Abstract
The precise control of quantum systems will play a major role in the realization of atomtronic devices. As in the case of electronic systems, a desirable property is the ability to implement switching. Here we show how to implement switching in a model of dipolar bosons confined to three coupled wells. The model describes interactions between bosons, tunneling of bosons between adjacent wells, and the effect of an external field. We conduct a study of the quantum dynamics of the system to probe the conditions under which switching behavior can occur. The analysis considers both integrable and nonintegrable regimes within the model. Through variation of the external field, we demonstrate how the system can be controlled between various “switchedon” and “switchedoff” configurations.
Introduction
The phenomenon of quantum tunneling is paramount in many studies of ultracold quantum gases. The twowell Bose–Hubbard Hamiltonian has been very successful in modeling quantum tunneling^{1,2}, displaying two principal dynamical scenarios. These are referred to as Josephson tunneling and selftrapping, and they have been experimentally observed^{3}. In the case of tunneling, the system can also be controlled to produce either alternating or direct currents^{4}. A threewell system opens up wider possibilities for physical behaviors^{5,6,7}, most notably as an ultracold version of a transistor^{8}, or similar type of switching device. The individual wells can be identified as the source, gate, and drain, potentially forming a building block in the emerging field of atomtronics^{9,10,11}. This prospect is driving research into transistorlike structures beyond the electronic domain^{12,13}.
Here we investigate the influence of integrability in the control of tunneling in a triplewell system. To do so, we must go beyond the familiar threewell Bose–Hubbard model^{14,15,16,17,18}, and consider a more general system which facilitates an integrable limit. Such a model has already been introduced into the literature. It models dipole–dipole interactions and the tunneling between adjacent sites for a population of ultracold dipolar bosons with large dipole moment, such as chromium or dysprosium, loaded in an aligned triplewell potential. The Hamiltonian has the general structure^{19}
The canonical creation and annihilation operators, \(a_i^\dagger\) and a_{i}, i = 1,2,3, represent the three bosonic degrees of freedom in the model, and \(N_i = a_i^\dagger a_i\), i = 1,2,3 is the number operator for each well. The parameters J_{i}, i = 1,3 are the couplings for the tunneling between wells, and U_{0} is the coupling constant for onsite interactions which results from contact interactions and dipole–dipole interactions (DDI). Both of these can be either attractive or repulsive, which in principle allows for the manufacture of weak net onsite interaction. The parameters U_{ij} = U_{ji}, i ≠ j characterize DDI between particles on different sites. Although the DDI follows an inverse cubic law, it is also dependent on the angle between dipole orientation and the displacement between dipoles. In combination with the geometry of the trap potential (viz. oblate versus prolate), it is entirely feasible to adjust the system parameters across a wide range of values. Importantly, this includes the possibility for the interwell couplings U_{ij} to have greater magnitude than the onsite coupling U_{0}. The experimental feasibility of this system for dipolar bosons was detailed by Lahaye et al.^{19}, using a triplewell potential. The wells are aligned along the yaxis, separated by a distance l, with bosons polarized by a sufficiently large external field along the zdirection. It was shown that U_{12} = U_{23} = αU_{13}, where the parameter 4 ≤ α ≤ 8 depends only on the ratio l/σ_{x}, where σ_{x} is the width of the Gaussian cloud along the xdirection. (See Methods for further details.)
In the case when U_{13} = U_{0}, the Hamiltonian (1) is integrable^{20}. In this limit there exists an additional conserved operator besides the Hamiltonian and the total particle number, such that the number of independent conserved operators is equal to the number of degrees of freedom. While for classical systems integrability is wellknown to prohibit chaotic behavior, the consequences for quantum system are less understood^{21,22}. Notwithstanding, it is recognized that quantum integrability has far reaching impacts. One route to characterize the degree of chaoticity in a quantum system is through energy level spacing distributions^{23}. Integrable systems tend to display Poissonian distributions^{24}, while nonintegrable systems generally observe the Wigner surmise^{25} following the Gaussian Orthogonal Ensemble, or similar^{26,27}. Another impact of quantum integrability is the absence of thermalization, observed in a quantum version of Newton’s cradle^{28} and similar systems^{29}. Here we demonstrate how integrability, and the breaking of it, can be utilized to investigate tunneling dynamics. This work contrasts the above mentioned studies in that it applies to a system with very low number of degrees of freedom.
A simple means of breaking integrability in the model is through an applied external field. Generally, it might be expected that this will drive the system into a chaotic dynamical regime. However it is shown that in certain circumstances the changing dynamics of the model, through tuning of the external field, can be predicted with remarkable accuracy. The result can be understood by revealing the structure of a hidden subsystem within the model. This level of control points towards the potential utility of a physical realization of the model as a quantum switch.
Results
Integrability
It has been established that the model (1) contains a family of integrable threewell tunneling models when U_{13} = U_{0}^{20}. In this case, we can write the Hamiltonian in the reduced form \(H_0 =  {\cal H} + (\alpha + 1)U_0N^2/4  U_0N/2\) yielding
where U = (α − 1)U_{0}/4. Note that (2) commutes with the total number operator N = N_{1} + N_{2} + N_{3}, and the interchange of the indices 1 and 3 leaves the Hamiltonian invariant. The Hamiltonian has, beyond the energy and the total number of particles N, another independent conserved quantity expressed through the operator^{20}
This conserved operator can alternatively be interpreted as a tunneling Hamiltonian for a twowell subsystem containing only wells 1 and 3. Because Q admits the factorization \(Q = {\mathrm{\Omega }}^\dagger {\mathrm{\Omega }}\), where Ω = J_{1}a_{3} − J_{3}a_{1}, the dynamical evolution governed by Q is harmonic for any initial state. Later, it will be shown that Q assumes a fundamental role in the analysis of resonant^{30} quantum dynamics of the system (2). This arises due to an unexpected connection with virtual processes. Details are provided in Methods.
As the model has three degrees of freedom and three independent conserved quantities satisfying
the model is integrable. Further details about the integrability, and associated exact solvability, have been established. This was achieved through the YangBaxter equation and Bethe Ansatz methods^{20}.
Breaking the integrability
In order to break the integrability, we add to the Hamiltonian (2) the operator H_{1} = ε(N_{3} − N_{1}), which acts as an external field for the wells labeled 1 and 3. This is schematically shown in Fig. 1. It is important to observe that the above Hamiltonian still commutes with the operator N. However, the operator Q is not conserved because the commutator \([H,Q] = 2\varepsilon J_1J_3(a_1^\dagger a_3  a_3^\dagger a_1)\) is nonzero when the parameters ε, J_{1} and J_{3} are all nonzero.
Structure of energy levels
The integrable threewell system (2) possesses many features in common with the twowell Bose–Hubbard model, which is also integrable because the total number operator is conserved and there are only two degrees of freedom. Set \(J = \sqrt {J_1^2 + J_3^2}\). Following Leggett^{2}, it is useful to define the regimes:

Rabi: U ≪ JN^{−1}.

Josephson: JN^{−1} ≪ U ≪ JN.

Fock: JN ≪ U.
where in the twowell case the “Fock regime corresponds to a strongly quantum pendulum, while in the Rabi and Josephson regimes the behavior is (semi)classical”^{2}.
Adopting the same classification for the integrable threewell system given by (2), numerical computation of the energy spectrum shows that transition from the Rabi to the Josephson regime is accompanied by the emergence of energy bands. Illustrative results are depicted in Fig. 2. Hereafter units are chosen such that ℏ = 1, and for all figures isotropic tunneling \(J_1/J = J_3/J = 1/\sqrt 2\) is adopted for simplicity. The Hamiltonian acts on the Fock space spanned by the normalized vectors \({\cal N}_1,{\cal N}_2,{\cal N}_3\rangle = {\cal C}^{  1}(a_1^\dagger )^{{\cal N}_1}(a_2^\dagger )^{{\cal N}_2}(a_3^\dagger )^{{\cal N}_3}0\rangle ,\) where \({\cal C} = \sqrt {{\cal N}_1!{\cal N}_2!{\cal N}_3!}\) and 0〉 ≡ 0,0,0〉 is the Fock vacuum. On each panel the line E = UN^{2} is depicted. This quantity is the expectation value of the state N,0,0〉. In the extreme Rabi regime with U = 0 the energy levels are uniformly distributed with spacing ΔE = J. The line E = UN^{2} emerges from the midpoint of the entire energy spectrum when U = 0, to lie on the lower edge of the uppermost energy band as U is increased to bring the system into the Josephson regime. Note that the separation into distinct energy bands becomes very evident once the system is deep into the Josephson regime. These features significantly influence the dynamical evolution of the system from the initial state N,0,0〉. In the Rabi regime, an accurate description of the initial state requires a linear combination over all eigenstates. However in the Josephson regime the state N,0,0〉 can be accurately approximated as a linear combination of a subset of eigenstates, due to the band structure. This conclusion applies for all particles numbers, with the result represented in Fig. 2 depicting the cases N = 30 and N = 60. Provided \(UN/J \gg 1\), the separation into bands is clearly identifiable. The consequences will be investigated at a deeper level in the next section, where we will fix N = 60. Moreover, it will be shown how the breaking of the integrability, through the application of an external field, allows for control of the dynamics in a predictable fashion.
Quantum dynamics
The time evolution of the expectation values for the number operators are computed using 〈N_{i}〉 = 〈Ψ(t)N_{i}Ψ(t)〉, i = 1,2,3, where Ψ(t)〉 = exp(−iHt)ϕ〉 and ϕ〉 represents an initial state. We adopt a protocol for which ϕ〉 = N,0,0〉, so the well labeled 1 is the source, the well labeled 2 is the gate, and the well labeled 3 is the drain.
We begin with the integrable model (2) and first consider variations in the interaction parameter U to manipulate the tunneling across the wells. Figure 3 presents results obtained for four choices of U. The dynamics typically display collapse and revival of oscillations in the Rabi regime, as in Fig. 3a. On increasing U, the period increases while the timeaverage of 〈N_{2}〉 decreases. Furthermore, the dynamics between wells 1 and 3 approach harmonic oscillations with \(\langle N_1\rangle + \langle N_3\rangle \simeq N\). The transition between the Rabi and Josephson regimes can be seen, qualitatively, in the passage from Fig. 3b and c. This change in behavior is in accord with the threshold point in Fig. 2.
In this latter regime, Fig. 3c and d, these dynamical features can be understood by observing that the integrable Hamiltonian possesses a hidden twowell algebraic structure, with an effective well given by the combined source and drain. As is known^{1,2,31}, the selftrapping regime is expected to occur in the twowell model in the Josephson regime. To be more precise, \(\langle N_2\rangle /N < \tilde \varepsilon\) when \(UN > J/(2\sqrt {\tilde \varepsilon  \tilde \varepsilon ^2} )\) if well 2 is initially empty. Thus, for \(UN \gg J\), we find \(\langle N_2\rangle /N \simeq 0\), and almost all bosons are distributed between the source and the drain if only a small fraction of bosons are initially in the gate.
On the other hand, it has been pointed out^{19} that for isotropic tunneling the source and the drain can form an effective noninteracting twowell system, by secondorder processes^{32,33,34} through the gate, such that \(\langle N_2\rangle \simeq 0\). For general tunneling, we find the remarkable result that the effective Hamiltonian is simply given by H_{eff} = −λQ, where Q is the conserved charge (3), and λ^{−1} = 4U(N − 1) (details are provided in Methods). This produces an effective tunneling coupling given by J_{eff} = λJ_{1}J_{3}, which decreases with increasing N, and therefore will only be observed in mesoscopic samples^{19}. In view of the above observations, we formally identify the resonant tunneling regime of the system to be determined by \(UN \gg J\), which contains the Josephson regime.
In Fig. 4, the time evolution of expectation values for number operators is displayed in a case of broken integrability. Increasing the value of ε suppresses the tunneling of particles into the drain, while increasing the timeaverage value of 〈N_{2}〉. For ε/J = 1.63 this suppression of tunneling into the drain is strong enough that its number expectation value is close to negligible, i.e. tunneling into the drain has been switchedoff.
Control of resonant tunneling
In Fig. 3d, the dynamics is seen to be remarkably close to being harmonic over sufficiently short time scales, with the period monotonically increasing with interaction coupling U. This behavior supports the conclusion that the effective Hamiltonian for the resonant tunneling regime is simply related to the conserved charge Q. The frequency of oscillation in this regime is given by ω_{J} = λJ^{2}, with the amplitude also being Udependent. When the initial state is N,0,0〉, the oscillations between the source and drain are coherent, with tunneling to the gate switchedoff. On the other hand, if the initial state is 0,N,0〉 the system will remain trapped in this initial state configuration, with tunneling from the gate switchedoff.
Next, we maintain the system in the resonant tunneling regime \(UN \gg J\) and study the nonintegrable dynamics using the parameter ε to control the behavior of the source and drain subsystem. The approach here, following the study above, is to choose the initial state N,0,0〉 and investigate the ability to control the frequency and amplitude of the populations oscillating between the source and the drain.
In Fig. 5a–c, the interaction coupling is fixed as U/J = 0.17, and results are shown for the expectation values of the populations using three choices for ε. It is seen that the presence of the external field does not significantly influence the gate, in the sense that it does not affect the negligible average population 〈N_{2}〉. Figure 5d shows how the amplitude decays while increasing the external field, as well as the dependence of the frequency. The three points highlighted in the curves correspond to the values of the amplitude and frequency of Fig. 5a (cyan circle), Fig. 5b (yellow triangle), and Fig. 5c (lime diamond).
In this nonintegrable regime the effective Hamiltonian is given by
For short time scales the dynamics exhibits Josephsonlike oscillation^{28} with frequency
where Δn = 1/(1 + γ^{2}) is the amplitude and \(\gamma = (\lambda (J_1^2  J_3^2)  2\varepsilon )/2\lambda J_1J_3\) (see Methods for details). Increasing the external field reduces the oscillation amplitude Δn and the period between the source and the drain, until the amplitude of oscillation is completely suppressed, i.e., all tunneling is switchedoff, demonstrating various levels of control, especially in the range 0 < ε < 0.2. Through semiclassical analysis, one can obtain analytic expressions for the expectation values of the relative populations, n_{i} ≡ N_{i}/N (i = 1, 3), in the wells 1 and 3, given by 〈n_{1}〉 = 1 − 〈n_{3}〉 and \(\langle n_3\rangle = {\mathrm{\Delta }}n\,{\mathrm{sin}}^2(\omega _{\mathrm{J}}t/2)\) (see details in Methods). In agreement with Chuang et al.^{35}, the maximum amplitude is obtained when the field is small.
Discussion
We have analyzed a model for boson tunneling in a triplewell system. This was conducted in both integrable and nonintegrable settings through variation of coupling parameters. The model draws an analogy with a transistor through identification of the wells as the source, gate, and drain. Our primary objective was to investigate how this model could be implemented as an atomtronic switching device.
In the integrable setting, we identified the resonant tunneling regime between the source and drain, for which expectation values of particle numbers in the gate are negligible. Moreover, it was found that a conserved operator of the integrable system acts as an effective Hamiltonian, which predicts coherent oscillations. This is in agreement with observations from numerical calculations.
We then broke integrability through application of an external field to the source and the drain. It was shown in Fig. 4 that the applied field to the system, in Rabi regime, was able to switchoff tunneling to the drain. On the other hand, in the resonant tunneling regime, the field did not destroy the harmonic nature of the oscillations, but did influence the amplitude and frequency. Increasing the applied field allowed for tuning the system from the switchedon configuration through to switchedoff (Fig. 5). Results from semiclassical analyses produced formulae for the amplitude and frequency, which proved to be remarkably accurate when compared to numerical calculations. This demonstrates the possibility to reliably control the harmonic dynamical behavior of the model in a particular regime. A surprising feature of this result is that the ability to control the system in a predictable manner arises through the breaking of integrability. Our results open possibilities for multilevel logic applications and consequently new avenues in the design of atomtronic devices.
It is important, finally, to comment on the limitations of a threemode Hamiltonian in the description of cold atom systems in a triplewell potential. Contributions from higher energy levels of the singleparticle spectrum cannot be ignored under certain coupling regimes. For example, the presence of the external field will ultimately lead to level crossings as the field strength is increased. In the case of the analogous doublewell system, estimates for when this may occur have been formulated^{36}. We have undertaken checks to confirm that it is indeed possible to avoid these undesired scenarios, within an experimentally feasible scenario. (See Supplementary Note 1, Supplementary Figure 1 for further details). However, it is also noteworthy that it is possible to include threebody, and higher, interaction terms as corrections^{37} to compensate for when the threemode approximation breaks down.
Methods
In this section, we provide the details concerning algebraic structures behind the model, and complementary semiclassical approximations, which were used to derive analytic expressions characterizing quantum control in the resonant tunneling regime. These expressions were found to give close agreement with results obtained by exact numerical diagonalization (see Supplementary Note 2, Supplementary Figures 2 and 3 for more details). We also discuss the feasibility of physical realization of the system.
Twomode structure and the resonant tunneling regime
The integrable threewell model can be structured through two modes, as follows. From Eq. 2, we define \(J = \sqrt {J_1^2 + J_3^2}\) and the operators N_{1,3} = N_{1} + N_{3}, a_{1,3} = J^{−1}(J_{1}a_{1} + J_{3}a_{3}) and \(a_{1,3}^\dagger = J^{  1}(J_1a_1^\dagger + J_3a_3^\dagger )\) satisfying the Heisenberg algebra
Then
such that the modes of wells 1 and 3 are now represented by the single mode “1,3”.
The twowell model exhibits a selftrapping regime, with onset in the vicinity of \(\chi \equiv UN/J \simeq 1\)^{1,31}. This translates to a resonant tunneling regime for the triplewell model. Here we follow the approach of using semiclassical analysis^{38}, such that this regime may be clearly identified. Using the usual numberphase correspondence, that is, \(a_2 = e^{i\theta _2}\sqrt {N_2}\), \(a_{1,3} = e^{i\theta _{1,3}}\sqrt {N_{1,3}}\) and the conservation of boson number N_{1,3} + N_{2} = N, we find
where n_{2} = N_{2}/N and ϕ = θ_{1,3} − θ_{2}. Consider the dynamics where the initial condition is n_{2} = 0. At the initial time t = 0 the system has the energy h = UN. By energy conservation at all times, we obtain the expression
The conditions χ > 1 and cosϕ = 1 (maximum value) imply that 0 ≤ n_{2} ≤ 0.5. From Eq. (7), we conclude that when \(\chi \gg {\mathrm{cos}}\,\phi \), n_{2} → 0, and the bosons are distributed between the wells labeled 1 and 3, producing the resonant tunneling regime.
Effective integrable Hamiltonian for resonant tunneling
In order to better understand the dynamics in the resonant tunneling regime, we first observe that the integrable Hamiltonian can be written, by using the conserved quantity N, as an effective Hamiltonian without onsite interaction (up to a global constant UN^{2}). Specifically H_{0} = H_{I} + V, where the interaction term H_{I} = −4UN_{2}(N_{1} + N_{3}) has eigenstate and eigenvalues given by
and the tunneling term \(V = (J_1a_1^\dagger + J_3a_3^\dagger )a_2 + {\mathrm{h}}.c.\) is treated as a perturbation. For the isotropic case \(J_1 = J_3 = J/\sqrt 2\)^{19}, since \(n_2 \simeq 0\) the interaction part is \(H_I \simeq 0\) and the wells 1 and 3 form an effective noninteracting twowell system coupled through well 2 by a secondorder process^{19,32,33,34} with the effective Hamiltonian \(H_{{\mathrm{eff}}} = J_{{\mathrm{eff}}}(a_1^\dagger a_3 + a_3^\dagger a_1)\). Recall that the transition rate from initial state s〉 to final state k〉 is expressed
where V^{(1)} = V for firstorder transition (Fermi’s golden rule), δ is the delta function and
for secondorder transitions. Equating secondorder transition of V with the firstorder transition of H_{eff} for the states N,0,0〉 and N−1,0,1〉, it is found that
Observing that \(J_3^2N_1 + J_1^2N_3\) is constant for isotropic tunneling in the regime \(\chi \gg 1\), then it does not affect the dynamics if we consider the linear combination \(J_{{\mathrm{eff}}}(a_1^\dagger a_3 + a_3^\dagger a_1) + \lambda \prime (J_3^2N_1 + J_1^2N_3)\). By numerical inspection, we conclude that the effective Hamiltonian for general tunneling, which includes the anisotropic tunneling J_{1} ≠ J_{3}, is given by
where \(Q = J_1^2N_3 + J_3^2N_1  J_1J_3(a_1a_3^\dagger + a_1^\dagger a_3)\) is conserved and λ^{−1} = 4U(N − 1).
Effective nonintegrable Hamiltonian and quantum control
For the nonintegrable case, the effective Hamiltonian in the resonant tunneling regime \(\chi \gg 1\) is given by H_{eff} = −λQ + ε(N_{3} − N_{1}). Returning to a semiclassical analysis it is found that, up to an irrelevant constant,
where n_{1} = N_{1}/N and φ = θ_{1} − θ_{3}. For initial condition n_{1} = 1 and n_{3} = 0, we have \(h =  \lambda J_3^2\), a constant. Applying energy conservation and the condition cosφ = ±1, we find that the amplitude of oscillation Δn (Fig. 5d) is given by Δn = 1/(1 + γ^{2}), where \(\gamma = [\lambda (J_1^2  J_3^2)  2\varepsilon ]/2\lambda J_1J_3\). Hamilton’s equation gives
Using the Ansatz \(n_1 = 1  {\mathrm{\Delta }}n\,{\mathrm{sin}}^2(\eta t)\), we can easily verify that the above results provide analytic expressions for the expectation values
where ω_{J} is the frequency given by Eq. (6). Results for similar types of investigation have been presented in the case of pairtunneling between two wells^{34}.
Physical realization
Here we discuss the feasibility of physical realization of the triplewell Hamiltonian (2), through use of Bose–Einstein condensates (BECs) of dipolar atoms.
Three parallel, tightly focused Gaussian beams, with waist of 1 μm and wavelength λ = 1.064 nm, separated by a distance l = 1.8 μm, form an optical triplewell potential aligned along the yaxis^{19}. A transverse beam, with waist of 6 μm, provides xzconfinement. For such a setup, in the harmonic approximation the potential of each well i = 1,2,3 is symmetrically cylindrical and is given by
where y_{i} = l,0,−l. The trap frequencies ω_{x} and ω_{r} can be controlled by the intensity of the laser beams. This configuration facilitates the formation of three cigarshaped wells. We consider a system of bosons with dipole–dipole interactions to provide longrange interactions, and weakly repulsive contact interactions to promote condensate stability^{39}. The dipoles are oriented along the zdirection (Fig. 6).
The transverse beam performs the function of the external field that controls the device. Its focus, when displaced along the yaxis by Δy, introduces the potential energy
This generates a potential difference, resulting in the external field strength \(\varepsilon = m\omega _y^2{\mathrm{\Delta }}yl\). The frequency of the transverse laser (ω_{y}) is much lower than the frequency of the parallel lasers (ω_{r}), so that displacement by Δy introduces a “tilting” of wells 1 and 3. These are the relevant wells in the resonant tunneling regime.
For the case of a dipolar BEC of ^{52}Cr^{40}, we numerically find that the integrability condition, with \(\alpha \sim 5.8\), is achieved for \(\omega _x\sim 2\pi \times 64\,{\mathrm{Hz}}\), \(\omega _r\sim 2\pi \times 220\,{\mathrm{Hz}}\), where we assumed the Gaussian approximation for the ground state. The value of U obtained in units of J is \(U/J\sim 7.5 \times 10^{  3}\), which means that the resonant tunneling regime can be achieved for \(N \gg 130\) atoms. In principle, this condition can be satisfied experimentally^{40}. As an example, for \(N\sim 5000\) atoms, with \(\omega _y\sim 2\pi \times 1.5\,{\mathrm{Hz}}\), translating the transverse laser by Δy = 1.8 μm we obtain \(\varepsilon /h\sim 3.6 \times 10^{  2}\,{\mathrm{Hz}}\). It results in a tunneling amplitude \({\mathrm{\Delta }}n\sim 0.25\), which means that 25% of the population of the source in the initial state N,0,0〉 switch to the drain, and back, through harmonic tunneling. This example approaches the case of Fig. 5b.
Another strongly dipolar BEC which can be considered is ^{164}Dy^{41}. In this case, we calculate \(\alpha \sim 5.9\) for \(\omega _x\sim 2\pi \times 22.7\,{\mathrm{Hz}}\), \(\omega _r\sim 2\pi \times 67.3\,{\mathrm{Hz}}\) with \(U/J\sim 2.2 \times 10^{  2}\). For N ~ 500 atoms, with \(\omega _y\sim 2\pi \times 0.76\,{\mathrm{Hz}}\), this yields \(\varepsilon /h\sim 3.0 \times 10^{  2}\,{\mathrm{Hz}}\) and \({\mathrm{\Delta }}n\sim 0.23\).
In both cases above, the parameter choices are such that higherorder interaction terms, such as correlated hopping, are negligible^{42}.
An analysis of the effects of perturbations is provided in Supplementary Note 3: Fidelity dynamics, Supplementary Figures 4 and 5.
Data availability
All relevant data are available on reasonable request from the authors.
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Acknowledgements
K.W.W. and A.F. were supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brazil. J.L. and A.F. were supported by the Australian Research Council through Discovery Project DP150101294. We thank Henri Boudinov, Ricardo R. B. Correia, Matt Davis, and Artem Volosniev for helpful discussions.
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All authors contributed to the conceptualization of the project, and actively engaged in the writing of the manuscript. K.W.W., L.H.Y., and A.P.T. implemented the theoretical analyses of the model, detailed the experimental feasibility, and processed the numerical computations. J.L. and A.F. designed the research framework, and directed the program of activities.
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Wilsmann, K.W., Ymai, L.H., Tonel, A.P. et al. Control of tunneling in an atomtronic switching device. Commun Phys 1, 91 (2018). https://doi.org/10.1038/s4200501800891
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