (22) |~q − ~q | 1 2 = k sin(θ). (23) 2 1 but only to a point. For a good summary, see (Ling et al., 2008). For a more detailed discussion on how focusing aﬀects the frac- Using the small-angle approximation, and substituting tion of downconverted light propagating near the optic axis, see one equation into the other, we ﬁnd: (Ljunggren and Tengner, 2005). For a more rigorous discussion 2 of how the rate of downconversion events (i.e., the signal/idler |~q1 − ~q2| power) changes with the crystal length, see (Loudon, 2000). ∆kz ≈ − . (24) 2kp 7

Finally, when we assume the transverse pump momentum IV. THE DOUBLE-GAUSSIAN APPROXIMATION profile is a Gaussian, In what follows here, we approximate the Sinc- 1 2 4 Gaussian biphoton momentum-space wavefunction (26) 2σ 2 2 p −σ |~qp| ν(~qp) = e p , (25) as a Double-Gaussian function (as seen in (Law and π Eberly, 2004) and(Fedorov et al., 2009)), by matching the second order moments in the sums and differences with σ being the pump radius in position space 13, we p of the transverse momenta. Transforming this approxi- renormalize, and find the biphoton wavefunction to be: mate wavefunction to position space, and computing the correlation width gives us an estimate of the true corre- Lzλp 2 2 ~ ~ 2 −σp|~q1+~q2| Φ(k1, k2) = N sinc |~q1 − ~q2| e , (26) lation width seen experimentally that we later compare 8πnp with more exact calculations and experimental data. In addition, we take a moment to explore the conveniences where the minus sign in the argument of the sinc function that come with the Double-Gaussian wavefunction. is eliminated since the sinc function is an even function. In this analysis, we consider only the horizontal com- Interestingly, we can use the radius to the first zero of the ponents of the transverse momenta, since the statistics Sinc function along with (22) to derive a simple formula are identical (with our approximations) in both trans- for the half-angle divergence of the degenerate collinear verse dimensions. The transverse pump profile is already SPDC light: assumed to be a Gaussian. Our first step is to trans- r form to a rotated set of coordinates to separate the Sinc 2λp function from the Gaussian. θSPDC ≈ . (27) Lz Let

In experiments where no filtering takes place to isolate k1x + k2x k1x − k2x k+ = √ and k− = √ . (28) the degenerate portion of the SPDC light, this angle will 2 2 be larger since the non-degenerate frequencies of SPDC light have a wider, ring-shaped distribution. With these rotated coordinates, the (horizontal) biphoton wavefunction becomes: To obtain a transverse correlation width from this biphoton wavefunction, we need to transform it to po- Lzλp 2 2 2 −2σpk+ φ(k+, k−) = N sinc k− e . (29) sition space. Fortunately, this biphoton wavefunction 4πnp is approximately 14 separable (subject to our paraxial approximation) into horizontal and vertical wavefunc- Taking the modulus-squared and integrating over k+, tions (i.e., into a product of functions, one dependent we isolate the probability density for k−: on only x-coordinates, and the other dependent on only r y-coordinates). In addition, we can find an orthogonal 3 a 2 2 ρ(k−) = sinc (ak−) (30) set of coordinates in terms of sums and differences of 4 π momenta that allows us to transform this wavefunction Lz λp where a ≡ , for convenience. ρ(k−) is an even func- by transforming the Sinc function and Gaussian indi- 4πnp vidually. While transforming the Gaussian is extremely tion, so its first-order moment, the expectation hk−i = 0. straightforward, transforming the concurrent Sinc-based The second-order moment is nonvanishing, with a value 2 3 function is more challenging, owing to that it is a Sinc hk−i = 4a . With this second order moment, we can fit function of the square of a momentum coordinate, and is ρ(k−) to a Gaussian by matching these moments. In do- not in most dictionaries of transforms. ing so, ρ(k−) is approximately a Gaussian with width q 3 σk− ≡ 4a . To see how good this Gaussian approximation of ρ(k−) is, we show in Fig. 2, both the Sinc-based probability den- 13 The pump radius σp in position space is defined as the standard sity (in momentum space) and the approximate Gaussian x1+x2 deviation of 2 . This is justified by noting that the pump density with matched moments. The overall scale of the radius in momentum space is explicitly given by the standard central peak is captured but the shape is significantly dif- deviation of (k1x + k2x), and using the properties of Fourier transformed Gaussian wavefunctions. ferent. However, a Gaussian probability density for the 14 For small values of x and y, sinc(x + y) ∼ sinc(x)sinc(y). For position difference density, ρ(x−), appropriately scaled is typical experimental parameters, the argument of the Sinc func- a good approximation for values near the central peak −3 tion is of the order 10 , even for transverse momenta as large (though the oscillatory behavior of the wings is still not as the pump momentum. With the paraxial approximation, the transverse momenta are much smaller than the pump momen- captured). In Fig. 3, we plot various choices of an approx- tum, and so the arguments of the Sinc functions are very small imate Gaussian density for our transformed Sinc-based indeed. position difference density function ρ(x−). We find that 8

Ρ k- Ρ x- 0.5 0.6

H L 0.4H L 0.5

0.4 0.3

0.3 0.2

0.2 0.1 0.1

k- x- -3 -2 -1 1 2 3 -6 -4 -2 2 4 6

FIG. 2: Plot comparing estimates of the momentum FIG. 3: Plot comparing different estimates of ρ(x−). difference probability density ρ(k−). The solid (blue) The solid blue wavy curve is our most accurate estimate curve with wavy side-bands gives our Sinc-based from the transformed Sinc-based distribution (52). The probability density estimate (30), where we set a = 2 tall dashed (magenta) curve is the Gaussian distribution for convenience. The dashed (magenta) Gaussian curve obtained from matching momentum means and gives our Gaussian-based probability density estimate variances, while the shallow dashed (red) curve is the with matching means and variances. Gaussian distribution obtained by matching position means and variances. The solid (green) curve gives us a refined Gaussian approximation, by setting the central simply setting the central maxima of both densities equal maximums equal to one another. The flat (gold) line, to each other works very well, as discussed in the next gives the height of the half maximum of the Sinc-based section. probability density (52) (blue curve). We see that the By approximating the Sinc-Gaussian wavefunction as a widths of half maximum are nearly identical (off by less Double-Gaussian wavefunction, the inverse Fourier trans- than 0.3%) for the Sinc-based and refined Gaussian form to position space becomes very straightforward. We distributions Again, we set a = 2 for convenience. note that k+ and k− form an orthogonal pair of coordinates, as k1x and k2x do. Because of this, the inverse Fourier transform is separable 15, and we find: x+ and x− in terms of x1 and x2, and take the magnitude- 2 2 x− x+ − 2 − 2 squared of ψ(x1, x2) to get a Double-Gaussian probability 1 4σx 4σx ψ(x+, x−) ≈ p e − e + . (31) density ρDG: 2πσx+ σx−

2 1 2 1 2 where σ = , σ = = 2σ , and where (x −x )2 (x +x )2 x− 2 x+ 2 p 1 2 1 2 4σk 4σk − 2 − 2 − + DG 1 4σx 4σx ρ (x1, x2) = e − e + . (33) 2πσx σx x1 + x2 x1 − x2 + − x+ = √ and x− = √ . (32) 2 2 Here, we deﬁne the transverse correlation width as the standard deviation of the distance between x1 and x2 A. Usefulness of the Double-Gaussian approximation (i.e., σ(x1−x2)). This is not deﬁned as a half-width, since √1 it represents the full width (at e of the maximum) of Here, we digress to discuss the usefulness of the the signal and idler photons’ position distributions con- Double-Gaussian approximation. To begin, we express ditioned on the location of the prior pump photon (see Section VI). For the Double-Gaussian√ density, the trans-

verse correlation width, σ(x1−x2), is 2σx− . 15 The Fourier transform convention we use is the unitary conven- Alternatively, the Double-Gaussian density can be put tion: ψ˜(k , k ) = 1 RR dx dx e−i(x1k1x+x2k2x)ψ(x , x ), 1x 2x 2π 1 2 1 2 into the standard form of a bi-variate Gaussian density 1 RR i(x1k1x+x2k2x) ˜ and ψ(x1, x2) = 2π dk1xdk2x e ψ(k1x, k2x). Since the Fourier transform is invariant under rotations (i.e., function; since the argument in the exponential can be thought of as an inner product between two vectors), we get identical formulas for √ 2 the Fourier transform in rotated coordinates. In particular, we ac − b 2 2 DG −(ax1+2bx1x2+cx2) ﬁnd ψ(x , x ) = 1 RR dk dk ei(x+k++x−k−)ψ˜(k , k ). ρ (x1, x2) = e , (34) + − 2π + − + − π 9

TABLE I: Statistics of the Double Gaussian

Name Value

marginal means hx1i = hx2i = 0 2 2x1σx σ(x1|x2) − conditioned mean hx2iρ(x2|x1) = x1 − σ2 +σ2 = rx1 x+ x− σx2 2 2 σx +σx x+ 2 2 + − σ marginal variance σx1 = σx2 = 2 2σ2 σ2 conditioned variance σ2 = σ2 = x+ x− (x1|x2) (x2|x1) σ2 +σ2 x+ x− σ2 −σ2 x+ x− co-variance hx1x2i − hx1ihx2i = 2 σ2 −σ2 σ x- hx1x2i−hx1ihx2i x+ x− Pearson r value r = = 2 2 σx σx σ +σ 1 2 x+ x−

joint entropy h(x1, x2) = log(2πeσx+ σx− ) 1 2 2 marginal entropy h(x1) = 2 log(πe(σx+ + σx− )); = h(x2)

mutual information h(x1:x2) = h(x1)+h(x2)−h(x1, x2); σ2 +σ2 x+ x− σx1 σx1 = log 2σ σ = log σ x+ x− (x1|x2) Probability notation σ2 ≡ σ2 (x1|x2) ρ(x1|x2) ρ(x1,x2) FIG. 4: Plot of the Double-Gaussian probability density ρ(x1|x2) = ρ(x2) for σx+ = 1 unit and σx− = 0.075 units. The horizontal dotted line indicates a particular value of x2, so that the half-width of the Gaussian along that dotted line is 1. Fourier-Transform Limited properties of the Double-Gaussian the conditioned half-width σ(x1|x2). The equivalent Sinc-Gaussian distribution will have subtle side bands Gaussian wavefunctions are minimum uncertainty (at about 5.0% the maximum intensity) parallel to the wavefunctions in that they are Fourier-transform limited; long axis of this Double-Gaussian as in Fig. 3. Heisenberg’s uncertainty relation; 1 σ σ ≥ , (37) x k 2 where is satisfied with equality. The Double-Gaussian wavefunction (33) factors into a product of two Gaussians (one in x and the other in x ), and so the standard σ2 + σ2 + − a = c = x+ x− , (35) deviations of these rotated coordinates also saturate the 4σ2 σ2 x+ x− Heisenberg relation: σ2 − σ2 x− x+ 1 1 b = 2 2 . (36) 4σ σ σx+ σk+ = : σx− σk− = . (38) x+ x− 2 2 Remarkably, the simple expressions for the statistics of This Double-Gaussian probability density has a number the double-Gaussian distribution (see Table I) show that of useful properties. First, it is separable into single these are not the only pairs that are related this way. Gaussians in rotated coordinates, making many integrals Since conditioning measurements on a single ensemble of straightforward to do analytically. Second, the marginal events λ doesn’t change the fact that those measurements and conditional probability densities of the Double Gaus- must satisfy an uncertainty relation, we find: sian density function are also Gaussian density functions. 1 Because of this, many statistics of the Double-Gaussian σ(x1|λ)σ(k1|λ) ≥ (39) density have particularly simple forms. For examples, 2 consider the statistics in Table 1. is still a valid uncertainty relation. In addition, since conditioning on average reduces the variance 16, we arrive 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 char- acterization of the overall shape of the Double-Gaussian 16 That conditioning on average reduces the variance can be seen distribution. from the law of total variance. Given two random variables X 10

at the relations: simply as a product of a horizontal and vertical transfer 1 1 function: σ σ ≥ : σ σ ≥ . (40) x1 (k1|k2) 2 k1 (x1|x2) 2 Tfs(z1, z2 : k1x, k1y, k2x, k2y) =

These relations are also useful for understanding how = Tfsx(z1, z2 : k1x, k2x)Tfsy(z1, z2 : k1y, k2y). (45) narrowband frequency ﬁlters undermine the resolution of temporal correlations as discussed in Appendix C. For where

the Double-Gaussian state, these relations are saturated 2 2 iz1k1x iz2k2x − k − k as well. From these properties, we may find many other Tfsx(z1, z2 : k1x, k2x) = e p e p , (46) useful identities for the double-Gaussian including: and Tfsy(z1, z2 : k1y, k2y) is similarly defined. rx = −rk (41) Because our position-space wavefunction is also (ap- σx σk σ σ(k |k ) proximately) separable into a product of vertical and + = − : x1 = 1 2 (42) horizontal wavefunctions, we can propagate ψDG(x , x ) σx− σk+ σ(x1|x2) σk1 1 2 (i.e., the Double-Gaussian approximation to ψ(x1, x2)), where rx and rk is the Pearson correlation coefficient without having to first propagate the entire transverse for the position and momentum statistics of the Double- wavefunction. Doing so, gives us: Gaussian, respectively. iz k2 iz k2 − 1 1x − 2 2x DG −1 ˜DG kp kp ψ (x1, x2 : z1, z2) = F [ψ (k1x, k2x)e e ], 2. Propagating the Double-Gaussian field (47) where F −1 is the inverse-Fourier transform operator. One especially useful aspect of the Double-Gaussian Perhaps surprisingly, propagating the double-Gaussian wavefunction, is that it is simple to propagate (in the field simply gives another bi-variate Gaussian density paraxial regime). Given the transverse momentum am- (see in appendix, (A1)). What changes through prop- plitude profile of a nearly monochromatic optical field in agation is the parameters defining the Double-Gaussian one transverse plane, we can find the transverse momen- field. Taking (31) to be the biphoton field backpropa- tum profile at another optical plane by multiplying it by gated to the center of the crystal (where z1 = z2 = 0, the paraxial free-space transfer function Tfs(z : kx, ky) and σx+ and σx− to be parameters defining the field at 17: z1 = z2 = 0, the transverse probability density of the photon pair when the propagation distances are equal iz 2 2 ikz− 2k (kx+ky ) Tfs(z : kx, ky) = e . (43) (i.e., z1 = z2 = z), as they would be if we measure both fields is the same image plane, gives us: For an entangled pair of optical fields at half the pump 2 2 frequency, the full transfer function becomes: − (x1−x2) − (x1+x2) DG 1 4˜σx (z)2 4˜σx (z)2 ρ (x1, x2; z) = e − e + 2πσ˜x (z)˜σx (z) Tfs(z1, z2 : k1x, k1y, k2x, k2y) = + − k (48) i p (z +z )− iz1 (k2 +k2 ) − iz2 (k2 +k2 ) = e 2 1 2 kp 1x 1y e kp 2x 2y . (44) s 2 2 z kp :σ ˜x+ (z) ≡ σx + , (49) i (z1+z2) + Since a global constant phase e 2 can come out- σx+ kp side the Fourier transform integral, and the relative s z 2 phases and amplitudes in position space will be inde- :σ ˜ (z) ≡ σ2 + . (50) x− x− pendent of this factor, we can remove it from the trans- σx− kp fer function, and express the remaining transfer function This particularly illustrates the convenience of working with the Double-Gaussian density, as we need only find

effective values for σx+ and σx− at one distance z to see and Y , the variance of X is equal to the mean over Y of the con- how the biphoton field might change under propagation ditioned variance V ar(X|Y ) plus the variance over Y of the con- 18 to the same imaging plane . In particular, where σx is ditioned mean E[X|Y ]. Both of these terms are non negative, so + the mean conditioned variance never exceeds the unconditioned much larger than σx− for highly entangled light, we can variance. see that the position correlations (say, as measured by 17 The free space transfer function comes about due to the momen- the Pearson correlation coefficient) decrease to zero, and tum decomposition of an optical field being a sum (or integral) gradually become strong anti-correlations as we move to over plane waves. For each plane wave defined by kx, ky, and kz, we add a phase corresponding to the plane wave translat- ing a total forward distance z. 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 18 On the other hand, propagating to independent imaging planes (Goodman et al., 1968). z1 and z2 is a more elaborate result discussed in the Appendix. 11

q the far field. This does not imply, however, that the pho- 9a simply 5 , which in turn gives us a transverse correla- ton pairs dis-entangle and re-entangle under propagation q 18a (Chan et al., 2007); the entanglement migrates to the tion width, σ(x1−x2), of 5 . In addition, matching this relative phase of the joint wavefunction and back again. exact variance to define an approximate Double-Gaussian Throughout the rest of this paper, all transverse corre- wavefunction also gives us the maximum likelihood esti- lation widths, probability densities, and biphoton ampli- mate (i.e., the estimated distribution with minimum rel- tudes will be assumed to be taken at z1 = z2 = 0 (or an ative entropy to the more accurate model) of a Double image plane conjugate to this plane), unless otherwise Gaussian distribution fitting our more exact results. As specified. a summary of our calculations, see the following: √ q (exact) (exact) 18a q 9Lz λp σ = 2σx = = , (x1−x2) − 5 10πnp V. ESTIMATING THE TRANSVERSE CORRELATION √ q (53) (PM) (PM) 16a q 4Lz λp σ = 2σx = = . WIDTH (x1−x2) − 9 9πnp

Using our earlier notation, the approximation to the Here, σPM refers to the peak-matching estimate that also Double-Gaussian wavefunction is expressed as follows: nearly matches the widths of the Gaussians, while σ(exact) is our more accurate calculation. Both estimates have (x −x )2 2 s 1 2 (x1+x2) − 2 − their uses when examining experimental data, as we shall 1 8σx 16σ2 ψ(x1, x2) ≈ √ e − e p , (51) see. 2 2πσpσx− Though explicitly calculating the variance of x− ac- p a cording to our accurate density function (52) gives us where σx− = 3 , making the transverse correlation the best possible estimate of σx− , it does not necessar- q 2a width, σ(x1−x2) = 3 . ily give us the best fitting Gaussian approximation to To see just how good (or not) this Gaussian-based esti- the Sinc-based distribution. The Gaussian obtained by explicitly matching position means and variances, gives mate of σ(x1−x2) is, we compare our Gaussian approximation to the the probability density of x− obtained when a distribution about 42.3% wider than the close fitting taking the Fourier transform of the Sinc-based function of distribution we obtain by matching peak values (see Fig. k− (30). The more accurate probability density obtained 3 for comparison). The resulting (overly wide) scaled from that Fourier transform is: Gaussian distribution (by matching variances) does not accurately reflect the probabilities of the most likely out- √ 3 x− x− comes (near x− = 0) (e.g., that within ± one ”sigma”, ρ(x−) = √ x− 2π S √ − C √ + 16 πa3 2πa 2πa we should get approximately 68% of the total data). In- 2 2 2 deed, by setting the central maximums equal to one an- √ x− x− + 2 a cos + sin , (52) other, we also find the Gaussian cumulative distribution 4a 4a function (CDF) that most accurately resembles the CDF where C(x) and S(x) are the Fresnel integrals, integrating of our more accurate distribution (52) near its median. π 2 π 2 As an example of the accuracy of this approximation, over cos( 2 t ) and sin( 2 t ), respectively. As seen in Fig.3, our peak-matching approximate Gaussian distribution, the Gaussian approximation obtained by matching hk−i 2 gives a total probability within one σx from the origin and hk−i gives a full width at half maximum (FWHM) − within an order of magnitude of the FWHM of the more of 68.3%, while the more accurate density function gives accurate approximation (52). a probability of 69.0%, (an absolute difference of only q 8a 0.7%) over the same interval. However, with a width σx− of 9 (i.e., by setting the maximums of our (Sinc-based and Gaussian-based) approximate density functions equal to one another), where A. Comparison with experimental data again, a ≡ Lz λp , one obtains a FWHM only 0.3% smaller 4πnp Though our estimate of σ follows from reason- than the FWHM from the more accurate case (52). In- (x1−x2) deed, numerical estimates based on fitting the widths of able approximations that work well within typical ex- the Sinc-based density to the Gaussian density have been perimental setups, it is important to show just how well performed (Chan et al., 2007; Law and Eberly, 2004) to (or not) this estimate of the transverse correlation width great effect (Edgar et al., 2012). Since choosing which corresponds with experimental data. This can be done width to fit is somewhat arbitrary, we point out that the in (at least) two ways. First, in (Howell et al., 2004), peak-matching fit also fits the full width at 48.2% of the they found a measure of the transverse correlation width maximum. However, the best estimate of σ is obtained by placing a 40µm slit in the signal beam, and scan- x− ning over the idler beam with another 40µm slit (see q 2 by an explicit calculation of σx− ≡ hx−i from the more Fig. 5 for a diagram of the idealized setup). By measur- accurate density (52). Remarkably, we find that σx− is ing coincident detections as they scan, and normalizing 12

an experiment was performed in which the joint position photon distribution was imaged with a camera. By fitting a Double-Gaussian to their empirical distribution, they found a correlation width of 10.9±0.7µm (for their 355nm pump beam and 5mm crystal). Surprisingly, this agrees more with our peak-matching estimate of 12.2µm than with our ostensibly more accurate estimate of 17.5µm (for these parameters). This however is to be expected, as the fitting by its very nature gives a result whose shape most closely resembles the shape the data gives, and low-level noise will mask information about the distribution beyond the central peak. Future experiments with higher-resolution measurements are needed to better explore the strength of this approximation. The second way that one can use experimental data FIG. 5: Idealized diagram of an experiment (Howell to place a limit on the transverse correlation width is to et al., 2004) to measure the transverse correlation use the comparatively larger amount of data about tem- width. The nonlinear crystal (NLC) is just after the poral correlation widths. As an example, if one knew laser, with a pump filter placed just after that. The that in a single downconversion event, the photons were beam is broken into signal and idler with a 50/50 generated no further than 100fs apart 90% of the time, beamsplitter. There is a loss of coincidences due to the then the speed of light assures us that the photon pair beamsplitter but this doesn’t affect the spatial intensity could be no farther than 30µm apart 90% of the time as profile. Two lenses are used to image the exiting face of well. Indeed, in (Ali Khan and Howell, 2006), they mea- the nonlinear crystal onto the image planes where two sured approximately a 50fs time-correlation width (using slits are placed. With one slit fixed, the other mobile, our convention) with downconverted photons from the and Avalanche Photo-Diodes (APDs) behind each slit, same 390nm pump laser incident on a 2mm long non- one can obtain the correlation width by comparing the linear crystal. With this value, we can place an upper width of the coincidence distribution to the width of the bound to the transverse correlation width of the light in slits. that setup by 15µm, which is not substantially above our 14.9µm estimate. the resulting histogram, they measured the conditional transverse probability distribution, and obtained a conditional width, which is approximately identical to the VI. THE BIPHOTON BIRTH ZONE correlation width 19. With their measurements, they obtained a transverse correlation width (adjusting for our When a photon pair is created in SPDC, the location conventions) of about 13.5µm (with an estimated error of the pair production can be essentially anywhere in larger than 10%), so that our theoretical estimate (53) the crystal illuminated by the pump beam. The uncer- of 11.6µm (using their pump wavelength of 390nm and tainty in where this pair-production takes place is limited crystal thickness of 2mm) underestimates this by 14.1%. by the uncertainty in the location of the pump photon. Another measurement with the same laser and crystal However, for any given photon pair created, the mean was taken in (Bennink et al., 2004), where they obtained separation between the two is generally much smaller a transverse correlation width of 17 ± 7µm. Given how than the uncertainty (i.e., standard deviation) in the ex- these experiments’ resolutions were limited both by fi- pected location of the downconversion event. It is useful nite slit widths 20 and a large statistical uncertainty in to conceptualize the region surrounding that mean position where the daughter photons are most likely to be σ(x1−x2), our approximation is accurate to within experimental uncertainty. More recently, (Edgar et al., 2012), found as what we shall call a birth zone. Given that momentum is very nearly conserved in SPDC, we define the expected (transverse) location of the downconversion event as the mean of the two pho- 19 When ρ(x1, x2) = ρ(x+)ρ(x−) and σx+ σx− , it follows that tons’ positions xm: σ(x1−x2) ≈ σ(x1|x2). 20 In order to obtain an estimate of 13.5µm for the transverse cor- x1 + x2 x+ relation width using slits 40µm wide, they deconvolved their co- xm ≡ = √ . (54) incidence histograms with the slit rectangle function. This gave 2 2 them more accurate estimates for the joint coincidence distributions at arbitrary resolution from which they could obtain more With the Double-Gaussian wavefunction (51) as our accurate estimates of the transverse correlation width. model for transverse position statistics in SPDC, we find 13

tively σ /σ ) to be a measure of the degree of corre- y1 x+ x− lation between the signal and idler photon ﬁelds 22. The birth zone number N is not to be confused with the Fe- dorov ratio R (Fedorov et al., 2004), which is the ratio

σx1 /σ(x1|x2). In a sense, each birth zone can be thought of as an independent source of photon pairs. In this way, ‘ σ1 we can develop an intuitive understanding of the coher- ΔBZ ence properties of the two-photon ﬁelds in SPDC 23. In particular, the birth zone width is useful in many calculations involving entangled photon pairs in SPDC. x1 Consider the mutual information of the Double-Gaussian distribution (33);

σ2 + σ2 x+ x− 1 1 h(x1 : x2) = log = log N + . 2σx+ σx− 2 N (57) Also for the double-Gaussian state, the birth zone σP number N is related to the Schmidt number K, a measure of entanglement 24 for pure continuous-variable states Δ P (Law and Eberly, 2004) (see appendix for more details). 1 For this state, K = 2 (N + 1/N), and the mutual infor- FIG. 6: Idealized diagram of the transverse intensity mation becomes: profile (in both x and y) of the downconverted light just as it exits the crystal. The blue circle encapsulates the h(x1 : x2) = log(K) ≈ log(N) − 1. (58) region within one standard deviation of the pump photon position (or also approximately the signal or where the approximation applies for large N. idler photon position) from the beam center. The red In addition, the birth zone number gives us a perspec- circle is centered on a particular downconversion event, tive in understanding the tradeoff between the first-order and encapsulates the region where the signal and idler spatial coherence and the measurable biphoton correla- photons are likely to be found given that their mean tions in the downconverted fields 25. For completeness, position is known to be at the canter of the circle (i.e., we also briefly discuss the second order spatial coherence, one birth zone). For a sense of scale, we let as it is related to the biphoton correlations. As a bit of (1) ∆p/∆BZ = 10. background, the first order coherence function, g (a, b), is a normalized correlation between the electric field at one point (a) and the electric field at another point (b) the standard deviation in the mean position σxm to be: (in space or in time). If the electric field is coherent be- 1 tween two points (so that the phase difference between σ = σ = σ . (55) xm 2 (x1+x2) p these two points is on average well-defined), then the coherence function will have a magnitude near unity. The In addition, we define the width of the pump, ∆P , to be twice this standard deviation. While the photon pair is expected to be created at xm with uncertainty σp, the signal and idler photons, conditioned on their mean posi- 22 Although the birth zone number N is one dimension is de- 0 tion being at some xm,√ will each have position uncertain- fined as ∆P /∆BZ , we can express it simply in terms of the ties σ0 = σ0 = σ / 2. As such, we define the birth measured full-width at half-maximum of the pump. Using the x1 x2 x− zone width, ∆ , to be twice these conditioned position peak-matching approximation for the birth zone width, we find: BZ FWHMp 21 N = r . uncertainties , giving us: 8 ln(2) Lz λp √ 9πnp 0 23 For an extensive reference on the relationship between first and ∆BZ ≡ 2σ = 2σx = σ . (56) x1 − (x1−x2) second-order coherence in biphoton fields in SPDC, see (Saleh et al., 2000). With these definitions, we take the birth zone num- 24 Given a pure continuous-variable density operatorρ ÂB = ber N (which in one dimension is ∆P /∆BZ or alterna- |ψAB ihψAB | describing the joint state shared by parties A and B, the Schmidt number K is the reciprocal of the trace of the 2 square of either reduced density operator; i.e., K ≡ 1/Tr[ˆρA] = 2 √ 1/Tr[ˆρB ]. The Schmidt number is also known as the inverse 21 The equation σ0 = 2σ holds only when ρ(x , x ) is sep- participation ratio. x1 x− + − 25 arable into the product ρ(x+)ρ(x−), as is the case for both the For a more thorough look into the first-order coherence properties Double-Gaussian and Sinc-Gaussian distributions. of downconverted fields, see (Dixon et al., 2010). 14 second-order coherence function, g(2)(a, b), is a normal- Gaussian probability density 26. In addition, we define ized correlation between the intensity (i.e., square of the the second order coherence width, ∆g(2), as the value of electric field) at one point with the intensity at another x where g(2) falls below unity, and the correlations can point. While g(1)(a, b) can be used to characterize the be treated as coming from a nonclassical source of light extent of interference effects in the signal/idler beams, 27. With these definitions, we find: g(2)(a, b) can be used to characterize the extent of sig- s 2 nal/idler photon correlations. N + 1 for large N ∆ ∆g(1) = ∆ ≈ −−−−−−−→ p = ∆ To examine the first and second-order coherence func- p (N 2 − 1)2 N BZ tions in SPDC for a qualitative understanding, we look (65) at the symmetric first order and second order spatial co- and herence functions, (g(1)(x, −x) and g(2)(x, −x), respec- s tively), as they have particularly simple forms in the ∆ 1 N 2 + 1 N 2 + 1 ∆g(2) = p log Double-Gaussian approximation. The first-order sym- N 2 N 2 − 1 2N metric spatial coherence is defined as: s for large N ∆p 1 N † −−−−−−−→ ≈ log . (66) (1) haˆ1(x)â1(−x)i N 2 2 g (x, −x) ≡ q , (59) haˆ†(x)â (x)ihaˆ†(−x)â (−x)i 1 1 1 1 We note that the errors in these approximations decrease monotonically, so that for N > 12.3, the error in our which in terms of our biphoton wavefunction ψ(x1, x2), approximation to ∆g(1) falls below 1%, and for N > 11.4, can be expressed as: the error in our approximation to ∆g(2) also falls below 1%. R dx ψ∗(x, x )ψ(−x, x ) (1) 2 2 2 Based on these calculations of the first and second- g (x, −x) = q , R 2 R 2 ( dx2|ψ(x, x2)| )( dx2|ψ(−x, x2)| ) order coherence widths, we see that for typical sources (60) of downconversion (where N ∼ 100) the general area We note that the expectation values taken here are taken over which the downconverted light will exhibit significant first-order coherence is approximately the same as with the state of the downconverted field |ΨSPDC i. 2 Similarly, the second-order symmetric spatial coher- the area of a birth zone ∆BZ . Because of this, down- ence is defined as converted light beams having a large degree of position or momentum correlation (i.e, a large N) can be con- † † sidered as a collection of many independent sources of (2) haˆ1(x)â2(−x)â2(−x)â1(x)i g (x, −x) ≡ , (61) photon pairs, each incoherent with one another (in the haˆ†(x)â (x)ihaˆ†(−x)â (−x)i 1 1 2 2 first-order sense). The second-order coherence width tells us a slightly different story since it only differs greatly which in terms of ψ(x , x ), is expressed as: 1 2 from the first-order coherence width in the limit of zero correlation, or N = 1. In this limit, we see that when |ψ(x, −x)|2 (2) the first-order coherence width is very large (implying the g (x, −x) = R 2 R 2 . ( dx2|ψ(x, x2)| )( dx1|ψ(x1, −x)| ) downconverted light can be described as a single coherent (62) beam), the second order coherence width is necessarily Using the Double-Gaussian approximation of the small. However, the reverse is not true, since for large biphoton wavefunction, these symmetric coherence func- N, one can have small first and second order coherence tions take simple Gaussian forms. For ψ(x1, x2) as de- widths. Since a small second order coherence width could fined in (51), we find: imply either a large or small first-order coherence width, the tradeoff between single photon coherence and bipho- 2 2 2 − x (N −1) (1) (1) 2∆2 N2+1 ton correlation is best understood by comparing ∆g g (x, −x) = e p (63) with N or the mutual information h(x1 :x2). and

x2 N2−1 2 − 2 2 ∆p N +1 (2) N + 1 2 26 Since ∆g(1) is the ”1σ” half-width of g(1)(x, −x), and g(1)(x, −x) g (x, −x) = e 2N . (64) 2N gives the coherence of photons separated a distance 2x away from one another, photons separated by less than ∆g(1) will be To find the range of values of x over which g(1) and approximately coherent. 27 Having a second order coherence g(2) below unity is a witness (2) g are significant, we define the first-order coherence that the source of light must be nonclassical (i.e., not a coherent (1) (1) width,√ ∆g , to be the value of x where g falls to or thermal state, as may generated from a collection of indepen- 1/ e. Note that this can be considered to be ”σ” in a dent atoms (Loudon, 2000)). 15

The relationship between the first-order spatial coher- We gratefully acknowledge helpful discussions with Dr. ence of the downconverted fields, and the biphoton corre- Gregory A. Howland, as well as the careful editing of lations as measured by the mutual information is a man- Daniel Lum, Sam Knarr, and Justin Winkler. In ad- ner of tradeoff. The mutual information of the position dition, we are thankful for the support from the Na- correlations (58) increases with N, while ∆g(1) (65) de- tional Research Council, DARPA-DSO InPho Grant No. creases with N. Thus, highly correlated downconverted W911NF-10-1-0404, DARPA-DSO Grant No. W31P4Q- fields can be treated as incoherent light, while highly co- 12-1-0015, and AFOSR Grant No. FA9550-13-1-0019. herent (in the first-order sense) downconverted fields can be treated as an uncorrelated source of downconverted light (i.e., as a single beam at the downconverted fre- Appendix A: The double-Gaussian field propagated to quency producing otherwise uncorrelated photon pairs). different distances As one final point on the relationship between first and The Double-Gaussian field (31), when propagated to second-order coherence, there are cases where the vis- different distances z and z has the same double- ibilities of first-order and second-order interference are 1 2 Gaussian form, though the coefficients are significantly very simply related to one another. In particular, it was more complex. shown in (Saleh et al., 2000) that for the two-slit ex- √ periment with downconverted biphotons, the first-order 2 DG ac − b −(ax2+2bx x +cx2) ρ (x , x ; z , z ) ≈ e 1 1 2 2 , visibility V1 and the second-order visibility V12 follow the 1 2 1 2 π relation: (A1) 2 2 2 2 2 2 2 2 2 kp(σx + σx )(z2 + kpσx σx ) V1 + V12 ≤ 1. (67) : a = + − + − , d 2 2 2 2 2 2 This can be understood both in terms of a tradeoff be- k (σ − σ )(z1z2 − k σ σ ) : b = p x+ x− p x+ x− , tween signal-idler entanglement and single-photon coher- d ence, as well as in terms of the monogamy of entangle- 2 2 2 2 2 2 2 kp(σx + σx )(z1 + kpσx σx ) ment between the signal photon, idler photon, and a mea- : c = + − + − , d surement device. Indeed, this tradeoff has been used suc- 2 2 2 2 2 2 2 2 2 2 : d = k (z + z )(σ + σ ) + 2k z1z2(σ − σ ) + cessfully to experimentally estimate the Schmidt number p 1 2 x+ x− p x+ x− + 4z2z2 + 4k4σ4 σ4 . (A2) (a measure of entanglement) of photon pairs in a beam 1 2 p x+ x− of down-converted light (Di Lorenzo Pires et al., 2009). However, we can still find some useful properties. For example, the Pearson correlation coefficient (see Table 1) has the simple expression: VII. CONCLUSION −b r = √ . (A3) The usefulness of spontaneous parametric downconver- ac sion (SPDC) as a source of entangled photon pairs is In more explicit terms, we get historically self-evident (see footnote 1). In this discussion, we have looked at the fundamental principles gov- σ2 − σ2 1 − z¯1z¯2 x+ x− erning SPDC in such a way as is often used in continuous- r = r0 : r0 = , (A4) p(¯z2 + 1)(¯z2 + 1) σ2 + σ2 variable quantum information experiments (namely, de- 1 2 x+ x− generate collinear Type-I SPDC). We paid particular at- where r0 is the correlation coefficient at z1 = z2 = 0, tention to how one can predict the transverse-correlation (see Table 1). In addition, the normalized propaga- width of photon pairs exiting a nonlinear crystal from tion distancesz ¯1 andz ¯2 are defined such thatz ¯1 =

first principles with accuracy matching current experi- z1/(kpσx+ σx− ), andz ¯2 = z2/(kpσx+ σx− ). Using this cor- mental data. Along the way, we digressed to explore relation function for the Double-Gaussian, (as mentioned how the double-Gaussian wavefunction used to describe previously), as we move from the near field (z1, z2 ≈ 0) to SPDC allows a straightforward analysis of its measure- the far field (z1, z2 0), the initially strong position cor- ment statistics even under free space propagation. In ad- relations gradually weaken and eventually become strong dition, we have developed further the concept of a bipho- position anti-correlations in the far field. In addition, we ton birth zone number, and have shown how it manifests also see, that when comparing a measurement of the sig- itself in the duality between the correlations within one nal photon in the near field to the idler photon in the far of the downconverted fields, and the correlations between field, the correlations approach zero. Since position mea- the downconverted fields. It is our hope that this discus- surements in the far field can be taken as measurements sion will inspire further interest in the transverse spatial of momentum (scaled accordingly), we see that the posi- correlations of entangled photon pairs, and in the relation of one photon is uncorrelated with the momentum tionship between intra- and inter-party coherence. of the other (and vise versa). 16

Appendix B: Schmidt decomposition and quantum ω1 and ω2 (Mikhailova et al., 2008a,b). In doing so, we entanglement of the Double-Gaussian state find that the frequency (and temporal) correlations between the signal/idler photon pairs differs significantly In order to measure the entanglement of a pair of par- whether they come from Type-I or Type-II SPDC. In ticles, one needs to know the complete density matrix de- Type-II SPDC, the polarizations of the signal-idler pho- scribing the pair of particles. To reckon with continuous- ton pairs are orthogonal to each other, and so experience variable states described by continuous wavefunctions (or a different index of refraction in birefringent nonlinear mixtures thereof), we need to decompose such wavefunc- crystals. In this case, the biphoton wavefunction (not tions into an orthogonal basis of eigenfunctions often counting the pump profile) depends to first order on the corresponding to measurably distinct outcomes of some difference between the signal and idler photon frequen- discrete observable. Occasionally, the density matrix in cies, resulting in a sinc function of the frequency differ- such a decomposition has a finite expression (with zero ence, which translates to a top-hat function of the time amplitudes for all other eigenfunctions in that infinite- difference. In this case, the temporal correlations can be dimensional basis), and the analysis is straightforward. characterized by the (full) width W(t1−t2) of the top-hat Other times, the density matrix has non-trivial compo- function, giving us: nents over its entire spectrum, and an exact determina- tion is impossible (though approximations may suffice). L |n(1) − n(2)| (Type-II) W = z g g , (C1) However, when the pair of particles can be described (t1−t2) c by a pure state (i.e., a single joint wavefunction), it is (1) sufficient to know just the eigenvalues of the reduced den- where ng is the group index of the signal photon at its sity matrix of either particle. These eigenvalues manifest central frequency ωp/2, and c is the speed of light in vac- themselves in the Schmidt decomposition of an entangled uum. This width amounts to the accumulated time lag pure state. between the signal and idler photons due to their experi- As discussed in (Fedorov et al., 2009) and (Law and encing different indices of refraction. Using the research Eberly, 2004), the Double-Gaussian state can be decom- in (Pittman et al., 1996b), where they used a 0.5mm BBO posed into Schmidt modes. crystal with a pump wavelength of 351.1nm, we find that W ≈ 125fs, which agrees within experimental un- ∞ (t1−t2) X p certainty with their results. ψDG(x , x ) = λ u (x )u (x ) (B1) 1 2 n n 1 n 2 In Type-I SPDC, the signal-idler photon pairs have n=0 parallel polarizations, and so they experience the same th Here, un(x) is the n “energy” eigenfunction of the quan- index of refraction in the nonlinear crystal. Conse- tum harmonic oscillator, and λn, in our notation, is: quently, the biphoton wavefunction (not counting pump profile) to lowest order depends on the square of the (σ − σ )2n 4N N − 12n signal-idler frequency difference. As a result, the tech- λ = 4σ σ x+ x− = n x+ x− 2n+2 2 niques used to estimate the transverse spatial correlation (σx+ + σx− ) (N + 1) N + 1

(B2) width σ(x1−x2) can also be applied to estimating the tem-

The Schmidt number K is expressed as the reciprocal of poral correlation width σ(t1−t2), giving us: the sum of the squares of the Schmidt eigenvalues: (exact) q σ = 9Lz κ1 , 1 1 1 (Type-I) (t1−t2) 10 (C2) K = = N + . (B3) (PM) q 4L κ P∞ 2 σ = z 1 . n=0 λn 2 N (t1−t2) 9

2 Interestingly, since the Schmidt eigenvalues of the d k1 ω where κ1 = 2 | p is the group velocity dispersion con- dω 2 Double-Gaussian state are geometrically distributed (a stant at half the pump frequency. maximum entropy distribution for constant N), the Because the signal and idler photons experience the Double-Gaussian state is the maximally entangled state same index of refection in Type-I SPDC, their tempo- for a constant birth zone number. ral correlation width can be much smaller than for pairs originating from a Type-II source. Indeed, for a 3mm Appendix C: Heisenberg limited temporal correlations in BiBO crystal cut for Type-I SPDC, with a pump wavelength of 775nm, the temporal correlation width σ(PM) SPDC (t1−t2) is predicted to be approximately 4.0fs, nearly two or- If instead of taking the small-angle approximation in ders of magnitude below the width for a typical Type- order to get the transverse biphoton wavefunction (26), II source. With a narrowband pump spectrum, the ra- 9 we look directly at the longitudinal component of the tio between σ(t1+t2) and σ(t1−t2) can be as large as 10 , wave vector mismatch ∆kz, we can express the biphoton a degree of temporal correlation outstripping any spa- state in terms of the signal and idler photon frequencies tial correlations discussed thus far. However, even with 17 negligible jitter times and noise in photon counting ap- 4 femtoseconds. However, if we perform frequency filter- paratuses, these temporal correlations are not accessible ing to look at the nearly degenerate part of the biphoton unless the experimental setup used to measure these cor- frequency spectrum, we increase the minimum resolvable relations accepts a relatively large range of frequencies; time correlation width by a factor inversely related to the the degree of measurable temporal correlations is limited fraction of frequencies allowed to pass through the filter. by the frequency-time Heisenberg relation (Mandel and If we consider a 2nm filter centered at 1550nm making 11 Wolf, 1995). In addition, due to dispersion in optical σω1 ≈ 7.8×10 (radians per second), then the minimum 2 fiber, the time correlation width measured by ideal pho- resolvable value of σ(t1−t2) would be about 6×10 fs, more ton detectors may still be significantly larger than what than two orders of magnitude wider that what is achiev- could be measured exiting the crystal. Indeed, just as able without filtering. there is a simple formula for the temporal spreading of Gaussian pulses due to dispersion, one can show that the

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