Journal of Optics
TUTORIAL • OPEN ACCESS Recent citations Introduction to the absolute brightness and - Simulating Correlations of Structured Spontaneously DownConverted Photon number statistics in spontaneous parametric Pairs down-conversion Sivan TrajtenbergMills et al
To cite this article: James Schneeloch et al 2019 J. Opt. 21 043501
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J. Opt. 21 (2019) 043501 (28pp) https://doi.org/10.1088/2040-8986/ab05a8
Tutorial Introduction to the absolute brightness and number statistics in spontaneous parametric down-conversion
James Schneeloch1 , Samuel H Knarr2,3, Daniela F Bogorin1, Mackenzie L Levangie4, Christopher C Tison1, Rebecca Frank4, Gregory A Howland1,5, Michael L Fanto1,5 and Paul M Alsing1
1 Air Force Research Laboratory, Information Directorate, Rome, NY 13441, United States of America 2 Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, United States of America 3 Center for Coherence and Quantum Optics, University of Rochester, Rochester, NY 14627, United States of America 4 Department of Physics, Northeastern University, Boston, MA 02115, United States of America 5 RIT Integrated Photonics group, Rochester Institute of Technology, Rochester, NY 14623, United States of America
E-mail: [email protected]
Received 3 August 2018, revised 16 January 2019 Accepted for publication 8 February 2019 Published 26 February 2019
Abstract As a tutorial, we examine the absolute brightness and number statistics of photon pairs generated in spontaneous parametric down-conversion (SPDC) from first principles. In doing so, we demonstrate how the diverse implementations of SPDC can be understood through a single common framework, and use this to derive straightforward formulas for the biphoton generation rate (pairs per second) in a variety of different circumstances. In particular, we consider the common cases of both collimated and focused Gaussian pump beams in a bulk nonlinear crystal, as well as in nonlinear waveguides and micro-ring resonators. Furthermore, we examine the number statistics of down-converted light using a non-perturbative approximation (the multi- mode squeezed vacuum), to provide quantitative formulas for the relative likelihood of multi- pair production events, and explore how the quantum state of the pump affects the subsequent statistics of the down-converted light. Following this, we consider the limits of the undepleted pump approximation, and conclude by performing experiments to test the effectiveness of our theoretical predictions for the biphoton generation rate in a variety of different sources. Keywords: photon pair, quantum optics, nonlinear optics, SPDC, down-conversion, quantum nonlinear optics
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2040-8978/19/043501+28$33.00 1 © 2019 IOP Publishing Ltd Printed in the UK J. Opt. 21 (2019) 043501 Tutorial 1. Introduction generally obtained from this Hamiltonian using first-order time-dependent perturbation theory. In section 3, we calculate Spontaneous parametric down-conversion (SPDC), first the generation rates for SPDC in both bulk and periodically- observed over five decades ago (Harris et al 1967)6 is the poled nonlinear crystals, for both collimated and focused premier workhorse in quantum optics, both as a source of pump beams, and for the collection of biphotons in a single entangled photon pairs as well as heralded single photons. As transverse (Gaussian) mode as well as over all modes. In single quantum events, pump photons inside a χ(2)-nonlinear section 4, we use a different (non-perturbative) approximation medium interact with the quantum vacuum via this medium to to obtain the number statistics of down-converted light, pro- down-convert into signal-idler photon pairs. This process is viding a quantitative description of the (approximate) multi- spontaneous because there is initially no field at the signal or mode squeezed vacuum state created by SPDC, and showing idler frequencies. When there are initial signal and idler fields, how one may optimize both the brightness and heralding the process is known as difference frequency generation, efficiency of down-converted light. In section 5, we use a stimulated parametric down-conversion, or parametric similar approach to examine the brightness of SPDC in amplification and is well-treated in classical nonlinear optics waveguides and micro-ring resonators (MRRs), where pump (Boyd 2007). SPDC was originally deemed an optical para- intensities may be substantially larger than in the bulk crystal metric process because in classical nonlinear optics, it can be regime. In section 6, we make a digression to consider SPDC interpreted as driving a resonant, periodic oscillation in the with a fully quantum, depletable pump, and examine the dielectric constant, in analogy to parametric oscillation of effect of the quantum state of the pump on the subsequent mechanical systems (Louisell et al 1961). In current termi- intensity of the down-converted light. Finally, we conclude nology, optical parametric processes are equated to those by performing experiments to test the formulas derived for the involving no net exchange of energy or momentum with the pair production rate in this tutorial, showing decent agreement nonlinear medium7. Because of this, we can treat SPDC as the relative to experimental design. quantum evolution of a closed system (i.e. the electro- magnetic field), where the Hamiltonian describing the non- linear interaction determines the state of the field. 2. Foundation: the Hamiltonian for the SPDC In this tutorial, we explore the fundamentals of SPDC process and rate calculation through a comprehensive derivation of the biphoton genera- fi tion rate for both type-I and type-II phase matching; for both The Hamiltonian for the electromagnetic eld, EM, is given single-mode and multi-mode pump illumination and biphoton by its total energy UEM up to a constant offset. This is a collection, and for both bulk crystals and nonlinear wave- common assumption, which is valid when the total energy guides, where some formulas (e.g. for type-I collinear SPDC) contains no explicit time dependence, or dissipative terms To fi are not found elsewhere in the literature. To accomplish this, nd the total energy, it is easier to work with the rate of fi we develop a general Hamiltonian describing all such SPDC change of the energy of the electromagnetic eld, UEM, since a constant term in the Hamiltonian will not alter the equations processes; show how to specialize it for each situation; and 9 derive the biphoton rates using techniques similar to Fermi’s of motion, and can be neglected . As discussed in Jackson (1999), the rate of change of the golden rule, as discussed in Ling et al (2008). Furthermore, energy of the electromagnetic field U is equal and opposite we discuss the number statistics of the down-converted light EM in sign to the rate of work done by the field on electric (described by the multi-mode squeezed vacuum8 state charges: (Lvovsky 2016)) in order to explore the tradeoff between the dU number of pairs produced, and the ability to herald single EM =- dr3 JE·().1 photons from coincidence counts due to multi-pair generation dt ò events. With the current emphasis of quantum nonlinear Here, the work done is exclusively due to the electric field, optics turning towards chip-scale implementations of quant- since the magnetic field produces force perpendicular to um information protocols, understanding the factors con- velocity, as shown in the Lorentz force law. Using Maxwell’s tributing to the brightness of photon-pair sources is critical for equations for arbitrary dielectrics, this can be re-expressed those entering the rich field of quantum optics. purely in terms of fields as: The rest of this reference is laid out as follows. In ⎛ ⎞ dU dB dD section 2, we derive the general Hamiltonian for SPDC pro- EM =+dr3 ⎜H ··E ⎟,2 () dt ò ⎝ dt dt ⎠ cesses and show how biphoton generation rates can be 6 where D =+0 EP, and m00HB=-m M. In early references on the subject, SPDC was known as parametric Here, we make our first three assumptions. First, we fluorescence. For a discussion on the historical development of SPDC, see Klyshko (1989). assume the material is non-magnetic so that m0 H = B. Sec- 7 Though the crystal absorbs some pump light according to ordinary linear ond, we assume the frequency spectrum of the light that will optics, this exchange of energy is an independent interaction not due to SPDC. 8 The multi-mode squeezed vacuum state of SPDC light is defined as the 9 For a more general method of deriving the Hamiltonian starting from the product of multiple simultaneous two-mode-squeezed vacuum states for each electromagnetic Lagrangian, see Hillery and Mlodinow (1984), Hillery correlated pair of modes. (2009).
2 J. Opt. 21 (2019) 043501 Tutorial be interacting with the material is far enough from any Note that here and throughout the paper, we use the interac- absorption bands (i.e. off-resonance) that the material is tion picture, where the nonlinear Hamiltonian shall be con- approximately lossless. Third, we assume that where the sidered a small contribution to the total Hamiltonian. In order fi pump light is weak compared to the electric eld binding to obtain the Hamiltonian for the quantum electromagnetic fi fi electrons to their atoms, the polarization eld P is expressible eld, we use the standard quantization procedure as discussed as a rapidly decaying power series in E: in Hillery and Mlodinow (1984), Mandel and Wolf (1995a) and Duan and Guo (1997). In a medium of index of refraction P=++ []()cc()12 E () EE c () 3 EEE +... . 3 i 0 ij j ijk jk ijkℓ jkℓ n, the electric displacement field operator Dˆ ()r is expressible Note that here, χ(1) and χ(2) are the first- and second-order optical as a sum over momentum and polarization modes in a rec- susceptibility tensors, and we use the Einstein summation con- tangular cavity of volume V with dimensions Lx, Ly, and Lz, 11 vention to simplify notation. Since most crystalline materials respectively . For convenience, Dˆ ()rt, is separated into + respond differently to fields polarized along its different principal positive and negative frequency components Dˆ ()rt, + - axes, the induced polarization will not always point in the same Dˆ (rt, ), where; direction as the applied electric field. These assumptions are 2 + 0n w k easily satisfied in most cases in nonlinear optics, as discussed in Drtˆ (), = i k atˆ () eik· r ,8 ( ) å ks, ks, Boyd (2007), and we use them throughout this tutorial. Since we ks, 2V are discussing SPDC, a second order process, we need only - + and Dˆ (rt, ) is the Hermitian conjugate of Dˆ ()rt, . Here, expand the polarization to second order in the electric field. aˆ is the annihilation operator of a photon12 with momentum Alternatively, we can express the electric field E as a rapidly ks, k and polarization in direction indexed by s (which can decaying power series of D: ks, take one of two values for each transverse direction), and V is ()12 () () 3 Ei =+[]()zzij Dj ijk DDjk + zijkℓ DDjkℓ D +... , 4 the quantization volume, which we may take to approach infinity in the continuum limit. With this, the quantum where, for example, z ()1 is the first-order inverse optical sus- ij Hamiltonian describing linear-optical effects becomes: ceptibility tensor. ⎛ 1 ⎞ When quantizing the electromagnetic fieldinmatter(Hillery ˆ ⎜⎟† HL =+å w() katatˆ ()ˆ ks, ().9 ( ) ) ⎝ ks, 2 ⎠ and Mlodinow 1984 , the Hamiltonian is best expressed in terms ks, of D and B instead of E and B. Indeed, if one were to substitute As one can see, the linear Hamiltonian cannot be responsible the quantum operator for the free electric fieldtocreatea for the creation of photon pairs, as it is only first-order in both quantum Hamiltonian, the equations of motion obtained are no the creation and annihilation operators. longer consistent with Maxwell’s equations, and lead to non- The nonlinear quantum Hamiltonian Hˆ , physical results. For a straightforward discussion of why this is NL so, see Quesada and Sipe (2017). ˆ 1 3 ()2 ˆ ˆ ˆ HdrrDrtDrtDrtNL = (()z ijℓ ijℓ(),,,,10()())() Expressing E in terms of D, we greatly simplify calcu- 3 ò lating the Hamiltonian using the prior assumption of a lossless has a deceptively simple form. With each field operator 10 medium . In this regime, the first and second order con- Dˆ ()rt, expressed as Dˆ+-()rt,,+ Drtˆ (), where Dˆ+()rt, tributions have full permutation symmetry, and the total depends only on annihilation operators, and Dˆ-(rt, ) on fi energy rate simpli es to: creation operators, the nonlinear Hamiltonian is actually a dU 1 d ()1 sum over eight distinct terms. Various combinations of EM =+dr3 (·HB DE · ) dt 2 ò dt these terms correspond to different basic nonlinear-optical processes, but only those processes that conserve energy 1 3 d ()2 + dr ()z DDDijk,5 ()contribute significantly to the probability-amplitude of down- 3 ò dt ijk conversion. For example, the two terms that are third-order which can be integrated to give our Hamiltonian for the in either photon creation or annihilation may be excluded, electromagnetic field in a second order nonlinear dielectric. as their contribution to the probability amplitude of photon ()1 Here, E is the electric field up to first order in the inverse pair generation is a rapidly varying phase that becomes susceptibility. negligible even over the small time it takes light to travel The Hamiltonian of the electromagnetic field is now through the nonlinear medium. Furthermore, for many non- expressible as a sum of two terms, one governing the linear- linear media, ζ(2) (or alternatively χ(2)) is only significant optical response, and one governing the nonlinear response: for one particular optical process (either by design or
EM=+ L NL,6() 11 To fully quantize the field in a bulk dielectric, one would take the where, continuum limit (V ¥) so that the displacement field operator is an integral over a continuum of momentum modes. For simplicity of our 1 3 ()2 calculation, we take the continuum limit later when we add up the NL = dr(()()()())z rDrDrDrijℓ.7 () 3 ò ijℓ contributions of the momentum modes of the fields in various configurations of SPDC. 12 Since we are quantizing the field in matter, the elementary excitations are 10 For a discussion on quantizing the electromagnetic field in a lossy collective excitations of both the electromagnetic and material degrees of medium, see Huttner and Barnett (1992). freedom. Even so, they are still regarded as photons.
3 J. Opt. 21 (2019) 043501 Tutorial happenstance)13. Even when ζ(2) is significant for multiple horizontal and vertical indices of a given Hermite–Gaussian nonlinear optical processes, simultaneously achieving phase- mode; matching (i.e. momentum conservation) for multiple pro- fi fi fi † † cesses is signi cantly more dif cult, but ef cient enough that aCaˆ = ˜q,m ˆ .12() ()qk,,zz s å ()m,,ks these multi-step optical parametric processes (Saltiel et al m 2003) have been demonstrated experimentally (Andrews et al 1970, Abu-Safe 2005), and have useful applications such as Since this change of basis preserves probability, and is enabling third-harmonic generation without needing a high therefore unitary, it follows that the boson commutation † relation [aaˆˆ()qk,,z s, ] = dqq, ¢ must be preserved in both third-order nonlinearity. ()qks¢,,z In the case of SPDC, either pump photons are destroyed representations. With this established, we can express the in exchange for signal-idler photon pairs or vice versa, so that displacement field operator Dˆ-(rt, ) in this new Hermite– Hˆ is well-approximated as: Gauss basis. NL The transverse spatial dependence of Dˆ-(rt, ) for a given ˆ 1 3 ()2 ˆ +--ˆ ˆ HNL =+dr(()z rDijℓ() rtDrtDrt,() ,() , H.c., ) Hermite–Gauss mode indexed by m relies on the sum: 3 ò ijℓ ()11 ˜ -iq· r Cem,q = LLgxyxy m (),, () 13 å where H.c. stands for Hermitian conjugate. Before we con- q tinue, we point out that we have conflated the polarization where we have defined gxy (), to be the normalized14 index s with the displacement field component index i. The m + Hermite–Gaussian wavefunction given by the index m. Here, operator Dˆ ()rt, is given by the sum in equation (8), but i m where the polarization unit vector is replaced by its is an ordered pair of non-negative integers corresponding to ks, the horizontal and vertical mode index, respectively. Because component parallel to the ith direction, ks, · x i. Throughout this paper, we will be working in the paraxial regime, where the momentum components can only take on values that are π the light is propagating primarily along a single direction (i.e. integer multiples of 2 divided by the respective length of the along the optic axis). In this situation, it is a valid approx- cavity in each direction, this relation is straightforward to fi imation to simply replace the displacement field component check through normalization. For nite size nonlinear crys- indices with polarization indices since the component of the tals, this relation is approximate, but accurate when the – displacement field parallel to the optic axis is negligible. Hermite Gaussian modes are encompassed by the crystal. Finally, using an element of the paraxial approximation (so that the frequency ω only depends on k ), the displacement – z 2.1. Transforming the Hamiltonian into the Hermite Gauss field operator becomes: basis fi 2 The canonical quantization of the electromagnetic eld into - 0nk w kz ˆ z -ikz z iw t † † fi Drt(), =- iå ksz, gxyem (),. eaˆm,,ks plane-wave modes with creation operators aˆ is the rst step 2L z ks, m,,ksz z in the standard quantum treatment of SPDC light. However, it ()14 will make subsequent calculations much simpler if we express fi the transverse momentum components of the eld in terms of Here, we point out that aˆ ††()tae= ˆ itw . – mm,,kszz ,, ks Hermite Gaussian modes, since Gaussian pump beams and With the displacement field operators expressed in the similar collection modes of down-converted light are valid Hermite–Gauss basis, we are ready to obtain the nonlinear descriptions of the light generated in SPDC experiments. Hamiltonian (11). In the standard approach for treating Moreover, expressing SPDC in terms of Laguerre-Gaussian SPDC, the pump field is treated as being bright enough that – and Hermite Gaussian modes of light is physically motivated classical electromagnetism is sufficient for its description, and because of their connection to orbital angular momentum that its intensity is not noticeably diminished due to down- ( ) Allen et al 1992 , and the conservation of the same quantity conversion events (also known as the undepleted pump ( ) Walborn et al 2010 . approximation). We use the classical pump approximation In order to do this, we first introduce some notation. Let throughout most of this paper, but use a more accurate q denote the projection of the momentum k onto the trans- description when discussing the number statistics of the verse plane, so that k =+qkzz ˆ, and zˆ is a unit vector down-converted light. pointing along the optic axis in the direction of propagation. Then, the creation operator aˆ † can be expressed as aˆ † . ks, ()qk,,z s 2.1.1. The classical pump field. Although arbitrary Since both plane waves and Hermite–Gaussian wavefunctions illumination of the nonlinear medium can be expressed as form a complete basis in 2D space, we can express the plane- an integral over all frequencies, SPDC occurs only in † wave creation operator aˆ()qk,, s as a sum over transverse mode narrow bands of pump frequencies where phase matching z creation operators aˆ † , where m is a vector denoting the (i.e. momentum conservation) can be achieved due to ()m,,ksz
13 Although not impossible, simultaneous generation of second harmonic 14 The Hermite–Gaussian mode functions are normalized so that the integral generation light and SPDC photon pairs in a single nonlinear medium with a over all space of their magnitude square gives unity (as with quantum single (pump) laser source has yet to be accomplished. wavefunctions).
4 J. Opt. 21 (2019) 043501 Tutorial dispersion15. In light of this, we limit ourselves to the With the parameters of a Gaussian pump beam, the ubiquitous case of a monochromatic pump beam with peak amount of energy per second delivered by such a beam (i.e. magnitude ∣∣D0 , frequency ω , polarization , and (non- its power) is expressed as: p p p normalized) spatial dependence fr()given by: p 0 2 ∣∣Dp 0 Pc= ps2 ,23() Drtp()∣∣,cos,15= Dp p frp ()()wp t () 3 p n 0 which can be separated into positive and negative frequency which equals the mean intensity of the beam times its components, giving us: effective area17. itwp - 0 e Drtpp()∣∣, = Dp frp ().16 () 2 2.1.2. Simplifying the nonlinear Hamiltonian. Incorporating 16 fi The (time averaged) pump intensity is then: our expressions for the displacement eld operators and the fi c classically bright pump eld, the nonlinear Hamiltonian I = Dfr0 22 p 3 ∣∣∣()∣p p .17 () becomes: 2 0 n
For later simplification, we factor out the linear phase due to HdrrDGreeˆ = 3 (()∣∣z ()20˜*() ikpz z- iw p t NL ò eff pp propagating the beam, and get: n 2 w -ik z 0 1 1 -ik11z z iw t † fr()= Gre˜p ()z ,18 ( ) ´-i gxye(), e aˆ p å m1 m1,k1z m ,k 2Lz 1 1z where G˜ ()r is implicitly defined. p 2 0n2 w2 -ik z iw t † Throughout most of this paper, G˜p ()r will describe the ´-i gxye(), 22z e aˆ å m2 m2,k2z 2Lz rest of a Gaussian pump beam, so that m2,k2z ⎛ 22⎞ ⎛ 22⎞ sp xy+ xy+ + H.c.) . Gr˜p()º-exp ⎜ ⎟ exp ⎜-ikz ⎟ ss()z ⎝ 4 ()z 2 ⎠ ⎝ 2Rz () ⎠ ()24 ⎛ ⎛ ⎞⎞ -1 z Here, we abbreviated n as n , and w as ω . Furthermore, ´ exp⎜i tan⎜ ⎟⎟ .() 19 k1z 1 k1z 1 ⎝ ⎠ ⎝ zR ⎠ we have already performed the sum over the components of the inverse susceptibility. The additional factor of 6=3! Except in section 3.4 where we consider SPDC using a comes from the permutation symmetry of the nonlinear focused pump beam, we use the simplifying approximation susceptibility where the total sum is 6 times the value of each that the pump beam is collimated, so that we may neglect the term, where all terms are added together. After simplifying, Guoy phase and curvature of the phase fronts in our we find: calculations. To condense notation, σ(z) is the evolving beam ( ) 0 radius as measured by standard deviation ; ∣∣Ep ww12 Hˆ NL = åå 2 2 ⎛ ⎞2 2Lz mm,,kknn1 2 z 1 1zz2 2 ss()z º+p 1.⎜ ⎟ ( 20 ) ⎝ ⎠ 3 ()2 -Dikzz zR ´ dr(()()()()c rGp* rg xyg,, xye ) ò eff mm12
R(z) is the evolving radius of curvature of the wavefronts: itDw †† ´+eaˆˆmm,,kk a H.c. ⎡ ⎤ 1 1zz2 2 ⎛ z ⎞2 ()25 Rz()º+ z⎢121⎜ R ⎟ ⎥ ( ) ⎣ ⎝ z ⎠ ⎦ ()2 Here we have switched from z eff to the effective nonlinear susceptibility c()2 using the approximation: and zR is the Rayleigh length, such that s()zRp= 2;s eff 2 -» 2 zc()22nnn2 2 ()2 ,26() 4ps p 0 eff p 1 2 eff zR º .22() lp which is satisfied under the same lossless media assumption that allowed us to invoke full permutation symmetry. Since As we can see, the first exponential governs the evolving χ(2) is what is measured experimentally, and tabulated in spatial amplitude of the beam; the second exponential handbooks of optical materials, the rest of this tutorial will describes the propagation and curvature of the phase fronts, be expressed in terms of the susceptibility, rather than its while the last exponential describes the Guoy phase. inverse. (2) 15 Since the χ nonlinearity is zero outside the nonlinear For example, in degenerate type-I SPDC, momentum conservation kp = 2k1 is achieved when np=n1. This is possible when the dispersion (i.e. medium, the spatial integration is carried over the dependency of index on frequency) is different for different polarizations in birefringent materials. It is also possible to achieve phase matching if the 17 The effective area of a probability distribution (as described by the dispersion is anomalous (e.g. near an absorption peak)(Cahill et al 1989), but transverse intensity distribution of light) is the reciprocal of the mean height birefringent phase matching is much more straightforward. of the probability density. This is the area that a uniform probability 16 The time averaged pump intensity may be taken as the magnitude of the distribution of such a mean height would have to have to be normalized. For Poynting vector ∣SEH∣∣=´1 * ∣, and in our approximations, ∣D0∣ = 2 p a two-dimensional radially symmetric Gaussian distribution, the effective 2 0 0 nE∣∣p . area is 4ps2.
5 J. Opt. 21 (2019) 043501 Tutorial dimensions of the medium. This Hamiltonian describes number is independent of its phase: fi SPDC in a nonlinear medium of length Lz,unspecied ⎛ ⎞ ( fi ⎜⎟Dwt transverse dimensions but signi cantly wider than the pump PWtkkmm= kk mm sinc 1zz,,1 2 ,2 1 zz ,,1 2 ,2 ⎝ ⎠ beam), and unspecified poling when illuminated by a 2 t monochromatic pump beam directed along the optic axis. ´¢dt eitD¢w Here, the sum over the polarization indices has already ò0 been carried out, giving the effective nonlinearity c()2 ()r . t eff WdtepdD¢ w itD¢w fi Δω ≡ ω( )+ω( )− ⟶()()kk1zz,,mm1 2 ,2 2.29 To simplify notation, we de ned k1z k2z large t ò0 ω(kpz),andΔkz ≡ k1z+k2z – kpz. Note that while the Hermite–Gauss modes of the down-converted light In practice, the interaction time t need not be arbitrarily large gxy (), are normalized to have unit norm, the pump spatial for this limit to apply. Instead, one only needs t to be sig- m nificantly longer than the inverse of Δω, which is achieved dependence G˜ ()r has a maximum magnitude of unity at p for times much longer than the picosecond time scales that r = 0. The peak pump intensity is fixed by the value of the light takes to travel the length of the nonlinear crystal, but not pump field strength ∣D ∣. p so large that multiple biphotons are likely to be generated in time t. The range of frequencies defining the width Δω (before this limit is invoked) is known as the phase-matching 2.2. Calculating the biphoton rate from the nonlinear bandwidth (and is of the order 1013–1014 for most materials). Hamiltonian ()2 Although ultimately limited by effective nonlinearity ceff, the With a general Hamiltonian describing most SPDC processes, phase-matching bandwidth is primarily determined by the we could calculate a general rate of biphoton generation using dispersion of the material where the condition ∣D 6 J. Opt. 21 (2019) 043501 Tutorial 2 2.2.1. Transition rate versus the rate of generated biphotons. so that ∣FD()∣kz becomes: The transition rate R is taken to be the average number of ⎛ ()2 ⎞2 2 c LZ 2 biphotons per second generated in the nonlinear medium. The ∣(FDkdze )∣2 =⎜ eff ⎟ -Dikzz z 2 ò reason this is so requires further explanation. The transition ⎝ 2ps1 ⎠ -LZ 2 fi rate R is de ned as the rate of change of the transition ⎡ ⎛ ⎞⎤ 2 ( + ) 1 2 probability. The transition probability P t dt is the ´-++dxdyexp⎢ () x22 y ⎜ ⎟⎥ .35 () ò ⎢ ⎜ 2 2 ⎟⎥ probability that the biphoton will be emitted either in the ⎣ ⎝ 4ssp 4 1 ⎠⎦ time interval tä[0, t] or in the interval tä[t, t+dt]. Since these intervals are disjoint, and the transition probability is Here, we have let the widths of the Hermite–Gaussian modes linear, the quantity Rdt is the probability that the biphoton of the signal-idler light be defined as σ1 in analogy with σp for will be emitted in an interval of length dt. Since the state of the pump beam. The value of σ1 is a free parameter in our the signal and idler field in the crystal is once again well- definition of the Hermite–Gaussian basis, but is best set using described by the vacuum state as soon as the biphoton exits the mode field diameter of the accepting single-mode fiber, the crystal, while the pump continues driving transitions, the and related collection optics that image the accepting mode to temporal statistics of biphotons generated in SPDC are well- the center of the crystal. To make the limits of the integral described as a Poisson point process. In particular, the over x and y arbitrarily large, it only suffices that the trans- probability of not generating a biphoton in the interval verse width of the crystal is larger than the dimensions of both t ä [0, t+dt] is given as the product of the same probability the Gaussian pump beam and of the signal and idler modes. over the interval t ä [0, t], and (1−Rdt). This defines a Even for crystals only a millimeter wide in x and y,itis differential equation, allowing one to obtain the exponential straightforward to have a well-collimated beam whose area is distribution for waiting times between biphoton generation contained within the crystal. Moreover, in single-mode non- events. One can then recursively obtain the probabilities of linear waveguides, the light can be confined to a much smaller one, two, or more transitions in an interval of length T from beam diameter without diverging. With these assumptions, this information as well. For example, the event of two the overlap integral simplifies significantly to: transitions in an interval of length T is broken down into one 2 2 ⎛DkL⎞ s p transition in the interval [0, t¢], a transition in the interval ∣(FDkL )∣2 = (2sincc()2 )22⎜⎟zz . zzeff ⎝ ⎠ 22 []tt¢¢+, dtand no transitions in the interval [tdtT¢+ , ]. 2 ss1 + 2 p Similar equations can be developed to describe the ()36 probability of detecting n biphotons over time T, Indeed, these statistics are described by a Poisson distribution with With this, the total rate for down-conversion from a colli- rate R such that the mean number of biphotons generated over mated Gaussian pump beam into Gaussian signal-idler time T is simply RT. For a more thorough discussion, see modes, RSM, is readily converted into an integral over the ω ω Hayat et al (1999), Ross (2010). signal and idler frequencies 1 and 2: 2 ∣∣()EL0 2 c()2 2 2 nn s2 Rdd= ww pzggpeff 12 SM ò 12 2 2 2 22 2pc nn1 2 ss1 + 2 p 3. The bulk crystal regime: photon-pair brightness ⎛ ⎞ 2⎜⎟DkLzz ´Dwwd12() wsinc . ⎝ 2 ⎠ Previously, we found a general form for the Hamiltonian ()37 describing SPDC (25) in a general bulk nonlinear crystal. All σ σ other parameters being fixed by experimental design, the The dependence of RSM on the widths p and 1 is subject to biphoton generation rate depends on the overlap integral these modes being both well-collimated and contained within the crystal. Φ(Δkz): To further simplify the total rate RSM, we express Δkz in 2 terms of the frequencies ω1 and ω2, and integrate over ω2 ∣(FDkz )∣ using the Dirac delta function to find: 2 3 ()2 -Dikzz º dr(()()()()c rGp* rg xyg,, xye ) . ò eff mm12 2 ∣∣()EL0 2 c()2 2 2 nn s2 ()34 Rd= w pzggpeff 12 SM ò 1 2 2 2 22 2pc nn1 2 ss1 + 2 p The simplest case to solve is that of the collimated Gaussian ⎛ ⎞ 2⎜⎟DkLzz pump beam incident on an isotropic rectangular crystal of ´-ww11()p wsinc ,38 () ⎝ 2 ⎠ dimensions Lx by Ly by Lz centered at the origin of a Cartesian coordinate system with z pointing along the optic axis. If we where make the additional assumption that we are collecting the fi down-converted light into single-mode bers, then only the D=kkzpp()wwww11 + k ( - ) - k ().39 () photons generated in the zeroth-order Hermite–Gaussian modes will contribute to the rate of detected events. In this Further simplification requires knowledge of the type of case, Gp ()r , gxy (), , and gxy ( , ) are all Gaussian functions, down-conversion being used. m1 m2 7 J. Opt. 21 (2019) 043501 Tutorial 3.1. Degenerate down-conversion order in frequency to describe the phase matching (Nasr et al 2005). In degenerate type-0/I SPDC, the first and third order Let us consider the case where the crystal is cut and tuned to terms of Δkz are zero. If we let the fourth-order dispersion optimize down-conversion such that the spectra of ω1 and ω2 dk4 constant ¡ º ∣∣4 wp 2, then the single-mode rate RSM is are both centered at half the pump frequency ωp. Then, the dw governed by: momentum mismatch Δkz can be Taylor-expanded (Fedorov et al 2009) about this central frequency so that: ⎛ ⎛ w ⎞4⎞ 2 Lz ¡ ⎜⎟p RdSMµ-ww11() w p w 1sinc ⎜ w1 - ⎟ ()43 ⎛ ⎞ 2 ò ⎝ ⎝ ⎠ ⎠ Dngp⎛ w ⎞ ⎛ w p⎞ 24 2 D»k ⎜ ⎟⎜⎟⎜⎟w -+kw -,40() z ⎝ ⎠⎝ 11⎠ ⎝ ⎠ 14 c 22 22()()()34 cosp 8G 1 4 ⎛ 24 ⎞ » w2 ⎜ ⎟ .44() p ⎝ ⎠ where κ is the group velocity dispersion constant at half the 21 Lz ¡ 2 pump frequency: ∣ dk∣ , and Δn is the group index mis- dw2 wp 2 g The resulting bandwidth and brightness of the down- match for the signal and idler photons ∣∣nngg12- at their converted light can exceed what occurs in ordinary central frequencies. degenerate SPDC by an order of magnitude, resulting in an In type-0 and type-I SPDC19, the group indices of the ultra-broadband source of entangled photon pairs spanning signal and idler light are identical because their polarizations hundreds of nanometers (Nasr et al 2005)20. are identical. Only the second-order contribution to Δkz is significant, and we find: 3.1.2. Multimode degenerate SPDC. Although many ⎛ 2⎞ experiments make use of photon pairs coupled into single- L k ⎛ wp ⎞ Rdµ-ww() w w sinc2⎜ z ⎜⎟w -⎟ mode fiber, this coupling destroys the transverse spatial SMò 11 p 1 ⎝ ⎝ 1 ⎠ ⎠ 22 correlations and the high-dimensional entanglement in that 2 2 ()Lz kwp - 62 p w p 2p degree of freedom. In experiments that involve coupling = » .41() 32 down-converted light into multi-mode fiber, or ones using a 33()LLz k z k large-area photon detector, the relevant rate of biphoton The approximation holds well for typical crystal parameters generation is the rate of generation into all transverse and crystal lengths longer than tenths of a millimeter (as is Hermite–Gaussian modes. Ordinarily, the total rate would typical). Here, we have also assumed that the portion of the be the sum of the single-mode rates over all pairs of signal generation rate formula dependent on the indices of refraction and idler modes (38). However, directly evaluating this sum is more or less constant over the bandwidth of the down- yields non-physical results, as the formula for the single-mode converted light, which is reasonable for most nonlinear rate is contingent on the paraxial approximation. For a given crystals. Making this final simplification, we arrive at the beam waist, Hermite–Gaussian beams with sufficiently large single-mode rate for degenerate type-0 and type-I SPDC: transverse momentum (or high mode index) are non-paraxial. Instead, it is much simpler to calculate the relative probability 2 2 22nngg12()deff w p R t1 = that the biphoton will be emitted into the zeroth order signal SM 3 3 2 2 p 30c nnn1 2 p k and idler Gaussian modes, and from this, determine the ratio 2 2 of the total rate to the single-mode rate. Where the idler mode s p P 32 σ fi – ´ L ,42() radius 1 de ning the Hermite Gauss basis is a free 222 z sss1 + 2 pp parameter, we set it equal to the pump radius σp to simplify calculation. For types 0 and I degenerate collinear down- ()2 where deff º ceff 2 is the more common convention for conversion, the ratio is given by: defining the effective nonlinear susceptibility, and we sub- stituted the relation for the power of the Gaussian pump 4as2 RSM 2 p beam (23). =á∣∣∣mm12 =0, = 0 Yñ = R 2 2 4 2 T ()sspp++a 3.1.1. Degenerate SPDC with negligible group velocity L l » zp,45() dispersion. Under ordinary circumstances, describing 2 4psnp p degenerate type-0/I SPDC only requires knowing the Δ L l momentum mismatch kz up to second order in the such that a = zp, and, frequency, which is governed by the group velocity 4pnp κ áñ=xyxy1,,,2 ∣()()()mm , gxygxy1 ,2 , ; 46 dispersion . However, it is possible to achieve degenerate 1 21 2 mm121 2 SPDC where the down-conversion wavelength happens to be and the approximation is valid for large pump beam widths fl κ close enough to an in ection point in the dispersion that is and thin crystals. at or near enough to zero, so that we must go out to fourth Deriving the transverse wavefunction of a biphoton generated in collinear SPDC is generally more involved than 19 Type-0 SPDC is where the pump, signal, and idler beam all have identical (typically vertical) polarization. In type-I SPDC, the signal and idler the case where we also consider only degenerate frequencies polarization are identical, but orthogonal to the pump polarization. In type-II SPDC, the pump polarization is identical to either the signal or idler 20 G(x) is the Gamma function of x, not to be confused with the attenuation polarization, but both signal and idler are mutually orthogonal. constant introduced in section 5.2. 8 J. Opt. 21 (2019) 043501 Tutorial (Schneeloch and Howell 2016). Instead, one must integrate single-mode rate for type-II degenerate SPDC: the biphoton wavefunction over the frequency spectrum of the 2 222 1 nngg12()deff wsp p P down-converted light, and renormalize accordingly, resulting R t2 = L . SM 2 2 2 222 z in a substantially broadened wavefunction. However, we may p0c nnn1 2 p Dng sss1 + 2 pp still approximate the accurate biphoton wavefunction as a ()50 scaled representation of the biphoton wavefunction in the degenerate frequency case. We scale a by a constant factor f, Interestingly, one may compare this to the corresponding and find for type-I SPDC: single-mode rate for collinear type-II SPDC derived in Ling et al (2008), and see that our formula differs by a near-unity 3 ⎛ ⎞ 2 factor of the ratio of the indices of refraction ng1ng2/n1n2, ()t1 32 2p nngg12 deff PLz R = ⎜ ⎟ .47() amounting to only a 3% difference in prediction using their T 27 c ⎝ nn2 2 ⎠ lk3 f 0 1 2 p experimental parameters. For a description of the absolute biphoton generation rate into non-collinear Gaussian modes, f≈ Here, we approximate 0.335 by matching the peaks of such as is useful when using type-II SPDC as a source of the degenerate and more accurate biphoton wavefunction in polarization-entangled photon pairs, their reference provides the same fashion as one can obtain a double-Gaussian an invaluable discussion. ( approximation to the biphoton wavefunction Schneeloch and In order to get the total rate for type-II SPDC, one can ) Howell 2016 . An interesting qualitative point here discussed use the inner product between the zeroth order Hermite– ( ) in other references Süzer and Goodson 2008 is that although Gaussian modes, and biphoton wavefunction for type-II ( the single-mode brightness increases with focusing i.e. SPDC as was done previously (47) for type-I SPDC. σ ) decreasing p , the overall brightness does not increase this However, the biphoton wavefunction for type-II SPDC is way, unless the focusing is strong enough that the curvature not as straightforward to derive or approximate, due to of the phase fronts of the pump beam affects phase matching. transverse walk-off between the signal and idler light21. For a thorough analysis of the biphoton wavefunction in type-II SPDC, see Walborn et al (2010). 3.1.3. Degenerate type-II SPDC. In type-II SPDC, the signal and idler photons are of orthogonal polarizations, and 3.1.4. Degenerate SPDC with narrow frequency filtering.In experience different indices of refraction. In this regime, the certain SPDC experiments where a pair of identical photons is preferable to a pair of highly correlated photons, one can linear contribution to Δkz about the signal and idler photons’ central frequencies (40) is nonzero, and cannot be ignored. In narrowly filter the frequency spectrum of the signal and idler this case: photons so that each is tightly clustered around half the pump frequency. Because the bandwidth of these frequency filters ⎛ Lnzg∣∣12- n g⎛ w p⎞ may be some orders of magnitude narrower than the natural 2 ⎜⎟ RdSMµ-´ww11() w p w 1 sinc ⎜ w1 - ò ⎝ 22c ⎝ ⎠ bandwidth of the down-converted light, the rate of biphotons fi 2⎞ generated passing through a narrowband frequency lter Lz k ⎛ wp ⎞ ⎜⎟⎟ +-⎝w1 ⎠ . behaves differently than the overall rate of biphoton 22⎠ generation. ()48 In particular, if we include a narrowband frequency filter, the integral over w1 for the rate (Helt et al 2012) simplifies For most nonlinear optical materials, the quadratic significantly, since the sinc function is essentially unity over contribution to the argument of the sinc function is the passband of the filter. Since the integral no longer depends negligible relative to the linear contribution because the on the width of the sinc function, the biphoton rate will not −2 group index difference Δng is large enough (of the order 10 depend on group velocity dispersion κ or group index ) or greater for most materials in comparison to the group- mismatch Δng. Moreover, the rate will scale as the square of κ velocity dispersion . This integral cannot be done the crystal length Lz, as one might expect when the analytically, but can be bounded from above. Because the probability amplitude for the SPDC event is obtained by square of the sinc function is a non-negative function, and integrating over the volume of the crystal. 2 w11()wwp - wp 4, the rate is bounded above by an integral that can be done analytically. Indeed: 3.2. Non-degenerate SPDC 2 cpw p By angle and temperature tuning the crystal, it is possible that RSM µ .49() the signal and idler frequency spectra no longer overlap, 2LnzgD The approximate proportionality is valid, when the width of 21 The ‘walk-off’ effect, where the signal and idler light have different mean ( ) the sinc function in ω1 is much less than the pump frequency momenta though still adding to the pump is due to the index of refraction in (typically, less than a quarter in most nonlinear media). birefringent crystals being dependent on direction of propagation. Because the group velocity depends on the gradient of the frequency with respect to Consequently, the approximation is an over-estimate momentum, the group velocity and mean phase velocity may point in (typically by less than 7%). From this, we can get the different directions. 9 J. Opt. 21 (2019) 043501 Tutorial grows further as though it were at a minimum. See figure 1 for comparison with and without periodic poling. By switching the poling periodically at these intervals such that the poling period Λpol=2π/Δkz, one can achieve significant brightness without perfect phase matching. This technique is known as quasi-phase matching. As shown in Boyd (2007), the poling profile c¯ (z) can be broken up into a Fourier series with fundamental momentum 2p : Lpol ik z 2p c¯ ()zX=+0 å Xen n : kn = n,53 ( ) Figure 1. Plot of the relative intensity of down-converted light with n¹0 Lpol ( ) quasi-phase matching 52 as a function of crystal length measured = in units of the poling period Λ. The sinusoidal blue curve is the where n runs from -¥ to ¥ excluding zero, X0 0, and relative brightness without periodic poling, while the oscillating ⎛ ⎞ ⎜⎟np ascending green curve gives the relative brightness when the crystal Xn = sinc⎝ ⎠.54() is periodically poled for first-order quasi-phase matching. The 2 parabolic orange curve is the approximate relative brightness with From this, we see the length dependence (52) simplifies to: first-order quasi-phase matching (56). 2 Lz 2 -D-ikkz()zn having different central frequencies that add up to the pump RdSMµ-ww11() w p w 1 L dzXeå n . òò- z 2 n frequency. The Taylor expansion for Δkz is taken with respect to the signal beam’s center frequency ω1(0). In this case: ()55 ⎛ ⎞ In performing this quasi-phase matching, typically only one Dng ()kk12+ 2 D»kz ⎜ ⎟()ww110-+()()ww110- () . Fourier component X will contribute to the brightness ⎝ c ⎠ 2 n because only one value of kn will be close enough to offset ()51 Δkz to achieve quasi-phase matching. The range of values of Δ Here, k and κ are the group velocity dispersion constants at kz over which phase-matching is favorable is approximately 1 2 π/ Δ the signal and idler central frequencies, respectively. When 4 Lz, while the shift in kz between different orders of π/Λ the central frequencies are different enough that the group quasi-phase matching is 4 pol, which is larger often by −2 X n fi index mismatch Δng is significant (e.g. greater than 10 ), the multiple orders of magnitude. Since n decreases with , rst- rate of biphoton generation is qualitatively identical for both order phase matching (i.e. n=1or−1), is most desirable for type-I and type-II SPDC. maximum brightness. The calculation for RSM follows the same steps with periodic poling as with an isotropic crystal. Δ 3.3. Periodic poling kz is still Taylor-expanded about the signal and idler central frequencies. The only difference is that the zero-order terms Thus far, we have examined the absolute brightness of SPDC for Δkz added to −km gives zero instead. As such, the single- (2) in isotropic crystals (i.e. where χ is a constant throughout mode rate of biphoton generation when periodic poling with ) fi PP() n the crystal volume . This is a ne regime when perfect phase nth order quasi-phase matching, RSM is multiplied by the matching is achievable (that is, where tuning the crystal 2 factor Xn : allows the indices of refraction to be such that Δkz=0). PP() n 4 However, this is not always possible. The general dependence R = RSM.56() SM n22p of biphoton brightness on crystal length Lz is given by: This correction holds for all types of down-conversion, and ¥ 2 -Dikzz will work for the multi-mode regime (discussed previously) RdSMµ-òòww11() w p w 1 dzze c¯ (),52 ( ) -¥ as well. It is important to note that where published values for where for an isotropic crystal c¯ (z) is unity inside the crystal, deff differ between isotropic and periodically poled nonlinear crystals of the same material, these factors of 2 are already and zero outside. When perfect phase matching is not np achievable (i.e. when the indices of refraction are not com- included. patible for SPDC at the desired pump and signal/idler fre- Although periodic poling is accomplished by switching quencies), the magnitude square of the integral over z the crystal orientation (and therefore the sign of c()2 ) peri- 2 oscillates with crystal length between zero and 4 Dkz . The odically over the length of the crystal, this is not the only value of Δkz is set by the frequencies of the pump, signal, and fashion in which quasi-phase matching can be achieved. If idler light, and the indices of refraction at their respective one instead periodically dopes the crystal, changing its frequencies. For a given set of pump, signal, and idler fre- composition periodically over its length, and therefore peri- ( ) quencies, imperfect phase matching can be ameliorated by odically changing χ 2 , quasi-phase matching may be periodically poling the nonlinear crystal. If one switches the achieved in precisely the same regimes. Alternatively, in a poling direction (changing c¯ from 1 to −1) just as the waveguide, one can produce a sinusoidal variation in the amplitude is maximum (i.e. when Lz=π/Δkz), the amplitude pump intensity by sinusoidally varying the width of the 10 J. Opt. 21 (2019) 043501 Tutorial waveguide, which can also be used to achieve quasi-phase 4. SPDC beyond the first-order approximation: the matching (Rao et al 2017). As one final note, poling periods two-mode squeezed vacuum in some materials can be made so small that the fundamental momentum completely offsets the pump momentum. In this In experimental studies of SPDC, it is only in the case of regime, it is possible to produce counter-propagating photon relatively low pump powers where SPDC is accurately pairs (Pasiskevicius et al 2008, 2012) in SPDC. described by first-order perturbation theory. In that approx- imation, the interaction of the pump beam with the quantum 3.4. SPDC with a focused pump beam vacuum either produces nothing, or yields a biphoton with low probability. However, the calculation to higher orders of In all the situations considered thus far, the pump beam was perturbation theory show the down-converted field to be in a considered collimated. However, if one wants to maximize superposition of not just the vacuum state and the single the number of biphotons generated per second that couple biphoton Fock state, but also of multi-biphoton Fock states as fi fi into a single-mode ber, a focused beam offers signi cant well. Although one could perform the perturbation theory ( ) improvement as discussed previously . In order to see how calculation to higher orders, it is actually possible in another the single-mode rate changes in the regime of tight focusing, approximation to solve the Schrödinger equation exactly for ( ) we turn to the work of Bennink Bennink 2010 , who treats SPDC (Zel’Dovich and Klyshko 1969, Mollow 1973, Lo and this situation in detail. Sollie 1993, Lvovsky 2016). The dependence of the rate of biphoton generation on the Here we consider the case of a collimated pump beam in spatial aspects of the pump beam is given by the overlap the zero-order transverse Gaussian mode, coupled to the zero- integral order signal and idler Gaussian modes. In this single-mode 2 approximation, we may solve for the time evolution of the RdrrErErErµ 3 (()()()())c()2 * ,57 () SM ò eff p 12 signal and idler creation operators using Heisenberg’s 2 equation of motion. This approximation is quite accurate for where in our approximations, D » 0 nE.Inorderto properly treat collinear SPDC into the zeroth-order signal/ experiments where the down-converted light is coupled into fi idler Gaussian modes when the pump beam is focused single-mode bers, as mentioned previously. strong enough that its width changes significantly over the Using the single-mode approximation, the nonlinear ( ) length of the crystal, Bennink uses a slightly different Hamiltonian 25 is given by: / expression for the signal idler spatial modes. Instead of HH.c.59ˆ =+ (iG aˆˆˆ a†† a )() NL å kkkp, 12 kp kk12 being Gaussian in transverse dimensions, and constant along kkkp,,12 the optic axis (i.e. collimated), Bennink considers the sig- nal/idler fields as focused Gaussian beams with their own such that beam parameters in addition to the pump beam. While a full www()()()kkk discussion of his calculations is beyond the scope of this G º- p 12eitDw kkkp,,12 3 222 tutorial (and redundant),hefinds the joint pair-collection 2Lz 0 nknknk()p ()()1 2 probability, which is proportional to the biphoton generation 3 ()2 * -Dikz ´ dr(()()()()ceff rgm xyg,,,mm xyg xye ) , rate. For type-II SPDC, and non-degenerate type-I SPDC, ò p 12 for near-perfect phase matching, and assuming identical ()60 beam focal parameters ξ for the pump, signal and idler where we let k1=k1z to simplify notation, and H.c. denotes modes, one can show: Hermitian conjugate. At this point we invoke the approx- d 23w imation that Δω≈0 over the time it takes light to propagate t2 eff p -1 RSM µ Tan()x P , ( 58 ) through the crystal. Dn g When the pump beam is narrowband enough that its ( ) ξ fi where the pump beam focal parameter is de ned as the coherence length is much longer than the crystal length Lz or ratio of the crystal length Lz divided by twice the Rayleigh alternatively that its longitudinal momentum bandwidth Δkp range, zR. Thus, a small focal parameter indicates a nearly is much smaller than 2π/Lz, we need only consider one value collimated beam. In the limit of a nearly collimated beam, of kp contributing to the general Hamiltonian because Gkkkp,,12 ’ Bennink s formula coincides with the single-mode formula is approximately constant over all values of kp. In this case, derived previously (50) up to constant factors. the pump is not truly monochromatic, but all values of pump To date, no calculations have obtained the absolute momentum can be grouped together and treated in unison as coincidence rates in the regime of focused pump beams, but the creation operator of a pump photon distributed over Bennink’s work captures the salient qualitative behavior of multiple pump modes. For typical lasers, this condition is the biphoton generation rate on changing pump focal para- easily satisfied, and makes subsequent calculations much ( ) meter. In addition, the work of Dixon et al 2014 expands on simpler. We make use of this approximation, and let Gkk, be fi 12 these results, and shows how one may sacri ce absolute substituted for Gkkkp,,12to condense notation. brightness in exchange for a greatly improved heralding Initially, the signal and idler fields are in the vacuum efficiency, as is useful in developing SPDC as a source of state. By solving Heisenberg’s equations of motion for the heralded single photons. annihilation operators of the fields, we can see what the 11 J. Opt. 21 (2019) 043501 Tutorial statistics of the signal and idler fields are as the light exits the Here, the mean pump photon number áNˆpñ will be the average nonlinear crystal. The evolution of the annihilation operator number of pump photons in the nonlinear medium at any aˆk1 is given by the equation: given time: daˆ -i P Lnzp k1 ˆ ˆ = [aHˆ kNL1,.]() 61 áñ=Np ·(),70 dt wp c Using the boson commutation relation: where P is pump power. † Considering the simple case of a collimated Gaussian [aaˆˆk ,62]()= dkk, 1 k1¢ 11¢ pump beam coupled to a pair of Gaussian signal and idler we find that: modes, and that the length of the crystal is much larger than the wavelength of the pump light, the only contributions to daˆ k1 † =-å Gaakk12ˆˆ kp k the sum over k and k are those such that Δk =0. This is dt 2 1 2 z k2 then a sum over one variable, which we may approximate as daˆ k2 † an integral, and express in terms of frequency. For a given =-å Gaakk12ˆˆ kp k ()63 dt 1 function f (k , k ): k1 1 2 ⎛ ⎞ and similarly, that ⎜2p ⎟ å fk()11, kp -» k ò dk 111 f () k, kp - k ⎝ Lz ⎠ daˆ kp k1 =-å Gaakk12ˆˆ k12 k.64() dt =+-dk dk f()( k, kd k k k ) kk12, ò 12 12 1 2 p If we take the undepleted pump approximation, then nn ⎛ nnwww+- n⎞ gg12 ⎜⎟11 2 2 pp daˆk = ddww12 f() wwd 12, p » 0, and aˆˆaN† =+»ˆˆ1 N, and we get a second- c2 ò ⎝ c ⎠ dt kp kp pp ⎛ ⎞ order differential equation for the annihilation operator aˆk1: nngg12 n2 n p =+-ddww12 f() wwdw 12,.⎜ 1 w2 wp⎟ 2 nc ò ⎝ n n ⎠ daˆ k1 † † 1 1 1 = (aGGaaˆkp() k k ˆ ) ˆ k .65() ()71 2 å 11¢ kp 1¢ dt k 1¢ For type-0 and type-I phase matching, the δ function sim- For all signal modes aˆk1, the corresponding linear system of plifies to δ(Δω), which gives us: second-order differential equations is expressible with vector 2 2 2 notation: Nt() 2 nngg12 s p Pd L SM » eff z 222 3 2 22 2 da2 ˆ tcp 0 nnp 1 n2 ss1 + 2 p sp 1 = ((NGGˆ ))·† aˆ ,66() 2 p 1 ⎛ ⎞ dt 2⎜⎟DkLzz ´-dww11() wp w 1sinc .72 () ò ⎝ 2 ⎠ where aˆk1 is a particular component of aˆ1. To solve this system † of equations, we can diagonalize GG and solve for the time These integrals over frequency can be evaluated or approxi- evolution of the eigenmodes of the Hamiltonian. This calc- mated with the same methods discussed in the previous fi ulation greatly simpli es assuming G is Hermitian, which it section. For type-I degenerate SPDC, is, under our current approximations. Using this, along with † † 22 NtSM () 22nngg12deff w p similar equations governing the evolution of aˆ1 , aˆ2, and aˆ2 , » 232 3 2 one can obtain the solution. tcp 30 nnp 1 n2 k † 2 2 ˆ =-ˆ ˆ ˆ ˆ s p P at11()cosh ( NGtapp)· i sinh( NGta)· 2 .() 67 ´ L 12.73() 222 z sss1 + 2 pp For type-II phase matching, the δ function does not simplify 4.1. The single-mode rate from the two-mode squeezed to δ(Δω), but the same upper bound approximation may be vacuum taken. The value of NSM(t) obtained will be the same as if one let the δ function be δ(Δω), but with an additional factor of Having found a formula for the time evolution of the anni- (2n – n )/n , which is of the order unity. hilation operators, the number of photon pairs can be calcu- p 1 2 Finally, to obtain the single-mode rate, we point out that lated by finding the expectation value áaaˆˆ† ñ and summing k1 k1 NSM(t) is the mean number of biphotons generated as a over all modes k1: function of time. We will use the time t=TDC as the time it † 2 NtSM ()=á aatˆˆk ñ=() ∣sinh ( NGtˆpkk)∣ , () 68 takes either the pump or down-converted light to travel the ååk1 1 12 22 k1 kk12 length of the crystal . Then, the rate RSM is the ratio of so that in the same limits where the first-order approximation is valid: 22 Phase matching occurs in degenerate type-0 and type-I SPDC when the 22ˆ NtSM ()»áñå ∣ Gkk12 ∣ Nt p .69() indices of refraction of the pump light and the down-converted light are kk12 identical. 12 J. Opt. 21 (2019) 043501 Tutorial NSM(TDC) over TDC, which is Lzn1/c, giving us: In order to serve as a viable source for heralded single 2 2 photons, the number of higher-order biphoton states gener- 22nngg12()deff w p R » ated must be small, relative to the single-biphoton state. SM 3 3 2 2 p 30c nnn1 2 p k Fortunately, the expression for the relative likelihood of 2 2 higher-order biphoton number states is quite simple: s p P 32 ´ Lz ()74 P()2ormore sss22+ 2 2 = sinh2()r . ( 78 ) 1 pp P()1 which agrees precisely with the formula for the rate of gen- When considering the SPDC state as a product of multiple erated biphotons we obtained earlier via first-order perturba- two-mode squeezed vacuum states, the ratio of events of tion theory. multi-biphoton generation to events of single biphoton gen- eration is straightforward to estimate. First, the total ratio of 4.2. The number statistics of the SPDC state multi-biphoton generation events to single biphoton genera- tion events is approximately the mean of the ratios of multi- Previously, we solved for the time evolution of the signal and biphoton to single biphoton events in each pair of modes. We idler annihilation operators. However, using that relation to can estimate this as the sum of the ratios over all modes obtain the actual quantum state of SPDC light takes one (which happens to equal N (T )) times the mean prob- additional step. SM DC ability over all mode pairs. For type-I SPDC, we find: If we define U as a unitary transformation diagonalizing the matrix G, the same transformation will define eigenmodes P()2ormore 12 c kp »-NTSM() DC * (42 ) ,79 () of the two-mode squeezing operator. P()1 35 nL1 z Let Λ be the diagonalized matrix of G: where, again, κ is the group velocity dispersion constant for L=UGU †.75() the down-converted light. In order to obtain this formula, we used the large signal-idler correlations to estimate the mar- ˆ fi Furthermore, let the annihilation operators b1 be de ned as ginal frequency probability density23, and converted it to ( U · aˆ1, i.e. the annihilation operators of the eigenmodes of the momentum to calculate the mean probability as the integral of SPDC Hamiltonian). Then, the linear system of equations for the square of the probability density times the mode spacing the annihilation operators separates into independent linear 2p Lz. For typical experimental parameters in bulk, this ratio equations for the annihilation operators of the eigenmodes: of multi-biphoton events to single biphoton events is of the − † order 10 8 per Watt of pump power. For CW beams of ˆ ˆ ˆ ˆ ˆ bt11()=L-Lcosh ( Ntbpp)· i sinh( Ntb)· 2 .() 76 typical intensities, multi-biphoton events would be exceed- The quantum state of SPDC light is obtained from these ingly rare. However, using pulsed lasers with a moderate eigenmodes (Lvovsky 2016), and is a product of multiple mean power, but small pulse length, it is possible to achieve two-mode squeezed states (one for each correlated pair of the high (peak) power levels necessary at the picosecond time ( eigenmodes) when the pump beam is in a coherent state. For scales near TDC i.e. how long light takes to travel through the reference, the two-mode squeezed state between modes 1 and crystal). Indeed, when using pulsed SPDC in improved her- 2 with squeezing amount r is given by: alded single photon sources, multi-photon events are sig- fi fi ¥ ni cant enough to limit the overall system ef ciency, so that 1 n ( ) ∣TMSVñ=å tanh()∣rn ñ12 ∣ n ñ . ( 77 ) new strategies such as in Broome et al 2011 are being cosh()r n=0 developed to reduce both the number and impact of these With the two-mode squeezed vacuum state properly events. scaled to fit experimental parameters, we can explore what we expect to measure as we increase the intensity of the pump. In a time interval equal to the length of time it takes light to pass 5. SPDC in waveguides and resonators through the nonlinear crystal, the state of the field has prob- abilities to be in a zero biphoton state, a one-biphoton state, a Although it is possible to couple entangled light into single- two-biphoton state, and so on. Light whose number statistics mode fibers, it is also possible to generate SPDC light inside a obey this exponentially decaying photon number distribution is waveguide made of the appropriate nonlinear material, so that known as thermal or super-Poissonian light because its var- the down-converted light is already propagating in spatial iance is larger than its mean. In contrast, coherent light modes easily coupled to fibers physically attached to the (as from dipole radiation or laser light) has Poissonian number nonlinear medium. With the intensity of the pump light being statistics. That said, it may seem surprising that coincidence large over the whole length of the waveguide, comparatively counting measurements show Poissonian statistics for the large pair generation rates can be achieved in a single spatial down-converted light (Avenhaus et al 2008).However,rea- mode compared to what has been done in the bulk regime. In listic experiments exhibit photodetection across multiple pairs this section, we will firstly consider the simple case of SPDC of modes; the empirical number statistics are those of a mixture of multiple exponentially distributed random variables, which 23 For a reference detailing the calculation of the joint frequency probability is better described with a Poisson distribution. distribution of biphotons in SPDC, see Mikhailova et al (2008). 13 J. Opt. 21 (2019) 043501 Tutorial in an antireflection (AR) coated nonlinear waveguide, and later calculation. While the unitary evolution of a closed follow this with the more sophisticated treatment of SPDC in quantum system does not permit any loss of energy (by say, a cavity (e.g. a waveguide without AR coatings) as in a MRR. absorption), it is straightforward to describe loss as a coupling Because the pump light intensity may be much larger inside a between modes of an extended quantum system-plus- cavity, it is possible to increase the efficiency of SPDC, environment, with a correspondingly extended unitary evol- though at the expense of increasing likelihood of multi- ution. In doing this, we remain able to treat SPDC in a lossy biphoton generation events. medium with our standard nonlinear Hamiltonian, but where the signal, idler, and pump creation and annihilation operators experience a continuous series of couplings (theoretically, 5.1. SPDC in a single-mode waveguide with generalized beamsplitters (BSs)) to scattering modes In a single-mode waveguide, the rate of biphoton generation in over the length of the medium, as discussed further in this SPDC would be straightforward to calculate if the pump were section. in the same single spatial mode as the signal and idler light. In our treatment of SPDC in cavities and resonators, we This would represent the ideal case of SPDC from a single- begin with a brief discussion for how the photon creation/ mode pump beam down-converting into a single spatial mode annihilation operators evolve when passing through a lossy calculated in section 3. Moreover, because high-intensity pump medium. Following this, we give an abbreviated introduction light is maintained over the entire length of the waveguide due describing how the modes in a single-bus MRR are coupled to to spatial mode confinement, SPDC can be made much more one another as a prototypical example of an optical cavity (see efficient than in bulk crystals, reaching record values in excess figure 4 for diagram). With this understanding, we then pro- of 109 pairs per second per mW of pump power (Jechow et al ceed to describe SPDC in a MRR, where the nonlinear 2008, Bock et al 2016). As in a single-mode fiber, one pump medium is the resonator itself. We find the Heisenberg transverse spatial mode can propagate through the waveguide, equation of motion for the photon creation/annihilation in addition to one transverse signal and idler mode. In the bulk operators in the lossy MRR, and use the relationship between crystal regime, we decomposed the down-converted light into the fields inside and outside the MRR to obtain the state of the Hermite–Gaussian spatial modes, but we could just as easily down-converted light in the output bus, where such light can decompose them into any basis of modes fitting a particular be directed and collected in a variety of experiments. With the waveguide. Indeed, we may approximate the spatial modes of state of the exiting SPDC light, we calculate the generation the waveguide with Hermite–Gaussian modes by setting the rate of exiting photon pairs, as well as isolated singles due to standard deviations σp and σ1 as equal to a fourth of the mode loss, among other factors, and compare the two to see what field diameters appropriate to those waveguides at the appro- factors impact the relative quality (i.e. heralding efficiency) of priate wavelengths. cavity-based SPDC photon sources. We conclude with a brief However, because the pump light and down-converted discussion on how the time correlations between photon pairs light are a full octave of frequency apart, a waveguide that is are affected by the MRR. For a thorough discussion of single-mode for the down-converted light will be multi-mode nonlinear optics in MRRs, we recommend the PhD theses at the much shorter pump wavelength. Ordinarily, the multi- (Vernon 2017) and (Gentry 2018). mode pump light adds a degree of complication due to modal To keep notation simple, we assume a ‘particle-in-a-box’ dispersion24, which makes phase matching more challenging mode expansion versus the more realistic Hermite–Gaussian with each spatial mode experiencing a different effective decomposition, as discussed above. For simplicity, we will index of refraction. However, with a graded index profile also assume near-perfect phase matching and negligible dis- (as is the case with waveguides produced by diffusing a persion. This is a valid approximation when the phase dopant into a nonlinear medium) this effect can be mitigated, matching bandwidth is much wider than the linewidth of the since the range of indices over the spatial modes can be made cavity, and where the optical properties of the material are small. In section 7, we test our theoretical prediction for also essentially constant over this linewidth. With this, we can type-II SPDC into a single-spatial mode using a periodically concentrate on the effects that the passive feedback of the poled potassium titanyl phosphate (PPKTP) waveguide. MRR cavity has on photon-pair generation. 5.2. SPDC in optical cavities and resonators 5.2.1. BS, propagation loss and cavities 5.2.1.1. Beam splitters. Before discussing cavities, let us first When considering SPDC in optical cavities and resonators, it discuss the simplest of all passive optical elements, the BS becomes necessary to accommodate loss (over possibly many through which fields will enter and exit a cavity. In figure 2, ) round trips to have even a qualitatively accurate description. we illustrate the standard BS with input modes aˆin, bˆ in and fi ˆ This is the case, even when the material is suf ciently lossless output modes aˆout, b out, related by the unitary matrix Ubs with to exploit the symmetries of the nonlinear susceptibility for transmission and refection coefficients τ, ρ such that ∣tr∣∣∣22+=1; 24 Modal dispersion is where the group velocity of light in higher-order spatial modes is slower than that of lower-order spatial modes. This is due to ⎛ ⎞ ⎛ ⎞ the larger transverse component of momentum taking away from the aˆout ⎛ tr⎞ aˆin ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟,.80aUaˆˆ= () longitudinal component of momentum for an otherwise monochro- ⎝ ˆ ⎠ ⎝-rt**⎠ ⎝ ˆ ⎠ out bs in matic beam. bout bin 14 J. Opt. 21 (2019) 043501 Tutorial Figure 3. Loudon’s propagation loss model based the continuum ( ) ˆ Figure 2. A beam splitter BS with input modes abˆin, in and output limit of a series of discrete beam splitters. modes abˆout, ˆ out. addition, we have defined the complex propagation constant Typically, one often encounters τ real with r =-i 1 t2 . as ξ(ω) ≡ β(ω)+iΓ(ω)/2. The significance of the unitarity of Ubs is that it preserves the To complete our treatment of loss in a continuous commutation relations between fields from input to output, so medium, we use (N−r)Δz=L−z, and convert from †† ˆ that [aaˆˆin,1in][= aˆˆout , aout] = 1, and similarly for the b discrete to continuous modes to obtain Loudon’s expression mode. This is just the statement of conservation of for an attenuated traveling beam (Loudon 2000): probability, i.e. that all signals have been accounted for, L iLxw()iLz xw ()(- ) and no parts of the signals have been lost. aeaiˆL ()www=+Gˆ0() () dzeszˆ(),. w ò0 ()87 5.2.1.2. Loss. To incorporate propagation (or scattering) loss in the system, one can use a model developed by Loudon For convenience, we have introduced the shorthand notation ( ) Loudon 2000, Alsing et al 2017 where in the frequency for the input field at z = 0 (aˆ1 in figure 3),asaˆ0 ()w = domain one has aˆ()z = 0, w and for the output field at z=L as aˆL (w).An ()in explicit computation (Alsing et al 2017) shows that aTaRsˆrr+1()wwwww=+ ()ˆ () ()ˆr (),81 ( a ) † [aaˆL ()wwdww,;ˆL ()]¢= ( -¢ )the expression for the attenu- ()out ()in fi sRaTsˆr ()wwwww=+ ()ˆr () ()ˆr (),81 ( b ) ated traveling wave aˆL (w) explicitly preserves the output eld commutation relations. as illustrated in figure 3. The attenuated signal (of interest) aˆr ( ‘ ’ ) To connect our expressions to alternative treatments of and the scattering sites unobserved, lost modes sˆr satisfy ( ) ( † lossy media, we can rewrite 87 in a Langevin form Walls the usual boson commutation relations [aaˆr ()ww, ˆr ()]¢= ) ()in,, out †()in out and Milburn 1994, Scully and Zubairy 1997, Orszag 2000 as, [ssˆr ()ww, ˆr ()¢ ] =-¢dw() w . iLxw()-G () w L Successive iteration of (81a) yields, aeaiefˆL ()ww=+-ˆ0() 1,88ˆ () w ( a ) N G()w L aTaRTsˆ ()wwww=+N ()ˆ () ()Nr- () wwˆ()in ().82 ( ) fˆ ()w º dz eiLzxw()(- ) sˆ()() z,,w 88 b N+11å r -G()w L ò r=1 1 - e 0 where the Langevin noise operators fˆ (w) satisfy the In the limit of having an infinite series of BSs with commutation relations, infinitesimal coupling, we obtain the relationship for how loss is † [ ffˆ ()wwdww,.89ˆ ()]¢= ( -¢ ) () treated in a continuous medium. We now take the continuum L limit: N ¥; Δz=L/N→0; and åN D()zdz-1 . r=1 ò0 ( ) fi fi One could deduce 88a by phenomenologically introdu- Because an individual BS in this in nite series has in nitesimal [()iLbw-G ()] w 2 ( 2 ) fi cing loss as aˆL ()ww~ eaˆ0 ( ), assuming that coupling i.e. ∣R()∣w 0 ,wede ne the independent attenua- ˆ 2 2 aˆL (w) takes the form of aˆL ()www=+afˆ0 () ( ), and tion constant G()ww=D ∣()∣Rz. Then, using ∣T ()∣w + † ∣R()∣w 2 = 1 we have, requiring by quantum mechanics that [aaˆL ()ww, ˆL ()]¢ = d (ww-¢). This deduction is the essence of the Langevin ∣()∣TRwww22NN=- (11 ∣()∣) =-G ( () LNe ) NL -G()w , approach, where the inclusion of loss requires the introduc- ()83 tion of additional noise operators fˆ (w) to ensure that the for which we define, quantum-mechanical commutation relations are preserved. This is also an embodiment of the fluctuation-dissipation 1 xwD in()(ww c )-G () w D z Te()w º=iz() e 2 ,84 ( ) theorem (Mandel and Wolf 1995b). What is not obtained ˆ xw()º+G bw ()i () w 2, ( 85 ) from this procedure is the actual form of f (w) as given by (88b). bw()º nc ()( w w ).86 ( ) Alternatively, one can treat loss in an optical medium as the Hamiltonian evolution of an extended quantum system. In (84), we have chosen the phase of T(ω) to incorporate If we consider the total Hamiltonian as the sum of the ˆˆ( ( )) the free propagation constant (i.e. wavenumber) β(ω) ≡ n(ω) system Hamiltonian (HHsys= L) see 9 , an environment ¥ † (ω/c) through a medium of index of refraction n(ω).In Hamiltonian of free photons Hˆenv = deeww ˆ () wˆ() w, ò-¥ 15 J. Opt. 21 (2019) 043501 Tutorial Fabry–Perot cavity with one input/output semitransparent mirror, and one fully reflecting mirror. In analogy with a classical field derivation (Alsing et al 2017), the output mode aˆout is a function of the sum over all possible trajectories from the input mode aˆin, as it makes an arbitrary number (including zero) of circulations around the cavity: aaˆˆout= r in ()91 a +-()tt* aaLaain00 (abˆ )() L aa L out ()91 +-()tr**aain0022 (acˆ2 Laa)() L t a L a out ()91 ( ) ( ) 2 Figure 4. A single-bus all-through micro-ring resonator MRR of +-()tr**aain0033 (()adˆ3 Laa)() L t a L a out ()91 length L=2π R with cavity field aˆ, coupled to a waveguide bus +¼,91()e with input field aˆin and output field aˆout. The constants ρ and τ are the self-coupling and cross-coupling coefficients, respectively, of the ¥ 2 * n bus to the MRR. The value z=0+ is the point P just inside the =-rtraaˆin ∣∣å ( ) ˆ()nL+1 ,91() f n=0 MRR that cross-couples to the input field aˆin, and z=L− is the point Q after one round trip in the MRR that cross-couples to the output ⎛ ¥ ⎞ iqq2 in field aˆ . =-⎜raeeag∣∣ t ( ra* )⎟ ˆin ()91 out ⎝ å ⎠ n=0 ¥ ()nL+1 2 n inLzxw()[(+1 )–] ¥ -Gidzesz∣∣trå (* ) ˆ(),, w and a coupling interaction between the two; Hˆint = i ò0 ò-¥ n=0 dwkw()(eaˆ††() wˆ() w- eaˆ() wˆ ()) w the Heisenberg equation ()91h of motion for this system in a reference frame rotating with ⎛ ra- eiq ⎞ respect to the central frequency of the light approximates to = ⎜ ⎟ aiˆin ()91 ⎝ * iq ⎠ the Heisenberg–Langevin equation (Walls and Milburn 1994, 1 - rae Orszag 2000): ¥ ()nL+1 -Gidzesz∣∣tr2 å (* )n ò inLzxw()[(+1 )–]ˆ(),. w n=0 0 i ga atˆ˙()=- [aHˆ, ˆsys] - atˆ() + ga fˆ () t,90 ( ) ()91j 2 a First, the output photon can arrive directly from the input bus by ‘ fl ’ ( ( )) ( where γ =Γ c/n , is the attenuation constant in time. The re ection offoftheMRR as described in 91a .Next, as a a ga ( )) solution to this equation also yields (88a). Here, it is also written diagrammatically in 91b , the photon can couple into understood that aˆ(t) is the time evolution of a single mode of the MRR, acquiring factor -t*, evolve through one circulation ( ) ( ) the electromagnetic field aˆ()w in a lossy medium. In the circumference L of the resonator as described in 87 ,and couple out of the resonator acquiring factor t. Successive paths lossless case (i.e. γa=0), the cavity mode evolves unitarily involve multiple circulations within the resonator, acquiring under the system Hamiltonian Hˆsys. When loss is present, the additional factors of r* from self-coupling (i.e. ‘reflection’) after mode is damped by the operator loss term -()ga 2 aˆ, but the total evolution remains unitary; it is preserved due to the each circulation. To simplify notation, we have used the fi iξL ≡ α iθ fi -G1 L additional noise term g fˆ. de nition e e de ning a = e 2 to be the internal a loss factor in one circulation of the resonator, and θ ≡ βL to be In the section where we specifically tackle the problem of the phase gained in free propagation over the same distance. SPDC in a lossy cavity, we use a Heisenberg–Langevin As derived in Alsing et al (2017), an explicit calculation equation similar to (90), but where Hˆ includes both Hˆ and sys L of the output field commutation relation yields, HˆNL. Moreover, we use a rotating frame of reference so that † the total time derivative of the propagating mode, aˆ˙, is given [aaˆout ()wwdww,.92ˆout ()]¢= ( -¢ ) () as (¶+tgz())cn ¶ aˆ. Once the equation of motion is solved to fi nd aˆ as a function of position in the MRR, this expression is This preservation of unitarity allows us to write incorporated into the interaction-picture Hamiltonian to find ˆ the state of the down-converted light. aGaHfˆout()ww=+ out,, in ()ˆ in out in () wwa (),93 ( a ) 2 ∣HGout,, in (ww )∣=-1,93 ∣ out in ( )∣ ( b ) 5.2.1.3. Cavities and MRR. We now use the above results to where Gout,in(ω) is the coefficient preceding aˆin in (91j), examine the output mode aˆout of a cavity subject to an input whose magnitude is always less than or equal to unity. This fi fi ˆ mode driving eld aˆin, with the internal cavity mode aˆ. de nes the Langevin quantum noise operator fa ()w from the Without loss of generality, we take the cavity to be a MRR unitary requirement of the preservation of the free field output as illustrated in figure 4, which also corresponds to a commutator. Interestingly, Gout,in(ω) is identical in form to the 16 J. Opt. 21 (2019) 043501 Tutorial classical transmission coefficient (Yariv 2000), as would be ‘particle-in-a-box’ modes) instead of the Hermite–Gaussian expected. It is important to note that in treating loss in a modes to describe gxym (), , and integrated over both MRR, we implicitly assumed the medium is isotropic. transverse dimensions. We let L=2πR, the circumference However, as shown in Alsing et al (2017), this assumption of the ring, essentially treating the ring as a conformal can be relaxed and the commutation relations (92) still hold mapping of a rectangular nonlinear waveguide26. With this, fi for multiple, piecewise de ned propagation wavevectors and we also let Vring ≡ LxLyL using the dimensions of the losses along the ring resonator of circumference L. deformed rectangular medium. Furthermore, where the pump is undepleted and in a coherent state, we have replaced the 5.2.2. Biphoton generation within the MRR. For biphoton pump annihilation operator aˆp with its corresponding coherent ()2 generation arising from either the c process of SPDC, or the state amplitude αp. We have taken the same steps used before χ(3) process of spontaneous four-wave mixing (SFWM), to express the Hamiltonian as an integral over frequency, and Alsing and Hach (Alsing and Hach 2017a) consider a signal we make the approximation that ww12is approximately mode aˆ, and an idler mode bˆ circulating within the MRR, and equal to the corresponding square root product of their central here, we do the same. values. In SFWM, we let α(ωp, z) represent the square of the In the non-depleted pump approximation, one can arrive pump coherent state amplitude. For the rest of this section, we at the Hamiltonian: will focus on SPDC, but it is instructive to be aware that besides issues related to different phase matching, depend- -Dikzi Dw t Hˆ NL= dzdww12 d g() www p 12 ez e ()3 ò ence on pump intensity, and the much smaller value of ceff † relative to c()2 , the physics of photon pair generation in a ´+((awzazbz, )ˆ†() , wˆ ()) , w H.c., () 94 eff p 12 cavity is very similar for both SPDC and SFWM. where for SPDC: For further simplification, and to arrive at the essential fi ⎛ ()2 ⎞ 3 aspects of SPDC in a MRR, we rst use the simplifying c nngg12 ()wp gi=-⎜ eff ⎟ FSPDC ()95 approximation of near-perfect phase matching, so that spdc 2 22 xy -Dikz ⎝ 4pcL⎠ nnn1 2 p 20 e z » 1. Next, we use the approximation of interaction 25 times long enough to enforce energy conservation so that and for SFWM : itDw eTD()(2pdwDC ). This judicious substitution ⎛ ()3 ⎞ ⎛ 2 ⎞ 3c nngg12 ()wp allows us to abbreviate the calculations done to calculate g =-⎜ eff ⎟ ⎜ ⎟ FSFWM.96() fi sfwm 2 24⎝ ⎠ xy the biphoton rate in rst-order perturbation theory as in ⎝ 4pcL ⎠ nnn1 2 p 0 previous sections. The interaction time, TDC, is the round-trip In more accurate treatments of SPDC and SFWM in a MRR, time of light at the signal/idler frequencies. With these substitutions, the Hamiltonian simplifies to: the mode functions grm ( ) would be calculated given the geometry of the material, and how the index of refraction ( ) ˆ 2p varies spatially e.g. step-index versus graded index . Here, HdzdNL =W-ò w111g()wwpp,, w fi SPDC SFWM TDC we de ne the spatial overlap integrals Fxy and Fxy as: † † ´W((awzazbz,pp )ˆ () ,11ˆ ()) , W-+ w H.c., SPDC * F=xy dxdy gm ()()() x,,, y gmm x y g x y () 97 ò p 12 ()101 SFWM F=xy dxdy g**()()()() x,,,,. y g x y gmm x y g x y ( ) ò mmpp1212 where W=ppww()alt. 2 pfor SPDC alt. SFWM such that Ω / +ν ()98 the signal frequency is at p 2 and the idler frequency is at Ωp/2−ν. Note that for later convenience, we define In the ‘particle-in-a-box’ mode basis, the coupling constants ( gTg)(º 2 p DC )spdc() sfwm . To simplify the Hamiltonian are given by: even further, we shift to a reference frame rotating at the ⎛ ()2 ⎞ 3 central frequency Ω /2. Then, in the following, the 32c nngg12 ()wp p gi=-⎜ eff ⎟ ()99 ν Ω / spdc 3 2 22 frequency represents an offset from p 2, so that ⎝ 92p c ⎠ nnn1 2 p 0Vring aˆ()W+p 2 nnaˆ( ) and bˆ()W--p 2 nnbˆ(). We will and further use the common quantum-optical shorthand notation † ⎛ ⎞ bˆ ()nnº- [bˆ()]† (Orszag 2000). Thus, in the non-depleted 27c()3 nn ⎛()w 2 ⎞ g =-⎜ eff ⎟ gg12⎜ p ⎟.() 100 pump approximation, we obtain the Hamiltonian: sfwm ⎜ ⎟ 2 24 ⎝ 16pc ⎠ nnn1 2 p ⎝ 0Vring ⎠ ˆ † ˆ† HNL=+òdzdna g( p azˆ () , n bz()) , n H.c ., () 102 In order to obtain this approximate Hamiltonian, we where we will take α ≡α (z, Ω /2) = constant throughout ( p p p have used the lowest-order plane-wave cavity modes i.e. the MRR. 25 As was discussed previously, the signal and idler modes The expression for gsfwm uses the additional (though common) assumption of a χ(3)-nonlinear medium with no χ(2) nonlinearity, such as satisfy the Heisenberg–Langevin equation of motion in the any material with a centro-symmetric structure (e.g. amorphous solids, liquids, gases, and any crystal whose unit cell is identical under reflection). 26 For a treatment of photon-pair generation in a MRR that does not rely on ()3 342 2 ()3 Under this assumption, we have the approximation: czeff »- 0 nnnp 1 2 eff . this conformal approximation, see Camacho (2012). 17 J. Opt. 21 (2019) 043501 Tutorial frequency domain (using ¶t atˆ()=- iatn ˆ())(Raymer and equal group index for signal and idler (as in type-I SPDC) to McKinstrie 2013, Alsing and Hach 2017a, 2017b), where this obtain the solution: ˆ ˆ time, HNL is included in Hsys. In the rotating reference frame, ⎛ ⎞ ⎛ ⎞ ⎛ iq ⎞ aˆ the equations for the signal and idler modes are given by: aˆ L- p 0+ ⎜ ⎟ iq cosh()rer- sinh ()⎜ ⎟ ⎜ ††⎟ » ae ⎜ ⎟⎜ ⎟ ⎛ ⎞ ⎝bˆ ⎠ ⎝-er-iqp sinh() cosh () r⎠⎝bˆ ⎠ c L- 0+ ⎜-+nnan ¶⎟ ˆ =- W ˆ† i zppaz(),,2, igL ( z ) b() z ⎛ ⎞ ⎝ nga ⎠ ⎛ ⎞ fˆ ⎜B11 0 ⎟ ⎜ a ⎟ g¢ + ⎜ †⎟,() 109 a ⎝ 0 B22⎠ ˆ - azˆ(),,,na+ polz fˆ () z n ⎝ f ⎠ 2 a b ()103a or in vector notation: ⎛ ⎞ ⎜ c ⎟ ˆ† -+i nnan ¶z bz(),,2, = igL*p ( z Wp ) azˆ() ˆ ⎝ ngb ⎠ aafˆ L-+=+RB· ˆ0 · ,() 110 g¢ † † b ˆ ˆ where R and B are defined implicitly. The coefficients B11 = - bz(),,,na+ polz fb () z n −Γ / 2 1 - a2 ,andB =-1 a2 ,whereα=e L 2.Theseco- ()103b 22 efficients are determined by requiring preservation of the co- † where g¢k is the internal propagation loss for mode k Î {}ab, , mmutation relations [aaˆˆ,,,† ][=== bbˆˆ][ aaˆˆ† ] ˆ L- L- L- L- 0+ 0+ and fk are corresponding Langevin noise operators added to ˆˆ† preserve the canonical form of the output commutators. The [bb0+, 0+]. Here, we have used the notation: r = ∣∣∣gLapDC ∣ T , iqp and aapp= ∣∣e ,andθp=(1/2)Ωp TDC,andΓ=γng/c.Itis constant αpolz is a Langevin coupling constant to the scattered modes required to preserve the unitary evolution of the fields interesting to point out that, in the limit of zero loss, the in the lossy MRR. squeezing transformation is essentially identical to that derived in By expressing the relations between the input, cavity, the previous section, though now expressed in terms of length and output fields in terms of matrices, we greatly simplify instead of time. the subsequent algebra used to find the state of the output In addition, we can consider many circulations within the fields. In particular, the input–output boundary conditions are resonator to examine the net relationship between gain and given by: loss. While G 2 represents the amplitude loss per unit length in the resonator, the quantity r/L represents the amplitude ⎛aˆ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ 0+ * aˆin r* 0 aˆ L- gain per unit length due to SPDC. Incorporating out-coupling ⎜ ⎟ ⎜-ta 0 ⎟⎜ ⎟ ⎜ a ⎟⎜ ⎟ ⎜ † ⎟ = ⎜ † ⎟ + ⎜ ⎟⎜ † ⎟,() 104a loss ∣r∣2 into the total loss per round trip, we find that in order ⎝bˆ ⎠ ⎝ 0 -tb⎠⎝bˆ ⎠ ⎝ 0 r ⎠⎝bˆ ⎠ 0+ in b L- to have a net exponential gain of SPDC light (i.e. parametric ⎛aˆ ⎞ ⎛ ⎞⎛aˆ ⎞ ⎛r 0 ⎞⎛aˆ ⎞ gain), the pump intensity must be high enough that r exceeds ⎜ out⎟ ta 0 ⎜ L-⎟ ⎜ a ⎟⎜ in⎟ ⎜ † ⎟ =+⎜ ⎟⎜ † ⎟ ⎜ ⎟⎜ † ⎟.() 104b the threshold: ⎝ ˆ ⎠ ⎝ 0 t**⎠⎝bˆ ⎠ ⎝ 0 r ⎠⎝ ˆ ⎠ bout b L- b bin GL ⎛ 1 ⎞ Defining vector and matrix notation implicitly, the boundary rthresh + ln ⎜ ⎟.() 111 2 ⎝ ∣∣r ⎠ conditions (104a) and (104b) may be written in simplified form: This is also known as the threshold for optical parametric oscillation, where over many cycles, intensities of down- aaaˆ0+-=-XT· ˆin + · ˆ L ()105 a converted light may be bright enough to be comparable to the aaaˆout=+TX**· ˆ in· ˆ L-.() 105 bpump. This is distinct from the pump power levels where multi-biphoton events become significant. When using SPDC Equations (103a) and (103b) in matrix notation are as a source of heralded single photons, we operate well below given by: this threshold, because multi-biphoton events would over- whelm the photon pair statistics at such high intensities. For nga polz ¶=zazˆ (),nnM · azˆ () , +fzˆ () ,n , () 106 typical MRR parameters, rthresh corresponds to input pump c powers of the order 1–10 milliwatts, though higher-Q where resonators will lower this threshold further. These approx- ⎛ ⎞ imations are liberal and numerous, as an accurate result ngn Ga ng iq ⎜ i - - ∣∣∣gLap ∣ e p⎟ requires knowing what the actual spatial modes of the c c M º ⎜ 2 ⎟.() 107 waveguide are, what the effective index of refraction of the ⎜ ng ngn Gb ⎟ ⎜ -iqp ⎟ propagating spatial modes are, and how much of the pump ⎝--∣∣∣gLap ∣ e i ⎠ c c 2 power in the MRR is in the lowest order spatial mode. In The solutions of (103a) and (103b) are then: particular, this is important because an MRR that is single- mode at the down-converted wavelength will be multi-mode n a L aeaˆ =+MML · ˆ g polz dz e()Lz- · fˆ () z . ( 108 ) at the pump wavelength. Due to conservation of momentum, L-+0 ò c 0 only the lowest-order pump mode in such an MRR can drive Although the solution requires taking the matrix exponential, photon pair generation if it is single-mode at the down- we use the approximation of equal loss (Γa=Γb=Γ), and converted wavelength. 18 J. Opt. 21 (2019) 043501 Tutorial 5.2.3. The output two-photon signal-idler state. To obtain the according to HˆNL, while the creation and annihilation state of the down-converted light outside the resonator, we operators evolve according to HˆL and the interaction may use the matrix expressions in (104a) and (104b),to Hamiltonian accounting for loss. In this case, we treat the express the output fields aˆ in terms of aˆ and fˆ .Todo evolution of the operators as in the Heisenberg–Langevin 0+ out ( ) ˆ this, we can express the output field operator as a sum over equation 108 , but without the contribution of HNL, the possible number of circulations in the MRR, as was done effectively setting r=0. In this picture, we can relax the (ν) previously in relating aˆ to aˆ . In this case, there are no assumption of near-perfect phase matching, so that rab out in (Δ / ) photons in the input field at the frequency of the down- acquires an additional factor of sinc kzL 2 after integrating over z. The integration over z is approximated under the converted light, as the down-converted light is being generated within the MRR. When relating aˆ to aˆ ,wefind: assumption that the damping over z is slow enough that the 0+ out fi ( exponential damping can be approximated to rst order i.e. ˆ ) = aafˆ0+()nn=+DJ ()·ˆout () nnn ()· (), ( 112 ) linearly from 0 to L−. For r 0, the output relation matrices D and J are greatly simplified to: where t 10 DRTX=-()* -1 ()113a D()n,0r == , () 119a areiq - ()01 JRTB=-() - * -1 . () 113b --1 a2 10 Assuming the parameters are the same for a and b, these J()n,0r == . () 119b areiq - ()01 matrices have relatively simple expressions: ⎛ ⎞ Although these matrices blow up in the limit of critical ariiiqqq- p a ⎜ (()())coshre e ()() e sinh r⎟ coupling (i.e. r a) where r=0, first-order perturbation D = t ⎜ ⎟ ()114 theory is no longer accurate in such regimes. When ⎜ -iiqqp i q ⎟ ⎜ ee()aarsinh ()(()() r cosh re- )⎟ calculating expectation values using the solutions to the ⎝ ⎠ Heisenberg–Langevin equation (where r>0) to get a more accurate estimate, the number of generated biphotons exiting 1 - a2 JD=- ,() 115 the resonator is maximum at critical coupling, but finite. t Now that the state of the SPDC light inside the resonator where has a straightforward form, the output state ∣Yñout is obtained º+-()arareeriiqq222 () cosh () , () 116 from the internal state ∣Yñ()Tab ab as the Heisenberg operators aˆ0+ evolve through the resonator and couple out, becoming and the dependence on ν is given by θ=νTDC. * ˆ For later convenience, we also define the notation: X R aˆ0+. The scattered light given by creation operator f has exited the system, and does not enter into the Heisenberg ⎛DD()nn ()⎞ ⎜ aa ab ⎟ propagation of the down-converted light from inside to D()n = ⎝ ⎠,() 117a DDba()nn bb () outside the resonator. Then, using our expression for aˆ0 in + ⎛ ⎞ terms of aˆ and fˆ . The output state of the fields has a JJaa()nn ab () out J()n = ⎜ ⎟.() 117b straightforward expression (with ν argument suppressed to ⎝JJba()nn bb ()⎠ save space): ¥ ⎛ ⎞ For weak but classically bright pump fields, the state of ⎜⎟DkLz ∣∣Yñout =vac ñ - drnaatt****ababab sinc the down-converted fields is well approximated to first order ò-¥ ⎝ 2 ⎠ ˆ † in HNL, and given by: iqqpp-i † ˆ ´+[(eDDaa**bb e DDab ba ) aˆ out b out ˆ NL ⎛ i NL ⎞ † -()iHT/ DC ⎜⎟ iqqpp-i † ˆ ∣(Yñ=TeDC ) ab ∣ Yñ»-in 1vacHTˆ DC ∣ ñ ++()eDJaa**bb e DJaba ab ˆout f ⎝ ⎠ b †† iqqpp**-i ˆˆ ()118a ++()eDJbb aa e DJab ba fba out ⎡ ¥ L †† † iqqpp-i ˆˆ iqnp() † ++()eJJ**bb e JJffab ]∣vac ñ , ( 120 ) =-⎢1,,dnann∣∣ gTDC dz ∣p ∣( e azˆ () bzˆ () aa ba ab ⎣ òò-¥ 0 where annihilation operators acting on the vacuum state yield +ñeazbz-iqnp()ˆ(),,vacnnˆ( ))]∣ a null result. In the previous matrix expressions, we let ()118b τa=τb=τ, and let τ be real to simplify notation. ¥ ⎡ † Interestingly, the phase of τ can be incorporated as a »-10,0,drnn()( eiqnp() aˆ †() n bˆ () n ⎣⎢ ab ++ iqp ò-¥ contribution to the phase e because although the previous expression contains terms associated to eiqp and e-iqp, closer +ñea-iqnp()ˆ()0 ,nn bˆ( 0 , ))]∣ vac , ( 118 c ) ++ examination of the coefficients associated to these terms NL iqp where Hˆ is given in (102), and for simplicity: rab ()n º reveals a global phase dependence of e . ∣gLT∣∣apDC ∣ . Note that in this estimation of the quantum With the state of the down-converted light exiting the state, we use the interaction picture, where the state evolves resonator ∣Yñout known, we see that it is readily decomposed 19 J. Opt. 21 (2019) 043501 Tutorial coincidence rate due to emission into a single pair of ()peak fi frequency peaks Rab , and nd: 22 8192 nngg12 deff w p R ()peak »-LBP()1,r4 ab 4 2 2 22 81p 0c nnn1 2 p LLxy ()123 where the approximation assumes α≈1 for the integration of ∣yn()∣2, and we are sufficiently far from critical coupling that ∣yn()∣2 is not significantly altered when assuming r≈0. In the limit of zero self-coupling (r 0) the down-converted light can only make one round trip around the MRR, and the formula becomes identical to the single-mode rate in the bulk 2 θ=ν Figure 5. Plot of ∣yq()∣ as a function of TDC, capturing the crystal (i.e. waveguide) regime. The pump buildup factor B frequency dependence of a single pair of signal/idler resonances in a approaches unity, ∣yn()∣2 grows wider than the phase- ρ= ab single-bus MRR. Here, we have assumed 0.5. The FWHM of matching bandwidth of the light so that it is near unity over the resonance is approximately ∣ar- ∣ 2 arwith a peak height of the bandwidth of the sinc function, and we must carry out the ()()11--rra24 4, so long as the coupling is non-critical (i.e. same phase-matching integrals as in previous sections. In this ()ar- r). same limit, we see that the effect of loss is that Rab scales as ∣a∣4, or by two factors of the power loss; one for the signal photon and one for the idler photon. into four elements. The amplitude for biphoton production It is interesting to point out that, here, the total rate R()peak † ˆ† ab precedes aˆoutb out, while the amplitude for a signal photon with scales linearly with L, even though the narrow frequency filtering † ˆ† of the MRR would suggest a quadratic dependence. This is due scattered idler precedes aˆout fb . The corresponding amplitudes for scattered idlers, and both scattered photons are straight- to the linewidth of the MRR itself depending on L, where longer forward as well. resonators have a corresponding narrower linewidth. As an example of the utility of this formula, consider the 5.2.4. Rate and heralding efficiency of biphotons exiting following. Let us assume type-I SPDC in a MRR of aluminum cavity. As was discussed previously for bulk crystals, the nitride with radius 30 μm, with transverse horizontal and rate of biphotons coupling out of the resonator is given by the vertical thicknesses of 1.0 μm and 0.3 μm, respectively. The ≈ −1 probability for the existence of the biphoton from ∣Yñout, effective nonlinearity deff 4.7 pm V . Let the quality factor 4 divided by the round-trip time TDC, where ∣Yñout is obtained at the pump wavelength be 10 which gives a buildup factor B λ = from ∣Yñ()TDC ab. As a function of ν, the biphoton rate per unit of about 12.3. Let the pump wavelength p 775 nm. We will = = = = = frequency ab(n) is given by: let ng1 ng2 2.19 and n1 n2 2.16 and np 2.14. With ()peak these parameters, we obtain an astonishingly high rate Rab 2pL2 ⎛DkL⎞ 7 ()n = ∣g ∣∣22ayn ∣∣ ()∣ 2sinc 2⎜⎟z , ( 121 ) of approximately 3.0×10 pairs per second per mW of pump ab 2 spdc pab ⎝ ⎠ TDC 2 power between correlated resonances in the cavity. In the limit (r ) B where of no self coupling 0 and no cavity buildup 1, ∣yn()∣2 » 1 over all frequency so that an accurate treatment ( ikD z ) yab() nº aatt****abab must explicitly consider phase-matching i.e. e »1 ,andan accurate treatment is well described in the bulk crystal regime. ´+(()()()())eDiqqpp**nn D e-i D nn D . aa bb ab ba While the ideality of our approximations (including our choice ()122 of basis modes) makes it unrealistic that this formula provides P is the input pump power (in the bus) and B is the cavity an accurate estimate of the number of exiting photon pairs per buildup factor at the pump wavelength, approximately27 equal second, it does illustrate the potential single-bus MRRs have as to the finesse divided by π/2. In figure 5, we have plotted a bright source of photon pairs via SPDC. ∣yn()∣2 to examine the shape of the spectrum of down- In more practical implementations of SPDC in MRRs, a ab fi converted light when the cavity linewidth is much narrower dual-bus con guration may be used so that one waveguide / than the phase matching bandwidth. Where we have assumed may be dedicated to coupling in out pump light, and the other strict energy conservation, this spectrum represents a subset for outcoupling SPDC photon pairs. Alternatively, in type-II of the detected biphotons, i.e. the spectrum of the signal light SPDC, the coupling between bus and MRR can be strongly over one linewidth of the cavity. When reflected about ν=0, polarization-dependent, and it may be possible to well- separate the signal and idler photons from one another instead this is the idler spectrum. With the rate ab(n) known, we integrate over the area of a single resonance to obtain the of tolerating the reduction in coincidences relative to singles that comes with separation with a non-polarizing BS. 27 Where optical cavities are also often rated by their Q ‘quality’ factor, it is In order to gauge the utility of the photon pairs exiting useful to know that in the low loss limit (and at the pump wave- 2l the resonator, it is not enough to simply know the photon pair length), BQ» p . nLp p rate. Because of loss in the resonator among other places, the 20 J. Opt. 21 (2019) 043501 Tutorial number of signal photons without matching idlers exiting the resonator is significant enough, that its dependence on experimental parameters is important to know. Although usually discussed in the context of spatial correlations, here, we shall define the resonator heralding efficiency ηR to be the ratio of the signal photon rate coming from exiting photon pairs (equal to the photon pair rate discussed previously), divided by the sum of this rate and the rate of signal photons exiting the resonator, where the idler has been lost. Where common factors in the ratio cancel out, we find: ∣()∣D n 2 hn()» bb ,() 124 R 22 ∣DJbb (nn )∣+ ∣ bb ( )∣ where we take the same approximations for calculating the individual rates as before. In this case, we find the heralding efficiency is nearly constant over the FSR of the resonator, and arrive at the approximation: 1 - r2 h » .() 125 R 2 --ra22 For the single-bus MRR studied here, we see a tradeoff between enhancing either brightness or heralding efficiency due to the ( parameters of the resonator. While lower intrinsic loss i.e. Figure 6. Plots showing reciprocal scaling of phase-matching (red) ) ( ) ( ) 2 a »1 is an absolute improvement, increasing the self- and resonance widths blue . a Plot of ∣ynab ()∣- , where the phase- coupling ρ only increases brightness at the expense of lowering matching ‘sinc-like’ function (red) is much wider than the resonance heralding efficiency. Indeed, ηR is maximized in the limit of no widths, and serves as an envelope function for the frequency ( ) ˜ 2 self coupling (i.e. where r 0), and only approaches 50% at difference spectrum. b Plot of ∣yab ()∣t- . When transforming from critical coupling. In the limit of strong self-coupling, where frequency to time, the resonance widths in time come from the α fi inverse transform of the phase-matching envelope in frequency, r 1 for constant loss , the heralding ef ciency decreases while the inverse transform of the resonance peaks in frequency towards zero, since it becomes progressively more and more becomes the exponential envelope function in time. likely that a photon in the resonator will be scattered out as loss rather than couple into the output bus. ()TDC ar , which serves as an envelope for a comb of 5.2.5. Time correlations of biphotons exiting cavity. In order inverse-transformed sinc resonances, with spacing equal to ‘ to accurately treat the time correlations between the signal TDC ()22p . The exact shape of the inverse-transformed sinc and idler photons exiting the cavity, it is necessary to resonances’ is determined by the type of phase matching, as include phase matching. Energy conservation allows us discussed in previous sections. Where TDC is on the order of a to say yab()nyn» ab (- 2 ), where n- º-()nn122 = few picoseconds, experimental measurements of the time (ww12- ) 2 . We are interested in this model only for correlations by coincidence counting are not yet capable of discussion of the behavior of the time correlations between resolving individual peaks, but may have sufficient range to biphotons exiting a MRR. For type-I SPDC, the overall phase capture the breadth of these time correlations. Indeed, the number − / matching function is given by sinc()()Lkn-- 8 yab n 2 , of tines in t until the exponential envelope decays to 1 e of its which can be broken up into three different terms The sinc peak value is directly proportional to the finesse of the resonator 28 function is a broad envelope function multiplying yab ()n- 2 , at the down-conversion frequency . As an example, when TDC is of the order of 2 picoseconds, coincidence counting setups and yab ()n- 2 is well approximated as the convolution of a with range of 20 nanoseconds will adequately capture the Dirac comb with spacing 2 2 p TDC with a Lorentzian ‘tine’ of ar- T ar( fi ( ) time correlations in single-bus micro ring resonators with a FWHM ∣ ∣(8 DC ) see gure 6 a for diagram of 3 2 finesse of the order 10 . ∣yn()∣- ). Because of the simplicity of our expression, we can readily take the inverse Fourier transform to examine the time correlations. Using the convolution theorem to our advantage, 6. SPDC with pump depletion we see that in time (as in figure 6(b)), the amplitude of t- has a similar breakdown to the corresponding function of ν−.The ‘envelope’ in time is given by the inverse transform of the ‘tine’ Throughout this paper, we have considered SPDC in the function in frequency, and the tine function in time is given by regime where the pump illumination is bright enough to be the inverse transform of the envelope function in frequency. The treated classically, but not so bright that multi-biphoton spacing of the comb in t− is given by TDC ()22p . In the time domain, the inverse transformed Lorenzian is an 28 The decay constant in number of tines is equal to 8p2, where is the exponential spike with decay constant in t− of ∣ar- ∣ 2 finesse of the resonator at the given wavelength. 21 J. Opt. 21 (2019) 043501 Tutorial fi ˆˆ()0 ˆ creation events become signi cant. We later explored a more áNN1ñ=p -á Np ñ. For simplicity, we let N11ºáNt() ñ. A fully fully quantum treatment of SPDC light (67), but only in the quantum treatment will account for the departure from a undepleted pump approximation. In this section, we will con- coherent state pump, as the down-converted photon pairs are sider SPDC in the regime of longer interaction times, where the later up-converted again in the reverse process, altering the pump light may be significantly depleted in exchange for bright pump statistics. For a fully quantum treatment, in which the intensities of the down-converted fields. We limit ourselves to complete number statistics of the pump, signal, and idler light the case of a simple waveguide, where a single pump mode is are considered, see Nation and Blencowe (2010), Alsing (2015). coupled to a single pair of signal and idler modes, and do not Although the depleted pump equation (130) is nonlinear, consider loss due either to absorption or coupling with other it is integrable using techniques similar to those used to solve modes. In the regime where the pump is undepeleted, we will the ordinary nonlinear pendulum. In doing so, we obtain an conclude by discussing how the number of generated bipho- implicit solution in the form of an integral: tons is affected when using different quantum states of pump Nt1() dx light as the source (e.g. Fock states). = 2∣∣gt . ( 131 ) ò0 3 ()0 2 ()0 When the pump light is dim enough that a fully quantum -+22xN()pp + xNx + descripton of the pump is necessary, it is also wise to consider ()0 For typical experimental parameters, Np 2; we may omit the when it is no longer possible to invoke the undepleted pump correction of 2 to the quadratic term in the integrand. Even so, approximation. In this section, we show how the mean number of this integral cannot be expressed in terms of elementary func- down-converted photon pairs changes with time when the pump tions, though certain definite integrals have straightforward can be depleted, and how in the limit of small times, we obtain expressions. In particular, the time to maximum depletion TD can the same result as in the undepeleted pump approximation. be found by taking the integral from zero to approximately The simplest Hamiltonian describing SPDC from a single ()0 Np 2, and solving for the time t. The approximation becomes pump mode to a single pair of signal and idler modes is given by: 29 ()0 exact in the limit of large Np . Values of N1 larger than the †† † HigaaagaaaNL=-( ˆˆ p 12 ˆ* ˆˆˆp 12)(), 126 critical value make the integrand imaginary, so that 50% pump depletion is the maximum amount allowed in this where g is the coupling constant between the pump mode, and coherent state model. Alternative derivations of the maximum ( ) the signal-idler mode pair as seen in 60 , albeit without incor- power conversion efficiency in SPDC, for this simple setup, porating a static pump power. Using the Heisenberg equation of also exhibit this approximate theoretical limit (Breitenbach motion, and the commutator algebra for the creation and anni- et al 1995), though more sophisticated experiments using hilation operators for each of the three modes, we can obtain a optical cavities give different values. The solution (simplified ˆ † differential equation for the photon number operator N1 º aaˆˆ1. ()0 ) 1 assuming Np > 1 can be expressed in terms of elliptic 2 ˆ integrals: dN1 2 =++-2∣∣(gNNˆp( ˆˆ12 N 1) NNˆˆ 12)() . 127 dt2 ⎛ ⎡ ⎛ ⎞ ⎤ ⎞ NN()00 ()⎡ N () 0⎤ 2csch,⎜⎢ii-1⎜ pp⎟ --⎥ ⎢ p⎥⎟ Since the Hamiltonian also guarantees that: ⎝ ⎣ ⎝ 22⎠ ⎦ ⎣ 2⎦⎠ ˆ TD »- , dNˆˆ12dN dNp ==-,() 128 ∣∣gN-+()000 NN () ( () +8 ) dt dt dt ppp the initial vacuum state of the down-converted fields also guar- ()132 antees that áNNˆˆ12ñ=á ñ,sothatNˆ1(t) isdescribedbythesimpler where ()ab, and ()a are incomplete and complete elliptic equation: integrals of the first kind, respectively. Here, ∣g∣2 takes the ( – ) 2 ˆ value using our Hermite Gauss quantization basis : dN1 2 ˆˆˆˆ2 =+-2∣∣(gNpp 2 NNN1 1 )() . 129 2 dt2 8 p23cd2 s2 ∣∣g 2 = eff p , ( 133 ) ( ) 2 22 3 2 22 From this, one can obtain a semi-classical differential equation 0nnnL1 2 p z lspp ss1 + 2 p ˆ describing the expectation value áNt1()ñ using the simplifying 6 assumptions that the signal and idler fields are in the vacuum which, for typical experimental values, is of the order 10 . state at time t=0, and that the thermal statistics of the photon Note that the absence of a sinc function in this expression is pairs described by the two-mode squeezed vacuum state for a due to our approximation of a single pump mode coupled to a coherent state pump follow the law for the geometric distribution: single pair of signal and idler modes, where the sinc function ˆˆˆ2 2 can be taken to be unity. For typical pump wavelengths and áNNN1 ñ=2 á1 ñ +á1 ñ: 0 4 crystal lengths, Np is of the order 10 , and TD is of the order 2 −5 30 dN1 2 ()0 10 s . At much larger pump powers, where multi-biphoton =+-+2∣∣(gN ( 2 N111 1 ) NN ( 6 4 )) . ( 130 ) dt2 p Here, N ()0 is the initial mean number of pump photons in the p 29 The maximum (critical) value of N expanded to first nontrivial order ()0 1 medium, which may be given by the (instantaneous) pump Np is ++1 ((N ()0 )-1 ). power, multiplied by the time it takes light to move through the 2 2 p 30 For a 1 mW pump, with 404 nm wavelength, incident on a BiBO crystal crystal, and divided by the energy of a pump photon. In addition, σ 2 × 6 ()0 3 mm long, and a pump radius p of 0.4 mm, ∣g∣ is about 8.136 10 , Np 4 −5 we used the fact from our initial conditions, that is about 3.71×10 , and TD is about 1.147×10 s. 22 J. Opt. 21 (2019) 043501 Tutorial indicating a valid approximation. Indeed, using the hyperbolic sine approximation for times less than TD/2, and the hyper- bolic secant approximation for times greater than TD/2, yields a maximum error of 0.7% for times between 0 and TD. 6.1. SPDC with different quantum pump statistics Regardless of the initial quantum state of the pump, we can use the differential equation for Nˆ1(t) (129) to find the rate of photon pair generation. For a given quantum state of the field rˆ, the mean number of photon pairs, also given by áNtˆ1()ñ is: NtSM ()= Tr [ˆr Ntˆ1()] . ( 136 ) fi Figure 7. Plots of the number of signal photons as a function of time, Because the signal and idler elds are initially in the fi for a coherent state pump, scaled with respect to TD. The blue curve vacuum state, for times before signi cant pump depletion, gives the exact solution obtained from numerically solving the this simplifies to: differential equation (130). The green curve gives the hyperbolic 2 ˆ sine approximation, which rapidly diverges for times beyond TD. NtSM()= Tr [ˆr sinh ( Ngt p )] . ( 137 ) The shallow orange curve gives the first-order approximation to fi N1(t), which agrees within 10% for times less than TD/12. The red At smaller pump powers or smaller times, this simpli es dotted curve gives the approximation as a hyperbolic secant, which further to: only differs noticeably from the exact numerical solution for times ˆ 22ˆ 22 less than TD/5. The fourth root of N1 is taken to allow better visual NtSM()»=áñTr [ˆ(r Ngt p)] Ngt p .() 138 comparison of the extreme variation in the approximations at large time values. Therefore, at small times, and pump powers, the average number of generated biphotons depends only on the mean pump power, regardless of whether it is in a coherent events become significant, near the optical damage threshold state, Fock state, or any other state. For higher pump powers, of the crystal, the depletion time can be less than a nano- where this approximation no longer applies, there is second. Instead of using unreasonably long nonlinear media, some qualitative difference between the efficiency of SPDC one could instead keep pump light in the crystal for micro- 31 with different pump photon statistics and same mean pump second-scale times with an optical cavity with a finesse in 6 power. excess of 5×10 , though an accurate description of this If we take the trace in the photon number basis, and requires us to treat SPDC in a cavity, as seen in section 5.2. let P(n ) be the probability of measuring n photons at time Of particular interest is the case of small times, where the p p t, then the number of generated biphotons N (t) is expres- cubic term can be neglected in the integrand. In this SM sible as: approximation, the integral has the form of an hyperbolic ¥ arcsine, which leads to the solution: 2 NtSM ()= å Pn (pp )sinh ( ngt ) . ( 139 ) 2 ()00 () 22 np=0 Nt1()»»sinh ( Npp ∣ gt ∣) N ∣ gt ∣ , ( 134 ) Since the function fx()= sinh2 ( x )is a convex, mono- which is in agreement with the undepleted pump approx- tonically increasing function32 of x for all positive values of x, ()0 imation where Np is the mean number of pump photons in the mean value of the function á fx()ñ is larger than the the crystal at any given time, given pump power and crystal function of the corresponding mean value of x, fx()áñ. length. See figure 7 for a side by-side comparison of the Consequently, pump beams with larger fluctuations of photon different approximations for N1(t). In the picosecond time number will have larger biphoton generation efficiency solely scales light takes to travel through nonlinear crystals, there is by the virtue of there being probable events of larger photon no meaningful distinction between these approximations, and number. Whether this is due to power instability in the pump, the simplest one will suffice. or a fundamental difference in the quantum number statistics In the limit of times on par with TD, the differential of the pump, the overall effect on biphoton generation rate equation (130) is such that the constant term contribution to will remain the same. Even so, comparing the mean number the second derivative may be neglected, and the approximate of biphotons generated for a Fock state pump, a coherent state solution has the form of the square of the hyperbolic secant: pump, and a thermal state pump with same mean photon number yields an inconsequential discrepency. Even at pump N ()0 p 2 ()0 2 intensities approaching the damage threshhold of many non- Nt1()~» TD sech (∣∣())()Ngtp - TD . 135 2 linear materials (e.g. 1 MW mm−2), the estimated difference ( ) Plotting this in figure 2 shows no significant departure from in NSM TDC between a Fock pump, a coherent pump, and a thermal pump is less than 1%. the exact numerical solution for times larger than TD 2, 31 For small round trip losses, the finesse is approximately 2π divided by the 32 A convex function is a function with non-negative second derivative (i.e. fraction of light lost in one round trip. ‘concave-up’). 23 J. Opt. 21 (2019) 043501 Tutorial When entering the regime of significant pump depletion and long interaction times, the efficiency of SPDC can vary significantly. Although we showed earlier that coherent state pumps incident on simple nonlinear media have a maximum down-conversion efficiency of approximately 50%, it has been shown (Niu et al 2017) that a 1-photon Fock state pump can have 100% down-conversion efficiency, while n-photon Fock states up to n=50 have maximum efficiencies above 77%. Efficiency aside, it is a very interesting question how the quantum state of the down-converted fields changes with the quantum state of the pump, and has been discussed since the early days of the field (Giallorenzi and Tang 1968), where in general the coherence of the pump field is mapped to the coherence of the down-converted field with the generated biphoton intensity remaining constant for constant pump power. The two-mode squeezed vacuum state for SPDC light assumes a coherent state pump, but the state of the down- converted fields for a Fock state pump, or a thermal state pump will differ greatly. The nature of the down-converted fields as a function of exotic quantum pump states remains a rich field for further development. 7. Comparisons with experiment Figure 8. Diagram of experiment used to obtain coincidence count rate for type-0 SPDC. The pump light exiting a single-mode fiber is In order to compare theoretical biphoton generation rates with focused to a given spot size at the center of the nonlinear crystal experimental data, we create a simple model accommodating (NLC), and is subsequently filtered out. The down-converted light is fi collimated, and collected into a single-mode fiber, and split by a loss and various ef ciencies throughout the experiment. Let / fi ( fi ) 50 50 ber BS, and sent to superconducting nanowire single-photon us consider the following setup see gure 8 . Here, we will detectors (SNSPDs). The coincidence counter records time intervals assume N biphotons per second are separated into the signal between detection events on each detector. The experiment allows us and idler arms, eventually arriving at the respective arm’s to directly measure the single-mode rate RSM with optics determing σ σ fi single-photon detector. In addition, we assume non-number p and 1 relative to the mode eld diameters of the input and output fi resolving detectors, so that a biphoton hitting one detector bers. registers as a single count. Here we define the coupling effi- When the photon pairs can be completely separated, such as ciencies into the collection modes as C1, C2, and C12 for signals, idlers, and coincidences, respectively. When we use a by polarization in type-II SPDC, the relative BS efficiencies non-polarizing BS, we define the BS efficiencies as β1, β2 and ()bbb1212,, can all be set equal to unity, with independent Φ ( Φ ) β12. Independent losses in the signal and idler channel due losses already being captured by E1 and E2. Here, 1 alt. 2 to, e.g. scattering, detector efficiency, and absorption, are is the count rate due to uncorrelated photons such as external given by the efficiencies E1 and E2. For a 50/50 BS, β1= noise, dark counts, and uncorrelated fluorescence stimulated β = / 2 3 4, since three out of four times, at least one photon of by the pump. Finally, A12 is the count rate of accidental the pair will exit a given output mode of the BS. Furthermore, coincidences due to a variety of sources, but nonetheless β = / 12 1 2 since half of the time, both photons exit the same detectable. Using straightforward algebra, one can show that port. When coupling down-converted light into a single-mode when the biphotons are separated by a 50/50 BS, the number fi fi ber, the coupling ef ciencies C1 and C2 are given as equal to of biphotons generated at the source N is given by: C, while the coincidence coupling efficiency C12=ηC, where η is the heralding efficiency. ⎛ ⎞ ()50 50 ()()NN1122-F -F b12 h For the experiments using type-0 and type-I SPDC, the N = ⎜ ⎟ ,() 143 ()NA12- 12 ⎝ bb12⎠ C down-converted light was separated with a 50/50 BS. In this situation N1 and N2 are related to the raw rate N and coin- where the fraction of BS efficiencies for a 50/50 BS is 8/9. cidence count rate N12 in the following way: The most challenging aspect of applying this formula in NNEC1111=+F·()b 140 general is to obtain the coupling efficiency C, and heralding efficiency η. When the BS is asymmetric, so that fraction γt of NNEC2222=+F·()b 141 the light is transmitted, and fraction γr is reflected (and nor- NNEECA12=+·() 1 2hb 12 12. 142 malized so that γt+γr=1), one finds: 24 J. Opt. 21 (2019) 043501 Tutorial Figure 9. Diagram of experiment used to obtain total coincidence count rate for type-I SPDC. The pump light is directed through a nonlinear crystal, and is subsequently filtered out. The down- converted light is split by a 50/50 BS and is focused onto large area single photon detectors. The experiment allows us to directly measure the total rate RT, though the relation between RSM and RT is determined by the overlap of the total biphoton spatial amplitude with the zero-order Gaussian modes used to compute RSM. 2 bg1 =+t 2 ggtr 2 bg2 =+r 2 ggtr bgg12 = 2tr .() 144 Figure 10. Diagram of experiment used to obtain coincidence count To test the validity of the generation rate formulas derived rate for type-II SPDC in a periodically poled, single-mode earlier in this paper, we performed three simple experiments, waveguide. The pump light is directed through an optical fiber whose parameters and results are also given in table A1 in the coupled to a nonlinear-optical waveguide, and is later filtered out. appendix. For convenience, we have included the corresp- Because this is type-II SPDC, the down-converted light is split efficiently with a polarizing BS (PBS) and is directed to a pair of onding rate formulas in table A2 in the appendix. superconducting nanowire single-photon detectors (SNSPDs), from which coincidence counts are recorded. 7.1. Type-0 SPDC in PPLN crystal coupled to single-mode fiber tip is in an image plane of the center of the crystal, the down- converted light is spatially correlated at the fiber tip, and we let The first experiment (figure 8) tests the single-mode rate for the coupling loss through the exiting fiber collimator to be the degenerate type-0 SPDC with a periodically poled nonlinear coupling efficiency C. Since the down-converted light was too crystal. We used a 40 mm periodically poled lithium niobate dim to be seen in free space with ordinary power meters, we (PPLN) crystal manufactured by Covesion with a 1 mm estimated C using laser light at 1564 nm shining through an (transverse) width, and 19.5m m poling period, temperature experiment with identical focusing optics and found C to be tuned to 107.2 °C for degenerate SPDC from 782.09 to ( ± ) 1564.18 nm. Our pump laser was an OBIS laser with mea- approximately 0.807 0.025 though the coupling to an ideal sured wavelength of 782.09 nm and bandwidth of approxi- mode-matched Gaussian beam may be higher. We estimate the fi η≈( ± ) mately 0.01 nm. The pump laser light was directed into the heralding ef ency 0.862 0.022 with our experimental fi crystal through a single-mode fiber, triplet fiber collimator, beam parameters, and the formula for the heralding ef ciency ( ) and focusing lens to obtain a well-approximated Gaussian for SPDC with focused Gaussian beams in Dixon et al 2014 . beam with spot size σ =52.6±2 μm at the center of the Per milliwatt of pump power per second, we measured singles p ± ± crystal. Using corresponding collection optics for the down- rates of 16.00 0.21 million and 17.99 0.22 million for the converted light, we obtain a mode-matched down-converted signal and idler detectors, and background noise levels Φ ≈ × 6 Φ ≈ × 6 beam radius of σ1=55.1±2 μm also at the center of the 1 0.05 10 and 2 0.06 10 . We measured a crystal. Using the Sellmeier equations for lithium niobate, and coincidence count rate of 2.93±0.05 million with accidentals ≈ × 6 published values for deff (Gayer et al 2008), we obtained the rate A12 0.02 10 , giving coincidence to singles ratios of necessary phase and group indices of refraction, as well as the 16.1% and 18.1%, respectively, which in turn gives us a raw group velocity dispersion constant κ. pair generation rate N of (95.63±2.71) million pairs per To simplify the initial alignment of our setup, we input second per mW of pump power. 1 564.18 nm light into the back end of the experiment, and With our experimental parameters, our formula (42) coupled the second harmonic generation light into the fiber that predicts a rate of (94.86±10.89)×106 coincidence counts would later be connected to the pump laser. Since the exit fiber per second per mW of pump power. The raw pair generation 25 J. Opt. 21 (2019) 043501 Tutorial rate obtained from our experiment was approximately 21.2 mm long, with values for σp and σ1 being (0.875± (95.63±2.71)×106 per second per mW of pump power, 0.125) μm, and (1.875±0.125) μm, respectively. Per mW of differing from our prediction by less than 1%, or 0.1 standard pump power, we measured singles rates of (3.71±0.05)× deviations. The relatively large uncertainty in the theoretical 106 s−1 and (4.51±0.05)×106 s−1, with a coincidence prediction is due to the propagation of uncertainties of mul- count rate of (4.71±0.07)×105 s−1. From these rates, we tiple variables. The individually large 5% uncertainty in deff is obtain a raw pair generation rate of approximately (35.5± due to imperfections between different manufacturing process 0.8)×106 s−1 per mW of pump power. of otherwise identical crystals. To have such a small dis- Using our single-mode formula for type-II SPDC in a agreement between theory and experiment is subject to mul- periodically poled medium and the given experimental para- tiple caveats, namely, that the true coupling efficiency is meters, we estimate a rate of (23.58´ 5.60) 106 per sec- unmeasured. Because we only measure the maximum cou- ond per mW of pump power. This differs from the pling in a parallel experiment, we can only assume that this experimental rate by as much as 33%, but due to asymetries in represents the coupling efficiency in the experiment if it too is the eigenmodes of the waveguide (Fiorentino et al 2007, optimally coupled. Though much effort was devoted to Shukhin et al 2015), a simple Gaussian mode of equal widths maximizing the coupling of the down-converted light into in both transverse dimensions cannot be assumed to be what the single-mode fiber, it is likely that the experimental couples into the exit fiber. Indeed, given the rubidium doping coupling efficiency is less than 0.805 by possibly as much as needed to create the waveguide, the waveguide itself has 10%–20%, which would then increase our estimate of N by different effective widths in each transverse dimension. 10%–20%, significantly exceeding the theoretical value. Assuming a 30% difference between the different transverse widths of the eigenmodes is reasonable (see diagram in Fiorentino et al (2007)), and is sufficient to produce a 7.2. Type-I SPDC in BiBO crystal incident on large area single- theoretical preciction that agrees well with experimental data. photon detectors The theoretical estimate is also subject to the relatively large In the next experiment (figure 9), we performed tests our uncertainties in the pump and signal/idler radii inside the formula for the total biphoton generation rate for collinear waveguide (of approximately 0.18 μm), whose value is gen- type-I SPDC in an isotropic crystal (47). Here, we used a erally more difficult to determine than in step-index optical 1 mm crystal of bismuth barium borate (BiBO) manufactured fibers. Moreover, the waveguide is small enough that modal by Newlight Photonics. We used a 405 nm OBIS laser to dispersion may noticeably change the effective index of produce down-converted photon pairs centered at 810 nm. refraction in comparison to bulk media. In addition, there is a We separated the photons with a 50/50 BS, and focused rather large (≈10%) uncertainty in deff, which varies sig- the light onto large-area single photon detectors. Because we nificantly between different PPKTP crystals, likely due to are sampling over all modes, extracting the raw pair genera- thermal stress patterns in the manufacturing process. For our tion rate N from the singles and coincidences is simpler; we theoretical prediction, we used the d24 coefficient responsible can set η and C equal to unity. Given our experimental for type-II SPDC given in Fiorentino et al (2007)(so that parameters, we predict a pair generation rate of (53.87± deff=d24 not counting quasi-phase matching factors), which 10.87)×106 per second per mW of pump power. The treats SPDC in a PPKTP waveguide. Where they list experiment measured singles rates per mW of pump power of d24=3.92 pm/V for SPDC for a 405 nm pump, we use 5 5 33 (6.16±0.05)×10 and (6.02±0.05)×10 per second, Miller’s rule to obtain d24≈3.18 pm/V for SPDC with a with respective background rates of (6.04±0.15)×104 and 773 nm pump. To describe our waveguide adequately, it is (6.40±0.10)×104 per second. We recorded a coincidence single-mode at the down-conversion wavelength, but it is rate of (2.71±0.06)×103 per second and an accidentals multi-mode at the pump wavelength. The mode field diameter rate of (4.39±2.79) per second. From these statistics, we at the pump wavelength is given as the diameter of the light obtain a raw pair generation rate of (64.68±1.69) million entering the crystal from a single-mode fiber fused to the pairs per second per mW of pump power, exceeding our waveguide, which is not the diameter of the TEM00 mode theoretical prediction by 20%, though this is still within the accepted by the waveguide. large range of uncertainty due to limited knowledge of the biphoton wavefunction, among other factors. 8. Conclusion 7.3. Type-II SPDC in single-mode PPKTP waveguide For our third experiment (figure 10), we used a waveguide of In this tutorial, we have shown the essential factors con- PPKTP manufactured by AdvR, for type-II SPDC from 773 tributing to the absolute photon-pair generation rate via SPDC to 1546 nm poled for first-order quasi-phase matching. This by deriving this rate from first principles. We began with experiment was pumped with a Newport NewFocus tunable deriving a general Hamiltonian for SPDC processes, and laser centered at 773 nm. Here, we separated the signal and 33 Miller’s rule is the approximation that the second order susceptibility idler photons completely with a polarizing BS. Moreover, we ()2 fi cwwweff ( p,,12) is proportional to the product of the rst-order suscept- may set C=1 since both pair-generation, and collection (1) (1) (1) ibilities χ (ωp)χ (ω1)χ (ω2). For transparent media with negligible occur in a single optical mode. The waveguide we used was absorption, χ(1)(ω)≈n(ω)2−1. 26 J. Opt. 21 (2019) 043501 Tutorial simplified it for the popular cases of bulk crystals, single-mode Acknowledgments waveguides and for generation in MRRs as a prototypical example of cavity-enhanced SPDC, and for its importance in We gratefully acknowledge support from the National integrated photonics. We examined the effect of focusing the Research Council Research Associate Programs, and funding pump beam, and of using periodically poled crystals. We from the OSD ARAP QSEP program, as well as insightful discussed how to describe the field without perturbation theory discussions with Dr A Matthew Smith. In addition, S H K via the two-mode squeezed vacuum state, and the behavior of acknowledges support from the Air Force Office of Scientific SPDC when the pump light can be depleted. We investigated Research Grant FA9550-16-1-0359 and from Northrop the number statistics of down-converted light and developed Grumman Grant 058264-002. Any opinions, findings and useful guidelines for optimizing the coincidence to accidentals conclusions or recommendations expressed in this material ratio, important in using SPDC as a heralded single photon are those of the author(s) and do not necessarily reflect the source, and in loophole-free quantum secure communication. views of AFRL. Most importantly, we compared our theoretical predictions with experimental data, and find that to the extent that the theoretical approximations resemble the reality of the experi- Appendix. Tables of experimental results and ment, the agreement improves correspondingly. parameters Table A1. Here, Rth and Rexp are the theoretically predicted and experimentally determined pair generation rates. Table of experimental parameters and results Type-0, SM in PPLN Type-I, MM in BiBO Type-II, SM in PPKTP λp 782.09±0.1 nm 405.0±1.0 nm 773.0±1.0 nm −1 −1 −1 deff 23.95±1.20 pm V 3.70±0.18 pm V 3.18±0.32 pm V Lz 40.0±0.001 mm 1.0±0.001 mm 21.2±0.01 mm σp 52.6±2.0 μmN/A 0.875±0.125 μm σ1 55.1±2.0 μmN/A 1.875±0.125 μm n1 2.155±0.001 1.822±0.001 1.736±0.002 n2 2.155±0.001 1.822±0.001 1.783±0.002 np 2.195±0.001 1.822±0.001 1.759±0.002 ng1 2.200±0.001 1.866±0.001 1.765±0.002 ng2 2.200±0.001 1.866±0.001 1.815±0.002 − − − − κ 96.75±0.2×10 27 s2 m 1 160.9±0.2×10 27 s2 m 1 N/A 6 −1 −1 6 −1 −1 6 −1 −1 Rth 94.86±10.89×10 s mW 53.87±10.87×10 s mW 23.58±5.60×10 s mW 6 −1 −1 6 −1 −1 6 −1 −1 Rexp 95.63±2.71×10 s mW 64.68±1.69×10 s mW 35.5±0.8×10 s mW Table A2. Here, f is approximately 0.335 and D=nnnggg∣∣12 - . Table of generation rate formulas for different types of SPDC Type Formula 2 222 Type-0/I, SM 22nngg12()deff w pps P 3 2 3 3 2 2 22 2Lz p 30c nnn1 2 p k ss1 + 2 pp s 3 2 Type-0/I, MM 32 2p nngg12 deff PLz 2 2 3 270c ()nn1 2 lkp f 2 222 Type-II, SM 1 nngg12()deff wsp p P 2 2 2 22 2Lz p0c nnn1 2 p Dng ss1 + 2 pp s 27 J. Opt. 21 (2019) 043501 Tutorial ORCID iDs Klyshko D 1989 Photons and Nonlinear Optics (London: Gordon and Breach) 77 James Schneeloch https://orcid.org/0000-0002-7296-9166 Ling A, Lamas-Linares A and Kurtsiefer C 2008 Phys. Rev. A 043834 Lo C and Sollie R 1993 Phys. Rev. A 47 733 References Loudon R 2000 Quantum Theory of Light 3rd edn (New York: Oxford University Press) ch 7.5 Louisell W H, Yariv A and Siegman A E 1961 Phys. Rev. 124 1646 Abu-Safe H H 2005 Appl. Opt. 44 7458 Lvovsky A I 2016 arXiv:1401.4118v2 Allen L, Beijersbergen M W, Spreeuw R J C and Woerdman J P Mandel L and Wolf E 1995a Optical Coherence and Quantum 1992 Phys. Rev. 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