Multidimensional entanglement transport through single-mode fiber

Concept and principle

The concept and principle of multidimensional spin-orbit entanglement transport through SMF are illustrated in Fig. 1. Light beams carrying OAM are characterized by a helical phase front of exp(iℓθ) (43), where θ is the azimuthal angle and ℓ ∈ [−∞, +∞] is the topological charge. This implies that OAM modes, in principle, form a complete basis in an infinitely large Hilbert space. However, the control of such high-dimensional states is complex, and their transport requires custom channels, e.g., specially designed custom multimode fiber. On the contrary, polarization is limited to just a two-level system but is easily transported down SMF. Here, we compromise between the two-level spin entanglement and the high-dimensional OAM entanglement to transport multidimensional spin-orbit hybrid entanglement. A consequence is that the entire high-dimensional OAM Hilbert space can be accessed but two dimensions at a time. We will demonstrate that, in doing so, we are able to transport multidimensional entanglement down conventional fiber.

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To see how this works, consider the generation of OAM-entangled pairs of photons by spontaneous parametric down-conversion (SPDC). The biphoton state produced from the type I SPDC process can be expressed in the OAM basis as∣ψ〉AB=∑ℓcℓ∣H〉A∣H〉B∣ℓ〉A∣−ℓ〉B(1)where ∣c_ℓ∣² is the probability of finding photons A and B in the eigenstates ∣±ℓ〉, respectively, and ∣H〉 is the horizontal polarization. Subsequently, one of the photons (e.g., photon A), from the N-dimensional OAM-entangled photon pair, is passed through a spin-orbit coupling (SOC) optics for OAM to spin conversion, resulting in a hybrid multidimensional polarization (spin) and OAM-entangled state∣Ψℓ〉AB=12(∣R〉A∣ℓ1〉B+∣L〉A∣ℓ2〉B)(2)

Here, ∣R〉 and ∣L〉 are the right and left circular polarization (spin) eigenstates on the qubit space HA,spin of photon A, and ∣ℓ₁〉 and ∣ℓ₂〉 denote the OAM eigenstates on the OAM subspace HB,orbit of photon B. The state in Eq. 2 represents a maximally entangled Bell state where the polarization degree of freedom of photon A is entangled with the OAM of photon B. Each prepared photon pair can be mapped onto a density operator ρ_ℓ = ∣Ψ^ℓ〉〈Ψ^ℓ∣ with ∣Ψ^ℓ〉 ∈ HA,spin ⊗ HB,orbit by actively switching between OAM modes sequentially in time: The larger Hilbert space is spanned by multiple two-dimensional subspaces, i.e., multidimensional states in the quantum channel. For simplicity, we will consider subspaces of ∣±ℓ〉 but stress that any OAM subspace is possible, only limited by the SOC device characteristics. We can represent the density matrix of this system asρAB=∑ℓ=1∞γℓρℓ(3)where γ_ℓ represents the probability of post-selecting the hybrid state ρ_ℓ. The density matrix pertaining to system related to photon B isρB=TrA(ρAB)=∑ℓ=0∞γℓ2(∣ℓ〉B〈ℓ∣B+∣−ℓ〉B〈−ℓ∣B)(4)where Tr_A(•) is the partial trace over photon A. While photon B is in a superposition of spatial modes (but a single spin state), photon A is in a superposition of spin states but only a single fundamental spatial mode (Gaussian), i.e., ∣ψ〉_A ∝ (∣L〉 + ∣R〉)∣0〉 and ∣ψ〉_B ∝ ∣H〉(∣ℓ〉 + ∣−ℓ〉). As a consequence, photon A can readily be transported down SMF while still maintaining the spatial mode entanglement with photon B. We stress that we use the term “multidimensional” as a proxy for multi-OAM states due to the variability of OAM modes in the reduced state of photon B: Any one of these infinite possibilities can be accessed by suitable SOC optics, offering distinct advantages over only 1 two-dimensional subspace as is the case with polarization.

Implementation

We prepare the state in Eq. 2 via postselection from the high-dimensional SPDC state. In this paper, the SOC optics is based on q plates (44). The q plate couples the spin and OAM degrees of freedom following∣R〉∣ℓ〉→q−plate∣L〉∣ℓ−2q〉,   ∣L〉∣ℓ〉→q−plate∣R〉∣ℓ+2q〉(5)where q is the topological charge of the q plate. Accordingly, the circular polarization eigenstates are inverted, and an OAM variation of ±2q is imparted on the photon depending on the handedness of the input circular polarization (spin) state. Transmission of photon A through the SMF together with a detection acts as a postselection, resulting in the desired hybrid state (see the Supplementary Materials). Hence, considering N different OAM states ±ℓ_n and N different q plates with different topological charge q_n (such as ℓ_n ± 2q_n = 0) would allow us to access 2 × N dimensions using a single SMF.

The experimental setup for multidimensional spin-orbit entanglement transport through SMF is shown in Fig. 2A. A continuous-wave pump laser (Cobolt MLD diode laser, λ = 405 nm) was spatially filtered by a pinhole with a diameter of 100 μm to deliver 118 mW of average power in a Gaussian beam at the nonlinear crystal [a temperature-controlled 10-mm-long periodically poled potassium titanyl phosphate (PPKTP) crystal], generating two lower-frequency photons by means of a degenerate type I SPDC process. By virtue of this, the signal and idler photons had the same wavelength (λ = 810 nm) and polarization (horizontal). The residual pump beam was filtered out by a band-pass filter with a center wavelength of 810 nm and a bandwidth of 10 nm. The two correlated photons, signal and idler, were spatially separated by a 50:50 beam splitter (BS), with the signal photon A interacting with the SOC optics, e.g., q plate, for orbit-to-spin conversion. Subsequently, photon A was coupled into the 250-m SMF by a 20× objective lens to transmit through the fiber and coupled out by another 20× objective lens. The idler photon B was imaged to the spatial light modulator (SLM) with lenses f₃ and f₄. After that, photon B was imaged again by f₅ and a collimator, being coupled into an SMF, and hence spatial filtered as well, for detection.

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The projective measurements were done by the quarter–wave plate (QWP) along with a polarizer for photon A and SLM along with an SMF for photon B. Photon A, encoded with polarization eigenstates, was transmitted through the SMF, while photon B, encoded with multidimensional OAM eigenstates, was transported through free space. Last, both photons were detected by the single-photon detectors, with the output pulses synchronized with a coincidence counter.

Hybrid entanglement transport

We first evaluate the SMF quantum channel by measuring the OAM mode spectrum after transmitting through 250 m of SMF. Figure 2 (B and C, respectively) shows the mode spectrum of the ℓ = ±1 and ℓ = ±2 subspaces. We project photon A onto right circular polarization (R), left circular polarization (L), horizontal polarization (H), and vertical polarization (V) by adjusting the QWP and polarizer at the output of the fiber while measuring the OAM of photon B holographically with an SLM and SMF. The results are in very good agreement with a channel that is impervious to OAM. When the orthogonal polarization states are selected on photon A (∣L〉 and ∣R〉), an OAM state of high purity is measured for photon B, approximately 93% for ℓ = ±1 and approximately 87% for ℓ = ±2. The slightly lower value for the ℓ = ±2 subspace is due to concatenation of two SOC optics for the hybrid entanglement step (see the Supplementary Materials and Methods).

To confirm the state and the entanglement conservation, we perform a full quantum state tomography on the hybrid state to reconstruct the density matrix. Figure 3 shows the state tomography measurements and resulting density matrices for both the ℓ = ±1 and ℓ = ±2 subspaces after 250 m of SMF, with the free space ℓ = ±1 shown as a point of comparison (see the Supplementary Materials for more results in free space and in 2 m of SMF). The fidelity against a maximally entangled state is calculated to be 95% for the ℓ = ±1 and 92% for the ℓ = ±2 subspaces. This confirms that the fiber largely maintains the fidelity of each state. Using concurrence (C) as our measure of entanglement (see Methods), we find C = 0.91 for free space, down slightly to C = 0.9 for ℓ = ±1 and C = 0.88 for ℓ = ±2.

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The state tomography results depicted for each of the subspaces have different average coincidence count rates due to several small imperfections during the experimental realization: different collection efficiency depending on the detected spatial mode, temperature fluctuations in the nonlinear crystal oven causing a slight reduction in the SPDC efficiency, and some error in the rotation of the wave plates may have caused differences in the state tomography outcome depending on the projections. Nonetheless, the fidelity of the free-space density matrix is still higher owing to the minimal cross-talk between the orthogonal modes (across the diagonal).

To carry out a nonlocality test in the hybrid regime, we define the two sets of dichotomic observables for A and B: the bases a and a˜ of Alice correspond to the linear polarization states {∣H〉, ∣V〉} and {∣A〉, ∣D〉}, respectively, while the bases b and b˜ of Bob correspond to the OAM states {cos(π8)∣ℓ〉−sin(π8)∣−ℓ〉,−sin(π8)∣ℓ〉+cos(π8)∣−ℓ〉} and {cos(π8)∣ℓ〉+sin(π8)∣−ℓ〉,sin(π8)∣ℓ〉−cos(π8)∣−ℓ〉}. From the data shown in Fig. 4, we calculate the Clauser-Horne-Shimony-Holt (CHSH) Bell parameters in free space and through SMF (see Methods). We find CHSH Bell parameters of S = 2.77 ± 0.06 and S = 2.47 ± 0.09 for the ℓ = ±1 subspace in free space and in 250 m of SMF, respectively, reducing to S = 2.51 ± 0.04 and S = 2.25 ± 0.19 for the ℓ = ±2 subspace. In all cases, we violate the Bell-like inequality.

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A hybrid quantum eraser

Next, we use the same experimental setup to demonstrate a hybrid quantum eraser across 250 m of SMF. We treat OAM as our “path” and the polarization as the “which path” marker to realize a quantum eraser with our hybrid entangled photons. We first distinguish the OAM (path) information in the system by marking the OAM eigenstates of photon B with linear polarizations of photon A. In this experiment, we achieve this by placing a QWP before a polarizer, transforming Eq. 2 to∣Ψ˜ℓ〉AB=12(∣H〉A∣ℓ1〉B+∣V〉A∣ℓ2〉B)(6)

By selecting either polarization states, ∣H〉 or ∣V〉, the distribution of photon B collapses on one of the OAM eigenstates, ℓ_1,2, having a uniform azimuthal distribution and fringe visibility of V≡(max−min)(max+min)=0, reminiscent of the smeared pattern that is observed from distinguishable (noninterfering) paths in the traditional quantum eraser (45). The OAM information can be erased by projecting photon A onto the complimentary basis of the OAM markers, i.e., ∣±〉=12(∣H〉±∣V〉), causing the previously distinguished OAM (paths) to interfere, thus creating azimuthal fringes that can be detected with an azimuthal pattern sensitive scanner. The fringes appear with a visibility of V = 1, indicative of OAM information reduction (46). The appearance in azimuthal fringes of photon B is indicative of OAM information being erased from photon B. It is noteworthy to point out that, here, the QWP acts as the path marker, while the polarizer acts as the eraser. Notably, complementarity between path information and fringe visibility (V) is essential to the quantum eraser. By defining the two distinct paths using the OAM degree of freedom, we find that it is possible to distinguish (V = 0.05 ± 0.01) and erase (V = 0.98 ± 0.002) the OAM path information of a photon through the polarization control of its entangled twin in free space, as shown in Fig. 5. With only marginal loss of visibility after transmitting through 250 m of SMF, the entanglement is conserved with the ability to distinguish (V = 0.11 ± 0.01) and erase (V = 0.93 ± 0.01) the OAM path information.

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Although the eraser procedure was performed on the ℓ = ±1 subspace, it can be extended to higher subspaces: Depending on the SOC device that is used for photon A, the SLM holograms implemented for scanning the azimuthal fringes of photon B can be adapted to the relevant subspace.