Recent progress in single-photon and entangled-photon generation and applications

Shigeki Takeuchi

doi:10.7567/JJAP.53.030101

1. Introduction

In the late 18th century, "light" was considered only as a form of propagating electro-magnetic waves with a wavelength of around 1 µm. In 1905, the concept of "light quanta", was first introduced by Einstein to explain black-body radiation.¹⁾ This concept was experimentally verified by the discovery of Compton scattering in 1923,²⁾ and the word "photon" was first used for light quanta by Lewis in 1926.³⁾ However, at this time, the technologies were far beyond the generation and observation of single photons.

The situation started to change in 1956. Hanbury-Brown and Twiss (HBT) proposed and measured the intensity correlation of light.⁴⁾ In their experiment, light was divided by a half mirror and detected by two photomultiplier tubes in a system that is now called the HBT interferometer (Fig. 1), and they observed the higher-order interference effect. They used this method to determine the angular size of the star Sirius.

**Fig. 1.** HBT interferometer. The number of coincidence detection events between the two photon detectors D₁ and D₂ is measured by changing the optical delay $c\tau$ .
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$c\tau $ — **Fig. 1.** HBT interferometer. The number of coincidence detection events between the two photon detectors D₁ and D₂ is measured by changing the optical delay $c\tau$ .
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In 1977, Kimble et al.⁵⁾ first observed evidence of single-photon emission from sodium atoms using the HBT interferometer. They observed anticorrelation, which is now commonly considered as the signature of single-photon generation.

In the late 1960s, the generation of "entangled photons" (quantum–mechanically correlated photons) attracted attention. Even though the nonlocality in quantum physics was first discussed by Einstein et al.⁶⁾ and Bohr,⁷⁾ it was unclear if it could be tested by experiments. In 1964, Bell showed that the nonlocality of quantum mechanics can be tested experimentally using a pair of entangled photons. The experimental verification of the violation of Bell's inequality was performed using a pair of photons generated by the cascade emission of photons from single atoms.⁸^,⁹⁾ However, there were two problems in this method. The first one was the small flux of entangled photons into photon detectors because the photons were emitted at a 4π solid angle. The second one was the imperfect correlation of photons due to the decoherence in the cascade emission process.

In the late 1980s, Mandel and his collaborators performed a series of important experiments on the generation of entangled photons using spontaneous parametric downconversion.¹⁰^,¹¹⁾ This technology markedly enhanced research using single and entangled photons, and has been widely used for quantum information experiments for more than two decades.

In this article, we review the recent progress in single-photon/entangled-photon sources and their applications, mainly based on the works of the author and his collaborators. In Sect. 2, we explain the basic concepts required for the understanding of this article: the HBT interferometer, intensity correlation function, photon number distribution, parametric downconversion, and Hong–Ou–Mandel interference. In Sect. 3, we examine heralded single-photon sources using parametric downconversion for quantum key distribution and quantum information processing. In Sect. 4, solid-state single-photon sources efficiently coupled to single-mode fibers (SMFs) will be discussed. In Sect. 5, recent developments of entangled-photon sources, especially ultrabroadband-frequency entanglement generation, are introduced. Finally, we summarize the topics with future prospects in Sect. 6. Quantum information processing and quantum metrology using these photon sources can also be found elsewhere.¹²⁾

2. Basic concepts

In this section, we briefly introduce some concepts required to understand this review article: the HBT interferometer, intensity correlation function, photon number distribution, parametric downconversion, and Hong–Ou–Mandel interference. For deeper understanding of these concepts, please read textbooks on quantum optics.¹³^–¹⁵⁾

2.1. HBT interferometer

Figure 1 shows the experimental setup used to measure the correlation of light intensities obtained at times t and $t + \tau$ . The incident light is divided by a half mirror and detected by two photon detectors, D1 and D2. By changing the optical path length difference $c\tau$ between the half mirror and the two detectors, the frequency of the coincidence detection of the photons in D1 and D2 is recorded. This experimental setup is called the HBT interferometer, named after HBT who first invented this setup. Note that $c\tau$ can be changed via the time delay for the electric pulses from the detectors instead of the optical path length difference.

The second-order correlation function $g^{(2)}(\tau )$ is given as

$\begin{equation} g^{(2)}(\tau) = \frac{\langle\hat{E}^{(-)}(t)\hat{E}^{(-)}(t + \tau)\hat{E}^{(+)}(t + \tau)\hat{E}^{(+)}(t)\rangle}{\langle\hat{E}^{(-)}(t)\hat{E}^{(+)}(t)\rangle\langle\hat{E}^{(-)}(t + \tau)\hat{E}^{(+)}(t + \tau)\rangle}, \end{equation} \tag{ 1 }$

where $\hat{E}^{( + )}(t)$ and $\hat{E}^{( - )}(t)$ are the electric field operators for propagating and counter-propagating components, respectively. $g^{(2)}(0)$ indicates how many photons tend to exist at the same time. For an n-photon number state $|n\rangle$ and $\tau \to 0$ ,

$\begin{equation} g^{(2)}(0) = 1 - \frac{1}{n}. \end{equation} \tag{ 2 }$

Note that $g^{(2)}(0)$ is smaller than 1 for an arbitrary n. In particular, $g^{(2)}(0) = 0$ for the single-photon state $(|1\rangle )$ . The state of light with $g^{(2)}(0) > 1$ ( $g^{(2)}(0) < 1$ ) is called photon bunching (photon antibunching). $g^{(2)}(0) = 1$ for a coherent state of light, and $g^{(2)}(0) = 2$ for thermal light.

2.2. Photon number distribution

The photon number distribution, which is a plot of the probabilities $P(n)$ of having n photons in an optical pulse, is important for evaluating the photon statistics. The coherent state with the average number of photons $|\alpha |^{2}$ is written by

$\begin{equation} |\alpha \rangle = e^{-|\alpha|^{2}/2}\sum_{n = 0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle. \end{equation} \tag{ 3 }$

The probability of having n photons in coherent states follows a Poisson distribution:

$\begin{equation} P_{\alpha}(n) = e^{-|\alpha|^{2}}\frac{|\alpha|^{2n}}{n!}. \end{equation} \tag{ 4 }$

Figure 2(a) shows the photon number distribution of weak coherent light with an average photon number $\bar{n} = 1$ . $P(1)$ , the probability of having one photon in a pulse, is 0.37, which is the same as $P(0)$ , the probability of having zero photons. The non-negligible probabilities of having two or more photons can also be seen. These components could be used by eavesdroppers in the case of quantum key distribution, or would result in errors in the case of photonic quantum computation. Figure 2(b) shows the photon number distribution of a single-photon source, which is ideal for quantum key distribution and quantum information processing. In this case, $P(n)$ is 1 only for $n = 1$ , otherwise it is 0.

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2.3. Parametric downconversion

When a pump laser beam is incident to a nonlinear crystal, a photon in the pump laser beam with angular frequency $\omega _{\text{p}}$ and wave vector $\boldsymbol{{k}}_{\text{p}}$ is converted to a pair of photons (a signal photon with wave vector k_s and angular frequency ω_s, and an idler photon with wave vector k_i and angular frequency ω_i) when low energy conservation and low momentum conservation (phase-matching condition),

$\begin{equation} \hbar\omega_{\text{p}} = \hbar\omega_{\text{s}} + \hbar \omega_{\text{i}}, \end{equation} \tag{ 5 }$

$\begin{equation} \hbar\boldsymbol{{k}}_{\text{p}} = \hbar\boldsymbol{{k}}_{\text{s}} + \hbar\boldsymbol{{k}}_{\text{s}}, \end{equation} \tag{ 6 }$

are satisfied (Fig. 3). This process is called spontaneous parametric downconversion (SPDC).

For SPDC using bulk nonlinear crystals, there are two types of phase matching according to the combination of polarizations of the pump beam and daughter photons. In the case of type-I phase matching, two daughter photons have the same polarization (ordinary or extraordinary ray). For type-II phase matching, the polarization of two daughter photons is orthogonal (ordinary and extraordinary rays). Historically, Burnham and Weinberg experimentally proved that the photons in a pair are generated simultaneously;¹⁶⁾ however, this technology did not attract much attention until the series of experiments by Mandel and his collaborators.

2.4. Hong–Ou–Mandel interference

The famous experiment of two-photon interference was performed using SPDC.¹⁰⁾ Suppose that two "indistinguishable" photons are incident to a beam splitter with a reflectivity of 50% (Fig. 4). If the photons behaved as classical particles, they would be output from each of the output ports with a probability of 50%. However, Hong, Ou, and Mandel theoretically suggested that this probability should be 0 owing to the quantum interference; the probability amplitudes of the two processes ("both of the photons are reflected" and "both of the photons are transmitted") have the same amplitude but opposite signs, and thus completely destructively interfere. They experimentally demonstrated this phenomenon using pairs of photons generated via SPDC. This phenomenon is called Hong–Ou–Mandel (HOM) interference and has become a general tool for quantum information processing.

**Fig. 4.** Hong–Ou–Mandel two-photon interference. When two indistinguishable photons are incident to a half mirror, the two cases shown on the right side do not occur owing to the quantum interference.
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An example of an HOM interference fringe is shown in Fig. 5. In an ideal case, the coincidence count is zero at $\tau = 0$ . However, in actual experiments, it cannot reach zero. The quality of two-photon interference is usually evaluated by the visibility $V = (I_{\text{max}} - I_{\text{min}})/I_{\text{max}}$ . $V = 1$ means perfect two-photon interference, V decreases as the photons become distinguishable, and $V = 0$ when the two-photon interference effect cannot be observed.

Now let us discuss the shape of the HOM dip between the photons in a pair generated via parametric downconversion. The two-photon state generated via degenerate parametric downconversion with a monochromatic pump (frequency $\omega _{0}$ ) can be written as¹⁰^,¹⁷⁾

$\begin{equation} |\Psi\rangle = \int\phi(\omega_{1}, \omega_{0} - \omega_{1})|\omega_{1}, \omega_{0} - \omega_{1}\rangle\,d\omega_{1}, \end{equation} \tag{ 7 }$

where $\phi (\omega _{1},\omega _{2})$ is a weight function with a peak at $\omega _{1} = \omega _{2} = \omega _{0}/2$ . Assuming that $\phi (\omega _{0}/2 + \omega ,\omega _{0}/2 - \omega )$ is a real and symmetric Gaussian in $\omega$ with bandwidth $\Delta \omega$ , the coincidence count is given as¹⁰^,¹⁷⁾

$\begin{equation} N_{\text{c}} = C(1 - e^{-(\Delta\omega\delta\tau)^{2}}), \end{equation} \tag{ 8 }$

where we assumed that the transmittance and reflectance of the beam splitter are 1/2, the coincidence resolving time is on the order of nanoseconds and much longer than the photon correlation time (typically femtoseconds–picoseconds), and C is a constant. This means that the width of the HOM interference dip is given by the inverse of the bandwidth of the parametric fluorescence spectrum.

One interesting feature of the HOM dip is that the width of the dip does not change with even-order dispersion in one of the optical paths of the interferometer.¹⁸⁾ Thus, it has been suggested that HOM interference can be used for optical coherence tomography robust against dispersion.¹⁹⁾ To achieve a higher resolution in $c\delta \tau$ , parametric fluorescence with a wider spectrum bandwidth $\Delta \omega$ is important. This issue will be discussed in Sect. 5. The analysis of the effect of the general higher-order dispersion on the shape of the HOM dip suggests that the HOM "dip" can be changed to the HOM "bump" when appropriate odd-order dispersion exists in the optical path.¹⁷⁾

3. Heralded single photons using parametric downconversion

Since a signal photon and an idler photon are generated at the same time in SPDC, the detection of the idler photon "heralds" the existence of the signal photon.²⁰⁾ Such a photon source is called a "heralded single-photon source (HSPS)" and is useful for quantum key distribution (QKD),²¹⁾ photonic quantum information, and metrology. Here, we introduce a pulsed HSPS with small higher-photon-number components for QKD and a highly indistinguishable HSPS for photonic quantum information processing.

3.1. HSPSs for QKD

QKD enables us to share a series of random numbers, which can be used as a secret key for cryptography, between distant parties without being observed by eavesdroppers. For the physical implementation of QKD protocols, weak coherent pulses (WCPs) have been used;²²^,²³⁾ however, the security is strictly limited even when the average number of photons per pulse is optimized, since the probability of having two photons in one pulse P(2) cannot be decreased without reducing P(1), the probability of having one photon in one pulse.²⁴⁾

QKD using HSPSs is one approach to solve this problem. There have been reports about such sources and QKD systems using CW pumping;²⁵^–²⁸⁾ however, there were some problems since the signal photons are produced randomly in time. First, it is very difficult to synchronize the sender and receiver stations. Second, such a system may not be used for future systems equipped with quantum relays or quantum repeaters, because the accuracy of the timing/position of photons is indispensable for Bell state analysis using two-photon interference.

We realized the first HSPS by pulsed SPDC at 1550 nm.²⁹⁾ The experimental setup used is shown in Fig. 6. A mode-locked Ti:sapphire laser (Spectra Physics Tsunami) associated with a second-harmonic generation unit produces 150 fs pulses at 390 nm at a rate of 82 MHz. A Pellin-Broca prism removes any remaining infrared radiation from the pump laser. The beam is focused into a β-barium borate (BBO) crystal cut for the type-I phase-matching condition. Signal photons with a wavelength of 520 nm and idler photons (1550 nm) are separated using a dichroic mirror oriented at 45°; the signal is reflected and the idler is transmitted. The signal photons are then coupled to a SMF and detected by a single-photon-counting module (SPCM), which creates the "trigger signal (heralding signal)". The idler photons are coupled to another SMF and evaluated by a laboratory-built photon detector [InGaAs avalanche photodiode (APD)], which was triggered by the trigger signal.

In this experiment, the single photons are created in a very narrow time window defined by the pump and coupled to an SMF, making this source directly usable for QKD. $P_{\text{h}}(1)$ for the source, the probability of a single photon existing in the idler pulse when a heralding signal was detected, was 0.187, and $P_{\text{h}}(2)$ , the probability of two photons existing in the idler optical pulse when a heralding signal was detected, was $2.4 \times 10^{ - 3}$ , which is much smaller than the Poisson distribution for the same $P_{\text{h}}(1)$ . The actual source rate was 2.16 × 10⁵ triggers s⁻¹.

3.2. QKD system using HSPS

Then, we applied the pulsed HSPS for the first demonstration of a BB84-based QKD experiment performed over 40 km on a fiber,³⁰⁾ which exceeds the achievable distance with QKD based on the original BB84 protocol with WCPs. An HSPS using nondegenerate parametric downconversion with femtosecond laser pumping was used in our experiment.²⁹⁾ Figure 7 shows the schematic of the QKD system. The photon source was coupled to a one-way QKD system realized with two unbalanced Mach–Zehnder interferometers using two planar light waveguides (PLCs) and phase modulators (PMA and PMB). Alice and Bob cryptographic systems are each driven by a computer-controlled module (EA and EB). Alice's and Bob's computers can communicate via an internet connection for reconciliation and quantum bit error rate (QBER) measurement. There are two optical channels made of dispersion-shifted fibers (DSF) dedicated to timing synchronization between these modules. The first channel is used to transmit the heralding signal from the HSPS source using a pulsed laser (idQuantique id300) and an optical receiver (Anritsu MH9S8). The HSPS signal from the SPCM passes through a digital delay line (Stanford Research DG535) to EA and then to another similar digital delay line. The first delay line imposes a dead time of 1 µs and the second one is used to adjust the delay between Alice and Bob. The second channel carries an 82 MHz clock signal from the femtosecond pump laser using a special optical emitter (Gravitron WSM-2) and a receiver (Gravitron WRM-2) with very low jitter. This clock is used as a time reference for our QKD system.

**Fig. 7.** QKD system using a HSPS. PC: polarization controller, PBS: polarizing beam splitter, PLC: planar light waveguide, APD: avalanche photodiode operated in the Geiger mode. Thick blue lines indicate the DSF channel, the red line indicates the HSPS heralding signal, and the green line indicates the 82 MHz clock signal. Black lines represent the driving signal for phase modulation and APD gated operation. Dotted black lines denote output signals from the APD, and dashed-dotted black lines denote the exchange of signals between personal computers and EA and EB.
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For this experiment, we used an improved HSPS from our previous source.²⁹⁾ $P_{\text{h}}(1)$ was increased from 0.18²⁹⁾ to 0.296³⁰⁾ by the improvement of the lens inserted in the setup and also by a better alignment procedure. With the strong suppression of the two-photon component [ $P_{\text{h}}(2) = 1.48 \times 10^{5}$ ] achieved after the sender station, we successfully performed a QKD experiment with a QBER of 4.23% and a secret key creation rate of 0.16 bit/s, suggesting unconditional security.³¹⁾ Since we adopted a pulsed HSPS, the system clocks (82 MHz) at the sender and receiver were well synchronized and are suitable for future quantum relays and quantum repeaters.

3.3. HSPSs for photonic quantum information

Photons are also one of the most useful particles for quantum information processing because they can be transmitted ("flying qubits") without decoherence, they can be easily detected by conventional technologies, and they can be manipulated with high precision using developed optical devices. However, the lack of interaction between photons has hindered their use in quantum information processing. To overcome this problem, Knill, Laflamme, and Milburn (KLM)³²⁾ found that it is possible to effectively generate the required interaction between single photons using HOM interference,¹⁰⁾ which is two-photon interference at a beam splitter. On the basis of this concept, controlled-NOT gates for photonic qubits were proposed³³^,³⁴⁾ and demonstrated.³⁵^–³⁸⁾ Furthermore, small-scale optical quantum circuits,³⁹^,⁴⁰⁾ including the original heralded controlled-NOT operation proposed by KLM,⁴¹⁾ have been demonstrated recently. HOM interference is also important for generating NOON states⁴²⁾ used in quantum metrology⁴³⁾ and quantum lithography.⁴⁴^,⁴⁵⁾

For these applications, it is very important to develop highly indistinguishable single-photon sources that can generate high-visibility HOM interference. In the previous demonstrations, the HOM visibilities (86³⁹⁾ and 90%⁴¹⁾) of the HSPSs strongly limited the performance of the quantum circuits.⁴⁶⁾

We theoretically and experimentally investigated the conditions for realizing highly indistinguishable HSPSs using SPDC.⁴⁷⁾ We considered a case in which photons are generated at the same wavelength by type-I downconversion.

Figure 8 shows the causes of the degradation in HOM visibility. For a two-photon-interference experiment between the photons in a pair [Fig. 8(a)], both of the photons are generated at the same time [Fig. 8(b)], and thus the temporal modes can be matched perfectly. However, in the cases where two HSPSs are used as the photon sources [Fig. 8(c)], the timing of photon pair generation fluctuates for two reasons: (1) The limited width of the pump laser pulses [Fig. 8(d)]. For example, the photon pairs may be generated at the back side (near the input surface of the crystal) of the pump pulse in BBO1 and at the front side (near the output) in BBO2. (2) The group velocity mismatch (GVM) between the pump pulse and the daughter photons. The daughter photons may be generated anywhere inside the crystal. For example, the daughter photons are generated near the input surface of the pump laser beam of BBO1 and near the output surface in BBO2 in Fig. 8(e). Since the daughter photons propagate faster than the pump light owing to GVM, the two photons generated from these crystals exhibit timing jitter.

In a previous analysis,⁴⁸⁾ photon pairs were assumed to be uniformly generated in a crystal; i.e., the conversion efficiency was assumed to remain constant as the pump pulse propagates through the crystal. Here, we extend the theory by considering that the number of photon pairs generated at a point inside the crystal increases linearly as the pump propagates within the crystal.⁴⁹⁾ We have found that this effect becomes significant for crystal lengths of about 1.5 mm or longer.

We also experimentally investigated the effect of GVM in crystals by using crystals with different lengths (0.7 and 1.5 mm); this has not been experimentally investigated previously (Fig. 9). The experimentally obtained HOM dips between the heralded single photons under four different conditions are shown in Fig. 10. We succeeded in obtaining a high visibility of 95.8 ± 2% after compensating for the reflectivity of the beam splitter (observed value: 95.2 ± 2%) using a 0.7-mm-long β-BBO crystal and 2 nm band-pass filters [Fig. 10(a)]. To the best of our knowledge, this visibility is equal to the highest visibility ever reported;⁵⁰⁾ however, in our case, the coincidence rates were higher by a factor of 4.

**Fig. 9.** Experimental setup of HOM interferometer between independent photons. SHG: second-harmonic generation, PMF: polarization maintaining fiber, SMF: single-mode fiber, HWP (QWP): half (quarter)-wave plate, SPCM: single-photon-counting module.
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**Fig. 10.** High-visibility HOM dips between daughter photons. The HOM dips are between Signal 1 and Idler 1 in Fig. 9 under four different conditions: (a) 0.7 mm BBO with 2 nm band-pass filters, (b) 0.7 mm BBO with 4 nm band-pass filters, (c) 1.5 mm BBO with 2 nm band-pass filters, and (d) 1.5 mm BBO with 4 nm band-pass filters.
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As an alternative approach, high-visibility heralded single photons were realized by controlling the modal structure of the photon pair emission.⁵¹⁾ By using a special phase-matching condition where the frequency correlation between idler and signal photons does not exist, in principle one can prepare heralded single photons directly in the pure state. A visibility of 94.4% for a HOM-interference dip between two heralded single photons was obtained by this method.⁵¹⁾

4. Solid-state single-photon generators efficiently coupled to SMFs

The collection of fluorescence photons from single nanoemitters is of fundamental importance in fields such as quantum information and biological sensing. For example, single nanoemitters such as quantum dots (QDs) and color defect centers in nanodiamonds can be employed as single-photon sources, which are crucial devices for realizing quantum cryptography in future secure communication networks.⁵²^–⁵⁸⁾ The fiber-based sensing of fluorescent nanoparticles with ultrahigh sensitivity has also been intensively studied.⁵⁹^,⁶⁰⁾ All these applications require the ability to collect as many photons as possible and to efficiently couple these photons into single-mode optical fibers.

To enhance the fluorescence collection efficiency of photons from these nanoemitters, various photon collection optics have been investigated, including solid immersion lenses,⁶¹^,⁶²⁾ photonic crystal fibers,⁶³⁾ and tapered fibers.⁶⁴^–⁶⁶⁾ Tapered fibers are particularly promising in view of their high collection efficiencies and their ability to directly couple fluorescence photons into a SMF. A theoretical study has predicted that it should be possible to couple 28% of the total emission from gas atoms around a tapered region into SMF outputs.⁶⁷⁾ Preliminary experimental results for coupling between tapered fibers and solid-state nanoemitters have been reported.⁶⁸^–⁷⁰⁾ However, the efficient coupling of fluorescence from single solid-state nanoemitters into tapered fibers was not reported. This is mainly due to the following two difficulties: (1) The efficient coupling of nanoemitters into tapered fibers requires ultrasmall taper diameters of the order of 300 nm (approximately half the emission wavelength) as in Ref. 67; however, it is currently difficult to fabricate such ultrathin tapered fibers with low transmission loss.⁷¹⁾ (2) Ultrathin tapered fibers suffer from rapid transmission degradation.⁷²⁾

Recently, we have demonstrated the highly efficient coupling of fluorescence from single QDs into SMFs by using ultrathin tapered fibers.⁷³⁾ We succeeded in producing tapered fibers with a diameter of 300 nm and a transmittance of 90%, and preserved their transmittance by conducting all experiments in a dust-free environment. We used CdSe/ZnS QDs (Evident; crystal size, 9.6 nm) as solid-state nanoemitters. The QDs were dissolved in toluene solution and the ultrathin tapered fibers with a diameter of 300 nm were dipped in these solutions to deposit these nanoemitters directly on their tapered surfaces. They were then mounted on a piezoelectric transducer (PZT).

Figure 11 schematically depicts the experimental setup. A green He–Ne laser beam (λ = 543.5 nm) was used as the excitation light. It was circularly polarized to eliminate the polarization dependence of the fluorescence intensity of single QDs. It was focused on the tapered fiber by a 100× objective whose numerical aperture (NA) was 0.8. The fluorescence was collected by the same objective and filtered from the excitation laser light using a dichroic beam splitter and additional filters. We used a single-mode optical fiber as a pinhole of a confocal setup. The fluorescence was then detected by an avalanche photodiode (denoted APD1). The fluorescence channeled into the tapered fiber was detected by another APD (APD2) after filtering the excitation light. The two APDs (APD1 and APD2) together with a time-correlated single-photon counting module (Becker & Hickl, SPC-130) were used for second-order photon correlation measurements in order to discriminate single emitters by observing the antibunching correlation. Figures 12(a) and 12(b) show scanning images of a single QD on a 300-nm-diameter tapered fiber, which were measured through the objective and tapered fiber, respectively. We observed single-step fluorescence blinking at this spot. The fluorescence detected through the tapered fiber [25 kcps in Fig. 12(b)] was larger than that detected through the objective [18 kcps in Fig. 12(a)]. The total photon fluorescence that could be detected from both ends of the tapered fiber was 50 kcps, which is almost three times greater than the photon count detected through the objective. Figure 12(c) shows a second-order photon correlation histogram of the single QD, where the coincidence events between APD1 (through the objective) and APD2 (through the fiber) were recorded in terms of time difference. It clearly shows antibunching at time 0 with a value of $g^{2}(0) = 0.096$ , demonstrating that the single QD is a single emitter. The excited-state lifetime of this QD was determined to be 35.4 ns from this second-order photon correlation data. Figure 12(d) shows an excitation intensity dependence plot of the fluorescence detected by APD2. It also shows that the fluorescence saturates as the excitation intensity increases. We measured the second-order photon correlation and fluorescence saturation for three other single QDs on the same 300 nm tapered fiber and found that the mean lifetime ( $\tau _{\text{qd}}$ ) and mean saturated photon count ( $C_{\infty }$ ) were 29.6 ns and 592 kcps, respectively. From these data, we concluded that we were able to couple 7.4 ± 1.2% of the total emitted photons from single CdSe/ZnS nanocrystals into tapered fibers.

**Fig. 11.** Schematic diagram of the experimental setup. OBJ: objective, BS: dichroic beam splitter, SMF: single-mode fiber, APD: avalanche photodiode. The tapered fiber was fixed to a glass substrate using UV adhesives and the substrate was mounted on a PZT. To measure fluorescence spectra, the fiber connections to the APDs were switched to a spectrometer.
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**Fig. 12.** Scanning images of QDs on tapered fiber measured through (a) objective and (b) tapered fiber. The excitation intensity was 28 W/cm². A single QD (indicated by white arrows) is visible in both images. The low-intensity blurred spots that appeared in the lower part of (a) are other QDs, which are not clear in (b) simply because their relative intensity to the photon count of the brightest spot is lower in (b). (c) Histogram of the second-order photon correlation of the single QD. (d) Excitation intensity dependence plot of the photon counts of this QD.
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This efficient photon collection technique is highly promising for nanoparticle sensing and single-photon sources. A recent report suggested that the coupling efficiency can reach 22% for a tapered fiber with a diameter of 350 nm.⁷⁴⁾

Using this technique, we have also recently reported an ultrabright fiber-coupled diamond-based single-photon source⁷⁵⁾ where 689 ± 12 kcps single photons are coupled directly to a SMF. In our experiment, an integrated nanodiamond-tapered-fiber system is investigated; this system consists of ultrathin single-mode tapered fibers with a diameter smaller than 273 nm that were equipped with nanodiamonds smaller than 150 nm containing nitrogen vacancy centers (NVs).

5. Entangled-photon generation

Quantum entanglement plays a key role in various systems, such as quantum computation⁷⁶⁾ and metrology.⁴³⁾

Mandel and his collaborators also succeeded in the generation of a polarization-entangled state using SPDC.¹¹⁾ In this experiment, the polarization of one of the photons in a pair (type-I phase matching) was rotated and the polarized photon was injected into a 50 : 50 beam splitter. When two photons were output from each of the output ports, the state was entangled: $(|H\rangle _{\text{s}}|V\rangle _{\text{i}} + |V\rangle _{\text{s}}|H\rangle _{\text{i}})/\sqrt{2}$ .

Later, improved sources of entangled photons in polarization were proposed that used SPDC. Examples of widely used schemes using bulk crystals proposed by Kwiat and his collaborators employed a type-II BBO crystal⁷⁷⁾ and a pair of type-I BBO crystals.⁷⁸⁾ More examples of methods used to generate two-photon polarization-entangled photon pairs can be found elsewhere.⁷⁹⁾

Furthermore, interest in multidimensional photonic quantum states has grown recently because of their potential for realizing new types of quantum communication protocols.⁸⁰^–⁸²⁾ As a means of realizing multidimensional states, photons in Laguerre–Gaussian (LG) modes have been attracting a lot of attention.⁸³^–⁸⁷⁾ LG modes form an orthogonal basis set of paraxial solutions to the wave equation and have screw phase dislocations $\exp (il\varphi )$ , where l is referred to as the azimuthal mode index. Photons in LG modes have orbital angular momentum $l\hbar$ .⁸⁸^,⁸⁹⁾

For this topic, we proposed a method for manipulating the quantum state of photons in LG modes using the Gouy phase shift⁹⁰^,⁹¹⁾ and experimentally realized a method to observe the quantum correlation in LG modes of photons.⁹²⁾

Another way to realize multidimensional photonic quantum states is to use polarization states of higher photon numbers, namely, n-photon polarization states with $n > 1$ . For example, the two-photon state $|2\rangle$ has three polarization states $|HH\rangle$ , $|VV\rangle$ , and $|HV\rangle$ , which form a set of orthogonal basis states. We have succeeded in the generation and verification of genuine entangled states between two-photon polarization states for the first time.⁹³⁾

Here, we introduce another example of multidimensional entangled states, an ultrabroadband entangled photon.

5.1. Broadband entangled photons toward monocycle biphotons

The temporal entanglement of two photons can show simultaneity in their arrival times to within their time-bandwidth product, which cannot occur for classical light.⁹⁴⁾ If two photons arrive within several femtoseconds of each other, with one cycle equal to the inverse of its center frequency (1 cycle = 1/ν_c), potential applications include an increase in the resolution of quantum optical coherence tomography¹⁹^,⁹⁵⁾ and in the rate of two-photon absorption or sum-frequency generation (SFG).⁹⁶^–⁹⁸⁾

To realize ultrashort temporally entangled two-photon states, the parametric fluorescence must have a bandwidth $\Delta \nu$ that is comparable to its center frequency $\nu _{\text{c}} = c/\lambda _{\text{c}}$ , e.g., 282 THz at a center wavelength of λ_c = 1064 nm. In bulk nonlinear crystals, broadband phase matching within a small solid angle of emission can only occur over a small interaction length, thereby degrading the photon pair generation rate. To date, such pair emission has been limited to a bandwidth of Δν = 154 THz (253 nm) at λ_c = 702 nm by introducing a temperature gradient in a crystal.⁹⁹⁾ We have recently achieved Δν = 73 THz (160 nm) at λ_c = 808 nm using two bulk crystals with their optic axes tilted relative to each other.¹⁰⁰⁾

Harris¹⁰¹⁾ proposed a method to compress the temporal correlation of biphotons to a single optical cycle of light (e.g., 3.5 fs at a wavelength of 1064 nm). The biphotons propagate collinearly with a large frequency bandwidth owing to an aperiodically poled nonlinear optical crystal, i.e., a chirped quasi-phase-matched (QPM) device.¹⁰²⁾ A photon pair that is phase-matched at each poling period possesses different combinations of two frequencies, resulting in an ultrabroadband two-photon state after passing through the crystal. After spatially splitting the photons and applying phase compensation to one path and a time delay to the other, the temporal width of the biphotons was measured by SFG. Nasr et al. have demonstrated a bandwidth increase using such a device at λ_c = 808 nm to achieve Δν = 188 THz in a two-photon interference experiment.¹⁰³⁾ Mohan et al. have shown spectral broadening at λ_c = 1064 nm, resulting in Δν = 166 THz under a collinear condition.⁹⁵⁾ Sensarn et al. have successfully performed a "chirp and compress" experiment at λ_c = 1064 nm and Δν = 40 THz at a nondegenerate lobe.¹⁰⁴⁾

However, there are some difficulties. Harris' original proposal involves the use of a collinear condition and requires a very large bandwidth (Δν = 3000 nm), i.e., a QPM device with large chirping (50%) for 1.2 cycle temporal correlation when a pump laser at 532 nm is used (see Sect. 2 for detail). The realization of such a device is still beyond current technologies. In addition, the dichroic mirror used to separate the fluorescence into lower and higher frequencies also causes additional problems, such as an anomalous dispersion near the cutoff wavelength.

Recently, we have proposed an alternative method using noncollinear SPDC.¹⁰⁵⁾ We showed theoretically that the chirp rate required to achieve monocycle temporal correlation can be substantially reduced compared with that in the case of collinear SPDC.¹⁰¹⁾ We have experimentally demonstrated octave-spanning (790–1610 nm) noncollinear parametric fluorescence from a 10% chirped MgSLT crystal using both a superconducting nanowire single-photon detector and a photomultiplier tube. Our numerical calculation suggested that the temporal correlation of two photons can be a 1.2 cycle with perfect chirp compensation and a 7.2 cycle with moderate chirp compensation using a prism pair, assuming the experimentally observed spectra.

Figure 13 shows the schematic of the chirped quasi-phase-matched device developed by Professor Kurimura at NIMS.¹⁰⁵⁾ The chirped-QPM device is based on 1.0 mol % magnesium-doped stoichiometric lithium tantalite (MgSLT). Its ferroelectric spontaneous polarization is aperiodically switched by electric field poling with patterned electrodes. Its cross section is 0.5 × 0.5 mm² and its lengths is 20 mm. This device is designed for a cw pump at a wavelength of 532 nm. The operation temperature is controlled by a Peltier unit and stabilized at 293 K.

**Fig. 13.** Schematic of the chirped quasi-phase-matched device. The poling period $\Lambda (z)$ is linearly chirped inside the device from $\Lambda _{0} = 8.000$ to 8.825 µm. The daughter photon pairs are emitted in a direction slightly tilted from the pump beam propagating direction.
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**Fig. 13.** Schematic of the chirped quasi-phase-matched device. The poling period $\Lambda (z)$ is linearly chirped inside the device from $\Lambda _{0} = 8.000$ to 8.825 µm. The daughter photon pairs are emitted in a direction slightly tilted from the pump beam propagating direction.
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Figure 14(a) shows the parametric fluorescence spectra obtained under the noncollinear condition.¹⁰⁵⁾ The fluorescence was measured using a superconducting nanowire single-photon detector (SNSPD) operated at 4.2 K, consisting of a meander structure of a NbN nanowire. Note that the detection efficiency of the SNSPD strongly depends on the wavelength. We calibrated the detection efficiency of the SNSPD and other losses at the optical components used in the experimental setup. The spectrum of the parametric fluorescence spans over the wavelength range from 790 to 1610 nm, which is in reasonable agreement with the numerical prediction (solid line). Figure 14(b) shows the spectrum in the longer-wavelength range in detail, confirming the emission even up to 1610 nm.

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6. Summary and future outlook

In this article, we have reviewed the recent progress in single-photon/entangled-photon emitters and their applications: HSPSs using parametric downconversion and their application to QKD, highly indistinguishable HSPSs, SMF-coupled solid-state single-photon sources, and ultrabroadband-frequency entanglement generation.

As we have already seen, these single-photon sources/entangled-photon sources are indispensable not only for photonic quantum circuits, photonic quantum simulators, and QKD, but also for innovations in the fields of quantum enhanced sensing and metrology.

The challenges that must be overcome for these sources strongly depend on the target applications. At the moment, the most important challenge for single-photon sources for photonic quantum circuits and simulators is to achieve high efficiency, high indistinguishability, and small high-photon-number components at the same time. To date, the HSPS using SPDC provides sufficient indistinguishability for experimental demonstrations of functional photonic quantum circuits; however, such a source cannot satisfy the requirement for both high efficiency and small high-photon-number components. People are trying to overcome this problem by integrating multiple HSPSs. As an alternative approach, considerable effort has been devoted to realize solid-state single-photon sources that generate indistinguishable single photons. When the visibility of two-photon interference using these photons becomes comparable to the best value (96%) obtained using HSPSs with parametric downconversion,⁴⁷⁾ the research on optical quantum technology will be markedly accelerated. For quantum-enhanced metrology and sensing, a broadband entangled-photon source with high photon-pair flux will be a key source.

Acknowledgements

I would like to thank the main co-authors of the works presented in this article, Keiji Sasaki, Jun-ichi Hotta, Hideki Fujiwara, Holger F. Hofmann, Ryo Okamoto, Masazumi Fujiwara, Alexandre Soujaeff, Masayuki Okano, Shanthi Subashchandran, Hong-Quan Zhao, Kenji Tsujino, Daisuke Kawase, Masato Tanida, Akira Tanaka, Kiyota Toubaru, Tetsuya Noda, Sunao Kurimura, Hwan Hong Lim, Mitsuo Takeda, Yoko Miyamoto, Mitsuru Matsui, Toshio Hasegawa, Tsuyoshi Nishioka, Toyohiro Tsurumaru, Toru Hirohata, Peiheng Wu, Jian Chen, Lin Kang, Labao Zhang, Oliver Benson, Tim Schröder, and Jeremy L. O'Brien for their contribution, assistance, and suggestions. I would also like to thank all other collaborators and colleagues, and friends for their kind support. The works reported here were supported in part by the Precursory Research for Embryonic Science and Technology (PRESTO) "Field and Reactions" and "Light and Control" Projects of Japan Science and Technology Corporation (JST), Core Research for Evolutionary Science and Technology (CREST) by JST, Grants-in-Aid from the Japan Society for the Promotion of Science (JSPS), the Strategic Information and Communications R&D Promotion Programme (SCOPE) by the Ministry of Internal Affairs and Communication, the Quantum Cybernetics Project by JSPS, the FIRST Program by JSPS, the Project for Developing Innovation Systems by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Global Center of Excellence Program by MEXT, Mitsubishi Electric Corporation, the International Communication Foundation, Daiwa Anglo-Japanese Foundation, and the Research Foundation for Opto-Science and Technology.

Recent progress in single-photon and entangled-photon generation and applications

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Abstract

1. Introduction