1 INTRODUCTION

Photon pairs (biphotons) are constitute a two-photon state of light with a high degree of nonclassical correlations between, e.g., their detection times and energies of photons in a pair [1]. Due to these properties, sources of photon pairs provide a reliable foundation for the development of new fields of quantum technologies including quantum visualization [2, 3], quantum optical coherence tomography [4], quantum spectroscopy and nonlinear microscopy [5, 6], and optical quantum memory and quantum communications [7, 8].

Carrier frequencies of correlated photon pairs can be strongly different, which significantly expands the capabilities of the applications listed above, in particular, in spectroscopy. For this reason, methods for the generation of photon pairs at frequencies in various spectral bands, e.g., in the visible and near infrared bands [9, 10], as well as in the optical and terahertz bands [11, 12], have been actively developed in recent times. Furthermore, the generation of biphotons with frequencies in the visible and telecommunication spectral bands is one of the important approaches in the creation of quantum entanglement between remote sites of a quantum network and in the implementation of distributed quantum computing [10].

The most widespread modern methods for the generation of photon pairs are based on the use of spontaneous parametric down-conversion in crystals with a quadratic nonlinear susceptibility \({{\chi }^{{(2)}}}\) [13–16] and spontaneous four-wave mixing in optical fibers with cubic nonlinear susceptibility \({{\chi }^{{(3)}}}\) [17–21]. An important step toward the application of developed methods is the implementation of such sources on an optical platform ensuring their smallness and scalability. The optical fiber used as a nonlinear medium for generation is promising for the practical implementation of this approach. In this case, the generation of biphotons occurs in a small core of the fiber over the entire its length, which assists to compensate a lower order of nonlinearity of the quartz core [22] and ensures direct matching with other optical fibers. Experimental capabilities of existing optical fiber solutions [23] will allow one to transfer sources beyond large experimental setups and to implement them in the form of miniature tools, which will obviously expand their practical applications.

The dispersion of the fiber in the biphoton source should ensure phase matching between photons of exciting laser radiation (pump) and generated photons. It is nontrivial to implement this matching for photons with spectral lines in the visible and infrared (telecommunication) bands; in particular, this matching cannot be ensured with standard single-mode optical fibers.

At the same time, micro/nanofibers [24, 25] and photonic-crystal fibers [26, 27] provide wide possibilities for phase matching and an increase in the efficiency of generation of photon pairs due to the controlled variation of the geometrical parameters of these fibers. The generation of photon pairs near wavelengths of 0.6 and 0.85 μm [24], as well as of 0.9 and 1.3 μm [25], was demonstrated in narrow fibers under pulsed pump. The authors of [19] demonstrated the possibility of generation of visible–telecommunication photon pairs with wavelengths near 0.5 and 1.6 μm in photonic-crystal fibers under pumping by femtosecond laser pulses and showed that a biphoton wave packet in this case is generated in a single spectral mode, i.e., is in a factorizable state. It is noteworthy that there are no experimental studies of the possibility of the generation of visible–telecommunication photon pairs in photonic-crystal fibers with cw pump, which is highly relevant for practical applications of sources on an integrated platform because low-power pump lasers can be of interest for use in small devices [28].

2 GENERATION OF PHOTON PAIRS IN OPTICAL FIBERS UNDER CONTINUOUS-WAVE PUMPING

Spontaneous four-wave mixing is a parametric process based on the third-order nonlinear susceptibility \({{\chi }^{{(3)}}}\) of the fiber core, where two pump photons are transformed into two daughter photons with different carrier frequencies. These daughter photons are conditionally called signal and idler photons. The appearance of this phenomenon requires the energy conservation and phase matching conditions, which are written in the form [29]

$${{\omega }_{{\text{s}}}} + {{\omega }_{{\text{i}}}} = 2{{\omega }_{{\text{p}}}},$$
(1)
$$\Delta k = {{k}_{{\text{i}}}}({{\omega }_{{\text{i}}}}) + {{k}_{{\text{s}}}}({{\omega }_{{\text{s}}}})-2{{k}_{{\text{p}}}}\left( {{{\omega }_{{\text{p}}}}} \right) + 2\gamma {{P}_{{\text{p}}}} = 0.$$
(2)

Here, ωp, ωs, and ωi (kp, ks, and ki) are the frequencies (wavenumbers) of the pump, signal, and idler waves, respectively; Pp is the peak pump power, γ = \(2\pi {{n}_{2}}{\text{/}}\lambda {{A}_{{{\text{eff}}}}}\) is the fiber nonlinearity coefficient, where \({{n}_{2}}\) is the nonlinear refractive index, \({{A}_{{{\text{eff}}}}}\) is the effective area of the fiber mode, and \(\lambda \) is the pump wavelength. According to Eq. (2), the dispersion of the optical fiber plays an important role in spontaneous four-wave mixing, controlling the spectral characteristics of generated photons. In this context, the possibility of creation of air microholes at the design of photonic-crystal fibers made it possible to fabricate waveguides with various group velocity dispersion profiles, which cannot be implemented in standard optical fibers.

Photon pairs generated at the output of the fiber are described as a biphoton state [30]

$${\text{|}}\Psi \rangle \propto \iint d{{\omega }_{{\text{s}}}}{\kern 1pt} d{{\omega }_{{\text{i}}}}F({{\omega }_{{\text{s}}}},{{\omega }_{{\text{i}}}})\hat {a}_{{\text{s}}}^{ + }({{\omega }_{{\text{s}}}})\hat {a}_{{\text{i}}}^{ + }({{\omega }_{{\text{i}}}}){\text{|}}{{0}_{{\text{s}}}}\rangle {\text{|}}{{0}_{{\text{i}}}}\rangle ,$$
(3)

where |ωs, ωi〉 is the state of a photon pair and F(ωs, ωi) is the joint spectral amplitude; the absolute value of this amplitude squared is the probability of the generation of a photon pair with frequencies ωs and ωi. The joint spectral amplitude depends on the pump spectral envelope and the phase matching condition as [30]

$$F({{\omega }_{{\text{s}}}},{{\omega }_{{\text{i}}}}) = \alpha ({{\omega }_{{\text{s}}}} + {{\omega }_{{\text{i}}}}) {\text{exp}}\left( {\frac{{i\Delta \beta L}}{2}} \right){\kern 1pt} {\text{sinc}}\left( {\frac{{\Delta \beta L}}{2}} \right),$$
(4)

where L is the length of the optical fiber and \(\alpha ({{\omega }_{{\text{s}}}}\, + \,{{\omega }_{{\text{i}}}})\) is the pump spectral envelope. The joint spectral amplitude determines both the spectral composition of generated photons and their frequency–time correlation properties. The inverse Fourier transform of the joint spectral amplitude allows one to obtain the second order correlation function in the time representation \({{G}^{{(2)}}}(\tau )\), where the width of the time profile specifies the correlation time Tc, which is important in practical applications.

Generation rate of photon pairs at the output of the optical fiber source r is proportional to the square of the peak pump power \(r \propto \gamma P_{{\text{p}}}^{2}\) [18]. The pulsed pump power Pp is usually several orders of magnitude higher than the cw pump power at the same average power \(\tilde {P}\). As a result, a much higher generation rate of photon pairs in the case of pulsed pumping could be expected according to the quadratic dependence. However, the generation rate of photon pairs in the case of pulsed pumping behaves \(r \propto P_{{\text{p}}}^{2}{\text{/}}S = {{\tilde {P}}^{2}}S\), where S is the pulse–pause ratio of radiation of used pulsed sources. Correspondingly, at the same average pump power, the generation rate under pulsed pumping increases linearly with the pulse–pause ratio of pump pulses rather than quadratically with the peak pump power. In particular, for laser pulses with the duration τp ≈ 1000 fs and the repetition frequency Rp = 76 MHz, S ~ 104. This pulse–pause ratio can be compensated by increasing the cw pump pulse by a factor of 100. However, in increase in the pump power is usually accompanied by the proportional increase in the number of noise photons in the output signal caused by spontaneous Raman scattering [18, 22] or by pumping radiation itself, which partially penetrates to measuring channels even after spectral filtering. Thus, the search for study of optical waveguides ensuring the efficient generation of photon pairs by femtosecond laser pump pulses with a microwatt average power open a way to the transfer of these waveguides to a platform with low-power diode pumping with a milliwatt power, which significantly reduces the dramatic effect of noise photons.

3 EXPERIMENTAL SETUP

A single-mode photonic-crystal fiber (NL-PM 750, NKT Photonics) with two zero dispersion wavelengths near 750 and 1270 nm was used as a nonlinear medium for the generation of biphotons. This photonic-crystal fiber has a small effective mode area Aeff ≈ 2 μm and, correspondingly, a high nonlinearity coefficient of γ ≈ 95 (W km)–1 [31]. The spectral dependence of the group-velocity dispersion \({{\beta }^{{(2)}}}(\lambda )\) is shown in Fig. 1a. To illustrate the possibilities listed above, Fig. 1b presents the dependence of wavelengths of generated photon pairs on the laser pump wavelength, which was calculated from the group-velocity dispersion. This dependence is often called the phase matching curve because it is determined from Eqs. (1) and (2). The preliminary numerical analysis of spontaneous four-wave mixing regimes in terms of the presented dispersion dependence showed that biphoton states with wavelengths near λi = 1.6 μm (idler photon) and λs = 0.5 μm (signal photon) can be generated in this photonic-crystal fiber under pumping at a wavelength of λp = 0.8 μm. The presented results clearly demonstrate the possibility of the generation of photon pairs in the visible and telecommunication bands, as well as the variation of their wavelengths in a wide spectral range by changing the pump field wavelength. Using this possibility, one can exactly place the wavelength of the idler photon to the standard telecommunication band near 1.55 μm by means of a small change in λp.

Fig. 1.
figure 1

(Color online) (а) Spectral dependence of the group-velocity dispersion used to generate photon pairs in the optical fiber. (b) Wavelengths of photon pairs at the output of the optical fiber versus the wavelength of input laser radiation (pumping). Red circles mark the wavelengths of photons for pumping with the wavelength near 800 nm.

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The sketch of the experimental setup for the generation of photon pairs is presented in Fig. 2. Generation was ensured either by femtosecond laser pulses from a Ti:sapphire laser (Mira-HP, Coherent) at a wavelength of 800 nm with the pulse repetition frequency Rp = 76 MHz (see Fig. 2a) or by a cw diode laser (LuxX 808-140, Omicron-Laserage) with almost the same wavelength (see Fig. 2b). Pump radiation was introduced in the photonic-crystal fiber through an aspherical lens and its power was controlled at the output of the fiber.

Fig. 2.
figure 2

(Color online) Sketch of the experimental setup for the generation of photon pairs: (Ti:sapphire laser) pulsed pump laser, (IF) Faraday insulator, (M) mirror, (DG) diffraction grating, (P) polarizer, (HW) half-wave plate, (PCF) photonic-crystal fiber, (L1, L2) aspherical lenses, (DM) system of optical filters ensuring the separation of biphotons into two channel and their filtering from undesired noise photons, (D1) single-photon detectors of the visible band, (D2) single-photon detectors of the infrared band, and (TDC) time-to-digital converter.

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The duration of femtosecond pulses from the Ti:sapphire laser was ~150 fs. Further, laser pulses passed through an optical system including a diffraction grating, lens, and several mirrors, where the pulse duration was increased to ~500 fs by means of the filtering of side components of the spectral band. The frequency band of laser pulses at the output of the Ti:sapphire laser was controlled by additional spectral filtering.

Signal and idler photons generated due to spontaneous four-wave mixing at the output of the fiber were separated into two channel using a long-wavelength filter with a cutoff wavelength of 850 nm, which operated as a dichroic mirror. A cascade of four short-wavelength optical filters with cutoff wavelengths of 600, 700, and 750 nm (two filters) was placed in the signal measurement channel in order to ensure a high optical density of about 21 at the pump wavelength λp = 800 nm. A cascade of three long-wavelength optical filters with cutoff wavelengths of 1050, 1100, and 1400 nm was placed in the idler measurement channel to ensure comparable optical densities at the pump wavelength taking into account the aforementioned filter separating biphotons into two channels. Single photons in the signal and idler channels were detected by a single-photon avalanche photodiode detector for the visible band (SPDM Count NIR, Laser Components) and a single-photon avalanche photodiode detector for the infrared band (ID230, IDquantique), respectively. Output signals from these detectors were guided to a time-to-digital converter (ID801, IDquantique), where they were numerically processed with a built-in programmable integrated logic circuit. As a result, the photon and coincidence rates in the detectors were calculated. Spectra of photons were measured by a monochromator with a detector based on a cooled CCD matrix (S7031-1006S, Hamamatsu).

4 RESULTS AND THEIR DISCUSSION

First, we measured spectra of signal photons λs, generated at the output of the photonic-crystal fiber for under pulsed and cw pumping; they are shown in Figs. 3a and 3b, respectively. The comparison of Figs. 3a and 3b shows that the spectral shape of signal photons is almost the same in both pumping regimes, except for a small decrease in the width of the spectral band from ≈1.0 nm in the pulsed regime to ≈0.8 nm in the cw pumping regime. Using the energy conservation law for the spontaneous four-wave mixing process, one can easily determine the wavelength of idler photons λi from the wavelengths λs and λp. In particular, at the pump wavelength λp = 800 nm and the wavelength of signal photons λs = 529 nm, idler photons will have a wavelength of λi = 1640 nm.

Fig. 3.
figure 3

(Color online) (a, b) Spectra of signal photons, (c, d) photon count rates in the signal and idler channels, and (e, f) histograms of coincidences for biphotons generated under (a, c, e) pulsed and (b, d, f) cw pumping.

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Further, we measured the photon count rates for output radiation (see Fig. 2d) for both pumping regimes. The photon count rates Ns and Ni in the signal and idler channels, respectively, and their coincidence rate Nsi at the output of the photonic-crystal fiber were measured as functions of the average pump power \(\tilde {P}\). The measured photon count rates Ns and Ni are presented in Figs. 3c and 3d, respectively. As shown in these figures, the dependence for signal photons is approximated with a high accuracy by the function \({{N}_{{\text{s}}}}(P) \propto {{\tilde {P}}^{2}}\), which is characteristic of the spontaneous four-wave mixing process. At the same time, the dependence for idler photons in the pulsed pumping regime is approximated by a different function \({{N}_{{\text{i}}}}(P) = a\tilde {P} + b{{\tilde {P}}^{2}}\) (a is a constant), where a comparatively small linear term \(a\tilde {P}\) (at \(\tilde {P}\) = 4 mW, it is one third of the entire signal) is most probably due to noise photons in the idler channel. The most probable source of noise can be residual radiation from spontaneous Raman scattering in the core of the waveguide. However, as clearly seen in Fig. 3d, the count rate in the idler channel Ni under cw pumping is described by an almost linear dependence \({{N}_{{\text{i}}}} \propto \tilde {P}\). Such a dependence indicates that the main contribution to the output signal from the idler channel in this regime comes from noise photons. An increase in the noise contribution in this case can be attributed to an increase in the average cw pump power necessary for the generation of photon pairs detected in the experiment.

The nonclassical properties of generated radiation are clearly manifested in histograms for the time dependence of coincidences. Figures 3e and 3f present these histograms measured in the pulsed and cw pumping regimes, respectively. In the former case, the coincidence histograms have the form of separate columns with a sharp peak, as seen in Fig. 3e. This peak corresponds to coincidences of detector counts caused by the generation of time-correlated photons. The ratio of the height of this peak to the value at the measurement time Texp gives the coincidence rate Nsi. The other peaks on the histogram correspond to random coincidences between detector counts. Although the level of noise in the idler channel in the case of cw pumping is higher, coincidences between channels are still clearly seen. However, the histogram in this case has the form of a single peak whose envelope is shown in Fig. 3f. Thus, the key differences of the generation of photon pairs in the photonic-crystal fiber under cw pumping from that under pulsed pumping are a higher noisiness in the idler (long-wavelength) measurement channel and the presence of the single peak in the coincidence histogram.

To characterize the performance and efficiency of our source, we calculated the second order cross-correlation function g(2)(τ) in terms of the measured coincidence rates Nsi between channels. In the case of pulsed pumping, the function g(2)(τ) can be determined by the formula [32, 33]

$${{g}^{{(2)}}}(\tau ) = \frac{{{{N}_{{{\text{si}}}}}(\tau ){{R}_{{\text{p}}}}}}{{{{N}_{{\text{s}}}}{{N}_{{\text{i}}}}}},$$
(5)

where Nsi(τ) is the rate of two-photon coincidences at a given time delay between two events τ = t1 – t2. We measured the function g(2)(τ) at τ = 0 when it reaches a maximum. Figure 4 presents the results of the measurements. The maximum value \(g_{0}^{{(2)}}\) = 550 is reached at the average pump power as low as 0.3 mW; in this case, the peak power in the pump pulse is ~8 W. We note that the generation rate of photon pairs at the output of the fiber in this case is much higher than the coincidence rates detected in the experiment (r \( \gg \) Nsi) because not all photon pairs are detected because of optical losses in the measurement channels of the experimental setup. These optical losses are due both to a lower efficiency of gathering of radiation with wavelengths in different spectral bands at the output of the photonic-crystal fiber and to a low quantum efficiency of several percent of the used detector in the idler channel. Since all measured count rates are proportional to the generation rate of biphotons at the output of the fiber, this generation rate r can be estimated as r = Rp/\(g_{0}^{{(2)}}\). Thus, the maximum coincidence rate in the pulsed pumping regime Nsi = 5 Hz (\(g_{0}^{{(2)}} = 20\)) corresponds to the generation rate at the output of the fiber r ≈ 4 MHz. It is worth noting that the photon detection rate in the proposed scheme can be further increased by increasing the gathering efficiency of interband radiation generated at the output of the fiber, by matching the fiber with standard fibers, and by using a detector with a higher quantum efficiency in the idler channel.

Fig. 4.
figure 4

(Color online) Coincidence rates of photon pairs and the second-order correlation function \({{g}^{{(2)}}}(0)\) versus the average femtosecond pump pulsed power (\(\tilde {P} \leqslant 4\) mW) and the average cw diode pump power (\(\tilde {P} \geqslant 5{\kern 1pt} \) mW).

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In the case of cw pumping, we calculated the correlation function g(2) by the formula [28]

$${{g}^{{(2)}}}(\tau ) = \frac{{{{N}_{{{\text{si}}}}}(\tau )}}{{{{N}_{{\text{s}}}}{{N}_{{\text{i}}}}{{T}_{{\text{b}}}}}},$$
(6)

where Tb is the time interval in which coincidences are detected.

According to Eq. (6), the function g(2)(τ) decreases proportionally to increasing Tb. Consequently, to maximize \(g_{0}^{{(2)}}\), it is necessary to ensure the detection of coincidences in the minimally accessible time interval. One of the factors limiting this time interval in the experiment is the time resolution (jitter) of the detectors used in the measurement channels. The single-photon detectors used in our experiments have the time resolutions Tj, s = 1.0 ns and Tj, i = 0.2 ns in the signal and idler channels, respectively, leading to the corresponding broadening of the time distribution Nsi(τ) in Fig. 3f. The use of detectors with smaller Tj values, e.g., ones based on superconducting nanowires with Tj ~ 20 ps, will make it possible to significantly increase \(g_{0}^{{(2)}}\) values. Thus, to exclude the influence of the detectors, we calculated \(g_{0}^{{(2)}}\) with Tb = 81 ps, which corresponds to the discretization time of the used time-to-digital converter.

Experimental dependences Nsi and \(g_{0}^{{(2)}}\) values calculated from them are also presented in Fig. 4. Coincidence rates Nsi were determined by summing counts in the time interval of 1.6 ns in the histograms. The maximum value \(g_{0}^{{(2)}}\) ≈ 2000 is reached at the cw pump power as low as 5 mW. The found \(g_{0}^{{(2)}}\) value is comparable with the data obtained with the use of superconducting detectors in [28] under cw pumping of a fiber with a suspended core. The maximum coincidence rate in the cw pumping regime is Nsi = 0.4 Hz (\(g_{0}^{{(2)}}\) = 560). Using the relation between Nsi and r obtained at the same setup in the pulsed pumping regime at a low noise signal, the maximum generation rate r for the cw pumping regime can be estimated at ~0.3 МHz.

To summarize, we have demonstrated the generation of interband biphotons in a photonic-crystal fiber under cw pumping. The photonic-crystal fiber with an unusual dispersion profile with two zero-dispersion wavelengths has ensured the generation of photon pairs in the visible and telecommunication bands at wavelengths near 0.5 and 1.6 μm. The silica core in combination with a small mode area of the fiber has simplified the use of a low-power cw laser for pumping and has ensured the generation rate comparable with those reached with pulsed femtosecond laser sources. The latter circumstance is of high interest for the fabrication of small integrated sources of biphotons.