Abstract
One promising technique for working toward practical photonic quantum technologies is to implement multiple operations on a monolithic chip, thereby improving stability, scalability and miniaturization. The onchip spatial control of entangled photons will certainly benefit numerous applications, including quantum imaging, quantum lithography, quantum metrology and quantum computation. However, external optical elements are usually required to spatially control the entangled photons. Here we present the first experimental demonstration of onchip spatial control of entangled photons, based on a domainengineered nonlinear photonic crystal. We manipulate the entangled photons using the inherent properties of the crystal during the parametric downconversion, demonstrating twophoton focusing and beamsplitting from a periodically poled lithium tantalate crystal with a parabolic phase profile. These experimental results indicate that versatile and precise spatial control of entangled photons is achievable. Because they may be operated independent of any bulk optical elements, domainengineered nonlinear photonic crystals may prove to be a valuable ingredient in onchip integrated quantum optics.
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Introduction
During parametric downconversion, two lowerfrequency photons (usually called the signal and idler) are generated from a pump photon via a nonlinear crystal. Due to the conservation of energy and momentum of the original pump photon, the frequency and momentum of the signal and idler are strongly correlated. In particular, the spatial entanglement of the photon pair has led to interesting research in many fields, including quantum imaging^{1,2,3,4,5,6}, quantum lithography^{7,8,9}, quantum metrology^{10,11,12,13} and quantum computation^{14}. To prepare specific twophoton states for various applications, the spatial entanglement of the signal and idler is tailored by manipulating the wavefront of the pump beam^{15,16} or modifying the entangled photons after their generation^{17,18,19}, using various optical elements, such as lenses, multislits or spatial light modulators. These bulk optical elements inevitably hinder the performance of the photonic quantum circuits during practical applications, which require more stability, scalability and miniaturization^{20,21,22,23,24}.
The aforementioned difficulty can be overcome with a different strategy that applies domain engineering in quadratic nonlinear photonic crystals, which is widely used in quasiphasematching (QPM) nonlinear optics^{25,26,27,28,29}. Although the spatial control of entangled photons via domain engineering has been already theoretically proposed^{30}, few related experiments, other than a recent experiment showing that the amplitude of the entangled photons can be modulated by a multistripe nonlinear photonic crystal^{31}, have been attempted. Comprehensive control of spatial entanglement (particularly the phase of entangled photons) has still not been experimentally demonstrated.
In this work, we experimentally demonstrate the steering of entangled photons (that is, the wavefront shaping of the entangled photons) via domain engineering in nonlinear photonic crystals. By introducing a transverse inhomogeneity into the crystal, the wavefront of the entangled photons can be shaped at will, and the propagation of the entangled photons can be steered. We investigate a periodically poled lithium tantalate (PPLT) crystal with a transverse parabolic phase profile and find that the generated entangled photons are focused at a fixed distance from the crystal. In this case, our engineered crystal is equivalent to a homogeneous nonlinear crystal and a focusing lens (Fig. 1a). Additionally, by translating the crystal, we realize a dualfocusing condition in which the twophoton is focused onto either of two symmetric directions. Under this condition, our engineered crystal serves as the entangled photon source, lens and beam splitter (Fig. 1b).
Results
Sample structure
The quadratic nonlinear coefficient of our crystal is expressed as
where d_{eff} is the effective quadratic susceptibility, G_{m}=2mπ/Λ is the mthorder reciprocal vector for the longitudinal modulation with period Λ in the beam's propagation direction z, F(G_{m}) represents the corresponding Fourier coefficient and φ(x) and U(x) describe the nonuniform transverse phase and amplitude distributions, respectively. Because φ(x)=ω_{p}x^{2}/(2cfG_{3}) is parabolic, a twophoton carrying the phase front ω_{p}x^{2}/(2cf) is focused at a distance f from the crystal, where ω_{p} is the angular pump frequency, c is the speed of light, G_{3} is the thirdorder reciprocal vector that compensates for the longitudinal phase mismatching in the QPM and f represents the equivalent focal length of the crystal for the pump wavelength. The focusing behaviour can be described by the HuygensFresnel principle: each point on the primary wavefront acts as a source of secondary wavelets of the pump, as well as a source of the entangled photon pair. Therefore, such an engineered nonlinear photonic crystal is termed a QPM twophoton lens. By following the Huygens–Fresnel principle, we may engineer this crystal for any arbitrary spatial manipulation of entangled photons, such as focusing, beamsplitting or even multifunction integration.
Figure 2a is a micrograph of our etched crystal. The transverse direction has a multistripe pattern that means that the amplitude modulation U(x) in equation (1) takes the form of a grating with a stripe width of 10 μm and a stripe interval Λ_{tr}=20 μm. The initial position of each stripe follows the function ω_{p}x^{2}/(2cfG_{3}), and the equivalent focal length of the crystal at the pump wavelength is designed to be f=33.3 mm.
Singlefocusing experiments
The layout of our experiment is shown in Figure 2b. With the input tip of the fibre scanning in the transverse (x) and longitudinal (z) direction, we obtain the spatial correlation of entangled photon pairs by performing coincidence counting using the two detectors. The pump coincides with the centre of the parabolic PPLT sample. Figure 3a shows a simulation of the twophoton propagation dynamics based on the Huygens–Fresnel principle. Experimentally we obtained the minimum twophoton spatial correlation at a distance of z=33 mm, as can be seen in Figure 3b. The full width of the correlation peak, at which the intensity drops to 1/e^{2} of its maximum value, was 28 μm. In contrast, we theoretically expected a 24μm focusing spot with a full pump width of 0.82 mm in the experiment. The experimental value was slightly larger than the theoretical calculation primarily because the pump has a greater divergence angle than the TEM_{00} mode. In addition to the primary focusing spot along the pump beam direction, which is located at x=0 mm, two other focusing spots were distributed symmetrically at x=+0.78 mm and −0.76 mm. The minor focusing spots resulted from multistripe interference. The theoretical separation between the two adjacent focusing spots was 2πcf/(ω_{p}Λ_{tr})=0.76 mm, which is consistent with the experimental value.
Figure 3c shows an enlarged view of the simulated twophoton propagation dynamics around the focusing spot. Figure 3d shows the corresponding measured results. Whenever the fibre tip deviates from the focal plane, the measured twophoton spatial correlation widens. For example, at distances of z=29 mm and z=37 mm, the spatial correlation peak widths are 92 and 120 μm, respectively. The Rayleigh range is measured to be 0.9 mm. We used a TEM_{00} Gaussian beam to fit the focusing behaviour and found that the twophoton behaves similar to a Gaussian beam with the pump wavelength (rather than the signal or idler wavelength).
Dualfocusing experiments
It is worth emphasizing that we engineered the parabolic PPLT with a transverse multistripe structure rather than a continuous structure. This introduces periodicity into the transverse direction of the crystal. For the structure that we fabricated, in any two stripes separated by a distance of 2πcf/(ω_{p}Λ_{tr})=0.76 mm, the added phases of the entangled photon pairs are approximately equivalent. Thus, the stripes at the 2jπcf/(ω_{p}Λ_{tr}) positions, where j is an integer, are all equivalent principal axes of the QPM twophoton lens, which are denoted by P_{j} in Figure 4a. In our experiment, we find that when the crystal is translated in the x direction by multiples of 0.76 mm, the spatial distributions of the single counts and the twophoton coincidence counts remain identical.
A new type of spatially entangled state is generated when the incident pump lies halfway between the two principal axes: this situation is illustrated in Figure 4a. The twophoton is deflected and focused onto either of the two symmetric directions around the pump beam. Figure 4b shows the simulated propagation dynamics of the dualfocused twophoton. Figure 4c shows the measured twophoton spatial correlation and single counts in the focal plane. There are two narrow correlation peaks with equivalent intensities, whereas the single counts follow a relatively smooth distribution. The measured peak interval of 0.77 mm agrees well with the theoretical prediction of 2πcf/(ω_{p}Λ_{tr})=0.76 mm. The phasematching of the dual focusing is actually achieved using two tilted reciprocal vectors as shown in Figure 4e. For such concurrent spontaneous parametric downconversion processes, it has been experimentally verified that their contributions to the twophoton state have a fixed phase relation^{32}. Therefore, if we collect the photons in the two foci described here, we obtain a NOON state with N=2. As we change the incident position of the pump between the two principal axes, the proportions of the photon pairs focused onto the two directions can be dynamically tuned. In this case, the crystal serves as both the beam splitter (with a tunable splitting ratio) in the pump, and the lens.
Phase matching analysis
To obtain a better understanding of the working principle of the engineered crystal, we analysed the Fourier spectra of the multistripe parabolic PPLT, which are the measured farfield diffraction patterns of the crystal illuminated by the pump laser beam. Figure 4d corresponds to the singlefocusing case, in which the pump coincides with the principal axis, whereas Figure 4e corresponds to the dualfocusing case, in which the pump beam lies halfway between the two principal axes. The thirdorder reciprocal vectors, which are used in the QPM, are marked by G_{a} in Figure 4d and by G_{b} and G_{c} in Figure 4e. In the singlefocusing case, G_{a} ensures an efficient collinear downconversion and the twophoton is focused along the pump direction. In the dual focusing case, as G_{b} and G_{c} are not along the pump direction, the downconversion occurs using a noncollinear geometry. Hence, the twophoton is focused onto either of two possible directions, as shown in Figure 4c.
Discussion
We experimentally realized the onchip steering of entangled photons, based on a domainengineered nonlinear photonic crystal. Using a transversely parabolic PPLT, we demonstrated twophoton focusing and beamsplitting. Our measured results agree well with the designed parameters of the domain structure, which shows that accurate spatial control of entangled photons can be achieved via domain engineering. Because this approach enables the control of the amplitude and phase of the twophoton up to lithographic precision, unique optical elements, such as lenses with extremely small focal lengths and high numerical apertures, can be engineered. In our experiment, the crystal served as an entangled photon source, lens and beam splitter, as illustrated in Figure 1. This multifunctionality shows the potential of integrating multiple optical transformations, such as a battery of lenses, into a single crystal. This multifunctional integration, which is inherent to the crystal, is free from any bulk optical elements and, therefore, may be exploited for onchip integrated quantum optics. However, for a fully integrated device, other necessary functions (such as spectral filtering) should also be realized on the chip. This requires further consideration.
The flexible, stateoftheart crystal poling technique enabled the fabrication of a wide variety of domain structures, such that we could spatially control the entangled photons at will. Combined with the temporal control of the entangled photons using longitudinal domain engineering^{33,34,35,36,37,38}, more interesting twophoton states may be prepared from nonlinear photonic crystals. This may attract interest in both fundamental physics and practical quantum technologies. Our technique might also find applications in research fields, such as quantum walk and continuousvariable encoding.
Methods
Calculation of the twophoton correlation
Assuming a monochromatic plane pump wave, the generated entangled twophoton state is
where Ψ_{0} is a normalization constant, Φ(Δk_{z}L)=sin(Δk_{z}L/2)/(Δk_{z}L/2) is the longitudinal detuning function (in which Δk_{z}=k_{p}−k_{sz}−k_{iz}−G_{3} and L is the stripe length), H_{tr} is the transverse mode function (which is the Fourier transform of the transverse inhomogeneity, including the pump profile and the transverse domain structure), and are the transverse wave vectors of the signal and idler photons, respectively. For a transversely infinite and homogeneous crystal, the momentum correlation H_{tr} is . However, in our experiment, a parabolic phase profile has been introduced, and therefore the momentum correlation of the photon pair is modified. According to Glauber's theory, the spatial correlation as a function of x and z can be derived to be
where E_{1} and E_{2} are the electric fields evaluated at the two detectors and E_{p} is the pump field. Equation (3) shows that the twophoton exhibits its minimum spatial correlation at a distance of z=f; therefore the corresponding plane is termed the twophoton focal plane. For a transversely infinite and homogeneous crystal (that is, U(x′)=const.) in the focal plane, the twophoton spatial correlation is expressed as
For a plane pump wave with an infinite beam width, the twophoton is focused onto an infinitesimal point. For a Gaussian pump beam, the twophoton is focused onto a spot, the size of which is determined by the pump beam size w_{0} and the pump wavelength (rather than the signal or idler wavelength).
Additional information
How to cite this article: Leng, H. Y. et al. Onchip steering of entangled photons in nonlinear photonic crystals. Nat. Commun. 2:429 doi: 10.1038/ncomms1439 (2011).
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Acknowledgements
We thank Y. H. Shih for helpful discussions and comments during the preparation of the manuscript. The authors also thank Y. Yuan for help in fabricating the crystal. This work was supported by the National Natural Science Foundations of China (contract nos 11021403, 10904066 and 11004030), the State Key Program for Basic Research in China (nos 2011CBA00205 and 2010CB630703) and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
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X.Q.Y. conceived the study. X.Q.Y. fabricated the parabolic PPLT sample, with assistance from C.Z. in designing the parabolic pattern of the crystal. P.X. and H.Y.L. completed the theoretical deduction. H.Y.L., P.X. and X.Q.Y. performed the experiment, with assistance from Y.X.G., Z.D.X. and H.J., H.Y.L. and P.X. analysed the results and wrote the paper. S.N.Z supervised the study and commented on the paper.
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Leng, H., Yu, X., Gong, Y. et al. Onchip steering of entangled photons in nonlinear photonic crystals. Nat Commun 2, 429 (2011). https://doi.org/10.1038/ncomms1439
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DOI: https://doi.org/10.1038/ncomms1439
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