Open Access
Add to BibSonomy
Abstract
We theoretically study the generation of photon pairs via spontaneous four-wave mixing (SFWM) in a liquid-filled microstructured suspended-core optical fiber. We show that it is possible to control the wavelength, group velocity, and bandwidths of the two-photon states. Our proposed fiber structure shows a large number of degrees of freedom to engineer the two-photon state. Here, we focus on the factorable state, which shows no spectral correlation in the two-photon components of the state, and allows the heralding of a single-photon pure state without the need for spectral post-filtering.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Photon pairs with specific spectral properties play a crucial role in quantum optical technologies. For instance, quantum optical coherence tomography [1], quantum teleportation [2], or quantum cryptography with entanglement-based quantum key distribution (QKD) protocols [3,4] require maximally entangled photon pairs. On the other hand, some other applications from metrology [5] to quantum information [6] need pure single-photon states. Therefore, tailorable photon-pair sources are a central part of any quantum optics experiment. Optical fiber sources based on spontaneous four-wave mixing (SFWM) are now one of the brightest photon-pair sources [7–9]. In this regard, silica-core fibers have attracted considerable attention due to the simplicity of their implementation [10–14]. Besides, fibers can be drawn in long length. They are also ideal for integration with optical-fiber networks because of the comparable mode shapes of the sources and networks. Photon-pair generation using photonic crystal fibers (PCF) [15], dispersion-shifted fibers [16,17], step-index multimode optical fibers (MMF) [18], graded-index MMF [19], and birefringent fibers [20] has been reported. Nevertheless, silica-core fibers only have a low nonlinearity and limited transmission bandwidth. Besides, the broadband spectrum of spontaneous Raman scattering (SRS) strongly contributes to uncorrelated noise photons and degrades the quality of the source.
One of the approaches that have been investigated to reduce the Raman noise is cooling down the device [16,21]. However, this method cannot suppress the Raman noise completely and it adds a layer of experimental difficulty. Another approach which is compatible with room temperature operation is generating photon pairs with a large spectral gap from the pump. This approach has been mostly investigated in specially designed microstructured fibers [22]. Although Raman noise becomes less detrimental, multiphonon Raman scattering can still be troublesome [23,24]. An ideal solution is to change the propagation medium to a material exhibiting a narrow-line Raman spectrum. Hollow-core photonic crystal fibers (HCPCF) allow to guide light in a gas or a liquid with a negligible optical overlap with the glass. Several demonstrations have shown the ability to explore Raman-free nonlinear optics [25–27].
A promising approach to overcome the limitations of silica-core fibers is generating photon-pairs in liquid-core optical fibers (Li-COF) [28], specifically with the liquid carbon disulfide (CS2) [29,30]. These liquid-core optical fibers have been also used for other nonlinear experiments [31] such as supercontinuum generation [32–37] and third-harmonic generation [38]. CS2 offers a significantly higher nonlinearity than silica, allowing for shorter length sources, narrow Raman lines, wider transmission windows, and a tunable dispersion through mixing with other liquids [39,40]. Wider transmission windows, particularly towards mid-infrared wavelengths, enable producing photons with larger wavelengths, useful for quantum spectroscopy applications [41–43]. Moreover, by choosing the proper fiber geometry, the linear refractive index of the liquid, and the pump wavelength, it becomes possible to generate photon pairs outside the narrow Raman lines of the liquid [28,44]. As a result, the Raman photons generated in the core of the fiber can be filtered out as their wavelengths are well-separated from those of the emitted correlated photons.
SFWM produces photons with a small probability but always in pairs. Hence, the detection of one photon from these sources heralds the presence of its twin. In this nonlinear parametric process, energy and momentum conservation (also know as phase-matching condition) result in spectral and spatial entanglements between the generated photon pairs. Therefore, the detection of one photon projects the heralded photon into a mixed quantum state [45]. However, many quantum information processing (QIP) protocols rely on Hong-Ou-Mandel (HOM) type interference [46] which requires the two interacting photons to be in pure states and indistinguishable [47]. Having photons in a mixed state reduces the visibility of the quantum interference. This problem can be solved by removing the entanglement between the two photons in a pair and producing uncorrelated photons, i.e., a factorable photon-pair state. Although this challenging task can be done using spatial and spectral filtering, doing so would decrease efficiency.
An alternative would be exploiting optical fibers that can minimize spatial correlations [48] and eliminating spectral correlation using group-velocity matching (GVM) [49]. Garay-Palmett et al. [50] have theoretically studied photon-pair-state preparation with tailored spectral properties by FWM in microstructured fibers (MSF). They showed, that the flexibility of engineering dispersion in MSF could provide a possible solution to generate a two-photon state with a specific spectral correlation suitable for QIP.
According to these previous works, engineering the dispersion of a microstructured fiber to satisfy the group-velocity matching results in producing uncorrelated signal and idler photons. Here, we will apply this concept to a novel fiber structure that offers many degrees of freedom to engineer its dispersion and consequently tailor the spectral properties of the generated pairs. Specifically, we propose and theoretically investigate the utilization of a novel CS2-filled microstructured suspended-core optical fiber for photon-pair generation. This structure offers expanded control over properties of the produced photon pairs, including wavelength, group velocity, bandwidths, and spectral correlations of the two-photon state, through the possibility of engineering the dispersion relation of the modes involved. The scanning electron microscope (SEM) of an example fiber is shown in Fig. 1, where the core can be filled with CS2.
This fiber design provides the following advantages: (i) a wide degree of freedom to engineer the dispersion relation by changing the transverse spatial structure such as the dimensions of the silica and the CS2 areas, (ii) strong transverse spatial mode confinement, caused by high dielectric contrast, in the small liquid-filled core, (iii) added control over the SFWM spectral state, which can be reached by fiber birefringence. We consider CS2 as the filling liquid because it has high nonlinearity, broad transmission spectrum, and allows a Raman-free photon source. Besides, it brings a strong index contrast, and its temperature-dependent refractive index adds an extra degree of freedom to tailor the dispersion relation. In particular, we show how this fiber can be used for the generation of spectrally uncorrelated photon pairs, where one of the photons is at a wavelength range optimal for detection with silicon single-photon detectors (SPD) and the other photon is in the telecom C-band between 1530nm – 1565nm and suited for fiber transmission. The remainder of this manuscript is structured in the following way: In Sec. 2. we will briefly introduce and review the previously developed concepts for generation of factorizable photon pairs, which we will apply in our proposed structure. In Sec. 3. we will discuss the modes and dispersion of our fiber design and show, that it can be indeed used for generation of factorizable photon pairs and in Sec. 4. we will summarize and conclude our results.
2. Theoretical description
Degenerate spontaneous four-wave mixing (SFWM) is a third-order nonlinear process in which two photons from the pump field with frequency $\omega$p are annihilated to create a pair of photons. One is called the signal photon with frequency $\omega$s, and the other one is the idler photon with frequency $\omega$i. This phase-sensitive process is most efficient when the energy and momentum conservation conditions are satisfied, i.e.
where $k(\omega )$ is the wave vector of each corresponding mode and $\Delta K$ is the wave-vector mismatch. We assume the degenerate pump case throughout this paper, where the two pump fields have the same spectrum. We also assume low pump powers such that self- and cross-phase modulation are negligible. Therefore, the phase-matching (PM) condition does not depend on the powers. The frequencies that fulfill the PM condition are called phase-matched frequencies and denoted as $\omega _j^0$, where j = s, i. We assume that all fields propagate in the same spatial mode of the fiber.
Using a standard perturbative approach [51], the biphoton state produced by SFWM in an optical fiber of length $L$ can then be written as [52]
Here, the quantity $\kappa$ is a normalizing constant which represents the generation efficiency and is dependent on the fiber length, pump intensity, third-order nonlinear coefficient of the fiber, and also the polarizations of the pump and the created pair. $\hat {a}_s^\dagger (\omega _s)$ and $\hat {a}_i^\dagger (\omega _i)$ are the creation operators and $F(\omega _s,\omega _i)$ is the joint spectral amplitude (JSA) function describing the spectral entanglement properties of the generated photon pair. However, using a linear approximation for the phase mismatch , in addition to modeling the pump spectral amplitudes as Gaussian functions centered at $\omega _p^0$ with bandwidth $\sigma _p$ ( half-width at 1/e maximum amplitude), makes it possible to carry out the integral analytically, yielding [50]
Where $\alpha (\omega _s,\omega _i)$ is the pump amplitude envelope function and $\phi (\omega _s,\omega _i)$ is the PM function, given as
Here, $\Delta K_{lin}$ is a first-order Taylor expansion of the wave-vector mismatch $\Delta K$, performed around the phase-matched frequencies
The shape of the mapped JSA in the {$\omega _s,\omega _i$} space contains information on the photon pair’s spectral correlation [49]: a shape resembling a tilted ellipse represents correlated states, while a circle (symmetric states) or a vertical or horizontal ellipse (asymmetric states) represent factorable/uncorrelated states. The overall shape of the JSA is determined by the pump bandwidth, the fiber length L, and the angle $\theta _{si} \ = \ - \tan ^{-1} (\tau _s / \tau _i)$ for the orientation of the phase-matching function with respect to the $\omega _s$ axis, as shown schematically in Fig. 2. Here, $\tau$ represents the group velocity mismatch between the pump and either of the signal and idler modes ($\tau _j \ = \ \beta _{1p} \ -\ \beta _{1j}$ ). This angle is related to the signal, idler, and pump modes’ group velocities and ranges from -90$^{\circ }$ to +90$^{\circ }$, such that in Fig. 2(a), it has a negative value. Figure 2 schematically shows how the pump envelope and the PM function construct the JSA.
Fig. 2. (a) PM function $\phi (\omega _s,\omega _i)$ (b) Pump envelope function $\alpha (\omega _s,\omega _i)$ (c) resulting joint spectral amplitude.
In order to herald pure single-photon states, the JSA must be factorable, i.e., $F(\omega _s,\omega _i) \ = \ g(\omega _s) h(\omega _i)$, which means the JSA is a product of two functions $g(\omega _s)$ and $h(\omega _i)$ depending only on the signal and idler frequencies, respectively. By approximating the PM function to a Gaussian function, $\textrm {sinc} (L\Delta k / 2) \approx \exp (-r (L\Delta k / 2)^2)$, with $r = 0.193$, practical information about the regime in which a factorable state can be achieved is obtainable [50]. This requires the group velocity mismatch terms to have opposite signs, or in other words, to have the group velocity of the pump in between the signal and idler’s. In this way, the PM angle $\theta _{si}$ lies in the range $[0,\pi /2]$. Hence the factorability region is bounded by the conditions $\tau _s = 0$ and $\tau _i = 0$. Under the symmetric group-velocity matching condition $\tau _s = - \tau _i$ we get $\theta _{si} = \pi /4$, allowing for retrieving a close to circular JSA. Under the asymmetric group-velocity matching condition $\tau _s = 0$ or $\tau _i = 0$, we get $\theta _{si} = 0$ or $\theta _{si} = \pi /2$, respectively, and we can get an elongated horizontal or vertical ellipse, respectively. Once the appropriate group velocity matching constraint has been applied, the JSA may be engineered by adjusting the relative widths of the pump envelope and phase-matching functions, which depend on the pump bandwidth and the fiber length.
It is clear that the pump envelope $\alpha (\omega _s,\omega _i)$ in {$\omega _s,\omega _i$} space has negative slope. Thus, the type of spectral correlations observed in a SFWM two-photon state is determined in part by the slope $\theta _{si}$ of the phase-matching function. Photonic crystal fibers often have two zero dispersion wavelengths (ZDWs) [53]. In this situation, the phase-matching curve takes the form of a closed loop [50]. When the phase matching is given by a closed loop , all phase matching orientation angles $\theta _{si}$ are possible. Thus, it becomes possible to generate factorable two-photon states for specific relative orientations and widths of the stripes representing the phase matching and pump envelop functions [50]. Building on this previously established feature and its advantage, we are seeking a fiber design that offers two ZDWs.
As we discussed, reaching different JSA shapes requires a control over the PM angle, the length of the fiber, and the central frequency and bandwidth of the pump pulse. Among these, controlling the pump pulse and the fiber length is more trivial, whereas tailoring the phase matching function is challenging and requires a strong control over the dispersion profile of the fiber. As we will show, our proposed liquid-filled fiber provides the needed control to allow for such JSA tailoring. The dispersion characteristics of this structure can be engineered by varying the size of the core, the air holes surrounding the core, and the working temperature. In particular, it becomes possible to control the group velocities, choose the zero dispersion wavelength(s), and tailor the FWM phase-matching properties [54] to satisfy the factorizability condition. Since our proposed structure shows two zero-dispersion wavelengths that make a closed-loop PM curve, it can cover all the PM angle orientations ($\theta _{si}$) and satisfy the factorizability condition. In the following, we show the source design for factorable photon-pair generation using this characteristic in a CS2-filled microstructured suspended-core optical fiber.
3. Results and discussion
The proposed fiber consists of a micrometer-sized silica core with a nano channel that can be filled with liquid CS2; the nano-channel core is surrounded by patterned air holes (Fig. 1). We show that it is feasible to tailor the PM function to design a source of uncorrelated photon pairs by changing the dimensions of the silica core and the liquid-filled nano channel.
To understand the effect of each parameter to engineer the dispersion relation of the structure, we apply some simplifications. First, assuming that the field profile for the wavelength ranges of interest does not penetrate too much into the strands, we can ignore the strands and replace our complicated structure with a simple combination shown in Fig. 3(a). To ensure that it is acceptable to replace the fiber structure with the simplified one shown in Fig. 3(a), we plot neff, group index (GI), and GVD of both structures for the fundamental mode. Due to the small birefringence in the structure, it supports two orthogonally polarized fundamental modes; HE 11x : x-polarized fundamental mode , HE 11y : y-polarized fundamental mode. Their mode profiles are shown in Fig. 3(a). The numerical modeling of the effective refractive index (neff) and mode profile for this fiber was carried out for both fundamental modes. For all the simulations throughout this work, we ignore the silica losses. The wavelength-dependent refractive indices of silica and CS2 are included through the Sellmeier equations [32,55–58]. Figure 3(b) shows the calculated effective refractive indices. This dispersion modeling shows that the shorter the wavelength is, the smaller the mode profile is, and then the higher neff is. When the wavelength is small enough, the mode is confined to the CS2 area. The modal characteristics are mainly determined by the CS2-filled channel, so neff follows the refractive index of CS2 because of the high index contrast and the resulting strong confinement. When the wavelength is longer, the mode expands more to the silica area. Although a part of the mode still propagates through the CS2, the main part is in the silica range. In such a case, the mode is influenced by both the CS2-filled channel and the silica core. This leads to a lower neff. If the wavelength is even longer, the main part of the mode propagates through the air, and neff will decrease even further and get closer to the refractive index of air. GI and GVD would also follow the same behavior. Comparing the characteristics of the simplified structure with the real one shows that the features of the simplified structure are almost the same as in the complete one for $\lambda \leq$3 $\mu$m. This is sufficient for our study as we investigate photon pairs below 2$\mu m$ wavelength. Although modal birefringence is an extra degree of freedom one can use to enhance the engineering capabilities further, we do not make use of it in this work so that we can clearly explain the origins of the physics of dispersion engineering in this structure. Therefore, we go one step further and replace the rectangle-ellipse with a square-circle structure (in Fig. 3(a) d=h and a=b).
Fig. 3. (a) The simplified fiber structure (a = 167.65 nm, b = 121.25 nm, d = 2204 nm and h = 1858 nm), profile of the x- and y-polarized fundamental mode. Comparison between (b) the effective refractive index (c) group index and (d) group velocity dispersion of the real fiber structure and the simplified one.
To investigate how we can control the dispersion relation, we perform the simulation first for the following structure: circular liquid core surrounded by large silica cladding, as shown in Fig. 4(a). This structure is a simple step-index fiber.
Fig. 4. (a) SEM image of a capillary fiber (up), profile of the fundamental mode (HE11) at 1.5 $\mu$m (down). (b) $\textrm {sinc}^2 [ \Delta K (\omega _s,\omega _i) {L}/{2}]$ for fundamental signal, idler, and pump modes. (c) zero-dispersion wavelength as a function of the core radius.The connecting line is to guide the eye and has no other significance.
Simulation of these two structures: i) square-circle structure and ii) step-index fiber, helps us understand how the performance depends on each part of the structure: the circle and the square, and the materials: CS2 and SiO2. First, we simulate the simplest structure, which is a CS2-silica step-index fiber with 4 $\mu$m core diameter and 125 $\mu$m cladding diameter. Figure 4(a) shows a SEM image of this structure and the simulated fundamental mode profile for it. Based on its dispersion properties for the fundamental mode, this structure indicates only one ZDW around $\lambda _{ZDW}= 1.9 \mu$m. The response of the FWM process in this structure was investigated through the phase-matching curve as a function of pump wavelengths shown in Fig. 4(b), considering a fiber length of L = 20 cm. We assume that the pumps, signal, and idler are all in the fundamental mode and linearly polarized. For efficient SFWM, the phase mismatch must be zero ($\Delta K = 0$), which maximizes the $\phi ^2(\omega _s,\omega _i)$ function. The PM curve comprises two parts, the diagonal line corresponding to degenerate SFWM and the parabolic curve corresponding to non-degenerate SFWM. The crossing point corresponds to the ZDW point.
In the non-degenerate case, generated photon pairs are well separated from both the residual pump photons and the spontaneous Raman scattering (SRS). The main Raman line of CS2 is at 656 cm$^{-1}$ [44]. Moreover, on the parabolic curve, signal photons are at shorter wavelengths than the pump, where single-photon detection is typically feasible with high efficiency using avalanche photodiodes. The idler photons are in the short- and mid-wavelength infrared, where the phase-matching curve offers a wide wavelength tunability based on the choice of the pump wavelength. This makes Li-COFs suitable sources for quantum spectroscopy schemes based on induced coherence [41].
Changing the core radius, temperature, and filling liquid can tune the dispersion of the structure and induce a wide frequency-shift in this fiber [28]. Figure 4(c) shows how we can tune the zero-dispersion wavelength point in this structure by varying the core radius. When the radius is small enough that most of the mode is in SiO2, the SiO2 dispersion property determines the ZDW. By increasing the radius, the GVD starts following the dispersion of the CS2. At this stage, the modal dispersion related to the refractive index difference between CS2 and SiO2 determines the ZDW. If we increase the radius even further, CS2 would be the dominant material, and its dispersion determines the ZDW.
In this fiber, we could not find two ZDWs using the fundamental mode. Still, the existence of different spatial modes (which have been used for higher-order mode supercontinuum generation [59]) gives us another degree of freedom to engineer the dispersion and also makes it possible to generate photon pairs with different wavelengths in the same structure. For instance, pump, signal, and idler can be all in the higher-order TM01 mode. Figure 5(a) shows the mode profile of the TM01 mode. This configuration exhibits two ZDWs, closed-loop PM curves, and achievable GVM. Figure 5(b) shows the closed-loop PM curve for the higher-order TM01 mode. In this case, we also tuned the working temperature and Fig. 5(b) and (c) show the results when the temperature is 41$^{\circ }$C. Applying heat makes the closed- loop in the phase-matching curve below 2.5 $\mu$m, which is the starting point of silica absorption. Thus, we can use temperature tuning to produce photon-pairs in silica’s transmission region. In Fig. 5(b) A and B correspond to the two special GVM cases: the horizontal case where the GV is matched between signal and pump ($\theta = 0$), and the vertical case where the GV is matched between idler and pump ($\theta = 90$). Using these two points makes it intuitive and easy to determine the uncorrelated photon pair generation range in the phase matching contour. The highlighted area indicates the regions where not only the phase-matching condition is satisfied, but also the condition for factorizability, i.e., the group velocity of the pump is between that of signal and idler. Although we can tune this region by changing the core radius, temperature, and filling liquid or even using intermodal processes, our control is restricted. Our simulation shows that only for a limited core radius range [1.7 $\mu$m - 2 $\mu$m] we can achieve two ZDWs. Besides, working with higher-order modes is not as efficient and straightforward as working with fundamental modes. Besides, to avoid silica loss, we also need to tune the temperature, which adds another layer of experimental difficulty. Therefore, we have to add more features to achieve two ZDWs in the fundamental mode.
Fig. 5. (a) Profile of the TM01 mode at 1.5 $\mu$m. (b) $\textrm {sinc}^2 [ \Delta K (\omega _s,\omega _i) {L}/{2}]$ as a function of pump wavelengths for TM01 signal, idler, and pump modes , (c) Part of the phase-matching curve in frequency domain.
Now, we analyse our fiber using the square-circle structure that has a circular liquid core in silica square with air cladding and is shown in the inset of Fig. 6(a). Using this structure can resolve the problem because it shows two ZDWs for the fundamental mode even without CS2 core. One of them comes from material dispersion, and the other one comes from modal dispersion. The dashed lines in Fig. 6(a) indicate ZDWs when there is no liquid core (r = 0).They can be tuned together but not independently. It means that we can have uncorrelated states without liquid core, but we can not fully control the two wavelengths of GVM. Therefore, our focus will be on the structure with CS2 core due to the high nonlinearity of CS2 and the extra degree of freedom to control the ZDW points independently, which means we can control where the A and B points are. Figure 6(a) shows how we can control the two ZDW points of the structure by changing the CS2 radius and SiO2 square width. This plot exhibits that the ZDW point with the longest wavelength is affected mainly by the width of the SiO2 square, and changing the core radius does not affect it much which is because the larger wavelength has its mode profile mainly in the SiO2. Its wavelength increases by increasing the square width. On the other hand, the ZDW point with the shortest wavelength is strongly affected by the core radius. By adding CS2, we can tune it to longer wavelengths that are not reachable without it.
Fig. 6. (a) The ZDW points of the square-circle structure with parameter sweeping: CS2-filled core radius r from 50 nm to 472 nm and SiO2 square width w from 1 $\mu$m to 2$\mu$m; dashed lines represent ZDW when there is no liquid core (inset: schematic of the fiber structure), (b) Profile of the fundamental mode at 1 $\mu$m (c) $\textrm {sinc}^2 [ \Delta K (\omega _s,\omega _i) {L}/{2}]$ as a function of pump wavelengths (r = 328 nm and w = 0.9 $\mu$m), (d) Part of the phase-matching curve in frequency domain.
As an example Fig. 6(c) exhibits the PM curve for a fiber with r = 328 nm and w = 0.9 $\mu$m. The highlighted area indicates the regions where not only the phase-matching condition is satisfied, but also the condition for factorizability, i.e., the group velocity of the pump is between that of signal and idler. This area that has been shown in the frequency domain in Fig. 6(d) covers all the range for $\theta _{si}$ betwen 0 and $\pi /2$. This structure generates the idler in the C-band telecom wavelength range and the signal in the optimal detection range of Silicon-based avalanche photodiodes. Such a combination is desired for heralded sources of single photons where the single-photon is meant for transfer in optical fibers [60].
The distance of the ZDW points marks the maximal extent of the phase-matching loop, which shape is determined by the complete fiber dispersion $k(\omega )$. In order to find a suitable structure that gives the desired factorizable photon pairs, we investigate PM for different geometrical design parameters of the fiber. Figure 7(a) shows wavelength ranges over which factorizable signal and idler photons are achievable. The blue dashed lines mark the 450nm - 900nm range, where the desired signal wavelength is. The red dashed lines mark the C-band telecom range, where the desired idler is. Each colored shading corresponds to a specific square width and has two edges: i) the upper edge of the idler part represents the GVM between signal and pump, which corresponds to the position of the A points for different core radii. The upper edge of the signal part represents the corresponding signal wavelengths for those points. ii) the bottom edge of the signal part represents the GVM between idler and pump, which corresponds to the position of B points for different core radii. The bottom edge of the idler part represents the corresponding idler wavelengths for those points. As an example, we marked 4 points in Fig. 7(a); Point Ai shows the group velocity matching between signal and pump. Point As represents the corresponding signal wavelength for point Ai. Point Bs shows the group velocity matching between idler and pump. Point Bi represents the corresponding idler wavelength for point Bs. Besides, the symmetric GVM (SGVM) condition can be satisfied for a point between these two edges.
Fig. 7. (a) The range over which factorable signal and idler photons are possible for different fiber designs. The red rectangle represents the C-band in which the idler is desired to be. The blue rectangle marks the desired signal range , (b) JSA (r = 328 nm and w = 0.9 $\mu$m) when $\lambda _p^0 = 1089$ nm with $\sigma _p = 1.8$ THz and purity of 0.96, (c) JSA (r = 328 nm and w = 1 $\mu$m) when $\lambda _p^0 = 1038$ nm with $\sigma _p = 3$ THz and purity of 0.95, (d) JSA (r = 352 nm and w = 0.9 $\mu$m) when $\lambda _p^0 = 992$ nm with $\sigma _p = 0.34$ THz and purity of 0.82; The red axis is frequency and the black axis is the corresponding wavelength.
In Fig. 7(a) the black point is in the bottom edge for the 1 $\mu$m square width and the upper edge for the 0.9 $\mu$m square width. These two structures satisfy asymmetric GVM conditions that create asymmetric factorable states in the C-band telecom range. Figure 7(b) and 7(c) show the JSA for these two structures. Figure 7(d) shows the JSA for a structure that is located between the two edges of the blue region in Fig. 7(a) marked by the red point, and satisfies the symmetric GVM condition. All of these three structures create idlers at/around 1550 nm and signals below 1 $\mu$m. Therefore, we manage to get a large range of possibilities for uncorrelated signal and idler wavelengths.
The factorizability of a state can be characterized by its purity [61], which can be calculated from the JSA. Our designed fiber shows high purity of up to 0.96, which can be improved even further using temporal walk off between two distinct pump pulses [62]. Importantly, this high purity is achieved directly through the generation process, no additional filtering is needed. This ensures that the intrinsic heralding efficiency, i.e. the probability of detecting the idler photon after successful detection of the signal photon without considering the influence of the detector or the setup outside the fiber, does only depend on the probability of the idler photon to leave the fiber. This probability can be inhibited by absorption in the fiber of Fresnel reflections at the outcoupling facet. To estimate the Fresnel reflection, we assume the fiber structure which results in the factorable JSA shown in Fig. 7(b), where the mode at 1550 nm has an effective refractive index of 1.25, leading to a transmission of more than 98% through the interface. The propagation losses will be dominated by the material absorption in CS2, which is 0.3 dB/m at 1550 nm [63]. Thus, the absorption probability in our assumed fiber of 20 cm length is at maximum 7%. The total probability for the idler photon to exit the fiber is then at worst 91%, which represents also the ideally attainable heralding efficiency.
An additional degree of freedom to tailor the spectral properties of the generated state is tuning the temperature. Figure 8 shows the JSA for a structure with three different temperatures. To keep the state factorable, the pump wavelength was changed such that $\lambda _p^0 = 1092$ nm in Fig. 8(a), $\lambda _p^0 = 1088$ nm in Fig. 8(b), and $\lambda _p^0 = 1084$ nm in Fig. 8(c). The temperature-dependent refractive index of CS2 gives us the ability to tune the wavelength of uncorrelated photon pairs by a few nanometers without changing the fiber structure. This feature can compensate for the deviations from the perfect design caused by fabrication errors. For example, in this structure, increasing the temperature by 10$^{\circ }$C changes the effective refractive index to 0.0058, which corresponds to 15 nm change in the hole radius if the square width is fixed. If we rescale the whole structure, tuning the temperature by 10$^{\circ }$C corresponds to 6 nm change in the hole radius and 16 nm change in the square width. However, one should notice that the boiling point of liquid CS2 is only 46$^{\circ }$C at ambient pressure. Heating the CS2 to the boiling point can lead to creating bubbles in the liquid core of the optical fiber, which destroys the transmission. Therefore, our limitation for temperature tunning is the liquid boiling point [37]. Nevertheless, if the system is encapsulated, the device can operate at much higher temperatures [64,65]. Besides, it can generate pairs with different spectral correlations without changing the structure or even the pump wavelength.
Fig. 8. JSA (r = 328 nm and w = 0.9 $\mu$m) for different temperature; (a) 20$^{\circ }$C, (b) 30$^{\circ }$C, (c) 40$^{\circ }$C.
4. Conclusion
We have theoretically studied SFWM in a microstructured suspended-core optical fiber with a liquid-filled nano channel that provides a source with diverse spectral properties, including correlated and uncorrelated states. Our proposed design shows two ZDWs and a closed-loop phase-matching curve that can satisfy the GVM condition to generate spectrally factorable photon-pair states, and it allows all the joint spectral intensity orientations for the produced photon pairs. Based on this, we have shown that it is possible to control the wavelength, group velocity, and bandwidths of the two-photon states and generate factorable photon pairs with high purity over a wide range of wavelengths, far away from the pump wavelength and in the desired spectral window. These results are achieved by tuning the dispersion relation of the involved modes via changing the geometrical design parameters of the fiber, controlling the pump frequency and its bandwidth.
Besides the mainly discussed variability in the fiber dispersion, our proposed fiber also has a number of advantageous features. First, the used CS2 core features a very high nonlinearity, resulting in much higher effective nonlinear coefficients than for commonly employed fibers. This is further improved by the strong localization in our fiber design, such that the effective nonlinearity for the design parameters of Fig. 7(b) is $\gamma \approx 11$ W-1m-1, several orders of magnitude higher than in conventional silica fibers and even higher than in CS2-filled fibers used for supercontinuum generation [32]. Furthermore, the CS2 core can contribute to a reduction of the Raman noise in photon-pair generation. As e.g. in the fibers discussed in Fig. 7(d) only 17% of the energy is interacting with silica, a reduction of the Raman noise of approximately a factor 5 with respect to an all-silica fiber with similar dimensions is possible. Finally, the C4-symmetric structure of the proposed fiber design leads to mode field distributions that facilitate high coupling efficiencies to Gaussian beams, in particular if large portions of the field is localized inside the circularly symmetric CS2 core. For example, for the fiber design of Fig. 7(b), field overlaps to Gaussian beams of well above 90% are attainable. This will facilitate the use of the proposed fibers in actual experiments. Our proposed fiber design for photon-pair generation can be implemented by selective filling of the central hole of the fiber with CS2. Although not yet demonstrated for this specific geometry, such selective filling of fibers has been experimentally shown for diameters down to 310 nm [66]. Similar techniques [66,67] can be applied for filling our proposed fiber with liquid or even other high index materials such as chalcogenide [65].
In summary, we proposed a novel CS2-filled microstructured fiber ideally suited to generate factorable photon pairs with non-degenerate frequencies, which can be easily used in a number of applications relying on heralded single photons.
Funding
Deutsche Forschungsgemeinschaft (259607349/GRK2101, 398816777-SFB 1375 (NOA), PE 1524/13-1 (NanoPair), SCHM 2655/11-1, SCHM 2655/12-1, SCHM 2655/3-2, SE 2749/1-1 (NanoSPDC)); Bundesministerium für Bildung und Forschung (13N14877 (QuanIm4Life), 13N16108 (PhoQuant)); Thüringer Ministerium für Wirtschaft, Wissenschaft und Digitale Gesellschaft (2021 FGI 0043 - Quantum Hub Thuringia); Horizon 2020 Framework Programme (Grant Agreement No. 899580).
Acknowledgments
We acknowledge support from Deutsche Forschungsgemeinschaf (DFG, German Research Foundation) within the international research training group 259607349/GRK2101 and under the project identifiers PE 1524/13-1 (NanoPair), SE 2749/1-1 (NanoSPDC), 398816777-SFB 1375 (NOA), the German Federal Ministry of Education and Research (BMBF) under the project identifiers 13N14877 (QuanIm4Life), 13N16108 (PhoQuant); European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 899580); and from the Thuringian Ministry for Economy, Science, and Digital Society and the European Social Funds (2021 FGI 0043 - Quantum Hub Thuringia). M.A.S. acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG) via the grants SCHM 2655/12-1, SCHM 2655/11-1 and SCHM 2655/3-2.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. M. B. Nasr, B. E. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of dispersion-canceled quantum-optical coherence tomography,” Phys. Rev. Lett. 91(8), 083601 (2003). [CrossRef]
2. Y.-H. Kim, S. P. Kulik, and Y. Shih, “Quantum teleportation of a polarization state with a complete bell state measurement,” Phys. Rev. Lett. 86(7), 1370–1373 (2001). [CrossRef]
3. J. Yin, Y. Cao, Y.-H. Li, et al., “Satellite-to-ground entanglement-based quantum key distribution,” Phys. Rev. Lett. 119(20), 200501 (2017). [CrossRef]
4. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144km,” Nat. Phys. 3(7), 481–486 (2007). [CrossRef]
5. S. V. Polyakov and A. L. Migdall, “Quantum radiometry,” J. Mod. Opt. 56(9), 1045–1052 (2009). [CrossRef]
6. J. L. O’brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3(12), 687–695 (2009). [CrossRef]
7. X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94(5), 053601 (2005). [CrossRef]
8. J. Fan, A. Migdall, and L. Wang, “Efficient generation of correlated photon pairs in a microstructure fiber,” Opt. Lett. 30(24), 3368–3370 (2005). [CrossRef]
9. J. Fulconis, O. Alibart, W. J. Wadsworth, and J. G. Rarity, “Quantum interference with photon pairs using two micro-structured fibres,” New J. Phys. 9(8), 276 (2007). [CrossRef]
10. L. Wang, C. Hong, and S. Friberg, “Generation of correlated photons via four-wave mixing in optical fibres,” J. Opt. B: Quantum Semiclassical Opt. 3(5), 346–352 (2001). [CrossRef]
11. X. Li, L. Cui, X. Guo, Z. Ou, and M. Vasilyev, “Fiber-based sources of quantum light for quantum information processing,” in Asia Communications and Photonics Conference, (Optical Society of America, 2021), pp. M4A–1.
12. K. Rottwitt, J. G. Koefoed, and E. N. Christensen, “Photon-pair sources based on intermodal four-wave mixing in few-mode fibers,” Fibers 6(2), 32 (2018). [CrossRef]
13. A. Shukhin, J. Keloth, K. Hakuta, and A. Kalachev, “Heralded single-photon and correlated-photon-pair generation via spontaneous four-wave mixing in tapered optical fibers,” Phys. Rev. A 101(5), 053822 (2020). [CrossRef]
14. N. Petrov, A. Fedotov, and A. Zheltikov, “High-brightness photon pairs and strongly antibunching heralded single photons from a highly nonlinear optical fiber,” Opt. Commun. 450, 304–307 (2019). [CrossRef]
15. O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102(12), 123603 (2009). [CrossRef]
16. K. F. Lee, J. Chen, C. Liang, X. Li, P. L. Voss, and P. Kumar, “Generation of high-purity telecom-band entangled photon pairs in dispersion-shifted fiber,” Opt. Lett. 31(12), 1905–1907 (2006). [CrossRef]
17. X. Li, X. Ma, Z. Y. Ou, L. Yang, L. Cui, and D. Yu, “Spectral study of photon pairs generated in dispersion shifted fiber with a pulsed pump,” Opt. Express 16(1), 32–44 (2008). [CrossRef]
18. H. Pourbeyram and A. Mafi, “Photon pair generation in multimode optical fibers via intermodal phase matching,” Phys. Rev. A 94(2), 023815 (2016). [CrossRef]
19. H. Pourbeyram and A. Mafi, “Photon pair generation with tailored frequency correlations in graded-index multimode fibers,” Opt. Lett. 43(9), 2018 (2018). [CrossRef]
20. B. J. Smith, P. Mahou, O. Cohen, J. Lundeen, and I. Walmsley, “Photon pair generation in birefringent optical fibers,” Opt. Express 17(26), 23589–23602 (2009). [CrossRef]
21. S. D. Dyer, B. Baek, and S. W. Nam, “High-brightness, low-noise, all-fiber photon pair source,” Opt. Express 17(12), 10290–10297 (2009). [CrossRef]
22. O. Alibart, J. Fulconis, G. Wong, S. Murdoch, W. Wadsworth, and J. Rarity, “Photon pair generation using four-wave mixing in a microstructured fibre: theory versus experiment,” New J. Phys. 8(5), 67 (2006). [CrossRef]
23. L. Cui, X. Li, C. Guo, Y. Li, Z. Xu, L. Wang, and W. Fang, “Generation of correlated photon pairs in micro/nano-fibers,” Opt. Lett. 38(23), 5063–5066 (2013). [CrossRef]
24. J. Rarity, J. Fulconis, J. Duligall, W. Wadsworth, and P. S. J. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13(2), 534–544 (2005). [CrossRef]
25. M. A. Finger, T. S. Iskhakov, N. Y. Joly, M. V. Chekhova, and P. S. J. Russell, “Raman-free, noble-gas-filled photonic-crystal fiber source for ultrafast, very bright twin-beam squeezed vacuum,” Phys. Rev. Lett. 115(14), 143602 (2015). [CrossRef]
26. M. Azhar, G. Wong, W. Chang, N. Joly, and P. S. J. Russell, “Raman-free nonlinear optical effects in high pressure gas-filled hollow core pcf,” Opt. Express 21(4), 4405–4410 (2013). [CrossRef]
27. K. Lynch-Klarup, E. Mondloch, M. Raymer, D. Arrestier, F. Gerome, and F. Benabid, “Supercritical xenon-filled hollow-core photonic bandgap fiber,” Opt. Express 21(11), 13726–13732 (2013). [CrossRef]
28. M. Barbier, I. Zaquine, and P. Delaye, “Spontaneous four-wave mixing in liquid-core fibers: towards fibered raman-free correlated photon sources,” New J. Phys. 17(5), 053031 (2015). [CrossRef]
29. Y. Zhu, X. Chen, Y. Xu, and Y. Xia, “Propagation properties of single-mode liquid-core optical fibers with subwavelength diameter,” J. Lightwave Technol. 25(10), 3051–3056 (2007). [CrossRef]
30. N. Karasawa, “Dispersion properties of liquid-core photonic crystal fibers,” Appl. Opt. 51(21), 5259–5265 (2012). [CrossRef]
31. K. Kieu, L. Schneebeli, R. Norwood, and N. Peyghambarian, “Integrated liquid-core optical fibers for ultra-efficient nonlinear liquid photonics,” Opt. Express 20(7), 8148–8154 (2012). [CrossRef]
32. M. Chemnitz, M. Gebhardt, C. Gaida, F. Stutzki, J. Kobelke, J. Limpert, A. Tünnermann, and M. A. Schmidt, “Hybrid soliton dynamics in liquid-core fibres,” Nat. Commun. 8(1), 42 (2017). [CrossRef]
33. F. Xu, J. Yuan, C. Mei, B. Yan, X. Zhou, Q. Wu, K. Wang, X. Sang, C. Yu, and G. Farrell, “Highly coherent supercontinuum generation in a polarization-maintaining cs 2-core photonic crystal fiber,” Appl. Opt. 58(6), 1386–1392 (2019). [CrossRef]
34. R. V. J. Raja, A. Husakou, J. Hermann, and K. Porsezian, “Supercontinuum generation in liquid-filled photonic crystal fiber with slow nonlinear response,” J. Opt. Soc. Am. B 27(9), 1763–1768 (2010). [CrossRef]
35. R. Zhang, J. Teipel, and H. Giessen, “Theoretical design of a liquid-core photonic crystal fiber for supercontinuum generation,” Opt. Express 14(15), 6800–6812 (2006). [CrossRef]
36. S. Kedenburg, T. Gissibl, T. Steinle, A. Steinmann, and H. Giessen, “Towards integration of a liquid-filled fiber capillary for supercontinuum generation in the 1.2–2.4 µm range,” Opt. Express 23(7), 8281–8289 (2015). [CrossRef]
37. D. Churin, T. Nguyen, K. Kieu, R. A. Norwood, and N. Peyghambarian, “Mid-ir supercontinuum generation in an integrated liquid-core optical fiber filled with cs 2,” Opt. Mater. Express 3(9), 1358–1364 (2013). [CrossRef]
38. K. Schaarschmidt, J. Kobelke, T. Meyer, M. Chemnitz, and M. A. Schmidt, “Third-harmonic generation with tailored modes in liquid core fibers with geometric birefringence,” Opt. Lett. 45(24), 6859–6862 (2020). [CrossRef]
39. S. Junaid, K. Schaarschmidt, M. Chemnitz, M. Chambonneau, S. Nolte, and M. A. Schmidt, “Tailoring modulation instabilities and four-wave mixing in dispersion-managed composite liquid-core fibers,” Opt. Express 28(3), 3097–3106 (2020). [CrossRef]
40. M. Chemnitz, S. Junaid, N. Walther, R. Scheibinger, K. Schaarschmidt, J. Kobelke, and M. A. Schmidt, “Tailoring soliton fission at telecom wavelengths using composite-liquid-core fibers,” Opt. Lett. 45(11), 2985–2988 (2020). [CrossRef]
41. D. A. Kalashnikov, A. V. Paterova, S. P. Kulik, and L. A. Krivitsky, “Infrared spectroscopy with visible light,” Nat. Photonics 10(2), 98–101 (2016). [CrossRef]
42. C. Lindner, S. Wolf, J. Kiessling, and F. Kühnemann, “Fourier transform infrared spectroscopy with visible light,” Opt. Express 28(4), 4426–4432 (2020). [CrossRef]
43. C. Lindner, J. Kunz, S. J. Herr, S. Wolf, J. Kießling, and F. Kühnemann, “Nonlinear interferometer for fourier-transform mid-infrared gas spectroscopy using near-infrared detection,” Opt. Express 29(3), 4035–4047 (2021). [CrossRef]
44. J. Laane and W. Kiefer, “Measurement of solvent shifts by raman difference spectroscopy,” Appl. Spectrosc. 35(3), 267–271 (1981). [CrossRef]
45. A. B. U’Ren, C. Silberhorn, R. Erdmann, K. Banaszek, W. P. Grice, I. A. Walmsley, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” arXiv preprint quant-ph/0611019 (2006).
46. C.-K. Hong, Z.-Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987). [CrossRef]
47. I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307(5716), 1733–1734 (2005). [CrossRef]
48. X. Li, J. Chen, P. Voss, J. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications: Improved generation of correlated photons,” Opt. Express 12(16), 3737–3744 (2004). [CrossRef]
49. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64(6), 063815 (2001). [CrossRef]
50. K. Garay-Palmett, H. McGuinness, O. Cohen, J. Lundeen, R. Rangel-Rojo, A. U’ren, M. Raymer, C. McKinstrie, S. Radic, and I. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15(22), 14870–14886 (2007). [CrossRef]
51. L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge university press, 1995).
52. J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Phys. Rev. A 72(3), 033801 (2005). [CrossRef]
53. R. Jiang, R. Saperstein, N. Alic, M. Nezhad, C. McKinstrie, J. Ford, Y. Fainman, and S. Radic, “Parametric wavelength conversion from conventional near-infrared to visible band,” IEEE Photonics Technol. Lett. 18(23), 2445–2447 (2006). [CrossRef]
54. K. P. Hansen, “Dispersion flattened hybrid-core nonlinear photonic crystal fiber,” Opt. Express 11(13), 1503–1509 (2003). [CrossRef]
55. G. P. Agrawal, “Nonlinear fiber optics,” in Nonlinear Science at the Dawn of the 21st Century, (Springer, 2000), pp. 195–211.
56. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55(10), 1205–1209 (1965). [CrossRef]
57. C. Tan, “Determination of refractive index of silica glass for infrared wavelengths by ir spectroscopy,” J. Non-Cryst. Solids 223(1-2), 158–163 (1998). [CrossRef]
58. M. Chemnitz, R. Scheibinger, C. Gaida, M. Gebhardt, F. Stutzki, S. Pumpe, J. Kobelke, A. Tünnermann, J. Limpert, and M. A. Schmidt, “Thermodynamic control of soliton dynamics in liquid-core fibers,” Optica 5(6), 695–703 (2018). [CrossRef]
59. S. Ramona, N. M. Lüpken, M. Chemnitz, K. Schaarschmidt, K. Jens, F. Carsten, and M. A. Schmidt, “Higher-order mode supercontinuum generation in dispersion-engineered liquid-core fibers,” Scientific Reports (Nature Publisher Group) 11, 5 (2021).
60. J. Su, L. Cui, Y. Li, and X. Li, “Micro/nano-fiber-based source of heralded single photons at the telecom band,” Chin. Opt. Lett. 16(4), 041903 (2018). [CrossRef]
61. C. Law, I. A. Walmsley, and J. Eberly, “Continuous frequency entanglement: effective finite hilbert space and entropy control,” Phys. Rev. Lett. 84(23), 5304–5307 (2000). [CrossRef]
62. B. Fang, O. Cohen, J. B. Moreno, and V. O. Lorenz, “State engineering of photon pairs produced through dual-pump spontaneous four-wave mixing,” Opt. Express 21(3), 2707–2717 (2013). [CrossRef]
63. S. Junaid, W. Huang, R. Scheibinger, K. Schaarschmidt, H. Schneidewind, P. Paradis, M. Bernier, R. Vallée, S.-E. Stanca, G. Zieger, and M. A. Schmidt, “Attenuation coefficients of selected organic and inorganic solvents in the mid-infrared spectral domain,” Opt. Mater. Express 12(4), 1754 (2022). [CrossRef]
64. S. Pumpe, M. Chemnitz, J. Kobelke, and M. A. Schmidt, “Monolithic optofluidic mode coupler for broadband thermo-and piezo-optical characterization of liquids,” Opt. Express 25(19), 22932–22946 (2017). [CrossRef]
65. S. Xie, F. Tani, J. C. Travers, P. Uebel, C. Caillaud, J. Troles, M. A. Schmidt, and P. S. J. Russell, “As 2 s 3–silica double-nanospike waveguide for mid-infrared supercontinuum generation,” Opt. Lett. 39(17), 5216–5219 (2014). [CrossRef]
66. P. Uebel, M. A. Schmidt, M. Scharrer, and P. S. J. Russell, “An azimuthally polarizing photonic crystal fibre with a central gold nanowire,” New J. Phys. 13(6), 063016 (2011). [CrossRef]
67. S. Junaid, J. Bierlich, A. Hartung, T. Meyer, M. Chemnitz, and M. A. Schmidt, “Supercontinuum generation in a carbon disulfide core microstructured optical fiber,” Opt. Express 29(13), 19891–19902 (2021). [CrossRef]
