The colored Hanbury Brown–Twiss effect

The Hanbury Brown–Twiss effect is one of the celebrated phenomenologies of modern physics that accommodates equally well classical (interferences of waves) and quantum (correlations between indistinguishable particles) interpretations. The effect was discovered in the late thirties with a basic observation of Hanbury Brown that radio-pulses from two distinct antennas generate signals on the oscilloscope that wiggle similarly to the naked eye. When Hanbury Brown and his mathematician colleague Twiss took the obvious step to propose bringing the effect in the optical range, they met with considerable opposition as single-photon interferences were deemed impossible. The Hanbury Brown–Twiss effect is nowadays universally accepted and, being so fundamental, embodies many subtleties of our understanding of the wave/particle dual nature of light. Thanks to a novel experimental technique, we report here a generalized version of the Hanbury Brown–Twiss effect to include the frequency of the detected light, or, from the particle point of view, the energy of the detected photons. Our source of light is a polariton condensate, that allows high-resolution filtering of a spectrally broad source with a high degree of coherence. In addition to the known tendencies of indistinguishable photons to arrive together on the detector, we find that photons of different colors present the opposite characteristic of avoiding each others. We postulate that fermions can be similarly brought to exhibit positive (boson-like) correlations by frequency filtering.

Photon correlations are the basic conveyor of optical information in the quantum information age 1 .By regulating a stream of photons and/or endowing them with quantum correlations such as entanglement, one can use them as qubits or to distribute cryptographic keys 2 .Every time that a platform demonstrates its ability to generate photons that have such correlations, new prospects arise to extend the frontiers of quantum technology.One rising star to engineer quantum states in solid state optics are microcavity polaritons 3 .These semiconductor heterostructures can trap both light (cavity photons) and matter (electronhole pairs) to keep them under strong interaction for a long time.The resulting eigenstates, polaritons, have had resounding successes in the investigation of quantum phases such as Bose-Einstein condensates 4 , superfluids 5 and nonlinear quantum fluids 6 .They also progress quickly to implement actual devices 7 .Lately, their quantum character was consolidated by the observation of squeezing 8 , indicating nonclassical features at a macroscopic level.Only missing to the polariton prize list is quantum correlation at the single particle level.In this Letter, we report the observation of single-photon antibunching from the emission of an out-of-equilibrium Bose-Eintein condensate.This is made possible thanks to generalized photon correlations that preserve the energy information as well as their time of detection 9 .The success of this technique and its results open the era of quantum polaritonics in microcavities and demonstrate the concept of a new class of quantum correlations that can be pursued in a wide range of optical platforms.
Since the seminal experiment of intensity correlations by Hanburry Brown and Twiss 10 , that revolutionized the field of quantum optics and led to Glauber's Nobel prize for the theory of optical coherence in a quantum setting 11 , the standard approach to quantify photon correlations is through correlators of the type: with a(t) the light field annihilation operator at time t and τ the time delay between detections.This quantitythe second-order correlation function-describes the statistical distribution between photons in their stream of temporal detection.Other properties of the photons can be included, e.g., their position 10,12 or polarization 13 , with applications spanning from subatomic interferometry 14 to entangled photon pairs generation 2 .One can also record the energy of the photon (or, equivalently, its frequency).This is a characterization of a different type than position or polarization since time and frequency are conjugate variables.It provides as much information as can be retrieved according to quantum mechanics 9,15,16 .Experimentally, such measurements have been performed for fixed sets of frequencies, by inserting filters in the paths of a standard Hanburry Brown-Twiss setup [17][18][19][20][21] .Considerably more information is gained, however, by spanning over all possible combinations of frequencies, as this reveals rich landscapes of correlations 22 that are washed out when averaging over the energy.
In this Letter, we introduce a technique using a streak camera setup that allows to image such a full mapping of time and frequency resolved photon correlations of an out-of-equilibrium condensate.We use the photon stream coming from a polariton condensate due to the intrinsic importance of Bose-Einstein condensation, which defines the fifth state of matter, as well as for the nontrivial-even unsuspected-features predicted by the theory.Both the principle and the technique are general and should allow, with optimized setups, to revisit systematically all of quantum optics in systems with stronger quantum optical features but weaker intensities of emission.The rich and varied correlation landscapes predicted by the theory 22 should be useful to power tomorrow's quantum technology 23 .
The measurement is sketched in Fig. 1(a) and is detailed in the Methods section.To summarize the basics: light is dispersed by a monochromator and is directed into a streak camera that operates at the singlephoton level, as has already been demonstrated with standard photon correlations under pulsed excitation 24  each detected photon in successive frames that are postprocessed to calculate intensity correlations.A scheme of the system is shown in Fig. 1(b): polaritons condense into the ground state from a reservoir of high energy polaritons injected by a CW off-resonant laser.There are constant losses, chiefly through the cavity mirror, which allow to study the steady-state correlations.The reservoir relaxation into the condensate is typically phononmediated and establishes a nontrivial quantum coherence between the two systems, 25 that is responsible for the spontaneous coherence buildup 26 .The main experimental signature of polariton condensation is line narrowing with increasing pumping as the system goes from a chaotic source to a coherent one.In standard photon correlation measurements, this results in a transition of g (2) (0) from above 1 to 1, 25,27 .However, none of these features guarantee quantum correlations at the single particle level.On the other hand, such an outof-equilibrium condensate is effectively able to sustain a quantum correlated steady state when the condensate is not fully formed and is in its stage of coherence buildup.This is demonstrated by, e.g., the Wigner function of the condensate, which is a representation of the state in the phase space of quadratures that proves its quantum character whenever it takes negative values.Its theoretical calculation for our regime of condensation is shown in Fig. 1(c) and does, indeed, correspond to a genuine quantum state.
The quantum character also manifests in direct twophoton correlations when keeping the energy degrees of freedom.The so-called "two-photon correlation spectrum" g Γ (ω 1 , ω 2 ; τ ) provides the tendency of a photon to be detected at a frequency ω 2 at a delay τ after another photon has been detected at frequency ω 1 9,22 .Each photon is detected within a frequency window of size Γ.The theoretical calculation is shown for photon coincidences (τ = 0) in Fig. 2(a) in the regime where the condensate is maintained in its nucleating stage in a steady state thanks to the out of equilibrium character of polaritons.One sees that on the diagonal of equal-frequency detection, the theory predicts bunching.This is true even for cases when the overall stream is uncorrelated, as such a bunching induced by frequency filtering is a general result linked to Purcell's argument of the Bose character of the field 28 .More interestingly, when correlating photons with opposite frequencies, the theory predicts correlation that exhibits a clear antibunching (g (2) (ω 1 , ω 2 ; 0) < 1), i.e., detecting a photon on one side of the spectrum makes it less likely to detect another photon on the other side.For stationary processes, such an antibunching of coloured photons is observed for systems with a nonlinearity at the single particle level 22 , e.g., a two-level system.For a state undergoing condensation, such an antibunching is a priori unexpected.It can be understood on physical grounds as a manifestation of the exchange energy term that makes bosons effectively attract each others.In the regime of their condensation, particles exhibit a transition towards a collective state with wich they share identical features, resulting in the antibunching of particles with too distinct characteris- (b) Experimental observation of g (2) (ω1, ω2; 0) in the same regime, displaying a remarkable agreement.(c) Time-resolved correlation for the three regions marked in the colour map: (i) on the diagonal (ω1 = ω2) exhibiting bunching, (ii) in the region of transition with no correlation, (iii) correlating opposing elbows, exhibiting antibunching.

tics.
With the procedure described previously, we are able to obtain g (2) (ω 1 , ω 2 ; τ ) experimentally.In Fig. 2b, we show the experimental τ = 0 two-photon correlations spectrum measured for the polariton condensate.An excellent agreement with the theory is obtained: same frequency photons are bunched and, remarkably, the antidiagonal also features two clear regions of antibunching where g (2) (ω 1 , ω 2 ; τ = 0) < 1.This is, to the best of our knowledge, the first time that a polariton system exhibits from a macroscopic phase a quantum behaviour at the single particle level.The data required prolonged acquisition times in extremely stable conditions to collect millions of events to reconstruct the landscape of quantum correlations, and although they have some level of noise, the features are unambiguous.In Fig. 2c, we show the temporal correlations for three points of the (ω 1 , ω 2 )-space, both for the experiment and the theoretical model.Again, there is an extremely good agreement and a clear evolution of the correlations from bunching (g (2) (ω, ω; 0) ≈ 1.5 in region 1) to antibunching (g (2) (−ω, ω; 0) ≈ 0.7 in region 3).Another fundamental feature of the theory is that correlations depend on the frequency windows that select which photons are correlated.Smaller windows lead to stronger correlations but, again, at the price of a smaller signal.In Fig. 3, we show the dependence of the two-photon correlation spectrum on the size of the frequency windows for a point that features antibunching.When the frequency window is very large, Γ γ a , both the experimental and theoretical g (2) (ω 1 , ω 2 ; τ ) recover as expected the results of standard photon correlations which has always been reported to be larger than 1 for these kinds of systems.As the size of the frequency window decreases, the system shows a transition from bunching to antibunching, demonstrating how single-photon antibunching can be uncovered simply by discriminating in frequency.The observation matches again extremely well with the theory.
At this point it is important to put these results in perspective with the canon of quantum optics and the pre-existing findings.There are various indicators of the quantum nature of a system, such as violation of Cauchy-Schwarz inequalities or Bell's inequalities, also for coloured photons 29 .In our case, the quantum character does not manifest in these terms, i.e., through a conventional antibunching g (2) (0) < g (2) (τ ) ≤ 1, or its multimode counterpart.Our findings are also different from squeezing, which is not present in the model of outof-equilibrium Bose-Einstein Condensation.The quantum character is however established theoretically from the unmistakable criterion provided by the Wigner function, Fig. 1c.In fact, frequency-resolved photon correlations provide a new indicator for the quantum nature of a system, that is straightforwardly implemented with standards optical laboratory equipment, and should allow to track quantum behaviours in a large family of systems where these are feeble, fragile or washed out by averaging.
As a conclusion, we have established that the photon statistics from the light emitted by a condensate of exciton-polariton in the dynamical process of condensation exhibits quantum correlations.These are not directly apparent if considering all photons indiscriminately as competing correlations exist between photons of different frequencies.This results in the washing out of a considerable amount of information.We have demonstrated the possibility to extract the correlation landscapes experimentally and found an excellent agreement with the theory.This shows that both the theoretical concepts and the experimental technique are robust and ripe to be deployed in a large range of quantum optical systems, with prospects of evidencing further classes of quantum correlations and optimizing those already known.Systems such as superconducting qubits, that have achieved an impressive control of conventional quantum correlations 30 , should be able to extend this technique to unravel the much richer quantum landscapes sculpted in the light field by a two-level system 22 .

Experiments:
The sample is a semiconductor microcavity with high reflective Bragg mirrors and 3 × 4 quan- tum wells at the antinodes of a 2λ cavity.It is pumped at normal incidence and non-resonantly with a single-mode laser at a wavelength of 725.2 nm.Each photon emitted by the lower polariton state is mapped to a pixel in the streak detector with a temporal and spectral resolution of 3.2 ps and 70 µeV respectively in a total window of 1536 ps × 456.7 µeV.The correlation landscapes are obtained from correlations between these clicks (≈ 1.69 per sweep in a total of 350 000 frames).We found it important not to use a chopper, that introduces long timescales correlations due to heating and cooling of the sample that result in non-stationarity of the photon statistics, although this is not apparent through single-photon observables such as the intensity.This, of course, limits the range of excitation power we can use.The widths in frequency are defined while computing the correlations by grouping consecutive pixels in windows of different sizes with a minimum step size of 10.6 µeV.Two-photon correlation spectra are obtained by spanning the windows to compute all possible correlations, including partial and full-overlapping of the windows.This shows the considerable improvement of a streak-camera over standard setups with APDs.All the computations are done with the raw data only: there is no normalization and the correlations go to 1 at long time self-consistently.

Theory:
The quantum dynamics of an out-ofequilibrium condensate maintained in a steady state through the interplay of pump and decay is described at its most elementary level through the master equation (see Supplementary Information online): Γ (ω 1 , T 1 ; ω 2 , T 2 ) = : T 2 i=1 A † ωi,Γ (T i )A ωi,Γ (T i ) : with T , (resp.:) the time (resp.normal) ordering and A ωi,Γ (T i ) = Ti −∞ e iωit e −Γ(Ti−t)/2 a(t) dt the field filtered at frequency ω i and width Γ at a time T i .

FIG. 1 :
FIG. 1: Principle and Setup of time-and frequency-resolved photon correlations.(a) Sketch of the experiment: the reflected light from a microcavity is dispersed onto a streak-camera detecting at the single-photon level and stored in individual frames which post-processing allows to build photon-correlation landscapes (b) Sketch of the theory: a laser excites non-resonantly the lower polariton dispersion, creating a reservoir of hot excitons b that condense into the ground state a at the minimum of the branch.(c) Theoretical Wigner function of the condensate in the steady state regime of out-of-equilibrium condensation: the Wigner function takes negative values (blue region), which proves the genuine quantum nature of the state.

FIG. 3 :
FIG.3: Effect of detectors linewidths.(a) two-photon correlation landscapes g(2) (ω1, ω2; 0) as a function of the filter width, from a fraction of the peak, 74.1 µeV (left), roughly half-peak width, 158.8 µeV (center), to full-peak filtering, corresponding to standard auto-correlations.The position of the two filters is shown explicitly on the spectral line as the red and yellow windows (orange when overlapping).(b) Timeresolved traces for the selected areas.The smallest frequency window available, 10.6 µeV, is shown in red: antibunching is even more pronounced although the signal becomes extremely noisy.
)with Lindblad terms L O (ρ) = 2OρO † − O † Oρ − ρO † O for any operator O, corresponding to the polariton condensate a and reservoir b lifetimes, reservoir pumping b † and particles transfer a † b from the reservoir to the condensate.Parameters are γ a = γ b throughout and P ba = 10γ a except for Fig.1where P ba = 4γ a (the Wigner is still negative at P ba = 4γ a ).The filter linewidth is Γ = γ a /2 throughout except in Fig.3where it spans Γ ∈ {0.5, 0.75, 1, 5}γ a .The reservoir pumping P b = γ b throughout except Fig.2where it is also 2γ b .The quantum state of the reduced density matrix ρ a = Tr b (ρ) can be computed in the Wigner function form, i.e., as a phase-space distribution in the quadratures X a = a † + a and P a = i(a † − a) of the field (cf.Supplementary Information online).The two-photon correlations resolved in frequency in windows of linewidth Γ reads 9 : g