Magic distances in the blockade mechanism of Rydberg p and d states
MMagic distances in the blockade mechanism of Rydberg p and d states B. Vermersch,
1, 2, ∗ A. W. Glaetzle,
1, 2 and P. Zoller
1, 2 Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria (Dated: November 9, 2018)We show that the Rydberg blockade mechanism, which is well known in the case of s states, can be sig-nificantly di ff erent for p and d states due to the van der Waals couplings between di ff erent Rydberg Zeemansublevels and the presence of a magnetic-field. We show, in particular, the existence of magic distances corre-sponding to the laser-excitation of a superposition of doubly excited states. I. INTRODUCTION
Rydberg states of alkali atoms have remarkable proper-ties [1] including long lifetimes and very high polarizabilitieswhich make them ideally suited to study a plethora of quan-tum phenomena. Due to the long-range nature of their inter-actions Rydberg atoms allow, in particular, to study variousspin models [2] and also have applications for quantum infor-mation processing [3].A central property of a laser-excited Rydberg gas is theblockade mechanism [4]: Two non-interacting ground-stateatoms can be simultaneously excited with a coherent laser toa Rydberg level. However, when the distance between theatoms becomes smaller than a typical length scale, the block-ade radius, the doubly excited state is shifted out of resonancedue to the strong interactions between Rydberg levels. Thismechanism can be generalized to the case of arbitrarily largenumber of atoms [5] leading to an e ff ective two-level system– the super-atom – consisting of the state with all atoms intheir ground state and the symmetric state with a single Ry-dberg excitation shared between all atoms. They are coupledwith an enhanced Rabi frequency which has been successfullydemonstrated in recent experiments [6–10].In typical experimental setups with optical lattices or dipoletraps [10–14], Rydberg atoms interact via van der Waals(vdW) interactions scaling as ∝ / r , with r the interparticledistance. For the familiar case of Rydberg s states these in-teractions are isotropic which result in a spherical symmetricblockade region. In contrast, for higher angular momentumstates, e.g., Rydberg p or d states, these interactions can bestrongly anisotropic [15] considering a specific Zeeman sub-level only.However, in general, the vdW interactions also mix pop-ulations in di ff erent Zeeman sublevels [16] and the blockademechanism cannot simply be described by treating the atomsas two-level systems consisting of a ground and a single Ry-dberg state. For Rydberg s states the strength of the vdWmixing matrix element is much smaller than the diagonal in-teraction matrix element and this treatment, considering onlya single Zeeman sublevel, is su ffi cient. On the contrary, forRydberg p and d states the vdW mixing matrix element isof the same order as the diagonal interaction matrix element ∗ [email protected] leading to much more complex blockade dynamics which wewill discuss in the following.Complementary to recent work on spin-spin interactionsbetween ground-state atoms with laser-admixed Rydberg in-teractions involving p states [17], we are interested here inthe dynamics of an ensemble of atoms, which are excited,resonantly with the bare atomic transition, to a p (or d ) Ry-dberg state, as in a typical Rydberg blockade experiment. Thegoal of the paper is to give a general theoretical descriptionof the blockade mechanism taking into account the vdW cou-plings between Zeeman levels but also the anisotropic charac-ter of the vdW interactions and the influence of a magnetic-field. We show, in particular, that the interplay between themagnetic-field and the vdW interactions leads to the existenceof magic distances where the laser can resonantly excite a su-perposition of doubly excited Rydberg states – even inside theblockade radius. Furthermore, our general treatment allows usto interpret a recent experiment demonstrating the anisotropicRydberg blockade using D / states [11].Our paper is organized as follows: In Sec. II we illustratethe Rydberg blockade in the presence of vdW coupling forthe simplest possible example of two Rydberg P / states. InSec. III we generalize our approach to Rydberg P / and D / states and provide a theoretical interpretation of the recent ex-perimental results of Barredo et al. [11]. Furthermore, wepropose parameters for a magic distance experiment with Ry-dberg D / states. Finally, in Sec. IV we describe the con-sequences of the presence of a magic distance at the many-body level and generalize the concept of a super-atom. InAppendix A we review the properties of the anisotropic vdWinteractions whereas in Appendix B we assess the e ff ect ofquadrupole-quadrupole interactions. II. GENERALIZED RYDBERG BLOCKADE IN THE P / MANIFOLD
In this section we illustrate the generalized blockade picturediscussing the conceptually simplest example of two atomslaser-excited to a specific Zeeman sublevel in the Rydberg P / manifold [see Fig. 1 (a)]. In this case the vdW inter-action Hamiltonian, H V , takes a simple form, which allowsus to analytically describe the dynamics of the system be-yond the familiar Rydberg S / state picture [3]. Figure 1(b)schematically illustrates the new features of the generalizedRydberg blockade mechanism: At large distance, r (cid:29) r b , a r X i v : . [ phy s i c s . a t o m - ph ] D ec (b)(a) . . . . . . . . . . . . r p | i| i ( r )⌦ ( r ) ⌦ ( r ) B ( r ) r r b Energy levels r r b p ( r ) (c) | g, g i| g, }⇠ | , i⇠ | , i r ⌦ B r z x B atom 1 atom 2 | g i| i| i ⌦ B S / n P / ✓ H V ( r, ✓ ) Figure 1. Setup: (a) Two atoms (green and violet cir-cles), separated by a distance r and an angle θ with respect tothe magnetic-field B (thick black arrow), are laser-excited to the | n P / , m j = (cid:105) Rydberg states which interact via vdW interac-tions H V ( r , θ ) of Eq. (5). (b) Principle of the generalized Rydbergblockade for the simplest case θ = π/
2: vdW interactions willmix the Rydberg states | , (cid:105) and | , (cid:105) yielding new eigenstates | λ (cid:105) and | λ (cid:105) which can be both laser-excited from the symmetricstate | g , } = (cid:16) | g , (cid:105) + | , g (cid:105) (cid:17) / √ Ω ( r ) and Ω ( r ) (red arrows), respectively, where the position dependency re-flects the spatially dependent populations of | , (cid:105) and | , (cid:105) in | λ i (cid:105) .In particular, for repulsive vdW interactions and Zeeman splitting ∆ B > | λ (cid:105) at the magic dis-tance r , whereas in the limits r → r → ∞ , we obtain the stan-dard blocked and non-interacting situation, respectively. (c) Sketchof the average probability of double excitation p ( r ) in presence of amagic distance. where vdW interactions are negligibly small, two atoms inthe atomic ground-state | g , g (cid:105) can be resonantly laser-excitedto a Rydberg state | , (cid:105) . At smaller distances, r ∼ r b , thedoubly excited Rydberg state | , (cid:105) will in general shift outof resonance because of the strong vdW interaction resultingin the well-known Rydberg blockade e ff ect [3]. However,due to the vdW couplings between doubly excited Rydbergstates, e.g. for θ = π/ | , (cid:105) and another Rydbergstate | , (cid:105) , new eigenstates | λ (cid:105) and | λ (cid:105) are formed whichasymptotically (for large distances) connect to the bare states | , (cid:105) and | , (cid:105) , respectively. In the presence of an externalmagnetic-field the latter two states are Zeeman split in energyby 2 (cid:126) ∆ B . In the generalized blockade picture (including vdWcouplings) both states | λ (cid:105) and | λ (cid:105) contain a contribution of | , (cid:105) and thus can be laser-excited from the ground-statewith detunings λ ( r ) and λ ( r ), respectively [18]. In the par-ticular configuration of Fig. 1(b) where ∆ B > | λ (cid:105) shifts into resonance at a magic distance r and can be reso-nantly laser-excited even within the blockade radius r b . Thiswill lead to a peak in the average probability of double ex-citation p ( r ) sketched in panel (c). In the following sectionwe will derive the position, height and width of this additionalpeak analytically for nP / Rydberg states. In Sec. III we gen- eralize our approach to Rydberg P / and D / where severalZeeman sublevels are coupled resulting in a multi-peak struc-ture of p ( r ) shown, for example, in Fig. 7. A. Description of the system
We consider the setup shown in Fig. 1. Two alkali atomslocated at positions r i ( i = ,
2) are laser-excited to aparticular Rydberg state in the nP / manifold, which interactvia vdW interactions H V ( r ), where r = r − r . Our goalis to study the competition between the laser excitation andthe vdW interactions as in a typical Rydberg blockade situa-tion. As a particular example we consider Rubidium atomsinitially in their electronic ground state, which we chooseas | g (cid:105) ≡ | S / , F = , m F = (cid:105) z = | S / , m j = (cid:105) z ⊗ | N (cid:105) .Here, | N (cid:105) = | I = , m I = (cid:105) z denotes the nuclear spin state. Aresonant laser with Rabi frequency Ω excites the atoms to oneof the | m j = ± (cid:105) ≡ | n P / , m j = ± (cid:105) z ⊗ | N (cid:105) Rydberg states[19]. The presence of the magnetic-field B = B z gives riseto a Zeeman splitting of the Rydberg energy levels describedby the Hamiltonian H ( i ) B = (cid:88) m j (cid:16) (cid:126) ω nP / + ∆ E m j − ∆ E g (cid:17) | m j (cid:105) i (cid:104) m j | , (1)where (cid:126) ω nP / is the energy of the transition between theground state | g (cid:105) and the Rydberg states | m j (cid:105) in the absenceof magnetic field. With ∆ E m j ≡ µ B g j Bm j , we denote the Ryd-berg Zeeman shift where µ B = . h MHz / G is the Bohr mag-neton and g j is the Landé Factor for the Rydberg level. Fi-nally, ∆ E g ≡ µ B g F Bm F , with g F the ground state Landé Fac-tor, is the ground-state Zeeman shift. We note that the valueof the magnetic field is chosen su ffi ciently small to neglecthigher order Zeeman e ff ects. Furthermore, the Zeeman shifts ∆ E m j should be much smaller than the fine structure splitting ∆ fs (typically several GHz for n = −
40) to neglect couplingsbetween fine structure manifolds (Paschen-Back e ff ect).We now consider excitation of the Rydberg level | (cid:105) fromthe ground-state | g (cid:105) with a laser propagating along the direc-tion of the magnetic-field k L = k L z with left circular po-larization. In the rotating wave approximation, the resultingHamiltonian is H ( i ) L = (cid:126) Ω (cid:104) e i ( k L . r i − ω L t ) | g (cid:105) i (cid:104) | + hc (cid:105) , (2)with ω L the laser frequency and Ω = −(cid:104) g | ε . d | (cid:105)E / (cid:126) the Rabifrequency. Here E is the electric-field field amplitude, (cid:15) itspolarization ( σ − here) and d is the dipole operator. In the ro-tating frame and absorbing the phase e i k L . r i term in the defi-nition of | g (cid:105) i , the single-particle Hamiltonian H ( i ) A = H ( i ) L + H ( i ) B reduces to H ( i ) A = (cid:126) Ω (cid:104) | g (cid:105) i (cid:104) | + | (cid:105) i (cid:104) g | (cid:105) − (cid:126) ∆ B | (cid:105) i (cid:104) | , (3)where the laser frequency ω L = ω nP / + ( ∆ E − / − ∆ E g ) / (cid:126) ischosen to be in resonance with the atomic transition and theZeeman splitting is (cid:126) ∆ B = ∆ E − / − ∆ E / .The two Rydberg atoms interact dominantly via dipole-dipole interactions. At large distances and away from Försterresonances [3] these interactions can be treated perturbativelyleading to the vdW Hamiltonian H V [16, 17]. We describein detail the derivation of H V in the various Rydberg fine-structure manifolds in Appendix A. Here, we are interestedin nP / Rydberg states where H V ( r ) = [ u I + c D ( θ )] / r (4)consists of a diagonal, isotropic, part, with I the 4 × D ( θ ) = − sin θ − sin 2 θ − sin 2 θ sin θ − sin 2 θ sin θ − sin θ −
43 12 sin 2 θ − sin 2 θ sin θ − sin θ −
23 12 sin 2 θ sin θ sin 2 θ sin 2 θ − sin θ , (5)written here in the basis ( | , (cid:105) , | , (cid:105) , | , (cid:105) , | , (cid:105) ) withthe notation | m , m (cid:105) ≡ | m (cid:105) ⊗ | m (cid:105) . Here, r = | r − r | and θ = ∠ ( r , B ) is the angle between the relative vector andthe magnetic field. The prefactors u , c [given by Eq. (A6)and (A7), respectively] are generalized vdW coe ffi cients. Wenote that this interaction Hamiltonian neglects the couplingsto the neighboring P / Rydberg manifold and is thereforeonly valid for distances r larger than a typical length scale r (cid:63) = ( ∆ fs / u ) / . The first term of (5) corresponds to the diag-onal part of the interactions, and does not depend on the angle θ , thus representing the typical and well-known vdW potential ∝ / r . The second term includes anisotropic diagonal matrixelements as well as couplings between the di ff erent m j levels.The values of u and c are shown in Fig. 2 for Rubidium P / states. They are of the same order of magnitude so that onecan expect the anisotropic characters of the interactions andthe vdW couplings to have a significant impact on the dynam-ics. We note that considering S / states, H V can be written inthe same form as (5). However, the existence of the couplingterm c is in this case only due to the fine-structure splitting ofthe neighboring p Rydberg levels [see (A6)] so that its valueis negligible compared to u .We also emphasize that the eigenvalues of (5) which corre-spond to the Born-Oppenheimer potential curves of the vdWinteractions are isotropic, i.e do not depend on the angle θ .However, in the presence of the magnetic-field along the axis z , the Zeeman shift between the m j levels leads to anisotropicinteractions.In the limiting case θ =
0, as a consequence of the con-servation of the total angular momentum, the doubly excitedstates | , (cid:105) , | , (cid:105) are eigenstates of the vdW Hamiltonian,resulting in a typical blockade situation. In contrast, for θ = π ,the vdW Hamiltonian reduces to H V = r u c u + c / − c / − c / u + c / c u , (6)with u ≡ u − c . In the following, we consider this particularorientation as it corresponds to the simplest configuration withvdW couplings.
20 30 40 50 60 70 80 90 n − . − . − . − . − . . . . . ( M H z µ m ) × − u /n u/n c/n Figure 2. u , u and c as a function of the principal quantum number n for Rubidium P / states. In contrast to S / states, the couplingterm has the same order of magnitude as the diagonal terms for P / states. Our goal is now to study the dynamics of the system mod-eled by the Hamiltonian H ≡ (cid:88) i = , H ( i ) A + H V (7)considering an initial state | Ψ ( t = (cid:105) = | g , g (cid:105) . We note thatthe forces corresponding to vdW interactions are responsiblefor mechanical e ff ects [20, 21]. Our model (7), which ne-glects the motion of the atoms but also the spontaneous decayof the Rydberg level and black body transitions is thereforeonly valid in the frozen gas regime [2] corresponding to shorttime scales, typically tens of micro seconds [22]. In the nextsubsection, we also comment on the possible influence of me-chanical e ff ects for our particular system.For symmetry reasons, the dynamics is restricted to fourstates | g , g (cid:105) , | g , } ≡ √ ( | g , (cid:105) + | , g (cid:105) ), | , (cid:105) , | , (cid:105) withthe corresponding Hamiltonian H = (cid:126) Ω / √ (cid:126) Ω / √ (cid:126) Ω / √ (cid:126) Ω / √ u / r c / r c / r u / r − (cid:126) ∆ B . . (8)In order to estimate the influence of the coupling c on the ef-ficiency of the blockade mechanism, we are interested in thevalue of the average probability of double excitation p ( r , θ ) = lim T →∞ T (cid:90) T (cid:16) |(cid:104) , | Ψ ( t ) (cid:105)| + |(cid:104) , | Ψ ( t ) (cid:105)| (cid:17) dt . (9)In order to solve the dynamics and as presented in [16] inthe absence of a magnetic-field, we now prediagonalize theHamiltonian in the doubly-excited subspace: H = (cid:126) Ω / √ (cid:126) Ω / √ (cid:126) Ω ( r ) / (cid:126) Ω ( r ) / (cid:126) Ω ( r ) / λ ( r ) 00 (cid:126) Ω ( r ) / λ ( r ) , . (10)written in the basis | g , g (cid:105) , | g , } , | λ (cid:105) , | λ (cid:105) where the vdWeigenstates | λ ( r ) (cid:105) = cos φ | , (cid:105) + sin φ | , (cid:105) , (11) | λ ( r ) (cid:105) = sin φ | , (cid:105) − cos φ | , (cid:105) , (12)with tan[2 φ ( r )] = c / ( r (cid:126) ∆ B ) are directly coupled to the singlylaser-excited state | g , } with space-dependent Rabi frequen-cies Ω ( r ) = √ Ω cos φ ( r ) , (13) Ω ( r ) = √ Ω sin φ ( r ) , (14)and have an energy λ , ( r ) = ur − (cid:126) ∆ B ∓ (cid:115)(cid:32) c (cid:126) ∆ B r (cid:33) + . (15)We illustrate the four level structure associated to theHamiltonian (10) in Fig. 1(b). The advantage of this repre-sentation is that one can directly infer from the values of Ω , and λ , whether the doubly excited state manifold can be pop-ulated. One can notice that in the limit of small distances r →
0, the doubly excited states are energetically excludedfrom the dynamics whereas the state | , (cid:105) can be resonantlyexcited at large distances r → ∞ where the coupling c / r be-comes negligible.This representation also illustrates the existence of a magicdistance at r = r at which the Zeeman shift compensates thevdW interactions resulting in a non-interacting vdW eigen-state | λ (cid:105) : λ ( r ) =
0. Around this distance, the double exci-tation probability p ( r ) will exhibit a resonance peak similarto the anti-blockade e ff ect [23] with the crucial di ff erence thatthe non-interacting vdW eigenstate | λ (cid:105) which is excited isa superposition of the two doubly excited states | , (cid:105) and | , (cid:105) . B. Properties at the magic distance r We now describe the magic distance peak and show that itcan be observed experimentally. First, its position, the magicdistance r , corresponds to a zero of one vdW eigenstate: λ i ( r ) = r = (cid:34) u (1 − α )2 (cid:126) ∆ B (cid:35) / , (16)with α = c / u . We note, that this solution only exists if ∆ B u (1 − α ) > , (17)which illustrates the fact that the magic distance only existswhen the magnetic-field competes with the vdW interactions( ∆ B and B have opposite signs). In the following, we considerthat the orientation of the magnetic-field sgn( B ) is chosen tosatisfy the condition (17) and derive analytical expressions de-scribing the magic distance peak. From Eq. (16), one can no-tice that the magic distance r , depends on the value of the Zeeman shift ∆ B : For small magnetic-fields, r is larger thanthe blockade radius r b ≡ ( u / (cid:126) Ω ) / so that we expect the emer-gence of a peak in the non-blocked region. However, for largemagnetic-fields, the magic distance leads to the formation ofa peak in the supposedly blocked region, i.e. r < r b .Let us first describe the case r < r b as the behavior ofthe system is in total contradiction with the standard blockadecriterion, i.e. r < r b = ⇒ p ( r ) → r = r only the vdW eigenstate | ˜ λ (cid:105) = | λ (cid:105) if α < | λ (cid:105) if α > , (18)whose eigenvalue ˜ λ vanishes at the magic distance:˜ λ ( r ) =
0, can be populated whereas the other vdWeigenstate is energetically excluded. Using the relation |(cid:104) ˜ λ ( r ) | , (cid:105)| = α/ (1 + α ) we then obtain the peak height p ( r ) = α + α (1 + α ) . (19)The analytical expression Eq. (19) illustrates the key role ofthe vdW coupling c in the magic distance phenomenon. Con-sidering P / states, α is typically of the order of unity (see forexample Fig. 2 for Rubidium atoms) so that the peak height p ( r ) is significant. We also note that the value of p ( r ) isindependent of the magnetic-field.We are now interested in the value of the peak width ∆ r associated to the range of distances ( r ) where | ˜ λ (cid:105) can be pop-ulated. This quantity allows us to assess the importance oflocalization e ff ects: In order to be observed, the peak width ∆ r should be larger than the uncertainty δ r in the distance r between the atoms. We estimate the value of ∆ r by consider-ing that | ˜ λ (cid:105) can be populated if its energy ˜ λ is in the excitationbandwidth [ − (cid:126) Ω , (cid:126) Ω ]: ∆ r ∼ (cid:126) Ω ˜ λ (cid:48) ( r ) , (20)with ˜ λ (cid:48) ( r ) ≡ (cid:32) ∂ ˜ λ∂ r (cid:33) r = r = − ur (cid:32) − α + α (cid:33) . (21)The value of the derivative ˜ λ (cid:48) ( r ) allows therefore to estimatethe possibility to observe the magic distance peak. UsingEqs. (16) and. (21), we note that the width ∆ r scales as B − / :For very large magnetic-fields corresponding to r (cid:28) r b , thepeak becomes narrow so that it might be di ffi cult to observe itexperimentally. However, in the limit of small magnetic-fieldswhere r (cid:46) r b , ∆ r can be larger than the uncertainty δ r and canbe therefore experimentally resolved. We also note that, as theRydberg atoms are not trapped, the vdW force F = λ (cid:48) ( r ) atthe magic distance may induce mechanical e ff ects [21]. How-ever, as in the case of the peak width, the strength of this forcecan be controlled via the value of the magnetic-field.In order to be more specific about the feasibility of a magicdistance experiment, we now consider the example of Rubid-ium atoms. In this case, the diagonal element of the vdW r ( µ m) . . . . . . p ( r ) r r b (a) r ( µ m) . . . . . . p ( r ) r r b (b) Figure 3. (a) Mean double excitation probability p ( r ) [Eq. (9)] for n = Ω / (2 π ) = . B = . ff ective Hamiltonian Eq. (24). Finally the red dotted linerepresents the usual blockade situation artificially obtained by setting c =
0. (b) p ( r ) for n =
70 corresponding to α ∼ .
45, a magnetic-field B = − . interactions u is always negative whereas α is smaller than 1for n <
43 and larger otherwise [see Fig. 2]. Consequently,the magic distance r exists for ∆ B < B >
0) if n <
43 and ∆ B < B <
0) otherwise [condition (17)]. We represent inFig. 3(a) the probability of double excitation, obtained numer-ically, as a function of the distance r for n = Ω / (2 π ) = . B = . p ≈ / c =
0, the value of p is not monotonousas around the magic distance r = . µ m [Eq. (16)], apeak emerges illustrating the crucial impact of the vdW cou-pling c on the Rydberg blockade. Its height p ( r ) = . ∆ r ∼ . µ m are in good agreement with the analyt-ical expressions Eqs. (19) and (20). Furthermore, the peakwidth ∆ r is larger than the uncertainty δ r for typical experi-mental setups [10–14] whereas the characteristic length scale˜ δ r ∼ FT / (2 µ ) ∼ . µ m ( T = π/ Ω is the typical time ofthe experiment and µ is the reduced mass of the atomic pair)corresponding to the vdW force is negligible compared to ∆ r .Consequently, for these parameters, the magic distance peakis accessible within state-of-the-art experiments. We note that,in this example, α = .
95 is close to 1 so that the derivative˜ λ (cid:48) ( r ) associated to the width of the peak and the vdW forceis strongly reduced compared to an anti-blockade configura-tion where ˜ λ (cid:48) ( r ) = − u / r , [see Eq. (21)]. The possibilityto observe the magic distance peak is however not restrictedto a particular value of α : Considering for example a largervalue of the principal quantum number n =
70 with α = . B = − . ∆ r ∼ µ m (cid:29) δ r and˜ δ r ∼ . µ m (cid:28) ∆ r.The expressions (19), (20) and (21) were derived assumingthat the peak emerges in the usually blocked region r < r b .Let us now for the sake of completeness describe the limit ofvery small magnetic-fields leading to r > r b . Around themagic distance r , | λ (cid:105) , | λ (cid:105) can in this case be excited si-multaneously. The resulting probability of double excitation, r ( µ m) . . . . . . p ( r ) r r b Figure 4. Mean double excitation probability p ( r ) [Eq. (9)] for n = Ω / (2 π ) = . B = r > r , r b and the an-alytical expression (22) [black dash-dotted line] correctly describesits emergence from the non-interacting limit r → ∞ . which can exceed the non-interaction limit 3 /
8, is obtained inthe limit (cid:126) Ω (cid:29) u / r , c / r , (cid:126) ∆ B from first-order perturbationtheory p ( r ) ≈ + c (cid:0) (cid:126) ∆ B r − u (cid:1) + c , (22)which shows in particular that p ( r ) ≤ .
5. An example ofthis situation is shown in Fig. 4 for n =
40 and B = p ≈ / r ≈ − µ m and is well described by the analyticalexpression Eq. (22) (black dash-dotted line) for r > r . Thewidth of this peak is in this example several microns, whichshows that for a such small magnetic-field the e ff ect of thevdW coupling on the dynamics is dominant. C. The e ff ective potential as a blockade criterion The peak corresponding to a magic distance r is due tothe vdW coupling c [see Eq. (19)] and is therefore a featurewhich goes beyond the usual Rydberg blockade. However, itis useful to define a generalized Rydberg blockade criterionthat would allow one to determine directly from the valuesof the parameters whether the system is Rydberg blocked, re-placing the usual criterion r < r b . To do so and following themethod of [16] given in the absence of a magnetic-field, weconsider the partially blocked regime | λ , | (cid:29) (cid:126) Ω and elimi-nate adiabatically the doubly excited states to obtain: p ( r ) = (cid:126) Ω (cid:88) i = , |(cid:104) , | λ i (cid:105)| λ i . (23)This expression is indeed very instructive as the comparisonwith the value (cid:126) Ω / V ff obtained from an e ff ective usual Ry-dberg blockade Hamiltonian H e ff = (cid:88) i = , H ( i ) A + V e ff | , (cid:105)(cid:104) , | (24)allows to define an e ff ective potential1 V ff ≡ (cid:88) i = , |(cid:104) , | λ i (cid:105)| λ i . (25)The e ff ective potential summarizes the interplay between thevdW interactions and the magnetic-field and is particularlyuseful to know whether the system is blocked, the criterion | V e ff | > (cid:126) Ω replacing the usual blockade criterion r < r b .Moreover, even if its expression was derived in the partiallyblocked regime, the e ff ective potential approach correctly de-scribes the dynamics in the non-interacting limit r → ∞ asin this case V e ff →
0. Finally, this quantity was recentlymeasured experimentally [11], giving us a starting point toassess the relevance of our work, see next section. Finally,from Eq. (25), one can immediately notice that the zero of thee ff ective potential indeed corresponds to the magic distance r . However, this approach cannot described properly the re-sulting peak p ( r ). This is shown in Fig. 3(a),(b) and 4 wherethe green dashed representing the e ff ective approach allowsto know whether the system is blocked or not but does notdescribe correctly the peak formed around r . III. GENERALIZATION TO OTHER ANGULARMOMENTUM STATES
The approach we used in the previous section to study theblockade dynamics in the particular configuration θ = π/ P / states can be generalized to any fine-structure manifold( S / , P / , P / , D / , ...) and any angle θ . In general, thevdW interactions within a given Zeeman manifold of Rydbergstates can we written as (see App.A) H V ( r , θ ) = (cid:88) αβγδ V αβ ; γδ ( r , θ ) | m α , m β (cid:105)(cid:104) m γ , m δ | , (26)where we used the notation | m α , m β (cid:105) = | n L j , m α (cid:105) z ⊗| n L j , m β (cid:105) z . The matrix V αβ ; γδ ( r , θ ), which has dimension(2 j + , accounts for the diagonal interactions and vdW cou-plings between the Zeeman sublevels. In the particular case of θ = M = m α + m β = m γ + m δ .For arbitrary angle θ the total angular momentum M canchange by 0 , ± , ± P / and D / Rydberg manifolds with j = / D / states. In the case of j = / H V can, in prin-ciple, be written as a 16 ×
16 matrix (see App. A). However, asthe anti-symmetric states of the type 1 / √ | m , m (cid:105)−| m , m (cid:105) ) r p r b | i| i ⌦ ( r )⌦ ( r ) r p r b | i| i ⌦ ( r )⌦ ( r ) r (10)0 r (2)0 (a) (b) Energy levels Energy levels | g, g i| g, }⇠ | , i⇠ | , i Figure 5. (a) Illustration of the blockade physics for laser-excited P / or D / Rydberg states with (a) | m j = (cid:105) and (b) | m j = (cid:105) anda magnetic field pointing in the positive z direction, i.e. B · z > | m j = (cid:105) with the same magnetic field gives rise to aset a magic distances r ( i )0 . with m (cid:44) m are not coupled to the laser-excited state | , (cid:105) ,we can reduce the basis to 10 states. Therefore, the two vdWeigenstates | λ (cid:105) , | λ (cid:105) for P / states of the previous section aresimply replaced by a set of vdW eigenstates {| λ i (cid:105)} i = of thevdW Hamiltonian of Eq. (26). For a particular magnetic fielddirection and a given angle θ , these 10 eigenstates will leadto a set of magic distances { r i } , illustrated in Fig. 5, and con-sequently to a multi-peak structure of the Rydberg excitationprobability (see Fig. 7). These eigenstates are then used toderive an e ff ective potential which can predict the behavior ofthe system apart from its zeros, which are the magic distances.As an illustration, let us now analyze how one can under-stand the recent experimental demonstration of anisotropicRydberg blockade [11] which were obtained using a laser ex-citation to the Rydberg state | D / , m j = (cid:105) ≡ | (cid:105) , with twoatoms separated by a distance r = µ m and an arbitrary an-gle θ with respect to the magnetic-field B = B z with B = H in the subspace formedby the doubly excited states and obtain 10 vdW eigenstates | λ i (cid:105) with Rabi frequencies Ω i = √ Ω (cid:104) λ i | , (cid:105) . The corre-sponding graphical representation, shown in Fig. 5 (a) illus-trates that as the interactions are attractive (see Appendix A)and the Zeeman shift between the laser-excited | , (cid:105) and theother doubly excited states is negative, this configuration doesnot allow for magic distances. Consequently, the competitionbetween the vdW interactions and the magnetic-field can bedescribed by an e ff ective potential:1 V ff = (cid:88) i = |(cid:104) , | λ i (cid:105)| λ i . (27)We note that by following this approach we neglect the exci-tation of the singly excited states which can only be excitedfrom the pair states (for example the excitation of | g , (cid:105) from | , (cid:105) ). However, we checked numerically that their e ff ect isnegligible in this situation.We now compare in Fig. 6(a), the probability of double ex-citation p ( θ ) obtained by the e ff ective potential approach andby the full Hamiltonian, for the experimental value of the Rabifrequency Ω / (2 π ) = . p < . ff ective potential is in good agreement with the exactresult. This allows us to compare the theoretical value of thee ff ective potential to the corresponding experimental quantityobtained by fitting the dynamics with an e ff ective Hamiltonian[11]. The result is shown in Fig. 6(b): The blue dotted line cor-responds to the naive diagonal element V rr ≡ (cid:104) , | H V | , (cid:105) of the vdW Hamiltonian and clearly disagrees with the exper-imental results. The green dashed line represents the e ff ec-tive potential calculated from the theoretical value of the vdWcoe ffi cients, defined in App. A. Finally, we obtained the redsolid line using an experimental correction for these coe ffi -cients which is obtained for θ = θ = ff ects [21]). The green curve cap-tures the qualitative influence of the mixing between Rydbergstates whereas the red curve gives a quantitative agreement.These results confirm the relevance of our approach in a sit-uation where the influence of the vdW coupling is not verysignificant as illustrated by the small di ff erence between thenaive and the e ff ective potential in Fig. 6(b).Let us now investigate the case of a laser-excitation to theRydberg state | (cid:105) , represented schematically in Fig. 5(b). Inthis situation, the magnetic-field competes with the attractivevdW interactions leading to the existence of a set of magicdistances. The corresponding e ff ective potential is shown inFig. 7(a) as a function of r and θ , where its zeros correspond tothe positions of the magic distances as a function of θ . Exceptfor θ = p ( r ) of double excitation for θ = π showing the emergence of a peak around each of these magicdistances. Furthermore, in order to minimize localization andmechanical e ff ects, the position of the magic distances and thewidth of the corresponding peaks can be increased by apply-ing a smaller magnetic-field (this behavior can be understoodgraphically with Fig.8(b), the width of the peaks being relatedto the slope of the energy levels where they cross the zero-energy line, see also Sec. II). IV. MANY-BODY EFFECTS AT THE MAGIC DISTANCE
Now that we have characterized the behavior of the systemat the magic distance for two atoms and have shown in partic-ular that the resulting peak can be observed with Rydberg p , d states, and in many geometric configurations, let us investi-gate the many-body scenario. In the absence of vdW coupling c , the Rydberg blockade mechanism is responsible for the ex-citation of a super-atom, which corresponds to a superpositionof singly excited states with a collective enhancement of the − −
45 0 45 90 θ (deg) . . . . p ( θ ) (a) − −
45 0 45 90 θ (deg) . . . . . . . V e ff ( M H z ) (b) Figure 6. (a) Average pair excitation probability obtained by the di-agonalization of the full Hamiltonian (blue solid line) and by the ef-fective potential (red circles). (b) V rr (blue dotted line), V e ff (greendashed lines) , V e ff (red solid line) which relies on the experimentaldetermination of C and experimental data [11] (dark circles). light-atom coupling [2, 5]. We now show that at the magicdistance, the many-body dynamics results in the collective ex-citation of a superposition of pair states, generalizing the con-cept of super-atom.In order to understand the many-body dynamics associatedto the magic distance phenomenon, it is instructive to first con-sider 4 atoms ( i = , , ,
4) placed at the corners of a squareof side r [see Fig. 8(a)] and laser-excited to a P / Rydbergstate | (cid:105) . The direction of the magnetic-field is perpendicularto the plane formed by the square so that for all pairs of atoms,the angle θ is fixed to π/ r < r b . Accordingto the results of Sec. II, for each pair of atoms ( i , j ) separatedby r , there is a non-interacting vdW eigenstate | ˜ λ ( i , j ) (cid:105) . How-ever, the double pair states such as | ˜ λ (1 , (cid:105) ⊗ | ˜ λ (3 , (cid:105) (shownschematically in Fig. 8(a)) are not vdW eigenstates of the to-tal interaction Hamiltonian and are therefore energetically ex-cluded from the dynamics [25]. Consequently, considering r ( µ m) − − θ ( d e g ) (a) log [ V eff / ( h MHz)] − . − . − . − . . . . . . . . r ( µ m) . . . . . p ( r ) (b) Figure 7. (a) V e ff in MHz and logarithmic scale for n =
82 and B =
3G for a laser-excitation to | (cid:105) . The positions of the magic distancescorrespond to the blue lines. (b) Corresponding probability of doubleexcitation for θ = π/
2, illustrating the emergence of the peaks aroundthe magic distances. The graphical conventions are defined in Fig. 3. that initially all atoms are in the ground-state | g (cid:105) and giventhe symmetries of the problem, the dynamics is restricted tothree states: The initial state | ˜ ψ g (cid:105) = | g ... g (cid:105) , the familiar super-atom state | ˜ ψ s (cid:105) = (cid:80) i | g ...
12 ( i ) ... g (cid:105) and the super pair state | ˜ ψ p (cid:105) = (cid:80) | r i − r j | = r | g ... ˜ λ ( i , j ) ... g (cid:105) .The above discussion can be easily extended to the caseof an arbitrary large number of atoms N , which is shownschematically in Fig. 8(b). First, as in the super-atom model[2], we can describe the many-body dynamics dividing thesystem into an ensemble of blockade spheres of radius r b ,which evolve independently. Second, as shown in the case offour atoms, inside each blockade sphere two non-interactingvdW states cannot be excited simultaneously and the dynam- ics is restricted to three symmetric states: | ˜ ψ g (cid:105) = | g ... g (cid:105) (28) | ˜ ψ s (cid:105) = √ N b (cid:88) i | g ...
12 ( i ) ... (cid:105) (29) | ˜ ψ p (cid:105) = √ M b (cid:88) || r i − r j |− r | < ∆ r | g ... λ ( i , j )1 ... g (cid:105) (30)where | ˜ ψ g (cid:105) is the initial state, | ˜ ψ s (cid:105) is the super-atom state with N b the number of atoms in the blockade sphere and | ˜ ψ p (cid:105) isthe “super pair state” which is the coherent superposition ofthe M b non-interacting vdW eigenstates formed at the magicdistance inside the blockade sphere.The many-body Hamiltonian associated to our three-statemodel reduces to H = (cid:126) Ω √ N b √ N b (cid:15) (cid:15) | ˜ ψ g (cid:105) , | ˜ ψ s (cid:105) , | ˜ ψ p (cid:105) , (31)with (cid:15) = M b α/ [ N b (1 + α )]. This Hamiltonian describes thecollective enhancement of the light-atom coupling by the vdWinteractions. Considering | Ψ ( t = (cid:105) = | ˜ ψ g (cid:105) , we can write thewave function at any time t : | Ψ ( t ) (cid:105) = Ω Ω s (cid:16) N b cos Ω s t + (cid:15) (cid:17) − i ΩΩ s √ N b sin Ω s t √ N b (cid:15) Ω s Ω (cid:16) cos Ω s t − (cid:17) | ˜ ψ g (cid:105) , | ˜ ψ s (cid:105) , | ˜ ψ p (cid:105) , (32)with the enhanced Rabi frequency: Ω s = Ω (cid:112) N b + (cid:15) . (33)In the absence of vdW coupling ( α = (cid:15) to the super pair state | ˜ ψ p (cid:105) vanishes and we obtainthe familiar super-atom limit. However, with P / states, α isof the order of unity so that the existence of the magic dis-tance is responsible for the excitation of the super pair state.We emphasize that the strength of the coupling (cid:15) to the su-per pair state depends on the number of magic distances M b per blockade sphere and thus on the geometric arrangementof the atoms. In a “lattice configuration” where the width ∆ r of the magic distance peak is smaller than the lattice spac-ing a [Fig.8(b)], (cid:15) can takes significant value when the magicdistance r is commensurate with the lattice and is negligibleotherwise. In the limit where the width ∆ r is larger than thetypical inter-atomic distance a , the value of (cid:15) , which can beobtained in the continuum limit, becomes proportional to ∆ r and can be easily controlled via the magnetic-field.Our three-state model shows the main e ff ect of the magicdistance at the many-body level, which is a collective exci-tation of the super pair state directly related to the geometricarrangement of the atoms. However, this model only gives afirst approximation of the dynamics because it neglects the in-fluence of finite-size e ff ects [26]. In order to estimate thesee ff ects, we now study numerically the dynamics of a smallsystem of 6 atoms placed on a 1D chain where the value of themagnetic-field is adjusted to satisfy: r ( B ) = a > ∆ r . Fig. 8(b) r (a) r b B | ˜ (1 , i| ˜ (3 , i a r (b) r b r B Ω t . . . . . . f p ( t ) (c) a = 3 µ m a = 2 µ m a = 1 µ m Figure 8. (a) Four atoms placed on a square of side r cannot be ex-cited simultaneously due to the non-additive properties of the vdWHamiltonian. This blockade mechanism is responsible for the ex-citation of a superposition of pair states. (b) For a larger numberof atoms, we can divide the system in an ensemble of independentblockade spheres in order to obtain the three state model. (c) Su-per pair oscillation f p ( t ) in the lattice regime a > ∆ r for N = P / states with Ω = π × . ff erentlattice spacings a = , , µ m and r ( B ) = a . shows the time-evolution of the super pair fraction f p which isdefined here as the probability to have two neighboring sitessimultaneously excited. As predicted by our model, the su-per pair fraction oscillates with a frequency and a mean valuewhich depends on the lattice spacing a , the progressive attenu-ation of the oscillations being due to finite-size e ff ects. As thevalue of the lattice spacing a decreases, we note that the fre-quency of the oscillations increases while the correspondingamplitude is reduced. This is in agreement with the predic-tion of our model [Eq. (30)] as the number of atoms N b perblockade sphere scales as a − whereas the geometric factor (cid:15) ∝ M b / N b does not vary significantly when the value of a changes.We have therefore successfully generalized the concept ofsuper-atom in presence of a magic distance. Due to the pres-ence of a non interacting vdW eigenstate, the doubly excitedstates can be populated with an enhanced Rabi frequency Ω s which depends on the number of atoms N b per blockadesphere but also on the spatial distribution of the atoms via thegeometric factor (cid:15) . V. CONCLUSION
In summary, in this paper we have described the e ff ect ofthe vdW couplings with p and d Rydberg states and gener-alized the fruitful concept of Rydberg blockade. We havediscussed the existence and have analyzed the properties ofmagic distances, and their e ff ect on many body quantum dy-namics. In particular, our three-state model introduces the keyfeatures of the magic distance at the many body level andpoints to a more detailed investigation regarding its conse-quences on the quantum simulation of many-body systems,such as, for example, the crystallization of Rydberg excita-tions in optical lattices [12, 27] or the statistical properties ofthese excitations in atomic clouds [28, 29].We thank T. Lahaye, A. Browaeys and J. Zeiher for inter-esting discussions. Work at Innsbruck is supported by theERC-Synergy Grant UQUAM and SFB FOQUS of the Aus-trian Science Fund. B. V. acknowledges the Marie Curie Ini-tial Training Network COHERENCE for financial support. Appendix A: Anisotropic van der Waals Hamiltonians1. General considerations
We present here the properties of the vdW interactions [16]and diagonalize the corresponding Hamiltonian to generalizethe results of [30]. We focus on distances comparable to theusual blockade Radius for which interactions are in the MHzregime, that is to say much smaller than the fine-structuresplitting. We also do not consider the case of a Förster res-onance [3]: as a consequence, interactions are ruled by vdWinteractions and the atoms are characterized by the same firstthree quantum numbers n , (cid:96), j . Within these two assumptions,we can treat the dipole-dipole operator in second-order per-turbation theory and obtain the interaction Hamiltonian in thefine-structure basis ( | m α , m β (cid:105) ) H V ( r , θ ) = (cid:88) αβγδ V αβ ; γδ ( r , θ ) | m α , m β (cid:105)(cid:104) m γ , m δ | , (A1)with V αβ ; γδ ( r , θ ) = r (cid:88) i C ( i )6 (cid:104) m α , m β |D i | m γ , m δ (cid:105) . (A2)The channel i refers to the second and third quantum number( (cid:96) s , j s ) and ( (cid:96) t , j t ) of the intermediary states, the correspond-ing vdW coe ffi cient C ( i )6 being: C ( i )6 = (cid:88) n s , n t e − δ α,β ( R n s (cid:96) s j s nl j R n t (cid:96) t j t n (cid:96) j ) where the radial matrix elements R are calculated using modelpotentials [31] combined with spectroscopy measurements[32, 33] and δ α,β is the Förster defect [3]. The matrices D i de-scribe the angular dependent part: Its (2 j + matrix elements0depend explicitly on θ , the angle between the quantization andthe molecular axis: D i = (cid:88) m s , m t M i | m s , m t (cid:105)(cid:104) m s , m t |M † i (A3) + δ ( (cid:96) s , j s ) (cid:44) ( (cid:96) t , j t ) (cid:88) m s , m t M i | m t , m s (cid:105)(cid:104) m t , m s |M † i (A4)where M i describes the angular momentum dependence of thedipole-dipole operator: (cid:104) m α , m β |M i | m s , m t (cid:105) = − (cid:114) π (cid:96) + (2 j + × ( − s − m α (cid:112) (2 (cid:96) s + j s + × ( − s − m β (cid:112) (2 (cid:96) t + j t + × (cid:40) (cid:96) (cid:96) s j s j s (cid:41) (cid:32) (cid:96) s (cid:96) (cid:33) × (cid:40) (cid:96) (cid:96) t j t j s (cid:41) (cid:32) (cid:96) t (cid:96) (cid:33) × (cid:88) µ,ν (cid:32) j t jm t ν − m β (cid:33) (cid:32) j s jm s µ − m α (cid:33) C , µ,ν ; µ + ν Y µ + ν ( ϑ, ϕ ) ∗ . (A5) j = / In the case of the P / and S / states, the vdW matrix iswritten in the basis | , (cid:105) , | , (cid:105) , | , (cid:105) , | , (cid:105) . The inter-mediary states are given in Table I whereas the dimensionless channel i l s j s l t j t for S / states l s j s l t j t for P / states1 P / P / P / S / P / P / P / D / P / P / D / D / Table I. Channels involved in the case of S / and P / Rydberg pairstates. matrices D i can be written in the form: D = − cos 2 θ + sin 2 θ sin 2 θ − θ sin 2 θ cos 2 θ + cos 2 θ + − sin 2 θ sin 2 θ cos 2 θ + cos 2 θ + − sin 2 θ − θ − sin 2 θ − sin 2 θ − cos 2 θ + D = − θ + − sin 2 θ − sin 2 θ θ − sin 2 θ θ + θ − sin 2 θ − sin 2 θ θ − θ + sin 2 θ θ sin 2 θ sin 2 θ − θ + D = − cos 2 θ + sin 2 θ sin 2 θ − θ sin 2 θ cos 2 θ + cos 2 θ + − sin 2 θ sin 2 θ cos 2 θ + cos 2 θ + − sin 2 θ − θ − sin 2 θ − sin 2 θ − cos 2 θ + . leading to Eq. (5) with u = C (1)6 + C (2)6 + C (3)6
81 (A6) c = − C (1)6 + C (2)6 − C (3)6
27 (A7)The values of u , u = u − c and c used in the main text areshown in Fig. 2 for Rubidium P / states. In contrast to S / states where the vdW coupling is negligible | c | (cid:28) | u | , the threeterms are of the same order of magnitude.It is also particularly instructive to diagonalize the vdWHamiltonian to know for example whether interactions are at-tractive or repulsive. The easiest way to do it is to considerthe basis where the quantization axis aligned with the direc-tions of the atoms ( θ = θ with an appropriate changeof basis. The eigenvalues are λ = r C (2)6 + C (3)6 (A8) λ = λ = r C (1)6 + C (2)6 + C (3)6 (A9) λ = r C (1)6 + C (2)6 + C (3)6 . (A10)with the corresponding eigenstates | λ (cid:105) = √ (cid:16) | , (cid:105) − | , (cid:105) (cid:17) (A11) | λ (cid:105) = | , (cid:105) (A12) | λ (cid:105) = | , (cid:105) (A13) | λ (cid:105) = √ (cid:16) | , (cid:105) + | , (cid:105) (cid:17) . (A14)The state | λ (cid:105) could be seen as a “Förster zero” [34] as thefirst channel does not contribute to its energy. λ is indeed ofthe same order of magnitude as the other energies as shownin Fig. 9 representing the distance-independent term r λ i for S / state (a) and P / states (b), as a function of the principalquantum number n . For S / states, the energies are almostdegenerate whereas P / states represent a promising optionto study e ff ects which go beyond the usual blockade pictureas the eigenenergies are di ff erent and do not even have thesame sign for n > j = / In the case of P / and D / states, six channels participateto the vdW interactions. They are shown in Table II. We donot write the 16 × D i matrices (which are identical for P / and D / states) as they can be obtained easily from standardsymbolic routines. However, let us diagonalize the total vdWHamiltonian in the particular case θ =
20 30 40 50 60 70 80 90 n r | λ i | n − ( M H z µ m ) × − (a)
20 30 40 50 60 70 80 90 n − − − − − r | λ i | n − ( M H z µ m ) (b) Figure 9. Energies of the vdW eigenstates for Rubidium S / (a)and P / (b) states as a function of the principal quantum number n .Empty (full) symbols represent negative (positive) values.channel i l s j s l t j t for P / states l s j s l t j t for D / states1 S / S / P / P / S / D / P / P / S / D / P / F / D / D / P / P / D / D / P / F / D / D / F / F / Table II. Channels involved in the case of P / and D / states. of block corresponding to each value of total angular momen-tum M = m α + m β . The symmetry of the problem with respectto the transformation | m α , m β (cid:105) → | m α , m β (cid:105) allows us to re-strict the study to the case M ≥
0. Its means in particular thatthe eigenvalues associated to M > j = / θ are obtained bythe appropriate change of basis. M = For M =
3, there is only one state | λ (cid:105) = | , (cid:105) (A15)with interaction energy (we omit the pre-factor r ) λ = C (3)6 + C (4)6 + C (5)6 + C (6)6 . (A16) M = In this case, the two eigenstates are simply | λ (cid:105) = √ (cid:104) | , (cid:105) + | , (cid:105) (cid:105) (A17) | λ (cid:105) = √ (cid:104) | , (cid:105) − | , (cid:105) (cid:105) (A18)with eigenvalues λ = C (2)6 + C (3)6 + C (4)6 + C (5)6 + C (6)6
625 (A19) λ = C (2)6 + C (3)6 + C (5)6 + C (6)6 . (A20) M = In this case, three interacting states are coupled | , (cid:105) , | , (cid:105) and | , (cid:105) . The eigenstates are | λ (cid:105) = cos α √ (cid:104) | , (cid:105) + | , (cid:105) (cid:105) + sin α | , (cid:105)| λ (cid:105) = − sin α √ (cid:104) | , (cid:105) + | , (cid:105) (cid:105) + cos α | , (cid:105)| λ (cid:105) = √ (cid:104) | , (cid:105) − | , (cid:105) (cid:105) with cos(2 α ) = δ α (cid:112) δ α + C α sin(2 α ) = C α (cid:112) δ α + C α δ α = − C (1)6 + C (2)6 − C (3)6 + C (4)6 + C (5)6 − C (6)6 C α = C (1)6 √ + C (2)6 √ − C (3)6 √ + C (4)6 √ − C (5)6 √ + C (6)6 √ λ = C (1)6 + C (2)6 + C (3)6 + C (4)6 + C (5)6 + C (6)6 − (cid:113) C α + δ α λ = C (1)6 + C (2)6 + C (3)6 + C (4)6 + C (5)6 + C (6)6 + (cid:113) C α + δ α λ = C (2)6 + C (3)6 + C (4)6 + C (5)6 + C (6)6 . M = The last 4 unknown eigenstates belong to the zero angularmomentum subspace: | λ (cid:105) = cos β √ (cid:104) | , (cid:105) + | , (cid:105) (cid:105) + sin β √ (cid:104) | , (cid:105) + | , (cid:105) (cid:105) | λ (cid:105) = − sin β √ (cid:104) | , (cid:105) + | , (cid:105) (cid:105) + cos β √ (cid:104) | , (cid:105) + | , (cid:105) (cid:105) | λ (cid:105) = cos γ √ (cid:104) | , (cid:105) − | , (cid:105) (cid:105) + sin γ √ (cid:104) | , (cid:105) − | , (cid:105) (cid:105) | λ (cid:105) = − sin γ √ (cid:104) | , (cid:105) − | , (cid:105) (cid:105) + cos γ √ (cid:104) | , (cid:105) − | , (cid:105) (cid:105) with cos(2 β ) = δ β (cid:113) δ β + C β sin(2 β ) = C β (cid:113) δ β + C β δ β = − C (1)6 − C (2)6 − C (3)6 + C (4)6 + C (5)6 − C (6)6 C β = C (1)6 − C (2)6 − C (3)6 + C (4)6 − C (5)6 + C (6)6 , cos(2 γ ) = δ γ (cid:113) δ γ + C γ sin(2 γ ) = C γ (cid:113) δ γ + C γ δ γ = C (2)6 − C (3)6 + C (4)6 + C (5)6 + C (6)6 C γ = C (1)6 − C (3)6 + C (4)6 − C (5)6 + C (6)6
625 and with energies λ = C (1)6 + C (2)6 + C (3)6 + C (4)6 + C (5)6 + C (6)6 − (cid:113) C β + δ β λ = C (1)6 + C (2)6 + C (3)6 + C (4)6 + C (5)6 + C (6)6 + (cid:113) C β + δ β λ = C (1)6 + C (2)6 + C (3)6 + C (4)6 + C (5)6 + C (6)6 − (cid:113) C γ + δ γ λ = C (1)6 + C (2)6 + C (3)6 + C (4)6 + C (5)6 + C (6)6 + (cid:113) C γ + δ γ . We represent in Fig. 10 the values of the eigenvalues for P / states as a function of the principal quantum number n . Wechecked that the values obtained with these analytical expres-sions coincide with the ones obtained by “brute” numericaldiagonalization of the vdW Hamiltonian. For P / and n < n >
38, some of the eigenstates be-come attractive. For D / states, the two Förster resonancesare also visible [Fig. 11]. Appendix B: E ff ects of the Quadrupole-Quadrupoleinteractions In order to show that the vdW interactions are dominant forthe range of parameters considered in this work, we now cal-culate the second term of the multipole expansion correspond-ing to quadrupole-quadrupole interactions. In a fine-structuremanifold ( n , (cid:96), j ), the matrix elements of the quadrupole-quadrupole Hamiltonian can be written in the form [35] H = C r (cid:88) m , m , m , m | m , m (cid:105) D (cid:104) m , m | (B1)with C = (cid:104) n , (cid:96), j | r | n , (cid:96), j (cid:105) is the radial part and D the an-gular part.The selection rules of the quadrupole-quadrupole operatorimpose j ≥ S / and P / states. We now consider thecase of P / and D / states. The eigenstates of H can beclassified in the case θ = M = m + m . Thereare given for M ≥ C coe ffi cient, isshown in Fig. 12. Our values, calculated from the model po-tential [31], including spin-orbit e ff ects are in good agreementwith previous calculations [30]. For the parameters of [11] the C coe ffi cient is 30 . µ m . Given that C = − µ m and for distances of the order of a few micrometers, thequadrupole-quadrupole interactions are therefore negligible.3
20 30 40 50 60 70 80 90 n − − − − − − − − r | λ i | n − ( M H z µ m ) (a)
20 30 40 50 60 70 80 90 n − − − − − − − − r | λ i | n − ( M H z µ m ) (b) Figure 10. (a) Degenerate and (b) non-degenerate eigenvalues r λ i for Rubidium P / states as a function of the principal quantum num-ber n . Empty (full) symbols represent negative (positive) values. M λ r / C | λ (cid:105) | , (cid:105) − √ (cid:16) | , (cid:105) + | , (cid:105) (cid:17)
225 1 √ (cid:16) | , (cid:105) − | , (cid:105) (cid:17) | , (cid:105) −
425 1 √ (cid:16) | , (cid:105) + | , (cid:105) (cid:17) −
825 1 √ (cid:16) | , (cid:105) − | , (cid:105) (cid:17) (cid:16) | , (cid:105) + | , (cid:105) + | , (cid:105) + | , (cid:105) (cid:17)
425 12 (cid:16) | , (cid:105) + | , (cid:105) − | , (cid:105) − | , (cid:105) (cid:17) (cid:16) | , (cid:105) − | , (cid:105) − | , (cid:105) + | , (cid:105) (cid:17) (cid:16) | , (cid:105) − | , (cid:105) + | , (cid:105) − | , (cid:105) (cid:17) Table III. Eigenstates and eigenvalues of the quadrupole-quadrupoleHamiltonian for j = /
20 30 40 50 60 70 80 90 n − − − − − − r | λ i | n − ( M H z µ m ) (a)
20 30 40 50 60 70 80 90 n − − − − − − r | λ i | n − ( M H z µ m ) (b) Figure 11. (a) Degenerate and (b) non-degenerate eigenvalues r λ i for Rubidium D / states as a function of the principal quantum num-ber n . Empty (full) symbols represent negative (positive) values andthe color code is the same as in Fig. 10.
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