Two-Nd Absorptions in NdGaO₃

• Two-Nd absorptions are optical transitions where two exchange-coupled Nd³⁺ ions are excited simultaneously, distinct from single-ion processes.
• They produce weak spectral lines shifted by approximately +50 GHz from primary transitions, serving as direct probes of nearest-neighbor exchange interactions.
• Field-dependent studies show linear and nonlinear splitting across antiferromagnetic, intermediate, and paramagnetic phases, elucidating collective quantum effects.

Two-Nd Absorptions refer to a family of optical transitions in rare-earth antiferromagnetic crystals—specifically exemplified by neodymium gallate (NdGaO₃)—in which two magnetically coupled Nd³⁺ ions are simultaneously excited through the absorption of a single photon. The phenomenon contrasts with conventional single-ion (“single-Nd”) absorptions, and its existence, spectral properties, and field dependencies serve as direct probes of nearest-neighbour exchange interactions and collective quantum effects in ordered magnetic materials. Two-Nd absorption features manifest as weak, distinct lines tens of gigahertz blue-shifted from the standard single-ion transitions, and they exhibit selection rules and field dependencies governed by both the pairwise exchange tensor and the magnetic phase of the host crystal (Hiraishi et al., 11 Nov 2025).

1. Physical Origin and Distinction from Single-Ion Absorptions

In stoichiometric NdGaO₃, single-Nd absorption corresponds to 4f–4f transitions on individual Nd³⁺ ions (e.g., the Z₁ → R₁ doublet). By contrast, two-Nd absorption involves the joint excitation of two exchange-coupled nearest-neighbour Nd³⁺ ions by a single photon. The transition energy of such a process is

$$\omega_{12} = \omega_1 + \omega_2 + \Delta E_{\rm ex}$$

where $\omega_{1,2}$ are the underlying single-ion transition frequencies and $\Delta E_{\rm ex}$ is an additional energy offset proportional to the pairwise exchange interaction.

Spectroscopically, two-Nd lines appear at $\approx$ +50 GHz above the main single-Nd ($\pi$- or $\sigma$-polarised) transitions. Their relative intensity is typically $\approx 1/5$ that of the main lines, and their field dependence, polarisation selection rules, and line splittings are not explainable by single-ion models.

2. Experimental Characteristics and Field-Dependent Behavior

With zero applied magnetic field, two-Nd spectral features present as lines offset by $\approx$ +50 GHz from the primary single-ion transitions. Upon applying a magnetic field $B_0$ along the crystalline $c$-axis (up to 3 T), the field evolution is as follows:

• Antiferromagnetic Phase (0–1.1 T): Four distinct two-Nd lines, corresponding to in-plane and out-of-plane Nd–Nd pairs, each show linear splitting with field. The slope sign and magnitude are determined by the ferro- or antiferro-aligned nature of the coupled pair.
• Intermediate Phase (1.1–2.3 T): The two-Nd transitions shift nonlinearly with $B_0$, mirroring the continuous rotation of local moments during the spin-reorientation crossover.
• Paramagnetic Phase ($>$2.3 T): The two magnetic sublattices become equivalent, halving the number of observable two-Nd features. Slopes approach those of two ferromagnetically aligned moments.

Two-Nd features also exhibit selection rules derived from the symmetry of combined irreducible representations of the pair (e.g., $\Gamma_3 \otimes \Gamma_4 = \Gamma_1$).

3. Theoretical Framework: Pair Hamiltonian and Exchange Interaction

Two-Nd absorption arises from quantum mechanical interactions described by an extended Hamiltonian that explicitly incorporates the exchange coupling of neighbouring Nd³⁺ ions. The total Hamiltonian is

$H_{\rm total} = H_1 + H_2 + H_{\rm ex}$

where $H_i$ is the effective single-ion Hamiltonian (crystal field, Zeeman, and mean-field interactions for site $i$) and $H_{\rm ex}$ is the exchange: $$H_{\rm ex} = -J_{\rm ex} \, \mathbf{J}_1 \cdot \mathbf{J}_2$$ In practice, an anisotropic exchange tensor $\tilde{J}_{\parallel,\perp}$ is employed, with

$$\tilde J_{\parallel,\perp} = \tfrac{1}{\mu_B^2}\mathrm{diag}\Bigl(\frac{J'_{\parallel,\perp}}{g_a^2}, \frac{J'_{\parallel,\perp}}{g_b^2}, \frac{J_{\parallel,\perp}}{g_c^2}\Bigr)$$

and fits yield $J_\perp/k_B = -0.65$ K, $J_\parallel/k_B = +0.07$ K, $J'_\perp = J'_\parallel = -0.1$ K.

The two-Nd optical transition frequency is set by the energy difference between the doubled ground and excited states, accounting for exchange-induced splitting: $$\omega_{12} = [E_e(1) + E_e(2) - E_{\rm ex}^e] - [E_g(1) + E_g(2) - E_{\rm ex}^g] = \omega_1 + \omega_2 + \Delta E_{\rm ex}$$ with $\Delta E_{\rm ex}$ typically $\pm |J_{\rm ex}|$ depending on the pair's alignment.

4. Numerical Agreement and Quantitative Results

Using no adjustable parameters (other than a minor refinement to $J_\perp$ vs. prior literature), the pair Hamiltonian reproduces the field dependence and positions of all observed two-Nd lines to within $\approx$1 GHz for fields 0–3 T. For $B_0=0$, single-Nd lines appear at 0 GHz ($\pi$) and $+20$ GHz ($\sigma$), while two-Nd lines are split by $\approx+50$ GHz with further exchange-related substructure $\approx 10$ GHz wide.

The measured field slopes $d\omega/dB$ for two-Nd features in the antiferromagnetic phase match the calculated $g_c\mu_B/h$ values, with $g_c\approx2.3$ and appropriate sign determined by sublattice symmetry.

5. Influence of Magnetic Phase Transitions

The occurrence and splitting of two-Nd lines are highly sensitive to the global magnetic phase:

• In the antiferromagnetic phase, the static ordered state supplies deterministic mean fields to each pair, maximizing the distinctiveness and sharpness of two-Nd transitions.
• The intermediate phase features nonlinear field evolution (“bending and merging”) of two-Nd lines, consistent with continuous, non-Ising (type-II) spin rotation.
• In the paramagnetic phase, equivalence of sublattices halves the number of pair lines and drives their field dependence toward simple Zeeman behaviour with offsets reflecting the residual exchange fields.

This phase-dependent spectroscopic signature provides direct evidence for the interplay of local pairwise exchange and collective magnetic order.

6. Selection Rules and Transition Intensities

Two-Nd transitions are governed by selection rules dictated by the combined symmetry of the pair:

• Pair states are classified according to the product of single-ion irreducible representations (e.g., $\Gamma_3 \otimes \Gamma_4$).
• Only specific photon polarisations (e.g., only $\sigma$ if the pair remains in $\Gamma_1$) are allowed for a given pair symmetry.
• Two-Nd features are weaker than single-Nd lines, with integrated intensities typically $\approx 1/5$ of the main transitions and distinct saturation behaviour.

7. Implications for Quantum Magnetism and Quantum Transduction

The observation and quantitative analysis of two-Nd absorptions in antiferromagnetic NdGaO₃ constitute a direct spectroscopic fingerprint of pairwise exchange in an ordered rare-earth magnet. These transitions enable experimental determination of exchange tensor components, mean-field contributions, and the microscopic nature of phase transitions.

The underlying physics is of particular interest for quantum technologies, including microwave-to-optical quantum transduction schemes in rare-earth crystals, where control and understanding of multi-ion interactions and collective electronic excitations are essential for optimized coherence and coupling.

Two-Nd absorptions remain relatively unexplored in comparison to single-ion spectroscopy; their accurate modelling and experimental resolution now provide a pathway for deeper exploration of quantum many-body effects and engineered quantum materials (Hiraishi et al., 11 Nov 2025).