We provide a mechanism to imprint local temporal correlations in photon streams with a character similar to spatial correlations in liquids. Typical single-photon light, such as that from an incoherently pumped two-level system, corresponds, in this picture, to a (temporal) gas, while uncorrelated light is the ideal gas. We argue that single-photon sources with good antibunching are those that exhibit temporal liquid features, i.e., a plateau for their short-time correlations (as opposed to a linear dependence) and oscillations at later times. This is a manifestation of photon time ordering that provides direct access to the single-photon purity, or probability of each emitted photon being detected, which is not usually available in stationary sources. We obtain general, closed-form analytical expressions for the second-order coherence function of a broad family of “liquid light,” which can be arbitrarily correlated, though never completely crystallized. These results invite us to reconsider what is understood as single-photon sources and how to implement them, as well as to deepen the analogies between time correlations of light and spatial correlations of matter.

A liquid is a condensed phase of matter whose definition at the microscopic level has been the subject of much debate.1 It consists of a dense, disordered assembly of molecules exhibiting short-range order in space. Simplest liquids like monatomic argon are well modeled as jumbled, closely packed spheres with an ordering governed by integer multiples of the molecular diameter.2 The molecular arrangement can be revealed experimentally by diffracting x-rays or neutrons in the fluid.3 The absence of a Bragg peak indicates that there is no long-range order, but oscillations in the radial diffracted intensities reveal short-range correlations, whereby each molecule is locally attached to a shell of its surrounding neighbors that remains free to move around and distort, but with more or less probability to be at a given distance. A gas, in contrast, presents no such correlations and has no short-range order. Two molecules still cannot sit in the same position, so there is a depletion of probabilities for close distances, but there are no oscillations. In condensed-matter physics, this is described by the structure factor, whose Fourier transform provides the so-called pair-correlation function g(r) that yields the probability of finding a molecule at a distance r from another molecule, relative to an uncorrelated—i.e., ideal—gas.4 All these correlations are in space.

Independently from these statistical considerations for correlations of distances between molecules, quantum optics arrived at the modern definition of quantum coherence of light through correlations of photons in time.5 This relies on the so-called second-order coherence function g(2)(τ), with a notation eerily reminiscent of the condensed-matter case, although there appears to be no trace of any connection from one field to the other. The g(2)(τ) function similarly quantifies the density of two-photons separated by a time delay τ, as compared to an uncorrelated (Poissonian or, as the optical terminology goes, “coherent”) photon stream.6 

There are obvious differences between the two cases: fluids are typically three-dimensional, and their correlations are in space for material objects, while quantum optics deals with one-dimensional correlations in time for detected photons. There are, however, more similarities than seem to have been previously appreciated. In quantum optics, the most studied type of quantum correlations is for single-photon sources,7 with a suppression of two-photon coincidences, i.e., two photons are never detected at exactly the same time. This is not trivial since photons, being bosons, have the natural tendency to exhibit the opposite behavior of bunching. Some order must be imbued into the photon stream to fight their urge to come together. The simplest way to achieve this is to recourse to a two-level system σ, put in its excited state at a rate Pσ, as sketched in Fig. 1(a). If the emitter has a radiative decay rate γσ, one finds for its second-order coherence,8 
g(2)(τ)=1exp(Pσ+γσ)|τ|.
(1)
This has coherence time (Pσ+γσ)1 with a linear τ short-time loss of coherence from perfect two-photon suppression g(2)(0) = 0 to uncorrelated emission limτg(2)(τ) = 1. This archetypal two-photon correlation function for a stationary single-photon source is shown in Fig. 2(ii) with its characteristic dip at τ = 0, in contrast to uncorrelated light in Fig. 2(i), which is that of, say, the stream of excitations, without passing through the two-level system.
FIG. 1.

Single-photon emission from (a) a two-level system under incoherent pumping Pσ and decay γσ and (b) with a prior cascade of intermediate transitions. (c) Photon emissions from successive random processes of excitation and emission, described by exponential distributions (bottom filled blue lines) or directly with the corresponding Erlang distributions for two events (top filled blue lines). (d) Case with N = 3, featuring one intermediate cascade and the corresponding Erlang distributions. (e) Case with N = 25 (with a rescaled time axis) with the corresponding Erlang distributions and only the final states (emission) of the cascade depicted. The photon stream is neatly ordered in time with the short-range correlations of a liquid, although there are no interactions and all processes are incoherent.

FIG. 1.

Single-photon emission from (a) a two-level system under incoherent pumping Pσ and decay γσ and (b) with a prior cascade of intermediate transitions. (c) Photon emissions from successive random processes of excitation and emission, described by exponential distributions (bottom filled blue lines) or directly with the corresponding Erlang distributions for two events (top filled blue lines). (d) Case with N = 3, featuring one intermediate cascade and the corresponding Erlang distributions. (e) Case with N = 25 (with a rescaled time axis) with the corresponding Erlang distributions and only the final states (emission) of the cascade depicted. The photon stream is neatly ordered in time with the short-range correlations of a liquid, although there are no interactions and all processes are incoherent.

Close modal
FIG. 2.

Two-photon correlation functions for (i) coherent light (ideal gas), (ii) an incoherently pumped two-level system (gas), (iii) an incoherently driven one-cascade two-level system (onset of liquid with a zoom in inset showing the g(2) = 1.0043 bunching), (iv) a strongly and coherently driven two-level system (Rabi liquid), (v) stronger liquefaction with five cascades, and (vi) liquid with 25 cascades. Note that the τ axis is rescaled in the latter case, as over the range ±10/γ of the other plots, g(2) remains below 2 × 10−3. We assume all rates to be equal.

FIG. 2.

Two-photon correlation functions for (i) coherent light (ideal gas), (ii) an incoherently pumped two-level system (gas), (iii) an incoherently driven one-cascade two-level system (onset of liquid with a zoom in inset showing the g(2) = 1.0043 bunching), (iv) a strongly and coherently driven two-level system (Rabi liquid), (v) stronger liquefaction with five cascades, and (vi) liquid with 25 cascades. Note that the τ axis is rescaled in the latter case, as over the range ±10/γ of the other plots, g(2) remains below 2 × 10−3. We assume all rates to be equal.

Close modal
Another paradigmatic type of excitation is coherent excitation, where a classical field (typically a laser) drives resonantly the two-level system, bringing it into another, much richer regime, including quantum dressing of the transitions and coherent scattering. All the regimes are covered by the same expression,9 
g(2)(τ)=1e34γσ|τ|coshγM|τ|4+3γσγMsinhγM|τ|4,
(2)
where γMγσ2(8Ωσ)2 is real at low-driving Ωσγσ/8 and becomes iΩM when Ωσ > γσ/8, with ΩM(8Ωσ)2γσ2 the (real) Mollow—also known as Rabi—splitting. In the low-driving, so-called Heitler regime, when Ωσγσ, the two-photon correlations take the simpler form
g(2)(τ)=1expγσ|τ|/22.
(3)
In both cases [Eqs. (2) and (3)], there is a qualitative change of the short-time correlations—where photon suppression occurs—from a linear τ dependence in the incoherent case of Eq. (1) to a quadratic τ2 one for the coherent cases, as is clear from Eq. (3). Such a change from linear to power dependence on time typically reflects a qualitative transformation of the response of a system. For fluids, such dependencies in an energy spectrum are, for instance, responsible for their diffusive or superfluid character.10 

Although not still compelling at this point of our discussion, we highlight that the two-photon correlation function (2) exhibits oscillations when γM becomes imaginary, i.e., when Ωσ > γσ/8, marking the onset of Mollow Physics [cf. Fig. 2(iv)]. In this case, they are understood as Rabi oscillations of the two-level system, which gets dressed by the laser.11 There is then a transition from a monotonous g(2)(τ)=1e3γσ|τ|/4(1+3γσ|τ|/4) at the threshold to one featuring all-time oscillations: g(2)(τ)=1e3γσ|τ|/4cos(ΩM|τ|/4)+3γσΩMsin(ΩM|τ|/4). The maximum g(2)(τM)=1+exp3πγσ/ΩM is obtained at τM = 4πM and, therefore, is at most 2, in the limit of Ωσ. Such oscillations are indeed Rabi oscillations of the populations since, in this case, g(2)(τ) = n0(τ)/nss, where n0(τ) is the dynamics of the system when starting from its ground state12 and nss is the steady state population, which is at most 12 for the coherently driven system since the stimulated emission prevents population inversion, thus explaining the maximum bunching of 2.

While such a Rabi interpretation is valid,13–15 in the following, we shall argue that such familiar oscillations are a particular case of a more general trend, namely, photon liquefaction in time, by which we mean temporal ordering of the photons similar to that in space when a gas becomes liquid. The terminology of “condensation” is more common to describe the gas-to-liquid transition, but given the predominance of Bose condensation for bosons, we prefer here to refer to that phenomenon with the alternative denomination of “liquefaction,” which, as we highlight again, further occurs in time. This approach is motivated by the notion of a “perfect single photon source,”16 understood as a source that suppresses photon coincidences not only at exactly τ = 0 but over a temporal window large enough or robust enough so that a physical detector will be resilient to the unavoidable time uncertainty associated with the photodetection process. Such temporal limitations of physical detectors were first highlighted for single-photon observables by Eberly and Wódkiewicz17 and later upgraded to multiphoton detection by del Valle et al.18 For two-photon suppression, this results in photon correlations of the type of Eqs. (2) and (3) being much more resilient to time-frequency uncertainties and, correspondingly, providing much better antibunching (i.e., a smaller probability of accidental two-photon coincidences), due to the flatter short-time correlation τ2.19 In Ref. 16, it was shown that in the mathematical idealization where the correlation is flattened so much as to actually open a non-analytic time-gap, i.e., forbidding completely two photons to be closer than a given time tG, then oscillations ensue in g(2)(τ) as a result of time-ordering, thus being a direct counterpart, but in time, of the transition from a gas to a liquid. In fact, in the case of a perfect, rigid time-gap, correlations are precisely those in space for a system of hard rods, as was first described by Zernike and Prins,20 who were also the first to derive the expression to compute diffracted intensities from molecular arrangements.

This mathematically perfect single-photon source16 was analyzed with no underlying physical mechanism to realize it. Here, we provide a broad class of photon temporal liquids based on a simple mechanism whereby the excitation undergoes a cascade of transitions between various states before ultimately emitting a photon. This is particularly relevant for solid-state systems21 or sufficiently complex ones22 where the two-level system is implemented by an artificial atom embedded in an environment that comes with various intermediate states, energy levels, shell structures, metastable transitions, etc., which could even be controlled or ultimately engineered.23 We find that even incoherent driving, insofar as it involves intermediate steps in the cascade, already features a two-photon correlation function that corresponds to the liquid phase, with oscillations and a power law dependence for the short-time correlations. The power law and visibility of the oscillations are furthermore directly related to the number of cascades that produce the flat plateau typical of hard-sphere repulsions in condensed matter, as well as the characteristic bunching elbow that marks the onset of local ordering and, ultimately, clear oscillations that endow the stationary source with the features of a pulsed single-photon emitter.16 

As opposed to solving a quantum optical master equation—which yields the same results, as we shall later show—we take a more insightful statistical and condensed-matter-inspired approach based on the waiting time distributions. The incoherent two-photon emission (1) can be understood as a two-step process with, first, an underlying, backbone Poisson (uncorrelated) stream of events corresponding to the incoherent excitation at rate Pσ that brings the system into its excited state. Each such event then draws another Poisson-distributed random number with parameter γσ, describing the spontaneous emission (second step).16 This is sketched in Fig. 1(c), where the exponential (Poisson) distributions alternate sampling of uncorrelated excitation T1 = 1/Pσ and spontaneous emission T2 = 1/γσ times. Since we are only interested in the emitted photons, we can equivalently consider directly the distribution for T1 + T2, which is given by w(τ)PσγσPσγσ(eγστePστ). If the emission involves N steps, first with parameter Pσ to excite, and each subsequent one with parameter γi (2 ≤ iN) to relax down a cascade of levels as shown in Fig. 1(b), then one needs to similarly replace the exponential decay by the distribution for the sum of N independent exponential random variables, which is one of the phase-type classes of distributions known as the hypoexponential distribution with N parameters. When those are all equal, the distribution is more popularly known as the Erlang distribution, and we shall focus on this case for conceptual simplicity. The waiting time distribution for N steps (one excitation plus N − 1 cascades) can then be simply obtained as
wN(τ)γNτN1eγτ(N1)!.
(4)
The case N = 2 is the limit of w(τ) above when Pσγσ. Two-photon correlation functions can be computed from the waiting time distribution wN(τ) by transiting to the Laplace space,24,
g̃N(2)(s)0gN(2)(τ)esτdτ=Nγw̃N(s)1w̃N(s),
(5)
with w̃N(s)=(1+sγ)N the Laplace transform of Eq. (4). This yields, by inverse Laplace transform,
gN(2)(τ)=1+p=1N1zNpexpγ(1zNp)|τ|,
(6)
where zNexpi2π/N are the Nth roots of unity. This succinct general expression can be easily made explicit for particular cases, e.g., with no cascade (N = 1 for the excitation alone), we have an uncorrelated (or coherent) photon stream, Fig. 2(i), while N = 2 in panel (ii) describes the excitation plus spontaneous emission of Fig. 1(c) and, therefore, recovers Eq. (1). We get new results with two cascades [N = 3, Fig. 1(d)], for which Eq. (6) simplifies to
g3(2)(τ)=12sin32γ|τ|+π6e32γ|τ|,
(7)
and for three cascades (N = 4),
g4(2)(τ)=1e2γ|τ|2eγ|τ|sin(γ|τ|),
(8)
with the possibility to derive similar closed-form expressions for other N. These cascaded chains of incoherent relaxation produce, interestingly, two-photon correlation functions that are more like in character those of the coherently driven case (2) than the incoherent case (1). Even with two cascades only, there is some onset of liquefaction with oscillations which, although not compelling numerically [cf. Fig. 2(iii)], are clear from the analytical expressions in Eq. (7). The maximum g3(2)(τM)=1+e3π1.0043 at γστM=2π/3, however, is only marginally different from unity. Higher g(2) are obtained for a higher number of cascades, at time delays which are zeros of transcendental equations [for instance, as a solution of expγτcos(γτ+π/4)=1/2 for N = 4], but they can be well approximated for large enough N when a clear plateau and oscillations develop in the gN(2)(τ) function, as shown in Fig. 2(vi). The kth peak of the oscillation is in very good approximation located at the mode (i.e., most likely value) of the waiting time distribution for the kth emission, itself occurring by construct after (k − 1)N full cycles of Poissonian excitations and subsequent cascades, followed by a final cycle of re-excitation and (N − 2) cascades, the (N − 1)th cascade being the sought event itself. This places the kth peak at the mode of the waiting time for the (kN − 1)th Poissonian event, or, if assuming all rates to be the same, the mode of the Erlang distribution, which is25,
τk±kN1γ,
(9)
as shown with grid lines in Fig. 2(vi) for negative τ. This makes a direct link between the bunching peaks and the photon time ordering and provides another manifestation of temporal liquefaction as a result of the “hardening” of photons that increasingly repulse each other. Their time ordering is the result of compound times averaging out the more extreme (Poissonian) fluctuations, producing a regular stream of photons. This is obvious in Fig. 1(e), where photons appear to be equidistantly spaced in time with small fluctuations, as would be expected from an externally controlled (pulsed) single-photon source.16 The strongest repulsion is at short times, where the Taylor expansion reveals a power-law plateau commensurate with the number of cascades,
gN(2)(τ)N2N!(γ|τ|)N1.
(10)
This extended photon repulsion in time means that a physical detector will be less affected, at a qualitative level, by the fundamental time uncertainty attached to photodetection. These qualities are inherited from the waiting time distribution (4) and the shape of the probability distribution for each photon emission. Their flattening at short times makes it possible to open an effective temporal gap. This paves the way toward a genuine perfect single-photon source with no multiphoton emission for detectors with a better temporal resolution than the time gap, just as a superconductor is a genuine perfect conductor with no loss whatsoever. Together, these two features (time gap and bunching peak) also allow for a straightforward estimation of the one-photon “purity” π1 of the source, defined here as how many emitted single photons are actually detected, i.e., not only photons are detected in isolation (suppressing multiple photons, as per the usual definition), but they must also be detected, thereby equally suppressing vacuum. Since the optimum emission rate is given by each cascade resulting in an event, so as
γN=γN,
(11)
this can be directly compared to the measured emission rate γeff (γ is that of one transition in the cascade and might be unknown). One thus obtains the purity π1γeff/γN as
π1=γeffτ1NN1.
(12)
This gives the percentage of photons actually detected. For N large enough, this is, in good approximation, simply γeffτ1, which is 1 when no photon is missing in the ordered stream. This can be compared to the perfect single-photon source16 whose purity π1γeff/γG—with γG the optimum emission rate for this source (explicited in Ref. 16)—is obtained as
π1=γefftGg(2)(tG)g(2)(tG)1,
(13)
that is, with the bunching g(2)(tG) at the gap τ = tG, replacing N in the cascade model. This suggests that the bunching peak relates to the number of cascades, although gN(2)(τ1) is much smaller than N due to the spreading in time on both sides of the peak that dilutes the bunching. In both cases, the time ordering gives access to the purity from the photon stream alone, which is otherwise not directly available in stationary single-photon sources. This is another manifestation of the superior and fundamental single-photon character of such liquid light, excluding not only multiple photons but also no photons. Such an order is local only, though, as fluctuations, however small, pile up and eventually wash out correlations for photons distant enough. This thus corresponds to a photon liquid and never a crystal, as the stream is intrinsically stationary; correlations fade away for long-enough times. This is, however, of little concern for the single-photon character. Clearly, the higher the N, the better the liquefaction in time, i.e., the closer one gets to a regularly spaced stream of exactly one photon. The limit N produces a Dirac comb and, therefore, a temporal crystal, but it is also unphysical, as the spacing between photons also becomes infinite since the excitation would have an infinite number of transitions to go through before it could be reloaded at the top of the (infinite) ladder. In this sense, this limit does not correspond to a harmonic oscillator, from which one does not expect quantum light anyway. Besides, if one collects the photons from all the transitions, then the quantum features disappear and uncorrelated light is produced in the limit of large N. The two-photon correlation in this case is obtained from similar statistical arguments as
ganyN(2)(τ)=11(N1)2expNN1γ|τ|,
(14)
which goes to ganyN(2)(τ)=1 for large N, while the case of N = 2 recovers the standard single-photon source result. Therefore, although it does not matter which transition is monitored for the single-photon emission, it must be unique.
Since incoherent pumping brings no coherence between the levels, our approach—which is tantamount to rate equations—yields exact results that would be obtained from a quantum treatment, i.e., with a master equation for a multilevel system. The case N = 3, for instance, with level structure |i⟩ for 0 ≤ i ≤ 2 and Hamiltonian Hi=1,2ωi|i⟩⟨i| could be described by the Lindbladian Lρi[H,ρ]+Pσ2L|20|+γ22L|12|+γσ2L|01|ρ, where for any operator Ω, LΩρ2ΩρΩΩΩρρΩΩ. One can then compute steady-state two-time correlators like G10(2)(τ)σ10(0)σ10(τ)σ10(τ)σ10(0) for σ10 ≡ |0⟩⟨1| from the quantum regression theorem, i.e., G10(2)(τ)=Trσ10σ10eLτ[σ10ρssσ10] on the steady-state density matrix, which is diagonal,
ρss=1Pσγσ+γσγ2+Pσγ2γσγ2Pσγ2Pσγσ.
(15)
The normalization of G10(2) yields g2(τ) for this transition, which recovers Eq. (7) for the case Pσ = γσ = γ2 = γ that we considered previously. So at this stage, we have no new result; we have only confirmed that the statistical derivation gives the same, exact expression as the quantum treatment for two-photon correlations. Such a quantum approach becomes necessary to compute dynamical observables that are not accessible, at least not straightforwardly, through such a statistical approach. An example is the photoluminescence spectrum, which we find to be always Lorentzian for all N, so the emission is always that of pure spontaneous emission from the observed cascade, with a linewidth of 2γ from the sum of pumping and decay that are taken, here, equal. There is thus no reshaping of the spectral shape, unlike the coherent driving which, in the Heitler regime, becomes the square of a Lorentzian. It is interesting that although the cascade alters considerably the two-photon properties, it leaves unaffected the single-photon ones. Indeed, although luminescence remains Lorentzian, antibunching is more robust in the cascade scheme, as can be demonstrated by considering the filtered two-photon correlation gΓ(2) that corresponds to the impact of photodetection as a temporal uncertainty 1/Γ imparted by a detector with bandwidth Γ.18 From the quantum model of the cascade, we find that the filtered two-photon correlations for large enough Γ (i.e., a good enough detector) are, for N = 3,
g3,Γ(2)(0)23Pσγσ+Pσγ2+γσγ2Γ2+O1Γ3,
(16)
which is to be compared to the case of coherent excitation of the two-level system,
gcoh,Γ(2)(0)γσ2+8Ωσ2Γ2+O1Γ3,
(17)
as well as the conventional incoherently excited two-level system,
g2LS,Γ(2)(0)23γσ+PσΓ+O1Γ2.
(18)
The coherent driving of the two-level system yields a much more robust antibunching (17) to detection than the incoherent pumping (18), since it recovers antibunching as Γ−2 with increasing filtering Γ from the detector as opposed to Γ−1 for the incoherent case. The one-cascade scheme N = 3, Eq. (16), also recovers antibunching as Γ−2, and one can furthermore check that, for identical emission rates, the one-cascade case can always provide a better coefficient for antibunching (i.e., a smaller prefactor) by adjusting the parameters of the cascade. The one-cascade antibunching is optimum for γ2/γσ=8(Ωσ/γσ)2/[1+4(Ωσ/γσ)2], with Pσ chosen to equate the emission rate to that of the coherently driven system, which is set by Ωσ. In this case, the one-cascade g(2)(0) is at most 2/3 that of the coherent case, a worst-case scenario when Ωσ = γσ, and is even smaller otherwise. While one-cascade emission already beats quantitatively the coherent driving regime, more cascades result in superior antibunching at a qualitative level, with a 1/ΓN−1 dependence,
gN,Γ(2)(0)23Pσγσk=2N1γkR1ΓN1+O1ΓN,
(19)
defined in terms of the emission rate of the cascade,
R1Pσ+1γσ+k=2N11γk1.
(20)
For comparison, we remind that the emission rate of the coherently driven 2LS is 4γσΩσ2/(γσ2+8Ωσ2). This shows that we have a candidate for the best known single-photon source to date that outperforms, at a qualitative level, the coherently driven standard.26 The loss of antibunching in τ is also interesting. Although we cannot provide it for all N, and full-τ expressions as well as general γk would be too bulky, it is interesting to consider as illustrations the leading-order in τ for N = 3 and 4 with equal rates γ2 = γ3 = Pσ = γ, in which case
gN=3,Γ(2)(τ)6γ2(2γ+Γ)(2γ+3Γ)(3γ2+3γΓ+Γ2)+34γ2(Γ2γ)(2γΓ+Γ2)(2γ+3Γ)(3γ2+3γΓ+Γ2)τ2+o(τ3),
(21)
and
gN=4,Γ(2)(τ)8γ3(2γ+3Γ)(2γ2+2γΓ+Γ2)+γ3Γ(Γ2γ)(2γ+3Γ)(2γ2+2γΓ+Γ2)τ2+o(τ3),
(22)
(note that the denominator is the same in both cases for the constant and τ dependence). These have various common features of interest. First, they provide exact Γ expressions for small τ, recovering the equal-rate version of the general Eq. (19) when γ ≪ Γ, i.e., gN,Γ(2)(0)2N3γσΓN1+O1ΓN. In addition, the case Γ = 0 yields
gN,Γ0(2)(0)=2,
(23)
i.e., the light gets thermalized. This is a physical result that is not obtained by the coherently driven system, which in the same limit tends to 1. The coherence of the driving laser is assumed to have no linewidth and, therefore, is not susceptible to being filtered. The cascaded emission, on the other hand, recovers the expected bosonic limit, showing again that it arises from basic spontaneous emission. This thermalization becomes particularly apparent for
Γ<γσ+Pσ,
(24)
(i.e., Γ < 2γ in our case), which one can define as the condition of subnatural filtering, in which case the τ correction produces bunching, i.e., g(2)(0) ≥ g(2)(τ) locally around τ = 0, even though one can still have g(2)(0) < 1 (sub-Poissonian). It is also notable that although the plateau flattens to order N with N cascades according to Eq. (10), filtering smoothens it back to τ2, translating the temporal plateau into a prefactor (19) for g(2)(0) instead. This suggests that characterizations of single-photon sources, although indeed relying on g(2)(0) as is the current practice, should, however, be more critical of their dependency on filtering, e.g., also considering quantities such as dgΓ(2)(0)/dΓ in the limit of large Γ. At exactly the threshold of Eq. (24), one recovers a flat plateau to order τ3, but with a finite density of two-photon coincidences, that one can obtain in full form through compact expressions which, nevertheless, will be enough and more instructive to provide as leading-order approximations, namely, gN=3,Γ=2γ(2)(τ)313+413γ3τ3+o(τ4) and gN=4,Γ=2γ(2)(τ)110+215γ3τ3+o(τ4). We provide these results not for the amusement of featuring an exact fundamental antibunching of 1/10 but to illustrate the sort of insights that a quantum model can add to our statistical treatment, as well as the limitations intrinsic to the detection process.
The exact cases, for all times τ, for both the coherent and incoherent pumping of the two-level system are provided in Ref. 19, but we can reproduce here their small-time approximations for the sake of comparison with the previous results. These are
gcoh,Γ(2)(τ)γγ+Γ2+γ2Γ24(γ+Γ)2τ2+o(τ3),
(25a)
g2LS,Γ(2)(τ)2(Pσ+γ)Pσ+γ+3Γ+Γ(Pσ+γ)(Γ[Pσ+γ])Pσ+γ+3Γτ2+o(τ3),
(25b)
where we see again that g2LS,0(2)(0)=2 but that gcoh,0(2)(0)=1 as previously commented, and also that while Eq. (25b) thermalizes according to the criterion of Eq. (24), Eq. (25a) does not and remains antibunched as long as it remains sub-Poissonian, suggesting a link between these two concepts through the Bose statistics of the thermal field. At the subnatural filtering threshold Eq. (24) with Pσ = γσ, we have g2LS,2γ(2)(τ)12+23γ3|τ|3+o(τ4), featuring the fabled criterion of 1/2 supposedly identifying single-photon emitters.27 In this particular context, it is seen to at least mark the transition from sub-Poissonian to super-Poissonian.

We have only briefly explored the realm of possibilities at the quantum level, and one should further study other parameter ranges (where pumping and decay rates are not simply equaled), driving regimes (coherent driving of the cascades), types of filtering (not Lorentzian), etc., so as to get a more complete picture of how such sources can be further optimized and exploited as quantum sources. However, even preliminary considerations show that the mechanism is far-reaching.

Conventional single-photon sources have now reached considerable two-photon suppression with g(2)(0) of the order of 10−5 in the pulsed emission regime.28–30 To what extent such performances can be further improved and/or maintained in the continuous emission regime is an open question. Our scheme unifies the pulsed and continuous emission regimes and is intrinsically robust to detection, which appears to be a key limiting factor in state-of-the-art implementations. It also upgrades the notion of single-photon purity to not only suppressing multiphoton emission but also suppressing no-photon emission, making a literal embodiment of the single photon: no more, no less. For those reasons, we expect multilevel schemes (or others able to realize similar types of single-photon temporal liquids) to eventually replace current approaches to the problem. There have been many studies of single-photon emission from multilevel schemes both theoretically and experimentally, but instead focusing on intersystem crossing, i.e., the disruption of one transition by others that couple to it. A typical example is the triplet spin configuration getting in the way of the singlet in a molecule.31 In this context, most models consider three-level systems only,32,33 as this is enough in most cases to account for the typical profiles of g(2)(τ) in the presence of such so-called shelving states that produce bunching elbows following the expected antibunching from g(2)(0) ≈ 0.34–40 More sophisticated models with more states can even identify the number of dark states (up to 7) through the best-fits of the correlation traces over 11 orders of magnitude in time.41 Importantly, however, all such models, although they produce the bunching, do so with merely a multi-exponential return to uncorrelated emission, that is, therefore, locally linear for small-τ correlations. That is, they do not result in either the flattening (gap opening) or the oscillations, however small, associated with liquefaction, so our scheme corresponds to an entirely different regime altogether. The oscillations that we report are conceptually noteworthy as they occur in a completely incoherent system, exposing the time-ordering as the system liquefies. There is a fairly broad range of parameters to produce them (which the Erlang particular case fulfills), but they are not systematic, and a more general study of the possible dynamics and links to self-oscillations should be undertaken. We have instead focused on the most basic configuration to highlight the conceptual novelty, but this could be both optimized and extended in several ways to tackle realistic experiments. Let alone that the Erlang (degenerate) distribution is a very special case (with only one parameter, whereas one naturally expects various transitions to come with possibly largely varying parameters), the number of steps in the cascade could also be a random variable, and the possibility for various carriers to undergo such cascades simultaneously could be included. Exciting such systems with a conventional single-photon source42 could be a way to enforce our mechanism in existing solid-state level structures. In another possible realization relying on, say, photon cascade emission,43 spectral filtering could be used to extract the single-photon character of a non-degenerate transition. One could also drive coherently, as doing so with the two-level system already makes a qualitative step forward, as is now clarified by our picture of the process. Regardless of its possible implementation, our basic picture of liquefaction should, however, remain and even apply to more general cases, e.g., interactions between several emitters that display curves similar to Eq. (6).44 

Our findings come with several conclusions. One is that the quest for perfect single-photon sources has been so far driven by technological improvements to reduce g(2)(0) as much as possible, but keeping the basic structure of a two-level system. This is a quantitative and asymptotic race that is doomed to imperfection as the limitations are fundamental: photodetection as a physical process will always make possible the simultaneous detection of two photons from a two-level system.19 To obtain perfect single-photon emission, one must open a gap somewhere.16 We have provided a straightforward mechanism, furthermore of relevance in solid-state platforms and that could otherwise be engineered, of a cascade process that results in strong repulsions between photons with the effect of imprinting strong correlations between them, similarly to how interactions order and correlate molecules in a fluid. The number N of cascades rules the magnitude of the effect, and we have given a general closed-form analytical expression for all N. A related conclusion is that g(2)(0) is not in itself the most relevant measure for two-photon suppression. One should instead consider g(2)(τ) locally around τ = 0, and consider both the power dependence of the short-time correlations as well as the presence of oscillations or at least bunching (elbows) past the first coherence time, as these mark the onset of short-time photon ordering. A quantum theory of the mechanism, however, shows that frequency filtering—which describes the effect of photodetection at a fundamental level—brings back the τN dependence to a quadratic τ2 one, with the benefit of the unfiltered plateau being translated to the magnitude of g(2)(0) which, interestingly, thus becomes again the central indicator of good single-photon emission. To unveil the temporal-plateau origin of this antibunching, one should undertake a closer inspection of the resilience of the correlations to detection. Our results also suggest that considerably more types of quantum lights are awaiting to be discovered and classified from such a perspective. Maybe the most far-reaching implication of our approach is that thermodynamic concepts, which are central to describing condensed matter, could also provide more systematic and deeper descriptions of quantum light, possibly relying on equations of states to describe the various phases as opposed to nth order coherence (almost always truncated to n = 2 anyway). Bunching, for instance, might be related to dense plasma.45 We also find it suggestive that detection behaves as a temperature with the effect of vaporizing the photon liquid into a gas. Other thermodynamic concepts, like entropy, would probably be usefully pursued. Intriguingly, this condensed-matter perspective requires trading space for time.

We acknowledge S. K. Kim, F. Sbresny, K. Boos, and K. Müller for interesting discussions and feedback as this work was being conceived during a stay of the authors in Munich. E.d.V. acknowledges support from the CAM Pricit Plan (Ayudas de Excelencia del Profesorado Universitario), the TUM-IAS Hans Fischer Fellowship, and Project Nos. AEI/10.13039/501100011033 (2DEnLight) and Sinérgico CAM 2020 Y2020/TCS-6545 (NanoQuCo-CM). F.P.L. acknowledges the HORIZON EIC-2022-PATHFINDERCHALLENGES-01 HEISINGBERG Project No. 101114978.

The authors have no conflicts to disclose.

Eduardo Zubizarreta Casalengua: Conceptualization (supporting); Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (lead); Software (lead); Validation (lead); Visualization (supporting); Writing – review & editing (equal). Elena del Valle: Conceptualization (supporting); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (supporting); Supervision (equal); Validation (supporting); Writing – review & editing (equal). Fabrice P. Laussy: Conceptualization (lead); Data curation (supporting); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

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