Characterization of the hyperfine interaction of the excited ${}^5\mathrm{D}_0$ state of ${\mathrm{Eu}}^{3+}:{\mathrm{Y}}_2{\mathrm{SiO}}_5$

We characterize the europium $\left({\mathrm{Eu}}^{3+}\right)$ hyperfine interaction of the excited state $\left({}^5\mathrm{D}_0\right)$ and determine its effective spin Hamiltonian parameters for the Zeeman and quadrupole tensors. An optical free induction decay method is used to measure all hyperfine splittings under a weak external magnetic field (up to 10 mT) for various field orientations. On the basis of the determined Hamiltonian, we discuss the possibility to predict optical transition probabilities between hyperfine levels for the ${}^7\mathrm{F}_0⟷{}^5\mathrm{D}_0$ transition. The obtained results provide necessary information to realize an optical quantum memory scheme which utilizes long spin coherence properties of ${}^{151}\mathrm{Eu}^{3+}:{\mathrm{Y}}_2{\mathrm{SiO}}_5$ material under external magnetic fields.

Figure 1

The energy level structure of ${}^{151}\mathrm{Eu}^{3+}:{\mathrm{Y}}_2{\mathrm{SiO}}_5$, without (left) and with (right) an external magnetic field $B$. The inhomogeneous broadening of the optical transition ${}^7\mathrm{F}_0⟷{}^5\mathrm{D}_0$ resonant at 580 nm contains hyperfine $m_I=-5/2\cdots +5/2$ sublevels both for the ground ${}^7\mathrm{F}_0$ and the excited ${}^5\mathrm{D}_0$ states. The ground state hyperfine splittings were characterized in [31].

Figure 2

(a) The class cleaning procedure at the quadrupole level (described in the main text) leaves four classes of atoms between the ground $|\pm \frac{k}{2}{\rangle }^{\left(g\right)}$ and excited $|\pm \frac{l}{2}{\rangle }^{\left(e\right)}$ Zeeman doublets. Under an external magnetic field, each doublet splits with energy difference $\delta e$ and $\delta g$ for the ground and excited state, respectively. (b) Using SHB technique, one can redistribute the population to reveal the energetic structure with contribution from the four different classes.

Figure 3

(a) Experimental setup. The laser is split into two beams and sent to two different paths: the signal, which prepares and probes the crystal and the local oscillator (LO). The amplitudes and frequencies of these beams are adjusted using acousto-optic modulators (AOM), which are used in double-pass configuration. PBS stands for polarizing beamsplitter. Coils around the cryostat provide the external magnetic field. (b) Experimental sequence, consisting of a preparation step, a spectral hole burning (SHB) step and the readout (R) step. The FID is measured with an oscilloscope right after the end of the readout pulse. (c) Oscilloscope trace for 10 mT magnetic field applied along $\vec{B}\parallel D_1$ : interference of the FID with the 4 MHz detuned local oscillator. The beatings reveal a complex absorption structure. (d) Imaginary part of the Fourier transform of the temporal trace presented in (c), which is proportional to the absorption of the spectral structure. Antiholes are indicated by a dashed vertical line, the solid vertical lines correspond to side holes.

Figure 4

Experimental SHB spectra obtained using the spiral scan of the magnetic field, for transitions connecting different hyperfine levels of the ${}^7\mathrm{F}_0$ ground state and the ${}^5\mathrm{D}_0$ excited state of ${}^{151}\mathrm{Eu}^{3+}:{\mathrm{Y}}_2{\mathrm{SiO}}_5$. Each vertical slice represents a hole burning spectrum obtained using the FID signal [an example is shown in Fig. 3]. White regions correspond to higher transmission (holes) while black regions represent increased absorption (antiholes). The positions of the side holes give directly the splittings of the excited state, while the positions of the strongest antiholes correspond to the ground-state splitting (cf. Fig. 2). The energy splittings predicted by the fitted spin Hamiltonian are shown as white (holes) and black (antiholes) lines. The strong central hole was removed to increase the contrast of the image. The color axis is nonlinear.

Figure 5

Experimental SHB spectra obtained by scanning the magnetic field of 10 mT in the $D_1\text{−}{D}_2$ plane (perpendicular to the $C_2$ symmetry axis). In this plane, the magnetic subsites are degenerate, such that fewer holes and antiholes are seen. The measured spectra are in good agreement with the energy splittings predicted by the fitted spin Hamiltonian, without any additional tuning of the parameters with respect to the fit shown in Fig. 4. All other experimental details are identical to Fig. 4.

Figure 6

Measured positions of the side holes (red squares) and main antiholes (black circles), which correspond to the nuclear Zeeman splittings of the excited ${}^5\mathrm{D}_0$ and ground ${}^7\mathrm{F}_0$ state, respectively. The dashed lines represent the result of the fitting of the spin Hamiltonian for each state. The extracted parameters are presented in Table 1. The bottom figure shows the average residual difference between the measured spectra and the fit (averaging over the three spectra).

Figure 7

Spherical plot (violet) of the energy splitting for the ground (three figures below) and excited (three figures above) states, ${\delta }_k/\left(2\right|\vec{B}\left|\right)$, in natural units for $k=1/2,3/2,\text{and}5/2$ (from left to right) as a function of $\vec{n}(\theta ,\varphi )=\vec{B}/|\vec{B}\left|\right)$. The orange plot is the hypothetical energy splitting if $M$ were isotropic (i.e., $M\propto \mathbb{1})$, which is almost the case for excited state (and, hence, there the orange plot is basically covered by the violet one). The coordinate system is the eigenbasis of ${\mathbf{Q}}^{\left(e\right)}$ or ${\mathbf{Q}}^{\left(g\right)}$, respectively.

Figure 8

Overlay of ${\delta }_k$ for $k=1/2,3/2,\text{and}5/2$ (from left to right) from the experimental data (red dots) and the perturbation theory based on the fitted parameters (blue surface) in the laboratory frame for the ground (three figures below) and excited (three figures above) states.