## Abstract

Photonic entanglement is one of the key resources in modern quantum optics. It opens the door to schemes such as quantum communication, quantum teleportation, and quantum-enhanced precision sensing. Sources based on parametric down-conversion or cascaded decays in atomic and atom-like emitters are limited because of their weak interaction with stationary qubits. This is due to their commonly broadband emission. Furthermore, these sources are commonly in the near-infrared such that quantum emitters in the blue spectral region, such as ions or many defect centers, cannot be addressed. Here, we present a sodium-resonant (589.0 nm) and narrow-band (14 MHz) degenerate entanglement source based on a single molecule. A beam-splitter renders two independently emitted photons into a polarization-entangled state. The quality of the entangled photon pairs is verified by the violation of Bell’s inequality. We measure a Bell parameter of $S=2.26\pm 0.05$. This attests that the detected photon pairs exceed the classical limit; it is reconfirmed by quantum-state tomography and an analysis of the raw detector counts, which result in a value of $S=2.24\pm 0.12$. The tomography shows fidelity of 82% to a maximally entangled Bell state. This work opens the route to background-free solid-state entanglement sources which surpass the probabilistic nature of the commonly used sources and are free from unwanted multi-photon events. The source is ideal for combination with stationary qubits such as atoms, ions, quantum dots, or defect centers.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Photon entanglement, besides its very fundamental relevance [1], now has become a crucial resource in quantum technology for the implementation of various protocols. Quantum communication [2], quantum teleportation [3], and quantum randomness [4,5] utilize entanglement as a key ingredient. Nowadays, certain quantum implementations can be rendered to be independent of their specific experimental implementation or measurement devices. The recent implementation of loophole-free tests of quantum mechanics [6,7] allows certification of, for example, the generation of random numbers [5] or device-independent quantum communication [8].

Early sources of photonic entanglement were based on the cascaded decay of atoms [9]. In the 1990s, sources based on non-linear crystals were developed [10]. Their handling is convenient, and their collection efficiency and spectral brightness have been optimized over the years [11,12]. Fundamentally, these sources suffer from their large spectral bandwidth and their probabilistic nature, which does not allow for the generation of a “push-button” photon pair. The latter problem cannot be addressed by increasing the pump power, as the probability to generate unwanted multi-photon events increases quadratically. Furthermore, sources based on parametric down-conversion are often limited to the (infra-)red region of the optical spectrum. Only a few sources in the visible region of the spectrum exist [13]. These are crucial for interfacing entangled photons to certain stationary quantum nodes, such as ions. Also, such sources are desirable for biological applications since the majority of the relevant optical transitions are in the band from 450–650 nm.

Entanglement sources based on single quantum emitters or other non-linearities have been experimentally investigated in the past decade [14–16]. These experiments are also commonly limited to the near-infrared region of the spectrum due to their implementation in semiconductor quantum dots. One alternative is the generation of photonic entanglement by mode-mixing of a pair of single photons on a beam-splitter (see Fig. 1). This has been proposed [17,18] and later experimentally realized with pair of photons from a non-linear process [19,20]. Later, single photons from individual emitters were entangled [21–23]. To date, this scheme has not yet become competitive with the just-mentioned sources in terms of brightness and measurements of the raw entanglement.

## 2. EXPERIMENT

The central building block for the entanglement source presented here is a single-photon source based on a single molecule. It consists of a single organic molecule which is embedded in a so-called Shpol’skiĭ matrix of $n$-tetradecane (Aldrich). The molecule under study was dibenzanthanthrene [DBATT, Fig. 2(a), CAS: 188-42-1, W. Schmidt, Greifenberg]. The sample is prepared by dissolving a small amount of dye in the solvent. Therefore, the sample is initially in a liquid form and sandwiched between a glass cover-slide and a solid-immersion lens made of ${\mathrm{ZrO}}_{2}$ ($n=2.18$, 3 mm $\varnothing $, A.W.I. Industries). Then the sample is shock-frozen in the cryostat, and the experiments are conducted at an operating temperature of 1.4 K. The molecule is excited with a narrow-band dye ring laser (Coherent 899-29) around 581 nm. The excitation power for the introduced experiments amounts to approximately 1 mW. The light emitted from the molecule is captured after atomic Faraday filtering [spectrum in Fig. 2(c)] by a single-mode fiber [24]. The coupling efficiency from a free-space single-photon stream to the fiber amounts to approximately 20%. To achieve two independent single-photon streams which are subsequently entangled, a beam-splitter and two fibers with a relative delay of $\mathrm{\Delta}t=75\text{\hspace{0.17em}}\mathrm{ns}$ are introduced. This temporal delay is more than 5 times longer than the coherence length of the photons, and allows the emission in both arms to be considered independent. One single molecule is laterally isolated out in an extended sample, and spectrally selected from an inhomogeneous distribution. Dibenzanthanthrene has been a workhorse in a variety of quantum optical experiments [25,26]. A simplified level scheme is depicted in Fig. 2(b). The system is excited into a higher-laying vibrational level and emits photons on the zero-phonon line. This excitation scheme increases neither spectral diffusion nor the observed linewidth [26].

One important advantage of molecules against other quantum emitters is their chemical tunability across the full visible spectrum. The combination with atomic systems such as sodium, potassium, or rubidium vapor has enabled the hybridization of single molecules and atoms [25]. Such experiments open the possibility to store single photons in an atomic vapor, and purify the single-photon emission utilizing an atomic-vapor-based Faraday filter [24,27]. To ensure the spectral alignment of the single photons to the sodium resonance, a Faraday filter based on atomic sodium is used [24]. This filter suppresses broadband background light which might be present due to neighboring molecules or spurious fluorescence in the experimental configuration. A 100 mm long evacuated glass cylinder with atomic sodium (Triad Technologies, CO, AR-coated windows) is heated to 153°C and placed in a solenoid which supplies a longitudinal magnetic field of approximately 220 mT. This configuration is placed between crossed polarizers. Their extinction ratio for off-resonant photons amounts to approximately 40 dB, which is limited by the wavefront distortion of the cell windows. The filter provides a Doppler-width limited passband with a near-unity transmittance ($>90\%$) for the selected molecular single photons [Fig. 2(d)].

Once a suitable molecule is selected, the photon statistics of the atomic
filtered emission is analyzed by a Hanbury Brown and Twiss recording. This
characterizes the *single particle nature* of
our source; the anti-bunching is presented as raw data in
Fig. 2(e), and shows
multi-photon contributions of less than 4%. Previous studies show that the
lifetime limited spectral width $\mathrm{\Delta}{\omega}_{0}$ amounts to $2\pi \times 12\u201350\text{\hspace{0.17em}}\mathrm{MHz}$ ([24]). This matches well to the recorded anti-bunching curve
(${\tau}_{\mathrm{sp}}=11.4\text{\hspace{0.17em}}\mathrm{ns}$, $\mathrm{\Delta}\omega =2\pi \times 14\text{\hspace{0.17em}}\mathrm{MHz}$), which has been acquired under low
excitation powers.

Due to the almost ideal spectral properties and the negligible spectral diffusion, the photons also undergo efficient Hong–Ou–Mandel (HOM) interference [28]. This is experimentally verified in the experimental configuration shown on the right of Fig. 2(c). The experiments are based on a free-space configuration with single-mode fiber coupling. The optical configuration is designed to be mechanically very stable (granite table). The entire configuration is aligned prior to the experiments with a laser in the HOM configuration. For this, the laser is locked to the sodium ${D}_{2}$-line; then the interference visibility is measured to be larger than 99.9%. With the single-photon stream, the photon-correlation function is analyzed among the detectors. This paper presents the raw clicks of all coincidence measurements. No background correction was performed. The anti-bunching curves in all plots are fitted with a two-level rate equation model and take the electric jitter by the detection system (photon counters and time tagger) into account. The electrical jitter was independently measured in the same configuration utilizing a short (ps) pulse laser. All fits were produced in the computer algebra system Mathematica (ver. 10.4.1, Wolfram Research). The relative delay among the time-tagged recordings on the detectors was determined by the laser pulses.

The single-molecule source exhibits a raw HOM visibility of 94% under
continuous-wave (CW) excitation, comparable to our previous results [26]. The data of the selected molecule is
presented in Fig. 2(f). This
curve fully resembles its anti-bunching curve shown in Fig. 2(e), but is based on the
anti-correlation carried by the so-called N00N state,
$\frac{1}{\sqrt{2}}(|20\u27e9+|02\u27e9)$ [26].
Such experiments have been performed with a variety of single-photon
sources [28,29], and allow us to quantify the suitability of these
sources in all-optical gates [30].
How well single photons can undergo interference depends to a good part on
the degree of first-order coherence of the involved photons,
$|{g}^{(1)}(\tau )|$. For two coinciding single photons, it is
introduced as the two-photon *coherentness*
${\mathcal{C}}^{2}={|{g}^{(1)}(0)|}^{2}$ [26].
This entity is usually measured as the HOM visibility of the photons.

## 3. THEORY

A closely related experiment to the well-studied HOM interference is the
supply of *orthogonal* polarized photons into
an entangling beam splitter. This is known as the Shih–Alley
configuration [19,20]. Here, no interference occurs, and
therefore this measurement is commonly used as a reference for the HOM
measurement. The state is defined by $|{\mathrm{\Psi}}_{\mathrm{out}}\u27e9=(i/2)(|HV\u27e9|0\u27e9-|0\u27e9|VH\u27e9)-(1/2)(|H\u27e9|V\u27e9-|V\u27e9|H\u27e9)$, where the latter term is the
$|{\mathrm{\Psi}}^{-}\u27e9$ Bell state. This is a polarization-entangled
photonic state, and is based on a coincidental photon detection at remote
locations with 50% success probability per trial. The entanglement is
formed, although the photons do not interact or interfere at the
beamsplitter, but rather due to the mode-mixing of the two incoming
photons. The final output state has also been analyzed in a quantum eraser
configuration [26]. A further
analysis as outlined in the following allows us to characterize the
polarization-entangled state which is formed from the molecular
photons.

One of the intriguing properties of the generated state is its “spooky action at a distance” [31]. This “non-locality” can be tested with Bell’s inequality [32], if a local hidden variable model is sufficient for its description. Here we use it in the rigid form of Clauser, Horne, Shimony, and Holt [33]. This is based on a correlation measurement which determines visibility among the parallel and orthogonal photon coincidences between two remote parties:

$N$ denotes the number of photon pairs measured according to their polarization ($h$ and $v$).Non-locality can be evaluated when the $E$ correlator is measured in appropriate bases or coordinate systems. In the Bell test, party $A$ measures in two bases labeled as $\alpha $ and ${\alpha}^{\prime}$. Correspondingly, the other party $B$ measures in $\beta $ and ${\beta}^{\prime}$. The Bell parameter $S$ is determined by these $2\times 2$ correlator measurements:

As for the experimental implementation, the input polarization of the single photons into the entangling beam splitter is changed from the prior HOM experiment. The photons now impinge orthogonally onto the central beam splitter. The output ports are monitored by single-mode fiber-coupled single-photon detectors in two orthogonal polarization bases ($h$ and $v$). With these $2\times 2$ attached detectors, four different types of photon coincidences occur between the output ports: $hh$, $hv$, $vh$, and $vv$. By utilizing Eq. (1), we determine the $E$-correlator. To allow this in any linear polarization bases, half-waveplates are introduced before detection [see Fig. 3(a)]. With two different waveplate configurations on each side, the 16 relevant correlation measurements are obtained.

## 4. RESULTS

Following this procedure to determine $S$, we perform the measurement in the four introduced polarization settings. The experimental outcomes are depicted in a matrix, where each quadrant represents one measurement setting and corresponds to one $E$-correlator. The raw recorded (cross-)correlation curves are shown in Fig. 3(b) for a time-range of $\pm 30\text{\hspace{0.17em}}\mathrm{ns}$. To achieve an identical signal-to-noise for all four quadrants, each of the polarization settings is recorded for 3 min. Then, the motor-controlled waveplates are rotated and the next quadrant is measured. This procedure was repeated for approximately 2 h. Subsequently, the $E$-correlators, and therefore the Bell parameter $S$, are computed [33].

As earlier reported in the literature [21], the detected photon coincidences can be corrected for background fluctuations, and a Bell parameter can be determined. Here, we fit the acquired correlation curves shown in Fig. 3(b). This is performed with an anti-bunched correlation function, which is based on a rate equation model. This function can be used due to the CW excitation, which takes the full temporal behavior into account. For an ideal $|{\mathrm{\Psi}}^{-}\u27e9$ state, we would expect the normalized coincidences as displayed in Fig. 3(c), which would result in a maximal violation of Bell’s inequality. We now determine the four $E$ correlators from the experimental fit. They are $-0.61$, 0.54, 0.57, and 0.54, respectively. This results in an overall Bell parameter of $S=2.26\pm 0.05$. This violates Bell’s inequality by more than five standard deviations, and proves the non-classicality of the photon pairs.

We now take the raw, fully unaltered recorded coincidences into account. This has not been performed before and presents a more rigid analysis for the usable photonic entanglement. For this, the coincidence window is set to a time bin of $\mathrm{\Delta}\tau =300\text{\hspace{0.17em}}\mathrm{ps}$. These raw coincidence counts are listed in Fig. 3(e). They allow us to calculate the raw Bell parameter, which amounts to $S=2.24\pm 0.12$. This implies that Bell’s inequality is violated by more than two standard deviations, which gives a 96% confidence that a local classical model is unable to explain the result. Interestingly, this resource can be used in quantum optical experiments as is; no further corrections have to be applied. The applied error propagation is equivalent to other recent references [7].

When the coincidence time window is increased from $\mathrm{\Delta}\tau =300\text{\hspace{0.17em}}\mathrm{ps}$, effectively more and more uncorrelated photon pairs are added to the outcome. The Bell parameter degrades, and from a certain point on, the raw data will not violate Bell’s inequality anymore. The relation of the Bell parameter versus the coincidence time window is discussed, and the experimental data is plotted in Supplement 1. The theoretical fit provides an alternative way to extract the Bell parameter. The value for coinciding photons ($\mathrm{\Delta}\tau =0$) amounts to $S=2.2\pm 0.05$.

Essential for this entanglement generation is that the involved photons are like single particles, and yet have the ability to undergo a two-photon interference—a property which represents the wave-particle duality [26]. If a source with such an optimal property is used, a maximally entangled state is generated and the Bell violation will result in an absolute $S$ value of $2/\sqrt{2}$. This is not the case for the presented photon source. One crucial influence is background contributions to the single-photon stream. Spurious photons will contaminate the entanglement among a photon pair, and a single photon source with ${g}^{(2)}(0)>{(\sqrt{2}-1)}^{2}\approx 0.17$ will not be able to reach a Bell parameter beyond $S=2.0$. On the other hand, for a perfect single-photon source, the photon interference ability should reach a value below ${g}_{\mathrm{HOM}}^{(2)}(0)<1-1/\sqrt{2}\approx 0.29$ in a HOM experiment. The corresponding characterization for our single-photon source is shown in Figs. 1(c) and 1(d). Subsequently, this source should violate Bell’s inequality with a value of $S=2.63$. Experimental factors, especially the quality of the involved beam splitters, or a beam shift originating from the waveplates, explain the deviation of the achieved Bell parameter from this theoretical maximum.

## 5. QUANTUM TOMOGRAPHY

This violation of Bell’s inequality proves that a classical theory is not able to explain the outcome. On the other hand, quantum mechanics provides the explanation for this counter-intuitive phenomenon. As for the quantum mechanical description of the Shih–Alley output state, a tomographic measurement characterizes the full polarization state of the photon pairs [34]. Thereby, it provides a comprehensive picture of its quantum mechanical state. For this, the measurements have to include circular polarization; this effectively extends the prior configuration of the Bell test.

The quantum tomography is performed in the same experimental configuration as in Fig. 3(a), which is equipped with additional quarter wave plates in the detection arms. The measurement consists of $3\times 3$ measurement settings. From these configurations, $4\times 4$ correlation curves are taken into account. The derivation of the density matrix is described by the following equations:

Figure 4 shows the real and imaginary parts of the measured density matrix. The real part is compared to an ideal $|{\mathrm{\Psi}}^{-}\u27e9$ state, which should only have components in the $HV$ and $VH$ bases. While the theory predicts a vanishing imaginary part for the $|{\mathrm{\Psi}}^{-}\u27e9$ state, we see that Fig. 4(b) shows some unwanted contributions. We account the deviations to the theoretical predictions to experimental subtleties, e.g., the optical components are non-ideal. Still, the fidelity of the quantum state exceeds 82%, calculated as $F=tr({\widehat{\rho}}_{{\mathrm{\Psi}}^{-}}.{\widehat{\rho}}_{\mathrm{M}})$.

Now that we have determined the density matrix, we can predict any experimental measurement outcome on the photonic quantum state. This spans the previously discussed Bell-type measurement, and it allows to estimate the Bell parameter $S$. It is calculated as $tr(\widehat{S}.{\widehat{\rho}}_{M})=2.2$. This value is fully consistent with the preceding measurements of the Bell test.

A more careful analysis shows that the deviations from $|{\mathrm{\Psi}}^{-}\u27e9$ which might be caused, for example, by imperfect wave plates, could be compensated by an adapted detection. This does not use the original selected linear polarization bases, but another basis which would be ideal with the utilized polarization optics. A theoretical estimation shows that this results in an only slightly larger Bell violation; it still does not match the predicted value, which can be derived from the single-photon nature and the HOM visibility as displayed in Figs. 1(e) and 1(f). This indicates that the optical elements are non-ideal and do not even perform a unitary transform of the photon’s polarization state.

The experimental entanglement generation from true single-photon sources with the aid of linear optics has been pioneered by others [21,23,35]. Usually, other less rigid entanglement criteria have been used. Still, a violation of Bell’s inequality presents a challenge since a fidelity above 78% is required [36]. Furthermore, the generation of photonic entanglement by semiconductor quantum dots often has the alternative of using parametric down-conversion (PDC) sources, because of their comparable emission spectrum. In this near-infrared region, the PDC sources are presently superior in terms of brightness, bandwidth, and ease of handling. Therefore, an entanglement source in the visible range, where the emission wavelength can be chosen deliberately, is unique.

The introduced scheme of entanglement generation forms an alternative to the down-conversion process by a non-linear crystal. The rate of entanglement generation is an important parameter for quantum information processing. As for parametric down-conversion sources, it can be increased with the pump rate, but only to a certain level, since this also increases the risk of generating multi-photon events. This degrades the entanglement purity, since more photon pairs are generated simultaneously. When now, for example, two pairs are generated, the detection of a photon coincidence might not necessarily originate from the same entangled photon pair, and the accidental rate increases. This limits—even from a fundamental point of view—the maximal meaningful rate for PDC sources. A meaningful limit is that, from a trigger pulse, maximally 0.1 photon pairs are generated. Then, already more than 10% multi-photon events occur, which lead to unwanted background. The mode-mixing on a beam splitter opens an interesting alternative: Two deterministic generated single photons will form with a 50% probability of an entangled pair. This scheme does not inherently suffer from background; it surpasses the maximal feasible rate of PDC sources by a factor of 5. Therefore, a source based on the mode-mixing of single photons has the potential to become the entanglement source with the highest spectral brightness and simultaneous negligible background contribution—a prerequisite for an efficient photon–matter interaction, where the photonic source matches to a stationary quantum node.

## 6. CONCLUSION

In conclusion, we have presented a single-molecule-based source of photonic entanglement. The source exhibits a 14 MHz wide emission, which is resonant to the ${\mathrm{D}}_{2}$-line of atomic sodium. It consists of two single photons which are polarization-entangled to a $|{\mathrm{\Psi}}^{-}\u27e9$ state by mode-mixing on a 50:50 beam splitter. These photons are then detected as raw coincidences at remote locations and violate a Bell test with a Bell parameter of $S=2.24\pm 0.12$, more than two standard deviations away from a classical explainable result. This violation is reconfirmed by various fits to the experimental data ($S=2.26$ and $S=2.2$), and by quantum state tomography which shows a fidelity of 82% with an expectation value of $S=2.2$ for the Bell parameter.

A crucial prerequisite for the described effect is an excellent interference ability of the single photons, which is usually proven by a high HOM visibility. For the presented single molecule, the HOM visibility amounts to 94%. The other important factor is the absence of background photons, which here are excellently suppressed with the aid of an atomic Faraday filter. This scheme of entanglement generation can be also implemented with other emitters [21], and will allow for bright, background-free sources of photonic entanglement.

Although entanglement sources based on parametric down-conversion are still more convenient to handle and relax the experimental demands, the introduced source has a number of inherent advantages: the brightness of the presented source, which is to date less than one coincidence per second, can be drastically increased. A first step is to trigger the single-photon emission, which allows for the generation of a few entangled photon pairs per second in our current configuration. Since in this scheme the entanglement rate increases quadratically with the count-rate of the single photons, a high single-photon count-rate is crucial. As for single emitters, this can be achieved using a dielectric cavity [37]. For such a single-molecule, single-photon source, we calculate that about $5\times {10}^{6}$ photon pairs per second will be generated—and still the source exhibits the same spectral width (14 MHz) as before. Multi-photon contamination, which occurs for strongly pumped down-conversion sources, is avoided. The narrow-band and spectral quiescent nature, and the option to modify the spectral position by chemical means, make single molecules an ideal photonic source which covers the entire visible spectrum. This extends to blue single-photon emitters, such as perylene around 440 nm ([38]), where sources based on parametric down-conversion will be limited.

The above evaluation with raw, unnormalized clicks proves that the introduced scheme can be directly applied to a number of experiments which are based on photonic entanglement. This spans implementations, such as quantum cryptography and entanglement distribution. Furthermore, fundamental experiments, including a loophole-free version of the Bell test, are enabled with the introduced source [39]. Due to the spectral alignment to other quantum systems—here with atomic sodium—the source enables memory-assisted quantum communication [40].

## Funding

Deutsche Forschungsgemeinschaft (DFG) (GE2737/5-1); Max-Planck-Gesellschaft (MPG); European Cooperation in Science and Technology (COST) (MP1403).

## Acknowledgment

M. R. conducted the experiment and analyzed the data. I. G. and M. R. wrote the manuscript. I. G. and J. W. supervised the team. All authors discussed the results and contributed to the final manuscript. The authors declare no competing financial interests.

See Supplement 1 for supporting content.

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