High-cooperativity coupling of a rare-earth spin ensemble to a superconducting resonator using yttrium orthosilicate as a substrate

TL;DR

This study demonstrates that yttrium orthosilicate (Y$_2$SiO$_5$) is an effective substrate for superconducting resonators, achieving high cooperativity coupling with rare-earth spins, suitable for quantum memory and microwave-optical conversion applications.

Contribution

We show that Y$_2$SiO$_5$ can host high-Q superconducting resonators with strong coupling to rare-earth spins, offering an alternative to traditional substrates like sapphire.

Findings

Resonator on Y$_2$SiO$_5$ exhibits a quality factor of approximately 400,000.

Coupling strength with Nd$^{145}$ spins reaches high cooperativity ($C=30$).

Dielectric loss tangent comparable to sapphire, indicating low loss.

Abstract

Yttrium orthosilicate (Y$_2$SiO$_5$, or YSO) has proved to be a convenient host for rare-earth ions used in demonstrations of microwave quantum memories and optical memories with microwave interfaces, and shows promise for coherent microwave--optical conversion owing to its favourable optical and spin properties. The strong coupling required by such microwave applications could be achieved using superconducting resonators patterned directly on Y$_2$SiO$_5$, and hence we investigate here the use of Y$_2$SiO$_5$ as an alternative to sapphire or silicon substrates for superconducting hybrid device fabrication. A NbN resonator with frequency 6.008 GHz and low power quality factor $Q \approx 400000$ was fabricated on a Y$_2$SiO$_5$ substrate doped with isotopically enriched Nd$^{145}$. Measurements of dielectric loss yield a loss-tangent $\tan\delta = 4 \times 10^{-6}$, comparable to sapphire. Electron spin resonance (ESR) measurements performed using the resonator show the characteristic angular dependence expected from the anisotropic Nd$^{145}$ spin, and the coupling strength between resonator and electron spins is in the high cooperativity regime ($C = 30$). These results demonstrate Y$_2$SiO$_5$ as an excellent substrate for low-loss, high-Q microwave resonators, especially in applications for coupling to optically-accessible rare earth spins.

Equations

$$\mathbf{g}$$

$$\mathbf{A}$$

$$\omega ={\omega }_{\mathrm{c}}-g_{\mathrm{ens}}^2\Delta /({\Delta }^2+{\gamma }_{\mathrm{s}}^2)$$

$$\kappa ={\kappa }_{\mathrm{c}}-g_{\mathrm{ens}}^2{\gamma }_{\mathrm{s}}/({\Delta }^2+{\gamma }_{\mathrm{s}}^2)$$

Full text

I Introduction

Rare-earth ions (REIs) in crystals are promising candidate systems for quantum information applications as they possess electron and nuclear spins as well as optical transitions at telecom wavelengths. Coherence times in these systems range from milliseconds Wolfowicz et al. (2015) for microwave excitations stored in an electron spin to hours Zhong et al. (2015); Rančić et al. (2017) for optically accessible nuclear excitations. REIs typically possess large g-factors (beneficial for strong coupling), a variety of nuclear spin states, and offer opportunities for optical pumping. Such properties have led to proposals for using rare-earth ions as a multimode microwave quantum memory Probst et al. (2015), a photonic memory for a quantum repeater Sangouard et al. (2011), and as a microwave–optical transducer Williamson et al. (2014); Fernandez-Gonzalvo et al. (2015) for use in quantum networks Kimble (2008).

Yttrium orthosilicate (Y${}_{\text{2}}$SiO${}_{\text{5}}$) is a widely used crystalline host for REIs in such quantum information applications, as its constituent elements provide an environment with a low background of nuclear magnetic moments that would otherwise contribute to spin decoherence and inhomogeneous broadening. The narrow homogeneous linewidths, for example down to $3.5\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ for the $883\text{\,}\mathrm{n}\mathrm{m}$ transition Usmani et al. (2010) in Nd${}^{\text{3+}}$:Y${}_{\text{2}}$SiO${}_{\text{5}}$, have been exploited as an optical quantum memory Hedges et al. (2010); Jobez et al. (2015) enabling storage of entangled states Clausen et al. (2011) and teleportation with $93\text{\,}\mathrm{\char 37\relax}$ fidelity Bussieres et al. (2014).

While optical storage experiments using REIs in Y${}_{\text{2}}$SiO${}_{\text{5}}$ can make use of a suitably optically dense medium, for microwave storage a cavity is employed to achieve a coupling between an ensemble of REIs and the microwave cavity field, so that excitations can be coherently exchanged between the two. This can be achieved with 3D microwave cavities Fernandez-Gonzalvo et al. (2015), which offer homogeneous $B_{1}$ fields and spatial mode-matching between microwave and optical fields, or planar superconducting resonators with small mode volumes and quality factors over $10^{5}$ to yield high spin-number sensitivities Bienfait et al. (2015) and act as an interface between superconducting qubits and spin ensembles Kubo et al. (2011). Dielectric resonators fabricated from rare-earth doped crystals such as YAG (Y3Al5O12) Farr et al. (2015) have used resonant modes with $Q\approx${10}^{4} to drive ESR transitions with cooperativity $C\approx 600$, while planar superconducting resonators have been coupled to rare-earths ($Q\approx${10}^{3}, $C=36$, Probst et al. (2013)), group V donors in silicon Bienfait et al. (2015); Zollitsch et al. (2015), NV centres in diamond Kubo et al. (2011), and ruby (Al${}_{\text{2}}$O${}_{\text{3}}$:Cr${}^{\text{3+}}$) Schuster et al. (2010).

Experiments coupling superconducting resonators to doped crystals have typically used a flip–chip approach with the sample glued or mechanically pressed onto the resonator chip Probst et al. (2013). This has the drawback of complicating the fabrication process, creates an additional interface layer between the resonator and sample increasing dielectric losses from spurious two-level systems (TLS) in the interface Burnett et al. (2014), as well as a variable gap between device and spins which is less controllable than fabrication on the substrate itself.

Common substrates for superconducting resonators include silicon and sapphire (Al${}_{\text{2}}$O${}_{\text{3}}$) Oliver and Welander (2013). Dielectric losses in these are usually quantified by the loss tangent $\tan\delta$, describing the ratio between the imaginary and real components of the complex permittivity of the material. This creates a limit on the quality factor achievable in superconducting resonators $Q=1/\tan\delta$. For high-Q devices, sapphire is often preferred for its low loss-tangent Lindström et al. (2009) $\tan\delta<${10}^{-5}$$, leading to quality factors approaching $10^{6}$.

In this work we investigate the suitability of using Y${}_{\text{2}}$SiO${}_{\text{5}}$ itself as a substrate for fabrication of planar superconducting devices. We fabricate a resonator on the polished surface of a Nd-doped Y${}_{\text{2}}$SiO${}_{\text{5}}$ sample, and find the device has a Q-factor $>10^{5}$ at low powers due to its low loss-tangent $\tan\delta=$4\text{\times}{10}^{-6}$$. We observe an anisotropic ESR spectrum matching that from simulations, and measure the coupling strength between resonator and electron spin to be in the high-cooperativity regime. These results suggest fabrication on doped Y${}_{\text{2}}$SiO${}_{\text{5}}$ is compatible with high-Q devices while simultaneously enabling a coupling to crystals doped with rare-earth ions, and could be a promising route toward scalable hybrid superconductor–spin quantum circuits.

II Device

The device is a lumped-element superconducting resonator, fabricated on a Czochralski-grown single crystal of Y${}_{\text{2}}$SiO${}_{\text{5}}$ Shoudu et al. (1999) doped with $10\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ isotopically purified ${}^{\text{145}}$Nd. The crystal was cut into a 5\text{\,}\mathrm{m}\mathrm{m}$$\times$$5\text{\,}\mathrm{m}\mathrm{m}$$\times$$460\text{\,}\mathrm{\SIUnitSymbolMicro m} chip along the principal dielectric axes (D${}_{\text{1}}$, D${}_{\text{2}}$, b) and a face perpendicular to b was polished for thin-film growth. Y${}_{\text{2}}$SiO${}_{\text{5}}$ has two inequivalent crystal sites where a Y{}^{\text{3+}} ion can be substituted by a rare-earth RE{}^{\text{3+}} ion. The large ionic radius of Nd${}^{\text{3+}}$ results in the larger crystal Site 1 being preferentially populatedWolfowicz et al. (2015), resulting in a stronger signal from Site 1 over Site 2. Due to the crystal’s C${}^{6}_{2\textrm{h}}$ (C2/c) space group, each site has two orientations related by a $\pi$ rotation around the crystal b axis. These two orientations are termed sub-sites, and their ESR properties are degenerate for $B_{0}$ fields applied in the D${}_{\text{1}}$–D${}_{\text{2}}$ plane or parallel to the b axis.

The resonator consists of a $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wire which functions as an inductor due to the constriction increasing the contribution of kinetic inductance, and a pair of $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick capacitive arms separated by a $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ gap. The use of narrow features increases the resilience of the resonator to applied magnetic field, and the large gap between the capacitive arms reduces the peak electric field of the resonator decreasing its susceptibility to dielectric losses from two-level systems. This forms a lumped-element resonator seen in the micrograph in Fig. 1 which generates an oscillating $B_{1}$ magnetic field around the inductor which can drive ESR transitions. Details of this resonator design’s properties in applied magnetic field will be described in a future publication.

Fabrication consisted of $40\text{\,}\mathrm{n}\mathrm{m}$ of sputtered NbN being patterned by photolithography and a SF${}_{\text{6}}$/Ar reactive ion etch process. The patterned chip was enclosed within a 3D copper cavity ($\mathrm{Q}\approx 100$) to suppress spontaneous emission from the resonator to the environment Bienfait et al. (2015). This was installed in the bore of a vector magnet in a dilution refrigerator at $10\text{\,}\mathrm{m}\mathrm{K}$ (measured at the mixing chamber plate) and probed using a vector network analyser (VNA) at zero applied magnetic field.

The lumped-element resonator has a frequency of $6.008\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ and an asymmetric lineshape due to interference with the background transmission of the 3D cavity. Fitting with a Fano resonance Bärnthaler et al. (2010) yields a quality factor $Q\approx$400,000 in the low-power limit in Fig. [1](#S1.F1). By tracking the centre frequency versus temperature Zmuidzinas ([2012](#bib.bib14)) we find the dielectric losses in the device are comparable to resonators fabricated on sapphire Pappas *et al.* ([2011](#bib.bib26)); Lindström *et al.* ([2009](#bib.bib15)) with a loss-tangent $\tan\delta=$4\text{\times}{10}^{-6}, where the filling factor $F\approx\tfrac{1}{2}$ representing the fraction of magnetic field penetrating the substrate has been factored out.

This demonstrates that Y${}_{\text{2}}$SiO${}_{\text{5}}$ is well-suited for devices incorporating resonators and superconducting qubits which are typically susceptible to dielectric losses from TLSs Faoro and Ioffe (2006), while also incorporating doped spins for cavity QED.

III Electron spin resonance

III.1 ESR spectrum

Having studied the resonator, we now use it to perform ESR to characterise the Nd spins in the Y${}_{\text{2}}$SiO${}_{\text{5}}$ that are coupled to the resonator. A magnetic field $B_{0}$ was applied in the plane of the superconducting thin film, roughly perpendicular to the Y${}_{\text{2}}$SiO${}_{\text{5}}$ b crystal axis, to minimise the magnetic flux threaded through the superconductor. The field was swept up to 360 mT and the resonator was used to observe ESR transitions, monitoring changes to its centre frequency tracked with a VNA. By this method a series of spectra were taken as a function of $B_{0}$ orientation in the D${}_{\text{1}}$–D${}_{\text{2}}$ plane to extract the angular dependence, or “roadmap”, of ESR line positions, in order to confirm the presence of Nd spins in the substrate and measure their coupling to the resonator.

For an electron spin $\bm{S}$ coupled to a nuclear spin $\bm{I}$ the Hamiltonian accounting for the electron Zeeman and hyperfine terms is $H=\mu_{\mathrm{B}}\bm{B}^{\mathrm{T}}\bm{g}\hat{\bm{S}}+\hat{\bm{S}}^{\mathrm{T}}\bm{A}\hat{\bm{I}}$. The anisotropy of Y${}_{\text{2}}$SiO${}_{\text{5}}$ results in the parameters $\bm{g}$ and $\bm{A}$ being tensors, previously calculated from spectroscopic studies of site 1 of 145${}^{\text{\!{}}}$Nd:Y${}_{\text{2}}$SiO${}_{\text{5}}$ at $9.4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ Maier-Flaig (2013) as

[TABLE]

A characteristic ESR spectrum ($B_{0}$ along the D${}_{\text{1}}$ axis) is plotted in Fig. 2(b). We observe a series of eight ESR transitions indicated in Fig. 2(a), corresponding to the ${}^{\text{145}}$Nd nuclear spin ($I=\tfrac{7}{2}$) in Site 1. A small misalignment between the D${}_{\text{1}}$–D${}_{\text{2}}$ and superconductor planes breaks the degeneracy between the crystal sub-sites and causes a further splitting of each line into two.

The roadmap of ESR spectrum with respect to $B_{0}$-field angle is plotted in Fig. 2(c), with transitions marked by circles. The angular dependence seen in these spectra matches well simulations for Nd spins in Site 1 (blue) while we also observe a signal at lower magnetic fields (higher g-factor) which we attribute to Site 2 (red). The full spin Hamiltonian of ${}^{\text{145}}$Nd in Site 2 including hyperfine terms has not been reported, therefore our simulations only account for Zeeman term (i.e. g tensor). Additional unidentified impurities are evident from the different angular dependences of 2–3 transitions marked in green and black.

These measurements enable ESR transitions to be positively identified by comparing their angular dependence to simulations Wisby et al. (2016) and bulk ESR data, ensuring further analysis of spin–resonator coupling properties can be clearly linked to a particular spin species.

III.2 High-cooperativity coupling

The highest intensity ESR transition for ${}^{\text{145}}$Nd in Site 1 with $m_{\mathrm{I}}=\tfrac{7}{2}$ at $B_{0}=$216\text{,}\mathrm{m}\mathrm{T}$$ along D${}_{\text{1}}$ was selected for an evaluation of the spin–resonator coupling strength. Measuring the resonator S${}_{\text{21}}$ with a VNA while sweeping magnetic field strength $B$, we observe the onset of an avoided crossing shown in Fig. 3. Fitting the resonator frequency $\omega$ and linewidth $\kappa$ at each field pointAbe et al. (2011) allows us to extract the coupling strength $g_{\textrm{ens}}$, inhomogeneous spin ensemble half-width $\gamma_{\mathrm{s}}$, and resonator half-width $\kappa_{\mathrm{c}}$ with

[TABLE]

where $\Delta=m_{0}(B-B_{\mathrm{0}})/\hbar$ is the field detuning calculated from the spin magnetic moment $m_{0}=\hbar\frac{\operatorname{d}\!{\omega}}{\operatorname{d}\!{B_{0}}}$. This accounts for cases where $\frac{\operatorname{d}\!{\omega}}{\operatorname{d}\!{B}}$ varies with $B$, as is the case for mixed spin systems in the low-field limit or near zero first-order Zeeman (ZEFOZ) points.

From this we extract $g_{\mathrm{ens}}=$1.5\text{,}\mathrm{M}\mathrm{H}\mathrm{z}, $\gamma_{\mathrm{s}}=$5.7\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}, and $\kappa=$13.2\text{,}\mathrm{k}\mathrm{H}\mathrm{z}$$. This corresponds to a high cooperativity $C=\tfrac{g_{\mathrm{ens}}^{2}}{\kappa\gamma}=30$.

For a bulk doped sample there is no clear single-spin coupling rate as the $\tfrac{1}{r}$ decay of the field strength from the wire results in a $B_{1}$ inhomogeneity spanning orders of magnitude while all spins in the sample contribute to the signal. However, a characteristic rate representing the average coupling strength over the spins contributing $50\text{\,}\mathrm{\char 37\relax}$ of the measured signal can be calculated. This corresponds to a region within $4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of the wire with $N\approx$6\text{\times}{10}^{8} resonant ${}^{\text{145}}$Nd spins in the same sub-site, accounting for the thermal population in the $m_{\mathrm{I}}=\tfrac{7}{2}$ ESR transition at $10\text{\,}\mathrm{m}\mathrm{K}$. From this we derive a characteristic single-spin coupling rate $g_{0}=\tfrac{g_{\textrm{ens}}}{\sqrt{N}}\approx$50\text{\,}\mathrm{H}\mathrm{z} leading to an expected Purcell enhanced emission rateBienfait et al. (2016) of $\Gamma_{\textrm{P}}=\tfrac{4g_{0}^{2}}{\kappa}\approx$1\text{,}\mathrm{H}\mathrm{z}$$.

These results indicate the coupling regime is limited by the spin linewidth $\gamma_{\mathrm{s}}>g_{\textrm{ens}}$, or equivalently that the Nd spin ensemble is decohering at a faster rate than excitations are being exchanged between it and the resonator. Steps toward improving this coupling strength could include using a more strongly doped rare-earth sample to increase $g_{\textrm{ens}}$ though this could also increase $\gamma_{\mathrm{s}}$, or exploiting the coherence-enhancing ZEFOZ transitions in REIs Ortu et al. (2018) to decrease $\gamma_{\mathrm{s}}$. The significant $B_{1}$ inhomogeneity in this resonator design also poses challenges for performing coherent operations on the entire spin ensemble with pulsed ESR. This could be mitigated by coupling to implanted layers of REIs Kukharchyk et al. (2014); Wisby et al. (2016), or by using resonators designed to generate a more homogeneous $B_{1}$ field.

IV Conclusion

We investigated the use of Y${}_{\text{2}}$SiO${}_{\text{5}}$ as an alternative to sapphire or silicon substrates for the fabrication of superconducting devices. The fabricated NbN lumped-element resonator had a quality factor $Q\approx$400,000 and a loss-tangent $\tan\delta=$4\text{\times}{10}^{-6} comparable to sapphire Lindström et al. (2009), and yielded a high-cooperativity coupling $C=30$ between the resonator and a 145${}^{\text{\!{}}}$Nd spin ensemble. These results demonstrate Y${}_{\text{2}}$SiO${}_{\text{5}}$ is well-suited for superconducting devices, while also enabling an interaction with optically-accessible rare earth spins

Further studies of this device include performing pulsed ESR measurements in the high-cooperativity regime and measuring coherence properties of the spin ensemble. The methods shown here are also applicable to other REIs in Y${}_{\text{2}}$SiO${}_{\text{5}}$, for example Er with its $1540\text{\,}\mathrm{n}\mathrm{m}$ telecom band optical transition Fernandez-Gonzalvo et al. (2015) and Yb which exhibits a large oscillator strength Welinski et al. (2016) and coherence-enhancing ZEFOZ and near-ZEFOZ transitions Ortu et al. (2018). It may also prove interesting to study the suitability as a substrate of other crystalline hosts for REIs, such as yttrium aluminium garnet (YAG), yttrium lithium fluoride (YLF), yttrium orthovanadate (YVO4), and calcium tungstate (CaWO4).

Fabricating superconducting devices on crystals doped with REIs as demonstrated here shows promise for integrating these optical elements alongside the fast information processing available from superconducting qubits. This is important step toward making a microwave–optical transducer capable of connecting such quantum processors within a quantum network. By exploiting long coherence times available from REIs this is also a route to integrating fast quantum electronics with a spin ensemble acting as a quantum memory.

Acknowledgements.

This work has received funding from the Engineering and Physical Sciences Research Council (EPSRC) through the Centre for Doctoral Training in Delivering Quantum Technologies (Grant No. EP/L015242/1), the European Community’s Seventh Framework Programme Grant No. 279781 (ASCENT), the IMTO Cancer AVIESAN (Cancer Plan, C16027HS, MALT), and the UK government’s Department for Business, Energy and Industrial Strategy.