Anisotropic optical trapping as a manifestation of the complex electronic structure of ultracold lanthanide atoms: the example of holmium

# Anisotropic optical trapping as a manifestation of the complex electronic structure of ultracold lanthanide atoms: the example of holmium

Hui Li, Jean-François Wyart, Olivier Dulieu and Maxence Lepers Laboratoire Aimé Cotton, CNRS, Université Paris-Sud, ENS Cachan, Université Paris-Saclay, 91405 Orsay, France LERMA, Observatoire de Paris-Meudon, PSL Research University, Sorbonne Universités, UPMC Univ. Paris 6, CNRS UMR8112, 92195 Meudon, France
###### Abstract

The efficiency of optical trapping is determined by the atomic dynamic dipole polarizability, whose real and imaginary parts are associated with the potential energy and photon-scattering rate respectively. In this article we develop a formalism to calculate analytically the real and imaginary parts of the scalar, vector and tensor polarizabilities of lanthanide atoms. We assume that the sum-over-state formula only comprises transitions involving electrons in the valence orbitals like , , or , while transitions involving core electrons are neglected. Applying this formalism to the ground level of configuration , we restrict the sum to transitions implying the configuration, which yields polarizabilities depending on two parameters: an effective transition energy and an effective transition dipole moment. Then, by introducing configuration-interaction mixing between and other configurations, we demonstrate that the imaginary part of the scalar, vector and tensor polarizabilities is very sensitive to configuration-interaction coefficients, whereas the real part is not. The magnitude and anisotropy of the photon-scattering rate is thus strongly related to the details of the atomic electronic structure. Those analytical results agree with our detailed electronic-structure calculations of energy levels, Landé -factors, transition probabilities, polarizabilities and van der Waals coefficients, previously performed on erbium and dysprosium, and presently performed on holmium. Our results show that, although the density of states decreases with increasing , the configuration interaction between , and is surprisingly stronger in erbium (), than in holmium (), itself stronger than in dysprosium ().

## I Introduction

The physics of ultracold gases has evolved rapidly and is poised to enter a new promising regime, where complex atomic and molecular species can be cooled and studied extensively. Lanthanide atoms, with a strong magnetic moment and a large orbital angular momentum, are extreme examples of such complex species. In fact, the interest for ultracold lanthanide atoms is motivated by several topics in current research, including ultracold collisions and quantum chaos Frisch et al. (2014); Maier et al. (2015a); Tang et al. (2015a), dipolar quantum gases with large magnetic moment and strong dipole-dipole interaction Lu et al. (2012); Nessi et al. (2014); Frisch et al. (2015); Yao et al. (2015); Baier et al. (2016), many-body quantum systems Maier et al. (2015b); Burdick et al. (2015), exotic quantum phases Fregoso et al. (2009); Aikawa et al. (2014a); Kadau et al. (2016) like stable quantum droplets Ferrier-Barbut et al. (2016); Xi and Saito (2016); Macia et al. (2016), synthetic gauge field Cui et al. (2013); Burdick et al. (2016), and optical clocks Kozlov et al. (2013); Vishnyakova et al. (2014); Sukachev et al. (2016). Recent progress in laser cooling and magneto-optical trapping of high-atomic-number (high-) lanthanides Hancox et al. (2004); Hemmerling et al. (2014), including dysprosium (Dy) Leefer et al. (2010); Lu et al. (2010); Maier et al. (2014); Dreon et al. (2016), erbium (Er) Ban et al. (2005); McClelland and Hanssen (2006); Frisch et al. (2012), holmium (Ho) Miao et al. (2014) and thulium (Tm) Sukachev et al. (2010) is paving the way towards these investigations. In addition, both Bose-Einstein condensates and quantum-degenerate Fermi gases have been produced in isotopes of Dy Lu et al. (2012, 2011); Tang et al. (2015b) and Er Aikawa et al. (2012, 2014b).

The ground level of holmium is characterized by the electronic configuration [Xe] and an electronic angular momentum . Due to the nuclear spin of its only stable (bosonic) isotope Ho, holmium is the atom possessing the largest number of hyperfine sublevels in the electronic ground level, namely . This rich structure is likely to be exploited in quantum information Saffman and Mølmer (2008); Hostetter et al. (2015). Like other lanthanides, the complex electronic structure of holmium induces a large magnetic dipole moment (9 ) that makes it an interesting candidate to investigate the anisotropic interactions between atoms Newman et al. (2011); Kao et al. (2016). Recently the holmium single magnetic atom and holmium molecular nanomagnet was also presented as a competing candidate for the realization of quantum bits Miyamachi et al. (2013); Shiddiq et al. (2016).

Many of the applications listed above involve optically trapped ultracold atoms. The trapping efficiency is determined by the interaction between the atoms and the electromagnetic field Manakov et al. (1986); Grimm et al. (2000). The microscopic property characterizing the atomic response is the (complex) dynamic dipole polarizability (DDP). On the one hand, the field induces a potential energy, i.e. an ac-Stark shift, on the atoms, which is proportional to the real part of the DDP. On the other hand, the field also induces photon-scattering, whose rate is proportional to the imaginary part of the DDP. In ultracold experiments, it is necessary to characterize the photon-scattering rate, as it provokes heating of the sample, and trap losses Grimm et al. (2000). Beyond trapping itself, the real part of the vector and tensor DDPs are also necessary to determine the Raman-coupling strengths between different Zeeman sublevels, which was proposed for the implementation of synthetic gauge fields Cui et al. (2013); Burdick et al. (2016). In our previous works on Er Lepers et al. (2014) and Dy Li et al. (), we have shown that, far from resonant frequencies, the ac-Stark shift only weakly depends on the field polarization and atomic Zeeman sublevel, despite the absence of spherical symmetry in the -electron wave functions. We have unveiled the inverse situation for photon-scattering, as the imaginary part of the vector and tensor DDPs represent significant fractions of the scalar one. This opens the possibility to control the trap heating and losses with an appropriate field polarization. However, the vector-to-scalar and tensor-to-scalar ratio vary strongly from Dy to Er, which is still unexplained.

Understanding the origin of that difference is a major motivation of the present work. Moreover, ultracold experiments may require to characterize the optical trapping of atomic excited levels with energies up to 25000 cm above the ground level. Calculating the DDP of such levels with the sum-over-state formula requires to model highly-excited levels, roughly up to 60000 cm above the ground level, which is a hard task for the most complex spectra of lanthanide atoms. Therefore, in this article, we present a simplified model of the DDP based on the sum-over-state formula, where we suppose that the only contributions come from transitions involving valence electrons like , , or , and where we ignore transitions involving core electrons. Assuming that all the levels of a given configuration have similar energies, we obtain analytical expressions of the DDPs of an arbitrary level, depending on a restricted number of effective parameters. Focusing on the ground level, of configuration [Xe] (, 11 and 12 for Dy, Ho and Er, respectively), we only take into account the excitation from the to the orbital, but not the excitation from the to the orbital. We demonstrate that the real part of the DDP is not influenced by the configuration interaction (CI) between [Xe] and other configurations like [Xe] and [Xe]. Our model also shows that the real part of the vector and tensor ground-level DDPs vanish. By contrast, the imaginary part of the DDPs is very sensitive to CI, and in particular to the weight of the [Xe] configuration in excited levels. We demonstrate that a strong CI mixing tends to increase the vector and tensor DDPs with respect to the scalar one. Surprisingly, CI mixing turns out to be larger for Er than for Ho, and for Ho than for Dy, although the energy spectrum of Dy is the densest one.

In order to check the validity of those conclusions, we perform a full numerical modeling of holmium spectrum, including energy levels, transition probabilities, polarizabilities and van der Waals coefficients, complementing our previous studies on erbium Lepers et al. (2014) and dysprosium Li et al. (). The DDPs and coefficients are calculated using the sum formula involving transition energies and transition dipole moments extracted from our computed transition probabilities. Following our previous work Wyart (2011); Lepers et al. (2014, 2016); Li et al. (), those quantities are calculated using a combination of ab initio and least-square fitting procedures provided by the Cowan suite of codes Cowan (1981) and extended in our group. Therefore we provide a theoretical interpretation of Ho even-parity levels, which especially results in the prediction of the widely unmeasured Landé -factors. Because the spectrum of high- lanthanide atoms in the ground level is composed of a few strong transitions emerging from a forest of weak ones, the sum-over-state formula is appropriate to calculate DDPs and coefficients. It offers the possibility to precisely calculate, with a single set of spectroscopic data, the real and imaginary parts of the scalar, vector and tensor DDPs, in a wide range of wavelengths, especially at 1064 nm, widely used experimentally for trapping purposes.

This article is outlined as follows. We develop our simplified model for the DDP in section II: we first recall useful formulas and especially the relationships between scalar, vector and tensor DDPs and tensor operators (see subsection II.1). Then we calculate the contribution from the levels of a single configuration (see subsection II.2) to the real and imaginary parts of the DDPs, while the two next subsections are devoted to the influence of CI mixing in the DDPs of the ground level of lanthanide atoms. The second part of the paper (section III) deals with the full numerical modeling of holmium spectrum – energy levels, transition probabilities, polarizabilities and van der Waals coefficients (see subsections III.1III.4 respectively). Section IV contains concluding remarks.

## Ii Dynamic dipole polarizability: a simplified model

### ii.1 Polarizability and tensor operators

For non-spherically-symmetric atoms like lanthanides, the ac-Stark shift is a linear combination of three terms, depending on the scalar, vector and tensor polarizabilities, taken at the angular frequency of the oscillating electric field (hereafter denoted “frequency”). The magnitude of each term is a function of the atomic Zeeman sublevel and of the electric-field polarization Manakov et al. (1986). The scalar , vector and tensor polarizabilities can be associated with the coupled polarizabilities , where , 1 and 2 respectively, is the rank of the corresponding irreducible tensor Manakov et al. (1986); Beloy (2009). Namely

 αscal(ω) = −α0(ω)√3(2J+1) (1) αvect(ω) = α1(ω)√2J(J+1)(2J+1) (2) αtens(ω) = α2(ω)√2J(2J−1)3(J+1)(2J+1)(2J+3). (3)

For an atomic level , where is the electronic-angular-momentum quantum number and stands for all the other quantum numbers, the general expression for is

 αk(ω)= √2k+1∑β′′J′′(−1)J+J′′ × {11kJJJ′′}∣∣⟨β′′J′′∥d∥βJ⟩∣∣2 × ⎛⎜ ⎜⎝(−1)kEβ′′J′′−EβJ−iℏγβ′′J′′2−ℏω +1Eβ′′J′′−EβJ−iℏγβ′′J′′2+ℏω⎞⎟ ⎟⎠ (4)

where () are the energies of the levels (), is the reduced transition dipole moment between these two levels, is the natural linewidth of the intermediate level , and the quantity between curly brackets is a Wigner 6-j symbol Varshalovich et al. (1988).

We consider frequencies far from any atomic resonances, i.e. , which is relevant for trapping purposes, and which greatly simplfies Eq. (4). We separate the real and imaginary parts ,

 R[αk(ω)] =2√2k+1∑β′′J′′(−1)J+J′′{11kJJJ′′}∣∣⟨β′′J′′∥d∥βJ⟩∣∣2(Eβ′′J′′−EβJ)δ(−1)k,1−ℏωδ(−1)k,−1(Eβ′′J′′−EβJ)2−ℏ2ω2 (5) I[αk(ω)] =√2k+1∑β′′J′′(−1)J+J′′{11kJJJ′′}ℏγβ′′J′′∣∣⟨β′′J′′∥d∥βJ⟩∣∣2 ×[(Eβ′′J′′−EβJ)2+ℏ2ω2]δ(−1)k,1−2ℏω(Eβ′′J′′−EβJ)δ(−1)k,−1[(Eβ′′J′′−EβJ)2−ℏ2ω2]2, (6)

where we used . Plugging Eqs. (5) and (6) into Eqs. (1)–(3), and introducing the explicit expressions of 6-j symbols (see Ref. Varshalovich et al. (1988), p. 302), we get to the real and imaginary parts of the scalar, vector and tensor contributions,

 R[αscal(ω)] (7) I[αscal(ω)] (8) R[αvect(ω)] (9) I[αvect(ω)] =∑β′′J′′J′′(J′′+1)−J(J+1)−2(J+1)(2J+1)×ℏ2ωγβ′′J′′|⟨β′′J′′∥d∥βJ⟩|2[(Eβ′′J′′−EβJ)2−ℏ2ω2]2 (10) R[αtens(ω)] =−∑β′′J′′3[J′′(J′′+1)−J(J+1)]2−9J′′(J′′+1)+J(J+1)+63(J+1)(2J+1)(2J+3)×(Eβ′′J′′−EβJ)|⟨β′′J′′∥d∥βJ⟩|2(Eβ′′J′′−EβJ)2−ℏ2ω2 (11) I[αtens(ω)] =−∑β′′J′′3[J′′(J′′+1)−J(J+1)]2−9J′′(J′′+1)+J(J+1)+66(J+1)(2J+1)(2J+3) ×ℏγβ′′J′′[(Eβ′′J′′−EβJ)2+ℏ2ω2]|⟨β′′J′′∥d∥βJ⟩|2[(Eβ′′J′′−EβJ)2−ℏ2ω2]2 (12)

Note that in Eqs. (7), (8) and (11) of Ref. Lepers et al. (2014), the sign of the vector polarizabiity is not correct; the error has been fixed in Eqs. (9) and (10) above.

### ii.2 Effect of a single intermediate configuration

In this subsection, we assume that the intermediate levels appearing in Eq. (4) all belong to the same configuration, and that their transition energies can be replaced by a single effective one. Moreover, we assume that the configurations of the and levels differ by the hopping of only one valence electron; in other words, we ignore transitions involving the core electrons. This will yield analytical expressions useful to estimate , and to understand the trapping in some relevant levels, like those belonging to the lowest or the [Xe] configurations.

Many levels of lanthanide atoms can be interpreted in the frame of the coupling scheme. The electronic core, containing the shell, is characterized by its orbital , spin and total electronic angular momentum . The valence electrons belong for instance to the , or shells. This group of electrons is characterized by their orbital , spin and total electronic angular momentum . Then and are coupled to give the total electronic angular momentum of the atomic level. In the present study, we focus on configurations (, 11, 12 for Dy, Ho, Er, respectively) with two valence electrons, including e.g.  or ; but our results can be extended to configurations with 3 valence electrons like or . The full label of the level is therefore , and its electronic parity is . In what follows, we will omit the xenon core [Xe] in electronic configurations.

It is worthwhile to note that the levels of the configuration are better described in the coupling scheme : is firstly coupled with to give , itself coupled with to give . In order to calculate the polarizability of such levels, it is necessary to apply the basis transformation from to coupling schemes Cowan (1981). However, if those levels appear in the sum over , the coupling scheme is sufficient, as all the levels of the configuration are assumed to have the same energy (see paragraph II.2.2).

#### ii.2.1 Transition dipole moment in jj coupling

In the electric-dipole (E1) approximation, the transitions with the strongest dipole moments are those for which one valence electron, say , is promoted to an orbital such that . The angular momenta of the atom must also satisfy the selection rules: or , , or , and or , excluding transitions between couples of angular momenta , whereas the quantum numbers of the core are not modified (, and ). In the frame of the coupling scheme, we can express the reduced transition dipole moment between the levels and as a function of the mono-electronic transition dipole moment expressed as the matrix element of the mono-electronic -operator. We apply the following successive steps Cowan (1981).

By writing atomic levels as the lists of quantum numbers (and similarly for double-primed quantum numbers), we start working with (, , ),

 ×∣∣⟨n1ℓ1n′′2ℓ′′2L′′vSvJ′′v∥d∥n1ℓ1n2ℓ2LvSvJv⟩∣∣2. (13)

Then we go one step further with (, , )

 ∣∣⟨n1ℓ1n′′2ℓ′′2L′′vSvJ′′v∥d∥n1ℓ1n2ℓ2LvSvJv⟩∣∣2 =(2Jv+1)(2J′′v+1){LvSvJvJ′′v1L′′v}2 ×∣∣⟨n1ℓ1n′′2ℓ′′2L′′v∥d∥n1ℓ1n2ℓ2Lv⟩∣∣2, (14)

and with (, , , , ),

 ∣∣⟨n1ℓ1n′′2ℓ′′2L′′v∥d∥n1ℓ1n2ℓ2Lv⟩∣∣2 ×(2L′′v+1){ℓ2ℓ1LvL′′v1ℓ′′2}2∣∣⟨n′′2ℓ′′2∥d∥n2ℓ2⟩∣∣2, (15)

where the ’s are Kronecker symbols, which bring a factor of 2 for equivalent electrons or . Finally,

 ∣∣⟨n′′2ℓ′′2∥d∥n2ℓ2⟩∣∣2 =e2r2n2ℓ2,n′′2ℓ′′2(2ℓ2+1) ×(2ℓ′′2+1)(ℓ′′21ℓ2000)2, (16)

where is a Wigner 3-j symbol, the absolute value of the electronic charge, and .

#### ii.2.2 Real part of the polarizability

We assume that the polarizability of the level , see Eq. (4), involves transitions towards levels belonging to configurations of the kind . By separating the contributions of those configurations, we can write

 αk(ω)=∑n′′2ℓ′′2αn′′2ℓ′′2k(ω), (17)

which relies on two main hypothesis: (i) Transitions to levels of configurations in which one core electron is excited, e.g.  are excluded, as they are often significantly weaker. (ii) Configuration interaction (CI) is totally neglected, both between different configurations of the kind , and with those of the kind . The effect of CI will be addressed in the next subsection.

The central assumption of this work is that the energy differences implying the levels of a given configuration can be replaced by a single effective energy ,

 Eβ′′J′′−EβJ≈ℏωn′′2ℓ′′2. (18)

The validity of this assumption depends on the frequency at which the DDPs are calculated, which should not “fall” into the levels of the configuration. If we denote and their smallest and largest energies, equation (18) is not applicable for

 min(Eβ′′J′′)−EβJ≲¯ω≲max(Eβ′′J′′)−EβJ (19)

where for and respectively. For ground-level Ho, the excluded frequencies, which correspond to the energies of the manifold, roughly range from 23000 to 24000 cm.

By consequence the sum in Eq. (4) is restricted to the quantum numbers , and allowed by electric-dipole transitions. (For configurations with at least one electron, there is obviously only one possible value.) Inserting Eq. (17) into Eq. (5), we can extract the real part

 R[αn′′2ℓ′′2k(ω)] =2(ωn′′2ℓ′′2δ(−1)k,1−ωδ(−1)k,−1)ℏ(ω2n′′2ℓ′′2−ω2) ×√2k+1∑L′′vJ′′vJ′′(−1)J+J′′{11kJJJ′′} ×∣∣⟨n1ℓ1n′′2ℓ′′2L′′vSvJ′′vJcJ′′∥d∥n1ℓ1n2ℓ2LvSvJvJcJ⟩∣∣2. (20)

Using Eq. (13), we obtain

 R[αn′′2ℓ′′2k(ω)] ×√2k+1∑L′′vJ′′vJ′′(−1)J+J′′{11kJJJ′′} ×(2J+1)(2J′′+1){JvJcJJ′′1J′′v}2 ×∣∣⟨n1ℓ1n′′2ℓ′′2L′′vSvJ′′v∥d∥n1ℓ1n2ℓ2LvSvJv⟩∣∣2. (21)

To calculate this expression, we note that the quantum number only appears in angular terms, so that we use the identity (see Ref. Varshalovich et al. (1988), p. 305)

with , as well as the invariance properties of Wigner 6-j symbols with respect to line and column permutations. Applying Eq. (22) with , , , , and , we can get rid of in Eq. (21)

 R[αn′′2ℓ′′2k(ω)] =2(ωn′′2ℓ′′2δ(−1)k,1−ωδ(−1)k,−1)ℏ(ω2n′′2ℓ′′2−ω2) ×√2k+1∑L′′vJ′′v(−1)Jc+2Jv+J′′v+J+k ×(2J+1){11kJvJvJ′′v}{JvJcJJkJv} ×∣∣⟨n1ℓ1n′′2ℓ′′2L′′vSvJ′′v∥d∥n1ℓ1n2ℓ2LvSvJv⟩∣∣2. (23)

At this point, it is worthwhile to note the following fact Angel and Sandars (1968). The definitions of the coupled polarizabilities and , given respectively by Eqs. (4) and (17), are such that they can be written as the reduced matrix elements of the operators and , which are tensors of rank . In particular, one can resort to the Wigner-Eckart theorem Varshalovich et al. (1988) to calculate the coupled polarizability of a level , namely , with a Clebsh-Gordan coefficient (and similarly for ). One can also apply the transformation of tensor operators regarding angular-momentum basis sets; in this respect, equation (23) can be seen as such a transformation,

 ⟨n1ℓ1n2ℓ2LvSvJvJcJ∥R[^αn′′2ℓ′′2k(ω)]∥n1ℓ1n2ℓ2LvSvJvJcJ⟩ ×⟨n1ℓ1n2ℓ2LvSvJv∥R[^αn′′2ℓ′′2k(ω)]∥n1ℓ1n2ℓ2LvSvJv⟩, (24)

 ×√2k+1∑J′′vL′′v(−1)Jv+J′′v{11kJvJvJ′′v} ×∣∣⟨n1ℓ1n′′2ℓ′′2L′′vSvJ′′v∥d∥n1ℓ1n2ℓ2LvSvJv⟩∣∣2. (25)

Coming back to our main purpose, we apply equation (22) twice more: firstly with Eq. (14) to express the sum over , and secondly with Eq. (15) to express the sum over . Doing so, we get to the final expression

 R[αn′′2ℓ′′2k(ω)] = 2√2k+1ℏ×ωn′′2ℓ′′2δ(−1)k,1−ωδ(−1)k,−1ω2n′′2ℓ′′2−ω2(1+δn1n2δℓ1ℓ2)(1+δn1n′′2δℓ1ℓ′′2) (26) × ×

which depends on two effective parameters: the transition frequency and the mono-electronic transition dipole moment .

The rest of Eq. (26) consists in very insightful angular terms. In particular, the 6-j symbols indicate that, if one of the quantum numbers , or is equal to 0, then the vector and tensor polarizabilities, proportional to and ) respectively, vanish. This is for instance the case for lanthanides in their ground level, which is characterized by . In our full numerical calculation of the polarizability Lepers et al. (2014); Li et al. (), we have shown that indeed the vector and tensor contributions are much weaker than the scalar one. Equation (26) tends to confirm that those weak contributions come from transitions in which one electron is excited. Such conclusions are also valid for any level belonging to the lowest configuration , as shown in our previous articles (see Ref. Li et al. () and in subsection III.3).

#### ii.2.3 Imaginary part of the polarizability

For the imaginary part to be relevant, we consider a metastable level , i.e. whose natural linewidth is negligible compared to the photon-scattering rate induced by the electromagnetic field Grimm et al. (2000); Lepers et al. (2014). In practice, this may concern excited levels of the lowest configuration or the levels , which have no decay channel in the E1 approximation (except for the level of Er) Kramida et al. (2015).

As Eq. (6) shows, the imaginary part of the polarizability involves the natural linewidth of intermediate levels ,

 γβ′′J′′ =∑~β~J,E~β~J

where is the transition probability characterizing the spontaneous emission from the level to the level . We focus on the influence of the levels belonging to the configuration . In addition, we assume that the latter levels only decay towards levels belonging to the configuration . Therefore the sum in Eq. (27) runs over the quantum numbers , and . If we express the squared reduced transition dipole moment as in Eq. (13), Equation (27) becomes

 γL′′vJ′′vJ′′ =ω3n′′2ℓ′′23πϵ0ℏc3∑~Lv~Jv~J(2~J+1){~JvJc~JJ′′1J′′v}2 (28)

Since only appears in angular factors, the sum over reduces to the orthogonalization relations of 6-j symbols,

 ∑~J(2~J+1){~JvJc~JJ′′1J′′v}2=12J′′v+1. (29)

By using Eqs. (14) and (15) for the transition dipole moment, we can calculate the sums over and in a similar way, and finally we get to the expression [see also Eq. (16)]

 γL′′vJ′′vJ′′ = ω3n′′2ℓ′′2r2n2ℓ2,n′′2ℓ′′23πϵ0ℏc3(2ℓ2+1)(ℓ′′21ℓ2000)2 (30) × (1+δn1n2δℓ1ℓ2)(1+δn1n′′