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

We achieve the strong-coupling regime between an ensemble of phosphorus donor spins in a highly enriched ${}^{28}\mathrm{Si}$ 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 $N=3.6\times 10^{13}$ unpaired spins in the ensemble. We observe a normal mode splitting of the spin-ensemble–cavity polariton resonances of $2g\sqrt{N}=580\mathrm{kHz}$ (where each spin is coupled with strength $g$) in a cavity with a quality factor of 75 000 ($\gamma \ll \kappa \approx 60\mathrm{kHz}$, where $\gamma$ and $\kappa$ are the spin dephasing and cavity loss rates, respectively). The spin ensemble has a long dephasing time ($T_2^{*}=9\mu \mathrm{s}$) 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.

Popular Summary

Electrons behave like microscopic magnets and can interact with one another via their magnetic fields. Unlike large magnets, however, the magnetic field of an electron can couple to microwaves, and electrons can also interact with one another through this collective coupling. Robert Dicke first studied the collective emission from an ensemble of microscopic magnets (usually called “spins”) through their common interaction with microwaves, typically referred to as superradiance, in the 1950s. This pioneering work demonstrated that an ensemble of electron spins can behave as a large collective pseudospin. These collective effects are particularly prominent when the spin ensemble is strongly coupled to the microwave field such that the interaction can occur before the energy is lost. Since the 1950s, several investigations have focused on superradiant effects in the strong coupling regime, but measurements of clearly resolved dynamics are lacking in the case of large ensembles. Here, we achieve strong coupling of a large ensemble of nearly identical spins that are uniformly coupled to one another through a common microwave field.

We focus on phosphorus donor spins in silicon, which have a long dephasing time, coupled to a three-dimensional cavity that provides uniform coupling across the ensemble. These essential properties allow the dynamics in our system to be interpreted directly as the motion of a single collective pseudospin. In particular, the spin decoherence rate is less than 1 Hz, significantly less than that of other candidate spin systems, making it an ideal memory for storing quantum information. We realize precise control over the initial state of the pseudospin and the resulting superradiant emission, which gives rise to novel experimental validations of the pseudospin model.

Such a highly correlated system is promising for studying entanglement in large quantum systems and is of wide interest in the fields of quantum computing and cavity quantum electrodynamics.

Figure 1

(a) Experimental scheme for delivering microwave and optical excitation to the phosphorus donor spin ensemble (${\widehat{s}}_i$, $i=1,\dots ,N_{\uparrow }+N_{\downarrow }$) positioned inside a cylindrical dielectric (sapphire) microwave cavity within a helium cryostat ($T=1.5\mathrm{K}$). The spins are placed in a $\sim 3500$-G dc magnetic field (${\overrightarrow{B}}_0$) and coupled magnetically to the cylindrical TE011 cavity mode (microwave magnetic field, ${\overrightarrow{B}}_{\mathrm{mw}}$) with individual spin-cavity coupling $g_i$ ($i=1,\dots ,N_{\uparrow }+N_{\downarrow }$). Current losses in the copper walls (${\kappa }_{\mathrm{int}}$) and coupling losses (${\kappa }_{\mathrm{ext}}$) through the antenna along with dephasing in the spin ensemble ($\gamma =1/T_2^{*}$) set the maximum window during which the spin-cavity dynamics can be probed. (b) In the case of small inhomogeneous broadening (${\omega }_i=\omega$) and uniform cavity coupling ($g_i=g$), the spin ensemble can be modeled as a single large pseudospin as shown in the Bloch sphere representation. The sphere surface represents the symmetric superradiant subspace ($|S,M⟩$). Strong correlations between spins in these states lead to an enhanced photon emission rate $\mathrm{\Gamma }\propto N^2$ near the equator ($M=0$). Uncorrelated spontaneous emission at a rate $\mathrm{\Gamma }\propto N$ dominates near the poles ($M=\pm S$).

Figure 2

(a) The number of spins interacting with the resonator ($N$) is controlled by tuning the laser frequency which is spin-selectively exciting a phosphorus donor no-phonon bound exciton transition [28]. (b) Reflected microwave power $S_{11}$ (color scale) spectrum of the cavity as a function of the spin-ensemble Zeeman splitting (${\omega }_s=g{\mu }_B{B}_0$) as the spin ensemble is brought into resonance with the bare cavity (${\omega }_c$). The clear avoided crossing shows that the system is in the strong-coupling regime with $2g\sqrt{N}$ (580 kHz) $\gg \kappa +\gamma$ (64 kHz). (c) The vacuum Rabi splitting ($2g\sqrt{N}$) of the polariton modes (shown at ${\omega }_s={\omega }_c$) can be controlled by detuning the laser frequency. Curves 1 and 2 were measured while the laser frequency was tuned to points 1 and 2 as indicated in (a).

Figure 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 $S=N/2$; the subsequent microwave pulse excites the pseudospin to $|S=N/2,M⟩$. The microwave emission from the pseudospin is recorded after the microwave pulse and a $3\text{−}\mu \mathrm{s}$ experimental dead time. (b) In-phase $\mathrm{Re}(\widehat{a})$ and quadrature $\mathrm{Im}(\widehat{a})$ components of the pseudospin microwave emission during the free-induction decay in the strong-coupling regime ($M\approx 0$, $N=3.6\times 10^{13}$). (c) Amplitude of the microwave emission $|\widehat{a}|$ during the FID (black solid line). The FID shows oscillations at rate $g\sqrt{N}=290\mathrm{kHz}$, many coherent cycles of energy exchange between the spin ensemble and the cavity are resolved. For comparison, the purely exponential decay in the weak-coupling regime is also shown (dashed red line) with $g\sqrt{N}=56\mathrm{kHz}$, $\kappa =960\mathrm{kHz}$. The decays are normalized to have the same magnitude at $t=0$.

Figure 4

Collective dynamics between the pseudospin and cavity. (a) Cavity photon number [$n(t)=⟨{\widehat{a}}^{†}\widehat{a}⟩$] during the FID experiment [the square of the FID amplitude shown in Fig. 3] with $N=3.6\times 10^{13}$ and $M\approx 0$. The simulated fit according to Eq. (3) is also shown (dashed curve). The tail of the decay is magnified $40\times$ and shown in green. (b) The pseudospin evolution on the Bloch sphere as determined from $⟨{\widehat{S}}_{-}(t)⟩$ and $⟨{\widehat{S}}_z(t)⟩$ in this simulation. The numbers (1–4) on the curves indicate equivalent points in time during the FID on the microwave emission plot (a) and the simulated pseudospin evolution plot (b). The dynamics of the pseudospin in this large $N$ limit is governed by the Maxwell-Bloch equations.

Figure 5

Fourier transform (plotted as magnitude) of the donor spin-ensemble free-induction decay as a function of the number of unpaired spins in the ensemble ($N$). The black dashed line shows a $g\sqrt{N}$ dependence of the pseudospin-cavity exchange rate as expected from the Tavis-Cummings model in the large $N$ limit. This gives a single-spin coupling of $g=48\mathrm{mHz}$ for a uniformly coupled ensemble. The side plot shows a vertical slice at $N=3.6\times 10^{13}{\mathrm{cm}}^{-3}$ for the maximum ensemble coupling that was achieved ($2g\sqrt{N}=580\mathrm{kHz}$).

Figure 6

Top: In-phase component ($\mathrm{Re}⟨\widehat{a}⟩$) 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 ($\theta =\pi$, $3\pi$, $5\pi$) indicated on the left correspond to the inverted state of the pseudospin $|S,S⟩$. The numbered arrows (1)–(4) shown on the right correspond to FID experiments shown separately in Fig. 7.

Figure 7

Comparison of individual FID traces taken as horizontal slices from Fig. 6 for selected rotation angles ($\theta$) of the microwave pulse. The initial pseudospin state is illustrated on the Bloch sphere for each case (inset). (a),(b) Traces (1) and (2) correspond to rotation angles of $\theta \approx \pi /2$ (initial pseudospin state $|S,0⟩$) and $\theta \approx \pi$ (initial pseudospin state $|S,S⟩$), respectively. All oscillations in trace (1) occur at a rate close to $g\sqrt{N}$. In contrast, trace (2) evolves more slowly with a long delay ($t_D$) in the FID oscillations occurring at the first minimum where the pseudospin reaches full inversion. The rest of the FID trace oscillates at $g\sqrt{N}$ so that this initial delay shows up as an overall phase shift. Inset in (b): A zoomed-in part of the two-dimensional FID plot from Fig. 6 with a fit (white dashed curve) to $t_D(M)$ according to Eq. (5). (c) Traces (3) and (4) correspond to slight under- or overrotation with respect to the fully inverted state of the pseudospin ($\theta \approx \pi \pm \delta$, with $\delta =9$ deg). The phases of the microwave emission are opposite in these two traces since the pseudospin precession reverses direction in the underrotated case but continues in the same direction in the overrotated case.