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, USA 2Department 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 communication to 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-idler pair. We propose to search for “dark SPDC” (dSPDC) events in which a new dark sector particle replaces the idler. Though it does not interact, the presence of a dark particle can be inferred by the properties of the signal photon. Examples of dark states include axion-like-particles and dark photons. 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 proposed which employ optical imaging and/or spectroscopy of the signal photons. The signal rates in our proposal 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 the characteristics of optical media needed to enhance dSPDC and estimate the rate. A bench-top demonstration 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 quantum information science. Among its many uses, it plays an enabling role in the areas of quantum networks and teleportation of quantum states as well as in quantum imaging. In quantum teleportation [1–4] the state of a distant quantum system, Alice, can be inferred with- out directly interacting with it, but rather by allowing it to interact with one of the photons in an entangled pair. The coherence of these optical systems has re- cently allowed teleportation over a large distance [5]. Figure 1: Pictorial representation of the dSPDC process. Quantum ghost imaging, or “interaction-free” imag- A dark particle ϕ is emitted in association with a signal ing [6], is used to discern (usually classical) informa- photon. The presence of ϕ can be inferred from the tion about an object without direct interaction. This distribution of the signal photon in angle and/or technique exploits the relationship among the emis- frequency. We consider both the colinear (θs = 0) and sion angles of a correlated photon pair to create an non-colinear (θs =6 0) cases. image with high angular resolution without placing the subject Alice in front of a high resolution detec- tor or allowing it to interact with intense light. These in capturing an image of Alice will occur at a rate methods of teleportation and imaging rely on the pro- ∝ 4 with standard methods, but at a rate ∝ 2 using duction of signal photons in association with an idler quantum optical methods. pair which is entangled (or at least correlated) in its direction, frequency, and sometimes polarization. A common method for generating entangled photon sponta- arXiv:2012.04707v2 [hep-ph] 17 Jan 2021 Quantum ghost imaging and teleportation both dif- pairs is the nonlinear optics process known as neous parametric down conversion fer parametrically from standard forms of information (SPDC). In SPDC transfer. This is because a system is probed, not by a pump photon decays, or down-converts, within a sending information to it and receiving information nonlinear optical medium into two other photons, a back, but rather by sending it half of an EPR pair, signal and an idler. The presence of the SPDC idler without need for a “response”. The difference is par- can be inferred by the detection of the signal [7]. ticularly apparent if Alice is an extremely weekly cou- SPDC is in wide use in quantum information and pro- pled system, say she is part of a dark sector, with a vides the seed for cavity enhanced setups such as op- coupling to photons. The rate of information flow tical parametric oscillators (OPO) [8]. In this work we propose to use quantum imag- ing and quantum communication tools to perform an interaction-free search for the emission of new parti- [email protected] cles beyond the standard model. The new tool we [email protected] present is dark SPDC, or dSPDC, an example sketch [email protected] of which is shown in Figure 1. A pump photon enters [email protected] an optical medium and down-converts to a signal pho- 2 ton and a dark particle, which can have a small mass, rate more carefully, focusing on the spectroscopic ap- and does not interact with the optical medium. Like proach and on colinear dSPDC in bulk crystals and SPDC, in dSPDC the presence of a dark state can waveguides. be inferred by the angle and frequency distribution of In this work we also present a proof-of-concept ex- the signal photon that was produced in association. periment using a quantum imaging setup with a Skip- The dSPDC process can occur either collinearly, with per CCD [18]. The high resolution is shown to pro- θs = θi = 0, or as shown in the sketch, in a non- duce a high resolution and low noise angular image of colinear way. an SPDC pattern. Such a detector may be employed The new dark particles which we propose to search both in imaging and spectroscopic dSPDC setups. for, low in mass and feebly interacting, are simple ex- This paper is structured as follows. In Section II tensions of the standard model (SM). Among the most we review axion-like particles and dark photons and well motivated are axions, or axion-like-particles, and present their Hamiltonians in the manner they are dark photons (see [9] for a review). Since these par- usually treated in nonlinear optics. In Section III ticles interact with SM photons, optical experiments we discuss energy and momentum conservation, which provide an opportunity to search for them. A canon- are called phase matching conditions in nonlinear op- ical setup is light-shining-through-wall (LSW) which tics, and the phase space for dSPDC. The usual con- set interesting limits on axions and dark photons us- servation rules will hold exactly in a thought experi- ing optical cavities [10]. In optical LSW experiments ment of an infinite optical medium. In Section III C a laser photon converts to an axion or dark photon, we discuss how the finite size of the optical medium enabling it to penetrate an opaque barrier. For detec- broadens the phase space distribution and plot it as a tion, however, the dark state must convert back to a function of the signal photon angle and energy. In Sec- photon and interact, which is a rare occurrence. As tion IV we discuss the choice of optical materials that a result, the signal rate in LSW scales as 4, where allow for the dSPDC phase matching to enhance its is a small coupling to dark states. This motivates rates and to suppress or eliminate SPDC backgrounds. the interaction-free approach of dSPDC, in which the In Section V we discuss the rates for dSPDC and its dark state is produced but does not interact again, scaling with the geometry of the experiment, showing yielding a rate ∝ 2. potential promising results for collinear waveguide se- From the perspective of particle physics, dSPDC, tups with long optical elements. In Section VI, we and also SPDC, may sound unusual1. In dSPDC, present an experimental proof-of-concept in which a for example, a massless photon is decaying to another Skipper CCD is used to image an SPDC pattern with massless photon plus a massive particle. In the lan- high resolution. We discuss future directions and con- guage of particle physics this would be a kinematically clude in Section VII. forbidden transition. The process, if it were to hap- pen in vacuum, violates energy and momentum con- servation and thus has no available phase space. In II. THE DARK SPDC HAMILTONIAN this paper we show how optics enables us to perform “engineering in phase space” and open kinematics to The SPDC process can be derived from an effective otherwise forbidden channels. This, in turn, allows to nonlinear optics Hamiltonian of the form [19, 20] design setups in which the dSPDC process is allowed (2) and can be used to search for dark particles. HSPDC = χ EiEsEp (1) Broadly speaking, the search strategies we propose where Ej is the electric field for pump, signal, and may be classified as employing either imaging or spec- idler photons and j = p, s, i respectively. We adopt troscopic tools, though methods employing both of the standard notation in nonlinear optics literature in these are also possible. In the first, the angular dis- which the vector nature of the field is implicit. Thus tribution of signal photons are measured, and in the the pump, signal and idler can each be of a particular second, the energy distribution is observed. For an polarization and χ(2) represents the coupling between imaging based search, a high angular resolution is re- the corresponding choices of polarization. quired, while the spectroscopic approach requires high We now consider the dark SPDC Hamiltonian frequency resolution. The later has the benefit that dSPDC can be implemented in a waveguide which will HdSPDC = gϕ ϕEsEp (2) enhance its rate for long optical elements [16, 17]. We present the basic ideas and formalism behind in which a pump and signal photon couple to a new ϕ m this method, and focus on the phase space for dSPDC. field which has a mass ϕ. The effective coupling g We compare the phase space distributions of SPDC ϕ, depends on the model and can depend on the fre- to dSPDC and discuss factors which may enhance the quencies in the process, the polarizations and the di- rate of dSPDC processes. To estimate the dSPDC rections of outgoing particles. This type of term arises rates in this work we re-scale known SPDC results. in well motivated dark sector models including axion- In a companion paper [17] we will derive the dSPDC like particles and dark photons.
A. Axion-like particles 1 Searches for missing energy and momentum are a common- place tool in the search for new physics at colliders [11–15]. Axion-like particles, also known as ALPs, are light However, the dSPDC kinematics are distinct as we show. pesudoscalar particles that couple to photons via a 3 term In addition to ALPs and dark photons, one can con- sider other new dark particles that are sufficiently ~ ~ Haxion = gϕ ϕE · B, (3) light and couples to photons. Examples include mil- licharged particle, which can be produced in pairs in which can be written in the scalar form of Equation (2) association with a signal photon. We leave these gen- with the pump and signal photons chosen to have or- eralizations for future work and proceed to discuss the thogonal polarization. The coupling gϕ in this case phase space for dSPDC in a model-independent way. is the usual axion-photon coupling ∼ 1/fa. ALPs are naturally light thanks to a spontaneously broken global symmetry at a scale fa. III. PHASE SPACE OF DARK SPDC Since we will not perform a detailed analysis of backgrounds and feasibility here, we will describe ex- As stated above, the effective energy and momen- isting limits on ALPs qualitatively and refer readers tum conservation rules in nonlinear optics are known to limit plots in [9, 21]. In this work we will con- as “phase matching conditions”. We begin by review- sider searches that can probe ALPs with a mass of ing their origin to make the connection with energy order 0.1 eV or lower. Existing limits in this mass and momentum conservation in particle physics and range come from stellar and supernova dynamics[22], then proceed to solve them for SPDC and dSPDC, to as well as from the CAST Heiloscope [23], which re- −10 −1 understand the phase space for these processes. quire gϕ . 10 GeV . 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- A. Phase matching and particle decay pling of order 10−6 GeV−1 are allowed by all lab-based searches. For axion masses around 10−3 −10−2 eV the PVLAS search[24] for magnetic birefringence places a We begin with a discussion of the phase space in −7 −6 −1 a two-body decay, or spontaneous down-conversion. limit of gϕ . 10 − 10 GeV . Bellow masses of an meV, LSW experiments such as OSQAR [25] and We use particle physics language, but will label the −8 particles as is usual in the nonlinear optics systems ALPS [26] set gϕ . 6 × 10 . A large scale LSW fa- cility, ALPS II [27], is proposed in order to improve which we will be discussing from the onset. A pump p s upon astrophysical limits and CAST. particle decays into a signal particle and another particle, either an idler photon i or a dark particle ϕ:
SPDC: γp → γs + γi B. Dark photons dSPDC: γp → γs + ϕ . (5) 0 A new U(1) gauge field, A , with mass mA0 that We would like to discuss the kinematics of the stan- mixes kinetically with the photon via the Hamiltonian dard model SPDC process, and the BSM dSPDC pro- cess together, to compare and contrast. For this, in µν 0 ~ ~ 0 ~ ~ 0 Hmix = F Fµν = E · E + B · B . (4) the discussion below we will use the idler label i to represent the idler photon in the SPDC case, and ϕ in It is possible to write the dark photon interaction in the dSPDC case. Hence ki and θi are the momentum a basis in which any electromagnetic current couples and emission angle of either the idler photon or of ϕ, to longitudinally polarized dark photons with an ef- depending on the process. fective coupling of (mA0 /ωA0 ) and to transversely po- The differential decay rate in the laboratory frame 2 larized dark photons with (mA0 /ωA0 ) [28, 29]. The is given by [21] same dynamics that leads to the nonlinear Hamilto- 4 dΓ δ (ppµ − psµ − piµ) nian HSPDC, will yield an effective coupling of two 2 3 3 = 2 |M| (6) photons to a dark photon as in Equation (2) with ϕ d ksd ki (2π) 2ωp 2ωs 2ωi representing a dark photon polarization state. For longitudinal polarization of the dark photon this can where M is the amplitude for the process (which be brought to the form of equation (2) with gϕ = carries a mass dimension of +1), and the energy- (2) momentum four-vectors χ mϕ. The coupling to transverse dark photons will be suppressed by an additional factor of mϕ/ωϕ. ωj Like axions, limits on dark photons come from pjµ = (7) kj astrophysical systems, such as energy loss in the Sun [28], as well as the lack of a detection of dark with j = p, s, i. Traditionally, one proceeds by inte- photons emitted by the Sun in dark matter direct de- grating over four degrees of freedom within the six di- tection experiments [29]. At a dark photon mass of or- −10 mensional phase space, leaving a differential rate with der 0.1 eV, the limit on the kinetic mixing is . 10 respect to the two dimensional phase space. This is with the limit becoming weaker linearly as the dark often chosen a solid angle for the decay dΩ, but in our photon mass decreases. Among purely lab-based ex- −7 case there will be non-trivial correlations of frequency periments, ALPS sets a limit of . 3 × 10 . In and angle as we will see in the next subsection. forthcoming sections of this work we will use these It is instructive, however, for our purpose to take a limits as benchmarks. step back and recall how the energy-momentum con- serving δ-functions come about. In calculating the 4 quantum amplitude for the transition, the fields in medium is finite. As a result, the sharp momentum the initial and final states are expanded in modes. conserving delta function will become a peaked distri- The plane-wave phases ei(ωt−k·x) are collected from bution of width L−1, where L is the crystal size. We each and a spacetime integral is performed will begin by solving the exact phase matching condi- tions in SPDC and dSPDC in the following subsection, d4x ei(∆ωt−∆k·x) = (2π)4 δ(3)(∆k) δ(∆ω) , (8) which correspond to the phase space distributions for ˆ an infinitely large crystal. Next, we will move on to the case in which the optical medium is finite. where frequency and momentum mismatch are defined as ∆ω = ωp − ωs − ωi B. Phase Matching in dSPDC , (9) ∆k = kp − ks − ki We now study the phase space for dSPDC to iden- and again, the label i can describe either an idler pho- tify the correlations between the emission angle and ton or a dark particle ϕ. In Equation (9), we sepa- the frequency of the signal photon. We will consider rated momentum and energy conserving delta func- in parallel the SPDC process as well, so in the end tions to pay homage to the optical systems which will we arrive to both results. We assume in this subsec- be the subject of upcoming discussion. The squared tion an optical medium with an infinite extent, such amplitude |M|2, and hence the rate, is proportional that the delta-functions enforce energy and momen- to a single power of the energy-momentum conserv- tum conservation. ing delta function times a space-time volume factor, The phase matching conditions ∆ω = 0 and ∆k = 0 which is absorbed for canonically normalized states —that must be strictly satisfied in the infinite extent (see e.g. [30]), giving Equation (6). optical medium scenario— imply The message from this is that energy and mo- mentum conservation, which are a consequence of ωp = ωs + ωi . (11) Neother’s theorem and space-time translation sym- kp = ks + ki metry, is enforced in quantum field theory by perfect destructive interference whenever there is a non-zero Using the k’s decomposition shown in Figure 1 the sec- mismatch in the momentum or energy of initial and ond equation can be expanded in coordinates parallel final states. In this sense, the term “phase matching” and perpendicular to the pump propagation direction captures the particle physicist’s notion of energy and momentum conservation well. kp = ks cos θs + ki cos θi (12) The dSPDC process shown in Figure 1 is a massless 0 = ks sin θs − ki sin θi pump photon decaying to a signal photon plus a mas- sive particle ϕ. This process clearly cannot occur in where the angles θs and θi are those indicated in the vacuum. For example, if we go to the rest frame of ϕ, cited figure. Since this process is happening in a mate- and define all quantities in this frame with a tilde˜, the rial medium, the photon dispersion relation is k = nω conservation of momentum implies k˜p = k˜s, while the where n is the refractive index. The refractive index conservation of energy implies ω˜p =ω ˜s + m. Combin- is in general a function of frequency, polarization, and ing these with the dispersion relation for a photon in the direction of propagation, and can thus be different vacuum ω˜p,s = |k˜p,s|, implies that energy and momen- for pump, signal, and idler photons. For dSPDC, in tum conservation cannot be satisfied and the process which the ϕ particle is massive and weekly interacting is kinematically forbidden. Another way to view this with matter the dispersion relation is the same as in is to recall that the energy-momentum four-vector for vacuum, the initial state (a photon) lies on a light cone i.e. it q pµp 2 2 is a null four-vector, µ = 0, while the final state kϕ = ωϕ − mϕ. (13) cannot accomplish this since there is a nonzero mass. In this work we show that optical systems will allow Thus we can explicitly write the dispersion relation us to open phase space for dSPDC. When a photon is for each particle in this process: inside an optical medium a different dispersion rela- tion holds, kp = npωp, (14)
ks = nsωs (15) n˜pω˜p = |k˜p| and n˜sω˜s = |k˜s| (10) and where n˜p and n˜s are indices of refraction of photons ϕ in the medium in the rest frame, which can be dif- niωi for SPDC ferent for pump and signal. As we will see later on, ki = q . (16) 2 2 under these conditions the conclusion that p → s + ϕ ωi − mϕ for dSPDC is kinematically forbidden can be evaded. Another effect that occurs in optical systems, but where, again, the label “i” refers to an idler photon not in decays in particle physics, is the breaking of for SPDC and to the dark field ϕ for dSPDC. Note spatial translation invariance by the finite extent of that ki for SPDC not only has m = 0 but also takes the optical medium. This allows for violation of mo- into account the “strong (electromagnetic) coupling” mentum conservation along the directions in which the with matter summarized by the refractive index ni. 5
As a result, for SPDC we always have ki > ωi while occur in a single-mode fiber or a waveguide. For this for dSPDC ki < ωi. Replacing these dispersion rela- case, so long as ns > np, an axion mass below some tions into (12) and using the first equation in (11) we threshold can be probed. Setting the emission angles obtain the phase matching relation of signal angle and to zero there are two solutions to the phase matching frequency equations which give signal photon energies of n2 + α2 n2 − Ξ2 r p ω s 2 m2 cos θs = (17) 2 ϕ (nsnp − 1) ± (ns − np) − (ns − 1) 2 2αωnpns ωp αω = 2 . where we define ns − 1 def ωs (21) αω = (18) ωp The + solution above corresponds to the case where the ϕ particle is emitted in the forward direction and whereas the − represents the case where it is emit- ni (1 − αω) for SPDC ted backwards. The signal frequency as a function def k ϕ i s 2 of mass is shown in Figure 3 for the calcite bench- Ξ = = 2 mϕ . (19) m ω ωp − α − mark. For the near-massless case, ϕ p, a phase (1 ω) 2 for dSPDC ωp matching solution exists for ns ≥ np, giving
Equation (17) defines the phase space for (d)SPDC, ωs np ∓ 1 φ = . (22) along with the azimuthal angle s. The idler angle, or ωp ns ∓ 1 that of the dark particle in the dSPDC case, is fixed in terms of the signal angle and frequency by requiring For calcite, we get ωs = 0.739ωp and ωs = 0.935ωp for conservation of transverse momentum forward and backward emitted axions respectively. ks Of course, not any combination of np and ns will sin θi = sin θs ki allow to achieve phase matching in dSPDC. We will state here that ns > np is a requirement for phase φi = φs + π (20) matching to be possible for a massless ϕ and that ns − where ki is evaluated at a frequency of ωi = ωp − ωs, np must grow as mϕ grows. We will discuss these and and ks at the frequency ωs according to the dispersion other requirements for dSPDC searches in Section IV. relation in Equations (15) and (16). In the left panel of Figure 2 we show the allowed phase space in the αω-θs plane for SPDC and for C. Thin planar layer of optical medium dSPDC with both a massless and massive ϕ. Here we have chosen np = 1.486, ns = 1.658 with ni = ns We now consider the effects of a finite crystal. in the SPDC case. These values were chosen to be Specifically, we will assume that the optical medium constant with frequency and propagation direction, is a planar thin layer2 of optical material of length L for simplicity and are inspired by the ordinary and along the pump propagation direction zˆ, and is infinite extraordinary refractive indices in calcite and will be in transverse directions. In this case, the integral in used as a benchmark in some of the examples below. Equation (8) is performed only over a finite range and We see that phase matching is achieved in different the delta function is replaced by a sinc x ≡ sin x/x regions of phase space for SPDC and its dark counter- function part. In dSPDC, signal emission angles are restricted L/2 to near the forward region and in a limited range of ∆kzL dz ei∆kz z = L sinc (23) signal frequencies. The need for more forward emis- ˆ−L/2 2 sion in dSPDC can be understood because ϕ effec- tively sees an index of refraction of 1. This implies at the level of the amplitude. As advertised, this al- it can carry less momentum for a given frequency, as lows for momentum non-conservation with a charac- compared to a photon which obeys k = nω. As a teristic width of order L−1. result, the signal photon must point nearly parallel The fully differential rate will be proportional to the to the pump, in order to conserve momentum (see squared sinc function Figure 2, right). An additional difference to notice is that, in contrast with the SPDC example shown, dΓ 2 2 ∆kzL 2 3 3 ∝ L sinc δ (∆kT )δ(∆ω)(24) for a fixed signal emission angle there are two differ- d kpd ki 2 ent signal frequencies that satisfy the phase matching dΓˆ ≡ (25) in dSPDC. This will always be the case for dSPDC. d3k d3k Although SPDC can be of this form as well, typical re- p i fractive indices usually favor single solutions as shown where ∆kT is the momentum mismatch vector in the for calcite. transverse directions. The constant of proportional- ity in Equation (24) will have frequency normalization
1. Phase matching for colinear dSPDC
One case of particular interest is that of colinear 2 The meaning of what constitutes the thin crystal layer limit dSPDC, in which the emission angle is zero, as would in our context will be discussed in Section V. 6
6
5 150 4
3
2
100 1 [deg] s 0 0.70 0.75 0.80 0.85 0.90 0.95 1.00
50
0 0.0 0.2 0.4 0.6 0.8 1.0
s/ p
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 here are 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 ∆k = 0 for SPDC and dSPDC, with massless ϕ. In both cases we take the same ωp and ωs. Due to the index of refraction for ϕ is essentially one, and that for the idler photon is larger (say ∼ 1.5), phase matching in dSPDC has a smaller signal emission angle than that of SPDC.
1.00 factors of the form (2ω)−1 and the matrix element M. Waveguide phase matching: The matrix element can also have non trivial angular θ φ 0.95 dependence in as well as in , depending on the model and the optical medium properties. Note, how- ever, that the current analysis for the phase space is 0.90 model independent. We thus postpone the discussion p
of overall rates to Section V and to [17] and here we / s 0.85 limit the study to the phase space distribution only. We will proceed with the defined phase space distri- bution dΓˆ in Equation (25), which we will now study. 0.80 3 One can trivially perform the integral over the d ki which will effectively enforce Equation (20) for trans- 0.75 verse momentum conservation, and set |ki| by conser- vation of energy. The argument of the sinc function is 0.70 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 ∆kz = kp − ks cos θs − ki cos θi (26) m / p q 2 2 2 2 = ωp np − nsαω cos θs ± Ξ − n α sin θs Figure 3: Solutions to the phase matching conditions for s ω the colinear dark SPDC process for the signal photon energy as a function of the dark particle ϕ mass. This where in the second step we have used Equation (20) phase space is relevant for waveguide-based experiments. and the definitions of αω and Ξ in Equations (18) Both axes are normalized to the pump frequency. The and (19). The ± accounts for the idler or dark particle two branches correspond to configurations in which ϕ is being emitted in the forward and backward direction emitted forward (bottom) and backward (top). respectively.
It is convenient to re-express the remaining three-dimensional phase space for (d)SPDC in terms of the signal emission angles θs and φs, and the signal frequency (or equivalently αω). Within our assumptions, the distribution in φs is flat. With respect to the polar angle and frequency we find 2 ˆ πω3α2 − α n3n2 q d Γ 2 p ω (1 ω) s ˜i X 2 2 2 2 2 2 ωpL = L sinc np − nsαω cos θs ± Ξ − n α sin θs (27) d θ dα p 2 2 2 2 s ω (cos s) ω Ξ − αωns sin θs ± 2
where In Figure 4 we show the double differential distri- def ni for SPDC n˜i = . (28) 1 for dSPDC 7
In Figure 6 we show two distributions, one for fixed signal frequency, and the other for a fixed signal an- gle, the latter in the forward direction. We see that both of these measurement schemes are able to distin- guish the signal produced by different values of mass mϕ. Furthermore, in cases where the standard model SPDC process is a source of background, it can be sep- arated from the dSPDC signal. As expected, in the forward measurement the dSPDC spectrum exhibits two peaks for the emission of ϕ in the forward or back- ward directions. It should be noted that the width of the highest peaks in these distributions decreases with crystal length L.
IV. OPTICAL MATERIALS FOR DARK SPDC 2 Figure 4: The phase space distribution d Γˆ/d(cos θs)dαω L for SPDC in arbitrary units. We use L = 1, λp = 10 , Having discussed the phase space distribution for np = 2 and ns = ni = 4. These parameters are dSPDC, we will now discuss which optical media are unrealistic in practice, but allow for clear visualization of needed to open this channel, to enhance its rates, and, the distribution. Dashed lines show slices of fixed signal if possible, to suppress the backgrounds. A more com- angle/energy. plete estimate of the dSPDC rate will be presented in [17]. Here we will discuss the various features qual- itatively to enable optimization of dSPDC searches. bution Γˆ for SPDC, and in Figure 5 it is shown for dSPDC both for a massless and a massive ϕ. The dis- tribution clearly peaks in regions of phase space that A. Refractive Indices satisfy the phase matching conditions, those shown in Figure 2. To have a clearer view of the qualitative features in the distribution we picked somewhat exag- As we discussed above, the refractive index of the gerated values for indices of refraction in these figures. optical medium plays a crucial role. It opens the phase space for the dSPDC decay and determines the kine- matics of the process. In order to enhance the dSPDC rate, it is always 1. dSPDC Searches with Imaging or Spectroscopy desirable to have a setup in which the dSPDC phase matching conditions which we discussed in Section III are accomplished. Since the left hand side of Equa- Though one can, in principle, measure both the an- tion (17) is the cosine of an angle, its right hand is gle and the energy of the signal photon, it is usu- restricted between ±1 so ally easier to measure either with a fixed angle or a 2 2 2 2 fixed energy. For example, a CCD detector with a np + αωns − Ξ monochromatic filter can easily measure a single en- − 1 < < 1. (31) 2αωnpns ergy slice of this distribution, as shown in the vertical dashed line in Figure 4. Likewise, a spectrometer with From this one finds that so long as ns > np, there is no spatial resolution can measure a fixed angle slice a range of ϕ mass which can take part in dSPDC. As of this distribution, as shown in the horizontal dashed the difference ns − np is taken to be larger, a greater line of Figure 4. An attractive choice for this is to look ϕ mass can be produced and larger opening angles θs θ in the forward region, with an emission angle s = 0, can be achieved. which is the case for waveguides. For this colinear Using the inequalities in Equation (31) we can ex- case we get a signal spectrum plore the range of desirable indices of refraction for 3 2 3 2 (0) ! a particular setup choice. For example, suppose we dΓˆ 2πωpαω (1 − αω) nsn˜i X ∆k L = L2sinc2 z± use monochromatic filter for the signal at half the en- dα 1 ω Ξ ± 2 ergy of the pump photons, αω = 2 . Figure 7 shows in (29) blue the region where the phase matching condition where is satisfied for the SPDC process and in orange the region where the phase matching condition is satisfied (0) ± kz± = kzp − kzs − kzi = ωp (np − nsαω ± Ξ) (30) for dSPDC. As can be seen, dSPDC is more restrictive in the refractive indices. Furthermore, typical SPDC is the momentum mismatch for colinear (d)SPDC. In experiments employ materials such as beta barium bo- some cases the colinear spectrum is dominated by just rate (BBO), potassium dideuterium phosphate (KDP) the forward emission of the idler/ϕ, while in others and lithium triborate (LBO) [19, 31–33] that do not al- there are two phase matching solutions which con- low the phase matching for dSPDC for this example. tribute similarly to the rate. 8
dSPDC dSPDC 180 100 180 100
160 160 2 2 10− 10− 140 140
4 4 120 10− 120 10− 100 100 6 6
(deg) 10− (deg) 10−
s 80 s 80 θ θ
8 8 60 10− 60 10− 40 40 10 10 10− 10− 20 20
0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
αω = ωs/ωp αω = ωs/ωp
2 Figure 5: Phase space distribution d Γˆ/d(cos θs)dαω in arbitrary units for the dSPDC process for m = 0 (left) and m = 0.5ωp. The other parameters are as in Figure 4, with somewhat exaggerated indices of refraction to allow seeing the 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 left we show angle distributions with fixed signal frequency, and on the right we show frequency distributions for fixed signal angle. These distributions allow to separate SPDC backgrounds from the dSPDC signal, as well as the dSPDC signal for different values of mϕ. The distributions become narrower for thicker crystals, enhancing the signal to background separation power. For these distributions we used L = 1 cm, λp = 400 nm and for the refractive indices np = 1.49 and ns = ni = 1.66 inspired by calcite.
A second example which is well motivated is the ordinary and extraordinary polarizations have indices forward region, namely θs = 0. In this case phase of refraction of no = 1.658 and ne = 1.486. Taking matching can be achieved so long as the former to be the signal and the later for the pump, phase matching can be met for mϕ < 0.16ωp. Bire- 2 2 (ns − np) mϕ fringence may also be achieved in single-mode fibers 2 ≥ 2 , (32) ns − 1 ωp and waveguides with non-circular cross section, e.g. polarization-maintaining optical fibers [36]. Of course, and ns > np. We will investigate colinear dSPDC in the dependence of the refractive index on frequency greater detail in [17]. may also be used to generate a signal-pump difference The dSPDC phase matching requirement ns > np in n. can be achieved in practice by several effects. The most common one, employed in the majority of SPDC experiments, is birefringence [31, 33–35]. In this case the polarization of each photon is used to obtain a different refractive index. For instance, in calcite, the 9
crystal
pump
dark
Figure 8: A sketch showing the overlap of the pump, signal, and dark particle (or idler) beams with the crystal. In general it is desirable to maximize the Figure 7: Regions in which the refractive indices np and integration length which is defined by the overlap of the 1 ns allow the phase matching with αω ≡ ωs/ωp = 2 for three beams and the crystal. the SPDC process (in this case assuming ni = ns) and m dSPDC (in this case assuming ωp 1). 0 AL have a longitudinal polarization, εl = (0, 0, 1). The crystal may be oriented to couple to this longitudinal B. Linear and Birefringent Media for Axions mode, giving an effective coupling of
0 Axion electrodynamics is itself a nonlinear theory. (2) (2) p s AL χA0 ≡ χjklεj εkεl (34) Therefore dSPDC with emission of an axion can occur L in a perfectly linear medium. The optical medium is Maximizing this coupling benefits the search both needed in order to satisfy the phase matching condi- by enhancing signal and reducing background. The tion that cannot be satisfied in vacuum. Because the background is reduced because the SPDC effective pump and signal photons in axion-dSPDC have dif- coupling to transverse modes, Equation (33) is sup- ferent polarizations, a birefringent material can sat- pressed. The signal is enhanced because the coupling n > n isfy the requirement of s p and Equation (32). to the longitudinal polarization of the dark photon is Because SPDC can be an important background to a 2 suppressed by mA0 /ω rather than (mA0 /ω) [17, 28, dSPDC axion search, using a linear or nearly linear 29, 38]. medium, where SPDC is absent is desirable. Inter- This motivates either non traditional orientation of estingly, materials with a crystalline structure that nonlinear crystals, or identifying nonlinear materials r → −r is invariant under a mirror transformation that would usually be ineffective for SPDC. As an χ(2) will have a vanishing from symmetry considera- example in the first category, any material which is tions [37]. uses for type II phase matching, in which the sig- nal and idler have orthogonal polarization, can be ro- C. Longitudinal Susceptibility for Dark Photon tated by 90 degrees to achieve a coupling of a longi- Searches tudinal polarization to a pump and an idler. In the second category, material with χ(2) tensors that are non-vanishing only in “maximally non-diaginal” xˆ-yˆ- As opposed to the axion case, a dSPDC process with zˆ elements, would obviously be discarded as standard a dark photon requires a nonlinear optical medium. SPDC sources, but can have an enhanced dSPDC cou- As a result, SPDC can be a background to dSPDC pling. dark 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- V. DARK SPDC SETUPS AND RATES gation. As a consequence, when one wants to enhance an SPDC process, the nonlinear medium is oriented in A precise estimate of the rate is beyond the scope of such a way that the second order susceptibility tensor this work, and will be discussed in more detail in [17]. χ(2) can appropriately couple to the transverse polar- Here we will re-scale known rate formulae in order to ization vectors of the pump, signal, and idler photons. examine the dependence of the rate on the geomet- The effective coupling between the modes in question ric factors such as the pump power and beam area, is given by the crystal length, and the area and angle from which (2) (2) p s i signal is collected. χSPDC ≡ χjklεj εkεl (33) Consider the geometry sketched in Figure 8. A with the ε being the (transverse) polarization vectors pump beam is incident on an optical element of for the pump, signal and idler photons. These are length L along the the pump direction. The width spanned by (1, 0, 0) and (0, 1, 0) in a frame in which of the pump beam is set by the laser parameters. The the respective photon is propagating in the zˆ direction. pump beam may consist of multiple modes, or can In dSPDC with a dark photon there is an important be guided in a fiber or waveguide [16] and in a single difference. Because the dark photon is massive, it can mode. The width and angle of the signal “beam” is set 10 by the apparatus used to collect and detect the signal. A. Dark Photon dSPDC Rate This too, can include collection of a single mode in a fiber (e.g. [19, 39]) or in multiple modes (as in the The dSPDC rate into a dark photon with longitu- CCD example in the next section. In SPDC, when 0 dinal polarization, AL, can similarly be written as a the idler photon may be collected the width and an- simple re-scaling of the expression above gle of the idler can also be set by a similar apparatus. However, in dSPDC (or in SPDC if we choose to only (2) 2 2 P χ 0 ω ω 0 L 0 m 0 p A s A collect the signal) the ϕ beam is not a parameter in (AL) 2 A L ΓdSPDC ∼ 2 (36) the problem. In this case we are interested in an in- ωA0 npnsAeff clusive rate, and would sum over a complete basis of (2) idler beams. Such a sum will be performed in [17], but where the effective coupling χ 0 is defined in Equa- AL usually the sum will be dominated by a set of modes tion (34). This is valid in regions where the dark pho- that are similar to those collected for the signal. ton mass is smaller than the pump frequency, such Generally, the signal rate will depend on all of the that the produced dark photons are relativistic and choices made above, but we can make some qualitative have a refractive index of 1. Using the optimistic observations. The rate for SPDC and dSPDC will be waveguide numbers above as a placeholder, assuming proportional to an integral over the volume defined an optimized setup with similar χ(2), the number of by the overlap of the three beams in Figure 8 and the events expected after integrating over a time tint are crystal. When the length of the volume is set by the crystal, the process is said to occur in the “thin crystal 2 0 0 (AL) 21 2 mA Pp L tint limit”. In this limit, the beam overlap is roughly a Nevents ∼ 10 2 . ω 0 Watt m year constant over the crystal length and thus the total rate A (37) will grow with L. Since dSPDC is a rare process, we The current strongest lab-based limit is for dark pho- observe that taking the collinear limit of the process, ton masses of order 0.1 eV is set by the ALPS exper- together with a thicker crystal may allow for a larger iment at ∼ 3 × 10−7. For this dark photon mass integration volume and an enhanced rate. mA0 ∼ 0.1ωp, the mA0 /ωA0 term does not represent The total rate for SPDC in a particular angle, in- a large suppression. In this case of order 10 dSPDC tegrated over frequencies, in the thin crystal limit events are produced in a day in a 1 cm crystal with is [17, 19], a Watt of pump power with the assumptions above. (2)2 This implies that a relatively small dSPDC experi- Ppχeff ωsωiL ΓSPDC ∼ (35) ment with an aggressive control on backgrounds could πn n n A p s i eff be used to push the current limits on dark photons. where L is the crystal length, Pp is the pump power, Improving the limits from solar cooling, for which (2) mA0 /ωA0 is constrained to be smaller, would repre- Aeff is the effective beam area, and χeff is the effective coupling of Equation (33). A parametrically similar sent an interesting challenge. Achieving ten events in rate formula applies to SPDC in waveguides [16, 17]. a year of running requires a Watt of power in a waveg- The pump power here is the effective power, which uide greater than 10 meters (or a shorter waveguide may be enhanced within a high finesse optical cavity, with higher stored power, perhaps using a Fabri-Perot e.g. [26]. The inverse dependence with effective area setup). Interestingly, in terms of system size, this can be understood by since the interaction hamilto- is still smaller than the ALPS-II experiment which nian is proportional to electric fields, which grow for would reach 100 meters in length and an effecting fixed power for a tighter spot. For “bulk crystal” se- power of a hundred kW. This is because LSW se- tups a pair production rate of order few times 106 per tups require both production and detection, with lim- 4 mW of pump power per second is achievable [39] in the its scaling as . forward direction. In waveguided setups, in which the beams remain confined along a length of the order of a cm, production rates of order 109 pairs per mW per B. Axion dSPDC Rate second are achievable in KTP crystals [16, 40], and 10 rates of order 10 were discussed for LN crystals [40]. A similar rate expression can be obtained for axion- Also noteworthy are cavity enhanced setups in which dSPDC the effective pump power is increased by placing the 2 device in an optical cavity, and/or OPOs, in which the (axion) Ppgaγγ ωsL ΓdSPDC ∼ (38) signal photon is also a resonant mode, including com- ωaxionnpnsAeff pact devices with high conversion efficiencies, e.g [41– 44]. where the different scaling with the frequency of the An important scaling of this rate is the L/Aeff de- dark field is due to the different structure of the pendence. This scaling applies for dSPDC rates dis- dSPDC interaction (recall that χ(2) carries a mass cussed below. Within the thin crystal limit one can dimension of −2 while the axion photon coupling’s thus achieve higher rates with: (a) a smaller beam dimension is −1). Optimal SPDC (dSPDC) rates are spot, and (b) a thicker crystal. It should be noted acieved in waveguides in which the effective area is of that for colinear SPDC, the crystal may be in the order the squared wavelength of the pump and signal “thin limit” even for thick crystals (see Figure 8 and light. Assuming a (linear) birefringent material that imagine zero signal and idler emission angle). can achieve dSPDC phase matching for an axion the 11
Dark Photon (mA0 = 0.1 eV) Axion-like particle (ma = 0.1 eV) −7 −6 −1 Current lab limit < 3 × 10 gaγ < 10 GeV Example dSPDC setup Pp = 1 W Pp = 1 kW L = 1 cm L = 10 m Γ = 10/day Γ = 10/day −10 −10 −1 Current Solar limit < 10 gaγ < 10 GeV Example dSPDC setup Pp = 1 W Pp = 100 kW L = 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 a benchmark mass of 0.1 eV. For each limit we show the parameters of an example dSPDC in a waveguide and the rate it would produce for couplings that would produce the specified benchmark rate with the corresponding coupling. The pump power is an effective power which can include an enhancement by an optical cavity setup. For dark photon rates we assume a nonlinearity of the same order found in KTP crystals.
number of signal event scales as without enhancing SPDC. This technique to re- duce SPDC may also be used in axion searches. 2 (axion) gaγ Pp L tint Nevents ∼ 40 . 10−6 GeV−1 Watt m year • Detector noise and optical impurities: Sources (39) of background which may be a limiting factor This rate suggests a dSPDC setup is promising in set- for dSPDC searches include detector noise, as ting new lab-based limits on ALPs. For example, a well as scattering of pump photons off of impu- 10 meter birefringent single mode fiber with kW of rities in the optical elements and surfaces. The pump power will generate of order 10 events per day Skipper CCD, one example of a detector tech- for couplings of order 10−7 GeV−1. To probe beyond nology with low noise, will be discussed in the the CAST limits in gaγ may be possible and requires next section. a larger setup, but not exceeding the scale, say, of ALPS-II. In a 100 meter length and an effective pump An optimal dSPDC based search for dark particles power of 100 kW, a few dSPDC signal events are ex- will likely consider these factors, and is left for future pected in a year. investigation. Maintaining a low background, would of course be crutial. We note, however, that optical fibers are rou- tinely used over much greater distances, maintaining VI. EXPERIMENTAL PROOF OF CONCEPT coherence (e.g. [5]), and an optimal setup should be identified. In this section we present an experimental SPDC angular imaging measurement with high resolution C. Backgrounds to dSPDC employing 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- There are several factors that should be considered sion angle. Instead, this experiment serves as a proof for the purpose of reducing backgrounds to SPDC: of concept for the high resolution imaging technique. • Crystal Length and Signal bandwidth: In addi- Imaging the dSPDC requires the detection of single tion to the growth of the signal rate, the signal photons with low noise and with high spatial and/or bandwidth in many setups will decrease with L. energy resolution. A technology that can achieve this If this is achieved the signal to background ra- is the Skipper CCD which is capable of measuring tio in a narrow band around the dSPDC phase the charge stored in each pixel with single electron matching solutions will scale as L2. resolution [18], ranging from very few electrons (0, 1, 2, ... ) up to more than a thousand (1000, 1001, • Timing: The dSPDC signal consists of a single 1002, ... ) [45]. This unique feature combined with photon whereas SPDC backgrounds will consist the high spatial resolution typical of a CCD detector of two coincident photons. Backgrounds can be makes this technology very promising for the detec- reduced using fast detectors and rejecting coin- tion of small optical signals with a very high spatial cidence events. resolution. • Optical material: As pointed out in Section IV, linear birefringent materials can be used to reduce SPDC backgrounds to axion searches. A. Description of the experimental setup Nonlinear materials with a χ(2) tensor which couples purely to longitudinal polarizations may With the aim of comparing the developed phase be used to enhance dark photon dSPDC events space model against real data, the system depicted 12
LL01-810 FGB37 + 405 nm 97SA 810 nm SPDC photons pump photons BBO SPDC
810 nm 810 nm Focusing Skipper reject filter pass filters lens CCD
Figure 9: Pictorial representation of the experimental setup implemented in the lab. A 405 nm diode laser was used as a source of pump photons. The beam passes pump through a 810 nm rejection filter, then through the BBO crystal, after this through two 810 nm band pass filters, a focusing lens and finally the SPDC photons reach the CCD detector. Figure 10: Picture of the source of entangled photons which is a part of a commercial system. The FGB37 filter shown in Figure 9 was placed just after the laser in Figure 9 was set up. A source of entangled pho- aperture. All the other components were placed outside tons that employs SPDC, which is part of a com- this device. Figure adapted from [46]. mercial system3, was used. Two type I nonlinear BBO crystals are used as a high efficient source of entangled photons [31]. This pre-assembled source B. Results can be seen in Figure 10. However, in this work, we did not take advantage of the polarization entan- Using the setup previously described, 400 images glement, we only use the energy-momentum conser- with 200 s exposure each were averaged to produce vation to get spatially correlated twin photons. In the data used for comparison with the model. Thus, addition to the original design, we added a Thorlabs we reduced a factor twenty the uncertainty in the ex- FGB37 filter after the laser to remove a small 810 nm pected number of photons in each pixel and drastically component coming along with the pump beam. Af- reduced the dark counts coming from random back- ter the BBO crystal, where the SPDC process occurs, ground. This significantly improved the identification two additional 810 nm band pass filters (one Semrock of minima between rings, which results to be crucial Brighline LL01-810 and one Asahi Spectra 97SA) were to compare the experimental data with the presented placed to prevent the 405 nm pump beam from reach- theory. ing the detector. We also placed a lens at its focal Figure 11 presents the averaged image of the SPDC distance from the BBO to prevent the SPDC cone ring coming out of the BBO crystal. The main ring from spreading during its travel to the CCD (see Fig- is clearly visible, as well as many secondary maxima ure 9). Thus, we got the SPDC ring projected on the and minima. The angular coordinates θ and φ were CCD surface with a radius of 110 pixel, which corre- indicated on the image. It has to be noted that the sponds to 1650 µm. This setup is also part of a re- angular coordinates used in the previous sections are search where the novel features of Skipper CCD [18] the ones inside the optical medium. Since the CCD are being tested for Quantum Imaging. The capa- detector is outside the BBO crystal, Snell’s law must bility of Skipper-CCD to reduce the readout noise as be used to relate the angles inside and outside the low as desired taking several samples of the charge in nonlinear crystal. Specifically, for the θ coordinate each pixel was not used in this work to acquire data, this is but it was for the calibration. Details about the same sin Θ detector used here can be found in reference [45]. θ = arcsin n We used an absolute calibration between the Ana- log to Digital Units (ADU) measured by the amplifier where θ is the angle inside the optical medium in from the CCD and the number of electrons in each which the SPDC process occurs, Θ is the angle out- pixel, possible with the Skipper-CCD, which was pre- side the optical medium and n is the refractive index. viously carried out for this system [45]. Thus, we re- In this work we apply the transformation to refer all constructed the number of photons per pixel, since at angles to those inside the optical medium in which the 810 nm it can be assumed that one photon creates at interaction occurs. most one electron, by reason of its energy is 0.43 eV As can be seen in the image, even though it is pos- above the silicon band gap (1.1 eV). The factor affect- sible to see many of the maxima and minima, there ing that relationship is the efficiency which -besides is a non uniform background component produced by being very high (∼ 90 %) at this wavelength- is pretty reflections and other imperfections of the experimen- uniform over the entire CCD surface [47]. tal configuration. The current implementation of our setup has some mechanical limitations that make hard to isolate, measure and remove this non-uniform back- ground component. Still, this image can be used to compare the theory with the experiment, so the profile of the intensity distribution along the blue dashed line 3 https://www.qutools.com/qued/ shown in Figure 11 was extracted and plotted against 13
• the refractive indices for the BBO crystal were taken from https://refractiveindex.info/ 4 • 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 (since signal and idler have the same frequency and angle then ns = ni). The model is very sensitive to this difference, and this quantity depends on factors such as the temperature of the BBO and its precise orien- tation in space, which were not measured. So it is not surprising that we had to perform this fine tun- ing on ns −np. The final values used for the refractive indices were np ≈ 1.66082 and ns = ni ≈ 1.66107. Al- Figure 11: Average of 400 images with 200 s exposure though this fine tuning had to be done, all parameters each to the SPDC photons each using the setup depicted are very well within expected values. in Figure 9. The angular coordinates θ and φ are indicated in the figure. The blue dashed line was used to As seen both in Figures 11 and 12 the phase space extract the data profile shown in Figure 12. factor, studied in previous sections, is the dominant modulation in the distribution of photons for the SPDC process in our setup. Even though there is
3 a non negligible background component, it is evident 10 2 2 Model the dependence I ∼ sinc θ for small values of θ Data predicted by Equation (27).
102 VII. DISCUSSION
Number of electrons We have presented a new method to search for new light and feebly coupled particles, such as axion-like 0 1 2 3 particles and dark photons. The dSPDC process al- θ (deg) lows to tag the production of a dark state as a pump photon down-converts to a signal photon and a “dark Figure 12: Experimental intensity profile for the SPDC idler” ϕ, in close analogy to SPDC. We have shown ring along the blue dashed line in Figure 11 compared that the presence of indices of refraction that are dif- with theoretical distribution. The y axis scale is the ferent than 1 open the phase space for the decay, or number of electrons in each pixel of the CCD detector down conversion, of the massless photon to the signal obtained via the absolute Skipper-CCD calibration plus the dark particle, even if ϕ has a mass. This type described. of search has a parametric advantage over light shining through wall setups since it only requires producing the axion or dark photon, without a need to detect it the model. This plot is shown in Figure 12. Some again. Precise sensitivity calculations for dSPDC with comments about this plot: Dark Photon and Axion cases that can be achieved • A uniform background component was added to through this method are ongoing [17]. the model. The commonplace use of optics in telecommunica- tions, imaging and in quantum information science, • The transmittance curve of the 810 nm filters as well as the development of advanced detectors, provided by the manufacturers was taken into can thus be harnessed to search for dark sector parti- account. cles. Increasing dSPDC signal rates will require high laser power, long optical elements. Enhancing the sig- So, to be specific, the plot in Figure 12 uses as model nal and suppressing SPDC backgrounds also requires the intensity profile given by
d2Γˆ Imodel (θ) = I0 F (αω)dαω + I1 ˆ d θ dα (cos s) ω 4 The extraordinary refractive index was taken from https: //refractiveindex.info/?shelf=main&book=BaB2O4&page= d2Γˆ where I0 and I1 are real numbers fitted, Tamosauskas-e (Tamošauskas et al., 2018) and the ordi- d(cos θs)dαω nary refractive index from https://refractiveindex.info/ is given by Equation (27) and F (αω) is the transmit- ?shelf=main&book=BaB2O4&page=Eimerl-o (Eimerl et al., tance of the filters as provided by the manufacturers, 1987). which is a sharp peaked function centered at 810 nm 5 It was not possible to obtain precise information of the dimen- with a pass band of 10 nm width. sions of the crystal from the manufacturer. Furthermore, the For the parameters of the distribution Γˆ, we used: length of the crystal is not easy to measure with our current implementation without breaking the SPDC source shown in • 405 nm for the pump wavelength as provided by Figure 10. So we decided to use a “reasonable value” and fine the manufacturer of the laser, tune it. 14 identifying the right optical media for the search. Ax- We also performed an experimental demonstration ion searches would benefit from optically linear and of the one of the setups discussed using a Skipper birefringent materials, with greater birefringence al- CCD for SPDC imaging. The setup we used for this lowing to search for higher axion masses. Searches demonstration was adapted from one designed to en- for dark photons would benefit from strongly nonlin- force SPDC and thus does not open dSPDC phase ear materials that are capable of coupling to a lon- space. We show that the Skipper technology allows gitudinal polarization. This, in turn, motivated ei- one to measure with high accuracy. Thus, quantum ther non-conventional optical media, or using conven- teleportation methods or ghost imaging of dark sector tional crystals that are oriented by a 90◦ rotation from particles achieved through phase space engineering is that which is desired for SPDC. Enhancing the effec- a very promising technique for future explorations of tive pump power with an optical cavity is a straight- dark sector parameter spaces. forward way to enhance the dSPDC rate. We leave the exploration of a “doubly resonant” OPO setup in which the signal is also a cavity eigenmode (in parallel Acknowledgments: We would like to thank Joe with [26]) for future work. Finally, detection of rare Chapman, Paul Kwiat, and Neal Sinclair for infor- signal events requires sensitive single photon detectors mative discussions. This work was funded by a DOE with high spatial and/or frequency resolution. QuantISED grant.
[1] A. Furusawa. Unconditional Quantum Teleportation. [12] Yang Bai, Patrick J. Fox, and Roni Harnik. The Teva- Science, 282(5389):706–709, October 1998. tron at the Frontier of Dark Matter Direct Detection. [2] D. Boschi, S. Branca, F. De Martini, L. Hardy, and JHEP, 12:048, 2010. S. Popescu. Experimental Realization of Teleporting [13] Jessica Goodman, Masahiro Ibe, Arvind Rajaraman, an Unknown Pure Quantum State via Dual Classi- William Shepherd, Tim M.P. Tait, and Hai-Bo Yu. cal and Einstein-Podolsky-Rosen Channels. Physical Constraints on Dark Matter from Colliders. Phys. Review Letters, 80(6):1121–1125, February 1998. Rev. D, 82:116010, 2010. [3] Dik Bouwmeester, Jian-Wei Pan, Klaus Mat- [14] Patrick J. Fox, Roni Harnik, Joachim Kopp, and tle, Manfred Eibl, Harald Weinfurter, and Anton Yuhsin Tsai. Missing Energy Signatures of Dark Mat- Zeilinger. Experimental quantum teleportation. Na- ter at the LHC. Phys. Rev. D, 85:056011, 2012. ture, 390(6660):575–579, December 1997. [15] T. Aaltonen et al. A Search for dark matter in events [4] Charles H. Bennett, Gilles Brassard, Claude with one√ jet and missing transverse energy in pp¯ colli- Crépeau, Richard Jozsa, Asher Peres, and sions at s = 1.96 TeV. Phys. Rev. Lett., 108:211804, William K. Wootters. Teleporting an unknown 2012. quantum state via dual classical and Einstein- [16] Marco Fiorentino, Sean M. Spillane, Raymond G. Podolsky-Rosen channels. Physical Review Letters, Beausoleil, Tony D. Roberts, Philip Battle, and 70(13):1895–1899, March 1993. Mark W. Munro. Spontaneous parametric down- [5] Raju Valivarthi, Samantha Davis, Cristian Pena, conversion in periodically poled KTP waveguides and Si Xie, Nikolai Lauk, Lautaro Narvaez, Jason P. All- bulk crystals. Optics Express, 15(12):7479–7488, June maras, Andrew D. Beyer, Yewon Gim, Meraj Hus- 2007. sein, George Iskander, Hyunseong Linus Kim, Boris [17] Juan Estrada, Roni Harnik, Dario Rodrigues, and Korzh, Andrew Mueller, Mandy Rominsky, Matthew Matias Senger. New Particle Searches with Nonlin- Shaw, Dawn Tang, Emma E. Wollman, Christoph ear Optics. In preparation. Simon, Panagiotis Spentzouris, Neil Sinclair, Daniel [18] Javier Tiffenberg, Miguel Sofo-Haro, Alex Drlica- Oblak, and Maria Spiropulu. Teleportation Systems Wagner, Rouven Essig, Yann Guardincerri, Steve Hol- Towards a Quantum Internet. arXiv e-prints, page land, Tomer Volansky, and Tien-Tien Yu. Single- arXiv:2007.11157, July 2020. electron and single-photon sensitivity with a sil- [6] Interaction. icon Skipper CCD. Physical Review Letters, [7] C. K. Hong and L. Mandel. Theory of parametric 119(13):131802, September 2017. arXiv: 1706.00028. frequency down conversion of light. Physical Review [19] Alexander Ling, Antia Lamas-Linares, and Christian A, 31(4):2409?2418, 1985. Kurtsiefer. Absolute emission rates of Spontaneous [8] J. A. Giordmaine and Robert C. Miller. Tunable Co- Parametric Down Conversion into single transverse herent Parametric Oscillation in LiNb O 3 at Optical Gaussian modes. Physical Review A, 77(4):043834, Frequencies. Physical Review Letters, 14(24):973–976, April 2008. arXiv: 0801.2220. June 1965. [20] James Schneeloch and John C Howell. Introduc- [9] R. Essig and et al. Dark Sectors and New, Light, tion to the transverse spatial correlations in sponta- Weakly-Coupled Particles. arXiv:1311.0029 [astro- neous parametric down-conversion through the bipho- ph, physics:hep-ex, physics:hep-ph], October 2013. ton birth zone. Journal of Optics, 18(5):053501, 2016. arXiv: 1311.0029. [21] P.A. Zyla et al. Review of Particle Physics. PTEP, [10] F. Hoogeveen and T. Ziegenhagen. Production and 2020(8):083C01, 2020. detection of light bosons using optical resonators. [22] Georg G. Raffelt. Astrophysical methods to constrain Nucl. Phys. B, 358:3–26, 1991. axions and other novel particle phenomena. Phys. [11] Jessica Goodman, Masahiro Ibe, Arvind Rajaraman, Rept., 198:1–113, 1990. William Shepherd, Tim M.P. Tait, and Hai-Bo Yu. [23] V. Anastassopoulos et al. New CAST Limit on the Constraints on Light Majorana dark Matter from Col- Axion-Photon Interaction. Nature Phys., 13:584–590, liders. Phys. Lett. B, 695:185–188, 2011. 2017. 15
[24] Federico Della Valle, Aldo Ejlli, Ugo Gastaldi, Amsterdam ; Boston, 3rd ed edition, 2008. Giuseppe Messineo, Edoardo Milotti, Ruggero Pengo, [38] Peter W. Graham, Jeremy Mardon, Surjeet Rajen- Giuseppe Ruoso, and Guido Zavattini. The PVLAS dran, and Yue Zhao. Parametrically enhanced hidden experiment: measuring vacuum magnetic birefrin- photon search. Phys. Rev. D, 90(7):075017, 2014. gence and dichroism with a birefringent Fabry–Perot [39] Ryan S. Bennink. Optimal Co-linear Gaussian Beams cavity. Eur. Phys. J. C, 76(1):24, 2016. for Spontaneous Parametric Down-Conversion. Phys- [25] R. Ballou et al. New exclusion limits on scalar and ical Review A, 81(5):053805, May 2010. arXiv: pseudoscalar axionlike particles from light shining 1003.3810. through a wall. Phys. Rev. D, 92(9):092002, 2015. [40] Sean M. Spillane, Marco Fiorentino, and Raymond G. [26] Klaus Ehret et al. Resonant laser power build-up Beausoleil. Spontaneous parametric down conver- in ALPS: A ’Light-shining-through-walls’ experiment. sion in a nanophotonic waveguide. Optics Express, Nucl. Instrum. Meth. A, 612:83–96, 2009. 15(14):8770–8780, July 2007. [27] Robin BÀhre et al. Any light particle search II — [41] J. U. FÃŒrst, D. V. Strekalov, D. Elser, A. Aiello, Technical Design Report. JINST, 8:T09001, 2013. U. L. Andersen, Ch. Marquardt, and G. Leuchs. Low- [28] Haipeng An, Maxim Pospelov, and Josef Pradler. Threshold Optical Parametric Oscillations in a Whis- New stellar constraints on dark photons. Phys. Lett. pering Gallery Mode Resonator. Physical Review Let- B, 725:190–195, 2013. ters, 105(26):263904, December 2010. [29] Haipeng An, Maxim Pospelov, and Josef Pradler. [42] T. Beckmann, H. Linnenbank, H. Steigerwald, Dark Matter Detectors as Dark Photon Helioscopes. B. Sturman, D. Haertle, K. Buse, and I. Breunig. Phys. Rev. Lett., 111:041302, 2013. Highly Tunable Low-Threshold Optical Parametric [30] David Tong. Quantum Field Theory. Oscillation in Radially Poled Whispering Gallery Res- http://www.damtp.cam.ac.uk/user/tong/qft.html. onators. Physical Review Letters, 106(14):143903, [31] Paul G. Kwiat, Edo Waks, Andrew G. White, Ian April 2011. Appelbaum, and Philippe H. Eberhard. Ultra-bright [43] Christoph S. Werner, Karsten Buse, and Ingo Bre- source of polarization-entangled photons. Physical unig. Continuous-wave whispering-gallery optical Review A, 60(2):R773–R776, August 1999. arXiv: parametric oscillator for high-resolution spectroscopy. quant-ph/9810003. Optics Letters, 40(5):772, March 2015. [32] Christophe Couteau. Spontaneous parametric down- [44] Juanjuan Lu, Ayed Al Sayem, Zheng Gong, Joshua B. conversion. Contemporary Physics, 59(3):291–304, Surya, Chang-Ling Zou, and Hong X. Tang. Ultralow- July 2018. threshold thin-film lithium niobate optical parametric [33] N. Boeuf. Calculating characteristics of noncollinear oscillator, 2021. phase matching in uniaxial and biaxial crystals. Op- [45] Dario Rodrigues, Kevin Andersson, Mariano Caba- tical Engineering, 39(4):1016, April 2000. bie, Andre Donadon, Gustavo Cancelo, Juan Estrada, [34] Paul-Antoine Moreau, Ermes Toninelli, Thomas Gre- Guillermo Fernandez-Moroni, Ricardo Piegaia, Ma- gory, and Miles J. Padgett. Imaging with quantum tias Senger, Miguel Sofo Haro, et al. Absolute mea- states of light. Nature Reviews Physics, 1(6):367–380, surement of the fano factor using a skipper-ccd. arXiv June 2019. preprint arXiv:2004.11499, 2020. [35] S. Magnitskiy, D. Frolovtsev, V. Firsov, P. Gostev, [46] qutools. quED. I. Protsenko, and M. Saygin. A SPDC-Based [47] C.J. Bebek, J.H. Emes, D.E. Groom, S. Haque, S.E. Source of Entangled Photons and its Characteriza- Holland, P.N. Jelinsky, A. Karcher, W.F. Kolbe, J.S. tion. Journal of Russian Laser Research, 36(6):618– Lee, N.P. Palaio, D.J. Schlegel, G. Wang, R. Groulx, 629, November 2015. R. Frost, J. Estrada, and M. Bonati. Status of [36] J. Noda, K. Okamoto, and Y. Sasaki. Polarization- the CCD development for the dark energy spec- maintaining fibers and their applications. Journal of troscopic instrument. Journal of Instrumentation, Lightwave Technology, 4(8):1071–1089, 1986. 12(04):C04018–C04018, apr 2017. [37] Robert W. Boyd. Nonlinear optics. Academic Press,