Flat bands and dynamical localization of spin-orbit coupled Bose-Einstein condensates

TL;DR

This paper explores how spin-orbit coupling and Zeeman field modulation enable flat bands and dynamical localization in shaken optical lattices with Bose-Einstein condensates, revealing conditions for complete tunneling suppression and localized states.

Contribution

It demonstrates that tuning spin-orbit coupling via Zeeman field modulation is essential for dynamical localization in BECs with shaken optical lattices, extending understanding beyond traditional models.

Findings

Energy bands become flat at specific Zeeman parameters.

Localization occurs on a dimer with components on different sites.

Localization persists even with nonlinearity in the system.

Abstract

Flat bands and dynamical localization of binary mixtures of Bose-Einstein condensates, with spin-orbit coupling subjected to a deep optical lattice which is shaking in time and to a periodic time modulation of the Zeeman field, are investigated. In contrast with usual dynamical localization in the absence of spin-orbit coupling, we find that to fully suppress the tunneling in the system the optical lattice shaking is not enough, and a proper tuning of the spin-orbit term, achievable via the Zeeman field modulation, is also required. This leads to a sequence of Zeeman parameter values where energy bands become flat, the tunneling in the system is suppressed, and the dynamical localization phenomenon occurs. Exact wave functions at the dynamical localization points show that the binary mixture localizes on a dimer with the two components occupying different sites. This type of localization occurs in exact form also for the ground state of the system at the dynamical localization points in the presence of nonlinearity and remains valid, although in approximate form, for a wide range of the Zeeman parameter around these points. The possibility of observing the above phenomena in real experiments is also briefly discussed.

Figures

Equations

$$\begin{array}{rl}i\frac{d{u}_n}{dt}=& -\Gamma (u_{n+1}+u_{n-1})+i\frac{\sigma }{2}(v_{n+1}-v_{n-1})\\ & +\Omega u_n+({\gamma }_1|u_n{|}^2+\gamma |v_n{|}^2)u_n+f(t)n{u}_n,\\ i\frac{d{v}_n}{dt}=& -\Gamma (v_{n+1}+v_{n-1})+i\frac{\sigma }{2}(u_{n+1}-u_{n-1})\\ & -\Omega v_n+(\gamma |u_n{|}^2+{\gamma }_2|v_n{|}^2)v_n+f(t)n{v}_n.\end{array}$$

$$\Omega (t)={\Omega }_0+\frac{1}{ϵ}{\Omega }_1\cos (\omega \tau ),f(t)=\frac{1}{ϵ}{f}_0\cos (\omega \tau ),$$

$$u_n=U_n{e}^{-i\frac{\sin (\omega t)}{\omega }({\Omega }_1+{\gamma }_{0}n)},v_n=V_n{e}^{i\frac{\sin (\omega t)}{\omega }({\Omega }_1-{\gamma }_{0}n)}.$$

$$i{\dot{U}}_n=$$

$$i{\dot{V}}_n=$$

$$J_0^{\pm }\equiv J_0^{\pm }(\eta )=J_0(\eta \pm \chi )$$

$$\chi =\frac{f_0}{\omega },\eta =2\frac{{\Omega }_1}{\omega }.$$

$$N=\sum _n(|U_n{|}^2+|V_n{|}^2)$$

$$H=\sum _n\{-J_0(\chi )\Gamma (U_n^{\ast }{U}_{n+1}+V_n^{\ast }{V}_{n+1})+$$

$$i\frac{\sigma }{2}{U}_n^{\ast }(J_0^{-}{V}_{n+1}-J_0^{+}{V}_{n-1})+$$

$$\frac{{\Omega }_0}{2}(|U_n{|}^2-|V_n{|}^2)+c.c.\}+E_{int},$$

$$E_{int}=\sum _n\{\frac{1}{2}({\gamma }_1|U_n{|}^4+{\gamma }_2|V_n{|}^4)+\gamma |U_n{|}^2|V_n{|}^2\},$$

$$U_n(t)=A\exp i(kn-ϵt),V_n(t)=B\exp i(kn-ϵt),$$

$${ϵ}_{\nu }(k)=-2\Gamma J_0(\chi )\cos (k)+$$

$$\nu \sqrt{{\Omega }_0^2+\frac{{\sigma }^2}{4}(J_0^{+}(\eta )-J_0^{-}(\eta ){)}^2+{\sigma }^2{J}_0^{+}(\eta )J_0^{-}(\eta )\sin (k{)}^2}$$

$${\eta }_n^{\pm }={\xi }_n\pm \bar{\xi },$$

$${ϵ}_{\nu }=\nu \sqrt{{\Omega }_0^2+\frac{{\sigma }^2}{4}{J}_0^{\pm }({\eta }_n^{\pm }{)}^2},$$

$$\{{\eta }_m^{-},{\eta }_{m+1}^{-},...,{\eta }_{2m}^{-},{\eta }_1^{+},{\eta }_{2m+1}^{-},{\eta }_2^{+},{\eta }_{2m+2}^{-},{\eta }_3^{+},{\eta }_{2m+3}^{-},...\}.$$

$$\{{\eta }_1^{-},{\eta }_2^{-},{\eta }_1^{+},{\eta }_3^{-},{\eta }_2^{+},{\eta }_4^{-},{\eta }_3^{+},{\eta }_5^{-},{\eta }_4^{+},{\eta }_6^{-},,{\eta }_5^{+},{\eta }_7^{-},...\}.$$

$$i\dot{A}\mp \frac{i\sigma }{2}{J}_0^{\mp }B-{\Omega }_{0}A=0,i\dot{B}\pm \frac{i\sigma }{2}{J}_0^{\mp }A+{\Omega }_{0}B=0,$$

$$A_{\nu }\equiv U_n=-2i\sqrt{\frac{({ϵ}_{\nu }+{\Omega }_0)({ϵ}_{\nu }^2-{\Omega }_0^2)}{2{ϵ}_{\nu }(\sigma J_0^{\mp }{)}^2}}{e}^{-i{ϵ}_{\nu }t}{\delta }_{n,n_0},$$

$$B_{\nu }\equiv V_n=-\nu \sqrt{\frac{{ϵ}_{\nu }-{\Omega }_0}{2{ϵ}_{\nu }}}{e}^{-i{ϵ}_{\nu }t}{\delta }_{n,n_0\pm 1},$$

$$\frac{1}{2}\sigma J_0^{+}b-(\mu -{\Omega }_0))a+{\gamma }_1{a}^3=0,$$

$$\frac{1}{2}\sigma J_0^{+}a-(\mu +{\Omega }_0))b+{\gamma }_2{b}^3=0,$$

Full text

I Introduction

Significant attention is presently devoted to the study of time-periodically driven many-body systems rev1 ; rev2 that exhibit interesting transport phenomena that resemble the ones observed in condensed matter physics under the action of static or time-periodic electric fields. In this context, it is well known that a constant electric field cannot induce transport in perfect crystals, due to the phenomenon of Bloch oscillations. In the presence of a time-periodic electric field, however, transport becomes generically possible, except for specific ratios of the field amplitude and frequency for which the phenomenon of dynamical localization (DL) appears.

As first demonstrated in Ref.DK for the Schrödinger tight-binding model of electrons in perfect crystals, DL emerges due to the tunneling suppression between adjacent sites induced by the periodic electric field. For harmonic fields this happens when the amplitude frequency ratio matches zeros of the Bessel function $J_{0}$. DL is not a peculiarity of linear systems but exists also in the presence of nonlinearity (interactions) as has been shown for the discrete nonlinear Schrödinger equation (DNLS) in Ref.cai and its quantum version (Bose-Hubbard model) in Ref.holthaus . To date, DL has been observed in many physical systems among which spin chains rev1 , periodically curved arrays of optical waveguides Longhi ; Sterke , cold atoms loaded in shaken optical lattices Lignier .

In Bose-Einstein (BEC) condensates, the analogues of periodic electric fields can be realized by means of shaking optical lattices. This leads to very interesting phenomena including the generation of synthetic gauge fields Struck , topological insulators Hauke , etc. In these systems it is also possible to modulate the interactions (scattering lengths) in time, a fact that allows to change the inter-site tunneling of BEC in optical lattices in a manner that depends not only on the amplitude and frequency of the modulation but also on the relative atomic imbalance between adjacent sites Haggi ; AKS . This leads to the appearance of new quantum phases Santos , density dependent gauge fields Greischner , matter wave excitations localized on a compact domain (compactons) AKS ; ASKA . Time-dependent modulations of the scattering lengths have been also shown to be very effective to suppress dynamical instabilities and to induce long-living Bloch oscillations SKB and DL BKS of matter wave gap-solitons.

Spin-orbit coupling (SOC) opens new possibilities for investigating the above phenomena in BEC systems. In particular, due to the interplay between SOC, periodicity and nonlinearity, DL could display interesting new features. In BEC systems the effective SOC stemming from internal atomic states which are coupled by Raman laser fields Galitskii-Nature can be tuned by means of fast and coherent modulations of the laser intensities Zhang . This can be achieved via modulations of the Raman term, as experimentally demonstrated in Ref.Garcia , by modulating gradient magnetic fields Luo , or by time periodic modulations of the Zeeman field SAGT .

In spite of this, however, very few investigations of flat bands and DL for SOC-BEC systems presently exist. In this context we mention the spin-dependent DL of a SOC single atom in a driven optical bipartite lattice Luo2016 , the DL in a SOC two-level atom trapped in periodic potential under the action of weak harmonically varying linear force KKZT , and the dynamical suppression of the tunneling in a double well potential for a non interacting (linear) SOC-BEC system KKV . Moreover, almost no studies exist on the effect of combined modulations on the DL phenomenon. In this respect we can mention only the work Greischner2 , where combined modulations of interactions and lattice shaking are used to generate the extended Hubbard models with asymmetric hopping which predict new quantum phases in BEC.

The aim of the present paper is to investigate DL phenomena in binary BEC mixtures subjected to optical lattice shaking, time periodic Zeeman field, and equal SOC contributions of Rashba and Dresselhaus type. For this we use a tight binding model for BEC-SOC mixture appropriate for deep optical lattices SA ; BGPMHM and treat the time modulations in the fast frequency limit. This leads to an effective time-averaged Hamiltonian system which can be analytically solved in the linear case and analytically (at DL points) and numerically investigated in the nonlinear case.

As a result we find that, in contrast with usual DL (e.g. in absence of SOC), the shaking of the optical lattice alone is not enough to fully suppress the tunneling, but suitable tunings of the SOC term with the optical lattice shaking, achieved via Zeeman field modulation, are also required. This leads to a sequence of Zeeman parameter values for which DL can occur (DL points). We show that at DL points the energy bands become flat and the tunneling is fully suppressed. In the linear case, exact wave functions derived at the DL points show that the localization occurs on a dimer with the BEC components occupying different sites. We show that this holds true, in exact form, also for the ground state wavefunctions at the DL points in the presence of intra-species contact interactions and remains valid, in approximate form, for a wide range around these points.

The paper is organized as follows. In Section II we introduce the model equations and derive the effective averaged Hamiltonian system. In section III we study the dispersion relation and the linear spectrum as a function of the system parameters. In Section IV exact analytical wavefunctions at the DL points of the linear case are derived and in Section V we extend results to the ground state of the system in the presence of nonlinearity. Finally, in section VI we discuss parameters design for possible experiments and briefly summarize the main results of the paper.

II Model and averaged equations

A BEC with equal Rashba and Dresselhaus SOC contributions loaded in a deep optical lattice can be described in the mean field approximation by the following DNLS equation SA ; BGPMHM :

[TABLE]

Here $\sigma$ and $\Omega$ denote the SOC and the Zeeman parameters while the linear ramp potential, modeling the optical lattice shaking, is assumed to be time-periodic with amplitude $f(t)=f_{0}\cos(\omega t)$. In order to have SOC tunability we also assume that the Zeeman term is varying periodically in time as $\Omega=\Omega(t)=\Omega_{0}+\Omega_{1}\cos(\omega t)$, where $\Omega_{0}$ is a constant fixed part and $\Omega_{1}$ is the amplitude of the part that is modulated with the same frequency $\omega$ of the lattice shaking. In the following we assume rapid and strongly varying fields, $\Omega(t),f(t)$, of the form

[TABLE]

with $\epsilon\ll 1$ and $\tau=\frac{t}{\epsilon}$ denoting a fast time variable. To remove the explicit fast time dependence from Eq. (1) it is convenient to perform the following transformation

[TABLE]

By substituting into Eq.(1) and averaging with respect to the fast time variable we obtain the following averaged system:

[TABLE]

where $J_{0}(\chi)$ denotes the zero-order Bessel function of a variable $\chi$, while $J_{0}^{\pm}$ stands for

[TABLE]

with

[TABLE]

Notice that Eq. (4) has two conserved quantities: the norm

[TABLE]

and the Hamiltonian (energy)

[TABLE]

where $E_{int}$ is the interaction energy,

[TABLE]

and c.c. denotes the complex-conjugate of the expression in the curly bracket. From Eq.(4) it is clear that the inter-well tunneling induced by the dispersive term $\Gamma$ can be suppressed if $\chi$ is taken as a zero of the Bessel function $J_{0}$. Notice that in absence of SOC this would be sufficient to fully suppress the tunneling in the system but in the presence of SOC this is not so because tunneling remains possible through the SOC (see the $\sigma$ term in Eq. (4)).

III Dispersion relations and linear energy spectrum

In the absence of contact interactions [$\gamma=\gamma_{1}=\gamma_{2}=0$ in Eq. (4)] it is possible to derive the dispersion relation by assuming a dependence of $U_{n},V_{n}$ on time and on the lattice site $n$ of the form

[TABLE]

where $A,B$ are real constants, $k$ is the crystal momentum varying in the first Brillouin zone, $k\in[-\pi,\pi]$, and $\epsilon$ has the physical meaning of chemical potential ($\equiv$ energy in linear case). The dispersion relation, e.g. the dependence of $\epsilon$ on $k$, directly follows from the compatibility condition of the resulting linear system, and one can easily show that it leads to

[TABLE]

with the index $\nu$ assuming the values $\nu=-1,1$, in correspondence of the lower and upper branches of the dispersion curve, respectively.

Notice that in the absence of modulations (e.g. $f_{0}=\Omega_{1}=\eta=\chi=0$) we have $J_{0}(\eta)=J_{0}^{\pm}=1$ and the above dispersion relation reduces the the one in Ref. SA for the case of a static optical lattice and constant Zeeman field. Similarly, in absence of the shaking of the optical lattice (e.g. for $f_{0}=\chi=0$), Eq. (III) reproduces the one considered in Ref. SAGT for the case of SOC tunability induced by time dependent Zeeman fields. Typical dispersion curves for different modulating parameter values are depicted in Fig.1. Notice from the bottom panel the occurrence of a flat band for the value $\eta=4.80965$ which is related to a zero of the Bessel function $J_{0}$ as we shall see in the following.

IV SOC dynamical localization: linear case

In this section we consider the effects of the optical lattice shaking and Zeeman modulation on the band flatness, suppression of tunneling and DL existence, in the absence of any contact interaction. To this end we fix the optical lattice shaking parameter $\chi$ to a zero of the Bessel function, say $\chi=\bar{\xi}$, so that the effective inter-well tunneling constant, $J_{0}(\chi)\Gamma$, vanishes. In this case the lower and upper bands are related by the symmetry $\epsilon_{1}(k)=-\epsilon_{-1}(k)$, and their dependence on $k$ is fully controlled by the Zeeman parameter $\eta$ through the factor $\sigma^{2}J_{0}^{+}J_{0}^{-}$ in Eq.(III).

Considering the SOC parameter different from zero, we have that the band flatness is achieved when one of the equations $J_{0}^{\pm}(\eta)\equiv J_{0}(\eta\pm\bar{\xi})=0$ is satisfied. This occurs for $\eta$ taken as:

[TABLE]

with $\xi_{n}$, $n=1,2,...$, denoting the n-th zero of $J_{0}$. Flat bands in $k$-space are found in correspondence of the N-fold degenerate eigenvalues

[TABLE]

which depend on $\sigma$ and with $\nu=-1,1,$ referring to the lower and upper bands, respectively. From this it follows that for $\chi=\bar{\xi}\equiv\xi_{m}$ the $m$-th zero of $J_{0}$, the band flatness occur in correspondence of the sequence of $\eta$ values (DL points), listed in increasing order,

[TABLE]

Thus, for $m=1$, e.g. $\chi=\xi_{1}=2.40483$, the first zero of $J_{0}$, the $\eta$-sequence is

[TABLE]

Notice that the flat band depicted in the bottom panel of Fig.1 just corresponds to the $\eta_{1}^{+}$ value in the above sequence. Also notice that at DL points the band velocity $v=d\epsilon(k)/dk$ vanishes for all values $k$, so the transport is fully suppressed and DL occurs.

In Fig. 2 we report the energy spectrum as a function of the modulation parameter $\eta$ for $\chi$ fixed as the first (top panel) and second (bottom panel) zero of $J_{0}$. In both cases the effective interwell tunneling $J_{0}(\chi)\Gamma$ vanishes and we see the shrinking of the spectrum into two degenerated points exactly at the values predicted by Eq. (12). Notice that the degenerate energies at these points correspond to different values of the crystal momentum $k$ and generate the upper and lower flat seen in the bottom panel of Fig.1, when plotted in the $k$-space.

In Fig.3 we show the energy spectrum as in Fig.2 but for $\chi$ slightly different from a zero of $J_{0}$. Since the effective interwell tunneling is not zero, no shrinking of the spectrum into single points can be observed in this case. Obviously, no flat bands, tunneling suppression or DL phenomena can arise for any value of $\eta$.

Notice that when the Zeeman field modulation is switched off, e.g. $\eta=0$, the flatness of the bands and the DL phenomenon occur when the intrawell tunneling is suppressed, e.g. when $\chi$ matches a zero of $J_{0}$. In this case, however, $J_{0}^{\pm}=J_{0}(\chi)=0$ so there are only the trivial flat bands $\epsilon_{\nu}=\nu\Omega_{0}$ and no SOC contribution to the DL at all, since $\sigma$ disappears from the dispersion relation. From this it is clear that in the presence of SOC the suppression of the interwell tunneling by means of the optical lattice shaking alone is not enough to induce DL, and the matching of the Zeeman field parameters with the values in Eq. (12) is absolutely necessary for DL and band flatness to exist.

Let us now investigate the eigenstates of the system at the DL points. In this respect, we remark from Eq. (4) that for $J_{0}(\chi)=0$ and $\eta$ satisfying one of the two equations: $J_{0}^{\pm}(\eta)=0$, the system reduces to a chain of uncoupled dimers described by the equations

[TABLE]

with $A,B$ standing for $U_{n},V_{n\pm 1}$ and the signs appearing in the equations related to which of the two equations $J_{0}^{\pm}=0$ is satisfied by $\eta$. In this case one can readily check that the following stationary dimer solution exists:

[TABLE]

with $\epsilon_{\nu}$ being the flat band energy at the DL point in Eq.(11).

Notice that the above eigenstates are normalized according to $|A|^{2}+|B|^{2}=1$ and they can exist, on any lattice point, for a total of $N$ orthonormal states per band ($N$ being the number of lattice sites in the chain). Actually, from this complete set of localized states one can construct extended (Bloch) states at the DL points in terms of a Fourier transform. The fact that the two components of the above dimer solution are localized on different sites implies that they exist also in the presence of the intra-species interactions, e.g. at the DL points the SOC system remains dimerized even in the presence of the inter-species nonlinearity if the nonlinearity intrasite scattering lengths are detuned to zero.

V SOC dynamical localization: nonlinear case

In the presence of contact interactions the spectrum cannot be computed analytically but it can be computed numerically, with high accuracy, by means of self-consistent diagonalization (SCD) procedure salerno . Quite surprisingly, we find that the results of the previous section for DL points survive in the presence of nonlinearity. This is shown in Fig.4 where the spectrum of the Hamiltonian versus $\eta$ is depicted for the cases of all attractive interactions (top panel) and all repulsive interactions (bottom panel). We see that, except for the presence of additional non degenerate bound state curves introduced by the interactions, both in the semi-infinite and interband gaps, the top and bottom bands curves are very similar to the ones depicted in Fig.2, shrinking into single points and leading to flat bands in $k$-space, at exactly the same values of $\eta$ derived in Eq.(12) for the linear case (similar results are found for other values of the nonlinearity parameters). The fact that the DL points are unaffected by the interaction is a consequence of the dimer localization and of the onsite nonlinearity.

From Fig.4 it is also clear that while in the attractive case the degeneracy of the ground state at the DL points is fully removed by the nonlinearity, in the repulsive case the ground state remains highly degenerated and almost unaffected by the interaction (see bottom panel of Fig.4). We remark that, in analogy with gap solitons of the continuous Gross-Pitaevskii equation KS2002 , the non degenerate localized states which appear in the band gaps originate from linear Bloch states at the center (resp. edges) of the Brillouin zone that become modulationally unstable when an attractive (resp. repulsive) interaction is switched on. The energies (chemical potentials) of these states, due to their negative (resp. positive) interaction energy contributions, are pulled just below (resp. above) the linear flat band value in the case of attractive (resp. repulsive) interactions. This explains why the ground state degeneracy at the DL points is fully removed for attractive interactions but not for repulsive interactions. The irrelevance of repulsive nonlinearities for flat band ground states also correlates with similar behaviors observed in the pure quantum regime of flat band interacting bosons in the small density limitquantumflatbands .

In the following we restrict to the case of all attractive interactions and concentrate on the ground state curve shown in the top panel of Fig. 4. In this respect we remark that the ground states at the DL points can be computed exactly by assuming the same type of dimerized localization found in the linear case. In this respect let us look for states localized on two sites of the form $U_{n_{0}}=ae^{-i\epsilon t}$ , $V_{n_{0}+1}=ibe^{-i\epsilon t}$ with a, b real constants and with $U_{n}=0$, and $V_{n}=0$ on all other sites different from $n_{0}$ and $n_{0}+1$, respectively. Substituting into Eq. (4) and looking for stationary solutions, we obtain the following cubic system:

[TABLE]

which can be solved, together with the normalization condition $a^{2}+b^{2}=1$, exactly for $a,b,\mu$, with $\mu=\epsilon+E_{int}$ denoting the chemical potential. Here we assumed $\eta$ to be a solution of the equation $J_{0}^{-}=0$, but results for the case $J_{0}^{+}=0$ follows in similar manner. From this we obtain exact nonlinear localized states at the DL points and, although the analytical expressions for $a,b$ are too involved to be reported, it is possible to obtain them numerically with high accuracy. Despite these results are strictly valid at the DL points where bands are flat, the above equations can be solved in general by considering $\eta$ a varying parameter, to see how results deviate (away from DL points) from the ones obtained by SCD in Fig.4. This is shown in Fig.5 for the ground state energy curve. Quite remarkably, we see that the comparison with the SCD curve in Fig.4 is exact at the DL points and very good for a wide interval, at least up to $\eta\approx 4$.

VI Discussion and conclusions

We briefly discuss possible parameters design and experimental setting to observe the above results presented in this paper. In this respect we refer to the SOC for the case of 87Rb atoms in the field of three laser beams implemented in a tripod scheme. The ground states from the $5S_{1/2}$ manifold are coupled via differently polarized light, by choosing $|1\rangle=|F=2,m_{F}=-1\rangle$, $|2\rangle=|F=2,m_{F}=+1\rangle$, and $|3\rangle=|F=1,m_{F}=0\rangle$ edmonds .

The optical lattice can be generated with two additional counterpropagating linearly polarized laser beams of wavelength $\lambda=2\pi/k_{L}=842$nm with a strength of the order of $10E_{R}$ with $E_{R}$ the recoil energy $E_{R}=\hbar^{2}k^{2}/(2m)$. These values guarantee the applicability of the tight-binding model we used. The passage of the optical lattice laser beams through an acousto-optic modulator permits one to introduce a frequency difference between them which can be used for the shaking of the lattice as discussed in Ref. arimondo . The strong modulation limit can be reached by considering a Zeeman field of normalized amplitude $>20$ and frequency of the modulation fixed by $\omega=2\Omega_{1}/\eta$. Under these circumstances, it should be possible to observe the described results and in particular the dimer localization properties of the ground state at the DL points.

In conclusion, we have discussed flat bands, tunneling suppression and DL in binary BEC mixtures with spin-orbit coupling subjected to a shaking optical lattice and periodic time modulations of the Zeeman field. For this we have used a tight binding model of the BEC mixture valid for deep optical lattices and considered the effects of the modulations by means of the averaging method. In particular, we showed that the suppression of the interwell tunneling is not enough to observe the DL phenomenon, and a suitable tuning of the SOC parameter with the optical lattice shaking is required. The SOC tuning, achieved via the Zeeman field, was shown to lead to a series of parameter values at which flat bands and DL can occur.

Exact analytical expressions of the BEC wave functions at the DL points have been derived, both in absence (linear case) and and in presence (nonlinear case) of interactions. In the latter case we have shown that the dimer localization occurs also for the ground state of the system, in exact form, at DL points, and in a very good approximation, around these points. Parameters design for possible experimental implementations of the above phenomena were also briefly discussed.

Acknowledgments

M.S. acknowledges partial support from the Ministero dell’Istruzione, dell’Universitá e della Ricerca (MIUR) through the PRIN (Programmi di Ricerca Scientifica di Rilevante Interesse Nazionale) grant on ”Statistical Mechanics and Complexity”: PRIN-2015-K7KK8L.