Abstract
Hybrid quantum systems based on spin ensembles coupled to superconducting microwave cavities are promising candidates for robust experiments in cavity quantum electrodynamics (QED) and for future technologies employing quantum mechanical effects^{1,2,3,4}. At present, the main source of decoherence in these systems is inhomogeneous spin broadening, which limits their performance for the coherent transfer and storage of quantum information^{5,6,7}. Here we study the dynamics of a superconducting cavity strongly coupled to an ensemble of nitrogen–vacancy centres in diamond. We experimentally observe how decoherence induced by inhomogeneous broadening can be suppressed in the strongcoupling regime—a phenomenon known as ‘cavity protection’^{5,7}. To demonstrate the potential of this effect for coherentcontrol schemes, we show how appropriately chosen microwave pulses can increase the amplitude of coherent oscillations between the cavity and spin ensemble by two orders of magnitude.
Main
The processing of quantum information requires special devices that can store and manipulate quantum bits. Hybrid quantum systems^{2} combine the advantages of different systems to overcome their individual physical limitations. In this context superconducting microwave cavities have emerged as ideal tools for realizing strong coupling to qubits ^{3,4,8,9,10,11,12} for the transfer of excitations on the singlephoton level^{13,14}. For the storage of quantum information the negatively charged nitrogen–vacancy (NV) centres in diamond show great potential, especially owing to their long coherence times (up to one second ^{15}) and to the combination of microwave and optical transitions which makes them an easily accessible and controllable qubit^{16}. Coherently passing quantum information between such a spin and a cavity requires that they are strongly coupled to each other. As has recently been shown^{4,10,11,12}, this limit can be reached by collective coupling to a large spin ensemble, in which case the coupling strength is increased by the square root of the ensemble size. However, this collective coupling comes with a considerable downside: in a solidstate environment a spin is always prone to inhomogeneous broadening. In particular, for an ensemble of NV centres, magnetic dipolar interaction with excess nuclear and electron spins in the diamond crystal leads to an inhomogeneous broadening of the spin transition^{17}, which acts as the dominant source of decoherence. Several approaches, including echotype refocusing techniques^{18,19}, have been suggested to overcome this limitation. Here we will concentrate on recent theoretical proposals which rely on the specific shape of the inhomogeneous spectral spin distribution ρ(ω) of the NV centre ensemble. In our explicitly timedependent study we will demonstrate the socalled ‘cavityprotection effect’^{5,7}, which was predicted in these proposals but has remained so far unobserved.
Our experiment is performed in a standard dilution refrigerator with the corresponding setup being sketched in Fig. 1a and a photograph of the resonator with a synthetic diamond on top shown in Fig. 1b. To avoid thermal excitations we cool the entire setup to a temperature of 25 mK, where the estimated thermal spin polarization is 99%. By applying an external magnetic field B = 9.4 mT through a set of two superconducting Helmholtz coils we Zeeman tune the NV spin ensemble into resonance with the cavity. Our resonator has a fundamental resonance at ω_{c}/2π = 2.6899 GHz with a quality factor of Q = 3,060. To excite and probe the coupled system we inject microwave pulses into the cavity and perform timeresolved transmission spectroscopy by a fast homodyne detection setup with subnanosecond time resolution. The number of microwave photons in the cavity remains at or below 10^{6}, which is very low compared with the number of 10^{12} NV spins involved in the coupling, ensuring that the Holstein–Primakoff^{20} approximation is valid for describing our experiments. As a result of this approximation, the excitations in the system can be treated as noninteracting quasiparticles^{5} and the dynamics for the 10^{6} photons in our system is equivalent to that of a corresponding singlephoton experiment as relevant for quantum information processing (see Supplementary Methods Section II).
Our starting point to account for the dynamics of a singlemode cavity coupled to a spin ensemble is the Tavis–Cummings Hamiltonian^{21}, which reads in the rotating wave approximation where a^{†} and a are standard creation and annihilation operators of the cavity mode and σ_{j}^{±}, σ_{j}^{z} are the Pauli operators associated with each individual spin. The first and second terms stand for the uncoupled resonator with frequency ω_{c} and for the spin ensemble with frequencies ω_{j}, centred around ω_{s}, respectively. The third and the last terms describe the cavity–spin interaction with coupling strength g_{j} as well as the driving electromagnetic field injected into the cavity with amplitude η(t) and frequency ω_{p}. The collective coupling to a large number of spins allows us to enter the strongcoupling regime of QED, for which the interaction term is commonly reduced to a collective term^{22} Ω(S^{−}a^{†} − S^{+}a), where the collective spin operators read . The prefactor stands for an effective coupling strength, which scales up a single cavity–spin interaction, typically on the order of g_{j}/2π 12 Hz, by a factor (refs 3, 4, 23). In this formulation the effective spin waves that are excited by the cavity mode can be identified as superradiant collective Dicke states which are effectively damped by the coupling to subradiant states in the ensemble^{13,24}. To accurately describe the corresponding dynamics we also need to take into account the specific profile of the spectral spin distribution^{7} . We achieve this by setting up a Volterra integral equation (see Supplementary Methods), , for the cavity amplitude A(t) = 〈a(t)〉. This includes a memorykernel , responsible for the nonMarkovian feedback of the NV ensemble on the cavity, and the function , which describes the contribution from an external drive and from the initial spin excitation. In the following, the cavity amplitude A(t)^{2}, calculated with this approach for stationary and pulsed driving fields, will be compared to its experimental counterpart—that is, the timeresolved microwave intensity measured in transmission through the cavity.
First, to demonstrate that our experiment is in the strongcoupling regime (having ω_{s} = ω_{c}) we apply a rectangular microwave pulse which is sufficiently long compared with the cavity decay rate κ, total decoherence rate Γ and coupling strength Ω (800 ns ≫ 1/2κ = 199 ns, 1/Γ = 53 ns, π/Ω = 58 ns) to drive the system into a steady state with varying probe frequency ω_{p}. Figure 2a shows that two effective eigenstates (polaritons) of the coupled system emerge in the transmission, , corresponding to the symmetric and antisymmetric superposition of the cavity and spin eigenstates, respectively. Strong coupling is secured because the Rabi splitting between these states Ω_{R}/2π = 19.2 MHz is substantially larger than the total decay rate of the system Γ/2π = 3.0 MHz (fullwidth at halfmaximum). The latter consists of a cavity decay rate, κ/2π = 0.4 MHz (halfwidth at halfmaximum), as well as of a spin decay rate which contains a negligibly small spin dissipation γ → 0 and a dominant contribution from the inhomogeneous broadening of the spin ensemble. Detailed spectroscopic measurements of the stationary transmission ^{6} reveal that the spectral function ρ(ω) which accurately captures the broadening is neither Lorentzian nor Gaussian, but has the intermediate form of a qGaussian^{6} (see Supplementary Methods Section I). As shown in Fig. 2b, our explicitly timedependent theoretical description yields excellent quantitative agreement with the experimental data, using such a qGaussian distribution function with a linewidth of γ_{q}/2π = 9.4 MHz (fullwidth at halfmaximum), a shape parameter q = 1.39 and an effective coupling strength 2 ⋅ Ω/2π = 17.2 MHz. After turning on and switching off the microwave pulse, coherent Rabi oscillations occur between the cavity and the spin ensemble, which we reproduce accurately, including their damping. Interestingly, the first Rabi peak shows a pronounced overshoot after switching off the microwave drive, at which the energy stored in the spin ensemble is coherently released back into the cavity. These oscillations are a hallmark of the nonMarkovian character of the system dynamics in the strongcoupling regime, for which an accurate knowledge of the memorykernel in our Volterra equation is essential.
Our hybrid cavity–spin system can not be modelled by two coupled damped harmonic oscillators as in the case of a purely Lorentzian spin distribution. The spectral profile of a qGaussian has marked consequences on the dynamics in the strongcoupling regime: in particular, for spectral distributions ρ(ω) with tails that fall off faster than a Lorentzian lineshape an increasing coupling strength was predicted to lead to a reduction of the decay rate Γ and to protect the system against decoherence—hence the name ‘cavityprotection effect’^{5,7}. In a nutshell, ‘cavity protection’ can be understood as follows: in the strongcoupling regime, the cavity couples to a set of superradiant states forming hybridized energy levels split by ℏΩ_{R}. The resulting polariton states are coupled to subradiant spinwave modes acting as the main source of decoherence, and the total decay rate depends on the energetic gap between the polaritons and the subradiant states. If the spectral profile of the inhomogeneous spin distribution decays sufficiently fast for increasing gap size, an energetic decoupling of the superradiant polaritons from the subradiant spinwave modes occurs, leading to a suppressed damping of the polaritons. Although the origin of this intriguing effect can also be understood on the basis of an ensemble of coupled classical oscillators^{5,7}, the important point to note is that it also survives on the level of singlephoton excitations (see Supplementary Methods Section II).
As the tails of our spin distribution satisfy the required fast decay, we now have the possibility to probe this exceptional behaviour in the experiment. We measure the decay rate Γ(Ω) of the cavity amplitude after driving the system for different coupling strengths Ω into steady state using a sufficiently long rectangular pulse (as in Fig. 2b, see Methods). The values of Γ(Ω), as determined by the slope of the temporal decay after switch off, are shown in Fig. 3. We find the decay rate to vary over almost one order of magnitude in a strongly nonmonotonic fashion: in the weakcoupling regime the decay rate Γ increases with growing coupling strength Ω as a result of the Purcell effect^{25} as the cavity mode increasingly couples to the spin ensemble. Entering the strongcoupling regime, this trend reverses and Γ decreases with growing Ω. To highlight this remarkable phenomenon, we also plot in Fig. 3 the behaviour for a Lorentzian spin distribution, for which Γ(Ω) is constant in the strongcoupling limit. Performing a Laplace transform of our Volterra equation we find that in the limit of very strong coupling (Ω → ∞) the decay rate takes the following closed analytical form Γ = κ + πΩ^{2}ρ(ω_{s} ± Ω) (in agreement with a stationary analysis^{7}). Whereas the maximally reachable value of Ω/2π = 8.6 MHz in our device already leads to a considerable reduction of Γ by 50% below its maximum, our numerical results (Fig. 3) suggest a further reduction of the decay rate with increasing coupling strength by an order of magnitude. We predict that a three times higher Ω could be achieved by filling up the cavity with diamond and by further increasing the NV density. Having the mode volume entirely occupied with NV centres, a factor of two enhanced NV concentration would already bring us close to exploiting the full potential of the cavityprotection effect. We estimate that such an increased NV concentration would cause a mean NV–NV interaction strength of 120 kHz, which is still small enough to have only a negligible influence on the dynamics of our system.
In a next step, we demonstrate that the ‘cavityprotection effect’ can also be employed for the realization of coherentcontrol schemes. In particular, we address a central question when dealing with coherently driven spin ensembles—namely, how to achieve high excitation levels in the spin ensemble with limited driving powers^{18,26}. In a simplified picture of two coupled harmonic oscillators this can be achieved by a drive modulated with the inverse of the effective coupling strength. To realize this for the nonLorentzian spectral spin distribution of our ensemble a pulsed driving is required to match the Rabi frequency Ω_{R} rather than the effective coupling strength 2 Ω, which quantities are here quite different from each other. We thus probe our setup by a driving field with a carrier frequency ω_{p} = ω_{c} = ω_{s} and a periodic modulation with tunable period τ. Realizing the latter with a simple periodic signchange of the carrier signal, we find that this driving scheme produces giant oscillations in the transmission (Fig. 4a) corresponding to a coherent exchange of energy between the cavity and the spin ensemble. A maximum oscillation amplitude occurs exactly at the point where the modulation period τ coincides with the inverse of the Rabi splitting 2π/Ω_{R}. Note that at this resonant driving the steadystate oscillation amplitude in the transmission signal (Fig. 4b) exceeds the stationary amplitude (Fig. 2b) by two orders of magnitude, although the net power applied to the cavity is exactly the same in both cases. Our approach demonstrates how to sustain coherent oscillations and how to reach considerably high excitation amplitudes of the spin ensemble without using strong driving powers. For comparison, we also plot in Fig. 4b the results both for a qGaussian as well as for a Lorentzian spin density, which clearly shows the substantially lower excitation amplitudes for the Lorentzian case. This clear signature of the ‘cavityprotection effect’ paves the way for the realization of sophisticated coherentcontrol schemes in the strongcoupling regime of QED.
Methods
The cavity is an overcoupled λ/2 resonator, made out of 200 nm thick sputtered niobium on a 330 μm thick sapphire substrate, structured by optical lithography and reactive ion etching. The diamond sample is a commercially available type Ib highpressure hightemperature diamond (HPHT) with a size of 4.5 × 2.25 × 0.5 mm^{3} and two polished (100) surfaces. The crystal contains an initial concentration of 200 ppm nitrogen and has a natural abundance of ^{13}C nuclear isotopes. We achieve a total density of 6 ppm NV centres by 50 h of neutron irradiation with a fluence of 5 × 10^{17} cm^{−2} and by annealing the crystal for 3 h at 900 °C, resulting in a conversion efficiency from initial nitrogen to NV centres of 3%. The diamond sample has been characterized by means of a roomtemperature confocal laser scanning microscope, allowing us to measure the NV density and the zerofield splitting parameters D and E by optically detected magnetic resonance^{27}. Residual nitrogen is incorporated in the diamond mainly as substitutional single defect centres, which also act as paramagnetic impurities with S = 1/2 and thus form the main source of inhomogeneous broadening.
The NV centre is a paramagnetic impurity which consists of a substitutional nitrogen atom and an adjacent vacancy in the diamond lattice. The energy level structure features an electron spin triplet (S = 1) as its ground state ^{28}. Owing to negligible direct NV–NV interactions, it can be described, to lowest order, by the Hamiltonian^{29}, H/h = DS_{z}^{2} + E(S_{x}^{2} − S_{y}^{2}) + μBS with S = (S_{x}, S_{y}, S_{z}), D = 2.877 GHz, E = 7.7 MHz and μ = 28 MHz mT^{−1}, providing us with an estimate for the mean spin transition frequency ω_{s}. The zerofield splitting term D = 2.877 GHz corresponds to 138 mK, which is high compared with the temperature of 25 mK at which the experiments are performed and allows us to thermally polarize the NV spins up to 99%. In the diamond lattice four crystallographic orientations of the NV defect are possible. Our diamond sample has a (100) surface orientation. We apply a d.c. magnetic field of 9.4 mT to Zeeman tune the spins into resonance with the cavity, which is applied in the plane of the resonator and therefore in the (100) plane of the crystal. We rotate the magnetic field direction by 22.5° in the plane, at which only two subensembles are degenerate. The external magnetic field leads, on the one hand, to a Zeeman splitting of subensembles such that the maximal possible ensemble–cavity coupling strength is reduced by a factor of . On the other hand, it narrows the width of the spectral spin distribution induced by a misalignment of the applied external magnetic field with respect to the crystallographic frame.
For the application of short microwave pulses with adjustable phase and power we use a frequency mixer controlled by an arbitrary waveform generator (AWG). The microwave signal passes the cryostat and is attenuated by −60 dB when reaching the resonator. The transmitted signal is fed into a circulator and amplified on the 4 K stage. Using a homodyne detection scheme, the transmitted and amplified microwave signal is mixed with the reference signal and both quadrature signals I and Q are recorded by a fast analogtodigital converter with 2 GS s^{−1} sampling frequency. The AWG repeats the pulse sequence and 10^{6} single traces are averaged. From the two quadratures I and Q the transmitted microwave intensity A(t)^{2} is calculated and plotted in Figs 2 and 4. Voltage fluctuations give a standard deviation of ±1.02 × 10^{−5} (V) on the quadratures I and Q. The squared steadystate amplitude of the cavity transmission (Fig. 2b) gives a mean signal of A^{2} = 1.44 × 10^{−7} ± 6.85 × 10^{−9} (V^{2}). To reduce the coupling strength (Fig. 3) we repeatedly pump the resonator with high power and a long microwave pulse, corresponding to 2.5 × 10^{7} microwave photons in the cavity. For this measurement power, a nonnegligible number of NV spins gets excited, leading to a reduced number of groundstate spins coupled to the cavity.
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Acknowledgements
We would like to thank C. Koller, F. Mintert, P. Rabl, H. Ritsch, K. Sandner and M. Trupke for helpful discussions, and D. Brasch/Terra Mater Magazin for the image shown in Fig. 1b. The experimental effort has been supported by the TOP grant of TU Vienna. S.P. acknowledges support by the Austrian Science Fund (FWF) in the framework of the Doctoral School ‘Building Solids for Function’ (Project W1243). D.O.K. and S.R. acknowledge funding by the FWF through Projects No. F25P14 (SFB IRON) and No. F49P10 (SFB NextLite).
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Author notes
 S. Putz
 & D. O. Krimer
These authors contributed equally to this work.
Affiliations
Vienna Center for Quantum Science and Technology, Atominstitut, Vienna University of Technology, Stadionallee 2, 1020 Vienna, Austria
 S. Putz
 , R. Amsüss
 , A. Valookaran
 , T. Nöbauer
 , J. Schmiedmayer
 & J. Majer
Zentrum für Mikro und Nanostrukturen, Vienna University of Technology, Floragasse 7, 1040 Vienna, Austria
 S. Putz
 & J. Majer
Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 810/136, 1040 Vienna, Austria
 D. O. Krimer
 & S. Rotter
Research Institute of Molecular Pathology and Max F. Perutz Laboratories, University of Vienna, Dr.BohrGasse 7, 1030 Vienna, Austria
 T. Nöbauer
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Contributions
S.P., R.A., A.V., T.N., J.S. and J.M. designed and set up the experiment. S.P., R.A. and A.V. carried out the measurements with supervision by J.M. D.O.K. and S.R. devised the theoretical framework and performed the calculations. S.P., D.O.K., S.R. and J.M. wrote the first manuscript draft, to which all authors suggested improvements.
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