Observation of the hyperfine structure and anticrossings of hyperfine levels in the luminescence spectra of LiYF₄:Ho³⁺

Abstract

Resolved hyperfine structure and narrow inhomogeneously broadened lines in the optical spectra of a rare-earth-doped crystal are favorable for the implementation of various sensors. Here, a well-resolved hyperfine structure in the photoluminescence spectra of LiYF₄:Ho single crystals and the anticrossings of hyperfine levels in a magnetic field are demonstrated using a self-made setup based on a Bruker 125HR high-resolution Fourier spectrometer. This is the first observation of the resolved hyperfine structure and anticrossing hyperfine levels in the luminescence spectra of a crystal. The narrowest spectral linewidth is only 0.0022 cm⁻¹. This fact together with a large value of the magnetic g factor of several crystal-field states creates prerequisites for developing magnetic field sensors, which can be in demand in modern quantum information technology devices operating at low temperatures. Very small random lattice strains characterizing the quality of a crystal can be detected using anticrossing points.

Introduction

Crystals doped with rare-earth (RE) ions exhibit very narrow homogeneous and inhomogeneous linewidths of the 4f^N–4f^N optical transitions, since the 4f^N electronic shell is well shielded from the crystalline environment by the filled 5s and 5p shells. The narrow-line spectra of transitions within the 4f^N shell of triply ionized RE elements cover the entire visible and infrared range. RE-doped materials are widely used as laser media, phosphors, scintillators, in solar cells, etc. Nowadays, RE-based luminescence thermometry is successfully developing, demonstrating a wide working temperature range, high thermal sensitivity, and spatial resolution^1,2. Over the past decade, significant progress has been achieved in the application of RE-doped crystals for quantum information processing, which is based on the use of electronic-nuclear hyperfine levels^3,4,5,6,7,8. Information on the hyperfine and superhyperfine interactions, isotopic effects, inhomogeneous line shapes, and random lattice strains in a crystal is essential for applications in modern quantum technologies. It can be acquired using high-resolution optical spectroscopy. High-resolution studies also reveal the narrowest spectral lines, which are favorable for sensor applications.

A series of high-resolution studies of RE materials has been performed using absorption or photoluminescence (PL) excitation spectroscopies. Hyperfine^{7,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}, superhyperfine²⁴, and isotope^{23,24,25,26,27,28,29,30} structures, as well as specific line shapes imposed by random lattice strains^23,31,32 and the hyperfine level anticrossings in a magnetic field³³ were observed in optical absorption spectra of a number of RE-doped crystals. We are not aware of any broadband high-resolution study of the luminescence spectra of RE-doped crystals, as well as of a resolved hyperfine structure in these spectra. However, the luminescence spectra open up additional possibilities, such as, e.g., the study of very small and/or highly diluted samples or the development of new types of remote sensors. Here, we present the results of a high-resolution study of broadband PL spectra of LiYF₄:Ho crystals in a magnetic field.

Results

Photoluminescence of LiYF₄:Ho³⁺ in zero magnetic field: hyperfine structure

Lithium yttrium fluoride crystals, LiYF₄, exhibit the luminescence from many optically excited levels of RE ions doped into this crystal, in a wide spectral range. PL of a LiYF₄:Ho³⁺ (0.1 at. %) crystal was excited by the wavelength 638.3 nm of a diode laser. This excitation wavelength corresponds to the transition from the ground state to the upper crystal-field (CF) level of the ⁵F₅ multiplet of the Ho³⁺ ion (see Supplementary Fig. S1). High-resolution PL spectra were acquired on an experimental setup built on the basis of a Bruker IFS 125HR high-resolution Fourier spectrometer (Fig. 1a and Methods).

Fig. 1: PL spectrum of LiYF₄:Ho³⁺ (0.1 at. %): T = 6 K, B = 0; λ_ex = 638.3 nm.

a Setup: 1—cryostat with magnetic coils (gray hatched), 1a—temperature controller, 1b—current source for magnetic coils; 2—temperature-stabilized diode laser, 2a—temperature controller and current source for laser; 3—PL module; 4—Fourier spectrometer; 5—PL registration module, 5a—preamplifier and an analog to digital converter; 6—workstation for spectra calculation and automatic control of magnetic field and temperature. b The whole spectral region studied. Identified intermultiplet transitions of Ho³⁺ as well as the ⁴I_13/2 → ⁴I_15/2 transition of Er³⁺ trace impurity are indicated. Inset presents the scheme of energy levels of Ho³⁺; the observed transitions are shown by arrows. c ⁵I₆ → ⁵I₇ and d ⁵I₇ → ⁵I₈ luminescent transitions. Insets show the spectral lines highlighted in blue on an enlarged scale. The numbers n_i → n_f indicate the initial and final levels of the corresponding transition, numbered from the lowest level in the CF multiplet (Table 1).

Figure 1b shows the low-temperature PL spectrum of LiYF₄:Ho³⁺ (0.1 at. %), in the whole spectral range studied. The observed optical manifolds are identified in accordance with the scheme given in the inset. Figure 1c, d presents the spectral manifolds ⁵I₆ → ⁵I₇ and ⁵I₇ → ⁵I₈, respectively, in extended scales as examples. At low temperatures, only lowest-energy levels of the ⁵I₆ and ⁵I₇ excited CF multiplets are populated after the laser excitation. The lines in the low-frequency part of the luminescent manifolds correspond to transitions to the upper levels of the final CF multiplets, broadened due to phonon relaxation to underlying levels⁹. In the high-frequency parts of the manifolds, many lines with well-resolved hyperfine structure are observed, as insets of Fig. 1c, d show.

The resolved hyperfine structure in optical absorption spectra of LiYF₄:Ho³⁺ was discovered⁹ and studied in detail^9,10,11,12 earlier. The CF energy levels of Ho³⁺ are characterized by the one-dimensional Γ₁ and Γ₂ and two-dimensional Γ₃₄ irreducible representations (IR) of the S₄ point symmetry group of the holmium site in LiYF₄. Table 1 lists the energies and symmetries (IRs) of several CF levels of Ho³⁺ in LiYF₄ (relevant for further discussion), determined from the analysis of polarized absorption and luminescence spectra, using the selection rules^11,12 (see Supplementary Table S1). An extended table of CF levels is presented in Supplementary Table S3. The data obtained are consistent with the results of previous studies^9,10,11,12, with one exception. Namely, we did not find the level 11,242.4 cm⁻¹ (⁵I₅)¹², but demonstrated the level 11,254.0 cm⁻¹, which is absent in ref. ¹². Figure 2 illustrates this. We note that the allowed ED transition ⁵I₈ (1, Γ₃₄) → ⁵I₅ (4, Γ₂) [11,254 cm⁻¹] is nevertheless very weak and can easily be missed if for some reason (higher holmium concentration, insufficient spectral resolution) the spectral lines are broadened.

Table 1 Information on CF levels of Ho³⁺ in LiYF₄:Ho³⁺.

Fig. 2: ⁵I₅ CF levels with resolved HFS.

The π- (k⊥c, E||c; black trace) and σ- (k⊥c, E⊥c; red trace) polarized absorption spectra of a LiYF₄:Ho³⁺ (0.1 at. %) single crystal at the temperature 3.5 K in zero magnetic field. δσ = 1/2 L = 0.01 cm⁻¹ (L is the maximal displacement of a moving mirror in the Fourier spectrometer).

In a zero magnetic field, the Γ₃₄ CF levels possess an eight-component equidistant magnetic hyperfine structure resulting from the interaction of 4f electrons with the magnetic moment of the holmium nucleus with spin I = 7/2. Each hyperfine component is doubly degenerate, the states |Γ₃, m> and |Γ₄, –m> have the same energy (here, m is the component of nuclear moment I along the crystallographic c axis, –7/2 ≤ m ≤ 7/2)^9,12. For the Γ₁ and Γ₂ non-degenerate electronic CF states magnetic HFS is forbidden in the first approximation. Electric quadrupole and pseudoquadrupole (magnetic dipole in the second approximation) hyperfine interactions split Γ₁ and Γ₂ singlets into four nonequidistant hyperfine sublevels and lead to nonequidistance in Γ₃₄ hyperfine manifolds¹⁰.

In the luminescence spectra, HFS is observed in other spectral regions compared to absorption spectra, in particular, at telecommunication wavelengths around 1.5 μm (transitions ⁵F₅ → ⁵I₆ and ⁵I₅ → ⁵I₇), which can be used, for example, to implement remote magnetic field sensors. For sensor applications, such parameters as the luminescence linewidth and sensitivity of the line position to perturbations are important.

Isotopic structure in the photoluminescence spectra of ⁷Li_1–x ⁶Li_xYF₄:Ho³⁺ and linewidths of hyperfine components

Figure 3a, b shows the absorption and PL spectra, respectively, of ⁷Li_1–x⁶Li_xYF₄:Ho³⁺ single crystals with different lithium isotope compositions, in the region of the optical transition between the ⁵I₇ Г₃₄ (5155.7 cm⁻¹) and ⁵I₈ Г₂ (6.85 cm⁻¹) CF levels of Ho³⁺. An identical fine structure of the hyperfine components is clearly observed in the absorption and PL spectra if x ≠ 0. Earlier, the structure observed in the absorption spectra was unambiguously ascribed to the isotopic disorder in the lithium sublattice^26,27. Because of the difference in masses of the lithium isotopes, the amplitudes of their zero-field vibrations differ, leading to different equilibrium positions of the nearest fluorine ions (through the anharmonicity of vibrations) and, thus, to the dependence of the crystal field for the Ho³⁺ ion on the lithium isotope composition in its nearest surrounding²⁶. Evidently, the origin of the fine structure of the hyperfine components in the PL spectra is the same. Figure 3c gives an example of such isotopic structure in the PL line corresponding to a singlet–singlet transition.

Fig. 3: Isotopic structure of hyperfine components in the spectra of crystals with different ⁶Li and ⁷Li isotope compositions.

Isotopic structure in the a absorption and b, c luminescence spectra of ⁷Li_1–x⁶Li_xYF₄:Ho³⁺ (0.1 at %) crystals with x = 0.07 (natural abundance of Li isotopes; green traces), x = 0.9 (red traces), and x = 0 (blue traces). a ⁵I₈ Г₂ (6.85) → ⁵I₇ Г₃₄ (5155.75), b ⁵I₇ Г₃₄ (5155.75) → ⁵I₈ Г₂ (6.85), and c ⁵I₆ Г₁ (8673.4) → ⁵I₇ Г₁ (5162.8) optical transitions. T = 6 K. δσ = 1/2 L = 0.001 cm⁻¹ (L is the maximal displacement of a moving mirror in the Fourier spectrometer).

In what follows, we study PL spectra of LiYF₄:Ho³⁺ in an external magnetic field. To avoid complexities associated with the isotopic structure, we use a monoisotopic ⁷LiYF₄:Ho³⁺ crystal. Preliminarily we investigated the widths of the PL lines in a zero magnetic field. Here, a remark is necessary concerning the resolution. The instrumental function of a Fourier spectrometer has the form $f_L(\sigma ) = \frac{{\sin x}}{x}$, $x = 4\pi L\sigma$, where σ is the wavenumber and L is the maximal displacement of a moving mirror in a Michelson interferometer. The distance between the first zeros of f_L(σ) equals δσ = 1/2 L (0.001 cm⁻¹ in our case) and is indicated as “resolution” in an instrument manual. The full width at half maximum (FWHM) of f_L is 0.6δσ. The measured line shape is a convolution of a real line shape and the instrumental function f_L. The line shape practically does not change down to W = 2δσ (W is FWHM of the studied spectral line). At W = δσ, weak additional maxima appear on the line wings, and in the case of the Loretzian line shape, FWHM of the convolution exceeds by 10% FWHM of the original line³⁴ (see Supplementary Fig. S3).

Figure 4 shows several PL lines of the ⁵I₆ → ⁵I₇ transition in Ho³⁺:⁷LiYF₄. The lines represented in Fig. 4a, b correspond to the transitions from the lowest in the ⁵I₆ multiplet Г₁ and Г₂ CF singlets, respectively, to the ⁵I₇ Г₃₄ (5155.75 cm⁻¹) doublet and reflect HFS of the latter. The four-component line of Fig. 4c originates from a singlet–singlet transition.

Fig. 4: Hyperfine structure of several photoluminescence lines in the ⁵I₆ → ⁵I₇ transition of Ho³⁺:⁷LiYF₄ at T = 6 K.

Lines correspond to the transitions a 2 → 2 [⁵I₆ Г₁ (8673.4) → ⁵I₇ Г₃₄ (5155.75)] and 3 → 3 [⁵I₆ Г₃₄ (8680.3) → ⁵I₇ Г₁ (5162.8)]; b 1 → 2 [⁵I₆ Г₂ (8670.9) → ⁵I₇ Г₃₄ (5155.75)]; c 2 → 3 [⁵I₆ Г₁ (8673.4) → ⁵I₇ Г₁ (5162.8)]. Red traces are experimental spectra registered with δσ = 0.001 cm⁻¹, blue curves represent the results of fit with sets of Loretzians. Note the stretched wavenumber scale at c.

To find the correct linewidths of the hyperfine components, we approximated the observed spectra with sets of Loretzians. The FWHM thus obtained range from 0.002 to 0.004 cm⁻¹ (for comparison, the smallest linewidth observed in the absorption spectra of this crystal is 0.012 cm⁻¹). The ⁵I₆ → ⁵I₇ transitions, inaccessible in absorption, exhibit the narrowest spectral lines. The PL linewidth of 0.002 cm⁻¹ observed here is, as far as we know, the smallest inhomogeneous linewidth ever observed in the luminescence spectra of crystals.

Photoluminescence of ⁷LiYF₄:Ho³⁺ in an external magnetic field B||c

The twofold degeneracy of the hyperfine states |Γ₃, m> and |Γ₄, –m> is lifted in an external magnetic field parallel to the c axis of the crystal, B||c. The electronic Zeeman energies are equal to ± g_||μ_BB/2, where μ_B = 0.4669 cm⁻¹T⁻¹ is the Bohr magneton and g_|| is the g factor of a doublet. Figure 5 shows HFS in a magnetic field B||c for two PL lines of ⁷LiYF₄:Ho³⁺ (0.1 at. %) starting from the ⁵I₅ Г₃₄ doublet at 11,241.6 cm⁻¹. The line 6089.3 cm⁻¹ has the ⁵I₇ Г₂ singlet at 5152.3 cm⁻¹ as a terminal level, its splitting is governed by the g factor g_|| = 8.1 of the ⁵I₅ Г₃₄ (11,241.6) doublet. Magnetic g factors of several Г₃₄ doublets given in Table 1 were determined experimentally by analyzing the splitting of PL lines corresponding to transitions between a given doublet and a singlet. We note that the g factor is proportional to the hyperfine interval Δ_HFS (ref. ⁹): | g_|| | = 2g₀Δ_HFS /A_J, where g₀ is the Lande factor and A_J is the magnetic hyperfine constant (see Supplementary information, Eqs. S5, S9, and S10).

Fig. 5: Evolution of the hyperfine structure with increasing magnetic field B||c.

Photoluminescence intensity map in the magnetic-field – wavenumber scale for two transitions from the ⁵I₅ Г₃₄ (11,241.6) CF level to the levels of the ⁵I₇ CF multiplet: ⁵I₇ Г₂ (5152.3) (top line) and ⁵I₇ Г₃₄ (5155.75) (bottom line). ⁷LiYF₄:Ho³⁺ (0.1 at. %), T = 10 K. B||c.

The line 6085.85 cm⁻¹ ends at the ⁵I₇ Г₃₄ (5155.75) doublet, and the hyperfine interval in HFS of this line is a sum of the hyperfine intervals in the initial ⁵I₅ Г₃₄ (11,241.6) and final ⁵I₇ Г₃₄ (5155.75) doublets of the transition but the splitting follows the sum of g factors of the involved levels, g_||(11,241.6) + g_||(5155.75) = 8.1 + 6.8 = 14.9 (see Supplementary information for more details; Supplementary Figs. S4–S34 show the magnetic-field-dependent PL spectra used to obtain the data given in Table 1).

Hyperfine levels’ anticrossing in the photoluminescence spectra of ⁷LiYF₄:Ho³⁺ in a magnetic field B||c

Figure 6a presents the PL intensity map for the radiative transition from the lowest electronic doublet in the ⁵I₇ CF multiplet, ⁵I₇ Г₃₄ (5155.75 cm⁻¹), to the first excited singlet of the ground ⁵I₈ CF multiplet, ⁵I₈ Г₂ (6.85 cm⁻¹). The presented picture coincides with the one observed in the absorption spectra at the corresponding transition³³ and demonstrates gaps at the anticrossings of the |Γ₄, m> and |Γ₃, m ± 2> states (|Δm| = 2) (see, e.g., a “huge” gap at 5149.0 cm⁻¹ in Fig. 6a) and of the |Γ₄, m> and |Γ₃, m> states (|Δm| = 0). However, it should be noted that when measuring luminescence, a better contrast is achieved.

Fig. 6: Anticrossings of the hyperfine levels in the luminescence spectra of ⁷LiYF₄:Ho³⁺ in a magnetic field B||c.

a, b Luminescence intensity maps in the magnetic-field – wavenumber scale for the a ⁵I₇ Г₃₄ (5155.75) → ⁵I₈ Г₂ (6.85) and b ⁵I₆ Г₂ (8670.9) → ⁵I₇ Г₃₄ (5155.75) transitions in ⁷LiYF₄:Ho³⁺ (0.1 at. %) at 6 K. Anticrossings of the hyperfine levels are observed. c Experimental and d simulated fragment of the spectrum corresponding to the ⁵I₆ Г₂ (8670.9) → ⁵I₇ Г₃₄ (5155.75) transition of ⁷LiYF₄:Ho³⁺ (0.1 at. %) in a zero magnetic field (in black) and in the field of 140 mT (in red), indicated by an arrow in b.

Narrowest PL lines of ⁷LiYF₄:Ho³⁺ (0.1 at. %) in a magnetic field were observed in the case of the ⁵I₆ → ⁵I₇ transitions, the same as in a zero magnetic field. Figure 6b shows a representative example. Here, the spectrum of the ⁵I₆ Г₂ (8670.9) → ⁵I₇ Г₃₄ (5155.75) transition is shown. This transition in the spectral region of 2.84 μm involves the same ⁵I₇ Г₃₄ (5155.75) CF doublet as the transition at 1.94 μm presented in Fig. 6a. Although the singlets participating in the transitions [⁵I₈ Г₂ (6.85 cm⁻¹) in the case of Fig. 6a and ⁵I₆ Г₂ (8670.9) in the case of Fig. 6b] are different, the spectra are similar.

Figure 6c shows a fragment of the spectrum corresponding to the ⁵I₆ Г₂ (8670.9) → ⁵I₇ Г₃₄ (5155.75) transition of ⁷LiYF₄:Ho³⁺ (0.1 at. %) in a zero magnetic field and in the field of 140 mT at one of the anticrossings. A four-component structure at the |Δm| = 2 anticrossings and a very specific line shape at the Δm = 0 anticrossing are observed.

The ⁵F₅ Г₃₄ (15,495.4) CF level demonstrates the largest Δm = 0 gaps, which are observed in all the lines starting from this level. As an example, Fig. 7a shows the spectrum of the ⁵F₅ Г₃₄ (15,495.4) → ⁵I₆ Г₂ (8670.9) transition. A large central gap is clearly seen in the hyperfine pattern in a magnetic field.

Fig. 7: The ⁵F₅ Г₃₄ (15,495.4) level demonstrates the largest deformation splitting.

a Luminescence intensity map in the magnetic-field – wavenumber scale and b calculated frequency vs magnetic field dependences for the ⁵F₅ Г₃₄ (15495.4) → ⁵I₆ Г₂ (8670.9) transition in ⁷LiYF₄:Ho³⁺ (0.1 at. %) at 10 K. c Experimental and d calculated spectra of this transitions at several values of the magnetic field. A strong horizontal line in a corresponds to the singlet–singlet transition ⁵F₅ Г₁ (15,512.7) → ⁵I₆ Г₂ (8687.75).

Discussion

Luminescence spectra complement absorption spectra, providing information on levels inaccessible to absorption. In the case of spectra consisting of narrow lines, the signal-to-noise ratio in PL spectra is much higher than in absorption spectra. Moreover, in some cases, the PL lines are noticeably narrower than the absorption ones, so that the HFS is resolved much better. A good example can be found in Fig. 4, where, in particular, quadrupole HFS of a Γ₁ → Γ₁ magnetic dipole transition is observed (Fig. 4c, see Supplementary Table S1 for selection rules). The use of luminescence instead of absorption makes it possible to work with micron- or even nano-sized samples, which is important, e.g., for sensor applications. The PL spectra of LiYF₄:Ho³⁺ can be used to control the external magnetic field. Here, the ⁵F₅ → ⁵I₆ and ⁵I₅ → ⁵I₇ luminescent transitions near 1.64 and 1.46 μm, respectively, fall into the transparency window of optical fibers and offer a possibility to implement, for example, remote magnetic field sensors. Let us consider the PL line at 6085.85 cm⁻¹ [⁵I₅ Г₃₄ (11241.6) → ⁵I₇ Г₃₄ (5155.75), see Fig. 5], the magnetic-field splitting of which, Δ = μ_B g_|| B (see Supplementary Eq. S4), follows the sum of g factors of the levels involved: g_|| = g_||(11,241.6) + g_||(5155.75) = 14.9. The position of a spectral line with the known shape can be determined with an accuracy δσ equal to 0.1 FWHM of the instrumental function³⁴ (in Fourier spectroscopy, where the wavenumber scale over the entire spectral region is set by a stabilized He-Ne laser, this is also the accuracy of the absolute wavenumber scale³⁵). In our case, the instrumental function is characterized by FWHM = 0.6/2 L. For L = 500 cm we find δσ = 0.6·10⁻⁴ cm⁻¹. Using the relation δΔ = 2δσ = μ_Bg_||δB, one obtains δB = 17 μT, i.e., the strength of an external magnetic field can be determined with the precision δB ≈ 17 μT. It is worth noting that the range of measured magnetic fields can be very broad (calibration curve σ(B) can be used for strong fields, for which Supplementary Eq. S4 is not valid). It is also possible to determine the direction of the magnetic field by installing two LiYF₄:Ho crystals with the mutually perpendicular c axes (g_⊥ = 0 for non-Kramers doublets). To control small deviations of the magnetic field strength, the method developed for magnetometry with nitrogen-vacancy (NV) centers in diamond³⁶ can be applied. In ref. ³⁶, a field-dependent position of a dip in the NV luminescence intensity was controlled with an accuracy of ~μT using AOM-modulated laser excitation and a lock-in amplifier. The authors of ref. ³⁶ noted that their “technique can be extended to other magnetically sensitive features in the NV PL or absorption, as well as features associated with other spin defects in solid-state systems”. In particular, a narrow hyperfine component in the luminescence spectrum of ⁷LiYF₄:Ho³⁺ can be used instead of a dip in a broad luminescence spectrum of NV. Unlike the case of NV centers³⁶, no additional field (102.4 mT, to set at the ground-state level anticrossing, and a secondary coil to apply small modulation of this field) is necessary in the case of LiYF₄:Ho, which can be favorable for quantum technology devices.

Similar to a ruby luminescent pressure sensor, a small piece of LiYF₄:Ho crystal can be placed nearby a quantum-memory crystal in the repeater, operating at low temperatures, to control an external magnetic field used to adjust the energy levels. However, the implementation of a practical and convenient sensor requires additional work. A high-resolution Fourier spectrometer, being the only spectral instrument able to deliver information on the HFS in a wide spectral range, is a bulky and very expensive device. A possible solution to the problem is to isolate the desired spectral line with an interference filter and use a Fabry–Perot interferometer. To avoid overlay of the interference orders, the free spectral range should be 2 cm⁻¹, which dictates a distance between Fabry–Perot mirrors 2.5 mm. In order to obtain a resolution of about 0.002 cm⁻¹ in this case, it is necessary to have a finesse of about 1000. The finesse depends on the reflection coefficient of the mirrors and the quality of their surface. Currently, firms offer Fabry–Perot interferometers with finesse up to 1500 in a wide spectral range³⁷.

Monoisotopic crystals are most favorable for realizing sensitive sensors, because they demonstrate the narrowest lines (as Fig. 3 convincingly shows). On the other hand, the isotopic structure in the spectra of LiYF₄:Ho³⁺ reflects the lithium isotope composition in a crystal. Under neutron irradiation, the ⁶Li isotopes can capture neutrons converting to ⁷Li. A change of the isotope composition in a crystal originally enriched with the ⁶Li isotope could be detected by PL spectra.

Next, we turn to the anticrossings observed in the PL spectra. As was shown in ref. ³³ on the basis of magnetic-field-dependent absorption spectra of ⁷LiYF₄:Ho³⁺, gaps observed at the anticrossings of the |Γ₄, m> and |Γ₃, m ± 2> states (|Δm| = 2) are caused by the transverse term in magnetic dipole hyperfine interaction but the four-component structure at the |Δm| = 2 anticrossings arises due to the mixing of the electron-nuclear wave functions of the crossing levels in the vicinity of the anticrossing point (at the anticrossing point, the wave functions of the crossing levels are present with equal weights). The Δm = 0 gaps have a different nature, they are due to random crystal-lattice deformations always present in a real crystal³³. Here, we focus on anticrossings of the hyperfine levels with equal nuclear spin projections, Δm = 0, which can be used to check crystal quality. In particular, by analyzing PL of Ho³⁺ (either introduced in a small amount or present as uncontrolled impurity) in ⁷LiYF₄:Nd, one can check the quality of this crystal proposed⁷ for a quantum memory based on off-resonant Raman interaction.

A very specific line shape with a sharp dip in the center and slopy wings is especially well seen just in the luminescence spectra (see Fig. 6c). Previously, such line shape was observed for electronic transitions between singlet and doublet CF levels in the spectra of LiYF₄:Tm³⁺ (ref. ²³), Cs₂NaYF₆:Yb³⁺ (ref. ³¹), and Tm³⁺ in ABO₄ (A = Y, Lu; B = P, V)³² and was shown to be the result of random lattice deformations^23,31,32. A theory has been developed that made it possible to extract information about the strain distribution function from the analysis of the line shape³². The measured shape of the PL line with a dip at the center in the region of the Δm = 0 anticrossings for the singlet-doublet transition ⁵I₆ Г₂ (8670.9) → ⁵I₇ Г₃₄ (5155.7) is modeled here on the basis of this theory.

The Hamiltonian of the electronic 4f¹⁰ shell of the Ho³⁺ impurity ion, $H = H_{{{{\mathrm{FI}}}}} + H_{{{{\mathrm{CF}}}}} + V$, where $V = H_{{{{\mathrm{MHF}}}}} + H_{{{{\mathrm{QHF}}}}} + H_Z + H_{{{{\mathrm{ElD}}}}}$, contains the free-ion energy H_FI, the interaction with the crystal-field H_CF, the magnetic (H_MHF) and quadrupole (H_QHF) hyperfine interactions, the electronic and nuclear Zeeman energies H_Z in an external magnetic field, and the electron-deformation interaction H_ElD, linear in components $e_{\alpha \beta }$ of the strain tensor e. Modeling of the spectral profiles involves consequent numerical diagonalizations of the Hamiltonian H₀ = H_FI + H_CF defined in the total space of 1001 electronic states and, at the next step, of the projection of the operator V onto the truncated basis of 76 × 8 = 608 electron-nuclear states corresponding to the lower 76 eigenvalues ($E_{\it{\Gamma }}$) of the Hamiltonian H₀. The obtained energies $E_{{\it{\Gamma }}j}$ and wave functions of the electron-nuclear sublevels of the ⁵I_J (J = 4,…8) and ⁵F₅ multiplets are used to calculate intensity distributions in spectral lines corresponding to magnetic (electric) dipole radiative transitions ${\it{\Gamma }} \to {\it{\Gamma }}^\prime$ at fixed values of the magnetic field strength and strain tensor components,

$$I_{\Gamma \Gamma ^\prime }(E,{{{\boldsymbol{e}}}}) = \mathop {\sum}\limits_{j \in {\it{\Gamma }}} {\mathop {\sum}\limits_{k \in {\it{\Gamma }}^\prime } {\mathop {\sum}\limits_\alpha {\left| { \langle {\it{\Gamma }}^\prime k} \right|M_\alpha \left| {{\it{\Gamma }}j \rangle } \right|^2} } } I_{0,\Gamma \Gamma ^\prime }(E_{{\it{\Gamma }}j} - E_{{\it{\Gamma }}^\prime k} - E)$$

(1)

Here M is the magnetic (electric) dipole moment operator of ten 4f electrons and $I_{0,\Gamma \Gamma ^\prime }(x) = (\delta E_{\Gamma \Gamma ^\prime }/\pi )(x^2 + \delta E_{\Gamma \Gamma ^\prime }^2)^{ - 1}$ is the form-function with FWHM 2$\delta E_{\Gamma \Gamma ^\prime }$ of individual transitions between hyperfine sublevels of the CF states ${\it{\Gamma }}$and ${\it{\Gamma }}^\prime$. Note that for the intraconfigurational transitions, the structure of the effective even electric dipole moment operator is determined by the odd component of the crystal field. The finite spectral envelope is modeled by averaging the distribution (1) with the distribution function g(e) of random strains induced by lattice point defects in LiYF₄ crystals, taking into account the elastic anisotropy of the crystal lattice³². The most pronounced spectral effect of random strains, namely, the formation of a dip in the envelope of the transition involving a CF non-Kramers doublet Γ₃₄, is caused by rhombic strains $e(B_g^1) = (e_{xx} - e_{yy})/2$ and $e(B_g^2) = e_{xy}$, which transform according to the B_g IR of the C_4h lattice factor group and split Γ₃₄ doublets. The corresponding distribution function can be written as follows:

$$g\left( {e\left( {B_g^1} \right),e\left( {B_g^2} \right)} \right) = \frac{{\nu _{B_g}\gamma _{B_g}}}{{2\pi }}\left( {\nu _{B_g}^2e_1^2 + e_2^2 + \gamma _{B_g}^2} \right)^{ - 3/2}$$

(2)

where $\nu _{B_g}$ = 2.44, $e_1 = e(B_g^1)\cos \varphi - e(B_g^2)\sin \varphi$, $e_2 = e(B_g^1)\sin \varphi + e(B_g^2)\cos \varphi$, $\varphi$ = 28.7° (ref. ³²), and $\gamma _{B_g}$, which is proportional to the concentration of defects and determines the distribution width, is considered as a fitting parameter.

Experimental shapes of spectral lines were satisfactorily reproduced by computations of integrals over a two-dimensional space of random strains ($x = e(B_g^1),y = e( {B_g^2} )$)

$$I_{\Gamma \Gamma ^\prime }\left( E \right) \sim {\int\!\!\!\!\!\int} {I_{\Gamma \Gamma ^\prime }} (E,x,y)g(x,y)dxdy$$

(3)

where we used $\gamma _{B_g}$ = 5$\cdot$10^–5 and parameters of the electron-deformation interaction presented in ref. ³³. Examples of the obtained spectral envelopes in external magnetic fields (including fields with strengths corresponding to the Δm = ±2 and Δm = 0 anticrossings) are shown in Figs. 6d and 7d.

In summary, we have presented the first observation of the resolved hyperfine structure in the luminescence spectra of a RE-doped crystal and demonstrated PL lines as narrow as 0.002–0.004 cm⁻¹ and with magnetic g factors as large as 10–15, including in the telecommunication spectral range. These PL lines are promising for creating remote magnetic field sensors that do not require an additional constant or variable magnetic field and/or microwave field and are capable of operating in a very wide range of measured magnetic fields. Our results pave the way for the development of a remote magnetic field sensor for, e.g., quantum repeaters installed in an extended quantum communication line. This work is also the first observation of hyperfine levels anticrossings in the luminescence spectra. The Δm = 0 anticrossings can be used to evaluate random lattice deformations in crystals for quantum information devices, i.e., the crystal quality.

Materials and methods

Crystal growth

Three single crystals of LiYF₄:Ho (0. 1 at. %) with different content of ⁷Li and ⁶Li isotopes were grown by the Stockbarger method. Li₂CO₃ with known lithium isotope composition was taken as a starting material. It was transformed into LiF by the “dry” method. The crystals were then grown from the mixture of appropriate fiuorides. Samples with dimensions 3 × 3 × 8 mm³ and containing the c axis in the 3 × 3 mm² plane were cut from the x-ray oriented crystals and polished.

Optical spectroscopy and low-temperature measurements

In the PL experiments displayed in Fig. 1a, a LiYF₄:Ho³⁺ sample was placed into a cryomagnetic system (1) of our own design based on a Sumitomo RP096 closed-cycle helium cryostat. The design feature was that the electromagnet together with the concentrating magnetic system was attached directly to the first stage of the cryostat, which reduced the system dimensions, eliminated the need for heat removal in a vacuum, and allowed the use of higher currents and, consequently, magnetic fields, as the resistivity of copper winding wires was significantly reduced. A computer-controlled multi-channel current source Korad KA3305P (1b) was used to change the magnitude of the applied magnetic field. The magnetic field was directed along the c axis of the crystal and could be varied from 0 to 500 mT. The sample temperature was chosen such that several lower CF levels of each excited CF multiplet were populated, but the spectral lines were not yet broadened due to the electron–phonon interaction, and were set in the range from 3.5 to 10 K. The temperature was measured with a Si-diode temperature sensor (Scientific Instruments Si-410-AA) and a temperature controller (Lakeshore 335) (1a) and was stabilized with the precision of ±0.05 K during the entire measurement period. Emission was excited by the linearly polarized light of a diode laser (Oclaro HL63193) with 100 mW power, incident perpendicular to the c axis of the crystal and polarized along the c axis; the spot on the crystal was 0.5 mm. The laser wavelength could be changed in the interval 635 ± 10 nm by changing the temperature in the range from t = –20 to +45 °C using a self-made thermoelectric cooler and resistive heater. The laser temperature +36.4 °C was chosen by searching for the maximum PL signal. A homemade temperature stabilization system with feedback (2a) based on a temperature controller Scientific Instruments M9700 was implemented. Thus selected laser wavelength 638.3 nm excited the upper CF level of the ⁵F₅ multiplet (Supplementary Fig. S1). Emission from the crystal was collimated by a mirror (diameter 90 mm, focal length 418 mm) in a homemade evacuated (10^–6 Torr) PL module (3) separated from a Fourier spectrometer (4) by a CaF₂ window W1. This gave a possibility of recording the spectra in the region of the atmospheric absorption in the middle and far infrared. Emission spectra were registered on a Bruker IFS 125HR high-resolution vacuum Fourier spectrometer (4) in the wavenumber range 2500–16,000 cm⁻¹, with the maximal displacement of a moving mirror in the Michelson interferometer up to L = 500 cm, which provided the instrumental function with FWHM = 0.6 δσ, where δσ = 1/2 L = 0.001 cm⁻¹ (see Supplementary Eq. S2). Thus, the minimal width of the instrumental function used to register PL spectra was 0.0006 cm⁻¹ (18 MHz). The smallest input diaphragm D used was 0.3 mm in diameter. Polarized PL spectra were measured using a BaF₂ crystal-based polarizer for the mid- and near-infrared ranges. Radiation from the output of the Michelson interferometer was directed to a homemade registration module (5) separated from the spectrometer by a CaF₂ window W2, with an optical cutoff filter F to eliminate excitation laser light [Semrock single notch filter NF03-633E or a germanium plate (red in Fig. 1a)]. Depending on the studied region of the spectrum, highly sensitive detectors based on InSb (1800–5500 cm⁻¹), high-gain InGaAs (5500–9000 cm⁻¹), or SiPM (9000–15,500 cm⁻¹) semiconductors were used. The signal was amplified by preamplifiers of our own design and then fed to the 16-bit ADC of the spectrometer (5a). The spectrum was calculated as a Fourier transform of the interferogram on a workstation (6) equipped with Bruker OPUS^TM software, LabView^TM software, and homemade software modules for COM ports that monitor and change sample and diode laser temperatures and electromagnet current. Polarized absorption spectra in zero magnetic field were acquired on the same experimental setup.