We begin by considering an ideal quantum two-level system that interacts with the outside world only through its electric dipole moment μ (ref. 10). Suppose the system is instantaneously prepared in the superposition of its ground |g〉 and excited |e〉 states

where P e is the probability of initializing the system in |e〉. From this point, spontaneous emission at a rate of Γ governs the remaining system dynamics and a single photon is coherently emitted with probability P e , while no photon is emitted with probability 1 − P e . As detected by an ideal photon counter, this results in the photocount distribution

where P n is the probability to detect n photons in the emitted pulse. It is on this principle that most indistinguishable single-photon sources based on solid-state quantum emitters operate4,5.

A popular mechanism for approximately preparing |ψ i 〉 is the optically driven Rabi oscillation4,11. Here, the system is initialized in its ground state and driven by a short Gaussian pulse from a coherent laser beam (of width τ FWHM ) that is resonant with the |g〉 ↔|e〉 transition. Short is relative to the lifetime of the excited state τ e = 1/Γ to minimize the number of spontaneous emissions that occur during the system–pulse interaction5,9. As a function of the integrated pulse area, that is, A = ∫ dtμ ⋅ E(t)/ℏ, where E(t) is the pulse’s electric field, the system undergoes coherent oscillations between its ground |g〉 and excited |e〉 states. For constant-area pulses of vanishing τ FWHM /τ e , the final state of the system after interaction with the laser field is arbitrarily close to the superposition

where φ is a phase set by the laser field. Examining P e (A) (Fig. 1a dotted line), we see Rabi oscillations that are perfectly sinusoidal, with the laser pulse capable of inducing an arbitrary number of rotations between |g〉 and |e〉. Because |ψ f (A)〉 looks very much like |ψ i 〉 for arbitrarily short pulses, it is commonly assumed that the photocount distribution P n always has P 1 ≫ P n>1 . However, we will use a quantum trajectory approach to show that, unexpectedly, P 2 > P 1 for any τ FWHM < τ e when A = 2nπ, with n ∈ {1,2,3, . . . }.

Figure 1: Simulations showing pulsed two-photon emission from an ideal quantum two-level system. a, Rabi oscillations in the excited-state population from an ideal two-level system (dashed line). Signatures of Rabi oscillations in the emitted photon number under excitation with a pulse of length τ FWHM = τ e /10, where τ e is the excited-state lifetime (solid line). b, System dynamics under excitation by a pulse of area π. Dashed grey line shows the driving pulse shape, blue line shows the ensemble-averaged excited-state probability, green line shows a typical quantum trajectory with one photon detection (denoted by the green triangle). Inset shows photocount histogram P n under π-pulse excitation, with single-photon emission P 1 dominating. c, System dynamics under excitation by a pulse of area 2π. Dashed grey line shows the driving pulse shape, blue line shows the ensemble-averaged excited-state probability, red line shows a typical quantum trajectory with two photon emissions (denoted by the red triangles). Inset shows photocount histogram P n under 2π-pulse excitation, with P 0 dominating but P 2 ≫ P 1 . d, Photocount distribution P n (solid lines) and photon number purities π n (dashed lines) 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. Full size image

To visually illustrate the process that is capable of generating two photons, we discuss the remainder of the theory section with a convenient pulse width of τ FWHM = τ e /10. Because of the finite pulse length, we expect that in roughly τ FWHM /τ e of the quantum trajectories a spontaneous emission occurs during the system–pulse interaction. Therefore, it is difficult to define |ψ f 〉, and the expected number of photons emitted by the system E[n] provides a better signature of the Rabi oscillations (Fig. 1a solid line). Notably, a consequence of the spontaneous emissions is that E[n] does not exactly follow the sinusoid of the ideal Rabi oscillations (difference highlighted with the shaded region). Because E[n] > P e (A), the system must be occasionally re-excited to emit additional photons during the system–pulse interaction.

We now examine this re-excitation process in detail, first by considering the commonly studied case of an on-demand single-photon source with A = π (Fig. 1b). By driving a half Rabi oscillation, known colloquially as a π-pulse11, the probability of single-photon generation is maximized. Because the excitation pulse is short (grey dashed line) compared to the excited-state lifetime, the emitted wavepacket has an exponential shape (blue line). To further understand the probabilistic elements of the photon emission, we study a typical quantum trajectory8,12 representing P e (t) (green line). The system is driven by the π-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. After computing thousands of such trajectories, P n is generated from the photodetection events and shows P 1 ≍ 1 (inset), indicating that the system acts as a good single-photon source. The small amount of two-photon emission (P 2 ) occurs due to re-excitation of the quantum system during interaction with the pulse. It roughly accounts for the disparity between E[n] and P e (A = π), and is an important but often overlooked source of error in on-demand single-photon sources.

As our first clue that re-excitation during the system–pulse interaction can yield interesting dynamics, the difference between E[n] and P e (A) is not constant as a function of A, and is maximized for A = 2nπ. Therefore, we now take a closer look at the system’s dynamics for a 2π-pulse (Fig. 1c) and find a photocount distribution where P 2 ≫ P 1 (inset). To understand why the two-level system counter-intuitively prefers to emit two photons over a single photon, consider a typical quantum trajectory (red line). The emission probability is proportional to P e (t) (blue line), which peaks halfway through the excitation pulse. Therefore, the first photon is most likely to be emitted after approximately π of the pulse area has been absorbed (first red triangle), and a remaining approximately π in area then re-excites the system with near-unity probability to emit a second photon (second red triangle). This two-photon process is triggered during the system–pulse interaction and, although these photons are emitted within a single excited-state lifetime, they have a temporal structure known as a photon bundle13. Signatures of the bundle can be found in P e (t): the emission shows a peak of width τ FWHM followed by a long tail of length τ e . This shows the conditional generation of a second photon based on a first emission during the system–pulse interaction, which means that the two-photon bundling effect dominates for arbitrarily short pulses and even for long pulses as well (Supplementary Fig. 4). Hence, although the efficiency as a pulsed two-photon source is given by P 2 ≍ 6%, an ideal two-level system could be operated in a much more efficient regime simply by choosing a longer pulse. We avoided this discussion in the main text because P 3 becomes non-negligible, which makes an intuitive interpretation of the dynamics more challenging.

To fully characterize the crossover where P 2 > P 1 , we simulated the photocount distributions as a function of pulse area (Fig. 1d). Clear oscillations can be seen between P 1 and P 2 (solid green and red lines, respectively), and P 2 is out of phase from the Rabi oscillations. Notably, the oscillations in E[n] make direct comparisons between the two probabilities difficult. To better illustrate the fraction of emission occurring by n-photon emission, we turn to a quantity called the photon number purity of the source13. By ignoring the vacuum component, P n is renormalized to

The purities (dashed lines) very clearly oscillate between emission dominated by single-photon processes π 1 for odd-π-pulses and two-photon processes π 2 for even-π-pulses. Quite remarkably, π 3 remains negligible for all pulse areas. Additionally, the purities reveal the limit of {π 1 , π 2 , π 3 } = {0.3,0.7,0} for arbitrarily short Gaussian pulses.

As suggested earlier, this two-photon emission comes as an ordered pair, where the first emission event within the pulse excitation window triggers the absorption and subsequent emission of a second photon. Unlike the first photon, the second photon has the entire excited-state lifetime to leave. This statement can be quantified by investigating the time-resolved probability mass functions for photodetection14 (see Methods for a derivation from system dynamics), defined by

The mass function p 1 (t 1 ) represents the probability density for emission of a single photon at time t 1 with no subsequent emissions, while p 2 (t 1 , τ) represents the joint probability density for emission of a single photon at time t 1 with a subsequent emission at time t 1 + τ. Additionally, p 2 (t 1 , τ) can be integrated along t 1 or τ to yield p 2 (τ) or p 2 (t 1 ), which give the probability density for waiting τ between the two emission events or detecting a photon pair with the first emission at time t 1 , respectively.

We explore these temporal dynamics for excitation by a 2π-,4π-, and 6π-pulse in Fig. 2a–c, respectively. First, consider excitation by the 2π-pulse: p 2 (t 1 , τ) captures the dynamics already discussed through having a high probability of the first emission at time t 1 only within the pulse window of 0.1/Γ, but the second emission occurs at a delay τ later within the spontaneous emission lifetime τ e . This effect is most clearly seen in p 1 (t 1 ), p 2 (t 1 ) and p 2 (τ) (traces to the left and top of the colour plots in Fig. 2), where the density of photon pair emission being triggered at time t 1 —that is, p 2 (t 1 )—is maximized after π of the pulse has been absorbed (t 1 = 0.15/Γ) and reaches nearly unity. The second photon of the pair then has the entire lifetime to leave, as seen in the long correlation time for p 2 (τ). Meanwhile, the enhancement in photon pair production leads to a corresponding decrease in the density of single-photon emissions p 1 (t 1 ) around t 1 = 0.15/Γ.

Figure 2: Simulations showing time-resolved single-photon and two-photon emission from an ideal quantum two-level system. a–c, Probability mass functions for single-photon (green) and two-photon (red) detection showing the internal temporal structure of the photon pairs, for excitation by a 2π-pulse (a), 4π-pulse (b), and 6π-pulse (c). Colour plots show p 2 (t 1 , τ), while the traces to the left show p 1 (t 1 ) and p 2 (t 1 ), and the traces to the top show p 2 (τ). Full size image

Next, consider excitation by the 4π-pulse: p 2 (t 1 , τ) shows the effects of an additional Rabi oscillation that the system undergoes during interaction with the pulse. If the first emission occurs after π of the pulse has been absorbed, a remaining 3π can result in a second emission in two different ways: either after absorption of π additional energy or after absorption of 3π additional energy. On the other hand, if the first emission occurs after 3π of the pulse has been absorbed, only a remaining π can be absorbed, resulting in a monotonic region of p 2 (t 1 , τ) just like for the 2π case. In either scenario, the probability of two emissions is most likely (but three emissions almost never happen) because a single emission converts an even-π-pulse into an odd-π-pulse, which anti-bunches the next emission. The high fidelity of this conversion process can clearly be observed in p 1 (t 1 ) and p 2 (t 1 ). The pair production is most likely after either π or 3π of the pulse has been absorbed (times t 1 = 0.1 and t 1 = 0.3, respectively), and it occurs with near-unity density. This means that if the first photon is emitted at time t 1 = 0.1 or t 1 = 0.3, then the conditional probability to emit a second photon is near unity. As a result, p 1 (t 1 ) and p 2 (t 1 ) almost look like they were just copied a second time from the 2π-pulse scenario, confirming our intuitive interpretation of the 4π-pulse scenario. These ideas trivially extrapolate to the 6π-pulse, where three complete Rabi oscillations occur, and the projections p 1 (t 1 ) and p 2 (t 1 ) are copied once more along t 1 .

Looking at the oscillations in p 2 (τ) for increasing pulse areas, one may notice a qualitative resemblance to the photon bunching15 behind a continuous-wave Mollow triplet11. In fact, the underlying process where a photon emission collapses the system into its ground state, restarting a Rabi rotation, is responsible for the dynamics in both cases. However, our observed phenomenon has a very important difference: after a photodetection the expected waiting time for the second, third, and nth photon emissions is identical in the continuous case, while our observed process dramatically suppresses P 3 .

Because this method of generating two-photon bundles requires the emission of the first photon to occur within a tightly defined time interval, as set by the pulse width, we explore the emission in the context of time–frequency uncertainty (Fig. 3). First consider the 2π-pulse case: we replot P e (t) as the light red shaded trace (panel a), which results in the first shaded emission spectrum16 (panel b). Compared to the natural linewidth of the system’s transition (dashed black line), the emission is spectrally broadened by the first emission of a super-natural linewidth photon of order 1/τ FWHM , which occurs during the laser pulse. We note that it would be interesting to explore the physics of super-natural linewidth photons that have been incoherently emitted as the result of a new many-body scattering process (having isolated them with a spectral notch filter to remove the second photon of a natural linewidth). As an effect of the increased number of Rabi oscillations (see for 4π- and 6π-pulses), the super-natural linewidth photons show oscillations in spectral power density that resemble an emerging dynamical Mollow triplet17.

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 2π-, 4π-, and 6π-pulse, denoted with increasing darkness of the shaded region for higher pulse area. Dashed lines show the natural emission linewidth. Full size image

Finally, we discuss the counting statistics of the emitted light in comparison to a coherent laser pulse to verify the nonclassical nature of the emission. Because the photocount distribution is fully described with just P 0 , P 1 and P 2 , its information is completely contained in the mean E[n] and the normalized second-order factorial moment8

which physically describes the relative probability to detect a correlated photon pair over randomly finding two uncorrelated photons in a Poissonian laser pulse of equivalent mean. Thus, the quantity g(2)[0] yields important information on how a beam of light deviates from the Poissonian counting statistics of a laser beam. For Poissonian statistics g(2)[0] = 1, but for sub-Poissonian statistics g(2)[0] < 1 (anti-bunching) and for super-Poissonian statistics g(2)[0] > 1 (bunching)5. In particular, because the emission under a 2π excitation is a weak two-photon pulse, we expect that the photons will arrive in ‘bunched’ pairs, where the first detection heralds the presence of a second photon in the pulse. This prediction is confirmed in Fig. 4, where emission for even-π-pulses strongly bunches, thus confirming the highly nonclassical nature of the emission and providing an experimentally accessible signature of the oscillations in P 2 .

Figure 4: Bunched photon pair emission from an ideal quantum two-level system. Normalized second-order factorial moment of the photocount distribution (blue), which measures the total degree of second-order coherence g(2)[0]. Dashed blue line indicates the Poissonian counting statistics of the laser pulse. Dotted black line again indicates Rabi oscillations for reference. Full size image

After having theoretically discovered that a quantum two-level system is able to preferentially emit two photons through a complex many-body scattering phenomenon, we found experimental signatures of this two-photon process using a single transition from an artificial atom. Our artificial atom of choice is an InGaAs quantum dot, due to its technological maturity and good optical quality18. The dot is embedded within a diode structure to minimize charge and spin noise (Methods), resulting in a nearly transform-limited optical transition19; luminescence experiments as a function of gate voltage (Fig. 5a) reveal the charge-stability region in which we operate20 (V g = 0.365 V). We used the X− transition due to its lack of fine structure, which results in a true two-level system (at zero magnetic field) with an excited-state lifetime of τ e = 602 ps (Supplementary Fig. 1). Exciting the system with laser pulses (τ FWHM = 80 ps), we drove Rabi oscillations between its ground and excited states (Fig. 5b). However, because the artificial atom resides in a solid-state environment, it possesses several non-idealities that slightly decrease the fidelity of the oscillations: a power-dependent dephasing rate arising from electron–phonon interaction21 and an excited-state dephasing due to spin or charge noise19. Additionally, the quantum systems are very sensitive to minimal pulse chirps arising due to optical set-up non-idealities22. Using these three effects as fitting parameters (Methods), we obtained near-perfect agreement between our quantum-optical model (blue) and the experimental Rabi oscillations.

Figure 5: Experiments showing two-photon emission from a single artificial atom’s transition. a, Photoluminescence versus applied gate bias and wavelength showing the charge-stability region of the X− transition. Inset: Voltage dependence of resonance fluorescence intensity under pulsed excitation, with the optimal resonance fluorescence signal occurring at V g = 0.365 V. b, Experimental resonance fluorescence signal showing Rabi oscillations, scaled to quantum simulations of emitted photon number (blue). c,d, Under excitation by π- and 2π-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 are g(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 area showing oscillations between anti-bunching (at odd π-pulses) and bunching (at even-π-pulses). Blue curve represents quantum simulations of the time-integrated correlations g(2)[0] from the experimental system. Dashed black line represents statistics of the incident laser pulse. f, Experimental second-order coherence measurements g(2)[0] versus pulse length (using 2π-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 of dephasing, 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 % chirp in 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 fluctuations in the photocount distribution5. The error in pulse area accounts for power drifts during the experiment, and the errors in pulse lengths are least-squares fitting errors to the pulse spectra. Full size image

Next, we measured the g(2)[0] values of the emitted wavepackets, to study the photon bunching effects outlined in Fig. 4. Two typical experiments are presented in Fig. 5c, d, showing g(2)[0] ≍ 0 (anti-bunching) and g(2)[0] > 1 (bunching) for π- and 2π-pulses, respectively. A complete data set is shown in Fig. 5e, with oscillations between anti-bunching at odd-π pulses and bunching at even-π-pulses. Using the same fitting parameters as in Fig. 5b, the correlation data are almost perfectly matched with our full quantum-optical model. Hence, we have found experimental evidence that suggests the artificial atom is affected by the predicted many-body two-photon scattering process that causes P 2 oscillations out of phase from the Rabi oscillations.