Magnetic Moments of Lanthanide van der Waals Dimers
MMagnetic Moments of Lanthanide van der Waals Dimers
Joseph McCann , John L. Bohn , and Lucie D. Augustoviˇcov´a JILA, NIST, and Department of Physics, University of Colorado, Boulder, CO 80309-0440, USA and Charles University, Faculty of Mathematics and Physics,Department of Chemical Physics and Optics, Ke Karlovu 3, CZ-12116 Prague 2, Czech Republic (Dated: February 4, 2021)Loosely bound van der Waals dimers of lanthanide atoms, as might be obtained in ultracold atomexperiments, are investigated. These molecules are known to exhibit a degree of quantum chaos,due to the strong anisotropic mixing of their angular spin and rotation degrees of freedom. Withina model of these molecules, we identify different realms of this anisotropic mixing, depending onwhether the spin, the rotation, or both, are significantly mixed by the anisotropy. These realms arein turn generally correlated with the resulting magnetic moments of the states.
I. INTRODUCTION: MOLECULARCOMPLEXITY
The harmony and quietude of ultracold atoms wereshattered in 2014, with the discovery that collisions oferbium atoms at 300 nK exhibited the telltale signs ofquantum chaos [1], followed by similar observations indysprosium [2, 3] and thulium [4], as well as dysprosium-erbium mixtures [5]. Magnetic field scans revealed notonly an unprecedented host of Fano-Feshbach resonances,but also that the magnetic field locations of these reso-nances appeared to be distributed according to the pre-dictions of random matrix theory, a finding suggestiveof quantum chaos. The observations attest to the un-expected complexity of weakly-bound lanthanide dimermolecules within several GHz of their dissociation thresh-old.This observation is a challenging one to interpretwithin the paradigms of quantum chaos. For one thing,it is not completely clear that the resolution in the ex-periments was capable of detecting all the resonances.Disregarding the very narrow ones may bias the spec-trum toward revealing less chaos than it truly possesses[6]. Further, while a common statistical analysis tool– the Brody function that characterizes the distributionof nearest-neighbor spacings in the spectrum – showedevidence of chaos, its appropriateness has been calledinto question. The claim was made that perhaps thedata were better fit by a semi-Poisson distribution, re-vealing the system to be quasi-integrable [7]. Anothercomplication arises in the unusual nature of the experi-ments. Typically, quantum chaos studies the spectrumand eigenfunctions of a given Hamiltonian. By contrast,the ultracold experiments determine a spectrum of mag-netic field values at which Fano-Feshbach resonances oc-cur. Thus the spectrum identifies a single, zero-energyeigenstate from each of an ensemble of different Hamilto-nians, one for each magnetic field. It is not immediatelyobvious how to view nearest-neighbor spacings in such acircumstance [8].These issues were clarified by a combination of experi-ment and theory that looked at the Fano-Feshbach spec-tra in both Er and Dy, comparing them to the results oflarge-scale scattering computations [3]. It was concluded that the observed Brody parameters corresponding tothe magnetic field spectra could indeed be accounted for.This required appreciating that, not only were the atomscomplex, with many internal spin states, but also thatthese states were strongly coupled by anisotropic inter-actions during their collision [9]. More significantly, thecalculation revealed that the degree to which the energyspectrum is chaotic is contingent on the value of the mag-netic field at which the spectrum is generated. Furtherevidence of order amid the chaos appears in measure-ments of molecular bound states by magnetic field modu-lation spectroscopy near broad Fano-Feshbach resonances[10, 11]. These measurements and subsequent analysisidentify these bound states as being of essentially single-channel, s -wave character, unencumbered by significantcoupling to other angular momentum states.Numerical models of the spectrum can of course re-veal information about the molecules that the exper-iment is not yet privy to. Thus Ref. [12] appliedmore sophisticated statistical tests, including a family ofinformation-theoretic entropies, to the numerical spec-trum from Ref. [3], looking at the energy range 0.5 GHzbelow threshold. Unlike the experiments, this calculationwas able to evaluate a significant region of the energyspectrum at any desired magnetic field, along with theenergy eigenstates. A main conclusion is that the spec-trum exhibits “multifractal” behavior, quantifying thedegree to which the spectrum appears chaotic. More-over, various measures of quantum chaos were seen toincrease as the magnetic field grew larger, and channelmixing increased.Generally, analysis in terms of entropies is useful in lo-cating complex systems along the complexity scale. Fortruly random systems, whose spectra are well-describedstatistically in random matrix theory, this theory pro-vides upper limits to the values of entropies [13]. Manyphysical systems fall short of this upper limit, and havesub-maximal entropies, suggesting that some order re-mains [13, 14]; this is the case for the model of Ref. [12].A consequence of the previous analyses is therefore thatlanthanide dimers just below their dissociation thresh-old are complicated, yes, but they are not thoroughlychaotic. There may be something orderly about them,at least in some eigenstates, that can be expressed in a r X i v : . [ phy s i c s . a t o m - ph ] F e b terms of the familiar ingredients of molecular physics,namely, vibrations, rotations, and spins, and their quan-tum numbers. In favorable cases, this would entail iden-tifying good quantum numbers, or nearly good quantumnumbers, so that an appropriate Hund’s case might beidentified. On the other hand, it may be the case thatin some states this is not possible, and the objects ofquantum chaos theory become the appropriate tools fordescribing the states, or at least specific ensembles ofstates.In this spirit, this article will examine weakly boundvan der Waals Dy dimers, and seek to understand which,if any, quantum numbers remain good, and under whatcircumstances. To give a specific context, we associatethis search to concrete observable quantities of the molec-ular states, namely, their magnetic moments. Theseweakly bound dimers can be created in the laboratory,say by magnetoassociation, and their magnetic dipolemoments can be measured [15]. They may also perhapsbe accessible by microwave spectroscopy. In any event,we pursue statistical aspects of a distribution of mag-netic moments, which in general are uncorrelated to theenergies of the states, but which provide insight into thecomposition of molecular wave functions.In the current article we focus for simplicity onmolecules in zero magnetic field, where the total angularmomentum is conserved. We find that the states withfairly well-defined angular momentum quantum numberstend to be those that are stretched, with near-maximalvalues of certain angular momenta. As a consequence,the more orderly versus more chaotic states can be dis-tinguished, on average, by the values of their magneticmoments. II. SCOPE OF THE PAPER
We contemplate a diatomic molecule composed of twoidentical lanthanide atoms, such as Dy, Er, Tm, etc.In practice, for the calculations below, we will use Dy,and explicitly consider bosonic isotopes with zero nuclearspin. Each of the atoms has spin j and magnetic moment µ = − gµ B j , where µ B = e (cid:126) / m e c is the Bohr magne-ton, expressed in cgs units; and g is the atom’s g -factor,here defined as a positive number. Thus in an appliedmagnetic field that defines the laboratory z axis, the en-ergies of the atom depend on the field as gµ B Bm , forstate | jm (cid:105) . This expression for energy shift is accuratefor fields small enough such that j remains a good quan-tum number.When two lanthanide atoms are combined into a vander Waals dimer, the resulting molecule is a compositeobject with total angular momentum J = j + j + L ,where j i is the spin of the i th atom and L is the or-bital angular momentum of the atoms about their centerof mass. We use L for this quantity to draw the anal-ogy with the partial wave angular momentum in ultra-cold collisions of the atoms. Since the atoms are electri- cally neutral, the orbital motion does not contribute tothe magnetic moment of the molecule, whose moment istherefore µ = − g j − g j . The extreme values of the mo-ment occur when m = m = ± j , whereby the molecularmagnetic moment must lie between ± gµ B (2 j ). For Dy,with j = 8 and g = 1 . ± . µ B .Various states of the molecule will have magnetic mo-ments between these two limits, depending on the detailsof how the state is constructed. The moment is thereforea probe of how the separate angular momenta work to-gether in the molecule.The number of possible energy eigenstates of thesemolecules is vast. To lend focus to the current investiga-tion, we strongly constrain its scope, to a set of molecularstates close to experimental reality. Specifically, we willconsider a pair of spin-stretched Dy atoms initially intheir lowest-energy state | jm (cid:105) = | − (cid:105) . A small fieldmay be applied to remove the degeneracy of the states.These atoms are assumed to collide via an s -wave col-lision with L = 0, hence the total angular momentumof the atom pair is | JM (cid:105) = | − (cid:105) , and this angularmomentum is considered to be conserved in sufficientlysmall magnetic field. It is conceivable, if not necessarilyeasy, to perform a microwave spectroscopy experimentthat associates the free atoms into weakly-bound statesof the Dy dimer. Measurements of the energy E α at twodistinct, small magnetic fields allow the determination ofthe magnetic moment of state | α (cid:105) via µ α = ∆ E α / ∆ B .The statistical distribution of the moments so defined arethe subject of our inquiry. III. MODEL
Given the complexity of the lanthanide van der Waalsdimers, a complete and accurate ab initio theory of theirstructure remains challenging. Nevertheless, guided byprevious models, some of the salient features of thedimers are apparent. The predominant features of theinteratomic interactions at large interatomic separationconsist of the magnetic dipole-dipole interaction and theanisotropic van der Waals interaction, the latter of whichis believed to be primarily responsible for the channelmixing that generates some degree of chaos in the molec-ular spectrum [3]. We will therefore include these in-teractions in some detail, with an emphasis on theirrepresentation in alternative angular momentum cou-pling schemes. By contrast, our representation of theBorn-Oppenheimer potentials will be somewhat moreschematic, as less important to the present analysis.
A. Basis Sets
To attain the total angular momentum J requires cou-pling the spins of the two atoms to their relative orbitalangular momentum. Formally, this can be done in eitherthe lab frame, or else in the body frame of the molecule.In either case we use as an intermediate the total spinangular momentum of the two atoms, j = j + j . Wecontemplate two basis sets for the molecule, resemblingthe Hund’s cases (a) and (b) familiar from the theory ofdiatomic molecules. These basis sets are as follows: Body-fixed frame (BF):
The individual spins j , j arecoupled to a total spin j , with projection Ω along theintermolecular (body frame) axis. The total angular mo-mentum is J and its projection on the lab axis is M .Suppressing the notation j and j , this basis is | j Ω; JM (cid:105) u = | j Ω (cid:105)| Ω JM (cid:105) , (1)where | j Ω (cid:105) = (cid:88) ω ω | j ω (cid:105)| j ω (cid:105)(cid:104) j ω j ω | j Ω (cid:105) , (2) | Ω JM (cid:105) = (cid:114) J + 18 π D J ∗ M Ω ( φ, θ, γ ) , (3) with ω i the projection of spin j i on the interatomic axis.Here the Wigner rotation matrix D is a function of theEuler angles ( φ, θ, γ ) that relate the body frame to the labframe. We include the subscript u for “unsymmetrized,”so that we don’t need to carry around an extra subscriptfor the symmetrized version below. The spins of theatoms are quantized along the body-frame axis, identify-ing this basis set as analogous to Hund’s case (a).Symmetrized according to even exchange of identicalbosons, the basis set becomes | j ¯Ω; JM (cid:105) = 1 (cid:112) δ ¯Ω0 ) (cid:2) | j ¯Ω; JM (cid:105) u + ( − J | j − ¯Ω; JM (cid:105) u (cid:3) . (4)Here ¯Ω = | Ω | is intrinsically non-negative and takes thevalues ¯Ω = 0 , , . . . , min( j , J ). Note that ¯Ω = 0 ispossible only when J is even. Coupled lab-frame (CLF):
The individual spins j , j are coupled to a total spin j , with projection m onthe lab axis: | j m (cid:105) = (cid:88) m m | j m (cid:105)| j m (cid:105)(cid:104) j m j m | j m (cid:105) . (5)The rotation of the molecule is described by the orbitalangular momentum L and its lab projections M L of the atoms about their center of mass, with wave function | LM L (cid:105) = Y LM L ( θ, φ ) = (cid:114) L + 14 π C LM L ( θ, φ ) , (6)where C LM L is a reduced spherical harmonic. These arecoupled into the total angular momentum J with labprojection M : | [ j L ] JM (cid:105) = (cid:88) m M L | j m (cid:105)| LM L (cid:105)(cid:104) j m LM L | JM (cid:105) . (7)This basis is already symmetric under exchange of theidentical bosons, provided that j + L is even. Quantiza-tion of the atomic spins in the laboratory frame identifiesthis basis set as analogous to Hund’s case (b).The two basis sets are related by a unitary transfor-mation with matrix elements (cid:104) j ¯Ω; JM | [ j (cid:48) L (cid:48) ] J (cid:48) M (cid:48) (cid:105) = 2 (cid:112) δ ¯Ω0 ) ( − M − ¯Ω √ L (cid:48) + 1 (cid:18) j L (cid:48) J ¯Ω 0 − ¯Ω (cid:19) δ j j (cid:48) δ JJ (cid:48) δ MM (cid:48) . (8)Thus the various pieces of the Hamiltonian can be castin either basis, as convenient, and easily transformed tothe other as necessary.For the examples considered in this paper, linked to apresumed initial state defined by s -wave scattering with L = 0, our identical bosons can only access states witheven values of j , that is, gerade states of the interatomicpotential energy surfaces. We will impose this restrictionon the results below. B. Hamiltonian
The Hamiltonian can be written as a sum of contribu-tions, H = T + H BO + H dd + H ad + H B , (9)which are, in order: kinetic energy; the Born-Oppenheimer potentials responsible primarily for short-range interactions; the long-range dipole-dipole interac-tion; the long-range anisotropic dispersion interaction;and the magnetic field Hamiltonian. Bearing in mind thetransformations between the basis sets, different parts ofthe Hamiltonian are easy to write in different bases andtransformed to the other as necessary. Thus the interac-tion terms H B O , H d d , and H a d are easily written in thebody frame, while T and H B take simple forms in the labframe.
1. Kinetic Energy
In the usual way, we write the total wave function inthe form Ψ(
R, σ ) = R − f ( R, σ ) where σ denotes all co-ordinates other than the interatomic spacing R . In thiscase the kinetic energy amounts to a radial componentand a centrifugal component, T = − (cid:126) m r d dR + T cent , (10)where m r is the reduced mass of the colliding pair, andthe centrifugal part is diagonal in the laboratory basis, T cent = − (cid:126) L ( L + 1)2 m r R δ j j (cid:48) δ LL (cid:48) δ JJ (cid:48) δ MM (cid:48) . (11)
2. Born-Oppenheimer potentials
The Born-Oppenheimer part is determined, in general,from detailed electronic structure calculations. These have been carried out for Dy and Er, but instead we findit convenient to use simpler, analytic forms as a stand-infor these potentials.The molecular axis is an axis of rotational symmetryfor the interactions among the electrons and nuclei thatmake up the molecule, whereby ¯Ω is a good quantumnumber, and the body frame basis makes sense. Thetotal spin angular momentum j need not be a goodquantum number and different values may be somehowcoupled together. However, the parity of j is good:the gerade states have even j values, and the ungerade states have odd j values. This Hamiltonian is moreoverindependent of the total rotational state of the molecule,hence independent of J and M .We simplify diagonal elements of H B O by employing aset of Lennard-Jones potentials, (cid:104) j ¯Ω; JM | H B O | j ¯Ω; JM (cid:105) = C ( ¯Ω , j ) R − C R . (12)Here each channel as assumed to have the same isotropicvan der Waals coefficient C , as the anisotropy is dealtwith separately. Each diagonal channel may have a dif-ferent C coefficient, which may be drawn from a sta-tistical ensemble so that these potentials have randomscattering lengths, if desired. However, in the results be-low, we employ a particular value of C in all channels.Likewise, it would be possible to generate random ma-trix elements that couple different values of j , but wehave not done so here. This approach exploits the ob-servation of Ref. [3] that the dominant channel couplingoccurs due to the anisotropic van der Waals interaction.For the calculations described below, we use C = 2274au [18], and artificially truncate the potentials at small R using the same value C = 1 . × in all channels.
3. Dipole-dipole Interaction
The dipole-dipole interaction is also naturally de-scribed in the body frame, where it is diagonal in ¯Ω, (cid:104) j ¯Ω; JM | V dd | j (cid:48) ¯Ω (cid:48) ; J (cid:48) M (cid:48) (cid:105) = − √ gµ B ) R − j + j (cid:48) δ ¯Ω¯Ω (cid:48) δ JJ (cid:48) δ MM (cid:48) (13) × ( − j − ¯Ω j ( j + 1)(2 j + 1) (cid:113) (2 j + 1)(2 j (cid:48) + 1) × j j (cid:48) j j j j (cid:18) j j (cid:48) ¯Ω 0 − ¯Ω (cid:19) , where j = j = j . The coupled spin j is not conservedby this interaction, but its parity is.
4. Anisotropic Dispersion Interaction
For large R , the atoms also exert anisotropic dispersionforces on each other. These are evaluated in detail inRef. [19]. The dominant dispersion term is of course theisotropic one described above. Second to this, and the only non-negligible correction in this context, is the termgiven explicitly in the uncoupled, body frame basis by[19] (cid:104) j ω j ω | V ad | j ω j ω (cid:105) = (cid:114) C , R (cid:16) (cid:104) j ω | j ω (cid:105) + (cid:104) j ω | j ω (cid:105) (cid:17) , (14)where C , is a numerical coefficient derived in pertur- bation theory. Adapting this to the coupled body framebasis gives the matrix elements (cid:104) j ¯Ω; JM | V ad | j (cid:48) ¯Ω (cid:48) ; J (cid:48) M (cid:48) (cid:105) = C , R − j + j (cid:48) δ ¯Ω¯Ω (cid:48) δ JJ (cid:48) δ MM (cid:48) × ( − j − ¯Ω √ (cid:113) (2 j + 1)(2 j + 1)(2 j (cid:48) + 1) × (cid:26) j j jj (cid:48) j (cid:27) (cid:18) j j (cid:48) ¯Ω 0 − ¯Ω (cid:19) . (15)Just as for dipoles, ¯Ω is conserved, as is the parity of j ,but j itself is not.The constant C , in front of this expression is subjectto considerable uncertainly in the literature. In general,the strength of the anisotropic dispersion contribution ischaracterized by diagonalizing the matrix (15), exclusiveof 1 /R , and defining ∆ C as the difference between themaximum and minimum eigenvalues. Reported values ofthis constant include ∆ C = 5 . C = 14 au[18], ∆ C = 25 au [9], and ∆ C = 174 au [12]. In theinterest of incorporating significant channel mixing in the model, we will use the last value, which corresponds to C , = − .
5. Magnetic Field Hamiltonian
The magnetic field acts on the magnetic moments ofthe atoms separately, H B = gµ B BT ( j ) + gµ B BT ( j ) . (16)Its matrix elements are conveniently written in the cou-pled laboratory frame as (cid:104) [ j L ] JM | H B | [ j (cid:48) L (cid:48) ] J (cid:48) M (cid:48) (cid:105) = gµ B B (cid:104) ( − j + ( − j (cid:48) (cid:105) δ LL (cid:48) δ MM (cid:48) ( − J − M + J (cid:48) + j + L (17) × (cid:113) j ( j + 1)(2 j + 1)(2 j + 1)(2 j (cid:48) + 1)(2 J + 1)(2 J (cid:48) + 1) × (cid:26) j J LJ (cid:48) j (cid:48) (cid:27) (cid:26) j j jj (cid:48) j (cid:27) (cid:18) J J (cid:48) − M M (cid:19) . This interaction is capable of mixing different values ofthe total angular momentum that differ by 1. j couldalso change by 1, except that its parity must be con-served. Therefore this matrix element is diagonal in j . C. Vibration
Each basis sets above defines a particular realizationof a set of R -dependent diabatic channels, which wouldbe suitable for scattering calculations. We denote forbrevity this set of quantum numbers by the collectiveket | d (cid:105) , which stands for either the body frame channelbasis (1) or else the lab frame channel basis (7). Thewave function R Ψ is acted upon by the Hamiltonian H = − (cid:126) m r d dR + V d ( R ) + V od ( R ) , (18)where V d is a set of diabatic potential curves, consisting ofthe diagonal matrix elements of the Hamiltonian T cent + V BO + V dd + V ad + H B as expressed in this basis, while V od contains all of the off-diagonal matrix elements.Each potential V d possesses a set of vibrational boundstates, given by − (cid:126) m r d f d,v d dR + V d f d,v d = E d,v d f d,v d . (19)The set of states | i (cid:105) ≡ | d, v d (cid:105) = | d (cid:105) f d,v d (20)therefore constitute an approximate set of molecularstates for our lanthanide diatom. These states repre-sent the molecular states as accurately as possible, whilestill retaining rigorously good values of the angular mo-mentum quantum numbers d of the body- or lab-frame,and a well-defined vibrational quantum number. We willrefer to these as the molecular basis states. If they areminimally mixed, then their quantum numbers are still avalid way to express the states of the molecule; if they arestrongly mixed, then they serve to identify what, exactly,is being mixed on the way toward making the moleculechaotic.The wave function can then be expanded in this basis, f = (cid:88) dv d c dv d | dv d (cid:105) . (21)Solving the Schr¨odinger equation amounts to diagonaliz-ing the Hamiltonian H in the extended basis | dv d (cid:105) . Thediagonal elements of this matrix are (cid:104) dv d |H| dv d (cid:105) = E dv d , (22)while those matrix elements explicitly off-diagonal in d are given by (cid:104) dv d |H| d (cid:48) v (cid:48) d (cid:105) = (cid:90) dRf dv d ( R ) (cid:104) d | V od | d (cid:48) (cid:105) f d (cid:48) v (cid:48) d ( R ) (23)and matrix elements of V od can be computed term byterm, knowing the explicit form of the various terms asgiven above.We obtain the molecular spectrum by diagonalizingthe Hamiltonian matrix in these terms. The vibrationalstates are computed in each diabatic channel by a Fouriergrid Hamiltonian method [20], subject to box boundaryconditions at the radius R = 400 a . This truncationmay alter those states within about (cid:126) / (2 m r R ) ≈ . D. Magnetic moments of the basis states: g -factors The basis states defined in the previous section affordthe simplest model of the magnetic moment distributionof the molecules. In the absence of channel coupling, andin the limit of zero magnetic field, all the quantum num-bers remain good. In this case, the magnetic moments ofthe states may be described by analytical formulas.We can write the magnetic moments in terms of g -factors as µ basis i = g ( i ) µ B M. (24)Expressions for the g -factors can then be derived by eval-uating diagonal matrix elements of the magnetic Hamil-tonian. In the body frame these are g ( j , ¯Ω , J ) = ¯Ω J ( J + 1) , (25)while in the lab frame they are g ( j , L, J ) = 12 (cid:20) j ( j + 1) − L ( L + 1) J ( J + 1) (cid:21) . (26)These are, of course, familiar expressions in molecularphysics [21].One can then define a statistical distribution of mag-netic moments for either of these forms, by simply givingthe occurrence of each possible quantum number equalweight. The statistical distribution of the moments inthe body frame would count each value of j from 0 to2 j , counting only even values for the gerade states weconsider here; and values of ¯Ω from j up to 2 j . For Dywith j = 8, this amounts to 81 possibilities. Countingeach such possibility equally would give a distributionwith the mean and standard deviation for the magneticmoments¯ µ b ody = gµ B M j + 312( j + 1) (27) σ ( µ ) b ody = gµ B | M | (cid:115) (28 j + 24 j − j + 2)(2 j + 3)720 j (2 j + 1)Likewise, in the lab frame the quantum number for gerade states will run even values of j from 0 to 2 j , while L runs, also in even values, from | J − j | to J + j , where J = 2 j in the examples considered here. Counting eachpossibility equally gives, for the lab frame,¯ µ l ab = gµ B M j + 312( j + 1) (28) σ ( µ ) l ab = gµ B | M | (cid:115) j + 548 j + 286 j − j (2 j + 1)The mean value of the magnetic moment is the same ineither basis [22], but the standard deviations are quitedifferent.For the j = 8 Dy atom in our examples, and in thestate where J = 16, M = −
16, we find that, in the bodyframe, the mean of the magnetic moment distribution is − . µ B , and the standard deviation of the distributionis 4 . µ B . By contrast, in the body frame the mean ofthe distribution is the same, but its standard deviationis 9 . µ B , significantly larger. Deviations of the distri-bution of the true magnetic moments from these valuescan be viewed as evidence of the mixing of basis states inthe true energy eigenstates. It will be recalled that theseresults are for the particular case of total angular mo-mentum equal to twice the atomic angular momentum, J = 2 j . Other manifolds of states will have analogousstatistical distributions, of course. E. Magnetic Moments of the Fully CoupledMolecule
Realistically, the distribution of magnetic moments canbe strongly modified by channel couplings in the physicalmodel of the molecule. Having the matrix representationin hand, we can compute the magnetic moments in themodel. Generically, at any value of the magnetic field B ,suppose the Hamiltonian is written H = H mol + M B, (29)where H mol is the complete molecular Hamiltonian inzero field and M is a magnetic moment matrix with M B = H B . Suppose we desire the magnetic moments ata magnetic field B . Then we contemplate a perturbationof the field ∆ B and write the Hamiltonian H = H mol + M B + M ∆ B. (30)Let U be the matrix whose columns are the eigenvectorsof H mol + M B , so that the energies of the molecule atfield B are the diagonal elements ofdiag( E α ( B )) = U T ( H mol + M B ) U. (31)Casting the full Hamiltonian in this basis, we get U T HU = diag E α ( B ) + U T M U ∆ B, (32)whereby, in the perturbative limit, the magnetic mo-ments of the states are given by µ α ≈ ∆ E α ∆ B = (cid:0) U T M U (cid:1) αα . (33)This expression is used to calculate the magnetic mo-ments, in the zero-field limit, in the examples below. IV. RESULTSA. Comparison of the basis states
Given the two standard basis sets, in the body and lab-oratory frames, the first question is to inquire which, if either, is a better representation of the full energy eigen-states of the molecule. To this end, we deploy the par-ticipation number, defined as follows. Any eigenstate | α (cid:105) of the Hamiltonian is expressed in a basis | i (cid:105) by | α (cid:105) = (cid:80) i | i (cid:105)(cid:104) i | α (cid:105) . Given this expansion, the participa-tion number is given by [14] D ( α ) = (cid:32)(cid:88) i |(cid:104) i | α (cid:105)| (cid:33) − . (34)This and related entropies, such as the Shannon entropy,serve to measure the deviation of the energy eigenstates | α (cid:105) from the basis states | i (cid:105) from which they are forged.For example, if the energy eigenstate is already uniquelyidentified by the basis state | i (cid:105) , i.e., if |(cid:104) i | α (cid:105)| = 1, then D ( α ) = 1; only a single basis state participates. Alterna-tively, if n states equally participate and |(cid:104) i | α (cid:105)| = 1 / √ n for each of them, then D ( α ) = n counts them. In thispaper we prefer the participation number to the Shannonentropy because of the significance of the value D ( α ) = 1in identifying states of good quantum number.We have calculated an exemplary spectrum, using themodel described in the previous section, in terms of boththe body and the laboratory basis set, assuming zeromagnetic field. These are converged so as to give thesame spectrum for both calculations. In Figure 1 we plotthe participation number of the states versus the energyof the state, for the part of the spectrum lying 10 GHzbelow the dissociation threshold. This is shown for boththe coupled body frame basis set (a) and the lab framebasis set (b). -10 -5 0 E (GHz) D () (a) -10 -5 0 E (GHz) D () (b) FIG. 1. Participation number D ( α ) of each energy eigen-state | α (cid:105) in the model versus the energy E α of the state. D ( α ) is computed with respect to the coupled body frame | j ¯Ω; JM (cid:105) in (a); and with respect to the coupled lab framebasis | [ j L ] JM (cid:105) in (b). It is immediately clear that D ( α ) is greater for thebody frame basis than for the lab frame basis, thus thelatter is more likely a reasonable description of the states.This comparison affords complementary perspectives onthe origin of chaos. In the body frame, potential in-teractions such as the model Born-Oppenheimer curves,the dipole-dipole interaction, and the anisotropic van derWaals interaction, are diagonal in the quantum number¯Ω, which ought to make this quantum number appro-priate for the description of the states. However, nearthreshold the molecule, rotating with high angular mo-mentum, is subject to strong Coriolis coupling, whichthoroughly mixes the different ¯Ω states. From this pointof view, chaos arises from couplings due to kinetic energy,From the other perspective, in the lab frame the ki-netic energy is already diagonal in the rotational quan-tum number L . The states of different L are mixed by thepotential interaction terms, primarily the anisotropic vander Waals interaction. This is a less significant mixing ofthe basis states, as evidenced by the smaller participationnumber. We may therefore try to identify the magneticmoments in terms of the laboratory-frame g -factors inEqn. (26). FIG. 2. Distributions of magnetic moments in the model Dy molecules, at various levels of approximation. Left panels:body frame. Right panels: lab frame. Top row: statisticalmoments. Middle row: basis moments (see text). Bottomrow: the final, physical moments, which are of course thesame in both calculations. B. Magnetic moments in the coupled states
The superiority of the lab frame over the body frameis shown in more detail by considering the distribution ofmagnetic moments. To see this, we present in Fig. 2 var-ious distributions of magnetic moments for the J = 16, M = −
16 state of Dy , computed at various levels ofapproximation. In the first column, panels (a), (b), (c),we have results calculated in the body frame basis. Inthe second column, panels (d), (e), (f), are results fromthe lab frame calculation. In each case, the first row represents the statistical distribution of magnetic mo-ments, as given by Equations (25) and (26), weightingthe occurrence of each possible quantum number equally.The middle row describes the distribution of “basis mo-ments,” those that belong to the ro-vibrational states de-fined in (20). Finally, the third row of Figure 2 includesthe physical distribution of magnetic moments, includingall channel couplings. It is of course the same in panels(c) and (f), as the physical result does not care for thebasis used to calculate it. Recall that the moments forstates of Dy with J = 16 should lie between ± . µ B .This entire range is represented in the actual moments,although they are biased toward negative values for the M = −
16 state considered.Panels (a) and (d) represent the statistical momentdistributions, which have the means and standard devi-ations given by (27,28), respectively. The body framemoments in (a) are heavily weighted near zero, since thevalue of ¯Ω = 0 occurs many times in this set of quantumnumbers, once for each value of j . The distribution in(a) is far from the physical distribution in (c) since, asnoted above, states with different ¯Ω get strongly mixedby Coriolis forces in the real molecule. By contrast, thestatistical distribution of magnetic moments in the labframe, (d), already resembles the final distribution in (f).The lab frame g -factors are already a good first guess atthe molecular moments, but differ in details.The second row adds a little bit to the physics ofthe molecules, by incorporating the vibrational structurewhile still assuming rigorously good quantum numbers ineither basis. Because vibrational motion in the diabaticpotential energy surfaces is considered, the energies shiftsomewhat and so do the magnetic moments, from the sta-tistical distribution. These shifts do not affect the bodyframe moments much, i.e., panels (a) and (b) are similar.In the lab frame, the main difference between the sta-tistical moments and the basis moments is that the basismoments in panel (e) tend to favor lower values thanthe statistical moments in panel (d). These basis mo-ments include vibrational motion in R , hence are influ-enced by (among other things) the centrifugal potential (cid:126) L ( L + 1) / m r R in each channel. For larger valuesof L , the distance between the inner and outer turningpoints of the diabatic potential V d are closer together,contributing to higher radial kinetic energy. As a con-sequence, the vibrational spacing is larger and there arefewer states of high- L to be found in the energy intervalconsidered. According to (26), these high- L states tendto correspond to lower g -factors, or higher magnetic mo-ments for the M = −
16 states we consider here [see (24)].Hence, states of higher L are less common in (e) than in(d), with the consequence that there are fewer positivemagnetic moments.The physical distributions of magnetic moments, in-cluding the full channel coupling, are given in panels(c) and (f) for this model. This calculation includesall the additional off-diagonal coupling between the ro-vibrational states used in panels (b) and (e). These cou- body frame lab frame¯ µ , statistical -3.49 -3.49 σ ( µ ), statistical 4.35 9.32¯ µ , basis -4.18 -7.37 σ ( µ ), basis 4.88 8.64¯ µ , full -7.23 -7.23 σ ( µ ), full 8.28 8.28TABLE I. Mean, ¯ µ , and standard deviation, σ ( µ ), for the distributions of magnetic moments shown in Figure2. plings influence the body frame moments dramatically,and the lab frame results less so. In either case, however,the distributions in the fully coupled calculations must bethe same, as seen by the means and standard deviationsof the moments presented in Table I. Taken together, weconclude that in the lab frame basis, states with goodvalues of the quantum numbers j and L , distributedas in Fig. (2e) already very nearly comprise the correctdistribution. C. Reduced density matrices
The laboratory frame is unambiguously the better setof quantum numbers to describe the molecules and theirmagnetic moments. The basis is not perfect, however;mixing of these states really does occur. A further lookinto the structure of the molecules would investigatewhich degrees of freedom are most strongly mixed andwhich are weakest, i.e., which of the several degrees offreedom possesses the best quantum numbers.To quantify the goodness of a given quantum number,we employ additional concepts from information theory.In the first step, we cast the problem in the language ofdensity matrices. In terms of the expansion coefficients (cid:104) α | i (cid:105) of the state | α (cid:105) in the basis | i (cid:105) , we construct a di-agonal density matrix with elements ρ ii (cid:48) ( α ) = |(cid:104) α | i (cid:105)| δ ii (cid:48) . (35)This has the essential property that a density matrixshould possess, namely, Tr( ρ ) = 1. This ρ would beanalogous to a pure state if it had only a single nonzeroelement, whereby we would have Tr( ρ ) = 1. More gen-erally, Tr( ρ ) falls short of unity, and the occurrence ofmultiple basis states in the eigenstate corresponds to thedensity matrix representing a mixed, as opposed to apure, state. This is indeed how one makes the intellec-tual transition to the entropy, given as S ( α ) = − Tr( ρ ln ρ ) . (36)Casting the state in terms of this apparent density ma-trix allows us to extract reduced density matrices for thedifferent degrees of freedom. For example, the laboratorybasis is indexed by its quantum numbers | i (cid:105) = | j , L, v (cid:105) (assuming fixed J , M ). Then we can extract the reduceddensity matrix in, say, the j quantum number via¯ ρ j ( α ) = (cid:88) L,v ρ j ,L,v ; j ,L,v ( α ) = (cid:88) L,v |(cid:104) α | j , L, v (cid:105)| . (37)Treating j as the only remaining degree of freedom, wecan assign a reduced entropy to the state, or in our case,a reduced participation number, given by¯ D j ( α ) = (cid:88) j ¯ ρ j ( α ) − = (cid:88) j (cid:88) L,ν |(cid:104) α | j , L, ν (cid:105)| − (38)Low values of ¯ D j ( α ) correspond to states where j is a nearly good quantum number in state | α (cid:105) , regard-less of whether the other quantum numbers are good ornot. The analogous reduced density matrices and partic-ipation numbers ¯ D L ( α ), ¯ D v ( α ) for the other degrees offreedom can be defined analogously. -10 -8 -6 -4 -2 002468 (a)-10 -8 -6 -4 -2 002468 (b)-10 -8 -6 -4 -2 0E (GHz)02468 (c) FIG. 3. Reduced participation numbers for the three relevantquantum numbers in the lab-frame basis set, for the J = 16, M = −
16 states of Dy near threshold. The participation number is shown for the three rel-evant quantum numbers of the lab frame basis set inFigure 3, for the same data as in Fig. 2. Panel (a) shows¯ D v ( α ) for the vibrational quantum number of the atoms.It is almost uniformly equal to unity, except perhaps verynear the dissociation threshold. We conclude that vibra-tional states are only weakly mixed, in this basis, upon0the introduction of potential coupling between the chan-nels.Figure (3b) shows the reduced participation number¯ D L ( α ) for the rotation of the atoms about their center ofmass. We note that for the current model with J = 16, L can take all the even values up to 32, or 7 values inall, whereby ¯ D L ( α ) could conceivably be as large as 7,for thorough mixing of all the L states. While ¯ D L ( α )occasionally approaches this limit for some states, never-theless it is nearly equal to unity for a large fraction ofthe states in this energy range. This is consistent withthe tale told above, that L in the laboratory frame basisset is appropriate for describing the states. The fact that¯ D L ( α ) is often close to one is evidence that the states arenot thoroughly chaotic, as they do not strongly mix thedifferent L states. It is significant, however, that somestates apparently do mix various L states.The real mixing of basis states occurs for the totalspin angular momentum j , whose reduced participationnumber ¯ D j ( α ) is shown in (3c). For the Dy modelconsidered, j can take even values form 0 to 2 j = 16,or nine values in all, setting an upper limit to the valueof ¯ D j ( α ). This limit is never quite achieved for thesestates, but there is certainly more scatter in the valuesof ¯ D j ( α ) than there is for ¯ D L ( α ). It appears, therefore,that the greatest channel mixing that contributes to thechaotic behavior of the molecule lies in the mixing of theatomic spins. Nevertheless, even in this case there existstates with good values of j , where ¯ D j ( α ) ≈ D. Regularities of the eigenstates
It is instructive to plot the participation numbers forthe angular momentum degrees of freedom in an alter-native way, as in Figure 4. Panels (b) and (c) show, re-spectively, the reduced participation numbers ¯ D L ( α ) and¯ D j ( α ) for each eigenstates, as a function the magneticmoment of that state. As a reference, panel (a) repeatsthe histogram of the magnetic moment distribution fromFig. 2.In this figure a semblance of order emerges in the corre-lation between participation number and magnetic mo-ment. Namely, as shown in Fig. 4(b), states with thehighest magnetic moments, down at least to the mean¯ µ = − . µ B , have low participation number ¯ D L ( α ) –they are states where L is a good quantum number. Thestates with lower magnetic moments are more often mix-tures of states with different L -values. Likewise, a cleartrend emerges in Fig. 4(c). States with extreme valuesof µ , either high near 20 µ B or low near − µ B , tend tocontain few j values. By contrast, states with interme-diate values, around the mean ¯ µ , mix together several j states.These results are qualitatively explained using semi-classical angular momentum coupling diagrams [23], asin Figure 5. In this diagram the ˆ z direction of the labo-ratory axis points upward, whereby the angular momen- FIG. 4. Reduced participation numbers for j and L , for the J = 16, M = −
16 states of Dy near threshold. These quan-tities are plotted versus the magnetic moment of the states,also counted in the histogram in panel (a). tum state J = 16, M = −
16 is represented by the arrowspointing straight down in all figures. (In the semiclassi-cal representation this vector would make a small anglecos − (16 / (cid:112) (16 × L and the total spin j , in-dicated in the figure for positive, negative, and near-zerovalues of magnetic moment in (a), (b), (c), respectively.In each case, these angular momenta determine the rela-tive orientation of the atoms (grey circles), and indicatethe kind of orientation of the spins (thick red arrows). Inthis picture, the more up the spins are allowed to point,the greater their magnetic moment.Consider first the case of large positive magnetic mo-ment Fig. 5(a). To achieve this result, the spin angularmomentum is near its maximum value j = 16. Thereare few comparable j states available to mix together,so in this limit so j remains a reasonably good quantumnumber. In order to attain the total angular momentum J , the rotational angular momentum must be near itsmaximum value L = 32, but pointing in the other di-rection. Consequently, it is not mixed with many other L -values either, and both j and L remain good quan-tum numbers for large magnetic moment.Next consider large negative magnetic moment,Fig. 5(b). To achieve this result, j must again be nearits maximum magnitude, but with classical vector point-1
Every chaotic system, governed by random matrix the-ory, is chaotic in the same manner. But each systemthat is only partially chaotic experiences chaos in itsown way. Here we have explored the zero-magnetic-fieldspectrum of a set of lanthanide dimer van der Waalsstates, to locate where their chaos resides. Among the | JM (cid:105) = | − (cid:105) states considered for Dy , we find thatstates belong to one of three realms of qualitatively differ-ent chaoticity, loosely correlated to the magnetic moment µ of the state. States with the highest values of µ tend tobe non-chaotic and described by the quantum numbers j and L ; states of the lowest µ are mildly chaotic dueto mixing of the orientation of the molecular rotation L ;and states with intermediate values of µ near the meanare “just right” for chaos, capable of mixing both thespin and rotation states.This report has dealt only with molecules associatedwith the spin-stretched atomic states | jm (cid:105) = | − (cid:105) mostclosely allied with experiment, but many other manifoldsof states exist. Future work should be able to find simi-lar systematics in the spectra and establish a zoology ofvan der Waals lanthanide dimers. More significantly, theresults remain to be extended to the case of nonzero mag-netic field. One presumes the appearance and pattern ofchaos may take different forms when states of differentangular momentum J are coupled and the molecules be-come overall more chaotic [3, 12]. In this context it isworth noting that even more exotic states of lanthanidedimers have been proposed, which possess large electricdipole moments as well as large magnetic dipole mo-ments, providing additional opportunities for introducingand probing chaos [24, 25].In the broader sense, these results imply the abilityto identify molecular states with qualitatively differentmanifestations of chaos by virtue of their magnetic mo-ment. This ability can be useful in dynamical studies ofthese chaotic molecules. For example, having preparedthe molecule in a particular state, a sudden quench to adifferent magnetic field value will project this state ontoa host of other energy eigenstates and will initiate dy-namics. Knowing what the states are likely to be like,one can imagine different quenches to and from moleculesthat are either rotationally chaotic, spin-chatoic, or both.The richness of the resulting dynamics remains to be con-templated. ACKNOWLEDGEMENTS
We gratefully acknowledge useful discussions with M.Lepers. This material is based upon work supported bythe National Science Foundation under Grant NumberPHY 1734006 and Grant Number PHY 1806971. L.D.A.acknowledges the financial support of the Czech ScienceFoundation (Grant No. 18-00918S).2 [1] A. Frisch et al. , Nature , 475 (2014).[2] K. Baumann, N. Q. Burdick, M. Lu, and B. L. Lev, Phys.Rev. A , 020701(R) (2014).[3] T. Maier, H. Kadau, M. Schmitt, M. Wenzel, I. Ferrier-Barbut, T. Pfau et al. , Phys. Rev. X , 041029 (2015).[4] V. A. Khlebnikov, D. A. Pershin, V. V. Tsyganok, E.T. Davletov, I. S. Cojocaru, E. S. Fedorova, A. A.Buchachenko, and A. V. Akimov, Phys. Rev. Lett. ,213402 (2019).[5] G. Durastante, C. Politi, M. Sohmen, P. Ilzh¨ofer, M. J.Mark, M. A. Norcia, and F. Ferlaino, Phys. Rev. A ,033330 (2020).[6] J. Mur-Petit and R. A. Molina, Phys. Rev. E , 042906(2015).[7] K. Roy et al , Europhys. Lett. , 46003 (2017).[8] L. D. Augustoviˇcov´a and J. L. Bohn, Phys. Rev. A ,023419 (2018).[9] S. Kotochigova and A. Petrov, Phys. Chem. Chem. Phys. , 19165 (2011).[10] T. Maier, I. Ferrier-Barbut, H. Kadau, M. Schmitt, M.Wenzel, C. Wink, T. Pfau, K. Jachymski, and P. S. Juli-enne, Phys. Rev. A , 060702(R) (2015).[11] E. Lucioni, L. Tanzi, A. Fregosi, J. Catani, S. Gozzini,M. Inguscio, A. Fioretti, C. Gabbanini, and G. Modugno,Phys. Rev. A , 060701(R) (2018).[12] C. Makrides, M. Li, E. Tiesinga, and S. Kotochigova, Sci.Adv. , eapp8308 (2018).[13] V. K. B. Kota, Embedded Random Matrix Ensembles inQuantum Physics: Lecture Notes in Physics, Vol 884 (Heidelberg, Springer, 2014). [14] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol,Adv. Phys. , 239 (2016).[15] A. Frisch, M. Mark, K. Aikawa, S. Baier, R. Grimm,A. Petrov, S. Kotochigova, G. Qu´em´ener, M. Lepers, O.Dulieu, and F. Ferlaino, Phys. Rev. Lett. , 203201(2015).[16] P. Jizba and T. Arimitsu, Ann. Phys. , 17 (2004).[17] H. Li, J.-F. Wyart, O. Dulieu, S. Nascimb`ene, and M.Lepers, J. Phys. B , 014005 (2017).[18] M. Lepers, private communication.[19] M. Lepers and O. Dulieu, in Cold Chemistry: Molecu-lar Scattering and Reactivity Near Absolute Zero , The-oretical and Computational chemistry Series No. 11, O.Dulieu and A. Osterwalder, eds, Royal Society of Chem-istry, 2018.[20] C. C. Marston and G. G. Balint-Kurti, J. Chem. Phys. , 3571 (1989).[21] J. Brown and A. Carrington, Rotational Spectroscopy ofDiatomic Molecules (Cambridge University Press, 2013).[22] Note that this mean value is given by the averageof the diagonal elements of the operator M , that is,(1 /N ) (cid:80) Ni =1 (cid:104) i |M| i (cid:105) = (1 /N )Tr( M ), and the trace of theoperator is basis-independent.[23] R. N. Zare, Angular Momentum: Understanding Spa-tial Aspects in Chemistry and Physics (New York, Wiley,1986).[24] M. Lepers, H. Li, J.-F. Wyart, G. Qu´em´ener, and O.Dulieu, Phys. Rev. Lett. , 063201 (2018).[25] H. Li, G. Qu´em´ener, J.-F. Wyart, O. Dulieu, and M.Lepers, Phys, Rev. A100