Coherent Storage of Microwave Excitations in Rare-Earth Nuclear Spins

Interfacing between various elements of a computer—from memory to processors to long range communication—will be as critical for quantum computers as it is for classical computers today. Paramagnetic rare-earth doped crystals, such as ${\mathrm{Nd}}^{3+}:{\mathrm{Y}}_2{\mathrm{SiO}}_5(\mathrm{YSO})$, are excellent candidates for such a quantum interface: they are known to exhibit long optical coherence lifetimes (for communication via optical photons), possess a nuclear spin (memory), and have in addition an electron spin that can offer hybrid coupling with superconducting qubits (processing). Here we study two of these three elements, demonstrating coherent storage and retrieval between electron and ${}^{145}\mathrm{Nd}$ nuclear spin states in ${\mathrm{Nd}}^{3+}:\mathrm{YSO}$. We find nuclear spin coherence times can reach 9 ms at $\sim 5\mathrm{K}$, about 2 orders of magnitude longer than the electron spin coherence, while quantum state and process tomography of the storage or retrieval operation between the electron and nuclear spin reveal an average state fidelity of 0.86. The times and fidelities are expected to further improve at lower temperatures and with more homogeneous radio-frequency excitation.

Figure 1

(a) ${\mathrm{Y}}_2{\mathrm{SiO}}_5$ crystal structure showing the coordination polyhedron corresponding to ${\mathrm{Nd}}^{3+}$ site within a unit cell. (b) Schematic energy level diagram of ${{}^{145}\mathrm{Nd}}^{3+}$ ions highlighting relevant electron and nuclear spin transitions in blue and red, respectively. (c) Field swept ESE spectrum for a magnetic field close to $D1$ ($T=6.5\mathrm{K}$) showing two sets of ESR transitions from two magnetically inequivalent ${\mathrm{Nd}}^{3+}$ classes. The electron spin transition at 561.5 mT was used for all ENDOR experiments, such as the spectrum shown in (d).

Figure 2

Electron and nuclear spin relaxation times as a function of temperature: $T_{1e}$ (squares), $T_{2e}$ (triangles), and $T_{2n}$ (circles). In the inset, corresponding decay curves for $T_{2n}$ from 5 to 7 K. $T_{1e}$ is modeled with an Orbach process $1/T_{1e}=A\exp (-\mathrm{\Delta }E/k_{B}T)$ with $A=6\times {10}^{10}{\mathrm{s}}^{-1}$ and $\mathrm{\Delta }E=77{\mathrm{cm}}^{-1}$ ($k_B$ is the Boltzmann constant). $T_{2n}$ is limited by $2T_{1e}$ giving the relation $1/T_{2n}=1/(2T_{1e})+1/T_{2n}^{\text{int}}$, where $T_{2n}^{\text{int}}$ is the decay time for the nucleus due to the spin environment only. As this decay has the same origin as for the electron, we can simply relate $T_{2n}^{\text{int}}=\kappa T_{2e}$.

Figure 3

(a) Sequence used for storing mw photons into nuclear spin coherences. (b) Input $+{\sigma }_X$, $+{\sigma }_Y$, and $+{\sigma }_Z$ (upper row) and corresponding output (lower row) density matrices are obtained by state tomography. Real and imaginary parts are shown in green and yellow, respectively. (c) Quantum process tomography matrix $\chi$ in the $(\mathbb{1},{\sigma }_X,{\sigma }_Y,{\sigma }_Z)$ basis. A perfect storage process would give only a $[\mathbb{1},\mathbb{1}]$ component. Left: Matrix reconstructed from experimental density matrices. Right: Simulated matrix considering pulse fidelity and spin relaxations.

Figure 4

(a) Electron Rabi oscillations obtained on the $|1⟩:|2⟩$ transition [Fig. 1]. The decay is Gaussian, typical of mw pulse inhomogeneity, with a standard deviation about 1.5% of the Rabi frequency. (b) Nuclear Rabi oscillations obtained on the $|2⟩:|3⟩$ transition. The oscillations do not decay completely and can be modeled by the use of a truncated Gaussian for the rf pulse inhomogeneity. Such behavior is likely due to the small sample size which only sees part of the magnetic field inhomogeneity (nearly 14% here) from the rf coil.