Coherent Rabi Dynamics of a Superradiant Spin Ensemble in a Microwave Cavity

Coherent Rabi Dynamics of a Superradiant Spin Ensemble in a Microwave Cavity PHYSICAL REVIEW X 7, 031002 (2017) 1 1 2 2 3 4 B. C. Rose, A. M. Tyryshkin, H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, 5 6 1 M. L. W. Thewalt, K. M. Itoh, and S. A. Lyon Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA Leibniz-Institut für Kristallzüchtung, 12489 Berlin, Germany PTB Braunschweig, 38116 Braunschweig, Germany VITCON Projectconsult GmbH, 07745 Jena, Germany Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 School of Fundamental Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohuku-ku, Yokohama 223-8522, Japan (Received 13 February 2017; revised manuscript received 18 May 2017; published 10 July 2017) We achieve the strong-coupling regime between an ensemble of phosphorus donor spins in a highly enriched Si crystal and a 3D dielectric resonator. Spins are polarized beyond Boltzmann equilibrium using spin-selective optical excitation of the no-phonon bound exciton transition resulting in N ¼ 3.6 × 10 unpaired spins in the ensemble. We observe a normal mode splitting of the spin-ensemble–cavity polariton pffiffiffiffi resonances of 2g N ¼ 580 kHz (where each spin is coupled with strength g) in a cavity with a quality factor of 75 000 (γ ≪ κ ≈ 60 kHz, where γ and κ are the spin dephasing and cavity loss rates, respectively). The spin ensemble has a long dephasing time (T ¼ 9 μs) providing a wide window for viewing the dynamics of the coupled spin-ensemble–cavity system. The free-induction decay shows up to a dozen collapses and revivals revealing a coherent exchange of excitations between the superradiant state of the spin ensemble and the pffiffiffiffi cavity at the rate g N. The ensemble is found to evolve as a single large pseudospin according to the Tavis-Cummings model due to minimal inhomogeneous broadening and uniform spin-cavity coupling. We demonstrate independent control of the total spin and the initial Z projection of the psuedospin using optical excitation and microwave manipulation, respectively. We vary the microwave excitation power to rotate the pseudospin on the Bloch sphere and observe a long delay in the onset of the superradiant emission as the pseudospin approaches full inversion. This delay is accompanied by an abrupt π-phase shift in the peusdospin microwave emission. The scaling of this delay with the initial angle and the sudden phase shift are explained by the Tavis-Cummings model. DOI: 10.1103/PhysRevX.7.031002 Subject Areas: Mesoscopics, Quantum Physics, Strongly Correlated Materials The enhanced collective emission from an ensemble of these collective effects are generally lacking in large ensem- atomlike systems due to coherent self-stimulated emission bles due to strong dephasing (short T ¼ 1=γ) [2,15,16]. was originally described by Dicke [1]. For this phenomenon Here, we demonstrate the dynamics of a strongly coupled he coined the term superradiance and showed that the atom ensemble of phosphorus donor spins in highly isotopically ensemble can behave as a large collective pseudospin. enriched Si with both a long dephasing time [17,18] and Superradiant emission has been observed in numerous uniform coupling to the radiation field (due to the use of a 3D physical systems [2–10]. These collective effects are par- microwave cavity), as shown schematically in Fig. 1(a).For ticularly prominent when the coupling between the spin the first time outside of ensembles of Rydberg atoms [15,16], pffiffiffiffi we study a spin-ensemble–cavity system with both of these ensemble and the radiation field (g N for N spins indi- essential properties allowing for it to be modeled accurately vidually coupled with strength g) is larger than any of the as a single large pseudospin [Fig. 1(b)] offering a simple losses in the system (κ þ γ, where κ is the radiative loss rate interpretation of the spin-ensemble–cavity evolution [1,19]. and γ is the spin dephasing rate). This strong-coupling regime pffiffiffiffi In particular, we are able to directly observe the dynamics (g N ≫ κ þ γ) has been extensively studied in both theory of superradiant emission under strong excitation, resolving and experiment [11–14], but clearly resolved dynamics of several cycles of coherent excitation transfer between the spin ensemble and the cavity [20]. In contrast to the more extensively studied low excitation limit [21,22], our experi- Published by the American Physical Society under the terms of ments performed under strong excitation cannot be modeled the Creative Commons Attribution 4.0 International license. as a linear system of two coupled harmonic oscillators. Further distribution of this work must maintain attribution to Instead, it must be treated with the Tavis-Cummings the author(s) and the published article’s title, journal citation, and DOI. model [19]. 2160-3308=17=7(3)=031002(10) 031002-1 Published by the American Physical Society B. C. ROSE et al. PHYS. REV. X 7, 031002 (2017) implementing a long-lived quantum memory. Both rates are 100–1000 times improvement compared to other candidate spin memories [23]. The key figure of merit for implementing a spin-ensemble quantum memory is pffiffiffiffi g NT ≫ 1, and we demonstrate that we are well beyond satisfying this criterion. Additionally, we achieve rela- tively uniform spin-cavity couplingwithlessthan5% variation across the spin ensemble. Both factors (long T and uniform spin-ensemble–cavity coupling) are vital for performing high-fidelity spin-ensemble manipulation while in the strong-coupling regime, which has been an outstanding problem in implementing a spin-ensemble quantum memory [24]. Initialization of the pseudospin is accomplished with a combination of optical excitation, which sets the size of the pseudospin, S ¼ N=2, and microwave manipulation, which sets the initial Z projection of the pseudospin M. Previous reports of large superradiant spin ensembles prepared only an initially inverted state jS; Si [15,16],while with our system we are able to prepare any initial pseudospin state jS; Mi in the superradiant subspace [Fig. 1(b)] with independent control of S and M. By controlling the size of pffiffiffiffi the pseudospin we demonstrate a g N dependence of the energy exchange rate between the pseudospin and cavity. By varying M, we are able to control a delay in the onset of the superradiant emission [25] and we report the first experimental observation of a log dependence of this delay when the pseudospin is near quasiequilibrium at M ¼ S. FIG. 1. (a) Experimental scheme for delivering microwave This log dependence is consistent with predictions from and optical excitation to the phosphorus donor spin ensemble the Tavis-Cummings model. We also observe an abrupt (s ˆ , i ¼ 1;…;N þ N ) positioned inside a cylindrical dielectric i ↑ ↓ π-phase shift in the pseudospin microwave emission around (sapphire) microwave cavity within a helium cryostat this fully inverted state M ¼ S. Observations of the log (T ¼ 1.5 K). The spins are placed in a ∼3500-G dc magnetic dependence in the delay and the abrupt phase shift near field (B ) and coupled magnetically to the cylindrical TE011 full pseudospin inversion has eluded previous implemen- cavity mode (microwave magnetic field, B ) with individual mw tations of strongly coupled spin ensembles due to their spin-cavity coupling g (i ¼ 1;…;N þ N ). Current losses in i ↑ ↓ much shorter T and the low fidelity of their pseudospin the copper walls (κ ) and coupling losses (κ ) through the int ext rotations (a direct consequence of nonuniform spin-cavity antenna along with dephasing in the spin ensemble (γ ¼ 1=T ) coupling) [26,27]. set the maximum window during which the spin-cavity dynamics The isotopically enriched silicon crystal (<50 ppm Si) can be probed. (b) In the case of small inhomogeneous broad- ening (ω ¼ ω) and uniform cavity coupling (g ¼ g), the spin we use in these experiments is phosphorus doped with a i i 15 −3 13 ensemble can be modeled as a single large pseudospin as shown density of 3.3 × 10 cm (5.7 × 10 total donors) and is in the Bloch sphere representation. The sphere surface represents 14 −3 otherwise highly pure (boron density less than 10 cm ) the symmetric superradiant subspace (jS; Mi). Strong correla- [18]. A tunable distributed Bragg reflector (DBR) laser tions between spins in these states lead to an enhanced photon (Eagleyard EYP-DBR-1080) is used to controllably polarize emission rate Γ ∝ N near the equator (M ¼ 0). Uncorrelated the phosphorus donor spin ensemble beyond Boltzmann spontaneous emission at a rate Γ ∝ N dominates near the poles equilibrium by spin-selective optical pumping of the phos- (M ¼S). phorus donor no-phonon bound exciton transitions [28]. The efficiency of the optical pumping and the resulting Donor electron spins in isotopically enriched Si form steady-state spin polarization (N ¼ N − N ) are controlled ↑ ↓ exceptionally well isolated qubits with the longest electron by detuning the laser from resonance with one of the no- spin coherence measured in a solid-state environment. phonon bound exciton transitions [Fig. 2(a)]. The laser is The spin decoherence rate 1=T is less than 1 Hz, and tunable between 1077–1081 nm (277.2–278.5 THz) and the ensemble dephasing rate 1=T can be less than 10 kHz fiber coupled to deliver∼10 mW of light to the sample. More [17]. These rates make the system of interest to the field than 95% electron spin polarization is achieved after 300 ms of quantum computation especially in the context of of resonant optical pumping [Fig. 2(a)]. The Pnucleiare 031002-2 COHERENT RABI DYNAMICS OF A SUPERRADIANT SPIN … PHYS. REV. X 7, 031002 (2017) (a) (b) (c) FIG. 3. (a) Pulse sequence of the free-induction decay (FID) experiment. The initial laser pulse sets the number of unpaired spins N and the total spin quantum number S ¼ N=2; the subsequent microwave pulse excites the pseudospin to jS ¼ N=2;Mi. The microwave emission from the pseudospin is recorded after the microwave pulse and a 3-μs experimental dead time. (b) In-phase Reða ˆÞ and quadrature Imða ˆÞ components of the pseudospin microwave emission during the free-induction decay in the strong-coupling regime (M ≈ 0, N ¼ 3.6 × 10 ). FIG. 2. (a) The number of spins interacting with the resonator (c) Amplitude of the microwave emission ja ˆj during the (N) is controlled by tuning the laser frequency which is FID (black solid line). The FID shows oscillations at rate pffiffiffiffi spin-selectively exciting a phosphorus donor no-phonon bound g N ¼ 290 kHz, many coherent cycles of energy exchange exciton transition [28]. (b) Reflected microwave power S (color 11 between the spin ensemble and the cavity are resolved. For scale) spectrum of the cavity as a function of the spin-ensemble comparison, the purely exponential decay in the weak-coupling pffiffiffiffi Zeeman splitting (ω ¼ gμ B ) as the spin ensemble is brought s B 0 regime is also shown (dashed red line) with g N ¼ 56 kHz, into resonance with the bare cavity (ω ). The clear avoided κ ¼ 960 kHz. The decays are normalized to have the same crossing shows that the system is in the strong-coupling regime magnitude at t ¼ 0. pffiffiffiffi with 2g N (580 kHz) ≫ κ þ γ (64 kHz). (c) The vacuum Rabi pffiffiffiffi splitting (2g N) of the polariton modes (shown at ω ¼ ω ) can s c be controlled by detuning the laser frequency. Curves 1 and 2 magnetic field (B ) to provide a Zeeman splitting to the spin were measured while the laser frequency was tuned to points 1 ensemble. A sapphire ring [Fig. 1(a)] is used inside of the and 2 as indicated in (a). copper resonator as a high dielectric material to concentrate the microwave magnetic field (B ) into the sample volume mw also polarized to ≈25% during the optical pumping. [30]. In the free-induction decay (FID) experiments, a single The mechanism behind this nuclear polarization remains 200-ns microwave pulse is used to tip the spins after unknown, but it is thought to be due to an enhanced cross- polarizing with the DBR laser [Fig. 3(a)]. The tipping angle relaxation rate of the donors under illumination [29]. X-band of the microwave pulse is controlled by varying the power of (9.6 GHz) ESR experiments are performed with a Bruker the microwave source (Agilent E8267D). ESR spectrometer (Elexsys E580) using a 3D sapphire The stationary states of the system are interrogated by dielectric resonator (ER-4118X-MD5) in a helium-flow measuring reflected microwave power (S ) with an Agilent cryostat (Oxford CF935) that is pumped to achieve 1.5 K. E5071C network analyzer [Figs. 2(b) and 2(c)]. Power A 1.4-T electromagnet (Bruker B-E 25) is used to apply a dc saturation is avoided by lowering the probe power to 031002-3 B. C. ROSE et al. PHYS. REV. X 7, 031002 (2017) −65 dBm (power incident on the resonator); below this excitations are exchanged between the spin ensemble and power the measured vacuum Rabi splitting is constant and the cavity over time. Spin relaxation (T ) of phosphorus no power broadening is observed. The S measurements donors at 1.5 K is slow and does not enter into the dynamics are performed under continuous illumination from the of the free-induction decay [17]. Even with this faster decay resonant DBR laser, which, in addition to polarizing the we are able to observe 12 collapses and revivals before spin ensemble, accelerates spin relaxation (T ) and reduces the free-induction decay falls below the noise level. power saturation from the microwave probe. An experimental dead time of 3 μs after the microwave The total ensemble coupling can be determined directly by pulse limits us from measuring the beginning of the FID looking at the eigenfrequencies of the coupled system while signal. The large signal-to-noise ratio and many oscillations the spin ensemble is tuned into resonance with the cavity we observe here offer clearly resolved dynamics of the [Fig. 2(b)]. This technique has been demonstrated for spin coupled spin-ensemble–cavity evolution. ensembles coupled to both 3D volume resonators [31] and A theoretical description of this experiment begins with a 2D superconducting microresonators [32]. The spin transi- general model for an ensemble of N spin-1=2 particles tion frequency (ω ) is varied through resonance with the interacting with a single-cavity mode: cavity (ω ) by changing the Zeeman splitting with a magnetic X X † † field (B ), which results in an avoided crossing with a clear H ¼ ω s ˆ þ ω a ˆ a ˆ þ 2 g ða ˆ þ a ˆÞs ˆ i Zi c i Xi splitting showing that the system is in the strong-coupling i i regime. In particular, with the laser tuned on resonance þ s ˆ · D · s ˆ ; ð1Þ 13 i ij j (N ¼ 3.6 × 10 ) the vacuum Rabi splitting of the polariton pffiffiffiffi hiji modes is 580 kHz (2g N), which is an order of magnitude larger than their 64-kHz linewidth [Fig. 2(c), curve 2]. where a ˆ , a ˆ are the creation and annihilation operators for The splitting can be controllably reduced by detuning the the cavity field photons of frequency ω , s ˆ (i ¼ 1;…;N c ki laser from resonance with the bound exciton transition [curve and k ¼ X, Y, Z) are the single spin (S ¼ 1=2) matrices for 1in Fig. 2(c)]. The large size of the ensemble is important to a spin with transition frequency ω and spin-cavity cou- achieve this large splitting since each spin is weakly coupled pling g . The last term (D ) describes dipolar interactions i ij to the 3D cavity with a single spin coupling g ¼ 48 mHz between spins. [33]. The narrow linewidth of the resonances is defined by Direct diagonalization of this Hamiltonian for large spin combined losses in the cavity (κ ¼ 60 kHz for Q ¼ 75000) ensembles is not possible; however, several simplifying and dephasing in the spin ensemble (γ ¼ 18 kHz, measured approximations can be made to good accuracy. The spin directly in the weak-coupling regime) [34]. Strong coupling pffiffiffiffi ensemble has a narrow inhomogeneous distribution in (2g N ≫ κ þ γ) implies that the system can efficiently and Zeeman frequencies compared to the spin-cavity coupling pffiffiffiffi coherently exchange excitations. The cooperativity param- (g NT ≫ 1) so that, for the duration of our FID experi- eter is C ¼ 2g N=κγ ¼ 91. ment, we can treat the spins as having identical transition The free evolution of the pseudospin in a cavity is most frequencies ω ≈ ω . The distribution in the individual i s directly studied in a single-pulse free-induction decay spin-cavity coupling (g) is mostly defined by the micro- experiment [Fig. 3(a)]. The pseudospin is first tuned into wave magnetic field inhomogeneity along the length of the resonance with the cavity (ω ¼ ω ) and polarized into its s c sample (5 mm). For our volume resonator this variation has ground state (M ¼ −S) with resonant laser pumping thus been measured to be less than 5% [35]. Thus, to a good defining N ¼ N − N and S ¼ N=2. This is followed by a ↑ ↓ approximation, the spin-ensemble–cavity system is in the single resonant microwave pulse (ω ¼ ω ¼ ω ) that drive s c small sample limit with a uniform spin-cavity coupling determines the initial state (M) of the pseudospin before the (g ≈ g) [19,36]. Finally, the dipolar coupling between FID. In the weak-coupling regime the envelope of the free- 15 −3 spins [last term in Eq. (1)] at the 3.3 × 10 cm donor induction decay is a simple exponential [Fig. 3(a)] with a density used in this experiment is ∼100 Hz and negligible characteristic decay time resulting from the inhomo- on the 30-μs time scale measured here [37]. With both the geneous broadening of the spin ensemble (T ). However, Zeeman frequency and the cavity coupling being constant in the strong-coupling limit we achieve here [Figs. 3(b) and across all of the spins, the ensemble can be treated as a 3(c)], the free-induction decay shows multiple oscillations single large pseudospin with a collective spin operator as energy is coherently exchanged between the spins and pffiffiffiffi S ¼ s . Including all of these considerations, ðz;Þ i ðz;Þi the resonator through their coupling at a rate of 2g N ¼ Eq. (1) reduces to the Tavis-Cummings model, 580 kHz corresponding to the vacuum Rabi splitting of the polariton modes. † † ˆ ˆ ˆ ˆ H ¼ ω S þ ω a ˆ a ˆ þ gða ˆ S þ a ˆS Þ; ð2Þ In the strong-coupling regime, the envelope decay of the TC s Z c − þ polariton modes [Fig. 3(c), dashed black line] is faster than which is a generalization of the Jaynes-Cummings model the decay in the weak-coupling regime, as it results from a combination of spin dephasing plus resonator losses as the for a single collective pseudospin S. From this model we 031002-4 COHERENT RABI DYNAMICS OF A SUPERRADIANT SPIN … PHYS. REV. X 7, 031002 (2017) derive the equations of motion for the pseudospin-cavity system: _ ˆ ha ˆðtÞi ¼ −κha ˆðtÞi − ighS ðtÞi − iVðtÞ; ˆ ˆ ˆ hS ðtÞi ¼ −γhS ðtÞi þ 2igha ˆðtÞihS ðtÞi; − − z ˆ ˆ ˆ ˆ ˆ hS ðtÞi ¼ igðha ðtÞihS ðtÞi − haðtÞihS ðtÞi Þ: ð3Þ z − − These are the Maxwell-Bloch equations for the expectation ˆ ˆ values ha ˆi, hS i, and hS i [38]. The dissipation factors, κ − z and γ, are introduced using the standard master equation formalism for the open system [39]. We also add a classical drive term with amplitude VðtÞ set by the microwave source. To simplify these equations we take the semi- classical limit by neglecting correlations between the ˆ ˆ spin-ensemble and spin-cavity photons (ha ˆS i¼ha ˆihS i i i for i ¼þ;−;z). This approximation is valid given the large size of our spin ensemble (N ∼ 10 spins). We use Eq. (3) to simulate the dynamics of the free-induction decay (Fig. 4). In the absence of losses (κ, γ ¼ 0) the total spin of the ˆ ˆ ensemble is conserved (½H ; S ¼ 0) and the system TC evolution is a trajectory on the surface of the Bloch sphere within the superradiant subspace [Fig. 1(b)]. Dephasing from inhomogeneous broadening (T ) and cavity photon FIG. 4. Collective dynamics between the pseudospin and dissipation (κ) open up a channel for mixing with sub- cavity. (a) Cavity photon number [nðtÞ¼ha ˆ a ˆi] during the radiant states so that S is no longer preserved and the FID experiment [the square of the FID amplitude shown in system enters states in the interior of the sphere [Fig. 4(b)] Fig. 3(c)] with N ¼ 3.6 × 10 and M ≈ 0. The simulated fit [40]. In our experiment this dephasing is not refocused and according to Eq. (3) is also shown (dashed curve). The tail of the thus is irreversible, so that it can be taken as a single loss decay is magnified 40× and shown in green. (b) The pseudospin rate γ for the transverse magnetization of the spin ensemble. evolution on the Bloch sphere as determined from hS ðtÞi and The value for γ (also used in the S measurements) is 11 ˆ hS ðtÞi in this simulation. The numbers (1–4) on the curves extracted directly from the free-induction decay in the indicate equivalent points in time during the FID on the micro- weak-coupling regime where spin dephasing is the only wave emission plot (a) and the simulated pseudospin evolution contribution. Photon losses are also taken into account with plot (b). The dynamics of the pseudospin in this large N limit is a single parameter κ ¼ κ þ κ , which represents both governed by the Maxwell-Bloch equations. ext int internal loss in the cavity (κ ) and external loss through the int antenna coupling (κ ). This parameter is extracted by measured FID curve [Fig. 4(a)] with the main difference ext measuring the linewidth of the cavity resonance under between the two curves being only the experimental dead low excitation where strong-coupling effects are negligible. time (initial 3 μs). The time evolution of the pseudospin state as determined from this fit is plotted in Fig. 4(b). The 64-kHz polariton linewidth in Fig. 2(c) is a combina- The motion of the pseudospin on the Bloch sphere is tion of both κ and γ, with κ ≫ γ. formally equivalent to a damped pendulum that is kicked A simulation of the coupled system according to these into motion by the microwave excitation [20]. The pseu- equations of motion is compared to the measured curve in Fig. 4(a). Here, we plot the intensity of the microwave dospin is initially polarized to its ground state at the south † † emission [cavity photon number, n ¼ha ˆ a ˆi¼ha ˆ iha ˆiþ pole [point 1 on the Bloch sphere in Fig. 4(b)]. A micro- Oðlog NÞ derived from the microwave amplitude, which is wave pulse is applied for 200 ns (black arrow), mainly measured directly and plotted in Fig. 3(c). This procedure is populating the cavity with photons but also starting the initial excitation of the pseudospin (point 2). The driving valid for a large spin ensemble since ðlog NÞ=N ≪ 1. The microwave pulse ends, but the remaining photons in the only fitting parameter we use in this simulation is a scaling cavity continue to be transferred to the pseudospin until no parameter for the microwave pulse [VðtÞ]. This parameter accounts for the resonator coupling as well as losses in the photons remain (point 3 on the Bloch sphere). The power of microwave excitation path. The same scaling parameter is the applied microwave pulse in this example is chosen so used consistently for all microwave powers set in our that this point would be close to the equator (jS; 0i). This point (3) corresponds to a minimum (n ¼ 0) in the experiments. The simulation shows an excellent fit to the 031002-5 B. C. ROSE et al. PHYS. REV. X 7, 031002 (2017) free-induction decay [Fig. 4(a), point 3] since the photons in the resonator have been fully absorbed by the pseudo- spin. After reaching this highest point, the excited pseu- dospin will start to emit photons back into the resonator, reversing the nutation on the Bloch sphere to transition towards its ground state [point 4 in Fig. 4(b)]. During the initial excitation the pseudospin (up until point 3) evolves with the same phase as the microwave pulse, but as the pseudospin reverses direction the photons it emits have a phase opposite to that of the initial pulse. The pseudospin eventually fully deexcites to jS;−Si (point 4), releasing FIG. 5. Fourier transform (plotted as magnitude) of the donor all of the photons back into the resonator. This point spin-ensemble free-induction decay as a function of the number of unpaired spins in the ensemble (N). The black dashed line corresponds to a maximum in the free-induction decay pffiffiffiffi shows a g N dependence of the pseudospin-cavity exchange rate [Fig. 4(a), point 4]. The pseudospin then reabsorbs these as expected from the Tavis-Cummings model in the large N limit. photons and is excited to the opposite side of the Bloch This gives a single-spin coupling of g ¼ 48 mHz for a uniformly sphere (point 5), since the emitted photons are of opposite coupled ensemble. The side plot shows a vertical slice at phase to the initial excitation. This coherent exchange of 13 −3 N ¼ 3.6 × 10 cm for the maximum ensemble coupling that excitations continues until dissipation and dephasing pffiffiffiffi was achieved (2g N ¼ 580 kHz). destroy the ensemble polarization. Near the maxima and minima of the FID, where n_ ¼ 0, the pseudospin evolution is dominated by the second reached because there are no excitations left in the system derivative (in time) of the cavity photon number (n). In (n ¼ 0) to be reabsorbed by the pseudospin. the strong-coupling regime (ignoring the dissipation terms) In order to better understand the dynamics in this strong- this can be approximated as coupling regime, we vary the initial state of the pseudospin to observe its effect on the subsequent evolution. The initial 2 2 ˆ ˆ ˆ ˆ ˆ n̈ ≈ g ðhS S iþhS S iÞ þ 4g ðn þ 1=2ÞhS i: ð4Þ state of the pseudospin jS; Mi can be accurately controlled þ − − þ z both by adjusting the detuning frequency of the laser, which With this expression we can explain the shape of the FID determines the total spin S [Fig. 2(a)], and by changing near each extrema. At minima (i.e., point 3) there are no the power of the microwave pulse, which determines M. photons in the cavity (n ≈ 0) and also hS i¼ 0; therefore, z The observed dependencies on S and M are plotted in the second term in Eq. (4) is small. However, the pseudo- Figs. 5 and 6, respectively. spin is near jS; 0i, where correlations between spins lead to Varying the laser detuning allows for fine control of the superradiant emission with an emission rate that is quad- net ensemble polarization (N) and the total spin-ensemble– pffiffiffiffi ratic in the total number of unpaired spins [1], so that the cavity coupling (g N). The magnitude of the Fourier 2 2 2 ˆ ˆ first term in Eq. (4) is large, n̈ ∝ g hS S i ∝ g N =2. þ − transform of the free-induction decay (Fig. 5) reveals that pffiffiffiffi At maxima (i.e., point 4) the pseudospin is near jS;−Si the pseudospin-cavity exchange frequency scales as g N (hS i≈−N=2), where the correlations between spins in the (dashed black), in agreement with the large pseudospin ˆ ˆ ensemble are negligible and hS S i ≈ N (emission of N − þ picture. The shapes of the polariton peaks in the Fourier uncorrelated emitters). However, most of the photons have transform of the free-induction decay (side plot in Fig. 5) transferred back into the cavity (n ≈ N), so that the second are less resolved than the shapes obtained by reflection term in Eq. (4) is now large, resulting in a similarly fast spectroscopy [Fig. 2(c)]. This is due to the fact that during 2 2 n̈ ∝−g N . The pseudospin is reexcited by the large the free-induction decay a coherence, created by the initial population of photons in the resonator (n ≈ N), but dis- microwave pulse, leaks to subradiant states (S<N=2)on sipation and dephasing eventually destroy the ensemble the time scale of T . This leakage gives rise to a time- polarization (N), terminating the FID. dependent change of N during the FID and decreases the The FID we observe is different from superradiance cavity coupling of the pseudospin, effectively smearing out phenomena in free space, where the system is far from the the peaks in the Fourier transforms. This leakage does not strong-coupling regime. In free space the emitted photons occur in the reflection measurements. leave the system much faster than they can be reabsorbed The initial Z projection (M) of the pseudospin is set by the pseudospin, which means that the second term in by the tipping angle (power) of the microwave pulse. Eq. (4) can be neglected. As a result, near M ¼ −S the This tipping angle [defined as polar angle θ with respect to pseudospin evolves slowly with n̈ ∝ g N. This is a factor of jS;−Si in Fig. 1(b)] is varied through several full rotations N smaller than the value of n̈ we observe in the strong- of the pseudospin, and the resulting free-induction decays coupling regime. Additionally, in free space the pseudospin (real components) are plotted in Fig. 6 (top) accompanied emission dies out entirely after the ground state jS;−Si is by Maxwell-Bloch simulations (bottom). Rotation angles 031002-6 COHERENT RABI DYNAMICS OF A SUPERRADIANT SPIN … PHYS. REV. X 7, 031002 (2017) FIG. 6. Top: In-phase component (Reha ˆi) of the pseudospin microwave emission during the FID experiment (time axis) plotted as a function of the rotation angle of the initial microwave pulse (microwave power axis). The signal intensity is plotted using a log color scale in order to emphasize the sign of the signal. The blue (yellow) colors correspond to negative (positive) signal intensities. Bottom: The corresponding FID simulations using the Maxwell-Bloch equations [Eq. (3)]. The rotation angles (θ ¼ π, 3π, 5π) indicated on the left correspond to the inverted state of the pseudospin jS; Si. The numbered arrows (1)–(4) shown on the right correspond to FID experiments shown separately in Fig. 7. FIG. 7. Comparison of individual FID traces taken as horizontal marked as π, 3π, and 5π on the left in Fig. 6 correspond slices from Fig. 6 for selected rotation angles (θ) of the micro- to the pseudospin being fully inverted (near jS; Si) after wave pulse. The initial pseudospin state is illustrated on the Bloch photons from the microwave pulse are fully absorbed by sphere for each case (inset). (a),(b) Traces (1) and (2) correspond the pseudospin. We observe that the period of the FID to rotation angles of θ ≈ π=2 (initial pseudospin state jS; 0i) and oscillations does not depend on the rotation angle (initial θ ≈ π (initial pseudospin state jS; Si), respectively. All oscilla- pffiffiffiffi pffiffiffiffi value of M) and is always g N. However, there is an tions in trace (1) occur at a rate close to g N. In contrast, trace overall phase delay in the onset of the oscillations when (2) evolves more slowly with a long delay (t ) in the FID oscillations occurring at the first minimum where the pseudospin approaching the fully inverted points at the top of the pffiffiffiffi reaches full inversion. The rest of the FID trace oscillates at g N Bloch sphere (Fig. 6). This shift is more clearly seen when so that this initial delay shows up as an overall phase shift. Inset in comparing the FID traces in Figs. 7(a) and 7(b) for (b): A zoomed-in part of the two-dimensional FID plot from θ ≈ π=2 and θ ≈ π, respectively. Fig. 6 with a fit (white dashed curve) to t ðMÞ according to This delay is explained by considering the pseudospin Eq. (5). (c) Traces (3) and (4) correspond to slight under- or evolution, starting from jS; Mi, as an avalanche process with overrotation with respect to the fully inverted state of the pseudo- the pseudospin transitioning down the ladder of states in the spin (θ ≈ π  δ, with δ ¼ 9 deg). The phases of the microwave superradiant subspace towards its ground state [Fig. 1(b)]. emission are opposite in these two traces since the pseudospin Initially there are no photons in the cavity and the emission precession reverses direction in the underrotated case but continues rate is slow. As the pseudospin deexcites, a larger population in the same direction in the overrotated case. of photons builds up in the cavity and the pseudospin emission becomes stimulated. The total time for this process to occur is dominated by the slow time it takes to emit the first to fit the delay observed in our experiment [white dashed line few photons in the avalanche, which is longest near full in inset of Fig. 7(a)]. We assume an initial state jS; Mi with inversion [Fig. 7(b)], since the transition probability of the M ≈ S. Γ is the total spontaneous emission rate of N pseudospin is the smallest there [1]. We use the expression independent spins (including Purcell enhancement). We derived in Ref. [2], extract Γ ¼ 2.9 MHz from this fit, which is close to the rate calculated directly from the measured coupling and loss 2 1 1 2 2S t ðMÞ ≈ þ  þ ≈ log D parameters (Γ ¼ 4g N=κ ¼ 3.2 MHz). The longest delay Γ S − M þ 1 N Γ S − M þ 1 we are able to achieve is t ≈ 3.5 μs, corresponding to an 2 2 initial Z projection of M ¼ 1.77 × 10 and to the tipping ¼ log ; ð5Þ Γ ð1 þ cos θÞ angle θ ¼ 171°. The theoretical limit of the time delay for 031002-7 B. C. ROSE et al. PHYS. REV. X 7, 031002 (2017) our system, assuming a perfect θ ¼ 180°rotation, is In conclusion, we demonstrate a strongly coupled spin t ðM ¼ SÞ¼ 2 log N=Γ ¼ 21 μs. This theoretical limit is ensemble with uniform spin-cavity coupling and small difficult to achieve experimentally since near full inversion inhomogeneous broadening allowing the system to be the delay becomes sensitive to small deviations of the modeled very accurately as a single large pseudospin. rotation angle. Even though our system is very uniform, This pseudospin evolves according to the Tavis- the delay we obtain is limited by the 5% variation in cavity Cummings model in the large N limit. The use of highly coupling across the sample, the 20-mG variation of the static enriched Si gives a small inhomogeneous broadening magnietc field across the sample (δB =B ¼ 6%), and low- 0 1 (long T ) providing a wide window for viewing the frequency fluctuations in the B field (∼20 mG) over a time 0 collective pseudospin-cavity dynamics. In particular, we scale of seconds. are able to observe the pseudospin-cavity dynamics with a The oscillation delay is also explained by Eq. (4). With large number of excitations where the Holstein-Primakoff the pseudospin near M ¼ S (hS i ≈ N) and no photons in approximation is no longer valid. We can prepare an the cavity (n ≪ N), the second term is linear in N. There arbitrary initial state of the pseudospin jS; Mi through a ˆ ˆ are also no correlations between spins here (hS S i ≈ N), þ − combination of optical polarization (determining S)and so that the first term is also linear in N, giving a slow total microwave manipulation (determining M). From this we evolution (n̈ ∝ g N) as compared to the other extrema in observe the coherent exchange of energy between the pffiffiffiffi 2 2 the FID (n̈ ∝ g N ). pseudospin and the cavity at a rate g N, being indepen- The phase of the microwave emission during the free- dent of M. Control over M allows us to view, for the first induction decay shifts abruptly by π as the pseudospin time, the pseudospin dynamics as it approaches a qua- state is varied through the quasiequilibrium at full inver- siequilibrium near full inversion. In particular, we observe sion around θ ¼ π, 3π,and 5π in Fig. 6.Thisphase shiftis (and can control) a delay in the pseudospin microwave more clearly seen in Fig. 7(c) where two traces are shown emission when it is near the quasiequilibrium point. that correspond to slight underrotation (trace 3) and slight We find that this delay scales with the initial state of overrotation (trace 4) of the pseudospin relative to full the pseudospin (jS; Mi)aslog½2S=ðS − M þ 1Þ,which inversion. This phase shift is explained by looking at the is consistent with an expression derived from the evolution of the pseudospin on the Bloch sphere. When Tavis-Cummings model. We also observe that the micro- the microwave drive excites the pseudospin short of the wave emission has an abrupt π-phase shift when the fully inverted state [θ ¼ π − δ,trace 3inFig. 7(c)], just pseudospin is at the quasiequilibrium, which is explained after all of the cavity photons have been absorbed, the by considering the motion of the pseudospin on the pseudospin reverses directions on the Bloch sphere in Bloch sphere. This level of pseudospin control is unprec- order to evolve towards its ground state. The phase of the edented and is vital for quantum memory applications. emitted photons in this case has an opposite phase to the These observations offer novel validations of the Tavis- original photons from the microwave drive. On the other Cummings model and show that donors in Si are a hand, when the cavity photons oversaturate the pseudo- promising platform for implementing a spin-ensemble spin [θ ¼ π þ δ, trace 4 in Fig. 7(c)] and drive it past the quantum memory. fully inverted state, then the pseudospin will continue to Integrating a superconducting qubit inside of a 3D evolve in the same direction as it transitions towards its cavity together with the spin ensemble is an appealing ground state. In this case the emitted photons are of the next step, and similar hybrid architectures are currently same phase as the initial microwave drive—opposite to under investigation [23]. Such a hybrid implementation the phase of the pseudospin emission in the underrotated with phosphorus donor electron spins could be challeng- experiment. ing due to the large magnetic fields that are required, The quasiequilibrium we observe when the pseudospin is which could drive the Josephson junctions of the super- at full inversion is a clear indication that the full dynamics conducting qubit normal, although junctions have been of the Maxwell-Bloch equations must be included to model shown to operate successfully in fields up to 7 T [44]. our system. This is different from the more extensively One alternative is to utilize a different donor, like bismuth, studied low-excitation limit where the pseudospin is with a large enough zero-field splitting to enable low-field always near M ¼ −S, and to good approximation S can measurements [45]. be considered to be a constant of motion (Holstein- Primakoff approximation) [26,41–43], With this approxi- This work was supported by the NSF and EPSRC mation the equations of motion are analogous to a system through the Materials World Network and NSF MRSEC of two coupled linear oscillators and cannot reproduce the Programs (Grants No. DMR-1107606, No. EP/I035536/1, behavior we observe near full inversion. In the low- and No. DMR-01420541), and the ARO (Grant excitation limit, the maxima and minima of the FID both No. W911NF-13-1-0179). M. L. W. T. was supported by evolve with n̈ ∝ g N, which is much slower than the the Natural Sciences and Engineering Research Council of 2 2 n ∝ g N behavior we observe here. Canada (NSERC). The work at Keio is supported by 031002-8 COHERENT RABI DYNAMICS OF A SUPERRADIANT SPIN … PHYS. REV. X 7, 031002 (2017) [17] A. M. Tyryshkin, S. Tojo, J. J. L. Morton, H. Riemann, N. V. KAKENHI (S) No. 26220602 and JSPS Core-to-Core Abrosimov, P. Becker, H.-J. Pohl, T. Schenkel, M. L. W. Program. The authors extend special thanks to Hakan Thewalt, K. M. Itoh, and S. A. Lyon, Electron Spin Coher- Türeci and Jonathan Keeling for helpful discussions of ence Exceeding Seconds in High-Purity Silicon, Nat. Mater. the Tavis-Cummings model. 11, 143 (2012). [18] P. Becker, H.-J. Pohl, H. Riemann, and N. Abrosimov, Enrichment of Silicon for a Better Kilogram, Phys. Status Solidi A 207, 49 (2010). [19] M. Tavis and F. W. Cummings, Exact Solution for an [1] R. H. 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Coherent Rabi Dynamics of a Superradiant Spin Ensemble in a Microwave Cavity

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PHYSICAL REVIEW X 7, 031002 (2017) 1 1 2 2 3 4 B. C. Rose, A. M. Tyryshkin, H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, 5 6 1 M. L. W. Thewalt, K. M. Itoh, and S. A. Lyon Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA Leibniz-Institut für Kristallzüchtung, 12489 Berlin, Germany PTB Braunschweig, 38116 Braunschweig, Germany VITCON Projectconsult GmbH, 07745 Jena, Germany Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 School of Fundamental Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohuku-ku, Yokohama 223-8522, Japan (Received 13 February 2017; revised manuscript received 18 May 2017; published 10 July 2017) We achieve the strong-coupling regime between an ensemble of phosphorus donor spins in a highly enriched Si crystal and a 3D dielectric resonator. Spins are polarized beyond Boltzmann equilibrium using spin-selective optical excitation of the no-phonon bound exciton transition resulting in N ¼ 3.6 × 10 unpaired spins in the ensemble. We observe a normal mode splitting of the spin-ensemble–cavity polariton pffiffiffiffi resonances of 2g N ¼ 580 kHz (where each spin is coupled with strength g) in a cavity with a quality factor of 75 000 (γ ≪ κ ≈ 60 kHz, where γ and κ are the spin dephasing and cavity loss rates, respectively). The spin ensemble has a long dephasing time (T ¼ 9 μs) providing a wide window for viewing the dynamics of the coupled spin-ensemble–cavity system. The free-induction decay shows up to a dozen collapses and revivals revealing a coherent exchange of excitations between the superradiant state of the spin ensemble and the pffiffiffiffi cavity at the rate g N. The ensemble is found to evolve as a single large pseudospin according to the Tavis-Cummings model due to minimal inhomogeneous broadening and uniform spin-cavity coupling. We demonstrate independent control of the total spin and the initial Z projection of the psuedospin using optical excitation and microwave manipulation, respectively. We vary the microwave excitation power to rotate the pseudospin on the Bloch sphere and observe a long delay in the onset of the superradiant emission as the pseudospin approaches full inversion. This delay is accompanied by an abrupt π-phase shift in the peusdospin microwave emission. The scaling of this delay with the initial angle and the sudden phase shift are explained by the Tavis-Cummings model. DOI: 10.1103/PhysRevX.7.031002 Subject Areas: Mesoscopics, Quantum Physics, Strongly Correlated Materials The enhanced collective emission from an ensemble of these collective effects are generally lacking in large ensem- atomlike systems due to coherent self-stimulated emission bles due to strong dephasing (short T ¼ 1=γ) [2,15,16]. was originally described by Dicke [1]. For this phenomenon Here, we demonstrate the dynamics of a strongly coupled he coined the term superradiance and showed that the atom ensemble of phosphorus donor spins in highly isotopically ensemble can behave as a large collective pseudospin. enriched Si with both a long dephasing time [17,18] and Superradiant emission has been observed in numerous uniform coupling to the radiation field (due to the use of a 3D physical systems [2–10]. These collective effects are par- microwave cavity), as shown schematically in Fig. 1(a).For ticularly prominent when the coupling between the spin the first time outside of ensembles of Rydberg atoms [15,16], pffiffiffiffi we study a spin-ensemble–cavity system with both of these ensemble and the radiation field (g N for N spins indi- essential properties allowing for it to be modeled accurately vidually coupled with strength g) is larger than any of the as a single large pseudospin [Fig. 1(b)] offering a simple losses in the system (κ þ γ, where κ is the radiative loss rate interpretation of the spin-ensemble–cavity evolution [1,19]. and γ is the spin dephasing rate). This strong-coupling regime pffiffiffiffi In particular, we are able to directly observe the dynamics (g N ≫ κ þ γ) has been extensively studied in both theory of superradiant emission under strong excitation, resolving and experiment [11–14], but clearly resolved dynamics of several cycles of coherent excitation transfer between the spin ensemble and the cavity [20]. In contrast to the more extensively studied low excitation limit [21,22], our experi- Published by the American Physical Society under the terms of ments performed under strong excitation cannot be modeled the Creative Commons Attribution 4.0 International license. as a linear system of two coupled harmonic oscillators. Further distribution of this work must maintain attribution to Instead, it must be treated with the Tavis-Cummings the author(s) and the published article’s title, journal citation, and DOI. model [19]. 2160-3308=17=7(3)=031002(10) 031002-1 Published by the American Physical Society B. C. ROSE et al. PHYS. REV. X 7, 031002 (2017) implementing a long-lived quantum memory. Both rates are 100–1000 times improvement compared to other candidate spin memories [23]. The key figure of merit for implementing a spin-ensemble quantum memory is pffiffiffiffi g NT ≫ 1, and we demonstrate that we are well beyond satisfying this criterion. Additionally, we achieve rela- tively uniform spin-cavity couplingwithlessthan5% variation across the spin ensemble. Both factors (long T and uniform spin-ensemble–cavity coupling) are vital for performing high-fidelity spin-ensemble manipulation while in the strong-coupling regime, which has been an outstanding problem in implementing a spin-ensemble quantum memory [24]. Initialization of the pseudospin is accomplished with a combination of optical excitation, which sets the size of the pseudospin, S ¼ N=2, and microwave manipulation, which sets the initial Z projection of the pseudospin M. Previous reports of large superradiant spin ensembles prepared only an initially inverted state jS; Si [15,16],while with our system we are able to prepare any initial pseudospin state jS; Mi in the superradiant subspace [Fig. 1(b)] with independent control of S and M. By controlling the size of pffiffiffiffi the pseudospin we demonstrate a g N dependence of the energy exchange rate between the pseudospin and cavity. By varying M, we are able to control a delay in the onset of the superradiant emission [25] and we report the first experimental observation of a log dependence of this delay when the pseudospin is near quasiequilibrium at M ¼ S. FIG. 1. (a) Experimental scheme for delivering microwave This log dependence is consistent with predictions from and optical excitation to the phosphorus donor spin ensemble the Tavis-Cummings model. We also observe an abrupt (s ˆ , i ¼ 1;…;N þ N ) positioned inside a cylindrical dielectric i ↑ ↓ π-phase shift in the pseudospin microwave emission around (sapphire) microwave cavity within a helium cryostat this fully inverted state M ¼ S. Observations of the log (T ¼ 1.5 K). The spins are placed in a ∼3500-G dc magnetic dependence in the delay and the abrupt phase shift near field (B ) and coupled magnetically to the cylindrical TE011 full pseudospin inversion has eluded previous implemen- cavity mode (microwave magnetic field, B ) with individual mw tations of strongly coupled spin ensembles due to their spin-cavity coupling g (i ¼ 1;…;N þ N ). Current losses in i ↑ ↓ much shorter T and the low fidelity of their pseudospin the copper walls (κ ) and coupling losses (κ ) through the int ext rotations (a direct consequence of nonuniform spin-cavity antenna along with dephasing in the spin ensemble (γ ¼ 1=T ) coupling) [26,27]. set the maximum window during which the spin-cavity dynamics The isotopically enriched silicon crystal (<50 ppm Si) can be probed. (b) In the case of small inhomogeneous broad- ening (ω ¼ ω) and uniform cavity coupling (g ¼ g), the spin we use in these experiments is phosphorus doped with a i i 15 −3 13 ensemble can be modeled as a single large pseudospin as shown density of 3.3 × 10 cm (5.7 × 10 total donors) and is in the Bloch sphere representation. The sphere surface represents 14 −3 otherwise highly pure (boron density less than 10 cm ) the symmetric superradiant subspace (jS; Mi). Strong correla- [18]. A tunable distributed Bragg reflector (DBR) laser tions between spins in these states lead to an enhanced photon (Eagleyard EYP-DBR-1080) is used to controllably polarize emission rate Γ ∝ N near the equator (M ¼ 0). Uncorrelated the phosphorus donor spin ensemble beyond Boltzmann spontaneous emission at a rate Γ ∝ N dominates near the poles equilibrium by spin-selective optical pumping of the phos- (M ¼S). phorus donor no-phonon bound exciton transitions [28]. The efficiency of the optical pumping and the resulting Donor electron spins in isotopically enriched Si form steady-state spin polarization (N ¼ N − N ) are controlled ↑ ↓ exceptionally well isolated qubits with the longest electron by detuning the laser from resonance with one of the no- spin coherence measured in a solid-state environment. phonon bound exciton transitions [Fig. 2(a)]. The laser is The spin decoherence rate 1=T is less than 1 Hz, and tunable between 1077–1081 nm (277.2–278.5 THz) and the ensemble dephasing rate 1=T can be less than 10 kHz fiber coupled to deliver∼10 mW of light to the sample. More [17]. These rates make the system of interest to the field than 95% electron spin polarization is achieved after 300 ms of quantum computation especially in the context of of resonant optical pumping [Fig. 2(a)]. The Pnucleiare 031002-2 COHERENT RABI DYNAMICS OF A SUPERRADIANT SPIN … PHYS. REV. X 7, 031002 (2017) (a) (b) (c) FIG. 3. (a) Pulse sequence of the free-induction decay (FID) experiment. The initial laser pulse sets the number of unpaired spins N and the total spin quantum number S ¼ N=2; the subsequent microwave pulse excites the pseudospin to jS ¼ N=2;Mi. The microwave emission from the pseudospin is recorded after the microwave pulse and a 3-μs experimental dead time. (b) In-phase Reða ˆÞ and quadrature Imða ˆÞ components of the pseudospin microwave emission during the free-induction decay in the strong-coupling regime (M ≈ 0, N ¼ 3.6 × 10 ). FIG. 2. (a) The number of spins interacting with the resonator (c) Amplitude of the microwave emission ja ˆj during the (N) is controlled by tuning the laser frequency which is FID (black solid line). The FID shows oscillations at rate pffiffiffiffi spin-selectively exciting a phosphorus donor no-phonon bound g N ¼ 290 kHz, many coherent cycles of energy exchange exciton transition [28]. (b) Reflected microwave power S (color 11 between the spin ensemble and the cavity are resolved. For scale) spectrum of the cavity as a function of the spin-ensemble comparison, the purely exponential decay in the weak-coupling pffiffiffiffi Zeeman splitting (ω ¼ gμ B ) as the spin ensemble is brought s B 0 regime is also shown (dashed red line) with g N ¼ 56 kHz, into resonance with the bare cavity (ω ). The clear avoided κ ¼ 960 kHz. The decays are normalized to have the same crossing shows that the system is in the strong-coupling regime magnitude at t ¼ 0. pffiffiffiffi with 2g N (580 kHz) ≫ κ þ γ (64 kHz). (c) The vacuum Rabi pffiffiffiffi splitting (2g N) of the polariton modes (shown at ω ¼ ω ) can s c be controlled by detuning the laser frequency. Curves 1 and 2 magnetic field (B ) to provide a Zeeman splitting to the spin were measured while the laser frequency was tuned to points 1 ensemble. A sapphire ring [Fig. 1(a)] is used inside of the and 2 as indicated in (a). copper resonator as a high dielectric material to concentrate the microwave magnetic field (B ) into the sample volume mw also polarized to ≈25% during the optical pumping. [30]. In the free-induction decay (FID) experiments, a single The mechanism behind this nuclear polarization remains 200-ns microwave pulse is used to tip the spins after unknown, but it is thought to be due to an enhanced cross- polarizing with the DBR laser [Fig. 3(a)]. The tipping angle relaxation rate of the donors under illumination [29]. X-band of the microwave pulse is controlled by varying the power of (9.6 GHz) ESR experiments are performed with a Bruker the microwave source (Agilent E8267D). ESR spectrometer (Elexsys E580) using a 3D sapphire The stationary states of the system are interrogated by dielectric resonator (ER-4118X-MD5) in a helium-flow measuring reflected microwave power (S ) with an Agilent cryostat (Oxford CF935) that is pumped to achieve 1.5 K. E5071C network analyzer [Figs. 2(b) and 2(c)]. Power A 1.4-T electromagnet (Bruker B-E 25) is used to apply a dc saturation is avoided by lowering the probe power to 031002-3 B. C. ROSE et al. PHYS. REV. X 7, 031002 (2017) −65 dBm (power incident on the resonator); below this excitations are exchanged between the spin ensemble and power the measured vacuum Rabi splitting is constant and the cavity over time. Spin relaxation (T ) of phosphorus no power broadening is observed. The S measurements donors at 1.5 K is slow and does not enter into the dynamics are performed under continuous illumination from the of the free-induction decay [17]. Even with this faster decay resonant DBR laser, which, in addition to polarizing the we are able to observe 12 collapses and revivals before spin ensemble, accelerates spin relaxation (T ) and reduces the free-induction decay falls below the noise level. power saturation from the microwave probe. An experimental dead time of 3 μs after the microwave The total ensemble coupling can be determined directly by pulse limits us from measuring the beginning of the FID looking at the eigenfrequencies of the coupled system while signal. The large signal-to-noise ratio and many oscillations the spin ensemble is tuned into resonance with the cavity we observe here offer clearly resolved dynamics of the [Fig. 2(b)]. This technique has been demonstrated for spin coupled spin-ensemble–cavity evolution. ensembles coupled to both 3D volume resonators [31] and A theoretical description of this experiment begins with a 2D superconducting microresonators [32]. The spin transi- general model for an ensemble of N spin-1=2 particles tion frequency (ω ) is varied through resonance with the interacting with a single-cavity mode: cavity (ω ) by changing the Zeeman splitting with a magnetic X X † † field (B ), which results in an avoided crossing with a clear H ¼ ω s ˆ þ ω a ˆ a ˆ þ 2 g ða ˆ þ a ˆÞs ˆ i Zi c i Xi splitting showing that the system is in the strong-coupling i i regime. In particular, with the laser tuned on resonance þ s ˆ · D · s ˆ ; ð1Þ 13 i ij j (N ¼ 3.6 × 10 ) the vacuum Rabi splitting of the polariton pffiffiffiffi hiji modes is 580 kHz (2g N), which is an order of magnitude larger than their 64-kHz linewidth [Fig. 2(c), curve 2]. where a ˆ , a ˆ are the creation and annihilation operators for The splitting can be controllably reduced by detuning the the cavity field photons of frequency ω , s ˆ (i ¼ 1;…;N c ki laser from resonance with the bound exciton transition [curve and k ¼ X, Y, Z) are the single spin (S ¼ 1=2) matrices for 1in Fig. 2(c)]. The large size of the ensemble is important to a spin with transition frequency ω and spin-cavity cou- achieve this large splitting since each spin is weakly coupled pling g . The last term (D ) describes dipolar interactions i ij to the 3D cavity with a single spin coupling g ¼ 48 mHz between spins. [33]. The narrow linewidth of the resonances is defined by Direct diagonalization of this Hamiltonian for large spin combined losses in the cavity (κ ¼ 60 kHz for Q ¼ 75000) ensembles is not possible; however, several simplifying and dephasing in the spin ensemble (γ ¼ 18 kHz, measured approximations can be made to good accuracy. The spin directly in the weak-coupling regime) [34]. Strong coupling pffiffiffiffi ensemble has a narrow inhomogeneous distribution in (2g N ≫ κ þ γ) implies that the system can efficiently and Zeeman frequencies compared to the spin-cavity coupling pffiffiffiffi coherently exchange excitations. The cooperativity param- (g NT ≫ 1) so that, for the duration of our FID experi- eter is C ¼ 2g N=κγ ¼ 91. ment, we can treat the spins as having identical transition The free evolution of the pseudospin in a cavity is most frequencies ω ≈ ω . The distribution in the individual i s directly studied in a single-pulse free-induction decay spin-cavity coupling (g) is mostly defined by the micro- experiment [Fig. 3(a)]. The pseudospin is first tuned into wave magnetic field inhomogeneity along the length of the resonance with the cavity (ω ¼ ω ) and polarized into its s c sample (5 mm). For our volume resonator this variation has ground state (M ¼ −S) with resonant laser pumping thus been measured to be less than 5% [35]. Thus, to a good defining N ¼ N − N and S ¼ N=2. This is followed by a ↑ ↓ approximation, the spin-ensemble–cavity system is in the single resonant microwave pulse (ω ¼ ω ¼ ω ) that drive s c small sample limit with a uniform spin-cavity coupling determines the initial state (M) of the pseudospin before the (g ≈ g) [19,36]. Finally, the dipolar coupling between FID. In the weak-coupling regime the envelope of the free- 15 −3 spins [last term in Eq. (1)] at the 3.3 × 10 cm donor induction decay is a simple exponential [Fig. 3(a)] with a density used in this experiment is ∼100 Hz and negligible characteristic decay time resulting from the inhomo- on the 30-μs time scale measured here [37]. With both the geneous broadening of the spin ensemble (T ). However, Zeeman frequency and the cavity coupling being constant in the strong-coupling limit we achieve here [Figs. 3(b) and across all of the spins, the ensemble can be treated as a 3(c)], the free-induction decay shows multiple oscillations single large pseudospin with a collective spin operator as energy is coherently exchanged between the spins and pffiffiffiffi S ¼ s . Including all of these considerations, ðz;Þ i ðz;Þi the resonator through their coupling at a rate of 2g N ¼ Eq. (1) reduces to the Tavis-Cummings model, 580 kHz corresponding to the vacuum Rabi splitting of the polariton modes. † † ˆ ˆ ˆ ˆ H ¼ ω S þ ω a ˆ a ˆ þ gða ˆ S þ a ˆS Þ; ð2Þ In the strong-coupling regime, the envelope decay of the TC s Z c − þ polariton modes [Fig. 3(c), dashed black line] is faster than which is a generalization of the Jaynes-Cummings model the decay in the weak-coupling regime, as it results from a combination of spin dephasing plus resonator losses as the for a single collective pseudospin S. From this model we 031002-4 COHERENT RABI DYNAMICS OF A SUPERRADIANT SPIN … PHYS. REV. X 7, 031002 (2017) derive the equations of motion for the pseudospin-cavity system: _ ˆ ha ˆðtÞi ¼ −κha ˆðtÞi − ighS ðtÞi − iVðtÞ; ˆ ˆ ˆ hS ðtÞi ¼ −γhS ðtÞi þ 2igha ˆðtÞihS ðtÞi; − − z ˆ ˆ ˆ ˆ ˆ hS ðtÞi ¼ igðha ðtÞihS ðtÞi − haðtÞihS ðtÞi Þ: ð3Þ z − − These are the Maxwell-Bloch equations for the expectation ˆ ˆ values ha ˆi, hS i, and hS i [38]. The dissipation factors, κ − z and γ, are introduced using the standard master equation formalism for the open system [39]. We also add a classical drive term with amplitude VðtÞ set by the microwave source. To simplify these equations we take the semi- classical limit by neglecting correlations between the ˆ ˆ spin-ensemble and spin-cavity photons (ha ˆS i¼ha ˆihS i i i for i ¼þ;−;z). This approximation is valid given the large size of our spin ensemble (N ∼ 10 spins). We use Eq. (3) to simulate the dynamics of the free-induction decay (Fig. 4). In the absence of losses (κ, γ ¼ 0) the total spin of the ˆ ˆ ensemble is conserved (½H ; S ¼ 0) and the system TC evolution is a trajectory on the surface of the Bloch sphere within the superradiant subspace [Fig. 1(b)]. Dephasing from inhomogeneous broadening (T ) and cavity photon FIG. 4. Collective dynamics between the pseudospin and dissipation (κ) open up a channel for mixing with sub- cavity. (a) Cavity photon number [nðtÞ¼ha ˆ a ˆi] during the radiant states so that S is no longer preserved and the FID experiment [the square of the FID amplitude shown in system enters states in the interior of the sphere [Fig. 4(b)] Fig. 3(c)] with N ¼ 3.6 × 10 and M ≈ 0. The simulated fit [40]. In our experiment this dephasing is not refocused and according to Eq. (3) is also shown (dashed curve). The tail of the thus is irreversible, so that it can be taken as a single loss decay is magnified 40× and shown in green. (b) The pseudospin rate γ for the transverse magnetization of the spin ensemble. evolution on the Bloch sphere as determined from hS ðtÞi and The value for γ (also used in the S measurements) is 11 ˆ hS ðtÞi in this simulation. The numbers (1–4) on the curves extracted directly from the free-induction decay in the indicate equivalent points in time during the FID on the micro- weak-coupling regime where spin dephasing is the only wave emission plot (a) and the simulated pseudospin evolution contribution. Photon losses are also taken into account with plot (b). The dynamics of the pseudospin in this large N limit is a single parameter κ ¼ κ þ κ , which represents both governed by the Maxwell-Bloch equations. ext int internal loss in the cavity (κ ) and external loss through the int antenna coupling (κ ). This parameter is extracted by measured FID curve [Fig. 4(a)] with the main difference ext measuring the linewidth of the cavity resonance under between the two curves being only the experimental dead low excitation where strong-coupling effects are negligible. time (initial 3 μs). The time evolution of the pseudospin state as determined from this fit is plotted in Fig. 4(b). The 64-kHz polariton linewidth in Fig. 2(c) is a combina- The motion of the pseudospin on the Bloch sphere is tion of both κ and γ, with κ ≫ γ. formally equivalent to a damped pendulum that is kicked A simulation of the coupled system according to these into motion by the microwave excitation [20]. The pseu- equations of motion is compared to the measured curve in Fig. 4(a). Here, we plot the intensity of the microwave dospin is initially polarized to its ground state at the south † † emission [cavity photon number, n ¼ha ˆ a ˆi¼ha ˆ iha ˆiþ pole [point 1 on the Bloch sphere in Fig. 4(b)]. A micro- Oðlog NÞ derived from the microwave amplitude, which is wave pulse is applied for 200 ns (black arrow), mainly measured directly and plotted in Fig. 3(c). This procedure is populating the cavity with photons but also starting the initial excitation of the pseudospin (point 2). The driving valid for a large spin ensemble since ðlog NÞ=N ≪ 1. The microwave pulse ends, but the remaining photons in the only fitting parameter we use in this simulation is a scaling cavity continue to be transferred to the pseudospin until no parameter for the microwave pulse [VðtÞ]. This parameter accounts for the resonator coupling as well as losses in the photons remain (point 3 on the Bloch sphere). The power of microwave excitation path. The same scaling parameter is the applied microwave pulse in this example is chosen so used consistently for all microwave powers set in our that this point would be close to the equator (jS; 0i). This point (3) corresponds to a minimum (n ¼ 0) in the experiments. The simulation shows an excellent fit to the 031002-5 B. C. ROSE et al. PHYS. REV. X 7, 031002 (2017) free-induction decay [Fig. 4(a), point 3] since the photons in the resonator have been fully absorbed by the pseudo- spin. After reaching this highest point, the excited pseu- dospin will start to emit photons back into the resonator, reversing the nutation on the Bloch sphere to transition towards its ground state [point 4 in Fig. 4(b)]. During the initial excitation the pseudospin (up until point 3) evolves with the same phase as the microwave pulse, but as the pseudospin reverses direction the photons it emits have a phase opposite to that of the initial pulse. The pseudospin eventually fully deexcites to jS;−Si (point 4), releasing FIG. 5. Fourier transform (plotted as magnitude) of the donor all of the photons back into the resonator. This point spin-ensemble free-induction decay as a function of the number of unpaired spins in the ensemble (N). The black dashed line corresponds to a maximum in the free-induction decay pffiffiffiffi shows a g N dependence of the pseudospin-cavity exchange rate [Fig. 4(a), point 4]. The pseudospin then reabsorbs these as expected from the Tavis-Cummings model in the large N limit. photons and is excited to the opposite side of the Bloch This gives a single-spin coupling of g ¼ 48 mHz for a uniformly sphere (point 5), since the emitted photons are of opposite coupled ensemble. The side plot shows a vertical slice at phase to the initial excitation. This coherent exchange of 13 −3 N ¼ 3.6 × 10 cm for the maximum ensemble coupling that excitations continues until dissipation and dephasing pffiffiffiffi was achieved (2g N ¼ 580 kHz). destroy the ensemble polarization. Near the maxima and minima of the FID, where n_ ¼ 0, the pseudospin evolution is dominated by the second reached because there are no excitations left in the system derivative (in time) of the cavity photon number (n). In (n ¼ 0) to be reabsorbed by the pseudospin. the strong-coupling regime (ignoring the dissipation terms) In order to better understand the dynamics in this strong- this can be approximated as coupling regime, we vary the initial state of the pseudospin to observe its effect on the subsequent evolution. The initial 2 2 ˆ ˆ ˆ ˆ ˆ n̈ ≈ g ðhS S iþhS S iÞ þ 4g ðn þ 1=2ÞhS i: ð4Þ state of the pseudospin jS; Mi can be accurately controlled þ − − þ z both by adjusting the detuning frequency of the laser, which With this expression we can explain the shape of the FID determines the total spin S [Fig. 2(a)], and by changing near each extrema. At minima (i.e., point 3) there are no the power of the microwave pulse, which determines M. photons in the cavity (n ≈ 0) and also hS i¼ 0; therefore, z The observed dependencies on S and M are plotted in the second term in Eq. (4) is small. However, the pseudo- Figs. 5 and 6, respectively. spin is near jS; 0i, where correlations between spins lead to Varying the laser detuning allows for fine control of the superradiant emission with an emission rate that is quad- net ensemble polarization (N) and the total spin-ensemble– pffiffiffiffi ratic in the total number of unpaired spins [1], so that the cavity coupling (g N). The magnitude of the Fourier 2 2 2 ˆ ˆ first term in Eq. (4) is large, n̈ ∝ g hS S i ∝ g N =2. þ − transform of the free-induction decay (Fig. 5) reveals that pffiffiffiffi At maxima (i.e., point 4) the pseudospin is near jS;−Si the pseudospin-cavity exchange frequency scales as g N (hS i≈−N=2), where the correlations between spins in the (dashed black), in agreement with the large pseudospin ˆ ˆ ensemble are negligible and hS S i ≈ N (emission of N − þ picture. The shapes of the polariton peaks in the Fourier uncorrelated emitters). However, most of the photons have transform of the free-induction decay (side plot in Fig. 5) transferred back into the cavity (n ≈ N), so that the second are less resolved than the shapes obtained by reflection term in Eq. (4) is now large, resulting in a similarly fast spectroscopy [Fig. 2(c)]. This is due to the fact that during 2 2 n̈ ∝−g N . The pseudospin is reexcited by the large the free-induction decay a coherence, created by the initial population of photons in the resonator (n ≈ N), but dis- microwave pulse, leaks to subradiant states (S<N=2)on sipation and dephasing eventually destroy the ensemble the time scale of T . This leakage gives rise to a time- polarization (N), terminating the FID. dependent change of N during the FID and decreases the The FID we observe is different from superradiance cavity coupling of the pseudospin, effectively smearing out phenomena in free space, where the system is far from the the peaks in the Fourier transforms. This leakage does not strong-coupling regime. In free space the emitted photons occur in the reflection measurements. leave the system much faster than they can be reabsorbed The initial Z projection (M) of the pseudospin is set by the pseudospin, which means that the second term in by the tipping angle (power) of the microwave pulse. Eq. (4) can be neglected. As a result, near M ¼ −S the This tipping angle [defined as polar angle θ with respect to pseudospin evolves slowly with n̈ ∝ g N. This is a factor of jS;−Si in Fig. 1(b)] is varied through several full rotations N smaller than the value of n̈ we observe in the strong- of the pseudospin, and the resulting free-induction decays coupling regime. Additionally, in free space the pseudospin (real components) are plotted in Fig. 6 (top) accompanied emission dies out entirely after the ground state jS;−Si is by Maxwell-Bloch simulations (bottom). Rotation angles 031002-6 COHERENT RABI DYNAMICS OF A SUPERRADIANT SPIN … PHYS. REV. X 7, 031002 (2017) FIG. 6. Top: In-phase component (Reha ˆi) of the pseudospin microwave emission during the FID experiment (time axis) plotted as a function of the rotation angle of the initial microwave pulse (microwave power axis). The signal intensity is plotted using a log color scale in order to emphasize the sign of the signal. The blue (yellow) colors correspond to negative (positive) signal intensities. Bottom: The corresponding FID simulations using the Maxwell-Bloch equations [Eq. (3)]. The rotation angles (θ ¼ π, 3π, 5π) indicated on the left correspond to the inverted state of the pseudospin jS; Si. The numbered arrows (1)–(4) shown on the right correspond to FID experiments shown separately in Fig. 7. FIG. 7. Comparison of individual FID traces taken as horizontal marked as π, 3π, and 5π on the left in Fig. 6 correspond slices from Fig. 6 for selected rotation angles (θ) of the micro- to the pseudospin being fully inverted (near jS; Si) after wave pulse. The initial pseudospin state is illustrated on the Bloch photons from the microwave pulse are fully absorbed by sphere for each case (inset). (a),(b) Traces (1) and (2) correspond the pseudospin. We observe that the period of the FID to rotation angles of θ ≈ π=2 (initial pseudospin state jS; 0i) and oscillations does not depend on the rotation angle (initial θ ≈ π (initial pseudospin state jS; Si), respectively. All oscilla- pffiffiffiffi pffiffiffiffi value of M) and is always g N. However, there is an tions in trace (1) occur at a rate close to g N. In contrast, trace overall phase delay in the onset of the oscillations when (2) evolves more slowly with a long delay (t ) in the FID oscillations occurring at the first minimum where the pseudospin approaching the fully inverted points at the top of the pffiffiffiffi reaches full inversion. The rest of the FID trace oscillates at g N Bloch sphere (Fig. 6). This shift is more clearly seen when so that this initial delay shows up as an overall phase shift. Inset in comparing the FID traces in Figs. 7(a) and 7(b) for (b): A zoomed-in part of the two-dimensional FID plot from θ ≈ π=2 and θ ≈ π, respectively. Fig. 6 with a fit (white dashed curve) to t ðMÞ according to This delay is explained by considering the pseudospin Eq. (5). (c) Traces (3) and (4) correspond to slight under- or evolution, starting from jS; Mi, as an avalanche process with overrotation with respect to the fully inverted state of the pseudo- the pseudospin transitioning down the ladder of states in the spin (θ ≈ π  δ, with δ ¼ 9 deg). The phases of the microwave superradiant subspace towards its ground state [Fig. 1(b)]. emission are opposite in these two traces since the pseudospin Initially there are no photons in the cavity and the emission precession reverses direction in the underrotated case but continues rate is slow. As the pseudospin deexcites, a larger population in the same direction in the overrotated case. of photons builds up in the cavity and the pseudospin emission becomes stimulated. The total time for this process to occur is dominated by the slow time it takes to emit the first to fit the delay observed in our experiment [white dashed line few photons in the avalanche, which is longest near full in inset of Fig. 7(a)]. We assume an initial state jS; Mi with inversion [Fig. 7(b)], since the transition probability of the M ≈ S. Γ is the total spontaneous emission rate of N pseudospin is the smallest there [1]. We use the expression independent spins (including Purcell enhancement). We derived in Ref. [2], extract Γ ¼ 2.9 MHz from this fit, which is close to the rate calculated directly from the measured coupling and loss 2 1 1 2 2S t ðMÞ ≈ þ  þ ≈ log D parameters (Γ ¼ 4g N=κ ¼ 3.2 MHz). The longest delay Γ S − M þ 1 N Γ S − M þ 1 we are able to achieve is t ≈ 3.5 μs, corresponding to an 2 2 initial Z projection of M ¼ 1.77 × 10 and to the tipping ¼ log ; ð5Þ Γ ð1 þ cos θÞ angle θ ¼ 171°. The theoretical limit of the time delay for 031002-7 B. C. ROSE et al. PHYS. REV. X 7, 031002 (2017) our system, assuming a perfect θ ¼ 180°rotation, is In conclusion, we demonstrate a strongly coupled spin t ðM ¼ SÞ¼ 2 log N=Γ ¼ 21 μs. This theoretical limit is ensemble with uniform spin-cavity coupling and small difficult to achieve experimentally since near full inversion inhomogeneous broadening allowing the system to be the delay becomes sensitive to small deviations of the modeled very accurately as a single large pseudospin. rotation angle. Even though our system is very uniform, This pseudospin evolves according to the Tavis- the delay we obtain is limited by the 5% variation in cavity Cummings model in the large N limit. The use of highly coupling across the sample, the 20-mG variation of the static enriched Si gives a small inhomogeneous broadening magnietc field across the sample (δB =B ¼ 6%), and low- 0 1 (long T ) providing a wide window for viewing the frequency fluctuations in the B field (∼20 mG) over a time 0 collective pseudospin-cavity dynamics. In particular, we scale of seconds. are able to observe the pseudospin-cavity dynamics with a The oscillation delay is also explained by Eq. (4). With large number of excitations where the Holstein-Primakoff the pseudospin near M ¼ S (hS i ≈ N) and no photons in approximation is no longer valid. We can prepare an the cavity (n ≪ N), the second term is linear in N. There arbitrary initial state of the pseudospin jS; Mi through a ˆ ˆ are also no correlations between spins here (hS S i ≈ N), þ − combination of optical polarization (determining S)and so that the first term is also linear in N, giving a slow total microwave manipulation (determining M). From this we evolution (n̈ ∝ g N) as compared to the other extrema in observe the coherent exchange of energy between the pffiffiffiffi 2 2 the FID (n̈ ∝ g N ). pseudospin and the cavity at a rate g N, being indepen- The phase of the microwave emission during the free- dent of M. Control over M allows us to view, for the first induction decay shifts abruptly by π as the pseudospin time, the pseudospin dynamics as it approaches a qua- state is varied through the quasiequilibrium at full inver- siequilibrium near full inversion. In particular, we observe sion around θ ¼ π, 3π,and 5π in Fig. 6.Thisphase shiftis (and can control) a delay in the pseudospin microwave more clearly seen in Fig. 7(c) where two traces are shown emission when it is near the quasiequilibrium point. that correspond to slight underrotation (trace 3) and slight We find that this delay scales with the initial state of overrotation (trace 4) of the pseudospin relative to full the pseudospin (jS; Mi)aslog½2S=ðS − M þ 1Þ,which inversion. This phase shift is explained by looking at the is consistent with an expression derived from the evolution of the pseudospin on the Bloch sphere. When Tavis-Cummings model. We also observe that the micro- the microwave drive excites the pseudospin short of the wave emission has an abrupt π-phase shift when the fully inverted state [θ ¼ π − δ,trace 3inFig. 7(c)], just pseudospin is at the quasiequilibrium, which is explained after all of the cavity photons have been absorbed, the by considering the motion of the pseudospin on the pseudospin reverses directions on the Bloch sphere in Bloch sphere. This level of pseudospin control is unprec- order to evolve towards its ground state. The phase of the edented and is vital for quantum memory applications. emitted photons in this case has an opposite phase to the These observations offer novel validations of the Tavis- original photons from the microwave drive. On the other Cummings model and show that donors in Si are a hand, when the cavity photons oversaturate the pseudo- promising platform for implementing a spin-ensemble spin [θ ¼ π þ δ, trace 4 in Fig. 7(c)] and drive it past the quantum memory. fully inverted state, then the pseudospin will continue to Integrating a superconducting qubit inside of a 3D evolve in the same direction as it transitions towards its cavity together with the spin ensemble is an appealing ground state. In this case the emitted photons are of the next step, and similar hybrid architectures are currently same phase as the initial microwave drive—opposite to under investigation [23]. Such a hybrid implementation the phase of the pseudospin emission in the underrotated with phosphorus donor electron spins could be challeng- experiment. ing due to the large magnetic fields that are required, The quasiequilibrium we observe when the pseudospin is which could drive the Josephson junctions of the super- at full inversion is a clear indication that the full dynamics conducting qubit normal, although junctions have been of the Maxwell-Bloch equations must be included to model shown to operate successfully in fields up to 7 T [44]. our system. This is different from the more extensively One alternative is to utilize a different donor, like bismuth, studied low-excitation limit where the pseudospin is with a large enough zero-field splitting to enable low-field always near M ¼ −S, and to good approximation S can measurements [45]. be considered to be a constant of motion (Holstein- Primakoff approximation) [26,41–43], With this approxi- This work was supported by the NSF and EPSRC mation the equations of motion are analogous to a system through the Materials World Network and NSF MRSEC of two coupled linear oscillators and cannot reproduce the Programs (Grants No. DMR-1107606, No. EP/I035536/1, behavior we observe near full inversion. In the low- and No. DMR-01420541), and the ARO (Grant excitation limit, the maxima and minima of the FID both No. W911NF-13-1-0179). M. L. W. T. was supported by evolve with n̈ ∝ g N, which is much slower than the the Natural Sciences and Engineering Research Council of 2 2 n ∝ g N behavior we observe here. Canada (NSERC). The work at Keio is supported by 031002-8 COHERENT RABI DYNAMICS OF A SUPERRADIANT SPIN … PHYS. REV. X 7, 031002 (2017) [17] A. M. Tyryshkin, S. Tojo, J. J. L. Morton, H. Riemann, N. V. KAKENHI (S) No. 26220602 and JSPS Core-to-Core Abrosimov, P. Becker, H.-J. Pohl, T. Schenkel, M. L. W. Program. The authors extend special thanks to Hakan Thewalt, K. M. Itoh, and S. A. Lyon, Electron Spin Coher- Türeci and Jonathan Keeling for helpful discussions of ence Exceeding Seconds in High-Purity Silicon, Nat. Mater. the Tavis-Cummings model. 11, 143 (2012). [18] P. Becker, H.-J. Pohl, H. Riemann, and N. Abrosimov, Enrichment of Silicon for a Better Kilogram, Phys. Status Solidi A 207, 49 (2010). [19] M. Tavis and F. W. Cummings, Exact Solution for an [1] R. H. 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