signatures of two-photon pulses from a quantum two-level...

LETTERSPUBLISHED ONLINE: 27 FEBRUARY 2017 | DOI: 10.1038/NPHYS4052

Signatures of two-photon pulses from a quantumtwo-level systemKevin A. Fischer1*†, Lukas Hanschke2†, JakobWierzbowski2, Tobias Simmet2, Constantin Dory1,Jonathan J. Finley2, Jelena Vučković1 and Kai Müller2*A two-level atom can generate a strongmany-body interactionwith light under pulsed excitation1–3. The best known e�ect issingle-photon generation, where a short Gaussian laser pulseis converted into a Lorentzian single-photon wavepacket4,5.However, recent studies suggested that scattering of intenselaser fields o� a two-level atom may generate oscillations intwo-photon emission that come out of phase with the Rabioscillations, as the power of the pulse increases6,7. Here,we provide an intuitive explanation for these oscillationsusing a quantum trajectory approach8 and show how theymay preferentially result in emission of two-photon pulses.Experimentally, we observe the signatures of these oscillationsby measuring the bunching of photon pulses scattered o�a two-level quantum system. Our theory and measurementsprovide insight into there-excitationprocess thatplagues5,9 on-demand single-photon sourceswhile suggesting the possibilityof producing new multi-photon states.

We begin by considering an ideal quantum two-level system thatinteracts with the outside world only through its electric dipolemomentµ (ref. 10). Suppose the system is instantaneously preparedin the superposition of its ground |gi and excited |ei states

| ii=p1�Pe|gi+

pPe|ei (1)

where Pe is the probability of initializing the system in |ei. From thispoint, spontaneous emission at a rate of � governs the remainingsystem dynamics and a single photon is coherently emittedwith probability Pe, while no photon is emitted with probability1�Pe. As detected by an ideal photon counter, this results in thephotocount distribution

Pn ={P0,P1,P2, ...}={1�Pe,Pe, 0, ...} (2)where Pn is the probability to detect n photons in the emitted pulse.It is on this principle that most indistinguishable single-photonsources based on solid-state quantum emitters operate4,5.

A popular mechanism for approximately preparing | ii is theoptically driven Rabi oscillation4,11. Here, the system is initializedin its ground state and driven by a short Gaussian pulse from acoherent laser beam (of width ⌧FWHM) that is resonant with the|gi$ |ei transition. Short is relative to the lifetime of the excitedstate ⌧e = 1/� to minimize the number of spontaneous emissionsthat occur during the system–pulse interaction5,9. As a functionof the integrated pulse area, that is, A = R dtµ ·E(t)/~, whereE(t) is the pulse’s electric field, the system undergoes coherent

oscillations between its ground |gi and excited |ei states. Forconstant-area pulses of vanishing ⌧FWHM/⌧e, the final state of thesystem after interaction with the laser field is arbitrarily close tothe superposition

| f (A)i=p1�Pe(A)|gi+e�i�

pPe(A)|ei (3)

where � is a phase set by the laser field. Examining Pe(A) (Fig. 1adotted line), we see Rabi oscillations that are perfectly sinusoidal,with the laser pulse capable of inducing an arbitrary number ofrotations between |gi and |ei. Because | f (A)i looks very muchlike | ii for arbitrarily short pulses, it is commonly assumed thatthe photocount distribution Pn always has P1 �Pn>1. However, wewill use a quantum trajectory approach to show that, unexpectedly,P2 >P1 for any ⌧FWHM Pe(A), thesystem must be occasionally re-excited to emit additional photonsduring the system–pulse interaction.

We now examine this re-excitation process in detail, first byconsidering the commonly studied case of an on-demand single-photon source with A = ⇡ (Fig. 1b). By driving a half Rabioscillation, known colloquially as a ⇡-pulse11, the probability ofsingle-photon generation ismaximized. Because the excitation pulseis short (grey dashed line) compared to the excited-state lifetime, theemitted wavepacket has an exponential shape (blue line). To furtherunderstand the probabilistic elements of the photon emission, westudy a typical quantum trajectory8,12 representing Pe(t) (greenline). The system is driven by the ⇡-pulse into its excited state,where it waits for a photon emission at some later random time(denoted by the green triangle) to return to its ground state. Aftercomputing thousands of such trajectories, Pn is generated fromthe photodetection events and shows P1 ⇡ 1 (inset), indicatingthat the system acts as a good single-photon source. The smallamount of two-photon emission (P2) occurs due to re-excitation ofthe quantum system during interaction with the pulse. It roughly

1E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA. 2Walter Schottky Institut and Physik Department, Technische UniversitätMünchen, 85748 Garching, Germany. †These authors contributed equally to this work. *e-mail: [email protected]; [email protected]

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Figure 1 | Simulations showing pulsed two-photon emission from an ideal quantum two-level system. a, Rabi oscillations in the excited-state populationfrom an ideal two-level system (dashed line). Signatures of Rabi oscillations in the emitted photon number under excitation with a pulse of length⌧FWHM =⌧e/10, where ⌧e is the excited-state lifetime (solid line). b, System dynamics under excitation by a pulse of area ⇡. Dashed grey line shows thedriving pulse shape, blue line shows the ensemble-averaged excited-state probability, green line shows a typical quantum trajectory with one photondetection (denoted by the green triangle). Inset shows photocount histogram Pn under ⇡-pulse excitation, with single-photon emission P1 dominating.c, System dynamics under excitation by a pulse of area 2⇡. Dashed grey line shows the driving pulse shape, blue line shows the ensemble-averagedexcited-state probability, red line shows a typical quantum trajectory with two photon emissions (denoted by the red triangles). Inset shows photocounthistogram Pn under 2⇡-pulse excitation, with P0 dominating but P2 �P1. d, Photocount distribution Pn (solid lines) and photon number purities ⇡n (dashedlines) versus pulse area (under same excitation conditions as solid line in a). n={1, 2, 3} shown in colours {green, red, and purple}, respectively.Note: green is associated with single-photon-related indicators and red is associated with two-photon-related indicators in all figures.

accounts for the disparity between E[n] and Pe(A = ⇡), and isan important but often overlooked source of error in on-demandsingle-photon sources.

As our first clue that re-excitation during the system–pulse inter-action can yield interesting dynamics, the di�erence between E[n]and Pe(A) is not constant as a function of A, and is maximizedfor A= 2n⇡. Therefore, we now take a closer look at the system’sdynamics for a 2⇡-pulse (Fig. 1c) and find a photocount distributionwhere P2 � P1 (inset). To understand why the two-level systemcounter-intuitively prefers to emit two photons over a single pho-ton, consider a typical quantum trajectory (red line). The emissionprobability is proportional to Pe(t) (blue line), which peaks halfwaythrough the excitation pulse. Therefore, the first photon is mostlikely to be emitted after approximately ⇡ of the pulse area has beenabsorbed (first red triangle), and a remaining approximately ⇡ inarea then re-excites the system with near-unity probability to emita second photon (second red triangle). This two-photon process istriggered during the system–pulse interaction and, although thesephotons are emitted within a single excited-state lifetime, they havea temporal structure known as a photon bundle13. Signatures ofthe bundle can be found in Pe(t): the emission shows a peak ofwidth ⌧FWHM followed by a long tail of length ⌧e. This shows theconditional generation of a second photon based on a first emissionduring the system–pulse interaction, which means that the two-photon bundling e�ect dominates for arbitrarily short pulses andeven for long pulses as well (Supplementary Fig. 4). Hence, although

the e�ciency as a pulsed two-photon source is given by P2 ⇡6%, anideal two-level system could be operated in a much more e�cientregime simply by choosing a longer pulse. We avoided this discus-sion in the main text because P3 becomes non-negligible, whichmakes an intuitive interpretation of the dynamics more challenging.

To fully characterize the crossover where P2 >P1, we simulatedthe photocount distributions as a function of pulse area (Fig. 1d).Clear oscillations can be seen betweenP1 and P2 (solid green and redlines, respectively), and P2 is out of phase from the Rabi oscillations.Notably, the oscillations in E[n] make direct comparisons betweenthe two probabilities di�cult. To better illustrate the fraction ofemission occurring by n-photon emission, we turn to a quantitycalled the photon number purity of the source13. By ignoring thevacuum component, Pn is renormalized to

⇡n =Pn/X

n>0

Pn (4)

The purities (dashed lines) very clearly oscillate between emissiondominated by single-photon processes ⇡1 for odd-⇡-pulses andtwo-photon processes ⇡2 for even-⇡-pulses. Quite remarkably, ⇡3remains negligible for all pulse areas. Additionally, the puritiesreveal the limit of {⇡1, ⇡2, ⇡3} = {0.3, 0.7, 0} for arbitrarily shortGaussian pulses.

As suggested earlier, this two-photon emission comes as anordered pair, where the first emission event within the pulse

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NATURE PHYSICS DOI: 10.1038/NPHYS4052 LETTERS

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Figure 2 | Simulations showing time-resolved single-photon and two-photon emission from an ideal quantum two-level system. a–c, Probability massfunctions for single-photon (green) and two-photon (red) detection showing the internal temporal structure of the photon pairs, for excitation by a2⇡-pulse (a), 4⇡-pulse (b), and 6⇡-pulse (c). Colour plots show p2(t1,⌧ ), while the traces to the left show p1(t1) and p2(t1), and the traces to the topshow p2(⌧ ).

excitation window triggers the absorption and subsequent emissionof a second photon. Unlike the first photon, the second photonhas the entire excited-state lifetime to leave. This statement canbe quantified by investigating the time-resolved probability massfunctions for photodetection14 (see Methods for a derivation fromsystem dynamics), defined by

P1 =Z

Rdt1p1(t1) and P2 =

ZZ

R2dt1d⌧p2(t1,⌧ ) (5)

The mass function p1(t1) represents the probability density foremission of a single photon at time t1 with no subsequent emissions,while p2(t1,⌧ ) represents the joint probability density for emission ofa single photon at time t1 with a subsequent emission at time t1 +⌧ .Additionally, p2(t1, ⌧ ) can be integrated along t1 or ⌧ to yield p2(⌧ )or p2(t1), which give the probability density for waiting ⌧ betweenthe two emission events or detecting a photon pair with the firstemission at time t1, respectively.

We explore these temporal dynamics for excitation by a 2⇡-, 4⇡-,and 6⇡-pulse in Fig. 2a–c, respectively. First, consider excitationby the 2⇡-pulse: p2(t1, ⌧ ) captures the dynamics already discussedthrough having a high probability of the first emission at time t1only within the pulse window of 0.1/� , but the second emissionoccurs at a delay ⌧ later within the spontaneous emission lifetime⌧e. This e�ect is most clearly seen in p1(t1), p2(t1) and p2(⌧ ) (tracesto the left and top of the colour plots in Fig. 2), where the density

of photon pair emission being triggered at time t1—that is, p2(t1)—is maximized after ⇡ of the pulse has been absorbed (t1 = 0.15/� )and reaches nearly unity. The second photon of the pair then has theentire lifetime to leave, as seen in the long correlation time for p2(⌧ ).Meanwhile, the enhancement in photon pair production leads to acorresponding decrease in the density of single-photon emissionsp1(t1) around t1 =0.15/� .

Next, consider excitation by the 4⇡-pulse: p2(t1, ⌧ ) shows thee�ects of an additional Rabi oscillation that the system undergoesduring interaction with the pulse. If the first emission occurs after⇡ of the pulse has been absorbed, a remaining 3⇡ can result in asecond emission in two di�erent ways: either after absorption of ⇡additional energy or after absorption of 3⇡ additional energy. Onthe other hand, if the first emission occurs after 3⇡ of the pulse hasbeen absorbed, only a remaining ⇡ can be absorbed, resulting ina monotonic region of p2(t1, ⌧ ) just like for the 2⇡ case. In eitherscenario, the probability of two emissions is most likely (but threeemissions almost never happen) because a single emission convertsan even-⇡-pulse into an odd-⇡-pulse, which anti-bunches the nextemission. The high fidelity of this conversion process can clearly beobserved in p1(t1) and p2(t1). The pair production ismost likely aftereither ⇡ or 3⇡ of the pulse has been absorbed (times t1 = 0.1 andt1 = 0.3, respectively), and it occurs with near-unity density. Thismeans that if the first photon is emitted at time t1 =0.1 or t1 =0.3,then the conditional probability to emit a second photon is nearunity. As a result, p1(t1) and p2(t1) almost look like they were just



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LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS4052

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Figure 3 | Super-natural linewidth photons from an ideal quantum two-level system. a,b, Rabi oscillations (a) and dynamical spectra of emission (b)under excitation by a 2⇡-, 4⇡-, and 6⇡-pulse, denoted with increasing darkness of the shaded region for higher pulse area. Dashed lines show the naturalemission linewidth.

copied a second time from the 2⇡-pulse scenario, confirming ourintuitive interpretation of the 4⇡-pulse scenario. These ideas triviallyextrapolate to the 6⇡-pulse, where three complete Rabi oscillationsoccur, and the projections p1(t1) and p2(t1) are copied once morealong t1.

Looking at the oscillations in p2(⌧ ) for increasing pulse areas,one may notice a qualitative resemblance to the photon bunching15behind a continuous-wave Mollow triplet11. In fact, the underly-ing process where a photon emission collapses the system intoits ground state, restarting a Rabi rotation, is responsible for thedynamics in both cases. However, our observed phenomenon hasa very important di�erence: after a photodetection the expectedwaiting time for the second, third, and nth photon emissions is iden-tical in the continuous case, while our observed process dramaticallysuppresses P3.

Because this method of generating two-photon bundles requiresthe emission of the first photon to occur within a tightly definedtime interval, as set by the pulse width, we explore the emission inthe context of time–frequency uncertainty (Fig. 3). First consider the2⇡-pulse case: we replot Pe(t) as the light red shaded trace (panela), which results in the first shaded emission spectrum16 (panelb). Compared to the natural linewidth of the system’s transition(dashed black line), the emission is spectrally broadened by thefirst emission of a super-natural linewidth photon of order 1/⌧FWHM,which occurs during the laser pulse. We note that it would beinteresting to explore the physics of super-natural linewidth photonsthat have been incoherently emitted as the result of a new many-body scattering process (having isolated them with a spectral notchfilter to remove the second photon of a natural linewidth). As ane�ect of the increased number of Rabi oscillations (see for 4⇡- and6⇡-pulses), the super-natural linewidth photons show oscillationsin spectral power density that resemble an emerging dynamicalMollow triplet17.

Finally, we discuss the counting statistics of the emitted lightin comparison to a coherent laser pulse to verify the nonclassicalnature of the emission. Because the photocount distribution is fullydescribed with just P0, P1 and P2, its information is completelycontained in the mean E[n] and the normalized second-orderfactorial moment8

g (2)[0]⌘ E[n (n�1)]E[n]2 (6)

which physically describes the relative probability to detect acorrelated photon pair over randomly finding two uncorrelatedphotons in a Poissonian laser pulse of equivalent mean. Thus, thequantity g (2)[0] yields important information on how a beamof lightdeviates from the Poissonian counting statistics of a laser beam.

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Figure 4 | Bunched photon pair emission from an ideal quantum two-levelsystem. Normalized second-order factorial moment of the photocountdistribution (blue), which measures the total degree of second-ordercoherence g(2)[0]. Dashed blue line indicates the Poissonian countingstatistics of the laser pulse. Dotted black line again indicates Rabioscillations for reference.

For Poissonian statistics g (2)[0]=1, but for sub-Poissonian statisticsg (2)[0] < 1 (anti-bunching) and for super-Poissonian statisticsg (2)[0]> 1 (bunching)5. In particular, because the emission undera 2⇡ excitation is a weak two-photon pulse, we expect that thephotons will arrive in ‘bunched’ pairs, where the first detectionheralds the presence of a second photon in the pulse. This predictionis confirmed in Fig. 4, where emission for even-⇡-pulses stronglybunches, thus confirming the highly nonclassical nature of theemission and providing an experimentally accessible signature ofthe oscillations in P2.

After having theoretically discovered that a quantum two-levelsystem is able to preferentially emit two photons through a complexmany-body scattering phenomenon, we found experimentalsignatures of this two-photon process using a single transitionfrom an artificial atom. Our artificial atom of choice is an InGaAsquantum dot, due to its technological maturity and good opticalquality18. The dot is embedded within a diode structure tominimizecharge and spin noise (Methods), resulting in a nearly transform-limited optical transition19; luminescence experiments as a functionof gate voltage (Fig. 5a) reveal the charge-stability region in whichwe operate20 (Vg = 0.365V). We used the X� transition due to itslack of fine structure, which results in a true two-level system (atzero magnetic field) with an excited-state lifetime of ⌧e = 602 ps(Supplementary Fig. 1). Exciting the system with laser pulses(⌧FWHM =80ps), we drove Rabi oscillations between its ground andexcited states (Fig. 5b). However, because the artificial atom residesin a solid-state environment, it possesses several non-idealities that

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Figure 5 | Experiments showing two-photon emission from a single artificial atom’s transition. a, Photoluminescence versus applied gate bias andwavelength showing the charge-stability region of the X� transition. Inset: Voltage dependence of resonance fluorescence intensity under pulsedexcitation, with the optimal resonance fluorescence signal occurring at Vg =0.365 V. b, Experimental resonance fluorescence signal showing Rabioscillations, scaled to quantum simulations of emitted photon number (blue). c,d, Under excitation by ⇡- and 2⇡-pulses (green and red triangles in b),Hanbury-Brown and Twiss data respectively show anti-bunching (g(2)[0]< 1) (c) and bunching (g(2)[0]> 1) (d). The measured values areg(2)[0]=0.096±0.009 and g(2)[0]=2.08±0.13, respectively. e, Experimental second-order coherence measurements g(2)[0] versus pulse areashowing oscillations between anti-bunching (at odd ⇡-pulses) and bunching (at even-⇡-pulses). Blue curve represents quantum simulations of thetime-integrated correlations g(2)[0] from the experimental system. Dashed black line represents statistics of the incident laser pulse. f, Experimentalsecond-order coherence measurements g(2)[0] versus pulse length (using 2⇡-pulses for excitation). Optimal bunching, and hence two-photon bundling,occurs for the 80 ps pulse. Solid blue line represents emission from an ideal two-level quantum system, long dashed blue line represents inclusion ofdephasing, short dashed blue line represents addition of a 2.7 % chirp in bandwidth, and short dotted blue line represents addition of a further 2.7 % chirpin bandwidth. Again, dashed black line represents statistics of the incident laser pulse. Note, the errors in g(2)[0] values for e and f are the standard

pn

fluctuations in the photocount distribution5. The error in pulse area accounts for power drifts during the experiment, and the errors in pulse lengths areleast-squares fitting errors to the pulse spectra.

slightly decrease the fidelity of the oscillations: a power-dependentdephasing rate arising from electron–phonon interaction21 and anexcited-state dephasing due to spin or charge noise19. Additionally,the quantum systems are very sensitive to minimal pulse chirpsarising due to optical set-up non-idealities22. Using these threee�ects as fitting parameters (Methods), we obtained near-perfectagreement between our quantum-optical model (blue) and theexperimental Rabi oscillations.

Next, we measured the g (2)[0] values of the emitted wavepack-ets, to study the photon bunching e�ects outlined in Fig. 4. Twotypical experiments are presented in Fig. 5c,d, showing g (2)[0]⇡0(anti-bunching) and g (2)[0]> 1 (bunching) for ⇡- and 2⇡-pulses,respectively. A complete data set is shown in Fig. 5e, with oscillationsbetween anti-bunching at odd-⇡ pulses and bunching at even-⇡-pulses. Using the same fitting parameters as in Fig. 5b, the corre-lation data are almost perfectly matched with our full quantum-optical model. Hence, we have found experimental evidence thatsuggests the artificial atom is a�ected by the predicted many-bodytwo-photon scattering process that causes P2 oscillations out ofphase from the Rabi oscillations.

Finally, we investigated how non-idealities a�ect the bunchingvalues (Fig. 5f) by experimentally characterizing the emission atfour pulse lengths. From our quantum-optical model, we see thatbunching is strongest for the ideal case (blue line), and decreasesfor every added non-ideality (long dashed blue for dephasing,short dashed blue for additional 2.7% chirp in bandwidth, andshort dotted blue for further 2.7% chirp in bandwidth), yieldingexcellent agreement with the data. Discussing these e�ects further,an enhanced pulse chirp decreases the fidelity of photon bunching

due to the function of a large chirp to adiabatically prepare thesystem in its excited state23, which decays with a single-photonemission. The minimal chirp that we observed can be removedwith pulse compressors in future experiments to achieve an evenbetter match with the ideal photon bunching curve. Additionally,at short pulse lengths the power-dependent dephasing results inanti-bunching. Due to the higher amplitude of shorter pulses (withfixed area), the dephasing rate diverges and the system acts as anincoherently pumped single-photon source. Thus, when includingnon-idealities we found the optimum bunching to occur at a pulselength of approximately 80 ps, which indicates where the two-photon process is strongest experimentally. Although we expect thetwo-photon emission is dominant, the non-idealities of the solid-state system could result in non-negligible P3 or higher Pn. Thisscenario is unfortunately not distinguishable through measuringg (2)[0] alone, but it is recently becoming possible to measure higher-order photon correlations that could help definitively identifyregimes of operation where P2 �P3 (refs 24,25). Finally, we expectfuture investigations on exploring optimal pulse shapes to enablemuch more e�cient and higher-purity two-photon emission bothfrom ideal and experimental two-level systems.

MethodsMethods, including statements of data availability and anyassociated accession codes and references, are available in theonline version of this paper.

Received 15 November 2016; accepted 27 January 2017;published online 27 February 2017



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AcknowledgementsThe authors thank A. Rasmussen for productive and helpful discussions in framing thecontext of this work. We gratefully acknowledge financial support from the NationalScience Foundation (Division of Materials Research—Grant No. 1503759), the DFG viathe Nanosystems Initiative Munich, the BMBF via Q.com (Project No. 16KIS0110) andBaCaTeC. K.A.F. acknowledges support from the Lu Stanford Graduate Fellowship andthe National Defense Science and Engineering Graduate Fellowship. J.W. acknowledgessupport from the PhD programme ExQM of the Elite Network of Bavaria. C.D.acknowledges support from the Andreas Bechtolsheim Stanford Graduate Fellowship. J.V.gratefully acknowledges support from the TUM Institute of Advanced Study.

Author contributionsK.A.F. performed the theoretical work and modelling. L.H., T.S., J.W. and K.M.performed the experiments. C.D. performed trial experiments. J.V. and J.J.F. providedexpertise. K.M. organized the collaboration and supervised the experiments. All authorsparticipated in the discussion and understanding of the results.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints.Correspondence and requests for materials should be addressed to K.A.F. or K.M.

Competing financial interestsThe authors declare no competing financial interests.

6



http://dx.doi.org/10.1038/nphys4052https://arxiv.org/abs/1302.2156http://go.nature.com/2ktxxkrhttp://dx.doi.org/10.1038/nphys4052http://www.nature.com/reprintswww.nature.com/naturephysics

NATURE PHYSICS DOI: 10.1038/NPHYS4052 LETTERSMethodsThe sample investigated is grown by molecular beam epitaxy (MBE). It consists of alayer of InGaAs quantum dots with low areal density (35db.

Second-order autocorrelation measurements were performed using aHanbury-Brown and Twiss (HBT) set-up consisting of one beamsplitter and twosingle-photon avalanche diodes. Note: their timing resolution (⇡100 ps) is too lowto measure the correlations predicted in Fig. 2. The detected photons were

correlated with a TimeHarp200 time-counting module. The integration times werebetween 30min and two hours.

Quantum-optical simulations were performed with the Quantum OpticsToolbox in Python (QuTiP)12, where the standard quantum two-level system wasused as a starting point. The dynamical calculations, especially those of themeasured degrees of second-order coherence g (2)[0], were calculated using adynamical form of the quantum regression theorem5. The driving laser wasmodelled as a Gaussian pulse, where the product of the transition dipole andelectric field is given by µ ·E(t)/~=A/

q⌧ 2p ⇡e

�t2/⌧2p , ⌧p =⌧FWHM/p2 ln2, and A is

the pulse area. The chirp22 was modelled by multiplying the driving field by anadditional exponential e�i↵t2 , where ↵ is the chirp parameter. As a function of thepercentage change in bandwidth due to the chirp1BW, then↵=p21BW +12BW/⌧ 2p . The phonon-induced dephasing21 was modelled as apower-dependent collapse operator in the quantum-optical master equation—thatis, with cphonon =

pB(µ ·E(t)/~)2� †� , where � is the atomic lowering operator and

the phonon parameter was fitted to be B=2⇥10�3 GHz�1. A phenomenologicaldephasing rate that accounted for the spin and charge noise was modelled with thecollapse operator cnoise =p�d� †� , where we fitted �d =1.3 ns�1. This dephasingrate is slightly lower than the spontaneous emission rate (�d =0.78� ), which isconsistent with state-of-the-art values for X� transitions in InGaAsquantum dots19.

Because the ideal quantum two-level system emits negligible P3 for short pulses,then the probability mass function for joint photodetection at two di�erent times issimply given by p2(t1,⌧ )=G(2)(t1,⌧ )/2. Here, G(2)(t1,⌧ ) is the standard Glaubersecond-order coherence function, which can be calculated for a pulsed two-levelsystem using a time-dependent form of the quantum regression theorem5. Next, wediscuss how to obtain p1(t1), which is slightly more nuanced. Consider a trajectoryfor the ideal two-level system under excitation by an even-⇡-pulse: Pe(A) alwaysreturns to zero if no emission events occur during the system–pulse interaction.Therefore, the probability density of a first detection at time t1 is given by � Pe(t1),and this density is the sum of emissions that yield only a single photon p1(t1) and ofemissions that yield two photons p2(t1). Hence, p1(t1)=� Pe(t1)�p2(t1).

Data availability. The data that support the plots within this paper and otherfindings of this study are available from the corresponding authors uponreasonable request.

References26. Kuhlmann, A. V. et al . A dark-field microscope for background-free detection

of resonance fluorescence from single semiconductor quantum dots operatingin a set-and-forget mode. Rev. Sci. Instrum. 84, 073905 (2013).

NATURE PHYSICS | www.nature.com/naturephysics



Signatures of two-photon pulses from a quantum two-level systemMethodsFigure 1 Simulations showing pulsed two-photon emission from an ideal quantum two-level system.Figure 2 Simulations showing time-resolved single-photon and two-photon emission from an ideal quantum two-level system.Figure 3 Super-natural linewidth photons from an ideal quantum two-level system.Figure 4 Bunched photon pair emission from an ideal quantum two-level system.Figure 5 Experiments showing two-photon emission from a single artificial atom's transition.ReferencesAcknowledgementsAuthor contributionsAdditional informationCompeting financial interestsMethodsData availability.

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