Characterization of the hyperfine interaction of the excited D state of Eu:YSiO
We characterize the Europium (Eu) hyperfine interaction of the excited state (D) and determine its effective spin Hamiltonian parameters for the Zeeman and quadrupole tensors. An optical free induction decay method is used to measure all hyperfine splittings under weak external magnetic field (up to 10 mT) for various field orientations. On the basis of the determined Hamiltonian we discuss the possibility to predict optical transition probabilities between hyperfine levels for the FD transition. The obtained results provide necessary information to realize an optical quantum memory scheme which utilizes long spin coherence properties of Eu:YSiO material under external magnetic fields.
Rare-earth-ion-doped crystals (REIC) have been actively studied during the last decade as promising solid-state materials for quantum information processing. In the field of quantum communication these compounds have been used as optical quantum memories: devices capable to store and release quantum states of light Lvovsky et al. (2009); Tittel et al. (2010); Bussières et al. (2013); de Riedmatten and Afzelius (2015). In this context different quantum memory protocols were utilized to demonstrate high-efficiency Hedges et al. (2010); Sabooni et al. (2013); Jobez et al. (2014), long storage time Lovrić et al. (2013); Heinze et al. (2013); Laplane et al. (2016), efficient temporal Usmani et al. (2010) and frequency multiplexing Saglamyurek et al. (2015), multiple-photon storage and entanglement storage Clausen et al. (2011); Saglamyurek et al. (2011) in various types of REICs.
Europium-doped yttrium orthosilicate Eu:YSiO is one of the most attractive solid-state systems to realize optical quantum memory for quantum repeater application. This is due to the long optical coherence times of a few milliseconds Yano et al. (1991); Equall et al. (1994) which together with excellent spin coherence properties of tens of milliseconds lifetime Alexander et al. (2007) offer the possibility to realize spin-wave storage of photonic states Afzelius et al. (2010). Storage times up to a few milliseconds have been demonstrated using different quantum memory schemes at zero magnetic field Jobez et al. (2015); Laplane et al. (2016, 2017).
Recently, the extension of the spin coherence lifetime in Eu:YSiO up to one minute has been demonstrated using the zero-first-order Zeeman shift (ZEFOZ) condition at high magnetic fields Zhong et al. (2015). The long coherence time is due to the decoupling of the hyperfine transition from magnetic-field fluctuations from the host spin flips Fraval et al. (2004, 2005). Further application of the dynamical decoupling technique using trains of rf-pulses resulted in solid-state hyperfine coherences up to 6 hours Zhong et al. (2015). This clearly demonstrates the potential of Eu:YSiO crystals to realize long-duration quantum light matter interface applicable for quantum communication.
Further use of the ZEFOZ transition for storing optical excitations requires knowledge of optical properties for this material under an applied magnetic field. This is important in order to find a proper energy level path where single photons can be efficiently transferred to spin-wave excitations. The excited state spin Hamiltonians have been previously characterized for other non-Kramers crystals Lovrić et al. (2011, 2012). This information allowed to predict optical transition probabilities between the ground and excited hyperfine levels. However, the magnetic properties of the D excited state of Eu:YSiO crystal have not been fully characterized so far.
In this work, we investigate the hyperfine properties of the excited state D of Eu:YSiO by fully reconstructing its effective spin Hamiltonian. To this end we use an optical free induction decay (FID) method on the optical FD transition, which allows us to measure all hyperfine splittings under weak external magnetic fields (up to 10 mT) applied in various directions. With this approach, all hyperfine splittings can be measured for both the ground and excited states at the same time, which is an efficient method to precisely characterize the relative orientation of the two spin Hamiltonians (for ground and excited states). This is crucial in order to predict optical branching ratios for various optical pumping tasks, like quantum memory applications. Using both Hamiltonians, we are able to calculate the optical transition probabilities for different hyperfine levels of the optical FD transition as a function of the external magnetic field.
The work is organized as follows. In Section II, we present the effective Hamiltonian describing the magnetic properties of hyperfine levels in Eu:YSiO. In Section III, we present the measurement method and the experimental details. Section IV shows the main results: the measurement of ground state and excited state hyperfine splittings as a function of the external magnetic field’s angle and the prediction of the transition probabilities using the fitted parameters. We finally discuss the implications of our findings and give an outlook in Section V.
Ii Hyperfine interaction for REICs
ii.1 Spin Hamiltonian
The hyperfine interaction of rare-earth centers is usually described using a complex Hamiltonian of the form Macfarlane and Shelby (1987)
where the first two terms describe the free ion and the crystal-field (cf), which together characterize the electronic coupling and determine the optical transitions. All other terms describe the hyperfine coupling, the nuclear quadrupole coupling, and the electronic and nuclear Zeeman Hamiltonians, respectively.
In the present work we consider the optical transition of Eu between the ground F (denoted as ) and the excited state D (denoted as ), which for YSiO material takes place at 580.04 nm wavelength. The energy level structure is displayed in Figure 1.
Due to the low symmetry of the YSiO crystal under study and the even number of electrons for the Eu ion, the net orbital angular momentum and the electron spin are quenched Macfarlane and Shelby (1987). This allows to efficiently represent the second group of terms in Eq. (1) as a perturbation for the electronic levels. Due to the quenching, the hyperfine coupling and the electronic Zeeman interactions are not present at the first order, which leads to the same order of magnitude for all the terms inside the second brackets of Eq. (1).
In this expression, the first term corresponds to the quadrupole interaction and is responsible for a partial lifting of the nuclear-spin states degeneracy in both the ground and the excited states for the nuclear spin of Europium (see Figure 1, left). The second term describes the Zeeman interaction and results in non-degenerate hyperfine levels in the presence of a magnetic field (see Figure 1, right).
As the energy splittings due to are very small compared to the optical transition, this term can be seen as a perturbation of the whole Hamiltonian. Two hyperfine Hamiltonians can be defined: one for the ground state and one for the excited state . The hyperfine ground state Hamiltonian has already been determined in a previous work Longdell et al. (2006). We are thus interested in the present work in characterizing the Hamiltonian of the excited state and its orientation with respect to the ground state Hamiltonian. This is done by determining experimentally and , that is, the quadrupole and Zeeman tensors of the excited state hyperfine Hamiltonian.
ii.2 Symmetry considerations in YSiO
In the present work we study only one of the stable Europium isotope, particularly Eu. While two isotopes Eu and Eu appear in approximately equal concentrations, their magnetic properties are slightly different. The larger electric quadrupole moment of Eu usually results in larger zero field splittings, while nuclear magnetic properties are usually stronger for the Eu isotope Erickson and Sharma (1981); Liu (2005). For quantum information applications the Eu isotope can offer a larger optical bandwidth and potentially longer coherence times, however the magnetic field requirements for ZEFOZ transitions for this isotope are more demanding Zhong et al. (2015).
YSiO is a monoclinic biaxial crystal of the space group. When Eu ions substitute Yttrium Y ions they can occupy two different crystallographic sites. Here we study the crystallographic site which offers a higher absorption coefficient and a longer optical coherence time Yano et al. (1991). For this site, Europium can also occupy two magnetically inequivalent subsites, and the Hamiltonians of these two subsites are related by a -rotation around the symmetry axis of the crystal. This means that two quadrupole tensors and and two Zeeman tensors and must be defined, one per magnetic subsite. Note that the two magnetic subsites become equivalent when an external magnetic field is applied along the symmetry axis or in the plane perpendicular to it. The crystal was cut along the polarization extinction axes and Li et al. (1992), where coincides with the crystallographic symmetry axis.
To summarize, in this work we determine the two tensors and in the (, , ) basis by measuring the splittings of the hyperfine excited state due to the presence of a magnetic field. The two tensors and are then deduced by a -rotation around the axis (see Appendix A for more details). Since the point symmetry at the site of Eu in YSiO crystal is , the tensor axes for each interaction type can be arbitrarily oriented with respect to each other and with respect to different electronic states. This makes the characterisation of their relative orientations in different electronic states a sophisticated problem.
Iii Experimental Methods
Several experimental methods can be used to measure the ground and excited state splittings. The most common techniques combine optical and radio-frequency (rf) fields, such as Raman Heterodyne Scattering (RHS) Longdell et al. (2006); Lovrić et al. (2012). This method requires an efficient coupling between rf-radiation and the spin transition under study. This is challenging for the excited state D due to the large quadrupole splittings and the weak Rabi frequencies. Preliminary RHS signals we recorded were weak and difficult to use for a quantitative analysis.
To overcome these technical limitations, we use spectral hole burning (SHB) in this work. With SHB, one can measure simultaneously the ground and excited state splittings with a single absorption measurement and without using rf fields. A difficulty using SHB is the interpretation of the complicated SHB spectrum. To solve this problem we use a technique called class cleaning, which we now describe in detail.
iii.1 Class cleaning for SHB at the Zeeman level
The general idea of SHB is the following: Given that the inhomogeneous broadening of the FD transition is large compared to the hyperfine splittings, sending a pump laser of fixed frequency on the ensemble for a much longer time than the radiative lifetime will cause the atoms to be redistributed among the hyperfine ground state levels. For a system with ground state levels and excited states, there will be a total of resonant transitions, corresponding to different classes of atoms. For instance, Figure 2(a) shows the four classes of resonant atoms in the case . The pumping process eventually leads to a spectral pattern of holes and antiholes in the absorption profile, shown in Figure 2(b).
We could try to directly use this technique to probe the different splittings we want to measure, but the spectral pattern would be composed of 31 holes and 930 anti-holes originating from the 36 classes of atoms for each magnetically inequivalent site. Retrieving the excited and ground state splittings would be a challenging task in this case.
Instead of using all the 36 classes of atoms, we perform a class cleaning of the atoms at the quadrupole level Nilsson et al. (2004). This means that by using an appropriate sequence presented in detail in Lauritzen et al. (2012); Laplane et al. (2016) we address a single transition of the kind , with . Since the class cleaning is only done at the quadrupole level, when sending light on a transition, we simultaneously address atoms on the four transitions associated with this system. Hence, we are left with 4 classes of atoms on the Zeeman structure instead of 36, as depicted in Figure 2(a). This leads to the appearance of 3 holes and 6 anti-holes, the positions of which directly give the excited and the ground state splittings Hastings-Simon et al. (2008) (see Figure 2(b)).
For typical splittings lower than 100 kHz the challenge is twofold: First the pumping laser should have a narrower linewidth than the energy splittings, and second the readout of the structure should be very precise in order to resolve it. We will see in the next section how we solve these issues in the present case.
iii.2 SHB spectrum measurement: heterodyne measurement of the FID
As explained previously, our goal is to measure SHB spectra, like the one shown in Figure 2(b), and extract the excited state splittings as a function of the direction of the magnetic field. A first simple idea is to use a readout pulse, whose frequency is chirped over time. The limitation with this solution is that the resolution of the measurement is strongly linked with the chirp rate: as the structure that we want to measure is only a few kilohertz wide, the chirp rate should be very slow. This tends to work with very weak readout amplitudes to avoid hole burning due to the readout pulse, implying measurements with low signal-to-noise ratios.
Instead of a frequency-resolved absorption measurement, we perform a temporal measurement of a signal emitted by the spectral structure we want to measure. In other terms, we excite the spectral structure with a short readout pulse, which will create an optical coherence on the atoms. These atoms will then emit light after the end of the readout pulse: This is the free induction decay (FID) Shelby et al. (1983); de Seze et al. (2005). As a temporal counterpart of the direct spectral absorption measurement, the absorption spectrum is simply the imaginary part of the Fourier transform of the measured FID. This requires that the spectrum of the readout pulse should be large compared to the probed spectral structure.
To measure the FID, we use an interferometric technique called balanced heterodyne detection: we mix the FID field with a 4 MHz-detuned optical local oscillator (LO) on a 50:50 beamsplitter, and measure the difference in photocurrent of two photodiodes placed in its two outputs. The advantage of this method is that the measurement is only limited by the shot-noise of the readout pulse.
iii.3 Experimental setup
In Figure 3(a) we show the experimental setup. Our laser source is a cavity-stabilized source with a sub-kHz linewidth, which emits 2 W of light at 580.04 nm. We use 40 mW for this experiment and split the power into two different beams. The first one, the signal beam, is used to prepare and excite the crystal sample. The second beam is used as the local oscillator for the heterodyne detection. In order to modulate the frequencies and the amplitudes of both the signal and the LO for the implementation of the sequence, acousto-optical modulators (AOMs) in a double pass configuration are used. The AOMs are driven by an analog generator card that performs both amplitude and phase modulation. The signal beam is then recombined on a 50:50 beamsplitter with the local oscillator for the heterodyne measurement, performed by a balanced photodiode detector.
For our study we use a 1 cm long isotopically pure Eu:YSiO crystal with a doping concentration of 1000 ppm. The crystal was grown by Czochralski method and cut from a boule.
To minimize the effect of decoherence processes, the crystal is cooled to 3 K in a commercial closed-cycle cooler from Cryomech, with a custom-made vibration-damping mount. In order to apply the magnetic field necessary to lift the Zeeman degeneracy, we use three pairs of copper coils close to a Helmholtz configuration. The magnetic field is limited to in the and directions and to in the direction, due to heating through the Joule effect. The axes of the coils , and define the lab frame in which the spin Hamiltonian is defined. The crystal axes , and are oriented closely to the , , and axes of the coils, respectively. Further possible misalignment is included in the fitting procedure discussed later.
Each transition that is probed requires a specific preparation procedure: As we want the FID signal to be the strongest possible, we additionally polarize all the spins to the state by optical pumping. These are simply variants of the basic class cleaning procedure discussed in Ref. Laplane et al. (2016).
Figure 3(b) shows the sequence that is used for the experiment. First the direction and amplitude of the magnetic field are set thanks to three independent current sources. Then the atomic preparation occurs, which consists in the class cleaning procedure (see Section III.1) and the pumping procedure previously mentioned. The preparation of the atoms is performed on a bandwidth of 5 MHz. Then, we perform SHB on the ensemble by sending a series of identical and spectrally narrow pulses. This sequence results in burning a structure of the type presented in Figure 2(b), where the holes and anti-holes have a typical width of the order of 10 kHz. We believe that this width is currently limited by the residual vibrations of the crystal during the SHB procedure. Eventually, a single 1.5 s long square pulse is sent as the readout pulse. The beginning of the FID measurement is triggered right after the end of this pulse. The LO is continuously sent to the heterodyne detection, with a detuning of 4 MHz with respect to the readout pulse.
Iv Experimental results
iv.1 Extraction of the absorption profiles
Figure 3(c) shows a typical trace recorded by the oscilloscope: The FID is beating with the LO at 4 MHz, and the slow modulations reveal the existence of a structure in the spectral domain. Nevertheless, if we consider directly the imaginary part of the Fourier transform of the measured signal, we do not recover the expected absorption spectrum: In close analogy to NMR Ernst (1969), we need to apply a linear phase correction to our data. The origin of this phase correction is twofold: first, for each FID measurement the relative phase of the LO is random. A constant phase should then be added for each measurement. Secondly, the measurement does not start right at the beginning of the FID emission. This shift in time implies a linear correction in frequency. Once these corrections have been applied, we obtain the absorption profile shown in Figure 3(d), which is of the same form as the one schematically presented in Figure 2(b).
iv.2 Scanning the magnetic field
In order to reconstruct the two and tensors, we have to know the splittings for several possible directions of the magnetic field. To scan the field homogeneously in space, we use the same method as the one presented in Longdell et al. (2002): we scan the magnetic field along a spiral parametrized by
where , . In our case, the scan of the space occurs along an ellipsoid, because . Since the – plane is roughly parallel to the plane, if we scan around the or axes we will cross the - plane several times. Outside this plane we observe two different SHB spectra as shown in Figure 2(b). Using this fact one can precisely identify the position of the - plane from the spiral measurement. In all of the spiral measurements we present in the article, was chosen to be 200.
In Figure 4, we show the SHB spectra for three transitions between the F and D manifolds, obtained with spiral scans. The hole positions were identified in these rotation patterns manually, by looking at the SHB spectrum for each orientation of the magnetic field along the spiral pattern individually. Whenever possible, the main antiholes would also be identified, however their amplitudes were generally smaller.
iv.3 Fitting procedure
To find the Hamiltonian which explains the observed spectra, we parametrize the effective Hamiltonian (Eq. (2)). Since the diagonal elements of the quadrupolar tensor are known Mitsunaga et al. (1991), we only fit the orientation of this tensor, using three Euler angles , and in the lab frame. Then, the Zeeman part is described by six parameters. They correspond to its three diagonal elements , and and three angles representing the orientation of the tensor in the basis , and . These angles are not the same as for the tensor due to the low site symmetry in the crystal. Eventually, two more parameters are used to identify the orientation of the symmetry axis connecting two magnetically inequivalent subsites: and defined in spherical coordinates in lab frame. A rotation of around this axis for both tensors is used to obtain the Hamiltonian for the second subsite containing and tensors as explained in Sec. II. The exact form of the Hamiltonian and details about the rotation transformations are given in the Appendix A.
In order to determine these 11 parameters we used a standard least squares fitting method.
Using the simulated annealing approach Kirkpatrick et al. (1983) it was possible to ensure that the fit corresponds to a global solution. In addition to this conventional method of analysing the data, in the Appendix C we develop a novel approach based on perturbation theory to facilitate the fitting procedure. Using this approach it is possible to estimate certain set of parameters of the Hamiltonian (specifically the orientation of the tensor and the symmetry axis) before performing a fitting. This in turn simplifies the search of a global solution by reducing the amount of numerical efforts for fitting procedure.
In Figure 4 we show the results of the fit for the ground and excited states along with the experimental data. The results for the ground state are in good agreement with the results obtained in Longdell et al. (2006) up to an additional rotation around the symmetry axis (in the plane). The extracted set of parameters for the ground and excited state Hamiltonians are summarized in Table 1.
The fitted value of confirms that the axis of the crystal was aligned close to the axis of the coils. To verify the validity of the fit parameters we performed the scan in the plane. The results (Figure 5) are in good agreement with predicted spectra and contain only one set of lines (holes and antiholes) due to the fact that both subsites in this plane are magnetically equivalent.
iv.4 Fitting ambiguities from the spin Hamiltonian symmetries
By fitting the recorded spectrum as a function of one cannot determine the spin Hamiltonian without ambiguity, as there is no unique solution. This is due to the fact that the measured spectrum is invariant under certain transformations of the Hamiltonian coming from its symmetries. Some type of the symmetries related, for example, to the global rotations of the interaction tensors and or the order of their diagonal elements is not physically meaningful. However, the type of the symmetry related to the relative signs of the diagonal elements (this transformation can be considered as a mirror reflection) does modify the relative orientations of the interaction tensors (for details see Appendix B).
In general, only absolute values of the diagonal elements of the effective and tensors can be extracted from the fit, which leads to the fact that relative signs of the eigenvalues can not be experimentally determined based on only such a measurement (Appendix B). For example, for each combination of the signs of , and , one obtains different solutions that lead to the same spectrum, but for which the orientation of the tensor is different (see Appendix B). Since the signs of the tensor for the ground and excited states have never been measured for this material we have possible combinations for each state, which means a total of 64 possible solutions.
iv.5 Reducing fitting ambiguities
|parameter||ground state, F||excited state, D|
|, MHz||, MHz||, MHz|
Some assumptions can be made to choose the global sign of both tensors. The nuclear magnetic moment of the ion can be substantially quenched or even inverted due to higher order hyperfine interaction Shelby and Macfarlane (1981). The tensor for the ground state is very anisotropic (see Table 1), so its eigenvalues might differ substantially from the value of the nuclear magnetic moment of the free ion, in particular some values could even be negative. For the excited state D, however, this effect is negligible Shelby and Macfarlane (1981). This is due to the much larger energy spacing for the closest energy level for excited state ( cm for D while only cm for F) which reduces the higher order perturbation effects on the nuclear magnetic moment for D.
The weak perturbation in the excited state is supported by the fact that the eigenvalues of are all similar (isotropic, see Table 1), and close to the magnetic moment of a free ion (1.389 MHz/T) up to the small quenching. Taking this into account we therefore assume that all eigenvalues of are positive. We are then left with 16 possible solutions.
iv.6 Identifying a unique solution from the optical branching ratios
To find a unique solution, one could measure the quadratic Zeeman interaction using SHB, as it is sensitive to the sign of the tensor Macfarlane and Shelby (1981); Silversmith et al. (1986). This approach requires measuring the shift of the spectral hole under strong magnetic fields (1 Tesla). One can also utilize optical branching ratios which are known to be sensitive to the sign and/or absolute value change of the nuclear projection between two electronic states. We use the latter to identify the proper solution.
The optical branching ratios at zero magnetic field were measured in a previous study using tailoring techniques Lauritzen et al. (2012) and are given in Table 2. We verified that the measured table of relative oscillator strengths is equivalent for the Eu isotope at least within the experimental erros given in Ref. Lauritzen et al. (2012). In order to calculate the relative oscillator strength for each transition , we write it as and overlap between nuclear eigenstates . In this expression, is the dipole moment of the optical transition defined by the electronic wavefunctions and is the same for each nuclear spin projection. This is done assuming that the electronic and the nuclear wavefunctions are separable for the ground and excited states, which was confirmed to be a good approximation for non-Kramers ions Mitsunaga et al. (1985); Bartholomew et al. (2016).
The branching ratio table is calculated for each magnetic subsite and the average values are considered (Table 2). By comparing experimental results with all possible combinations (deduced from the assumptions discussed above) obtained from the fitted Hamiltonians we found the solution which gives the best agreement. We note that among the remaining combinations the solution given in Table 1 is the only one which gives the relative oscillator strengths close to the experimental error bars. Other possible solutions are listed in Appendix B.
V Discussions and conclusions
Our analysis gives a solution which inverts the sign for one of the eigenvalues of the tensor (see Table 2). Such a sign change for the nuclear magnetic moment has been observed previously Silversmith et al. (1986), and originates from the well established effect of nuclear magnetic moment quenching Elliott (1957). This effect is caused by the interaction with nearby electronic levels giving rise to the pseudoquadrupole interaction and reduced magnetic moment which can be written as , where is the nuclear magnetic moment of the free Europium ion and is a quenching parameter Macfarlane and Shelby (1987).
By comparing the spectral properties of two different Eu isotopes (Eu and Eu) (see Table 3) one can notice that the ratios between their hyperfine resonances are different for various electronic states, indicating that the pseudoquadrupole contribution is larger for the ground F state Elliott (1957). Indeed, the ratio for the excited state of 2.62 is closer to the quadrupolar moment ratio for free isotopes 2.67 Stone (2015). This is due to the fact that the pseudoquadrupole interaction is proportional to the nuclear magnetic moment of each isotope rather than the nuclear electric quadrupole moment Blok and Shirley (1966).
However, this type of contribution is not big enough to explain the change of the sign for the parameter at the ground state, which leads to the inverted order of energy levels for zero field splittings (see Figure 1(a)). The simple estimation of the pseudoquarupole interaction based on this ratio gives values lower than MHz. For this reason, in order to explain the sign change of one has to take into account the electric field gradient created by the 4f electronic configuration in each electronic state Sharma and Erickson (1985). This type of contribution for Eu ion is defined by the mixing with the second electronic level () but not due to the fact that . This effect will be negligible for the excited D state again due to the much higher energy for D levels ( cm for D Yano et al. (1992) and cm for F levels Könz et al. (2003)).
The spin Hamiltonian that we found is consistent with previous observations for this optical transition in other crystals. However, in order to fully quantify different contributions, one has to know all the crystal field parameters for this material. Due to the low point symmetry of the crystallographic site they are still unknown. This fact leads to additional problems for characterizing anisotropic magnetic properties of different rare-earth dopants in the YSiO crystal.
In conclusion, we have characterized the spin Hamiltonian of the excited state of Eu:YSiO. We have determined all relevant parameters of the nuclear spin Hamiltonian in the electronic excited-state D and characterized its orientation with respect to the ground state spin Hamiltonian. This is particularly important to be able to predict the behavior of optical transitions under external magnetic fields.
Our characterization of Eu:YSiO is in good agreement with previously obtained results for relative optical strengths at zero magnetic fields. We characterized the relative signs between the hyperfine parameters for electronic ground and excited states and identified unique solution compatible with previous results from other crystals. Our results allow the calculation of transition frequencies and relative oscillator strengths for arbitrary magnetic field vectors. This is a crucial requirement in order to use highly coherent spin transitions in this material for the implementation of long lived optical quantum memories combined with extended spin coherence properties for spin transitions.
The authors thank Nuala Timoney and Cyril Laplane for useful discussions, as well as Claudio Barreiro for technical support.
We acknowledge funding from the Swiss FNS NCCR programme Quantum Science Technology (QSIT) and FNS Research Project No 172590, EUs H2020 programme under the Marie Skłodowska-Curie project QCALL (GA 675662) and EUâs FP7 programme under the ERC AdG project MEC (GA 339198).
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Appendix A The Q and M tensors
The tensors and can be diagonalized in their respective principle axis systems. In order express them in the lab frame we apply a rotation with the usual Euler angle convention:
where is the rotation matrix with Euler angles for ZYZ convention
The interaction tensors for the other magnetic subsite are defined using an additional -rotation around the symmetry axis of the crystal and given by
where is the rotation of angle around the axis:
The total rotation from the crystal frame to the lab frame is given by the rotation , where is an additional rotation angle in the plane. It was measured separately using polarization dependent absorption of the crystal.
In this work, we extract the parameters , , , , , , , , and from the measurement of the excited state splittings, in the presence of an external magnetic field. The interaction tensors in the basis can be calculated based on the fitted parameters (Table 1) and are found to be
Appendix B On the symmetry of the Hamiltonian
The main problem is to determine the right orientations of the magnetic tensors of the considered Hamiltonians: quadrupole interaction tensor and nuclear Zeeman interaction tensor . It is the relative orientations of the considered tensors for the ground and excited state which determines the optical transition strength behavior under an external magnetic field. In the case of low symmetry of the crystal site, the orientation of the interaction tensors for different energy levels can be very different. This makes the separate study of the energetic spectra of two states to be insufficient to fully predict optical transition strengths.
Here we show that without information about the signs of the Zeeman nuclear interaction tensor , the recorded eigenspectra can be fitted with different orientations of the quadruple interaction tensor . Since the same is applicable to the excited state, this leads to an uncertainty on the behavior of transition probabilities. Assuming that the signs of the eigenvalues of the tensor are not known, we define the transformation which changes the sign of the -th eigenvalue as
where is a reflection in the plane perpendicular to the -th direction. When applied to one side of , this transformation maps to , that is, the same tensor with same eigenvalues but with different signs. By doing a change of coordinates via , the Hamiltonian now reads
with . Since is also symmetric, the coordinate change does not have a physical influence on the Hamiltonian but leads to a different tensor. As a consequence, for every change of sign for the eigenvalues of the tensor the rotation of the tensor can be found, which in total gives the same experimental spectra. The list of possible solutions for the ground and excited states is given in Table 4. In general, eight different combinations of the signs lead to eight different solutions for each state (some of them can be equivalent). This leads to possible ways to connect each pair of solutions to calculate the branching ratio table.
|Solution||tensor signs||tensor angles||tensor angles|
Appendix C Perturbation theory approach
While the search for the parameters of (Eq. (2)) as presented in Table 1 was done numerically for the exact Hamiltonian, a perturbation theory approach helps to better understand the energy splittings as a function of the magnetic field orientation. It can also be used to facilitate the fitting procedure of the measured spectra involving 11 parameters for another material. The perturbation approach can be used to estimate certain number of parameters which can be further used as an initial guess for the nonlinear fitting. This can substantially decrease the computational time and verify its consistency. In our case, the quadrupole interaction is dominant over the Zeeman term , Eq. (2) (again we only consider one subsite). Then, at the first order, the energy splitting of each degenerate level can be seen as an isolated two-level system. Let us denote the eigenstates of by . The energy splitting is approximately
Note that the degeneracy in leads to an ambiguity in the choice of . Perturbation theory tells us to maximize over all possible eigenbases for each subspace . In other words, one finds that , where and are the maximal and minimal eigenvalues, respectively, of reduced to the subspace spanned by .
Let us discuss the special case of isotropic coupling, . In addition, we work in a reference frame where is diagonal and denote the direction of in this frame by . Then, it turns out that
where the only depends on and the eigenvalues of : . In other words, the energy splitting is proportional to the distance from the origin to the surface of an ellipsoid with principal axes aligned to the eigenbases of and length . Some examples, which were calculated for the extracted spin Hamiltonians, are depicted in Figure 6.
In the laboratory frame, the principal axes of the ellipsoids are rotated by . If we could assume , we could directly identify the unknown angles from the orientation of the ellipsoids in the laboratory frame. This does no longer hold in the case of general . In our case, it turns out that , which means that the orientation of (see Appendix B) is close to the orientation given by the measured values of (see Figure 7).
In summary the procedure to fit the spin Hamiltonian of the form can be described in different steps:
The parameters and of the tensor can be determined from broadband SHB or RHS.
Measuring all the splittings in different directions for each energy level and using perturbation approach, one can estimate the tensor angles from the orientation of the ellipsoids in the laboratory frame as described above. From this, all the required parameters of the tensor are found.
For our crystal, due to the presence of two magnetic subsites, it was necessary to deduce the orientation of the symmetry axis . This orientation can be estimated precisely by looking at the measured spectras and choosing directions of the magnetic field where two splittings coincide or are very close to each other. By extracting their positions, it is possible to get the orientation of the symmetry axis.
From this point, the only parameters which are unknown correspond to the tensor. Six parameters representing three eigenvalues and three rotation angles can be used to fit the data assuming that all its eigenvalues are in the order of magnitude of the nuclear magneton . In this way only six parameters can be used for the first fit which highly simplifies the overall task.
This receipt was verified for our case and in general can substantially decrease the number of numerical efforts in order to realize the search of the proper solution given by the high number of free parameters.