Orthogonal quasi-phase-matched superlattice for generation of hyperentangled photons

A crystal superlattice structure featuring nonlinear layers with alternating orthogonal optic axes interleaved with orthogonal poling directions, is shown to generate high-quality hyperentangled photon pairs via orthogonal quasi-phase-matched spontaneous parametric downconversion. We demonstrate that orthogonal quasi-phase matching (QPM) processes in a single nonlinear domain structure correct phase and group-velocity mismatches concurrently. Compared with the conventional two-orthogonal-crystals source and the double-nonlinearity single-crystal source, the orthogonal QPM superlattice is shown to suppress the spatial and temporal distinguishability of the generated photon pairs by several orders of magnitude, depending on the number of layers. This enhanced all-over-the-cone indistinguishability enables the generation of higher fluxes of photon-pairs by means of the combined use of (a) long nonlinear crystal in noncollinear geometry, (b) low coherence-time pumping and ultra-wide-band spectral detection, and (c) focused pumping and over-the-cone detection. While each of these three features is challenging by itself, it is remarkable that the orthogonal QPM superlattice meets all of these challenges without the need for separate spatial or temporal compensation.


Results
Quantum state. The nonlinear domain structure proposed in this paper is made of M contiguous parallel layers constructed by interleaving two sets of layers with orthogonal optic axes pointing along the x and y axes, as illustrated in Fig. 1a, and referred heretofore as the H and V layers. Elements of the H and V layers have thicknesses d H and d V , respectively, so that the overall length of the structure is = + L d d M ( ) 1 2 H V . We consider the parametric downconversion of an intense linearly polarized pump beam propagating along the axis of the structure (the z axis) in the type-0 (eee) regime. The interacting component of the pump field, the signal, and the idler fields are linearly polarized along the optic axes of each of the layers. The SPDC process is enhanced by means of QPM using periodic poling of the H and V layers along their respective optic axes.
This alternating structure may also be regarded as a unit set made of a pair of successive layers, forming a cascade of crossed layers, repeated M/2 times. The thickness of this two-layer unit set is d = d H + d V , and the overall length is L = Md/2. Each two-layer unit set supports nonlinear interactions in both the vertical (V) and the horizontal (H) directions, in sequence. Embedded in this overall structure is a periodic reversal of the nonlinearity sign, which enables concurrent QPM for both the H and V polarizations, in a distributed coherent fashion.
The alternation of the H and V layers suppresses the temporal which-layer information by realizing group-velocity matching between the HH and VV biphotons, thereby limiting their longitudinal walkoffs, as illustrated in the space-time diagrams in Fig. 1b, which demonstrates the clear superiority of the superlattice structure. In the space domain, the suppression of which-layer information is realized when the emission angle is much smaller than the ratio of the pump waist to the domain thickness 23 . This is also applicable in the presence of transverse walkoff. For the superlattice structure, this ratio is extremely large (thickness is in the order of few micrometers) compared with the cascaded implementations, where the thickness is the crystal length L itself.
The pump beam is treated classically as a superposition of monochromatic plane waves and its H and V components are expressed as , with angular frequencies ω p and wave vectors k p = (q p , κ p ) where q p are the components along the transverse coordinates x = (x, y), κ p are the longitudinal components, and σ H,V are the relative amplitudes of H and V polarization components.
The downconverted waves are also expressed in similar spectral and spatial expansions in terms of monochromatic planar waves with angular frequencies ω 1,2 and wave vectors k 1,2 = (q 1,2 , κ 1,2 ), traveling through the sequence of uniaxial anisotropic layers of the domain. Maxwell's equations dictate that κ ω , for the extraordinary (e) polarized field 46 . The horizontal and vertical components of the pump generate H and V downconverted photon pairs from the successive H and V layers of the structure, respectively, and the design of the domain structure is such that it is equally probable for a pump photon to downconvert in any of the M layers in one of the two polarizations. The superimposed SPDC radiation creates an entangled state described at small emission angles 48  where Φ H,V (ω 1 , ω 2 ; q 1 , q 2 ) are normalized weight functions representing the two-photon amplitudes over all spatial and spectral modes. Each pump plane-wave component is responsible for the generation of a pair of downconverted waves, perfectly satisfying the conservation of energy (ω 1 + ω 2 = ω p ) and transverse momentum (q 1 + q 2 = q p ). The two-photon amplitudes can therefore be written as (see Methods) where χ (2) is the second-order susceptibility of the bulk domain material, κ is the phased-array function, which has a maximum value of M/2 for x = 0. This corresponds to satisfying the QPM conditions  (3) determine the layers thicknesses d H,V required to obtain superimposed H and V SPDC cones centered at some particular emission angle.
The polarization, spatial, and spectral variables in the biphoton quantum state described by equation (1) are inherently coupled so that its hyperentangled nature cannot be directly observed. Nevertheless, the entanglement of one degree of freedom can be assessed by tracing out the two other degrees of freedom, as we will subsequently show.
Ideally, for maximally entangled polarization, the complex superposition weights Φ H and Φ V should be equal for all frequencies and transverse wave vectors. In reality, they are not, and their relative phase plays a major role in limiting the indistinguishability of the two components of the state. Equations similar to equation (2) apply to the CC source, which has two cascaded periodically-poled orthogonal crystals featuring type-0 downconversion, albeit with a different QPM condition Note that the first term in equation (6) is much greater than the second (phase-matching) term, and generally dominates within the significant spectral and spatial regions. For the DN source 36 , the concurrent type-0 and type-I interactions in a single periodically poled crystal produce the superposed HH and VV possibilities, respectively. The magnitude of the VV amplitude is therefore similar to the cascaded-crystals case [including the QPM condition ϕ ∆ V ], and the HH amplitude is given by with the relative phase be where Δκ ooe is the longitudinal wave vector mismatch for type-I (ooe) interaction, which is canceled out in the central emission direction via temperature tuning. We now compare the polarization and spatiotemporal characteristics of the hyperentangled photons emitted by the three sources. For comparison on an equal footing, all structures are made of LiNbO 3 crystals of 5-mm length featuring third-order QPM. We consider a typical experiment with a pump centered at 532 nm illuminating each of the three structures and producing nondegenerate and noncollinear SPDC light centered at 810 nm (signal) and 1550 nm (idler). This example is of practical importance because the range of the signal photon is appropriate for either quantum memory via atomic ensemble interaction 49,50 or low-noise highly efficient detection using commercial silicon avalanche photodiode, while the idler photon features low-loss long-distance fiber communication within the third telecom window 51,52 . The QPM conditions in equations (3) and (5) give the design thicknesses d H ≈ d V with values 10.8 μm for the CC and DN sources, and 5 μm for the superlattice, all at a phase-matching temperature T = 114.41 °C. The pump is assumed to be pulsed with the normalized amplitude 53 which has separable spectral and spatial dependence. Here, ω p 0 is the central frequency, τ p is the pulse duration, and W p is the pump beam waist. For a planar pump we take W p to be very large.
Polarization characteristics. The quantum state (1) is a pure state in multiple degrees of freedom (polarization, spatial, and spectral) described by a density operator ρ with Tr(ρ 2 ) = 1. If the spatial and spectral degrees of freedom are traced out, the result is a mixed polarization state described by a reduced density operator ρ σ where Tr[(ρ σ ) 2 ] < 1, only if correlations exist between polarization and the removed degrees of freedom 54 26 , the off-diagonal element of the reduced polarization density matrix ρ σ HH,VV is a measure of the partial purity or the indistinguishability of the HH and VV photon pairs. This element is given by HH,VV VV,HH 1 2 1 2 H 1 2 1 2 V 1 2 1 2 so that it represents the degree of spatial and spectral overlap of Φ H and Φ V . This purity measure is plotted in Fig. 2 as a function of the structure length L for each of the three sources. The integrals in equation (10) cover the entire SPDC emission with no spatial or spectral windows used. At its maximum of 1 2 , the off-diagonal element describes a pure (completely indistinguishable) polarization state, while at zero it corresponds to a mixed and entirely distinguishable polarization state. It is evident that the superlattice is far superior to both the CC and DN sources since it relatively maintains indistinguishability over longer structures, thereby enabling the generation of much higher biphoton fluxes with greater purity.
Scientific RepoRts | 7: 4169 | DOI:10.1038/s41598-017-03023-1 Angular characteristics. Equation (1) describes the entire SPDC field emerging from the nonlinear structure into free space. To delineate the angular (directional) characteristics, we use two spectral filters with Gaussian and spectral width σ ω centered at the signal and idler central frequencies ω 1,2 0 and placed directly after the nonlinear structure. The polarization-analyzed outcomes have the angular distributions which represent the probability densities of photon coincidence at points in the far field corresponding to the transverse wave vectors q 1 and q 2 . Under a planar pump assumption q 2 ≈ −q 1 , so that P H,V (q 1 , q 2 ) are completely defined by the functions p H,V (q 1 ) = P H,V (q 1 , −q 1 ), which are plotted in Fig. 3a,b in the xz plane for the three structures. The angular distribution of the relative phase ϑ = arg{Φ V /Φ H } is plotted in Fig. 3d. Ideally, this surface is flat. The plot shows that both the superlattice and the double-nonlinearity crystals have relative-phase angular distributions that are much flatter than that of the cascaded crystals.
Also shown in Fig. 3c is the distribution of the off-diagonal element of the reduced density matrix, HH,VV 1 2 HH,VV 1 HH,VV 1 1 . For the cascaded and double-nonlinearity crystals, this parameter is tiny (averages are 4.4 × 10 −7 and 4.9 × 10 −2 , respectively), indicating a strong correlation between polarization and other degrees of freedom, which are particularly manifested under the challenging conditions considered here (long crystal, noncollinear geometry, wide spectral filter, and low-coherence-time pump). The higher values of ρ | | σ q ( ) HH,VV 1 near the edges of the SPDC cone may be attributed to the relatively weaker birefringence, but these are not relevant since the rates of emission are negligible in this region.
In contrast, the superlattice exhibits ρ | | σ q ( ) HH,VV 1 that is very close to its maximum value of 0.5 over a wide range, yielding an average about 0.4 over the entire emission cone. This remarkably high purity, particularly, near the central and diagonal regions, is attributed to the temporal indistinguishability resulting from group velocity matching, and the spatial indistinguishability, which is due to the approximately symmetric condition κ κ ≈ j e H j e V ( ) ( ) implicit in the dispersion relations for q j,x ≈ q j,y (j = 1, 2, p). Analogous symmetries can then emerge for other anisotropy-affected quantities like κ ∆ e H,V and Δϕ H,V , assuring the almost full symmetry of the two-photon wave functions Φ H,V (ω 1 , ω 2 ; q 1 , q 2 ). Realizing such a high value of ρ | | q ( ) HH,VV 1 reveals, as will be seen later, an underlying near-perfect temporal indistinguishability.
The angular characteristics are compared in Fig. 4 for a focused pump. The angular distributions of the coincidence probabilities, along with the relative-phase at the central wavelengths, are shown. It is evident that the phase gradients of the superlattice structure and the double-nonlinearity crystal are lower by more than one order of magnitude, compared with that of the cascaded crystals. Spatial characteristics. The biphoton angular spectrum described by equation (11) corresponds to a spatial distribution in the exit plane of the crystal (z = 0) 56 given by   This distribution, which may be observed by measuring photon coincidence at pairs of positions in an image of the crystal's exit plane, is depicted in Fig. 5 for points on the x axis. As expected, the relative-phase gradient in the angular (transverse momentum) domain has a significant effect on the distribution in the conjugate position domain. This relative phase exhibits a strong quadratic dependence on (q 1x − q 2x ) as depicted in Fig. 4d, which corresponds to a quadratic phase function of (x 1 − x 2 ) in the spatial domain, introducing distortion much like Fresnel diffraction along the position difference (x 1 − x 2 ).
For the cascaded-crystals, relative to the HH pairs (which have coincidences more likely happen near momentum-conserved modes), the VV pairs can have coincidences quite far from such a tight mapping. By contrast, it is obvious how the two-photon wavepackets of the superlattice and double-nonlinearity structures are well indistinguishable in space, owing to the low phase gradient compared with the cascaded crystals. Figure 4a,b exhibit the anti-correlation nature of the two-photon directions in the far field originating from the phase matching in the nonlinear domains. In the conjugate near-field zone, Fig. 5a,b display the correlation in the two-photon positions because of their co-located birth from a downconverted pump photon. Here, the VV biphotons produced by the cascaded crystal show weaker positional correlations, which denotes the migration of the transverse-spatial entanglement out of the wavefunction modulus to the associated phase in the coordinate space 57 . The migration-to-phase process is then reversed after arbitrary propagation 58 so that correlations are restored in the far field, as Fig. 4b shows.
We now consider entanglement in the spatial degree of freedom and its possible reduction as a result of distinguishing information from the polarization and frequency degrees of freedom. Several measures can be adopted, including Schmidt analysis of the biphoton wavefunction 59,60 . We use here the purity of the spatial biphoton state, which is obtained by tracing out other degrees of freedom (see Methods): and (x 1 − x 2 ), normalized to their respective maximum values. For the cascaded crystals (first row), the high quadratic relative-phase function in (q 1x − q 2x ) axis in Fig. 4 introduces a magnifying distortion to the joint spatial correlations along (x 1 − x 2 ) axis. The superlattice and double-nonlinearity crystal have nearly perfect spatial indistinguishability. (d) Spatial purity ρ λ ∆ Tr[ ( ) ] q 2 as a function of the spectral filter bandwidth Δλ (full width at half maximum). At Δλ = 0, the purity is unity for both the DN and SL structures, since the distinguishing frequency information is absent. As Δλ increases, the spatial purity is reduced, indicating that the spectral degree of freedom plays an increasing distinguishing role, together with the polarization degree of freedom.
Scientific RepoRts | 7: 4169 | DOI:10.1038/s41598-017-03023-1 For monochromatic detection, the frequency information is suppressed so that the distinguishability is only due to polarization. Note that emission from the cascaded crystals exhibits a strong effect of polarization on the spatial degree of freedom. The superlattice and double-nonlinearity structures avoid such effect by virtue of the interleaved and co-located downconversion, respectively, producing HH and VV biphotons. However, the spectral information appears to be a more dominant distinguishing effect for wider detection bandwidth. This effect originates from the inherent coupling between the spectral and spatial degrees of freedom in the SPDC two-photon state 55    are aperture functions, taken to be Gaussian with widths σ q 1,2 , assumed to be fixed within the bandwidths of interest. These spectral distributions are shown in Fig. 6.
It is useful here to think about the two-photon wavefunctions as being formulated due to two contributions; one of them is by the spatial-spectral modes of the interacting fields, while the other is for the geometry of the illuminated structure itself 46 . Along our comparisons, the contribution due to the interacting fields is always the same while the geometries of the nonlinear structures lead to the different outcomes. Also shown in Fig. 6 is the distribution of the relative phase between P V and P H evaluated at the central directions q 1,2 0 . Here too, the gradient of the relative phase is significantly smaller for the superlattice structure in comparison with double-nonlinearity and cascaded structures. Since the relative phase is approximately linear within the range for which the magnitudes of the functions are significant, this effect amounts to a corresponding shift in the time domain, as we will see next.
Temporal characteristics. In the time domain, the wavepackets of the HH and VV biphotons are described by the probability densities where t 1 and t 2 are the times at which the signal and idler photons cross the structure's exit plane (z = 0) 56 . These functions are displayed in Fig. 7 along with their projections along the time difference (t 1 − t 2 ) and the time average + t t ( ) 2 is mainly due to the group-velocity mismatch between the pump and downconverted photons, that in the time difference (t 1 − t 2 ) is due to the group-velocity mismatch between the nondegenerate signal and idler photons. Such time delay renders the two superposed terms of the state entirely distinguishable, so that the entangled area 26 effectively shrinks to zero. It is required that these relative delays be much less than the coherence time of the pump field in order to suppress the temporal distinguishing information. and P V (t 1 , t 2 ) along the diagonal and off-diagonal directions as functions of the average arrival time (t 1 + t 2 )/2 and the relative delay (t 1 − t 2 ) between the signal and idler, normalized to their respective maximum values. The relative time delay is due to first-order dispersion, while dispersion of higher order contribute to the biphoton wavepacket distortion. (d) Normalized coincidence rate C(τ), after HOM interferometer, expected for a monochromatic (gray line) and a 133-fs pulsed (red line) pump beam. Here, the apertures are pinholes placed in the central emission directions. The sinusoidal oscillation within the dip envelope is at frequency (ω 1 − ω 2 ).
For the double-nonlinearity crystal source, the delay between the two signals and two idlers of the HH and VV possibilities depends on their birth position, with the average time delay being [u o (ω 1,2 ) − u e (ω 1,2 )]L/2 ≈ {0. 73 ps, 0.64 ps}, where u e (ω j ) is the reciprocal group velocity for the extraordinary polarization at angular frequency ω j . The distinguishing dispersive delay between the faster idler and the slower signal belonging to the same pair is {[u o (ω 1 ) − u o (ω 2 )] − [u e (ω 1 ) − u e (ω 2 )]}L/2 ≈ 88 fs on average. These delays introduce partial distinguishability as there is always a region emitting temporally indistinguishable pairs at the end of the DN crystal.
For the superlattice source, emission has double the coherence time of the cascaded crystals due to the uncertainty regarding the emission origin along the whole structure length. The signals and the idlers of the HH and VV possibilities emerge with an average delay of [u o (ω p ) − u o (ω 1,2 )]d/2 ≈ {3.9 fs, 5.4 fs}. The average dispersive delay between the faster idler and the slower signal of the same pair is [u o (ω 1 ) − u o (ω 2 )]d/2 ≈ 1.5 fs. Both types of delay are proportional to the two-layer thickness d instead of the total length L of the structure, as in the case of the other two structures (see space-time diagrams in Fig. 1b).
Temporal (spectral) entanglement is often assessed via Hong-Ou-Mandel (HOM) interferometry 61 . For degenerate SPDC photons, the spectral symmetry of the joint two-photon amplitude, i.e., Φ(ω 1 , ω 2 ) = Φ(ω 2 , ω 1 ), leads to perfect visibility of HOM interference. For nondegenerate photons, a HOM-type two-color interferometer has been implemented 62 . For our nondegenerate system, which features spectral, spatial, and polarization degrees of freedom, we use light emitted in four directions falling on two rings within the emission cone and centered at modes With a time delay τ introduced at one port of the beam splitter, the probability of coincidence, i.e., detecting one photon at each port, is given by Therefore, the normalized coincidence rate C(τ) is given by and Re denotes the real part. The relative delay τ corresponds to translating one of the wavefunctions in Fig. 7a,b, with respect to the other, introducing a time difference (t 1 − t 2 ) = 2τ. The normalized coincidence rate C(τ) measures their degree of overlap. Note that the wavepackets are indistinguishable in terms of the mean time + t t ( ) 1 2 1 2 in case of monochromatic pumping. Figure 7d is a plot of C(τ) for the three SPDC sources in two cases: a monochromatic pump, and a pulsed pump producing a train of 133-fs pulses. Since the signal and idler are highly nondegenerate, the familiar HOM dips are modulated by sinusoidal oscillations at the beat frequency (ω 1 − ω 2 ). For the CC and DN sources, the HOM dips are time-shifted from τ = 0. This time offset represents the required temporal compensation, which is generally introduced by use of birefringent element(s). The dip depth (visibility) is a measure of the degree of temporal indistinguishability after optimal compensation. For the DN crystal, it is impossible to realize full visibility, even with a monochromatic pump, due to the discrepancy in the temporal biphoton wavepacket from type-0 and type-I interactions, as highlighted in Fig. 7a,b. When the HH and VV possibilities are due to type-0 interaction (CC and superlattice cases), the dip width is the wavepacket length (coherence time) of the downconverted biphoton. Note that the dip width for the superlattice source is twice that for the cascaded crystals. However, when a pulsed pump is used, only the superlattice source, which features continuous group-velocity matching (see Fig. 1), can realize nearly perfect visibility without temporal compensation.

Discussion
We have introduced a new superlattice structure with unit cell composed of two orthogonal nonlinear thin layers, one producing horizontally polarized biphotons and the other producing vertically polarized biphotons. QPM is integrated into the same superlattice by poling the periodic layers in alternating orthogonal directions. The SPDC biphotons are emitted in a maximally entangled superposition of the HH and VV states over an emission cone significantly wider than for other sources, and with a broader spectral width. The distinguishing information is limited by the thickness of the layers. Within the emission cone, the magnitudes of the quantum state superposition weights are almost identical and their spatial-spectral relative-phase gradient is smaller by several orders of magnitude compared to other sources, depending on the thickness of the layers and their number. Spectral and angular (transverse-momentum) entanglement are also preserved, so that the overall quantum state is hyperentangled.
The higher degree of spatial and temporal indistinguishability of the biphotons generated by the superlattice structure enables tight focusing of the pump into a long crystal, even in the noncollinear configuration -a combination that severely limits biphoton flux rates in ordinary sources. The superlattice SPDC source also supports the use of a pump with low coherence time and enables all-over-the-cone two-photon collection, all without the need for compensation.
The parameters and/or configuration of the superlattice structure presented in this paper may be modified in several ways to obtain quantum states with different features. For example, instead of using third-order QPM, which leads to a delay of about {3.9 fs, 5.4 fs} between the HH and VV possibilities, in a chirped structure with first-order QPM this delay may be compressed to less than a monocycle of light. This offers a hyperentangled version of the original proposal by Harris 43 to generate temporally entangled photon pairs correlated to less than one cycle of light. A superlattice chirp-and-compress design will generate noncollinear ultra-broadband biphotons with no need for large chirping 44,45 and no need for self-compensation to eliminate temporal distinguishability, and hence no need for an additional time-delay element.
The superlattice source can be readily adapted to produce a tailored two-photon polarization state similar to the one first suggested by Klyshko 63 . By use of two or three nonlinear layers per each set of the sequence an arbitrary biphoton polarization state α β γ δ 1 2 may be created with angular biphoton spectrum controlled by the geometry of the layered structure. While preserving the spectral and momentum entanglement, the output polarization state can surf the S 4 sphere in a five-dimensional space. The feasibility to create such arbitrary polarization state is of fundamental importance as there is no way to generate it by the use of polarization transformers (characterized by two parameters) to move between two arbitrary points on the S 4 sphere.
The fabrication of the proposed superlattice structure will be challenging. The implementation of interleaved periodic poling in alternating orthogonal directions will require the development of new poling processes. Fabrication of a periodic thin-layered structure that alternatingly emits biphotons of orthogonal polarization without orthogonal poling is feasible by engineering the grating periods. However, such structure will require additional elements to suppress temporal walkoff.

Methods
Two-photon amplitude. In our analysis, we assume weak scattering so that the SPDC process remains spontaneous and the pump field is undepleted. We also neglect reflected waves at all interfaces and assume that the nonlinear layers have infinite transverse extent so that all fields do not interact with the transverse boundaries. The two-photon wavefunctions of the hyperentangled state (1)  Here  is the number of the two-layer sets, counted from the entrance plane z = −L (to the right). The output state is thus a coherent superposition of all two-photon amplitudes along the structure layers which can be expressed as  where σ, q, and ω refer to the polarization, momentum, and frequency degrees of freedom, respectively. The nonzero elements of the reduced state are thus ρ ρ = =  The corresponding state purity after frequency filters is then as given by equation (14).

HOM-type interferometer for orthogonal amplitudes of nondegenerate, noncollinear
which leads directly to the coincidence probability in equation (17).