Abstract
A twolevel atom can generate a strong manybody interaction with light under pulsed excitation^{1,2,3}. The best known effect is singlephoton generation, where a short Gaussian laser pulse is converted into a Lorentzian singlephoton wavepacket^{4,5}. However, recent studies suggested that scattering of intense laser fields off a twolevel atom may generate oscillations in twophoton emission that come out of phase with the Rabi oscillations, as the power of the pulse increases^{6,7}. Here, we provide an intuitive explanation for these oscillations using a quantum trajectory approach^{8} and show how they may preferentially result in emission of twophoton pulses. Experimentally, we observe the signatures of these oscillations by measuring the bunching of photon pulses scattered off a twolevel quantum system. Our theory and measurements provide insight into the reexcitation process that plagues^{5,9} ondemand singlephoton sources while suggesting the possibility of producing new multiphoton states.
Main
We begin by considering an ideal quantum twolevel 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 singlephoton sources based on solidstate quantum emitters operate^{4,5}.
A popular mechanism for approximately preparing ψ_{i}〉 is the optically driven Rabi oscillation^{4,11}. 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 interaction^{5,9}. 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 constantarea 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, . . . }.
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 reexcited to emit additional photons during the system–pulse interaction.
We now examine this reexcitation process in detail, first by considering the commonly studied case of an ondemand singlephoton source with A = π (Fig. 1b). By driving a half Rabi oscillation, known colloquially as a πpulse^{11}, the probability of singlephoton generation is maximized. Because the excitation pulse is short (grey dashed line) compared to the excitedstate 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 trajectory^{8,12} representing P_{e}(t) (green line). The system is driven by the πpulse 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 singlephoton source. The small amount of twophoton emission (P_{2}) occurs due to reexcitation 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 ondemand singlephoton sources.
As our first clue that reexcitation 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 2πpulse (Fig. 1c) and find a photocount distribution where P_{2} ≫ P_{1} (inset). To understand why the twolevel system counterintuitively 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 reexcites the system with nearunity probability to emit a second photon (second red triangle). This twophoton process is triggered during the system–pulse interaction and, although these photons are emitted within a single excitedstate lifetime, they have a temporal structure known as a photon bundle^{13}. 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 twophoton bundling effect dominates for arbitrarily short pulses and even for long pulses as well (Supplementary Fig. 4). Hence, although the efficiency as a pulsed twophoton source is given by P_{2} ≍ 6%, an ideal twolevel 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 nonnegligible, 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 nphoton emission, we turn to a quantity called the photon number purity of the source^{13}. By ignoring the vacuum component, P_{n} is renormalized to
The purities (dashed lines) very clearly oscillate between emission dominated by singlephoton processes π_{1} for oddπpulses and twophoton processes π_{2} for evenπpulses. 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 twophoton 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 excitedstate lifetime to leave. This statement can be quantified by investigating the timeresolved probability mass functions for photodetection^{14} (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 6πpulse in Fig. 2a–c, respectively. First, consider excitation by the 2πpulse: 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 singlephoton emissions p_{1}(t_{1}) around t_{1} = 0.15/Γ.
Next, consider excitation by the 4πpulse: 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 evenπpulse into an oddπpulse, which antibunches 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 nearunity 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 2πpulse scenario, confirming our intuitive interpretation of the 4πpulse scenario. These ideas trivially extrapolate to the 6πpulse, 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 bunching^{15} behind a continuouswave Mollow triplet^{11}. 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 twophoton 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 2πpulse case: we replot P_{e}(t) as the light red shaded trace (panel a), which results in the first shaded emission spectrum^{16} (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 supernatural 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 supernatural linewidth photons that have been incoherently emitted as the result of a new manybody 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 6πpulses), the supernatural linewidth photons show oscillations in spectral power density that resemble an emerging dynamical Mollow triplet^{17}.
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 secondorder factorial moment^{8}
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 subPoissonian statistics g^{(2)}[0] < 1 (antibunching) and for superPoissonian statistics g^{(2)}[0] > 1 (bunching)^{5}. In particular, because the emission under a 2π excitation is a weak twophoton 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 evenπpulses strongly bunches, thus confirming the highly nonclassical nature of the emission and providing an experimentally accessible signature of the oscillations in P_{2}.
After having theoretically discovered that a quantum twolevel system is able to preferentially emit two photons through a complex manybody scattering phenomenon, we found experimental signatures of this twophoton 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 quality^{18}. The dot is embedded within a diode structure to minimize charge and spin noise (Methods), resulting in a nearly transformlimited optical transition^{19}; luminescence experiments as a function of gate voltage (Fig. 5a) reveal the chargestability region in which we operate^{20} (V_{g} = 0.365 V). We used the X^{−} transition due to its lack of fine structure, which results in a true twolevel system (at zero magnetic field) with an excitedstate 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 solidstate environment, it possesses several nonidealities that slightly decrease the fidelity of the oscillations: a powerdependent dephasing rate arising from electron–phonon interaction^{21} and an excitedstate dephasing due to spin or charge noise^{19}. Additionally, the quantum systems are very sensitive to minimal pulse chirps arising due to optical setup nonidealities^{22}. Using these three effects as fitting parameters (Methods), we obtained nearperfect agreement between our quantumoptical model (blue) and the experimental Rabi oscillations.
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 (antibunching) and g^{(2)}[0] > 1 (bunching) for π and 2πpulses, respectively. A complete data set is shown in Fig. 5e, with oscillations between antibunching at oddπ pulses and bunching at evenπpulses. Using the same fitting parameters as in Fig. 5b, the correlation data are almost perfectly matched with our full quantumoptical model. Hence, we have found experimental evidence that suggests the artificial atom is affected by the predicted manybody twophoton scattering process that causes P_{2} oscillations out of phase from the Rabi oscillations.
Finally, we investigated how nonidealities affect the bunching values (Fig. 5f) by experimentally characterizing the emission at four pulse lengths. From our quantumoptical model, we see that bunching is strongest for the ideal case (blue line), and decreases for every added nonideality (long dashed blue for dephasing, short dashed blue for additional 2.7% chirp in bandwidth, and short dotted blue for further 2.7% chirp in bandwidth), yielding excellent agreement with the data. Discussing these effects further, an enhanced pulse chirp decreases the fidelity of photon bunching due to the function of a large chirp to adiabatically prepare the system in its excited state^{23}, which decays with a singlephoton emission. The minimal chirp that we observed can be removed with pulse compressors in future experiments to achieve an even better match with the ideal photon bunching curve. Additionally, at short pulse lengths the powerdependent dephasing results in antibunching. Due to the higher amplitude of shorter pulses (with fixed area), the dephasing rate diverges and the system acts as an incoherently pumped singlephoton source. Thus, when including nonidealities we found the optimum bunching to occur at a pulse length of approximately 80 ps, which indicates where the twophoton process is strongest experimentally. Although we expect the twophoton emission is dominant, the nonidealities of the solidstate system could result in nonnegligible P_{3} or higher P_{n}. This scenario is unfortunately not distinguishable through measuring g^{(2)}[0] alone, but it is recently becoming possible to measure higherorder photon correlations that could help definitively identify regimes of operation where P_{2} ≫ P_{3} (refs 24,25). Finally, we expect future investigations on exploring optimal pulse shapes to enable much more efficient and higherpurity twophoton emission both from ideal and experimental twolevel systems.
Methods
The sample investigated is grown by molecular beam epitaxy (MBE). It consists of a layer of InGaAs quantum dots with low areal density (<1 μm^{−2}), embedded within the intrinsic region of a Schottky photodiode formed from an ndoped layer below the quantum dots and a semitransparent titanium gold front contact. The distance between the doped layer and the quantum dots is 35 nm, which enables control over the charge status of the dot^{20}. A weak planar microcavity with an optical thickness of one wavelength (λ) is formed from a buried 18pair GaAs/AlAs distributive Bragg reflector (DBR) and the semitransparent top contact, which enhances the in and outcoupling of light.
All optical measurements were performed at 4.2 K in a liquid helium dipstick setup. For excitation and detection, a microscope objective with a numerical aperture of NA = 0.68 was used. Crosspolarized measurements were performed using a polarizing beam splitter. To further enhance the extinction ratio, additional thin film linear polarizers were placed in the excitation/detection pathways and a singlemode fibre was used to spatially filter the detection signal. Furthermore, a quarterwave plate was placed between the beamsplitter and the microscope objective to correct for birefringence of the optics and the sample itself^{26}. For the Rabi oscillations, a very weak laser background (due to an imperfect suppression of the excitation laser) was subtracted. This linearly increasing background was directly measured through electrically tuning the quantum dot out of resonance, and typically amounted to less than 10% of the signal by 5π pulse area; the quantum statistic g^{(2)}[0] is dependent on the square of the signal’s power, which means that the background (at a maximum) contributed to less than 1% of those measurements.
The 20 and 80pslong excitation pulses were derived from a fspulsed titanium sapphire laser (Coherent Mira 900) through pulse shaping. For the 20pslong pulses, a 4f pulse shaper with a focal length of 1 m and an 1,800 l mm^{−1} grating was used. For the 80pslong pulses a spectrometerlike filter with a focal length of 1 m and an 1,800 l mm^{−1} grating was used. Longer pulses were obtained through modulating a continuous wave laser. For the modulation, a fibrecoupled and EOMcontrolled lithium niobate Mach–Zehnder (MZ) interferometer with a bandwidth of 10 GHz (Photline NIRMXLN10) was used. Such modulators allow control of the output intensity through a d.c. bias and a radiofrequency input. The radiofrequency pulses were generated by a 3.35 GHz pulsepattern generator (Agilent 81133A). To obtain a high extinction ratio, the temperature of the modulator was stabilized and precisely controlled (1 mK) using a Peltier element, thermistor, and TEC controller. This enabled a static extinction ratio >35 db.
Secondorder autocorrelation measurements were performed using a HanburyBrown and Twiss (HBT) setup consisting of one beamsplitter and two singlephoton avalanche diodes. Note: their timing resolution (≍100 ps) is too low to measure the correlations predicted in Fig. 2. The detected photons were correlated with a TimeHarp200 timecounting module. The integration times were between 30 min and two hours.
Quantumoptical simulations were performed with the Quantum Optics Toolbox in Python (QuTiP)^{12}, where the standard quantum twolevel system was used as a starting point. The dynamical calculations, especially those of the measured degrees of secondorder coherence g^{(2)}[0], were calculated using a dynamical form of the quantum regression theorem^{5}. The driving laser was modelled as a Gaussian pulse, where the product of the transition dipole and electric field is given by , , and A is the pulse area. The chirp^{22} was modelled by multiplying the driving field by an additional exponential ${\text{e}}^{\text{i}\alpha {t}^{2}}$, where α is the chirp parameter. As a function of the percentage change in bandwidth due to the chirp Δ_{BW}, then . The phononinduced dephasing^{21} was modelled as a powerdependent collapse operator in the quantumoptical master equation—that is, with , where σ is the atomic lowering operator and the phonon parameter was fitted to be B = 2 × 10^{−3} GHz^{−1}. A phenomenological dephasing rate that accounted for the spin and charge noise was modelled with the collapse operator , where we fitted γ_{d} = 1.3 ns^{−1}. This dephasing rate is slightly lower than the spontaneous emission rate (γ_{d} = 0.78Γ), which is consistent with stateoftheart values for X^{−} transitions in InGaAs quantum dots^{19}.
Because the ideal quantum twolevel system emits negligible P_{3} for short pulses, then the probability mass function for joint photodetection at two different times is simply given by p_{2}(t_{1}, τ) = G^{(2)}(t_{1}, τ)/2. Here, G^{(2)}(t_{1}, τ) is the standard Glauber secondorder coherence function, which can be calculated for a pulsed twolevel system using a timedependent form of the quantum regression theorem^{5}. Next, we discuss how to obtain p_{1}(t_{1}), which is slightly more nuanced. Consider a trajectory for the ideal twolevel system under excitation by an evenπpulse: P_{e}(A) always returns to zero if no emission events occur during the system–pulse interaction. Therefore, the probability density of a first detection at time t_{1} is given by ΓP_{e}(t_{1}), and this density is the sum of emissions that yield only a single photon p_{1}(t_{1}) and of emissions that yield two photons p_{2}(t_{1}). Hence, p_{1}(t_{1}) = ΓP_{e}(t_{1}) − p_{2}(t_{1}).
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
The authors thank A. Rasmussen for productive and helpful discussions in framing the context of this work. We gratefully acknowledge financial support from the National Science Foundation (Division of Materials Research—Grant No. 1503759), the DFG via the Nanosystems Initiative Munich, the BMBF via Q.com (Project No. 16KIS0110) and BaCaTeC. K.A.F. acknowledges support from the Lu Stanford Graduate Fellowship and the National Defense Science and Engineering Graduate Fellowship. J.W. acknowledges support 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.
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K.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. provided expertise. K.M. organized the collaboration and supervised the experiments. All authors participated in the discussion and understanding of the results.
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Fischer, K., Hanschke, L., Wierzbowski, J. et al. Signatures of twophoton pulses from a quantum twolevel system. Nature Phys 13, 649–654 (2017). https://doi.org/10.1038/nphys4052
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