Quantum simulation of extended polaron models using compound atom-ion systems

We consider the prospects for quantum simulation of condensed matter models exhibiting strong electron-phonon coupling using a hybrid platform of trapped laser-cooled ions interacting with an ultracold atomic gas. This system naturally posesses a phonon structure, in contrast to the standard optical lattice scenarios usually employed with ultracold atoms in which the lattice is generated by laser light and thus it remains static. We derive the effective Hamiltonian describing the general system and discuss the arising energy scales, relating the results to commonly employed extended Hubbard-Holstein models. Although for a typical experimentally realistic system the coupling to phonons turns out to be small, we provide the means to enhance its role and reach interesting regimes with competing orders. Extended Lang-Firsov transformation reveals the emergence of phonon-induced long-range interactions between the atoms, which can give rise to both localized and extended bipolaron states with low effective mass, indicating the possibility of fermion pairing.

We consider the prospects for quantum simulation of condensed matter models exhibiting strong electron-phonon coupling using a hybrid platform of trapped laser-cooled ions interacting with an ultracold atomic gas.This system naturally possesses a phonon structure, in contrast to the standard optical lattice scenarios usually employed with ultracold atoms in which the lattice is generated by laser light and thus it remains static.We derive the effective Hamiltonian describing the general system and discuss the arising energy scales, relating the results to commonly employed extended Hubbard-Holstein models.Although for a typical experimentally realistic system the coupling to phonons turns out to be small, we provide the means to enhance its role and reach interesting regimes with competing orders.Extended Lang-Firsov transformation reveals the emergence of phonon-induced long-range interactions between the atoms, which can give rise to both localized and extended bipolaron states with low effective mass, indicating the possibility of fermion pairing.

I. INTRODUCTION.
Strong interactions in quantum many-body systems can lead to exotic collective effects that are difficult to characterize and understand at the microscopic level.The combination of the complexity arising from the Hilbert space exponentially growing with the particle number and the inherent entanglement of the many-body wave functions limit the power of even state-of-the-art computational methods.For these reasons, quantum simulation [1] has emerged as an alternative approach, aiming to create highly controllable artificial systems which would be well understood microscopically and easy to scale in the number of qubits.Multiple physical platforms have been developed with this task in mind, including superconducting circuits, photonic systems, trapped ions and ultracold atoms [2][3][4][5].
One major challenge in condensed matter physics is connected with the interplay of strong electron correlations and large, possibly finite-range, electron-phonon coupling [6].Such systems have been theoretically studied for a long time [7][8][9][10] and are believed to play an essential role in the formation of the superconducting state in certain materials [11][12][13].The physical picture behind the phenomenon can be provided by introducing polarons, which are quasiparticles composed of electrons dressed with lattice phonons.Their mutual interaction can lead to the formation of bound states with low effective mass that would thus be mobile and Bose condense at high temperature, leading possibly to high-T c superconductivity.However, this might require carefully tuned system parameters, and in general polaron models can feature rich physics depending on the system geometry and the type of interactions [14].In the strong coupling limit, analytical predictions for the polaron and bipolaron properties such as their effective mass can be derived [15,16].While the research on the static properties of different polaron models is still active and fruitful [17][18][19][20], there is a growing interest in polaron dynamics out of equilibrium.A particularly interesting scenario to consider in this context is the light-induced superconducting response of the system, which has been observed in several materials [21][22][23][24][25][26][27].The understanding of high-T c superconductivity remains incomplete, in particular on the microscopic level.Nonetheless, enormous advances in providing insight into the system properties via photonic spectroscopy of correlated materials in-and outof-equilibrium have been attained both theoretically and experimentally [28].
Quantum simulations of polaron physics have been proposed theoretically using ultracold atomic mixtures [29][30][31][32][33], cold molecules [34][35][36], trapped ions [37] and atom-ion systems [38].Particularly the latter opens intriguing perspectives, as ions confined in radiofrequency traps form crystal structures that combined with fermionic atoms emulate a solid-state material naturally, contrarily to atoms trapped in optical lattices [39,40], for which the back-action of the atoms on the lattice potential is very weak.Given the exceptional control of preparation and measurement of trapped ion systems [41], especially of their motion, the compound atom-ion system represents a promising candidate for quantum simulation of solid state physics [42], including extended Hubbard models [43] and lattice gauge theories [44] as well as charge transport [45,46] (see Ref. [47] for a detailed review).On the experimental side, various strategies have been developed in order to reach the quantum regime with these systems.In the most standard state-of-theart setting utilizing radiofrequency ion traps, a small atom-to-ion mass ratio (e.g., lithium atoms and ytterbium ions) enables one to reach the s-wave collision energy [48].Successful sympathetic cooling of the ion has also been reported in optical traps [49].Furthermore, it has been shown that sub-microkelvin temperatures can be attained when ionizing a Rydberg atom inside a Bose-Einstein condensate [50].These pioneering experiments open the door to study the aforementioned electron-phonon physics in the near future.
In this paper, we investigate the formation of bipolarons in extended Hubbard-Holstein models (HHM) that can be engineered with atom-ion systems.Specifically, we consider a linear ion crystal superimposed with a degenerate Fermi gas.Even in such a one-dimensional setting interesting features are expected.For instance, it has been predicted that at half filling a metallic phase in the HHM emerges as a result of the competition of the charge-and spin-density wave orders [51][52][53][54][55]. Furthermore, in the strong electron-phonon coupling regime, a long-ranged (i.e., nonlocal) electron-phonon coupling decreases the effective mass of polarons and bipolarons, thus enhancing the mobility of those quasiparticles [16,18].We show that the atom-phonon coupling in a compound atom-ion system can be made tunable with experimentally realistic techniques.Compared to previous studies [42,43,56], here we treat the ions quantum mechanically and show how the resulting atomphonon coupling can be exploited to form bipolarons. Our findings pave the way towards quantum simulation of extended HHM with tunable long-ranged atom-phonon couplings, and therefore realization of interesting quantum phases in the laboratory.

II. SYSTEM AND EFFECTIVE HAMILTONIAN
We consider an ensemble of identical ions in an external trap and a gas of fermionic atoms overlapping with it, schematically presented in Fig. 1(a).To characterize the ionic part, we first minimize the classical energy functional consisting of the trapping potential and Coulomb interaction with respect to the ions' positions.In general, the interplay between the trap and the interactions can lead to various system geometries with structural phase transitions between them.Here, we assume that the ion chain is linear and thermodynamically stable so that small displacements of the ions from the equilibrium only give rise to phonon excitations.A convenient approach to calculate the phonon spectrum in the general case has been provided, e.g., in Ref. [57] (see Appendix A for more details).Each ion can be associated with its local harmonic oscillator frequency defined as Ω j = Vjj M , where is the second derivative of the total potential energy calculated at equilibrium and M denotes the mass of the ion, while δR j is the displacement of the j-th ion from its equilibrium position.In the next step one introduces local ladder operators corresponding to these local oscillators and rewrites the Hamiltonian.The latter acquires quadratic form and can be diagonalized using a generalized Bogoliubov transformation, leading to the phonon mode structure with ω m being the energy of the m-th collective mode, and bm , b † m denoting the phonon creation and annihila-tion operators that fulfill the usual bosonic commutation relations.For a finite ion number the spectrum is discrete and thus gapped, as shown in Fig. 1(c), while for an infinitely long chain it becomes continuous with an acoustic branch along the chain and an optical one in the transverse directions [58].Crucially, the ions can be individually addressed and driven using additional optical pulses [3], allowing for some degree of manipulation of the phonon structure and dynamics, including creation of squeezed states [59,60].
The gas of neutral atoms of mass m which can be either bosonic or fermionic but in this work we focus on the fermionic case, is assumed to be confined in a quasione-dimensional waveguide, with transverse confinement being sufficiently strong to freeze the atomic motion in the ground state.The dynamics in the axial direction is governed by the ionic lattice.The atom-ion interaction at large distances from the ion core (typically above a few nanometers) is given by the polarization potential with r = |r| being the separation between the atom and the ion and C 4 = αe 2 /(8π 0 ) -in SI units -with α being the static atom polarizability, e the electron charge, and 0 the vacuum permittivity.The potential is characterized by the length R and energy E scales where µ is the reduced mass µ = M m/(m + M ).A possible choice for the atom-ion pair is 6 Li / 174 Yb + , which due to the low mass ratio is the most favorable to attain the ultracold regime in radio-frequency traps [48,61,62].
For this pair we have E /h 178.6 kHz, R 69.8 nm, and the mass ratio of about 0.035.In the lowest order, one can assume the ions to be static and the atoms thus move in a periodic potential resulting from the interaction with the ions depicted as a black line in Fig. 1(a).As long as the spacing between the ions is much bigger than R , which is a reasonable assumption as in a typical experimental setup the ion spacing reaches a few µm, it is sufficient to use the one-dimensional pseudopotential approximation for with coefficients g e , g o describing the interaction in the even and odd partial waves (one-dimensional analogues of the three-dimensional case).The periodic potential gives rise to a band structure which we calculate numerically, showing an example in Fig. 1(b).In addition, the atoms are interacting with each other via van der Waals forces which have local character described by a pseudopotential similar to (4).Having calculated the band structure and the corresponding Bloch states, one can switch to the basis consisting of maximally localized Wannier states in which the atomic part of the Hamiltonian takes the familiar form of the extended fermionic Hubbard model [43] ] Ĥa = ijσ Here the ĉiσ are atomic annihilation operators for lattice site i, with indices σ, ↑, ↓ denoting the two atomic spin states.The next-neighbour terms omitted here are smaller than the leading ones, but typically not completely negligible due to less localized Wannier functions with respect to the case of an optical lattice potential.

III. ATOM-PHONON COUPLING
The crucial element for the simulation of polaron models is the coupling of the atoms to the phonons which results from the ion-atom interaction beyond the static ion approximation.By expanding to the first order the atom-ion interaction (2) with respect to the ions' equilibrium positions, one arrives at the following textbook expression for the atom-phonon coupling Here, N is the number of ions, V ai (k) is the lattice Fourier transform of the atom-ion interaction, k is the lattice quasi-momentum, and A k is defined as being the lattice Fourier transform of the ion displacement operator with R n denoting the equilibrium position of the nth ion.The atom-phonon coupling term in the Hamiltonian is by definition given as with ρ(r) denoting the atomic density operator, which we expand in terms of the lattice Wannier states w n and perform a Fourier transform, such that where, assuming an effectively one-dimensional system, α nn (q) = dy w * n (y − R nn )w n (y)e iqy .This results in Here is the equilibrium distance between the ions and a summation over the phonon modes has been performed to shorten the notation, leading to new local phonon operators connected to a single lattice site (the summation over j cannot be performed in general, since the Ω j values may depend on the site index and the system is not translationally invariant) The terms leading to tunneling of particles between the lattice sites will in general be suppressed due to the small Wannier function overlap.We can now introduce the local atom-phonon coupling coefficient describing the term that does not involve tunneling (dropping the n, n indices in the α parameter) It is convenient to rewrite this in dimensionless form, using R , E as length and energy units.At this point we also restrict the consideration to the even part of the interaction (4), introducing the even scattering length a e = 2 /µg e , which gives where ζ denotes the fine structure constant and ξ = /mc has the dimension of length.For standard ion-atom pairs and realistic distance d between the ions the number resulting from Eq. ( 14) is on the order of 20.At the same time, the tunneling and interaction scales as well as M nj are only a fraction of E .This means that typically the phonon dynamics is largely decoupled from the atoms, i.e. the ion chain is stiff.In order to tune the system towards the more interesting regimes, one needs to bring the energy scales closer to each other.The analysis of the M nj / ω ratio suggests that the most effective solution is to utilize ion-atom Feshbach resonances to increase the R /a e ratio in Eq. ( 13).Further possible control knob is to lower the phonon mode frequencies by shaping the ion trap.One can, for instance, place the ions in an additional optical lattice potential [63] antialigned with the ions' equilibrium positions V opt (x) = A opt cos 2 (x/Λ) with the appropriate wavelength Λ.As long as this does not destabilize the chain, such potential only shifts the mode frequencies to lower energies, which are more compatible with the other terms in the Hamiltonian.The exemplary case showing the mode tuning as a function of the lattice depth A opt for fixed distance between the ions is shown in Fig. 1(c).We have checked that the lattice has negligible impact on the other parameters, as the ion equilibrium positions are not displaced, so the atomic Wannier functions remain intact.Another reasonable way of tuning the energy scales is to vary the ion separation d using again the external trap structure, as from Eq. ( 13) one obtains M nj / ω ∝ d 5/4 .Figure 2 shows two exemplary cases of the atomphonon coupling strength M ij in a finite chain consisting of N = 11 ions.For the presentation we have chosen experimentally realistic parameters corresponding to Yb + ions and Li atoms with ion separation d = 15 R .The difference between the two plots lies in the value of the atom-ion scattering length a e , which for the case depicted in panel (a) takes the value a e = 0.1 R , while for the case (b) a e = 0.008 R , corresponding to strong, resonant interactions.In the latter case we need to include the energy dependence of the scattering length to obtain reliable results.We observe that for weak atom-ion repulsion the resulting coupling is rather local, while for strong interactions it extends over the whole chain, allowing for realization of different regimes.

IV. LANG-FIRSOV TRANSFORMATION
Having discussed the system parameters, we now proceed to the analysis of the connection between the ionatom simulator and the theoretical models of lattice polarons.As outlined above, the system can be described with the following Hamiltonian with xl = âl + â † l .Note also the presence of the two types of phonon operators â, b left for brevity.We have neglected the impact of the terms that involve phononinduced tunneling, as they are suppressed in the same way as the nearest-neighbor interaction terms due to small Wannier function overlap.
In order to get more insight into the physics of the ion chain, it is convenient to perform the generalized Lang-Firsov transformation H = e S He −S [8,64] defined by the generator where λ ij are for now arbitrary complex numbers.The transformation, detailed in Appendix B, introduces longrange phonon-mediated interactions and dresses the tunneling term with the lattice distortions.We choose the values of λ parameters in such a way that the atomphonon coupling term is canceled, which requires solving a system of linear equations due to nontrivial phonon mode structure of the chain.In contrast, for the case of purely local phonons and translational invariance one can eliminate the coupling term with a single λ parameter that does not depend on the site index, namely λ ij = δ ij M ii /Ω.After the transformation one obtains a new interaction term W ij n i n j with . This is the long-ranged interaction mediated by the phonons, containing a direct coupling term, but also an additional one which arises due to the nontrivial mode structure of the crystal.Both terms can turn out to be important depending on the system parameters.Interestingly, the first term in Eq. ( 17) contains the information about the coupling coefficient, while the second one involves the decomposition of the phonon modes, thus potentially leading to a difference between the bare coupling and the total induced interaction.In Figure 3, we show the effective interaction for the same parameters as in Fig. 2. For small scattering length shown in panel (b), corresponding to strong interactions, we find that the effective term has the form of a decaying sinusoid, as found also in Ref. [35] for atoms moving in a molecular crystal.Quite strikingly, in panel (a) the interactions have a completely different form, attracting each other weakly on long scales.This is due to the fact that the effective interaction is mediated mainly by the lowest phonon mode of the ionic lattice, as the other modes have much higher energies, and as a consequence can reflect the shape of a single collective mode.This is in contrast to the models employing local phonons, which always lead to the effective interactions of the shape shown in panel (b) of Fig. 3.

V. POLARON PROPERTIES
The model described by Eq. ( 15) has a considerably rich structure and represents a numerical challenge even in one dimension.Let us first briefly discuss the physics of a simpler model with local interactions and phonon modes, namely the Hubbard-Holstein model given by the Hamiltonian (18) This Hamiltonian can be viewed as the simplest possible extension of the Hubbard model taking into account the coupling to phonons and has been widely studied in the literature [8, 15, 19, 20, 51-54, 65, 66].Its main feature is the competition between the coupling term and the interaction, which drive the system towards two different phases, the Mott insulator and charge density wave.Furthermore, an additional phase can emerge at the interface of the two insulators.Surprisingly, this intervening phase has been shown to be conducting [51].Extending the model ( 18) e.g. by including long-range interactions can lead to even richer physics, e.g.induce pairing between the polarons and potentially turn the metallic phase into a superconducting one [55].
Here we are more interested in the prospects for efficient quantum simulation of models with strong atomphonon coupling rather than numerical solutions.Let us then only briefly discuss the emerging physics on the level of one or two fermions, neglecting the phonon dynamics.Already at this level the model turns out to be interesting.The tunneling constant can be used as energy measure.Then one can distinguish the antiadiabatic regime in which ω 0 /J 1 such that the phonons can be formally integrated out.Furthermore, the strong coupling regime can be identified in which the coupling to phonons dominates over the tunneling.Naturally, the most interesting and computationally challenging problem emerges when all the terms in the Hamiltonian compete with each other.
In order to demonstrate the impact of phonon-induced interactions, let us discuss the effective mass of the bipolaron, which is a bound state of two dressed fermions.For a lattice model with purely local interactions, the only possibility for the bound state to exist is in the spin singlet state when the effective onsite interaction is attractive.In contrast, adding a finite range interaction leads to two additional states where the fermions are bound in neighboring sites, regardless of their spin state.This intuitively provides an additional pairing mechanism which can enhance the conductivity [55].Furthermore, the additional interaction also renormalizes the effective mass of the singlet localized bipolaron [16].In Appendix C we show perturbatively that the singlet bipolaron indeed becomes exponentially lighter once nearest neighbor interactions are included.
Our calculations indicate that observation of such effects in a hybrid ion-atom system can be possible.Indeed, the typical kinetic energy J can take values of the order of 0.1 E as shown in Fig. 1(b), the phonon frequency ω 0 as well as the bare atomic interaction U can be manipulated via the external potential and Feshbach resonances, and the induced terms can reach 0.1 E as it can be seen in Fig. 3, leading to high tunability of the ratio between the local and nonlocal interactions.

VI. CONCLUSIONS
In this work we outlined a possibility for quantum simulation of solid state models in which the coupling to phonons competes with strong interactions between fermions.Our proposal relies on preparing a hybrid ionatom system with a chain of ions acting as a lattice for the atoms.While ion chains are now routinely prepared in laboratories, so far ultracold ion-atom hybrid systems have only been created using a single ion [47].From the atomic side, state-of-the-art atomic setups provide the possibility to control the number of atoms to a high degree [67].In order to gain access to the phonon spectrum, one needs an exceptional level of control over the system, but in principle the microscopic parameters of the model can be tuned via manipulating the trap geometry and the interparticle interactions.The resulting parameters such as induced interactions can reach a fraction of the characteristic energy E , meaning that observation of interesting quantum phases will require cooling the system to nanokelvin temperatures.Recent experimental advancements [48][49][50] indicate that this can be achieved in the near future.
We have considered a simple system consisting of a stable linear ion chain.It would also be interesting to investigate the transition to a zig-zag geometry, which would lead to realization of a two-leg ladder lattice.Furthermore, out-of-equilibrium scenarios involving preparation of nontrivial phonon states as well as driving the phonons can also lie within reach of experiments.
The characteristic features of the model in this reference frame are: the modification of the tunneling term (dressing of atoms by the lattice distortion), the emergence of long-range interactions due to the nonlocal atom-phonon coupling term (both directly and indirectly from the collective character of the phonon modes), and a modified atom-phonon coupling term.
It is now possible to choose the λ ij coefficients in such a way that the atom-phonon coupling term is completely eliminated, leaving behind a potentially long-range effective interaction between the atoms.However, it is also possible to optimize the values of λ in a different way, e.g. to minimize the energy of some variational wavefunction.
In order to get more insight here, it is useful to consider the simplified version of the model with translational invariance, local phonons with frequency ω 0 and coupling strength M 0 and no long-range coupling terms [8].In this case one has λ = M 0 /ω 0 and the transformed Hamiltonian reads with the onsite interaction shifted by E P = M 2 0 /ω 0 .Also in the case of the full model the induced interactions are expected to be of the order of M 2 0 /ω 0 with the lowest phonon frequency inducing the strongest interaction.
Let us discuss several regimes which can simplify the situation.Firstly, the adiabaticity parameter = ω 0 /J can be defined.In the antiadiabatic regime 1 the phonons are "fast" compared to the atoms and they can be integrated out leading to an extended Bose-Hubbard model with renormalized hopping and long-range interactions.In general, one can assume the phonons to decouple from the atoms and use a simple wavefunction such as the vacuum state, a thermal state or a coherent state for the phononic part.This approximation gets worse as the phonon energy scales become comparable to the atomic part of the Hamiltonian.It is also worth noting that in the case of a discrete phonon spectrum the lowest modes are decisive for the shape and strength of the interaction.In the antiadiabatic scenario, if M 2 0 /ω 0 1, the corrections induced by the phonons are negligible.In our setup one can tune the lowest mode frequency in a wide range, allowing to switch between different regimes.Furthermore, in a short chain already the second lowest mode is separated in energy scale from the other quantities, making the lowest mode the only one coupled to the atomic dynamics.This can be seen in the spatial profile of the induced interaction, which is reminiscent of the ionic displacement amplitudes of the lowest mode as demonstrated in Fig. 3.

Appendix C: Strong coupling expansion
Here, we briefly discuss the calculation of the effective mass in the strong coupling regime based on Refs.[15,16].After performing the Lang-Firsov transformation as in the previous section, we consider the case of a single atom or two atoms in the chain.For simplicity we neglect finite size effects and work with translationally invariant model described by local phonons with frequency ω 0 and coupling strength ω 0 g 2 with dimensionless g parameter.Here we also take = 1 and the lattice spacing d = 1 to simplify the notation.We assume that the wave function separates into an atomic and phononic part, and take the phonon state to be the vacuum.This leads to a general model with the interaction term described by and the kinetic part by ) Here f l (i) describes the coupling of the atom at site i to the oscillator at site l and can be nonlocal.For the Holstein-Hubbard model we have f l (i) = δ li .In our system the coupling to the next nearest neighbours is nonnegligible.We will consider a simplified model in which with 0 < κ < 1 being the model parameter.The band narrowing factor e −g 2 in Eq. (C2) is given by g2 and results in g 2 for the Holstein case, but for our model is equal to g 2 (2κ 2 − 2κ + 1).The polaron shift (the correction to the onsite interaction from the last term in Eq. (C1)) P = ω 0 g 2 for the Holstein model, while in the present model we have This gives the ratio ω 0 g2 / P = 1−2κ+2κ 2 1+4κ+4κ 2 , in contrast to 1 for the Holstein case.
In the strong coupling limit g 1, the kinetic term can be regarded as a perturbation.The energy can then be calculated using perturbation theory.For a single atom in the first order we have E p = − P − 2Je −g 2 cos k.The second order correction to this expression reads E (2)  p = −J 2 e −2g , which results in with γ denoting Euler's gamma constant.At this level the model does not really differ from the Holstein polaron mass apart from the modification of the coupling strengths g and g.In the limit g → ∞ the mass approaches the limit e g2 /2J 1 − 8J/ωg 2 (1 − κ) 2 .
The case of the bipolaron is more involved.We expect that two polarons can create a bound state if there is some effective attraction in the system.For the Holstein model this arises when the effective onsite interaction Ũ = U − 2ωg 2 < 0, and a localized bipolaron is the ground state.Close to the Ũ = 0 point, a state extending over two lattice sites localized due to the tunneling exchange can also exist.Nonlocal atom-phonon coupling changes the situation significantly, as due to the induced interaction polarons can bind across the adjacent lattice sites.Furthermore, a bound state can also exist even if the atoms are in a triplet spin state so that they cannot occupy the same site, which is not possible in the Holstein model.The important difference is that due to the coupling to adjacent phonons, the mass of the bipolaron is much lower than in the Holstein case.To see this, let us consider the simplest case of the singlet localized bipolaron, for which up to the first order we have E b = U − 2 P .The second order correction is given as n,m (g(1 − κ)) 2(n+m) (1 + (−1) n+m cos k) 2  n!m!(n + m)(ω − U + 2 P − ωg 2 κ) , (C8) where the denominator is modified by the interaction of polarons in adjacent lattice sites.The leading term in the singlet bipolaron mass in the g → ∞ limit reads where we introduced u = U − 2 P + ωg 2 κ to shorten the notation.It is interesting to note that while for the Holstein bipolaron its mass is proportional to exp 4g 2 , which is equivalent to exp 4 P /ω, here we have m b ∝ exp 4ξ P /ω with ξ = (1 − 2κ + 3κ 2 /2)/(1 + 2κ) 2 being a correction coefficient.The bipolaron in our model can thus be much lighter, i.e. by a factor exp 2 P /ω already at κ ≈ 0.12 for which ξ = 0.5.

FIG. 1 :
FIG. 1: (a) Schematic drawing of the system consisting of an ion chain (big orange balls) in an external trap (blue line) and a repulsive optical lattice potential (orange line).A neutral atom (small green ball) is moving in a periodic potential stemming from the interaction with the ion chain (black line below).(b) Typical band structure of a single atom moving in the ionic lattice with strong ion-atom interactions.(c) Phonon spectrum of a finite linear ion chain consisting of N = 11 ions as a function of the external lattice depth A opt .

2 4πε0M d 3
) Let us now discuss the interplay of the different quantities present in the model.Expressing the characteristic phonon energy scale ω = 2 e in the chosen units leads to ω

FIG. 2 :
FIG. 2: Atom-phonon coupling strength in a lattice composed of N = 11 ions for two different values of the ion-atom scattering length a e = 0.1R (a) and a e = 0.008R (b).

-FIG. 3 :
FIG.3:The effective interaction between the atoms after the Lang-Firsov transformation for two different values of the ion-atom scattering length a e = 0.1R (a)and a e = 0.008R (b).