Self-Stimulated Pulse Echo Trains from Inhomogeneously Broadened Spin Ensembles

We show experimentally and describe theoretically how a conventional magnetic resonance Hahn echo sequence can lead to a self-stimulated pulse echo train when an inhomogeneously broadened spin ensemble is coupled to a resonator. Effective strong coupling between the subsystems assures that the first Hahn echo can act as a refocusing pulse on the spins, leading to self-stimulated secondary echoes. Within the framework of mean field theory, we show that this process can continue multiple times leading to a train of echoes. We introduce an analytical model that explains the shape of the first echo and numerical results that account well for the experimentally observed shape and strength of the echo train and provides insights into the collective effects involved.

Figure 1

(a) Experimentally observed emission from an inhomogeneously broadened spin ensemble (${}^{145}\mathrm{Nd}$ ions doped at 200 ppm in a ${\mathrm{Y}}_2{\mathrm{SiO}}_5$ host) in a cavity subject to a conventional Hahn echo sequence with $\tau =30\mu \mathrm{s}$ pulse delay. The spin-cavity coupling exhibits high cooperativity $C=153$ (see the Supplemental Material [31] for full details). For reference, the $\pi /2$ and $\pi$ pulses are shown in green on a different intensity scale. (b) Schematic of the refocusing mechanism leading to a self-stimulated spin echo train (see text).

Figure 2

Numerical results (a) Top: Intracavity photon number $|\alpha {|}^2$ as a function of time for a numerical calculation with $\tau =60\mu \mathrm{s}$ and $\mathrm{\Gamma }=2\pi \times 0.5\mathrm{kHz}$ (for other parameters, see text). Center: $⟨{\sigma }_x⟩$ for the spins in a small detuning region as a function of time. Bottom: $⟨{\sigma }_z⟩$ for selected frequency classes. (b) Theoretical results averaged over 45 different samples of the detuning distributions. We consider $\tau =45\mu \mathrm{s}$, $\mathrm{\Gamma }=2\pi \times 0.5\mathrm{kHz}$ and we observe that the echoes change shape over time. (c) Averaged $|\alpha {|}^2$ for $\tau =30\mu \mathrm{s}$ and different values of the spin dephasing rate $\mathrm{\Gamma }$. (d) Emitted number of photons in the echoes as a function of their order of arrival according to theory (dotted) and experiment (dot dashed, arbitrary units). The results are shown for $\tau =30\mu \mathrm{s}$ and $\mathrm{\Gamma }=2\pi \times 2.5\mathrm{kHz}$. An exponential fit $y=a{e}^{-bx}$ reveals a decay rate of $b=2\pi \times \{6.29\mathrm{kHz}(\mathrm{theory}),6.84\mathrm{kHz}(\mathrm{experiment})\}>\mathrm{\Gamma }$. (e) The first spin echo signal for different values of the Lorentzian inhomogeneous width ${\mathrm{\Gamma }}_L$ as computed from Eq. (7), assuming an (overestimated) Holstein-Primakoff spin excitation amplitude $\beta =1$.