Hyperfine characterization and spin coherence lifetime extension in Pr${}^{3+}$:La${}_2$(WO${}_4$)${}_3$

Rare-earth ions in dielectric crystals are interesting candidates for storing quantum states of photons. A limiting factor on the optical density and thus the conversion efficiency is the distortion introduced in the crystal by doping elements of one type into a crystal matrix of another type. Here we investigate the system Pr${}^{3+}$:La${}_2$(WO${}_4$)${}_3$, where the similarity of the ionic radii of Pr and La minimizes distortions due to doping. We characterize the praseodymium hyperfine interaction of the ground-state (${}^3$$H$${}_4$) and one excited state (${}^1$$D$${}_2$) and determine the spin Hamiltonian parameters by numerical analysis of Raman-heterodyne spectra, which were collected for a range of static external magnetic-field strengths and orientations. On the basis of a crystal-field analysis, we discuss the physical origin of the experimentally determined quadrupole and Zeeman tensor characteristics. We show the potential for quantum memory applications by measuring the spin coherence lifetime in a magnetic field that is chosen such that additional magnetic fields do not shift the transition frequency in first order. Experimental results demonstrate a spin coherence lifetime of 158 ms — almost 3 orders of magnitude longer than in zero field.

Figure 1

Pr${}^{3+}$:La${}_2$(WO${}_4$)${}_3$ level structure of the lowest (0) ${}^3$$H$${}_4$ and ${}^1$$D$${}_2$ crystal-field manifolds, including their hyperfine structure. The order of the energy levels follows from previous work,[19] whereas the transition frequencies given above could be measured with higher precision utilizing zero-field Raman-heterodyne scattering within the present work. The arrows on the right indicate the range of hyperfine transitions excited by rf in the separate experiments.

Figure 2

RHS pulse sequences. Black lines indicate frequencies and gray areas the applied optical and rf powers. (a) Continuous-wave sequence for the ground-state measurements. A low-power laser probe ($P_p=$ 0.3 mW) and a scanning frequency rf ($P_{\mathrm{RF}}\approx$ 1 W, ${\tau }_s=$ 50 ms, ${\nu }_1=$ 7.4, and ${\nu }_2=$ 22.4 MHz for $|\pm \frac{1}{2}\rangle \leftrightarrow |\pm \frac{3}{2}\rangle$ or, respectively, 17.1 and 32.1 MHz for $|\pm \frac{3}{2}\rangle \leftrightarrow |\pm \frac{5}{2}\rangle$) were applied to the sample. The optimum temperature of the cold finger for this scheme was 4.5 K, since still lower temperatures gave such slow hyperfine level relaxation rates that it would be necessary to repump the hyperfine level population. (b) Pulsed RHS sequence for the excited state. Probe and erase beam were overlapped in the sample at an angle of 0.6${}^{\circ }$. To allow for higher repetition rate the chirped erase laser ($\Delta =$ 64 MHz, ${\tau }_e=$ 10 ms, average power $P_e\approx 20$ mW) redistributed the populations. An initial population difference between hyperfine sublevels was created by the probe beam ($P_p=$ 1.2 mW, ${\tau }_p=$ 100 $\mu$s) and converted to coherences by an rf pulse (${\nu }_{\mathrm{RF}}=$ 4.94/7.23 MHz ($|\pm \frac{1}{2}\rangle \leftrightarrow |\pm \frac{3}{2}\rangle$ resp. $|\pm \frac{3}{2}\rangle \leftrightarrow |\pm \frac{5}{2}\rangle$), $P_{\mathrm{RF}}=$ 214/287 W, ${\tau }_{\mathrm{RF}}=$ 4 $\mu$s). For optical heterodyne detection the same probe beam was left active for an additional time ${\tau }_{\mathrm{dec}}$. The temperature of the cold finger for this scheme was 2.4 K.

Figure 3

Representative ground- (cw) and exited-state (pulsed) RHS spectra. Both spectra from the ground-state ($|g,i\leftrightarrow j\rangle$) are recorded at $\vec{B}=(2.60,-5.27,-5.76)$ mT. For the excited state spectra, we used $\vec{B}=(4.62,1.19,4.42)$ mT. The normalization is relative to the largest line from ground- or excited state spectra, respectively. The $|g,i\leftrightarrow j\rangle$ spectra shown here resolve all $8+8$ possible RHS transitions, while in the $|e,i\leftrightarrow j\rangle$ spectra not all lines are resolved. The histograms on the right show the distributions of fitted FWHM RHS linewidths for all recorded data (see Sec. IA in Ref. 25), plotted separately for the two hyperfine transitions ($|g/e,i\leftrightarrow j\rangle$, $i/j=\pm \frac{1}{2}/\pm \frac{3}{2}$ or $\pm \frac{3}{2}/\pm \frac{5}{2}$). For both the ground- and the excited state, the $\pm \frac{3}{2}\leftrightarrow \pm \frac{5}{2}$ linewidths are bigger (see Sec. 5c). The mean linewidths are $|g,\pm \frac{1}{2}\leftrightarrow \pm \frac{3}{2}\rangle \approx$ 105 kHz, $|g,\pm \frac{3}{2}\leftrightarrow \pm \frac{5}{2}\rangle \approx$ 301 kHz, $|g,\text{all}\rangle \approx$ 196 kHz and $|e,\pm \frac{1}{2}\leftrightarrow \pm \frac{3}{2}\rangle \approx$ 18.3 kHz, $|e,\pm \frac{3}{2}\leftrightarrow \pm \frac{5}{2}\rangle \approx$ 26.3 kHz, $|e,\text{all}\rangle \approx$ 22.3 kHz.

Figure 4

RHS spectra of the $|\pm \frac{1}{2}\rangle \leftrightarrow |\pm \frac{3}{2}\rangle$ and $|\pm \frac{3}{2}\rangle \leftrightarrow |\pm \frac{5}{2}\rangle$ hyperfine transitions. (a) ${}^3$$H$${}_4$(0) ground-state cw RHS spectra. (b) ${}^1$$D$${}_2$(0) excited state pulsed RHS spectra. In both cases the solid lines represent the fit results and the shaded background the absolute value of the experimental spectra. For each field orientation, the spectrum was normalized to the maximum signal amplitude.

Figure 5

Raman-echo decays at the ZEFOZ point and with magnetic detunings of $-0.2$ and $-0.5$ mT for the $z$ component.