Orbital angular momentum multiplexed deterministic all-optical quantum teleportation

Quantum teleportation is one of the most essential protocol in quantum information. In addition to increasing the scale of teleportation distance, improving its information transmission capacity is also vital importance for its practical applications. Recently, the orbital angular momentum (OAM) of light has attracted wide attention as an important degree of freedom for realizing multiplexing to increase information transmission capacity. Here we show that by utilizing the OAM multiplexed continuous variable entanglement, 9 OAM multiplexed channels of parallel all-optical quantum teleportation can be deterministically established in experiment. More importantly, our parallel all-optical quantum teleportation scheme can teleport OAM-superposition-mode coded coherent state, which demonstrates the teleportation of more than one optical mode with fidelity beating the classical limit and thus ensures the increase of information transmission capacity. Our results open the avenue for deterministically implementing parallel quantum communication protocols and provide a promising paradigm for constructing high-capacity all-optical quantum communication networks.

Multiplexing can largely improve the information transmission capacity of optical communication system by integrating several channels into one. Multiplexing has been experimentally realized in both classical and quantum systems by using different degrees of freedom (dofs) of light, such as wavelength [25][26][27][28] , polarization 29,30 , space 31,32 , and time 33,34 . It is well-known that the orbital angular momentum (OAM) of light 35 could support infinite number of optical modes in principle. Due to this advantage, the OAM of light 35 has attracted wide attention as an important dof for realizing multiplexing. Recently, OAM multiplexing has been experimentally realized in free space [36][37][38][39] , optical fibers 40 and on chip 41 .
In this article, by utilizing the OAM multiplexed continuous variables (CV) entanglement and OAM mode-matched parametric amplifier (PA) based on the four-wave mixing process (FWM) 39,[42][43][44][45] , we experimentally demonstrate a 9-channel parallel all-optical quantum teleportation 4 . Different from the wellknown CV teleportation protocol 3,6 which requires Bell-type measurement and feed-forward techniques in classical channels, the all-optical teleportation (AOT) protocol is measurement-free and its classical channel is all-optical 4 . These features guarantee that OAM multiplexing technology in optical communication [36][37][38][39][40][41] can be directly applied in AOT. More importantly, we show that our OAM multiplexed quantum AOT scheme can teleport OAMsuperposition-mode coded coherent state with fidelity beating the classical limit, which demonstrates the teleportation of more than one optical mode and thus ensures the increase of information transmission capacity 36,46 .

Results
OAM multiplexed AOT architecture. Figure 1 shows the scheme for parallel AOT by multiplexing OAM channels. Our OAM multiplexed Einstein-Podolsky-Rosen (EPR) entangled state 39 is generated from the double-Λ configuration FWM process. Since the pump beam is very strong in this FWM process, it can be regarded as classical field. Therefore, the pump noise will not add extra noise into the system. In this way, the interaction Hamiltonian for the OAM-multiplexed FWM process can be described asĤ whereb y 1;' andb y 2;À' are the creation operators associated with an OAM mode in EPR 1 and EPR 2 beams, respectively. ℓ and −ℓ are the corresponding topological charges. h. c. is the Hermitian conjugate. γ ℓ is the interaction strength of each pair of OAM modes in FWM process. Based on Eq. (1), each inputoutput relation of the OAM-multiplexed FWM process can be expressed asb where G 1,ℓ = cosh 2 (γ ℓ τ) = cosh 2 (γ −ℓ τ) is the FWM gain of the corresponding OAM mode. τ is the time scale of interaction.b in1;' andb in2;À' are vacuum inputs. Then, for each OAM-multiplexed channel,b 2;À' is distributed to Alice (sender), andb 1;' is distributed to Bob (receiver). Alice amplifies the input stateâ in;' carrying OAM mode through an OAM mode-matched PA 39 with the help ofb 2;À' . The intensity gain of the corresponding OAM-mode codedâ in;' is G 2,ℓ . When the pump beam is a Gaussian beam, this PA can match its input states with opposite topological charges of ℓ and −ℓ due to OAM conservation. After such amplification,â c;' (i.e., the amplifiedâ in;' ) can be expressed asâ According to the definition given in other works 4,47 , this OAM mode-matched PA based on double-Λ configuration FWM process 45 , as described by Eq. (4), is a linear amplifier. Then, Alice sendsâ c;' to Bob through an all-optical classical channel. To retrieve the input state, receiver Bob couplesâ c;' andb 1;' by a beam splitter with a transmission of ε = 1/G 2,ℓ . Then, the output under the approximation G 2,ℓ ≫ 1. The two last terms in Eq. (5) vanish in the limit of an EPR entangled state (G 1,ℓ ≫ 1) and very high parametric gain (G 2,ℓ ≫ 1), leaving as an output state only the first term, which is equal to the input stateâ in;' , thus we achieve teleportation. In other words, although the amplification process will certainly add noise to the input state as indicated by Eq. (4), such added noise will be greatly reduced by the appropriate attenuation ratio of the beam splitter at Bob station and the coupling of two entangled EPR beams on the same beam splitter as shown by Eq. (5). In this way, the noise cancellation is realized by using the quantum correlation between the two entangled EPR beams, thus ensuring the realization of quantum teleportation. In a word, the noise introduced by this amplification process is indispensable for realizing quantum AOT. It plays the role of introducing EPR 2 into the system for implementing the subsequent noise cancellation operation at Bob station, then makes the fidelity beating the classical limit and thus realizing the quantum AOT. In teleportation, the quality of the retrieved state can be quantified by its teleportation fidelity F ∈ [0, 1]. If the fidelity F is higher than the classical limit of AOT without the help of entanglement, the quantum teleportation succeeds. The fidelity F is defined as the overlap between the input state and the output state characterized by the density matrix. For a coherent state carrying OAM, the fidelity can be expressed as 48 where σ Q is the variance of the teleported state in representation of the Q function and σ Q ¼ is the fluctuation variance of amplitude (phase) quadrature. β in and β out are the amplitudes of the input state at Alice station and the output state at Bob station, respectively. Based on Eq. (5), σ Then, the fidelity of deterministic quantum AOT for a coherent input state carrying OAM can be expressed as under the approximation G 2,ℓ ≫ 1.
Our detailed experimental setup for parallel AOT by multiplexing OAM channels is shown in Fig. 2a. A cavity stabilized Ti:sapphire laser emits a Gaussian laser beam. Its frequency is 0.9 GHz blue detuned from the 85 Rb D1 line (5S 1/2 , F = 2 → 5P 1/2 ). A polarization beam splitter (PBS) is used to divide the laser into two. One beam, which is vertically polarized and has a power of 100 mW, is served as the pump beam of the FWM process in a 85 Rb vapor cell for producing OAM multiplexed EPR entangled state 39 . Such pump power is strong enough to maximize the squeezing of our system. As shown in Fig. 2b, in this double-Λ configuration FWM process, two pump photons convert to one photon for EPR 1 (blueshifted from the pump beam) and one photon for EPR 2 (redshifted from the pump beam). Reflected by a Glan-Laser polarizer (GL), this pump beam is seeded into the 85 Rb vapor cell which is 12 mm long and stabilized at 113 ∘ C. This pump beam has a large waist of about 930 μm at the center of vapor cell. This pump beam with strong enough power of 100 mW and large waist of 930 μm ensures a sufficient number of OAM multiplexed quantum AOT channels 39 . The residual pump beam after the FWM process is eliminated by a Glan-Thompson polarizer (GT) with an extinction ratio of 10 5 :1. In this way, we obtain the OAM-multiplexed EPR entangled state for realizing quantum AOT. After that, the EPR 2 is distributed to Alice (sender), and EPR 1 is distributed to Bob (receiver). The other beam from the first PBS, which is horizontally polarized, is also divided into two by another PBS. The weak one passes through an acousto-optic modulator  (AOM) and a spatial light modulator (SLM) to generate the OAM-mode coded input stateâ in which is blue-shifted about 3.04 GHz from the pump beam and has a power of 0.4 μW. The strong one is served as the pump beam of OAM mode-matched PA at Alice station. This OAM mode-matched PA is also based on the double-Λ configuration FWM process in another 85 Rb vapor cell which is 12 mm long and stabilized at 110 ∘ C. Alice amplifies the input stateâ in carrying OAM modes through this OAM mode-matched PA with the help of EPR 2 . Combined by a GL, the pump, input stateâ in , and EPR 2 are crossed in the center of the 85 Rb vapor cell. Due to OAM conservation, only the input stateâ in and EPR 2 with opposite topological charges can be coupled when the pump is a Gaussian beam. The angle between input stateâ in and EPR 2 is about 14 mrad and the pump beam is symmetrically crossed withâ in and EPR 2 beams in the same plane. We set the intensity gain ofâ in at 10 ≫ 1, which ensures that the amplifiedâ in can be regarded as a classical field 4 . The residual pump beam after this modematched PA is eliminated by another GT. After such amplification, Alice sends the amplifiedâ in to Bob through an all-optical classical channel. To retrieve the input state, receiver Bob couples the amplifiedâ in and EPR 1 by a PBS and then attenuates this coupled beam with a beam splitter which consists of a half wave plate (HWP) and another PBS. Here, a piezo-electric transducer (PZT) is placed in the path of EPR 1 to change the relative phase between EPR 1 and EPR 2 . The transmission of this beam splitter is ε = 1/10, which can be precisely set by the photodetector (D 1 ) connecting to an oscilloscope (OS). Then, Victor measures the amplitude (phase) quadrature variance of the teleported OAM mode (â out ) by balanced homodyne detection (BHD) and calculates its teleportation fidelity F. Another PZT is placed in the path of a out to change the relative phase betweenâ out and local oscillator (LO). The LO beam with a power of 650 μW is  ig. 3 The quadrature variances of the retrieved state for ℓ = 1. a, b The variance of amplitude (phase) quadrature of the retrieved state. c, d The amplitude (phase) quadrature measurement of the orbital angular momentum (OAM) multiplexed entanglement source. The yellow trace is the variance of amplitude (phase) quadrature difference. The red trace is the variance of amplitude (phase) quadrature sum. The blue curve is noise power of the photocurrents output from balanced homodyne detection (BHD) versus the scanning phase. The black trace represents the corresponding shot noise limit (SNL). The insets show the transverse pattern of the output field when the four-wave mixing (FWM) is seeded by a bright beam. The left, center, and right beams areb 1;1 , Gaussian pump beam, andb 2;À1 , respectively. e The results of OAM mode analysis. The computer-generated hologram for ℓ = 1 case and the corresponding images of input and output fields are shown on the right inset, respectively. obtained by setting up a similar FWM process in the OAMmode matched PA at Alice station, which is a few mm above the current beams 43 . Its pump is split from the pump beam of OAM-mode matched PA, while its seed is split from the beam right after the SLM and beforeâ in . In this way, the frequency of LO beam naturally matches the one ofâ out . Our balanced homodyne detector has a transimpedance gain of 10 5 V A −1 and a quantum efficiency of 97%.
Teleporting a single OAM mode. The amplitude and phase quadrature variances of the teleported state carrying OAM mode with ℓ = 1 measured by Victor's BHD at 2 MHz sideband are shown in Fig. 3a (locking the relative phase between LO beam andâ out to 0 by micro-control unit 49,50 ) and Fig. 3b (locking the relative phase between LO beam andâ out to π/2 by proportionalintegral-differential circuit 49  ) of the EPR entanglement is 2.51 ± 0.16 dB (2.58 ± 0.18 dB) below the corresponding shot noise limit (SNL) as shown in Fig. 3c (Fig. 3d), indicating the presence of EPR entanglement between these two OAM modes. As shown in Fig. 3a, b, the black traces, measured by blocking the EPR entanglement and pump beams, are the quadrature variances of the input state. To realize AOT, the EPR entanglement is distributed to Alice and Bob. The relative phase between EPR 1 and EPR 2 is scanned by a PZT. The measured quadrature variances of the retrieved state by Victor are shown as the red traces in Fig. 3a  ). Therefore, we can treat the minima of each red trace as the variance ofXâ out orŶâ out quadrature of the output retrieved state under the help of EPR entanglement. Both the variances ofXâ out andŶâ out quadratures are almost equal and 3.58 ± 0.21 dB above the corresponding quadrature variances of the input state. This indicates that the corresponding fidelity of the retrieved state by quantum AOT with the help of EPR entanglement is 0.61 ± 0.02. To have the corresponding classical AOT for comparison, we block the EPR entanglement. The measured quadrature variances are shown as blue traces in Fig. 3a and b, which are 4.66 ± 0.12 dB above the corresponding quadrature variances of the input state. This gives a fidelity of 0.51 ± 0.01 for such classical AOT as the classical limit. Hence, it shows that such classical limit is beaten by the quantum AOT with the help of EPR entanglement. In addition, we find that the OAM mode carried by the retrieved state is almost the same with input state as shown in Fig. 3e.
OAM multiplexed teleportation channels. To illustrate the parallelism of our OAM multiplexed AOT, we do a series of the above measurements while changing the topological charge ℓ carried by an input state from −5 to 5, and keeping G 2,ℓ = 10. As shown in Fig. 4a, the red dot trace is the fidelity of quantum AOT with EPR entanglement for teleporting different OAM modes, while the blue dot trace is the fidelity of classical AOT with only two vacuum states. It shows that the fidelity of quantum AOT for topological charge in the range of −4 to 4 is better than the corresponding fidelity of the classical AOT. In addition, we can see that for the quantum AOT with EPR entanglement, the larger the |ℓ| is, the lower the fidelity will be. This is because the inseparability of OAM-multiplexed entanglement source gets worse with the increase of |ℓ| as shown in Fig. 4b. We calculate the inseparability of OAM-multiplexed entanglement source by Ib Þ 51-53 , which is directly related to the two-mode squeezing of OAM multiplexed entanglement. The corresponding mode analyses for the input and retrieved states are shown in Fig. 5. These results clearly show that we have successfully constructed 9 OAM multiplexed channels of parallel quantum AOT with fidelities beating the classical limit.
To directly show the advantage of our OAM multiplexed AOT scheme in terms of increasing the information transmission capacity, we implement the teleportation of OAM-superpositionmode coded coherent state (â in ¼â in;' þâ in;À' ; ' ¼ 1; 2; 3; 4). Firstly, in order to prove that the OAM superposition mode carried by input coherent state remains unchanged after AOT, we make a series of OAM mode analyses of the input and the retrieved output states as shown in Fig. 6. It shows that our AOT can well maintain the OAM superposition mode distribution of the input states. Secondly, we measure the fidelity of quantum AOT for teleporting OAM-superposition-mode coded coherent state. Forâ in ¼â in;' þâ in;À' ; ' ¼ 1; 2; 3; 4, the fidelity of such quantum AOT as indicated on the top of each subfigure is better than the measured fidelity of corresponding classical AOT (0.51 ± 0.01). In principle, the fidelity for teleportingâ in ¼â in;' þâ in;À' is same as the one for teleportinĝ a in ¼â in;' (â in;À' ) because the squeezing levels of EPR entanglement for these two cases are equal 39 . In our experiment, the slight difference between the fidelities for these two cases is caused by measurement error. These results clearly illustrate that our OAM multiplexed AOT can simultaneously teleport more than one OAM mode with fidelity beating the classical limit.

Discussion
Although the OAM modes themselves are a subset of a discrete Hilbert space, what we teleported is the amplitude quadrature and phase quadrature of the coherent state, which are described by a continuous Hilbert space. From this point of view, our experiment demonstrates 9 parallel channels of deterministic CV quantum teleportation of coherent states carrying different OAM modes. Moreover, we have shown that our OAM multiplexed AOT scheme can teleport OAM-superposition-mode coded coherent state, which implements the simultaneous teleportation of more than one optical mode. The successful implementation of such OAM multiplexed quantum teleportation benefits from the measurement-free AOT architecture, which makes it possible to implant the OAM multiplexing technique of the modern optical communication into its all-optical channels. Our results open the avenue for deterministically implementing parallel quantum communication protocols and provide a promising paradigm for   constructing high-capacity all-optical quantum communication networks.

Methods
The OAM mode analysis of teleported state. We use the same method as mentioned in other work 54 to analyze the OAM mode of the state. As shown in Fig. 7, the analyzed state is sent to the SLM. Only the OAM mode carrying topological charge of ℓ can be transformed into a Gaussian mode by loading a computer-generated hologram with a topological charge −ℓ on the SLM. Due to the fact that only the Gaussian mode can be coupled into the mono-mode fiber, the output beam power of the mono-mode fiber almost only contains the power of the corresponding teleported OAM mode carrying topological charge of ℓ under high coupling efficiency of the fiber. We load the computer-generated hologram with a topological charge ranging from −6 to 6 on the SLM. In this way, we can obtain the proportion of the OAM mode with a topological charge from −6 to 6.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.