Superradiant Searches for Dark Photons in Two Stage Atomic Transitions

We study a new mechanism to discover dark photon fields, by resonantly triggering two photon transitions in cold gas preparations. Using coherently prepared cold parahydrogen, coupling sensitivity for sub-meV mass dark photon fields can be advanced by orders of magnitude, with a modified light-shining-through-wall setup. We calculate the effect of a background dark photon field on the dipole moment and corresponding transition rate of cold parahydrogen pumped into its first vibrational excited state by counter-propagating laser beams. The nonlinear amplification of two photon emission triggered by dark photons in a cold parahydrogen sample is numerically simulated to obtain the expected dark photon coupling sensitivity. 1 ar X iv :1 90 9. 07 38 7v 1 [ he pph ] 1 6 Se p 20 19

Using two stage atomic transitions, this paper proposes a new method to enhance the detection of dark photons produced by lasers shining through walls. Our proposal involves a dark photon field produced in a laser cavity, then passed through a sample of quasi-stable coherently excited atoms whose E1 dipole transitions are parity-forbidden. Under these conditions, during the brief time that the excited atoms are coherent, the diminutive field of the dark photon can resonantly trigger two-photon electronic transitions. As compared to traditional light-shining-through-wall experiments, we project a large gain in sensitivity to dark photons with µeV-meV masses. These sensitivity gains appear within reach using preparations of parahydrogen (pH 2 ) coherently excited by counter-propagating nanosecond laser pulses [32,33].
The use of two stage superradiant atomic transitions for the production and detection of weakly coupled particles was proposed and studied extensively by Yoshimura et al. [32][33][34][35][36][37][38][39][40]. These authors have pointed out that macroscopic quantities of coherently excited atoms may be employed to measure neutrino properties [36,39,41]. The use of multi-stage atomic transitions for the discovery of axion dark matter has recently been considered in [42,43]. In contrast, the experiment we propose is sensitive to any U(1) vector bosons kinetically-mixed with the Standard Model photon, whether or not dark matter is comprised of a dark photon.
Coherent superradiant emission by atomic systems was formalized by Dicke in [44]. However, the possibility that superradiance might be observed in macroscopic amalgams of material has received increased attention in the last decade after being proposed as a method to measure certain neutrino properties [35,36]. Before proceeding further, we will develop some physical intuition about classic (aka Dicke) versus macro (aka Yoshimura) superradiance. A formal derivation can be found in Appendix A. Let us consider a group of atomic emitters with number density n occupying volume V , which have been prepared in excited states, such that each excited atomic emitter is indistinguishable from the next (we want them to have the same phase). Let us suppose that some atom in volume V de-excites and emits a photon with momentum k 1 . Then if a single, isolated atom has a photon emission rate Γ 0 , and for the moment neglecting superradiant effects (superradiant effects would indeed be negligible if the spacing between atomic emitters is much greater than wavelength of photons emitted, n −1/3 k −1 1 ) the emission rate of photons from volume V follows trivially, Γ tot = nV Γ 0 .
However, if the wavelength of the emitted photon is larger than the inter-atomic spacing, there will be a superradiant enhancement to the rate of photon emission. In fact, in the case that the emitted photon's wavelength is much greater than the volume itself (k −3 1 V ) the total rate of photon emission will be Γ tot = n 2 V 2 Γ 0 , because of superradiance. This superradiant enhancement can be understood from basic quantum principles as follows. Firstly, momentum conservation tells us that an emitting atom in its final state will have a momentum of size ∼ k 1 . The uncertainty principle tells us that the atom is only localized over a distance ∼ 1 k 1 . Altogether these imply that it is not possi-

Dicke Superradiance
Macro Superradiance k1 k2 → → -Δk = k1+k2 = → → → k1 → -Δk = k1 → → Figure 1: Illustration of single photon emission, aka Dicke superradiance and two photon emission, aka Yoshimura superradiance. Crucially, the volume for superradiant emission is determined by the final state momentum ∆k of the atomic emitters. For classic Dicke superradiance, this is just the momentum of the emitted photons ∆k = |k 1 |. For two photons emitted back-to-back, the final state momentum can be tiny, ∆k ≈ |k 1 − k 2 | → 0. Of course, superradiant emission will also depend on the linewidth of lasers used to excite the atoms, material dephasing effects, and other factors, see Section 2.
ble to determine which atom in the volume k −3 1 emitted the photon. But we know that quantum mechanics tells us that the probability for an event to occur is the squared sum of all ways for the event to occur, and if we cannot distinguish between atoms, then we must sum over all atoms in the coherence volume k −3 1 , and square that sum to obtain the probability for emission. From this we obtain an extra factor of nV in our superradiantly enhanced emission rate. This is illustrated in Figure 1.
In the preceding heuristic argument, it is crucial to realize that it is the final state momentum and corresponding spatial uncertainty of the emitting atoms that determines the volume over which superradiant emission can occur. Therefore, a process that somehow reduces the final state momentum of atomic emitters, can potentially result in superradiant emission over volumes larger than the wavelength of emitted photons. Perhaps the simplest such process is two photon emission. In two photon emission, the final state momentum of the emitting atom will be the sum of emitted momenta ∆k = k 1 + k 2 . For back-to-back two photon emission where k 1 ≈ −k 2 , the superradiant emission volume ∆k −3 can in principle be arbitrarily large -it is only limited by the difference in momentum of the emitted photons. This is illustrated in Figure 1. In practice, dephasing of atoms, the related decoherence time of the atomic medium, and the linewidth of lasers used to excited the atoms will also limit the superradiant emission volume. It is interesting to note that much of the preceding logic about cooperative emission of photons could be equally applied to cooperative absorption.
The remainder of this paper proceeds as follows. In Section 2, we calculate coherence in two stage electronic transitions, and study how much coherence is attained in cold preparations of parahydrogen excited by counter-propagating lasers. We find that the coherence necessary to begin realizing our proposal has been obtained in a number of experiments. The interaction of a dark photon with coherently excited two stage atomic systems is derived in Section 3.2. Enthusiastic readers may wish to skip to Section 4, which includes a schematic description of the experiment, along with its sensitivity to a kinetically mixed dark photon. Determining this dark photon sensitivity requires numerically integrating the dark and visible photon field equations in a background of coherently excited atoms. Conclusions are presented in Section 5. Throughout we use natural units where = c = k B = 1.

Coherence in two stage atomic transitions
Pulses from high-power lasers allow for the preparation of atoms in coherent excited states, from which they can be cooperatively de-excited. Before investigating how the weak electromagnetic field sourced by a dark photon can be detected by cooperatively de-exciting coherently prepared atoms, it will be useful to examine under what conditions counter-propagating lasers excite highly coherent atoms in the first place. After deriving the coherence of atomic states excited by counter-propagating pairs of photons, we will examine how laser power, atomic density, and temperature alter this coherence. The derivation given below can be found in many prior references [45,46]. Our aim here is to quantify the experimental capability, in terms of coherently excited atoms, that will be needed to detect dark photons.

Quasi-stable excited states
We first consider an atomic system with ground state |g and excited state |e . For the atomic systems we are interested in, for example vibrational modes of parahydrogen, and electronic states of Ytterbium, or Xenon [41], both states |g and |e will have even parity, meaning that E1 dipole transitions between the two states are forbidden. However, it will be possible to excite state |g to state |e through multiple E1 dipole transitions, and similarly de-excite |e to |g . So besides states |g and |e , we consider intermediate states, |j + and |j − , where +(−) will indicate excitation into a positive (negative) angular momentum state by a circularly polarized photon. Figure 2 illustrates the basic setup. In physical realizations, there will be many j states to transition through, for example the = 0, 1, 2, 3... electronic angular momentum states of hydrogen. Since by design these excited states will lie at energies beyond those provided by the input lasers, transitions |e> |g> |e> |g> |j ± > X Figure 2: Illustration of the energy levels of an atomic system, with ground state |g and excited state |e . E1 dipole transitions between |g and |e are forbidden; the two-step process of transtioning from |g to |e through a virtual state |j ± is shown.
through these states will be virtual. Defining our atomic Hamiltonian as where H I is the interaction Hamiltonian and H 0 is defined by H 0 |g = ω g |g , H 0 |e = ω e |e , H 0 |j ± = ω j |j ± . With our states specified, we define the wavefunction for this simplified atomic system We have added a phase δ to account for detuning of the lasers; in other words, the laser beams exciting the atoms will be off resonance by a factor ∼ δ. The laser-atomic interaction Hamiltonian will depend on the orientation and quality of the impinging laser beams. Experimental setups similar to the one we are outlining (for example [33]), employ counter-propagating beams which have been circularly polarized. Therefore, we will consider two counter-propagating laser beams propagating along the z direction with electric fields given as where r , l are unit normalized right-and left-handed polarization vectors for the laser beams. Then the laser-atom interaction Hamiltonian is where d is the polarization of the atom. The actual dipole coupling and transition rate are experimental inputs in these formulae. Here we define the expectation value of the dipole transitions, with the assumption that both counter-propagating pump lasers will have left-handed circular polarization, using the convention that left-handedness is defined along the direction of the beam propagation. More explicitly, sinceẼ 2 is electric field of a laser beam propagating in the +z direction, while also forẼ 2 The same relations hold forẼ 1 , except with l ↔ r , sinceẼ 1 is the electric field of the laser beam propagating in the −z direction. Finally, assuming that the two lasers will carry the same frequency (up to a detuning factor δ), we define ω ≡ ω 1 = ω 2 = ω eg /2, and we use the convention throughout this paper that ω ik for any states i, k is defined as ω ik = ω i − ω k , to arrive at the following Schrödinger equations for this multi-state atomic system where we have incorporated the spatial part of the electric fields into "barred" quantities, The sum over all intermediate states j is implicit in Eqs. (8)- (11). To find the time evolution of this system, we first integrate Eq. (8) and Eq. (9) over t. We will be using the so-called Markov approximation, treating c g , c e as constant in the resulting integral. This standard approximation is justified, so long as we expect virtual transitions through c j+ and c j− to be sufficiently rapid compared to changes in c g , c e , which should be satisfied so long as the frequency of the |g → |e transition is substantially smaller than the frequency of higher energy atomic states, ω eg ω je , ω jg . For example, in the case of pH 2 the frequency of the first vibrational state, ω eg ∼ 0.5 eV, can be compared to the lowest lying electronic excitations, ω je , ω jg ∼ 10 eV, from which we conclude that the Markov approximation is justified. Using similar logic, we approximate the electric fields of the laser beams as being constant in this integral, since the laser frequency is also small compared to the transition frequencies to intermediate j states. Setting the initial condition c j ± ,0 = 0, we find the time evolution of c j+ and c j− , Substituting these solutions for c j+ and c j− into the Schrodinger equations for c g and c e , we invoke the slowly varying envelope approximation, i.e. we assume that since the development of the electric fields around the atoms is slow compared to the frequencies of all transitions, all time-dependent exponentials of the form can be set to zero. With the slowly varying envelope approximation, the two state system can be compactly expressed as with the effective Hamiltonian where Ω ge is the Rabi frequency of the system, and we have defined interstate dipole couplings as in [46], Applying the density matrix of the atomic system ρ = |e e| |e g| |g e| |g g| = ρ ee ρ eg ρ ge ρ gg (23) to the von Neumann equation i∂ t ρ = [H ef f , ρ], leads to the Maxwell-Bloch equations The final terms in Eqs. (24), (25), and (26) have been added to account for spontaneous |e → |g transitions and the decoherence time of the mixed state. As such, T 1 is the excited state lifetime and T 2 is the decoherence time.
To better quantify the coherence of this system, we define the Bloch vector r = Tr(σρ) where σ are the Pauli matrices. This implies By construction, r 1 and r 2 quantify the degree to which the atoms are coherently excited of the system, with maximum coherence attained when r 1 = 1 and r 3 = 0. The Bloch vector direction r 3 indicates the population difference between the excitation and the ground states. Note that r 1 , r 2 , and r 3 are all real numbers. Applying the Bloch vector basis to Eqs. (24), (25), and (26) we obtain where we note that a eg is assumed to be real.

Quantifying coherence in quasi-stable excited states
Using the Bloch vector time evolution given by Eqs. (30)-(32), we can now determine the degree and duration of coherence in cold atomic preparations excited by two counterpropagating lasers with electric fieldsẼ 1 andẼ 2 . We consider an excited set of atoms with an expected spontaneous deexcitation time (not including superradiant enhancement) of T 1 and a decoherence time T 2 . In the case of the first vibrationally excited state of pH 2 , the total lifetime has been observed to be T 1 ∼ 10 µs at ∼ 10 K temperatures [47][48][49], which will be appreciably longer than the decoherence time of the first pH 2 vibrational excitation at these temperatures, where this decoherence time will be of order ∼ 1 − 100 ns [50]. In more detail, the decoherence time (T 2 ) of pH 2 has been studied extensively for a variety of temperatures and densities [33,36,40,50]. In some regimes, it is accurate to use the mean interaction time of hydrogen atoms as an estimate of the decoherence time resulting from pH 2 collisions at number density n, where this expression has approximated the velocity of pH 2 using the temperature (T ) and mass (m H ) of hydrogen, and the cross-section using the Bohr radius, σ H ≈ πr 2 bohr . While Eq. (33) is remarkably close to the measured decoherence time for a sample of pH 2 prepared at T ∼ 80 Kelvin and density n ∼ 3 × 10 19 cm −3 , this approximation will break down for sufficiently cold and dense pH 2 , which will not behave like an ideal gas. In addition, we should note that the Raman linewidth, or full-width-half-at-half-maximum of pH 2 's first vibrational emission line, is often used to determine the decoherence time. However, this linewidth also has a contribution from Doppler broadening of pH 2  [33,38], the decoherence time is an estimate using results from Ref. [50].
where here ω 0 is the first vibrational mode frequency. A total decoherence determination for the first vibrational mode of pH 2 , for temperatures ranging from 77 − 500 K, was approximated by fitting a phenomenological formula [50] where, for example, it was found that for T = 80 K, the collisional term A ≈ 100 MHz cm −3 , and the broadening term B ≈ 20 MHz cm 3 , which implies a 10 ns decoherence time for n ∼ 10 19 cm −3 , as previously noted. Given the theoretical expectations and experimental results detailed above, it is safe to assume that T 2 ≈ 10 ns is an achievable decoherence time for cold parahydrogen. In terms of Bloch vector r 1 , the largest coherence reported in a similar setup was r 1 0.07 for parahydrogen at density n ∼ 5 × 10 19 cm −3 [32]. In the remainder of this article, we will find that advancing coupling sensitivity to dark photons (assuming a roughly 30 cm cylindrical chamber and 1 cm laser beam diameter) requires parahydrogen number densities nearer to n ∼ 10 21 cm −3 . As noted in Figure 3, a higher-power laser than that used in [32] is also required. In Figure 3, we have shown how coherence of pH 2 can be expected to develop in time for n ∼ 10 21 cm −3 , by solving Eqs. (30)-(32), assuming a ∼ 10 nanosecond decoherence time, and intrinsic detuning by experimental effects like Doppler broadening, of both δ = 10 and δ = 100 MHz. We will see that in this ∼ ten nanosecond timeframe, a dark photon field applied to the cold atoms can greatly enhance the two photon transition rate.  32), where we take a gg = 0.90 × 10 −24 cm 3 , a ee = 0.87 × 10 −24 cm 3 , a eg = 0.0275 × 10 −24 cm 3 [36]. The intensity of the pump lasers is indicated. For comparison, we note that a coherence of r 1 0.07 has been achieved for parahydrogen at density 5 × 10 19 cm −3 , using lasers less powerful than those assumed here [32]. However, the nanosecond pulse gigawatt power lasers required are commerically available [52]. (Indeed, even continuous gigawatt lasers as powerful as we require have been demonstrated in recent years [53].) The left panel assumes experimental detuning δ = 100 MHz, as achieved in recent counter-propagating pulsed laser experiments [33]. The right panel assumes δ = 10 MHz, a linewidth that has been achieved in solid parahydrogen [51]. of a dark photon field, the rate for two photon de-excitation can be resonantly enhanced. Suitably applied to coherently excited atoms, we will find that very weakly coupled dark photon fields can trigger two photon transitions, during the ∼ 10 nanosecond window of time that the atoms are coherently excited.

Two photon transitions with kinetic mixing
We begin with the dark photon. The dark photon field is a new massive U (1) gauge field that kinetically mixes with the Standard Model photon. Its Lagrangian has the general form where A µ and A µ are the four vector potential of the ordinary photon and dark photon field, and F µν and F µν describe their field strength separately. Additionally, the dark photon is characterized by a mass m A and the kinetic mixing is suppressed by a constant χ. Here J µ em =ψγ µ ψ corresponds to the electromagnetic charged current with charged fermions ψ.
There is no direct coupling between the dark photon and charged fermions in Eq. (36). Rather, an effective interaction is introduced through kinetic mixing between the photon and dark photon, so long as m A > 0. Equivalently, one can diagonalize the kinetic mixing term by redefinition of the photon field A µ → A µ + χA µ . To first order in χ we obtain the Lagrangian To find the dark photon absorption and emission amplitude in atomic transitions, it will be convenient to work with the effective Hamiltonian for electrons in the non-relativistic limit in the presence of the dark photon field. Substituting the interaction terms in Eq. (37) into the Dirac Lagrangian, we arrive at the Dirac equation for the electron where m e is the electron mass. It will be convenient to work in the Dirac basis and divide the spinor into a dominant component ψ d and a subdominant component ψ s , i.e. ψ = (ψ d , ψ s ) T . Separating out the time derivative from the Dirac equation, we find the Hamiltonian for the system where Here σ are the Pauli spin matrices and the electric potential Φ = A 0 . The non-relativistic Hamiltonian for this system is obtained by subtracting m e from both sides of Eq. (40) and combining the equations of motion for ψ d and ψ s , where this expression is valid in the non-relativistic limit where |H nr | m e and similarly e(A 0 + A 0 ) m e . Eq. (42) gives the effective Hamiltonian for an electron in the presence of electromagnetic and dark photon fields. Subtracting from it the standard QED Hamiltonian we single out the components introduced by the new dark photon field The first line of Eq. (43) reminds us of the standard QED Hamiltonian, with an additional gauge field, whereas the second line can be dropped if we work to first order in e and χ.
With the effective Hamiltonian in hand we are now prepared to compute the transition amplitude from initial atomic state |i to final state |f with the absorption or emission of a dark photon. This transition has the general form We start with the first term in Eq. (43), which describes an E1 (electric-dipole) type transition. Using the relation ∂ µ A µ = 0, which can be readily obtained from the Euler-Lagrange equation (38) for the dark photon, we find where p e is the momentum operator for the electron. Using the relation where H 0 = p 2 e /2m e is the unperturbed atomic Hamiltonian, we obtain where again we note that ω ik ≡ ω i − ω k as the energy difference between the initial and final atomic states. The first term in Eq. (46) is suppressed by a factor ∼ ω/m e and is therefore negligible compared to the second term. Hence we drop this first term for simplicity.
To evaluate the second term, we define the vector component of the dark photon field as A = |A | exp(iωt − ik · r), which will have energy ω = ω if . Because we will be considering dipole moments substantially smaller than the wavelength of the applied laser (or the wavelength of the dark photon), the dipole approximation exp(−ik · r) 1 applies. With this approximation where the d = er is the dipole operator. Following standard electromagnetic conventions we define the dark electric field as where Assuming |A | varies slowly in space and time, we obtain Decomposing the dark electric field into a transverse component E T and longitudinal component E L , Note that our proposed experiment is only sensitive to dark photons with sub-meV masses, since this is a necessary condition for coherent excitation of two stage atomic transitions in the target sample (see Section 4). While the dark photon masses will be m A meV, the transition energy ω ∼ eV , therefore we expect the effect of longitudinal component of the dark electric field to be subdominant since |E L |/|E T | m 2 A /ω 2 . Therefore, we only focus on the transverse component in computing the transition amplitude.
We could proceed to evaluate the transition amplitude induced by the second and third terms of Eq. (43). However, we note that the second term is characterized by M1 (magnetic dipole) type transition which is suppressed by 1/m e compared with M E1 . The third term vanishes in the leading order expansion of exp(−ik · r), and so we also neglect it hereafter.

Dark photon induced superradiance
Now that we have obtained the dark photon dipole transition amplitude, we are ready to study dark photon induced superradiance. We will focus on the transition between the excitation state |e and ground state |g of a pH 2 target. As previously noted, |g and |e will denote the 0th and 1st vibrational state for pH 2 where both of these have J = 0. Since |e and |g share the same parity, an E1 dipole transition is forbidden, but the transition between them can take place via two E1 transitions, by transitioning through an intermediate virtual state |j . Hence we will compute E1 × E1 transitions for which two particles are emitted, as shown in Figures 4a and 4b. As we mentioned in the previous section the interaction between dark photon and electron allows for this E1×E1 transition to occur via the emission of a dark photon and standard model photon |e → |g + γ + γ along with the standard two photon emission process |e → |g + γ + γ, illustrated in Figures 4a and 4b. These two processes will reinforce each other in a coherently excited atomic medium, since the emission of a dark photon can trigger and amplify the two photon emission process, and vice versa. To demonstrate this mutual reinforcement, we shall derive the evolution equations of the dark photon and photon fields during deexcitation.

Maxwell-Bloch equations
First we will reformulate the Maxwell-Bloch equations as they were derived out in Section 2, now including the dark photon's effect on the electric dippole. As before, we denote the spin m J = ±1 states as |j ± . In addition to the two photon fields E 1 and E 2 , we define a dark photon E propagating in the positive z directioñ Because we are only treating the transverse component of the dark photon field, we take = T . We expect that to good approximation ω 1 = ω 2 = ω = ω = ω eg /2, since this is already required for coherence of the excited atomic state. We again write the pH 2 wave function as the superposition of atomic states For the sake of simplicity, we will set δ = 0 in the remainder of this treatment, which amounts to assuming that the atoms, lasers, and dark photon field are in phase over the target volume for timescales shorter than the decoherence time, T 2 ∼ 10 ns. For a full discussion of the physical requirements for ∼ 10 ns decoherence times, and the loss of coherence as δ is increased, see Section 2. The full interaction Hamiltonian is then The Schrödinger equations will now include terms proportional to the dark photon field, where as in Section 2 we absorb spatial dependence into overbarred fields,Ē 1 = E 1 e −iωz , E 2 = E 2 e iωz andĒ = E e ikz . Note also that we have left implicit the sum over all intermediate states j in Eq. (59) and Eq. (60). Integrating Eq. (57) and Eq. (58) over t, using the Markovian approximation, and imposing the initial condition c j ± ,0 = 0, Applying Eq. (61) and Eq. (62) in Eq. (59) and Eq. (60) and using the slowly varying envelope approximation, we obtain the equation for the two-state system in the presence of a dark photon, where the 2 × 2 matrix is the effective Hamiltonian (H ef f ), and its components are where we have defined the dipole couplings a ee , a gg , and a ge as before. In contrast to Section 2, we now also define which quantifies the relative phase between the polarization of the photon field and the dark photon field. As before we introduce the density matrix and add relaxation terms to obtain the Maxwell-Bloch equations where T 1 and T 2 are relaxation and decoherence time, respectively. We can expand Eq. (66) to make manifest oscillations in Ω eg here assuming ω k. From Eq. (69) we also need to decompose ρ ge correspondingly We note that ρ − ge only comes from the atomic excitation due to the absorption of E 2 and E or two dark photons, and the coherence developed in these processes is small. Thus, to leading order we can drop the second term in Eq. (71) and assume no spatial phase in ρ ge .

Field equations
The Bloch equations we have derived in the previous section show the evolution of the population of ground state and the excitation state in the presence of electric and dark electric fields. Now we would like to see how these fields evolve as the population changes. It is straightforward to obtain from Eq. (37) the field equations There is no free electric charge in the target and J em can be identified as the polarization current density determined by the polarization field where n is the number density of pH 2 . We recall the definition of E in Eq. (48) and take the time derivative on both sides of Eq. (72) and Eq. (73) to obtain where i = 1, 2 represent different electric fields. The polarization field arises from the dipole moment in the atomic transition wherẽ Note thatẼ 1 andẼ 2 propagate in opposite directions with opposite spin angular momenta, the microscopic polarization that sources these fields is also different. Accounting for the conservation of angular momentum we have We work in the limit where m A ω so approximately ω k. We can substitute c j± as given in Eq. (61) and Eq. (62) into Eqs. (78) to (80) and keep only the terms containing e ±iωt (to match the left hand side of Eq. (75) and Eq. (76)) to obtain By matching the oscillation phases of the electric fields and the microscopic polarization and using the slowly varying envelope approximation, we arrive at the field equations for The first terms on the right hand sides of these equations, which are proportional to a ee and a gg , do not affect the transition from excited to ground states, but rather describe absorption and reemission of photon or dark photons propagating in the medium. More importantly, the second terms on the right hand sides of the above equations, proportional to a eg , describe the production of electromagnetic fields via excited to ground state transitions of the atoms. Altogether, E 1 can be amplified by seed E 2 and E fields, and correspondingly, E 2 and E are amplified by the E 1 field through transitions. For our purposes, we are most interested in the fact that E will amplify E 1 and E 2 in these equations, which forms the basis for our dark photon detection proposal.

Bloch vector
Defining the Bloch vector as in Section 2, from Eq. (68) and Eq. (69) we obtain where the spatially averaged visible and dark photon fields are together defined as and we assume a eg is real. Note that the expression above has assumed thatĒ 1 andĒ 2 are in phase, which is appropriate for atoms pumped by phase-matched lasers. Due to the smallness of the mixing parameter χ, the dark photon field itself will not drive the evolution of the state population in the system. However, the dark photon can trigger the production of E 1 and E 2 , which in turn trigger additional photon production. Therefore, while it would be safe to drop the dark photon component in Eqs. (86) to (88), we retain it in numeric computations for the sake of rigor. We must retain the dark photon component in the field equations. Using Eqs. (83) to (85) we obtain In the experimental setup we soon describe, after the atoms are pumped into their excited states, the laser fields will be shut off so that |Ẽ 1 | = |Ẽ 2 | ≈ 0. It is clear from Eq. (89) that in this circumstance, a non-zero dark electric field E will be essential to develop the E 1 field, which will in turn trigger additional two photon emission.

Experimental setup
Our proposed experimental setup is schematically illustrated in Figure 5. A continuous laser beam is injected into a resonant cavity, which enhances the laser's probability to oscillate into dark photons. After hitting the wall photons are stopped and only dark photons are allowed through. The target (pH 2 for example) is pumped into a coherently excited state as detailed in Section 2. As it propagates through the target, the dark photon field triggers atomic deexcitation via the emission of a dark photon and a standard model photon. The electric field generated from the first deexcitation subsequently triggers two photon emission, producing back-to-back photons with the same frequency. These photons trigger further deexcitations, detected at both ends of the target vessel.  Figure 5: Schematic view of the proposed experiment. First, the pH 2 sample is coherently excited to energy ω eg by back-to-back pump lasers (pump lasers not shown). The excited atoms' E1 dipole transitions are parity forbidden, meaning the atoms are metastable over the ∼ ten nanosecond integration time of the experiment. On the other hand, the emission of two ω eg /2 energy photons in an E1 × E1 transition is allowed. As in lightshining-through wall experiments, a laser is fed into a resonant cavity to increase the dark photon conversion probability. In this case, the laser will operate at energy ω eg /2, so that after passing through the wall, dark photons act as a trigger field for the emission of back-to-back photons which are then observed by detectors labeled D1 and D2.
There are two primary advantages to conducting the experiment in the manner described above. First, the pH 2 sample's response to a dark photon field can be precisely determined by passing a very weak laser field through the sample, where low-power lasers can directly test the response to weakly coupled dark photons. Then, in discovery mode, where visible photons are prevented from passing through the wall, the two photon emission process would presumably only occur, if triggered by a dark photon over the ∼ 10 nanosecond coherence time, because the spontaneous deexcitation process is negligibly slow, as detailed in Section 4.4. Photons produced by dark photon induced transitions would be emitted back-to-back and at the frequency, ω = ω eg /2. Altogether this pro-vides a powerful background rejection method, since the rate for spontaneous two photon emission is very small. This can be contrasted with more ambitious experiments utilizing two photon emission processes [36,[41][42][43]. In these experiments, the signal (one photon and either two neutrinos or an axion) will need to be distinguished from a sizable two photon emission background, since both processes are triggered. Therefore it is plausible that the experiment we have outlined is an intermediate step that could be reached while working towards the proposals laid out in [36,[41][42][43].

Dark photon induced transition rate
To begin with, let us quote the estimated rate for the emission of γ 1 and γ 2 in our proposed experiment. First, we note that without both coherent enhancement and exponential amplification of photon fields by the atomic medium that will be discussed shortly, the |e → |g + γ + γ transition rate depicted in Figure 4a is rather slow. To satisfy the coherent amplification condition, we require where L is the length of the target, which is the longest dimension of the target volume. Under these conditions (see Appendix B for a full derivation) the naive rate for dark photon-induced two stage transitions is where ω 1 is the cavity laser frequency, equal to the dark photon frequency ω , N pass is the number of cavity reflections, P L is the cavity laser power, l is the cavity length, A is the area of the excited atomic target (limited by the pump lasers' beam width) and n is the target number density. Using this naive estimate results in an unobservably small rate, because it does not account for the development of electromagnetic fields in the atomic medium (c.f. the field equations given in Eqs. (89) through (91)). The predicted rate for our benchmark experimental and model parameters, using a dark photon mixing χ = 10 −9 , m A = 10 −4 eV, ω = ω 1 = 0.26 eV, N pass = 2 × 10 4 , l = 50 cm, η = 1, pH 2 number density n = 10 21 cm −3 , target area A = 1 cm 2 , target length L = 30 cm, and for parahydrogen dipole coupling a eg = 0.0275 × 10 −24 cm 3 , is Γ ≈ 10 −5 s −1 . This emission rate is unobservably low considering that each experimental run is expected to last about 10 ns.
However, even a small production rate for E 1 can be exponentially enhanced in coherently prepared atoms. As detailed in Appendix A, the transition rate for producing two photon pairs is exponentially enhanced as the electromagnetic field strength grows, The dependence on E 2 1 E 2 2 in Eq. (94) shows that the growth of signal fields will be exponential after the dark photon establishes a small E 1 seed field. A similar amplification has been observed to be as large as 10 18 compared with spontaneous emission [32]. We expect an even larger amplification factor to be achieved in our benchmark experimental setup.

Numerically simulating field development
When simulating the development of electric fields in coherently prepared atoms, it will be convenient to rescale the spacetime coordinates and the electric fields to be dimensionless. We define where β represents the typical length and time scale for the evolution of the system and ω eg n is the energy stored in excited atoms. The Bloch equations and field equations can be written in terms of these new variables As mentioned before, dipole couplings of parahydrogen have been measured to be a gg = 0.90 × 10 −24 cm 3 , a ee = 0.87 × 10 −24 cm 3 , a eg = 0.0275 × 10 −24 cm 3 [37]. For the relaxation and decoherence times, we take T 1 = 10 3 ns and T 2 = 10 ns respectively; for extended discussion of coherence in preparations of pH 2 , see Section 2. The photon and dark photon energies are ω = ω eg /2 ≈ 0.26 eV. Altogether this gives β = 0.092 10 21 cm −3 N ns = 2.8 10 21 cm −3 N cm , ω eg n = 2.5 * 10 10 n 10 21 cm −3 W/mm 2 .
A typical target vessel is 10 to 100 cm long. Here we assume a vessel that is 30 cm long, which is smaller than the expected length scale over which the pH 2 is coherent. If we assume all atoms are initially prepared in the coherent state, then r 1 = 1 across the target. We will also consider smaller values of r 1 = 0.1, 0.5, 0.9, which correspond to fewer atoms in the coherent state. With the aid of a resonant cavity, the transmission probability for a dark photon to shine through the wall is given by [54,55] p trans = 2(N pass + 1) where l is the size of the cavity and N pass is the number of reflections the laser undergoes in the dark photon generating cavity. We assume l = 50 cm and N pass = 2 × 10 4 in our benchmark setup. These values are in line with what is currently attainable at the ALPS II experiment [17]. The initial dark photon field power in the target volume is estimated to be where for our benchmark setup we assume a laser power P L = 1 Wmm −2 . As mentioned in Section 3.2.1 η is determined by the relative phase between the polarization of the photon and dark photon fields. Without loss of generality we set it to be unity here. We show in Figure 6 through Figure 8 the time evolution of the system. In these figures we assume all the pH 2 atoms are initially prepared in the coherent state, i.e. r 1 = 1, r 2 = 0 and r 3 = 0, across the target. We also assume a dark photon mass m A = 0.1 meV.
As shown in Figure 6 r 1 and r 3 decay exponentially when no laser is present. In this case no initial dark photon field is pumped through the wall and so spontaneous deexcitation dominates the evolution of the system. We note that Figure 7, which shows no substantial E 1 , E 2 , or E field developing when P L = 0, has not included the effect of spontaneous two photon deexcitations, which are expected to be negligibly small, see Section 4.4.
This scenario changes dramatically in the presence of dark photons produced by a laser. Assuming the laser power P L = 1 Wmm −2 and the mixing χ = 10 −3 , a sudden drop takes place in r 1 and r 3 around 10 ns. This drop corresponds to decay and release of the target's energy through production of E 1 and E 2 as well as a minor enhancement of the dark photon field E . The dynamics can be explained as follows. The initial dark photon fields induces a deexcitation via E 1 and E (Figure 4a illustrated this process), then this E 1 field triggers additional two photon deexcitation producing E 1 and E 2 symmetrically (see Figure 4b). The growing E 1 and E 2 , when large enough, cause abrupt decoherence and deexcitation, which in turn gives rise to additional energy release in the form of E 1 and E 2 . As can be identified from Eq. (85) E will also be generated by E 1 induced transitions, at a rate suppressed by χ.   Figure 6 no laser is present for dotted lines and a 1 Wmm −2 laser with χ = 10 −3 is assumed for the dashed lines. We also assume the same initial Bloch vectors as Figure 6. |E 1 | 2 (red) is taken at the left end of the target with z = 0 cm while |E 2 | 2 (blue) and |E | 2 (black) are taken at the right end of the target with z = 30 cm.
The transitions are less explosive when χ = 10 −9 , as illustrated in Figure 6 and Figure 8. The deviations of r 1 and r 3 from spontaneous decay are barely observed and the peak intensity of E 1 and E 2 are relatively low compared to χ = 10 −3 . In this case, the dark photon has induced the generation of an observable but small quantity of E 1 and E 2 photons. The dark photon field remains essentially constant since E regenerated from E 1 is too small to be observed.

Spontaneous two photon emission background
We now consider a possible background from spontaneous deexcitation and emission of photons from cold atoms over the runtime of our experiment (around 10 ns). We will find that this background is negligible. Since the transition from excitation state |e to the ground state |g is E1 forbidden, single photon deexcitation is only viable through higher order transitions. Note that we are only looking for signal photons with energy around ω = 1 2 ω eg , because our signal photons are expected at this frequency. The background from spontaneous two photon emission has a rate given by (see Appendix A) where z = ω 1 /ω eg is the fraction of the energy for one of the two photons in the transition. We assume an uncertainty ∆ν = 100 MHz in the frequency measurement, which translates to ∆z = 8.0 × 10 −7 . For a sample target with length L = 30 cm and cross section area A = 1 cm 2 the uncertainty in the emission solid angle is ∆Ω/4π = A/4π(L/2) 2 = 3.5 × 10 −4 . These two photons from spontaneous decay process can be emitted in any direction. Since we only detect photons at the ends of the atomic sample, the fraction of background photons which reach the detector is 2∆Ω/4π. Given the target number density n = 10 21 cm −3 and complete coherence (ρ eg = 0.5) the total number of pH 2 We see that over the course of any reasonable number of experimental repetitions, we should not expect a single background event from spontaneous two photon deexcitation processes.

Results and sensitivity
The signature of the proposed dark photon search is the symmetric emission of photons with frequency ω = ω eg /2 at both ends of the target. The number of signal photons emitted during one experimental trial run (of ∼ 10 ns) is Figure 10: Sensitivity of our proposed experiment assuming the benchmark setup with light-shining-through wall laser power P L = 1 Wmm −2 , target area A = 1 cm 2 , number of cavity reflections N pass = 2 × 10 4 , cavity length l = 50 cm, target length L = 30 cm, pH 2 target number density as indicated, and where we assume N rep = 10 3 exposures, each lasting ∼ ten nanoseconds. The constraints from other dark photon experiments, astrophysics, and cosmology are shown for comparison, e.g. Coulomb [56,57], CMB [58,59], CROWS [60], GammeV [61], ALPS [16,17], and stellar constraints [10,11,15,17], see [62] for a summary of these bounds. The black lines show the sensitivity of our proposed experiment, for pH 2 number densities indicated, and coherence factors as indicated to the right of each sensitivity (see Section 2 for discussion of coherence in parahydrogen).
where A is the area of the target and t is the time duration of the experiment. The experiment can be repeated many times to accumulate signal photons. The Bloch equations and field equations derived in Section 3.2 are highly nonlinear, but we infer from Eq. (89) that E 1 ∝ χE , and therefore the number of photons emitted is where N rep is the number of repetition of the experiment. To see in what regime this scaling holds, we show in Figure 9 the number of photons produced as a function of the mixing, χ assuming different laser powers. There is an upper bound on the number of signal photons, which is saturated if all of the excited atoms are deexcited. It is clear from the figure that before saturation N s is proportional to P L and χ 4 . As χ becomes large enough a significant amount of energy stored in the target is released and one gains very little by increasing the mixing or laser power. We note also that N rep , N pass + 1 and sin 2 (m 2 A l/ω) are will have the same scaling as P L when determining the number of signal photons emitted.
To estimate the sensitivity of our proposed experiment, we require the emission of at least ten photon pairs after a certain number of excitation/deexcitation repetitions. As a benchmark we take the laser power P L = 1 Wmm −2 , the target area A = 1 cm 2 , the number of cavity reflections N pass = 2 × 10 4 , the cavity size l = 50 cm, and the number of repetitions as N rep = 10 3 . In the regime that a fraction of the pH 2 deexcites, the number of emitted photons can be estimated as where this expression has been normalized assuming n = 10 21 cm −3 .
We show the sensitivity of our proposed experiment in Figure 10. Also shown in the figure are the light-shining-through-wall experiments, cosmological, and astrophysical bounds reviewed in [62]. The coherent amplification condition we have assumed throughout ((ω − k)L 1 )) requires that our dark photon mass not be too large. This restricts m A 0.6 meV. As a consequence we have truncated the mass sensitivity at one meV. As seen from the figure, over the mass range 10 −5 ∼ 10 −3 eV our proposed experiment appears rather sensitive to dark photon kinetic mixing.

Conclusions
We have studied a new method to detect dark photon fields using resonant two photon de-excitation of coherently excited atoms. Our proposed experiment combines dark photon production techniques demonstrated by light-shining-through-wall experiments with a new detection method: dark photons triggering two photon transitions in a gas of parahydrogen coherently excited into its first vibrational state. The potential coupling sensitivity to dark photons we project in our benchmark setup is orders of magnitudes beyond present limits for µeV − meV mass dark photon fields.
A major technical hurdle to realizing our proposal will be the preparation of suitably coherent samples of cold parahydrogen using counter-propagating laser beams. As we examined in Section 2, the coherence times and pH 2 densities necessary have already been achieved in laboratory conditions. It remains to suitably increase the fraction of coherently excited pH 2 by using more powerful lasers and colder parahydrogen, as we explored in Section 2. However, even if complete parahydrogen sample coherence is not attained, it would still be possible to realize our proposal by increasing the density of parahydrogen, as explore in Section 4. Indeed, although we have not shown it in Figure  10, the setup we propose with an increased pH 2 number density (2 × 10 21 ), assuming completely coherent atoms (r 1 = 1) can probe kinetic mixings χ 10 −15 . It may also be possible to realize a similar proposal to the one laid out here, using two photon nuclear transitions and free electron lasers. This might permit detecting dark photons at masses greater than an eV.
Our setup relies on the nonlinear development of electromagnetic fields in coherent atoms, and so our sensitivity estimates have relied on numerical simulations of dark photon and photon cascades in pH 2 . However as explained in Section 4, the proposed experiment will allow for dark photon detection to be directly calibrated using a low power trigger laser, as an equivalent stand-in for the dark photon field itself. While for this reason, we have focused on the detection of dark photons in this article, very similar methods could be used to detect axions and other light, electromagneticallycoupled particles. We leave this and other uses of multi stage atomic transitions to future work.

A Coherence and nonlinearity in two photon emission
Let us estimate the transition rate of two photon emission process, as illustrated in Figure 4b. The transition matrix for |e → |g transition is where T is the time-ordering operator and we write the electric fields in as in general, where ω m and k m are the energy and momentum of the emitted photons.
Integrating over t yields where as before we have defined ω ik = ω i − ω k and d ik = i| − d · ( * ) |k . k a ej is the change in the momentum of a specific pH 2 after the transition and r a is the spatial position of the pH 2 . We can perform the second time integral and obtain with M a = d gj d je ω je + ω 1 E 1 E 2 4 e −i(k 1 +k 2 −k a eg )·(r−ra) = a eg 4 E 1 E 2 e −i(k 1 +k 2 −k a eg )(r−ra) .
First we consider the case that the pH 2 is not emitting coherently, which we will call spontaneous two-photon deexcitation. In the case of spontaneous two-photon deexcitation, each pH 2 emits two photons with frequencies that are not necessarily ∼ ω eg /2, in contrast with two photon emission induced by a trigger laser (where the trigger laser frequency used in earlier sections of this document matched the pump laser frequencies, all of these being ω eg /2). In the spontaneous emission case, we sum up the contribution from all pH 2 which gives the emission rate where we have explicitly replaced E m by 2ω m /V for m = 1, 2, and N e is the number of spontaneous emitters. Since the exponential phase is random for each molecule, the product of the phases from different molecules will sum up to zero in the expansion of the square in Eq. (115). This gives Carrying out the integral we find dΓ sp dω 1 = 1 (2π) 3 N e |a eg | 2 ω 3 1 ω 3 2 .
If we define z ≡ ω 1 /ω eg , Eq. (117) can be written as We have used this equation to estimate the two photon spontaneous emission background in Sec. 4.4. Second we will estimate the rate for two photon emission for pH 2 pumped and triggered in a manner which allows for macro superradiance. In the presence of appropriately applied background fields, the pH 2 molecules will tend to emit photons collectively with the same momenta. If the phase k a eg is random for every molecule, the product of the phases would still cancel as we have derived before and the rate would be proportional to N e ; however, if the molecules are pumped into the excitation state coherently (by counter propagating lasers, in the setup we have considered), we can drop the superscript a in k a eg and turn the sum into a spatial integral, i.e.
where n is the number density of the target and ρ ge is the fraction of molecules in the coherent state. In the special case where we use two counter propagating lasers with the same frequency to pump the molecules, k eg ≈ 0, although of course this can be spoiled by the lasers' linewidth and other experimental factors discussed in Section 2. For a dense and large enough target, the spatial integral in r a turns into a delta function, which gives where N is the total number of pH 2 in the target. In the case k eg ≈ 0, the delta function forces k 1 + k 2 = 0, meaning that the two photons emitted superradiantly have to be back-to-back and have equal frequency. Since the delta function is squared we replace one by the target volume V . Evaluating the integrals yields Eq. (121) shows the transition rate in two photon superradiance is proportional to N 2 if coherence conditions are met. This can be compared with (out-of-phase) spontaneous two photon emission described in Eq. (118) where the rate is proportional to N instead. We also see that the rate grows nonlinearly with E 1 and E 2 , the strength of the background fields. At the onset of superradiance, the emission rate is determined by the power of the trigger laser fields. As the photons from the deexcitation increase the strength of the electric fields, the deexcitation rate becomes larger and larger. This exponential growth is clearly seen in Figure 7.

B Estimate of dark photon triggered two stage transitions
Let us now move on to estimate the emission rate of γ 1 and γ 2 in our proposed experiment, as depicted in Figure 5. Consider the process illustrated in Figure 4a. First, the transition matrix for the deexcitation from |e to |g via the emission of a dark photon and a photon in the dark photon background is given by For χ = 10 −9 , m A = 10 −4 eV, ω = ω 1 = 0.26 eV, N pass = 2 × 10 4 , l = 0.5 m, η = 1, pH 2 number density n = 10 21 cm −3 , target area A = 1 cm 2 , length L = 30 cm, laser power P L = 1W/mm 2 , a eg = 0.0275 × 10 −24 cm 3 , This emission rate is relatively low considering the the experimental trial time (determined by the decoherence time) of about 10 ns. But the signal is enhanced by stimulating nonlinearly growing two photon superradiant transitions. This is discussed in Sec. 4.2.