Spin coherence lifetime extension in Tm${}^{3+}$:YAG through dynamical decoupling

We report on spin coherence lifetime extension in Tm${}^{3+}$:YAG obtained through dynamically decoupling the thulium spins from their magnetic environment. The coherence lifetime reached with a Carr-Purcell-Meiboom-Gill sequence revealed a 450-fold extension [$\sim$$(230\pm 30)$ ms] with respect to previously measured values. Comparison to a simple theoretical model allowed us to estimate the correlation time of the fluctuations of the ground-level transition frequency to $(172\pm 30)$ $\mu$s at 1.7 K. For attaining efficient decoupling sequences, we developed a strategy inspired by the zero-first-order Zeeman effect to minimize the large inhomogeneous broadening of the ground-level spin transition.

Figure 1

Orientations of the six orientationally distinct sites of the Y${}^{3+}$ ion in the YAG crystal lattice.[36] Each parallelepiped represents the local $D_2$ symmetry for one of the sites. The $\widehat{\mathbf{x}}$, $\widehat{\mathbf{y}}$, $\widehat{\mathbf{z}}$ axis are the local axis for site 4.

Figure 2

(a) Spin sublevel splitting per tesla (T) and (b) Rabi frequency per millitesla (mT) as a function of the orientation of the static magnetic field $\mathbf{B}$ with respect to the $\left\{\right[100],[010],[001\left]\right\}$ reference frame of the YAG crystal. The rf field ${\mathbf{B}}_1$ is always orthogonal to $\mathbf{B}$, and contained in the $\left(001\right)$ plane. The lines indicate the orientation chosen for the experiments: $(\theta =54.8^{\circ },\varphi ={45}^{\circ })$, which ensures low ${\Gamma }_{\mathrm{inh}}$ and high $\Omega$.

Figure 3

(a) Hole-burning spectrum. Ground level splitting antiholes (${\Delta }_g$), excited level splitting holes (${\Delta }_e$), and mixed antiholes (${\Delta }_g-{\Delta }_e$) are observed in addition to the central hole. The ${\Delta }_g$ antihole is 105 kHz wide. (b) Spin nutation experiment. The rf field is turned on at $t=0$. The spins nutate at a Rabi angular frequency of $2\pi \times 264$ kHz.

Figure 4

Spin-echo experiment. $|{\rho }_{\mathrm{ab}}|$ is obtained from the quantity $\ln (I_f/I_{\mathrm{ref}})/\ln (I_i/I_{\mathrm{ref}})$, where $I_i$ and $I_f$ are the transmitted probe intensities before and after the rf sequence, respectively, and $I_{\mathrm{ref}}$ is a reference intensity for a known population difference (typically 0). The fit returns the parameters ${\sigma }_{\Delta }/\left(2\pi \right)=(2.3\pm 0.2)$ kHz and ${\tau }_c=(172\pm 30)$ $\mu$s.

Figure 5

(a) Scheme of the CPMG sequence. The signs “$+$” and “$-$” that precede the label of the $\pi$ pulses refer to the pulse phase, corresponding to 0 or $\pi$, respectively. $n$ and $\tau$ stand for the number of $\pi$ pulses and the time interval between them. (b), (c) Coherence recovered after a CPMG sequence with (b) fixed free evolution time $n\tau =2$ ms and increasing $n$, or (c) fixed $n=4$ and increasing $\tau$. The calculated data are obtained from Eq. (A15) with ${\sigma }_{\Delta }/\left(2\pi \right)=2.3$ kHz and ${\tau }_c=172$ $\mu$s.

Figure 6

Measurement of $T_2$ through CPMG sequences with different $\tau$ values. For data sets labeled $\tau =150$, 100, 10, and 3 $\mu$s, $\tau$ is kept fixed and $n$ is increased. The data set labeled $n=1$ is a standard spin-echo experiment ($n$ is fixed and $\tau$ is increased). The dashed lines result from least-squares fits.

Figure 7

CPMG-extended $T_2$ as a function of $\tau$. The experimental values (solid black symbols) result from the least-squares fit of the data of Fig. 6. The theoretical prediction (solid red line) represents Eq. (A16). The green dashed line indicates the lower bound ${\left(2T_1\right)}^{-1}$ ($T_1\sim 1$ min). The ratio between theoretical and experimental $T_2$ values is also plotted (hollow blue symbols; these data refer to the right vertical axis).