Long-range multi-body interactions and three-body anti-blockade in a trapped Rydberg ion chain

Trapped Rydberg ions represent a flexible platform for quantum simulation and information processing which combines a high degree of control over electronic and vibrational degrees of freedom. The possibility to individually excite ions to high-lying Rydberg levels provides a system where strong and long-range interactions between pairs of excited ions can be engineered and tuned via external laser fields. We show that the coupling between Rydberg pair interactions and collective motional modes gives rise to effective long-range multi-body interactions, consisting of two, three, and four-body terms. Their shape, strength, and range can be controlled via the ion trap parameters and strongly depends on both the equilibrium configuration and vibrational modes of the ion crystal. By focusing on an experimentally feasible quasi one-dimensional setup of $ {}^{88}\mathrm{Sr}^+ $ Rydberg ions, we demonstrate that multi-body interactions are enhanced by the emergence of a soft mode associated, e.g., with a structural phase transition. This has a striking impact on many-body electronic states and results, for example, in a three-body anti-blockade effect. Our study shows that trapped Rydberg ions offer new opportunities to study exotic many-body quantum dynamics driven by enhanced multi-body interactions.

Introduction.-The coupling between internal atomic states and collective vibrational modes is the hallmark of trapped ion setups. The possibility to engineer phonon-mediated two-body interactions, which can be tuned via laser fields and trapping parameters, combined with single-ion control and high fidelity state preparation, makes them a powerful platform for quantum simulation and information processing [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. A further enhancement of this setup can be achieved in trapped Rydberg ions, where each ion can be individually excited to a high-lying Rydberg level [15][16][17][18][19][20][21][22][23]. The strong dipole-dipole interactions characterizing this system can be exploited to simulate equilibrium and non-equilibrium quantum many-body spin models [24][25][26] and for quantum information processing beyond the scalability limitations of conventional ion settings [27][28][29]. Furthermore, the interplay between electronic and vibrational degrees of freedom allows to devise non-classical motional states and sets the stage for the study of complex quantum phenomena arising from electron-phonon coupling [30].
In this work we demonstrate that the intertwining between Rydberg interactions and non-local vibrational modes provides a mechanism to engineer long-range multi-body interactions in state-of-the-art setups of trapped Rydberg ions. This, combined with the precise control over system parameters, makes trapped Rydberg ions a flexible platform for quantum simulation with enhanced tunable spin-phonon coupling, allowing for the study of non-trivial kinetically constrained models [31], long-lived quantum information storage [32] and exotic quantum states of matter [33][34][35][36][37][38][39][40]. Importantly, ions can be conveniently trapped via state-independent electric potentials and, therefore, trapped Rydberg ions do not require magic trapping conditions as neutral Rydberg atoms [41][42][43]. Focusing on a chain of 88 Sr + Rydberg ions confined by harmonic potentials [see Fig. 1(a)], which has been recently experimentally realized [29], we demonstrate that multi-body interactions are significantly enhanced by the emergence of soft vibrational modes. These occur, e.g., at the linear-to-zigzag transition + + +

FIG. 1. Setup and phonon eigenfrequencies of a three-ion crystal.
(a) Each ion is modeled as a two-level system whose ground state, |↓ , is coupled to a Rydberg excited state, |↑ , by a laser with Rabi frequency Ω and detuning ∆. The state |↑ spontaneously decays to |↓ with rate γ. The equilibrium positions of the ions r 0 n are determined by the interplay between Coulomb repulsion and harmonic confinement. Ions m and n interact through the interaction potential V R (r n , r m ) when both are excited to Rydberg states. (b) Eigenfrequencies Ω l,λ as a function of the trap aspect ratio α = ω y /ω x . For α > α * , top panel (λ = 1) corresponds to longitudinal phonon modes while bottom panel (λ = 2) displays the transverse ones. The gray line highlights the structural phase transition at α * = 12/5. The two configurations of a three-ion chain are sketched on the top. Shaded blue areas represent a sketch of the trapping potential in the x − y plane. in a chain of few ions [44][45][46][47][48][49][50] and in infinite chains, which provide a good qualitative description of the central part of finite chains with a few dozens of ions. Here, the presence of strong spin-phonon coupling induced interactions gives rise to non-trivial many-body phenomena, such as a three-body anti-blockade effect which can be detected via Rydberg state spectroscopy.
Spin-phonon coupling induced multi-body interactions.-We consider a quasi one-dimensional chain of N two-level Rydberg ions confined by a harmonic potential, as sketched in Fig. 1(a). Each ion is modeled as a two-level system (with |↓ arXiv:2005.05726v2 [cond-mat.quant-gas] 18 May 2020 and |↑ being the ground state and the excited Rydberg state, respectively). The two levels are coupled by a laser field with Rabi frequency Ω and detuning ∆. The overall Hamiltonian is with H ions = l,λ Ω l,λ (a † l,λ a l,λ + 1/2) describing the vibrational dynamics of an ion crystal confined in a threedimensional harmonic potential v µ (r n,µ ) = Mω 2 µ r 2 n,µ /2. Here, r n is the n−th ion position, M its mass, and ω µ the trapping frequency along direction µ = {x, y, z} [50][51][52]. The bosonic operators a ( †) l,λ are associated with the phonon mode (l, λ) with eigenfrequency Ω l,λ , where λ labels the phonon branches [53]. Without loss of generality, we assume ω x,y ω z , so that the motion of the ions is effectively confined to the x − y plane. In Eq. (1), H L = N n=1 Ωσ x n + ∆n n describes the laser excitation of the ions to the Rydberg state. Here, n n = (1 + σ z n )/2 and σ µ n are the Rydberg number operator and Pauli matrices acting on the n−th ion, respectively. The electrostatic dipole-dipole interaction between pairs of Rydberg excited ions is modeled by H int = m,n V R (r m , r n )n m n n , where V R (r m , r n ) = V R (|r m − r n |) and the prime denotes m n. Under typical experimental conditions, the displacements of the ions from their equilibrium positions r 0 n are much smaller than typical inter-ion distances. Hence, we expand the two ion potential V R (r m , r n ) to the first order around r 0 n obtaining [43,53] Here, H 0 int = m,n V R (r 0 m , r 0 n )n m n n is the bare Rydberg interaction Hamiltonian. The second term describes an effective coupling between vibrational modes and Rydberg states, whose strength is quantified by the parameters W λ mnl = 2 l,λ µ G mnR 0 mn;µ M µλ ml , with l,λ = /(2MΩ l,λ ) the characteristic length of the harmonic confinement. The normal mode matrices M µλ ml connect local ion displacements to the chain normal modes [53] and we introduced ∇ r m, µ V R (r m , r 0 n )| r 0 m = G mnR 0 mn , with G mn being magnitude of the gradient of the Rydberg potential andR 0 mn = (r 0 m − r 0 n )/|r 0 m − r 0 n |. The interaction Hamiltonian H int in Eq. (2) couples electronic and vibrational degrees of freedom. They can be approximately decoupled through a polaron transformation U = e −S [5,6,43,53] which leads to the transformed Hamiltonian Here, the polaron transformation generates in the spin Hamiltonian H spin = H L +H 0 int +H eff int the additional Rydberg interaction contribution H eff int = − m,n i, jṼ mni j n m n n n i n j .
This consists of long-range multi-body effective interactions coupling two, three, and four spins. Their strength is encoded in the interaction coefficients with coupling parameters F µν mi = l,λ Ω −2 l,λ M µλ ml M νλ il , and, hence, it is determined by the gradient of the ion-ion Rydberg potential G mn and by the vibrational mode structure of the ion chain.
The term H res in Eq. (3) contains a residual spin-phonon interaction which, in general, can be neglected in the limit max[|W λ mnl |/ Ω λ,l ] 1 [5,6,53]. However, in the strong interaction regime which will be the focus of next sections, the contribution of H res to the spin dynamics is negligible when Ω Ω * ∼ min(Ω −1/2 l,λ ) [53]. In this case, it is possible to decouple electronic and vibrational degrees of freedom via a proper choice of Ω.
Three-ion chain.-We first demonstrate the onset of the effective interactions in a minimal setting of three ions. We notice that, due to the respective factorsR 0 mn;µ in Eq. (5), transverse modes do not contribute toṼ mni j in a linear chain. Starting from the latter, the system features a second-order phase transition to a zigzag configuration, where both longitudinal and transverse modes contribute to Eq. (5), at the critical value α * = 12/5 of the trap aspect ratio α = ω y /ω x [50]. As shown in Fig. 1(b), the critical point is associated with the emergence of a soft mode with vanishing eigenfrequency, i.e. Ω l,λ → 0. Since F µν mi ∝ Ω −2 l,λ , we expect that the presence of such soft modes results in an increasing of the effective interaction strength. As shown in Fig. 2(a), in the zigzag regime and close to structural phase transition, the collective nature of the phonon modes, encoded in normal mode matrices M µν mn , leads to coupling parameters F µν mi with a non-trivial pattern over the whole chain. For a three-ion chain, taking into account interactions between both nearest neighbors (NNs) and next-to-nearest neighbors (NNNs), H eff int can be explicitly written as H eff int = −C 2b NN (n 1 n 2 + n 2 n 3 ) − C 2b NNN n 1 n 3 − C 3b n 1 n 2 n 3 . (6) Here, C 2b NN and C 2b NNN parameterize effective two-body interactions between NNs and NNNs, respectively, and C 3b the three-body one.
In trapped Rydberg ions van der Waals interactions are generally weaker than their atomic counterparts and do not give access to large gradients, which are essential to maximize effects of multi-body interactions [see Eq. (5)]. To enhance the ion-ion Rydberg interactions, we exploit the interplay between dipole-dipole interactions of microwave (MW) dressed states [19,27] and MW tuned Förster resonance in a setup of 88 Sr + trapped Rydberg ions [21,22,29]. This allows us to obtain the effective ion-ion interaction potential shown in Fig. 2(b) [53]. The corresponding effective interaction coefficients are shown in Fig. 2(c,d) as a function of the trap aspect ratio α. We note that for α > α * their values are fixed. In this case, indeed, only the term ∝ F xx mi contributes toṼ mni j in Eq. (5). Hence, spins are coupled via Eq. (2) to longitudinal phonons only, whose eigenfrequencies are constant, as shown The sign of the interaction coefficients depends on the gradient of the Rydberg potential at NNs and NNNs, G NN and G NNN , respectively. The potential chosen in Fig. 2 , corresponding to attractive interactions. Interestingly, for α 1.5, C 2b NN > 0, C 2b NNN 0, and C 3b < 0, implying a competition between attractive two-body and repulsive three-body effective interactions [see Fig. 2 We now investigate an interaction induced three-body Rydberg anti-blockade regime, a generalization of the well-studied facilitation mechanism in the presence of two-body Rydberg interactions [55][56][57][58][59]. By denoting with V NN and V NNN the bare Rydberg interactions between NNs and NNNs contained in H 0 int , respectively, this regime is achieved when [see the level structure in Fig. 3 If the ions are prepared in state |↓↓↓ at time t = 0, an enhancement in the projector on state |↑↑↑ at subsequent times, P ↑↑↑ (t), is expected for values of ∆ satisfying Eq. (7). The behavior of the time-integrated expectation value of P ↑↑↑ (t), denoted by P ↑↑↑ , is shown in Fig. 3(b, c). Panel (b) shows the case with bare Rydberg interactions only (i.e., with   7). This results in a shift of the position of the peak of P ↑↑↑ and to an enhancement of its value due to the stronger coupling between spins. As can be seen in Fig. 3(d), which shows the difference δ P ↑↑↑ between the time-integrated expectation values of P ↑↑↑ (t) with and without the contributions of H eff int , the presence of phonon-mediated multi-body interactions results in a clear spectroscopic signature.
Infinite linear chain.-In the previous section we showed that spin-phonon coupling induced interactions are strongly amplified by the emergence of a soft mode close to a structural phase transition in a three-ion chain. It is now natural to inspect the behavior of H eff int in longer chains, where the vibrational spectrum becomes dense and long wavelength soft modes naturally occur as the particle number increases. To gain insights into the phenomenology of this case, we consider the infinite chain limit, i.e. N → ∞, which provides a good description of the central region of long yet finite chains [49,60]. In the linear regime [50], the equilibrium positions of the ions are r 0 n = (na, 0), with a being the fixed inter-ion distance and n ∈ {0, ±1, ..., ±∞}. To break the translational symmetry of the infinite chain and mimic the effect of a longitudinal confinement, we replace the harmonic trapping potential along the x axis with a periodic one commensurate with the lattice spacing, i.e., v x (r n, x ) = −Mω x (a/2π) 2 cos(2πr 0 n, x /a).
In this case, expanding the ions' coordinates in Fourier modes and generalizing the steps leading to Eqs. (2) and (3) [Ω λ (k)] −2 e −iks , with s = m − i, −π ≤ k < π defining the wavevector of the first Brillouin zone, and Ω λ (k) the eigenfrequency of phonon mode (k, λ). In the linear configuration,R 0 mn;y = 0 and only longitudinal modes contribute toṼ mni j via F x (s). The shape, strength, and range of effective multi-body interactions can be controlled via the trap parameter η Fig. 4(a). In the stiff limit (η x 1) F x (s) is peaked around s = 0. This implies that the dominant contributions to H eff int consist of connected strings of neighboring two-, three-, and four-body terms. On the other hand, in the intermediate, (η x ∼ 1) and soft (η x 1) limit cases, F x (s) has a broader distribution and is non-negligible also for values |s| > 0. As a consequence, in H eff int a rich structure with exotic long-range and multi-body interaction terms delocalized through the whole chain arises, as shown, e.g., in the bottom row of the inset in Fig. 4(a). This broad spectrum of possible interaction patterns, allowed by the collective nature of phonon modes and the precise control over the chain trapping parameters, is in contrast with the case of Rydberg atom tweezer arrays, where only short-range two-and three-body interactions can be engineered [43].
Three-body spectroscopy of a long chain.-The previous discussion allows to gain an understanding of the three-body spectroscopy of a long yet finite chain, which can be experimentally investigated in trapped Rydberg ion simulators. Indeed, in a long enough chain, long wavelength soft modes, which give the largest contribution to H eff int , coincide with good approximation with the ones of the corresponding infinite chain limit [49,60]. Moreover, due to the finite lifetime of Rydberg excitations τ = γ −1 , with γ being the spontaneous decay rate [see Fig. 1(a)], one can effectively consider the vibrational spectrum of the chain as continuous when the energy gaps between phonon modes are smaller than γ. For 88 Sr + ions (with τ ∼ 30 µs) in a trap with longitudinal frequency ω x = 2π × 0.5 MHz, the above conditions are both met for chains with N 20, which are within the reach of current state-of-the-art setups [29].
In a long yet finite chain, system properties depend on the number of ions, N [46,47,61]. In particular, by increasing N, one can decrease the inter-ion distance in the central region even in the presence of a weak longitudinal confinement. Larger Rydberg potential gradients can hence be achieved which, combined with the fact that H eff int ∝ ω −2 x [see Eq. (5)] and a denser spectrum, results in an overall enhancement of the effective interactions. This effect can be seen in Fig. 4(b). Here, we plot the ratio between the effective two and threebody coefficients for an infinite chain and the corresponding ones for the three-ion setup shown in Fig. 2(c). Due to the different vibrational structure, in an infinite chain the signs of two-and three-body terms are opposite, allowing to investigate They are all ∝ F x (s = m − i). Note that, in the stiff limit, the 4b contribution is strongly suppressed with respect to the 4b one. (b) Ratios between the effective interaction coefficients in a linear infinite chain and the corresponding ones for the three-ion case as a function of the infinite chain trapping frequency, (ω x ) ∞ (units 2π×MHz). Here, ion distance is fixed as a = 3.1 µm, Rydberg potential parameters are as in Fig. 2, and η x > 1 throughout the whole range of (ω x ) ∞ considered. the competition between attractive and repulsive interactions. Multi-body coefficients can be further enhanced by reducing the inter-ion distance while keeping ω x constant. We therefore expect that trapped Rydberg ions will give access to different regimes of the effective interactions, allowing to detect signatures of the linear-to-zigzag transition via ion spectroscopy, and eventually paving the way for the study of quantum magnetism and frustration phenomena in the presence of exotic multi-body effects.
Conclusions.-We investigated the emergence of longrange multi-body interactions in a chain of trapped Rydberg ions induced by the coupling between phonon modes and ionion Rydberg interactions. We showed that these interactions are extremely sensitive to the chain equilibrium configuration and vibrational regimes, such as the emergence of soft modes. By employing realistic parameters from a state-of-the-art setup of 88 Sr + Rydberg ions, we demonstrated that they result in a neat signature of the linear-to-zigzag transition in ion Rydberg state spectroscopic signal. The intertwining between chain configuration, vibrational structure, and effective interactions illustrated in this work provides a versatile mechanism to investigate quantum dynamics in the presence of non-trivial multi-body interactions and exotic constraints.

Supplemental Material for "Long-range multi-body interactions and three-body anti-blockade in a trapped Rydberg ion chain"
In this Supplemental Material we provide additional details on the dynamics of a system of singly charged ions trapped in a three-dimensional harmonic potential. We then comment on the derivation of Eqs. (2), (3), and (4) of the main text and provide details of the polaron transformation employed to decouple spin and phonon degrees of freedom. Finally, we discuss the microwave dressing scheme leading to the ion-ion potential shown in Fig. 2(b) of the main text in a state-of-the-art setup of trapped Rydberg ions.

I. DYNAMICS OF A TRAPPED ION CHAIN
In this section we provide additional details on the dynamics of a chain of singly charged ions interacting through Coulomb repulsion and trapped in a three-dimensional harmonic potential. In the spirit of the Born-Oppenheimer approximation, we neglect effects of the Rydberg interactions on the dynamics of the ions. The Hamiltonian of the system is with kinetic energy term T ions = µ n p 2 n,µ /(2M), with p n ions' momenta, and potential energy The latter consists of the contributions due to the harmonic trap, with v µ (r n,µ ) = Mω 2 µ r 2 n,µ /2 (being r n the position of the n−th ion, M its mass, and ω µ the trapping frequency along direction µ = {x, y, z} ≡ {1, 2, 3}), and of the Coulomb repulsion between the ions, with V 0 = e 2 /(4π 0 ). For the sake of simplicity, we assume ω z ω x,y so that the motion of the ions is confined to the x − y plane. The equilibrium positions of the ions, r 0 n , are determined by solving ∇ r n V ions | r n =r 0 n = 0, ∀n. Of particular interest is the case of the three-ion chain considered in main text. Here, from symmetry considerations, one obtains r 0 1, x = −r 0 3, x = X , r 0 2,x = 0, r 0 1,y = −2r 0 2,y = r 0 3,y = Y with X, Y > 0 and 3 = V 0 /(Mω 2 x ) a characteristic length scale of the system [50,51]. As discussed in the main text, the chain undergoes a linear-to-zigzag phase transition at the critical value of the trap aspect ratio α * = 12/5, being α = ω y /ω x . In the linear regime, for α < α * , we get X = (5/4) 1/3 and Y = 0, while in transverse one, i.e. when α < α * , X = [4(1 − α 2 /3)] −1/3 and Y = [(3/α 2 ) 2/3 − X 2 ] 1/2 /3 [50].
At low temperature, ions perform small oscillations around their equilibrium positions r 0 n [52]. In this limit, V ions can be expanded to second order in the displacements q n = r n − r 0 n , leading to with K µν mn = M −1 ∂ 2 V ions /(∂r m,µ ∂r n,ν )| r m, n =r 0 m, n the 2N × 2N ions' dynamical matrix [62] and µ, ν = {x, y}. Phonon modes and vibrational eigenfrequencies Ω m,µ of the system are obtained by solving the eigensystem associated to K in Eq. (S3), i.e., where δ m,n denotes the Kronecker delta and in the normal modes matrix M the k-th column corresponds to the (normalized) k-th eigenvector of K. The annihilation and creation operators a l,λ and a † l,λ of the phonon mode (l, λ), with λ = {1, 2} labeling the two phonon branches, are then introduced as q m,µ = n,ν M µν mn n,ν (a † n,ν + a n,ν ) and p m,µ = n,ν M µν mn i℘ n,ν (a † n,ν − a n,ν ), with n,ν = /(2MΩ n,ν ) and ℘ n,ν = M Ω n,ν /2 being the characteristic length and momentum associated with the harmonic trapping potential, respectively. In terms of the phonon mode operators, H ions can be written as in Eq. (1) of the main text, i.e., In the case of an infinite chain, ions' coordinates q m,µ and p m,µ can be expanded in Fourier modes as with −π < k < π defining the wavevector of the first Brillouin zone, µ (k) = /[2MΩ µ (k)], and ℘ µ (k) = M Ω µ (k)/2. Here, a λ (k) and a † λ (k) are annihilation and creation operators of the phonon mode (l, λ), respectively. In terms of the latter, the generalization of Eq. (S6) to an infinite chain is where normal mode eigenfrequencies are given by [50] with (c x , c y ) = (−2, 1) and the parameter η λ = V 0 /(Ma 3 ω 2 λ ) characterizing the trapping regime of the system.

II. EFFECTIVE HAMILTONIAN AND POLARON TRANSFORMATION
Here, we discuss in more details the approximations leading to Eqs. (2), (3), and (4) of the main text. Starting from Eq. (1), in the first step we expand the Rydberg interaction potential V R (r m , r n ) around ions' equilibrium positions r 0 n to the first order in the displacements q n /a = (r n − r 0 n )/a, being a the typical distance between ions. We obtain, Here, H ions is given in Eq. (S6) and H spin describes the dynamics of a (decoupled) system of spins interacting both through bare Rydberg interactions and effective multi-body interactions described by In Eq. (S13), H res represents a residual spin-phonon coupling arising as a consequence of the polaron transformation and is given by [5,6,43] In general, the contribution of H res to the system dynamics can be neglected when max[|W λ mnl |/( Ω l,λ )] 1 [5,6]. This condition can be relaxed in the classical limit Ω → 0, where the polaron transformation of Eq. (S12) results in a perfect spin-phonon decoupling [43]. Moreover, in the strong effective interaction regime reached when soft modes emerge, such as, e.g., close to the structural phase transition in the three-ion chain considered in the main text, effects of H res are negligible with respect to H eff int when Ω min(| Ṽ mni j Ω l,λ /W λ mnl |). In the presence of a single soft mode, with corresponding eigenfrequency Ω soft → 0, the latter condition can be further simplified by noticing that, in this case, the dominant contribution to both H res and H eff int is given by the soft mode only while all the other modes can be neglected.  Fig. 3 of the main text, one gets Ω * /h ∼ 1 MHz.
We now briefly comment on the infinite ion chain case. Here, by substituting the expanded potential of Eq. (S10) in H int , we get with W λ mn (k) = (2/π) 1/2 λ (k)G mnR 0 mn;λ e ikm , where G mn andR 0 mn;λ are defined as in Eq. (S11). As in the finite chain case, the spin-phonon coupling term in the right hand side can be eliminated via the polaron transformation U = e −S [5,6,43], with This leads to a transformed Hamiltonian H = UHU −1 with the same form as Eq. (S13), in which the effective multi-body interaction coefficients contained of H spin are given bỹ . (S18)

III. MICROWAVE DRESSED POTENTIAL IN A TRAPPED RYDBERG IONS CHAIN
Here, we provide further details about the derivation of the tailored microwave (MW) dressed interaction potential, shown in Fig. 2(b) of the main text, allowing to maximize the strength of effective interactions.
We consider a setup of 88 Sr + trapped Rydberg ions and focus on the four Rydberg states mS 1/2 , mP 3/2 , nS 1/2 , and nP 1/2 (with m = n + 1) shown in Fig. S1(a). For a pair of Rydberg excited ions, all these levels are coupled by the dipole-dipole interaction V dd = 1 4π 0 µ 1 ·µ 2 −3(µ 1 ·n)(µ 2 ·n) |r| 3 , with µ i the electric dipole operator of ion i, r the distance between the two ions, and n = r/|r|. Moreover, two of the Rydberg states, namely mS 1/2 and nP 1/2 , are coupled by a MW field. For a single two-level ion, the MW coupling Hamiltonian is (S19) We denote the lowest energy dressed eigenstate of H MW by |− = mS 1/2 − nP 1/2 / √ 2, with corresponding eigenenergy E |− = −Ω MW /2 (we set E |mS 1/2 = 0). For a two ion system, in the regime Ω MW V dd , V dd causes a distance-dependent energy shift −−| V dd |−− = C 3e R 3 on the two-ion state |−− , whose eigenenergy becomes E |−− = −Ω MW + C 3e R 3 . Note that the energy offset can be tuned via the MW Rabi frequency.
By acting on the MW Rabi frequency, the energy of the |−− state can be tuned close to that of |SP + . If we use the energy of |SP + for R → ∞ as the zero energy reference, the energy of |−− for R → ∞ becomes ∆E, being the energy difference ∆E between the states |−− and |SP + tunable via the MW field. Furthermore, these two two-ion states are coupled by the dipole-dipole interaction V dd , with −−| V dd |SP + = C 3f R 3 . Hence, the effective overall Hamiltonian for a two-ion system in the |−− , |SP + basis is The corresponding eigenenergies as a function of the ion-ion distance R are plotted in Fig S1(b). The combination of dipole-dipole interactions and MW-tuned Förster resonance allows to conveniently tailor the effective ion-ion interaction energy, given by the difference between the eigenenergies of Eq. (S21) and the bare energies 0 and ∆E of the states |SP + and |−− , respectively. In particular, as discussed in the main text, effects of the multi-body interactions arising from the interplay between electronic and vibrational degrees of freedom are maximized in the presence of a Rydberg interaction potential with weak NNs and NNNs interactions (V NN and V NNN , respectively) but large values of the corresponding gradients (i.e., G NN and G NNN ). As shown in Fig S1(c) and (d), such a potential can be obtained by the dressing scheme discussed so far in state-of-the-art setups of trapped Rydberg ions.