of dipolar bosons in a four-well potential
We study interacting dipolar atomic bosons in a four-well potential within a ring geometry and outline how a four-site Bose-Hubbard (BH) model including next-nearest-neighbor interaction terms can be derived for the above four-well system. We analyze the ground state of dipolar bosons by varying the strength of the interaction between particles in next-nearest-neighbor wells. We perform this analysis both numerically and analytically by reformulating the dipolar-boson model within the continuous variable picture applied in [Phys. Rev. A 84, 061601(R) (2011)]. By using this approach we obtain an effective description of the transition mechanism and show that when the next-nearest-neighbor interaction crosses a precise value of the on-site interaction, the ground state exhibits a change from the uniform state (delocalization regime) to a macroscopic two-pulse state, with strongly localized bosons (localization regime). These predictions are confirmed by the results obtained by diagonalizing numerically the four-site BH Hamiltonian.
Dipolar quantum gases  confined in a multiple-well geometry are attracting growing attention - due to the considerably rich scenario of novel properties and effects that emerges from the interplay of anisotropic dipole-dipole interactions (coupling the magnetic/electric moments of dipolar bosons) with two-boson contact interactions and the interwell boson tunneling.
In this class of systems, special interest has been focused in the last decade on the simple model where bosons are trapped by a triple-well potential. This system, effectively described by a 3-site extended Bose-Hubbard (BH) model, combines the effect of long-range dipolar interactions ,  with the nontrivial, highly nonlinear dynamics of BH models -. Note that the description by means of the BH model for bosons in multi-well systems is reliable within certain conditions on the number of particles and the strength of the dipolar interaction which has to not dominate the contact interaction [8, 9].
More specifically, in the presence of the open-chain geometry, the BH triple well has made evident the non-local character of dipolar interactions within the Josephson-like dynamics  (in the supplemental material of the latter reference, Lahaye and co-workers have considered a four-site square system to discuss the realization of interferometric arrangements) and the possibility to induce macroscopic interwell coherence independent from the tunneling parameter . This system has revealed as well a complex ground-state phase diagram where unstable regimes can be controlled through the dipolar and contact interactions . However, it is worth to observe that this is true as long as the s-wave scattering length (which characterizes the contact interaction) is larger than a critical value depending on the geometry of the external potential and the strength of the dipolar interaction .
By adopting instead the ring geometry (closed chain with periodic boundary conditions) one finds that, in addition to translational invariant vortex-like states ,  arising when the system includes only contact interactions, the presence of dipolar interactions shows the formation of different density-wave states  and the possibility to observe the transition between them. Recently, the ground-state phase diagram of the closed BH triple well has been explored to show the influence of the possible anisotropy of dipolar interactions  while the coherent control of boson tunneling through dipolar interactions has been studied in the presence of high-frequency time-periodic local potentials . In the recent paper  the ground-state properties of dipolar bosons trapped in a 3-well potential have been investigated when the on-site interaction and the dipolar interaction are varied. The ring geometry assumed for this model has been used to show its complete equivalence with the symmetric 3-site BH model. The nice result was that the term representing dipolar interactions can be absorbed in the on-site interaction term of the equivalent BH model whose strength has the form . This equivalence has allowed one to exploit the considerable amount of information about the low-energy properties of BH model to investigate the 3-well dipolar model. In particular, by varying and , the ground state of dipolar bosons has been found to involve dramatic changes of their space distribution which, within the equivalent BH picture, are caused by the change of the on-site interaction from attractive () to repulsive (). The corresponding entanglement properties have been explored.
In this work, we consider interacting dipolar bosons at zero temperature confined in a 4-well potential forming an equilateral square. The microscopic dynamics of this system is still described by a 4-site extended Bose-Hubbard (EBH) Hamiltonian which includes the hopping processes through the amplitude and the same on-site effective interaction used for the 3-well dipolar model. The novel aspect is that the apparently harmless addition of the fourth well totally changes the symmetry properties of the dipolar model. This causes the occurrence in the equivalent BH model of an extra-term in which non adjacent sites feature a dipolar-like interaction term depending on . Such a term does not occur in the BH model related to the 3-well dipolar model. Its presence dramatically changes the properties of the ground state.
To explore the new scenario we resort to the semiquantum approach applied in several papers (see, e.g., Refs. -) which reduces the Schrödinger problem of many-boson models to a diagonalizable form. A “dual” version of this method is also known which has been developed for spin models and applied to two-mode bosonic systems . In the sequel, we refer to this method as the continuous variable picture (CVP). The latter allows one to derive a suitable set of equations describing the boson populations of low-energy states and to exploit their solution to reconstruct the low-energy eigenstates of the system. This diagonalization scheme has been successfully applied to highlight the inner mechanism governing the localization-delocalization transition characterizing the BH models with attractive interaction , .
Within the CVP framework the ground state structure is predicted to exhibit a change when becomes larger than . The regime characterized by the uniform-boson distribution (delocalization regime) transforms into a non uniform distribution (localization regime) where the ground state features the almost complete boson-localization in two non adjacent wells. The so predicted delocalization-localization transition is corroborated by the results deriving from the numerical diagonalization of the four-site BH Hamiltonian. Thanks to the numerical approach one observes that the delocalization regime corresponds to a Fock state with the bosons equally shared among the four wells. The localization regime corresponds, instead, to a symmetric superposition of two Fock states each one characterized by non adjacent wells occupied by half of the total boson-population. It is interesting to observe that the emergence of the two-pulse state as ground state allows us to establish an immediate link with the mechanism responsible for the occurrence of the checkerboard-insulator in optical lattices (that, actually, can be regarded as multi-well systems) .
2 The model Hamiltonian
The model describing dipolar interacting bosons trapped by a potential can be derived from the bosonic-field Hamiltonian
where is the bosonic field, , , and is the boson mass. The trapping potential
characterized by the trapping frequency in the axial direction, represents the superposition of a strong harmonic confinement along axis with (planar) potential wells placed at the equidistant sites of a ring lattice. In the presence of four potential wells the lattice is a square with side and vertices and . is the depth of each well. The bosonic field can be expanded in terms of the annihilation operators
obeying the standard bosonic commutators . Owing to the form of the trapping potential, single-particle wave functions exhibit a factorized form
in which represents the ground-state wave function of harmonic potential , and the planar wave function ( is the center of the th well) describes the localization at the th well.
Note that we are assuming that the planar part of the potential is strong enough compared to other energies (the interaction energies in particular) such that the on-site wave functions () are fixed, being independent on the number of bosons in each well. We shall work under the hypothesis that the four minima of the potential are well separated. In such a way, the on-site wave function may be described by a single function , where is the center of the th well. The condition ( is the width of each Gaussian in the potential of Eq. (2)) entails that bosons are strongly localized in the proximity of sites in the plane. The functions and are orthogonal for so that one easily proves the orthonormality condition .
Potential , describing boson-boson interactions, is the sum of a short-range (sr) -dependent contact potential (with and the interatomic s-wave scattering length) and of a long-range dipole-dipole (dd) potential
The coupling of dipoles through the relevant magnetic moment (electric moment ) is embodied in () in which () is the vacuum magnetic susceptibility (vacuum dielectric constant). The relative position of the particles is given by the vector . For external (electric or magnetic) fields large enough the boson dipoles are aligned along the same direction, so that is the angle between the vector and the dipole orientation.
2.1 4-well dipolar-boson model
By assuming symmetric wells, the resulting dipolar-boson model is described by the 4-site extended Bose-Hubbard (EBH) Hamiltonian
where is the hopping amplitude, due to the ring geometry, and the interaction Hamiltonian
in addition to the standard boson-boson interaction , includes the -dependent term describing dipolar interactions. In , the local number operator counts the number of particles in the th well of the ring.
The three macroscopic parameters , and are defined as follows. The hopping amplitude is given by
where and () are two nearest-neighbor sites, while the on-site interaction combines the contributions of short-range and dipole-dipole interactions 
We write the nearest-neighbor interaction amplitude in the form
since the main contribution is due to the dipolar potential .
The comparison with the 3-site extended BH model, describing dipolar bosons trapped in three wells, shows how the presence of more than three wells induces significant changes in the interaction processes. In the triple-well case, the relevant Hamiltonian reads
Thanks to the equality
where is such that , the Hamiltonian reduces to the simpler 3-site BH model (the so-called BH trimer)
In the latter formula, shows that the nearest-neighbor (dipolar) interactions have been absorbed by the effective on-site interaction while only appears in the constant term . Based on this result, the ground-state structure of has been thoroughly investigated in Ref.  by exploiting the well-known properties of the BH-trimer ground state both in the attractive () and repulsive () interaction regime.
The application of the same scheme to the 4-well dipolar model, where
shows that, in addition to , the nonlocal term depending on now includes the contribution involving the coupling of non adjacent opertors . This leads to recast model (6) into the form
where and , characterized by the extra term coupling next nearest-neighbor sites in the 4-site lattice. Then the new form (13) of Hamiltonian (6) is that of the 4-site BH model where the effective on-site interaction parameter is once more but, unlike the case of dipolar bosons in three-well potential, a new interaction term modify the spectral properties of the model with respect to the case of the BH Hamiltonian.
3 The 4-well dipolar-boson model within the continuous variable picture
The CVP is obtained by observing that physical quantities depending on the local boson populations can be equivalently described in terms of densities . For large enough, the latter can be seen as continuous variables. This assumption leads to reformulate in terms of densities both Fock states and, accordingly, the action of bosonic operators on such states. After setting and observing that creation (destruction) processes () entail that
one determines the effect of the action of Hamiltonian on a generic quantum state where . The corresponding calculations describing the essence of this approach are discussed in A. Within the new formalism the eigenvalue problem for the BH Hamiltonian
takes the CVP form 
including the generalized Laplacian
with , and the effective potential
where assumes the value () in the presence of an effective on-site repulsive (attractive) interaction ().
The solutions to this problem (and the relevant eigenvalues ) are found by observing that it can be reduced to a multidimensional harmonic-oscillator problem in the proximity of the extremal points of . The essential information concerning the ground-state configuration is thus obtained by imposing the stationarity condition of . In the following, we use this condition to determine the ground state of model (13) and its dependence on the model parameters.
3.1 Bosonic-population equations charaterizing the ground state
The application of the CVP to the 4-well dipolar-boson model (13) yields the new eigenvalue equation
where the effective Hamiltonian contains the generalized Laplacian defined on the squared ring
and the potential
in which and owing to the conservation of the total boson number . At this point, we observe that due to ring geometry and for large enough the parameter is sufficiently small. We can thus write a more useful version of which underlines its symmetric character under exchanges of boson populations, that is
In general, two main regimes (the repulsive and attractive ones) can be identified by considering the interplay between parameters and occurring in the effective potential or, equivalently, in Hamiltonian . For a given , one has
The derivation of the equations for variables discussed below shows how the first case actually splits into two independent, significantly different, regimes.
To derive the equations for the ’s one must consider the constraint , implying that one of the coordinates can be seen as a dependent variable. By assuming, for example, , the equations with gives
determining the configurations for which the stationarity condition of is realized. Since densities describe the bosonic populations (BP), we will refer to such equations as the BP equations. In general, such equations can be shown  to describe different weakly-excited states in addition to the ground state. The solutions of the BP equations can be found analytically when one consideres the special class of solutions for which . This reflects the exchange symmetry characterizing such equations. Then, by setting , one finds
showing how two of the three BP equations (the first and the third) are reduced to a unique equation. The resulting system still contains the ground state. One easily checks that, with and for any value of and , the uniform solution satisfies the previous equations and reproduces the same ground state of the 4-site BH model in the absence of dipolar interaction.
3.2 Solutions of the BP equations
A large amount of information can be extracted from the reduced BP equations. By rewriting equation (20) in the form
with , one easily identifies the solution entailing that the equation (19) becomes
Such equations show that three different regimes characterize the low-energy scenario relevant to and, more in general, to the 4-well dipolar model. For (corresponding to ) one has two cases
in addition to the case
where . Note that, in the proximity of (namely, in the limit ), parameter can assume arbitrarily large values.
Apart from and , equations (21) and (22) do not provide further solutions for and in that the factors contained in the squared brackets is always positive. Summarizing, the uniform solution is the unique solution for . This is the ultraweak dipolar-interaction regime.
which, combined with the constraint , gives
in the interval . Then, by observing that and , the configuration of the system appears to be completely determined. The range of where these solutions are defined depends on and . In view of definitions and ), after rewriting the latter inequality for in the form
one finds that for the range of tends to zero while, for , the range is . This case represents the weak dipolar-interaction regime.
Since , the first equation is solved once more by while the second one is solved either by
or by setting
One immediately gets the relevant solutions given by
where the range of is determined by . Note that given by equation (24). The inequality defining the upper bound can be rewritten in the more explicit form showing that for , and for a generic, arbitrarily large . This case represents the strong dipolar-interaction regime.
3.3 The low-energy scenario
In the CVP form (15) of the dipolar-boson model the energy of the system in the proximity of minimum-energy configurations is described by potential (16). The comparison of the energies corresponding to the BP configurations analyzed in the previous section reveals the change of structure of the minimum when the parameter is varied with respect to . Numerical calculations confirm that both the uniform solution and solution (23) represent minimum-energy configurations in the corresponding regimes.
The energy of the uniform solution with obtained from (16) is easily found to be
This represents the ground-state energy for , namely, when and (ultraweak dipolar interaction).
For any value in one has
By observing that inequality defines the range of , the two energies are found to coincide for , consistently with the fact that, in this case, the solution described by (23) and (24) reduce to the uniform solution. For one has which, owing to , implies that . As expected, in this limit one obtains that . Going to the opposite extreme one easily checks that
where inequality is now substituted by . The condition is reached for . Note that the diverging factor in the previous expressions is in fact irrelevant since the effective energies are defined by (see the energy eigenvalue in (15)).
The transition from the regime to the one with thus entails the change of the ground state structure, which from the uniform-boson distribution relevant to transforms into a non uniform distribution with two separated peaks such that either or .
For (strong dipolar-iteraction regime) one has and for any . As a consequence, potential (16) takes the form
giving the minimum-energy formula
when the corresponding to solution (28) are substituted. Since
in the interval and , respectively, then and describe the same function of . Hence, simply represents the continuation of in the upper interval with for . Once more, one easily checks that
for essentially any value of where is the uniform-solution energy corresponding to the choice . Note that solution (28) is defined provided is satisfied (the value is the unique case for which ). Not surprisingly, this inequality reproduces the more explicit one already found for solution (23). For one has while for one has .
The fact that for suggests that when the previous inequality is violated the minimum energy becomes that described by the uniform solution. This circumstance is confirmed by the fact that both solution (23) and solution (28) reproduce the uniform solution whenever tends to its extreme permitted value.
Concluding, this analysis shows that, rather counterintuitively, the crucial change in the ground-state structure takes place when crosses (transition from the ultraweak-interaction to the weak-interaction regime) while the change of the effective dipolar interaction from repulsive () to attractive () is completely irrelevant. The emerging ground state significantly differs from that of the dipolar-boson model in a triple well. In the latter case the transition from the weakly-attractive regime, characterized by the uniform solution (full boson delocalization), to the strongly-attractive regime () shows that the ground state becomes a symmetric superposition of three macroscopic states each one describing the almost complete localization of bosons in one of the three wells (Schrödinger-cat state)
Such a state manifestly reflects the equivalence of the 3-well dipolar-boson system with the attractive BH trimer. The 4-well dipolar model instead features a strongly-attractive regime where bosons are macroscopically localized in non adjacent wells (for example, and ) and the ground state will be a symmetric superposition
the second Fock state corresponding to the equivalent configuration and ). Even for , states involving the full localization of bosons in one of the four wells (, , … ) are in no way involved in the ground state. The latter in turn reflects the crucial role played by the extra term in the interaction Hamiltonian in .
4 Dipolar-boson ground state
The ground state of the Hamiltonian (13) can be written in the form of superposition of different Fock states which, due to the conservation of total boson number reads
where we have omitted the occupation number of the fourth well .
Different ground states are sustained by the Hamiltonian (13) depending on the relative magnitude of the parameters and . To show how this causes changes in the ground state structure, we have studied the probabilities on varying of