# Beltrami States in Two-dimensional Electron Magnetohydrodynamics

B. K. Shivamoggi
University of Central Florida
Orlando, FL 32816-1364, U.S.A.

Abstract

In this paper, the Hamiltonian formulations along with the Poisson brackets
for two-dimensional (2D) electron magnetohydrodynamics (EMHD) flows are
developed. These formulations are used to deduce the Beltrami states for 2D
EMHD flows. In the massless electron limit, the EMHD Beltrami states reduce to
the force-free states, though there is no force-free Beltrami state in the
general EMHD case.

1\. Introduction

In the magnetohydrodynamics (MHD) model, the dynamics is dominated by ions
with electrons serving to shield out rapidly any charge imbalances. In
electron MHD (EMHD), with $\ell\ll\rho_{s_{i}},~{}s=i,e,\rho_{s}$ being the
gyro-radius, on the other hand, the dynamics is dominated by electrons with
the demagnetized ions serving to provide the neutralizing static background
(Kingsep et al. [1], Gordeev et al. [2]). The assumptions underlying the EMHD
model are $\ell\ll d_{i}$, where $d_{s}\equiv c/\omega_{p_{s}}$ is the skin
depth, and that the frequencies involved are greater than $\omega_{c_{i}}$ and
$\omega_{p_{i}}$, $\omega_{c}$ being the cyclotron frequency. Some theoretical
developments including two invariants for the 2D EMHD equations were given by
Biskamp et al. [3] .

On the other hand, equations governing several plasma dynamics models are
known to admit a significant class of exact solutions (Shivamoggi [4]) under
the Beltrami condition - the local current density is proportional to the
magnetic field - the force-free state (Lundquist [5], Lust and Schluter [6]).
These Beltrami solutions are known to correlate well with real plasma behavior
(Priest and Forbes [7], Schindler [8]). The purpose of this paper is to
develop the Hamiltonian formulation along with the Poisson bracket for two-
dimensional (2D) EMHD flows. This formulation is used to deduce the Beltrami
states for 2D EMHD flows. These considerations are extended to the massless-
electron limit (Huba and Rudakov [9], Rheinhardt and Geppert [10], Wood et al.
[11]).

2\. Governing Equations of 2D EMHD

On assuming that the displacement current $\partial{\bf{E}}/\partial t$ is
negligible, which is valid if
$\omega\ll\omega_{p_{e}}^{2}/\omega_{ce},$ the continuty of the electron flow
implies

$\frac{\partial n_{e}}{\partial
t}=-\nabla\cdot\left(n_{e}\bf{v}_{e}\right)=\nabla\cdot\left(\frac{\bf{J}}{e}\right)=\frac{c}{e}~{}\nabla\cdot\left(\nabla\times\bf{B}\right)=0$

or

$n_{e}=const.$ (1)

The 2D EMHD system of equations can then be written in terms of two scalar
potentials (Biskamp et al. [3]) - the magnetic flux function $A$ describing
the in-plane magnetic field ${\bf B}=\nabla\times A\ {\bf\hat{i}}_{z}$ and the
stream function $\psi$ describing the in-plane solenoidal electron flow
velocity ${\bf v}_{e}=\nabla\times\psi\ {\bf\hat{i}}_{z}$, which is
proportional to the in-plane current density (so $\psi$ also represents the
out-of-plane magnetic field):
* the equation of generalized vorticity:

$\frac{\partial}{\partial
t}\left(\omega+\frac{1}{d_{e}^{2}}\psi\right)+\left({\bf
v}_{e}\cdot\nabla\right)\omega-\frac{1}{m_{e}n_{e}c}\left({\bf
B}\cdot\nabla\right)J=0$ (2)

* the generalized Ohm’s law:

$\frac{\partial}{\partial t}\left(A+\frac{d_{e}^{2}}{c}J\right)+\left({\bf
v}_{e}\cdot\nabla\right)\left(A+\frac{d_{e}^{2}}{c}J\right)=0$ (3)

where,

$\frac{1}{c}J=-\nabla^{2}A~{},~{}\omega=-\nabla^{2}\psi.$ (4)

Equations (1) and (2) may be rewritten as

$\frac{\partial\psi_{e}}{\partial
t}=\left[\psi,\psi_{e}\right]+\frac{1}{c}\left[J,A_{e}\right]$ (5)

$\frac{\partial A_{e}}{\partial t}=\left[\psi,A_{e}\right]$ (6)

where,

$\left.\begin{aligned} \psi_{e}&\equiv
m_{e}n_{e}\left(\psi/d_{e}^{2}+\omega\right)\\\ A_{e}&\equiv
A+\left(d_{e}^{2}/c^{2}\right)J\\\ \left[A,B\right]&\equiv\nabla A\times\nabla
B\cdot{\bf\hat{i}}_{z}.\end{aligned}\right\\}$ (7)

In the linear limit, on assuming a constant magnetic field ${\bf B}_{0}$ in
the $x$-direction and a space-time dependence $e^{i\left(kx-\omega t\right)}$
for the perturbed quantities, equations (5) and (6) give the whistler waves
($\psi_{k}=\pm kA_{k}$ \- the whistler condition),

$\omega\approx k^{2}d_{e}V_{A_{e}}$ (8)

which are dispersive unlike the Alfvén waves. Here, $V_{A_{e}}\equiv
B_{0}/\sqrt{m_{e}n_{e}}$.

3\. The Hamiltonian Formulation

Equations (5) and (6) have the Hamiltonian formulation (Biskamp et al. [3]),

$H=\frac{1}{2}\int\limits_{\mathscr{A}}\left(\psi\psi_{e}+\frac{1}{c}A_{e}J\right)d\mathscr{A}$
(9)

$\mathscr{A}$ being the area occupied by the plasma.

The Hamilton’s equations are, on taking $(\psi_{e},A_{e})$ to be the canonical
variables, given by

$\left(\begin{matrix}\displaystyle\frac{\partial\psi_{e}}{\partial t}\\\ \\\
\displaystyle\frac{\partial A_{e}}{\partial
t}\end{matrix}\right)=\mathscr{J}\left(\begin{matrix}\displaystyle\frac{\delta
H}{\delta\psi_{e}}\\\ \\\ \displaystyle\frac{\delta H}{\delta
A_{e}}\end{matrix}\right)$ (10)

where $\mathscr{J}$ is a $\left(\psi_{e},A_{e}\right)$-dependent differential
operator which produces a skew-symmetric transformation of vector functions
vanishing on $\partial\mathscr{A}$ and satisfies a closure condition on an
associated symplectic two-form,

$\mathscr{J}\equiv\left(\begin{matrix}\left[\cdot,\psi_{e}\right]&\left[\cdot,A_{e}\right]\\\
\\\ \left[\cdot,A_{e}\right]&0\end{matrix}\right)$ (11)

and $\delta H/\delta q$ is the variational derivative. Equation (10) is just
equations (5) and (6).

The operator $\mathscr{J}$ induces a Poisson bracket as follows,

$\displaystyle\left[F,G\right]$ $\displaystyle\equiv\left(\left\\{\frac{\delta
F}{\delta\psi_{e}},\frac{\delta F}{\delta
A_{e}}\right\\},~{}\mathscr{J}\left\\{\frac{\delta
G}{\delta\psi_{e}},\frac{\delta G}{\delta A_{e}}\right\\}^{T}\right)$ (12)
$\displaystyle=\int\limits_{\mathscr{A}}\left\\{\frac{\delta
F}{\delta\psi_{e}},\frac{\delta F}{\delta
A_{e}}\right\\}\left(\begin{matrix}\left[\displaystyle\frac{\delta
G}{\delta\psi_{e}},\psi_{e}\right]+\left[\displaystyle\frac{\delta G}{\delta
A_{e}},A_{e}\right]\\\ \left[\displaystyle\frac{\delta
G}{\delta\psi_{e}},A_{e}\right]\end{matrix}\right)d\mathscr{A}$
$\displaystyle=\int\limits_{\mathscr{A}}\left\\{\psi_{e}\left[\frac{\delta
F}{\delta\psi_{e}},\frac{\delta
G}{\delta\psi_{e}}\right]+A_{e}\left(\left[\frac{\delta
F}{\delta\psi_{e}},\frac{\delta G}{\delta A_{e}}\right]-\left[\frac{\delta
G}{\delta\psi_{e}},\frac{\delta F}{\delta
A_{e}}\right]\right)\right\\}d\mathscr{A}$

which is a bilinear function defined on admissible functionals
$F\left[\psi_{e},A_{e}\right]$ and $G\left[\psi_{e},A_{e}\right]$ satisfying
the gauge condition,

$\nabla\cdot\left(\begin{matrix}\displaystyle\frac{\delta
F}{\delta\psi_{e}}\\\ \\\ \displaystyle\frac{\delta F}{\delta
A_{e}}\end{matrix}\right)=0,~{}\nabla\cdot\left(\begin{matrix}\displaystyle\frac{\delta
G}{\delta\psi_{e}}\\\ \\\ \displaystyle\frac{\delta G}{\delta
A_{e}}\end{matrix}\right)=0~{}\text{in}~{}\mathscr{A}$ (13a)

and the boundary conditions,

$\left\\{\left|\frac{\delta F}{\delta\psi_{e}}\right|,\left|\frac{\delta
F}{\delta A_{e}}\right|\right\\},\left\\{\left|\frac{\delta
G}{\delta\psi_{e}}\right|,\left|\frac{\delta G}{\delta
A_{e}}\right|\right\\}={\bf 0}~{}\text{on}~{}\partial\mathscr{A}.$ (13b)

The Poisson bracket (12) satisfies the anti-symmetry property,

$\left[F,G\right]=-\left[G,F\right]$ (14a)

and the Jacobi identity,

$\left[\left[F,G\right],K\right]+\left[\left[G,K\right],F\right]+\left[\left[K,F\right],G\right]=0.$
(14b)

In the absence of elecron inertia ($d_{e}\to 0$), (12) reduces to the MHD
result given by Holm et al. [12].

4\. Beltrami States

The Casimir invariants for this problem are annihilators (with respect to any
pairing functionals) of the Poisson brackets, which become degenerate when
expressed in terms of these natural quantities. The Casimir invariants are
therefore solutions of the equations,

$\mathscr{J}\left(\begin{matrix}\displaystyle\frac{\delta\mathscr{C}}{\delta\psi_{e}}\\\
\\\ \displaystyle\frac{\delta\mathscr{C}}{\delta
A_{e}}\end{matrix}\right)=\left(\begin{matrix}0\\\ \\\ 0\end{matrix}\right).$
(15)

It may be verified that two such solutions are,

$\mathscr{C}_{(1)}=\int\limits_{\mathscr{A}}f\left(A_{e}\right)d\mathscr{A}$
(16)

$\mathscr{C}_{(2)}=\int\limits_{\mathscr{A}}\psi_{e}g\left(A_{e}\right)d\mathscr{A}$
(17)

for any functions $f$ and $g$. (16) and (17) were deduced by Biskamp et al.
[3] by working directly with equations (5) and (6). $\mathscr{C}_{(1)}$
signifies the conservation of generalized magnetic flux in EMHD.

Minimization of $H$, keeping $\mathscr{C}_{(1)}$ fixed, gives

$\frac{\delta H}{\delta
A_{e}}=\lambda_{(1)}\frac{\delta\mathscr{C}_{(1)}}{\delta A_{e}}$ (18)

or

$\frac{1}{c}J=\lambda_{(1)}f^{\prime}\left(A_{e}\right)$ (19)

which prescribes the Grad-Shafranov equilibria for 2D EMHD. There is no force-
free state in 2D EMHD. This situation changes, however, in the massless-
electron limit (see Section 5).

On the other hand, minimization of $H$, keeping $\mathscr{C}_{(2)}$ fixed,
gives

$\frac{\delta
H}{\delta\psi_{e}}=\lambda_{(2)}\frac{\delta\mathscr{C}_{(2)}}{\delta\psi_{e}}$
(20)

or

$\psi=\lambda_{(2)}g\left(A_{e}\right)$ (21)

or

${\bf v}_{e}=\lambda_{(2)}g^{\prime}\left(A_{e}\right){\bf B}_{e}\equiv a{\bf
B_{e}}$ (22)

where,

${\bf B}_{e}\equiv\nabla\times A_{e}\ \hat{{\bf i}}_{z}={\bf
B}-d_{e}^{2}\nabla^{2}{\bf B}.$ (23)

(21) describes the generalized Alfvénic state in 2D EMHD.

5\. EMHD in the Massless Electron Limit

In the massless electron limit, the Ohm’s law is

${\bf{E}}+\frac{1}{c}~{}{\bf{v}}_{e}\times{\bf{B}}={\bf{0}}$ (24)

which corresponds to the constraint of zero electric field in the reference
frame moving with the electron fluid.

Using,

${\bf{v}}_{e}=-\frac{\bf{J}}{n_{e}e}=-\frac{c}{n_{e}e}\nabla\times{\bf{B}}$
(25)

and Faraday’s Law,

$-\frac{1}{c}~{}\frac{\partial{\bf{B}}}{\partial t}=\nabla\times{\bf{E}}$ (26)

equation (24) leads to (Huba and Rudakov [9]),

$\frac{\partial{\bf{B}}}{\partial
t}+\frac{c}{n_{e}e}\nabla\times\left[\left(\nabla\times{\bf{B}}\right)\times{\bf{B}}\right]={\bf{0}}$
(27)

or

$\frac{\partial A_{i}}{\partial
t}+\frac{c}{n_{e}e}B_{j}\left(\partial_{j}B_{i}\right)=\frac{c}{n_{e}e}B_{j}\left(\partial_{i}B_{j}\right).$
(28)

In 2D111See Appendix for the 3D formulation in the massless-electron limit.,
where ${\bf A}=A\hat{\bf i}_{z}$, equation (28) becomes (upon normalizing $t$
appropriately),

$\frac{\partial A}{\partial t}=\left[B_{z},A\right].$ (29)

Equation (29) implies that the out-of-plane magnetic field $B_{z}$ plays the
role of the stream function for the in-plane electron flow, as to be expected.
Equation (29) has the Hamiltonian formulation,

$\mathscr{H}=\underset{\mathscr{A}}{\int}AB_{z}~{}d\mathscr{A}.$ (30)

The Hamilton’s equation, on taking $A$ to be the canonical variable, is given
by

$\frac{\partial A}{\partial
t}~{}=\mathscr{J}~{}\frac{\delta\mathscr{H}}{\delta A}$ (31)

where $\mathscr{J}$ is an $A$-dependent differential operator which produces a
skew-symmetric transformation of scalar functions vanishing on
$\partial\mathscr{A}$ and satisfies a closure condition on an associated
symplectic two-form,

$\mathscr{J}=\left[\cdot,A\right].$ (32)

Equation (31) is then just equation (29).
The operator $\mathscr{J}$ induces a Poisson bracket as follows,

$\displaystyle\left[F,G\right]$ $\displaystyle=\left(\frac{\delta F}{\delta
A},~{}\mathscr{J}\frac{\delta G}{\delta A}\right)$ (33)
$\displaystyle=\underset{\mathscr{A}}{\int}\left(\frac{\delta F}{\delta
A}\cdot\left[\frac{\delta G}{\delta A},A\right]\right)d\mathscr{A}$
$\displaystyle=\underset{\mathscr{A}}{\int}\left(A\left[\frac{\delta F}{\delta
A},\frac{\delta G}{\delta A}\right]\right)d\mathscr{A}$

which is a bilinear function defined on admissible functionals $F[A]$ and
$G[A]$, satisfying the conditions,

$\nabla\cdot\begin{pmatrix}\delta F/\delta A\\\ \delta G/\delta
A\end{pmatrix}=0~{}\text{in}~{}A~{}\text{and}~{}|\frac{\delta F}{\delta
A}|,|\frac{\delta G}{\delta A}|=0~{}\text{on}~{}\partial\mathscr{A}.$ (34)

The Casimir invariants are then solutions of the equation,

$\mathscr{J}\frac{\delta\mathscr{C}}{\delta A}=0$ (35)

which leads to

$\mathscr{C}=\int f\left(A\right)d\mathscr{A}$ (36)

$f(A)$ being an arbitrary function.

Minimization of $\mathscr{H}$, keeping $\mathscr{C}$ fixed, leads to the
Beltrami state,

$\frac{\delta\mathscr{H}}{\delta A}=\lambda\frac{\delta\mathscr{C}}{\delta A}$
(37)

or

$B_{z}=\lambda f^{\prime}\left(A\right)$ (38)

which leads to,

${\bf\mathscr{J}}=c\lambda f^{\prime\prime}\left(A\right){\bf B}\equiv a{\bf
B}$ (39)

describing a force-free state in the massless-electron limit (contrary to the
general EMHD case). So, EMHD, in the massless-electron limit, mimics usual MHD
(this result holds in 3D as well - see Appendix).

6\. Discussion

The intrinsic tendency of real plasmas (or fluids) toward Beltramization is a
long standing issue awaiting full clarification. Work to date has indicated
that the Beltramization process provides the means via which plasma systems
can accomplish,

  * *

an effective depletion of non-linearities,

  * *

ergodicity of the streamlines of the plasma flow in question (Moffatt [13]),

  * *

selective dissipation of total energy (Woltjer [14]).

In this paper, the Hamiltonian formulations along with the Poisson brackets
for 2D EMHD flows are developed. These formulations are used to deduce the
Beltrami states for 2D EMHD flows. Though there is no force-free Beltrami
state in 2D EMHD flows, in the massless-electron limit Beltrami states become
force-free. So, EMHD, in the massless-electron limit, mimics usual MHD (this
result holds in 3D as well - see Appendix).

Acknowledgments

Part of this work was carried out during the course of my participation in the
Turbulence Workshop at the Kavli Institute of Theoretical Physics, Santa
Barbara. I am thankful to Dr. Maxim Lyutikov for a helpful correspondence.
This research was supported in part by NSF grant No. PHY05-51164.

Appendix

In the massless electron limit, upon appropriately renormalizing $t$, equation
(27) can be rewritten as

$\frac{\partial{\bf B}}{\partial t}+\nabla\times\\{\left(\nabla\times{\bf
B}\right)\times{\bf B}\\}={\bf 0}$ (A$\cdot$1)

Equation (A$\cdot$1) has the Hamiltonian characterization,

$\mathscr{H}=\frac{1}{2}\underset{\mathscr{V}}{\int}{\bf B}^{2}d\mathscr{V}$
(A$\cdot$2)

where $\mathscr{V}$ is the volume occupied by the plasma. Hamilton’s equation,
on taking B to be the canonical variable, is given by

$\frac{\partial{\bf B}}{\partial
t}=\mathscr{J}\frac{\delta\mathscr{H}}{\delta{\bf B}}$ (A$\cdot$3)

where $\mathscr{J}$ is a B-dependent differential operator, which produces a
skew-symmetric transformation of vector functions vanishing on
$\partial\mathscr{V}$ and satisfies a closure condition on an associated
symplectic two-form222(A$\cdot$4) is similar to the one given by Olver [15]
for incompressible hydrodynamics (with the flow vorticity $\mathbf{\omega}$
taking the place of ${\bf B}).$,

$\mathscr{J}=\nabla\times\\{{\bf
B}\times\left(\nabla\times\left(\cdot\right)\right)\\}.$ (A$\cdot$4)

Equation (A$\cdot$3) is then just equation (A$\cdot$1).

The operator $\mathscr{J}$ may be seen to induce the Poisson bracket (similar
to the one given by Kuznetsov and Mikhailov [16]),

$\left[F,G\right]=\left(\frac{\delta F}{\delta{\bf B}},\mathscr{J}\frac{\delta
G}{\delta{\bf B}}\right)=\underset{\mathscr{V}}{\int}{\bf
B}\cdot\left[\left(\nabla\times\frac{\delta G}{\delta{\bf
B}}\right)\times\left(\nabla\times\frac{\delta F}{\delta{\bf
B}}\right)\right]d\mathscr{V}$ (A$\cdot$5)

which is a bilinear operation defined on the admissible functionals satisfying
the conditions,

$\nabla\cdot\begin{pmatrix}\delta F/\delta{\bf B}\\\ \delta G/\delta{\bf
B}\end{pmatrix}={\bf 0}~{}\text{in}~{}\mathscr{V}$

and

$|\frac{\delta F}{\delta{\bf B}}|,|\frac{\delta G}{\delta{\bf
B}}|=0~{}\text{on}~{}\partial\mathscr{V}.$ (A$\cdot$6)

The Casimir invariant is then given by the solution of the equation,

$\mathscr{J}\frac{\delta\mathscr{C}}{\delta{\bf B}}={\bf 0}$ (A$\cdot$7)

which leads to the magnetic helicity,

$\mathscr{C}=\underset{\mathscr{V}}{\int}\textbf{A}\cdot\textbf{B}~{}d\mathscr{V}.$
(A$\cdot$8)

Minimization of $\mathscr{H}$, keeping $\mathscr{C}$ fixed, leads to the
Beltrami state,

$\frac{\delta\mathscr{H}}{\delta\textbf{B}}=\lambda\frac{\delta\mathscr{C}}{\delta\textbf{B}}$
(A$\cdot$9)

or

$\textbf{B}=\lambda\textbf{A}$ (A$\cdot$10)

which leads to

$\textbf{J}=c\lambda\textbf{B}$ (A$\cdot$11)

describing again a force-free state in the massless-electron limit. So, EMHD,
in the massless-electron limit, mimics usual MHD.

## References

  * [1] A. S. Kingsep, K. V. Chukbar and V. V. Yankov: Rev. Plasma Phys. 16, 243, (1990).
  * [2] A. V. Gordeev, A. S. Kingsep and L. I. Rudakov: Phys. Reports 243, 215, (1994).
  * [3] D. Biskamp, E. Schwarz and J. F. Drake: Phys. Rev. Lett. 76, 1264, (1996).
  * [4] B. K. Shivamoggi: Euro. Phys. J D 64, 393, (2011).
  * [5] S. Lundquist: Arkiv. Fysik 2, 361, (1950).
  * [6] R. Lust and A. Schluter: Z. Astrophys. 34, 263, (1954).
  * [7] E. R. Priest and T. Forbes: Magnetic Reconnection, Cambridge Univ. Press, (2000).
  * [8] K. Schindler: Physics of Space Plasma Activity, Cambrdige Univ. Press, (2007).
  * [9] J. D. Huba and L. I. Rudakov: Phys. Scripta T107, 20 (2004).
  * [10] M. Rheinhardt and U. Geppert: Phys. Rev. Lett. 88, 101103, (2002).
  * [11] T. S. Wood, R. Hollerbach and M. Lyutikov: Physics Plasmas 21, 052110, (2014).
  * [12] D. D. Holm, J. E. Marsden, I. Ratiu and A. Weinstein: Phys. Rep. 123, 1, (1985).
  * [13] H. K. Moffatt: inWhither Turbulence, Ed. J. L. Lumley, p. 250, Springer Verlag, (1990).
  * [14] L. Woltjer: Proc. Nat. Acad. Sci. 44, 489, (1958).
  * [15] P. J. Olver: J. Math. Anal. Appl. 89, 233, (1982).
  * [16] E. A. Kuznetsov and A. V. Mikhailov: Phys. Lett. A 77, 37, (1980).


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

# Ghost Imaging of Dark Particles

J. Estrada ${}^{\pumpkin}$ Fermi National Accelerator Laboratory, PO Box 500,
Batavia IL, 60510, USA  R. Harnik ${}^{\mathrightghost}$ Fermi National
Accelerator Laboratory, PO Box 500, Batavia IL, 60510, USA  D. Rodrigues
${}^{\mathrightbat}$ Department of Physics, FCEN, University of Buenos Aires
and IFIBA, CONICET, Buenos Aires, Argentina  Fermi National Accelerator
Laboratory, PO Box 500, Batavia IL, 60510, USA  M. Senger ${}^{\mathwitch*}$
Department of Physics, FCEN, University of Buenos Aires and IFIBA, CONICET,
Buenos Aires, Argentina  Physik-Institut der Universität Zürich

(October 31, 2020)

###### Abstract

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.††${}^{\pumpkin}$<EMAIL_ADDRESS><EMAIL_ADDRESS>††${}^{\mathrightbat}$
<EMAIL_ADDRESS>In quantum teleportation furusawa_unconditional_1998 ;
boschi_experimental_1998 ; bouwmeester_experimental_1997 ;
bennett_teleporting_1993  the state of a distant quantum system, Alice, can be
inferred without 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 recently allowed teleportation over a large distance
2020arXiv200711157V . Quantum ghost imaging, or “interaction-free” imaging
white_interaction-free_1998 , is used to discern (usually classical)
information about an object without direct interaction. This technique
exploits the relationship among the emission angles of a correlated photon
pair to create an image with high angular resolution without placing the
subject Alice in front of a high resolution detector or allowing it to
interact with intense light.††${}^{\mathwitch*}$<EMAIL_ADDRESS>These methods of teleportation and imaging rely on the production of signal
photons in association with an idler pair which is entangled (or at least
correlated) in its direction, frequency, and sometimes polarization.

Quantum ghost imaging and teleportation both differ parametrically from
standard forms of information transfer. This is because a system is probed,
not by sending information to it and receiving information back, but rather by
sending it half of an EPR pair, without need for a “response”. The difference
is particularly apparent if Alice is an extremely weekly coupled system, say
she is part of a dark sector, with a coupling $\epsilon$ to photons. The rate
of information flow in capturing an image of Alice will occur at a rate
$\propto\epsilon^{4}$ with standard methods, but at a rate
$\propto\epsilon^{2}$ using quantum optical methods.

A common method for generating entangled photon pairs is the nonlinear optics
process known as _spontaneous parametric down conversion_ (SPDC). In SPDC a
pump photon decays, or down-converts, within a nonlinear optical medium into
two other photons, a signal and an idler. The presence of the SPDC idler can
be inferred by the detection of the signal hong_mandel_1985 . SPDC is in wide
use in quantum information and provides the seed for cavity enhanced setups
such as optical parametric oscillators (OPO) giordmaine_tunable_1965 .

Figure 1: Pictorial representation of the dSPDC process. A dark particle
$\varphi$ is emitted in association with a signal photon. The presence of
$\varphi$ can be inferred from the distribution of the signal photon in angle
and/or frequency. We consider both the colinear ($\theta_{s}=0$) and non-
colinear ($\theta_{s}\neq 0$) cases.

In this work we propose to use quantum imaging and quantum communication tools
to perform an interaction-free search for the emission of new particles beyond
the standard model. The new tool we present is _dark SPDC_ , or dSPDC, an
example sketch of which is shown in Figure 1. A pump photon enters an optical
medium and down-converts to a signal photon and a dark particle, which can
have a small mass, and does not interact with the optical medium. Like SPDC,
in dSPDC the presence of a dark state can be inferred by the angle and
frequency distribution of the signal photon that was produced in association.
The dSPDC process can occur either collinearly, with
$\theta_{s}=\theta_{i}=0$, or as shown in the sketch, in a non-colinear way.

The new dark particles which we propose to search for, low in mass and feebly
interacting, are simple extensions of the standard model (SM). Among the most
well motivated are axions, or axion-like-particles, and dark photons (see
essig_dark_2013  for a review). Since these particles interact with SM
photons, optical experiments provide an opportunity to search for them. A
canonical setup is light-shining-through-wall (LSW) which set interesting
limits on axions and dark photons using optical cavities Hoogeveen:1990vq . In
optical LSW experiments a laser photon converts to an axion or dark photon,
enabling it to penetrate an opaque barrier. For detection, however, the dark
state must convert back to a photon and interact, which is a rare occurrence.
As a result, the signal rate in LSW scales as $\epsilon^{4}$, where $\epsilon$
is a small coupling to dark states. This motivates the interaction-free
approach of dSPDC, in which the dark state is produced but does not interact
again, yielding a rate $\propto\epsilon^{2}$.

From the perspective of particle physics, dSPDC, and also SPDC, may sound
unusual111Searches for missing energy and momentum are a commonplace tool in
the search for new physics at colliders Goodman:2010yf ; Bai:2010hh ;
Goodman:2010ku ; Fox:2011pm ; Aaltonen:2012jb . However, the dSPDC kinematics
are distinct as we show.. In dSPDC, for example, a massless photon is decaying
to another massless photon plus a massive particle. In the language of
particle physics this would be a kinematically forbidden transition. The
process, if it were to happen in vacuum, violates energy and momentum
conservation and thus has no available phase space. In this paper we show how
optics enables us to perform “engineering in phase space” and open kinematics
to otherwise forbidden channels. This, in turn, allows to design setups in
which the dSPDC process is allowed and can be used to search for dark
particles.

Broadly speaking, the search strategies we propose may be classified as
employing either imaging or spectroscopic tools, though methods employing both
of these are also possible. In the first, the angular distribution of signal
photons are measured, and in the second, the energy distribution is observed.
For an imaging based search, a high angular resolution is required, while the
spectroscopic approach requires high frequency resolution. The later has the
benefit that dSPDC can be implemented in a waveguide which will enhance its
rate for long optical elements fiorentino_spontaneous_2007 ; ourpaper .

We present the basic ideas and formalism behind this method, and focus on the
phase space for dSPDC. We compare the phase space distributions of SPDC to
dSPDC and discuss factors which may enhance the rate of dSPDC processes. To
estimate the dSPDC rates in this work we re-scale known SPDC results. In a
companion paper ourpaper  we will derive the dSPDC rate more carefully,
focusing on the spectroscopic approach and on _colinear_ dSPDC in bulk
crystals and waveguides.

In this work we also present a proof-of-concept experiment using a quantum
imaging setup with a Skipper CCD tiffenberg_single-electron_2017 . The high
resolution is shown to produce a high resolution and low noise angular image
of an SPDC pattern. Such a detector may be employed both in imaging and
spectroscopic dSPDC setups.

This paper is structured as follows. In Section II we review axion-like
particles and dark photons and present their Hamiltonians in the manner they
are usually treated in nonlinear optics. In Section III we discuss energy and
momentum conservation, which are called _phase matching conditions_ in
nonlinear optics, and the phase space for dSPDC. The usual conservation rules
will hold exactly in a thought experiment of an infinite optical medium. In
Section III.3 we discuss how the finite size of the optical medium broadens
the phase space distribution and plot it as a function of the signal photon
angle and energy. In Section IV we discuss the choice of optical materials
that allow for the dSPDC phase matching to enhance its rates and to suppress
or eliminate SPDC backgrounds. In Section V we discuss the rates for dSPDC and
its scaling with the geometry of the experiment, showing potential promising
results for collinear waveguide setups with long optical elements. In Section
VI, we present an experimental proof-of-concept in which a Skipper CCD is used
to image an SPDC pattern with high resolution. We discuss future directions
and conclude in Section VII.

##  II The Dark SPDC Hamiltonian

The SPDC process can be derived from an effective nonlinear optics Hamiltonian
of the form ling_absolute_2008 ; Schneeloch2016

$\mathcal{H}_{\mathrm{SPDC}}=\chi^{(2)}E_{i}E_{s}E_{p}$ (1)

where $E_{j}$ is the electric field for pump, signal, and idler photons and
$j=p,s,i$ respectively. We adopt the standard notation in nonlinear optics
literature in which the vector nature of the field is implicit. Thus the pump,
signal and idler can each be of a particular polarization and $\chi^{(2)}$
represents the coupling between the corresponding choices of polarization.

We now consider the dark SPDC Hamiltonian

$\mathcal{H}_{\mathrm{dSPDC}}=g_{\varphi}\,\varphi E_{s}E_{p}$ (2)

in which a pump and signal photon couple to a new field $\varphi$ which has a
mass $m_{\varphi}$. The effective coupling $g_{\varphi}$, depends on the model
and can depend on the frequencies in the process, the polarizations and the
directions of outgoing particles. This type of term arises in well motivated
dark sector models including axion-like particles and dark photons.

###  II.1 Axion-like particles

Axion-like particles, also known as ALPs, are light pesudoscalar particles
that couple to photons via a term

$\mathcal{H}_{\mathrm{axion}}=g_{\varphi}\,\varphi\vec{E}\cdot\vec{B}\,,$ (3)

which can be written in the scalar form of Equation (2) with the pump and
signal photons chosen to have orthogonal polarization. The coupling
$g_{\varphi}$ in this case is the usual axion-photon coupling $\sim 1/f_{a}$.
ALPs are naturally light thanks to a spontaneously broken global symmetry at a
scale $f_{a}$.

Since we will not perform a detailed analysis of backgrounds and feasibility
here, we will describe existing limits on ALPs qualitatively and refer readers
to limit plots in Zyla:2020zbs ; essig_dark_2013 . In this work we will
consider searches that can probe ALPs with a mass of order 0.1 eV or lower.
Existing limits in this mass range come from stellar and supernova
dynamicsRaffelt:1990yz , as well as from the CAST Heiloscope
Anastassopoulos:2017ftl , which require $g_{\varphi}\lesssim
10^{-10}\leavevmode\nobreak\ \mathrm{GeV}^{-1}$. For comparison with our
setups, we will also consider limits that are entirely lab-based, which are
much weaker. At $\sim 0.1$ eV ALP coupling of order
$10^{-6}\leavevmode\nobreak\ \mathrm{GeV}^{-1}$ are allowed by all lab-based
searches. For axion masses around $10^{-3}-10^{-2}$ eV the PVLAS
searchDellaValle:2015xxa  for magnetic birefringence places a limit of
$g_{\varphi}\lesssim 10^{-7}-10^{-6}\leavevmode\nobreak\ \mathrm{GeV}^{-1}$.
Bellow masses of an meV, LSW experiments such as OSQAR Ballou:2015cka  and
ALPS Ehret:2009sq  set $g_{\varphi}\lesssim 6\times 10^{-8}$. A large scale
LSW facility, ALPS II Bahre:2013ywa , is proposed in order to improve upon
astrophysical limits and CAST.

###  II.2 Dark photons

A new U(1) gauge field, $A^{\prime}$, with mass $m_{A^{\prime}}$ that mixes
kinetically with the photon via the Hamiltonian

$\mathcal{H}_{\mathrm{mix}}=\epsilon
F^{\mu\nu}F^{\prime}_{\mu\nu}=\epsilon\left(\vec{E}\cdot\vec{E}^{\prime}+\vec{B}\cdot\vec{B}^{\prime}\right)\,.$
(4)

It is possible to write the dark photon interaction in a basis in which any
electromagnetic current couples to longitudinally polarized dark photons with
an effective coupling of $\epsilon(m_{A^{\prime}}/\omega_{A^{\prime}})$ and to
transversely polarized dark photons with
$\epsilon(m_{A^{\prime}}/\omega_{A^{\prime}})^{2}$ An:2013yfc ; An:2013yua .
The same dynamics that leads to the nonlinear Hamiltonian
$\mathcal{H}_{\mathrm{SPDC}}$, will yield an effective coupling of two photons
to a dark photon as in Equation (2) with $\varphi$ representing a dark photon
polarization state. For longitudinal polarization of the dark photon this can
be brought to the form of equation (2) with
$g_{\varphi}=\epsilon\chi^{(2)}m_{\varphi}$. The coupling to transverse dark
photons will be suppressed by an additional factor of
$m_{\varphi}/\omega_{\varphi}$.

Like axions, limits on dark photons come from astrophysical systems, such as
energy loss in the Sun An:2013yfc , as well as the lack of a detection of dark
photons emitted by the Sun in dark matter direct detection experiments
An:2013yua . At a dark photon mass of order 0.1 eV, the limit on the kinetic
mixing is $\epsilon\lesssim 10^{-10}$ with the limit becoming weaker linearly
as the dark photon mass decreases. Among purely lab-based experiments, ALPS
sets a limit of $\epsilon\lesssim 3\times 10^{-7}$. In forthcoming sections of
this work we will use these limits as benchmarks.

$\mathghost\mathghost\mathghost\mathghost$

In addition to ALPs and dark photons, one can consider other new dark
particles that are sufficiently light and couples to photons. Examples include
millicharged particle, which can be produced in pairs in association with a
signal photon. We leave these generalizations for future work and proceed to
discuss the phase space for dSPDC in a model-independent way.

##  III Phase space of Dark SPDC

As stated above, the effective energy and momentum conservation rules in
nonlinear optics are known as “phase matching conditions”. We begin by
reviewing their origin to make the connection with energy and momentum
conservation in particle physics and then proceed to solve them for SPDC and
dSPDC, to understand the phase space for these processes.

###  III.1 Phase matching and particle decay

We begin with a discussion of the phase space in a two-body decay, or
spontaneous down-conversion. We use particle physics language, but will label
the particles as is usual in the nonlinear optics systems which we will be
discussing from the onset. A pump particle $p$ decays into a signal particle
$s$ and another particle, either an idler photon $i$ or a dark particle
$\varphi$:

SPDC: $\displaystyle\gamma_{p}\to\gamma_{s}+\gamma_{i}$ dSPDC:
$\displaystyle\gamma_{p}\to\gamma_{s}+\varphi\,.$ (5)

We would like to discuss the kinematics of the standard model SPDC process,
and the BSM dSPDC process together, to compare and contrast. For this, in the
discussion below we will use the idler label $i$ to represent the idler photon
in the SPDC case, and $\varphi$ in the dSPDC case. Hence $\boldsymbol{k}_{i}$
and $\theta_{i}$ are the momentum and emission angle of either the idler
photon or of $\varphi$, depending on the process.

The differential decay rate in the laboratory frame is given by Zyla:2020zbs

$\frac{d\Gamma}{d^{3}k_{s}d^{3}k_{i}}=\frac{\delta^{4}\left(p_{p\mu}-p_{s\mu}-p_{i\mu}\right)}{(2\pi)^{2}\,2\omega_{p}\,2\omega_{s}\,2\omega_{i}}|\mathcal{M}|^{2}$
(6)

where $\mathcal{M}$ is the amplitude for the process (which carries a mass
dimension of $+1$), and the energy-momentum four-vectors

$p_{j\mu}=\left(\begin{array}[]{c}\omega_{j}\\\
\boldsymbol{k}_{j}\end{array}\right)$ (7)

with $j=p,s,i$. Traditionally, one proceeds by integrating over four degrees
of freedom within the six dimensional phase space, leaving a differential rate
with respect to the two dimensional phase space. This is often chosen a solid
angle for the decay $d\Omega$, but in our case there will be non-trivial
correlations of frequency and angle as we will see in the next subsection.

It is instructive, however, for our purpose to take a step back and recall how
the energy-momentum conserving $\delta$-functions come about. In calculating
the quantum amplitude for the transition, the fields in the initial and final
states are expanded in modes. The plane-wave phases $e^{i\left(\omega
t-\boldsymbol{k}\cdot\boldsymbol{x}\right)}$ are collected from each and a
spacetime integral is performed

$\int d^{4}x\,e^{i\left(\Delta\omega
t-\Delta\boldsymbol{k}\cdot\boldsymbol{x}\right)}=(2\pi)^{4}\,\delta^{(3)}(\Delta\boldsymbol{k})\,\delta(\Delta\omega)\,,$
(8)

where frequency and momentum mismatch are defined as

$\left\\{\begin{aligned} &\Delta\omega=\omega_{p}-\omega_{s}-\omega_{i}\\\
&\Delta\boldsymbol{k}=\boldsymbol{k}_{p}-\boldsymbol{k}_{s}-\boldsymbol{k}_{i}\end{aligned}\text{,}\right.$
(9)

and again, the label $i$ can describe either an idler photon or a dark
particle $\varphi$. In Equation (9), we separated momentum and energy
conserving delta functions to pay homage to the optical systems which will be
the subject of upcoming discussion. The squared amplitude $|\mathcal{M}|^{2}$,
and hence the rate, is proportional to a single power of the energy-momentum
conserving delta function times a space-time volume factor, which is absorbed
for canonically normalized states (see e.g. TongNotes ), giving Equation (6).

The message from this is that energy and momentum conservation, which are a
consequence of Neother’s theorem and space-time translation symmetry, is
enforced in quantum field theory by perfect destructive interference whenever
there is a non-zero mismatch in the momentum or energy of initial and final
states. In this sense, the term “phase matching” captures the particle
physicist’s notion of energy and momentum conservation well.

The dSPDC process shown in Figure 1 is a massless pump photon decaying to a
signal photon plus a massive particle $\varphi$. This process clearly cannot
occur in vacuum. For example, if we go to the rest frame of $\varphi$, and
define all quantities in this frame with a tilde $\tilde{\,}$, the
conservation of momentum implies
$\boldsymbol{\tilde{k}}_{p}=\boldsymbol{\tilde{k}}_{s}$, while the
conservation of energy implies $\tilde{\omega}_{p}=\tilde{\omega}_{s}+m$.
Combining these with the dispersion relation for a photon in vacuum
$\tilde{\omega}_{p,s}=|\boldsymbol{\tilde{k}}_{p,s}|$, implies that energy and
momentum conservation cannot be satisfied and the process is kinematically
forbidden. Another way to view this is to recall that the energy-momentum
four-vector for the initial state (a photon) lies on a light cone i.e. it is a
null four-vector, $p^{\mu}p_{\mu}=0$, while the final state cannot accomplish
this since there is a nonzero mass.

In this work we show that optical systems will allow us to open phase space
for dSPDC. When a photon is inside an optical medium a different dispersion
relation holds,

$\tilde{n}_{p}\tilde{\omega}_{p}=|\boldsymbol{\tilde{k}}_{p}|\qquad\mbox{and}\qquad\tilde{n}_{s}\tilde{\omega}_{s}=|\boldsymbol{\tilde{k}_{s}}|$
(10)

where $\tilde{n}_{p}$ and $\tilde{n}_{s}$ are indices of refraction of photons
in the medium in the $\varphi$ rest frame, which can be different for pump and
signal. As we will see later on, under these conditions the conclusion that
$p\to s+\varphi$ is kinematically forbidden can be evaded.

Another effect that occurs in optical systems, but not in decays in particle
physics, is the breaking of spatial translation invariance by the finite
extent of the optical medium. This allows for violation of momentum
conservation along the directions in which the medium is finite. As a result,
the sharp momentum conserving delta function will become a peaked distribution
of width $L^{-1}$, where $L$ is the crystal size. We will begin by solving the
exact phase matching conditions in SPDC and dSPDC in the following subsection,
which correspond to the phase space distributions for an infinitely large
crystal. Next, we will move on to the case in which the optical medium is
finite.

###  III.2 Phase Matching in dSPDC

We now study the phase space for dSPDC to identify the correlations between
the emission angle and the frequency of the signal photon. We will consider in
parallel the SPDC process as well, so in the end we arrive to both results. We
assume in this subsection an optical medium with an infinite extent, such that
the delta-functions enforce energy and momentum conservation.

The phase matching conditions $\Delta\omega=0$ and $\Delta\boldsymbol{k}=0$
—that must be strictly satisfied in the infinite extent optical medium
scenario— imply

$\left\\{\begin{aligned} &\omega_{p}=\omega_{s}+\omega_{i}\\\
&\boldsymbol{k}_{p}=\boldsymbol{k}_{s}+\boldsymbol{k}_{i}\end{aligned}\,.\right.$
(11)

Using the $\boldsymbol{k}$’s decomposition shown in Figure 1 the second
equation can be expanded in coordinates parallel and perpendicular to the pump
propagation direction

$\left\\{\begin{aligned} &k_{p}=k_{s}\cos\theta_{s}+k_{i}\cos\theta_{i}\\\
&0=k_{s}\sin\theta_{s}-k_{i}\sin\theta_{i}\end{aligned}\right.$ (12)

where the angles $\theta_{s}$ and $\theta_{i}$ are those indicated in the
cited figure. Since this process is happening in a material medium, the photon
dispersion relation is $k=n\omega$ where $n$ is the refractive index. The
refractive index is in general a function of frequency, polarization, and the
direction of propagation, and can thus be different for pump, signal, and
idler photons. For dSPDC, in which the $\varphi$ particle is massive and
weekly interacting with matter the dispersion relation is the same as in
vacuum,

$k_{\varphi}=\sqrt{\omega_{\varphi}^{2}-m_{\varphi}^{2}}\text{.}$ (13)

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

$\displaystyle k_{p}=n_{p}\omega_{p}\text{,}$ (14) $\displaystyle
k_{s}=n_{s}\omega_{s}$ (15)

and

$k_{i}=\left\\{\begin{aligned} &n_{i}\omega_{i}&&\text{for SPDC}\\\
&\sqrt{\omega_{i}^{2}-m_{\varphi}^{2}}&&\text{for
dSPDC}\end{aligned}\text{.}\right.$ (16)

where, again, the label “$i$” refers to an idler photon for SPDC and to the
dark field $\varphi$ for dSPDC. Note that $k_{i}$ for SPDC not only has $m=0$
but also takes into account the “strong (electromagnetic) coupling” with
matter summarized by the refractive index $n_{i}$. As a result, for SPDC we
always have $k_{i}>\omega_{i}$ while for dSPDC $k_{i}<\omega_{i}$. Replacing
these dispersion relations into (12) and using the first equation in (11) we
obtain the phase matching relation of signal angle and frequency

$\cos\theta_{s}=\frac{n_{p}^{2}+\alpha_{\omega}^{2}n_{s}^{2}-\Xi^{2}}{2\alpha_{\omega}n_{p}n_{s}}$
(17)

where we define

$\alpha_{\omega}\overset{{\scriptscriptstyle\text{def}}}{=}\frac{\omega_{s}}{\omega_{p}}$
(18)

and

$\Xi\overset{{\scriptscriptstyle\text{def}}}{=}\frac{k_{i}}{\omega_{p}}=\left\\{\begin{aligned}
&n_{i}\left(1-\alpha_{\omega}\right)&&\text{for SPDC}\\\
&\sqrt{\left(1-\alpha_{\omega}\right)^{2}-\frac{m_{\varphi}^{2}}{\omega_{p}^{2}}}&&\text{for
dSPDC}\end{aligned}\right.\text{.}$ (19)

Equation (17) defines the phase space for (d)SPDC, along with the azimuthal
angle $\phi_{s}$.

Figure 2: _Left:_ The allowed phase space for SPDC (black), dSPDC with $m=0$
(red), and dSPDC with $m=0.1\omega_{p}$ (blue) shown as in the plane of signal
emission angle as a function of frequency ratio $\alpha_{\omega}$. The indices
of refraction here are $n_{p}=1.658$ and $n_{s}=1.486$ 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
$\Delta\boldsymbol{k}=0$ for SPDC and dSPDC, with massless $\varphi$. In both
cases we take the same $\omega_{p}$ and $\omega_{s}$. Due to the index of
refraction for $\varphi$ is essentially one, and that for the idler photon is
larger (say $\sim 1.5$), phase matching in dSPDC has a smaller signal emission
angle than that of SPDC.

The idler angle, or that of the dark particle in the dSPDC case, is fixed in
terms of the signal angle and frequency by requiring conservation of
transverse momentum

$\displaystyle\sin\theta_{i}$ $\displaystyle=$
$\displaystyle\frac{k_{s}}{k_{i}}\sin\theta_{s}$ $\displaystyle\phi_{i}$
$\displaystyle=$ $\displaystyle\phi_{s}+\pi$ (20)

where $k_{i}$ is evaluated at a frequency of
$\omega_{i}=\omega_{p}-\omega_{s}$, and $k_{s}$ at the frequency $\omega_{s}$
according to the dispersion relation in Equations (15) and (16).

In the left panel of Figure 2 we show the allowed phase space in the
$\alpha_{\omega}$-$\theta_{s}$ plane for SPDC and for dSPDC with both a
massless and massive $\varphi$. Here we have chosen $n_{p}=1.486$,
$n_{s}=1.658$ with $n_{i}=n_{s}$ in the SPDC case. These values were chosen to
be constant with frequency and propagation direction, for simplicity and are
inspired by the ordinary and extraordinary refractive indices in calcite and
will be used as a benchmark in some of the examples below.

We see that phase matching is achieved in different regions of phase space for
SPDC and its dark counterpart. In dSPDC, signal emission angles are restricted
to near the forward region and in a limited range of signal frequencies. The
need for more forward emission in dSPDC can be understood because $\varphi$
effectively sees an index of refraction of 1. This implies it can carry less
momentum for a given frequency, as compared to a photon which obeys
$k=n\omega$. As a result, the signal photon must point nearly parallel to the
pump, in order to conserve momentum (see Figure 2, right). An additional
difference to notice is that, in contrast with the SPDC example shown, for a
fixed signal emission angle there are two different signal frequencies that
satisfy the phase matching in dSPDC. This will always be the case for dSPDC.
Although SPDC can be of this form as well, typical refractive indices usually
favor single solutions as shown for calcite.

####  III.2.1 Phase matching for colinear dSPDC

One case of particular interest is that of colinear dSPDC, in which the
emission angle is zero, as would occur in a single-mode fiber or a waveguide.
For this case, so long as $n_{s}>n_{p}$, an axion mass below some threshold
can be probed.

Figure 3: Solutions to the phase matching conditions for the colinear dark
SPDC process for the signal photon energy as a function of the dark particle
$\varphi$ mass. This phase space is relevant for waveguide-based experiments.
Both axes are normalized to the pump frequency. The two branches correspond to
configurations in which $\varphi$ is emitted forward (bottom) and backward
(top).

Setting the emission angles to zero there are two solutions to the phase
matching equations which give signal photon energies of

$\alpha_{\omega}=\frac{\left(n_{s}n_{p}-1\right)\pm\sqrt{\left(n_{s}-n_{p}\right)^{2}-\left(n_{s}^{2}-1\right)\frac{m_{\varphi}^{2}}{\omega_{p}^{2}}}}{n_{s}^{2}-1}\,.$
(21)

The $+$ solution above corresponds to the case where the $\varphi$ particle is
emitted in the forward direction whereas the $-$ represents the case where it
is emitted backwards. The signal frequency as a function of $\varphi$ mass is
shown in Figure 3 for the calcite benchmark. For the near-massless case,
$m_{\varphi}\ll\omega_{p}$, a phase matching solution exists for $n_{s}\geq
n_{p}$, giving

$\frac{\omega_{s}}{\omega_{p}}=\frac{n_{p}\mp 1}{n_{s}\mp 1}\,.$ (22)

For calcite, we get $\omega_{s}=0.739\omega_{p}$ and
$\omega_{s}=0.935\omega_{p}$ for forward and backward emitted axions
respectively.

Of course, not any combination of $n_{p}$ and $n_{s}$ will allow to achieve
phase matching in dSPDC. We will state here that $n_{s}>n_{p}$ is a
requirement for phase matching to be possible for a massless $\varphi$ and
that $n_{s}-n_{p}$ must grow as $m_{\varphi}$ grows. We will discuss these and
other requirements for dSPDC searches in Section IV.

###  III.3 Thin planar layer of optical medium

We now consider the effects of a finite crystal. Specifically, we will assume
that the optical medium is a planar thin layer222The meaning of what
constitutes the thin crystal layer limit in our context will be discussed in
Section V. of optical material of length $L$ along the pump propagation
direction $\hat{z}$, and is infinite in transverse directions. In this case,
the integral in Equation (8) is performed only over a finite range and the
delta function is replaced by a $\text{sinc}\ x\equiv\sin x/x$ function

$\int_{-L/2}^{L/2}dz\,e^{i\Delta k_{z}z}=L\,\mathrm{sinc}\left(\frac{\Delta
k_{z}L}{2}\right)$ (23)

at the level of the amplitude. As advertised, this allows for momentum non-
conservation with a characteristic width of order $L^{-1}$.

The fully differential rate will be proportional to the squared sinc function

$\displaystyle\frac{d\Gamma}{d^{3}k_{p}d^{3}k_{i}}$ $\displaystyle\propto$
$\displaystyle L^{2}\mathrm{sinc}^{2}\left(\frac{\Delta
k_{z}L}{2}\right)\delta^{2}(\Delta\boldsymbol{k}_{T})\delta(\Delta\omega)$
(24) $\displaystyle\equiv$
$\displaystyle\frac{d\hat{\Gamma}}{d^{3}k_{p}d^{3}k_{i}}$ (25)

where $\Delta\boldsymbol{k}_{T}$ is the momentum mismatch vector in the
transverse directions. The constant of proportionality in Equation (24) will
have frequency normalization factors of the form $(2\omega)^{-1}$ and the
matrix element $\mathcal{M}$. The matrix element can also have non trivial
angular dependence in $\theta$ as well as in $\phi$, depending on the model
and the optical medium properties. Note, however, that the current analysis
for the phase space is model independent. We thus postpone the discussion of
overall rates to Section V and to ourpaper  and here we limit the study to the
phase space distribution only. We will proceed with the defined phase space
distribution $d\hat{\Gamma}$ in Equation (25), which we will now study.

One can trivially perform the integral over the $d^{3}k_{i}$ which will
effectively enforce Equation (20) for transverse momentum conservation, and
set $|k_{i}|$ by conservation of energy. The argument of the sinc function is

$\displaystyle\Delta k_{z}$ $\displaystyle=$ $\displaystyle
k_{p}-k_{s}\cos\theta_{s}-k_{i}\cos\theta_{i}$ $\displaystyle=$
$\displaystyle\omega_{p}\left(n_{p}-n_{s}\alpha_{\omega}\cos\theta_{s}\pm\sqrt{\Xi^{2}-n_{s}^{2}\alpha_{\omega}^{2}\sin^{2}\theta_{s}}\right)$

where in the second step we have used Equation (20) and the definitions of
$\alpha_{\omega}$ and $\Xi$ in Equations (18) and (19). The $\pm$ accounts for
the idler or dark particle being emitted in the forward and backward direction
respectively.

It is convenient to re-express the remaining three-dimensional phase space for
(d)SPDC in terms of the signal emission angles $\theta_{s}$ and $\phi_{s}$,
and the signal frequency (or equivalently $\alpha_{\omega}$). Within our
assumptions, the distribution in $\phi_{s}$ is flat. With respect to the polar
angle and frequency we find

$\frac{d^{2}\hat{\Gamma}}{d(\cos\theta_{s})d\alpha_{\omega}}=\frac{2\pi\omega_{p}^{3}\alpha_{\omega}^{2}\left(1-\alpha_{\omega}\right)n_{s}^{3}\tilde{n}_{i}^{2}}{\sqrt{\Xi^{2}-\alpha_{\omega}^{2}n_{s}^{2}\sin^{2}\theta_{s}}}\sum_{\pm}L^{2}\text{sinc}^{2}\left[\left(n_{p}-n_{s}\alpha_{\omega}\cos\theta_{s}\pm\sqrt{\Xi^{2}-n_{s}^{2}\alpha_{\omega}^{2}\sin^{2}\theta_{s}}\right)\frac{\omega_{p}L}{2}\right]$
(27)

where

$\tilde{n}_{i}\overset{{\scriptscriptstyle\text{def}}}{=}\left\\{\begin{aligned}
&n_{i}&&\text{for SPDC}\\\ &1&&\text{for dSPDC}\end{aligned}\right.\text{.}$
(28)

In Figure 4 we show the double differential distribution $\hat{\Gamma}$ for
SPDC, and in Figure 5 it is shown for dSPDC both for a massless and a massive
$\varphi$. The distribution clearly peaks in regions of phase space that
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 exaggerated values for indices of refraction in these figures.

Figure 4: The phase space distribution
$d^{2}\hat{\Gamma}/d(\cos\theta_{s})d\alpha_{\omega}$ for SPDC in arbitrary
units. We use $L=1$, $\lambda_{p}=\frac{L}{10}$, $n_{p}=2$ and
$n_{s}=n_{i}=4$. These parameters are unrealistic in practice, but allow for
clear visualization of the distribution. Dashed lines show slices of fixed
signal angle/energy.

Figure 5: Phase space distribution
$d^{2}\hat{\Gamma}/d(\cos\theta_{s})d\alpha_{\omega}$ in arbitrary units for
the dSPDC process for $m=0$ (left) and $m=0.5\omega_{p}$. The other parameters
are as in Figure 4, with somewhat exaggerated indices of refraction to allow
seeing the features in the distribution.

####  III.3.1 dSPDC Searches with Imaging or Spectroscopy

Though one can, in principle, measure both the angle and the energy of the
signal photon, it is usually easier to measure either with a fixed angle or a
fixed energy. For example, a CCD detector with a monochromatic filter can
easily measure a single energy slice of this distribution, as shown in the
vertical dashed line in Figure 4. Likewise, a spectrometer with no spatial
resolution can measure a fixed angle slice of this distribution, as shown in
the horizontal dashed line of Figure 4. An attractive choice for this is to
look in the forward region, with an emission angle $\theta_{s}=0$, which is
the case for waveguides. For this colinear case we get a signal spectrum

$\frac{d\hat{\Gamma}}{d\alpha_{\omega}}=\frac{2\pi\omega_{p}^{3}\alpha_{\omega}^{2}\left(1-\alpha_{\omega}\right)n_{s}^{3}\tilde{n}_{i}^{2}}{\Xi}\sum_{\pm}L^{2}\mathrm{sinc}^{2}\left(\frac{\Delta
k_{z\pm}^{(0)}L}{2}\right)$ (29)

where

$k_{z\pm}^{(0)}=k_{zp}-k_{zs}-k_{zi}^{\pm}=\omega_{p}\left(n_{p}-n_{s}\alpha_{\omega}\pm\Xi\right)$
(30)

is the momentum mismatch for colinear (d)SPDC. In some cases the colinear
spectrum is dominated by just the forward emission of the idler/$\varphi$,
while in others there are two phase matching solutions which contribute
similarly to the rate.

In Figure 6 we show two distributions, one for fixed signal frequency, and the
other for a fixed signal angle, the latter in the forward direction. We see
that both of these measurement schemes are able to distinguish the signal
produced by different values of mass $m_{\varphi}$. Furthermore, in cases
where the standard model SPDC process is a source of background, it can be
separated from the dSPDC signal. As expected, in the forward measurement the
dSPDC spectrum exhibits two peaks for the emission of $\varphi$ in the forward
or backward directions. It should be noted that the width of the highest peaks
in these distributions decreases with crystal length $L$.

Figure 6: Slices of the phase space distributions for SPDC and for dSPDC with
different choices of $\varphi$ 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
$m_{\varphi}$. The distributions become narrower for thicker crystals,
enhancing the signal to background separation power. For these distributions
we used $L=1\text{ cm}$, $\lambda_{p}=400\text{ nm}$ and for the refractive
indices $n_{p}=1.49$ and $n_{s}=n_{i}=1.66$ inspired by calcite.

##  IV Optical Materials for Dark SPDC

Having discussed the phase space distribution for dSPDC, we will now discuss
which optical media are needed to open this channel, to enhance its rates,
and, if possible, to suppress the backgrounds. A more complete estimate of the
dSPDC rate will be presented in ourpaper . Here we will discuss the various
features qualitatively to enable optimization of dSPDC searches.

###  IV.1 Refractive Indices

As we discussed above, the refractive index of the optical medium plays a
crucial role. It opens the phase space for the dSPDC decay and determines the
kinematics of the process.

In order to enhance the dSPDC rate, it is always 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 Equation (17) is the cosine of
an angle, its right hand is restricted between $\pm 1$ so

$-1<\frac{n_{p}^{2}+\alpha_{\omega}^{2}n_{s}^{2}-\Xi^{2}}{2\alpha_{\omega}n_{p}n_{s}}<1\text{.}$
(31)

From this one finds that so long as $n_{s}>n_{p}$, there is a range of
$\varphi$ mass which can take part in dSPDC. As the difference $n_{s}-n_{p}$
is taken to be larger, a greater $\varphi$ mass can be produced and larger
opening angles $\theta_{s}$ can be achieved.

Using the inequalities in Equation (31) we can explore the range of desirable
indices of refraction for a particular setup choice. For example, suppose we
use monochromatic filter for the signal at half the energy of the pump
photons, $\alpha_{\omega}=\frac{1}{2}$. Figure 7 shows in blue the region
where the phase matching condition is satisfied for the SPDC process and in
orange the region where the phase matching condition is satisfied for dSPDC.
As can be seen, dSPDC is more restrictive in the refractive indices.
Furthermore, typical SPDC experiments employ materials such as _beta barium
borate_(BBO), _potassium dideuterium phosphate_ (KDP) and _lithium triborate_
(LBO) kwiat_ultra-bright_1999 ; couteau_spontaneous_2018 ; ling_absolute_2008
; boeuf_calculating_2000  that do not allow the phase matching for dSPDC for
this example.

Figure 7: Regions in which the refractive indices $n_{p}$ and $n_{s}$ allow
the phase matching with
$\alpha_{\omega}\equiv\omega_{s}/\omega_{p}=\frac{1}{2}$ for the SPDC process
(in this case assuming $n_{i}=n_{s}$) and dSPDC (in this case assuming
$\frac{m}{\omega_{p}}\ll 1$).

A second example which is well motivated is the forward region, namely
$\theta_{s}=0$. In this case phase matching can be achieved so long as

$\frac{\left(n_{s}-n_{p}\right)^{2}}{n_{s}^{2}-1}\geq\frac{m_{\varphi}^{2}}{\omega_{p}^{2}}\,,$
(32)

and $n_{s}>n_{p}$. We will investigate colinear dSPDC in greater detail in
ourpaper .

The dSPDC phase matching requirement $n_{s}>n_{p}$ can be achieved in practice
by several effects. The most common one, employed in the majority of SPDC
experiments, is birefringence moreau_imaging_2019 ; kwiat_ultra-bright_1999 ;
magnitskiy_spdc-based_2015 ; boeuf_calculating_2000 . In this case the
polarization of each photon is used to obtain a different refractive index.
For instance, in calcite, the ordinary and extraordinary polarizations have
indices of refraction of $n_{o}=1.658$ and $n_{e}=1.486$. Taking the former to
be the signal and the later for the pump, phase matching can be met for
$m_{\varphi}<0.16\omega_{p}$. Birefringence may also be achieved in single-
mode fibers and waveguides with non-circular cross section, e.g. polarization-
maintaining optical fibers 1074847 . Of course, the dependence of the
refractive index on frequency may also be used to generate a signal-pump
difference in $n$.

###  IV.2 Linear and Birefringent Media for Axions

Axion electrodynamics is itself a nonlinear theory. Therefore dSPDC with
emission of an axion can occur in a perfectly linear medium. The optical
medium is needed in order to satisfy the phase matching condition that cannot
be satisfied in vacuum. Because the pump and signal photons in axion-dSPDC
have different polarizations, a birefringent material can satisfy the
requirement of $n_{s}>n_{p}$ and Equation (32). Because SPDC can be an
important background to a dSPDC axion search, using a linear or nearly linear
medium, where SPDC is absent is desirable. Interestingly, materials with a
crystalline structure that is invariant under a mirror transformation
$\boldsymbol{r}\to-\boldsymbol{r}$ will have a vanishing
$\chi^{\left(2\right)}$ from symmetry considerations boyd_nonlinear_2008 .

###  IV.3 Longitudinal Susceptibility for Dark Photon Searches

As opposed to the axion case, a dSPDC process with a dark photon requires a
nonlinear optical medium. As a result, SPDC can be a background to dSPDC dark
photon searches. In SPDC, the pump, signal, and idler are standard model
photons and their polarization is restricted to be orthogonal to their
propagation. As a consequence, when one wants to enhance an SPDC process, the
nonlinear medium is oriented in such a way that the second order
susceptibility tensor $\chi^{(2)}$ can appropriately couple to the transverse
polarization vectors of the pump, signal, and idler photons. The effective
coupling between the modes in question is given by

$\chi^{(2)}_{\mathrm{SPDC}}\equiv\chi^{(2)}_{jkl}\varepsilon^{p}_{j}\varepsilon^{s}_{k}\varepsilon^{i}_{l}$
(33)

with the $\varepsilon$ being the (transverse) polarization vectors for the
pump, signal and idler photons. These are spanned by $(1,0,0)$ and $(0,1,0)$
in a frame in which the respective photon is propagating in the $\hat{z}$
direction.

In dSPDC with a dark photon there is an important difference. Because the dark
photon is massive, it can have a longitudinal polarization,
$\varepsilon^{A_{L}^{\prime}}_{l}=(0,0,1)$. The crystal may be oriented to
couple to this longitudinal mode, giving an effective coupling of

$\chi^{(2)}_{A_{L}^{\prime}}\equiv\chi^{(2)}_{jkl}\varepsilon^{p}_{j}\varepsilon^{s}_{k}\varepsilon^{A_{L}^{\prime}}_{l}$
(34)

Maximizing this coupling benefits the search both by enhancing signal and
reducing background. The background is reduced because the SPDC effective
coupling to transverse modes, Equation (33) is suppressed. The signal is
enhanced because the coupling to the longitudinal polarization of the dark
photon is suppressed by $m_{A^{\prime}}/\omega$ rather than
$(m_{A^{\prime}}/\omega)^{2}$ An:2013yfc ; An:2013yua ; Graham:2014sha ;
ourpaper .

This motivates either non traditional orientation of nonlinear crystals, or
identifying nonlinear materials that would usually be ineffective for SPDC. As
an example in the first category, any material which is uses for type II phase
matching, in which the signal and idler have orthogonal polarization, can be
rotated by 90 degrees to achieve a coupling of a longitudinal polarization to
a pump and an idler. In the second category, material with $\chi^{(2)}$
tensors that are non-vanishing only in “maximally non-diaginal”
$\hat{x}$-$\hat{y}$-$\hat{z}$ elements, would obviously be discarded as
standard SPDC sources, but can have an enhanced dSPDC coupling.

##  V Dark SPDC Setups and Rates

A precise estimate of the rate is beyond the scope of this work, and will be
discussed in more detail in ourpaper . Here we will re-scale known rate
formulae in order to examine the dependence of the rate on the geometric
factors such as the pump power and beam area, the crystal length, and the area
and angle from which signal is collected.

Consider the geometry sketched in Figure 8. A pump beam is incident on an
optical element of length $L$ along the the pump direction.

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
integration length which is defined by the overlap of the three beams and the
crystal.

The width of the pump beam is set by the laser parameters. The pump beam may
consist of multiple modes, or can be guided in a fiber or waveguide
fiorentino_spontaneous_2007  and in a single mode. The width and angle of the
signal “beam” is set by the apparatus used to collect and detect the signal.
This too, can include collection of a single mode in a fiber (e.g.
ling_absolute_2008 ; bennink_optimal_2010 ) or in multiple modes (as in the
CCD example in the next section. In SPDC, when the idler photon may be
collected the width and angle of the idler can also be set by a similar
apparatus. However, in dSPDC (or in SPDC if we choose to only collect the
signal) the $\varphi$ beam is not a parameter in the problem. In this case we
are interested in an inclusive rate, and would sum over a complete basis of
idler beams. Such a sum will be performed in ourpaper , but usually the sum
will be dominated by a set of modes that are similar to those collected for
the signal.

Generally, the signal rate will depend on all of the choices made above, but
we can make some qualitative observations. The rate for SPDC and dSPDC will be
proportional to an integral over the volume defined by the overlap of the
three beams in Figure 8 and the crystal. When the length of the volume is set
by the crystal, the process is said to occur in the “thin crystal limit”. In
this limit, the beam overlap is roughly a constant over the crystal length and
thus the total rate will grow with $L$. Since dSPDC is a rare process, we
observe that taking the collinear limit of the process, together with a
thicker crystal may allow for a larger integration volume and an enhanced
rate.

The total rate for SPDC in a particular angle, integrated over frequencies, in
the thin crystal limit is ling_absolute_2008 ; ourpaper ,

$\Gamma_{\mathrm{SPDC}}\sim\frac{P_{p}{\chi^{(2)}_{\mathrm{eff}}}^{2}\omega_{s}\omega_{i}L}{\pi
n_{p}n_{s}n_{i}A_{\mathrm{eff}}}$ (35)

where $L$ is the crystal length, $P_{p}$ is the pump power, $A_{\mathrm{eff}}$
is the effective beam area, and ${\chi^{(2)}_{\mathrm{eff}}}$ is the effective
coupling of Equation (33). A parametrically similar rate formula applies to
SPDC in waveguides fiorentino_spontaneous_2007 ; ourpaper . The pump power
here is the effective power, which may be enhanced within a high finesse
optical cavity, e.g. Ehret:2009sq . The inverse dependence with effective area
can be understood by since the interaction hamiltonian is proportional to
electric fields, which grow for fixed power for a tighter spot. For “bulk
crystal” setups a pair production rate of order few times $10^{6}$ per mW of
pump power per second is achievable bennink_optimal_2010  in the forward
direction. In waveguided setups, in which the beams remain confined along a
length of the order of a cm, production rates of order $10^{9}$ pairs per mW
per second are achievable in KTP crystals fiorentino_spontaneous_2007 ;
spillane_spontaneous_2007 , and rates of order $10^{10}$ were discussed for LN
crystals spillane_spontaneous_2007 . Also noteworthy are cavity enhanced
setups in which the effective pump power is increased by placing the device in
an optical cavity, and/or OPOs, in which the signal photon is also a resonant
mode, including compact devices with high conversion efficiencies, e.g
furst_low-threshold_2010 ; beckmann_highly_2011 ; werner_continuous-wave_2015
; lu2021ultralowthreshold .

An important scaling of this rate is the $L/A_{\mathrm{eff}}$ dependence. This
scaling applies for dSPDC rates discussed below. Within the thin crystal limit
one can thus achieve higher rates with: (a) a smaller beam spot, and (b) a
thicker crystal. It should be noted that for colinear SPDC, the crystal may be
in the “thin limit” even for thick crystals (see Figure 8 and imagine zero
signal and idler emission angle).

###  V.1 Dark Photon dSPDC Rate

The dSPDC rate into a dark photon with longitudinal polarization,
$A_{L}^{\prime}$, can similarly be written as a simple re-scaling of the
expression above

$\Gamma^{(A^{\prime}_{L})}_{\mathrm{dSPDC}}\sim\epsilon^{2}\frac{m_{A^{\prime}}^{2}}{\omega_{A^{\prime}}^{2}}\frac{P_{p}{\chi^{(2)}_{A_{L}^{\prime}}}^{2}\omega_{s}\omega_{A^{\prime}}L}{n_{p}n_{s}A_{\mathrm{eff}}}$
(36)

where the effective coupling $\chi^{(2)}_{A_{L}^{\prime}}$ is defined in
Equation (34). This is valid in regions where the dark photon mass is smaller
than the pump frequency, such that the produced dark photons are relativistic
and have a refractive index of 1. Using the optimistic waveguide numbers above
as a placeholder, assuming an optimized setup with similar $\chi^{(2)}$, the
number of events expected after integrating over a time $t_{\mathrm{int}}$ are

$N^{(A^{\prime}_{L})}_{\mathrm{events}}\sim
10^{21}\,\left(\epsilon^{2}\frac{m_{A^{\prime}}^{2}}{\omega_{A^{\prime}}^{2}}\right)\left(\frac{P_{p}}{\mathrm{Watt}}\right)\left(\frac{L}{\mathrm{m}}\right)\left(\frac{t_{\mathrm{int}}}{\mathrm{year}}\right)\,.$
(37)

The current strongest lab-based limit is for dark photon masses of order 0.1
eV is set by the ALPS experiment at $\epsilon\sim 3\times 10^{-7}$. For this
dark photon mass $m_{A^{\prime}}\sim 0.1\omega_{p}$, the
$m_{A^{\prime}}/\omega_{A^{\prime}}$ term does not represent a large
suppression. In this case of order 10 dSPDC events are produced in a day in a
1 cm crystal with a Watt of pump power with the assumptions above. This
implies that a relatively small dSPDC experiment with an aggressive control on
backgrounds could be used to push the current limits on dark photons.

Improving the limits from solar cooling, for which $\epsilon
m_{A^{\prime}}/\omega_{A^{\prime}}$ is constrained to be smaller, would
represent an interesting challenge. Achieving ten events in a year of running
requires a Watt of power in a waveguide greater than 10 meters (or a shorter
waveguide with higher stored power, perhaps using a Fabri-Perot setup).
Interestingly, in terms of system size, this is still smaller than the ALPS-II
experiment which would reach 100 meters in length and an effecting power of a
hundred kW. This is because LSW setups require both production and detection,
with limits scaling as $\epsilon^{4}$.

|  Dark Photon ($m_{A^{\prime}}=0.1$ eV) | Axion-like particle ($m_{a}=0.1$ eV)
---|---|---
Current lab limit | $\epsilon<3\times 10^{-7}$ |  $g_{a\gamma}<10^{-6}$ GeV-1
Example dSPDC setup |  $P_{p}=1$ W |  $P_{p}=1$ kW
|  $L=1$ cm |  $L=10$ m
|  $\Gamma=10$/day |  $\Gamma=10$/day
Current Solar limit | $\epsilon<10^{-10}$ |  $g_{a\gamma}<10^{-10}$ GeV-1
Example dSPDC setup |  $P_{p}=1$ W |  $P_{p}=100$ kW
|  $L=10$ m |  $L=100$ m
|  $\Gamma=10$/year |  $\Gamma=10$/year
Table 1:  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.

###  V.2 Axion dSPDC Rate

A similar rate expression can be obtained for axion-dSPDC

$\Gamma^{\mathrm{(axion)}}_{\mathrm{dSPDC}}\sim\frac{P_{p}g_{a\gamma\gamma}^{2}\omega_{s}L}{\omega_{\mathrm{axion}}n_{p}n_{s}A_{\mathrm{eff}}}$
(38)

where the different scaling with the frequency of the dark field is due to the
different structure of the dSPDC interaction (recall that $\chi^{(2)}$ carries
a mass dimension of $-2$ while the axion photon coupling’s dimension is $-1$).
Optimal SPDC (dSPDC) rates are acieved in waveguides in which the effective
area is of order the squared wavelength of the pump and signal light. Assuming
a (linear) birefringent material that can achieve dSPDC phase matching for an
axion the number of signal event scales as

$N^{(\mathrm{axion)}}_{\mathrm{events}}\sim
40\,\left(\frac{g_{a\gamma}}{10^{-6}\leavevmode\nobreak\
\mathrm{GeV}^{-1}}\right)^{2}\left(\frac{P_{p}}{\mathrm{Watt}}\right)\left(\frac{L}{\mathrm{m}}\right)\left(\frac{t_{\mathrm{int}}}{\mathrm{year}}\right)\,.$
(39)

This rate suggests a dSPDC setup is promising in setting new lab-based limits
on ALPs. For example, a 10 meter birefringent single mode fiber with kW of
pump power will generate of order 10 events per day for couplings of order
$10^{-7}$ GeV-1. To probe beyond the CAST limits in $g_{a\gamma}$ may be
possible and requires a larger setup, but not exceeding the scale, say, of
ALPS-II. In a 100 meter length and an effective pump power of 100 kW, a few
dSPDC signal events are expected in a year.

Maintaining a low background, would of course be crutial. We note, however,
that optical fibers are routinely used over much greater distances,
maintaining coherence (e.g. 2020arXiv200711157V ), and an optimal setup should
be identified.

###  V.3 Backgrounds to dSPDC

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

  * •

Crystal Length and Signal bandwidth: In addition to the growth of the signal
rate, the signal bandwidth in many setups will decrease with $L$. If this is
achieved the signal to background ratio in a narrow band around the dSPDC
phase matching solutions will scale as $L^{2}$.

  * •

Timing: The dSPDC signal consists of a single photon whereas SPDC backgrounds
will consist of two coincident photons. Backgrounds can be reduced using fast
detectors and rejecting coincidence events.

  * •

Optical material: As pointed out in Section IV, linear birefringent materials
can be used to reduce SPDC backgrounds to axion searches. Nonlinear materials
with a $\chi^{(2)}$ tensor which couples purely to longitudinal polarizations
may be used to enhance dark photon dSPDC events without enhancing SPDC. This
technique to reduce SPDC may also be used in axion searches.

  * •

Detector noise and optical impurities: Sources of background which may be a
limiting factor for dSPDC searches include detector noise, as well as
scattering of pump photons off of impurities in the optical elements and
surfaces. The Skipper CCD, one example of a detector technology with low
noise, will be discussed in the next section.

An optimal dSPDC based search for dark particles will likely consider these
factors, and is left for future investigation.

##  VI Experimental proof of concept

In this section we present an experimental SPDC angular imaging measurement
with high resolution employing a Skipper CCD and a BBO nonlinear crystal. In
this setup and for the chosen frequencies, dSPDC phase matching is not
achievable at any emission angle. Instead, this experiment serves as a proof
of concept for the high resolution imaging technique.

Imaging the dSPDC requires the detection of single photons with low noise and
with high spatial and/or energy resolution. A technology that can achieve this
is the Skipper CCD which is capable of measuring the charge stored in each
pixel with single electron resolution tiffenberg_single-electron_2017 ,
ranging from very few electrons (0, 1, 2, $\dots$) up to more than a thousand
(1000, 1001, 1002, $\dots$) Rodrigues2020 . This unique feature combined with
the high spatial resolution typical of a CCD detector makes this technology
very promising for the detection of small optical signals with a very high
spatial resolution.

###  VI.1 Description of the experimental setup

With the aim of comparing the developed phase space model against real data,
the system depicted in Figure 9 was set up. A source of entangled photons that
employs SPDC, which is part of a commercial
system333https://www.qutools.com/qued/, was used. Two type I nonlinear BBO
crystals are used as a high efficient source of entangled photons kwiat_ultra-
bright_1999 . This pre-assembled source can be seen in Figure 10. However, in
this work, we did not take advantage of the polarization entanglement, we only
use the energy-momentum conservation to get spatially correlated twin photons.
In addition to the original design, we added a Thorlabs FGB37 filter after the
laser to remove a small $810\text{ n}\mathrm{m}$ component coming along with
the pump beam. After the BBO crystal, where the SPDC process occurs, two
additional $810\text{ n}\mathrm{m}$ band pass filters (one Semrock Brighline
LL01-810 and one Asahi Spectra 97SA) were placed to prevent the $405\text{
n}\mathrm{m}$ pump beam from reaching the detector. We also placed a lens at
its focal distance from the BBO to prevent the SPDC cone from spreading during
its travel to the CCD (see Figure 9). Thus, we got the SPDC ring projected on
the CCD surface with a radius of $110\,\mathrm{pixel}$, which corresponds to
$1650\text{ }\mu\mathrm{m}$. This setup is also part of a research where the
novel features of Skipper CCD tiffenberg_single-electron_2017  are being
tested for Quantum Imaging. The capability of Skipper-CCD to reduce the
readout noise as low as desired taking several samples of the charge in each
pixel was not used in this work to acquire data, but it was for the
calibration. Details about the same detector used here can be found in
reference Rodrigues2020 .

Figure 9: Pictorial representation of the experimental setup implemented in
the lab. A $405\text{ n}\mathrm{m}$ diode laser was used as a source of pump
photons. The beam passes through a $810\text{ n}\mathrm{m}$ rejection filter,
then through the BBO crystal, after this through two $810\text{ n}\mathrm{m}$
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 aperture. All the other components were placed outside
this device. Figure adapted from qutools_qued_nodate .

We used an absolute calibration between the Analog to Digital Units (ADU)
measured by the amplifier from the CCD and the number of electrons in each
pixel, possible with the Skipper-CCD, which was previously carried out for
this system Rodrigues2020 . Thus, we reconstructed the number of photons per
pixel, since at $810\text{ n}\mathrm{m}$ it can be assumed that one photon
creates at most one electron, by reason of its energy is $0.43\,\mathrm{eV}$
above the silicon band gap ($1.1\,\mathrm{eV}$). The factor affecting that
relationship is the efficiency which -besides being very high ($\sim
90\,\mathrm{\%}$) at this wavelength- is pretty uniform over the entire CCD
surface Bebek_2017 .

###  VI.2 Results

Using the setup previously described, 400 images with $200\,\mathrm{s}$
exposure each were averaged to produce the data used for comparison with the
model. Thus, we reduced a factor twenty the uncertainty in the expected number
of photons in each pixel and drastically reduced the dark counts coming from
random background. This significantly improved the identification of minima
between rings, which results to be crucial to compare the experimental data
with the presented theory.

Figure 11 presents the averaged image of the SPDC ring coming out of the BBO
crystal. The main ring is clearly visible, as well as many secondary maxima
and minima. The angular coordinates $\theta$ and $\phi$ were indicated on the
image. It has to be noted that the angular coordinates used in the previous
sections are the ones inside the optical medium. Since the CCD detector is
outside the BBO crystal, Snell’s law must be used to relate the angles inside
and outside the nonlinear crystal. Specifically, for the $\theta$ coordinate
this is

$\theta=\arcsin\left(\frac{\sin\Theta}{n}\right)$

where $\theta$ is the angle inside the optical medium in which the SPDC
process occurs, $\Theta$ is the angle outside the optical medium and $n$ is
the refractive index. In this work we apply the transformation to refer all
angles to those inside the optical medium in which the interaction occurs.

Figure 11: Average of 400 images with $200\,\mathrm{s}$ exposure each to the
SPDC photons each using the setup depicted in Figure 9. The angular
coordinates $\theta$ and $\phi$ are indicated in the figure. The blue dashed
line was used to extract the data profile shown in Figure 12.

As can be seen in the image, even though it is possible to see many of the
maxima and minima, there is a non uniform background component produced by
reflections and other imperfections of the experimental configuration. The
current implementation of our setup has some mechanical limitations that make
hard to isolate, measure and remove this non-uniform background 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 shown in
Figure 11 was extracted and plotted against the model. This plot is shown in
Figure 12. Some comments about this plot:

  * •

A uniform background component was added to the model.

  * •

The transmittance curve of the $810\text{ n}\mathrm{m}$ filters provided by
the manufacturers was taken into account.

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

$I_{\text{model}}\left(\theta\right)=I_{0}\int\frac{d^{2}\hat{\Gamma}}{d(\cos\theta_{s})d\alpha_{\omega}}F(\alpha_{\omega})d\alpha_{\omega}+I_{1}$

where $I_{0}$ and $I_{1}$ are real numbers fitted,
$\frac{d^{2}\hat{\Gamma}}{d(\cos\theta_{s})d\alpha_{\omega}}$ is given by
Equation (27) and $F(\alpha_{\omega})$ is the transmittance of the filters as
provided by the manufacturers, which is a sharp peaked function centered at
$810\text{ n}\mathrm{m}$ with a pass band of $10\text{ n}\mathrm{m}$ width.

Figure 12: Experimental intensity profile for the SPDC ring along the blue
dashed line in Figure 11 compared with theoretical distribution. The $y$ axis
scale is the number of electrons in each pixel of the CCD detector obtained
via the absolute Skipper-CCD calibration described.

For the parameters of the distribution $\hat{\Gamma}$, we used:

  * •

$405\text{ n}\mathrm{m}$ for the pump wavelength as provided by the
manufacturer of the laser,

  * •

the refractive indices for the BBO crystal were taken from
https://refractiveindex.info/ 444The extraordinary refractive index was taken
from https://refractiveindex.info/?shelf=main&book=BaB2O4&page=Tamosauskas-e
(TamoÂšauskas et al., 2018) and the ordinary refractive index from
https://refractiveindex.info/?shelf=main&book=BaB2O4&page=Eimerl-o (Eimerl et
al., 1987).

  * •

and the crystal length was assumed to be $L=1.14\text{ m}\mathrm{m}$ after
fine tuning 555It was not possible to obtain precise information of the
dimensions of the crystal from the manufacturer. Furthermore, the length of
the crystal is not easy to measure with our current implementation without
breaking the SPDC source shown in Figure 10. So we decided to use a
“reasonable value” and fine tune it..

We had to perform a fine tuning of $n_{s}-n_{p}$ too (since signal and idler
have the same frequency and angle then $n_{s}=n_{i}$). The model is very
sensitive to this difference, and this quantity depends on factors such as the
temperature of the BBO and its precise orientation in space, which were not
measured. So it is not surprising that we had to perform this fine tuning on
$n_{s}-n_{p}$. The final values used for the refractive indices were
$n_{p}\approx 1.66082$ and $n_{s}=n_{i}\approx 1.66107$. Although this fine
tuning had to be done, all parameters are very well within expected values.

As seen both in Figures 11 and 12 the phase space 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 a non negligible background
component, it is evident the dependence
$I\sim\text{sinc}^{2}\left(\theta^{2}\right)$ for small values of $\theta$
predicted by Equation (27).

##  VII Discussion

We have presented a new method to search for new light and feebly coupled
particles, such as axion-like particles and dark photons. The dSPDC process
allows to tag the production of a dark state as a pump photon down-converts to
a signal photon and a “dark idler” $\varphi$, in close analogy to SPDC. We
have shown that the presence of indices of refraction that are different than
1 open the phase space for the decay, or down conversion, of the massless
photon to the signal plus the dark particle, even if $\varphi$ has a mass.
This type 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 again. Precise sensitivity calculations for dSPDC with Dark
Photon and Axion cases that can be achieved through this method are ongoing
ourpaper .

The commonplace use of optics in telecommunications, imaging and in quantum
information science, as well as the development of advanced detectors, can
thus be harnessed to search for dark sector particles. Increasing dSPDC signal
rates will require high laser power, long optical elements. Enhancing the
signal and suppressing SPDC backgrounds also requires identifying the right
optical media for the search. Axion searches would benefit from optically
linear and birefringent materials, with greater birefringence allowing to
search for higher axion masses. Searches for dark photons would benefit from
strongly nonlinear materials that are capable of coupling to a longitudinal
polarization. This, in turn, motivated either non-conventional optical media,
or using conventional crystals that are oriented by a $90^{\circ}$ rotation
from that which is desired for SPDC. Enhancing the effective pump power with
an optical cavity is a straightforward 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 with Ehret:2009sq ) for future work. Finally,
detection of rare signal events requires sensitive single photon detectors
with high spatial and/or frequency resolution.

We also performed an experimental demonstration of the one of the setups
discussed using a Skipper CCD for SPDC imaging. The setup we used for this
demonstration was adapted from one designed to enforce SPDC and thus does not
open dSPDC phase space. We show that the Skipper technology allows one to
measure with high accuracy. Thus, quantum teleportation methods or ghost
imaging of dark sector particles achieved through phase space engineering is a
very promising technique for future explorations of dark sector parameter
spaces.

$\mathghost\mathghost\mathghost\mathghost$

Acknowledgments: We would like to thank Joe Chapman, Paul Kwiat, and Neal
Sinclair for informative discussions. This work was funded by a DOE QuantISED
grant.

## References

  * [1] A. Furusawa.  Unconditional Quantum Teleportation.  Science, 282(5389):706–709, October 1998.
  * [2] D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu.  Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels.  Physical Review Letters, 80(6):1121–1125, February 1998.
  * [3] Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter, and Anton Zeilinger.  Experimental quantum teleportation.  Nature, 390(6660):575–579, December 1997.
  * [4] Charles H. Bennett, Gilles Brassard, Claude CrÃ©peau, Richard Jozsa, Asher Peres, and William K. Wootters.  Teleporting an unknown quantum 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. Allmaras, Andrew D. Beyer, Yewon Gim, Meraj Hussein, George Iskander, Hyunseong Linus Kim, Boris Korzh, Andrew Mueller, Mandy Rominsky, Matthew Shaw, Dawn Tang, Emma E. Wollman, Christoph Simon, Panagiotis Spentzouris, Neil Sinclair, Daniel Oblak, and Maria Spiropulu.  Teleportation Systems Towards a Quantum Internet.  arXiv e-prints, page arXiv:2007.11157, July 2020.
  * [6] Interaction.
  * [7] C. K. Hong and L. Mandel.  Theory of parametric frequency down conversion of light.  Physical Review A, 31(4):2409?2418, 1985.
  * [8] J. A. Giordmaine and Robert C. Miller.  Tunable Coherent Parametric Oscillation in LiNb O 3 at Optical Frequencies.  Physical Review Letters, 14(24):973–976, June 1965.
  * [9] 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.
  * [10] F. Hoogeveen and T. Ziegenhagen.  Production and detection of light bosons using optical resonators.  Nucl. Phys. B, 358:3–26, 1991.
  * [11] Jessica Goodman, Masahiro Ibe, Arvind Rajaraman, William Shepherd, Tim M.P. Tait, and Hai-Bo Yu.  Constraints on Light Majorana dark Matter from Colliders.  Phys. Lett. B, 695:185–188, 2011.
  * [12] Yang Bai, Patrick J. Fox, and Roni Harnik.  The Tevatron at the Frontier of Dark Matter Direct Detection.  JHEP, 12:048, 2010.
  * [13] 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.
  * [14] Patrick J. Fox, Roni Harnik, Joachim Kopp, and Yuhsin Tsai.  Missing Energy Signatures of Dark Matter at the LHC.  Phys. Rev. D, 85:056011, 2012.
  * [15] T. Aaltonen et al.  A Search for dark matter in events with one jet and missing transverse energy in $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV.  Phys. Rev. Lett., 108:211804, 2012.
  * [16] Marco Fiorentino, Sean M. Spillane, Raymond G. Beausoleil, Tony D. Roberts, Philip Battle, and Mark W. Munro.  Spontaneous parametric down-conversion in periodically poled KTP waveguides and bulk crystals.  Optics Express, 15(12):7479–7488, June 2007.
  * [17] Juan Estrada, Roni Harnik, Dario Rodrigues, and Matias Senger.  New Particle Searches with Nonlinear Optics.  In preparation.
  * [18] Javier Tiffenberg, Miguel Sofo-Haro, Alex Drlica-Wagner, Rouven Essig, Yann Guardincerri, Steve Holland, Tomer Volansky, and Tien-Tien Yu.  Single-electron and single-photon sensitivity with a silicon Skipper CCD.  Physical Review Letters, 119(13):131802, September 2017.  arXiv: 1706.00028.
  * [19] Alexander Ling, Antia Lamas-Linares, and Christian Kurtsiefer.  Absolute emission rates of Spontaneous Parametric Down Conversion into single transverse Gaussian modes.  Physical Review A, 77(4):043834, April 2008.  arXiv: 0801.2220.
  * [20] James Schneeloch and John C Howell.  Introduction to the transverse spatial correlations in spontaneous parametric down-conversion through the biphoton birth zone.  Journal of Optics, 18(5):053501, 2016.
  * [21] P.A. Zyla et al.  Review of Particle Physics.  PTEP, 2020(8):083C01, 2020.
  * [22] Georg G. Raffelt.  Astrophysical methods to constrain axions and other novel particle phenomena.  Phys. Rept., 198:1–113, 1990.
  * [23] V. Anastassopoulos et al.  New CAST Limit on the Axion-Photon Interaction.  Nature Phys., 13:584–590, 2017.
  * [24] Federico Della Valle, Aldo Ejlli, Ugo Gastaldi, Giuseppe Messineo, Edoardo Milotti, Ruggero Pengo, Giuseppe Ruoso, and Guido Zavattini.  The PVLAS experiment: measuring vacuum magnetic birefringence and dichroism with a birefringent Fabry–Perot cavity.  Eur. Phys. J. C, 76(1):24, 2016.
  * [25] R. Ballou et al.  New exclusion limits on scalar and pseudoscalar axionlike particles from light shining through a wall.  Phys. Rev. D, 92(9):092002, 2015.
  * [26] 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.
  * [27] Robin BÃ€hre et al.  Any light particle search II —Technical Design Report.  JINST, 8:T09001, 2013.
  * [28] Haipeng An, Maxim Pospelov, and Josef Pradler.  New stellar constraints on dark photons.  Phys. Lett. B, 725:190–195, 2013.
  * [29] Haipeng An, Maxim Pospelov, and Josef Pradler.  Dark Matter Detectors as Dark Photon Helioscopes.  Phys. Rev. Lett., 111:041302, 2013.
  * [30] David Tong.  Quantum Field Theory.  http://www.damtp.cam.ac.uk/user/tong/qft.html.
  * [31] Paul G. Kwiat, Edo Waks, Andrew G. White, Ian Appelbaum, and Philippe H. Eberhard.  Ultra-bright source of polarization-entangled photons.  Physical Review A, 60(2):R773–R776, August 1999.  arXiv: quant-ph/9810003.
  * [32] Christophe Couteau.  Spontaneous parametric down-conversion.  Contemporary Physics, 59(3):291–304, July 2018.
  * [33] N. Boeuf.  Calculating characteristics of noncollinear phase matching in uniaxial and biaxial crystals.  Optical Engineering, 39(4):1016, April 2000.
  * [34] Paul-Antoine Moreau, Ermes Toninelli, Thomas Gregory, and Miles J. Padgett.  Imaging with quantum states of light.  Nature Reviews Physics, 1(6):367–380, June 2019.
  * [35] S. Magnitskiy, D. Frolovtsev, V. Firsov, P. Gostev, I. Protsenko, and M. Saygin.  A SPDC-Based Source of Entangled Photons and its Characterization.  Journal of Russian Laser Research, 36(6):618–629, November 2015\.
  * [36] J. Noda, K. Okamoto, and Y. Sasaki.  Polarization-maintaining fibers and their applications.  Journal of Lightwave Technology, 4(8):1071–1089, 1986.
  * [37] Robert W. Boyd.  Nonlinear optics.  Academic Press, Amsterdam ; Boston, 3rd ed edition, 2008.
  * [38] Peter W. Graham, Jeremy Mardon, Surjeet Rajendran, and Yue Zhao.  Parametrically enhanced hidden photon search.  Phys. Rev. D, 90(7):075017, 2014.
  * [39] Ryan S. Bennink.  Optimal Co-linear Gaussian Beams for Spontaneous Parametric Down-Conversion.  Physical Review A, 81(5):053805, May 2010.  arXiv: 1003.3810.
  * [40] Sean M. Spillane, Marco Fiorentino, and Raymond G. Beausoleil.  Spontaneous parametric down conversion in a nanophotonic waveguide.  Optics Express, 15(14):8770–8780, July 2007.
  * [41] J. U. FÃŒrst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, Ch. Marquardt, and G. Leuchs.  Low-Threshold Optical Parametric Oscillations in a Whispering Gallery Mode Resonator.  Physical Review Letters, 105(26):263904, December 2010.
  * [42] T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig.  Highly Tunable Low-Threshold Optical Parametric Oscillation in Radially Poled Whispering Gallery Resonators.  Physical Review Letters, 106(14):143903, April 2011.
  * [43] Christoph S. Werner, Karsten Buse, and Ingo Breunig.  Continuous-wave whispering-gallery optical parametric oscillator for high-resolution spectroscopy.  Optics Letters, 40(5):772, March 2015.
  * [44] Juanjuan Lu, Ayed Al Sayem, Zheng Gong, Joshua B. Surya, Chang-Ling Zou, and Hong X. Tang.  Ultralow-threshold thin-film lithium niobate optical parametric oscillator, 2021.
  * [45] Dario Rodrigues, Kevin Andersson, Mariano Cababie, Andre Donadon, Gustavo Cancelo, Juan Estrada, Guillermo Fernandez-Moroni, Ricardo Piegaia, Matias Senger, Miguel Sofo Haro, et al.  Absolute measurement of the fano factor using a skipper-ccd.  arXiv preprint arXiv:2004.11499, 2020.
  * [46] qutools.  quED.
  * [47] 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 of the CCD development for the dark energy spectroscopic instrument.  Journal of Instrumentation, 12(04):C04018–C04018, apr 2017.