Tuning of interaction potentials

Stark tuned Förster resonances

The energy defects of the pair-states ∆ control the distance RvdWat which cross-over

between long-range, resonant dipole-dipole V (R) ∼ R−3_{and off-resonant dipole-}

dipole (i.e. van der Waals) interactions will occur, as discussed earlier in Sec. (2.5.1). Static external electric fields can be used to offset pair-state energies via induced Stark shifts [Sec. (2.4.1)]. The special case when a given pair of dipole-dipole coupled pair-states has negligible energy defect ∆/V (R) → 0 is called a Förster resonance, and corresponds to the case when resonant long-range interactions (∝ R−3_{) are obtained [71, 178–182].}

Finding resonant states and values of electric field for which these resonances occur can be done with the

StarkMapResonances

arguments two atom types9_{, their initial target states, and energy window, and then}

performs a Stark map calculation in the pair-state basis. Since pair-state interactions

V (R) [Eq. (2.38)] can couple target pair-state to states that differ in projection of

total angular momentum by ∆m_{j= 0, ±1, it is necessary to calculate Stark maps for} up to nine different manifolds corresponding to all possible combinations (m0

j1, m0j2) for dipole-coupled states. After diagonalization, only pair states which are in the

8_{This analysis assumes that C}

6for all states is of the similar order of magnitude for all dipole coupled

states, which is usually true.

Electric field, Ez(V/cm) Pair-state relative energy ,δ Eµ / h (GHz) =(Rb-85,Rb-85) = (−1.00|64 P1/21/2〉 + . . . , 1.00|65 P3/21/2〉 + . . .)

Figure 2.20: Automatic search for Förster res- onance. ARC automatically searches within a set range of electric fields and pair-state ener- gies for resonances with the given pair-state. An example calculation here shows pair-state energies relative to the unperturbed pair-state |66 S1/2 mj = 1/2, 64 S1/2 mj = 1/2〉 of two rubidium-85 atoms. This pair-state (solid red) in electric-field Ez≈ 0.165 V/cm becomes resonant with another Stark-shifted state. In interactive use of ARC users can select resonant states to see their composition, here shown marked by a blue square, that corresponds to an almost pure |64 P1/2mj= 1/2, 65 P3/2mj= 1/2〉 pair-state. These resonances have been detected in Ref. [71].

specified energy window are considered, discarding the pair-states that are not dipole coupled. Since electric field leads to strong admixing of basis pair-states, the algorithm identifies states as dipole coupled if the basis state with dominant contribution in the obtained eigen-state is coupled to the target pair-state provided in the initialization. That pair-state is also admixed by the electric field, but can be found as the state with the largest initial state fraction. Finally the interactive routine allows users to select states, see plots with resonances, and identify state composition (Fig. 2.20).

Dressing

Coupling of states with AC fields, discussed in the single atom context in Sec. (2.4.3), can be used for tuning pair-state interaction potentials. In this context, we can highlight two distinct cases of off-resonant dressing and resonant dressing.

Off-resonant dressing can be used to admix the Rydberg state into the ground state, introducing interactions between new ground eigenstates [183]. A driving field with Rabi frequency Ω tuned ∆  Ω off-resonance from the |g〉 ↔ |r〉 resonance, acting on a single atom, gives rise to a ground eigen-state |˜g〉 ∼ |g〉 + "|r〉, where

" = Ω/(2∆) is the admixture of the Rydberg state |r〉. This admixing causes the

usual AC Stark shift [Sec. (2.4.3)], given here for red-detuned driving (∆ < 0) as

δAC=−∆ −

p

∆2+ Ω2

2 . (2.41)

Note that even when the admixed Rydberg state has a finite lifetime τ_r, it has a small impact on the ground eigen-state lifetime τ˜g= "2τr, as typically "  1.

Interactions between the Rydberg states will cause changes to this AC Stark shift. For example, this can be easily calculated in the simple case of two two-level atoms interacting with van der Waals interactions V = −C6/R6[Fig. 2.21(a)]. The system

Hamiltonian in the pair-state basis {|g, g〉, |r, g〉, |g, r〉, |r, r〉} is (h = 1, RWA)

H =      0 Ω/2 Ω/2 0 Ω/2 −∆ 0 Ω/2 Ω/2 0 −∆ Ω/2 0 Ω/2 Ω/2 −(2∆ − V )     . (2.42)

Inter-atomic distance, R/R_B Pair-state |˜g ˜ g〉 Stark shift, δ /[ 2× δAC (Ω )] Inter-atomic distance, R Pair-state energy |g, g〉 |g, r〉, |r, g〉 |r, r〉 (a) (b) ∼ C6/R6 Ω Ω ∼1×δAC Ωp2
∼2×δAC(Ω) ∼ "4_C 6/R6 ∆ ∆ _{|g, r〉 + |r, g〉} |g, g〉 Ωp2 Ω Ω |g〉 |r〉 |g〉 |r〉

Figure 2.21: Effective potential between ground-state atoms due to Rydberg state ad- mixing in off-resonant dressing. (a) Pair-state energy diagram where two ground |g〉 state atoms at distance R are dressed by ∆ off-resonant field with Rabi frequency Ω that admixes Rydberg state |r〉 into new ground eigen-state |˜g〉 ∼ |g〉 + "|r〉. Rydberg states are interact- ing with van der Waals interactions C6/R6. The new ground pair-state will have AC Stark

shift δ = E(˜g, ˜g) − E(g, g) relative to the unperturbed ground-state [solid red (b)]. At large distances R  RB≡ 6

p

|C6|/Ω AC Stark shift approaches Stark shift value 2 × δAC(Ω) (dotted

line) for two independently dressed atoms [(b) right inset ] with Rabi driving field Ω. Below

R ® RBatoms start feeling repulsive van der Waals interactions (dashed line), with interaction

strength scaled down by "4_{, the probability that both atoms are simultaneously in the excited}

state. However, deeply in the blockaded region R  RBonly one atom can be excited, at the

same time we don’t know which one. This limit can be seen as dressing of the superatom consisting of two atoms [(b) left inset], with enhanced driving Ω ×p2 between ground and singly excited, symmetric collective state. Indeed the Stark shift δ saturates in this limit at value of a single superatom Stark shift δAC(Ωp2) (dotted line). Calculation parameters

∆ = 20 Ω.

Diagonalizing this Hamiltonian, we obtain the AC Stark shift of the |˜g, ˜g〉 pair-state, shown in Fig. 2.21(b). The effect of the interactions can be seen as an effective pair-state interaction soft-core potential V_{D(R)|˜g˜g〉〈˜g˜g| whose amplitude is}

Max[VD(R)] = 2δAC(Ω) − δAC(Ωp2) ≈ Ω 4

8∆3. (2.43)

More complex Rydberg level energies, arising e.g. due to avoided resonances (Fig. 2.19), can cause reduction of ∆ over a range of inter-atomic distances R, leading to localised stronger dressing, as discussed in Ref. [75].

Dressing, both resonant and off-resonant, can also be done in the Rydberg state manifold with microwave and terahertz fields, where it can be exploited for fine- tuning of Rydberg interaction potentials [160, 161].