Optical characterization of excited-state spin Hamiltonians and clock transitions at telecommunication wavelength in Tm3+:YAlO3

AI summary

Overview

Research area: Quantum photonics and rare-earth-doped solids — specifically optical spectroscopy and spin-Hamiltonian characterization of thulium ions in a crystalline host for telecom-wavelength quantum technologies.

Technical level: Intermediate to Advanced. The paper is written for readers comfortable with effective spin Hamiltonians, crystal-field/Stark level structure, Zeeman and hyperfine interactions, and spectral hole burning. The core physics is presented through perturbation theory with explicit tensor notation, but the experimental narrative is accessible to a motivated graduate student.

Scope (one sentence): The paper uses spectral hole burning and magnetic-field-dependent optical line-shift measurements to determine the spin Hamiltonians of the two lowest Stark levels of the excited ³F₄ and ³H₄ manifolds of Tm³⁺:YAlO₃ at the ~1451 nm zero-phonon line, and from those Hamiltonians predicts orientations and field magnitudes at which optical clock transitions occur.

What This Paper Is About

The 1451 nm zero-phonon line between the ³F₄ and ³H₄ excited-state manifolds of Tm³⁺ in YAlO₃ sits in the telecom E-band, but its optical coherence time was previously measured at only 4.75 μs, limited by magnetic-field fluctuations from spin flip-flops. To find operating points where the transition frequency is insensitive to those fluctuations — optical clock transitions, the optical analogue of zero-first-order-Zeeman points — the authors needed the full spin Hamiltonians of both levels. This paper supplies them, entirely through optical measurements, and then uses them to predict specific clock points.

Key Contributions

1. Establishes the spin Hamiltonians of two optically excited Stark levels purely by optical means. The lowest Stark levels of ³H₄ (X1) and ³F₄ (Y1) are characterized through spectral hole burning and inhomogeneous line-shift measurements, with no electron paramagnetic resonance or microwave/radio-frequency resonance required — an approach the authors state can be generalized to other crystals.

2. Separates and quantifies the enhanced nuclear Zeeman effect and the quadratic Zeeman effect. Side-hole detunings give the enhanced nuclear Zeeman tensor Γ, while field-dependent shifts of the inhomogeneous line center give the quadratic shift; because the splitting depends on products A·Λ and the quadratic shift on Λ alone, combining them yields both the hyperfine constants and the Λ tensor principal values.

3. Resolves site symmetry and magnetic equivalence in the Pbnm crystal. The four 4c Wyckoff sites are grouped into two magnetically equivalent pairs (site 1 and site 2), which become magnetically equivalent to each other when the field lies in the ac or bc planes — exploited to define the field reference frame and to constrain the fits.

4. Predicts a catalogue of optical clock points for the Z1↔X1, Z1↔Y1, and Y1↔X1 transitions, with quantified first- and second-order magnetic sensitivities, identifying the excited-state Y1↔X1 transition as the most promising for coherence improvement.

Main Findings

• Enhanced nuclear Zeeman tensor principal values (X1, ³H₄): γxx/h = 24.30(16) MHz/T, γyy/h = 8.66(20) MHz/T, γzz/h = 50.44(8) MHz/T; in-plane principal-axis angle α = −35.08(10)° for site 1 and +35.08(10)° for site 2.

• Enhanced nuclear Zeeman tensor principal values (Y1, ³F₄): γxx/h = 22.98(16) MHz/T, γyy/h = 8.23(25) MHz/T, γzz/h = 50.05(11) MHz/T; α = 61.52(11)° for site 1 and −61.52(11)° for site 2.

• Quadratic Zeeman shift rates of the Y1–X1 transition: Δk_a = 559 ± 4 MHz/T², Δk_b = −93 ± 2 MHz/T², Δk_c = 172 ± 2 MHz/T² for B parallel to the a, b, and c axes respectively. The negative value along b is notable.

• Magnetic hyperfine constants: A(³H₄)/h = −465.54 ± 4.43 MHz and A(³F₄)/h = −430.05 ± 2.33 MHz. The ³H₄ value differs from the previously calculated value of −678.3 MHz, which the authors attribute to the very low Cs point symmetry of the Tm³⁺ site, where the radial 4f wavefunction assumption underlying the calculation may not hold.

• Full Hamiltonian parameters: Λxx = 3.01(3), Λyy = 0.743(7), Λzz = 6.79(7) in units of 10²¹ J⁻¹ for X1; 1.95(1), 0.472(3), 4.67(3) for Y1; with g_J = 0.80 (X1) and 1.25 (Y1) from Russell-Saunders coupling. The Z1 ground-state values are included from literature for completeness.

• Best predicted clock point for the Y1↑↔X1↑ transition: |B| = 30.88 mT, θ = 131.0°, φ = 61.8°, with second-order sensitivity S2 = 8.9 Hz/G². This is 13 times smaller than the S2 = 1.2·10² Hz/G² at the |B| = 11.75 mT, θ = 90°, φ = 145.1° point, implying a coherence time more than 13 times larger under the model used.

• Additional low-sensitivity Y1↔X1 points: |B| = 27.58 mT (θ = 90°, φ = 32.4°) and |B| = 30.19 mT (θ = 90°, φ = 87.2°), both with S2 = 16 Hz/G²; and |B| = 30.88 mT at θ = 49.0°, φ = 61.8° with S2 = 8.9 Hz/G².

• First-order sensitivity heatmap: At B = 30.88 mT, the points (θ, φ) = (49.0°, 61.8°) and (131.0°, 61.8°) show first-order sensitivities below 10⁻² Hz/T, confirming genuine clock transitions.

• Z1-involving transitions are less useful: All optical clock transitions involving the Z1 level of the ³H₆ manifold have much larger S2, making them less suitable for coherence applications.

Methodology in Plain English

The team used a 0.1% Tm³⁺-doped YAlO₃ crystal measuring 12.0 × 2.2 × 2.0 mm³ along the a, b, and c axes, cooled to about 800 mK in a cryogen-free dilution refrigerator and placed inside a superconducting split-coil vector magnet capable of 2 T in arbitrary directions (field homogeneity better than 1.8% over the crystal).

Two lasers did the work. A laser near 1685 nm pumped ions from the ground ³H₆ manifold up to the 7th Stark level of ³F₄, from which they rapidly decay into Y1. A second, tunable laser near 1450 nm then either scanned across the Y1↔X1 absorption line to find its center, or burned a narrow spectral hole and later read it out after a wait shorter than the 0.85 ms lifetime of X1. Acousto-optic modulators made the pulses, a phase modulator with a serrodyne signal scanned the probe frequency, and a 10 MHz-bandwidth InGaAs photoreceiver detected the transmitted light. A wavemeter (Bristol Instruments Model 671) provided the frequency reference for the line-center measurements.

The two measurement types probe complementary physics. The splitting of a Stark level into its two nuclear-spin sublevels grows linearly with field, and its rate (in MHz/T) depends on the product of the hyperfine constant A and the Λ tensor components. The overall shift of a level grows quadratically with field and depends only on Λ. Measuring both, for many field orientations, lets the team solve for A and Λ separately. Side holes appear in pairs — inner and outer — and time-resolved hole burning assigns each pair to the X1 or Y1 level. Angular scans in the ab plane (where the two sites are inequivalent, giving distinct splittings) and in the bc and ac planes (where they coincide) were fitted simultaneously with a single parameter set per level, imposing the mirror symmetry that forces site 2 to have angle −α.

With the Hamiltonians in hand, the authors numerically computed the first-order sensitivity |∇_B ν_t| on 1° angular grids and field amplitudes from 0 to 10 T in 1 mT steps, extracted local minima, and used a root-finding algorithm to confirm that all three partial derivatives vanish simultaneously. Second-order sensitivity was taken as the largest absolute eigenvalue of the Hessian of the transition frequency, giving a simple estimate 1/T₂ ≈ S2(ΔB)².

Why This Matters

Clock transitions convert a fragile optical coherence into one protected from the dominant noise source. For Tm³⁺:YAlO₃ at 1451 nm, the measured 4.75 μs coherence time is currently limited by spin flip-flop magnetic noise; moving to a clock point is the obvious route to extending it without changing the material. The result is a concrete, experimentally actionable recipe: specific field magnitudes around 11–31 mT at specific crystal orientations, derived from measured — not assumed — Hamiltonian parameters.

Real-world applications:

• Telecom-compatible quantum memories. A long-lived optical memory at 1451 nm operates directly in the low-loss E-band of standard fiber, avoiding frequency conversion.
• Indistinguishable single-photon sources. Coherent, narrow-linewidth photons at telecom wavelength are a prerequisite for photon-interference-based repeater protocols.
• Quantum repeaters and repeater infrastructure. Ensemble memories with extended coherence are the building block for entanglement distribution over deployed fiber.
• Precision magnetometry and frequency metrology. The same clock-point machinery identifies field orientations where a transition is first-order insensitive to field noise, useful for compact solid-state sensors and references.

Industry relevance: the transition lies at ~1451 nm, inside the telecom E-band and accessible with commercial semiconductor lasers; the pump and probe are standard tunable CW sources. The characterization is entirely optical, so it can be performed on packaged devices without microwave access or EPR equipment.

Future Directions

• Measure the coherence time at the predicted clock points. The paper predicts, but does not report, an extended optical coherence time at any clock point; the claim that T₂ should improve by more than 13 times follows from the dephasing model, not from measurement. Verifying this experimentally is the immediate next step.
• Resolve the hyperfine constant discrepancy. The measured A(³H₄)/h = −465.5 MHz departs substantially from the calculated −678.3 MHz; testing whether the low Cs symmetry invalidates the radial-wavefunction assumption would refine Hamiltonian predictions across hosts.
• Extend the all-optical characterization recipe to other crystals and ions. The authors explicitly state the method can be generalized, which would allow rapid screening of candidate hosts for clock transitions without resonance-based techniques.
• Assess clock points for site 2 and for single ions or nanostructures. The predicted points are tabulated for site 1 (with φ → −φ for site 2), and the practical behavior of these points in the presence of site disorder, crystal strain, or reduced-mode-volume device geometries is not addressed.

Target Audience

This paper is most valuable to researchers in rare-earth quantum photonics and solid-state quantum memories, particularly groups working on telecom-wavelength interfaces, spectral hole burning, and ZEFOZ/clock-transition engineering. It also serves materials scientists and spectroscopists interested in optically characterizing spin Hamiltonians of low-symmetry non-Kramers sites, and quantum-device engineers who need concrete field-and-orientation operating points rather than purely theoretical predictions. Graduate students entering the field will find the site-symmetry and tensor-diagonalization discussion a useful template for similar work.

Authors’ abstract

The zero-phonon line (ZPL) between the two excited-state manifolds 3F4 and 3H4 of the trivalent thulium ion in yttrium aluminum perovskite (Tm3+:YAlO3) lies within the telecom E-band at around 1451 nm and exhibits an optical coherence time of a few μs at cryogenic temperatures. To improve this time in view of potential applications, e.g. single-photon sources and optical quantum memory, we characterized the magnetic properties of the lowest Stark levels of these two manifolds. Varying the external magnetic field amplitude and orientation, we measured the enhanced nuclear Zeeman splitting by probing the field-dependent frequency detuning of side holes created by means of spectral hole burning, and we characterized the inhomogeneous line shift caused by the quadratic Zeeman interaction. Together, this allowed us to establish the spin Hamiltonians for the lowest Stark level of the 3F4 and 3H4 manifolds, and to predict optical clock transitions for particular orientations and magnitudes of the magnetic field