Polarisation-preserving photon frequency conversion from a trapped-ion-compatible wavelength to the telecom C-band

We demonstrate polarisation-preserving frequency conversion of single-photon-level light at 854 nm, resonant with a trapped-ion transition and qubit, to the 1550-nm telecom C band. A total photon in / fiber-coupled photon out efficiency of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim$$\end{document}∼30% is achieved, for a free-running photon noise rate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim$$\end{document}∼60 Hz. This performance would enable telecom conversion of 854 nm polarisation qubits, produced in existing trapped-ion systems, with a signal-to-noise ratio greater than 1. In combination with near-future trapped-ion systems, our converter would enable the observation of entanglement between an ion and a photon that has travelled more than 100 km in optical fiber: three orders of magnitude further than the state-of-the-art.


I. INTRODUCTION
similar and dissimilar quantum matter to interface over both short and long distances.
Telecom frequency conversion of photons connected to several examples of quantum matter has recently been demonstrated, including quantum dots [20][21][22][23], cold gas atomic ensembles [24][25][26] and solid-state ensembles [27]. Applying QFC to trapped ions is challenging. The comparatively low rate and efficiency with which photons have been collected from / absorbed by an ion demands a highly efficient and low noise conversion process. Readily accessible photonic transitions in ions also lie in the ultraviolet or visible regime, which suffer high absorption and strong dispersion in nonlinear crystals. Furthermore, direct (single step) conversion of those photons to telecom in the so called 'long pump wavelength regime' is not possible, leading to additional noise processes during conversion [12]. Nevertheless, significant progress has been made in overcoming these challenges [28][29][30]. In [30], for example, the authors convert attenuated laser light at 369.5 nm (a transition in Yb + ) to 1311 nm, achieving a waveguide efficiency of ∼ 5 % (including coupling losses) and a total efficiency for fiber-coupled output photons of 0.4 %.
In this paper, we present experiments that demonstrate photon conversion from λ s = 854 nm (s = signal) to λ t = 1550 nm (t = target, Telecom C band), via difference-frequency generation (DFG) in a waveguide-integrated χ (2) crystal, using a strong pump laser at λ p = 1902 nm. The signal wavelength corresponds to the P 3/2 to D 5/2 dipole transition in singularly ionised atomic Ca + , which can be efficiently collected from the ion in a cavity quantum electrodynamic (CQED) setting. Our experiments use laser light, resonant with the ionic transition and attenuated to the single-photon level. We refer the reader to work studying conversion from 854 nm to 1310 nm [31].
The content of the paper is as follows. First, the importance of the 854-nm transition in Ca + is briefly discussed and motivated. Section II presents a scheme that enables telecom conversion of one polarisation component of an 854-nm photon. Here, the limits on the efficiency and photon noise are presented. Section III presents a scheme that preserves the polarisation during conversion, allowing translation of a polarisation qubit from 854 nm to 1550 nm with high fidelity. The achieved performance brings QFC experiments within reach of existing 854-nm trapped-ion photon sources. Section IV discusses the potential for our conversion scheme, combined with future trapped-ion sources, to enable the observation of entanglement between an ion and a photon that has travelled more than 100 km of optical fiber. Section V presents our discussion and conclusion.
A. The 854 nm transition in Ca + From the perspective of achieving efficient, low-noise photon frequency conversion from a trapped-ion wavelength to telecom, the 854-nm transition in Ca + represents an ideal choice. Transmission losses at this wavelength are low in non-linear conversion crystals and the required poling period to overcome the phase mismatch can be precisely manufactured, allowing for efficient firstorder quasi phase matching and long interaction lengths. Futhermore, single-step conversion to 1550 nm requires a pump laser in the so-called long-pump-wavelength regime: the 1902-nm pump photons have lower energy than the target 1550-nm wavelength, such that spontaneous parametric down conversion (SPDC) of the pump cannot produce noise photons at 1550 nm [12]. The threshold input photon wavelength for this condition is λ > 1550/2 nm = 775 nm (1310/2 nm = 655 nm), which can be found in very few ionic species and in each case is branching-ratio unfavored.
From the perspective of the ion, the 854-nm transition is directly connected to a leading qubit transition [32]: emission of an 854-nm photon by Ca + leaves the ion's valence electron in the metastable excited state |D 5/2 of the well known 729-nm optical qubit. A broad array of techniques are readily available for manipulating the quantum state on this optical qubit transition [32]. It is a challenge to efficiently collect 854-nm photons from the ion (or have them absorbed by the ion): in free space, resonant excitation to the excited |P 3/2 leads to the emission of an 854-nm photon in only ∼1/17 of cases [33], in most cases a 393-nm photon is emitted. However, an optical cavity around the ion can enhance emission on this transition [34,35]. In such a CQED system, both near-maximal entanglement between a travelling 854-nm polarisation qubit and the ion qubit [35], and state mapping from ion qubit to photon [36] have been achieved with high fidelity. The bandwidth of an 854-nm photon in a CQED setup can be significantly narrower than atomic transition linewidth in free space (23 MHz), as the photon leaks slowly out of a high-finesse cavity (e.g. ∼ 50 kHz [35]). Such narrowband photons present challenges and opportunities for low noise QFC. On the one hand, noise photons introduced by the QFC process will be integrated over the temporally long photon wave packets (many microseconds). On the other hand, the narrowband photons can be strongly spectrally filtered to reduce such noise, without compromising transmission.
We perform frequency conversion by DFG using the χ (2) nonlinearity in a LiNbO 3 waveguideintegrated chip. Each chip is 48 mm long and contains ridge waveguides (LiNbO 3 layer on LiTaO 3 substrate) milled out along its length, with dimensions of approximately 11.0 µm by 12.1 µm (fabricated by NTT electronics). To achieve first-order quasi-phase matching, the guides are poled with a period of approximately 22 µm. While each chip has 12 individual waveguides, a single guide in any one chip is used for each experiment. The waveguides enable a continuous high spatial mode overlap between the three optical fields, are single mode for 1550 nm and 1902 nm, multimode for 854 nm and are anti-reflection coated for all those wavelengths on each facet (R ≤ 1%). The conversion process is phase matched when all three optical fields have the same linear polarisation; the orthogonal polarisation is supported in the guide but remains unconverted.
We now provide a summary of the basic experimental setup, detailed in Figure 1. For the pump we use a Tm-doped fiber laser (AdValue Photonics AP-SF1) and for the 854 nm input a diode laser (Toptica DLPRO), stabilised to within a few MHz of the ionic transition using a wavemeter lock (High Finesse WSU10). Both signal and pump are delivered to the photon conversion setup using polarisation-maintaining single-mode optical fiber, spatially overlapped using a dichroic mirror and free-space coupled into a ridge waveguide via an aspheric lens. A second asphere at the waveguide output collimates the output fields, which are then sent to various filtering and analysis stages. The chip is temperature-stabilised and a waveguide is chosen with a quasi-phase matching temperature of 38 • C. The spectral acceptance bandwidth of the phase matched conversion process centred at 854 nm is measured to be ∼ 0.2 nm (82 GHz), which agrees with theoretical calculations based on refractive indices of bulk LiNbO 3 at the corresponding wavelengths [37]. Note that this acceptance bandwidth for photon conversion is orders of magnitude broader than 854 nm photons from the ion, and does therefore not act as a filter. The temperature bandwidth (FWHM) of the phase matched conversion process is measured to be 2.4 ± 0.2 • C.

Efficiency.
QFC in a material with a χ (2) nonlinearity is a three-wave mixing process in which the quantum states of light can be coherently interchanged between two frequencies ω 1 and ω 2 via the interaction with a strong (undepleted) pump field at frequency ω p , where ω 1 + ω p = ω 2 . In our experiments, ω 1 and ω 2 are the frequencies of the 1550 nm and 854 nm photons, respectively. For QFC in a waveguide-integrated lossless material, and in the case of perfect phase matching, one can show that the efficiency of conversion for an interaction (waveguide) length L is given by [38] η = N 2 /N 1 = sin 2 ( η nor P p L). Here, N 2 is the number of output 1550 nm photons, N 1 the number of input 854 nm photons, P p is the pump power. The normalised efficiency η nor depends on the material nonlinear strength and the three-wave mode-overlap integrals. For a given waveguide length L, there is a pump power P max that achieves complete conversion. For larger pump powers, conversion back to the initial frequency takes place.  nm, one obtains a maximum external classical waveguide conversion efficiency of η waveguide class = 0.59 ± 0.03, which still contains losses due to in-coupling into the waveguide and waveguide propagation losses [49].
Assuming that the waveguide transmission losses are equal at 854 nm and 1550 nm, 0.89±0.04 ) is the maximum conversion efficiency. The value that we obtain, η waveguide class , is 0.30±0.05 lower than this maximum. Figure 2 shows that, at the point of maximum conversion efficiency, a fraction 0.22±0.01 of unconverted 854 nm light remains at the waveguide output (when accounting for known passive optical losses). One sees, therefore, that the majority of 'missing conversion efficiency' lies in unconverted 854 nm light.
In the case of perfect phase matching and no losses, the DFG process for three optical modes in a waveguide allows for complete depletion (conversion) of the signal wave. However, the situation is more complicated when several spatial (or axial [39]) modes are involved. Indeed, our waveguides are multimode at 854 nm and a fraction of 854 nm populates higher-order modes due to imperfect mode matching at the input. The different modes have different effective refractive indices than the fundamental mode and are not therefore simultaneously phase matched. As a consequence, the conversion efficiency of higher order modes is weak and they remain largely unconverted. Beam profile measurements of the unconverted 854 nm light exiting the waveguide reveal that it consists of spatial modes of much higher order than the fundamental.
We conclude from this experiment that a waveguide conversion efficiency of 0.59 ±0.03 was achieved, limited by the unintentional excitation of higher-order waveguide modes at 854 nm. Any way in which the strong pump laser field can introduce photons directly at the output telecom wavelength introduces noise that can dominate the single-photon QFC signal. We call the rate of such noise the noise photon rate (NPR) to distinguish it from the intrinsic detector dark count rate (DCR).
At 200 mW, the pump photon flux in the waveguide is vast ( 2 × 10 18 Hz) such that even extremely weak processes in the waveguide, through which pump photons are converted directly to telecom, can overwhelm the output. The key process through which this happens on propagation through the crystal is anti-Stokes Raman scattering, where pump photons receive energy from phonons in the crystal. Anti-stokes Raman spectra for similar ridge waveguides to ours are presented in [42] and show that the large spectral separation between our 1902 nm pump and 1550 nm target is far from any Raman resonance peaks. Nevertheless, significant photon noise has been observed at spectral separations well beyond that expected by theory [12], covering even the large spectral separation in our experiments.
We study the noise photons at the waveguide output for 200 mW input pump light only, using the single-photon analysis path at the waveguide output in Figure 1. That analysis path consists of For room temperature operation and a 12 nm filtering bandwidth, we observe a total detector click rate of 1.4 kHz, which is completely dominated by noise photons. When accounting for our detector efficiency of 0.1, that noise rate corresponds to 14 kHz of photon noise in the filtering bandwidth before detection (Fig. 3). The NPR is seen to reduce with crystal temperature T , as one would expect for anti-Stokes Raman scattering, approximately following the phonon occupation number given by the Boltzmann distribution NPR = Ae − ∆ω k · 1 T , where ∆ω is pump-target frequency difference; , k are Planck and Boltzmann constants, A is a fitting parameter. By extrapolation of the theoretical fit, operation at -50 • C could provide a total noise reduction by a factor of 9. This offers a way to reduce photon noise, without the need for narrowband filtering.
The NPR at the quasi-phase matching temperature is seen to reduce in proportion to the filtering bandwidth (Figure 3), consistent with the noise source being broadband and white. Using our narrowest filtering bandwidth of 2 pm at 1550 nm (250 MHz bandwidth, transmission 0.26 ± 0.01 at 1550 nm), the NPR before detection is reduced to 4 ±2 Hz. Note, in Section III we employ a few picometer filtering stage with a greatly improved transmission at 1550 nm. 3. Efficiency and signal-to-noise ratio, at the single photon level.
In order to determine a signal-to-noise ratio (SNR) one has to determine an appropriate signal: the rate of 854 nm photons one can expect to be available for converting in experiments involving a trapped Ca + ion. We consider the case where 854 nm photons are to be generated on-demand by (e.g. triggered by a laser pulse), and are entangled with, the Ca + . In [35], the repetition rate for attempting to generate photons was ∼1 kHz, while photons were actually collected into optical fiber at a rate of ∼100 Hz. In future CQED ion trap systems, that use state-of-the-art mirror coatings with losses of only a few parts per million [43] (to enable higher collection efficiencies), and faster cooling schemes for state reinitialisation [44] (to enable higher repetition rates), it is feasible that triggered 854 nm photons could be collected at a rate of 10 kHz from the ion.
Using calibrated neutral density filters, the input 854 nm light power in our setup is attenuated to a value corresponding to an average photon rate of 10 kHz (2 fW) before the input aspheric lens, to replicate a future trapped-ion source. Figure 4 presents the photon count rate of the 1550 nm detector as a function of pump power and for our 2 pm bandwidth telecom filter. At the peak conversion efficiency (again at around 200 mW pump) 136 ±3 Hz counts are recorded, corresponding to a total detected conversion efficiency of 0.0136±0.0004. When removing the 0.10 efficiency of our detector this corresponds to a photon in/out conversion efficiency of η out/in = 0.136: the probability that an incoming 854 nm photon is converted to a single-mode-fiber-coupled 1550 nm photon that has passed the filtering stage. This result is consistent with our classical light measurements, leading to an external waveguide conversion efficiency of 0.62 ± 0.03 when removing the filtering losses.
Since the detected total noise rate is only a few Hz, the SNR profile shown in Figure 4 closely follows the conversion efficiency curve, peaking at 66±6 at 200 mW. A SNR > 1 would therefore be possible for an input 854 nm photon rate 60 times lower than was used (∼ 160 Hz) bringing QFC experiments using 854 nm photons from an ion within reach of existing experimental systems.

III. POLARISATION-PRESERVING CONVERSION
For applications in quantum networking, the conversion process should preserve any photonic degree of freedom used to encode quantum information. While the single-waveguide conversion scheme presented in Figure 1 is suitable for e.g. time-bin encoded photonic qubits, only one polarisation component is converted. Polarisation qubits are appealing as they are straightforward to manipulate, analyse and can be preserved through long optical fibers [45]. Furthermore, polarisation entanglement between ion and 854 nm photon, as well as state mapping to photonic polarisation have been achieved experimentally [35,36].
Our polarisation-preserving conversion scheme, which employs two independent waveguide crystals in series ( Figure 5), is now briefly summarised. The conversion process in each waveguide is phase matched when all three optical fields have the same linear polarisation (vertical |V ); the orthogonal polarisation (horizontal, |H ) is supported in the guide but remains unconverted.
Consider an arbitrary input 854 nm single-photon polarisation state α|H 854 + β|V 854 and input classical pump polarisation state δ|H pump + γ|V pump . The first waveguide in Figure 5  All errors bars represent one standard deviation and derive from Poissonian photon counting statistics.

Polarsation-preservation.
To determine the extent to which the polarisation is preserved, as it is converted from 854 nm to 1550 nm, we perform process tomography [46]. Specifically, the 6 standard basis states for polarisation qubits are injected (horizontal, vertical, diagonal, antidiagonal, right-and leftcircular). For each input, sufficient measurements are performed to reconstruct the output 1550 nm polarisation state, via single-photon counting. From these measurements, the process matrix χ i j is reconstructed via an optimisation to find the most likely process to have generated the data (maximum likelihood estimation). The process matrix describes the process ε applied to any in- where O k are a set of operators which form a basis for the set of operators on the polarisation qubit state space. We choose the basis corresponding to the identity and standard Pauli operators, respectively. To quantify polarisation-preservation, we maximise the χ 11 (identity) element of the reconstructed process matrix when allowing for arbitrary unitary operations (via a simple numerical search), obtaining a value of 0.93 ±0.01: this is the minimum fidelity with which any input polarisation state is translated through the device and detected, up to a fixed known unitary operation, and is more than sufficient to preserve polarisation entanglement between ion and photon, as discussed in the following section. We find that a fidelity of 0.95 ±0.01 would be achieved when accounting for the measured NPR and DCR. The remaining infidelity is attributed to errors in the angular settings of waveplates by less than a degree.

IV. FUTURE APPLICATIONS: LONG-DISTANCE TRANSMISSION.
Converting photons from 854 nm to 1550 nm offers a reduction in attenuation in optical fiber from 3 dB/km to 0.2 dB/km, respectively. When accounting for the finite 30% efficiency, one finds that our conversion system offers an improved rate of photon transmission for all fiber lengths greater than 1.9 km. The advantage becomes profound for longer distances: over 50 km (100 km) of fiber, the transmission probability using our device would be 3% (0.3%), compared to 1 × 10 −13 % (1 × 10 −28 %) at 854 nm.
Using photons to distribute entanglement between remote network nodes is an important task in quantum networking. We are interested therefore in assessing the distance over which our conversion and detection setup could allow for entanglement to be detected between a photon and an ion. To answer this question, we consider the case where the ion emits 854 nm photons on demand, at a rate of 10 kHz and in the maximally entangled state (|g, H 854nm + |e, V 854nm )/ where g (e) are orthogonal electronic states of the ion [35]. Next, we apply a modified version of the process matrix describing our converter to the photon part of the entangled state. The process matrix is modified in the sense that it is reconstructed from experimental data after subtracting detector dark counts, leaving imperfection due to photon noise. Photon noise is treated separately from dark counts at this point, since the former attenuate at the same rate as photons from the ion through a subsequent optical fiber, while detector dark counts do not. Finally, we apply a second process to the photon state which accounts for the 30% conversion efficiency, transmission probability through optical fiber of length L, detection efficiency (10%) and dark counts at 1.8 Hz (modelled by a depolarisation channel weighted in proportion to transmitted signal). Entanglement in the final ion-photon state is quantified by the negativity [47], although other measures give equivalent results. The result is that entanglement between ion and photon is present up until 84 km of optical fiber. Beyond this distance, detector dark counts overwhelm the (imperfect) entanglement in the converted ion-photon state. For reference, after 84 km, telecom photons from the conversion process should be detected at a rate of 6.3 Hz, compared with the dark counts of 1.8 Hz, yielding a SNR of (6.3+1.8)/1.8 = 4.5 (photon noise is negligible).
The achievable distance for ion-photon entanglement in our setup could be significantly extended by considering only those detection events that occur when the ion could have generated a photon. A generation rate of 10 kHz for photons from the ion allows for 100 µs per photon, of which less than 20 µs consists of the photon wave packet itself [35]. The remaining 4/5 of the time is allocated for reinitialising the ion after each attempt, during which time counts at the detector can be ignored, allowing the dark counts to be reduced to 1.8/5 = 0.4 Hz. With such a reduced dark count rate, the maximum achievable distance for the observation of ion-photon entanglement in our setup is 122 km.
Note that, under ambient conditions the polarisation in a 25 km telecom fiber spool is known to be passively stable over timescales of several minutes [45]. To correct for polarisation rotations in long fibers between remote locations, one could consider periodic calibrations with classical fields, or even calibrating continuously using classical fields in parallel with the quantum signal: exploiting the narrowband nature of trapped-ion networking photons for filtering.
Transmission of light-matter networking photons through optical fibers many tens of kilometres long poses an interesting problem: the photon travel time eventually becomes longer than the minimum time between photons set by the generation rate. In our example we considered a 10 kHz generation rate, corresponding to a photon every 100 µs, whereas the travel time over 84 km (100 km) is 420 µs (610 µs). With a single ionic-qubit in a trap, one must wait to see if a generated photon is detected after the fiber, before attempting to generate another photon, or entanglement with the first photon will typically be lost. For single ion qubit experiments, the maximum generation rate is therefore limited by the photon travel time. With multiple ionic qubits in a trap, however, one could envisage different ways to overcome the photon travel time limit. For example, after generating a photon with which it is entangled, the state of the ion could be swapped into one of a collection of co-trapped ions, freeing up the original ion to generate a new photon without destroying entanglement with the first photon. First-step experiments in this direction have recently been performed [48]. stages. The first waveguide rests on an X translation stage and the second on XZ stages. Each crystal is independently temperature stabilized for optimal phase matching.

V. CONCLUSION AND DISCUSSION
We have demonstrated a polarisation-preserving photonic interface between the 854 nm transition in trapped Ca + and the 1550 nm telecom C band. A total photon in / fiber-coupled telecom photon out efficiency of 30% was achieved, for a free-running photon noise rate of ∼ 60 Hz. This highly efficient and low noise converter will enable telecom conversion using existing trapped-ion systems with a SNR > 1. In combination with near-future trapped-ion systems, our converter allows for the distribution of ion-photon entanglement over more than 100 km of optical fiber, opening up the possibility of building large-scale light-matter quantum networks.
In principle, our device should function equally as well in reverse: allowing 1550 nm photons to be converted to 854 nm via sum-frequency generation with the pump laser.
For the experiments presented in this work it was not necessary to stabilise the frequency of the pump laser, since the conversion process bandwidth (several tens of GHz) is large compared to the frequency drift rate of the pump laser (few hundred MHz per hour). However, the spectral properties of the pump laser will be transferred onto the converted photon: whether that is important depends on the particular experiment that one wishes to perform with frequency converted photons. For example, for the observation of ion-photon polarisation entanglement, frequency broadening of the converted photon should play no role. For entanglement swapping between remote nodes via one-or two-photon detection [5], frequency distinguishability between photons becomes important. For schemes where remote nodes absorb photons that have been frequency converted (e.g. conversion to telecom then back to 854 nm for absorption by a remote ion), special care may need to be taken to stabilise the pump laser frequency to minimise the spectral footprint on the photons. One could also consider using the temporal and spectral properties of the pump as a way to coherently modify those properties of the converted photons, to overcome bandwidth mismatches between remote quantum matter.
During the preparation of this manuscript we became aware of complimentary work to ours, in which polarisation-preserving conversion from 854 nm to 1310 nm (Telecom O band) was achieved using a single ridge waveguide scheme in a cavity [19].

VI. ACKNOWLEDGEMENTS
We thank Carsten Langrock, Christian Roos and Petar Jurcevic for comments on the manuscript and the staff at IQOQI for technical and administrative support. We thank Rainer Blatt for providing the laboratory space, environment and group support in which to develop our work. This work was supported by the START prize of the Austrian FWF project Y 849-N20, the Army Research Laboratory Center for Distributed Quantum Information via the project SciNet and the Institute for Quantum Optics and Quantum Information.