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Optimization of photon correlations by frequency filtering Alejandro González-Tudela,1 Elena del Valle,2 and Fabrice P. Laussy2,3 1Max–Planck Institut fu¨r Quantenoptik, 85748 Garching, Germany 2Departamento de F´ısica Teo´rica de la Materia Condensada, Universidad Autońoma de Madrid, 28049 Madrid, Spain 3Russian Quantum Center, Novaya 100, 143025 Skolkovo, Moscow Region, Russia (Dated: January 16, 2015) PhotoncorrelationsareacornerstoneofQuantumOptics. Recentworks[NJP15025019&033036 (2013), PRA 90 052111 (2014)] have shown that by keeping track of the frequency of the photons, 5 richlandscapesofcorrelationsarerevealed. Strongercorrelationsareusuallyfoundwherethesystem 1 emission is weak. Here, we characterize both the strength and signal of such correlations, through 0 the introduction of the “frequency resolved Mandel parameter”. We study a plethora of nonlinear 2 quantumsystems,showinghowonecansubstantiallyoptimizecorrelationsbycombiningparameters such as pumping, filtering windows and time delay. n a PACSnumbers: 42.50.Ar,42.50.Lc,05.10.Gg,42.25.Kb,42.79.Pw J 4 1 I. INTRODUCTION a monochromator in a streak camera set-up [14]. The- oretically, the Glauber correlator must be upgraded to ] l the so-called time and frequency resolved photon corre- l The quantum theory of optical coherence developed a lations, [8, 15, 16]: by Glauber in the 60s [1, 2] revolutionized the field of h - Quantum Optics by identifying photon correlations as g(2) (ω ,ω ;τ)= s the fundamental characterization of light, instead of fre- Γ1,Γ2 1 2 e quency [3]. This is a great insight since coherence had (cid:104)A† (t)A† (t+τ)A (t+τ)A (t)(cid:105) m lim ω1,Γ1 ω2,Γ2 ω2,Γ2 ω1,Γ1 , t. bmeaetnicuitnydwerhsitloeoidt ifsornocewntuunrdieesrsatsooadfeinattuerremosfomfofnacotcohrrioz-- t→∞ (cid:104)(A†ω1,Γ1Aω1,Γ1)(t)(cid:105)(cid:104)(A†ω2,Γ2Aω2,Γ2)(t+τ)(cid:105) (2) a ing correlators. This has been confirmed experimentally -m woritshintghlee-pahdovtaonncesooufrnceesw[4li]g)hatsswoeulrlcaessp(sruogchresassitnhpehlaosteor- wfiehledredeAteωcit,eΓdi(ta)t =freq(cid:82)u−te∞ncdyt1ωe(,iωwi−itΓhi/in2)(at−ftr1e)qau(te1n)cyiswtihne- d i n detection. In the quantum picture, the frequency ν of dow Γi, at time t. We have recently developed a theory o light is linked to the energy E of its constituting parti- tocomputesuchcorrelations[17]andintroducedthecon- c cles through Planck constant: E = hν. The standard cept of “two-photon correlation spectrum” (2PS) which, [ approach of photon correlations has consisted so far es- beyond correlating merely peaks, spans over all the pos- 2 sentially in detecting photons from a light source as a sible combinations of photon frequencies [18, 19]. Land- v function of time, disregarding their frequency. Experi- scapes of correlations of unsuspected complexity are re- 9 mentallythisisachievedeitherwiththeoriginalHanbury vealed as a result, which are averaged out in standard 9 Brown–Twiss[5]configurationorbydetectingindividual photon detection or remain hidden when constraining to 7 photons with a streakcamera[6]. For stationary signals, particular (fixed) sets of frequencies. The 2PS enlarges 1 the most important photon correlation is measured by the set of tools in multidimensional spectroscopy [20–24] 0 the second order correlation function: and reveals a new class of correlated emission, that can . 1 be useful for quantum information processing [25, 26], 0 (cid:104)a†(t)a†(t+τ)a(t+τ)a(t)(cid:105) enhance squeezing [27] or for the study of the founda- 5 g(2)(τ)= lim (1) 1 t→∞ (cid:104)(a†a)(t)(cid:105)(cid:104)(a†a)(t+τ)(cid:105) tions of quantum mechanics [14, 28]. When looking at : the full picture, strong correlations turn out to originate v with a(t) the light field annihilation operator of the sys- fromphotonsnotpartofthespectralpeaks,sinceapeak i X tem under study at time t. If the corresponding spectral results from a single-photon transition between two real shape is singled-peak, the question of frequency correla- states. Various such photons have weak correlations and r a tions of the emitted photons may appear a moot point. even when they do, they are of a classical character. In We will see shortly that it is not. In many cases, nev- contrast, collective transitions that require two photons ertheless, the emission is multi-peaked and it is then toundertaketheemissionarestronglyandnon-classically clear that Eq. (1), which correlates photons regardless correlated. Since they involve a virtual state whose en- of which peak they originate from, is leaving some in- ergy is not fixed, unlike for real states, they are emitted formation aside. It is natural to inquire what are the at other frequencies than those of the peaks [17–19, 25]. correlations of each peak in isolation, or what are the Recently, the 2PS of a nontrivial quantum emitter cross-correlations between peaks [7, 8]. Experimentally, has been experimentally observed [29], with spectacular this is readily achieved by inserted filters in the arms of agreement with the theory and positively identifying in the Hanbury Brown-Twiss configurations [9–13] or using a rich landscape of correlations the “leapfrog emission”, 2 i.e., between two real states separated by an intermedi- in Section V. In the JC case, we study the optimiza- ate virtual one, as well as their violation of the Cauchy tion with the intrinsic system parameters, namely the Schwarz inequalities. The emitter was a semiconductor cavity-photon lifetime and the pumping rate, while in quantum dot and the physical picture that of resonance the biexciton case, we study the dependence on extrin- fluorescenceintheMollowtripletregime[30]. Shortlybe- sic parameters, namely, the filters linewidth and/or time fore that, a 2PS of spontaneous emission was measured delay. Clearly, a comprehensive analysis could be given from a polariton condensate [14], which features, how- along such lines to any system of interest. The present ever, no quantum correlated emission and presents in- text aims at illustrating such points in particular cases steadasimplerandsmoothlandscapealternatingbunch- andleavesittofutureworkstocombinetheminthecases ing and antibunching as frequencies get similar or far where they will be needed. apart, due to the fundamental Boson indistinguishabil- ity. These experiments confirm that the theory is sound and robust and that the physics of photon correlation is II. FREQUENCY-RESOLVED MANDEL ripe to take advantage of the effects uncovered by their PARAMETER tagging with a frequency. For instance, the mere Purcell enhancement of leapfrog processes results in N-photon Mandel introduced for standard photon-correlations emitters [31]. (that is, without the frequency information) a param- A central theme of frequency engineering is the inter- eter [32], now bearing his name, intended to correct for play between signal and correlations. Correlated emis- the previously mentioned normalization issue: the bal- sion transiting by virtual states is a high-order process ancing of two vanishing quantities that provide a finite- and is therefore much less frequent than direct emission. value correlation which is, for all practical purposes, not This brings the concern of the practical measurement of measurable since these quantities are those accessible to a 2PS, since this requires measuring coincidences from the experiment. The “Mandel parameter” is defined, for spectral windows where the system already emits very a stationary signal, as: little. Mathematically, this difficulty is concealed for both classes of correlations, Eqs. (1) and (2) alike, by Q(τ)=n (g(2)(τ)−1), (3) a the normalization (denominator) which balances the intensity of the coincidence emission (numerator), turning where na = limt→∞(cid:104)a†(t)a(t)(cid:105) is the steady state popu- two vanishing numbers into a finite ratio. In this text, lation of the detected mode. The offset by unity makes we address this problem and revisit the 2PS to take into the Mandel parameter negative when the light is quan- account the available amount of signal. To do so, in tum (in the sense that it is sub-Poissonian and as such Section II, we introduce the frequency-resolved Mandel has no classical counterpart). The product by na makes parameter Q (ω ,ω ;τ) that combines both correla- the normalization of coincidences to the average signal Γ1,Γ2 1 2 tions and emission intensity. In the light of this new insteadof,aspreviously,touncorrelatedcoincidences. It parameter, we revisit the two-photon correlation map at conveys,therefore,ameaningfulinformationonthemag- τ = 0 [19] for several paradigmatic examples in non- nitudeoftheavailablesignal. NotethatQ(0)=0results linear quantum optics built around a two-level system. either from the lack of correlated emission (g(2)(0) = 1) Namely, we consider both its incoherent and coherent or from too little emission (na → 0). In this way, the driving, the latter bringing the system into the Mollow Mandelparameterreallycharacterizestheamountofcor- triplet regime, already graced with its experimental ob- related emission. servation[29]. Wealsoconsideritscouplingtoacavityto Following the spirit of Mandel, we introduce a fre- realizetheJaynes-Cummings(JC)physics,aswellasthe quency resolved version: biexciton configuration found in, typically, quantum dot systems. These systems are briefly introduced all along Q (ω ,ω ;τ)= Γ1,Γ2 1 2 the paper, but mainly to settle notations and we refer to (cid:113) the literature for the concepts attached to them as well S(1)(ω )S(1)(ω )(g(2) (ω ,ω ;τ)−1), (4) Γ1 1 Γ2 2 Γ1,Γ2 1 2 as for their relevance to our problem. EventheMandelparameterdoesnotfullycapturethe where S(1)(ω ) = lim (cid:104)(A† A )(t)(cid:105) is now the problematic of the signal, since some correlations are so steady sΓtaite sipectrumt→, ∞whichωrie,pΓriesωein,tΓsi, physically, the strong that they dominate over the scarcity of emission. amountofphotonspassingthroughthefilteroflinewidth In Section III, we complement the information of the Γ centeredatω . Hereitmustbeemphasizedthatwhile i i available signal with an estimate of the measuring time Q(τ) < 0 is a sufficient condition to establish the quan- this supposes, defining a notion of valleys of accessible tum character of the emission, as it corresponds to a correlations. There is considerable freedom added by Cauchy-Schwarz inequality (CSI) violation, there is not filtering photons when studying their correlations, and such a straightforward interpretation for the frequency we explore various ways to optimize them. Various ap- resolved version that would read: proaches are illustrated for various systems, focusing on the JC model in Section IV and the biexciton cascade [g(2)(ω ,ω ,0)]2 <g(2)(ω ,ω ,τ)g(2)(ω ,ω ,τ). (5) Γ 1 2 Γ 1 1 Γ 2 2 3 Such violation of classical inequalities gives rise to their (a) own landscape of correlations [25]. In contrast, the anticorrelation in frequency, which we will qualify as “frequency antibunching” in agreement with the liter- 20 3 (b) ature [33], only reflects anti-correlations of intensities, which can be, or not, linked to a quantum character of 2 the emission. 15 1 Our main theme in this text is illustrated in Fig. 1, starting with the 2PS of the JC (b) under weak in- 0 10 coherent pumping in the regime of spontaneous emission[34,35],inwhichcaseitsspectrallineshapeissimply - 1 the Rabi doublet (a). The physical meaning of this cor- 5 relationmapisamplydiscussedinRef.[19]. Itisenough - 2 for our discussion to highlight the main phenomenology, 1 namelythesetofhorizontalandverticallines,thatcorre- - 3 spondtotransitionsbetweenrealstates,andantidiagonal 3 (c) linesω +ω =E −E thatcorrespondtotwo-photon 1 2 2,± 0 2 0.02 “leapfrog”emissionfromthesecondmanifoldwithlevels atenergiesE andthegroundstateatenergyE . The 2,± 0 1 transitions at the Rabi frequency ±R ≈ ±g are all an- 1 tibunched (blue on the figure), since they are dominated 0 0 by the decay of one polariton from the lower manifold and one excitation√cannot be split into two polaritons, - 1 while the lines at ( 2±1)g are mainly bunched (red on -0.02 the figure), since they correspond to a cascade from the - 2 second manifold. The presence of such cascade correlations in a regime of low excitations, where the second - 3 3 manifold has a vanishing probability to be excited, illus- (d) trates the somewhat artificial character of the 2PS. The 2 problemreallypertainstophoton-correlationsingeneral rather than to the inclusion of frequency, since they sim- 1 ilarly predicts g(2)(0) = 0 regardless of the pumping intensity, that is to say, the system exhibits antibunch- 0 ing of its overall emission however small is the probability for two excitations to be present simultaneously, and - 1 therefore for photon blockade to enforce the antibunching [36]. The same holds for the harmonic oscillator at - 2 vanishing pumping which still generates bunched statis- tics g(2)(0) = 2 regardless of the probability to reach - 3 - 3 - 2 - 1 0 1 2 3 two excitations in the system. The paradox arises from the fact that in the limit where the probability of two- photon effects vanishes, so does the possibility to per- FIG. 1. (Colour online) (a) One-photon spectrum for the JC form a measurement, since there is no signal. Instead, systematlowpumping,exhibitingtheRabidoubletofstrong if one considers the Mandel correlations, Eq. (4), that coupling. (b) The corresponding 2PS obtained by span- are shown in Fig. 1(c), one sees how the result makes ning g(2)(ω ,ω ) over all frequencies (c) The corresponding more physical sense: most of the nonlinear features have γa 1 2 frequency-resolved Mandel parameter Q (ω ,ω ), retaining disappeared or are considerably weakened in the regions γa 1 2 onlythewellobservablefeatures. (d)Q (ω ,ω )withablack wherethereisastrongsignal(thecorrelationstendtodie (grey) mask superimposed to let appγeaar 1only2 areas where more slowly than the signal), and the remaining features N ≥ 105 coincidence are obtained in a time T = 12 h c exp are concentrated on antibunching between the peaks, as (1 h) for γ =1/ns. Parameters: γ =0.1g, γ =0.001g and a a σ well as a trace of the bunching cascades. The leapfrog Pσ =0.01γa. correlations are extremely strong, which is a general result in all systems, while the antibunching background that dominates the 2PS profile has now disappeared. It tureless, and therefore neater, antibunching. This could isalsoworthnotinghowtheautocorrelationofeachpeak be of interest for single-photon emitters [33]. (alongthediagonal)hasthebutterflyshapeduetoindis- Although the Mandel correlation spectrum, Fig. 1(c), tinguishability bunching enforced by filtering [19], while appears more physical than the underlying 2PS, cross-correlation between the two peaks feature a struc- Fig. 1(b), the latter still presents us with a more funda- 4 mental physical picture. Indeed, we have merely tamed confirmationoftheseresultscouldbetokeeponebranch down the features, not removed them, and it is useful of the setup on one peak, and correlate its input with to keep track of correlations even though they are out of that of the other branch sweeping the entire spectrum. reachofanactualexperiment. Inanycase,the2PScould This should display transitions from antibunching, no stillbemeasuredideallyandshouldbetterberegardedas correlation, strong bunching, no correlation again and a theoretical limiting case. The 2PS indeed converges to a weaker antibunching in the autocorrelation, due to in- auniqueresultinthelimitofvanishingpumping,thereby distinguishability bunching. For this set of parameters, defininganunambiguouscorrelationmap,whileitsMan- the experiment would need to run stably for a longer delcounterparttendstozeroandtherelativeimportance time in order to collect the same amount of signal to ofbunchingversusantibunchingareasinFig.1(c)depend observe also the leapfrog processes without intersecting on one’s choice of the pumping rate. Finally, it is worth with the peaks. There is a non-trivial interplay of the mentioningthatbythetimeofwritingthistext,the2PS system parameters that helps/hinders the observation of of resonance fluorescence has already been measured in correlations which we will be explored in Section IV. its entirety [29] even for a large splitting of the satel- Beforemovingontotheoptimization,wereviewother lite peaks with spectral windows of little emission. It examples of nonlinear systems explored in Ref. [19] in seems therefore reasonable that with the ever-increasing thelightofthefrequencyresolvedMandelparameterand technological progress, all fundamental quantum optical the estimated time to resolve it. We start by the most emitters, even those with much smaller emission rates, basic system that displays a non-trivial map of correla- will be likewise characterized. tions, namely, theincoherentlypumpedtwo-levelsystem which we recover by setting g =0 in the JC model. The one-photonspectrumofthissystemisasingleLorentzian III. VALLEYS OF ACCESSIBLE peakwithbroadeningΓ =γ +P asshowninFig.2(c). σ σ σ CORRELATIONS Itstwo-photonMandelspectrumshowsabutterfly shape of anticorrelation, typical of two-level systems [19]. By While QΓ(ω1,ω2) provides a physically sound picture choosing Γ = γσ and γσ = 0.1 ns−1, the analysis of the of which regions of the 2PS are the most favorable for measuring time shows that a small region of frequency observation, it also suffers from its own shortcomings. antibunching with N > 106 would be observed within c The arbitrary scale of Q makes it difficult to attach to it one hour, whereas most of the butterfly would be ob- aquantitativefigureofmerit. InthisSection, wefurther served within 12 hours for the same threshold of counts. delineate the valleys of accessible correlations based on While it is much more binding to observe, if filtering far a down-to-earth estimate of the numbers of coincidences in the tail of a two-level system, one should indeed ob- that can be extracted from the emission. serve bunching, against naive expectations. Assuming no correlations, the possibility of detecting Next, we consider a resonant coherent driving of the at least one coincidence in a time window ∆t at frequen- two-levelsystem,describedbyHamiltonianH =Ω (σ+ cies ω , within the frequency windows Γ , is given by: d σ i i σ†). In the weak driving regime, the system has been recently exploited to design ultra-narrow single-photon p=(1−e−n1)(1−e−n2), (6) sources [37? –40]. In the strong-driving regime, which is the one that we focus on in this paper, the spectrum is where n =S (ω )γ∆t represents the number of filtered thewellknownMollow triplet [30],asshowninFig.2(f). photonsifromΓia soiurce that emits at a rate γ. For sim- Frequency-resolvedcorrelationsforthissystemhavebeen plicity we always consider symmetric filters in this Sec- theoreticallyinvestigatedinthepast[16,41–44]andeven tion, Γ = Γ . Using that definition and assuming no measured [13, 45, 46] before the concept of the 2PS was 1 2 further inefficiencies in detection, we can estimate the put forward. But in both the theoretical and experi- experimental time required to obtain a given number of mental contexts, this was at the particular frequencies random coincidences, N , as follows: of the three peaks. Notwithstanding, interesting corre- c lations arise mainly outside the peaks [19, 25], at the T =N ∆t/p. (7) cost of a weaker signal. This has been confirmed exper- exp c imentally [29] with the full reconstruction of the Mollow With this, we plot the regions that would be resolved 2PS. Resonance fluorescence is indeed a system ideally with increasing experimental time, T . For exam- suited to pioneer a comprehensive analysis of frequency exp ple, assuming γ = 1 ns−1 as a quantum dot figure of photon correlations, since it is obtained in the strong- a merit [10], we show in Fig. 1(d) the frequency-resolved driving regime of an extremely quantum emitter, which Mandelparameterwithamaskovertheregionsforwhich implies a large emission of strongly correlated photons. the number of coincidences is N < 105 for a detec- Figure 2(e) shows that with γ = 0.1 ns−1 and Γ = γ , c σ σ tion time of T = 1 (gray) and 12 hours (black) for onlywithin1hourofexperimentaltime,theregionswith exp ∆t = 1/Γ. This shows how the regions with a sizable N > 5×107 unveils all the horizontal and vertical grid c numberofcoincidencesreducetothoseinvolvingatleast of correlations and great part of the leapfrogs. It only one peak, as expected. Therefore, a first experimental takes 12 hours to reveal the complete two-photon Man- 5 0 -0.001 -0.002 -0.003 andabiexcitonicstatewhoseenergy(ω )differsfromthe B 4 (a) (b) (c) sumofitsexcitonicconstituentsbyχduetoCoulombin- 3 teraction. The one-photon spectrum is then composed 2 of two peaks (at energies ωH and ωBH = ωB − ωH) 1 which give rise to an interesting and rich landscape of 0 two-photon correlations. The most prominent feature is -1 the antidiagonal corresponding to the leapfrog between -2 ground and biexciton state, ω1 +ω2 = ωB = −χ (as- -3 suming ωH = 0 as the reference energy), with potential -4 -1 for applications in the generation of entangled photon -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 pairsbyfrequencyfiltering[18]. Withγ =0.1ns−1 and σ 0.05 0 -0.05 setting the threshold at Nc = 105 random coincidences, within one hour, the anticorrelation area and bunching (d) (e) (f) of the one-photon transition peaks would be observable, 20 whereas in 12 hours most of its leapfrog structure would 10 be revealed as well, especially by filtering on the sides of the one-photon peaks. Due to both its fundamental 0 importance and practical applications, we will return to -10 theproblemofoptimizingtheobservationoftheleapfrog processes by changing both the intrinsic parameters as -1 -20 well as the filtering ones in Section V. -20 -10 0 10 20 -20 -10 0 10 20 0.04 0.02 0 -0.02 IV. OPTIMIZATION OF CORRELATIONS IN 50 (g) (h) (i) THE JAYNES-CUMMINGS MODEL 0 Wenowillustratehowtooptimizephotoncorrelations thankstofrequencyfiltering,intheparticularcaseofthe -50 JCmodel. Wedoaqualitativeanalysistoavoidfocusing the discussion on a particular set of experimental figures -100 of merit. We consider the system parameters are vari- -1 ables for the optimization and postpone to next Section -150 -150 -100 -50 0 50-150 -100 -50 0 50 (andtoothersystems)theoptimizationthroughextrinsic parameters, e.g., the filters and detection time. One parameter that can be easily modified is the in- FIG.2. (Colouronline)Panel(a-b)[(c)]: Mandeltwo-photon coherent pump rate, P . The example of the previous σ spectra [one-photon spectrum] for an incoherently pumped Section was chosen to be well into the linear regime, i.e., two-levelsystemwithP =0.01γ ,γ =0.1ns−1andΓ=γ . σ σ σ σ with a very small pumping rate, namely P = 0.01γ . In Panel (b) we introduce a black (grey) mask for the points σ a where N < 106 in a time T = 12 h (1 h). Panel (d-e) Increasing pumping has two interesting consequences for c exp the observation of correlations: [(f)]: Mandeltwo-photonspectra[one-photonspectrum]fora coherently driven two-level system with Ω = 5γ , γ = 0.1 σ σ σ ns−1 and Γ = γ . In Panel (e) we introduce a black (grey) 1. signal increases, σ mask for the points where Nc < 5×107 in a time Texp = 2. the system enters the nonlinear regime. 12 h (1 h). Panel (g-h) [(i)]: Mandel two-photon spectra [one-photon spectrum] for an incoherently pumped biexciton In Fig. 3, upper row, the effect of increasing pump- systemwithP =γ ,χ=100γ ,γ =0.1ns−1andΓ=5γ . σ σ σ σ σ ing is shown for both the spectral shape (a–d) and the In Panel (h) we introduce a black (grey) mask for the points Mandelparameterresolvedinfrequency(e–h). Thespec- where N <5×105 in a time T =12 h (1 h). c exp tra let appear inner peaks between the Rabi doublet, corresponding to transitions from the higher manifolds. The corresponding Mandel 2PS also develops new fea- del spectrum. tures at the same time as the overall intensity of the Finally, we consider a biexciton level scheme as de- correlations increases, from ∼ 0.02 with P = 0.01γ to σ a scribed in Refs. [18, 47, 48] which is relevant in semicon- ∼ 0.5 with P = 0.5γ (note the change of the color σ a ductor quantum optics as it describes the typical level scales). In particular, the higher manifolds become vis- structure of quantum dots beyond the simplest two-level ible in antibunching only as they get populated, while systempicture[49–51]. Focusingonasinglepolarization, they manifest themselves in bunching more clearly at it consist of a three-level scheme as depicted in the inset low pumping. The Jaynes–Cummings fork provides a of Fig. 2(h), with a ground, an excitonic (at energy ω ) well-structuredsetofcorrelationsbetweenthepeaks: the H 6 (a) (b) (c) (d) 3 (e) (f) 0.2 (g) (h) 2 0.02 0.2 0.50 1 0 0 0 0 0 -1 -2 0.02- 0.2- 0.2- 0.25- -3 (i) (j) (k) (l) 123 (m) 0.02 (n) 0.005 (o) 0.001 (p) 00.0001 0 0 0 0 -1 --32 0.02- 0.005- 0.001- 0.0003- -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 FIG.3. (Colouronline)Panels(a-d): One-photonspectraforaJCsystemwithγ =0.1g,γ =0.001gandincreasingpumping a σ P as depicted in the legend. Panels (e-f): Mandel two-photon spectra for a JC system with the same parameters as in (a-d). σ Panels(i-l): One-photonspectraforaJCsystemwithP =0.01g,γ =0.001gandincreasingcavitydecayrate,γ asdepicted σ σ a in the legend. Panels (m-p): Mandel two-photon spectra for a JC system with the same parameters as in (i-l). innerpeaksemitbunchedphotonsbutareotherwiseanti- glecurvesincetherearenolongerpolaritonmodesinthe bunchedwitheachother,orwiththeremotestRabipeak, system. Itisinstructiveinthiscasetoplotg(2)(R ,±R ) γa 1 1 andareuncorrelatedwiththeother—closer—Rabipeak. togetherwiththestandardg(2) (dottedblue)asshownin When various manifolds are well populated, correlations theinsetofFig.4(a). Thedifferenceinthiscasebetween are dominated by real-state transitions and virtual pro- filteringthesamepeakorcross-correlatingthembecomes cesses shy away in comparison. significant. As already observed, frequency filtering the Rabi peaks improves antibunching as compared to non- Another parameter that is less easily tuned but that filtered correlations, as we are discarding the frequency determinesthestrong-couplingpropertyisthecavityde- regions with bunched photons. This is another instance cayrate. InthebottomrowofFig.3,theeffectofincreas- ofhowfrequencyfilteringcanbeusedtoharnesscorrela- ingγ isshown,alwayskeepingthesysteminthestrong- a tions. Note also that worsening the cavity quality factor coupling regime (γ < 4g). This results in the structure a betters the overall antibunching while it spoils the peaks smoothingoutaswellastheintensityofcorrelationsdy- antibunching. ing (note, again, the color scales). The absolute scale of theMandelcorrelationsindeeddecreasesbyoneorderof magnitude, but partly because the cavity gets less populated,andincreasingpumpingcouldcompensateforthat. V. OPTIMIZATION OF CORRELATIONS IN A Forγ =0.5g theleapfrogantidiagonalhavedisappeared BIEXCITON CASCADE a andforγ =g,onlytheanticorrelationbetweentheRabi a peaks have survived, surrounding a region of indistin- We now study photon correlations in a biexciton cas- guishabilitybunching. Totrackmorequantitativelyhow cade. To link with the previous discussion on the JC, we the frequency-resolved correlations evolve with the cav- show in Fig. 4(b) the dependence on the pumping rate ity decay rate, we show in Fig. 4(a) the value of Q for for two configurations, letting here the frequency win- γa pairs of frequencies corresponding to filtering the Rabi dow grow with the pumping as Γ=2(γ +P ). The two σ σ peaks. To show that there is some difference due to in- configurations are filtering the leapfrog transition at the distinguishabilitybunchinginthecaseofautocorrelation, biexciton frequency (in solid black) and the biexciton- we present both the filtering for the same Rabi peak (in exciton cascade (in dashed red). Both transitions are solid black) and for the two Rabi peaks (dashed red). bunched, Q >0, due to their two-photon cascade char- Γ For γ ≥4g, when the system reaches the weak-coupling acter. However, theonethatcorrespondstotheleapfrog a regime,frequency-filteredcorrelationscollapseintoasin- process exhibits a clear optimal pumping intensity, at 7 0 (a) 0.006 1000 JC 0.6 100 0.004 0.4 0.2 10 0.002 0.0 0.1 1.0 10.0 1 0 0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0 0.1 -0.002 0.05 (b) Biexciton 0.04 0.01 0.01 0.1 1 10 100 1000 0.03 0.02 0.01 FIG. 5. (Colour online) Mandel parameter Q (ω ,ω ) Γ1,Γ2 BH H 0.00 when filteringthe two peaks of the biexciton-excitoncascade 0.001 0.01 0.1 1 10 100 as a function of the widths of the spectral windows. The diagonal corresponds to the typical case of identical filters. (c) Stronger correlations and of a varying nature are however 0.05 obtained for asymmetric filters. Parameters are the same as Biexciton 0.04 for Fig. 2. 0.03 0.02 cascade (dashed red). The leapfrog correlations strongly depend on the filter linewidth, due to the virtual nature 0.01 of their transitions, with an optimum value at around 0.00 4γ . Thebiexciton-excitoncascadealsodisplaysamaxi- σ 0.1 1.0 10.0 mumΓ∼γ ,butitsdependenceismuchweaker. Inboth σ cases, and in contrast with g(2)(ω ,ω ) where smaller Γ 1 2 linewidths optimize leapfrog correlations, the compro- FIG. 4. (Colour online) (a) Q (ω ,ω ) as a function of γ foraJCsystemwithP =0.01γγa,γ1 =20.001gforpairoffrea- miseforsignalrequireslargerfilterlinewidthstooptimize σ a σ quencies (R ,R ) (solid black), (R ,−R ) (dashed red). In- the intensity of correlations. 1 1 1 1 set: g(2)(ω ,ω )forthesameparameters,togetherwithstan- In the analysis so far, we have always considered sym- γa 1 2 dardphotoncorrelationg(2)(0)(dottedblue). (b)QΓ(ω1,ω2) metricfiltersforthetwofrequencies: Γ1 =Γ2 =Γ. How- as a function of P for a biexciton cascade with the same ever, for a cascaded emission such as biexciton → exci- σ parameters as in Fig. 2 and Γ = 2(γσ +Pσ), for pair a of ton → ground state, it is worth exploring the situation frequenciesasindicatedininset. (c)QΓ(ω1,ω2)asafunction wherethefiltersareasymmetric,Γ1 (cid:54)=Γ2. Thisisshown of Γ, with the same parameter of Fig. 2. inFig.5,filteringthebiexciton/excitonpeaksforasitua- tion where they have the same broadening and intensity (P = γ ). Two areas of bunching/antibunching op- σ σ P ∼ 1.5γ , whereas the one-photon transitions exhibit σ σ pose each other depending on the relative value of Γ vs 1 two local maxima at around P ∼ 0.2γ and P ∼ 8γ , σ σ σ σ Γ ,separatedbyafrontierofno-correlationthatroughly 2 thatfollowthesuccessivegrowthoftheexcitonandbiex- correspond to the case of symmetric filters, showing the citon populations. The low pumping regime leads to interestinliftingthislimitationevenwhenbothspectral Q → 0 due to the small emission whereas in the high Γ peaks are equal. Such a structure is typical of photon pumping case this is because one recovers the standard cascades. From the level structure, the natural order of photon correlations g(2)(0)≈1. thecascademakesitindeedlikelytodetectaphotonfirst of frequency ω then of ω . Since an ω [ω ] pho- BH H BH H ton, filtered with Γ [Γ ], is the first [second] photon in 1 2 A. Asymmetric filters the cascade, if Γ > Γ the time spent by the photon in 2 1 filter ω is larger than the one ω which is favouring the 1 2 We discuss in more details how the filter linewidth af- simultaneousdetectionofthetwophotonsofthecascade, fects the correlations. In Fig. 4(c), we show the depen- and therefore, yields a strong bunching. In the opposite dence with the filters linewidth Γ for correlations of the regime, when Γ >Γ , the ω photon spends less time 1 2 BH leapfrog cascade (solid black) and the biexciton-exciton in the filter preventing the simultaneous detection of the 8 100 2 (a) 0.03 50 1 0.04 0 0 0.02 0 - 50 - 1 -0.01 -100 - 2 0 - 2 - 1 0 1 2 2 (b) 1 FIG. 6. (Colour online) Mandel parameter Q (ω /2 + 0.02 Γ B ω,ω /2−ω;τ)whenfilteringtheleapfrogprocesses(antidiag- B onal)ofthebiexciton-excitoncascadeandasafunctionofau- tocorrelationtime. Thisshowsthecontrastbetweenleapfrog 0 (virtual) processes where the correlations are symmetric in τ 0 and with a fast decay, and real processes that, when in- tercepted by the filters, lead to characteristic antibunching- bunching transitions with a slow decay. Parameters are the - 1 same as in Fig. 2 except for Γ=10γ . σ -0.02 - 2 two photons of the cascade and therefore yielding strong 0.5 1 5 10 50 antibunchingintheMandelparameter. Thereisanopti- mumvaluetoobserveeitherantibunching/bunching—as a rule of thumb, an order of magnitude difference—since FIG. 7. (Colour online) (a) Mandel parameter ultimately the observations of correlations quenches for Q (ω /2,ω /2;τ) when filtering the leapfrog processes Γ B B verybroad/asymmetricfilters. Thislossofcorrelationsis of the biexciton-exciton cascade and (b) Q (ω ,ω ;τ) Γ BH H due,interestingly,totheoverlapofthefilteringwindows. when filtering the two peaks, as a function of correlation time τ and filters linewidth Γ. This highlights again the difference between real and virtual processes. An optimum valueofthefilterslinewidthcanbefoundforthecaseofreal B. Delayed correlations state transitions. Parameters are the same as for Fig. 2. We have also restricted our attention to τ = 0, i.e., coincidences. However, particularly for cascaded emis- serve strong bunching/antibunching is τ ∼ 1/Γ and the sion, it is to be expected that correlations are maximum correlations ultimately decay in the intrinsic timescale at nonzero delay [17]. To condense the bulk of the in- of the system given by 1/γ . The different timescales σ formation into a single figure (Fig. 6), we consider the where correlations survive between leapfrog and normal jointfrequency-andtime-resolvedMandel2PSalongthe cascadedemissionsisanotherconsequenceoftheirdiffer- leapfrogantidiagonalofFig.2(e-f),Q (ω /2+ω,ω /2− Γ B B ent physical origin. ω;τ)) for Γ = 10γ . In this line lies the information of σ boththeleapfrogsandtheone-photoncascade. Leapfrog emission is maximum at ω = 0 [18] and is symmetric in τ, which is typical of second-order processes where C. Combining the parameters the photons, being virtual, have no time order. Due to this symmetry, the optimal delay to observe correla- Finally, after having explored separately both the de- tions in this case is τ =0, and they decay with the filter pendenceonthefilterlinewidth,Γ,andthetemporalde- timescale 1/Γ. Contrarily, the biexciton-exciton photon layofthephotons, τ, onecannaturallythinkofcombin- cascade, appearing at ω ={ω ,ω }, is strongly asym- ingthemtooptimizecorrelationscumulatively. InFig.7, BH H metric, as clearly shown in Fig. 6. It shows a bunch- we show the joint τ and Γ dependence of the frequency- ing/antibunching behaviour as expected for cascades of resolved Mandel parameter for the two more relevant real transitions, where there is a definite temporal order cases: the leapfrog two-photon cascade in (a) and the in the emission. In this case, the optimal value to ob- consecutive one-photon transitions in (b). As previously 9 discussed, the temporal shape of the leapfrog cascade is Strongcorrelationsareoftenfoundinregionsofthespec- a symmetric decay of correlations within timescale 1/Γ, trum where there is a weak emission, making their ex- as shown in the figure. As Γ increases, so does the decay perimental detection particularly difficult, since this im- rate of the correlations (the correlation time and the fil- plies coincidences of rare events. We have introduced terslinewidthareroughlyininverseproportion,asshown a “frequency-resolved Mandel parameter” as well as a by the dotted lines which are 1/(Γ/γ )). The τ = 0 quantitative estimate of the time required to accumulate σ correlation also strongly decreases and is eventually sur- a given number of coincidences, to address this prob- rounded by antibunching oscillations in τ, that corre- lematicforseveralparadigmaticnon-linearquantumsys- spond to the fast off-resonance one-photon transitions. tems. We have shown the considerable flexibility opened Forthesetofparameterschosenhere, theoptimalcorre- by frequency-filtering, either in energy or in time, with lationsaretobefoundatτ =0andforΓ∼3γ . Alsoas possibilities to enhance correlations by varying filters σ discussed previously, in the consecutive one-photon cas- linewidths (possibly asymmetrically), temporal windows cade in Fig. 7(b), the temporal shape exhibits a typical of detections and system parameters (such as pumping). asymmetric bunching/antibunching shape. The pattern Depending on whether the correlations originate from is fairly robust but can indeed be magnified by the ap- real states transitions or involve virtual processes, differ- propriate choice of filters (we consider here symmetric ent strategies should be adapted, corresponding to their filters for simplicity). The temporal decay occurs this intrinsically different character. With the recent experi- time approximately within a time scale of 1/γ , while mental demonstration of the underlying physics [29], the σ the maximum value for the correlations, both bunching fieldoftwo-photonspectroscopyisnowripetopowerap- and antibunching, is obtained at τ ∼ 1/Γ. In this case, plicationsandoptimizeresourcesbasedonphotoncorre- the maximum is found at Γ ∼ 3γ and τ ∼ 1/Γ. Note lations. σ that correlations are optimised for filters with a width equaltothespectralpeaks(3γ forourchoiceofparam- σ eters). ACKNOWLEDGMENTS VI. SUMMARY AND CONCLUSIONS This work has been funded by the ERC Grant PO- LAFLOW. EdV acknowledges support from the IEF Summing up, filtering the photons emitted by a quan- projectSQUIRREL(623708)andAGTfromtheAlexan- tum source has a dramatic impact on their correlations. der Von Humboldt Foundation. [1] R. J. Glauber, Phys. Rev. Lett. 10, 84 (1963). J.-P. Poizat, Nat. Photon. 4, 696 (2010). [2] R. J. Glauber, Phys. Rev. 130, 2529 (1963). [13] A.Ulhaq,S.Weiler,S.M.Ulrich,R.Roßbach,M.Jetter, [3] R. J. Glauber, Rev. Mod. Phys. 78, 1267 (2006). and P. Michler, Nat. Photon. 6, 238 (2012). [4] B. Lounis and M. Orrit, Reports on Progress in Physics [14] B.Silva,A.Gonza´lezTudela,C.SańchezMunõz,D.Bal- 68, 1129 (2005). larini, G. Gigli, K. W. West, L. Pfeiffer, E. del Valle, [5] R. Hanbury Brown and R. Q. Twiss, Nature 178, 1046 D. Sanvitto, and F. P. Laussy, arXiv:1406.0964 (2014). (1956). [15] L. Kno¨ll and G. Weber, J. Phys. B.: At. Mol. Phys. 19, [6] J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berster- 2817 (1986). mann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schnei- [16] G. Nienhuis, Phys. Rev. A 47, 510 (1993). der, S. Ho¨fling, A. Forchel, C. Kruse, J. Kalden, and [17] E. del Valle, A. Gonzalez-Tudela, F. P. Laussy, C. Teje- D. Homme, Nature 460, 245 (2009). dor, andM.J.Hartmann,Phys.Rev.Lett.109,183601 [7] C.Cohen-TannoudjiandS.Reynaud,Phil.Trans.R.Soc. (2012). Lond. A 293, 223 (1979). [18] E. del Valle, New J. Phys. 15, 025019 (2013). [8] J. Dalibard and S. Reynaud, J. Phys. France 44, 1337 [19] A.Gonzalez-Tudela,F.P.Laussy,C.Tejedor,M.J.Hart- (1983). mann, andE.delValle,NewJ.Phys.15,033036(2013). [9] N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, [20] G. Nardin, T. M. Autry, K. L. Silverman, and S. T. J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Cundiff, Opt. Express 21, 28617 (2013). Petroff, Phys. Rev. Lett. 96, 130501 (2006). [21] Y.-S.Ra, M.C. Tichy, H.-T.Lim, O. Kwon, F.Mintert, [10] K. Hennessy, A. Badolato, M. Winger, D. Gerace, A.Buchleitner, andY.-H.Kim,Naturecommunications M. Atature, S. Gulde, S. Fa˘lt, E. L. Hu, and 4 (2013). A.˘Imamog¯lu, Nature 445, 896 (2007). [22] G. Nardin, G. Moody, R. Singh, T. M. Autry, H. Li, [11] M. Kaniber, A. Laucht, A.Neumann, J.M. Villas-Boâs, F. Morier-Genoud, and S. T. Cundiff, Phys. Rev. Lett. M. Bichler, M.-C. Amann, and J. J. Finley, Phys. Rev. 112, 046402 (2014). B 77, 161303(R) (2008). [23] M. Gessner, F. Schlawin, H. Hoffner, S. Mukamel, and [12] G. Sallen, A. Tribu, T. Aichele, R. André, L. Besombes, A. Buchleitner, New Journal of Physics 16, 092001 C. Bougerol, M. Richard, S. Tatarenko, K. Kheng, and (2014). 10 [24] F. Schlawin, M. Gessner, S. Mukamel, and A. Buchleit- [39] Y.-M. He, Y. He, Y.-J. Wei, D. Wu, M. Atatu¨re, ner, Phys. Rev. A 90, 023603 (2014). C.Schneider,S.Ho¨fling,M.Kamp,C.-Y.Lu, andJ.-W. [25] C. Sańchez-Munõz, E. del Valle, C. Tejedor, and F. P. Pan, Nature nanotechnology 8, 213 (2013). Laussy, Physical Review A 90 (2014). [40] O. chip resonantly-driven quantum emitter with en- [26] H. Flayac and V. Savona, Phys. Rev. Lett. 113, 143603 hanced coherence, arXiv:1404.3967 (2014). (2014). [41] K. Wo´dkiewicz, Phys. Lett. A 77, 315 (1980). [27] P. Gru¨nwald, D. Vasylyev, J. Ha¨ggblad, and W. Vogel, [42] H. F. Arnoldus and G. Nienhuis, J. Phys. B.: At. Mol. Phys. Rev. A 91, 013816 (2015). Phys. 16, 2325 (1983). [28] R. Folman, arXiv:1305.3083 (2013). [43] H. F. Arnoldus and G. Nienhuis, J. Phys. B.: At. Mol. [29] M.Peiris,B.Petrak,K.Konthasinghe,Y.Yu,Z.C.Niu, Phys. 17, 963 (1984). and A. Muller, arXiv:1501.00898 (2015). [44] V. Shatokhin and S. Kilin, Phys. Rev. A 63, 023803 [30] B. R. Mollow, Phys. Rev. 188, 1969 (1969). (2001). [31] C.SanchezMunõz,E.delValle,A.G.Tudela,K.Mu¨ller, [45] A. Aspect, G. Roger, S. Reynaud, J. Dalibard, and S. Lichtmannecker, M. Kaniber, C. Tejedor, J. Finley, C. Cohen-Tannoudji, Phys. Rev. Lett. 45, 617 (1980). and F. Laussy, Nat. Photon. 8, 550 (2014). [46] S. Weiler, D. Stojanovic, S. Ulrich, M. Jetter, and [32] L.MandelandE.Wolf,Rev.Mod.Phys.37,231(1965). P. Michler, Phys. Rev. B 87, 241302 (2013). [33] Z. Deutsch, O. Schwartz, R. Tenne, R. Popovitz-Biro, [47] E.delValle,S.Zippilli,F.P.Laussy,A.Gonzalez-Tudela, and D. Oron, Nano Lett. 12, 2948 (2012). G. Morigi, and C. Tejedor, Phys. Rev. B 81, 035302 [34] E.delValle,F.P.Laussy, andC.Tejedor,Phys.Rev.B (2010). 79, 235326 (2009). [48] E. del Valle, A. Gonzalez-Tudela, E. Cancellieri, F. P. [35] A. V. Poshakinskiy and A. N. Poddubny, Sov. Phys. Laussy, andC.Tejedor,NewJ.Phys.13,113014(2011). JETP 118, 205 (2014). [49] G. Chen, T. H. Stievater, E. T. Batteh, X. Li, D. G. [36] K.Birnbaum,A.Boca,R.Miller,A.Boozer,T.Northup, Steel, D. Gammon, D. S. Katzer, D. Park, and L. J. and H. Kimble, Nature 436, 87 (2005). Sham, Phys. Rev. Lett. 88, 117901 (2002). [37] H. S. Nguyen, G. Sallen, C. Voisin, P. Roussignol, [50] I.A.Akimov,J.T.Andrews, andF.Henneberger,Phys. C. Diederichs, and G. Cassabois, Appl. Phys. Lett. 99, Rev. Lett. 96, 067401 (2006). 261904 (2011). [51] Y.Ota,S.Iwamoto,N.Kumagai, andY.Arakawa,Phys. [38] C.Matthiesen,A.N.Vamivakas, andM.Atatu¨re,Phys. Rev. Lett. 107, 233602 (2011). Rev. Lett. 108, 093602 (2012).