Rare-Earth Ion Qubit

Figure

Description

Rare-earth ion qubits encode quantum information in the electronic and nuclear spin states of trivalent rare-earth ions (lanthanides) doped into crystalline hosts, most commonly yttrium orthosilicate (Y₂SiO₅, or YSO). The three most studied species for quantum applications are:

• Erbium (Er³⁺): Possesses a ${}^4{I}_{15/2}\to {}^4{I}_{13/2}$ optical transition at 1536 nm, directly in the telecom C-band. This is the only solid-state qubit with a native telecom-wavelength optical interface, eliminating the need for quantum frequency conversion in fiber-based quantum networks. The ${}^{167}{\text{Er}}^{3+}$ isotope ($I=7/2$) provides hyperfine structure for long-lived nuclear spin storage.

• Europium (Eu³⁺): Holds the record for the longest measured coherence time in any solid-state system: $T_2=6$ hours for the ${}^{151}{\text{Eu}}^{3+}$ nuclear spin transition in Y₂SiO₅ at 2 K using a zero-first-order Zeeman (ZEFOZ) magnetic field and dynamical decoupling (Zhong et al. 2015).

• Praseodymium (Pr³⁺): Used in early demonstrations of quantum memory protocols including electromagnetically induced transparency (EIT), atomic frequency comb (AFC) storage, and controlled reversible inhomogeneous broadening (CRIB).

The fundamental challenge for all rare-earth species is the weak oscillator strength of the 4f–4f optical transitions (shielded by outer 5s and 5p electrons), resulting in long radiative lifetimes (~ms) and low photon emission rates. This is addressed through Purcell enhancement using optical cavities — nanophotonic resonators in silicon (for Er³⁺) or Fabry-Perot microcavities coupled to the crystal. Cavity enhancement factors exceeding 100× have been demonstrated, enabling single-ion detection and single-shot spin readout.

Hamiltonian

For a Kramers ion (e.g., Er³⁺) in a crystal field with an applied magnetic field:

$H=H_{\text{CF}}+{\mu }_B(\mathbf{L}+2\mathbf{S})\cdot \mathbf{B}+A_{\text{hfs}}\mathbf{I}\cdot \mathbf{J}$

where $H_{\text{CF}}$ is the crystal field Hamiltonian splitting the $4f^N$ multiplets, ${\mu }_B$ is the Bohr magneton, and $A_{\text{hfs}}$ is the hyperfine constant.

For a non-Kramers ion (e.g., Eu³⁺, Pr³⁺), the ground state is a singlet in zero field, and the qubit is encoded in hyperfine levels split by the nuclear quadrupole and second-order hyperfine interactions:

$H_{\text{nuc}}=A_{\text{hfs}}\mathbf{I}\cdot {\mathbf{J}}_{\text{eff}}+P[I_z^2-I(I+1)/3]+{\mu }_n{g}_n\mathbf{I}\cdot \mathbf{B}$

At a ZEFOZ (zero-first-order Zeeman) point, the transition frequency is insensitive to magnetic field fluctuations to first order, analogous to clock transitions in trapped ions, enabling the extraordinary 6-hour coherence time.

Motivation

Rare-earth ions in crystals offer a unique combination: extremely long spin coherence times (hours), optical transitions at telecom wavelengths (Er³⁺), and the ability to frequency-multiplex many spectrally distinct ions within a single device. This makes them leading candidates for quantum network nodes — interfacing long-lived spin quantum memories with photonic channels. The crystalline host provides a reproducible, scalable environment compared to individually fabricated defects, and the narrow inhomogeneous optical lines (~GHz) contain thousands of individually addressable ions via frequency selection.

Experimental Status

Hours-long nuclear spin coherence — Zhong et al. (2015):

• Demonstrated $T_2=6$ hours for ${}^{151}{\text{Eu}}^{3+}$ nuclear spins in Y₂SiO₅ at 2 K.
• Used a ZEFOZ magnetic field (1.28 T applied at a specific crystal orientation) and dynamical decoupling.
• Measured decoherence rate of $8\times 10^{-5}{\text{s}}^{-1}$ over 100 ms evaluation windows.
• Record solid-state coherence time by orders of magnitude.

Single Er³⁺ ion detection — Dibos et al. (2018):

• First optical observation of individual Er³⁺ ions in a silicon nanophotonic crystal cavity.
• Purcell-enhanced emission rate >650× bulk, enabling identification of dozens of individual ions.
• Demonstrated spin-dependent optical transitions via Zeeman splitting.

Single-shot spin readout — Raha et al. (2020):

• Quantum nondemolition measurement of a single Er³⁺ ion qubit spin state.
• Achieved 94.6% single-shot readout fidelity via cavity-enhanced cycling transition.
• Demonstrated coherent spin dynamics and quantum jumps from optical pumping.

Single Yb³⁺ cavity QED — Kindem et al. (2020):

• Control and single-shot readout of a single ${}^{171}{\text{Yb}}^{3+}$ ion in a YVO₄ nanophotonic cavity.
• Spin coherence $T_2>30\text{ms}$ for a cavity-coupled ion.
• Single-shot readout fidelity >95%.

Key Metrics

Metric	Value	Notes	Fidelity reference
$T_2$ (nuclear spin)	6 hours	${}^{151}{\text{Eu}}^{3+}$ in Y₂SiO₅, ZEFOZ + DD	Zhong et al. 2015
$T_2$ (electronic spin)	~ms	Er³⁺ Kramers doublet; host-dependent	—
Optical wavelength (Er³⁺)	1536 nm	Telecom C-band; native fiber compatibility	—
Single-shot readout fidelity	94.6%	Single Er³⁺ via cavity QND	Raha et al. 2020
Purcell enhancement	>650×	Er³⁺ in Si nanophotonic cavity	Dibos et al. 2018
Operating temperature	2–4 K	Closed-cycle cryostat sufficient	—
Spectral multiplexing	>1000 ions	Individually addressable via frequency in single device	—

Scaling Considerations

• Weak oscillator strength: the fundamental 4f–4f transition rates are slow (~ms radiative lifetimes), requiring high-finesse optical cavities for practical single-ion operation. Achieving sufficient Purcell enhancement while maintaining low loss is an active engineering challenge.
• Inhomogeneous broadening: while useful for multiplexing, the ~GHz inhomogeneous linewidth means that most ions in the ensemble are spectrally detuned from the cavity resonance. Strain tuning or spectral hole burning is needed to isolate and optimize individual ions.
• Two-qubit gates: no high-fidelity two-qubit gate between individual rare-earth ions has been demonstrated; proposals include dipole-dipole interactions between nearby ions, cavity-mediated coupling, and photonic Bell-state measurements.
• Integration with silicon photonics: Er³⁺ in silicon is particularly promising for CMOS-compatible quantum networks, but erbium in silicon suffers from broad optical lines unless specific crystalline sites are engineered.
• Quantum memory vs. processor: the extremely long coherence times make rare-earth ions exceptional quantum memories, but gate operations are comparatively slow and two-qubit interactions remain challenging — the platform is better suited to quantum repeater nodes than general-purpose processors.

References

Coherence record

• M. Zhong et al., “Optically addressable nuclear spins in a solid with a six-hour coherence time,” Nature 517, 177 (2015)

Single-ion detection and readout

• A. M. Dibos et al., “Atomic Source of Single Photons in the Telecom Band,” Phys. Rev. Lett. 120, 243601 (2018) | arXiv:1711.10368
• M. Raha et al., “Optical quantum nondemolition measurement of a single rare earth ion qubit,” Nat. Commun. 11, 1605 (2020)

Cavity QED

• J. M. Kindem et al., “Control and single-shot readout of an ion embedded in a nanophotonic cavity,” Nature 580, 201 (2020) | arXiv:1907.12161

Linked Papers

• zhong-2015-six-hour-coherence
• dibos-2018-telecom-single-photon
• raha-2020-rare-earth-qnd-readout
• kindem-2020-single-ion-nanophotonic

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