Mid-infrared Spectrally Pure Single-Photon States Generation from 22 Nonlinear Optical Crystals

. We theoretically investigate the preparation of pure-state single-photon source from 14 birefringent crystals (CMTC, THI, LiIO 3 , AAS, HGS, CGA, TAS, AGS, AGSe, GaSe, LIS, LISe, LGS, and LGSe) and 8 periodic poling crystals (LT, LN, KTP, KN, BaTiO 3 , MgBaF 4 , PMN-0.38PT, and OP-ZnSe) in a wavelength range from 1224nm to 11650nm. Te three kinds of group-velocity-matching (GVM) conditions, the phase-matching conditions, the spectral purity, and the Hong-Ou-Mandel interference are calculated for each crystal. Tis study may provide high-quality single-photon sources for quantum sensing, quantum imaging, and quantum communication applications at the mid-infrared wavelength range.


I. INTRODUCTION
Single-photon source at mid-infrared (MIR) wavelength range (approximately 2-20 µm) has important potential applications in quantum sensing, quantum imaging, and quantum communication [1][2][3][4].Firstly, the 3-5 µm band contains absorption peaks of many gases, such as H 2 O, CO, CO 2 , SO 2 , and SO 3 [5], so this band is important for sensing of these gases in environmental monitoring [6,7]; the 7-10 µm band contains absorption peaks of H 2 O 2 , CH 4 , O 3 , TAT, Acetone, and sarin [5], etc, therefore this band is important for the sensing chemical or explosive materials in the applications of industrial production or defense security.The single-photon source in these gas sensing applications may provide an ultrahigh sensitivity [8].Secondly, 3-5 µm and 8-14 µm are two widely used ranges for MIR thermal infrared imaging camera for medical and forensic usage [9], since the room temperature objects emit light at these wavelength ranges.And MIR single-photon sources can provide a diagnosis in a non-invasive manner, which is important for medical or biological samples [10].Thirdly, 3-5 µm is also an atmospheric transmission window with relatively high transparency, which is useful for large-scale free-space quantum communications, such as entanglement distribution [11], quantum key distribution [12], or quantum direct communication [13].
However, the previous studies are still insufficient for the need of MIR applications.On one hand, the previous experimental work is mainly focused on PPLN crystal and the wavelength range is below 5 µm.SO, the range from 5-20 µm is still needs further exploration.On the other hand, the spectrally-pure single photon source is proved to be a good resource [33], but this source is still rare because the group-velocity matching (GVM) conditions can only be matched at very limited wavelengths in a crystal.Therefore, it is still necessary to explore more nonlinear optical crystals to fully meet the need of MIR band applications.For this purpose, in this work, we investigate MIR spectrally-pure single-photon generation from 14 crystals by the birefringence phase matching (BPM) method, and 8 crystals by the quasiphase-matching (QPM) method.They can meet three kinds of GVM conditions and prepare spectrally uncorrelated biphotons so as to generate spectrally pure heralded single-photon state.The calculated result is from [22] according to the method from [50].

II. THEORY A. The characteristics of 22 kinds of nonlinear crystals
We investigate 22 kinds of nonlinear crystals in this work.Table I summarizes them from several perspectives: axial type, point group, transparency range, and the maximal nonlinear coefficient.We separate these crystals into three categories by their axial type or phasematching form in order to illustrate the result more clearly in the next section.Most Sellmeier equations were obtained from Refs.[48,49], and we have updated some Sellmeier equations with the latest references and concluded in Tab.I.
In the first category, 9 birefringent uniaxial crystals are listed in the table.They have a very large transparency range up to 20 µm except CMTC and LiIO 3 .We show 4 birefringent biaxial crystals in the second category, these four crystals can be written as LiM X (M = In, Ga and X = S, Se).Here, In and Ga are elements of group IIIA in the periodic table; S and Se are elements of group VIA.They are all mm2 point group.With their transparency range up to 14 µm, they have similar properties and can perform many applications in the MIR band.The other 8 periodic poling crystals usually realize their phase-matching by the QPM method, so we discuss them in the section III B and IV in detail.

B. The GVM theory of spectrally-pure single-photon states generation
The SPDC process generates the biphoton state |ψ , which can be written as where â † is the creation operator; ω is the angular frequency, and the subscripts s and i denote the signal and idler photon.The joint spectral amplitude (JSA) f (ω s , ω i ) can be obtained by multiplying the pump envelope function (PEF) α(ω s , ω i ) and the phase matching function (PMF) φ(ω s , ω i ), i.e., f For PEF, it is usuaaly a Gaussian distribution and can be expressed as [51], where σ p is the bandwidth of the pump; ω p0 is the center frequency of the pump; the full-width at half-maximum (FWHM) is FWHM ω =2 ln(2)σ p ≈ 1.67σ p .
If we use wavelengths as the variable by ω = 2πc/λ for ease of calculation, the PEF can be rewritten as (3) where λ 0 /2 is the central wavelength of the pump; ∆λ is the bandwidth of wavelength and σ p = 2πc ∆λ/[(λ0) 2 −(∆λ/2) 2 ] , where c is the light speed.
By assuming a flat phase distribution, the PMF can be written as an sinc function shape [51], where L is the length of crystal, ∆k is the wave vector mismatch.For QPM case, ∆k According to the refractive index coordinate in Appendix of Ref. [52], θ is the polar angle between the optical axis of the crystal and the light propagation direction, ϕ is the azimuth angle in the xy plane.For uniaxial crystals, θ is the cutting angle of the crystals.For biaxial crystals, when light propagates in the xz plane, ϕ=0 • , θ is the cutting angle; when light propagates in the yz plane, ϕ=90 • , θ is the cutting angle; when light propagates in the xy plane, θ=90 • , ϕ is the cutting angle.
When the ∆k=0, the phase-matching condition is satisfied.Under on this precondition, we consider the GVM condition to prepare an intrinsic spectrally-pure state.The angle θ P M F between the positive direction of the horizontal axis and the ridge direction of the PEF is determined by [53]: where ω) , (µ = p, s, i) is the group velocity of the pump, the signal and the idler.
We consider three kinds of GVM conditions [54].The GVM 1 condition (θ The GVM 2 condition (θ The GVM 3 condition (θ  The pure state not only can be prepared through these three GVM conditions but also all the conditions that the θ P M F angles are between 0 and 90 • [55,56].Since these three GVM conditions are listed in Eqs.(6)(7)(8) are the most widely-used cases in the experiment, we mainly consider these three conditions within this work.Besides, the degenerate or nondegenerate case of other θ P M F under type-II and type-0 phase-matching conditions will be illustrated in section III C.

A. Birefringent crystals
Firstly, we consider the birefringent crystals with the BPM method.We assume the wavelength is degenerated, i.e., 2λ p = λ s = λ i .For uniaxial crystals, negative uniaxial crystals satisfy the type-II SPDC with e→o+e phase-matching interaction.Here the pump and idler are extraordinary (e) beams, while the signal is ordinary  II.Three kinds of GVM conditions for 10 uniaxial BPM crystals.λ p(s,i) is the GVM wavelength for the pump (signal, idler).θ is the phase-matching angle and d eff is the effective nonlinear coefficient.The Sellmeier equations are obtained from Refs.[48,49].Most of the d eff values can be obtained from the SNLO v78 software package, developed by AS-Photonics, LLC [57].*The d eff value for CMTC is not available from SNLO, we have calculated the d eff using the method in the Appendix of Ref. [52] and considering Miller's rule [58].† The d eff value for THI is unknown.(o) beam.In contrast, the positive uniaxial crystals can meet the Type-II SPDC with o→o+e phase-matching interaction.Note all the simulations are based on collinear figuration.10 kinds of uniaxial crystals are investigated in this work.Taking AgSe crystal as an example, we plot the PMF and GVM 1(2,3) conditions for different wavelengths and phase-matched angles in Fig. 1 (a).The PMF (red) crosses the GVM 1(2,3) (blue and green) at three black points, which meet the PMF and three kinds of GVM conditions simultaneously.The three points associated with wavelengths of 4914, 8158, and 6272 nm, respectively, and angles of 79.9 • , 81.9 • , and 67.2 • .Further, we calculate the other uniaxial crystals with the same method, and then we summarize the result in Table II.In Table II, the down-converted photons have a wavelength range from 1298 to 11650 nm, which is in the near-infrared (NIR) and MIR bands.The corresponding spectral purity at GVM 1(2,3) wavelengths is 0.97, 0.97, and 0.82, respectively.
For biaxial crystals, all the crystals we investigated can only satisfy the GVM conditions in the xy plane, with polar angle θ = 90 • .We study 4 kinds of biaxial crystals and choose the LISe crystal as an example, which represents the mm2 point group.The assignment of dielectric and crystallographic axes are X, Y, Z ⇒ b, a, c.As shown in Fig. 1 (b), in the xy plane the cross points reflect the GVM condition can be fulfilled at the wavelength of 3824, 6410, and 4982 nm, respectively.
dition (also shown in Fig. 4), which can meet the different application demand in NIR, MIR, and telecom wavelengths.

B. Periodic poling crystals
In this section, we consider 8 periodic poling crystals with the QPM method, which has several advantages.For example, the largest component of the nonlinear coefficient matrix (usually d 33 ) can be utilized; there is no walk-off angle so as to achieve good spatial mode; it allows phase-matching interaction in isotropic media, in which the BPM is not applicable [59].The GVM wavelengths, the poling period Λ, and the effective nonlinear coefficient d ef f are calculated and listed in Tab.IV.The LT, LN, KTP, and KN crystals are traditionally oftenused QPM crystals [21,[60][61][62].Here, we find the GVM 2 wavelengths are all above 2 µm.
The BaTiO 3 crystal shows a low birefringence, thus, only suitable for QPM method.With its high transmission in the IR range, it is possible to prepare purestate at 3036, 3986, and 3480 nm, respectively.The MgBaF 4 crystal can meet the GVM 1 condition at 1978 nm, and the GVM 3 condition at 2780 nm.Note that this crystal does not satisfy the GVM 2 condition.The PMN-0.38PT is a functional ferroelectric material.The GVM condition only can be fulfilled at two wavelengths, i.e., 5620 nm and 7944 nm for GVM 1 and GVM 3 conditions.The orientation-patterned zinc selenide (OP-ZnSe) is an isotropic semiconductor material, therefore, the QPM rather than BPM is applicable.OP-ZnSe has extremely high nonlinear coefficients.Since the crystal possesses only one refractive index, it can only perform type-0 SPDC, i.e., e→e+e interaction, which will be discussed in the next section.All the QPM crystals can prepare pure-state at the range from 1224 to 7944 nm, as listed in Tab.IV.

C. Wavelength nondegenerate case
In this section, we focus on the wavelength nondegenerate case using QPM method.We take OP-ZnSe as an example and calculate the θ P M F in the range of 0 and 90 degrees and the corresponding poling period Λ in Fig. 2. The dashed black line in Fig. 2 indicates the degenerate case, i.e., 2λ p = λ s .For one fixed pump wavelength, we can find θ P M F and Λ of different signal wavelengths.
The OP-ZnSe crystal can only perform type-0 SPDC, i.e., e→e+e interaction.Under the degenerate condition, the signal and the idler have the same group velocity, so pure state can not be prepared.The lines of all the angles θ P M F converge in one point.At this point, all the GVM conditions are satisfied synchronously.Due to the singularity caused by these GVM conditions, the degenerate case at this point does not provide high purity, while the pure state can be prepared in the other area of different θ P M F .

Maximal purity
Poling period Λ (μm) The result of all the QPM crystals for three kinds of GVM conditions.GVM 1(2) can achieve a purity of 0.97.GVM3 can achieve a purity of 0.82.The wavelength versus achievable maximal spectral purity and poling period Λ can be reflected on the scale of the left and right Y-axis.The wavelength range is from 1224 nm to 7944 nm, and the poling period Λ is from 6.1 µm to 1301.38 µm

D. HOM interference simulation
The quality of the spectrally uncorrelated biphoton state can be tested by Hong-Ou-Mandel (HOM) interference.There are two kinds of HOM interference, the first one is the HOM interferences using signal and idler photons from the same SPDC source, with a typical setup shown in [63].In this case, the two-fold coincidence probability P 2 (τ ) as a function of the time delay τ is given by [64][65][66]: The second one is the HOM interference with two independent heralded single-photon sources, with a typical experimental setup shown in Refs.[67,68].In this interference, two signals s 1 and s 2 are sent to a beamsplitter for interference, and two idlers i 1 and i 2 are detected by single-photon detectors for heralding the signals.The four-fold coincidence counts P 4 as a function of τ can be described by [65, 66] where f 1 and f 2 are the JSAs from the first and the second crystals.We choose BiTaO 3 , LGSe, and PMN-0.38PT as examples to test the HOM interference.Figure 3 (a) shows JSA figure is generated from BiTaO 3 crystal, which is under the GVM 1 condition.BiTaO 3 crystal is a uniaxial QPM crystal.The JSA is obtained by using a pump laser with a bandwidth of ∆λ = 4 nm, and a crystal length L of 100 mm.The JSA has a long stripe shape along the horizontal axis.As for the spectral distributions of the signal and the idler photons, we can obtain them by projecting the joint spectral intensity onto the horizontal and vertical axes.The FWHM of the signal (idler) is 27.02 nm (0.92 nm). Figure 3 (c) shows the HOM pattern of two signals heralded by two idlers, the FWHM is 726.87 fs with visibility of 96.68%.Figure 3 (d) shows the HOM pattern of two heralded idlers with an FWHM of 8.74 ps and a visibility of 96.68%.
For the GVM 2 condition, the result is on the second row of Fig. 3.We investigate a biaxial BPM crystal LGSe.The JSA shape is also a long stripe, but it locates along the vertical axis.The pump bandwidth ∆λ and the crystal length L of Fig. 3 (e) are 8 nm and 200 mm.The FWHM of the signal (idler) is 5.17 nm (75.40 nm) for Fig. 3 (f).The FWHM of the HOM pattern by two heralded signals (idler) is 12.46 ps (1.17 ps), and the visibility is 97.05%, as shown in Fig. 3

(g)(h).
For the GVM 3 condition, the result is on the third row of Fig. 3.We concentrate on PMN-0.38PTcrystal.This crystal has been studied before, however, it only focuses on the GVM 1 and GVM 2 conditions [22].Here we make a thorough study of the GVM 3 condition.In this case, the JSA shape is near-round, and the spectra of the signal and idler are almost equal.Figure 3 (i) is obtained by using a pump bandwidth of ∆λ = 11 nm, and a crystal length L of 100 mm.The spectra of the signal and idler have the same FWHM of 54.64 nm.The HOM interference from two independent signal or idler sources manifests the same performance with the FWHM of 12.24 ps and visibility of 82.33%, which is much lower than the GVM 1 and GVM 2 case.In the case of two-fold HOM interference, the visibility is 100% and the FWHM of the HOM pattern is 2.20 ps for Fig. 3 (l).

IV. DISCUSSION
We summarize the result of all the BPM crystals in Fig. 4. The left vertical axis of the figure denotes the GVM condition and the corresponding PMF angle θ P M F .The right vertical axis shows the maximal purity.The horizontal axis shows a wavelength range from 0 to 12 µm.Most of the crystals are located on the MIR band, from 2 µm to 12 µm.There are three cases on the NIR band.We also conclude the results of QPM crystals in Figure 5, which shows the down-converted wavelength, the poling period, and the maximal purity for all the results we calculated above.
It is important to discuss the detection of the single photons in the MIR region.Recent work shows that superconducting nanowire single-photon detectors (SNSPD) which have the best performance (98%) in the NIR band [69], while having a detection efficiency at MIR band of 70% at 2 µm [70], 40% at 2.5 µm and 10% at 3 µm [71], 1.64% for free-space communication [30]; upconversion detectors module combine with SAPD method demonstrates the efficiency of 6.5% in room temperature [14]; semiconductor photodiodes likes Cd admixture, graphene, black arsenic phosphorus, black phosphorus carbide, tellurene, PtSe 2 and PdSe 2 are good candidates for wide detection range [72].See a recent review about MIR single-Photon detection in [73].In the future, developing new material for SNSPD and a more effective nonlinear process of upconversion for MIR detection will be promising.
For the GVM 3 condition, the purity can be further improved from 0.82 to near 1 using the custom poling crystal scheme, for example by machine learning method or metaheuristic algorithm [25,86].

V. CONCLUSION
In conclusion, we have theoretically investigated 22 nonlinear optical crystals for MIR photon generation.The down-converted photons wavelength range is from 1298 nm (1224 nm) to 11650 nm (7944 nm) for the BPM (QPM) crystals.The corresponding purity for three kinds of GVM conditions are around 0.97, 0.97, and 0.82, respectively.The wavelength nondegenerated condition, the 4-fold HOM interference, and the 2-fold HOM interference are calculated in detail.This study may be helpful to the study of quantum communication, quantum imaging, and quantum metrology at MIR range.

FIG. 2 .
FIG. 2. (a, b): The GVM angle θP M F and the corresponding poling period Λ for QPM crystal OP-ZnSe.The solid black line indicates θP M F = 45 • , and the dashed black line indicates the degenerate case 2λp=λs.In this figure, the QPM nondegenerate wavelengths case is under type-0 phase-matching condition.

FIG. 3 .
FIG.3.(a, e, i) are the JSAs; (b, f, j) are the spectra; and (c, d, g, h, k, l) are the HOM interference patterns.The FWHM of the spectra (∆) for the signal and the idler, the visibility (V) and the FWHM (∆) of two-fold and four-fold HOM interference are shown in the figures.The parameters of L = 100 mm and ∆λ = 4 nm (FWHM = 6.66 nm), L = 200 mm and ∆λ = 8 nm (FWHM = 13.32 nm), L = 100 mm and ∆λ = 11 nm (FWHM = 18.32 nm) are adopted for BaTiO3, LGSe, and PMN-0.38PT,respecitvely.Note that for all the calculations of purity, we use a grid size of 200×200 for all the JSAs.
FIG.5.The result of all the QPM crystals for three kinds of GVM conditions.GVM 1(2) can achieve a purity of 0.97.GVM3 can achieve a purity of 0.82.The wavelength versus achievable maximal spectral purity and poling period Λ can be reflected on the scale of the left and right Y-axis.The wavelength range is from 1224 nm to 7944 nm, and the poling period Λ is from 6.1 µm to 1301.38 µm

TABLE I
[48,49]properties of the 10 uniaxial birefringent crystals (part I), 4 biaxial birefringent crystals (part II), and 8 periodic poling crystals (part III) discussed in this work, including the chemical formula, the axis (uniaxial, biaxial, or isotropic), the point group, the transparency range λtransp., and the maximal nonlinear coefficient dmax at different wavelength (in µm in the bracket).Most of the data were obtained from Refs.[48,49].The BaTiO3 crystal has different types of point group at different temperatures.*d+=|d31sinθ| + |d22cosθ|.†

TABLE IV .
[58]49]inds of GVM conditions for 8 QPM crystals.λp(s,i) is the GVM wavelength for the pump (signal, idler).Λ is the poling period and d eff is the effective nonlinear coefficient.The Sellmeier equations are obtained from Refs.[48,49].Most of the d eff values can be obtained from the SNLO v78 software package, developed by AS-Photonics, LLC[57].*Thedeffvalues for BaTiO3 and MgBaF4 are not available from SNLO, so we have calculated them using the method in the Appendix of Ref.[52]and considering Miller's rule[58].† The d eff value for PMN-0.38PT is unknown.