An L-band emitter with quantum memory in silicon

Fluorescent centres in silicon have recently attracted great interest, owing to their remarkable properties for quantum technology. Here, we demonstrate that the C-centre in silicon can realise an optically readable quantum register in the L-band wavelength region where the transmission losses in commercial optical fibres are minimal. Our in-depth theoretical characterisation confirms the assignment of the C-centre to the carbon-oxygen interstitial pair defect. We further explore its magneto-optical properties, such as hyperfine and spin-orbit coupling constants from first principles calculations, which are crucial for tight control of the quantum states of the triplet electron spin. Based on this data, we set up quantum optics protocols to initialise and read out the quantum states of the electron spin, and realise a quantum memory by transferring quantum information from the electron spin to proximate 29Si nuclear spins. Our findings establish an optically readable long-living quantum memory in silicon where the scalability of qubits may be achieved by CMOS-compatible technology.


INTRODUCTION
Silicon is a mature platform for quantum technology, offering multiple advantages for the integration of defect quantum emitters. As compared to other host materials 1 , high-purity isotopically enriched single crystals of silicon 2,3 and integrated photonic devices 4,5 are readily available. These characteristics of the host material enable the improvement of optical and spin properties of the quantum defects. Recent experimental advances allowed for the isolation of single quantum emitters in this material, such as G-, W-and T-centres, with a great promise for quantum communication [6][7][8][9][10][11][12][13][14][15][16][17] . However, to further reduce the transmission losses in optical fibres, it is essential to find an active defect with the emission wavelength in either the L-or the C-band and an optically addressable spin state. This caused a revival of interest in the magneto-optical properties of deep defects in silicon, including those configurations that have been discovered decades ago 11,18-21 . Carbon and oxygen atoms are the most common impurities and therefore are important sources of colour centres in silicon crystals. When both are present in a sample, the oxygen interstitial is able to trap a mobile carbon atom, forming a carbon-oxygen interstitial pair (C i O i ) 22,23 . This stable defect complex was thoroughly characterised by various experimental techniques. Its photoluminescence (PL) signal was associated with the C-line at 789 meV 24,25 in the spin singlet, neutral charge state 26 and with the Si-G15 electron paramagnetic resonance (EPR) centre in the spin doublet, positive charge state 27 . Local vibrational modes (LVM) in the PL spectrum were identified at 64.5, 72.6, 138.1, and 145.3 meV energies 24,25 . The uniaxial stress perturbation effect on the PL line was consistent with C 1h point symmetry 26 . Deep-level transient spectroscopy (DLTS) measurements identify a deep donor level at E VBM + 360 meV position 28 . Time-resolved PL measurements reveal the excited state lifetime of 2.3 μs at cryogenic temperatures and a second slower decay activated at elevated temperatures with an activation energy of 3.1 meV 29 . The latter was associated with a reverse intersystem crossing (rISC) from a lower-lying triplet excited state. From the measurements, radiative and ISC lifetimes of 14 and 2.8 μs were obtained, respectively. PL excitation measurements reported a series of excited states, revealing the bound exciton origin of the C-line 30 . It was assumed, that the triplet state is dark owing to the quenched orbital momentum of the hole by the defect strain field. However, this phosphorescence was detected at low temperatures despite the measured isotropic g-factor in the triplet state 31 .
The defect structure was tentatively identified 32,33 using density functional theory (DFT) calculation within the local density approximation (LDA) 34,35 . The identification was based on the calculated LVMs and the position of the donor level, where the latter could be compromised by the underestimation of the band gap by DFT-LDA method. The atomic structure of the defect is shown in Fig. 1a. It represents a pair of neighbouring Si-C and Si-O defects, connected into a square-like structure by a mutual displacement from the initial split-interstitial positions towards the opposite directions (up-down configuration) 36 .
The C-centre in silicon exhibits emission in the telecom L-band with a confirmed high electron spin state which has a great potential in quantum communication with the possibility of scalable creation of the quantum emitters in carbon and oxygencontaminated silicon by CMOS compatible technology (e.g., ref. 12 ). However, the lack of understanding the nature of the electronic structure and the origin of the emission of the C-centre precludes it from being considered as a qubit. In this work, we aim to significantly advance the magneto-optical characterisation of the C i O i 0 defect by means of high-level DFT calculations, in order to elaborate on the feasible quantum optics protocols for quantum technology applications. To this end, we determine the key zero-field splitting (ZFS) parameters of the defect, taking into account the effects of hyperfine and spin-orbit-interaction. We propose a spin-selective optical cycle for the coherent manipulation and sensitive readout of the triplet electron spin state, suitable for the optically detected magnetic resonance (ODMR) measurements. Hyperfine interaction between the electron and proximate nuclear spins mediates to realise a quantum register with long-living memory as the ground state singlet does not disturb the proximate nuclear spin states.

Electronic structure
The calculated electronic structure of the C i O i 0 defect is shown in Fig. 1b. The only in-gap state is induced by the carbon impurity dangling bond resulting in a deep donor level. The calculated (+/0) charge transition level of E VBM + 408 meV is in good agreement with the DLTS level (360 meV) 28 . Note that we label the states by their irreducible representations in the C 1h point symmetry group. The 1 A 0 is the ground state with two electrons occupying the deep defect level. Promoting one electron from the deep level to the conduction band minimum in the same or opposite spin channel results in the singlet and triplet excited states, respectively. In the double group notation, the irreducible representations of these semi-localised defect states, formulated in the two-electron representation in the orbital and spin product basis, are as follows The calculated zero-phonon line (ZPL) energy in the singlet-singlet transition is 750 meV after the same finite size correction as for the charge transition level calculation was applied (see Methods). This correction is justified by the excitonic nature of the excited states, where the electrostatic potential of the localised hole is corrected similarly to the ionised ground state 37 . The calculated ZPL energy is smaller than the calculated donor transition energy E CBM À E ðþ=0Þ ¼ 780 meV À Á , leading to a stable neutral emitter. We obtain the energy of the triplet level to be 2.8 meV below the singlet excited state, in good agreement with the temperature activation energy (3.1 meV) for the rISC process 29 . We validate our HSE06 DFT results by comparison with G0W0 and Bethe-Salpeter-equation (G0W0+BSE) calculations in 216-atom supercell model [38][39][40] (see  Supplementary Note 3). To this end, we use the HSE06 relaxed ground state atomic positions as the computation of the quantum mechanical forces at the G0W0 + BSE level has not yet been implemented. Thus, we can compare the ground state electronic structure and the vertical excitation energies in both calculation methods. We obtain energy differences between the HSE06 ΔSCF and G0W0+BSE calculations of 11 and 15 meV for the vertical excitation energy and the excited state singlet-triplet splitting, respectively. The G0W0+BSE exciton wavefunction is dominated by a single Slater determinant in complete agreement with the ΔSCF electronic configuration. This justifies the ΔSCF method for the excitation calculations of the defect.

Spin properties
After obtaining the electronic states, we explore the spin interactions crucial for the tight control of the quantum bit defect. We start with the calculation of the dipolar spin-spin interaction of the electron spin in the triplet excited state, governed by the Hamiltonian operator Our calculations result in axial D = −771 MHz and rhombic E = 94 MHz ZFS parameters. In order to describe the new eigenstates, we introduce the magnetic quantum number m S in the notation. The D parameter lifts the degeneracy of the triplet state to 2Γ 1 (±) + Γ 2 (0), while E further splits the ±1 spin states resulting in Γ 1 (−), Γ 1 (+), Γ 2 (0), in ascending energy order. Spin-orbit coupling is described by the Hamiltonian operator which belongs to the totally symmetric Γ 1 irreducible representation in the double group. The selection rules for spin-orbit coupling are depicted in Fig. 3a. We further express the two-particle operators in (6) with single-particle ones (l zŝz ,l ±ŝ ± ) and expand the matrix elements between the wavefunctions to a combination of matrix elements between one-particle spin-orbitals (see Supplementary Note 1). Furthermore, we substitute the DFT-calculated matrix elements Λ ij of the spin-orbit Hamiltonian expanded on the ordered basis functions of the one-particle spin-orbitals in the above expansion, namely, This way, the spin-orbit couplings to each sublevel are obtained as follows The spin-orbit coupling of the singlet and triplet states reshapes the relative energies of the spin sublevels within the excited triplet state. It provides a large stabilisation for the m S = 0 level, but is negligible for the other two sublevels, owing to the differences in the energy gaps. Thus, the axial ZFS parameter is modified to D = 5.7 GHz, which is in reasonable agreement with the~3 GHz readout from the broad peaks in the magneto-PL spectrum 31 . The re-normalisation also leads to a strong mixing of the excited state singlet and triplet m S = 0 levels, manifesting as a 1% contribution of the triplet state to the excited singlet state.
In addition, we calculate the hyperfine interaction at 13 C, 17 O, and 29 Si isotopes surrounding the defect (see Supplementary Note 2). We conclude that the hyperfine interaction in the neutral triplet excited state of the defect is very similar to that of the positive doublet ground state, only incorporating a scaling factor of 2 of the two spin quantum numbers in the neutral and positive charge states. It can be explained by the delocalised nature of the excitonic excited state, so the similar hole spin density governs the interaction as for the positively charged defect. We find good agreement for the calculated positively charged defect 13 Table I and Supplementary Fig. 1a].

Optical properties
Having characterised the spin properties of the defect, we turn to the detailed description of the optical transitions. First, we calculate the phonon spectrum and simulate the PL spectrum [see Fig. 2a] according to the Franck-Condon principle. We obtain a sideband with LVMs at 63, 70 and 134 meV energies closely resembling the experimental spectrum 24,25,31 . We also observe an LVM at 87 meV, giving rise to a respective weak experimental signal [see Fig. 2b]. The calculated Huang-Rhys factor is 2.1, corresponding to a Debye-Waller factor of 12%. The bound exciton nature of the excitation results in a relatively delocalised electron orbital, accounting for the observed long optical lifetime in experiments 29 .
Owing to the spin-orbit coupling, ISC and phosphorescence processes can also connect the singlet ground and excited states in a spin-selective optical cycle. The strong mixing with λ z spin-orbit coupling in the excited state enables a triplet signal with considerable intensity. According to our spin-orbit calculations, the intensity ratio of the triplet phosphorescence is 1% for the m S = 0 state in the triplet manifold, in good agreement with the experimental findings 31 . This results in about 1.4 ms radiative lifetime for this state. On the other hand, allowing the intersystem crossing to the ground state can be formulated as a first-order perturbation for the m S = −1 and m S = +1 branches of the triplet (see Methods). Our calculations yield weak transition dipoles of 0.004D and 0.002D for the m S = − 1 and m S = + 1 phosphorescence branches, respectively, which have very long (>10 ms) radiative lifetimes.
The optical properties of the C-centre can be compared to other promising quantum defects in silicon. The G-and T-centres emit in the telecommunication O-band 9,16,17 and the W-centre emits near the short wavelength edge of this band 11 . The emission of the Er +3 ion lies in the C-band 41 , whereas the C-centre exhibits ZPL in the most favourable L-band 24,25 , regarding the optical fibre transmission loss. However, the former shows a very long radiative lifetime of about 1−2 ms 42,43 , which can be decreased 10-fold with integration inside silicon waveguides 44 . Moreover, emission enhancement up to 70-fold was demonstrated by embedding erbium ions in a Fabry-Perot resonator 45 . The radiative lifetime of ensemble G-centres was reported to be 5.9 ns 46 , and a~40 times enhancement in the PL intensity was achieved by resonant coupling to a dielectric meta-surface 47 . The emission of the excitonic excited states is much dimmer, as reported for the T-centre with 940 ns excited state lifetime 16 ; however, Purcell enhancement was demonstrated for up to~58-fold increase in the emission intensity 17 . Since the nature of the C-centre optical excitation is reminiscent of that of the W-and T-centres, we can assume that the integration in optical resonators could lead to a large Purcell-enhancement, as was already demonstrated for the G-and T-centres, making the relatively dim PL and phosphorescence of the C-centre significantly brighter.

Quantum protocols
We propose a quantum optical protocol to utilise the metastable triplet state and its phosphorescence for ODMR and its hyperfine coupling to nearby nuclear spins for quantum register applications. Optical initialisation of the triplet spin state is feasible owing to this spin-selective ISC to m S = 0 state. The m S = ±1 states are dark in the phosphorescence, suggested by their calculated very long lifetime. We predict an ODMR contrast detected in the phosphorescence signal, where the spin rotation between the bright 0 j i and the dark ± 1 j istates decrease the phosphorescence intensity [see Fig. 3a]. The ODMR contrast diminishes at elevated temperatures together with the phosphorescence signal itself, owing to the activated rISC rate.
A long-lived quantum memory can be achieved in the defect by transferring the coherent electron spin state to the nuclear spin manifold. This can be achieved by Landau-Zener transitions 48,49 at the triplet state level anti-crossing (LAC) condition where we note the electron and nuclear spin magnetic levels as m S ; m I j i. The nuclear spin flip-flop process is feasible utilising the perpendicular hyperfine couplings (A xx , A yy ) to nearby nuclei. We calculate such couplings to be larger than 10MHz for several positions of the 29 Si nuclei (see Supplementary Table 1), allowing for short ramp time of the magnetic field sweep across the LAC [see Fig. 3b], however, decreasing the electron spin coherence time. As a compromise between operation speed and decoherence due to the hyperfine interaction itself, we suggest further lying atoms with hyperfine couplings in the order of 1 MHz as the quantum registers [see Supplementary Fig. 1b]. Alternatively, a weak magnetic field perpendicular to the nuclear spin quantisation axis (set by A zz ) and a resonant microwave excitation pulse can also be utilised to map the electron spin superposition onto the nuclear spin as demonstrated for NV centre and a single 13 C in diamond in ref. 50 . The former allows for Larmor precession in the degenerate nuclear spin manifold of the m S = 0 state, while the latter is tuned to the hyperfine resonance at D À E À Azz 2 energy.
where π L and π MW are π pulses in the nuclear and electron spin manifolds, respectively. The readout of the nuclear spin memory into the electron spin manifold utilises the same magnetic field sweep as the storage process, however, it is the inverse process for the above pulse sequence. The initialisation of the nuclear spin is feasible with subsequent memory readout and optical pumping processes. We note that there are methods for the fast manipulation of multi-qubit registers with a combination of microwave and radio frequency pulses as well 51,52 . Based on the above initialisation, storage, and readout protocols, the individual control of the electron spin of the C-centre and a nearby single nuclear spin is feasible for the realisation of long-living quantum memory. Additionally, as the excited triplet state relaxes to the singlet ground state of the defect, further decoherence by the hyperfine coupling to the electron spin is eliminated, increasing the coherence time of the nuclear spin register. We note that isotope engineering of silicon is viable to find the optimum operation conditions, similar to the recent results for divacancy qubits in silicon carbide 53 .
There are quantum memories already investigated in silicon, the spin doublet T-centre 54 and the Er +3 ion 41 , showing favourable spin coherence properties. For the former one, ODMR was demonstrated using spin selective resonant optical excitation (hyperpolarization method) reporting T 1 > 16 s and Hahn-echo T 2 times exceeding a millisecond and a second for the electron and hydrogen nuclear spins, respectively 16 . The erbium ion shows shorter coherence times of T 1~1 ms and T 2 = 7.5 μs 55 . However, the advantage of the singlet ground state for the increased nuclear spin coherence was proposed for the ST1 and TR12 centres in diamond as well 56,57 . These successful applications call for the need for an in-depth experimental study on the spin dynamics in the C-centre as well for a direct comparison to the above contenders.

DISCUSSION
The C i O i 0 defect in silicon, identified as the C-centre, was investigated for quantum technology applications. We obtain electronic structures in-line with experimental observations. We explore the spin interactions in the defect and report rhombic ZFS values crucial for the coherent manipulation of the electron spin qubit. We determine the symmetry-allowed spin-orbit couplings, revealing ISC transition with λ z coupling between the excited states, and with different λ ± coupling strength from the m S = ±1 substates of the triplet excited state is coupled to the ground state. These phosphorescence transitions correspond to the observed triplet radiative transition at cryogenic temperatures 31 . The lack of phosphorescence at elevated temperatures can be explained by a thermally activated rISC process, thermalising the excited states. On the other hand, population trapping in the triplet state is feasible at cryogenic temperatures, increasing the relative intensity of the phosphorescence assisted by the spin-orbit mixing with the close-lying excited singlet state. We conclude that C-centre in silicon is not only a promising single photon source for quantum communication applications via telecommunication optical fibres, but optical readout of the spin state is feasible in its triplet excited state. This opens the door for several quantum technology applications utilising this defect, including cryogenic temperature nanoscale sensing and quantum communication applications.

Details on DFT calculations
The atomic structure of the defect is modelled in a 512-atom silicon supercell.
To this end, we used the Heyd-Scuseria-Ernzerhof (HSE06) functional 58 and Γ-point sampling of the Brillouin zone, as implemented in the VASP plane wave-based code [59][60][61][62] . For the description of core electrons, projector augmented wave pseudo-potentials are used 63,64 . Fig. 3 Proposed quantum protocols. a Quantum protocol for ODMR in C-centre. The red and blue arrows show fluorescence and phosphorescence, respectively. The blue dashed arrow highlights the spin-selective ISC leading to the main contribution to phosphorescence. The dotted lines represent the weak ISC to the ground state. Orange lines correspond to coherent spin control using microwave excitation. The spin-spin interaction incorporates both dipolar spin coupling and the re-normalisation effect from the spin-orbit interaction. The spinselective ISC initialises the bright 0 j i spin state. Applying spin rotation leads to a contrast in the phosphorescence emission. b Magnetic field dependence of the triplet electronic and doublet nuclear spin levels labelled by m S ; m I j i . Procedure for the Landau-Zener transition is highlighted for negative hyperfine coupling parameters. The initial superposition (green and red) of electronic states α 0; þ 1 2 þ β À1; þ 1 2 is transferred to the coherent superposition of nuclear states α 0; þ 1 2 þ β 0; À 1 2 . The degenerate 0; ± 1 2 hyperfine levels are separated for the sake of clarity.
Excited states were calculated with the ΔSCF method 65 . The finite size correction of the charged supercells is obtained from selfconsistent potential correction method 66 to be 56 meV. The spin-spin contribution to the ZFS parameters was calculated by the approach of Bodrog and Gali 67 using the VASP implementation of Martijn Marsman. The Perdew-Burke-Ernzerhof 68 functional was employed to compute the phonon spectrum as well as in the non-collinear DFT calculations in order to manage the computational cost.
Details on GW calculations G0W0 38 and BSE 39,40 calculations were performed using VASP in a 216-atom supercell model, keeping the Γ-point sampling. The orbital basis was calculated with DFT HSE06 functional, with a number of unoccupied orbitals more than 10 times the occupied ones. The energy cutoff for the calculation of the response function was limited to 150 eV to manage the computational workload. BSE was calculated beyond the Tamm-Dancoff approximation 69 , including 100 occupied and 100 unoccupied orbitals.

Calculation of phosphorescence rates
The transition dipole of the phosphorescence process between the triplet excited and singlet ground state can be calculated from perturbation theory as 70 where Δμ ¼ 3 A 00 h jerj 3 A 00 i À 1 A 0 h jerj 1 A 0 i ¼ 13:7 D is the change of the macroscopic dipole moment in the transition.

DATA AVAILABILITY
The authors declare that the main data supporting the findings of this study are available within the paper and its Supplementary files. Part of the source data is provided in this paper. The data that support the findings of this study are available from the corresponding author upon reasonable request.