Dipole-dipole frequency shifts in multilevel atoms
A. Cidrim, A. Piñeiro Orioli, C. Sanner, R. B. Hutson, J. Ye, R. Bachelard, A. M. Rey
DDipole-dipole frequency shifts in multilevel atoms
A. Cidrim,
1, 2, 3
A. Pi˜neiro Orioli,
2, 3
C. Sanner, R. B. Hutson, J. Ye, R. Bachelard, and A. M. Rey
2, 3 Departamento de F´ısica, Universidade Federal de S˜ao Carlos, 13565-905 S˜ao Carlos, S˜ao Paulo, Brazil JILA, NIST, Department of Physics, University of Colorado, Boulder, CO 80309, USA Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA
Dipole-dipole interactions lead to frequency shifts that are expected to limit the performance ofnext-generation atomic clocks. In this work, we compute dipolar frequency shifts accounting forthe intrinsic atomic multilevel structure in standard Ramsey spectroscopy. When interrogating thetransitions featuring the smallest Clebsch-Gordan coefficients, we find that a simplified two-leveltreatment becomes inappropriate, even in the presence of large Zeeman shifts. For these cases,we show a net suppression of dipolar frequency shifts and the emergence of dominant non-classicaleffects for experimentally relevant parameters. Our findings are pertinent to current generations ofoptical lattice and optical tweezer clocks, opening a way to further increase their current accuracy,and thus their potential to probe fundamental and many-body physics.
Introduction.—
Current optical atomic clocks havereached unprecedented precision and accuracy [1–10],making them cutting-edge platforms for many techno-logical applications and for the exploration of many-body [11–20] and fundamental physics [21–25]. The re-duction of noise in atomic detection and laser stabiliza-tion in such systems has allowed measurements of theatomic transition with submillihertz resolution [6, 26, 27].At this point, dipole-dipole interactions between theatoms are expected to play an important role, in theform of induced density dependent shifts in the mea-sured atomic transition frequency. Simple two-level mod-els have been applied to quantitatively determine thesedipolar shifts [28–34], but, in reality, atoms have a com-plex internal multilevel structure which has to be takeninto account. This calls for a deeper understanding of therole of multiple internal levels in dipolar systems [35–39],which is also relevant for applications in quantum simu-lators [11, 17, 18, 40] and quantum computing [41–43].In this work, we investigate dipolar frequency shiftsexperienced by arrays of multilevel atoms in a Ramseyspectroscopy protocol. In general, the strength of dipo-lar interactions is set by the magnitude of the transi-tion’s dipole moment, which is proportional to a Clebsch-Gordan coefficient (CGC). However, in multilevel atomsthe dependence of the dipolar shift on the choice of tran-sition is more complex. This is because the CGC betweentwo specific states not only sets the strength of the dipolecouplings, but also affects the coupling strength to nearbylevels. Specifically, transitions with low (high) CGC fea-ture a stronger (weaker) decay to and interactions withtheir neighbouring states.Our results show that the magnitude of the dipolarfrequency shift is mainly controlled by the CGC of the interrogated levels. Therefore, one can strongly suppressdipolar shifts by selectively choosing the levels with thesmallest CGC. We also find that interactions with nearbylevels can significantly modify the shift. Specifically, weshow that a full multilevel calculation is necessary whenthe CGC of the interrogated transition is small, whereas simplified two-level models are accurate when the CGC islarge. Surprisingly, the relevance of the multilevel struc-ture holds even in the presence of strong magnetic fields,under which the large Zeeman shifts suppress exchangewith nearby levels. Moreover, we find that the suppres-sion of the shift from small CGC leads to an increased rel-ative importance of beyond-mean-field effects for specificexperimentally relevant array geometries and laser wavevector configurations. In short, our work offers a simpleway for current experiments to reduce dipolar shifts byalmost two orders of magnitude, while at the same timedrawing theorists’ attention to the important yet largelyneglected role of internal levels in many-body dipolar sys-tems.
Multilevel coupled dipole model.—
We consider a sys-tem of N point-like atoms pinned in a deep optical lat-tice or a tweezer array with unity occupation, always intheir motional ground state. We assume that each atom i has a multilevel internal structure of ground and ex-cited manifolds, g and e , with respective total angularmomenta F g and F e . There are thus (2 F a + 1) hyper-fine states | a m (cid:105) i ≡ | a, F a , m (cid:105) i with angular momentumprojections m ∈ [ − F a , F a ], for each manifold a ∈ { g, e } .The photon-mediated interaction between the atoms oc-curs via both coherent exchange and incoherent decay ofexcitations [see Fig. 1(a)], and the dipole dynamics canbe modelled by a multilevel coupled dipole master equa-tion [38, 39, 44–46] ˙ˆ ρ = − i (cid:104) ˆ H, ˆ ρ ( t ) (cid:105) + L (ˆ ρ ) ( (cid:126) = 1),where ˆ H = − (cid:88) i,j ∆ ijg m e n ,g m (cid:48) e n (cid:48) ˆ σ e n g m i ˆ σ g m (cid:48) e n (cid:48) j , (1) L (ˆ ρ ) = (cid:88) i,j Γ ijg m e n ,g m (cid:48) e n (cid:48) (cid:16) σ g m (cid:48) e n (cid:48) j ˆ ρ ˆ σ e n g m i − (cid:8) ˆ σ e n g m i ˆ σ g m (cid:48) e n (cid:48) j , ˆ ρ (cid:9) (cid:17) , (2)and ˆ σ a m b n i = | a m (cid:105) i (cid:104) b n | i . For a two-level atom, theseoperators become the usual raising/lowering Pauli opera-tors. For clarity, we have used Einstein notation for levels a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Γt δ / Γ × − g − / → e − / g − / → e − / g − / → e − / k -7/2 -5/2 -3/2 -1/2 +3/2 +5/2+7/2 +9/2-9/2 +1/2 | e ⟩ | g ⟩ (b)(a) Mean fieldPerturbative ω -5/2 -1/2-9/2 Γ ij Δ ij δ ∝ | C eg | Dipolar shift
FIG. 1. (a) Ramsey spectroscopy for multilevel atoms withinternal level structure F g = F e = 9 / g α and e β by a laser withwave vector k and pulse area θ . During a dark time t , theatoms interact via coherent and incoherent dipole-dipole pro-cesses, ∆ ij and Γ ij , respectively. This induces a frequencyshift δ g α e β ∼ | C e β g α | controlled by the CGC of the interro-gated transition. The schematic form of the dipolar shift cor-responding to interrogating π -polarized transitions (coloredaccording to their CGC) is depicted here . (b) Shift from theRamsey protocol addressing three different π -polarized tran-sitions for a 3D lattice of spacing d = 7 λ/
12 with N = 10 atoms and θ = π/
2. The dipolar shift is highly suppressed forthe transition with smallest CGC. in the equations above (i.e., repeated indices m, m (cid:48) , n , or n (cid:48) are summed). The terms proportional to ∆ ijg m e n ,g m (cid:48) e n (cid:48) and Γ ijg m e n ,g m (cid:48) e n (cid:48) characterize the elastic and dissipativecomponents of the dipolar interactions and their ampli-tudes relate to the free-space electromagnetic Green’stensor G ij ≡ G ( r i − r j ) of an oscillating point dipoleat position r j according to∆ ijg m e n ,g m (cid:48) e n (cid:48) ≡ C e n g m e ∗ n − m · Re { G ij } · C e n (cid:48) g m (cid:48) e n (cid:48) − m (cid:48) , Γ ijg m e n ,g m (cid:48) e n (cid:48) ≡ C e n g m e ∗ n − m · Im { G ij } · C e n (cid:48) g m (cid:48) e n (cid:48) − m (cid:48) , (3)where C e n g m ≡ (cid:104) F g , m ; 1 , n − m | F e , n (cid:105) is the CGCof the transition g m ↔ e n with polarization vector e n − m . We define the spherical basis e = ˆ z , e ± = ∓ (ˆ x ± i ˆ y ) / √
2. The vacuum Green’s tensor is givenby G ( r ) = (3Γ / e ik r / ( k r ) ) (cid:104) (cid:0) k r + ik r − (cid:1) − (cid:0) k r + i k r − (cid:1) ˆ r ⊗ ˆ r (cid:105) , where ˆ r = r /r , r = | r | .Γ = | d eg | k / [3 π (cid:126) (cid:15) (2 F e + 1)] is the total spontaneousdecay rate, d eg the radial dipole matrix element, k = ω /c = 2 π/λ the atomic transition wavenumber and (cid:15) the vacuum permittivity. For i = j , the coherent in-teraction coefficient is ∆ iig m e n ,g m (cid:48) e n (cid:48) = 0 and the incoher-ent term reduces to the single-particle spontaneous decayterm Γ iig m e n ,g m (cid:48) e n (cid:48) = δ n − m,n (cid:48) − m (cid:48) C e n g m C e n (cid:48) g m (cid:48) Γ /
2. Note thatthe total decay rate Γ e n ≡ (cid:80) m Γ iig m e n ,g m e n = Γ is thesame for any excited state e n because of the sum rule (cid:80) m | C e n g m | = 1. Ramsey spectroscopy with multilevel atoms.—
We in-vestigate the effect of the atomic multilevel nature onthe following Ramsey spectroscopy protocol assuming, at first, zero external magnetic field. We start by se-lecting a pair of states g α and e β , driving the transitionbetween them with a resonant laser of pulse area θ , wavevector k , and polarization (cid:15) . The laser drive is assumedto be much stronger than the interaction energies, suchthat it creates an uncorrelated coherent superposition | Ψ g α ,e β (cid:105) = (cid:78) j (cid:16) cos( θ/ | g α (cid:105) j + e i k · r j sin( θ/ | e β (cid:105) j (cid:17) .We hereafter consider θ = π/
2, as generally used inclock experiments, or θ = π/
4, as the latter can leadto more pronounced and thus easily observable dipo-lar shifts. Then, the system evolves freely for a darktime t . By analogy with two-level systems, we define (cid:104) ˆ S y (cid:105) ≡ Im {(cid:104) ˆ S e β g α (cid:105)} and (cid:104) ˆ S x (cid:105) ≡ Re {(cid:104) ˆ S e β g α (cid:105)} , wherethe multilevel collective spin operator (under the ap-propriate gauge transformation that removes the phase k · r j imprinted by the laser on atom j ) reads ˆ S e β g α = (cid:80) j e i k · r j ˆ σ e β g α j . The collective vector precesses aroundthe z -direction of the Bloch sphere and accumulates anazimuthal phase as a result of the dipole-dipole interac-tions. The corresponding time-dependent frequency shiftis defined as δ g α e β ( t ) ≡ πt arctan (cid:104) ˆ S y (cid:105) ( t ) (cid:104) ˆ S x (cid:105) ( t ) . (4)Dipolar interactions also lead to a reduction of the con-trast C g α e β ( t ) ≡ N (cid:113) (cid:104) ˆ S x (cid:105) ( t ) + (cid:104) ˆ S y (cid:105) ( t ).We employ three different types of approximations toinvestigate this multilevel many-body system:i) a short-time perturbative expansion, valid for t (cid:28) Γ − ,where operators are expanded as (cid:104) ˆ O(cid:105) ≈ (cid:104) ˆ O(cid:105) + (cid:104) ˆ O(cid:105) t + (cid:104) ˆ O(cid:105) t /
2, allowing us to compute the dipolar frequencyshift at first order in time, i.e., δ g α e β ( t ) ≈ δ g α e β + δ g α e β t ;ii) a mean-field (MF) approximation, which neglects thebuild up of quantum correlations by approximating two-atom correlators as (cid:104) ˆ σ abi ˆ σ cdj (cid:105) ≈ (cid:104) ˆ σ abi (cid:105)(cid:104) ˆ σ cdj (cid:105) (for i (cid:54) = j );iii) a second-order cumulant expansion, which fac-torizes three-point (and higher-order) correlations interms of one- and two-point functions (cid:104) ˆ σ abi ˆ σ cdj ˆ σ efk (cid:105) ≈− (cid:104) ˆ σ abi (cid:105)(cid:104) ˆ σ cdj (cid:105)(cid:104) ˆ σ efk (cid:105) + (cid:104) ˆ σ abi (cid:105)(cid:104) ˆ σ cdj ˆ σ efk (cid:105) + (cid:104) ˆ σ cdj (cid:105)(cid:104) ˆ σ abi ˆ σ efk (cid:105) + (cid:104) ˆ σ efk (cid:105)(cid:104) ˆ σ abi ˆ σ cdj (cid:105) (with i , j , k all different).Due to the large number of equations to solve, in theMF and cumulant calculations we further assume that,when addressing a transition g α ↔ e β , only g α , e β , andtheir adjacent levels (i.e., g α ± and e β ± ) play a rele-vant role in the dynamics. We have checked on smallersystems that the neglected levels have no significant ef-fect on the frequency shift over the dark times considered(Γ t <
1) [47].
Short-time perturbative expansion.—
To gain physicalintuition of the problem, we analytically derive short-time expressions for the shift. The zero-order shift reads δ g α e β = − cos θ πN (cid:88) i,j (cid:54) = i U jig α e β , (5)where we have defined U jig α e β ≡ Γ jig α e β ,g α e β sin( k · r ij ) +∆ jig α e β ,g α e β cos( k · r ij ). Physically, the term U jig α e β de-scribes the classical interaction energy between two os-cillating dipoles at positions r i and r j [28], where bothcoherent and incoherent processes contribute. At thisorder, only the transition between g α and e β , directlydriven by the pulse, is involved and the MF treatment isexact. Furthermore, δ g α e β is proportional to | C e β g α | [seeEq. (3)], so that the multilevel system differs from two-level atoms [28] via a renormalization by the CGC. Notethat the zero-order shift vanishes for a θ = π/ (cid:104) ˆ S z (cid:105) component.The next-order correction does involve other levels andis given by δ g α e β = − πN (cid:88) i,j (cid:54) = i (cid:40) U jig α e β (cid:101) Γ g α e β ( θ )+ (cid:88) p (cid:32) (cid:88) k (cid:54) = i,j W kjip ( θ ) + (cid:88) p (cid:48) Q jip,p (cid:48) ( θ ) (cid:33)(cid:41) , (6)with p and p (cid:48) referring to polarizations [47].The first contribution in Eq. (6) is similar to the zero-order shift. The cos θ , however, is replaced by (cid:101) Γ g α e β ( θ ),which contains a collective contribution and an explicitdependence on the CGC of the transition interrogated.The W kjip ( θ ) are two-photon coherent and incoherentprocesses between three different atoms, where one of thecontributing transitions is always g α ↔ e β . Thus, theseterms are proportional to at least | C e β g α | . The Q jip,p (cid:48) ( θ )terms correspond to processes involving two atoms only,yet not necessarily from the g α ↔ e β transition. As two-photon processes, they nevertheless contain the productof four CGCs and, as we shall discuss later, they carrybeyond-mean-field contributions. Suppression of the frequency shift.—
Although our con-clusions are valid for generic multilevel systems, in thiswork we focus our analysis on the case of Sr, givenits metrological relevance for atomic clocks [1, 5, 11, 17].More specifically, we assume multilevel atoms with F g = F e = 9 /
2, organized in a 2D or 3D array with magic-wavelength spacing d = 7 λ/
12 [48], see Fig. 1(a).For simplicity, we will hereafter consider addressing π -polarized transitions (i.e., α = β ), where the quantizationaxis is defined by the laser polarization (cid:15) . For this sys-tem it is important to know that the CGC for π -polarizedtransitions scales as C e m g m ∝ m , i.e., it is largest for ± / ± / α = − / , − /
2, and − /
2. As a consequence × -2 -12 δ i / ( Γ | C e α g α | − ) k ϵ α = − 9/2 × -2 -25 ˛˛ α = − 1/2 α = − 1/2 × -2 -47 k ϵ − − − − − | − | | δ m a x i − δ m i n i | / Γ | δ | /Γ -1/2-9/2 (d) y i μ | B | ≫ Γ (e) x i x i x i -9/2 -7/2
911 211 -1/2-3/2 +1/2 | C eg | (c)(b)(a) FIG. 2. Local frequency shifts δ i for a 3D lattice with N = 10 atoms, interrogated by a laser (as shown in the scheme tothe left of the first panel) with a pulse area of π/ t = 0 . g − / ↔ e − / , (b) g − / ↔ e − / , and (c) g − / ↔ e − / in the presence of alarge magnetic field B . The shift is calculated using Eq. (6)and is averaged along (cid:15) , with the resulting contribution atpositions ( x i , y i ). Shifts are rescaled by the correspondingCGC squared and by an overall 10 − factor. (d) Absolutevalue of the difference between maximum and minimum of thelocal shift ( δ max i and δ min i ) versus magnitude of the global shift | δ | for different geometries with N ∼ . Symbols representconfigurations shown in the inset. The light-blue symbolscorrespond to the large- | B | limit for α = − /
2. (e) CGCsquared for different σ ± and π -transitions. of the scaling with the CGC, the shift is reduced by afactor 81 for the α = − / α = − /
2. Note that the suppression remains valid evenat longer times beyond the regime of validity of the short-time expansion. The decay of the contrast C g α e β ( t ) alsoshows a scaling with the CGC, which leads to suppressedsub/superradiance effects for α = − / t ≈ . Q in Eq. (6)] do not con-tribute substantially.Further insight is provided by the local dipolar shiftpatterns δ g α e β i ≡ πt arctan( (cid:104) ˆ s yi (cid:105) / (cid:104) ˆ s xi (cid:105) ) [single-particlecounterpart of Eq. (4)]. Local density shifts directly en-code the anisotropic and geometry-dependent characterof dipolar interactions, so they can provide further in-formation on the importance of the multilevel structurein experimentally relevant array geometries. Moreover,local density shifts are amenable for experimental obser-vation via imaging spectroscopy [27], since they are in-sensitive to laser drifts which are common for all atomsin the array. In Fig. 2(a-c), we present the local shiftsobtained with π/ N = 10 for α = − / − / | C e α g α | , which we emphasize by rescal-ing the plots as δ i / (Γ | C e α g α | − ). However, the dipolarpatterns of the − / − / | C e α g α | scaling.The local shifts of the − / − / − / − / π -transition has a small CGC com-pared to the adjacent σ ± -transitions, whereas for the − / π -transition the opposite is true, see Fig. 2(e).Therefore, nearby levels play a more important role inthe − / N in all cases shown, except in 2D when the laser wave vec-tor is parallel to the atomic plane, see Figs. 3(a) and (b).This is because in the latter configuration, all the dipolesalign perpendicular to the plane and the correspondingdipolar interactions depend only on the distance betweenatoms, and not on their orientation [47]. Role of magnetic fields.—
Optical clock experimentsare typically conducted under a bias magnetic field B (along the quantization axis) that allows to spectroscop-ically address specific transitions. This leads to a Zeemanshift of order µ | B | (with µ ≡ µ B / (cid:126) and µ B the Bohrmagneton) for the g α ↔ e α transition considered, whichtrivially adds to the zero-order expression of Eq. (5) andcan be removed in the appropriate rotating frame. How-ever, magnetic fields can non-trivially affect dipolar shiftsat higher orders.If the magnetic field is weak (i.e., µ | B | (cid:46) Γ), we findthe above results on the dipolar shift are only weaklyaffected at late times. This is because the first-ordercorrection, Eq. (6), turns out to be independent of themagnetic field [47]. In contrast, strong magnetic fields( µ | B | (cid:29) Γ) can significantly alter the short-time behav-ior of the shift. Large Zeeman shifts effectively suppressexchange interactions involving off-resonant transitions.In other words, ∆ ijg m e n ,g m (cid:48) e n (cid:48) = Γ ijg m e n ,g m (cid:48) e n (cid:48) = 0 unless m = m (cid:48) and n = n (cid:48) (assuming different g -factors for theground and excited manifolds). This leads to an effec-tive 4-level (or 3-level) system composed of e α , g α , andthe ground levels adjacent to it. In this limit, almostall terms in the first-order expression, Eq. (6), involvingtransitions different from g α ↔ e α are suppressed, exceptfor terms with p = p (cid:48) appearing in Q jip,p (cid:48) [47].Consistently with the discussion above, we find thatthe modification of the shift strongly depends on the N − − δ / Γ ( Γ t = . ) × − -9/2 δδ BMF
Large-B limit
50 75 100 − × − N − × − δδ BMF
Large-B limit -1/2 Γt − − − δ / Γ × − g − / → e − / g − / → e − / g − / → e − / ϵ k -5/2 -1/2-9/2 CumulantMean field Γt − × − (b)(a)(c) θ = π /2 θ = π /4 (d) FIG. 3. Global shift for 2D arrays of atoms with laser config-uration shown in (c). (a,b) N -scaling of the total shift fromthe short-time expansion (red/blue) and beyond-MF contri-bution (black) at Γ t = 0 .
3: (a) shows the g − / ↔ e − / tran-sition with π/
2, and (b) the g − / ↔ e − / transition with π/
4. The dotted, light-colored lines correspond to the large-magnetic-field limit. The inset of (a) shows the zoomed-inregion where beyond-MF corrections become comparable tothe total shift. (c,d) Dipolar shift δ as a function of the darktime: cumulant (full lines) against MF (dashed lines) approx-imations for the g − / ↔ e − / (red), g − / ↔ e − / (yellow),and g − / ↔ e − / (blue) transitions. Simulations performedfor N = 8 atoms and using (c) a π/ π/ CGC of the addressed transition. For − / − / | B | . Despite this, the global shift remains suppressed bythe small CGC as found for small | B | . Beyond-mean-field effects.—
An important conse-quence of the strong shift suppression is that higher-order, non-classical terms can have a contributioncomparable to the lowest-order, semi-classical ones. Thezero-order shift, Eq. (5), is perfectly described by the MFapproach, yet the Q terms of the first-order, Eq. (6), arenot. More specifically, the difference between the shiftgiven by the exact, first-order perturbative equationsand the MF approximation reads δ g α e β BMF ≡ πN (cid:88) i,j (cid:54) = i (cid:88) p (cid:40) (cid:88) p (cid:48) Q jip,p (cid:48) ( θ ) − W ji MF ,p ( θ ) (cid:41) , (7)where W ji MF ,p ( θ ) is a MF-only term related to W kji fromEq. (6) [47]. In general, we find beyond-MF effects to berelevant in cases (but not in every case) where either thesystem is small or when a transition with small CGC isaddressed. Therefore beyond-MF effects could be rele-vant for recent tweezer clocks experiments which operatewith relative small systems and enjoy almost a minute-long coherence time [8, 9].Figure 3(c) shows the effect of beyond-MF terms fora small 8 lattice, driven by a π/ N , when the MF contributions are no longer suppressed,the beyond-MF term becomes negligible.Although beyond-MF corrections do not scale up with N , we find cases where they can be relevant even forlarge systems because of a strong suppression of the totalshift by the multilevel structure. An example is the casewith a pulse area of θ = π/ − / ∼ atoms. Note, however,that in this case the beyond-MF term is suppressed inthe large B-field limit. Conclusion.—
We have shown that dipolar frequencyshifts are strongly modified in systems featuring a mul-tilevel structure. The predicted two orders of magni-tude suppression obtained by properly addressing spe-cific transitions can lead to the improved accuracy neces-sary for the exploration of fundamental physics [21–25],providing new insights on the behavior of strongly andlong-range interacting many-body systems.
Acknowledgments.—
We thank C. Qu, L. Sonderhouse,and N. Schine for helpful discussions and feedback. A.C.and R.B. are supported by FAPESP through GrantsNo. 2017/09390-7, 2018/18353-0, 2019/13143-0, and2018/15554-5. R.B. benefited from Grants from theNational Council for Scientific and Technological De-velopment (CNPq, Grant Nos. 302981/2017-9 and409946/2018-4). C.S. thanks the Humboldt Foundationfor support. This work is supported by the AFOSRGrant No. FA9550-18-1-0319 and its MURI Initiative,by the DARPA and ARO Grant No. W911NF-16-1-0576, the ARO single investigator Grant No. W911NF-19-1-0210, the NSF PHY1820885, NSF JILA-PFC PHY-1734006 Grants, NSF QLCI-2016244 grant, and by NIST. [1] T. Nicholson, S. Campbell, R. Hutson, G. Marti,B. Bloom, R. McNally, W. Zhang, M. Barrett, M. Safronova, G. Strouse, W. Tew, and J. Ye, NatureCommunications , 1 (2015).[2] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O.Schmidt, Rev. Mod. Phys. , 637 (2015).[3] M. Schioppo, R. C. Brown, W. F. McGrew, N. Hinkley,R. 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FIRST-ORDER FREQUENCY SHIFT: DEFINITIONS
The coefficients for the short-time expansion of the dipolar shift [Eqs. (5) and (6) in the main text] are given by δ g α e β ( t ) ≈ δ g α e β + δ g α e β t with δ g α e β = 12 π (cid:104) ˆ S y (cid:105) (cid:104) ˆ S x (cid:105) = − cos θ πN (cid:88) i,j (cid:54) = i U jig α e β , (S1)and δ g α e β = 14 π (cid:32) (cid:104) ˆ S y (cid:105) (cid:104) ˆ S x (cid:105) − (cid:104) ˆ S x (cid:105) (cid:104) ˆ S y (cid:105) (cid:104) ˆ S x (cid:105) (cid:33) = − πN (cid:88) i,j (cid:54) = i (cid:40) U jig α e β (cid:101) Γ g α e β ( θ ) + (cid:88) p (cid:32) (cid:88) k (cid:54) = i,j W kjip ( θ ) + (cid:88) p (cid:48) Q jip,p (cid:48) ( θ ) (cid:33)(cid:41) . (S2)Recall that we expand expectation values of operators as (cid:104) ˆ O(cid:105) ≈ (cid:104) ˆ O(cid:105) + (cid:104) ˆ O(cid:105) t + (cid:104) ˆ O(cid:105) t /
2. In the following, we providethe detailed definitions of the terms contained in the above expressions, starting with U jig m e n ,g m (cid:48) e n (cid:48) ( r ik ) ≡ Γ jig m e n ,g m (cid:48) e n (cid:48) sin( k · r ik ) + ∆ jig m e n ,g m (cid:48) e n (cid:48) cos( k · r ik ) ,E jig m e n ,g m (cid:48) e n (cid:48) ( r ik ) ≡ Γ jig m e n ,g m (cid:48) e n (cid:48) cos( k · r ik ) − ∆ jig m e n ,g m (cid:48) e n (cid:48) sin( k · r ik ) . (S3)For the sake of simplicity, we have further defined (as used in the main text) U jig α e β ≡ U jig α e β ,g α e β ( r ij ) and E jig α e β ≡ E jig α e β ,g α e β ( r ij ) . (S4)The first-order shift contains the term (cid:101) Γ g α e β ( θ ) ≡ sin ( θ/ | C e β g α | ) Γ2 + (cos θ/N ) (cid:88) i,j (cid:54) = i E jig α e β . (S5)The two-body terms that involve two-photon transitions between levels stem exclusively from beyond-MF contribu-tions and are given by Q jip,p (cid:48) ( θ ) ≡ (cid:16) sin ( θ/ A − ji ; ijg β + p e β ,g β + p (cid:48) e β − cos ( θ/ A − ji ; ijg α e α − p (cid:48) ,g β + p e β (cid:17) , (S6)while the three-body terms read W kjip ( θ ) ≡ sin θ (cid:16) V − ji ; ikg α e α − p ,g α e β + V − ji ; ikg β + p e β ,g α e β + 2 δ p, ( α − β ) V + ji ; ikg α e β ,g α e β (cid:17) − cos θ (cid:16) cos ( θ/ D ij ; jkg α e α − p ,g α e β − sin ( θ/ D ij ; jkg β + p e β ,g α e β (cid:17) , (S7)with V ± ji ; ikg m e n ,g m (cid:48) e n (cid:48) ≡ A ∓ ji ; ikg m e n ,g m (cid:48) e n (cid:48) a ∓ ( r ij , r ik ) + B ± ji ; ikg m e n ,g m (cid:48) e n (cid:48) b ± ( r ij , r ik ) , (S8) A ± ji ; ikg m e n ,g m (cid:48) e n (cid:48) ≡ ∆ jig m e n ,g m (cid:48) e n (cid:48) Γ ikg m (cid:48) e n (cid:48) ,g m e n ∓ Γ jig m e n ,g m (cid:48) e n (cid:48) ∆ ikg m (cid:48) e n (cid:48) ,g m e n ,B ± ji ; ikg m e n ,g m (cid:48) e n (cid:48) ≡ ∆ jig m e n ,g m (cid:48) e n (cid:48) ∆ ikg m (cid:48) e n (cid:48) ,g m e n ∓ Γ jig m e n ,g m (cid:48) e n (cid:48) Γ ikg m (cid:48) e n (cid:48) ,g m e n , (S9) a ± ( r ij , r ik ) ≡ cos( k · r ij ) cos( k · r ik ) ± sin( k · r ij ) sin( k · r ik ) ,b ± ( r ij , r ik ) ≡ sin( k · r ij ) cos( k · r ik ) ± cos( k · r ij ) sin( k · r ik ) , (S10)and D ij ; jkg m e n ,g m (cid:48) e n (cid:48) ≡ ∆ ijg m e n ,g m (cid:48) e n (cid:48) E jkg m (cid:48) e n (cid:48) ,g m e n ( r ik ) + Γ ijg m e n ,g m (cid:48) e n (cid:48) U jkg m (cid:48) e n (cid:48) ,g m e n ( r ik ) . (S11)As mentioned in the main text, the dipolar shift computed in MF approximation deviates from the exact result at firstorder in time. Compared to Eq. (S2), the MF expression does not have the Q jip,p (cid:48) terms and it contains an additionalterm which is equal to the three-body term of Eq. (S7) after setting k = j and k = i in the first and second lines,respectively. Specifically, the MF shift is given by δ g α e β , MF = δ g α e β and δ g α e β , MF = − πN (cid:88) i,j (cid:54) = i (cid:40) U jig α e β (cid:101) Γ g α e β ( θ ) + (cid:88) p (cid:32) W ji MF ,p ( θ ) + (cid:88) k (cid:54) = i,j W kjip ( θ ) (cid:33)(cid:41) , (S12)with W ji MF ,p ( θ ) ≡ sin θ δ p, ( α − β ) V + ji ; ijg α e β ,g α e β − cos θ (cid:16) cos ( θ/ D ij ; jig α e α − p ,g α e β − sin ( θ/ D ij ; jig β + p e β ,g α e β (cid:17) . (S13) EQUATIONS OF MOTION
Here we provide the equations of motion for the expectation values derived from the multilevel coupled dipole masterequation introduced in the main text. We also include in the equations the Zeeman shifts induced by a magnetic fieldparallel to the quantization axis. The Zeeman Hamiltonian is given by ˆ H B = − (cid:80) n,i ∆ e n ˆ σ e n e n i − (cid:80) m,i ∆ g m ˆ σ g m g m i ,where ∆ e n = nδ e , ∆ g m = mδ g , and δ g (cid:54) = δ e due to a differential g -factor between ground and excited manifolds.We use Einstein notation for sums over level indices (i.e., repeated indices m, m (cid:48) , n, n (cid:48) are summed if they do notappear on the left hand side of the equation). Additionally, note that for the sake of simplicity in notation we havechosen to simplify G ijg m e n ,g m (cid:48) e n (cid:48) → G ijmn,m (cid:48) n (cid:48) .The single-point equations read ddt (cid:104) ˆ σ e β g α i (cid:105) = − (cid:32) Γ2 + i (∆ e β − ∆ g α ) (cid:33) (cid:104) ˆ σ e β g α i (cid:105) + (cid:88) j (cid:54) = i (cid:16) G jimn,αn (cid:48) (cid:104) ˆ σ e n g m j ˆ σ e β e n (cid:48) i (cid:105) − G jimn,m (cid:48) β (cid:104) ˆ σ e n g m j ˆ σ g m (cid:48) g α i (cid:105) (cid:17) , (S14) ddt (cid:104) ˆ σ e β e γ i (cid:105) = − i (∆ e β − ∆ e γ ) (cid:104) ˆ σ e β e γ i (cid:105) − (cid:0) G ∗ iimγ,mn (cid:104) ˆ σ e β e n i (cid:105) + G iimn,mβ (cid:104) ˆ σ e n e γ i (cid:105) (cid:1) − (cid:88) j (cid:54) = i (cid:16) G ∗ ijm (cid:48) γ,mn (cid:104) ˆ σ e β g m (cid:48) i ˆ σ g m e n j (cid:105) + G jimn,m (cid:48) β (cid:104) ˆ σ e n g m j ˆ σ g m (cid:48) e γ i (cid:105) (cid:17) , (S15) ddt (cid:104) ˆ σ g α g γ i (cid:105) = − i (∆ g α − ∆ g γ ) (cid:104) ˆ σ g α g γ i (cid:105) + (cid:0) G ∗ iiαn (cid:48) ,γn + G iiαn (cid:48) ,γn (cid:1) (cid:104) ˆ σ e n e n (cid:48) i (cid:105) + (cid:88) j (cid:54) = i (cid:16) G jimn (cid:48) ,γn (cid:104) ˆ σ e n (cid:48) g m j ˆ σ g α e n i (cid:105) + G ∗ ijαn (cid:48) ,mn (cid:104) ˆ σ e n (cid:48) g γ i ˆ σ g m e n j (cid:105) (cid:17) . (S16)The two-point equations read ddt (cid:104) ˆ σ e β g α i ˆ σ e γ e η j (cid:105) = − i (∆ e β − ∆ g α + ∆ e γ − ∆ e η ) (cid:104) ˆ σ e β g α i ˆ σ e γ e η j (cid:105)− (cid:16) G ∗ jimη,αn (cid:104) ˆ σ e β e n i ˆ σ e γ g m j (cid:105) + G iimn,mβ (cid:104) ˆ σ e n g α i ˆ σ e γ e η j (cid:105) + G ∗ jjmη,mn (cid:104) ˆ σ e β g α i ˆ σ e γ e n j (cid:105) + G jjmn,mγ (cid:104) ˆ σ e β g α i ˆ σ e n e η j (cid:105) (cid:17) − (cid:88) k (cid:54) = i,j (cid:16) G ∗ jkm (cid:48) η,mn (cid:104) ˆ σ e β g α i ˆ σ e γ g m (cid:48) j ˆ σ g m e n k (cid:105) + G kim (cid:48) n,mβ (cid:104) ˆ σ g m g α i ˆ σ e γ e η j ˆ σ e n g m (cid:48) k (cid:105) + G kjm (cid:48) n,mγ (cid:104) ˆ σ e β g α i ˆ σ g m e η j ˆ σ e n g m (cid:48) k (cid:105) − G kimn (cid:48) ,αn (cid:104) ˆ σ e β e n i ˆ σ e γ e η j ˆ σ e n (cid:48) g m k (cid:105) (cid:17) , (S17) ddt (cid:104) ˆ σ e β g α i ˆ σ g γ g η j (cid:105) = − i (∆ e β − ∆ g α + ∆ g γ − ∆ g η ) (cid:104) ˆ σ e β g α i ˆ σ g γ g η j (cid:105) + (cid:16) G jjγn (cid:48) ,ηn + G ∗ jjγn (cid:48) ,ηn (cid:17) (cid:104) ˆ σ e β g α i ˆ σ e n (cid:48) e n j (cid:105) + (cid:16) G jiγn (cid:48) ,αn + G ∗ jiγn (cid:48) ,αn (cid:17) (cid:104) ˆ σ e β e n i ˆ σ e n (cid:48) g η j (cid:105)− (cid:16) G iimn,mβ (cid:104) ˆ σ e n g α i ˆ σ g γ g η j (cid:105) + G jiγn,mβ (cid:104) ˆ σ g m g α i ˆ σ e n g η j (cid:105) (cid:17) + (cid:88) k (cid:54) = i,j (cid:16) G kjmn,ηn (cid:48) (cid:104) ˆ σ e β g α i ˆ σ g γ e n (cid:48) j ˆ σ e n g m k (cid:105) + G ∗ jkγn (cid:48) ,mn (cid:104) ˆ σ e β g α i ˆ σ e n (cid:48) g η j ˆ σ g m e n k (cid:105) + G kimn,αn (cid:48) (cid:104) ˆ σ e β e n (cid:48) i ˆ σ g γ g η j ˆ σ e n g m k (cid:105) − G kimn,m (cid:48) β (cid:104) ˆ σ g m (cid:48) g α i ˆ σ g γ g η j ˆ σ e n g m k (cid:105) (cid:17) , (S18) ddt (cid:104) ˆ σ e β e γ i ˆ σ e η e ζ j (cid:105) = − i (∆ e β − ∆ e γ + ∆ e η − ∆ e ζ ) (cid:104) ˆ σ e β e γ i ˆ σ e η e ζ j (cid:105)− (cid:16) G ∗ iimγ,mn (cid:104) ˆ σ e β e n i ˆ σ e η e ζ j (cid:105) + G iimn,mβ (cid:104) ˆ σ e n e γ i ˆ σ e η e ζ j (cid:105) + G ∗ jjmζ,mn (cid:104) ˆ σ e β e γ i ˆ σ e η e n j (cid:105) + G jjmn,mη (cid:104) ˆ σ e β e γ i ˆ σ e n e ζ j (cid:105) (cid:17) − (cid:88) k (cid:54) = i,j (cid:16) G ∗ jkm (cid:48) ζ,mn (cid:104) ˆ σ e β e γ i ˆ σ e η g m (cid:48) j ˆ σ g m e n k (cid:105) + G kjm (cid:48) n,mη (cid:104) ˆ σ e β e γ i ˆ σ g m e ζ j ˆ σ e n g m (cid:48) k (cid:105) + G ∗ ikm (cid:48) γ,mn (cid:104) ˆ σ e β g m (cid:48) i ˆ σ e η e ζ j ˆ σ g m e n k (cid:105) + G kim (cid:48) n,me β (cid:104) ˆ σ g m e γ i ˆ σ e η e ζ j ˆ σ e n g m (cid:48) k (cid:105) (cid:17) , (S19) ddt (cid:104) ˆ σ g α g γ i ˆ σ g η g ζ j (cid:105) = − i (∆ g α − ∆ g γ + ∆ g η − ∆ g ζ ) (cid:104) ˆ σ g α g γ i ˆ σ g η g ζ j (cid:105) + (cid:104) (cid:16) G ijαn (cid:48) ,ζn + G ∗ ijαn (cid:48) ,ζn (cid:17) (cid:104) ˆ σ e n (cid:48) g γ i ˆ σ g η e n j (cid:105) + (cid:0) G iiαn (cid:48) ,γn + G ∗ iiαn (cid:48) ,γn (cid:1) (cid:104) ˆ σ e n (cid:48) e n i ˆ σ g η g ζ j (cid:105) + (cid:16) G jjηn (cid:48) ,ζn + G ∗ jjηn (cid:48) ,ζn (cid:17) (cid:104) ˆ σ g α g γ i ˆ σ e n (cid:48) e n j (cid:105) + (cid:16) G jiηn (cid:48) ,γn + G ∗ jiηn (cid:48) ,γn (cid:17) (cid:104) ˆ σ g α e n i ˆ σ e n (cid:48) g ζ j (cid:105) (cid:105) + (cid:88) k (cid:54) = i,j (cid:16) G kjmn (cid:48) ,ζn (cid:104) ˆ σ g α g γ i ˆ σ g η e n j ˆ σ e n (cid:48) g m k (cid:105) + G ∗ jkηn (cid:48) ,mn (cid:104) ˆ σ g α g γ i ˆ σ e n (cid:48) g ζ j ˆ σ g m e n k (cid:105) + G kimn (cid:48) ,γn (cid:104) ˆ σ g α e n i ˆ σ g η g ζ j ˆ σ e n (cid:48) g m k (cid:105) + G ∗ ikαn (cid:48) ,mn (cid:104) ˆ σ e n (cid:48) g γ i ˆ σ g η g ζ j ˆ σ g m e n k (cid:105) (cid:17) , (S20) ddt (cid:104) ˆ σ e β e γ i ˆ σ g α g η j (cid:105) = − i (∆ e β − ∆ e γ + ∆ g α − ∆ g η ) (cid:104) ˆ σ e β e γ i ˆ σ g α g η j (cid:105) + (cid:104) (cid:16) G jjαn (cid:48) ,ηn + G ∗ jjαn (cid:48) ,ηn (cid:17) (cid:104) ˆ σ e β e γ i ˆ σ e n (cid:48) e n j (cid:105)− (cid:16) G ∗ iimγ,mn (cid:104) ˆ σ e β e n i ˆ σ g α g η j (cid:105) + G iimn,mβ (cid:104) ˆ σ e n e γ i ˆ σ g α g η j (cid:105) + G ∗ ijmγ,ηn (cid:104) ˆ σ e β g m i ˆ σ g α e n j (cid:105) + G jiαn,mβ (cid:104) ˆ σ g m e γ i ˆ σ e n g η j (cid:105) (cid:17)(cid:105) + (cid:88) k (cid:54) = i,j (cid:16) G kjmn (cid:48) ,ηn (cid:104) ˆ σ e β e γ i ˆ σ g α e n j ˆ σ e n (cid:48) g m k (cid:105) + G ∗ jkαn (cid:48) ,mn (cid:104) ˆ σ e β e γ i ˆ σ e n (cid:48) g η j ˆ σ g m e n k (cid:105)−G ∗ ikm (cid:48) γ,mn (cid:104) ˆ σ e β g m (cid:48) i ˆ σ g α g η j ˆ σ g m e n k (cid:105) − G kim (cid:48) n,mβ (cid:104) ˆ σ g m e γ i ˆ σ g α g η j ˆ σ e n g m (cid:48) k (cid:105) (cid:17) , (S21) ddt (cid:104) ˆ σ e β g α i ˆ σ g η e γ j (cid:105) = − i (∆ e β − ∆ g α + ∆ g η − ∆ e γ ) (cid:104) ˆ σ e β g α i ˆ σ g η e γ j (cid:105) + (cid:104) (cid:16) G jiηn (cid:48) ,αn + G ∗ jiηn (cid:48) ,αn (cid:17) (cid:104) ˆ σ e β e n i ˆ σ e n (cid:48) e γ j (cid:105)− (cid:16) G ∗ jimγ,αn (cid:104) ˆ σ e β e n i ˆ σ g η g m j (cid:105) + G iimn,mβ (cid:104) ˆ σ e n g α i ˆ σ g η e γ j (cid:105) + G ∗ jjmγ,mn (cid:104) ˆ σ e β g α i ˆ σ g η e n j (cid:105) + G jiηn,mβ (cid:104) ˆ σ g m g α i ˆ σ e n e γ j (cid:105) (cid:17)(cid:105) + (cid:88) k (cid:54) = i,j (cid:16) G ∗ jkηn (cid:48) ,mn (cid:104) ˆ σ e β g α i ˆ σ e n (cid:48) e γ j ˆ σ g m e n k (cid:105) + G kimn (cid:48) ,αn (cid:104) ˆ σ e β e n i ˆ σ g η e γ j ˆ σ e n (cid:48) g m k (cid:105)−G ∗ jkm (cid:48) γ,mn (cid:104) ˆ σ e β g α i ˆ σ g η g m (cid:48) j ˆ σ g m e n k (cid:105) − G kim (cid:48) n,mβ (cid:104) ˆ σ g m g α i ˆ σ g η e γ j ˆ σ e n g m (cid:48) k (cid:105) (cid:17) , (S22) ddt (cid:104) ˆ σ e β g α i ˆ σ e γ g η j (cid:105) = − i (∆ e β − ∆ g α + ∆ e γ − ∆ g η ) (cid:104) ˆ σ e β g α i ˆ σ e γ g η j (cid:105)− (cid:16) G iimn,mβ (cid:104) ˆ σ e n g α i ˆ σ e γ g η j (cid:105) + G jjmn,mγ (cid:104) ˆ σ e β g α i ˆ σ e n g η j (cid:105) (cid:17) + (cid:88) k (cid:54) = i,j (cid:16) G kjmn (cid:48) ,ηn (cid:104) ˆ σ e β g α i ˆ σ e γ e n j ˆ σ e n (cid:48) g m k (cid:105) + G kimn (cid:48) ,αn (cid:104) ˆ σ e β e n i ˆ σ e γ g η j ˆ σ e n (cid:48) g m k (cid:105)−G kjm (cid:48) n,mγ (cid:104) ˆ σ e β g α i ˆ σ g m g η j ˆ σ e n g m (cid:48) k (cid:105) − G kim (cid:48) n,mβ (cid:104) ˆ σ g m g α i ˆ σ e γ g η j ˆ σ e n g m (cid:48) k (cid:105) (cid:17) . (S23)We have defined G ijmn,m (cid:48) n (cid:48) ≡ Γ ijmn,m (cid:48) n (cid:48) + i ∆ ijmn,m (cid:48) n (cid:48) and G ∗ ijmn,m (cid:48) n (cid:48) ≡ Γ ijmn,m (cid:48) n (cid:48) − i ∆ ijmn,m (cid:48) n (cid:48) . SHORT-TIME EXPANSION WITH MAGNETIC FIELDS
Here we provide details of the short-time perturbative expansion in the presence of a magnetic field. We show (1)that the first-order term of the dipolar shift is independent from Zeeman shifts of any size, and (2) we derive simplifiedexpressions for the shift in the large Zeeman shift limit.
Lab-frame calculation
Zero-order term
The Zeeman shifts contribute only trivially to the frequency shift at zero order. The only difference comes from (cid:104) ˆ σ e β g α i (cid:105) which acquires an extra term − i (∆ e β − ∆ g α ) (cid:104) ˆ σ e β g α i (cid:105) . This translates into an extra contribution to thezero-order shift of the form δ g α e β = − cos θ πN (cid:32) (∆ e β − ∆ g α ) + (cid:88) i,j (cid:54) = i U jig α e β (cid:33) . (S24)In other words, the dipolar shift is simply shifted by the energy difference between g α and e β due to the Zeemansplitting, as expected. First-order term
There are two independent contributions to the first-order term of the shift coming from the Zeeman splitting. Onecontribution comes from the product of two first-order coherences (cid:104) ˆ S x (cid:105) (cid:104) ˆ S y (cid:105) (cid:104) ˆ S x (cid:105) = (cid:104) ˆ S x (cid:105) (cid:104) ˆ S y (cid:105) (cid:104) ˆ S x (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B =0 + (∆ e β − ∆ g α ) (cid:32) Γ2 + cos θN (cid:88) i,j (cid:54) = i E jig α e β (cid:33) . (S25)Here, the subscript ( · ) | B =0 indicates the expression ( · ) evaluated for zero magnetic field. The other contribution comesfrom the second order coherence (cid:104) ˆ σ e β g α i (cid:105) = (cid:104) ˆ σ e β g α i (cid:105) (cid:12)(cid:12) B =0 + 2 i (∆ e β − ∆ g α ) (cid:32) Γ (cid:104) ˆ σ e β g α i (cid:105) − (cid:16) (cid:104) ˆ σ e β e β i (cid:105) − (cid:104) ˆ σ g α g α i (cid:105) (cid:17) (cid:88) j (cid:54) = i G jiαβ,αβ (cid:104) ˆ σ e β g α j (cid:105) (cid:33) . (S26)Together, this yields (cid:104) ˆ S y (cid:105) (cid:104) ˆ S x (cid:105) − (cid:104) ˆ S y (cid:105) (cid:104) ˆ S x (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B =0 = 2 (cid:32) (cid:104) ˆ S x (cid:105) (cid:104) ˆ S y (cid:105) (cid:104) ˆ S x (cid:105) − (cid:104) ˆ S x (cid:105) (cid:104) ˆ S y (cid:105) (cid:104) ˆ S x (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B =0 (cid:33) = ⇒ δ g α e β = δ g α e β | B =0 . (S27)Thus, the magnetic field does not contribute to the dipolar shift at first order in the short-time expansion. However,note that the first-order approximation to the shift quickly becomes invalid as we increase the size of the Zeemanshifts. In the limit of large magnetic field it is better to switch from the lab to the rotating frame, as we show in thenext section. Large Zeeman shift limit
To take the strong magnetic field limit (i.e., µ | B | (cid:29) Γ) it is convenient to rewrite the above equations of motionin the rotating frame of the field. Specifically, we introduce rotated variables as (cid:104) ˆ¯ σ a m b n i (cid:105) = e i (∆ am − ∆ bn ) t (cid:104) ˆ σ a m b n i (cid:105) with a, b ∈ { g, e } . Let us assume a strong magnetic field which generates large Zeeman shifts, such that (∆ e n − ∆ e β ) (cid:29) Γ and(∆ g m − ∆ g α ) (cid:29) Γ ∀ m (cid:54) = α and n (cid:54) = β . Under the rotating-wave approximation, we disregard any fast oscillating termsappearing in the equations of motion of the rotated variables, i.e., e i (∆ en − ∆ en (cid:48) ) t ≈ δ n,n (cid:48) and e i (∆ gm − ∆ gm (cid:48) ) t ≈ δ m,m (cid:48) (where here δ stands for the Kronecker delta). Note that this type of oscillating phases usually appear multiplying eachother. However, we can use this approximation because we are assuming that the ground and excited manifolds havedifferent g -factors (i.e., δ e (cid:54) = δ g ), such that e i (∆ en − ∆ en (cid:48) ) t and e i (∆ gm − ∆ gm (cid:48) ) t oscillate at vastly different frequencies.The strong-field suppression of off-resonant processes in these equations implies that, compared to Eqs. (S14-S18),fewer processes involving levels different from g α and e β contribute. As a consequence, the first-order coefficient ofthe dipolar shift becomes¯ δ g α e β = lim µ | B |(cid:29) Γ δ g α e β = − πN (cid:88) i,j (cid:54) = i (cid:40) U jig α e β (cid:101) Γ g α e β ( θ ) + (cid:32) (cid:88) k (cid:54) = i,j ¯ W kji ( θ ) + ¯ Q ji ( θ ) (cid:33)(cid:41) , (S28)with the following definitions: ¯ W kji ( θ ) ≡ W kjip =( α − β ) ( θ ) , (S29)¯ Q ji ( θ ) ≡ (cid:32) sin ( θ/ (cid:88) p A − ji ; ijg β + p e β ,g β + p e β − cos ( θ/ A − ji ; ijg α e β ,g α e β (cid:33) . (S30)In this limit, the three-body processes ¯ W kji ( θ ) involve only the transition interrogated (no sums over polarizations).The beyond-MF terms ¯ Q ji ( θ ) do maintain a summation over polarizations p (cid:54) = ( β − α ) and hence connect to otherlevels. Note, however, that all processes involve the excited state e β and no other excited state.The above expressions show that to first order in time the dipolar shift of a multilevel system under a strongmagnetic field still deviates from a naive two-level model (in which quantities are simply rescaled by the appropriatepower of the CGC). The main difference is the first term in ¯ Q ji ( θ ) [Eq. (S30)], which describes both coherent andincoherent processes that involve e β and all ground states accessible through a single-photon transition of polarization p . THE ROLE OF NON-ADJACENT LEVELS
In the main text we have pointed out that, due to the large number of equations to solve, only levels adjacent tothe selected π -transitions were considered in the dynamics. We discuss here the validity of this approximation.For a given transition g α ↔ e α the adjacent levels would be g α ± and e α ± . All other levels, g α ± p and e α ± p with p (cid:54) = 0 ,
1, will be called ‘non-adjacent’. Figure S1 presents a comparison of the dipolar shift obtained in cumulantsimulations with ( p = 0 , ,
2: dashed lines) and without ( p = 0 ,
1: solid lines) non-adjacent levels for a 2D system of N = 3 atoms. Results for the − / − / Γ t δ / Γ × − ϵ k p = 0,1 p = 0,1,2 g α ± p , e α ± p α = − 9/2 α = − 1/2 Γt δ / Γ × − Γ t × − Γt δ / Γ × − Γ t × − ϵ k Γ t δ / Γ × − Γt δ / Γ × − ϵ k p = 0,1 p = 0,1,2 g α ± p , e α ± p α = − 9/2 α = − 1/2 ϵ k Γt δ / Γ × − (a) (b) θ = π /2 θ = π /4 FIG. S1. Dipolar shift δ for different approximations of the internal level structure, with dynamics solved using cumulantsimulations [approximation iii) in the main text]. We consider a system size of N = 3 and the cartoons in each subplotshow the direction of the interrogating laser and its polarization. We show simulations for a Ramsey pulse that interrogatesa π -transition, g α → e α , with pulse area (a) θ = π/
2, and (b) θ = π/
4. The simulations are performed for two differentsets of levels: including only adjacent levels, { g α ± p , e α ± p : p = 0 , } (full lines) and, for comparison, also including additionalbeyond-adjacent levels, { g α ± p , e α ± p : p = 0 , , } (dashed lines). Note that we are considering a system with angular momenta F e = F g = 9 /
2, such that the levels included have to fulfill | α ± p | ≤ /
2. The curves are shown for α = − / − / The results of Fig. S1 show that neglecting non-adjacent levels has no significant effect on the frequency shiftover the dark times considered (Γ t < − / − / − / LOCAL SHIFTS COMPARISON: TWO-LEVEL SYSTEMS AND LARGE-B LIMIT
We show in Fig. S2 the local shifts obtained for a more complete set of cases than in Fig. 2 in the main text. Inparticular, we added results for a two-level system (first column), for α = − / k parallel (top row) and orthogonal (second row) to theatomic plane. The 3D plots that were shown in the main text are displayed here along with the new ones for ease ofcomparison.The most important takeaway from these plots is the close similarity of the first three columns. This substantiatesthe claim made in the main text that the − / π/ δ g α e β i ≡ πt arctan( (cid:104) ˆ s yi (cid:105) / (cid:104) ˆ s xi (cid:105) ), yields the same short-time expression as the one given in Eq. (S2) butwithout the sum over i . In other words, for each atom i , δ g α e β i is obtained by simply evaluating the sums over j (cid:54) = i in Eq. (S2). This is not generally true for pulses with θ (cid:54) = π/ × -2 × -48 × -2 × -2 -30 × -2 -12 (b) δ i / ( Γ | C e α g α | − ) (a) × -112 k ϵϵ k k ϵϵ k k ϵ y i y i x i y i k ϵ × -101 × -2 ˛˛ × -2 -101 ˛˛ μ | B | ≫ Γ × -2 × -2 -47 μ | B | ≫ Γ μ | B | ≫ Γ α = − 9/2 α = − 9/2 × -2 -3.9-1.0 x i x i × -2 -25 ˛˛ (c) × -2 -21 α = − 1/2 α = − 1/2 x i x i × -2 -47 ˛˛ μ | B | ≫ Γ μ | B | ≫ Γ μ | B | ≫ Γ FIG. S2. Extended version of Fig. 2 from the main text (see corresponding caption). The additional columns offer thecomparison with a two-level system, i.e., | C e α g α | = 1, and with the large magnetic field limit for transition α = − / − / FIRST-ORDER EXPRESSION FOR RAMSEY FRINGES CONTRAST
The Ramsey fringes contrast was defined in the main text as C g α e β ( t ) ≡ N (cid:113) (cid:104) ˆ S x (cid:105) ( t ) + (cid:104) ˆ S y (cid:105) ( t ) . (S31)Following the same expansion in Γ t as performed for the calculation of the frequency shift, the contrast up to firstorder in time is given by C g α e β ( θ, t ) ≈ sin θ (cid:34) − (cid:32) Γ2 + cos θN (cid:88) i,j (cid:54) = i E jig α e β (cid:33) t (cid:35) . (S32)According to this expression, for a π/ θ (cid:54) = π/
2. In these cases, the decay rate is modified by the collective term E , which accounts for phenomena such assubradiant or superradiant emission. Similarly to the classical-dipole energy U , the E jig α e β terms [see Eq. (S3)] arealso proportional to the CGC of the selected transition g α ↔ e β , and can thus be highly suppressed when | C e β g α | (cid:28) − / − /
2, and − / − / − / − / E jig α e β in Eq. (S32). However, this sign is hard to predict from first prin-ciples, because the sum generally contains both positive and negative terms of different magnitudes which sensitivelydepend on details of the geometry. FREQUENCY SHIFT SCALING WITH SYSTEM SIZE
In this section, we present results for the scaling of the dipolar shift with the total atom number (or system size) N , including more cases than shown in the main text. We provide results for the − / Γ t C g − / → e − / g − / → e − / g − / → e − / Single particle -5/2 -1/2-9/2 − → − Single particle k ϵ θ = π /2 (a) (b) Γ t C ϵ k θ = π /4 Γ t C ϵ k θ = π /4 (c) FIG. S3. Contrast decay of the Ramsey fringes. (a) 3D lattice with N = 10 atoms and a π/ π/ N = 8 and π/ − / S5, and for the − / θ = π/ θ = π/ U (cid:101) Γ term is insensitive to magnetic fields, so dashed lines for this quantity are omitted). Note that in the bottomrows of Fig. S5(a) and (c) the lines corresponding to the W terms (yellow) cannot be seen, because their values areapproximately zero and hence they are hidden behind other lines.The main takeaways from these figures are the following. First, all figures show that the 2D array configurationwith k parallel to the plane (center column) is the only case where the total shift appears to keep growing with N for the range of values of N considered. All other cases (left and right columns) seem to saturate. Second, for the − / δ g α e β BMF (black lines in top row) accounts for a large share of thetotal shift in many of the cases presented. This shows the relevance of beyond-MF processes in this genuine multilevelcase. Third, for π/ t = 0 . − / N for the 2D configuration with k parallel to the atomicplane can be qualitatively understood from geometrical considerations. The key element is that the polarization (cid:15) ,i.e. the quantization axis, is perpendicular to the atomic plane. Because of this, the dipolar interaction coefficientsinvolving π -transitions become isotropic within the 2D plane (which is not the case when (cid:15) is parallel to the atomicplane). In the far-field limit ( k r ij (cid:29)
1) they are given by∆ ijg α e α ,g α e α = 3Γ4 (cid:0) C e α g α (cid:1) Re e ik r ij k r ij , Γ ijg α e α ,g α e α = 3Γ4 (cid:0) C e α g α (cid:1) Im e ik r ij k r ij . (S33)In the short-time expansion presented above, these interaction coefficients appear in convoluted sums over the whole2D array. The asymptotic behavior of these sums can be studied by approximating them through integrals. Usually, a3D integral over the dipolar interactions vanishes due to the anisotropy of the dipolar interactions. This explains thesaturation of the shift observed for the 3D case in Figs. S4, S5, S6, and S7. However, since the interaction coefficientsin Eq. (S33) are isotropic within the 2D atomic plane, the 2D integrals can show nontrivial scaling with N due to thelong-range tail 1 / ( k r ij ) of the dipolar interactions. Note, however, that the system sizes employed in the numericalresults of Figs. S4, S5, S6, and S7 are not sufficiently large to observe a clean scaling with N .As an example, we can consider the classical energy term U α ≡ N (cid:80) i,j (cid:54) = i U jig α e α , which appears in both the zeroand the first-order shift expressions, see Eqs. (S1) and (S2). For the 2D configuration with k parallel to the atomicplane we have, in the k r ij (cid:29) U jig α e α ∝ Re[ e ik r ij − i k · r ij / ( k r ij )]. In the large- N limit we can assume that (cid:80) j (cid:54) = i U jig α e α is approximately independent of i , and substitute the remaining sum by a 2D integral as (up to constants) U α ∼ Re (cid:82) π dϕ (cid:82) Lε dre ir (1 − cos ϕ ) , where ε > L ∼ N . A simple analysis then predicts U ∼ N / . An analogous calculation can be performed for E α ≡ N (cid:80) i,j (cid:54) = i E jig α e α , which also yields E ∼ N / . N − × − δ g α e β δ g α e β BMF N D i p o l a r s h i f t ( Γ t = . ) × − δ g α e β δ g α e β BMF (a) (b) N × − δ g α e β δ g α e β BMF (c) θ = π /4 F i r s t - o r d e r t e r m s s h i f t ( Γ t = . ) ϵ k ϵ k k ϵ α = β = − 1/2 N δ / × − WU g ΓQ N × − WU g ΓQ N − × − WU g ΓQ FIG. S4. Scaling with system size N at time Γ t = 0 . π -polarized − / θ = π/
4. Without magnetic field, the total shift for the times considered is dominatedfor all geometries by a combination of the zero-order shift and the beyond-mean-field, first-order term Q . All other first-orderterms appear negligible. With a strong magnetic field, the Q term is strongly modified, evidencing the multilevel nature of thesystem. N × − δ g α e β δ g α e β BMF N − × − WU g ΓQ N − × − δ g α e β δ g α e β BMF N − D i p o l a r s h i f t ( Γ t = . ) × − δ g α e β δ g α e β BMF N − × − WU g ΓQ θ = π /2 N − − × − WU g ΓQ (a) (b) (c) F i r s t - o r d e r t e r m s s h i f t ( Γ t = . ) ϵ k ϵ k k ϵ α = β = − 1/2 FIG. S5. Scaling with system size N at time Γ t = 0 . π -polarized − / θ = π/
2. The leading term in the absence of a magnetic field is (cid:101)
Γ times thetotal classical dipole energy U [first line in Eq. (6)]. The application of a strong magnetic field, on the other hand, makes thebeyond-mean-field, two-body correlation term Q the dominant one. N × − δ g α e β δ g α e β BMF N − × − δ g α e β δ g α e β BMF N D i p o l a r s h i f t ( Γ t = . ) × − δ g α e β δ g α e β BMF (a) (b) (c) θ = π /4 F i r s t - o r d e r t e r m s s h i f t ( Γ t = . ) ϵ k ϵ k k ϵ α = β = − 9/2 N × − WU g ΓQ N − × − WU g ΓQ N × − WU g ΓQ FIG. S6. Scaling with system size N at time Γ t = 0 . π -polarized − / θ = π/
4. For the dark time considered (Γ t = 0 . − / π/
4. Here, however, the first-order terms are mostly dominated by U (cid:101) Γand the three-body contributions W , since they are not strongly suppressed as in the − / − / N × − δ g α e β δ g α e β BMF N − × − δ g α e β δ g α e β BMF N − × − WU g ΓQ N D i p o l a r s h i f t ( Γ t = . ) × − δ g α e β δ g α e β BMF N × − WU g ΓQ (a) N × − WU g ΓQ (b) (c) θ = π /2 F i r s t - o r d e r t e r m s s h i f t ( Γ t = . ) ϵ k ϵ k k ϵ α = β = − 9/2 FIG. S7. Scaling with system size N at time Γ t = 0 . π -polarized − / θ = π/
2. This case is dominated by U (cid:101) Γ [first line in Eq. (6)] and the typically comparablethree-body contributions W [except for the two-dimensional geometry considered in (a)]. In (b), for N = 8 , the total shift( ≈ . × − ) is comparable to δ BMF , which justifies the large deviation between MF and cumulant simulations in Fig. 3(c)of the main text. All dashed curves (with a strong bias magnetic field) lie on top of the corresponding full-line curves (nomagnetic field), confirming the two-level nature of the − //