Management of the correlations of UltracoldBosons in triple wells

The N-boson state $| {\rm{\Psi }}\rangle$ is governed by the time-dependent Schrödinger equation

$$\begin{eqnarray}&&{\rm{i}}{\partial }_{t}| {\rm{\Psi }}\rangle =\hat{H}| {\rm{\Psi }}\rangle ,\end{eqnarray} \tag{ 1 }$$

with the Hamiltonian

$$\begin{eqnarray}&&\hat{H}({x}_{1},{x}_{2},\,\ldots ,\,{x}_{N})=\displaystyle \sum _{j=1}^{N}\hat{h}({x}_{j})+\displaystyle \sum _{k\gt j=1}^{N}\hat{W}(| {x}_{j}-{x}_{k}| ).\end{eqnarray} \tag{ 2 }$$

We compute the ground state of the Hamiltonian in equation (2) by propagating equation (1) in imaginary time to damp out any excitation in the one-dimensional many-body system. Here, x_j represents the position of the jth boson, $\hat{h}$ is the single-particle Hamiltonian $\hat{h}(x)=\hat{T}(x)+{\hat{V}}_{{\rm{trap}}}(x);$ $\hat{T}(x)\left(=-\tfrac{{{\hslash }}^{2}}{2m}\tfrac{{\partial }^{2}}{\partial {x}^{2}}\right)$ and ${\hat{V}}_{{\rm{trap}}}(x)$ are the usual kinetic and external potential energy, respectively. Interactions of ultracold dilute bosonic gases are typically modeled using a Dirac-delta distribution: $\hat{W}({x}_{j}-{x}_{k})={\lambda }_{0}\delta ({x}_{j}-{x}_{k})$. Here, λ₀ is referred to as the strength of interactions. We scale λ₀ with the particle number as $\lambda ={\lambda }_{0}(N-1)$. In equations (1), (2) and the remainder of this work dimensionless units are employed. To define dimensionless units, we divide the Hamiltonian by ${{\hslash }}^{2}/({{mL}}^{2})$, where m is the mass of the considered boson and L a convenient length scale. We first choose a length scale of L = 1 μm. The scale of energy for the mass of ⁸⁷Rb is ${{\hslash }}^{2}/({{mL}}^{2})=2\pi {\hslash }\times 116$ Hz and the scale of time is ${{mL}}^{2}/{\hslash }=1.37$ ms. The one-dimensional scattering parameter λ is related to the three-dimensional scattering length a_3D by $\lambda =2{Lm}{\omega }_{\perp }{a}_{3D}/{\hslash }$, where ω_⊥is the frequency of the transversal confinement [43]. Using ${a}_{3D}=100.4{a}_{0}$ , where a₀ is the Bohr radius, and λ = 6 (λ = 20), one obtains ω_⊥ = 41.3 kHz (ω_⊥ = 1376 kHz).

We consider N = 90 interacting bosons in a trap of the form

$$\begin{eqnarray}&&{V}_{{\rm{trap}}}(x)=-\alpha x+{V}_{0}{\sin }^{4}({kx})+{f}_{w}(x).\end{eqnarray} \tag{ 3 }$$

Here α is the tilt and V₀ the barrier height. We fix k = 2 for the lattice spacing. The term f_w(x) introduces quasi-hard-wall boundary conditions. The tilt αx renders the trapping potential similar to that of charged particles in a constant electric field and can be realized by applying a magnetic field gradient to ultracold neutral bosons in a lattice. We note the possibility of achieving, virtually, any periodic lattice in the experiment [44]. The potential is plotted in figure 1(a) for α ∈ [0,16].

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The key idea of the MCTDHB approach is the use of time-adaptive basis states. The bosonic field operator which annihilates a particle at position x is represented by a set of M orthonormal, time-dependent functions (orbitals) {$\varphi$_i(x, t)}

$$\begin{eqnarray}&&\hat{{\rm{\Psi }}}(x,t)=\displaystyle \sum _{i=1}^{M}{\hat{a}}_{i}{\varphi }_{i}(x,t).\end{eqnarray} \tag{ 4 }$$

Here, the bosonic creation and annihilation operators ${\hat{a}}_{i},{\hat{a}}_{i}^{\dagger }$ obey the usual commutation relations at any instant t.

The ansatz for the many-boson wave function assumed in MCTDHB is

$$\begin{eqnarray}&&| {\rm{\Psi }}(t)\rangle =\displaystyle \sum _{n}{C}_{\vec{n}}(t)| \vec{n};t\rangle .\end{eqnarray} \tag{ 5 }$$

Here, the summation runs over all possible configurations $\{\vec{n}=({n}_{1},\,\ldots ,\,{n}_{M})\}$, for which ${\sum }_{i}{n}_{i}=N$. Thus N bosons are distributed over M accessible orbitals. Using the bosonic creation operators $\{{\hat{a}}_{k}^{\dagger }\}$, the time-dependent configurations can be written as

$$\begin{eqnarray}\begin{array}{rcl}| \vec{n};t\rangle & = & \displaystyle \frac{{[{a}_{1}^{\dagger }(t)]}^{{n}_{1}}{[{a}_{2}^{\dagger }(t)]}^{{n}_{2}}...{[{a}_{M}^{\dagger }(t)]}^{{n}_{M}}}{\sqrt{{n}_{1}!{n}_{2}!{n}_{3}!...{n}_{M}!}}| 0\rangle \\ & \equiv & | {n}_{1},{n}_{2},{n}_{3},\ldots {n}_{M};t\rangle .\end{array}\end{eqnarray} \tag{ 6 }$$

Here, $| 0\rangle$ is the vacuum state. Since the permanents $| {n}_{1},\,\ldots ,\,{n}_{M};t\rangle$ are a complete basis set of N-body Hilbert space for $M\to \infty$, the variational principle [45] guarantees that the solutions of the time-dependent many-body problem provided by the MCTDHB method gradually improve towards exactness when the number of considered creation operators M in the ansatz, equation (5), is increased [36, 39]. To derive the MCTDHB equations, the time-dependent variational principle [45] is employed to determine the time-evolution of the expansion coefficients $\{{C}_{\vec{n}}(t)\}$ and the orbitals $\{{\varphi }_{i}({\bf{r}},t)\}$ [37, 38]. The MCTDHB equations are obtained by fixing a gauge freedom in the choice of the orbitals by the condition $\langle \tfrac{{\rm{\partial }}{\varphi }_{j}}{{\rm{\partial }}t}| {\varphi }_{i} \rangle =0,{\rm{\forall }}j,i\in [1,M]$.

In Lagrangian formulation, the functional action of the time-dependent Schrödinger equation [with many-body ansatz, equation (5)] reads as [37, 38]

$$\begin{eqnarray}&&S[\{{C}_{\vec{n}}(t)\},\{{\varphi }_{i}({\bf{r}},t)\}]=\int {\rm{d}}t\{ \langle {\rm{\Psi }}| \hat{H}-i\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}t}| {\rm{\Psi }} \rangle -\displaystyle \displaystyle \sum _{i,j=1}^{M}{\mu }_{{ij}}(t)[ \langle {\varphi }_{i}| {\varphi }_{j} \rangle -{\delta }_{{ij}}]\}.\end{eqnarray} \tag{ 7 }$$

To ensure that the time-dependent orbitals remain orthonormal during the propagation, time-dependent Lagrange multipliers μ_ij(t) have been introduced here.

The variation of the action with respect to the expansion coefficients ${C}_{\vec{n}}(t)$ yields the equations of motion for the expansion coefficients [37, 38]

$$\begin{eqnarray}&&\displaystyle \sum _{\vec{n}}{{ \mathcal H }}_{\vec{n}\vec{n}^{\prime} }(t){C}_{\vec{n}}=i\displaystyle \frac{\partial {C}_{\vec{n}^{\prime} (t)}}{\partial t}.\end{eqnarray} \tag{ 8 }$$

This equation of motion is a first-order differential equation and the matrix ${{ \mathcal H }}_{\vec{n}\vec{n}^{\prime} }(t)=\langle \vec{n};t| \hat{H}| \vec{n}^{\prime} ;t\rangle$ is time-dependent as the permanents $| \vec{n};t\rangle$ and $\hat{H}$ itself are functions of time.

The variation of the action with respect to the orbitals {${\varphi }_{i}$ (x, t)} yields the equation of motion for the orbitals [37, 38]:

$$\begin{eqnarray}&&i| \dot{{\varphi }_{i}} \rangle =\hat{{\bf{P}}}\left[\hat{h}| {\varphi }_{i} \rangle +\displaystyle \sum _{k,s,q,l=1}^{M}\{{\boldsymbol{\rho }}(t)\}{}_{ik}^{-1}{\rho }_{ksql}{\hat{W}}_{sl}| {\varphi }_{q} \rangle \right],\end{eqnarray} \tag{ 9 }$$

where

$$\begin{eqnarray}&&\hat{{\bf{P}}}=1-\displaystyle \sum _{{i}^{{\rm{{\prime} }}}=1}^{M}| {\varphi }_{{i}^{{\rm{{\prime} }}}} \rangle \langle {\varphi }_{{i}^{{\rm{{\prime} }}}}| \end{eqnarray} \tag{ 10 }$$

is a projector, ${\rho }_{{jk}}=\langle {\rm{\Psi }}| {\hat{a}}_{j}^{\dagger }{\hat{a}}_{k}| {\rm{\Psi }}\rangle$ are the matrix elements of the one-body RDM, and ${\rho }_{{ksql}}=\langle {\rm{\Psi }}| {\hat{a}}_{k}^{\dagger }{\hat{a}}_{s}^{\dagger }{\hat{a}}_{l}{\hat{a}}_{q}| {\rm{\Psi }}\rangle$ are the matrix elements of the two-body RDM. The local interaction potentials read ${\hat{W}}_{{sl}}=\int {\rm{d}}{x}^{{\rm{{\prime} }}}{\varphi }_{s}^{\ast }({x}^{{\rm{{\prime} }}},t)W(x,{x}^{{\rm{{\prime} }}}){\varphi }_{l}({x}^{{\rm{{\prime} }}},t)$. For the contact interactions that we consider in the main part of this paper, the ${\hat{W}}_{{sl}}$ are given by ${\hat{W}}_{{sl}}={\lambda }_{0}{\varphi }_{s}^{\ast }(x,t){\varphi }_{l}(x,t)$.

The MCTDHB thus yields descriptions of many-boson systems that allow for correlations to be intrinsically described without any a priori requirements. Coherent systems (states whose one-body RDM has a single contributing eigenvalue) [27] and fragmented systems (states whose one-body RDM has several macroscopic eigenvalues) [28, 29] can be described by MCTDHB alike [46–50].

Notably, when M = 1 is set in equation (5) the MCTDHB ansatz becomes identical to the wavefunction ansatz of the time-dependent Gross–Pitaevskii (TDGP) theory and, consequently, the MCTDHB equations of motion boil down to the TDGP equation. For further details about the MCTDHB method see [37, 38, 41].

MCTDHB represents a generalization beyond exact diagonalization approaches with static basis sets like the Bose–Hubbard approach that uses Wannier functions. Necessarily, the self-consistent basis of MCTDHB is superior to a fixed static (Wannier) basis as shown directly in [36, 39, 51–54].

Since our focus in this work is on the physics of the ground state of interacting bosons in a tilted triple well potential, we will, for the sake of brevity, omit indicating the explicit time-dependence of quantities in the following. The ground states that we discuss in the following were obtained by propagating the coupled MCTDHB equations [equations (8), (9)] in imaginary time to damp out all excitations.

The one-body RDM of the N-boson state $| {\rm{\Psi }}(t)\rangle$ is defined as:

$$\begin{eqnarray}&&{\rho }^{(1)}(x,x^{\prime} )=\langle {\rm{\Psi }}| {\hat{{\rm{\Psi }}}}^{\dagger }(x^{\prime} )\hat{{\rm{\Psi }}}(x)| {\rm{\Psi }}\rangle =\displaystyle \sum _{i}{n}_{i}{\phi }_{i}^{* }(x^{\prime} ){\phi }_{i}(x),\end{eqnarray} \tag{ 11 }$$

in its eigenbasis {ϕ_i(x)} [55, 56]. Here n_i is the ith eigenvalue and ϕ_i(x) the corresponding eigenfunction, also known as natural occupation and natural orbital, respectively. The diagonal ${\rho }^{(1)}(x,x)$ is the single-particle probability distribution ρ(x). A BEC is condensed if its RDM has only a single macroscopic eigenvalue [27] and k-fold fragmented, if its RDM has k macroscopic eigenvalues [28, 29]. The first-order coherence of a condensed state is maintained everywhere in space. Therefore, the value of the first occupation $\tfrac{{n}_{1}}{N}\approx 1$ ($\tfrac{{n}_{1}}{N}\lt 1$) is also indicative of the (loss of) coherence of the state [see equation (12) below].

To obtain a spatially resolved picture of the correlations between the atoms in the many-body state that are triggered by a specific trap geometry, we study the behavior of the first-order correlation function,

$$\begin{eqnarray}&&| {g}^{(1)}(x,x^{\prime} ){| }^{2}={\left|\displaystyle \frac{{\rho }^{(1)}(x,x^{\prime} )}{\sqrt{{\rho }^{(1)}(x,x){\rho }^{(1)}(x^{\prime} ,x^{\prime} )}}\right|}^{2}.\end{eqnarray} \tag{ 12 }$$

The value $| {g}^{(1)}(x,x^{\prime} ){| }^{2}$ marks the first-order coherence between the points x and $x^{\prime}$ ($| {g}^{(1)}(x,x^{\prime} ){| }^{2}\approx 1$) or its absence ($| {g}^{(1)}(x,x^{\prime} ){| }^{2}\approx 0$) in the state $| {\rm{\Psi }}\rangle$ [30]. Here, the system of atoms is said to be in a coherent state if $| {g}^{(1)}(x,x^{\prime} ){| }^{2}\approx 1$, similarly it is said to be in an incoherent state when $| {g}^{(1)}(x,x^{\prime} ){| }^{2}\approx 0$.

In the following discussion for first-order correlation (see section 3.2), we use the term inter-well coherence if x is in the vicinity of a different minimum of ${V}_{{\rm{trap}}}$ than $x^{\prime}$ and $| {g}^{(1)}(x,x^{\prime} ){| }^{2}\approx 1$ holds. Moreover, we use the term intra-well coherence if $| {g}^{(1)}(x,x^{\prime} ){| }^{2}\approx 1$ holds for coordinates x and $x^{\prime}$ that are both in the vicinity of the same minimum.

To quantify the fragmentation, coherence, and correlation properties of the many-boson system we discuss the behavior of the natural occupations, $\tfrac{{n}_{i}}{N}$, as a function of the tilt α [equation (3)], see figure 2.

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For moderate barrier height, V₀ = 80, and no tilt, α = 0, the bosons are not completely fragmented, i.e. $\tfrac{{n}_{1}}{N}\gt 60 \%$ and $\tfrac{{n}_{\mathrm{2,3}}}{N}\lt 20 \%$, for both small and large interaction strengths (λ = 6 and λ = 20). This is in contrast to the entirely threefold fragmented state found for V₀ = 180 with $\tfrac{{n}_{1}}{N}\approx \tfrac{{n}_{2}}{N}\approx \tfrac{{n}_{3}}{N}\approx 33.33 \%$ [see figure 2, panels (a), (b) and figure 2, panels (c), (d)]. We conclude that, at zero tilt, fragmentation can be tuned by the barrier height alone. As α grows larger so does the first natural occupation, $\tfrac{{n}_{1}}{N}\to 1$, while the other two natural occupations decrease, i.e. $\tfrac{{n}_{\mathrm{2,3}}}{N}\to 0$, see figures 2(a) and (b).

At large barriers, V₀ = 180, and moderate interactions λ = 6, the state exhibits threefold fragmentation at α = 0. As α increases past a threshold value of α ≈ 7.5, the state becomes coherent with $\tfrac{{n}_{1}}{N}\approx 1,\tfrac{{n}_{2}}{N}\approx \tfrac{{n}_{3}}{N}\approx 0$, see figure 2(c). Interestingly, the second natural occupation $\tfrac{{n}_{2}}{N}$ remains constant up to tilts as large as α ≈ 2, while $\tfrac{{n}_{3}}{N}$ starts to drop from $\tfrac{1}{3}$ to 0 already at α ≈ 0. For α > 2, n₂ falls off gradually and vanishes at α ≈ 7.5 [see figure 2(c)]. Beyond this tilt the density is almost exclusively localized in the rightmost well. For larger barriers, V₀ = 180, and moderate interactions, λ = 6, an increasing tilt α thus triggers a transition from a fully threefold fragmented to a fully condensed state, i.e. the tilt can be used to control fragmentation.

For larger interactions, λ = 20, and a large barrier height, V₀ = 180, the transition between a fragmented and a depleted state is still found, however, at larger tilts α [compare figures 2(c) and (d)].

We have verified that the above findings for the natural occupations and the fragmentation of the state also hold for the case of long-range interactions of the form $\hat{W}({r}_{j}-{r}_{k})={\lambda }_{0}/(| {x}_{j}-{x}_{k}{| }^{3}+{{\rm{\Delta }}}^{3})$ [58–60, 61]. The natural occupations follow the same pattern as their contact-interaction counterparts, but the restoration of coherence seems to happen at even larger values α as compared to the case of contact interactions. This demonstrates the sharper effect of long-range interactions on the fragmentation, see supplemental material⁵. We thus conclude that the tilt of the triple well can be used to tune the many-body state from fragmented to condensed.

To get a spatially resolved picture of the correlations between the atoms, we plot the first-order correlation function, $| {g}^{(1)}{| }^{2}$ [as defined in equation (12)] for various tilts (α = 0, 2.5, 6.5, 16), barrier heights (V₀ = 80, 180) and interaction strengths (λ = 6, 20) in figure 3.

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We first discuss the correlation function for a moderate barrier height, V₀ = 80, in figures 3(a)–(h). At a small interaction strength (λ = 6), coherence between different wells persists since $| {g}^{(1)}(x,x^{\prime} ){| }^{2}$ is significantly larger than zero at off-diagonal values $x\ne x^{\prime}$ for all tilts, figures 3(a)–(d). For larger interaction strengths (λ = 20), inter-well coherence is absent for no tilt (α = 0), figure 3(e). As the tilt increases, inter-well coherence between populated neighboring wells is gradually restored, see figures 3(a)–(d) for λ = 6 and figures 3(f)–(h) for λ = 20: $| {g}^{(1)}(x,x^{\prime} )|$ gradually grows towards unity for values $x\ne x^{\prime}$. We note that a tilt-driven localization takes place for larger tilts: for tilts α ≳ 10, the left well contains almost no particles [see figure 1(b)].

The effect of interactions is to merely diminish the inter-well coherence, see figures 3(a)–(d) and (e)–(h): the value of $| {g}^{(1)}(x,x^{\prime} ){| }^{2}$ is generally closer to unity on the off-diagonal $x\ne x^{\prime}$ for small interactions [figures 3(a)–(d)] as compared to larger interactions [figure 3(e)–(h)].

We now turn to an analysis of the details of the spreading of the coherence. Two competing tendencies are observed in the evolution of coherence as a function of the tilt α that can be illustrated with panels (a), (b), (e), (f), (g) of figures 3: tendency I. is an increase of the coherence between the center and the right well with the tilt see figures 3(a) to (b), and figures 3(e) to (f), respectively. Tendency II. is the restoration of coherence between the central and the left well. Both, tendency I. and II. can be understood by the following consideration about the behavior of interaction and interaction energy as a function of increasing tilt α: naturally, an increase of α forces the particles downhill and the density gradually accumulates more in the right well, depleting the central and the left well (see also figure 1). This leads to an interaction induced broadening of the density in the right well, because the presence of more particles implies a larger local portion of interaction energy; thus, due to the tilt, the density in the right well penetrates more into the potential barrier between central and the right well and, thereby, increases the tunneling between these wells—hence the partial revival of coherence between the central and right well is seen going from figure 3(a) to (b) and going from figure 3 (e) to (f). Tendency II., the restoration of coherence between the left and central wells going from figure 3(a) to (b) or, equivalently, from figure 3(f) to (g), can also be understood as a consequence of the tilt-driven migration of interaction energy towards the right well: since the central and left wells are gradually depleted, the local contribution to the interaction energy is decreasing there. The coherence between the left and central well is restored [figures 3(b) and (g)], when the sub-system remaining in these wells is effectively non-interacting and its 'local' state is well-described by a product of a single complex-valued function.

We note here that we verified that the above tendency I. starts affecting the correlation patterns at smaller tilts than the tendency II. not only for λ = 20, V₀ = 80 [see change from figure 3(e) to (f)], but also for λ = 6, V₀ = 80: for tilts $\alpha \lesssim 1.5$ with V₀ = 80 and λ = 6, the correlation pattern (not shown) closely resembles the one depicted in panel (f) of figure 3 for λ = 20 and V₀ = 80.

We thus demonstrate that an increase of the tilt, at a fixed interaction strength, assists inter-well coherence of bosons in neighboring wells, while an increase of the interaction strength, for fixed moderate barrier heights, diminishes inter-well coherence (section 3.3).

We now analyze the correlations for larger barrier heights (${V}_{0}=180$), figures 3(i)–(p). For zero tilt and in comparison to moderate barrier heights, inter-well coherence is completely lost at large barrier heights, $| {g}^{(1)}(x,x^{\prime} )\approx 0{| }^{2}$ for $x\ne x^{\prime}$ in figures 3(i), (m).

By comparing the correlations at moderate barrier heights to the correlations at larger barrier heights, we find—as expected—that a larger value of V₀ increases the degree of localization of the system. This is true, independently of the interparticle interaction strength; compare first and third as well as second and fourth row of figure 3.

Similarly to moderate barrier heights, a restoration of coherence is also seen as the tilt α is increased for larger barriers V₀. This restoration of coherence is followed by a revival of next-to-nearest-neighbor-coherence to a smaller degree [see figures 3(b) and 3(g)] in the case of moderate barriers. For large barriers, however, the revival of the next-to-nearest-neighbor-coherence is much more prominent, while the nearest neighbors remain incoherent, see figure 3(j) for weak interactions and figure 3(o) for strong interactions.

Note that we have checked the persistence of the revival of next-to-nearest-neighbor coherence at a larger accuracy, i.e. for M = 4 orbitals. We found that the effect appears at larger tilts (α = 7.1) for the case of M = 4. We therefore speculate that the next-to-nearest neighbor coherence results from a resonance condition involving two- or many-particle correlated tunneling processes [39, 62, 63]; the lowered energy at the M = 4-level of MCTDHB (footnote 5) seems to cause the resonance condition to be fulfilled at a different tilt.

The effect of stronger interactions is, one, to defer the restoration of coherence to larger tilts (from α = 2.5 for λ = 6 to α = 6.5 for λ = 20) and, two, to shift the tilt-driven localization of the bosons to larger tilts. For strong interactions, λ = 20, at α = 16 two wells are populated and at α = 6.5 three wells are populated. For weak interactions, λ = 6, in contrast, only one well is populated at α = 16 and two wells are populated at α = 6.5.

We assess the generality of our findings for the coherence properties for long-range interactions in the supplemental material (footnote 5). The inclusion of long-range interactions favors the fragmentation of the BEC for a larger barrier height. We find that our main conclusions for short-ranged interactions hold also for the case of long-ranged interactions.

The left-right inter-well correlation can be defined as:

$$\begin{eqnarray}&&| {g}^{(1)}({x}_{l},{x}_{r}){| }^{2}={\left|\displaystyle \frac{{\rho }^{(1)}({x}_{r},{x}_{l})}{\sqrt{{\rho }^{(1)}({x}_{r},{x}_{r}){\rho }^{(1)}({x}_{l},{x}_{l})}}\right|}^{2}.\end{eqnarray} \tag{ 13 }$$

The quantity $| {g}^{(1)}({x}_{l}(\alpha ),{x}_{r}(\alpha )){| }^{2}$ gives the degree of first-order correlation between points x = x_r(α) and $x^{\prime} ={x}_{l}(\alpha )$. For our potential, equation (3), the position x_r(α) [x_l(α)] of the right [left] well minimum is weakly dependent on the tilt α. The right [left] well extends from about 1.5–2.1 [−1.5 to −2.1] for the considered tilts α. We plot the inter-well correlations as a function of the tilt α for various barrier heights V₀ and interaction strengths λ in figure 4.

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For moderate barrier height (and smaller interaction strength), significant inter-well correlations persists for a smaller α window (note the different ranges of the panels in figure 4). However, it disappears with a further increase of α at $\lambda =6$ [see figure 4(a)]. Further increasing the interaction strength (λ = 20) leads to left-right correlations that persist out to larger tilts α [figure 4(b)]; this behavior is a consequence of the delay of the tilt-driven localization by the increased repulsion.

In the case of larger barrier height, the left-right correlation is observed only for certain values of the tilting parameter [figures 4(c) and (d)]. Hence, by tuning the barrier height, interaction strength and tilt of the triple well, the left-right coherence can be adjusted.

The behavior of the natural orbitals ϕ_i(x) is shown in figure 5 as a function of the tilt α for a fixed barrier height V₀ and a fixed interaction strength λ for contact interactions. Without a tilt (α = 0), the first natural orbital ϕ₁(x) has three maxima which are centered at positions of the wells. The second natural orbital ϕ₂(x) has two maxima that are localized in the first and the third wells and the third natural orbital ϕ₃(x) has three maxima which are localized at the positions of the minima of the triple well similar to ϕ₁(x) [see figure 5(a)]. The behavior of the natural orbitals complements the nearly equal population in the three natural orbitals, i.e., the threefold fragmentation of the condensate [see figures 2(d) and 3(m)].

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For α > 0, the natural orbitals ϕ_i(x) adapt their shape to fit the new form of the external trapping potential. The orbitals ϕ_i(x) now have a single maximum and are localized independently in the three different wells [figure 5(b)]. With a further increase in α [figure 5(c)], two maxima emerge in ϕ₁(x) that are localized in the first and the third wells with different amplitudes. This structure of ϕ₁ is responsible for the next-to-nearest-neighbor correlations, i.e. correlations between the first and the third wells [see figure 3(o)]. The third orbital, ϕ₃(x), shows a node at the center of the third well leading to a higher kinetic energy. The occupation of the third natural orbital becomes therefore the least energetically favorable. For large values of the tilt α, the natural orbitals are mainly localized in the second and the third wells [figure 5(d)].

We note here, that—similar to figure 3(o)—we also found the origin of the next-to-nearest-neighbor correlations in figure 3(j) for ${V}_{0}=180,\lambda =6,\alpha =2.5$ to be the delocalization of the first natural orbital between the left and right wells.

We also perform an analysis (footnote ⁵, section 2) that suggest that the Hubbard model may not be applicable. Direct comparisons of MCTDHB and the Bose–Hubbard model can be found in [64–66].