FERMILAB-PUB-20-620-E-QIS-T

Ghost Imaging of Dark Particles

J. Estrada ,1 R. Harnik ,1 D. Rodrigues ,2, 1 and M. Senger 2, 3

1Fermi National Accelerator Laboratory, PO Box 500, Batavia IL, 60510, USA2Department of Physics, FCEN, University of Buenos Aires and IFIBA, CONICET, Buenos Aires, Argentina

3Physik-Institut der Universität Zürich(Dated: October 31, 2020)

We propose a new way to use optical tools from quantum imaging and quantum communicationto search for physics beyond the standard model. Spontaneous parametric down conversion (SPDC)is a commonly used source of entangled photons in which pump photons convert to a signal-idlerpair. We propose to search for “dark SPDC” (dSPDC) events in which a new dark sector particlereplaces the idler. Though it does not interact, the presence of a dark particle can be inferred bythe properties of the signal photon. Examples of dark states include axion-like-particles and darkphotons. We show that the presence of an optical medium opens the phase space of the down-conversion process, or decay, which would be forbidden in vacuum. Search schemes are proposedwhich employ optical imaging and/or spectroscopy of the signal photons. The signal rates in ourproposal scales with the second power of the feeble coupling to new physics, as opposed to light-shining-through-wall experiments whose signal scales with coupling to the fourth. We analyze thecharacteristics of optical media needed to enhance dSPDC and estimate the rate. A bench-topdemonstration of a high resolution ghost imaging measurement is performed employing a Skipper-CCD to demonstrate its utility in a dSPDC search.

I. INTRODUCTION

Nonlinear optics is a powerful new tool for quantuminformation science. Among its many uses, it plays anenabling role in the areas of quantum networks andteleportation of quantum states as well as in quantumimaging. In quantum teleportation [1–4] the state ofa distant quantum system, Alice, can be inferred with-out directly interacting with it, but rather by allowingit to interact with one of the photons in an entangledpair. The coherence of these optical systems has re-cently allowed teleportation over a large distance [5].Quantum ghost imaging, or “interaction-free” imag-ing [6], is used to discern (usually classical) informa-tion about an object without direct interaction. Thistechnique exploits the relationship among the emis-sion angles of a correlated photon pair to create animage with high angular resolution without placingthe subject Alice in front of a high resolution detec-tor or allowing it to interact with intense light. Thesemethods of teleportation and imaging rely on the pro-duction of signal photons in association with an idlerpair which is entangled (or at least correlated) in itsdirection, frequency, and sometimes polarization.

Quantum ghost imaging and teleportation both dif-fer parametrically from standard forms of informationtransfer. This is because a system is probed, not bysending information to it and receiving informationback, but rather by sending it half of an EPR pair,without need for a “response”. The difference is par-ticularly apparent if Alice is an extremely weekly cou-pled system, say she is part of a dark sector, with acoupling ε to photons. The rate of information flow

estrada@fnal.govroni@fnal.govrodriguesfm@df.uba.armatias.senger@physik.uzh.ch

Figure 1: Pictorial representation of the dSPDC process.A dark particle ϕ is emitted in association with a signalphoton. The presence of ϕ can be inferred from thedistribution of the signal photon in angle and/orfrequency. We consider both the colinear (θs = 0) andnon-colinear (θs �= 0) cases.

in capturing an image of Alice will occur at a rate∝ ε4 with standard methods, but at a rate ∝ ε2 usingquantum optical methods.

A common method for generating entangled photonpairs is the nonlinear optics process known as sponta-neous parametric down conversion (SPDC). In SPDCa pump photon decays, or down-converts, within anonlinear optical medium into two other photons, asignal and an idler. The presence of the SPDC idlercan be inferred by the detection of the signal [7].

In this work we propose to use quantum imag-ing and quantum communication tools to perform aninteraction-free search for the emission of new parti-cles beyond the standard model. The new tool wepresent is dark SPDC, or dSPDC, an example sketchof which is shown in Figure 1. A pump photon entersan optical medium and down-converts to a signal pho-ton and a dark particle, which can have a small mass,and does not interact with the optical medium. LikeSPDC, in dSPDC the presence of a dark state can

arX

iv:2

012.

0470

7v1

[hep

-ph]

8 D

ec 2

020

2

be inferred by the angle and frequency distribution ofthe signal photon that was produced in association.The dSPDC process can occur either collinearly, withθs = θi = 0, or as shown in the sketch, in a non-colinear way.

The new dark particles which we propose to searchfor, low in mass and feebly interacting, are simple ex-tensions of the standard model (SM). Among the mostwell motivated are axions, or axion-like-particles, anddark photons (see [8] for a review). Since these par-ticles interact with SM photons, optical experimentsprovide an opportunity to search for them. A canon-ical setup is light-shining-through-wall (LSW) whichset interesting limits on axions and dark photons us-ing optical cavities [9]. In optical LSW experimentsa laser photon converts to an axion or dark photon,enabling it to penetrate an opaque barrier. For detec-tion, however, the dark state must convert back to aphoton and interact, which is a rare occurrence. Asa result, the signal rate in LSW scales as ε4, whereε is a small coupling to dark states. This motivatesthe interaction-free approach of dSPDC, in which thedark state is produced but does not interact again,yielding a rate ∝ ε2.

From the perspective of particle physics, dSPDC,and also SPDC, may sound unusual1. In dSPDC,for example, a massless photon is decaying to anothermassless photon plus a massive particle. In the lan-guage of particle physics this would be a kinematicallyforbidden transition. The process, if it were to hap-pen in vacuum, violates energy and momentum con-servation and thus has no available phase space. Inthis paper we show how optics enables us to perform“engineering in phase space” and open kinematics tootherwise forbidden channels. This, in turn, allows todesign setups in which the dSPDC process is allowedand can be used to search for dark particles.

Broadly speaking, the search strategies we proposemay be classified as employing either imaging or spec-troscopic tools, though methods employing both ofthese are also possible. In the first, the angular dis-tribution of signal photons are measured, and in thesecond, the energy distribution is observed. For animaging based search, a high angular resolution is re-quired, while the spectroscopic approach requires highfrequency resolution. The later has the benefit thatdSPDC can be implemented in a waveguide which willenhance its rate for long optical elements [15, 16].

We present the basic ideas and formalism behindthis method, and focus on the phase space for dSPDC.We compare the phase space distributions of SPDCto dSPDC and discuss factors which may enhance therate of dSPDC processes. To estimate the dSPDCrates in this work we re-scale known SPDC results.In a companion paper [16] we will derive the dSPDCrate more carefully, focusing on the spectroscopic ap-proach and on colinear dSPDC in bulk crystals andwaveguides.

1 Searches for missing energy and momentum are a common-place tool in the search for new physics at colliders [10–14].However, the dSPDC kinematics are distinct as we show.

In this work we also present a proof-of-concept ex-periment using a quantum imaging setup with a Skip-per CCD [17]. The high resolution is shown to pro-duce a high resolution and low noise angular image ofan SPDC pattern. Such a detector may be employedboth in imaging and spectroscopic dSPDC setups.

This paper is structured as follows. In Section IIwe review axion-like particles and dark photons andpresent their Hamiltonians in the manner they areusually treated in nonlinear optics. In Section IIIwe discuss energy and momentum conservation, whichare called phase matching conditions in nonlinear op-tics, and the phase space for dSPDC. The usual con-servation rules will hold exactly in a thought experi-ment of an infinite optical medium. In Section III Cwe discuss how the finite size of the optical mediumbroadens the phase space distribution and plot it as afunction of the signal photon angle and energy. In Sec-tion IV we discuss the choice of optical materials thatallow for the dSPDC phase matching to enhance itsrates and to suppress or eliminate SPDC backgrounds.In Section V we discuss the rates for dSPDC and itsscaling with the geometry of the experiment, showingpotential promising results for collinear waveguide se-tups with long optical elements. In Section VI, wepresent an experimental proof-of-concept in which aSkipper CCD is used to image an SPDC pattern withhigh resolution. We discuss future directions and con-clude in Section VII.

II. THE DARK SPDC HAMILTONIAN

The SPDC process can be derived from an effectivenonlinear optics Hamiltonian of the form [18, 19]

HSPDC = χ(2)EiEsEp (1)

where Ej is the electric field for pump, signal, andidler photons and j = p, s, i respectively. We adoptthe standard notation in nonlinear optics literature inwhich the vector nature of the field is implicit. Thusthe pump, signal and idler can each be of a particularpolarization and χ(2) represents the coupling betweenthe corresponding choices of polarization.

We now consider the dark SPDC Hamiltonian

HdSPDC = gϕ ϕEsEp (2)

in which a pump and signal photon couple to a newfield ϕ which has a mass mϕ. The effective couplinggϕ, depends on the model and can depend on the fre-quencies in the process, the polarizations and the di-rections of outgoing particles. This type of term arisesin well motivated dark sector models including axion-like particles and dark photons.

A. Axion-like particles

Axion-like particles, also known as ALPs, are lightpesudoscalar particles that couple to photons via aterm

Haxion = gϕ ϕ �E · �B , (3)

3

which can be written in the scalar form of Equation (2)with the pump and signal photons chosen to have or-thogonal polarization. The coupling gϕ in this caseis the usual axion-photon coupling ∼ 1/fa. ALPsare naturally light thanks to a spontaneously brokenglobal symmetry at a scale fa.

Since we will not perform a detailed analysis ofbackgrounds and feasibility here, we will describe ex-isting limits on ALPs qualitatively and refer readersto limit plots in [8, 20]. In this work we will con-sider searches that can probe ALPs with a mass oforder 0.1 eV or lower. Existing limits in this massrange come from stellar and supernova dynamics[21],as well as from the CAST Heiloscope [22], which re-quire gϕ � 10−10 GeV−1. For comparison with our se-tups, we will also consider limits that are entirely lab-based, which are much weaker. At ∼ 0.1 eV ALP cou-pling of order 10−6 GeV−1 are allowed by all lab-basedsearches. For axion masses around 10−3 −10−2 eV thePVLAS search[23] for magnetic birefringence places alimit of gϕ � 10−7 − 10−6 GeV−1. Bellow masses ofan meV, LSW experiments such as OSQAR [24] andALPS [25] set gϕ � 6 × 10−8. A large scale LSW fa-cility, ALPS II [26], is proposed in order to improveupon astrophysical limits and CAST.

B. Dark photons

A new U(1) gauge field, A′, with mass mA′ thatmixes kinetically with the photon via the Hamiltonian

Hmix = εF μνF ′μν = ε

(�E · �E′ + �B · �B′

). (4)

It is possible to write the dark photon interaction ina basis in which any electromagnetic current couplesto longitudinally polarized dark photons with an ef-fective coupling of ε(mA′/ωA′) and to transversely po-larized dark photons with ε(mA′/ωA′)2 [27, 28]. Thesame dynamics that leads to the nonlinear Hamilto-nian HSPDC, will yield an effective coupling of twophotons to a dark photon as in Equation (2) with ϕrepresenting a dark photon polarization state. Forlongitudinal polarization of the dark photon this canbe brought to the form of equation (2) with gϕ =εχ(2)mϕ. The coupling to transverse dark photonswill be suppressed by an additional factor of mϕ/ωϕ.

Like axions, limits on dark photons come fromastrophysical systems, such as energy loss in theSun [27], as well as the lack of a detection of darkphotons emitted by the Sun in dark matter direct de-tection experiments [28]. At a dark photon mass of or-der 0.1 eV, the limit on the kinetic mixing is ε � 10−10

with the limit becoming weaker linearly as the darkphoton mass decreases. Among purely lab-based ex-periments, ALPS sets a limit of ε � 3 × 10−7. Inforthcoming sections of this work we will use theselimits as benchmarks.

In addition to ALPs and dark photons, one can con-sider other new dark particles that are sufficientlylight and couples to photons. Examples include mil-licharged particle, which can be produced in pairs in

association with a signal photon. We leave these gen-eralizations for future work and proceed to discuss thephase space for dSPDC in a model-independent way.

III. PHASE SPACE OF DARK SPDC

As stated above, the effective energy and momen-tum conservation rules in nonlinear optics are knownas “phase matching conditions”. We begin by review-ing their origin to make the connection with energyand momentum conservation in particle physics andthen proceed to solve them for SPDC and dSPDC, tounderstand the phase space for these processes.

A. Phase matching and particle decay

We begin with a discussion of the phase space ina two-body decay, or spontaneous down-conversion.We use particle physics language, but will label theparticles as is usual in the nonlinear optics systemswhich we will be discussing from the onset. A pumpparticle p decays into a signal particle s and anotherparticle, either an idler photon i or a dark particle ϕ:

SPDC: γp → γs + γi

dSPDC: γp → γs + ϕ . (5)We would like to discuss the kinematics of the stan-dard model SPDC process, and the BSM dSPDC pro-cess together, to compare and contrast. For this, inthe discussion below we will use the idler label i torepresent the idler photon in the SPDC case, and ϕ inthe dSPDC case. Hence ki and θi are the momentumand emission angle of either the idler photon or of ϕ,depending on the process.

The differential decay rate in the laboratory frameis given by [20]

dΓd3ksd3ki

=δ4 (ppμ − psμ − piμ)(2π)2 2ωp 2ωs 2ωi

|M|2 (6)

where M is the amplitude for the process (whichcarries a mass dimension of +1), and the energy-momentum four-vectors

pjμ =(

ωj

kj

)(7)

with j = p, s, i. Traditionally, one proceeds by inte-grating over four degrees of freedom within the six di-mensional phase space, leaving a differential rate withrespect to the two dimensional phase space. This isoften chosen a solid angle for the decay dΩ, but in ourcase there will be non-trivial correlations of frequencyand angle as we will see in the next subsection.

It is instructive, however, for our purpose to take astep back and recall how the energy-momentum con-serving δ-functions come about. In calculating thequantum amplitude for the transition, the fields inthe initial and final states are expanded in modes.The plane-wave phases ei(ωt−k·x) are collected fromeach and a spacetime integral is performedˆ

d4x ei(Δωt−Δk·x) = (2π)4 δ(3)(Δk) δ(Δω) , (8)

4

where frequency and momentum mismatch are definedas {

Δω = ωp − ωs − ωi

Δk = kp − ks − ki, (9)

and again, the label i can describe either an idler pho-ton or a dark particle ϕ. In Equation (9), we sepa-rated momentum and energy conserving delta func-tions to pay homage to the optical systems which willbe the subject of upcoming discussion. The squaredamplitude |M|2, and hence the rate, is proportionalto a single power of the energy-momentum conserv-ing delta function times a space-time volume factor,which is absorbed for canonically normalized states(see e.g. [29]), giving Equation (6).

The message from this is that energy and mo-mentum conservation, which are a consequence ofNeother’s theorem and space-time translation sym-metry, is enforced in quantum field theory by perfectdestructive interference whenever there is a non-zeromismatch in the momentum or energy of initial andfinal states. In this sense, the term “phase matching”captures the particle physicist’s notion of energy andmomentum conservation well.

The dSPDC process shown in Figure 1 is a masslesspump photon decaying to a signal photon plus a mas-sive particle ϕ. This process clearly cannot occur invacuum. For example, if we go to the rest frame of ϕ,and define all quantities in this frame with a tilde , theconservation of momentum implies kp = ks, while theconservation of energy implies ωp = ωs + m. Combin-ing these with the dispersion relation for a photon invacuum ωp,s = |kp,s|, implies that energy and momen-tum conservation cannot be satisfied and the processis kinematically forbidden. Another way to view thisis to recall that the energy-momentum four-vector forthe initial state (a photon) lies on a light cone i.e. itis a null four-vector, pμpμ = 0, while the final statecannot accomplish this since there is a nonzero mass.

In this work we show that optical systems will allowus to open phase space for dSPDC. When a photon isinside an optical medium a different dispersion rela-tion holds,

npωp = |kp| and nsωs = |ks| (10)

where np and ns are indices of refraction of photonsin the medium in the ϕ rest frame, which can be dif-ferent for pump and signal. As we will see later on,under these conditions the conclusion that p → s + ϕis kinematically forbidden can be evaded.

Another effect that occurs in optical systems, butnot in decays in particle physics, is the breaking ofspatial translation invariance by the finite extent ofthe optical medium. This allows for violation of mo-mentum conservation along the directions in which themedium is finite. As a result, the sharp momentumconserving delta function will become a peaked distri-bution of width L−1, where L is the crystal size. Wewill begin by solving the exact phase matching condi-tions in SPDC and dSPDC in the following subsection,which correspond to the phase space distributions foran infinitely large crystal. Next, we will move on tothe case in which the optical medium is finite.

B. Phase Matching in dSPDC

We now study the phase space for dSPDC to iden-tify the correlations between the emission angle andthe frequency of the signal photon. We will considerin parallel the SPDC process as well, so in the endwe arrive to both results. We assume in this subsec-tion an optical medium with an infinite extent, suchthat the delta-functions enforce energy and momen-tum conservation.

The phase matching conditions Δω = 0 and Δk = 0—that must be strictly satisfied in the infinite extentoptical medium scenario— imply{

ωp = ωs + ωi

kp = ks + ki. (11)

Using the k’s decomposition shown in Figure 1 the sec-ond equation can be expanded in coordinates paralleland perpendicular to the pump propagation direction{

kp = ks cos θs + ki cos θi

0 = ks sin θs − ki sin θi(12)

where the angles θs and θi are those indicated in thecited figure. Since this process is happening in a mate-rial medium, the photon dispersion relation is k = nωwhere n is the refractive index. The refractive indexis in general a function of frequency, polarization, andthe direction of propagation, and can thus be differentfor pump, signal, and idler photons. For dSPDC, inwhich the ϕ particle is massive and weekly interactingwith matter the dispersion relation is the same as invacuum,

kϕ =√

ω2ϕ − m2

ϕ. (13)

Thus we can explicitly write the dispersion relationfor each particle in this process:

kp = npωp, (14)ks = nsωs (15)

and

ki =

⎧⎨⎩

niωi for SPDC√ω2

i − m2ϕ for dSPDC

. (16)

where, again, the label “i” refers to an idler photonfor SPDC and to the dark field ϕ for dSPDC. Notethat ki for SPDC not only has m = 0 but also takesinto account the “strong (electromagnetic) coupling”with matter summarized by the refractive index ni.As a result, for SPDC we always have ki > ωi whilefor dSPDC ki < ωi. Replacing these dispersion rela-tions into (12) and using the first equation in (11) weobtain the phase matching relation of signal angle andfrequency

cos θs =n2

p + α2ωn2

s − Ξ2

2αωnpns(17)

where we define

αωdef=

ωs

ωp(18)

5

and

Ξ def=ki

ωp=

⎧⎪⎨⎪⎩

ni (1 − αω) for SPDC√(1 − αω)2 − m2

ϕ

ω2p

for dSPDC. (19)

Equation (17) defines the phase space for (d)SPDC,along with the azimuthal angle φs. The idler angle, orthat of the dark particle in the dSPDC case, is fixed interms of the signal angle and frequency by requiringconservation of transverse momentum

sin θi =ks

kisin θs

φi = φs + π (20)

where ki is evaluated at a frequency of ωi = ωp − ωs,and ks at the frequency ωs according to the dispersionrelation in Equations (15) and (16).

In the left panel of Figure 2 we show the allowedphase space in the αω-θs plane for SPDC and fordSPDC with both a massless and massive ϕ. Herewe have chosen np = 1.486, ns = 1.658 with ni = ns

in the SPDC case. These values were chosen to beconstant with frequency and propagation direction,for simplicity and are inspired by the ordinary andextraordinary refractive indices in calcite and will beused as a benchmark in some of the examples below.

We see that phase matching is achieved in differentregions of phase space for SPDC and its dark counter-part. In dSPDC, signal emission angles are restrictedto near the forward region and in a limited range ofsignal frequencies. The need for more forward emis-sion in dSPDC can be understood because ϕ effec-tively sees an index of refraction of 1. This impliesit can carry less momentum for a given frequency, ascompared to a photon which obeys k = nω. As aresult, the signal photon must point nearly parallelto the pump, in order to conserve momentum (seeFigure 2, right). An additional difference to noticeis that, in contrast with the SPDC example shown,for a fixed signal emission angle there are two differ-ent signal frequencies that satisfy the phase matchingin dSPDC. This will always be the case for dSPDC.Although SPDC can be of this form as well, typical re-fractive indices usually favor single solutions as shownfor calcite.

1. Phase matching for colinear dSPDC

One case of particular interest is that of colineardSPDC, in which the emission angle is zero, as wouldoccur in a single-mode fiber or a waveguide. For thiscase, so long as ns > np, an axion mass below somethreshold can be probed. Setting the emission anglesto zero there are two solutions to the phase matchingequations which give signal photon energies of

αω =(nsnp − 1) ±

√(ns − np)2 − (n2

s − 1) m2ϕ

ω2p

n2s − 1

.

(21)

The + solution above corresponds to the case wherethe ϕ particle is emitted in the forward directionwhereas the − represents the case where it is emit-ted backwards. The signal frequency as a functionof ϕ mass is shown in Figure 3 for the calcite bench-mark. For the near-massless case, mϕ � ωp, a phasematching solution exists for ns ≥ np, giving

ωs

ωp=

np ∓ 1ns ∓ 1

. (22)

For calcite, we get ωs = 0.739ωp and ωs = 0.935ωp forforward and backward emitted axions respectively.

Of course, not any combination of np and ns willallow to achieve phase matching in dSPDC. We willstate here that ns > np is a requirement for phasematching to be possible for a massless ϕ and that ns −np must grow as mϕ grows. We will discuss these andother requirements for dSPDC searches in Section IV.

C. Thin planar layer of optical medium

We now consider the effects of a finite crystal.Specifically, we will assume that the optical mediumis a planar thin layer2 of optical material of length Lalong the pump propagation direction z, and is infinitein transverse directions. In this case, the integral inEquation (8) is performed only over a finite range andthe delta function is replaced by a sinc x ≡ sin x/xfunction

ˆ L/2

−L/2dz eiΔkzz = L sinc

(ΔkzL

2

)(23)

at the level of the amplitude. As advertised, this al-lows for momentum non-conservation with a charac-teristic width of order L−1.

The fully differential rate will be proportional to thesquared sinc function

dΓd3kpd3ki

∝ L2sinc2(

ΔkzL

2

)δ2(ΔkT )δ(Δω)(24)

≡ dΓd3kpd3ki

(25)

where ΔkT is the momentum mismatch vector in thetransverse directions. The constant of proportional-ity in Equation (24) will have frequency normalizationfactors of the form (2ω)−1 and the matrix element M.The matrix element can also have non trivial angulardependence in θ as well as in φ, depending on themodel and the optical medium properties. Note, how-ever, that the current analysis for the phase space ismodel independent. We thus postpone the discussionof overall rates to Section V and to [16] and here welimit the study to the phase space distribution only.We will proceed with the defined phase space distri-bution dΓ in Equation (25), which we will now study.

2 The meaning of what constitutes the thin crystal layer limitin our context will be discussed in Section V.

6

[deg

]

Figure 2: Left: The allowed phase space for SPDC (black), dSPDC with m = 0 (red), and dSPDC with m = 0.1ωp

(blue) shown as in the plane of signal emission angle as a function of frequency ratio αω. The indices of refraction hereare np = 1.658 and ns = 1.486 as an example, inspired by calcite. The inset shows a zoom-in of the dSPDC phase space.Right: Sketches depicting the momentum phase matching condition Δk = 0 for SPDC and dSPDC, with massless ϕ. Inboth cases we take the same ωp and ωs. Due to the index of refraction for ϕ is essentially one, and that for the idlerphoton is larger (say ∼ 1.5), phase matching in dSPDC has a smaller signal emission angle than that of SPDC.

Waveguide phase matching:

Figure 3: Solutions to the phase matching conditions forthe colinear dark SPDC process for the signal photonenergy as a function of the dark particle ϕ mass. Thisphase space is relevant for waveguide-based experiments.Both axes are normalized to the pump frequency. Thetwo branches correspond to configurations in which ϕ isemitted forward (bottom) and backward (top).

One can trivially perform the integral over the d3ki

which will effectively enforce Equation (20) for trans-verse momentum conservation, and set |ki| by conser-vation of energy. The argument of the sinc functionis

Δkz = kp − ks cos θs − ki cos θi (26)

= ωp

(np − nsαω cos θs ±

√Ξ2 − n2

sα2ω sin2 θs

)

where in the second step we have used Equation (20)and the definitions of αω and Ξ in Equations (18)and (19). The ± accounts for the idler or dark particlebeing emitted in the forward and backward directionrespectively.

It is convenient to re-express the remaining three-dimensional phase space for (d)SPDC in terms of thesignal emission angles θs and φs, and the signal frequency (or equivalently αω). Within our assumptions, thedistribution in φs is flat. With respect to the polar angle and frequency we find

d2Γd(cos θs)dαω

=2πω3

pα2ω (1 − αω) n3

sn2i√

Ξ2 − α2ωn2

s sin2 θs

∑±

L2sinc2[(

np − nsαω cos θs ±√

Ξ2 − n2sα2

ω sin2 θs

)ωpL

2

](27)

where

nidef=

{ni for SPDC1 for dSPDC

. (28)

In Figure 4 we show the double differential distri-

7

Figure 4: The phase space distribution d2Γ/d(cos θs)dαω

for SPDC in arbitrary units. We use L = 1, λp = L10 ,

np = 2 and ns = ni = 4. These parameters areunrealistic in practice, but allow for clear visualization ofthe distribution. Dashed lines show slices of fixed signalangle/energy.

bution Γ for SPDC, and in Figure 5 it is shown fordSPDC both for a massless and a massive ϕ. The dis-tribution clearly peaks in regions of phase space thatsatisfy the phase matching conditions, those shown inFigure 2. To have a clearer view of the qualitativefeatures in the distribution we picked somewhat exag-gerated values for indices of refraction in these figures.

1. dSPDC Searches with Imaging or Spectroscopy

Though one can, in principle, measure both the an-gle and the energy of the signal photon, it is usu-ally easier to measure either with a fixed angle or afixed energy. For example, a CCD detector with amonochromatic filter can easily measure a single en-ergy slice of this distribution, as shown in the verticaldashed line in Figure 4. Likewise, a spectrometer withno spatial resolution can measure a fixed angle sliceof this distribution, as shown in the horizontal dashedline of Figure 4. An attractive choice for this is to lookin the forward region, with an emission angle θs = 0,which is the case for waveguides. For this colinearcase we get a signal spectrum

dΓdαω

=2πω3

pα2ω (1 − αω) n3

sn2i

Ξ∑

±L2sinc2

(Δk

(0)z±L

2

)

(29)where

k(0)z± = kzp − kzs − k±

zi = ωp (np − nsαω ± Ξ) (30)

is the momentum mismatch for colinear (d)SPDC. Insome cases the colinear spectrum is dominated by justthe forward emission of the idler/ϕ, while in othersthere are two phase matching solutions which con-tribute similarly to the rate.

In Figure 6 we show two distributions, one for fixedsignal frequency, and the other for a fixed signal an-gle, the latter in the forward direction. We see thatboth of these measurement schemes are able to distin-guish the signal produced by different values of massmϕ. Furthermore, in cases where the standard modelSPDC process is a source of background, it can be sep-arated from the dSPDC signal. As expected, in theforward measurement the dSPDC spectrum exhibitstwo peaks for the emission of ϕ in the forward or back-ward directions. It should be noted that the width ofthe highest peaks in these distributions decreases withcrystal length L.

IV. OPTICAL MATERIALS FOR DARKSPDC

Having discussed the phase space distribution fordSPDC, we will now discuss which optical media areneeded to open this channel, to enhance its rates, and,if possible, to suppress the backgrounds. A more com-plete estimate of the dSPDC rate will be presentedin [16]. Here we will discuss the various features qual-itatively to enable optimization of dSPDC searches.

A. Refractive Indices

As we discussed above, the refractive index of theoptical medium plays a crucial role. It opens the phasespace for the dSPDC decay and determines the kine-matics of the process.

In order to enhance the dSPDC rate, it is alwaysdesirable to have a setup in which the dSPDC phasematching conditions which we discussed in Section IIIare accomplished. Since the left hand side of Equa-tion (17) is the cosine of an angle, its right hand isrestricted between ±1 so

− 1 <n2

p + α2ωn2

s − Ξ2

2αωnpns< 1. (31)

From this one finds that so long as ns > np, there isa range of ϕ mass which can take part in dSPDC. Asthe difference ns − np is taken to be larger, a greaterϕ mass can be produced and larger opening angles θs

can be achieved.Using the inequalities in Equation (31) we can ex-

plore the range of desirable indices of refraction fora particular setup choice. For example, suppose weuse monochromatic filter for the signal at half the en-ergy of the pump photons, αω = 1

2 . Figure 7 shows inblue the region where the phase matching conditionis satisfied for the SPDC process and in orange theregion where the phase matching condition is satisfiedfor dSPDC. As can be seen, dSPDC is more restrictivein the refractive indices. Furthermore, typical SPDCexperiments employ materials such as beta barium bo-rate (BBO), potassium dideuterium phosphate (KDP)and lithium triborate (LBO) [18, 30–32] that do not al-low the phase matching for dSPDC for this example.

8

0.0 0.2 0.4 0.6 0.8 1.0

αω = ωs/ωp

0

20

40

60

80

100

120

140

160

180

θ s(deg)

dSPDC

10−10

10−8

10−6

10−4

10−2

100

0.0 0.2 0.4 0.6 0.8 1.0

αω = ωs/ωp

0

20

40

60

80

100

120

140

160

180

θ s(deg)

dSPDC

10−10

10−8

10−6

10−4

10−2

100

Figure 5: Phase space distribution d2Γ/d(cos θs)dαω in arbitrary units for the dSPDC process for m = 0 (left) andm = 0.5ωp. The other parameters are as in Figure 4, with somewhat exaggerated indices of refraction to allow seeingthe features in the distribution.

Figure 6: Slices of the phase space distributions for SPDC and for dSPDC with different choices of ϕ mass. On the leftwe show angle distributions with fixed signal frequency, and on the right we show frequency distributions for fixedsignal angle. These distributions allow to separate SPDC backgrounds from the dSPDC signal, as well as the dSPDCsignal for different values of mϕ. The distributions become narrower for thicker crystals, enhancing the signal tobackground separation power. For these distributions we used L = 1 cm, λp = 400 nm and for the refractive indicesnp = 1.49 and ns = ni = 1.66 inspired by calcite.

A second example which is well motivated is theforward region, namely θs = 0. In this case phasematching can be achieved so long as

(ns − np)2

n2s − 1

≥ m2ϕ

ω2p

, (32)

and ns > np. We will investigate colinear dSPDC ingreater detail in [16].

The dSPDC phase matching requirement ns > np

can be achieved in practice by several effects. Themost common one, employed in the majority of SPDCexperiments, is birefringence [30, 32–34]. In this casethe polarization of each photon is used to obtain adifferent refractive index. For instance, in calcite, the

ordinary and extraordinary polarizations have indicesof refraction of no = 1.658 and ne = 1.486. Takingthe former to be the signal and the later for the pump,phase matching can be met for mϕ < 0.16ωp. Bire-fringence may also be achieved in single-mode fibersand waveguides with non-circular cross section, e.g.polarization-maintaining optical fibers [35]. Of course,the dependence of the refractive index on frequencymay also be used to generate a signal-pump differencein n.

9

Figure 7: Regions in which the refractive indices np andns allow the phase matching with αω ≡ ωs/ωp = 1

2 forthe SPDC process (in this case assuming ni = ns) anddSPDC (in this case assuming m

ωp� 1).

B. Linear and Birefringent Media for Axions

Axion electrodynamics is itself a nonlinear theory.Therefore dSPDC with emission of an axion can occurin a perfectly linear medium. The optical medium isneeded in order to satisfy the phase matching condi-tion that cannot be satisfied in vacuum. Because thepump and signal photons in axion-dSPDC have dif-ferent polarizations, a birefringent material can sat-isfy the requirement of ns > np and Equation (32).Because SPDC can be an important background to adSPDC axion search, using a linear or nearly linearmedium, where SPDC is absent is desirable. Inter-estingly, materials with a crystalline structure thatis invariant under a mirror transformation r → −rwill have a vanishing χ(2) from symmetry considera-tions [36].

C. Longitudinal Susceptibility for Dark PhotonSearches

As opposed to the axion case, a dSPDC process witha dark photon requires a nonlinear optical medium.As a result, SPDC can be a background to dSPDCdark photon searches. In SPDC, the pump, signal,and idler are standard model photons and their polar-ization is restricted to be orthogonal to their propa-gation. As a consequence, when one wants to enhancean SPDC process, the nonlinear medium is oriented insuch a way that the second order susceptibility tensorχ(2) can appropriately couple to the transverse polar-ization vectors of the pump, signal, and idler photons.The effective coupling between the modes in questionis given by

χ(2)SPDC ≡ χ

(2)jklε

pj εs

kεil (33)

with the ε being the (transverse) polarization vectorsfor the pump, signal and idler photons. These arespanned by (1, 0, 0) and (0, 1, 0) in a frame in whichthe respective photon is propagating in the z direction.

In dSPDC with a dark photon there is an importantdifference. Because the dark photon is massive, it can

crystal

pump

signal

dark particle(or idler)L

{

Figure 8: A sketch showing the overlap of the pump,signal, and dark particle (or idler) beams with thecrystal. In general it is desirable to maximize theintegration length which is defined by the overlap of thethree beams and the crystal.

have a longitudinal polarization, εA′

L

l = (0, 0, 1). Thecrystal may be oriented to couple to this longitudinalmode, giving an effective coupling of

χ(2)A′

L≡ χ

(2)jklε

pj εs

kεA′

L

l (34)

Maximizing this coupling benefits the search bothby enhancing signal and reducing background. Thebackground is reduced because the SPDC effectivecoupling to transverse modes, Equation (33) is sup-pressed. The signal is enhanced because the couplingto the longitudinal polarization of the dark photon issuppressed by mA′/ω rather than (mA′/ω)2 [16, 27,28, 37].

This motivates either non traditional orientation ofnonlinear crystals, or identifying nonlinear materialsthat would usually be ineffective for SPDC. As anexample in the first category, any material which isuses for type II phase matching, in which the sig-nal and idler have orthogonal polarization, can be ro-tated by 90 degrees to achieve a coupling of a longi-tudinal polarization to a pump and an idler. In thesecond category, material with χ(2) tensors that arenon-vanishing only in “maximally non-diaginal” x-y-z elements, would obviously be discarded as standardSPDC sources, but can have an enhanced dSPDC cou-pling.

V. DARK SPDC SETUPS AND RATES

A precise estimate of the rate is beyond the scope ofthis work, and will be discussed in more detail in [16].Here we will re-scale known rate formulae in order toexamine the dependence of the rate on the geomet-ric factors such as the pump power and beam area,the crystal length, and the area and angle from whichsignal is collected.

Consider the geometry sketched in Figure 8. Apump beam is incident on an optical element oflength L along the the pump direction. The widthof the pump beam is set by the laser parameters. Thepump beam may consist of multiple modes, or canbe guided in a fiber or waveguide [15] and in a singlemode. The width and angle of the signal “beam” is set

10

by the apparatus used to collect and detect the signal.This too, can include collection of a single mode in afiber (e.g. [18, 38]) or in multiple modes (as in theCCD example in the next section. In SPDC, whenthe idler photon may be collected the width and an-gle of the idler can also be set by a similar apparatus.However, in dSPDC (or in SPDC if we choose to onlycollect the signal) the ϕ beam is not a parameter inthe problem. In this case we are interested in an in-clusive rate, and would sum over a complete basis ofidler beams. Such a sum will be performed in [16], butusually the sum will be dominated by a set of modesthat are similar to those collected for the signal.

Generally, the signal rate will depend on all of thechoices made above, but we can make some qualitativeobservations. The rate for SPDC and dSPDC will beproportional to an integral over the volume definedby the overlap of the three beams in Figure 8 and thecrystal. When the length of the volume is set by thecrystal, the process is said to occur in the “thin crystallimit”. In this limit, the beam overlap is roughly aconstant over the crystal length and thus the total ratewill grow with L. Since dSPDC is a rare process, weobserve that taking the collinear limit of the process,together with a thicker crystal may allow for a largerintegration volume and an enhanced rate.

The total rate for SPDC in a particular angle, in-tegrated over frequencies, in the thin crystal limitis [16, 18],

ΓSPDC ∼ Ppχ(2)eff

2ωsωiL

πnpnsniAeff(35)

where L is the crystal length, Pp is the pump power,Aeff is the effective beam area, and χ

(2)eff is the effective

coupling of Equation (33). A parametrically similarrate formula applies to SPDC in waveguides [15, 16].The pump power here is the effective power, whichmay be enhanced within a high finesse optical cavity,e.g. [25]. The inverse dependence with effective areacan be understood by since the interaction hamilto-nian is proportional to electric fields, which grow forfixed power for a tighter spot. For “bulk crystal” se-tups a pair production rate of order few times 106 permW of pump power per second is achievable [38] in theforward direction. In waveguided setups, in which thebeams remain confined along a length of the order ofa cm, production rates of order 109 pairs per mW persecond are achievable in KTP crystals [15, 39], andrates of order 1010 were discussed for LN crystals [39].

An important scaling of this rate is the L/Aeff de-pendence. This scaling applies for dSPDC rates dis-cussed below. Within the thin crystal limit one canthus achieve higher rates with: (a) a smaller beamspot, and (b) a thicker crystal. It should be notedthat for colinear SPDC, the crystal may be in the“thin limit” even for thick crystals (see Figure 8 andimagine zero signal and idler emission angle).

A. Dark Photon dSPDC Rate

The dSPDC rate into a dark photon with longitu-dinal polarization, A′

L, can similarly be written as a

simple re-scaling of the expression above

Γ(A′L)

dSPDC ∼ ε2 m2A′

ω2A′

Ppχ(2)A′

L

2ωsωA′L

npnsAeff(36)

where the effective coupling χ(2)A′

Lis defined in Equa-

tion (34). This is valid in regions where the dark pho-ton mass is smaller than the pump frequency, suchthat the produced dark photons are relativistic andhave a refractive index of 1. Using the optimisticwaveguide numbers above as a placeholder, assumingan optimized setup with similar χ(2), the number ofevents expected after integrating over a time tint are

N(A′

L)events ∼ 1021

(ε2 m2

A′

ω2A′

) (Pp

Watt

) (L

m

) (tintyear

).

(37)The current strongest lab-based limit is for dark pho-ton masses of order 0.1 eV is set by the ALPS exper-iment at ε ∼ 3 × 10−7. For this dark photon massmA′ ∼ 0.1ωp, the mA′/ωA′ term does not representa large suppression. In this case of order 10 dSPDCevents are produced in a day in a 1 cm crystal witha Watt of pump power with the assumptions above.This implies that a relatively small dSPDC experi-ment with an aggressive control on backgrounds couldbe used to push the current limits on dark photons.

Improving the limits from solar cooling, for whichεmA′/ωA′ is constrained to be smaller, would repre-sent an interesting challenge. Achieving ten events ina year of running requires a Watt of power in a waveg-uide greater than 10 meters (or a shorter waveguidewith higher stored power, perhaps using a Fabri-Perotsetup). Interestingly, in terms of system size, thisis still smaller than the ALPS-II experiment whichwould reach 100 meters in length and an effectingpower of a hundred kW. This is because LSW se-tups require both production and detection, with lim-its scaling as ε4.

B. Axion dSPDC Rate

A similar rate expression can be obtained for axion-dSPDC

Γ(axion)dSPDC ∼ Ppg2

aγγωsL

ωaxionnpnsAeff(38)

where the different scaling with the frequency of thedark field is due to the different structure of thedSPDC interaction (recall that χ(2) carries a massdimension of −2 while the axion photon coupling’sdimension is −1). Optimal SPDC (dSPDC) rates areacieved in waveguides in which the effective area is oforder the squared wavelength of the pump and signallight. Assuming a (linear) birefringent material thatcan achieve dSPDC phase matching for an axion thenumber of signal event scales as

N(axion)events ∼ 40

(gaγ

10−6 GeV−1

)2 (Pp

Watt

) (L

m

) (tintyear

).

(39)

11

Dark Photon (mA′ = 0.1 eV) Axion-like particle (ma = 0.1 eV)Current lab limit ε < 3 × 10−7 gaγ < 10−6 GeV−1

Example dSPDC setup Pp = 1 W Pp = 1 kWL = 1 cm L = 10 m

Γ = 10/day Γ = 10/dayCurrent Solar limit ε < 10−10 gaγ < 10−10 GeV−1

Example dSPDC setup Pp = 1 W Pp = 100 kWL = 10 m L = 100 m

Γ = 10/year Γ = 10/year

Table I: Current lab-based and Solar-based based limits on the couplings of dark photns and axion-like particles with abenchmark mass of 0.1 eV. For each limit we show the parameters of an example dSPDC in a waveguide and the rate itwould produce for couplings that would produce the specified benchmark rate with the corresponding coupling. Thepump power is an effective power which can include an enhancement by an optical cavity setup. For dark photon rateswe assume a nonlinearity of the same order found in KTP crystals.

This rate suggests a dSPDC setup is promising in set-ting new lab-based limits on ALPs. For example, a10 meter birefringent single mode fiber with kW ofpump power will generate of order 10 events per dayfor couplings of order 10−7 GeV−1. To probe beyondthe CAST limits in gaγ may be possible and requiresa larger setup, but not exceeding the scale, say, ofALPS-II. In a 100 meter length and an effective pumppower of 100 kW, a few dSPDC signal events are ex-pected in a year.

Maintaining a low background, would of course becrutial. We note, however, that optical fibers are rou-tinely used over much greater distances, maintainingcoherence (e.g. [5]), and an optimal setup should beidentified.

C. Backgrounds to dSPDC

There are several factors that should be consideredfor the purpose of reducing backgrounds to SPDC:

• Crystal Length and Signal bandwidth: In addi-tion to the growth of the signal rate, the signalbandwidth in many setups will decrease with L.If this is achieved the signal to background ra-tio in a narrow band around the dSPDC phasematching solutions will scale as L2.

• Timing: The dSPDC signal consists of a singlephoton whereas SPDC backgrounds will consistof two coincident photons. Backgrounds can bereduced using fast detectors and rejecting coin-cidence events.

• Optical material: As pointed out in Section IV,linear birefringent materials can be used toreduce SPDC backgrounds to axion searches.Nonlinear materials with a χ(2) tensor whichcouples purely to longitudinal polarizations maybe used to enhance dark photon dSPDC eventswithout enhancing SPDC. This technique to re-duce SPDC may also be used in axion searches.

• Detector noise and optical impurities: Sourcesof background which may be a limiting factor

for dSPDC searches include detector noise, aswell as scattering of pump photons off of impu-rities in the optical elements and surfaces. TheSkipper CCD, one example of a detector tech-nology with low noise, will be discussed in thenext section.

An optimal dSPDC based search for dark particleswill likely consider these factors, and is left for futureinvestigation.

VI. EXPERIMENTAL PROOF OF CONCEPT

In this section we present an experimental SPDCangular imaging measurement with high resolutionemploying a Skipper CCD and a BBO nonlinear crys-tal. In this setup and for the chosen frequencies,dSPDC phase matching is not achievable at any emis-sion angle. Instead, this experiment serves as a proofof concept for the high resolution imaging technique.

Imaging the dSPDC requires the detection of singlephotons with low noise and with high spatial and/orenergy resolution. A technology that can achieve thisis the Skipper CCD which is capable of measuringthe charge stored in each pixel with single electronresolution [17], ranging from very few electrons (0,1, 2, . . . ) up to more than a thousand (1000, 1001,1002, . . . ) [40]. This unique feature combined withthe high spatial resolution typical of a CCD detectormakes this technology very promising for the detec-tion of small optical signals with a very high spatialresolution.

A. Description of the experimental setup

With the aim of comparing the developed phasespace model against real data, the system depictedin Figure 9 was set up. A source of entangled pho-tons that employs SPDC, which is part of a com-mercial system3, was used. Two type I nonlinear

3 https://www.qutools.com/qued/

12

405 nmpump

photons BBO

810 nmreject lter

810 nmpass lters

SkipperCCD

810 nm SPDC photonsFGB37LL01-810

+97SA

Focusinglens

Figure 9: Pictorial representation of the experimentalsetup implemented in the lab. A 405 nm diode laser wasused as a source of pump photons. The beam passesthrough a 810 nm rejection filter, then through the BBOcrystal, after this through two 810 nm band pass filters, afocusing lens and finally the SPDC photons reach theCCD detector.

pump

SPDC

Figure 10: Picture of the source of entangled photonswhich is a part of a commercial system. The FGB37filter shown in Figure 9 was placed just after the laseraperture. All the other components were placed outsidethis device. Figure adapted from [41].

BBO crystals are used as a high efficient source ofentangled photons [30]. This pre-assembled sourcecan be seen in Figure 10. However, in this work,we did not take advantage of the polarization entan-glement, we only use the energy-momentum conser-vation to get spatially correlated twin photons. Inaddition to the original design, we added a ThorlabsFGB37 filter after the laser to remove a small 810 nmcomponent coming along with the pump beam. Af-ter the BBO crystal, where the SPDC process occurs,two additional 810 nm band pass filters (one SemrockBrighline LL01-810 and one Asahi Spectra 97SA) wereplaced to prevent the 405 nm pump beam from reach-ing the detector. We also placed a lens at its focaldistance from the BBO to prevent the SPDC conefrom spreading during its travel to the CCD (see Fig-ure 9). Thus, we got the SPDC ring projected on theCCD surface with a radius of 110 pixel, which corre-sponds to 1650 μm. This setup is also part of a re-search where the novel features of Skipper CCD [17]are being tested for Quantum Imaging. The capa-bility of Skipper-CCD to reduce the readout noise aslow as desired taking several samples of the charge ineach pixel was not used in this work to acquire data,but it was for the calibration. Details about the samedetector used here can be found in reference [40].

We used an absolute calibration between the Ana-log to Digital Units (ADU) measured by the amplifierfrom the CCD and the number of electrons in each

pixel, possible with the Skipper-CCD, which was pre-viously carried out for this system [40]. Thus, we re-constructed the number of photons per pixel, since at810 nm it can be assumed that one photon creates atmost one electron, by reason of its energy is 0.43 eVabove the silicon band gap (1.1 eV). The factor affect-ing that relationship is the efficiency which -besidesbeing very high (∼ 90 %) at this wavelength- is prettyuniform over the entire CCD surface [42].

B. Results

Using the setup previously described, 400 imageswith 200 s exposure each were averaged to producethe data used for comparison with the model. Thus,we reduced a factor twenty the uncertainty in the ex-pected number of photons in each pixel and drasticallyreduced the dark counts coming from random back-ground. This significantly improved the identificationof minima between rings, which results to be crucialto compare the experimental data with the presentedtheory.

Figure 11 presents the averaged image of the SPDCring coming out of the BBO crystal. The main ringis clearly visible, as well as many secondary maximaand minima. The angular coordinates θ and φ wereindicated on the image. It has to be noted that theangular coordinates used in the previous sections arethe ones inside the optical medium. Since the CCDdetector is outside the BBO crystal, Snell’s law mustbe used to relate the angles inside and outside thenonlinear crystal. Specifically, for the θ coordinatethis is

θ = arcsin(

sin Θn

)

where θ is the angle inside the optical medium inwhich the SPDC process occurs, Θ is the angle out-side the optical medium and n is the refractive index.In this work we apply the transformation to refer allangles to those inside the optical medium in which theinteraction occurs.

As can be seen in the image, even though it is pos-sible to see many of the maxima and minima, thereis a non uniform background component produced byreflections and other imperfections of the experimen-tal configuration. The current implementation of oursetup has some mechanical limitations that make hardto isolate, measure and remove this non-uniform back-ground component. Still, this image can be used tocompare the theory with the experiment, so the profileof the intensity distribution along the blue dashed lineshown in Figure 11 was extracted and plotted againstthe model. This plot is shown in Figure 12. Somecomments about this plot:

• A uniform background component was added tothe model.

• The transmittance curve of the 810 nm filtersprovided by the manufacturers was taken intoaccount.

13

Figure 11: Average of 400 images with 200 s exposureeach to the SPDC photons each using the setup depictedin Figure 9. The angular coordinates θ and φ areindicated in the figure. The blue dashed line was used toextract the data profile shown in Figure 12.

0 1 2 3

θ (deg)

102

103

Number

ofelectron

s Model

Data

Figure 12: Experimental intensity profile for the SPDCring along the blue dashed line in Figure 11 comparedwith theoretical distribution. The y axis scale is thenumber of electrons in each pixel of the CCD detectorobtained via the absolute Skipper-CCD calibrationdescribed.

So, to be specific, the plot in Figure 12 uses as modelthe intensity profile given by

Imodel (θ) = I0

ˆd2Γ

d(cos θs)dαωF (αω)dαω + I1

where I0 and I1 are real numbers fitted, d2Γd(cos θs)dαω

is given by Equation (27) and F (αω) is the transmit-tance of the filters as provided by the manufacturers,which is a sharp peaked function centered at 810 nmwith a pass band of 10 nm width.

For the parameters of the distribution Γ, we used:

• 405 nm for the pump wavelength as provided bythe manufacturer of the laser,

• the refractive indices for the BBO crystal weretaken from https://refractiveindex.info/ 4

4 The extraordinary refractive index was taken from https://refractiveindex.info/?shelf=main&book=BaB2O4&page=

• and the crystal length was assumed to be L =1.14 mm after fine tuning 5.

We had to perform a fine tuning of ns − np too (sincesignal and idler have the same frequency and anglethen ns = ni). The model is very sensitive to thisdifference, and this quantity depends on factors suchas the temperature of the BBO and its precise orien-tation in space, which were not measured. So it isnot surprising that we had to perform this fine tun-ing on ns −np. The final values used for the refractiveindices were np ≈ 1.66082 and ns = ni ≈ 1.66107. Al-though this fine tuning had to be done, all parametersare very well within expected values.

As seen both in Figures 11 and 12 the phase spacefactor, studied in previous sections, is the dominantmodulation in the distribution of photons for theSPDC process in our setup. Even though there isa non negligible background component, it is evidentthe dependence I ∼ sinc2 (

θ2)for small values of θ

predicted by Equation (27).

VII. DISCUSSION

We have presented a new method to search for newlight and feebly coupled particles, such as axion-likeparticles and dark photons. The dSPDC process al-lows to tag the production of a dark state as a pumpphoton down-converts to a signal photon and a “darkidler” ϕ, in close analogy to SPDC. We have shownthat the presence of indices of refraction that are dif-ferent than 1 open the phase space for the decay, ordown conversion, of the massless photon to the signalplus the dark particle, even if ϕ has a mass. This typeof search has a parametric advantage over light shiningthrough wall setups since it only requires producingthe axion or dark photon, without a need to detect itagain. Precise sensitivity calculations for dSPDC withDark Photon and Axion cases that can be achievedthrough this method are ongoing [16].

The commonplace use of optics in telecommunica-tions, imaging and in quantum information science,as well as the development of advanced detectors,can thus be harnessed to search for dark sector parti-cles. Increasing dSPDC signal rates will require highlaser power, long optical elements. Enhancing the sig-nal and suppressing SPDC backgrounds also requiresidentifying the right optical media for the search. Ax-ion searches would benefit from optically linear andbirefringent materials, with greater birefringence al-lowing to search for higher axion masses. Searches

Tamosauskas-e (Tamošauskas et al., 2018) and the ordi-nary refractive index from https://refractiveindex.info/?shelf=main&book=BaB2O4&page=Eimerl-o (Eimerl et al.,1987).

5 It was not possible to obtain precise information of the dimen-sions of the crystal from the manufacturer. Furthermore, thelength of the crystal is not easy to measure with our currentimplementation without breaking the SPDC source shown inFigure 10. So we decided to use a “reasonable value” and finetune it.

14

for dark photons would benefit from strongly nonlin-ear materials that are capable of coupling to a longi-tudinal polarization. This, in turn, motivated eithernon-conventional optical media, or using conventionalcrystals that are oriented by a 90◦ rotation from thatwhich is desired for SPDC. Enhancing the effectivepump power with an optical cavity is a straightfor-ward way to enhance the dSPDC rate. We leave theexploration of a “doubly resonant” setup in which thesignal is also a cavity eigenmode (in parallel with [25])for future work. Finally, detection of rare signal eventsrequires sensitive single photon detectors with highspatial and/or frequency resolution.

We also performed an experimental demonstrationof the one of the setups discussed using a Skipper

CCD for SPDC imaging. The setup we used for thisdemonstration was adapted from one designed to en-force SPDC and thus does not open dSPDC phasespace. We show that the Skipper technology allowsone to measure with high accuracy. Thus, quantumteleportation methods or ghost imaging of dark sectorparticles achieved through phase space engineering isa very promising technique for future explorations ofdark sector parameter spaces.

Acknowledgments: We would like to thank JoeChapman, Paul Kwiat, and Neal Sinclair for infor-mative discussions. This work was funded by a DOEQuantISED grant.

[1] A. Furusawa. Unconditional Quantum Teleportation.Science, 282(5389):706–709, October 1998.

[2] D. Boschi, S. Branca, F. De Martini, L. Hardy, andS. Popescu. Experimental Realization of Teleportingan Unknown Pure Quantum State via Dual Classi-cal and Einstein-Podolsky-Rosen Channels. PhysicalReview Letters, 80(6):1121–1125, February 1998.

[3] Dik Bouwmeester, Jian-Wei Pan, Klaus Mat-tle, Manfred Eibl, Harald Weinfurter, and AntonZeilinger. Experimental quantum teleportation. Na-ture, 390(6660):575–579, December 1997.

[4] Charles H. Bennett, Gilles Brassard, ClaudeCrépeau, Richard Jozsa, Asher Peres, andWilliam K. Wootters. Teleporting an unknownquantum state via dual classical and Einstein-Podolsky-Rosen channels. Physical Review Letters,70(13):1895–1899, March 1993.

[5] Raju Valivarthi, Samantha Davis, Cristian Pena,Si Xie, Nikolai Lauk, Lautaro Narvaez, Jason P. All-maras, Andrew D. Beyer, Yewon Gim, Meraj Hus-sein, George Iskander, Hyunseong Linus Kim, BorisKorzh, Andrew Mueller, Mandy Rominsky, MatthewShaw, Dawn Tang, Emma E. Wollman, ChristophSimon, Panagiotis Spentzouris, Neil Sinclair, DanielOblak, and Maria Spiropulu. Teleportation SystemsTowards a Quantum Internet. arXiv e-prints, pagearXiv:2007.11157, July 2020.

[6] Interaction.[7] C. K. Hong and L. Mandel. Theory of parametric

frequency down conversion of light. Physical ReviewA, 31(4):2409?2418, 1985.

[8] R. Essig and et al. Dark Sectors and New, Light,Weakly-Coupled Particles. arXiv:1311.0029 [astro-ph, physics:hep-ex, physics:hep-ph], October 2013.arXiv: 1311.0029.

[9] F. Hoogeveen and T. Ziegenhagen. Production anddetection of light bosons using optical resonators.Nucl. Phys. B, 358:3–26, 1991.

[10] Jessica Goodman, Masahiro Ibe, Arvind Rajaraman,William Shepherd, Tim M.P. Tait, and Hai-Bo Yu.Constraints on Light Majorana dark Matter from Col-liders. Phys. Lett. B, 695:185–188, 2011.

[11] Yang Bai, Patrick J. Fox, and Roni Harnik. The Teva-tron at the Frontier of Dark Matter Direct Detection.JHEP, 12:048, 2010.

[12] Jessica Goodman, Masahiro Ibe, Arvind Rajaraman,William Shepherd, Tim M.P. Tait, and Hai-Bo Yu.Constraints on Dark Matter from Colliders. Phys.

Rev. D, 82:116010, 2010.[13] Patrick J. Fox, Roni Harnik, Joachim Kopp, and

Yuhsin Tsai. Missing Energy Signatures of Dark Mat-ter at the LHC. Phys. Rev. D, 85:056011, 2012.

[14] T. Aaltonen et al. A Search for dark matter in eventswith one jet and missing transverse energy in pp colli-sions at

√s = 1.96 TeV. Phys. Rev. Lett., 108:211804,

2012.[15] Marco Fiorentino, Sean M. Spillane, Raymond G.

Beausoleil, Tony D. Roberts, Philip Battle, andMark W. Munro. Spontaneous parametric down-conversion in periodically poled KTP waveguides andbulk crystals. Optics Express, 15(12):7479–7488, June2007.

[16] Juan Estrada, Roni Harnik, Dario Rodrigues, andMatias Senger. New Particle Searches with Nonlin-ear Optics. In preparation.

[17] Javier Tiffenberg, Miguel Sofo-Haro, Alex Drlica-Wagner, Rouven Essig, Yann Guardincerri, Steve Hol-land, Tomer Volansky, and Tien-Tien Yu. Single-electron and single-photon sensitivity with a sil-icon Skipper CCD. Physical Review Letters,119(13):131802, September 2017. arXiv: 1706.00028.

[18] Alexander Ling, Antia Lamas-Linares, and ChristianKurtsiefer. Absolute emission rates of SpontaneousParametric Down Conversion into single transverseGaussian modes. Physical Review A, 77(4):043834,April 2008. arXiv: 0801.2220.

[19] James Schneeloch and John C Howell. Introduc-tion to the transverse spatial correlations in sponta-neous parametric down-conversion through the bipho-ton birth zone. Journal of Optics, 18(5):053501, 2016.

[20] P.A. Zyla et al. Review of Particle Physics. PTEP,2020(8):083C01, 2020.

[21] Georg G. Raffelt. Astrophysical methods to constrainaxions and other novel particle phenomena. Phys.Rept., 198:1–113, 1990.

[22] V. Anastassopoulos et al. New CAST Limit on theAxion-Photon Interaction. Nature Phys., 13:584–590,2017.

[23] Federico Della Valle, Aldo Ejlli, Ugo Gastaldi,Giuseppe Messineo, Edoardo Milotti, Ruggero Pengo,Giuseppe Ruoso, and Guido Zavattini. The PVLASexperiment: measuring vacuum magnetic birefrin-gence and dichroism with a birefringent Fabry–Perotcavity. Eur. Phys. J. C, 76(1):24, 2016.

[24] R. Ballou et al. New exclusion limits on scalar andpseudoscalar axionlike particles from light shining

15

through a wall. Phys. Rev. D, 92(9):092002, 2015.[25] Klaus Ehret et al. Resonant laser power build-up

in ALPS: A ’Light-shining-through-walls’ experiment.Nucl. Instrum. Meth. A, 612:83–96, 2009.

[26] Robin BÀhre et al. Any light particle search II —Technical Design Report. JINST, 8:T09001, 2013.

[27] Haipeng An, Maxim Pospelov, and Josef Pradler.New stellar constraints on dark photons. Phys. Lett.B, 725:190–195, 2013.

[28] Haipeng An, Maxim Pospelov, and Josef Pradler.Dark Matter Detectors as Dark Photon Helioscopes.Phys. Rev. Lett., 111:041302, 2013.

[29] David Tong. Quantum Field Theory.http://www.damtp.cam.ac.uk/user/tong/qft.html.

[30] Paul G. Kwiat, Edo Waks, Andrew G. White, IanAppelbaum, and Philippe H. Eberhard. Ultra-brightsource of polarization-entangled photons. PhysicalReview A, 60(2):R773–R776, August 1999. arXiv:quant-ph/9810003.

[31] Christophe Couteau. Spontaneous parametric down-conversion. Contemporary Physics, 59(3):291–304,July 2018.

[32] N. Boeuf. Calculating characteristics of noncollinearphase matching in uniaxial and biaxial crystals. Op-tical Engineering, 39(4):1016, April 2000.

[33] Paul-Antoine Moreau, Ermes Toninelli, Thomas Gre-gory, and Miles J. Padgett. Imaging with quantumstates of light. Nature Reviews Physics, 1(6):367–380,June 2019.

[34] S. Magnitskiy, D. Frolovtsev, V. Firsov, P. Gostev,I. Protsenko, and M. Saygin. A SPDC-BasedSource of Entangled Photons and its Characteriza-

tion. Journal of Russian Laser Research, 36(6):618–629, November 2015.

[35] J. Noda, K. Okamoto, and Y. Sasaki. Polarization-maintaining fibers and their applications. Journal ofLightwave Technology, 4(8):1071–1089, 1986.

[36] Robert W. Boyd. Nonlinear optics. Academic Press,Amsterdam ; Boston, 3rd ed edition, 2008.

[37] Peter W. Graham, Jeremy Mardon, Surjeet Rajen-dran, and Yue Zhao. Parametrically enhanced hiddenphoton search. Phys. Rev. D, 90(7):075017, 2014.

[38] Ryan S. Bennink. Optimal Co-linear Gaussian Beamsfor Spontaneous Parametric Down-Conversion. Phys-ical Review A, 81(5):053805, May 2010. arXiv:1003.3810.

[39] Sean M. Spillane, Marco Fiorentino, and Raymond G.Beausoleil. Spontaneous parametric down conver-sion in a nanophotonic waveguide. Optics Express,15(14):8770–8780, July 2007.

[40] Dario Rodrigues, Kevin Andersson, Mariano Caba-bie, Andre Donadon, Gustavo Cancelo, Juan Estrada,Guillermo Fernandez-Moroni, Ricardo Piegaia, Ma-tias Senger, Miguel Sofo Haro, et al. Absolute mea-surement of the fano factor using a skipper-ccd. arXivpreprint arXiv:2004.11499, 2020.

[41] qutools. quED.[42] C.J. Bebek, J.H. Emes, D.E. Groom, S. Haque, S.E.

Holland, P.N. Jelinsky, A. Karcher, W.F. Kolbe, J.S.Lee, N.P. Palaio, D.J. Schlegel, G. Wang, R. Groulx,R. Frost, J. Estrada, and M. Bonati. Status ofthe CCD development for the dark energy spec-troscopic instrument. Journal of Instrumentation,12(04):C04018–C04018, apr 2017.