1. INTRODUCTION

The spin and orbital angular momentum (OAM) of photons represent two important degrees of freedom for optical information processing [1,2]. The spin angular momentum (SAM), associated with the left and right circular polarization (LCP and RCP) states of light, has been widely studied in the optical spin Hall effect [3–5], rotational Doppler effects [6,7], laser cooling [8,9], quantum interference [10,11], etc. On the other hand, as the OAM has an unlimited upper bound of mode numbers, it has been successfully used for multi-channel communications [12–14], optical encryptions and holography [15–17], high-dimensional quantum information processing [18,19], and so on.

The spin–orbit interaction (SOI) represents an important route for controlling both the SAM and the trajectory of light [20–23]. In addition, the SOI effect can be used to implement SAM-to-OAM conversion and develop advanced OAM generators [24–27]. Light with both SAM and OAM has great potential for applications in classical and quantum information processing [28–30]. When the SAM and OAM of light have been involved during nonlinear optical processes, their evolutions have attracted a lot of interest. For example, the selection rules of harmonic generations for circularly polarized light were established from a symmetry perspective [31–34]. A similar concept has been proposed for controlling the angular momentum states of entangled pairs of photons [35,36]; however, the idea has not been experimentally verified. On the other hand, the conservation of OAM in harmonic generations has been verified [37], and was extended to the spatiotemporal regime [38]. Recently, the SOI effects in harmonic generation were also observed on optical metasurfaces with phase singularities [39,40]. Moreover, the harmonic spin–orbit angular momentum cascade phenomenon was also discovered by considering the SOI effects of both pump and harmonic waves in a nonlinear crystal [41]. However, previous works on the optical SOI effect in nonlinear optics are still within the framework of the classical regime, and its quantum counterpart remains unexplored. As a typical quantum nonlinear optical process, spontaneous parametric downconversion (SPDC) [42–44] is one of the most versatile techniques for generating entangled photons and plays an important role in quantum information processing. The study of SOI in SPDC might provide new routes for preparing angular-momentum-based multi-dimensional entangled photon states [45], which will benefit areas such as quantum computation, rotational-invariant quantum communication [46], and quantum metrology [47].

Here, we experimentally demonstrate that the optical SOIs in a nonlinear crystal with threefold rotational (C3) symmetry can be used to generate two-photon quantum states with controlled angular momentum states through the SPDC process. In Fig. 1, the angular momentum states of the pump wave and the signal/idler photons generated from the SPDC process are denoted as $(\sigma ,\ell)$, where $\sigma = \pm 1$ refers to the SAM state, which is related to the LCP and RCP components, and the integer $\ell$ refers to the OAM value of photons. After the SOI process, the pump waves carry two angular momentum states, ${(\sigma ,\ell)_{{\rm pump}}}$ and ${(- \sigma ,\ell + 2\sigma)_{{\rm pump}}}$. It is worth noting that the generated angular momentum of the SPDC photons should have a broadband spectrum, which has been extensively studied through the mode expansion method [18,48–50]. Based on the SAM-dependent symmetry selection rules and the OAM conservation law, the angular momentum modes of the SPDC twin photons can be described as $\sum\limits_{\delta = 0, \pm 1, \pm 2, \ldots} {{c_\delta}| - \sigma ,\ell /2 + \delta {\rangle _s}| - \sigma ,\ell /2 - \delta {\rangle _i}}$ and $\sum\limits_{\delta = 0, \pm 1, \pm 2, \ldots} {{c_\delta}|\sigma ,\ell /2 + \sigma + \delta {\rangle _s}|\sigma ,\ell /2 + \sigma - \delta {\rangle _i}}$, respectively, which are different from that of the general SPDC process.

 

Fig. 1. Schematic diagram of the optical spin–orbit interaction (SOI) in the SPDC process in a nonlinear crystal with threefold rotational symmetry. The pump wave with angular momentum state ${(\sigma ,\ell)_{{\rm pump}}}$ is focused into the nonlinear crystal along its C3 axis. After the SOI process, the pump waves carry two angular momentum states ${(\sigma ,\ell)_{{\rm pump}}}$ and ${(- \sigma ,\ell + 2\sigma)_{{\rm pump}}}$. Then SPDC twin photons with two kinds of angular momentum states are generated.

Download Full Size | PDF

 

Fig. 2. Linear optical spin–orbit interaction process in the thin LN and thick BBO crystals. The incident light is at a wavelength of 775 nm, with angular momentum states of ${(1,0)_{775}}$ and ${(1, - 2)_{775}}$. The fringes in the cylindrical lens images show the OAM values of light. (a)–(d) LCP and RCP components of the transmitted light from (a), (b) the thin LN crystal (250 µm) and (c), (d) the thick BBO crystal (5 mm). After the strong SOI effect in the thick crystal, the cross-polarized light has angular momentum states of ${(- 1,2)_{775}}$ and ${(-1,0)_{775}}$.

Download Full Size | PDF

2. RESULTS

A. Linear Optical Spin–Orbit Interaction Process in Thin and Thick Crystals

The nonlinear crystals utilized to investigate the SOI effect in the SPDC process are a 250 µm thick lithium niobate (LN) film and a 5 mm thick beta-barium borate (BBO) [42,51]. The propagation direction of the pump wave is along the C3 rotational axis of the nonlinear crystals to introduce the intrinsic SOI effect [41]. First, the linear SOI property in the nonlinear crystals was experimentally studied. A femtosecond laser at a wavelength of 775 nm (repetition frequency: 80 MHz) was focused into the LN [Fig. 2(a)] and BBO crystals [Fig. 2(b)] by an objective (NA = 0.25). The intensity distributions of the polarization-resolved transmitted light were then captured by a CCD camera.

Figure 2 shows the experimental results under the illumination of a left circularly polarized Gaussian beam and a vortex beam. Their angular momentum states ${(\sigma ,\ell)_{{\rm pump}}}$ are ${(1,0)_{775}}$ and ${(1, - 2)_{775}}$, respectively. The transmitted intensity patterns with LCP and RCP components were imaged by a spherical lens and a cylindrical lens. The cylindrical lens imaged pattern can be used to identify the OAM value $\ell$ and its sign of the transmitted light [52]. In the case of thin LN crystal, it can be seen that the result of using the co-polarization components (LCP–LCP) is much stronger than that of the cross-polarization components (LCP–RCP). This should be due to the fact that the SOI effect in the 250 µm thick LN crystal is negligible. In comparison, for the thick BBO crystal, the result of using the cross-polarization components (LCP–RCP) is comparable to that of the co-polarization components (LCP–LCP). The intensity ratio ${I_{{\rm{RCP}}}}/{I_{{\rm{LCP}}}} \approx 1$ of the two components is close to the theoretical upper limit (50%) of a SOI process [41]. From the OAM measurement, we can see that the transmitted light carries angular momentum states of ${(1,0)_{775}}$ and ${(- 1,2)_{775}}$ for the case of Gaussian beam incidence, and ${(1, - 2)_{775}}$ and ${(- 1,0)_{775}}$ for the case of vortex beam incidence.

B. Symmetry-Selective Polarization Properties of the SPDC Twin Photons from the Thin LN Crystal

Next, we will discuss the SPDC process. From previous studies [31–34], it is known that the non-zero second-order nonlinear susceptibility tensors of a nonlinear crystal with threefold symmetry are $\chi _{\textit{yyy}}^{(2)} = - \chi _{\textit{yxx}}^{(2)} = - \chi _{\textit{xxy}}^{(2)} = - \chi _{\textit{xyx}}^{(2)} = \chi _1^{(2)}$ (${{\rm{C}}_{3v}}$) and $\chi _{\textit{xxx}}^{(2)} = - \chi _{\textit{xyy}}^{(2)} = - \chi _{\textit{yyx}}^{(2)} = - \chi _{\textit{yxy}}^{(2)} = \chi _2^{(2)}$ (${{\rm{C}}_{3h}}$). When a circularly polarized pump wave is propagating along the C3 rotational axis of a nonlinear crystal, the twin photons should have the same circular polarization state, which is the opposite of that of the pump wave, i.e., $|\sigma ,0{\rangle _{{\rm pump}}} \to \;\sum\limits_{\delta = 0, \pm 1, \pm 2, \ldots} {{c_\delta}| - \sigma ,\delta {\rangle _s}| - \sigma , - \delta {\rangle _i}}$, where the subscripts $s$ and $i$ represent the generated signal and idler photons and ${c_\delta}$ represents the complex amplitude. This can be predicted from the conservation law of SAM in nonlinear optical processes, ${\sigma _{{\rm pump}}}\hbar + pm\hbar = {\sigma _s}\hbar + {\sigma _i}\hbar$. In this formula, $m = 3$ describes the rotational symmetry of the nonlinear crystal and $p$ is an integer (see more details in Section S3 of Supplement 1). To verify the SAM-dependent symmetry-selective properties of the SPDC twin photons, a thin LN crystal with a negligible linear SOI effect [Fig. 2(a)] was used. A schematic diagram of the experimental setup used to characterize the properties of the SPDC twin photons is shown in Fig. 3(a). The generated SPDC twin photons were separated and individually projected to LCP and RCP states, and finally coupled to the 1550 nm single-mode fibers. The second-order correlation function ${g^{(2)}}$ of the collected photons was measured and shows typical characteristics of a two-photon state, indicating the SPDC process (see Section S4 of Supplement 1). Under the illumination of a pump wave with LCP ${(1,0)_{775}}$ and RCP ${(- 1,0)_{775}}$ states, the coincidence properties of the four polarization states of the SPDC twin photons $|RR{\rangle _{s,i}}$, $|RL{\rangle _{s,i}}$, $|LR{\rangle _{s,i}}$, and $|LL{\rangle _{s,i}}$ were measured and are shown in Figs. 3(b) and 3(c), respectively. The clear contrasts of the polarization-dependent coincidence of the twin photons strongly support the aforementioned symmetry selection rules in the SPDC process.

 

Fig. 3. Symmetry-selective polarization properties of the SPDC twin photons from the thin LN crystal. (a) Illustration of the experimental setup. First, the 250 µm thick LN was pumped by a focused Gaussian beam (green) with LCP and RCP states. Then the generated twin photons in the crystal (red) were separated and projected to different polarization states. (b), (c) The normalized coincidence of the polarization-resolved SPDC twin photon states at a wavelength of 1550 nm. When the LN crystal was pumped with (b) LCP and (c) RCP states, the polarization components of the twin photons were analyzed. HWP, half-wave plate; QWP, quarter-wave plate; PBS, polarization beam splitter; BS, beam splitter; CF, color filter; SF, single-mode fiber; TCSPC, time-correlated single-photon counting.

Download Full Size | PDF

To further verify the polarization quantum correlation in this process, we measured the polarization characteristic curves and the quantum state tomography of the twin photons. The coincidence of the photons with different polarization states was measured by rotating the half-wave plates (HWPs) in the two collection arms. The experimental results can be fitted well by a sine curve, confirming that the polarization states of the twin photons are $|RR{\rangle _{s,i}}$ or $|LL{\rangle _{s,i}}$ (see Fig. S6 of Supplement 1). In addition, quantum state tomography tests were performed to obtain the density matrix of the SPDC photon pairs, which contains the full polarization information of the two-photon state [53]. The reconstructed density matrices of the twin photons, corresponding to the pump waves with LCP and RCP states, are given in Fig. S7 of Supplement 1. It is found that the density matrices, which were retrieved from the experimental results, agree well with those of the ideal two-photon polarization states $|RR{\rangle _{s,i}}$ and $|LL{\rangle _{s,i}}$. Measurement fidelities of $0.94 \pm 0.04$ and $0.91 \pm 0.02$ were achieved for the pump waves with LCP and RCP states, respectively, clearly confirming our prediction on the polarization properties of the twin photons in the SPDC process in a nonlinear crystal with C3 rotational symmetry.

C. Optical Spin–Orbit Interaction in the SPDC Process from the Thick BBO Crystal

Next, we plan to explore the properties of SPDC twin photons generated in a thick BBO crystal. As shown in Fig. 4(a), when a pump wave with an angular momentum state of ${(\sigma ,\ell)_{775}}$ is focused onto a BBO crystal along its C3 rotational axis, the pump wave after the SOI process consists of two angular momentum components of ${(\sigma ,\ell)_{775}}$ (direct channel) and ${(- \sigma ,\ell + 2\sigma)_{775}}$ (SOI channel) [Fig. 2(b)]. Based on the symmetry selection rules and the conservation law of OAM, the pump waves generate twin photons with angular momentum modes of $\sum\limits_{\delta = 0, \pm 1, \pm 2, \ldots} {{c_\delta}| - \sigma ,\ell /2 + \delta {\rangle _s}| - \sigma ,\ell /2 - \delta {\rangle _i}}$ and $\sum\limits_{\delta = 0, \pm 1, \pm 2, \ldots} {{c_\delta}|\sigma ,\ell /2 + \sigma + \delta {\rangle _s}|\sigma ,\ell /2 + \sigma - \delta {\rangle _i}}$, respectively. In principle, the SPDC twin photons can experience a cascaded SOI effect in the BBO crystal, as was demonstrated in the second-harmonic generation process [41]. Therefore, SPDC twin photons with the new angular momentum states of $\sum\limits_{\delta = 0, \pm 1, \pm 2, \ldots} {{c_\delta}} |\sigma ,\ell /2 - 2\sigma + \delta {\rangle _s}|\sigma ,\ell /2 - 2\sigma - \delta {\rangle _i}$ and $\sum\limits_{\delta = 0, \pm 1, \pm 2, \ldots} {{c_\delta}} | - \sigma ,\ell /2 + 3\sigma + \delta {\rangle _s}| - \sigma ,\ell /2 + 3\sigma - \delta {\rangle _i}$, respectively, can also be generated. Due to the fact that the coincidence of the twin photons is relatively low, we will only pay attention to the first SOI effect and its impact on the SPDC process.

 

Fig. 4. Optical spin–orbit interaction in the SPDC process from the thick BBO crystal. (a) Two-step depiction of the optical spin–orbit interaction in the SPDC process. For a focused pump wave with an angular momentum state of ${(\sigma ,\ell)_{775}}$, two kinds of angular momentum states ${(\sigma ,\ell)_{775}}$ and ${(- \sigma ,\ell + 2\sigma)_{775}}$ will be generated through the direct and SOI channels. Then the angular momentum states of the SPDC twin photons are $\sum\limits_{\delta = 0, \pm 1, \pm 2, \ldots} {{c_\delta}| - \sigma ,\ell /2 + \delta {\rangle _s}| - \sigma ,\ell /2 - \delta {\rangle _i}}$ and $\sum\limits_{\delta = 0, \pm 1, \pm 2, \ldots} {{c_\delta}|\sigma ,\ell /2 + \sigma + \delta {\rangle _s}|\sigma ,\ell /2 + \sigma - \delta {\rangle _i}}$. (b) Illustration of the experimental setup. The OAM states of the pump waves were prepared through the $q$-plate and the HWP. Then the generated twin photons in the crystal (red) were collected by a single-mode fiber to measure the second-order correlation functions and coincidence. (c), (d) The normalized coincidence of the SPDC twin photon states for six different angular momentum states $\{{{{(\pm 1,0)}_{775}},{{(\pm 1, + 2)}_{775}},{{(\pm 1, - 2)}_{775}}} \}$ of the pump waves. The polarization states of the SPDC twin photons are fixed at (c) $|LL{\rangle _{s,i}}$ and (d) $|RR{\rangle _{s,i}}$. HWP, half-wave plate; QWP, quarter-wave plate; PBS, polarization beam splitter; BS, beam splitter; CF, color filter; SF, single-mode fiber; TCSPC, time-correlated single-photon counting.

Download Full Size | PDF

To verify the SOI effect in the SPDC process from the thick BBO crystal, we focus on the first two angular momentum states $\sum\limits_{\delta = 0, \pm 1, \pm 2, \ldots} {{c_\delta}| - \sigma ,\ell /2 + \delta {\rangle _s}| - \sigma ,\ell /2 - \delta {\rangle _i}}$ and $\sum\limits_{\delta = 0, \pm 1, \pm 2, \ldots} {{c_\delta}|\sigma ,\ell /2 + \sigma + \delta {\rangle _s}|\sigma ,\ell /2 + \sigma - \delta {\rangle _i}}$ of the twin photons. As the symmetry selection rules of the SPDC process have been proved in thin LN crystals, the experimental setup [Fig. 4(b)] was simplified to keep the polarization of the twin photons the same as each other during the measurements. Six kinds of angular momentum states $\{{({\rm{\pm 1}},{{0}})_{775}},\;{({\rm{\pm 1}},{\rm{+ 2}})_{775}},\;{({\rm{\pm 1}},{\rm{- 2}})_{775}}\}$ of the pump waves were used to excite the BBO crystal, and only photon pairs with ${\ell _{s,i}} = 0$ were efficiently collected to give coincidence. The OAM states (${\ell _{775}} = \pm 2$) of the pump waves were generated by using a $q$-plate (see the Methods section in Supplement 1). As shown in Figs. 4(c) and 4(d), the normalized coincidence of the SPDC twin photons was characterized at the polarization states of $|LL{\rangle _{s,i}}$ and $|RR{\rangle _{s,i}}$, respectively. In the two histograms, it can be seen that the normalized coincidence of the SPDC twin photons is much higher for the pump waves with angular momentum states of ${(\pm 1,0)_{775}}$ and ${(\pm 1, \mp 2)_{775}}$. The coincidence peak located at ${(\pm 1,0)_{775}}$ corresponds to the direct channel $(\pm 1,{0_{775}}\to \ \sum\limits_{\delta =0,\pm 1,\pm 2,\ldots }{{{c}_{\delta }}|\mp 1,\delta {{\rangle }_{s}}|\mp 1,-\delta {{\rangle }_{i}}}$, which has been confirmed in the thin crystal experiment. In comparison, the coincidence peak for ${(\pm 1, \mp 2)_{775}}$ represents the SOI channel ${(\pm 1, \mp 2)_{775}}\mathop \to \limits^{{\rm{SOI}}} \;{(\mp 1,0)_{775}} \to \;\sum\limits_{\delta = 0, \pm 1, \pm 2, \ldots} {{c_\delta}| \pm 1,\delta {\rangle _s}| \pm 1, - \delta {\rangle _i}}$, in which the new angular momentum states of the pump waves are generated through the SOI process. The observed results can be easily understood because the OAM mode with strongest amplitude within a broadband OAM spectrum of the SPDC twin photons is shifted to ${\ell _{s,i}} = 0$, which can be efficiently collected by using a single-mode fiber.

The two angular momentum states ${(\pm 1,0)_{775}}$ and ${(\pm 1, \mp 2)_{775}}$ of the pump waves were chosen to further confirm the SOI effect and the polarization properties of the SPDC twin photons. While the polarization states of the twin photons were continuously changed through the HWPs in the collection arm, the corresponding coincidence values were measured and are given in Section S5 of Supplement 1. It is shown that the experimental results agree well with the theoretical predictions, from which the polarization states of the twin photons were confirmed to be $|LL{\rangle _{s,i}}$ and $|RR{\rangle _{s,i}}$. In addition, the circular polarization states of the twin photons are opposite to each other for the pump waves with angular momentum states ${(\pm 1,0)_{775}}$ and ${(\pm 1, \mp 2)_{775}}$, further confirming the SOI process of the pump wave with OAM values of ${\ell _{775}} = \pm 2$.

3. DISCUSSION

In summary, we have investigated the optical SOI within the quantum framework of the SPDC process in nonlinear crystals with C3 symmetry. It is found that the SAM states of the twin photons are opposite to that of the pump wave, verifying the SAM-dependent symmetry selection rules in the SPDC process. In the direct channel, the OAM is conserved during the SPDC process. However, it is not conserved in the SOI channel, going from $\ell$ to $\ell + 2\sigma$ for the pump wave $(\sigma ,\ell)$. The total angular momentum of the photons within the two channels experiences a crystal-imparted shift of $\mp 3\sigma$. By judiciously choosing the angular momentum states of the pump waves, the SOI pathways of the SPDC process were experimentally verified. The symmetry selection rules and the optical SOI provide a novel route for controlling and generating various kinds of angular-momentum-based entangled photon states, such as polarization entanglement, high-dimensional OAM entanglement, and spin–orbital hyperentanglement. The yield of the entangled photons in the crystal could be greatly enhanced by introducing artificial structures into the crystal or the design of a Fabry–Perot cavity. An efficient angular-momentum-based entangled photon source would definitely find more practical applications in high-dimensional quantum information processing, including quantum communication, quantum cryptography, etc.