Scalable spin-photon entanglement by time-to-polarization conversion

Spin-photon interfaces are strong candidates for building blocks of future quantum networks and quantum computers. Several systems currently under examination present promising features, but none of them yet fulfil all requirements for these aims. A particularly attractive strategy for the realization of these applications is the creation of strings of entangled photons, where quantum correlations among the photons are mediated by operations on the spin of the emitter. Here, we demonstrate for the first time the creation of spin-photon entanglement within the fundamental unit of a novel, scalable protocol based on time-to-polarization conversion. This principle allows us to bypass many of the imperfections of currently available photon sources and can therefore be utilized with a large variety of emitters. We execute the protocol using a nitrogen-vacancy centre in diamond, which possesses a long coherence lifetime and multiple spin degrees of freedom, thereby offering an outlook towards the creation of large entangled states.


INTRODUCTION
The generation of entangled photon states is of central importance in linear optical quantum computing (LOQC) 8,9 and optical quantum communication 10,11 , and has potential applications in quantum sensing and metrology 12 . Currently, entangled photon sources rely mostly on spontaneous parametric down-conversion, which is robust and offers high purity, but is limited by intrinsically probabilistic entanglement generation 13 . For most applications in quantum technology, large entangled states are necessary in order to reach performance levels which exceed those of classical devices. The generation of such states therefore remains an outstanding challenge.
Cluster states are particularly desirable resources as they enable measurement-based quantum computation and have an in-built resilience to noise and loss [14][15][16] . Spin-based protocols have been developed for the generation of entangled photon strings, the most prominent of these being the "cluster state machine gun" (CSMG) of Lindner and Rudolph 5 . This protocol is appealingly simple and robust, and can be scaled to higher-dimensional cluster states using multiple spins 7,17 . Its requirements are however quite stringent. Ideally, it necessitates two orthogonally polarized, energydegenerate optical transitions with negligible cross-decay terms (see Fig. 1a)). A first demonstration of the CSMG protocol with quantum dots was shown by Schwartz et al. 18 , where the length of the cluster state was limited to three photons due to the short lifetime of the qubit.
Here we employ a nitrogen-vacancy (NV) defect in diamond, which is known for its excellent quantum coherence properties, and is therefore a promising candidate for quantum technology 6,19 . As a CSMG source, however, it presents several drawbacks. In particular, the optical transition energies connecting different spin states require careful tuning into resonance while maintaining negligible state mixing, and several of its excited-state levels can decay to multiple ground states and have a strong decay channel into a long-lived metastable state 20 (see Fig. 1b and 1c). It is therefore challenging to adhere to the requirements of the original CSMG scheme 21 .
In this work we develop and demonstrate a novel, scalable scheme based on time-to-polarization conversion (TPC). Its main advantage is that it requires the use of just one optical transition (Fig. 1d). Therefore the most favourable spin properties of the strongest transition available can be used, and tuning of energy levels is no longer required. With these simplifications, it is applicable to a large variety of emitters. Figure 1: Conceptual overview. a) Ideal level scheme for the protocol of Lindner and Rudolph. Simultaneous excitation of two energy-degenerate transitions creates photons of orthogonal polarizations. Starting the cycle in an equal superposition of the states |0⟩ and |1⟩ creates a spin-photon entangled state of the type (|0, ⟩ + |1, ⟩)/√2 in the first iteration. b) Photoluminescence excitation spectrum of the NV centre used herein. Inset: electron microscope image of a solid-immersion lens milled into diamond. c) Simplified level scheme corresponding to the optical NV transitions. The emitter does not present the required energy-degenerate transitions with orthogonal photon polarization and undesired decay channels occur, particularly from the level |±1 ⟩, which can non-radiatively decay to the metastable state (MS). Spin readout is performed by observing phonon sideband (PSB) photons. d) Ideal time-to-polarization entangling (TPC) scheme. The two ground states |0 and |-1 constitute the matter qubit. The red laser is resonant with the |0|0e transition and far-detuned (by 0.87 GHz in this experiment) from the (|±1⟩ ↔ | ± 1 〉) transition, ensuring negligible cross excitation (see panel c). The two paths "A" and "B" display the evolution for initialization of the spin in the states |0⟩ and |−1⟩, respectively, as we run through our entangling sequence (from left to right). We start with the spin in the superposition (|0⟩ + |−1⟩)/√2 such that paths "A" and "B" also occur in superposition, forming the basis of the entanglement generation. In panels , a laser pulse is applied which leads to the generation of a photon in path "A". The photon is guided via a switch into a channel in which its polarization state is rotated to a horizontal orientation. No photon is generated in path "B" since no resonant transition is available from the state |−1⟩. In panels , the spin state is inverted in both paths by a microwave pulse and the classical optical routing is switched to a vertically polarized channel. In panels , a further excitation cycle is triggered with a second laser pulse. Due to the route switching, this excitation results in the generation of a vertically polarized photon in the evolution path "B". This time, no photon can be generated in path "A". Although the photon is originally emitted with the same polarization state in both evolution paths "A" and "B", the classical time-dependent photonic routing and polarization elements enable the entanglement of the spin with the photon polarization. The path information is erased with a polarizing beamsplitter at which the possible photon trajectories are overlapped simultaneously, resulting again in the desired entangled state of the type (|0, ⟩ + |1, ⟩)/√2 (see text). Together, the threelevel system and the photonic routing elements therefore act as the desired four level system. The procedure can thus be iterated to generate a string of entangled photons, analogously to the CSMG protocol.
Briefly, our protocol starts with an equal spin superposition between the ms=0 and ms=-1 optical ground states. Upon excitation, the emitter produces a photon with 50% probability, which is stored in an H-polarized channel. During the storage time the superposition of the spin is inverted by a πrotation. A second excitation pulse can then launch a photon, again with 50% probability, into a Vpolarized channel. The sequence therefore generates a single photon in a superposition of polarizations. At this stage, it is however not possible to verify entanglement on the spin-photon state as the initial spin state is revealed directly by the photon's position. Experimental sequence to generate spin-photon entanglement by time-to-polarization qubit conversion (TPC). The dashed rectangle highlights the entangling operations, which can be iterated at will to scale the output to a multiphoton entangled state. The probing field is used to measure the phase of the interferometer. The microwave operations are rotations around the spin's y-axis. Events and states are shown on a time axis.
The path information, and with it the emission time information, is therefore erased by matching the delay time between excitation pulses to the storage time of the H-arm. After the TPC process, the time-separated wavepackets merge into a single polarization qubit entangled with the NV centre spin.
In more detail, the experimental sequence used to demonstrate the method unfolds in three steps: preparation, entanglement generation and tomography (  (Fig. 2b).
We experimentally demonstrate the process using a NV centre in an artificial diamond created by chemical vapour deposition. A microlens (Fig. 1b) is machined over a pre-allocated NV centre by focused-ion beam milling for improved photon collection efficiency. We manipulate the spin using a microwave field radiated from two bond wires. The diamond is cooled to ~ 4.5 K in a closed-cycle cryostat and photons are collected through a window using a microscope objective.

RESULTS
The fundamental unit of the proposed protocol is experimentally demonstrated by performing partial tomography on the resulting spin-photon system and quantifying the respective entanglement. The resulting measured correlations in the ⊗ basis are represented in Figure 3a, with a correlation value of = 〈 ⊗ 〉 = (83.7±1.6)%. To extract the = 〈 ⊗ 〉 correlations, spin-photon entanglement is generated for two different initial spin superposition states = (|−1 〉 − |0〉)/√2 and = (|−1 〉 + |0〉)√2, corresponding to the spin measurement projection states | ± 〉. Following established procedure 22 , we now observe the spin-photon correlations by measuring their dependence on the interferometer phase, . The photon detection events in the quadrature ports (D, A, R and L) are sorted according to  at the time of detection and combined taking into account each port's phase offset. Fig. 3b shows the conditional probability of projecting the spin state onto | + 〉 and | − 〉 , given the measurement of a photon in the equatorial basis |〉 = | 〉 + | ñ. The resulting curves correspond to a correlation =(40.7±2.9) %, showing the entanglement signature expected for the |y 〉 Bell-state (as opposed to =0 for a statistical mixture). In order to probe the quality of our source directly, the presented results have an accurately calibrated measure of the background light present in our ZPL detection window deducted (see Supplementary Information).
From the retrieved correlations, we estimate a lower bound on the entanglement fidelity, with respect to the ideal |y 〉 Bell-state, of ≥(64.7±1.3) % after (≥(56.0±0.9) % before) background subtraction. This value is significantly above the bound for a classical state ( ≤ 50 %), thereby demonstrating the entanglement in our spin-photon protocol, by over 11 standard deviations (over 6 without background subtraction). The fidelity is currently limited by a variety of imperfections (such as spin mixing in the excited state manifold and imperfect spin readout) which can be minimized by improvements to the setup and system (see Supplementary information).

DISCUSSION
The generation of entangled photon strings from quantum emitters will enable a multitude of applications in quantum technology. The TPC technique presented here relaxes many of the requirements placed on the emitter and will therefore broaden the range of systems for which such generation schemes are possible. We have performed the first demonstration of all key elements of the method, paving the path for future efforts towards generating multiphoton states. The emission and collection of photons from the optical transition can be drastically improved using an optical resonator. The most advanced results suggest that an enhancement of ZPL emission by a factor approaching 20 is possible with current technology 20 . Replacing the input PBS by a switch can further double the success probability per photon. Our initialization routine can be drastically shortened by including an additional laser on the |±1〉 ↔ | ± 1 〉 and implementing single-shot readout. With a feasible system efficiency of 40 % and a sequence duration of 10 µs, it will become possible to generate 3-photon cluster states at a rate exceeding 6 kHz, or 10-photon states at a rate of 10 Hz. NV centres are furthermore well-suited to the direct generation of two-dimensional cluster states, which are required for universal quantum computation 17 . The necessary quantum operations using ancillary spins 7 or remote centres have been impressively demonstrated in previous work 24,25 .

Sample and fabrication:
An artificial, single-crystal diamond of natural isotopic abundance and with a {1, 1, 1} surface orientation hosts the NV centre. We surveyed the diamond for shallow defects and created solid immersion lenses using focussed ion-beam milling over several defects with the desired N-V axis orientation (perpendicular to the surface). We then coated the surface with 110 nm of SiO2 in order to reduce Fresnel reflection losses and laser backscatter at the high-index interface.

Experimental details:
Our resonant optical pulses at around 637.2 nm are delivered from a narrowband external-cavity diode laser (Toptica DL Pro HP 637) and switched with two electro-optic amplitude modulators (Jenoptik AM635) in series. The fluorescence is split at a laser line filter into the resonant ZPL portion and the far off-resonant PSB portion. The latter was used to perform efficient spin readout. Since most of the photons decay into the PSB 26 and the system has significant photon losses, the measured ZPL efficiency (source to click) is ~2 × 10 -5 .The average probability of getting a spin readout click when prepared in = 0 is 16.7±0.1%. We therefore observe an average of 36 PSB-ZPL coincidence events per hour.
The time-to-polarization conversion was performed by directing the ZPL part of the NV emission into a polarization-maintaining, fibre-based Mach-Zehnder interferometer. We matched the time between the two optical π-pulses in our sequence to the propagation delay between the arms of the interferometer. The interferometer was passively stabilized to minimize path length changes occurring during each entanglement cycle. The phase was furthermore tracked during each entanglement cycle by sending resonant laser pulses through the TPC between entanglement sequences and measuring the intensity on four quadrature detectors corresponding to the photon states D, A, R and L.
Passive routing of photons: The emitted photons are diagonally polarised with respect to the unbalanced interferometer´s arms and therefore split in two possible propagation paths (50% take the H(V)-polarised long (short) arm) and two corresponding propagation times. Therefore two excitation/emission cycles result in four arrival times at the output of the interferometer: 1) Early emission takes the short (V) path; 2) Early emission, long (H) path; 3) Late emission, short (V) path; 4) Late emission, long (H) path.
If the separation between the two emissions matches the time difference in the propagation, then events of type 2) and 3) have the same arrival time. Detection within this time window erases the path information and heralds the intended function of the passive switch (50% success probability). Events outside this window, 1) and 4), are path-revealing and are not conducive to the creation of the photonic polarisation qubit.

Electron and nuclear spin initialization:
The initialization sequence relies on electron and nuclear spin flips in the optically excited state of the NV centre. A low-power laser pulse (5 µs long), resonant with the |0〉 ↔ |0 〉 transition, results in an electron (nuclear) spin flip with high (low) probability. Subsequently, nuclear spin selective microwave pulses are applied to the ground state, which drive the population in the undesired states back to |0〉. This sequence is repeated several times in order to enhance the probability of initializing the electron (nuclear) spin in state | = −1〉 (| = 0〉).

Fidelity estimation:
Following a well-established method 22,25 we calculate the lower bound on the fidelity as: Further details of all methods are provided in the supplementary information.
Note: During preparation of the manuscript, we became aware of preliminary efforts towards the results achieved in this work using a quantum dot 27 .