Mid-infrared spectrally-uncorrelated biphotons generation from doped PPLN: a theoretical investigation

The mid-infrared (MIR) wavelength range (approximately 2-20 $\mu$m) is of great interest to a variety of scientific and technological applications in sensing, imaging, and communications Tournie and Cerutti (2019); Ebrahim-Zadeh and Sorokina (2008). Firstly, this range contains strong absorption bands of a variety of gases, leads to essential applications in gas sensing El Shamy et al. (2020) and environmental monitoring Chen et al. (2020). Secondly, the MIR region covers the atmospheric transmission window between 3 $\mu$m and 5 $\mu$m. This region with relatively high transparency is beneficial for atmospheric monitoring and free-space communications Bellei et al. (2016). Thirdly, room temperature objects emit light at MIR wavelengths, resulting in novel applications in infrared thermal imaging Tittl et al. (2015); Mancinelli et al. (2017). Fourthly, with the rapid development of optical fiber communications, there is a growing demand to expand the communication wavelengths into the MIR region to increase the communication bandwidth Soref (2015).

All these practical applications mentioned above are based on strong light sources in the MIR region. However, the advantages of the MIR range have not been fully exploited using quantum technology. In order to further improve the sensitivity in those applications, one promising approach is to utilize a single-photon source or entangled photon source in the MIR region. Spontaneous parametric down-conversion (SPDC) is one of the widely used methods to prepare biphotons, which can be used to produce single-photon source and entangled photon source. Previously, several theoretical and experimental works have been dedicated to the development of single or entangled photon source in MIR range from an SPDC process. In 2016, Lee et al. calculated the extended phase-matching properties of periodically poled potassium niobate (PPKN) for single-photon generation at the MIR range Lee et al. (2016). In 2017, Mancinelli et al. performed coincidence measurements on twin photons generated from periodically poled lithium niobate (PPLN) at 3.1 $\mu$m Mancinelli et al. (2017). In 2018, McCracken, et al. numerically investigated the mid-infrared single-photon generation in a range of novel nonlinear materials, including PPLN, PPKTP, GaP, GaAs, CdSiP₂, and ZnGeP₂ McCracken et al. (2018). In 2019, Rosenfeld et al. experimentally prepared MIR photon pairs in a four-wave mixing process from a silicon-on-insulator waveguide at around 2.1 $\mu$m Rosenfeld et al. . In 2020 Kundys et al. numerically studied the reconfigurable MIR single-photon sources based on functional ferroelectrics, i.e., PMN-0.38PT crystal at 5.6 $\mu$m Kundys et al. (2020). In the same year, Prabhakar et al. experimentally demonstrated the entangled photons generation and Hong-Ou-Mandel interference at 2.1 $\mu$m from PPLN crystals Prabhakar et al. (2020).

Many biphoton sources have been demonstrated in the previous studies Lee et al. (2016); Mancinelli et al. (2017); McCracken et al. (2018); Kundys et al. (2020); Prabhakar et al. (2020); however, from the experimental point of view, it is still lack of high-quality biphotons source at the MIR range, especially the spectrally uncorrelated biphotonsMosley et al. (2008a); Grice et al. (2001); U’Ren et al. (2006); Graffitti et al. (2018); Laudenbach et al. (2016). The biphotons generated from an SPDC process are generally spectrally correlated due to the energy and momentum conservation laws. However, it is necessary to utilize biphotons with no spectral correlations to obtain the heralded pure-state single photon for many quantum information processing (QIP) protocols, such as quantum computation Walmsley and Raymer (2005), boson sampling Broome et al. (2013), and quantum teleportation Valivarthi et al. (2016), measurement-device-independent quantum key distribution Valivarthi et al. (2017). In quantum sensing, a high-NOON state, which provides high accuracy for phase-sensitive measurements, should be prepared from spectrally uncorrelated biphotons Nagata et al. (2007); Yabuno et al. (2012). Therefore, spectrally uncorrelation of photon pairs from SPDC is indispensable for the future development of quantum information at the MIR range.

In this work, we investigate the generation of spectrally uncorrelated biphotons from PPLN crystals at MIR range. PPLN has the merits of large nonlinear coefficient and wide transparency range (0.4 $\mu$m $\sim$ 5 $\mu$m) Nikogosyan (2005); Liu et al. (2017); Yang et al. (2020); Wang et al. (2020). Another unique advantage of PPLN is that its intrinsical group-velocity-matched (GVM) wavelengths are in the MIR range, and the spectrally uncorrelated biphoton state can be engineered at the GVM wavelengths Mosley et al. (2008a); Jin et al. (2018).

In addition, we also consider the manipulation of the biphoton state by adding different dopants in PPLN. Traditionally, PPLN is doped with different dopants to improve its properties. For example, by doping Mg and Zr, the damage resistance can be improved from visible to ultraviolet region Kong et al. (2020); by doping Zn, the electro-optical coefficients can be improved Abdi et al. (1999); by doping Fe, the photorefractive properties can be improved Zhang et al. (1998). Especially, doping rare-earth elements (e.g., Tm, Er, Dy, Tb, Gd, Pr) makes PPLN a right candidate for lasers and quantum memories Palatnikov et al. (2006); Dutta et al. (2020). It is natural to deduce that doping can also be utilized as a degree of freedom to manipulate the single photons state at the MIR range. Therefore, we study the spectrally uncorrelated biphotons generation from doped PPLN in this work. Specifically, we investigated the GVM wavelengths, the tilt angle, the poling period, the thermal properties and the HOM interferences of biphtons generated from three kinds of doped LN crystals, including MgLN [MgO($x$ mol%)LiNbO₃], ZnLN [ZnO($x$ mol%)LiNbO₃], and InZnLN [In₂O₃($x$ mol%)ZnO(5.5 mol%)LiNbO₃, with $x$ from 0 to 7].

The biphoton state $|\psi\rangle$ generated from SPDC can be expressed as Mosley et al. (2008b)

$$|\psi\rangle=\int_{0}^{\infty}\int_{0}^{\infty}\,\mathrm{d}\omega_{s}\,\mathrm{d}\omega_{i}f(\omega_{s},\omega_{i})\hat{a}_{s}^{\dagger}(\omega_{s})\hat{a}_{i}^{\dagger}(\omega_{i})|0\rangle|0\rangle,$$

(1)

where $\omega$ is the angular frequency, $\hat{a}^{\dagger}$ is the creation operator, and the subscripts $s$ and $i$ indicate the signal and idler photon. The joint spectral amplitude (JSA) $f(\omega_{s},\omega_{i})$ can be calculated as the product of the pump envelope function (PEF) $\alpha(\omega_{s},\omega_{i})$ and the phase-matching function (PMF) $\phi(\omega_{s},\omega_{i})$:

$$f(\omega_{s},\omega_{i})=\alpha(\omega_{s},\omega_{i})\times\phi(\omega_{s},\omega_{i}).$$

(2)

A PEF with a Gaussian-distribution can be written as

$$\alpha(\omega_{s},\omega_{i})=\exp[-\frac{1}{2}\left(\frac{\omega_{s}+\omega_{i}-\omega_{p_{0}}}{\sigma_{p}}\right)^{2}],$$

(3)

where $\omega_{p_{0}}$ is the center angular frequency of the pump; $\sigma_{p}$ is the bandwidth of the pump, and the subscript $p$ indicates the pump photon. The full-width at half-maximum (FWHM) is FWHM_ω = 2$\sqrt{\ln(2)}\sigma_{p}\approx 1.67\sigma_{p}$.

If we choose wavelengths as the variables, the PEF can be rewritten as

$$\alpha(\lambda_{s},\lambda_{i})=\exp\left(-\frac{1}{2}\left\{\frac{{1/\lambda_{s}+1/\lambda_{i}-1/(\lambda_{0}/2)}}{{\Delta\lambda/[(\lambda_{0}/2)^{2}-(\Delta\lambda/2)^{2}]}}\right\}^{2}\right),$$

(4)

where $\lambda_{0}/2$ is the central wavelength of the pump; The FWHM of the pump at intensity level is FWHM_λ = $\frac{2\sqrt{\ln(2)}{\lambda_{0}}^{2}\Delta\lambda\left({\lambda_{0}}^{2}-\Delta\lambda^{2}\right)}{{\lambda_{0}}^{4}+\Delta\lambda^{4}-2{\lambda_{0}}^{2}\Delta\lambda^{2}[1+\ln(4)]}$. For $\Delta\lambda<<\lambda_{0}$, FWHM${}_{\lambda}\approx 2\sqrt{\ln(2)}\Delta\lambda\approx 1.67\Delta\lambda$.

The PMF function can be written as

$$\phi(\omega_{s},\omega_{i})=\operatorname{Sinc}\left(\frac{\Delta kL}{2}\right),$$

(5)

where $L$ is the length of the crystal, $\Delta k=k_{p}-k_{i}-k_{s}\pm\frac{2\pi}{\Lambda}$ and $k=\frac{2\pi n(\lambda)}{\lambda}$ is the wave vector. $\Lambda$ is the poling period and can be calculated by

$$\Lambda=\frac{2\pi}{|k_{p}-k_{i}-k_{s}|}.$$

(6)

The angle $\theta$ between the ridge direction of PMF and the horizontal axis is determined by Jin et al. (2013a)

$$\tan\theta=-\left(\frac{V_{g,p}^{-1}(\omega_{p})-V_{g,s}^{-1}(\omega_{s})}{V_{g,p}^{-1}(\omega_{p})-V_{g,i}^{-1}(\omega_{i})}\right),$$

(7)

where $V_{g,\mu}=\frac{d\omega}{dk_{\mu}(\omega)}=\frac{1}{k_{\mu}^{\prime}(\omega)},(\mu=p,s,i)$ is the group velocity. The shape of the JSA is determined by the following three GVM conditions Edamatsu et al. (2011); Jin et al. (2019).
The GVM₁ condition ($\theta=$ 0^∘):

$$V_{g,p}^{-1}(\omega_{p})=V_{g,s}^{-1}(\omega_{s}).$$

(8)

The GVM₂ condition ($\theta=$ 90^∘):

$$V_{g,p}^{-1}(\omega_{p})=V_{g,i}^{-1}(\omega_{i}).$$

(9)

The GVM₃ condition ($\theta=$ 45^∘):

$$2V_{g,p}^{-1}(\omega_{p})=V_{g,s}^{-1}(\omega_{s})+V_{g,i}^{-1}(\omega_{i}).$$

(10)

The purity can be calculated by performing Schmidt decomposition on $f(\omega_{s},\omega_{i})$:

$$f(\omega_{s},\omega_{i})=\sum_{j}c_{j}\phi_{j}(\omega_{s})\varphi(\omega_{i}),$$

(11)

where $\phi_{j}(\omega_{s})$ and $\varphi_{j}(\omega_{i})$ are the two orthogonal basis vectors in the frequency domain, and $c_{j}$ is a set of non-negative real numbers that satisfy the normalization condition $\sum_{j}c_{j}^{2}=1$. The purity p is defined as:

$$p=\sum_{j}c_{j}^{4}.$$

(12)

GVM wavelengths are important parameters for nonlinear crystals. The maximal purities under GVM₁, GVM₂, and GVM₃ conditions are around 0.97, 0.97, and 0.82, respectively Edamatsu et al. (2011).

Note the three GVM conditions in Eqs.(8-10) are also called fully-asymmetric GVM or symmetric GVM conditions in Ref. Graffitti et al. (2018). In fact, the spectrally-pure state can be prepared for any $\theta$ angle between 0 and 90 degrees, independently from the PDC-type (type-0, type-I, or type-II) and wavelength degeneracy (degenerated or non-degenerated). Perfectly separable JSA can be achieved only with Gaussian pump and Gaussian PMF (via domain engineering techniques), as proven in Ref. Quesada and Brańczyk (2018). However, the three GVM conditions listed in Eqs.(8-10) with the degenerated wavelengths are the most widely used cases in the experiments Jin et al. (2020); Cai et al. (2020); Duan et al. (2020). Especially, the GVM₃ condition has a symmetric distribution along the anti-diagonal direction, i.e., $f(\omega_{s},\omega_{i})=f(\omega_{i},\omega_{s})$, which is the prerequisite condition for high-visibility HOM interferences. Therefore, we mainly focus on the type-II phase-matching condition in the calculation below.

In this section, we first calculate the angle $\theta$ in the range of 0 and 90 degrees, and the corresponding poling period $\Lambda$ under type-II, type-I, and type-0 phase-matching conditions. Then, we consider the thermal properties, and the HOM interferences for the doped PPLN crystals under the type-II phase-matching condition. The Sellmeier equations for these crystals are chosen from Refs. Schlarb and Betzler (1994); Schlarb et al. (1995, 1996), where the Sellmeier equations are parameterized by the dopant concentration. The PPLN crystals are negative uniaxial crystals $(n_{o}>n_{e})$, where $n_{o(e)}$ is the refractive index of the ordinary (extraordinary) ray.

We notice that LN crystal has several different growing methods, and each method has a different effect on the Sellmeier equations. The MgLN, ZnLN, and InZnLN crystals investigated in the Calculation and Simulation section were grown using the Czochralski technique from a congruent melt (mole ratio Li/Nb $\approx$ 0.942) Schlarb and Betzler (1994); Schlarb et al. (1995, 1996). As shown in Fig. 1, 0 mol% Zn doped LN (i.e., pure LN) has the GVM₁, GVM₂, GVM₃ wavelengths of 2478.6 nm, 3931.4 nm , and 3163.6 nm, respectively. In contrast, the pure LN grown from the stoichiometric melt (mole ratio Li/Nb $\approx$ 1) Hobden and Warner (1966) has the GVM₁, GVM₂, GVM₃ wavelengths of 2681.0 nm, 4653.6 nm, and 3595.8 nm, respectively. Further, using the Sellmeier equations from Ref. Zelmon et al. (1997), the pure LN grown from the congruent melt (mole ratio Li/Nb $\approx$ 0.937) has the GVM₁, GVM₂, GVM₃ wavelengths of 2678.5 nm, 4361.7 nm, and 3499.1 nm, respectively. These differences are mainly caused by different growing methods. In addition, the LN crystal can also be doped with other elements, which also affect the Sellmeier equations. For example, NdMgLN crystal (grown by the Czochralski technique from a congruent melt in Ref. Kitaeva et al. (1998) has the GVM₁, GVM₂, GVM₃ wavelengths of 2615.3 nm, 4562.5 nm, and 3511.5 nm, respectively. We also investigated ErMgLN and MgTiLN crystal Zhang et al. (2009); Yongmao et al. (1992). However, they do not satisfy the three kinds of GVM conditions. In this calculation, the maximal doping ratio we considered is 7 mol%. In fact, this ratio can still be improved, and we anticipate the wavelength tunability can be further improved at a larger doping ratio.

In this work, limited by the available Sellmeier equations, we only investigate the LN crystals doped with Mg, Zn, In, and Nd. In the future, it is promising to explore more LN crystals doped with different chemical elements and with different doping ratios. Another promising direction is investigating the spectrally pure single-photon state generation from doped PPLN waveguide Cheng et al. (2019); Sun et al. (2019); Niu et al. (2020) or doped PPLN film Ge et al. (2018), which has the merits of higher nonlinear efficiency, easier for integration and microminiaturization. In Fig. 7, the purities are 0.968, 0.963, and 0.826, respectively, and this purity can be further improved to near 1 using the custom poling technique Brańczyk et al. (2011); Cui et al. (2019).

In the section of Calculation and Simulation, we only show the configuration of $o\to o+e$ in the type-II phase-matching condition. In fact, we also investigate the configuration of $e\to o+e$. Although the GVM₂ wavelength under this configuration is in the telecom wavelength, the effective nonlinear coefficient, unfortunately, is zero. The reason is analyzed in detail in the Appendix.

For future applications, spectrally uncorrelated biphotons can be used to prepare pure single-photons and entangled photons which can be applied for sensing, imaging, and communication with a quantum-enhanced performance at MIR region. As shown in the section of Calculation and simulation, all the poling periods of the doped PPLN crystals are above 5 $\mu$m, which can be easily fabricated using the off-the-shelf technology. So, the MIR-band single-photon source calculated in the work is ready for fundamental research and industry applications, so as to promote the second quantum revolution.

In conclusion, we have theoretically studied the preparation of spectrally uncorrelated biphotons at MIR wavelengths from doped LN crystals, including MgO doped LN, ZnO doped LN, and In₂O₃ doped ZnLN with doping ratio from 0 to 7 mol%. The tilt angle, poling period, thermal properties, and HOM interference of the biphotons are calculated under type-II, type-I, and type-0 phase-matching conditions. For 5 mol% MgO doped LN, the three GVM wavelengths are 2474.7 nm, 4016.6 nm, and 3207.6 nm, and the corresponding purities are 0.968, 0.963, and 0.826, respectively. In the calculation of the thermal properties, we found that thermal-related GVM wavelength is not dominated by the doped elements, but by LiNbO₃. When the temperature was increased from 20 ${}^{\circ}C$ to 120 ${}^{\circ}C$, the three GVM wavelengths are increased by about 2.8 nm, -78 nm, and -40 nm, respectively. In the simulation of HOM interference using the 5 mol% MgO doped PPLN, visibility of 96.8% was achieved in a HOM interference between two independent sources, and visibility of 100% was achieved in a HOM interference between the signal and idler from the same SPDC source. The interference patterns oscillate by changing the doping ratio or temperature. The spectrally uncorrelated biphotons can be used to prepare pure single-photon source and entangled photon source at MIR wavelengths.

This work is partially supported by the National Key R$\&$D Program of China (Grant No. 2018YFA0307400), the National Natural Science Foundations of China (Grant Nos. 91836102, 11704290, 12074299, 61775025, 61405030), and the Program of State Key Laboratory of Quantum Optics and Quantum Optics Devices (No: KF201813). We thank Profs. Zhi-Yuan Zhou, Yan-Xiao Gong, and Ping Xu for helpful discussions.

The effective nonlinear coefficient $d_{\textrm{eff}}$ for collinear phase-matching in uniaxial crystals can be deduced as follow Smith (2016).
For $o\to o+e$ phase-matching condition,

$$\begin{array}[]{l}d_{\textrm{eff}}\textrm{(ooe)}=\\ -d_{xxx}C_{\theta}C_{\phi}S_{\phi}^{2}+d_{xyy}C_{\theta}C_{\phi}S_{\phi}^{2}-d_{xyz}S_{\theta}C_{\phi}S_{\phi}\\ +d_{xxz}S_{\theta}S_{\phi}^{2}+d_{xxy}(C_{\theta}C_{\phi}^{2}S_{\phi}-C_{\theta}S_{\phi}^{3})\\ +d_{yxx}C_{\theta}C_{\phi}^{2}S_{\phi}-d_{yyy}C_{\theta}C_{\phi}^{2}S_{\phi}+d_{yyz}S_{\theta}C_{\phi}^{2}\\ -d_{yxz}S_{\theta}C_{\phi}S_{\phi}+d_{yxy}(C_{\theta}C_{\phi}S_{\phi}^{2}-C_{\theta}C_{\phi}^{3}).\\ \end{array}$$

(15)

For $e\to o+e$ phase-matching condition,

$$\begin{array}[]{l}d_{\textrm{eff}}\textrm{(eoe)}=\\ d_{xxx}C_{\theta}^{2}C_{\phi}^{2}S_{\phi}-d_{xyy}C_{\theta}^{2}C_{\phi}^{2}S_{\phi}+d_{xyz}C_{\theta}S_{\theta}C_{\phi}^{2}\\ -d_{xxz}C_{\theta}S_{\theta}C_{\phi}S_{\phi}+d_{xxy}(C_{\theta}^{2}C_{\phi}S_{\phi}^{2}-C_{\theta}^{2}C_{\phi}^{3})\\ +d_{yxx}C_{\theta}^{2}C_{\phi}S_{\phi}^{2}-d_{yyy}C_{\theta}^{2}C_{\phi}S_{\phi}^{2}+d_{yyz}C_{\theta}S_{\theta}C_{\phi}S_{\phi}\\ -d_{yxz}C_{\theta}S_{\theta}S_{\phi}^{2}+d_{yxy}(C_{\theta}^{2}S_{\phi}^{3}-C_{\theta}^{2}C_{\phi}^{2}S_{\phi})\\ -d_{zxx}C_{\theta}S_{\theta}C_{\phi}S_{\phi}+d_{zyy}C_{\theta}S_{\theta}C_{\phi}S_{\phi}-d_{zyz}S_{\theta}^{2}C_{\phi}\\ +d_{zxz}S_{\theta}^{2}S_{\phi}+d_{zxy}(C_{\theta}S_{\theta}C_{\phi}^{2}-C_{\theta}S_{\theta}S_{\phi}^{2}),\\ \end{array}$$

(16)

where $S_{\theta}=\sin(\theta+\rho)$, $C_{\theta}=\cos(\theta+\rho)$, $S_{\phi}=\sin\phi$, $C_{\phi}=\cos\phi$, and $\rho$ is the walk-off angle. $\theta$ is the polar angle, $\phi$ is the azimuth angle, and Z is the optical axis, as defined in Fig. 9(a). For PPLN under $o\to o+e$ or $e\to o+e$ phase-matching conditions, $\rho=0^{\circ}$, $\theta=90^{\circ}$, and $\phi=90^{\circ}$, as shown in Fig. 9(b, c). Therefore,

$$d_{\textrm{eff}}\textrm{(ooe)}=d_{xxz}$$

(17)

and

$$d_{\textrm{eff}}\textrm{(eoe)}=d_{zxz}.$$

(18)

The second-order nonlinear coefficient matrix for LN is Smith :

$$\begin{array}[]{ll}d&=\left(\begin{array}[]{cccccc}0&0&0&0&d_{xxz}&d_{xxy}\\ d_{yxx}&d_{yyy}&0&d_{yyz}&0&0\\ d_{zxx}&d_{zyy}&d_{zzz}&0&0&0\\ \end{array}\right)\\ \\ &=\left(\begin{array}[]{cccccc}0&0&0&0&d_{15}&d_{16}\\ d_{21}&d_{22}&0&d_{24}&0&0\\ d_{31}&d_{32}&d_{33}&0&0&0\\ \end{array}\right)\\ \\ &=\left(\begin{array}[]{cccccc}0&0&0&0&-4.6&-2.2\\ -2.2&2.2&0&-4.6&0&0\\ -4.6&-4.6&-25.0&0&0&0\\ \end{array}\right).\\ \end{array}$$

(19)

So,

$$d_{\textrm{eff}}\textrm{(ooe)}=d_{xxz}=-4.6\,\textrm{pm/V}$$

(20)

and

$$d_{\textrm{eff}}\textrm{(eoe)}=d_{zxz}=0.$$

(21)

Therefore, the configuration of $e\to o+e$ does not exist for PPLN.