Control of tunneling in an atomtronic switching device

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 non-integrable regimes within the model. Through variation of the external field, we demonstrate how the system can be controlled between various “switched-on” and “switched-off” configurations. Atomtronics uses ultracold atoms to construct quantum analogues of electronic devices such as diodes and transistors. The authors report an atomtronic switching device by controlling boson tunnelling in a triple well system.

T he phenomenon of quantum tunneling is paramount in many studies of ultracold quantum gases. The two-well 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 self-trapping, 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 three-well 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 transistor-like structures beyond the electronic domain 12,13 .
Here we investigate the influence of integrability in the control of tunneling in a triple-well system. To do so, we must go beyond the familiar three-well 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 triple-well potential. The Hamiltonian has the general structure 19 The canonical creation and annihilation operators, a y i and a i , i = 1,2,3, represent the three bosonic degrees of freedom in the model, and N i ¼ a y i 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 on-site 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 on-site 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 inter-well couplings U ij to have greater magnitude than the on-site coupling U 0 . The experimental feasibility of this system for dipolar bosons was detailed by Lahaye et al. 19 , using a triple-well potential. The wells are aligned along the y-axis, separated by a distance l, with bosons polarized by a sufficiently large external field along the z-direction. 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 x-direction. (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 well-known 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 non-integrable 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 three-well tunneling models when U 13 = U 0 20 . In this case, we can write the Hamiltonian in the reduced 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 two-well subsystem containing only wells 1 and 3. Because Q admits the factorization Q ¼ Ω y Ω, 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 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 ½H; Q ¼ 2εJ 1 J 3 ða y 1 a 3 À a y 3 a 1 Þ is non-zero when the parameters ε, J 1 and J 3 are all non-zero.
Structure of energy levels. The integrable three-well system (2) possesses many features in common with the two-well Bose-Hubbard model, which is also integrable because the total number operator is conserved and there are only two degrees of freedom. Set J ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi . Following Leggett 2 , it is useful to define the regimes: where in the two-well 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 three-well 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 ffiffi ffi 2 p is adopted for simplicity. The Hamiltonian acts on the Fock space spanned by the normalized vectors 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 ) 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 time-average of 〈N 2 〉 decreases. Furthermore, the dynamics between wells 1 and 3 Fig. 1 Schematic representation of the system. With reference to the Hamiltonian H = H 0 + H 1 , the arrows J 1 and J 3 represent the tunneling couplings between the wells, U characterizes inter-well and intra-well interaction between bosons, while ε is the coupling strength for an external field  In this latter regime, Fig. 3c and d, these dynamical features can be understood by observing that the integrable Hamiltonian possesses a hidden two-well algebraic structure, with an effective well given by the combined source and drain. As is known 1,2,31 , the self-trapping regime is expected to occur in the two-well model in the Josephson regime. To be more precise, hN 2 i=N<ε when UN>J=ð2 ffiffiffiffiffiffiffiffiffiffiffi f ε Àε 2 p Þ if well 2 is initially empty. Thus, for UN ) J, we find hN 2 i=N ' 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 two-well system, by second-order processes [32][33][34] through the gate, such that hN 2 i ' 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 ) 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 time-average 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 switched-off.
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 U-dependent. When the initial state is |N,0,0〉, the oscillations between the source and drain are coherent, with tunneling to the gate switched-off. 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 switched-off.
Next, we maintain the system in the resonant tunneling regime UN ) J and study the non-integrable 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 non-integrable regime the effective Hamiltonian is given by For short time scales the dynamics exhibits Josephson-like oscillation 28 with frequency where Δn = 1/(1 + γ 2 ) is the amplitude and γ ¼ ðλðJ 2 1 À J 2 3 Þ À 2εÞ=2λJ 1 J 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 switched-off, 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 hn 3 i ¼ Δn sin 2 ðω J t=2Þ (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 triple-well system. This was conducted in both integrable and non-integrable 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 switch-off 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 switched-on configuration through to switched-off (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 multi-level logic applications and consequently new avenues in the design of atomtronic devices.
It is important, finally, to comment on the limitations of a three-mode Hamiltonian in the description of cold atom systems in a triple-well potential. Contributions from higher energy levels of the single-particle 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 double-well 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 three-body, and higher, interaction terms as corrections 37 to compensate for when the three-mode 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. Two-mode structure and the resonant tunneling regime. The integrable threewell model can be structured through two modes, as follows. From Eq. 2, we define J ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi J 2 1 þ J 2 3 p 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 y 1;3 ¼ J À1 ðJ 1 a y 1 þ J 3 a y 3 Þ satisfying the Heisenberg algebra ½N 1;3 ; a 1;3 ¼ Àa 1;3 ; ½N 1;3 ; a y 1;3 ¼ a y 1;3 ; ½a 1;3 ; a y 1;3 ¼ 1: Then such that the modes of wells 1 and 3 are now represented by the single mode "1,3".
The two-well model exhibits a self-trapping regime, with onset in the vicinity of χ UN=J ' 1 1,31 . This translates to a resonant tunneling regime for the triple-well model. Here we follow the approach of using semiclassical analysis 38 , such that this regime may be clearly identified. Using the usual number-phase correspondence, that is, a 2 ¼ e iθ 2 ffiffiffiffiffi ffi N 2 p , a 1;3 ¼ e iθ 1;3 ffiffiffiffiffiffiffiffi N 1;3 p and the conservation of boson number 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 χ ) jcos ϕj, 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 on-site 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 ¼ ðJ 1 a y 1 þ J 3 a y 3 Þa 2 þ h:c: is treated as a perturbation. For the isotropic case J 1 ¼ J 3 ¼ J= ffiffi ffi 2 p 19 , since n 2 ' 0 the interaction part is H I ' 0 and the wells 1 and 3 form an effective non-interacting two-well system coupled through well 2 by a second-order process 19,[32][33][34] with the effective Hamiltonian H eff ¼ J eff ða y 1 a 3 þ a y 3 a 1 Þ. Recall that the transition rate from initial state |s〉 to final state |k〉 is expressed where V (1) = V for first-order transition (Fermi's golden rule), δ is the delta function and for second-order transitions. Equating second-order 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 2 3 N 1 þ J 2 1 N 3 is constant for isotropic tunneling in the regime χ ) 1, then it does not affect the dynamics if we consider the linear combination J eff ða y 1 a 3 þ a y 3 a 1 Þ þ λ′ðJ 2 3 N 1 þ J 2 1 N 3 Þ. 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 n 1 = N 1 /N and φ = θ 1 − θ 3 . For initial condition n 1 = 1 and n 3 = 0, we have h ¼ ÀλJ 2 3 , 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 γ ¼ ½λðJ 2 1 À J 2 3 Þ À 2ε=2λJ 1 J 3 . Hamilton's equation gives Using the Ansatz n 1 ¼ 1 À Δn sin 2 ðηtÞ, 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 pair-tunneling between two wells 34 .
Physical realization. Here we discuss the feasibility of physical realization of the triple-well 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 triple-well potential aligned along the y-axis 19 . A transverse beam, with waist of 6 μm, provides xz-confinement. 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 long-range interactions, and weakly repulsive contact interactions to promote condensate stability 39 . The dipoles are oriented along the z-direction (Fig. 6). The transverse beam performs the function of the external field that controls the device. Its focus, when displaced along the y-axis by Δy, introduces the potential energy This generates a potential difference, resulting in the external field strength ε ¼ mω 2 y Δyl. 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 α $ 5:8, is achieved for ω x $ 2π 64 Hz, ω r $ 2π 220 Hz, where we assumed the Gaussian approximation for the ground state. The value of U obtained in units of J is U=J $ 7:5 10 À3 , which means that the resonant tunneling regime can be achieved for N ) 130 atoms. In principle, this condition can be satisfied experimentally 40 . As an example, for N $ 5000 atoms, with ω y $ 2π 1:5 Hz, translating the transverse laser by Δy = 1.8 μm we obtain ε=h $ 3:6 10 À2 Hz. It results in a tunneling amplitude Δn $ 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.
In both cases above, the parameter choices are such that higher-order 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. z x y y z x Fig. 6 Schematic representation of the trap geometry. Three parallel lasers (blue) are crossed by a transverse beam (green). The cigar-shapes, in red, represent a dipolar Bose-Einstein condensate trapped in a triple-well potential, and the green internal arrows depict the orientation of the dipoles