SUPPLEMENTARY INFORMATION

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1 DOI: 1.138/NPHOTON A quantum memory for orbital angular momentum photonic qubits - Supplementary Information - A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, J. Laurat Laboratoire Kastler Brossel, Université Pierre et Marie Curie, Ecole Normale Supérieure, CNRS, 4 place Jussieu, 755 Paris Cedex 5, France (Dated: November 5, 13 Supporting documentation is provided for our manuscript Ref. [1]. I. STATE CHARACTERISATION SETUP It is well known that the quantum state tomography of a qubit encoded in polarization can be performed by using a quarter-wave plate, a half-wave plate and a polarizing beam splitter. In our case of OAM encoding, the tomography can be realized using two mode projectors inserted into an interferometer with a tunable phase (denoted ϕ in the main text. This section details the characterisation setup, as shown in Fig. S1. LG l= p= A. Tomography of the qubit states We consider qubits encoded in the Hilbert space spanned by the two Laguerre-Gaussian (LG states R = = and L = 1 = LGl= 1 p=. A qubit ψ in a pure state can be written as: ψ = α + β 1 with α, β C and α + β = 1 (1 It can also be expressed in other bases, in particular { H =( R + L /, V =( R L / } and { D = ( R +i L /, A =( R i L / } as ψ = α+β ψ = α iβ H + α β V ( D + α+iβ A. (3 The state to characterize is split on a 5/5 non-polarising beam splitter (BS into two beams, which are respectively diffracted by forked holograms to change their OAM states: in the Right branch (Holo R, one unit of orbital angular momentum is subtracted from each state (l l 1, while one unit is added in the Left branch (Holo L, l l + 1. The diffracted light is then coupled in each path into a single-mode fibre. Only the TEM mode, corresponding to = LG l= p= in the LG basis, can be coupled into the fibre, therefore this coupling acts as a projection onto that state. In summary, the input state ψ transforms as follows in the Right and Left branch, neglecting common amplitude losses: α + β 1 Holo R α + β 1 Holo L α + β Fibre α α + + β Fibre β. Bringing the two fibres together in the fibre beam splitter FBS1, the following states arrive on the single photon detectors APD1 and APD: APD1 ( α +e iϕ β ( APD α +e i(ϕ+π β. The phase factor e iϕ comes from the difference in phase accumulated between the two branches when traversing the interferometer. Electronic address: julien.laurat@upmc.fr NATURE PHOTONICS Macmillan Publishers Limited. All rights reserved.

2 DOI: 1.138/NPHOTON FIG. S1: State characterisation setup. Sig: signal light input; BS: non-polarising beam splitter; Holo R and Holo L: fork holograms; PZT: piezo translator; FBS1 and : fiber beam splitters (5/5, non-polarising, but polarisation-maintaining; APD1 and : single-photon counting modules; Ref: reference light input; T: light-absorbing fibre terminator; Cam: digital camera; ϕ: interferometer phase. For the measurement, we consider three different cases. First, we block one arm of the interferometer and leave the other one open, giving for instance on APD1: APD1, L blocked α APD1, R blocked e iϕ β. The count rates measured on APD1 give thus acess to the R and L components of the input state ψ, i.e. α and β. Second, both interferometer arms are open and we assume an interferometer phase ϕ =. The control of this phase will be discussed below. In this case, this leads to APD1,ϕ= (α + β APD,ϕ= (α β. The coefficients of the states arriving on the two detectors are proportional to those of the H and V components when expressing ψ in that basis (cf. Eq. and the measurements of the two detectors give us the corresponding weights α + β and α β. Third, choosing an interferometer phase of ϕ = π/, the measurements give access to the weights in the { D, A } basis. If we are thus able to a block each arm of the interferometer and b control or select the interferometer phase ϕ, then our setup enables us to measure the state in each of the three bases { R, L }, { H, V }, and { D, A }. The relative normalized probabilities p i of the outcomes i, extracted from the count rates in these different bases give us the Stokes parameters []: S 1 = p R p L ; S = p H p V ; S 3 = p D p A in terms of which the density matrix is expressed, as mentioned in the main text: ρ = 1 ( 1+S1 S is 3 S +is 3 1 S 1 NATURE PHOTONICS 14 Macmillan Publishers Limited. All rights reserved.

3 DOI: 1.138/NPHOTON SUPPLEMENTARY INFORMATION B. Determination of the interferometer phase As explained in the previous section, the interferometer phase ϕ determines in which basis the incoming state is measured. Due to mechanical and temperature changes, this phase drifts on the timescale of a few seconds. It has therefore to be measured continuously, in parallel with the state measurements. To this end, reference light is coupled backwards into the interferometer, using the fibre beam splitter FBS, see Fig. S1. The reference light has a power of about 1 nw and the same wavelength and polarisation as the signal. To avoid scattered reference light to arrive in the APDs, the reference is only on while no signal measurement is performed, i.e., about 1 out of 15 ms for each MOT cycle, and blocked by AOMs otherwise. Since the detection of the reference averages over several MOT cycles, these short interruptions are not relevant. The two TEM reference beams emerging from the fibres are, by diffraction at the holograms, transformed into LG and LG 1 modes, respectively. They leave the entrance beam splitter BS together and are superimposed on a digital camera connected to the measurement computer, with the interferometer phase ϕ as the relative phase between them: E Cam E LG +e iϕ E LG 1 Here, E j are the electric fields of the light arriving on the camera and of the two contributing LG modes. Changing to a coordinate system (x, y (x r,y r rotated by ϕ/ around the propagation axis z corresponds to a phase advancement by l ϕ/ = ϕ/ for the mode R, and ϕ/ for the mode L: ECam (E r r ( LG = e iϕ/ E LG = e iϕ/ (E LG = e iϕ/ E HG. + e iϕ (E r LG 1. + e iϕ (e iϕ/ E LG 1 + E LG 1 In the rotated system, we obtain a Hermite-Gaussian 1-mode, i.e., an H mode. Thus, in the original coordinates of our camera, we see the intensity pattern of an H mode rotated by the angle ϕ/. Thus, the angle between the dark line in the recorded image and the vertical direction is directly half the interferometer phase. One could use this information to stabilise the interferometer to a desired angle in real time. However, here we want to show the measurements for the whole angle range. Therefore, we modulate the phase slowly using a piezo translator PZT to cover all angles evenly. During the measurements, images are taken about every 15 ms by the computer. They are timestamped based on the same clock as the timestamps for the signal single photon events, allowing for the later correlation. After the measurement, the angular information is extracted from the images by the following algorithm. Each image is first processed using a median filter replacing each pixel s value by the median of itself and its surrounding pixels, eliminating dead pixel spikes. To determine the rotation centre of the mode, the average image of all images in the series is then calculated, resembling a doughnut. A -dimensional fit of a ring to this shape results in the centre and size. They are used to define a circular area of interest for the following analysis. Pixels outside this area will be ignored, while the pixels inside are grouped in n pie slices (angle bins around the determined centre. (The parameter n is always divisible by 4 and is equal to 1 for the data presented here. Then, for each image the brightness Īi for each pie slice i {,..., n 1} is calculated. The slice i maximizing the value of (Ī(i+n/4mod n + Ī(i n/4mod n (Īi + Ī(i+n/mod n gives the axis of the dark line in the image and thus the phase ϕ for the timestamp corresponding to this image. C. Alignment of the setup The detection setup needs to be carefully aligned. This procedure is done using bright input states instead of weak pulses and classical photodiodes in place of APDs. In a first step, the Left and Right holograms are centred visually onto the beam and a camera is placed behind each hologram. LG and LG 1 modes are sent in and the position of each hologram is tuned to visually obtain a TEM -like mode for the correct input. This mode is then coupled into the respective single mode fibre. A fine tuning of the hologram position and the fibre coupling is performed; switching between different input modes, we aim for a maximum coupling of the desired mode in the respective branch and a maximum rejection of the neighbouring ones. For example, the two direct neighbours of the LG mode are the modes LG + and TEM. For these direct NATURE PHOTONICS Macmillan Publishers Limited. All rights reserved.

4 DOI: 1.138/NPHOTON FIG. S: Classical limit on the fidelity and efficiency for various relevant photon numbers n and corresponding experimental data points. n =. for the Red curve,.6 for the Orange curve, 1. for the Green curve and 6 for the Blue curve. Fidelity Efficiency neighbours, we obtain a rejection always better than 15 db. For the next-to-next neighbours, like LG mode in the Left branch, we obtain a suppression of about 7 db. Having optimised each input branch, the interferometer is closed with the fibre beam splitter FBS1 and completed as depicted in the Fig. S1. To test the convention between the interferometer phase and the image detection algorithm, we modulate the interferometer phase using the PZT, send in bright H, V, D, and A states as signals and observe the respective fringe phasing in terms of the phase deduced from the images. This measurement also serves to check the good visibility of the observed interference and the independence of the coupling efficiency into the concerned fibre of the PZT position, i.e., the phase modulation. After these alignments and verifications, the detection system is regarded as a black box and not adapted or adjusted any more during the measurements on weak light states. It should be emphasised that the alignment procedure is general and in no way optimised for specific states to be measured with the system afterwards. We note that extending our tomography technique to more modes would be possible but challenging. Indeed, due to the use of additional beam splitters, the signal to noise ratio would drop rapidly with the increasing number of modes. An interesting solution can consist in adapting our protocol with the recently proposed efficient OAM sorting techniques [3, 4]. II. CLASSICAL BOUND ON THE FIDELITY FOR A NON-UNIT EFFICIENCY MEMORY We remind here the classical thresholds for demonstrating the quantum functioning of an optical memory when probing by weak coherent state pulses, as expressed in [5] and [6]. If we were probing our memory with single photons, there would be a well-known limit of /3 on the fidelity that could be obtained with a classical protocol. The limit in the case of coherent states can be computed as an average over the Poissonian distribution of the bound on multiphoton states. This bound was computed in [7] and found to be (N + 1/(N +, giving then the following average value for a coherent state with a mean photon number n: ( N e n n F N 1 =. (4 N + N! 1 e n N=1 However, this value is only valid for a 1% efficient process. In the case of non-unit efficiency, [5] and [6] describe an optimal classical strategy that allows increasing the fidelity even more at the expense of the efficiency. The idea is to perform the measure-and-resend attacks only if the photon number is higher than a certain threshold, and induce losses otherwise. This threshold depends on the mean photon number per pulse and on the efficiency the eavesdropper is aiming at. The higher the mean photon number, the higher this threshold can be. In this case, the limit can be written as: F = ( Nmin N min+ p + N N min ( N N+ p + N N min e n n N N! where N min is the aforementioned threshold (computed from n and the target efficiency, and p is the probability to perform a measurement if the photon number N is precisely equal to N min (this probability is also obtained from n e n n N N! (5 4 NATURE PHOTONICS 14 Macmillan Publishers Limited. All rights reserved.

5 DOI: 1.138/NPHOTON SUPPLEMENTARY INFORMATION FIG. S3: Storage and retrieval efficiency as a function of the storage time. Experimental points are fitted by a Gaussian distribution, with a 1/e decay time of 15 µs. Inset: the dotted curve corresponds to the reference input pulse, while the blue curves correspond to the leakage and to the retrieval. Efficiency (% 1 5 Evetns Time (μs Storage time (μs and the target efficiency. Figure 4 in the main text gives this classical limit with our measured value of the memory efficiency. The kicks in the figure are the points where this threshold increases by one photon. Figure S shows the limit on the fidelity and efficiency that can be achieved classically for different mean photon numbers, and the experimental points (without correction at these respective photon numbers. Except at the lowest photon number, where the background noise (attributed to the APD dark counts and the contamination from the control field starts to degrade the measured fidelity, our data points are all several standard deviations above the classical threshold. III. EIT-BASED MEMORY When the pulse to store propagates through the atomic ensemble under EIT condition, in the presence of a control with a Rabi frequency Ω.3Γ in our case, with Γ the linewidth of the excited level, the group velocity v g is reduced by a factor 3 1 4, leading to a delay of ns. In the same time, the pulse is compressed by the same factor v g /c, and its spatial length becomes 3 mm, to be compared with the length of our atomic ensemble of mm. Thus, the pulse can not be contained entirely in the atomic ensemble, and a leakage is observed, before we switch off the control field. Figure S3 displays in the inset a typical storage histogram, where a leakage of 8 ± 1 % can be observed. The experimental control Rabi frequency actually results from a trade-off between the delay and the width of the transparency window, which increases with Ω. Indeed, the spectrum of the 3 ns long signal pulse must fit inside the transparency window. The EIT bandwidth, given by 1/d Ω /Γ with d the optical depth [8], is equal to 3 MHz for Ω =.3Γ. A way to overstep the current delay-bandwidth product is to operate at larger optical depth. Based on collective effects, the EIT memory protocol is very sensitive to dephasing due to inhomogeneous broadening. Atomic motion and residual magnetic field are the two main factors. The magnetic field is canceled down to 5 mg via microwave spectroscopy. By assuming a gradient of magnetic field and a quadratic atomic distribution, one can find a dephasing constant of 1 µs, defined as the 1/e decay time. The motional dephasing is more limiting in our case, as the temperature of the atoms is estimated around 1 mk. Thus, the dephasing constant is given by τ = λ/(π sin θ m/(k B T, where λ is the wavelength of the control and signal fields, θ the angle between the two beams, m the atomic mass, k B the Boltzmann constant, and T the temperature. In our experimental case, we find τ = 15 µs, in good agreement with our experimental results, as displayed in Fig. S3. [1] A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, J. Laurat, A quantum memory for orbital angular momentum photonic qubits, Nature Photon. [] D.F.V. James, P.G. Kwiat, W.J. Munro, A.G. White, Measurements of qubits, Phys. Rev. A 64, 5313 (1. [3] M.P.J. Lavery et al., Refractive elements for the measurement of the orbital angular momentum of a single photon, Opt. Express, (1. [4] M. Mirhosseini, M. Malik, Z. Shi, R. Boyd, Efficient separation of light s orbital angular momentum, Nature Com. 4, 781 (13. [5] H.P. Specht et al., A single-atom quantum memory, Nature 473, 19 (11. [6] M. Gündogan, P.M. Ledingham, A. Almasi, M. Cristiani, H. de Riedmatten, Quantum storage of a photonic polarization qubit in a solid, Phys. Rev. Lett. 18, 1954 (1. [7] S. Massar and S. Popescu, Optimal extraction of information from finite quantum ensembles, Phys. Rev. Lett. 74, 159 (1995. [8] M. Lukin, Colloquium: Trapping and manipulating photon states in atomic ensembles, Rev. Mod. Phys. 75, (3. NATURE PHOTONICS Macmillan Publishers Limited. All rights reserved.