Divalent Rydberg atoms in optical lattices: intensity landscape and magic trapping

# Divalent Rydberg atoms in optical lattices: intensity landscape and magic trapping

Turker Topcu and Andrei Derevianko Department of Physics, University of Nevada, Reno, NV 89557, USA
July 11, 2019
###### Abstract

We develop a theoretical understanding of trapping divalent Rydberg atoms in optical lattices. Because the size of the Rydberg electron cloud can be comparable to the scale of spatial variations of laser intensity, we pay special attention to averaging optical fields over the atomic wavefunctions. Optical potential is proportional to the ac Stark polarizability. We find that in the independent particle approximation for the valence electrons, this polarizability breaks into two contributions: the singly ionized core polarizability and the contribution from the Rydberg electron. Unlike the usually employed free electron polarizability, the Rydberg contribution depends both on laser intensity profile and the rotational symmetry of the total electronic wavefunction. We focus on the Rydberg states of Sr and evaluate the dynamic polarizabilities of the 5ss() and 5sp() Rydberg states. We specifically choose Sr atom for its optical lattice clock applications. We find that there are several magic wavelengths in the infrared region of the spectrum at which the differential Stark shift between the clock states (5s() and 5s5p()) and the Rydberg states, 5ss() and 5sp(), vanishes. We tabulate these wavelengths as a function of the principal quantum number of the Rydberg electron. We find that because the contribution to the total polarizability from the Rydberg electron vanishes at short wavelengths, magic wavelengths below 1000 nm are “universal” as they do not depend on the principal quantum number .

###### pacs:
37.10.Jk, 32.10.Dk, 32.80.Qk, 32.80.Rm

## I Introduction

Quantum information processing (QIP) with neutral atoms has a number of distinct and appealing advantages, such as scalability, massive parallelism, long coherence times and reliance on well-established experimental techniques. Historically, the dominant fraction of QIP schemes with neutral atoms has focused on alkali-metal atoms, which possess a single valence electron outside a tightly-bound atomic core. The most successful experimental demonstration UrbJohHen09 (); GaeMirWil09 () of quantum two-qubit logic gate with neutral atoms has been carried out using Rydberg gates, originally proposed in Ref. JakCirZol00 (). These experiments employed Rydberg excitations of Rb, an alkali-metal atom. In recent years, there have been important new developments with cooling and trapping divalent atoms, such as group-II atoms (e.g., Mg, Ca, Sr) and group-II-like atoms such as Yb, Hg, Cd, and Zn, which can greatly benefit experiments. Considering the experimental success of Rydberg gates with alkali-metal atoms, it is natural to ask if the distinct properties of divalent atoms could improve the experimental feasibility GilMukBri13 (); LocBodSad13 (); ValJonPot12 (); MukMilNat11 (); MilLocCor11 ().

Optical trapping is essential for QIP experiments due to long coherence times that can be achieved. In QIP experiments, the size of Rydberg atoms can easily be larger than the lattice constant of the optical lattice. As we recently demonstrated in TopDer13 (); TopDerTuneOut13 (), one-dimensional trapping potential for an alkali-metal Rydberg atom is proportional to the expectation value , where is the wavenumber of the optical lattice laser whose wavevector is aligned with -axis. Here is the lattice laser wavelength, is frequency and is the speed of light. We have termed “landscaping polarizability” as it modulates the free-electron polarizability according to the intensity profile of the optical lattice. The factor has a universal dependence on and in the short wavelength (or high principal quantum number ) limit, and . In this limit, the Rydberg atom is no longer trapped as the optical trapping potential is directly proportional to the landscaping polarizability. Divalent Rydberg atoms can still remain trapped even in this limit.

Indeed, the divalent atoms have the advantage of a second optically active valence electron, which contributes to the total polarizability, making it easier to trap the atoms in a tight optical lattice. Optical trapping of Rydberg atoms also faces challenges associated with the small polarizability of Rydberg states. In this sense, utilization of divalent atoms can greatly simplify the trapping of Rydberg states because the second (non-Rydberg) valence electron sizably contributes to the trapping potential, which is proportional to the atomic polarizability. Due to the resonant structure of polarizability contributed by the non-Rydberg valence electron, the lattice wavelength can be also tuned so as to make the total trapping potential large MukMilNat11 ().

QIP with neutral trapped atoms comes at a price: trapping optical fields strongly perturbs atomic energy levels such that there are uncontrollable accumulations of differential phase between qubit states as atoms move in the traps SafWal03 (); SafWal10 (). In addition, the underlying Stark shift is proportional to the local intensity of the trapping lasers; the shift is non-uniform across the atomic ensemble and it is also sensitive to laser intensity fluctuations. This problem is elegantly mitigated using the so-called “magic” traps DerKat11 (). At the “magic” trapping conditions, two atomic levels of interest are shifted by exactly same amount by the trapping fields; therefore the differential effect of trapping fields simply vanishes for that qubit transition. The idea of such âmagicâ trapping has also been crucial for establishing a new class of atomic clocks: the optical lattice clocks DerKat11 ().

When we go back to neutral alkali-metal atoms and review the literature, we easily find a large body of work on optical trapping and utilization of Rydberg states in quantum information experiments BreCavJes99 (); JakBriCir99 (); YouCha2000 (); LukHem2000 (); MomEckErt03 (). However, the situation is different for divalent atoms and the ideas on involving them in experiments are in their infancy. To our knowledge, two papers so far have considered divalent systems in this setting. Mukherjee et al. in Ref. MukMilNat11 () considered the possibility of studying many-body physics using alkaline-earth atoms trapped in optical lattices. They found magic wavelengths for simultaneously trapping () Rydberg states with the ground state for the Sr atom. In their treatment, the polarizability in the Rydberg state is evaluated by adding the polarizability of the Sr ion to the free electron polarizability, representing the contribution from the Rydberg electron. This treatment neglects averaging of laser intensity profile over the Rydberg wavefunction. Also in Ref. MukMilNat11 (), the splitting into core (Sr) and Rydberg polarizabilities has been carried out in an ad hoc manner. Our rigorous derivation presents below shows that indeed for the () states, the total polarizability splits into independent contributions from the valence and the Rydberg electrons. We will show in this paper that the situation is more complicated in the more general case, and even in the independent-particle approximation, the extent to which the Rydberg electron can contribute to the total polarizability is dictated by the angular momentum of the entire atom. Furthermore, in a recent paper TopDer13 (), we demonstrated that the free-electron polarizability of the Rydberg electron is modulated by the intensity distribution in an optical lattice, and it is this “intensity landscape modulated polarizability” that plays a role in trapping the Rydberg state. It can have both positive and negative values and, in contrast to the treatment of Ref. MukMilNat11 (), is not simply the free-electron polarizability. In another paper OvsDerGib11 (), Ovsiannikov et al. proposed using optically trapped Sr atoms in Rydberg states to probe ambient temperature at 10 mK level in clock experiments. To this end, they showed that the ac Stark shift experienced by the Rydberg electron is modulated by the intensity distribution. In their treatment, the contribution from the Sr ionic core was neglected. Here we combine the complimentary treatments of Refs. MukMilNat11 () and OvsDerGib11 () and rigorously derive and evaluate ac polarizabilities of divalent Rydberg atoms.

In alkali-metal atoms, one can always find magic wavelengths above a certain  TopDer13 () and we find that the same holds true for divalent atoms. We find and tabulate several wavelengths at which magic trapping conditions can be attained for the clock states () and () of Sr with Rydberg states.

The paper is organized as follows: in Sec. II we start by briefly reviewing optical trapping of alkali-metal Rydberg atoms and develop a theoretical understanding using the second order perturbation theory for divalent Rydberg atoms. We break down atomic polarizability individual contributions from the valence and the Rydberg electrons taking the rotational symmetry of the many-body state into account. In Sec. III.1, we evaluate the landscaping polarizability of the Rydberg electron in an state and contrast it with the 5s ground state polarizability of the Sr ion. In order to find magic wavelengths for , , and transitions, we need to accurately calculate the 5s and 5s5p() state polarizabilities. We perform these calculations in Sec. III.2 and show that we can recover the well known magic wavelength at 814 nm at which Sr optical lattice clocks are operated. We then discuss and calculate the Rydberg state polarizabilities for the 5ss() and 5sp() states using that for the Sr ion and the contributions from the Rydberg electron individually in Sec. III.3. Finally, we demonstrate that the magic trapping conditions for these Rydberg states with the clock states can be satisfied at several wavelengths in the IR range and below nm. Universal (-independent) magic trapping is shown to exist for nm. We conclude in Sec. IV with final remarks. Unless specified otherwise, atomic units, are used throughout the paper. We also use the Gaussian system of units for electromagnetic quantities.

## Ii Formalism

In this section, we develop a formalism for computing adiabatic trapping potentials for divalent atoms. The formalism requires both understanding of how Rydberg electron wavefunction averages over lattice laser intensity (landscaping polarizability) and the many-body character of the two-electron states of specific rotational symmetry. In Sec. II.1, we start by reviewing main ideas behind landscaping polarizability for alkali-metal Rydberg atoms introduced in Refs. TopDer13 (); TopDerTuneOut13 (). In Sec. II.2, we discuss atomic structure for two-electron states and then move onto deriving a second order perturbative expression for the polarizability of divalent Rydberg atoms in the velocity gauge. We particularly pick Sr atom due to its wide spread use in optical lattice clocks and well developed experimental techniques for its cooling and trapping in the ground state.

### ii.1 Optical trapping and alkali metal atoms

In a recent paper TopDer13 (), we demonstrated that the trapping potential seen by alkali-metal Rydberg atoms in an optical lattice formed along the -axis can be decomposed into a position dependent term which varies as the position of the atom changes along the optical lattice, and an offset term , which shifts the potential energy by a fixed amount everywhere along the lattice:

 Ur(Z)=U0r+UZrsin2(kZ). (1)

Here is the lattice laser wave vector. It is the -dependent piece of the potential that provides confinement in the -direction, because only this part of the potential can exert force on the atom. For a Rydberg state , the position dependent term and the offset in the trapping potential can be written as

 UZr = F204ω2⟨nlmz|cos(2kz)|nlmz⟩≡−αlscnlmz(ω)F204, (2) U0r = F204ω2⟨nlmz|sin2(kz)|nlmz⟩, (3)

where is the position of the Rydberg electron relative to the nucleus and is the laser field strength. We termed the landscaping polarizability because it convolutes the free electron polarizability with the laser intensity profile:

 αlscr(ω)=⟨cos(2kz)⟩αe(ω). (4)

In the limit , and . Away from this limit, however, exhibits oscillatory behavior and changes sign several times as is increased before eventually the free electron character takes over.

The landscaping polarizability can be evaluated by expanding in terms of irreducible tensor operators (ITO) of rank :

 cos(2kz) =∑K=event(K)MK=0(r), (5) t(K)MK(r) =(−1)K/2(2K+1)jK(2kr)C(K)MK(^r). (6)

Here are the normalized spherical harmonics and are the spherical Bessel functions. Application of the Wigner-Eckart theorem results in

 αlscnlmz(ω) ≡ −1ω2⟨nlmz|∑K=event(K)MK=0|nlmz⟩ (7) = (8)

where the reduced matrix element is given by

 ⟨nl||t(K)||nl⟩=⟨nl||CK||nl⟩∫∞0drP2nl(r)jK(2kr). (9)

Here are radial orbitals and the reduced matrix elements can be expressed in terms of the 3- symbols VarMosKhe89 (). We have shown in TopDer13 () that as a function of , the landscaping polarizability for Rydberg states exhibits sizable oscillations in the infrared (IR) region, changing sign several times before the free-electron character starts to dominate it. This enables magic trapping conditions for the Rydberg and the ground states of alkali-metal atoms, where and the ground state polarizabilities match. Since has to vanish in order to change sign, there are also wavelengths at which vanishes, which are referred to as the tune-out wavelength TopDerTuneOut13 (). The Rydberg atom does not feel the optical trap in lattices tuned to these “tune-out” wavelengths.

### ii.2 Divalent atoms

As we move onto multi-valent Rydberg atoms, a new effect appears. The non-Rydberg (spectator) valence electron can sizably contribute to the total polarizability of the atom. This contribution to the total polarizability of divalent atoms has been taken into account in an ad hoc manner in Ref. MukMilNat11 (). Also, Ref. MukMilNat11 () used the free electron polarizability to represent the contribution from the Rydberg electron. In this section, we employ the more rigorous concept of landscaping polarizability to evaluate the contribution from the Rydberg electron. Moreover, we develop a theoretical framework which accounts for the overall rotational symmetry of the wavefunction when the atom is in a given -state. It turns out that for states with a Rydberg -electron (e.g. for Sr), individual contributions from the ground () and the Rydberg () states simply add to give the total polarizability of the atom. However, for states with a -electron in the Rydberg state (e.g. for Sr), the total rotational symmetry () imposes some restrictions.

#### ii.2.1 Atomic structure

We begin with the two-electron wave function: for a particular rotational symmetry the wavefunction can be expanded in terms of two-particle basis functions as

 Ψ(πJMJ)=∑k≥lcklΦkl(πJMJ). (10)

Here is the total angular momentum with projection , and is the parity of the state . The basis functions are defined in the subspace of virtual orbitals

 Φkl(πJMJ)=ηkl∑mk,mlCJMJjkmkjlmla†nkjkmka†nljlml|0core⟩≡|kl(J,MJ)⟩, (11)

where is a normalization factor, are creation operators, and the quasi-vacuum state corresponds to the closed-shell core. The Clebsch-Gordan coefficients mix single-particle orbitals to form a wavefunction with a well defined rotational symmetry . In general, the coefficients in Eq. (10) are determined from a CI (configuration-interaction) procedure involving diagonalization of the entire atomic Hamiltonian. For two-particle wavefunctions constructed in this way (Eq. (11)), the reduced matrix elements of an ITO of rank can be written in terms of single-particle orbitals as JohPlaSap95 ()

 ⟨rs (JF)||T(J)||mn(JI)⟩=√(2JI+1)(2JF+1)(−1)J∑m≤nr≤sηrsηmncrscmn (12) ×[(−1)jr+js+JI{JJIJFjsjrjm}⟨r||t(J)||m⟩δns+(−1)jr+jn{JJIJFjsjrjn}⟨r||t(J)||n⟩δms +(−1)JF+JI+1{JJIJFjrjsjm}⟨s||t(J)||m⟩δnr+(−1)jr+jn+JF{JJIJFjrjsjn}⟨s||t(J)||n⟩δmr].

Here and are the total angular momenta of the two-particle states and and are the single-particle operators related to by . For Rydberg excitations, we will employ a simplified “independent particle approximation”, where only one of the CI coefficients remains non-zero.

#### ii.2.2 Polarizability

Full interaction potential for the electrons in the electromagnetic field in the velocity gauge (also transverse or Coulomb gauge) is

 V=−∑jA\textscVG(rj,t)⋅pjc+∑jA2\textscVG(rj,t)2c2. (13)

Here and are the linear momentum operator and the coordinate of atomic electron , and we used the Gaussian system of units. The vector potential of an optical lattice in the velocity gauge is

 A\textscVG(rj,t)=−2cF0ω^ϵsin(k(Z+zj))sin(ωt), (14)

where we separated out the nuclear coordinate . In divalent atoms, there are two optically-active electrons. Let us assume that one of the electrons is in the Rydberg state and the other is in the ground state of the remaining singly-charged core (by and we refer to the -component of ). In the equations that follow, we will denote the quantum numbers by to simplify the notation. Then the energy shift due to the optical lattice (second order in the field strength) for the two electron state becomes

 δEgr(ω)=14c2∑g′r′2ΔEg′r′ΔE2g′r′−ω2×⟨γgjg;γrjr(JM)∣∣ ∣∣∑jA\textscVG(rj)⋅pj∣∣ ∣∣γg′jg′;γr′jr′(J′M′)⟩×⟨γg′jg′;γr′jr′(J′M′)∣∣ ∣∣∑j′A\textscVG(rj′)⋅pj′∣∣ ∣∣γgjg;γrjr(JM)⟩+12c2∑j⟨γgjg;γrjr(JM)∣∣(A\textscVG(rj))2∣∣γgjg;γrjr(JM)⟩, (15)

where the summation is over the intermediate states and are their energies. We will express in terms of the conventional AC polarizability for a standing wave described by (14), . Eq. (15) omits dynamic polarizability of the closed-shell core (e.g. Sr for Sr). This contribution is negligibly small in the differential Stark shifts as its contribution is nearly identical for optically excited levels DerPor11 ().

We will proceed as follows: First we will expand the two-particle operators in Eq. (15) in ITOs. We will then assume the independent particle approximation and break up the two-particle matrix elements into linear combinations of single-particle matrix elements, while retaining the original overall rotational symmetry . This results in an expression for which looks like the sum of the polarizabilities for the individual one-electron systems: the ground and the Rydberg state electrons. The second order term that results from Eq. (15) for the Rydberg electron is small compared to the term TopDer13 (), therefore we will ignore it. This is not the case for the ground state electron, and we will keep both the second order term and the term to evaluate its polarizability. Retaining of the original two-electron state will determine the extent to which the Rydberg electron polarizability contributes to .

Now we focus on the first term in Eq. (15) (which we also refer to as the second-order term). In order to cast this term into a tractable form, we expand the operators , Eq. (14), in terms of ITOs. To this end, we first express in ITOs: , where is the polarization vector and is an ITO of rank . An explicit expression for is derived in the Appendix. are proportional to the spherical Bessel functions , and the limitation comes from the axial symmetry of Eq. (14).

Realizing that the momentum vector is a tensor of rank 1, we can express as an expansion in terms of composite tensor operator

 A\textscVG(rj)⋅pj = F02∑μ(−1)μϵ−μ∑L,MLCLML1μK0{S(K)⊗p(1)}(L)ML (16) ≡ F02∑L,μ(−1)μϵ−μB(L)μ(K), (17)

where we defined by realizing that the Clebsch-Gordan coefficient in (16) forces . Notation stands for a tensor of rank obtained by coupling tensors of rank and of rank .

Due to the small spatial extent of the ground state wavefunction of the singly charged-ion, we take only the term in the multipolar expansion of ; this corresponds to the leading E1 multipole. Furthermore, we assume the long wavelength approximation for the compact ground state, which yields the usual dipole approximation: . For the Rydberg state, however, we will go beyond the long wavelength dipole approximation (see below). Keeping only the term for the ground state collapses to the term alone, and the operator becomes

 A\textscVG(rj)⋅pj = F02∑μ(−1)μϵ−μB(1)μ(0) (18) = 2cωF0(^ϵ⋅^p). (19)

Inserting Eq. (18) into Eq. (15) yields the product , where denotes the resolvent operator (it is a scalar). Recoupling this product, we obtain ITOs of ranks , 1, and 2, that can be recognized as the conventional scalar, vector, and tensor polarizabilities. Since we focus on the states, only the scalar contribution to the polarizability remains, which is identified by and its only component . We will evaluate this contribution later.

We now turn to the second term in Eq. (15) involving the expectation value of . In alkali-metal Rydberg atoms, this is the dominant term and it gives rise to the landscaping polarizability  TopDer13 () reviewed in Sec. II.1. Similar to the one-electron case in Eq. (5), we expand in the single-particle operators

 (A\textscVG(rj))2=F204∑Kt(K)MK=0(rj), (20)

which are related to the many-body operator through . We use the Wigner-Eckart theorem to integrate over the magnetic quantum numbers so that the matrix element in the second term in Eq. (15) is expressed in terms of the reduced matrix element

 ⟨γgγrJM|T(K)0|γgγrJM⟩ = (−1)J−M(JKJ−M0M)⟨γgγrJ||T(K)||γgγrJ⟩. (21)

Because the term is the expectation value in the state , the Wigner-Eckart theorem limits to . Moreover, since we are only interested in the states, . This restricts the representation of in term of ITO in Eq. (20) to the term alone.

In evaluating the many-body matrix elements, we will assume the independent electron approximation. In this approximation, the reduced matrix element in Eq. (21) can be broken up into reduced matrix elements involving only the one-electron orbitals using Eq. (12)

 ⟨jgjr(J)||T(K) ||jgjr(J)⟩=(2J+1)(−1)K+jg+jr+J ×[{KJJjrjgjg}⟨g||t(K)||g⟩+{KJJjrjgjr}⟨r||t(K)||r⟩]. (22)

Two of the terms in Eq. (12) dropped out because we assume that and are two distinct one-electron states, hence .

We now put together the second-order term and the term expressed in terms of one-particle reduced matrix elements to obtain an expression for following from Eq. (15). We are particularly interested in the states and . In the independent electron approximation, the energy of the two-electron state in can be separated into individual contributions from the one-electron states: . These approximations simplify Eq. (15) greatly, and the dynamic polarizability separates into individual contributions from the ground and the Rydberg states

 αgr(ω)=−1ω2∑g′≠g2ΔEg′ΔE2g′−ω2∑L,J′,μ(−1)μϵ−μ(0LJ′0μM′){LJ′0\sfrac12\sfrac12\sfrac12}{L0J′\sfrac12\sfrac12\sfrac12}×⟨nglg\sfrac12||b(L)μ(r)||ng′lg′\sfrac12⟩⟨ng′lg′\sfrac12||b(L)μ(r)||nglg\sfrac12⟩−1ω2∑r′≠r2ΔEr′ΔE2r′−ω2∑L,J′,μ(−1)μϵ−μ(0LJ′0μM′){LJ′0\sfrac12\sfrac12\sfrac12}{L0J′\sfrac12\sfrac12\sfrac12}×⟨nrlr\sfrac12||b(L)μ(r)||nr′lr′\sfrac12⟩⟨nr′lr′\sfrac12||b(L)μ(r)||nrlr\sfrac12⟩−1ω2∑K(0K0000){K00\sfrac12\sfrac12\sfrac12}⟨nglg\sfrac12||t(K)(r)||nglg\sfrac12⟩−1ω2∑K(0K0000){K00\sfrac12\sfrac12\sfrac12}⟨nrlr\sfrac12||t(K)(r)||nrlr\sfrac12⟩. (23)

Here we have expressed in terms of the one-electron operators: . To split the second-order term in (15) into individual contributions, we used the fact that the main contribution to for a given state comes from states that are nearby in energy, and the overlaps between the ground and Rydberg states are small, i.e. and . Therefore we neglected terms involving and and arrived at the first two terms in Eq. (23).

The first and the thirds terms can be consolidated together to give the ground state polarizability for the singly-charged ionic core in the velocity gauge. For the Sr atom, this would be the polarizability of the state of the Sr ion, . Because of the gauge invariance, can be calculated either in the velocity or the length gauges, and we will evaluate using the length gauge below. The second and the fourth terms in (23) result from the Rydberg electron. The 3- symbol in the last term, which came from integrating over the magnetic quantum numbers using the Wigner-Eckart theorem in (21), collapses the sum to the term alone. Furthermore, and forces in the second term. With these simplifications and using the closed form expressions for the 3- and 6- symbols which appear in (23), the dynamic polarizability can be rewritten as

 αgr(ω)=αSr+5s(ω)−1ω2∑L,μr′≠r(−1)μϵ−μ2ΔEr′ΔE2r′−ω2(−1)L−μδL,J′δμ,−M′2(2L+1)3/2×|⟨nrlr\sfrac12||b(L)μ(r)||nr′lr′\sfrac12⟩|2−1ω2√2⟨nrlr\sfrac12||t(0)(r)||nrlr\sfrac12⟩. (24)

The dominant contribution for the Rydberg electron comes from the last term as discussed in TopDerGauge13 (). The correction to the term is given by the second term in (25) and is negligibly small for Rydberg states as demonstrated in TopDerGauge13 (). The term is what gives rise to the landscaping polarizability in alkali metal atoms, and is proportional to . Therefore the total polarizability of the divalent Rydberg atom can be expressed as

 αJ=0nglg;nrlr(ω)=αion(ω)+αlsc,J=0nrlr(ω)+αcore(ω)+αcv(ω), (25)

where is the polarizability of the residual ion (e.g. Sr) and is the contribution from the Rydberg landscaping polarizability to the total polarizability of the two-electron state. The notation refers to the total polarizability defined in (23) and (24) (). The polarizability comes from the contributions from core-excited states of doubly ionized atom (e.g. Sr) to the total polarizability DerPor11 (). We will neglect this term because it is almost identical for both valence levels and it vanishes when only the differential contribution is considered. We also neglect , which is a small term counteracting . It arises from excitations to occupied valence orbitals and is much smaller than  DerPor11 ().

The contribution can be explicitly written as

 αlsc,J=0nrlr(ω)=−1ω2√2⟨nrlr\sfrac12||t(0)(r)||nrlr\sfrac12⟩. (26)

Therefore the consequence of the overall rotational symmetry of the many-body state is that only the term in (Eq. (20)) contributes to the total polarizability. To calculate the reduced matrix element in , we use the same expansion used for the alkali-metal atoms in (5)

 ⟨nrlr\sfrac12||t(0)(r)||nrlr\sfrac12⟩ = ⟨nrlr\sfrac12||C(0)(^r)||nrlr\sfrac12⟩∫∞0P2nrlr(r)j0(2kr)dr (27) = √2∫∞0P2nrlr(r)j0(2kr)dr. (28)

Thus,

 αlsc,J=0nrlr(ω)=−αe(ω)∫∞0P2nrlr(r)j0(2kr)dr. (29)

For the 5s ground state of the Sr ion, the trapping potential reads where the dynamic polarizability fo the Sr ion is given by

 αion(ω)=∑jE5s−Ej(E5s−Ej)2−ω2|⟨ψ5s|D|ψj⟩|2, (30)

where is the electric dipole operator and are the ionic energy levels. We evaluate using a high-accuracy method detailed in Ref. DerJohSaf99 ().

## Iii Numerical results for Sr

This section is organized as follows: in Sec. III.1, we evaluate the ground state polarizability for the Sr ion and the landscaping polarizabilities for a few Rydberg states and discuss their general features. To find magic wavelengths at which clock-to-Rydberg state transition frequencies do not change, we then calculate the dynamic polarizabilities for the and clock states of Sr in Sec. III.2. Finally in Sec. III.3, we evaluate the total divalent Rydberg state polarizabilities by combining these as described in Sec. II.2 to find the magic wavelengths.

### iii.1 Residual ion and the Rydberg electron

We start by calculating the ground state polarizability for the Sr ion and the landscaping polarizabilities for a few s Rydberg states. The main features of p Rydberg states are essentially the same as discussed in Sec. II.2.2. These individual polarizabilities can be added to obtain the total polarizability of the Sr atom in the Rydberg state 5ss() as we demonstrated in the previous section. Fig. 1 shows the polarizability for the Sr ion in the 5s ground state and the landscaping polarizabilities for the Rydberg electron in the 5ss() state of Sr for and 130. The main feature of is that it converges to its static value after 1000 nm at a.u.. On the other hand, the Rydberg landscaping polarizabilities start out essentially at zero at small , and oscillate with increasing amplitude towards larger , before the free electron character of the polarizability takes over and drops off as .

The radial wave functions needed to evaluate are computed by numerical integration of the time-independent Schrödinger equation using a model for the ionic core seen by the Rydberg electron in state . Both in () and () singly excited Rydberg states of Sr, the Rydberg electron moves under the influence of the Sr potential. We model this potential as

 V(r)=−1r−(Za−1)e−arr+be−cr, (31)

where is the atomic number (38 for Sr) and , and are fitting parameters, which depend on the particular symmetry of the many-body Rydberg state. The coefficients , and for Sr are listed in Ref. Millen11 () for a variety of symmetries. For the () states of Sr, we use , and , and for the () states we use , and .

### iii.2 The clock states

In order to determine magic wavelengths, we need to calculate the dynamic polarizabilities for the 5s() and 5s5p() states of Sr. We use the relativistic formulation of the second order perturbation theory JohPlaSap95 (),

 αJ=0γ(ω)=−1√3∑γ′{<