Coherent Rabi dynamics of a superradiant spin ensemble in a microwave cavity

TL;DR

This paper demonstrates strong coupling and coherent Rabi oscillations in a superradiant spin ensemble within a microwave cavity, revealing detailed quantum dynamics and control of collective spin states.

Contribution

It reports the first observation of superradiant Rabi dynamics in a large, optically polarized spin ensemble coupled to a microwave cavity, with detailed control and modeling.

Findings

Normal mode splitting of 580 kHz observed

Long spin dephasing time T2* = 9 μs

Clear collapses and revivals in free induction decay

Abstract

We achieve the strong coupling regime between an ensemble of phosphorus donor spins in a highly enriched $^{28}$Si crystal and a 3D dielectric resonator. Spins were polarized beyond Boltzmann equilibrium using spin selective optical excitation of the no-phonon bound exciton transition resulting in $N$ = $3.6\cdot10^{13}$ unpaired spins in the ensemble. We observed a normal mode splitting of the spin ensemble-cavity polariton resonances of 2$g\sqrt{N}$ = 580 kHz (where each spin is coupled with strength $g$) in a cavity with a quality factor of 75,000 ($\gamma \ll \kappa \approx$ 60 kHz where $\gamma$ and $\kappa$ are the spin dephasing and cavity loss rates, respectively). The spin ensemble has a long dephasing time (T$_2^*$ = 9 $\mu$s) 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 cavity at the rate $g\sqrt{N}$. 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 $\pi$ 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.

Equations

$$\begin{array}{rl}\hat{H}=\sum _i{\omega }_i{\hat{s}}_{Zi}& +{\omega }_c{\hat{a}}^{†}\hat{a}+2\sum _i{g}_i({\hat{a}}^{†}+\hat{a}){\hat{s}}_{Xi}\\ & +\sum _{<ij>}{\hat{\mathbf{s}}}_{\mathbf{i}}\cdot {\hat{\mathbf{D}}}_{\mathbf{ij}}\cdot {\hat{\mathbf{s}}}_{\mathbf{j}}\end{array}$$

$${\hat{H}}_{TC}={\omega }_s{\hat{S}}_Z+{\omega }_c{\hat{a}}^{†}\hat{a}+g({\hat{a}}^{†}{\hat{S}}_{-}+\hat{a}{\hat{S}}_{+})$$

$$\begin{array}{rl}& ⟨\dot{\hat{a}}(t)⟩=-\kappa ⟨\hat{a}(t)⟩-ig⟨{\hat{S}}_{-}(t)⟩-iV(t)\\ & ⟨{\dot{\hat{S}}}_{-}(t)⟩=-\gamma ⟨{\hat{S}}_z(t)⟩+2ig⟨\hat{a}(t)⟩⟨{\hat{S}}_z(t)⟩\\ & ⟨{\dot{\hat{S}}}_z(t)⟩=ig\left(⟨a^{†}(t)⟩⟨S_{-}(t)⟩-⟨a(t)⟩⟨S_{-}(t){⟩}^{\ast }\right)\end{array}$$

$$\ddot{n}\approx g^2\left(⟨{\hat{S}}_{+}{\hat{S}}_{-}⟩+⟨{\hat{S}}_{-}{\hat{S}}_{+}⟩\right)+4g^2\left(n+1/2\right)⟨{\hat{S}}_z⟩$$

$$\begin{array}{rl}{t}_D(M)& \approx \frac{2}{\Gamma }\left[\frac{1}{S-M}+\dots +\frac{1}{N}\right]\approx \frac{2}{\Gamma }\log \frac{2S}{S-M}\\ & =\frac{2}{\Gamma }\log \frac{2}{(1+\cos \theta )},\end{array}$$

Full text

The enhanced collective emission from an ensemble of atom-like systems due to coherent self-stimulated emission was originally described by DickeDicke (1954). For this phenomenon he coined the term superradiance and showed that the atom ensemble can behave as a large collective pseudospin. Superradiant emission has been observed in numerous physical systems.Gross and Haroche (1982); Haroche and Raimond (1985); Skribanowitz et al. (1973); Gross et al. (1976); Scheibner et al. (2007); Chalupczak and Josephs-Franks (2015) These collective effects are particularily prominent when the coupling between the spin-ensemble and the radiation field ($g\sqrt{N}$ for $N$ spins individually coupled with strength $g$) is larger than any of the losses in the system ($\kappa+\gamma$ where $\kappa$ is the radiative loss rate and $\gamma$ is the spin dephasing rate). This strong coupling regime has been extensively studied in both theory and experiment,Zhu et al. (1990a); Hettich et al. (2002); Baumann et al. (2010); Diniz et al. (2011) but clearly resolved dynamics of these collective effects are generally lacking in large ensembles due to strong dephasing (short T${}_{2}^{*}=2\pi/\gamma$).Mielke et al. (1997); Brune et al. (1996); Gross and Haroche (1982) Here we demonstrate the dynamics of a strongly coupled ensemble of phosphorus donor spins in highly isotopically enriched 28Si with both a long dephasing timeTyryshkin et al. (2012); Becker et al. (2010) and uniform coupling to the radiation field (due to the use of a 3D microwave cavity) as shown schematically in Fig. 1A. For the first time outside of ensembles of Rydberg atomsBrune et al. (1996); Mielke et al. (1997), we have studied a spin ensemble-cavity system with both of these essential properties allowing for it to be modeled accurately as a single large pseudospin (Fig. 1B) offering a simple interpretation of the spin ensemble-cavity evolution.Dicke (1954); Tavis and Cummings (1968) In particular, we are able to directly observe the dynamics of superradiant emissionKaluzny et al. (1983) under strong excitation. In contrast to the more extensively studied low excitation limit, with strong excitation the system cannot be modeled as a linear system of two coupled harmonic oscillators.Thompson et al. (1992); Zhu et al. (1990b) Instead it must be treated with the Tavis-Cummings model.Tavis and Cummings (1968)

Initialization of the pseudospin was accomplished with a combination of optical excitation which sets the size of the pseudospin, $S=N/2$, and microwave manipulation which sets the initial z-projection of the pseudospin, $M$. Previous reports of large superradiant ensembles only prepared an initially inverted state $\mid S,S\rangle$,Mielke et al. (1997); Brune et al. (1996) while with our system we are able to prepare any initial pseudospin state $\mid S,M\rangle$ in the superradiant subspace (Fig. 1C) with independent control of S and M. By controlling the size of the pseudospin we demonstrate a $g\sqrt{N}$ 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 emissionHaake et al. (1981) and we report the first experimental observation of a log dependence of this delay when the pseudospin is near quasi-equilibrium at $M=S$. This log dependence is consistent with predictions from the Tavis-Cummings model. We also observe an abrupt $\pi$ phase shift in the pseudospin microwave emission around this fully inverted state $M=S$.

The isotopically enriched silicon crystal ($<$50 ppm 29Si) used in these experiments is phosphorus doped with a density of 3.3$\cdot$1015 cm*-3* (5.7$\cdot$1013 total donors) and is otherwise highly pure (boron density less than 1014 cm*-3*).Becker et al. (2010) A tunable distributed Bragg reflector (DBR) laser (Eagleyard EYP-DBR-1080) was used to controllably polarize the phosphorus donor spin ensemble beyond Boltzmann equilibrium by spin-selective optical pumping of the phosphorus donor no-phonon bound exciton transitions.Yang et al. (2009) The efficiency of the optical pumping and the resulting steady state spin polarization ($N=N_{\uparrow}-N_{\downarrow}$) was controlled by detuning the laser from resonance with one of the no-phonon bound exciton transitions (Fig. 2A). The laser was tunable between 1077-1081nm (277.2-278.5 THz) and fiber coupled to deliver $\sim$10 mW of light to the sample. More than 95$\%$ electron spin polarization was achieved after 300 ms of resonant optical pumping (Fig. 2A). The 31P nuclei were also polarized to $\approx 25\%$ during the optical pumping. The mechanism behind this nuclear polarization remains unknown but it is thought to be due to an enhanced cross relaxation rate of the donors under illumination.Gumann et al. (2014) X-band (9.6 GHz) ESR experiments were performed with a Bruker ESR spectrometer (Elexsys E580) using a dielectric 3D resonator (ER-4118X-MD5) in a helium-flow cryostat (Oxford CF935). In the free induction decay experiments a single 200 ns microwave pulse was used to tip the spins after polarizing with the DBR laser (Fig. 3A). The tipping angle of the microwave pulse was controlled by varying the power of the microwave source (Agilent E8267D).

The stationary states of the system were interrogated by measuring reflected microwave power ($S_{11}$) with an Agilent E5071C network analyzer (Figs. 2B-C). Power saturation was avoided by lowering the probe power to -65 dBm (power incident on the resonator); below this power the measured vacuum Rabi splitting was constant and no power broadening was observed. The $S_{11}$ measurements were performed under continuous illumination from the resonant DBR laser which in addition to polarizing the spin ensemble accelerates spin relaxation (T1) and reduces power saturation from the microwave probe.

The total ensemble coupling can be determined directly by looking at the eigenfrequencies of the coupled system while the spin ensemble is tuned into resonance with the cavity (Fig. 2B). This technique has been demonstrated for spin ensembles coupled to both 3D volume resonators Abe et al. (2011) and 2D superconducting microresonatorsSchuster et al. (2010). The spin transition frequency ($\omega_{s}$) is varied through resonance with the cavity ($\omega_{c}$) by changing the Zeeman splitting with a magnetic field (B0) which resulted in an avoided crossing with a clear splitting showing that the system is in the strong coupling regime. In particular, with the laser tuned on resonance ($N$ = 3.6$\cdot 10^{13}$) the vacuum Rabi splitting of the polariton modes is 580 kHz ($2g\sqrt{N}$) which is an order of magnitude larger than their 64 kHz linewidth (Fig. 2C, curve 2). The splitting can be controllably reduced by detuning the laser from resonance with the bound exciton transition (curve 1 in Fig. 2C). The large size of the ensemble is important to achieve this large splitting since each spin is weakly coupled to the 3D cavity with a single spin coupling $g$ = 38 mHz. The narrow linewidth of the resonances is defined by combined losses in the cavity ($\kappa$ = 60 kHz for Q = 75,000) and dephasing in the spin ensemble ($\gamma$ = 18 kHz). The ensemble is in the high cooperativity limit ($C=g^{2}N/\kappa\gamma\approx 280$) where energy can be exchanged efficiently between the spins and the cavity.

The free evolution of the pseudospin in a cavity is most directly studied in a single pulse free induction decay (FID) experiment (Fig. 3A). The pseudospin is first polarized into its ground state ($M=-S$) with resonant laser pumping thus defining $N=N_{\uparrow}-N_{\downarrow}$ and $S=N/2$. This is followed by a single microwave pulse that determines the initial state (M) of the pseudospin before the FID. In the weak coupling regime the envelope of the free induction decay is a simple exponential (Fig. 3A) with a characteristic decay time resulting from the inhomogeneous broadening of the spin ensemble (T${}_{2}^{*}$). However, in the strong coupling limit achieved here (Fig. 3B-C) the free induction decay shows multiple oscillations as energy is coherently exchanged between the spins and the resonator through their coupling at a rate of $2g\sqrt{N}=$ 580 kHz corresponding to the vacuum Rabi splitting of the polariton modes.

In the strong coupling regime, the envelope decay of the polariton modes (Fig. 3C, dashed black line) is faster than the decay in the weak coupling regime as it results from a combination of spin dephasing plus resonator losses as the excitations are exchanged between the spin ensemble and the cavity over time. Spin relaxation (T1) of phosphorus donors at 1.5 K is slow and does not enter into the dynamics of the free induction decay.Tyryshkin et al. (2012) Even with this faster decay we are able to observe 12 collapses and revivals before the free induction decay falls below the noise level. An experimental dead time of 3 $\mu$s after the microwave pulse limits us from measuring the beginning of the FID signal. The large signal to noise and many oscillations observed here offer clearly resolved dynamics of the coupled spin-ensemble/cavity evolution.

A theoretical description of this experiment begins with a general model for an ensemble of $N$ spin 1/2 particles interacting with a single cavity mode:

[TABLE]

where $\hat{a}^{\dagger}$, $\hat{a}$ are the creation and annihilation operators for the cavity field photons of frequency $\omega_{c}$, $s_{ki}$ ($i$ = 1,…,$N$ and $k$ = X,Y,Z) are the single spin ($S=1/2$) matrices for a spin with transition frequency $\omega_{i}$ and spin-cavity coupling $g_{i}$. The last term ($\bf{\hat{D}}_{ij}$) describes dipolar interactions between spins.

Direct diagonalization of this Hamiltonian for large spin ensembles is not possible, however several simplifying approximations can be made to good accuracy. The spin ensemble has a narrow inhomogeneous distribution in Zeeman frequencies compared to the spin-cavity coupling ($g\sqrt{N}$T${}_{2}^{*}$ $\gg 1$) so that, for the duration of our FID experiment, we can treat the spins as having identical transition frequencies $\omega_{i}\approx$ $\omega_{s}$. The distribution in the individual spin-cavity coupling ($g$) is mostly defined by the microwave magnetic field inhomogeneity along the length of the sample (5 mm). For our volume resonator this variation has been measured to be less than 5$\%$.Morton et al. (2005) Thus to a good approximation the spin ensemble-cavity system is in the small sample limit with a uniform spin-cavity coupling ($g_{i}\approx g$).Raimond et al. (1982); Tavis and Cummings (1968) Finally, the dipolar coupling between spins (last term in Eq. 1) at the 3.3$\cdot$1015 cm*-3* donor density used in this experiment is $\sim$100 Hz and negligible on the 30 $\mu$s timescale measured here.Witzel et al. (2012) With both the Zeeman frequency and the cavity coupling being constant across all of the spins, the ensemble can be treated as a single large pseudospin with a collective spin operator $\hat{S}_{(z,\pm)}=\sum\limits_{i}\hat{s}_{(z,\pm)i}$. Including all of these considerations, Eq. 1 reduces to the Tavis-Cummings model:

[TABLE]

which is a generalization of the Jaynes-Cummings model for a single collective pseudospin $\hat{S}$. From this model we derive the equations of motion for the pseudospin-cavity system:

[TABLE]

These are the Maxwell-Bloch equations for the expectation values $\langle\hat{a}\rangle$, $\langle S_{-}\rangle$, and $\langle S_{z}\rangle$.Arecchi and Bonifacio (1965) The dissipation factors, $\kappa$ and $\gamma$, were introduced using the standard master equation formalism for the open system.Tarasov (2008) We have also added a classical drive term with amplitude $V(t)$ set by the microwave source. To simplify these equations we take the semiclassical limit by neglecting correlations between the spin ensemble and cavity photons ($\langle aS_{i}\rangle=\langle a\rangle\langle S_{i}\rangle$ for $i=+,-,z$). This approximation is valid given the large size of our spin ensemble ($N\sim 10^{13}$ spins). We use Eq. 3 to simulate the dynamics of the free induction decay (Fig. 4).

In the absence of dephasing ($\gamma=0$) the total spin of the ensemble is conserved ($\left[\hat{H}_{TC},\hat{S}^{2}\right]=0$) and the system evolution is a trajectory on the surface of the Bloch sphere within the superradiant subspace (Fig. 1C). Dephasing from inhomogeneous broadening (T${}_{2}^{*}$) opens up a channel for mixing with subradiant states so that $S$ is no longer preserved and the system enters states in the interior of the sphere (Fig. 4B).Temnov and Woggon (2005) In our experiment this dephasing is not refocused and thus is irreversible, so that it can be taken as a single loss rate, $\gamma$, for the transverse magnetization of the spin ensemble. The value for $\gamma$ (also used in the $S_{11}$ measurements) was extracted directly from the free induction decay in the weak coupling regime where spin dephasing is the only contribution. Photon losses are also taken into account with a single parameter $\kappa=\kappa_{ext}+\kappa_{int}$ which represents both internal loss in the cavity ($\kappa_{int}$) and external loss through the antenna coupling ($\kappa_{ext}$). This parameter was extracted by measuring the linewidth of the cavity resonance under low excitation where strong coupling effects were negligible. The 64 kHz polariton linewidth in Fig. 2C is a combination of both $\kappa$ and $\gamma$, with $\kappa\gg\gamma$.

A simulation of the coupled system according to these equations of motion is compared to the measured curve in Figure 4A. Here we plot the intensity of the microwave emission (cavity photon number, $n=\langle\hat{a}^{\dagger}\hat{a}\rangle$) derived from the microwave amplitude which was measured directly and plotted in Fig. 3C. The only fitting parameter used in this simulation was a scaling parameter for the microwave pulse ($V(t)$). This parameter accounts for the resonator coupling as well as losses in the microwave excitation path. The same scaling parameter was used consistently for all microwave powers set in our experiments. The simulation shows an excellent fit to the measured FID curve (Fig. 4A) with the main difference between the two curves being only the experimental dead time (initial 3 $\mu$s). The time evolution of the pseudospin state as determined from this fit is plotted in Figure 4B.

The motion of the pseudospin on the Bloch sphere is formally equivalent to a damped pendulum that is kicked into motion by the microwave excitation.Kaluzny et al. (1983) The pseudospin is initially polarized to its ground state at the south pole (point 1 on the Bloch sphere in Fig. 4B). A microwave pulse is applied for 200 ns (black arrow), mainly populating the cavity with photons but also starting the initial excitation of the pseudospin (point 2). The driving microwave pulse ends but the remaining photons in the cavity continue to be transfered to the pseudospin until no photons remain (point 3 on the Bloch sphere). The power of the applied microwave pulse in this example was chosen so that this point would be close to the equator ($\mid S,0\rangle$). This point (3) corresponds to a minimum ($n=0$) in the free induction decay (Fig. 4A, point 3) since the photons in the resonator have been fully absorbed by the pseudospin. After reaching this highest point, the excited pseudospin 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. 4B). 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 de-excites to $\mid S,-S\rangle$ (point 4), releasing all of the photons back into the resonator. This point corresponds to a maximum in the free induction decay (Fig. 4A, point 4). The pseudospin then reabsorbs these photons and is excited to the opposite side of the Bloch sphere (point 5), since the emitted photons are of opposite phase to the initial excitation. This coherent exchange of excitations continues until dissipation and dephasing destroy the ensemble polarization.

Near the maxima and minima of the FID, where $\dot{n}=0$, the pseudospin evolution is dominated by the second derivative (in time) of the cavity photon number ($\ddot{n}$). In the strong coupling regime (ignoring the dissipation terms) this can be approximated as:

[TABLE]

With this expression we can explain the shape of the FID near each extrema. At minima (i.e. point 3) there are no photons in the cavity ($n\approx 0$) and also $\langle\hat{S}_{z}\rangle=0$, therefore the second term in Eq. 4 is small. However, the pseudospin is near $\mid S,0\rangle$ where correlations between spins lead to superradiant emission with an emission rate that is quadratic in the total number of unpaired spinsDicke (1954) so that the first term in Eq. 4 is large, $\ddot{n}\propto g^{2}\langle\hat{S}_{+}\hat{S}_{-}\rangle\propto g^{2}N^{2}/2$. At maxima (i.e. point 4) the pseudospin is near $\mid S,-S\rangle$ ($\langle\hat{S}_{z}\rangle\approx-N/2$) where the correlations between spins in the ensemble are negligible and $\langle\hat{S}_{-}\hat{S}_{+}\rangle\approx N$ (similar to spontaneous emission of N uncorrelated emitters). However, most of the photons have transfered back into the cavity ($n\approx N$) so that the second term in Eq. 4 is now large, resulting in a similarly fast $\ddot{n}\propto-g^{2}N^{2}$. The pseudospin is re-excited by the large population of photons in the resonator ($n\approx N$), but dissipation and dephasing eventually destroy the ensemble polarization ($N$), terminating the FID.

The FID we observed is different from superradiance phenomena in free space where the system is far from the strong coupling regime. In free space the emitted photons leave the system much faster than they can be reabsorbed by the pseudospin which means that the second term in Eq. 4 can be neglected. As a result, near $M=-S$ the pseudospin evolves slowly with $\ddot{n}\propto g^{2}N$. This is a factor of N smaller than the value of $\ddot{n}$ we observe in the strong coupling regime. Additionally, in free space the pseudospin emission dies out entirely after the ground state $\mid S,-S\rangle$ is reached because there are no excitations left in the system ($n=0$) to be reabsorbed by the pseudospin.

In order to better understand the dynamics in this strong coupling regime we varied the initial state of the pseudospin to observe its effect on the subsequent evolution. The initial state of the pseudospin, $\mid S,M\rangle$, can be accurately controlled both by adjusting the detuning frequency of the laser which determines the total spin $S$ (Fig. 2A), and by changing the power of the microwave pulse, which determines $M$. The observed dependencies on $S$ and $M$ are plotted in Figs. 5 and 6, respectively.

Varying the laser detuning allows for fine control of the net ensemble polarization ($N$) and the total spin ensemble-cavity coupling ($g\sqrt{N}$). The magnitude of the Fourier transform of the free induction decay (Fig. 5) reveals that the pseudospin-cavity exchange frequency scales as $g\sqrt{N}$ (dashed black) in agreement with the large pseudospin picture. The shapes of the polariton peaks in the Fourier transform of the free induction decay (side plot in Fig. 5) are less resolved than the shapes obtained by reflection spectroscopy (Fig. 2C). This is due to the fact that during the free induction decay a coherence, created by the initial microwave pulse, leaks to subradiant states ($S<N/2$) on the timescale of $T_{2}^{*}$. This leakage gives rise to a time dependent change of $N$ during the FID and decreases the cavity coupling of the pseudospin, effectively smearing out the peaks in the Fourier transforms. This leakage does not occur in the reflection measurements.

The initial z-projection ($M$) of the pseudospin is set by the tipping angle (power) of the microwave pulse. This tipping angle (defined as polar angle $\theta$ with respect to $\mid S,-S\rangle$ in Fig. 1C) was varied through several full rotations of the pseudospin and the resulting free induction decays (real components) are plotted in Fig. 6 (top) accompanied by Maxwell-Bloch simulations (bottom). Rotation angles marked as $\pi$, 3$\pi$, and 5$\pi$ on the left in Fig. 6 correspond to the pseudospin being fully inverted (near $\mid S,S\rangle$) after photons from the microwave pulse are fully absorbed by the pseudospin. We observe that the period of the FID oscillations does not depend on the rotation angle (initial value of $M$) and is always $g\sqrt{N}$. However there is an overall phase delay in the onset of the oscillations when approaching the fully inverted points at the top of the Bloch sphere (Fig. 6). This shift is more clearly seen when comparing the FID traces in Fig. 7A and B for $\theta\approx\pi/2$ and $\theta\approx\pi$, respectively.

This delay is explained by considering the pseudospin evolution, starting from $\mid S,M\rangle$, as an avalanche process with the pseudospin transitioning down the ladder of states in the superradiant subspace towards its ground state (Fig. 1C). Initially there are no photons in the cavity and the emission is spontaneous. As the pseudospin de-excites, a larger population 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 spontaneously emit the first few photons in the avalanche which is longest near full inversion (Fig. 7B), since the transition probability of the pseudospin is the smallest there.Dicke (1954) We use the expression derived in Gross and Haroche (1982):

[TABLE]

to fit the delay observed in our experiment (white dashed line in inset of Fig. 7A). We assume an initial state $\mid S,M\rangle$ with $M\approx S$. $\Gamma$ is the total spontaneous emission rate from $N$ independent spins. We extract $\Gamma=1.0$ MHz from this fit. The longest delay we were able to achieve was $t_{D}\approx 3.5~{}\mu$s corresponding to an initial Z projection of $M=8.8\cdot 10^{12}$ and to the tipping angle $\theta=177^{\circ}$. The deviation from $180^{\circ}$ was determined by the resolution of the microwave power sweep experiment in Fig. 6.

The oscillation delay is also explained by Eq. 4. With the pseudospin near $M=S$ ($\langle\hat{S}_{z}\rangle\approx N$) and no photons in the cavity ($n\ll N$), the second term is linear in $N$. There are also no correlations between spins here ($\langle\hat{S}_{+}\hat{S}_{-}\rangle\approx N$) so that the first term is also linear in $N$ giving a slow total evolution ($\ddot{n}\propto g^{2}N$) as compared to the other extrema in the FID ($\ddot{n}\propto g^{2}N^{2}$).

The phase of the microwave emission during the free induction decay shifts abruptly by $\pi$ as the pseudospin state is varied through the quasi-equilibrium at full inversion around $\theta=\pi$, 3$\pi$, and 5$\pi$ in Fig. 6. This phase shift is more clearly seen in Fig. 7C where two traces are shown that correspond to slight underrotation (trace 3) and slight overrotation (trace 4) of the pseudospin relative to full inversion. This phase shift is explained by looking at the evolution of the pseudospin on the Bloch sphere. When the microwave drive excites the pseudospin short of the fully inverted state ($\theta=\pi-\delta$, trace 3 in Fig. 7C), just after all of the cavity photons have been absorbed, the pseudospin reverses directions on the Bloch sphere in order to evolve towards its ground state. The phase of the emitted photons in this case have an opposite phase to the original photons from the microwave drive. On the other hand, when the cavity photons oversaturate the pseudospin ($\theta=\pi+\delta$, trace 4 in Fig. 7C) and drive it past the fully inverted state, then the pseudospin will continue to evolve in the same direction as it transitions towards its ground state. In this case the emitted photons are of the same phase as the initial microwave drive - opposite to the phase of the pseudospin emission in the under-rotated experiment.

The quasi-equilibrium we observe when the pseudospin is at full inversion is a clear indication that the full dynamics of the Maxwell-Bloch equations must be included to model our system. This is different from the more extensively studied low excitation limit where the pseudospin is always near $M=-S$ and to good approximation $\hat{S}_{z}$ can be considered to be a constant of motion (Holstein-Primakoff approximation). Primakoff and Holstein (1939); Raizen et al. (1989); Childs et al. (1996); Putz et al. (2014) With this approximation the equations of motion are analogous to a system of two coupled linear oscillators and cannot reproduce the behavior we observe near full inversion. In the low excitation limit, the maxima and minima of the FID both evolve with $\ddot{n}\propto g^{2}N$ which is much slower than the $\ddot{n}\propto g^{2}N^{2}$ behavior we observe here.

In conclusion, we have demonstrated a strongly coupled spin ensemble with uniform spin-cavity coupling and small inhomogeneous broadening allowing the system to be modeled very accurately as a single large pseudospin. This pseudospin evolves according to the Tavis-Cummings model in the large $N$ limit. The use of highly enriched 28Si gives a small inhomogeneous broadening (long T${}_{2}^{*}$) providing a wide window for viewing the collective pseudospin-cavity dynamics. In particular we are able to observe the pseudospin-cavity dynamics with a large number of excitations where the Holstein-Primakoff approximation is no longer valid. We can prepare an arbitrary initial state of the pseudospin $\mid S,M\rangle$ through a combination of optical polarization (determining $S$) and microwave manipulation (determining $M$). From this we observe the coherent exchange of energy between the pseudospin and the cavity at a rate $g\sqrt{N}$, being independent of M. Control over M allows us to view, for the first time, the pseudospin dynamics as it approaches a quasi-equilibrium near full inversion. In particular we observe (and can control) a delay in the pseudospin microwave emission when it is near the quasi-equilibrium point. We find that this delay scales with the initial state of the pseudospin ($\mid S,M\rangle$) as $\log{\frac{2S}{S-M}}$ which is consistent with an expression derived from the Tavis-Cummings model. We also observe that the microwave emission has an abrupt $\pi$ phase shift when the pseudospin is at the quasi-equilibrium which is explained by considering the motion of the pseudospin on the Bloch sphere. These observations offer novel validations of the Tavis-Cummings model.

I Acknowledgments

This work was supported by the NSF and EPSRC through the Materials World Network and NSF MRSEC Programs (Grant No. DMR-1107606, EP/I035536/1, and DMR-01420541), and the ARO (Grant No. W911NF-13-1-0179). MLWT was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The work at keio is supported by KAKENHI (S) No. 26220602 and JSPS Core-to-Core Program. The authors extend special thanks to Hakan Türeci and Jonathan Keeling for helpful discussions of the Tavis-Cummings model.