Simulation of the hyperfine-resolved Zeeman spectrum of Eu atoms in a magnetic trap

We present a simulation of the laser absorption spectra of Eu atoms in the a 8 S 7/2 2 y 8 P 7/2 band at 462.7 nm measured previously in a buffer-gas-loaded magnetic trap @ J. Kim et al. , Phys. Rev. Lett. 78 , 3665 ~ 1997 !# . The simulations of the hyperﬁne-resolved Zeeman spectra are based on exact magnetic eigenproperties of the states involved, and yield a complete assignment of all the features observed. This reveals that apart from the M J 5 7/2 state and its hyperﬁne substates, the M J 5 5/2 and 3/2 states are also trapped at B max 5 0.52 T and a temperature of about 250 mK.


I. INTRODUCTION
In a recent experiment, we trapped up to 10 12 ground state Eu atoms at a temperature close to 250 mK in a buffer-gasloaded magnetic trap ͓1͔. The Eu atoms were detected by laser absorption spectroscopy with a resolution sufficient to distinguish many of the hyperfine Zeeman transitions. The assignment of the spectra was based on a simple model calculation which made use of the analytic magnetic eigenproperties of the Eu atoms in the strong-field limit. Here we present a simulation of the Eu trap spectra based on exact eigenproperties. This simulation provides an assignment of all the observed features in the trap spectrum and yields an improved fit to the data.
The principle of buffer-gas loading and a detailed description of our apparatus are given in our previous papers ͓1-3͔. Briefly, our magnetic trap is a linear quadrupole field formed by two superconducting coils arranged in an anti-Helmholtz configuration. In the center between the two coils is a zero field point from which the magnitude of the field increases linearly over the trapping region in any direction. This configuration confines atoms that are in the low-field-seeking states ͑i.e., states whose energy increases with increasing field strength; as a result, they seek regions of minimum field strength where their energy is lowest͒. For a magnetic dipole moment of 7 B ͑Bohr magneton͒, corresponding to the M J ϭ7/2 state of Eu(a 8 S 7/2 ), and a magnetic field strength at the trap edge B max Ϸ0.52 T, the maximum depth of the magnetic trap is 2.4 K. Eu atoms are produced and trapped within a copper cell located in the bore of the magnet. The cell is filled with 3 He buffer gas and its temperature is maintained by thermal contact with a dilution refrigerator. The bottom of the cell is outfitted with a fused silica window to enable optical access. The Eu atoms are created by ablating a solid sample of Eu metal ͑99.9% pure, with the 151 Eu and 153 Eu isotopes in their natural abundances͒ placed within the cell at the edge of the trapping region with a 10 mJ, 7 ns pulse of a frequency doubled yttrium aluminum garnet ͑YAG͒ laser. The trapping procedure starts by raising the temperature of the cell to about 250 mK by a resistive heater ͑resulting in a 3 He number density of about 10 16 cm Ϫ3 ) and firing the ablation laser ͑at which point the heater is turned off͒. The initially hot Eu atoms diffuse through the helium gas and quickly thermalize with it via elastic collisions ͓1͔. The cell temperature first rises over about 1 s due to heating by the ablation pulse but then drops to roughly 170 mK over 20 s. The corresponding 3 He number density decreases to about 4ϫ10 13 cm Ϫ3 as a result of which the trapped Eu ensemble thermally uncouples from the buffer gas and from the cell.
The trapped Eu atoms are detected by laser absorption spectroscopy in the a 8 S 7/2 -y 8 P 7/2 band at 462.7 nm (6 ns lifetime ͓5͔͒. A probe beam is produced using a KNbO 3 crystal to double the output of an actively stabilized Ti:sapphire laser. The typical ͑doubled͒ power used to probe the atoms is 0.1 W. The 3.5Ϯ1.5 mm diameter beam enters at an angle of about 2°with respect to the cell axis and passes through the cell center with an offset of 3Ϯ1.5 mm. The beam then reflects from a mirror at the top of the cell, passes the cell center again, exits the cell, and is detected by a photomultiplier tube. The laser is scanned over the entire absorption band of about 13 GHz at a rate of about 15 GHz/s; this allows for ample data averaging at nearly fixed delay times with respect to the ablation pulse.

II. EXACT HYPERFINE ZEEMAN EIGENPROPERTIES
The total Hamiltonian H of an atom with total electronic angular momentum J and nuclear spin I subject to a magnetic field B takes the form ͓4͔ the field-free Hamiltonian, and the Zeeman Hamiltonian. In the above, J and I are the electronic and nuclear angular momentum quantum numbers, g J and g I the corresponding g factors ͑see Appendix͒, a and b the hyperfine magnetic dipole and electric quadrupole coupling constants, and B the Bohr magneton. The magnetic field has a magnitude Bϵ͉B͉ and its direction determines the space-fixed Z axis. We neglect higher-order interactions ͑such as the magnetic octupole interaction͒. that pertain to that same state but in the highfield limit. In the case of ground state europium atoms ͑see Table I͒, Jϭ7/2 and Iϭ5/2. This leads to (2Jϩ1)(2Iϩ1) ϭ48 wave functions and hence a 48ϫ48 matrix. Diagonalization of this matrix at a given field strength yields the exact hyperfine Zeeman eigenproperties of the atoms.

III. TRANSITION PROBABILITIES
The transition amplitude between states ͉M J Ј ,M I Ј͘ ←͉M J ,M I ͘ due to an electric dipole operator d q is given by where ͗J,M J ,JЈ,M J Ј ;1,q͘ is a Clebsch-Gordan coefficient.
The transition probability ͑line strength͒ is S q ϭA q 2 with qϭ0 or Ϯ1 for parallel and perpendicular transitions, respectively. The relative weights of the parallel and perpendicular transitions are determined by the angle between the local magnetic field vector and the electric vector of the probe laser beam. For the arrangement used the two types of transitions were found to be nearly equiprobable. Figure 1 gives a comparison, for the Fϭ2 family of states of 153 Eu, of the exact eigenenergies with the eigenenergies obtained in the low-and high-field limits, Eqs. ͑A5͒ and ͑A6͒. One can see that the low-field limit applies only at field strengths of up to about 0.001 T while the strong-field limit takes over at about 0.005 T. This is in keeping with the nature of the two limiting cases. The low-and high-field limits obtain respectively from the coupled and uncoupled basis representation of H by neglecting the off-diagonal matrix elements. In the uncoupled basis ͑used here͒, the offdiagonal elements decrease as B increases, thus allowing the high-field limit to be reached. The magnitudes of the matrix elements at a given field strength reflect the interplay between the magnetic term, proportional to B , and the hyperfine constants a and b. Since 1 B ϭ14 GHz/T and both a and b are on the order of MHz, the high-field limit is bound to prevail at BϷ10 Ϫ2 T, as observed. The correlation between the low-and high-field limits for either Eu(a 8 S 7/2 ) or Eu(y 8 P 7/2 ) states is shown in Fig. 2. Figure 3 shows a profile of the magnetic field strength along the probe beam. The underlying magnetic field strength distribution was calculated from the known parameters of the solenoids. The field distribution provides a basis for the evaluation of the Boltzmann factors of the initial atomic states. The distribution of the number density n(B) of atoms in a low-field-seeking state with eigenenergy E(B) у0 at the loading temperature T is given by where n 0 is the number density at the center of the trap. The differential number dN(B) of atoms exposed to a field strength between B and BϩdB is then given by IϭNS. Figure 4 shows a sample spectrum of the trapped ensemble of Eu atoms at 0.52 T trap depth and a loading temperature of 250 mK. The spectrum was measured at delay times with respect to the ablation pulse ranging between 20 and 40 s, i.e., after the 3 He buffer gas was cryopumped away.

IV. RESULTS AND DISCUSSION
An essentially identical spectrum was obtained at delay times ranging between 40 and 60 s after the ablation pulse, just with a poorer signal-to-noise ratio "the atoms were previously determined to leave the trap at an effective two-body loss rate of (2.5Ϯ1.5)ϫ10 Ϫ13 cm 3 /s ͓1͔…. The best signalto-noise ratio was attained for the spectrum at delay times of up to 20 s ͓1͔. However, the relative contributions to the structured features in the spectrum from the 151 Eu and 153 Eu isotopes as well as the linewidths are different from the spec- FIG. 2. Correlation diagram between the low-and high-field limits for the states of either of the stable isotopes of Eu(a 8 S 7/2 ). Also shown is the transition to the y 8 P 7/2 state used to probe the atoms. tra at 40 s and 60 s delay ͑and from the simulation͒. We ascribe this difference to the presence of the He buffer gas, although we are unable to identify the mechanism responsible for it.
A diagnostic spectral feature is the plateau region between Ϫ10 and Ϫ5 GHz, which is mainly composed of the ⌬M J ϭϪ1 transition from the M J ϭ7/2 state. Because this transition has a relatively large magnetic broadening, its shape is indicative of the distribution of atoms at different field strengths. The greater the shift, the higher the field to which the atoms are exposed, as seen in the 20 s spectra of Ref. ͓1͔. In the 40 s spectra, Fig. 4, the plateau is less frequency-shifted, indicating that the Eu atoms have cooled and concentrated in the low-field region at the center of the trap. Figure 5 shows the contributions to the spectrum from the M J ϭ7/2 ͑b͒, 5/2 ͑c͒, and 3/2 ͑d͒ initial states; ͑a͒ provides a summary where the contributions are plotted on the same scale. One can see that most of the spectrum arises from Eu(M J ϭ7/2). However, there is a sizable contribution from the M J ϭ5/2 state, which gives rise to a prominent peak at a shift of about Ϫ2 GHz due to a transition from the M I ϭ5/2 hyperfine state.

V. CONCLUSIONS
We simulated the laser absorption spectra of magnetically trapped Eu atoms in the a 8 S 7/2 -y 8 P 7/2 band using exact eigenenergies and transition probabilities. These were obtained from a numerical diagonalization of the corresponding 48ϫ48 Hamiltonian matrix. The simulation yields a complete assignment of all the features observed in our previous experiment and provides an improved fit to the data. Apart from the M J ϭ7/2 state and its hyperfine substates, the M J ϭ5/2 and 3/2 states are also found to be trapped at B max ϭ0.52 T and a temperature of about 250 mK.  Fig. 4 from the initial states with M J ϭ7/2 ͑b͒, 5/2 ͑c͒, and 3/2 ͑d͒. ͑a͒ shows all three contributions on the same scale (M J ϭ7/2 by full line, 5/2 by dashed line, and 3/2 by dotted line͒.
The transformation between the coupled and uncoupled basis sets is given by the Clebsch-Gordan coefficients The Hamiltonian matrix consists of 13 block-diagonal matrices whose dimensions range between 1 and 6. Each block pertains to a given value of the good quantum number M ϭM J ϩM I .

Low-field limit
In the low-field limit, the angular momenta J and I couple to a resultant total angular momentum FϭJϩI characterized by a quantum number FϭJϩI,JϩIϪ1, . . . ,͉JϪI͉; each value of F has 2Fϩ1 projections M ϭϪF,ϪFϩ1, . . . , ϩF on the space-fixed axis defined by the direction of B. Note that there are ͚ F (2Fϩ1)ϭ(2Jϩ1)(2Iϩ1) coupled states ͉F,M ͘. The eigenenergy is given by ͓4͔ with KϵF͑Fϩ1 ͒ϪJ͑ Jϩ1 ͒ϪI͑ Iϩ1 ͒.

Strong-field limit
In the strong-field limit, the angular momenta J and I are no longer integrals of motion but their projections M J and M I on the space-fixed axis are. Since M J and M I take respectively (2Jϩ1) and (2Iϩ1) The g factors are given by S and L the total electronic spin and orbital angular momentum quantum numbers, and g S the gyromagnetic ratio of the electron.