with CN=2N/N!⁢(e−i⁢𝐤2⁢𝐝2⁢α)Nsubscript𝐶𝑁superscript2𝑁𝑁superscriptsuperscript𝑒𝑖subscript𝐤2subscript𝐝2𝛼𝑁C_{N}=\sqrt{2^{N}/N!}(e^{-i\textbf{k}_{2}\textbf{d}_{2}}\alpha)^{N}italic_C start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = square-root start_ARG 2 start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT / italic_N ! end_ARG ( italic_e start_POSTSUPERSCRIPT - italic_i k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_α ) start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT and the phase-dependent state

As we have just seen, dark states are eigenstates of the operator 𝐄(+)⁢(𝐫,t)superscript𝐄𝐫𝑡\textbf{E}^{(+)}(\textbf{r},t)E start_POSTSUPERSCRIPT ( + ) end_POSTSUPERSCRIPT ( r , italic_t ) with null eigenvalue, which implies that they are undetectable by usual sensors such as those described by two-level atoms. On the other hand, the bright states couple stronger than in the single-mode case. Thus, a natural question arises concerning the connection between the quantum mechanical dark (bright) states and the classical effects of destructive (constructive) interference between radiation fields, as in regions of destructive interference no light is detected, while in regions of constructive interference light scattering is enhanced. To address this question we consider, for simplicity, the case of two modes with θ=0𝜃0\theta=0italic_θ = 0. Then, one can easily show that in-phase coherent states decompose exclusively on the MSS subspace:

On the other hand, when a coherent state is sent to a double slit, part of the light goes through slit 1 at position 𝐝1subscript𝐝1\textbf{d}_{1}d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, and part of it goes through slit 2 at position 𝐝2subscript𝐝2\textbf{d}_{2}d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. In this case, we do not have a superposition state, but rather a product state of the two modes 𝐤1subscript𝐤1\textbf{k}_{1}k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and 𝐤2subscript𝐤2\textbf{k}_{2}k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in coherent states with amplitude α𝛼\alphaitalic_α (assumed the same for both slits) and a relative phase which depends on the position of the slit, i.e.

where δ⁢ϕ=−(𝐤2⋅𝐝2−𝐤1⋅𝐝1)+θ=𝐤2⋅𝐫2−𝐤1⋅𝐫1𝛿italic-ϕ⋅subscript𝐤2subscript𝐝2⋅subscript𝐤1subscript𝐝1𝜃⋅subscript𝐤2subscript𝐫2⋅subscript𝐤1subscript𝐫1\delta\phi=-(\textbf{k}_{2}\cdot\textbf{d}_{2}-\textbf{k}_{1}\cdot\textbf{d}_{% 1})+\theta=\textbf{k}_{2}\cdot\textbf{r}_{2}-\textbf{k}_{1}\cdot\textbf{r}_{1}italic_δ italic_ϕ = - ( k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋅ d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋅ d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_θ = k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋅ r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋅ r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, i.e., δ⁢ϕ𝛿italic-ϕ\delta\phiitalic_δ italic_ϕ represents the phase difference of the two light paths from the slits to the sensor atom. Clearly, at any detector position (δ⁢ϕ𝛿italic-ϕ\delta\phiitalic_δ italic_ϕ) we may have a bright, a dark, or a superposition of bright and dark states, which implies that the photon can be at any position. In other words, apart from the g⁢(𝐤)𝑔𝐤g(\textbf{k})italic_g ( k ) distribution, the average number of photons as a function of δ⁢ϕ𝛿italic-ϕ\delta\phiitalic_δ italic_ϕ is constant, i.e., ⟨a†⁢a+b†⁢b⟩⁢(δ⁢ϕ)=1delimited-⟨⟩superscript𝑎†𝑎superscript𝑏†𝑏𝛿italic-ϕ1\langle a^{\dagger}a+b^{\dagger}b\rangle(\delta\phi)=1⟨ italic_a start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_a + italic_b start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_b ⟩ ( italic_δ italic_ϕ ) = 1. However, the sensor atom can detect the photon only at positions where the bright state survives.

A key phenomenon for evidencing the wave nature of light comes from Young’s double slit experiment. The fundamental result is that both classical and single-photon coherent sources produce the same fringe pattern, despite the very different nature of these fields. To revisit this experiment using the collective basis, one can consider two equally weighted light modes emerging from two slits and in the far-field limit. Without loss of generality, we assume both waves with wave vectors 𝐤1subscript𝐤1\textbf{k}_{1}k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (mode a𝑎aitalic_a) and 𝐤2subscript𝐤2\textbf{k}_{2}k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (mode b𝑏bitalic_b), with |𝐤1|=|𝐤2|=ksubscript𝐤1subscript𝐤2𝑘|\textbf{k}_{1}|=|\textbf{k}_{2}|=k| k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | = | k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | = italic_k. Then, 𝐤1⋅𝐫1⋅subscript𝐤1subscript𝐫1\textbf{k}_{1}\cdot\textbf{r}_{1}k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋅ r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and 𝐤2⋅𝐫2⋅subscript𝐤2subscript𝐫2\textbf{k}_{2}\cdot\textbf{r}_{2}k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋅ r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, with 𝐫isubscript𝐫𝑖\textbf{r}_{i}r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT the vector connecting the ithsuperscript𝑖thi^{\text{th}}italic_i start_POSTSUPERSCRIPT th end_POSTSUPERSCRIPT slit with the sensor position, are the phases acquired by the respective fields when propagating from slits 1 and 2, respectively, to the detection point (see Fig. 1).

From an experimental perspective, the two-mode light-matter interaction discussed in the present work suggests an implementation in optical cavities coupled to a two-level atom [22, 23], trapped ions where a single emitter can be coupled to its two vibrational modes [24, 25], or superconducting circuits [26]. We visualize many possibilities, drawing inspiration from the diverse applications that appear in the context of super- and subradiance in atomic systems. For example, one could employ photonic superradiant states to further enhance light emission in high-brightness light sources. On the other hand, as the dark states do not interact with matter, they could in principle be employed as decoherence-free photonic quantum memories. Finally, by taking advantage of the fact that bright states do interact with atoms and dark states do not, one could use such states to imprint a conditional phase on an atomic system. With this, one could employ such collective mode states to implement single-shot logic operations in crossed-cavity setups [23], such as two- and three-qubit CNOT and Fredkin gates, thus allowing for universal quantum computing with traveling modes [27].

In light of our discussion, we can conclude that the single-mode case has a unique PDS, namely the vacuum state |0⟩delimited-|⟩0\lvert 0\rangle| 0 ⟩. It may sound trivial since there is no photon to excite the detector, yet its interest lies in its uniqueness: any other state excites the sensor atom. Thus, our interpretation in terms of dark and bright states provides a new way to explain why single-mode Fock states |N⟩delimited-|⟩𝑁\lvert N\rangle| italic_N ⟩ with N>1𝑁1N>1italic_N > 1 do excite the sensor atom, even for zero mean electric field. On the other hand, the multi-mode case is fundamentally different since it possesses an infinite family of dark states with an arbitrarily large number of photons, which do not couple to the sensor atom in the ground state. This scenario is analogous to the case of two-level atoms where single-atom (spontaneous) emission always occurs, whereas the emission from a couple of atoms in the dark state |−⟩=(|e,g⟩−|g,e⟩)/2\lvert-\rangle=(\lvert e,g\rangle-\lvert g,e\rangle)/\sqrt{2}| - ⟩ = ( | italic_e , italic_g ⟩ - | italic_g , italic_e ⟩ ) / square-root start_ARG 2 end_ARG is suppressed. Alternatively, dark states can be produced using single emitters with a multilevel structure [20], in particular in electromagnetically induced transparency [21]. In addition, the two-mode case also predicts bright and intermediate states, the latter having no correspondence in classical physics. Although the discussions above were developed for only two modes of the radiation field, their generalization to the case N𝑁Nitalic_N modes of the field is straightforward (see SM).

meaning that photons are present at every point on the screen, contrary to the standard classical description of interference, which states that no light arrives at points of destructive interference. The single-photon state |S⟩ket𝑆\left|S\right\rangle| italic_S ⟩, which decomposes as a sum of PDSs or MSSs only (see Eq. (12)), has the same feature, as discussed above. Therefore, the decomposition either in only PDSs or only MSSs explains why single-photon fields and classical fields exhibit the same fringe pattern.

In each of the modes we have a non-zero average electric field but, as they are out of phase with each other, the average resulting electric field is zero. According to the classical interpretation of interference, such a field would be undetectable but, according to Glauber’s theory [16], it is detectable, in the sense that it will induce a non-trivial dynamics for the sensor/atom. This can be easily explained using the description in terms of bright and dark states, since such state can be written in the form |Υ⟩=[|ψ00⟩−2|ψ01⟩+(|ψ02⟩−|ψ22⟩)/2]/2\lvert\Upsilon\rangle=\left[\lvert\psi_{0}^{0}\rangle-\sqrt{2}\lvert\psi_{0}^{% 1}\rangle+\left(\lvert\psi_{0}^{2}\rangle-\lvert\psi_{2}^{2}\rangle\right)/% \sqrt{2}\right]/2| roman_Υ ⟩ = [ | italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ⟩ - square-root start_ARG 2 end_ARG | italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ⟩ + ( | italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ - | italic_ψ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ ) / square-root start_ARG 2 end_ARG ] / 2, which shows a projection onto the detectable subspace of bright states.

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with δ⁢ϕ𝛿italic-ϕ\delta\phiitalic_δ italic_ϕ defined right after Eq. (12). Such state corresponds to a MSS when the two modes are in phase, |χN(2lπ)⟩=ei⁢N⁢θ|ψNN(θ)⟩\left|\chi^{N}\left(2l\pi\right)\right\rangle=e^{iN\theta}\lvert\psi_{N}^{N}(% \theta)\rangle| italic_χ start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ( 2 italic_l italic_π ) ⟩ = italic_e start_POSTSUPERSCRIPT italic_i italic_N italic_θ end_POSTSUPERSCRIPT | italic_ψ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ( italic_θ ) ⟩, and to a PDS when in opposite phases, |χN((2l+1)π)⟩=|ψ0N(θ)⟩\left|\chi^{N}\left((2l+1)\pi\right)\right\rangle=\lvert\psi_{0}^{N}(\theta)\rangle| italic_χ start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ( ( 2 italic_l + 1 ) italic_π ) ⟩ = | italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ( italic_θ ) ⟩, with l=0,±1,±2,…𝑙0plus-or-minus1plus-or-minus2…l=0,\pm 1,\pm 2,...italic_l = 0 , ± 1 , ± 2 , … . In this case, the average number of photons is again independent of the phase difference, i.e.,

This implies an enhanced absorption by a factor of 2222 (H|α,α⟩|g⟩=2αg|α,α⟩|e⟩H\lvert\alpha,\alpha\rangle\lvert g\rangle=2\alpha g\lvert\alpha,\alpha\rangle% \lvert e\rangleitalic_H | italic_α , italic_α ⟩ | italic_g ⟩ = 2 italic_α italic_g | italic_α , italic_α ⟩ | italic_e ⟩) as compared to a single coherent state (H|α⟩|g⟩=gα|α⟩|e⟩H\lvert\alpha\rangle\lvert g\rangle=g\alpha\lvert\alpha\rangle\lvert e\rangleitalic_H | italic_α ⟩ | italic_g ⟩ = italic_g italic_α | italic_α ⟩ | italic_e ⟩). The two-mode MSS quantum state therefore corresponds to the constructive interference of classical in-phase fields.

and, consequently, the Hamiltonian (3) becomes H=g⁢(a+b⁢ei⁢θ)⁢σ++g⁢(a†+b†⁢e−i⁢θ)⁢σ−𝐻??𝑎𝑏superscript𝑒𝑖𝜃superscript𝜎𝑔superscript𝑎†superscript𝑏†superscript𝑒𝑖𝜃superscript𝜎H=g(a+be^{i\theta})\sigma^{+}+g(a^{\dagger}+b^{\dagger}e^{-i\theta})\sigma^{-}italic_H = italic_g ( italic_a + italic_b italic_e start_POSTSUPERSCRIPT italic_i italic_θ end_POSTSUPERSCRIPT ) italic_σ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT + italic_g ( italic_a start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT + italic_b start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_i italic_θ end_POSTSUPERSCRIPT ) italic_σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT. Here, g=g⁢(𝐫)𝑔𝑔𝐫g=g(\textbf{r})italic_g = italic_g ( r ) denotes the coupling constant between the sensor and the two modes, assumed equal for both modes for simplicity, and r is the sensor (atom) position. Then, we introduce the symmetric and antisymmetric collective operators c=(a+b⁢ei⁢θ)/2𝑐𝑎𝑏superscript𝑒𝑖𝜃2c=(a+be^{i\theta})/\sqrt{2}italic_c = ( italic_a + italic_b italic_e start_POSTSUPERSCRIPT italic_i italic_θ end_POSTSUPERSCRIPT ) / square-root start_ARG 2 end_ARG and d=(−a⁢e−i⁢θ+b)/2𝑑𝑎superscript𝑒𝑖𝜃𝑏2d=(-ae^{-i\theta}+b)/\sqrt{2}italic_d = ( - italic_a italic_e start_POSTSUPERSCRIPT - italic_i italic_θ end_POSTSUPERSCRIPT + italic_b ) / square-root start_ARG 2 end_ARG [12], respectively. For any number n≤N𝑛𝑁n\leq Nitalic_n ≤ italic_N, we also define states with a total number of N𝑁Nitalic_N excitations |ψnN⁢(θ)⟩delimited-|⟩subscriptsuperscript𝜓𝑁𝑛𝜃\lvert\psi^{N}_{n}(\theta)\rangle| italic_ψ start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_θ ) ⟩ in the collective ({c,d}𝑐𝑑\{c,d\}{ italic_c , italic_d }) and in the bare ({a,b}𝑎𝑏\{a,b\}{ italic_a , italic_b }) basis as

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In conclusion, we have discussed how a description of multi-mode light in terms of maximally superradiant or perfectly dark collective states offers a natural interpretation for constructive and destructive interference. Remarkably, this Dicke-like bosonic basis applies to classical and non-classical states of light, thus going beyond the simple classical approach of average fields. We have shown that, from a quantum perspective, interference is intimately related to the coupling of light and matter which differs for the bright and dark states. This is completely different in the classical description where no assumption on the matter is necessary to describe the sum of electromagnetic fields. One can interpret this as a manifestation of the quantum-measurement process where the expectation value of an observable depends on the properties of the measuring apparatus [28, 29]. Within this framework, we have interpreted the double-slit experiment and the interference of light waves in general in terms of bright and dark states, i.e., using only the corpuscular description of the light and the quantum-superposition principle.

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with σ+superscript𝜎\sigma^{+}italic_σ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT (σ−superscript𝜎\sigma^{-}italic_σ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT) the raising (lowering) operators that induce transitions between ground |g⟩delimited-|⟩𝑔\lvert g\rangle| italic_g ⟩ and excited |e⟩delimited-|⟩𝑒\lvert e\rangle| italic_e ⟩ states of the sensor atom. From Eq. (2) one can easily see that, for single mode fields, only zero-intensity (vacuum) fields are unable to excite the sensor.

The quest for understanding what light is, and what its properties are, comes from the Greek school of philosophy a few centuries BC. Since then the subject has been extensively studied, with prominent contributions from Newton and Huygens, the former defending the corpuscular and the latter advocating the wave character of light  [1, 2]. This dispute remained unresolved until, among others, Young performed experiments on light diffraction  [3, 4], and Maxwell developed a unified theory of electromagnetism which includes a wave equation for the light field [5]. This effectively removed any doubts about the wave aspect of light. However, in 1905, Einstein explained the photoelectric effect by reintroducing the idea of light particles [6] (later reinterpreted by Lamb and Scully 111In 1968, Lamb and Scully provided a semiclassical model, which treats the electromagnetic field classically and the matter quantum mechanically, to explain the photoelectric effect without the need for the corpuscular aspect of light [30].). Since then, and with the advent of quantum physics, light is associated with both properties, wave and particle. Depending on the experiment, one or the other aspect manifests itself: the interference of delocalized waves on the one hand and the propagation of particles along well-defined trajectories on the other hand. Although this is textbook knowledge by now, it was highly debated at its time. For example, Millikan argued that the particle aspect “flies in the face of the thoroughly established facts of interference” [8].

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This state gives a suppressed interaction H|α,−α⟩|g⟩=0H\lvert\alpha,-\alpha\rangle\lvert g\rangle=0italic_H | italic_α , - italic_α ⟩ | italic_g ⟩ = 0, which can be interpreted either as belonging to the PDS subspace or, classically, as a destructive interference for two fields with the same amplitude but opposite phases. However, not every destructively interfering field is undetectable. This can be seen by considering two modes in the state

Here we resolve Millikan’s objection and show that the interference between independent radiation modes, usually taken as an undoubted signature of the wave character of radiation, has in fact a purely corpuscular explanation. For this we adopt a microscopic description of the measurement process in terms of energy exchange between the radiation and a sensor atom, as formally derived by Glauber  [9]. We then introduce the entangled dark and bright states of light, i.e., a new kind of subradiant and superradiant states for collective modes of light fields, analogous to those which appear in atomic systems [10]. With this new model we are able to interpret the optical double-slit experiment in terms of particle states only. We discuss the rather counterintuitive result that a vanishing photon-detection probability at locations of destructive interference does not prove the absence of photons but instead indicates a photonic state that is perfectly dark for the employed sensor atom. We also show that the collective states of the light fall into three distinct classes: perfectly dark, maximally superradiant, and intermediate. Interestingly, classical interference, fully destructive or constructive, is described by a superposition of perfectly dark or maximally superradiant states only (see the illustration in Fig.1(a)), but any intermediate quantum state has no counterpart in classical theory.

Classical theory asserts that several electromagnetic waves cannot interact with matter if they interfere destructively to zero, whereas quantum mechanics predicts a nontrivial light-matter dynamics even when the average electric field vanishes. Here we show that in quantum optics classical interference emerges from collective bright and dark states of light, i.e., entangled superpositions of multi-mode photon-number states. This makes it possible to explain wave interference using the particle description of light and the superposition principle for linear systems.

This expression comes from the energy-exchange interaction between the field and the sensor, described by the Hamiltonian (in the rotating-wave approximation)

Of particular interest is the case n=0𝑛0n=0italic_n = 0 for which 𝐄(+)(𝐫,t)|ψ0N⟩=0\textbf{E}^{(+)}(\textbf{r},t)\lvert\psi_{0}^{N}\rangle=0E start_POSTSUPERSCRIPT ( + ) end_POSTSUPERSCRIPT ( r , italic_t ) | italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ⟩ = 0. The n=0𝑛0n=0italic_n = 0 state is therefore unable to excite the sensor atom, for any N𝑁Nitalic_N. We name it the perfectly dark state (PDS) for the subspace of N𝑁Nitalic_N photons

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To address the multi-mode case, we consider two modes, A𝐴Aitalic_A and B𝐵Bitalic_B, represented by their respective annihilation (creation) operators a𝑎aitalic_a and b𝑏bitalic_b (a†superscript𝑎†a^{\dagger}italic_a start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT and b†superscript𝑏†b^{\dagger}italic_b start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT), and a relative phase between them given by θ𝜃\thetaitalic_θ. In this case, the positive frequency operator can be written as

Then, the probability for a photon at position r is given by the first-order intensity correlation function G(1)(𝐫,𝐫,0)=⟨ψ|E(−)E(+)|ψ⟩=|⟨0|E(+)|ψ⟩|2G^{(1)}(\textbf{r},\textbf{r},0)=\langle\psi\rvert E^{(-)}E^{(+)}\lvert\psi% \rangle=\left|\langle 0\rvert E^{(+)}\lvert\psi\rangle\right|^{2}italic_G start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( r , r , 0 ) = ⟨ italic_ψ | italic_E start_POSTSUPERSCRIPT ( - ) end_POSTSUPERSCRIPT italic_E start_POSTSUPERSCRIPT ( + ) end_POSTSUPERSCRIPT | italic_ψ ⟩ = | ⟨ 0 | italic_E start_POSTSUPERSCRIPT ( + ) end_POSTSUPERSCRIPT | italic_ψ ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, with E(+)∝a⁢ei⁢𝐤1⋅𝐫+b⁢ei⁢𝐤2⋅𝐫∝a+b⁢ei⁢θproportional-tosuperscript𝐸𝑎superscript𝑒⋅𝑖subscript𝐤1𝐫𝑏superscript𝑒⋅𝑖subscript𝐤2𝐫proportional-to𝑎𝑏superscript𝑒𝑖𝜃E^{(+)}\propto ae^{i\textbf{k}_{1}\cdot\textbf{r}}+be^{i\textbf{k}_{2}\cdot% \textbf{r}}\propto a+be^{i\theta}italic_E start_POSTSUPERSCRIPT ( + ) end_POSTSUPERSCRIPT ∝ italic_a italic_e start_POSTSUPERSCRIPT italic_i k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋅ r end_POSTSUPERSCRIPT + italic_b italic_e start_POSTSUPERSCRIPT italic_i k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⋅ r end_POSTSUPERSCRIPT ∝ italic_a + italic_b italic_e start_POSTSUPERSCRIPT italic_i italic_θ end_POSTSUPERSCRIPT, and θ=(𝐤2−𝐤1)⋅𝐫𝜃⋅subscript𝐤2subscript𝐤1𝐫\theta=(\textbf{k}_{2}-\textbf{k}_{1})\cdot\textbf{r}italic_θ = ( k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ⋅ r. As described above, for this collective measurement operator the dark and bright states can be written as |ψ01(θ)⟩=(|1,0⟩a,b−e−i⁢θ|0,1⟩a,b)/2\lvert\psi^{1}_{0}(\theta)\rangle=\left(\lvert 1,0\rangle_{a,b}-e^{-i\theta}% \lvert 0,1\rangle_{a,b}\right)/\sqrt{2}| italic_ψ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_θ ) ⟩ = ( | 1 , 0 ⟩ start_POSTSUBSCRIPT italic_a , italic_b end_POSTSUBSCRIPT - italic_e start_POSTSUPERSCRIPT - italic_i italic_θ end_POSTSUPERSCRIPT | 0 , 1 ⟩ start_POSTSUBSCRIPT italic_a , italic_b end_POSTSUBSCRIPT ) / square-root start_ARG 2 end_ARG and |ψ11(θ)⟩=(|1,0⟩a,b+e−i⁢θ|0,1⟩a,b)/2\lvert\psi^{1}_{1}(\theta)\rangle=\left(\lvert 1,0\rangle_{a,b}+e^{-i\theta}% \lvert 0,1\rangle_{a,b}\right)/\sqrt{2}| italic_ψ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_θ ) ⟩ = ( | 1 , 0 ⟩ start_POSTSUBSCRIPT italic_a , italic_b end_POSTSUBSCRIPT + italic_e start_POSTSUPERSCRIPT - italic_i italic_θ end_POSTSUPERSCRIPT | 0 , 1 ⟩ start_POSTSUBSCRIPT italic_a , italic_b end_POSTSUBSCRIPT ) / square-root start_ARG 2 end_ARG, respectively (apart from the other vacuum modes of the electromagnetic field). With such equations we can rewrite Eq. (11), up to a global phase factor, as

For a single photon impinging on a double slit, the field in the plane of interest and the detection process can be described following Scully and Zubairy [17], i.e., by replacing the two slits with two source atoms, the first at the position 𝐝1subscript𝐝1\textbf{d}_{1}d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and the second at the position 𝐝2subscript𝐝2\textbf{d}_{2}d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (the positions of the slits). Apart from a normalization factor which depends on the radiation pattern of the two ‘slit’ atoms, g⁢(𝐤)𝑔𝐤g(\textbf{k})italic_g ( k ), the field is given by the state (see SM for details)

According to the quantum theory of optical coherence introduced by Glauber [9], the statistical properties of a given field can be derived from its electrical field operator

As shown in the supplemental material (SM), the states |ψnN⟩delimited-|⟩subscriptsuperscript𝜓𝑁𝑛\lvert\psi^{N}_{n}\rangle| italic_ψ start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ constitute a complete basis satisfying H|ψnN⟩|g⟩=g2⁢n|ψn−1N−1⟩|e⟩H\left|\psi_{n}^{N}\right\rangle\lvert g\rangle=g\sqrt{2n}\left|\psi_{n-1}^{N-% 1}\right\rangle\lvert e\rangleitalic_H | italic_ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ⟩ | italic_g ⟩ = italic_g square-root start_ARG 2 italic_n end_ARG | italic_ψ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N - 1 end_POSTSUPERSCRIPT ⟩ | italic_e ⟩. This describes the excitation of the atom accompanied by a transition from state |ψnN⟩delimited-|⟩superscriptsubscript𝜓𝑛𝑁\lvert\psi_{n}^{N}\rangle| italic_ψ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ⟩ to |ψn−1N−1⟩delimited-|⟩superscriptsubscript𝜓𝑛1𝑁1\lvert\psi_{n-1}^{N-1}\rangle| italic_ψ start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N - 1 end_POSTSUPERSCRIPT ⟩, analogue to the N𝑁Nitalic_N-excitation Dicke basis for multi-atom systems [10, 13]. In analogy to the “cooperation number” for Dicke states, the 2⁢n2𝑛\sqrt{2n}square-root start_ARG 2 italic_n end_ARG factor represents the cooperativity of the absorption, where the 22\sqrt{2}square-root start_ARG 2 end_ARG appears due to the two modes.

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In atomic system it was coined subradiant by Dicke since H|ψ0N⟩|g⟩=0H\lvert\psi_{0}^{N}\rangle\lvert g\rangle=0italic_H | italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ⟩ | italic_g ⟩ = 0. Oppositely, the n=N𝑛𝑁n=Nitalic_n = italic_N state

with 𝐄(+)⁢(𝐫,t)superscript𝐄𝐫𝑡\textbf{E}^{(+)}(\textbf{r},t)E start_POSTSUPERSCRIPT ( + ) end_POSTSUPERSCRIPT ( r , italic_t ) and 𝐄(−)⁢(𝐫,t)superscript𝐄𝐫𝑡\textbf{E}^{(-)}(\textbf{r},t)E start_POSTSUPERSCRIPT ( - ) end_POSTSUPERSCRIPT ( r , italic_t ) the positive and negative frequency parts, respectively. Quantum mechanically, these parts are proportional to the photon annihilation and creation operators, that is, 𝐄(+)⁢(𝐫,t)∝aproportional-tosuperscript𝐄𝐫𝑡𝑎\textbf{E}^{(+)}(\textbf{r},t)\propto aE start_POSTSUPERSCRIPT ( + ) end_POSTSUPERSCRIPT ( r , italic_t ) ∝ italic_a and 𝐄(−)⁢(𝐫,t)∝a†proportional-tosuperscript𝐄𝐫𝑡superscript𝑎†\textbf{E}^{(-)}(\textbf{r},t)\propto a^{\dagger}E start_POSTSUPERSCRIPT ( - ) end_POSTSUPERSCRIPT ( r , italic_t ) ∝ italic_a start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT. Still according to [9], the probability of a photon from a single mode in a given state |Ψ⟩delimited-|⟩Ψ\lvert\Psi\rangle| roman_Ψ ⟩ being absorbed by a sensor atom is proportional to

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The particular case just discussed actually points towards a more general result: as proven in the SM, states of light composed of only PDSs or MSSs exhibit the same interference patterns as those derived from linear (classical) optics or, equivalently, for coherent states. However, this is not the case for the general collective states |ψnN⁢(θ)⟩delimited-|⟩subscriptsuperscript𝜓𝑁𝑛𝜃\lvert\psi^{N}_{n}(\theta)\rangle| italic_ψ start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_θ ) ⟩, Eq. (5), or superpostions with both dark and bright states, an effect which could be useful to discriminate between quantum and classical states of light without the need of employing field-field correlations [18, 17]. This can be exemplified by considering a Mach-Zehnder interferometer (MZI), see SM for details, with the input state as the intermediate entangled state |ψ12⟩=(|0,2⟩a,b−|2,0⟩a,b)/2\lvert\psi^{2}_{1}\rangle=(\lvert 0,2\rangle_{a,b}-\lvert 2,0\rangle_{a,b})/% \sqrt{2}| italic_ψ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ = ( | 0 , 2 ⟩ start_POSTSUBSCRIPT italic_a , italic_b end_POSTSUBSCRIPT - | 2 , 0 ⟩ start_POSTSUBSCRIPT italic_a , italic_b end_POSTSUBSCRIPT ) / square-root start_ARG 2 end_ARG (mode a𝑎aitalic_a (b)b)italic_b ) as the first (second) input port of the MZI), a state which is generated by impinging two photons on the two input ports of a 50/50505050/5050 / 50 beam splitter [19]. With this input, the two output ports of the MZI contain the same average number of photons independent of the relative phase φ𝜑\varphiitalic_φ within the MZI, i.e., ⟨na⟩|ψ12⟩=⟨nb⟩|ψ12⟩=1subscriptdelimited-⟨⟩subscript𝑛𝑎ketsuperscriptsubscript𝜓12subscriptdelimited-⟨⟩subscript𝑛𝑏ketsuperscriptsubscript𝜓121\left\langle n_{a}\right\rangle_{\left|\psi_{1}^{2}\right\rangle}=\left\langle n% _{b}\right\rangle_{\left|\psi_{1}^{2}\right\rangle}=1⟨ italic_n start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT | italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ end_POSTSUBSCRIPT = ⟨ italic_n start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT | italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ end_POSTSUBSCRIPT = 1 (see SM). This means that the fringes completely disappear, and the visibility of an interference patters goes to zero. The situation is completely different for classical fields, where the intensity in one of the output ports of the MZI goes to zero depending on the relative phase φ𝜑\varphiitalic_φ. In this sense, the reduction of the visibility in the quantum case stems from the quantum nature of the field.

comes with a transition rate g⁢2⁢N𝑔2𝑁g\sqrt{2N}italic_g square-root start_ARG 2 italic_N end_ARG, which is 22\sqrt{2}square-root start_ARG 2 end_ARG times that of the single-mode result: HJ⁢C|g⟩|N⟩=gN|e⟩|N−1⟩H_{JC}\lvert g\rangle\lvert N\rangle=g\sqrt{N}\lvert e\rangle\lvert N-1\rangleitalic_H start_POSTSUBSCRIPT italic_J italic_C end_POSTSUBSCRIPT | italic_g ⟩ | italic_N ⟩ = italic_g square-root start_ARG italic_N end_ARG | italic_e ⟩ | italic_N - 1 ⟩, with HJ⁢Csubscript𝐻𝐽𝐶H_{JC}italic_H start_POSTSUBSCRIPT italic_J italic_C end_POSTSUBSCRIPT denoting the standard Jaynes-Cummings Hamiltonian [14]. State (7) is the analogue of the symmetric superradiant mode, studied by Dicke in the atomic decay cascade [10, 15], and represents the most superradiant of the states with N𝑁Nitalic_N photons. We here refer to this state as a maximally superradiant state (MSS) or bright state. Finally, states within the range 0< italic_n < italic_N have intermediate transition rates. In contrast to the two-level-atom case, the present Hilbert space is unbounded, even for a finite number of field modes, since each one can support an arbitrary number of photons [13] (see Fig.1(b)).

For two-mode fields, and from a quantum perspective, the situation is much more interesting. To see this, we evaluate the probability of exciting the sensor using the eigenstates of the positive-frequency operator with null eigenvalues, that is, 𝐄(+)(𝐫,t)|Ψ⟩=0\textbf{E}^{(+)}(\textbf{r},t)\lvert\Psi\rangle=0E start_POSTSUPERSCRIPT ( + ) end_POSTSUPERSCRIPT ( r , italic_t ) | roman_Ψ ⟩ = 0 or, equivalently, H|Ψ⟩|g⟩=0H\lvert\Psi\rangle\lvert g\rangle=0italic_H | roman_Ψ ⟩ | italic_g ⟩ = 0. For a single mode, only the vacuum state satisfies this condition. But as shown below, multi-mode fields allow for a plethora of such states, even states with many photons in each mode. In the context of cavity quantum electrodynamics, the states which carry photons but are unable to excite an atom were dubbed “generalized ground states” by Alsing, Cardimona, and Carmichael [11], but here we decide to name them perfectly dark states (PDSs) since the sensor cannot see the field whenever it is in such a state.