Multiplexed storage and real-time manipulation based on amultiple-degree-of-freedom quantum memory

Tian-Shu Yang,1, 2 Zong-Quan Zhou∗,1, 2 Yi-Lin Hua,1, 2 Xiao Liu,1, 2 Zong-Feng Li,1, 2 Pei-Yun Li,1, 2 Yu Ma,1, 2

Chao Liu,1, 2 Peng-Jun Liang,1, 2 Xue Li,1, 2 Yi-Xin Xiao,1, 2 Jun Hu,1, 2 Chuan-Feng Li†,1, 2 and Guang-Can Guo1, 2

1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China2Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, 230026, P.R. China

(Dated: September 6, 2018)

The faithful storage and coherent manipulationof quantum states with matter-systems enablethe construction of large-scale quantum networksbased on quantum repeater. To achieve usefulcommunication rates, highly multimode quantummemories will be required to construct a multi-plexed quantum repeater. Here, we present thefirst demonstration of the on-demand storage oforbital-angular-momentum states with weak co-herent pulses at the single-photon-level in a rare-earth-ion doped crystal. Through the combina-tion of this spatial degree-of-freedom with tem-poral and spectral degrees of freedom, we createa multiple-degree-of-freedom memory with highmultimode capacity. This device can serve as aquantum mode converter with high fidelity, whichis the fundamental requirements for the construc-tion of a multiplexed quantum repeater. Thisdevice further enables essentially arbitrary spec-tral and temporal manipulations of spatial-qutrit-encoded photonic pulses in real-time. Therefore,the developed quantum memory can serve as abuilding block for scalable photonic quantum in-formation processing architectures.

IntroductionLarge-scale quantum networks enable the long-distancequantum communication and optical quantum comput-ing [1–3]. Due to the exponential photon loss in opticalfibers [4], quantum communication via ground based op-tical fibers is currently limited to distances on hundred ofkilometers. To overcome this problem, the idea of quan-tum repeater [5, 6] has been proposed to establish quan-tum entanglement over long distances based on quantummemories and entanglement swapping. It has been shownthat to reach practical data rates using this approach, themost significant improvements can be achieved throughthe use of multiplexed quantum memories [7–9].

The multiplexing of quantum memory can be imple-mented in any degree-of-freedom (DOF) of photons, suchas those in the temporal [10], spectral [11] and spatial [12]domains. Rare-earth-ion doped crystals (REIC) offer in-

∗email:zq zhou@ustc.edu.cn†email:cfli@ustc.edu.cn

teresting possibilities as multiple-DOF quantum memo-ries for photons by virtue of their large inhomogeneousbandwidths [11, 13, 14] and long coherence time [15]at cryogenic temperatures. Recently, there have beenseveral important demonstrations using REIC, such asthe simultaneous storage of 100 temporal modes [16]by atomic frequency comb (AFC) [17] featuring pre-programmed delays, the storage of tens of temporalmodes by spin-wave AFC with on-demand readout [18–22] and the storage of 26 frequency modes with feed-forward controlled readout [11]. The orbital-angular-momentum (OAM) of a photon receives much attentionbecause of the high capacity of OAM states for informa-tion transmission and spatial multimode operations [23].Tremendous developments have recently been achieved inquantum memories for OAM states [24–26], paving theway to quantum networks and scalable communicationarchitectures based on this DOF.

To date, most previous experiments with quantummemories have been confined to the storage of multi-ple modes using only one DOF, e.g. temporal, spec-tral or spatial. To significantly improve the communi-cation capacity of quantum memory and quantum chan-nel, we consider a quantum memory using more thanone DOF simultaneously [11, 27–29]. Here, we reporton the experimental realization of an on-demand quan-tum memory storing single photons encoded with three-dimensional OAM states in a REIC. We present the re-sults of a multiplexed spin-wave memory operating si-multaneously in temporal, spectral and spatial DOF. Inaddition to expanding the number of modes in the mem-ory through parallel multiplexing, a quantum mode con-verter (QMC) [30] can also be realized that can performmode conversion in the temporal and spectral domainssimultaneously and independently. Indeed, our quan-tum memory enables arbitrary temporal and spectral ma-nipulations of spatial-qutrit-encoded photonic pulses andthus can serve as a real-time sequencer [14], a real-timemultiplexer/demultiplexer [31], a real-time beam splitter[32], a real-time frequency shifter [33], a real-time tem-poral/spectral filter [31], among other functionalities.

Results

Experimental SetupA schematic drawn of our experimental set-up and rel-evant atomic level structure of Pr3+ ions are presented

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Fig. 1 Experimental setup and atomic levels. a, Schematic illustrations of the experiment. The AFC is prepared ina memory crystal (MC) and a narrow spectral filter has been prepared in a filter crystal (FC). The beam waist of theinput light is 65 µm at the middle of the MC. The pump/control light has a beam waist of 300 µm inside the MC toensure good overlap with the high-dimensional input light. The input pulses are attenuated to single-photon level byneutral density filters (NDF). The spatial modes of these photons are converted into OAM superposition states by aspatial light modulator (SLM1). After storage in the MC, the retrieved signal passes through two consecutiveacousto-optical modulators (AOM) [35], which act as a temporal gate and a frequency shifter [36]. The AOM areused in double-pass configuration to ensure the photons’ spatial mode unchanged when the frequency of photons isswept over tens of MHz. The SLM2 and a single mode fiber (SMF) are employed to analyze the OAM states of theretrieved photons. The FC is double-passed with help of a polarization beam splitter (PBS), a half-wave plate(HWP) and a Faraday rotator (FR). Two band-pass filters (BPF) centered at 606 nm are employed to furthersuppress noises before the final detection of signal photons using single-photon detector (SPD). QWP: quarter-waveplate. b, Hyperfine states of the first sublevels of the ground and the excited states of Pr3+ in Y2SiO5. The inputfield is resonant with 1/2g - 3/2e, and the control field is resonant with 3/2g - 3/2e (See Methods section for details).

in Fig. 1. The memory crystal (MC) and filter crys-tal (FC) used in this set-up are 3 × 6 × 3 mm crystalsof 0.05% doped Pr3+:Y2SiO5, which are cooled to 3.2K using a cryogen-free cryostat (Montana InstrumentsCryostation). In order to maximize absorption, the po-larization of input light is close to the D2 axis of Y2SiO5

crystal. To realize reliable quantum storage with highmultimode capacity, we created a high-contrast AFC inMC (the AFC structure is shown in Supplementary Note1). Spin-wave storage is employed to enable on-demandretrieval and extend storage time [17]. The control andinput light are steered towards the MC in opposite di-rections with an angular offset ∼ 4◦ to reject the strongcontrol field and avoid the detection of free induction de-cay noise [34]. To achieve a low noise floor, we increasethe absorption depth of the FC by employing a double-pass configuration. Fig. 2a presents the time histogramsof the input photons (blue) and the photons retrieved at12.68 µs (green) with a spin-wave storage efficiency ηSW= 5.51%. For an input with a mean photon number µ= 1.12 per pulse, we have measured a signal-to-noise ra-tio (SNR) ∼ 39.7 ± 6.7 with the input photons in theGaussian mode.

Quantum process tomographyThe ability to realize the on-demand storage of photonicOAM superposition states in solid-state systems is cru-cial for the construction of OAM based high dimensionalquantum networks [24]. The quantum process tomogra-phy for qutrit operations [37] is benchmark the storageperformance for OAM qutrit in our solid-state quantummemory. The qutrit states are prepared in the followingbasis of OAM states: |LGl=−1

p=0 〉, |LGl=0p=0〉, |LGl=1

p=0〉. Here,

|LGlp〉 corresponds to OAM states defined as Laguerre-

Gaussian (LG) modes, where l and p are the azimuthaland radial indices, respectively. In the following, we usethe kets |L〉, |G〉 and |R〉 to denote the OAM states

|LGl=1p=0〉, |LGl=0

p=0〉 and |LGl=−1p=0 〉, respectively. For µ =

1.12, we first characterized the input states before thequantum memory using quantum state tomography (seeMethods section for details). The reconstructed densitymatrices of input are not ideal because of the prepara-tion and measurements based on spatial light modulators(SLM) and single-mode fiber (SMF) are not perfect [26].We then characterized the memory process using quan-tum process tomography. Fig. 2b presents the real part

3

of the experimentally reconstructed process matrix χ. Itis found to have a fidelity of 0.909 ± 0.010 to Identityoperation. This fidelity exceeds the classical bound of0.831 (see Supplementary Note 4 for details), therebyconfirming the quantum nature of the memory opera-tion. The non-unity value of the memory fidelity may becaused by the limited beam waist of the pump/controllight, which may result in imperfect overlap with differentOAM modes. Moreover, we noted that the memory per-formance for superposition states of |L〉 and |R〉 is muchbetter than that achieved here (as detailed in Supplemen-tary Note 2). The visibility of such two-dimensional su-perposition states is higher than the fidelity of the mem-ory process in all three dimensions. This result indicatesthat the storage efficiency is balanced for the symmetri-cal LG modes but is not balanced for all three consideredspatial modes.

Multiplexing storage in multiple DOFCarrying information in multiple DOF on photons canexpand the channel capacity of quantum communicationprotocols [11, 38]. Here, we show that our solid-statememory can be simultaneously multiplexed in temporal,spectral and spatial DOF. As shown in Fig. 3a, twoAFC are created in the MC with an interval of 80 MHzbetween them to achieve spectral multiplexing. The twoAFC have the same peak spacing of ∆ = 200 kHz and thesame bandwidth of ΓAFC = 2 MHz. The spin-wave stor-age efficiencies are 5.05% and 5.13%, for the first AFCand the second AFC, respectively. The temporal mul-timode capacity of an AFC is limited by ΓAFC/∆ [17].However, increasing the number of modes, the time in-terval between the last control pulse and the first outputsignal pulse will be reduced. Therefore, we employed onlytwo temporal modes to reduce the noise caused by thelast control pulse. The spatial multiplexing is realized byusing three independent paths as input as shown in Fig.3b. These paths, s1, s2 and s3, correspond to the OAMstates as |L〉, |R〉, and |G〉 defined above. By combiningall three DOF together, we obtain 2 × 2 × 3 = 12 modesin total. The FC is employed to select out the desiredspectral modes. Fig. 3c illustrates the results of multi-mode storage over these three DOF for µ = 1.04. Theminimum crosstalk as obtained from mode crosstalk foreach mode is 19.7 ± 3.41, which is calculated as one takesthe counts in the diagonal term as the signal and thenlocates the large peaks over the range of output modesas the noise.

Here, the temporal, spectral and spatial DOF are em-ployed as classical DOF for multiplexing. One can chooseany DOF to carry quantum information. As a typical ex-ample, now we use the temporal and spectral DOF formultiplexing and each channel is encoded with spatialqutrit state of |ψ1〉 = (|L〉 + |G〉 + |R〉)/

√3. Each chan-

nel is labeled as fitj , where fi represent spectral modesi and tj similarly represent temporal modes j. Fig. 4ashows the experimental results for µ = 1.04. The min-imum crosstalk as obtained from the mode crosstalk is

b

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Fig. 2 Time histograms and the reconstructed processmatrix of quantum storage process in three dimensionalOAM space. a, Time histograms of the input photons(blue), the photons retrieved at 12.68 µs for µ = 1.12(green) and the unconditional noise for µ = 0 (black).b, Graphical representation of the process matrix χ ofmemory process as estimated via quantum processtomography. Details of operators λi are shown in theMethods. Only the real part of the experimentallyreconstructed process matrix is shown. All values are inthe imaginary part are smaller than 0.090.

approximately 15.2. We measured the fidelities of thespatial qutrit state for each channel as shown in Fig. 4b.

A quantum mode converter (QMC) can transfer pho-tonic pulses to a target temporal or spectral mode with-out distorting the photonic quantum states. A real-timeQMC that can operate on many DOF is essential for link-ing the components of a quantum network [30, 39]. Byadjusting the timing of the control pulse, one can spec-ify the recall time in an on-demand manner to realizethe temporal mode conversion. The two-AOM gate in

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735 757691

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Fig. 3 Multiplexed spin-wave storage in three degrees of freedom at the single photon level. a, The double AFCstructure (red) in the MC and the double filter structure (black) in the FC. b, Three independent spatial modescarrying different OAM states are employed for spatial multiplexing. The spatial mode s1 is converted into |L〉 statein SLM1; The spatial mode s2 is converted into |R〉 state in a spiral phase plate (SSP); the spatial mode s3 is aGaussian mode which is used as |G〉 state. They are combined by two pellicle beam splitters (BS). c, Ademonstration of temporal, spectral and spatial multiplexed storage for single-photon level inputs.

our system can be used as a high-speed frequency shifterby tuning its driving frequency. Therefore, spectral andtemporal mode conversion can be realized independentlyand simultaneously. Fig. 4c presents the results of QMCoperation for µ = 1.04. We can convert from fi to fjand from ti to tj , where these notations represent all dif-ferent spectral modes and temporal modes. The noiselevel is significantly weaker than the strength of the con-verted signal, which indicates that the QMC operatesquietly enough to avoid introducing any mode crosstalk.All these modes are encoded with OAM spatial qutrit

of |ψ1〉. To demonstrate that the qutrit state coher-ence is well preserved after QMC operation, we measuredthe fidelities (see Methods for details) between the inputand converted states. The results are presented in Fig.4d, which indicate the QMC can convert arbitrary tem-poral and spectral modes in real-time while preservingtheir quantum properties. Our device is expected to findapplications in quantum networks comprising two quan-tum memories, in which mismatched spectral or tempo-ral photon modes may need to be converted [30]. Thisdevice can ensure the indistinguishability of the photons

5

which are retrieved from any quantum memories. Thisdevice is required for many photonic information process-ing protocols, e.g., a Bell-state measurement [11], andquantum-memory-assisted multi-photon generation [9].

Arbitrary manipulations in real timeThe precise and arbitrary manipulation of photonicpulses while preserving photonic coherence is an impor-tant requirement for many proposed photonic technolo-gies [31]. In addition to the QMC functionality demon-strated above, the developed quantum memory can en-able arbitrary manipulations of photonic pulses in thetemporal and spectral domains in real-time. As an ex-ample, we prepared the OAM qutrit state |ψ1〉 in thef1t1 and f2t2 modes (Fig. 5a) as the input. Four typ-ical operations were demonstrated, i.e., exchange of thereadout times for the f1 and f2 photons, the simultane-ous retrieval of the f1 and f2 photons at t1, shifting thefrequency of f1 photons to f2 but keeping the frequencyof f2 photons unchanged and temporal beam splitting thef1 photons but filtering out the f2 photons. These opera-tions correspond to output of |ψ1〉f1 t2 , f2 t1 , |ψ1〉f1 t1 , f2 t1 ,|ψ1〉f2 t1 , f2 t2 and |ψ1〉f1 t1 , f1 t2 , respectively. Another ex-ample was implemented with the OAM qutrit state |ψ2〉= (|L〉 + |G〉 - i|R〉)/

√3 encoded in the f1t2 and f2t2

modes as the input, as shown in Fig. 5b with same out-put. The retrieved states were then characterized viaquantum state tomography as usual (see Methods). Ta-ble 1 shows the fidelities between output states and inputstates.

DiscussionIn conclusion, we have experimentally demonstrated amultiplexed solid-state quantum memory that operatessimultaneously in three DOF. The currently achievedmultimode capacity is certainly not the fundamentallimit for the physical system. Pr3+:Y2SiO5 has an in-homogeneous linewidth of 5 GHz, which can supportmore than 60 independent spectral modes. The numberof temporal modes that can be achieved using the AFCprotocol [17] is proportional to the number of absorptionin the comb, which has already been improved to 50 inEu3+:Y2SiO5 [22]. There is no fundamental limit on themultimode capacity in the OAM DOF since it is indepen-dent on the AFC bandwidth. The capacity in this DOFis simply determined by the useful size of the memoryin practice. We have recently demonstrated the faith-ful storage of 51 OAM spatial modes in a Nd3+:YVO4

crystal [26]. The combination of these start-of-the-arttechnologies could result in a multimode capacity of 60× 50 × 51 = 153000 modes. This large capacity couldgreatly enhance the data rate in memory-based quantumnetworks and in portable quantum hard drives with ex-tremely long lifetimes [15].

The developed multiple-DOF quantum memory canserve as a QMC, which is the fundamental requirementfor the construction of scalable networks based on mul-tiplexed quantum repeaters. Although it is not demon-

Input Mode Output Mode Fidelity

|ψ1〉f1 t1 , f2 t2

|ψ1〉f1 t2 , f2 t1 0.881±0.022

|ψ1〉f1 t1 , f2 t1 0.876±0.022

|ψ1〉f2 t1 , f2 t2 0.897±0.020

|ψ1〉f1 t1 , f1 t2 0.828±0.019

|ψ2〉f1 t2 , f2 t2

|ψ2〉f1 t2 , f2 t1 0.896±0.017

|ψ2〉f1 t1 , f2 t1 0.898±0.013

|ψ2〉f2 t1 , f2 t2 0.898±0.016

|ψ2〉f1 t1 , f1 t2 0.829±0.025

Table 1 Fidelities of qutrit states after temporaland spectral manipulations.The fidelity for the

output mode of |ψ〉f1 t1 , f1 t2 is a little lower than theothers because of the less photon counts in each outputcaused by the temporal splitting operation. The error

bars for the fidelities correspond to one standarddeviation caused by the statistical uncertainty of

photon counts.

strated in the current work, mode conversion in the spa-tial domain should also be feasible using a high-speeddigital micro-mirror device [40]. QMC can also find ap-plications in linear optical quantum computations. Onetypical example is to solve the mode mismatch caused byfiber-loop length effects and the time jitter of the photonsources in a boson sampling protocol [41, 42].

Quantum communication and quantum computationin a large-scale quantum network rely on the ability offaithful storage and manipulation of photonic pulses car-rying quantum information. The presented quantummemory can apply arbitrary temporal and spectral ma-nipulations to photonic pulses in real-time, which indi-cates that this single device can serve as a variable tem-poral beam splitter [32, 43] and a relative phase shifter[35] that enables arbitrary control of splitting ratio andphase for each output. Therefore, this device can per-form arbitrary single-qubit operations [44]. Combiningwith the recent achievements on generation of heraldedsingle photons [45, 46], this device should provide the suf-ficient set of operations to allow for universal quantumcomputing in the KLM scheme [47]. Our results are ex-pected to find applications in large-scale memory-basedquantum networks and advanced photonic informationprocessing architectures.

Methods

AFC preparation.We tailored the absorption spectrum of Pr3+ ions to pre-pare the AFC using spectral hole burning [35]. The fre-quency of the pump light was first scanned over 16 MHzto create a wide transparent window in the Pr3+ absorp-tion line. Then, a 1.6 MHz sweep was performed outsidethe pit to prepare the atoms into the 1/2g state. Theburn-back procedure created an absorbing feature of 2MHz in width resonant with the 1/2g - 3/2e transition

6

a

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Input Modes

Outp

ut Modes

Outp

ut Modes

f t11

f t11

f t1

2

f t22

f t21

ft2

2

f t22

f t21

ft

21 f t1

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2

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f t1

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1

f t1

1

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d

Channel Fidelity

f1t1 0.881 ± 0.021

f1t2 0.891 ± 0.016

f2t1 0.896 ± 0.010

f2t2 0.885 ± 0.014

Mode Conversion Fidelity

f1t1 to f2t2 0.879 ± 0.025

f1t2 to f2t1 0.887 ± 0.023

f2t1 to f1t2 0.886 ± 0.022

f2t2 to f1t1 0.877 ± 0.014

698

621

650

597

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0 0

200

400

600713

746 753786

Fig. 4 Multiplexed storage and quantum mode conversion for spatial encoded qutrit state using four temporal andspectral channels. a, The multiplexed storage for spatial qutrit state of |ψ1〉 in the temporal and spectral DOF. b,Fidelities for the qutrit states after storage. c, The performance of the QMC with an encoded state of |ψ1〉. d,Fidelities for the qutrit state after mode conversion. The error bars for the fidelities correspond to one standarddeviation caused by the statistical uncertainty of photon counts.

but simultaneously populated the 3/2g state, which, inprinciple, must be empty for spin-wave storage. Thus,a clean pulse was applied at the 3/2g - 3/2e transitionto empty this ground state. After the successful prepa-ration of absorbing band in the 1/2g state, a stream ofhole-burning pulses was applied on the 1/2g-3/2e transi-tion. An AFC structure with a periodicity of ∆ = 200kHz is prepared in this step. These pulses burned thedesired spectral comb of ions on the 1/2g - 3/2e transi-tion and anti-holes at the 3/2g - 3/2e transition; thus, ashort burst of clean pulses was applied to maintain theemptiness of the 3/2g state. For AFC preparation, theremaining 5/2g ground state is used as an auxiliary statewhich stores those atoms which don’t contribute to theAFC components. To reduce the noise generated by thecontrol pulses during spin-wave storage, we applied 100control pulses separated by 25 µs and another 50 controlpulses with a separation of 100 µs after the preparationof the comb [21]. An example of the AFC with a pe-riodicity ∆ = 200 kHz is illustrated in Fig. 3a. A de-tailed estimation of the structure and storage efficiencyof the AFC memory is presented in Supplementary Note1. The signal photons are mapped onto the AFC, lead-ing to an AFC echo after a time 1/∆. Spin-wave storageis achieved by applying two on-resonance control pulsesto induce reversible transfer between the 3/2e state and

3/2g state before the AFC echo emission. The completestorage time is 12.68 µs in our experiment which includesan AFC storage time of 5 µs and a spin-wave storage timeof 7.68 µs.

Filtering the noise.In order to achieve a low noise floor, temporal, spectraland spatial filter methods are employed. The input andcontrol beams are sent to the MC in opposite directionswith a small angular offset for spatial filtering. Tem-poral filtering is achieved by means of a temporal gateimplemented with two AOM. This AOM gate temporallyblocked the strong control pulses. This is important toavoid burning a spectral hole in the FC and to avoidblinding the single-photon detector. We used two 2-nmbandpass filters at 606 nm to filter out incoherent fluores-cence noise. The spectral of the filter mode was achievedby narrow-band spectral filter in the FC (shown by thedashed black line in Fig. 3a), which is created by 0.8MHz sweep around the input light frequency, leading toa transparent window of approximately 1.84 MHz due tothe power broadening effect. Furthermore, the FC is im-plemented in a double-pass configuration to achieve highabsorption.

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Fig. 5 Arbitrary temporal and spectral manipulationsin real time. Four typical operations are presented fortwo different input states. a, The OAM qutrit state|ψ1〉 is encoded on the f1t1 and f2t2 modes. The f1photons are marked as red color and the f2 photons aremarked as blue color. These operations, from up todown, correspond to a pulses sequencer, a multiplexer, aselective spectral shifter and a configurable beamsplitter (the f2 photons are filtered out), respectively.The little “×2” indicates that the integration time oftemporal beam splitting is two times of the otheroperations. b, The OAM qutrit state |ψ2〉 is encoded onthe f1t2 and f2t2 modes. These operations, from up todown, correspond to a demultiplexer, a pulsessequencer, a selective spectral shifter (the f1 photons isfrequency shifted to f2 and retrieved at t1 while the f2photons are retrieved at t2) and a configurable beamsplitter (the f2 photons are filtered out), respectively.All of these operations can be determined after thephotons have been absorbed into the quantum memory.The inputs are shifted earlier by 3 µs in the histogramsfor visual effects.

Quantum tomography.To characterize the memory performance for three di-mensional OAM states, a quantum process tomographyfor the quantum memory operation is performed. Recon-structing the process matrix χ of any three-dimensionalstate requires nine linearly independent measurements.We chose three OAM eigenstates and six OAM superpo-

sition states as our nine input states, which are listed asfollows: |L〉, |G〉, |R〉, (|L〉 + |G〉)/

√2, (|R〉 + |G〉)/

√2,

(i|L〉 + |G〉)/√

2, (-i|R〉 + |G〉)/√

2, (|L〉 + |R〉)/√

2, and

(|L〉 - i|R〉)/√

2 [26]. The complete operators for the re-construction of the matrix χ are as follows [26, 48]:

λ1 =

1 0 0

0 1 0

0 0 1

(1)

λ2 =

0 1 0

1 0 0

0 0 0

(2)

λ3 =

0 −i 0

i 0 0

0 0 0

(3)

λ4 =

1 0 0

0 −1 0

0 0 0

(4)

λ5 =

0 0 1

0 0 0

1 0 0

(5)

λ6 =

0 0 −i0 0 0

i 0 0

(6)

λ7 =

0 0 0

0 0 1

0 1 0

(7)

λ8 =

0 0 0

0 0 −i0 i 0

(8)

λ9 =1√3

1 0 0

0 1 0

0 0 −2

(9)

Here, λ1 is the identity operation. The process ma-trix χ can be expressed on the basis of λi and maps aninput matrix ρin onto the output matrix ρout [48, 49].

8

Density matrices of the input states ρin and density ma-trices of the output states ρout are reconstructed usingquantum state tomography [48, 49]. For a given qutritstate (|ψ〉), we used ρ=|ψ〉〈ψ| to reconstruct the den-sity matrix ρ from the measurement results. As exam-ples, we presented graphical representations of the re-constructed density matrices for the OAM qutrit states|ψ1〉 and |ψ2〉 in Supplementary Note 3.The fidelities ofthe output states with respect to the input states can be

calculated based on the reconstructed density matricesusing the formula Tr(

√√ρoutputρinput

√ρoutput)

2.

Data availabilityThe data that support the findings of this study are avail-able from the corresponding authors on request.

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AcknowledgmentsThis work was supported by the Na-tional Key R&D Program of China (No.2017YFA0304100,2016YFA0302700), the Na-tional Natural Science Foundation of China (Nos.61327901,11774331,11774335,61490711,11504362,11654002),Anhui Initiative in Quantum Information Technologies(No. AHY020100),Key Research Program of FrontierSciences, CAS (No. QYZDY-SSW-SLH003), the Fun-damental Research Funds for the Central Universities(Nos. WK2470000023, WK2470000026).

Author contributionsZ.Q.Z. and C.F.L. designed experiment. T.S.Y. andZ.Q.Z. carried out the experiment assisted by Y.L.H.,X.L., Z.F.L., P.Y.L., Y.M., C.L., P.J.L., Y.X.X., J.H.and X.L. T.S.Y. and Z.Q.Z. wrote the paper with inputfrom other authors. C.F.L. and G.C.G supervised theproject. All authors discussed the experimental proce-dures and results.

Additional informationCompeting interests: The authors declare no compet-ing interests.