Triggered emission of indistinguishable photons from an organic dye molecule

Two-particle interference is one of the most striking consequences of quantum physics. After the first seminal experiments with photons,^1–4 the effect has been demonstrated also for massive particles and quasi-particles such as electrons,^5,6 plasmons,^7,8 and atoms.^9,10 Glauber well captured this phenomenon intuitively within a generalized picture of interference, affecting all measurement outcomes which result from multiple indistinguishable stories.¹¹ Two-particle interference though does not have any classical counterpart, not even in the case of light.¹² Indeed two-photon interference (TPI) is at the heart of many quantum technologies (see Ref. 13), potentially outperforming classical computation schemes,^14,15 offering physics-protected information,¹⁶ simulating highly complex physical systems,^17–19 and enhancing the sensitivity of precise measurements.^20,21 The so-called Hong–Ou–Mandel (HOM) interferometer, originally developed to measure small time shifts^2,3 nowadays reaching attosecond resolution,²² is a well-known experimental scheme unveiling TPI. Essentially it consists of two single photon detectors at the output ports of a beam splitter (BS). Due to the bosonic nature of photons, in the case of two indistinguishable photons impinging separately on the BS ports, interference forces them to coalesce at one of the outputs, suppressing the rate of simultaneous coincidences and hence yielding the two-photon path-entangled state²³ $12(|2,0⟩+|0,2⟩)$⁠. In order to enable effective quantum interference, the involved particles have to be prepared in the very same quantum state, so as to determine indistinguishability among different possible events with the same output state. It is also well established that quantum emitters in the solid state play a major role in the generation of single photons on a deterministic basis,²⁴ which is key to the preparation of more complex states of light^25,26 that are useful for many protocols in quantum technologies. However, the ability of such systems to provide photons in the same quantum state remains one of the most elusive requirements to meet, in particular in the sub-micrometric environment of integrated photonics.^27,28

Single molecules of polyaromatic hydrocarbons (PAH) in suitable host matrices are known for emitting with high quantum efficiency in very narrow (few-tens-of-MHz wide) and stable zero phonon lines (ZPL). Molecular quantum emitters can also be integrated inside complex and hybrid photonic structures, yielding almost 100% collection efficiency,²⁹ 99% light extinction³⁰ and allowing for on-chip photon manipulation,^31,32 (for a complete review see, e.g., Ref. 33). Very promising results have been reported in HOM interference experiments with single photon streams, obtained from molecular emitters after continuous wave (CW) laser pumping,^34–37 including the case of photons emitted by remote molecules.³⁸ However, triggered single photon pulses appear necessary in any kind of logic operation with photons. In this work, we demonstrate that single, indistinguishable photons can be generated by single PAH molecules, triggered by the laser excitation pulses. Notably, the results are obtained for non-resonant pumping of the ZPL and without coupling to optical cavities. Hong–Ou–Mandel experiments are reported, based on triggered single photons from single dibenzoterrylene (DBT) molecules in anthracene nanocrystals.³⁹ Such molecular source is particularly suitable to isolate single emitters with respect to neighboring ones and to the background, even when a spectrally broad pulsed laser is employed in excitation. It also proves to be photostable over a timescale of at least one month. For this experiment, we employ the sample geometry already presented in Ref. 40, where nanocrystals are deposited on a gold mirror and then stabilized with a 200 nm-thick layer of Polyvinyl alcohol (PVA).

In order to model the outcome of the HOM measurements, we independently characterize also the multi-photon emission probability, the emission linewidth, the molecule lifetime, and the setup parameters. The model that is briefly outlined in the supplementary material correctly describes our data and suggests that the main limitation to the observed raw-data visibility is to be found in the residual dephasing observed at 3 K, which is currently our lowest temperature of operation. Upon cooling down to 1.4 K, an intrinsic wave-function overlap of up to 95% can be estimated.

In Fig. 1(a), the experimental setup used for characterization and for the HOM measurements is outlined. Single-molecule fluorescence is excited and collected in an epi-fluorescence microscope through a 0.7-N.A. objective, using either a tunable CW laser (⁠$λ=783.5$ nm) or a pulsed laser with central wavelength $λ=766.0$ nm, adjustable repetition rate, and 50-ps long pulses (Picoquant LDH-D-FA-765L, linewidth of about 0.3 nm), that can be operated in CW as well. A Longpass filter (Semrock LP02–785RE) is employed for rejection of the back-scattered pump light. The ZPL component at 783.5 nm is separated from the phonon sideband (PSB) exploiting a 0.4 nm-wide reflective Notch filter (OptiGrate BNF-785-OD4–12.5M) and coupled into a single-mode fiber. In order to probe the indistinguishability of consecutively emitted photons, the photon stream is split in two by a polarizing beam splitter cube (PBS), with a fiber polarization control in front (FPC, Thorlabs FPC023). One of the paths is then delayed by $Δt≃40$ ns (8 m-long fiber), much longer than the excited state lifetime (⁠$≃4$ ns), as required to avoid temporal correlations between the photons in the two arms. Quantum interference in the train of single photon pulses is then measured by an unbalanced fiber-based Mach–Zehnder interferometer (MZI). In particular, a FPC unit is added on one path to switch between parallel and orthogonal configuration, before the MZI is closed on a fibered 50/50 BS (Thorlabs TN785R5A2). The two output ports are finally connected to a couple of single photon counting modules (Excelitas SPCM-NIR-14, quantum efficiency 70%). According to semi-analytical simulations,⁴¹ the collection efficiency at first lens for the sample under investigation (and N.A. limited to 0.7) is estimated to be around 25%. Concerning the optical apparatus, the transmission of the free-space part of the setup is about 40%, the fiber coupling efficiency amounts to 60%, and the transmission of the fiber circuit to the SPADs is about 70%, corresponding to an optical setup efficiency of 17% and an overall efficiency of about 3%. The temporal resolution of the whole electronic chain involved in the start–stop measurement is $230$ ps (see below and also supplementary material for details). When one of the interferometer arms is blocked, the setup operates in the so-called Hanbury–Brown and Twiss⁴² (HBT) configuration, allowing for the characterization of the photon statistics. Instead, when the photon flux is equally distributed on the two arms, we can probe the photon indistinguishability through HOM interference. In Figs. 1(c) and 1(d), the histogram of the inter-photon arrival times for small time delays, normalized to the background average value, is reported for photon streams collected in the HBT and HOM configuration, respectively (saturation parameter $s≃0.2$⁠). In both cases, the dataset well reproduces the second order correlation function $g2(τ)$⁠, probing multi-photon emission probability and indistinguishability of the photon stream. From the best fit to the data in Fig. 1(c) with equation $gHBT(2)(τ)=(1−b* exp (−|τ|/τHBT))$⁠, we obtain $gHBT(2)(0)=0.03±0.02$⁠.

Moreover, the effect of non-classical interference is clearly visible in panel d around zero time delay, where the data set for parallel polarization is well below the one for photons which are made distinguishable in polarization, i.e., to say, orthogonal. The other two dips occurring $Δt$-away from the main one correspond to the suppressed probability of having two photons closer than the emitter lifetime, in the case of one photon being transmitted/reflected when the other is reflected/transmitted. From the best fit to this measurement with the following equation, derived from Ref. 43 in the low saturation regime

with $g(t)=gHBT(2)(t)$ and V = 0 for orthogonal polarizations, we deduce $Δt=40.3±0.2$ ns and the unbalance between reflectance $|r|2$ and transmittance $|t|2$ of the second beam splitter to be $|r|2/|t|2=1.10±0.07$⁠. The latter estimation is in agreement with direct measurements implemented with laser light at 783.5 nm, from which we evaluate both the overall losses of the fiber connector + BS (0.06 ± 0.01), and $|r|2$ and $|t|2$ values to be 0.50 ± 0.01 and 0.44 ± 0.01, respectively. These parameters will be used in analyzing the data for pulsed excitation, as they can be considered unchanged.

The degree of coherence of the molecule emission can be estimated as $|g(1)(0)|2=g⊥(2)(0)−g∥(2)(0)g⊥(2)(0)=0.89±0.06$⁠. This value results compatible with an estimate of the first order coherence from the second order autocorrelation function measured in HBT configuration, following the relationship:³⁶ $|g(1)(0)|2=1−2gHBT(2)(0)=0.94±0.04$⁠. It is important to notice that such visibility corresponds to the post-selected photons colliding at the beam splitter within a time-window much smaller than the photon wave packet duration (in our case 256 ps and 4 ns, respectively). As such, it is effective only at the price of a reduced emitter brightness. In other words, suppressed coincidences at zero time delay are expected also in a HOM experiment with detuned sources or in the presence of dephasing, with spectral fluctuations smaller than the inverse of the overall setup temporal resolution.³⁸ A non-perfect visibility in the case of severe post-selection can be hence ascribed only to the detector time-jitter, or to optical misalignment of the setup. The HOM interference profile in CW operation still gives some insight about the indistinguishability of the considered photon stream. Indeed, while the minimum of the dip is related to suppression of multi-photon emission events and other technical aspects, the characteristic time of the exponential profile depends on the coherence of the emission.³⁶ In detail, we find from the fits that the timescale of the dip for parallel configuration is shorter (⁠$τ∥=2.7±0.2$ ns) than that for orthogonal polarization (⁠$τ⊥=3.6(5)± 0.3(0)$ ns) and in the HBT case of Fig. 1(c) (⁠$τHBT=3.5±0.2$ ns). This mismatch is expected in the presence of pure dephasing and corresponds to an estimation of the coherence time of the emitter equal to $τc=2τ∥=5.4±0.5$ ns. It is interesting to compare this value with the direct measurement of the ZPL linewidth obtained via excitation spectroscopy that means recording the fluorescence signal in the PSB as a function of the excitation laser frequency, scanning across the ZPL [see Fig. 1(b)]. Indeed, in the latter case a FWHM of the Lorentzian fit to data yields a value of $=55.1±0.5$ MHz, compatible with the estimate from HOM interference yielding $FWHM=1/(πτc)=59±10$ MHz. We can conclude that the system shows negligible spectral diffusion under CW pumping, as the two estimate for the dephasing broadening are consistent within the error bars, although sensitive to frequency fluctuations on very different time scales (40 ns in the HOM measurements and seconds-long integration time for the excitation spectroscopy).

A more direct estimate of the photon wave packet overlap can be obtained addressing with laser pulses the molecule previously characterized in CW. First the fluorescence lifetime is measured, recording the histogram of the photon arrival times with respect to the laser excitation pulse. A typical dataset is reported in Fig. 2(c) and from the best fit with a single exponential decay, a lifetime $τ1=4.01±0.01$ ns is obtained, with uncertainty given by the standard deviation over 10 measurements.⁴⁴ The discrepancy with the timescale obtained for the anti-bunching profiles under CW operation $τ⊥∼τHBT∼3.5$ ns can be ascribed to the dependency of τ_HBT on both the excited state lifetime and pumping rate (see²⁹ the supplementary material). In order to perform HOM interference, the MZI delay and the temporal distance among neighboring pulses have to coincide. To this aim, the repetition rate is adjusted according to the result for $Δt$ from the HOM fit in CW. The uncertainty associated with this measurement is estimated from the error in the fit parameter and amounts to about 200 ps. This directly impacts the uncertainty on the visibility, as it will be shown later. In Fig. 2(a), the $gHOM(2)(τ)$ under pulsed excitation is compared for parallel and orthogonal polarization in red and blue solid lines, respectively, with the second one artificially delayed for sake of clarity. All curves are normalized with respect to the maximum peak amplitude, calculated averaging over more than ten pulses, excluding the ones in zero and $±Δt$⁠. The latter pulses are associated with coincidences between photons that are one or two periods apart, leading to areas $Ak=−1=N(1−R2), Ak=+1=N(1−T2)$⁠, whereas all other peaks can be due to 3 classes of events, which sum up to a constant probability, $Ak≤−2=Ak≥2=N$⁠, with N being proportional to the integration time (supplementary material Ref. 45).

One way of defining the TPI visibility is through the comparison of the peak area around zero delay for the case of indistinguishable vs distinguishable photons, calculating the ratio, $V=A⊥−A∥A⊥$⁠. In our case, integrating within a time window $ΔT=26$ ns (corresponding to about 96% of the photon wave packet area), we estimate a visibility $V=78%±4%$⁠.⁴⁶ The results are further analyzed with a model based on Ref. 47, extrapolated for a train of infinite pulses (for details see the supplementary material), with the introduction of a phenomenological parameter v. This factor takes into account a non-perfect spatial alignment and polarization control, the residual multiphoton probability, as well as the effect of emission of distinguishable photons within the filter range but outside the ZPL. It amounts to the visibility that would be measured in the limit of negligible dephasing. As it can be appreciated in Fig. 2(b), the model correctly describes the data, considering the unbalance between $|r|2$ and $|t|2$ of the second beam splitter, the temporal resolution of the electronic system (230 ps, obtained as standard deviation of the $gHBT(2)(τ)$ for laser pulses, details in the supplementary material) and a delay given by the inverse of the laser nominal repetition rate (24.79 MHz). The only free parameters in the model are the excited state lifetime τ₁, the pure dephasing $Γ*$ and v. A least squares fitting algorithm yields the following values: $τ1=4.04±0.02$ ns, $Γ*=55±10$ MHz, and $v=0.95±0.02$⁠. The first two parameters allow an independent estimation of the linewidth as $FWHM=Γ/π=1/(2πτ1)+Γ*/π=57±4$ MHz, in substantial agreement with the direct measurement reported above.

Interestingly, the factor v is the visibility that could be obtained just cooling the system further to 1.4 K,⁴⁸ in order to get truly lifetime-limited linewidths $(FWHM=1/(2πτ1)≃40$ $MHz)$⁠. It is, therefore, important to understand what is limiting v to a value smaller than 1. Although the polarization optics have been characterized with an extinction ratio of 1/1600, the experimenter introduces an error in setting the relative orientation between the two arms. Another contribution could stem from the not-perfect purity of the single photon Fock state. This can be inferred from a measurement of the $gHBT(2)(τ)$ in the HBT, reported, e.g., in Fig. 2(d), yielding $g̃HBT(2)(0)=A0/AN=0.008±0.008$⁠. Based on Ref. 49, we then estimate the mean wave packet overlap of the single photon component M_S, in the case of negligible dephasing and assuming full distinguishability for the rest of the input state,⁴⁹ as $MS=v+14RT(1−g̃(2)(0))−1≃97%$⁠. Such value obviously depends on the filtering efficiency. In fact, the presence of a residual component of distinguishable photons $(1−α)$⁠, given by the portion of the emission in the PSB leaking through the notch filter, degrades the expected maximum visibility to⁵⁰ $α2$⁠. According to the fluorescence spectrum measured before and after the filter (reported in the supplementary material), roughly 98% of the light overlaps with the ZPL. In the optimal condition of zero dephasing but with the same filtering window, one can hence expect a visibility of 96%, which is just what has been estimated above.

Scanning the laser repetition rate and hence effectively the delay between the two photon wave packets, different sets of data are recorded, and the corresponding values for the visibility are plotted in Fig. 3(a), together with the derived theoretical curve. Using our model to estimate the error on the visibility, we observe that the main contribution to its uncertainty is given by the error in estimating the delay line (⁠$±0.2 ns$⁠), that determines an uncertainty on V equal to about 4%. TPI has been measured also as a function of the delay in the unbalanced MZI and the distance among interfering particles in number of photons. In Fig. 3(b), the $gHOM(2)(τ)$ using parallel polarizations for a delay of 125.0 ± 0.2 ns (around 25 m) is reported. It is worth noticing that the visibility starts to decrease only around such time delay [see Fig. 3(c)], which corresponds to about 30 times the photon wave packet extension. Remarkably, as reported in the visibility graph in Fig. 3(c), we observe no relevant drop in TPI also between photons separated by up to 10 fluorescence cycles. As an example, Fig. 3(d) shows the coincidences histogram for the case of interference between photons separated by five fluorescence cycles. Here a delay $Δt$ of 125 ns was employed with a laser repetition rate of 40 MHz. Such a remarkable spectral stability is in agreement with the negligible spectral diffusion reported for CW operation both here and in the literature.^38,39

In conclusion, in this paper we demonstrate triggered generation of highly indistinguishable single photons from a single organic dye molecule under non-resonant pulsed excitation. The results are obtained for emitters in a sub-micrometric environment, without the help of any photonic resonance, and using only a 0.4 nm-wide spectral filter to select the emission. A HOM interference visibility of more than 75% is reported, limited by the residual dephasing present at the operating temperature of 3 K. A visibility of 96% is expected for the very same experiment in the case of 1.5 kelvin operation. We also find that HOM visibility does not show relevant reduction for photons separated by up to 125 ns (equivalent to 30 times the wave packet duration) and by up to 10 fluorescence cycles. Such a spectral stability is of major interest for applications involving multiple photons, such as linear optical quantum computing, where temporal demultiplexing is a typical strategy to increase the number of available resources. The source presented in this paper shows a brightness at detector limited to around 2%, corresponding to a brightness at first lens (N.A. = 0.67) of around 5%. Hence, in the perspective of implementation for quantum applications, integration of the emitter with photonic devices becomes mandatory. A photonic resonance can modify both the radiation pattern and the spectral distribution of the emission, in order to bring the source brightness to the state-of-the-art level.⁵¹ In this respect, the same type of system has been shown to be particularly suitable for the integration in hybrid photonic structures.^52,53

See the supplementary material that contains (i) theoretical calculation of the function used to fit the coincidences histograms under pulsed operation; (ii) calibration of the response function of the electronic chain for the collection of the coincidences histograms; (iii) emission spectrum of the molecule under investigation and considerations concerning PSB contribution in the photon stream selected by the Notch filter.