Dual epitaxial telecom spin-photon interfaces with correlated long-lived coherence

Superconducting resonator fabrication and flip-chip mounting The resonators were fabricated on a Y₂O₃ wafer or a high-resistivity silicon substrate. The substrate was solvent-cleaned and then transferred to the e-beam deposition chamber and baked in a vacuum (<1e-7 mBar) at 300 ^∘C for 1 hour, followed by an overnight vacuum pump. A 75 nm niobium layer was subsequently deposited with e-beam evaporation, followed by optical lithography on a positive photoresist mask (Az 703 MIR) and reactive-ion etching (RIE) with $\mathrm{CF_{4}}$, $\mathrm{CHF_{3}}$ and Ar plasma for 4 min 30 sec for etching a 75 nm niobium layer.

Continuous-wave ESR CW ESR was performed by measuring the resonator transmission spectrum $\mathrm{S_{21}}$ as a function of an in-plane magnetic field with a vector network analyzer (Keysight E5071B). The estimated on-chip power was $\mathrm{\approx}$ -80 dBm, and the magnetic field was ramped in steps of 0.1 mT. The resonator spectrum was fitted to the model for LC superconducting resonators coupled to a transmission line from  [87] Eq. (23) to obtain the center frequency $\mathrm{\omega_{0}}$, quality factors $\mathrm{Q_{i}}$, $\mathrm{Q_{e}}$, and $\mathrm{Q_{\alpha}}$,

where $\mathrm{\omega_{0}}$ is the resonator frequency, $\mathrm{Q_{i}}$ is the intrinsic quality factor, $\mathrm{Q_{e}}$ is the external quality factor, and $\mathrm{Q_{\alpha}}$ is introduced to account for the asymmetry of the transmission spectrum [87]. The fitted intrinsic linewidth $\mathrm{k_{i}}$ and frequency $\mathrm{\omega}$ was plotted against the B field, and a slow-varying background was subtracted to obtain the spectrum in Fig. 2(b) in the main text.

Pulsed ESR The pulsed spin echo measurement setup is shown in Fig. S1. An arbitrary waveform generator (Zurich Instrument HDAWG) was used to generate 200 ns long I and Q pulses at 200 MHz frequency and upconverted to 5.7 GHz using an IQ mixer (MX1: Marki MMIQ 0218LXPC) with the estimated on-chip $\mathrm{\pi}$ pulse power of -65 dBm (300 pW) at the sample after amplification (G1:Mini-Circuits ZX60-83-LNS+) and 45 dB cryogenic attenuation. A pulsed pump generated using IQ upconversion (S2: Signal Core SC5511A and Marki MMIQ 0307LXP) was sent to the TWPA during spin echo acquisition time to turn on the gain of the TWPA. The TWPA pump frequency and power were tuned to obtain a maximum signal-to-noise (SNR) gain of 10 dB while avoiding spurious pump tones interfering with the spin echo frequency. The spin echo signal was amplified by TWPA (+15 dB) at the 8 mK stage, followed by HEMT (Low noise factory $\mathrm{LNF-LNC4\_8F}$ +44 dB) at the 4K stage. Dual Junction cryogenic isolators (I1: Low noise factory $\mathrm{LNF-CICIC4\_8A}$) were placed in between the resonator and TWPA and the TPWA and HEMT to prevent signal reflection and thermal noise leakage to the 8 mK stage. A diode limiter (PIN1: Pasternack PE8023) and an isolator (I2: Fairview Microwave SFC0206S) were added to protect the room temperature readout electronics during pulses. Bandpass filters with 4.9-6 GHz passband (BP1: Mini-Circuits VBFZ-5500-S+) were placed in the pulse generation and readout chain to avoid amplifier saturation. The spin echo was further amplified with room temperature amplifier (G1:ZX60-83-LNS+,+21 dB) followed by demodulation (Marki MMIQ0218LXPC mixer) to obtain I^′, Q^′ signals at 200 MHz. The local oscillator (LO) signal to the demodulation mixer was supplied by the same source used for pulse generation (S1:Keysight N5181A). The demodulated spin echo was further amplified by an amplifier (Mini-Circuits ZX-60-P103LN+, +23 dB) and bandpass filtered (BP2: Mini-Circuits ZX75BP-204-S+) to obtain I^′, Q^′ traces on a fast oscilloscope (OSC1:Tektronix TDS5104). We note that the I^′, Q^′ signals obtained after demodulation have a phase offset with respect to the input I and Q signal used in pulses. Nevertheless, we measure only the amplitude of the spin echo signal, and therefore, we ignore the phase offset.

where $\mathrm{S_{1}}$ is the magnetic noise sensitivity of the transition given as $\mathrm{g_{eff}\mu_{B}}$, $\mathrm{\Gamma_{res}}$ is the residual decoherence due to instantaneous diffusion and $\mathrm{T_{1}}$ limit, which is estimated from the inverse of CPMG coherence time $\mathrm{\Gamma_{res}}=1/\mathrm{T_{2}^{CPMG}}$. $\mathrm{\delta B}$ is the RMS magnetic noise fluctuation.

where $\mathrm{\Gamma_{eff}}$ is the effective linewidth given as,

The term $\mathrm{\Gamma_{SD}}$ represents the spectral diffusion linewidth due to dipolar coupling with environmental spin bath, R represents the effective spin-flip rate, and $\mathrm{\Gamma_{0}}$ represents instantaneous diffusion and homogeneous linewidth. In the limit of $\mathrm{R\tau}<<$ 1, we can simplify $\mathrm{\Gamma_{eff}=\Gamma_{0}+\frac{1}{2}\Gamma_{SD}(1-e^{-RT_{W}})}$ [25] and fit effective linewidth $\mathrm{\Gamma_{Eff}}$ for fixed $\mathrm{T_{W}}$ by sweeping $\mathrm{\tau}$ (inset of Fig. 4(c) in the main text). We obtained $\mathrm{\Gamma_{SD}}$ = 4.4(0.7) kHz and R= 287(140) Hz, which is comparable with slow spectral diffusion previously reported in bulk rare-earth doped crystals [92]. The 287 Hz spin-flip rate is indicative of residual flip-flops due to dipolar interaction in the paramagnetic spin bath as a source of spectral diffusion on a 10 ms timescale. However, this slow spectral diffusion does not significantly affect the linewidth on a short time scale, and the short-time-scale linewidth $\mathrm{\Gamma_{0}}$ of 0.7(0.3) kHz is close to the homogeneous linewidth of 0.8(0.03) kHz from two pulse-echo measurements. The spectral diffusion parameters $\mathrm{\Gamma_{SD}}$ and R predict a spectral diffusion limited coherence time $\mathrm{T_{2,SD}=\sqrt{\pi\Gamma_{SD}R}/2}$ of 1.01 ms. The fitted spectral diffusion parameters allow us to predict the optical linewidth broadening in the C_3i site due to magnetic spectral diffusion in a 10 ms timescale as,

where $\mathrm{A_{max}}$ is a normalization factor we set equal to 1, $\mathrm{T_{Ze}}$ is the spin Zeeman temperature set to 0.2788 K (5.8 GHz spin transition splitting), and $\mathrm{T_{Corr}}$ is the corrected spin temperature obtained by multiplying the measured sample stage temperature by a fitting parameter C, with $\mathrm{T_{Corr}=CT}$. The fit of data in Fig. S6a to Eq. (9) gives us a correction factor C = 1.45(0.06). A good agreement between the data and model validates the fitting and provides a corrected base temperature of 12.5 mK. At this temperature, we expect a unity spin polarization (p) at 5.8 GHz: $\mathrm{p=tanh(hf/2k_{B}T)}$ and 1-p $\approx\mathrm{9\times 10^{-20}}$.

where $\mathrm{\Gamma_{0}}$ is a temperature-independent part of decoherence such as instantaneous diffusion, $\mathrm{\xi}$ is a temperature-independent parameter with a strength proportional to the dipolar coupling to paramagnetic spin bath, and $\mathrm{T_{Ze}}$ is the Zeeman temperature of the spin bath. We fit the model to the data in Fig. S6b to obtain $\mathrm{\xi}$= 96.8(9.7) kHz, $\mathrm{\Gamma_{0}}$= 2.9(0.1) kHz and $\mathrm{T_{Ze}}$ = 0.34 K. The Zeeman temperature $\mathrm{T_{Ze}}$ = 0.34 K corresponds to a spin bath splitting of 6.9 GHz (equivalently g=3.8) and supports $\mathrm{Er^{3+}}$ in $\mathrm{C_{2}}$ sites with g $\approx$ 3.8 as the primary source of decoherence at higher temperatures (T$\geq$ 0.1 K). Indeed, out of the six $\mathrm{C_{2}}$ sub-sites, two orientations have a g= 3.6, and four orientations have g= 8.6 for the in-plane magnetic field direction $\mathrm{\theta}$=40 degrees in this measurement. We also note that the theoretical dipolar coupling of the Er³⁺ spins in $\mathrm{C_{3i}}$ site to the $\mathrm{C_{2}}$ site spin-bath with a total concentration of $\mathrm{n_{C_{2}}}$ of 81.4 ppm and g=3.8 is $\mathrm{\nu_{dd}=(h\mu_{0}\gamma_{C_{3i}}\gamma_{C_{2}}n_{C_{2}})/(4\pi)}$ = 295 kHz, which is close to experimentally estimated parameter $\xi$ = 96.8 kHz within a factor of three [92]. The discrepancy of a factor of 3 can be accounted for given that $\xi$ does not have a closed-form expression and anisotropy of g factor leads to a complex dipolar interaction [95] [98].

Instantaneous diffusion limit under dynamical decoupling To study the instantaneous diffusion limit, we performed CPMG dynamical decoupling measurements as a function of the number of pulses N for samples taken from different radial positions (r) along the wafer (Fig. S8a) at 12.5 mK. While all samples follow the trend of an initial linear increase in $\mathrm{T_{2}^{CPMG}}$ with the number of $\pi$ pulses followed by a saturation after N=200, the samples towards the edge of the wafer (large r) show a greater $\mathrm{T_{2}^{CPMG}}$. We also note a slight decrease in $\mathrm{T_{2}^{CPMG}}$ for N= 400 and 500 pulses, which could be attributed to the accumulation of pulse errors [63]. Increasing the number of pulses can lead to efficient decoupling from the spin bath due to a narrower filter function. However, the saturation of $\mathrm{T_{2}^{CPMG}}$ with N suggests instantaneous diffusion between resonant spins as the limiting source of decoherence [99]. Further, repeating dynamical decoupling measurement with a more robust XY8 sequence (Fig. S8b, red) produces similar coherence times as CPMG sequences (Fig. S8b, green), thus further evidencing instantaneous diffusion as the limiting decoherence mechanism. This allows us to estimate the effective spin concentration $\mathrm{n_{eff}}$ from the ID-limited CPMG coherence time as,

The higher saturation value of $\mathrm{T_{2}^{CPMG}}$ for large r implies a decrease in instantaneous diffusion for the samples closer to the edge. This could be explained by an observed increase in spin inhomogeneous linewidth towards the edge, leading to a decrease in spin spectral density, in close agreement with the increase in T₁ towards the wafer edge.

Inhomogeneous linewidth on wafer-scale Figure S10 compares the spin inhomogeneous linewidth for both low and high field transitions between samples taken from different points of the wafer (r). The low field transition (Fig. S10a) measured using CW ESR shows 3.4 times broader linewidth for the sample farther from the center (r=20 mm, red) than the sample close to the center (r=14 mm, blue). The high field transition measured using spin echo detection (Fig. S10b) also shows a broader inhomogeneous linewidth by a factor of 5 for the sample at r=33 mm than the sample at r=14 mm. Therefore, a significant increase in spin inhomogeneous linewidth with distance (r) for both low and high field transitions explains a reduced spin spectral density and hence reduced instantaneous diffusion, which is in close agreement with measurements in Fig. S8. This increase in inhomogeneous linewidth towards the edge, along with the modeled strain-induced angle dependence of linewidth in Fig. S5, indicates an increase in lattice strain towards the edge.

Theoretical instantaneous diffusion limit To estimate the theoretical ID limit, we first estimate the number of photons corresponding to -65 dBm on-chip power as $\mathrm{\langle n\rangle=(P/hf)\times(ke/k^{2})}$ = 41085 using f = $\mathrm{5.8\times 10^{9}}$ Hz, $\mathrm{k\approx k_{e}=}$ 1.9 MHz for the device used in Fig. 2 in the main text. We then estimate the Rabi frequency as $\mathrm{\Omega=g\sqrt{\langle n\rangle}}$ $\sim 0.78$ MHz using $g\sim 128$ Hz. The ID limit can, therefore, be estimated as,

where $\mathrm{\langle sin^{2}\frac{\theta}{2}\rangle}$ is the average spin-flip probability evaluated by integrating the spin-flip probability over the inhomogeneous linewidth [95]. Using inhomogeneous linewidth of 100 MHz, Rabi frequency $\mathrm{\sim}$ 0.78 MHz, we get $\mathrm{\langle sin^{2}\frac{\theta}{2}\rangle}\sim$ 0.016. Using these values and the estimated $\mathrm{C_{3i}}$ density of $\mathrm{32.4\times 10^{16}spins/cm^{3}}$ we get $\mathrm{T_{2,ID}}\sim$ 0.50 ms. The measured $\mathrm{T_{2}^{CPMG}}$ $\approx$ 1.1 ms is consistent with the theoretical lower bound on ID, and the largest source of uncertainty in this calculation is the estimated single spin coupling g.

Direct-process limited relaxation rate scales as $\mathrm{B^{-5}}$ at low temperatures while cross-relaxation limited spin relaxation scales as the square of density ($\mathrm{n^{2}}$). Therefore, long spin lifetimes can be expected by operating at lower magnetic fields and/or lower $\mathrm{Er^{3+}}$ density[101]. Indeed, $\mathrm{T_{1}}$ of 15 s limited by the direct process was observed for a low density(0.7 ppb) of $\mathrm{Er^{3+}}$ in $\mathrm{CaWO_{4}}$ (10 mK, 500 mT)[23]. Low magnetic field measurements of $\mathrm{Er^{3+}}$ in $\mathrm{Y_{2}SiO_{5}}$ have demonstrated direct process limited $\mathrm{T_{1}}$ of 45 s (0.5 K, 11 mT) [28] and even longer phonon-bottlenecked direct process limited $\mathrm{T_{1}}$ of 10 hours at lower fields and temperature (20 mK, 6.5 mT) [102]. This indicates a possibility of significantly increasing the spin lifetime by operating at lower $\mathrm{Er^{3+}}$ densities and magnetic fields, typically used in experiments on optically addressed single RE qubits [35] [28].

Magnetic TLS Figure S12a shows a broad g $\mathrm{\approx}$ 2.07 background signal observed in the form of an increase in resonator linewidth when performing CW ESR on the device used in Fig. 2 in the main text. The signal had a broad asymmetric lineshape, which could be explained by the presence of a broad background of multiple paramagnetic sub-species with g $\mathrm{\approx}$ 2. Fitting the model $\kappa_{i}=\kappa_{i,0}+\Omega^{2}\gamma_{s}/(\gamma_{s}^{2}+\Delta^{2})$ provides an ensemble coupling $\mathrm{\Omega}$ of 2.68 MHz and a spin half-width of $\mathcal{O}$(800 MHz). This allows us to estimate a total number of 15 $\mathrm{\times 10^{9}}$ paramagnetic spins distributed over the inductor area of $\mathrm{\approx}$ 1256 $\mathrm{\mu m^{2}}$. Therefore, the magnetic TLS has an area density of 12.5 $\times\mathrm{10^{6}}$ spins/$\mathrm{\mu m^{2}}$. Assuming the TLS are localized in the thin film with 1.5 $\mathrm{\mu m}$ thickness, we get a total volume density of magnetic TLS as $\mathrm{8.32\times 10^{18}\,spin/cm^{3}}$. These defects could be paramagnetic impurities or defects like charged oxygen vacancies ($\mathrm{F^{+}}$ center) and interstitial defects (O^2-) which have been previously reported in $\mathrm{Y_{2}O_{3}}$ thin-films as well as ceramics [103] [24]. Another possible contribution to the magnetic TLS signal could be from silicon substrate or niobium surface oxides. Therefore, the estimated density is the upper limit on magnetic defect density in the thin film.

Electrical TLS Power-dependent quality factor measurements of superconducting resonators can be used to study the density and dynamics of electrical TLS. We performed power dependence measurements on the three resonators used in Fig. 3 of the main text. This allows us to obtain the variation of TLS density and interaction strength throughout the thin-film sample. Fig. S12b shows the inverse of intrinsic quality factor ($\mathrm{Q_{i}}$) as a function of the number of photons in the microwave resonator $\mathrm{\langle n\rangle}$. The data were fitted to the model [104],

where $\mathrm{Q_{i0}}$ is the quality factor due to power-independent losses. In the low power limit, $\mathrm{Ftan\delta}$ is proportional to the TLS loss tangent $\mathrm{\delta}$ times the filling factor F in the resonator, $\mathrm{nc}$ is the critical photon number for saturation and $\mathrm{\alpha}$ is the exponent of the power dependence. The fitting parameters for all 3 devices are shown in Table S2, and the data with fit in Fig. S12b. All devices exhibited a similar power-independent internal quality factor ($\mathrm{Q_{i0}}$) between 13,000-14000 as well as similar $\mathrm{Ftan\delta}$, suggesting a similar density of electrical TLS on RE/superconducting surfaces across the wafer scale. A decrease in $\mathrm{\alpha}$ to 0.36 for Dev 1 is indicative of interacting TLS leading to weaker power dependence of the quality factor for that device.

As shown in Fig. S13, the optical power of $\approx$30 mW provided by a continuously tunable diode laser (Toptica CTL1500) was split into three paths, one for laser locking, one for wavelength monitoring, and the remaining (10 % power) was sent to the optical spectroscopy setup.

where $\eta_{cav}=\kappa_{e}/\kappa$ is the collection efficiency from the cavity, which is equal to 4 % for a cavity that is under coupled, $\eta_{fiber}=$ 51 % is the total fiber transmission efficiency including mode-mismatch with the cavity, $\eta_{passive}$ = 72.4 % accounts for the loss from couplers and other components and $\eta_{snspd}$ $\approx$ 70 % is the efficiency of the SNSPD. Thus, the total photon counting efficiency of the setup $\eta_{total}$ is 1.9 %.

The Purcell factor $\mathit{F_{P}}$ obtained here presents a lower limit on the maximum Purcell factor $\mathit{F_{P,Max}}$ due to misalignment between the dipole moment of $\mathrm{Er^{3+}}$ in the $\mathrm{C_{2}}$ sub-site and the cavity mode polarization. Nevertheless, this could be used to estimate the cavity mode volume ($\mathrm{V_{m}}$) following equation 18. Substituting for the cavity quality factor Q, $\mathrm{F_{P}=271.4}$, wavelength $\mathrm{\lambda}$ and the local correction factor to the electric field [45] $\mathrm{\chi_{L}=\frac{3n^{2}}{2n^{2}+1}}$, we get $\mathrm{V_{m}}=8.9\Big{(}\frac{\lambda}{n}\Big{)}^{3}$.

The mode volume can be used to obtain the single ion coupling strength g as [107],

Subsituting $\mathrm{V_{m}}=8.9\Big{(}\frac{\lambda}{n}\Big{)}^{3}$, $\omega_{a}/2\pi=195,100$ GHz, and dipole moment for Er³⁺ Y₁-Z₁ optical transition in C₂ site $\mu=\frac{3\hbar e^{2}nf}{2m_{e}\omega\chi_{L}}=$ 9.7 $\times$ 10^-33 Cm [30], we get a single ion coupling strength g₀=0.95 MHz. This is excellent agreement with the g₀=1.0 MHz obtained from $4g^{2}/\kappa=1/2\pi T_{1,{\rm{cav}}}$, where $T_{1,{\rm{cav}}}$ is the Purcell enhanced lifetime of 0.14 ms.

The Purcell enhanced $\mathrm{T_{1}}$ depends on the detuning between the laser and the center of the cavity, following the expression,

where $\mathrm{T_{1,cav}}$ is the cavity-enhanced emitter decay, $\mathrm{T_{1,0}}$ is the intrinsic lifetime in absence of cavity enhancement, $\mathrm{\gamma}$ is the cavity linewidth, $\mathrm{\zeta}$ = 0.22 is the branching ratio, $\mathrm{F_{P}}$ is the measured Purcell enhancement and $\mathrm{\delta}$ is the laser detuning from the cavity center. Figure S15 shows the measured 1/$\mathrm{T_{1,cav}}$ as a function of the laser detuning from cavity center fitted to the model in equation 20.

Optical homogeneous linewidth in C₂ site versus T₁ Application of magnetic field leads to a reduction of the optical linewidth along with a weak dependence on $\mathrm{T_{1}}$ as shown in Fig. 4(b) of the main text. The linewidth broadening trend can be modeled by a combination of broadening mechanisms,

where $\Gamma_{0}$ is the linewidth in short timescale (t$\mathrm{<150\,\mu s}$) including homogeneous linewidth, fast spectral diffusion, superhyperfine broadening, and drift of laser during the measurement timescale, $\mathrm{\Gamma_{SD}}$ is the linewidth due to dipolar coupling to spin bath inducing spectral diffusion, R is the effective spin-bath flip rate and $\mathrm{\Gamma_{TLS}}$ is the dipolar coupling to electrical TLS. Further, $\mathrm{\Gamma_{SD,eff}}$ has an explicit dependence on the nature and splitting of the spin bath and can be described as,

where $\mathrm{\Gamma_{SD}}$ is the maximum dipolar coupling strength to the spin bath, $\mathrm{g_{sB}}$ is the effective g factor of the spin bath, B is the magnetic field, and T is the temperature. We assumed $\mathrm{g_{env}}$ =1.2, T= 100 mK (this combination is not unique and is chosen to obtain the best fit) to fit equation 21 to obtain the parameters $\mathrm{\Gamma_{SD}}$ = 4.32 (1.1), $\mathrm{\Gamma_{0}}$ = 0.88 (0.11), $\mathrm{\Gamma_{TLS}}$ <0.008 MHz and R = 0.83 (0.32) kHz. The fitted parameter values indicate spectral diffusion induced by paramagnetic spin bath as the primary source of spectral diffusion at B=0, which is frozen by application of B $\approx$ 250 mT. The contribution from TLS is <0.01 MHz, indicating the absence of slow spectral diffusion from TLS. The short timescale linewidth $\mathrm{\Gamma_{0}}$ of 0.95 MHz indicates broadening mechanisms such as fast spectral diffusion by non-magnetic noise. A similar narrowing of homogeneous linewidth with magnetic was also observed for C_3i site, as discussed below.

Optical homogeneous linewidth in C_3i site versus B field Homogeneous linewidth measured at 5 nW power using transient spectral holeburning (Fig. S16) shows a dependence on the magnetic field, which indicates the freezing of the spectral diffusion noise from a paramagnetic spin bath. The zero magnetic field linewidth of 800 kHz is in good agreement with the single ion linewidth measured at zero field and with the same power. The magnetic field dependence is fitted with a spectral diffusion-induced linewidth broadening model,

where $\mathrm{\Gamma_{SD}}$ is the dipolar linewidth due to coupling to a paramagnetic spin bath with an effective g-factor g_env and $\Gamma_{0}$ is the magnetic field independent part of the linewidth, which includes broadening due to laser power, laser linewidth, instantaneous spectral diffusion. We obtain $\Gamma_{0}$= 158.6 (92.9) kHz, $\mathrm{\Gamma_{SD}}$= 635.5 (121.4) kHz, $\mathrm{g_{env}}$= 2.02 and T=0.22 K. The fitted parameters $\mathrm{g_{env}}$ and T had a strong correlation (C=1), and they are not unique to obtain a good fit. In the future, this could be improved by performing separate measurements to estimate the sample temperature T. Nevertheless, the model suggests freezing of paramagnetic impurities as the mechanism of homogeneous linewidth narrowing at higher B field.