Long-range quantum tunneling via matter waves

Abstract

Quantum tunneling is a quantum phenomenon in which a microscopic object crosses through a potential barrier even if its energy cannot overcome the barrier. A general belief is that tunneling occurs only when the barrier width is comparable to, or smaller than the de Broglie’s wavelength of the object. Here, we study the tunneling of an ultracold atom among N far-separated trapping potentials in a state-selective optical lattice and present a mechanism to realize long-range tunneling. We find that, mediated by the propagating matter wave emitted from the atom, coherent tunneling of the atom to the remote lattices occurs as long as bound states are present in the energy spectrum of the system formed by the atom and its matter-wave. Going beyond the Markovian approximation, and breaking through the conventional distance constraint, our result opens another avenue to realizing tunneling and gives a guideline for developing tunneling devices.

Introduction

As one of the weirdest effects in the microscopic world, quantum tunneling is a phenomenon consisting of a subatomic particle passing through a potential barrier with a potential energy greater than the energy of the particle^{1,2,3,4,5,6,7,8}. It explains various radioactive decay of nuclei and how two nuclei overcome their mutual repulsion and fuse to generate huge energies^9,10. Quantum tunneling at metallic surfaces is the physical principle behind the operation of the scanning tunneling microscope^11,12,13. It also lays a foundation for nanotechnologies, such as the tunnel diode¹⁴ and resonant tunneling¹⁵, and some interdisciplinary sciences including quantum biology^16,17,18 and quantum chemistry^19,20. As a ubiquitous behavior of microscopic matters, such as electron²¹, proton²², nucleon⁹, photon^23,24, and superconducting Cooper pairs²⁵, quantum tunneling has been observed in many experiments^{20,26,27,28,29,30,31}.

In parallel with its successful applications, the question of how long a particle takes to tunnel through a barrier has remained contentious since the early days of quantum mechanics^{1,2,3,4,5,6,7,8}. The progress in ultracold-atom physics and attosecond science makes the direct measurement of the tunneling time possible^{32,33,34,35,36,37,38}. There is consensus on the fact that quantum tunneling is a consequence of the wave-particle duality and is more likely to occur when the width of the barrier is comparable to or smaller than the de Broglie wavelength of the particle. With the increase of the width of barriers, the tunneling rate experiences an exponential decrease^39,40. Recently, how to realize long-range tunneling has attracted much attention. For this purpose, the second-order quantum tunneling assisted by photons in the many-body lattice system^41,42, the virtual occupation of an intermediate site in the quantum-dot system⁴³, and the molecular bridge in the chemical reaction paradigm^44,45 have been proposed. Higher-order resonant tunneling achieved by the interplay between the atomic interactions and the tilted force of the lattice can also realize long-range tunneling up to five lattice sites^46,47. The chaos in a periodically driven lattice system assists the long-range tunneling too⁴⁸. However, a common necessity of the above works is the nearest-neighbor hopping of the particle, which is equal to the overlap integral of the wave functions in the two sites. It indicates that they still do not essentially go beyond the distance constraint of quantum tunneling.

Inspired by the experimental simulation of the atomic spontaneous emission by the matter-wave of an ultracold atom in a state-selective optical lattice, which is dubbed “quantum optics without photons”^49,50,51, we propose a scheme to realize long-range tunneling of the atom among the lattice sites. The separation between the lattice sites is so large that the spatial wave functions of the internal ground-state atom confined in the sites have a physically negligible overlap and thus direct tunneling cannot occur. By applying a laser, the atom jumps to the excited state and is converted into a propagating matter-wave. Mediated by this matter wave, long-range tunneling among the lattice sites occurs. It is interesting to find that such tunneling sensitively depends on the features of the energy spectrum of the total system formed by the laser-controlled atom and its matter-wave. As long as bound states out of the continuum (BOCs) or in the continuum (BICs) are present in the energy spectrum, a dynamically periodic tunneling, with frequencies equal to the differences of the energies of the bound states divided by ℏ, occurs among the lattice sites. This periodicity makes the tunneling time in our system well-defined and experimentally measurable. Breaking through the conventional distance constraint on quantum tunneling, our result is beneficial to designing quantum tunneling devices.

Results and discussion

Model

The ultracold-atom system offers us an ideal platform to simulate quantum optics and many-body physics^{52,53,54,55,56,57,58}. Inspired by the experiments of the emission of atomic matter waves^49,50,51, we investigate the tunneling dynamics of an ultracold atom in an optical lattice. Having two hyperfine states $\vert a\rangle$ and $\left\vert b\right\rangle$ and a mass m, the atom is confined in a state-selective optical lattice with N unoccupied sites⁵⁸. The lattice is embedded in an isolated tube, see Fig. 1. When the atomic internal state is $\left\vert a\right\rangle$, it is trapped in the spatial ground state ${\Phi }_{a}(z)={(\sqrt{\pi }\bar{z})}^{-1/2}\exp [-{z}^{2}/(2{\bar{z}}^{2})]$, with an eigenenergy $\tilde{\omega }/2$ and $\bar{z}=\sqrt{\hslash /(m\tilde{\omega })}$, by the lattice potential $V(z)=m{\tilde{\omega }}^{2}{z}^{2}/2$. Thus, the movement of the $\left\vert a\right\rangle$-state atom to other lattice sites is classically forbidden. Introducing an operator ${\hat{\sigma }}_{j}\equiv {\left\vert 0\right\rangle }_{j}\left\langle 1\right\vert$, with ${\left\vert 0\right\rangle }_{j}$ and ${\left\vert 1\right\rangle }_{j}$ denoting the unoccupied and occupied states of the atom in the j th lattice site, we rewrite the Hamiltonian of the $\left\vert a\right\rangle$-state atom ${\hat{H}}_{a}={\sum }_{j = 1}^{N}\hslash ({\omega }_{a}+\tilde{\omega }/2){\hat{\sigma }}_{j}^{{{\dagger}} }{\hat{\sigma }}_{j}$ with ω_a being the bare frequency of $\left\vert a\right\rangle$. On the contrary, when the atom is in the excited state $\left\vert b\right\rangle$ with a bare frequency ω_b, it is set free from the site and propagates as a matter wave ${\Phi }_{b,k}(z)={e}^{ikz}/\sqrt{L}$ along the tube, with L being the tube length and k = 2πn/L ($n\in {\mathbb{Z}}$). Without feeling the optical lattice, the $\left\vert b\right\rangle$-state atom also has no chance to move to other lattice sites. The Hamiltonian of the $\left\vert b\right\rangle$-state atom is ${\hat{H}}_{b}={\sum }_{k}\hslash [{\omega }_{b}+\hslash {k}^{2}/(2m)]{\hat{b}}_{k}^{{{\dagger}} }{\hat{b}}_{k}$, where ${\hat{b}}_{k}$ is the annihilation operator of the k th mode of the emitted matter-wave quanta. The states $\left\vert a\right\rangle$ and $\left\vert b\right\rangle$ are coupled by a microwave field such that the atom performs the Rabi oscillation described by ${\hat{H}}_{ab}=\frac{\hslash \Omega }{2}{\sum }_{j = 1}^{N}{\sum }_{k}({\gamma }_{jk}{e}^{i{\omega }_{L}t}{\hat{\sigma }}_{j}^{{{\dagger}} }{\hat{b}}_{k}+\,{\mbox{h.c.}})$, where Ω and ω_L are the strength and the frequency of the driving field, and ${\gamma }_{jk}=\int\,dz{\Phi }_{a}(z-{z}_{j}){\Phi }_{b,k}^{* }(z)$ is the Franck-Condon overlap between the trapped wave function at the j th site and the k th mode of the matter wave. The lattice sites are assumed to be so far separated that the overlap of the wave functions of the atom in different sites is much smaller than ∣γ_jk∣ and thus the direct tunneling among them is negligible. This endows a substantial difference in our scheme from the ones in refs. ^{41,42,43,44,45,46,47,48}, where the nearest-neighbor hopping is necessary. In a rotating frame ${\hat{H}}_{0}={\sum }_{j}\hslash ({\omega }_{b}-{\omega }_{L}){\hat{\sigma }}_{j}^{{{\dagger}} }{\hat{\sigma }}_{j}+{\sum }_{k}\hslash {\omega }_{b}{\hat{b}}_{k}^{{{\dagger}} }{\hat{b}}_{k}$, the total Hamiltonian becomes⁵⁹

$$\frac{\hat{H}}{\hslash }={\omega }_{0}\mathop{\sum }_{j=1}^{N}{\hat{\sigma }}_{j}^{{{\dagger}} }{\hat{\sigma }}_{j}+\mathop{\sum}_{k}[{\omega }_{k}{\hat{b}}_{k}^{{{\dagger}} }{\hat{b}}_{k}+{\sum}_{j=1}^{N}({g}_{jk}{\hat{\sigma }}_{j}^{{{\dagger}} }{\hat{b}}_{k}+\,{\mbox{h.c.}})],$$

(1)

where ${\omega }_{0}={\omega }_{a}+\frac{\tilde{\omega }}{2}-{\omega }_{b}+{\omega }_{L}$, ${\omega }_{k}=\frac{\hslash {k}^{2}}{2m}$, and ${g}_{jk}=\frac{\Omega {\gamma }_{jk}}{2}$. Having a rotating-wave approximate form, Eq. (1) guarantees the number of the atom always to be one. It rules out the nonphysical scenarios that a propagating $\left\vert b\right\rangle$-state atom and a trapped $\left\vert a\right\rangle$-state atom are created and annihilated simultaneously, which, respectively, are described by ${\hat{\sigma }}_{j}^{{{\dagger}} }{\hat{b}}_{k}^{{{\dagger}} }$ and ${\hat{\sigma }}_{j}{\hat{b}}_{k}$. The single-site case of Eq. (1) has been experimentally realized in refs. ^49,50. It is noted that Eq. (1) takes the same form as the Tavis–Cummings model with a multimode boson field⁶⁰. Here, the N sites of the optical lattice play the role of N identical two-level systems, each being separately coupled with the field.

Fig. 1: Scheme of the system.

An ultracold atom is confined in a state-selective optical lattice with different sites labeled by z_j (j = 1, ⋯, N) wrapped by an isolated tube. When it is in the ground state $\left\vert a\right\rangle$, the atom is trapped in the sites. When it is in the excited state $\left\vert b\right\rangle$, it propagates in the tube as a matter wave. The two states are coupled by a laser. The separation between the sites is so large that atomic direct tunneling among the sites is impossible.

Tunneling dynamics

Considering the atom is initially in the first lattice site, we have $\vert \Psi (0)\rangle =\vert {a}_{1},\{{0}_{k}\}\rangle$. Its evolved state can be expanded as $\vert \Psi (t)\rangle ={\sum }_{j = 1}^{N}{c}_{j}(t)\vert {a}_{j},\{{0}_{k}\}\rangle +{\sum }_{k}{d}_{k}(t)\vert b,{1}_{k}\rangle$, where $\vert {a}_{j},\{{0}_{k}\}\rangle$ denotes that the atom in the state $\left\vert a\right\rangle$ is confined in the j th lattice site and no wave-matter quanta is triggered in the isolated tube and $\left\vert b,{1}_{k}\right\rangle$ denotes that the atom in the state $\left\vert b\right\rangle$ propagates in the isolated tube as a matter-wave quanta in the k th mode. From the Schrödinger equation, we derive the equation of motion satisfied by the column vector ${{{\bf{c}}}}(t)={({c}_{1}(t)\cdots {c}_{N}(t))}^{{{{\rm{T}}}}}$ formed by the probability amplitudes of the atom in the N sites as

$$\dot{{{{\bf{c}}}}}(t)+i{\omega }_{0}{{{\bf{c}}}}(t)+\int_{\! 0}^{t}d\tau {{{\bf{f}}}}(t-\tau ){{{\bf{c}}}}(\tau )=0.$$

(2)

Here ${{{\bf{f}}}}(t-\tau )=\int_{0}^{\infty }d\omega {e}^{-i\omega (t-\tau )}{{{\bf{J}}}}(\omega )$ is a N-by-N correlation-function matrix of the matter wave and the element of the matrix J(ω) is defined as ${J}_{jl}(\omega )={\sum }_{k}{g}_{jk}^{* }{g}_{lk}\delta (\omega -{\omega }_{k})$ characterizing the correlated spectral density of the atom in the j th and l th lattice sites. Such a correlation indicates that, although direct tunneling of the atom among different lattice sites is absent, the mediated tunneling via the $\left\vert b\right\rangle$-state matter wave is triggered by the Rabi oscillation. One can readily derive

$${J}_{jl}(\omega )={J}_{| j-l| }(\omega )=\frac{{\Omega }^{2}{e}^{\frac{-2\omega }{\tilde{\omega }}}}{\sqrt{8\pi \tilde{\omega }\omega }}\cos \left[{\left(\frac{2\omega }{\tilde{\omega }}\right)}^{\frac{1}{2}}\frac{{z}_{j}-{z}_{l}}{\bar{z}}\right].$$

(3)

Reflecting the memory effect, the convolution in Eq. (2) renders the tunneling dynamics non-Markovian. Its dominant role in the dynamics has been observed^49,51.

In the special case of the small-Ω limit, we can make the Markovian approximation to Eq. (2) by replacing c(τ) with c(t) and extending the upper limit of the time integral to infinity^61,62. This approximation is also applicable when the spectral density reduces to a constant. The obtained solution is

$${{{{\bf{c}}}}}_{{{{\rm{MA}}}}}(t)=\exp [-({{{\boldsymbol{\kappa }}}}+i{\omega }_{0}+i{{{\mathbf{\Delta }}}})t]{{{\bf{c}}}}(0),$$

(4)

where κ = πJ(ω₀) and ${{{\mathbf{\Delta }}}}={{{\mathcal{P}}}}\int\frac{{{{\bf{J}}}}(\omega )d\omega }{{\omega }_{0}-\omega }$, with ${{{\mathcal{P}}}}$ being the Cauchy’s principal value. We see that ∣c_MA(t)∣² exponentially decays to zero due to the positivity of κ. Thus, the atom asymptotically relaxes to the tube as a matter wave and no tunneling to other lattice sites occurs in the long-time condition of the Markovian dynamics. An exception occurs when ω₀ < 0, where κ = 0 but Δ ≠ 0, and thus a lossless tunneling can happen via exchanging the virtual matter waves⁶³.

The strong driving invalidates the Markovian approximation. In the general non-Markovian case, Eq. (2) can only be solved numerically. However, its asymptotic form is solvable by the Laplace transform method. It converts Eq. (2) into $[s+i{\omega }_{0}+\tilde{{{{\bf{f}}}}}(s)]\tilde{{{{\bf{c}}}}}(s)={{{\bf{c}}}}(0)$, where $\tilde{{{{\bf{f}}}}}(s)=\int_{0}^{\infty }d\omega \frac{{{{\bf{J}}}}(\omega )}{s+i\omega }$. Using the Jordan decomposition ${{{\bf{D}}}}(s)={{{{\bf{V}}}}}_{s}^{-1}\tilde{{{{\bf{f}}}}}(s){{{{\bf{V}}}}}_{s}={\mbox{diag}}\,[{D}_{1}(s),\cdots \,,{D}_{N}(s)]$, we obtain $\tilde{{{{\bf{c}}}}}(s)={{{{\bf{V}}}}}_{s}{\left[s+i{\omega }_{0}+{{{\bf{D}}}}(s)\right]}^{-1}{{{{\bf{V}}}}}_{s}^{-1}{{{\bf{c}}}}(0)$. c(t) is calculated by making the inverse Laplace transform $\tilde{{{{\bf{c}}}}}(s)$, which needs finding the poles of $\tilde{{{{\bf{c}}}}}(s)$ from (ϖ = is)

$${Y}_{j}(\varpi )\equiv {\omega }_{0}-i{D}_{j}(-i\varpi )=\varpi ,\,(j=1,\cdots \,,N).$$

(5)

It is found that the root ϖ multiplied by ℏ is just the eigenenergy of Eq. (1). It indicates that the atomic tunneling dynamics described by c(t) is intrinsically determined by the features of the energy spectrum of the total system formed by the confined atom and its propagating matter-wave. We find that Eq. (5) has three types of roots. First, Y_j(ϖ) is ill-defined in the region of ϖ > 0 due to the poles in D_j(−iϖ), and thus Eq. (5) has an infinite number of roots, which form a continuous energy band. Second, because Y_j(ϖ) is a decreasing function in the region out of the continuum, i.e., ϖ < 0, Eq. (5) has an isolated root ${\varpi }_{j}^{{{{\rm{boc}}}}}$ provided Y_j(0) < 0. The eigenstate corresponding to $\hslash {\varpi }_{j}^{{{{\rm{boc}}}}}$ is called a BOC. Third, a removable singularity ${\varpi }_{j}^{\,{{\mathrm{bic}}}\,}$ exists for D_j(−iϖ) when ${\omega }_{0}={\varpi }_{j}^{\,{{\mathrm{bic}}}}+i{D}_{j}(-i{\varpi }_{j}^{{{\mathrm{bic}}}\,})$ also fulfills Eq. (5). The eigenstate corresponding to $\hslash {\varpi }_{j}^{\,{{\mathrm{bic}}}\,}$ is called a BIC. The formation of the BOC and BIC has profound consequences on the non-Markovian dynamics^{64,65,66,67,68}. According to the Cauchy’s residue theorem and contour integration, we have⁶⁹

$${{{\bf{c}}}}(t)={{{\bf{Z}}}}(t)+\int_{0}^{\infty }\frac{d\varpi }{2\pi }[\tilde{{{{\bf{c}}}}}({0}^{+}-i\varpi )-\tilde{{{{\bf{c}}}}}({0}^{-}-i\varpi )]{e}^{-i\varpi t},$$

(6)

where ${{{\bf{Z}}}}(t)={\sum} _{\alpha = {{\mathrm{bic,boc}}}}{\sum }_{j = 1}^{N}$Res$[\tilde{{{{\bf{c}}}}}(-i{\varpi }_{j}^{\alpha })]{e}^{-i{\varpi }_{j}^{\alpha }t}$, with $\,{{\mbox{Res}}}\,[\tilde{{{{\bf{c}}}}}(-i{\varpi }_{j}^{\alpha })]$ is the residue contributed by the j th bound state obtained from Y_j(ϖ) = ϖ, and the second term comes from the continuous energy band. Containing an infinite number of superposition components oscillating with time in continuously changing frequencies ϖ, the second term tends to zero in the long-time limit due to the out-of-phase interference. Thus, if the bound state is absent, then ${\lim }_{t\to \infty }{{{\bf{c}}}}(t)={{{\bf{0}}}}$ characterizes a complete relaxation of the atom in the isolated tube, while if the bound states are formed, then ${\lim }_{t\to \infty }{{{\bf{c}}}}(t)={{{\bf{Z}}}}(t)$ implies a stable tunneling of the atom among the lattice sites. The result reveals that we can achieve a persistent matter-wave mediated long-range tunneling of the atom among the optical lattice by manipulating the formation of the bound states even when the lattice sites are so separated that direct tunneling cannot happen. It inspires that, parallel to the rapidly developing electromagnetic-wave-based waveguide QED⁷⁰, our matter-wave-based waveguide supplies another realization of the efficient interconnect among well-separated spatial nodes⁷¹. Note that, besides the real-valued eigenenergies of the BIC and BOC, Eq. (5) also has complex-value eigenvalues. They can be found by analytically extending the function to the second Riemann sheet⁷². These complex-valued eigenenergies contribute unstable oscillations and thus do not survive in the long-time tunneling dynamics.

First, we assume that the lattice has two sites, i.e., N = 2. It is easy to derive ${D}_{1/2}(s)={\tilde{f}}_{0}(s)\pm {\tilde{f}}_{1}(s)$ and ${{{{\bf{V}}}}}_{s}=\left(\begin{array}{cc}1&1\\ 1&-1\end{array}\right)$. We thus have $\tilde{{{{\bf{c}}}}}(s)={{{\bf{M}}}}{[s+i{\omega }_{0}+{{{\bf{D}}}}(s)]}^{-1}{\left(\begin{array}{cc}1&1\end{array}\right)}^{{{{\rm{T}}}}}$, where ${{{\bf{M}}}}=\left(\begin{array}{cc}1/2&1/2\\ 1/2&-1/2\end{array}\right)$. If the bound states are present, then

$${c}_{j}(\infty )=\mathop{\sum}_{\alpha \in \,{{\mathrm{bic,boc}}}\,}{\sum }_{l=1}^{2}{M}_{jl}{Z}_{l}^{\alpha }{e}^{-i{\varpi }_{l}^{\alpha }t},$$

(7)

where ${Z}_{l}^{\alpha }={[1+{\partial }_{s}{D}_{l}(s)]}^{-1}{| }_{s\to -i{\varpi }_{l}^{\alpha }}$. The l th BOC is formed when Y_l(0) < 0. The divergence of D_1/2(s) is removed by the BIC frequency determined ${J}_{0}({\omega }_{1/2}^{\,{{\mathrm{bic}}}})\pm {J}_{1}({\omega }_{1/2}^{{{\mathrm{bic}}}\,})=0$ as

$${\varpi }_{1/2}^{\,{{\mathrm{bic}}}\,}=\tilde{\omega }{(\bar{z}{n}_{1/2}\pi /d)}^{2}/2,$$

(8)

with d = ∣z₁ − z₂∣, n₁ and n₂ being odd and even numbers, under the condition ${\omega }_{0}={\varpi }_{l}^{\,{{\mathrm{bic}}}}+i{D}_{l}(-i{\varpi }_{l}^{{{\mathrm{bic}}}\,})$. Equation (7) clearly shows that, in sharp contrast to complete relaxation to the isolated tube as a matter wave under the Markovian approximation in Eq. (4), a stable distribution and a persistently oscillating tunneling of the atom between the two sites occur when one and two bound states are formed, respectively. Such tunneling mediated by the matter wave has not been reported before.

We plot in Fig. 2a the energy spectrum of the system. It is found that two branches of BOCs are present and one BIC is formed at ${\omega }_{0}=0.18\tilde{\omega }$. The evolution of the modulus of the atomic probability amplitudes in the two lattice sites in Fig. 2b, c shows that the atom completely relaxes into the isolated tube when no bound state is formed in the regime ${\omega }_{0} > 0.25\tilde{\omega }$, which has no difference from the Markovian result in Eq. (4). In the regime ${\omega }_{0}\in (0.058,0.25)\tilde{\omega }$, one BOC is formed and thus both ∣c₁(t)∣ and ∣c₂(t)∣ tend to equal constants, see the orange squares in Fig. 2d, e. This corresponds to a stable distribution of the atom in the two lattice sites. An exception occurs when ${\omega }_{0}=0.18\tilde{\omega }$, where a BIC and a BOC coexist, and thus the atom experiences a lossless oscillation between the two lattice sites in a frequency $| {\varpi }_{1}^{{{{\rm{boc}}}}}-{\varpi }_{2}^{\,{{\mathrm{bic}}}\,}|$, see the green rhombuses in Fig. 2d, e. Such a lossless oscillation between the two lattice sites can also occur in a frequency $| {\varpi }_{1}^{{{{\rm{boc}}}}}-{\varpi }_{2}^{{{{\rm{boc}}}}}|$ when two BOCs are present in the regime ${\omega }_{0} < 0.058\tilde{\omega }$, see the blue circles in Fig. 2d, e. The matching of the numerical results with the analytic ones in Eq. (7) for the three typical parameter regimes confirms the dominant role of the bound states in the long-time dynamics. This lossless oscillation characterizes coherent tunneling of the atom between the two sites mediated by the propagating matter-wave. Thus, we achieve long-range coherent tunneling even when the sites are so separated that the spatial wave functions of the atom have a negligible overlap.

Fig. 2: Energy spectrum and dynamics for different values of atomic frequency detuning in the case of two lattice sites.

a Energy spectrum and evolution of the absolute values of the probability amplitudes (b) ∣c₁(t)∣ in the first site and (c) ∣c₂(t)∣ in the second site for different values of atomic frequency detuning ω₀. The red dot in a  marks the energy of the bound state in the continuum. T in b and c defined as $2\pi /| {\varpi }_{1}^{{{{\rm{boc}}}}}-{\varpi }_{2}^{\,{{\rm{boc/bic}}}\,}|$ is the oscillation period in the steady state. Evolution of (d) ∣c₁(t)∣ and (e) ∣c₂(t)∣ when ${\omega }_{0}=-0.02\tilde{\omega }$ (blue dots), $0.06\tilde{\omega }$ (orange squares), and $0.18\tilde{\omega }$ (green rhombuses). Their long-time behaviors evaluated from Eq. (7) are shown by the solid lines in the same colors. We use $d=5\bar{z}$, $\Omega =0.13\tilde{\omega }$, $\bar{z}=0.065$ nm, and $\tilde{\omega }=2\pi \times 40$ kHz.

Figure 3a shows the energy spectrum of the system in different site separations d. Two branches of BOCs separate the spectrum into two regimes: one BOC when $d\le 5\bar{z}$ and two BOCs when $d \, > \, 5\bar{z}$. A BIC is formed at $d=8.67\bar{z}$ and $17.36\bar{z}$, respectively. The evolution of ∣c_j(t)∣ in Fig. 3b, c indicates that ∣c₁(t)∣ and ∣c₂(t)∣ evolve to an equal constant in the single-BOC regime and experience a lossless oscillation with a common period $T=2\pi /| {\varpi }_{1}^{{{{\rm{boc}}}}}-{\varpi }_{2}^{{{{\rm{boc}}}}}|$ and a π/2 phase difference in the two-BOC regime. The latter reveals that the atom coherently goes back and forth between the two sites. With increasing d, ${\varpi }_{1}^{{{{\rm{boc}}}}}$ tends closer and closer to ${\varpi }_{2}^{{{{\rm{boc}}}}}$, and thus the period T for the atom to finish one cyclic tunneling between the two sites becomes longer and longer. At the two distances in the presence of the BIC, ∣c_j(t)∣ behaves as a periodic oscillation with three frequencies, i.e., $| {\varpi }_{1}^{{{{\rm{boc}}}}}-{\varpi }_{2}^{{{{\rm{boc}}}}}|$ and $| {\varpi }_{j}^{{{{\rm{boc}}}}}-{\varpi }^{{{{\rm{bic}}}}}|$ (j = 1, 2). The phase diagram for forming different numbers of bound states in the d-ω₀ plane is shown in Fig. 3d. The formed zero, one, and two BOCs separate the diagram into three parts. Several discrete curves supporting the formation of the BIC in different n_l are added above the diagram. It gives a global picture of the features of atomic tunneling. If no bound state is present, then the atom relaxes as a matter wave in the tube. If one bound state is formed, then it tends to a stable equal probability distribution in the two sites. If two or three bound states are formed, then the atom periodically and losslessly oscillates between the two sites in frequencies proportional to the eigenenergy difference of any pair of the bound states.

Fig. 3: Energy spectrum and dynamics for different values of the site separations in the case of two lattice sites.

a Energy spectrum and evolution of the absolute values of the probability amplitudes (b) ∣c₁(t)∣ in the first site and (c) ∣c₂(t)∣ in the second site for different values of the lattice-site separation d when ${\omega }_{0}=0.05\tilde{\omega }$. Red dots in a mark the energies of the bound states in the continuum. d Phase diagram for forming different numbers of bound states out of the continuum denoted by ${{{{\mathcal{N}}}}}_{{{{\rm{boc}}}}}$ in the d − ω₀ plane. The solid lines evaluated from Eq. (8) for different n_l in c mark the positions forming the bound states in the continuum. Other parameters are the same as in Fig. 2.

Our result can be generalized to the multiple-site case. When N = 3, we have ${D}_{1}(s)={\tilde{f}}_{0}-{\tilde{f}}_{2}$, ${D}_{2/3}(s)={\tilde{f}}_{0}+\frac{1}{2}({\tilde{f}}_{2}\mp \tilde{e})$, and ${{{{\bf{V}}}}}_{s}=\left(\begin{array}{ccc}-1&1&1\\ 0&\frac{{\tilde{f}}_{1}(\tilde{e}-3{\tilde{f}}_{2})}{{\tilde{f}}_{2}\tilde{e}-2{\tilde{f}}_{1}^{2}-{\tilde{f}}_{2}^{2}}&\frac{{\tilde{f}}_{1}(\tilde{e}+3{\tilde{f}}_{2})}{{\tilde{f}}_{2}\tilde{e}+2{\tilde{f}}_{1}^{2}+{\tilde{f}}_{2}^{2}}\\ 1&1&1\end{array}\right)$, where the argument s of ${\tilde{f}}_{j}(s)$ has been omitted for brevity and $\tilde{e}={(8{\tilde{f}}_{1}^{2}+{\tilde{f}}_{2}^{2})}^{\frac{1}{2}}$. It is obtained that $\tilde{{{{\bf{c}}}}}(s)={{{\bf{M}}}}{[s+i{\omega }_{0}+{{{\bf{D}}}}(s)]}^{-1}{\left(\begin{array}{ccc}1&1&1\end{array}\right)}^{{{{\rm{T}}}}}$, where ${{{\bf{M}}}}=\left(\begin{array}{ccc}1/2&\frac{1-{\tilde{f}}_{2}/\tilde{e}}{4}&\frac{1+{\tilde{f}}_{2}/\tilde{e}}{4}\\ 0&-{\tilde{f}}_{1}/\tilde{e}&{\tilde{f}}_{1}/\tilde{e}\\ -1/2&\frac{1-{\tilde{f}}_{2}/\tilde{e}}{4}&\frac{1+{\tilde{f}}_{2}/\tilde{e}}{4}\end{array}\right)$. The BIC for D₁(s) is formed at ${\varpi }_{1}^{\,{{\mathrm{bic}}}\,}=\tilde{\omega }{(\bar{z}n\pi /d)}^{2}/8$, with n being an even number. According to the residue theorem, we have

$${c}_{j}(\infty )={\mathop{\sum}_{\alpha =\,{{{\mathrm{bic,boc}}}}}\,}{{\sum}_{l=1}^{3}}\left.{\frac{{M}_{jl}{e}^{st}}{1+{\partial }_{s}{D}_{l}(s)}}\right |_{{s} \to -{i}{{\varpi }_{l}^{\alpha }}}.$$

(9)

The energy spectrum in Fig. 4a shows that three BOCs and one BIC at most are formed. Without the bound state, the atom relaxes as a matter wave in the tube and no occupation in the sites survives, see Fig. 4b. When three BOCs are present, ∣c₁(t)∣ and ∣c₂(t)∣ tend to a lossless oscillation in three frequencies $| {\varpi }_{i}^{{{{\rm{boc}}}}}-{\varpi }_{j}^{{{{\rm{boc}}}}}|$ (i, j = 1, 2, 3), and ∣c₂(t)∣ behaves as an oscillation in a frequency $| {\varpi }_{2}^{{{{\rm{boc}}}}}-{\varpi }_{3}^{{{{\rm{boc}}}}}|$, see Fig. 4c. They match our analytical result (9) and verify the distinguished bound-state role in the tunneling dynamics. The result confirms again that we can realize long-range tunneling among the lattice sites by the mediation of the matter wave.

Fig. 4: Energy spectrum and dynamics for different values of the atomic frequency detuning in the case of three lattice sites.

a Energy spectrum and evolution of the absolute values of the probability amplitudes ∣c₁(t)∣ (blue dots), ∣c₂(t)∣ (orange squares), and ∣c₃(t)∣ (green rhombuses) when the atomic frequency detuning ${\omega }_{0}=0.4\tilde{\omega }$ in b and $-0.05\tilde{\omega }$ in c for the atom tunneling among three lattice sites. The red dot in a marks the energy of the bound state in the continuum. The long-time behaviors evaluated from Eq. (31) are shown by the solid lines in the same colors in c. We use $d=5\bar{z}$ and $\Omega =0.13\tilde{\omega }$.

Conclusions

We have studied the tunneling dynamics of an ultracold atom in a state-selective optical lattice embedded in an isolated tube. A mechanism to realize long-range tunneling of the atom among the lattice sites via the mediation of its propagating matter wave is found. In contrast to one’s general belief that tunneling occurs when the atomic wavelength exceeds the width of the confined potential, our result reveals an alternative tunneling by converting the atom into a matter wave. We find that such tunneling can be controlled by engineering the features of the energy spectrum of the system. When zero, one, and more BOCs or BICs are present in the energy spectrum, the tunneling exhibits complete relaxation in the tube, stable distribution, and lossless oscillation among the lattice sites, respectively. The observation of the bound-state effect in the single-lattice-site case^49,51 gives support to realize our scheme. Supplying insightful instruction to realize controllable long-range tunneling, our result is helpful not only to simulating many-body physics with long-range orders^58,73, but also to designing quantum-tunneling⁷⁴ and -interconnect^71,75 devices. In parallel to cavity^76,77, circuit^78,79, and waveguide QED systems^80,81, our matter-wave setup supplies a perfect realization of quantum interconnect among different quantum nodes. The matter-wave version is advantageous over the photonic platforms because there is no loss for the matter waves.

Methods

Derivation of dynamical evolution

The initial state of the total system is $\vert \Psi (0)\rangle =\vert {a}_{1},\{{0}_{k}\}\rangle$. Its evolved state is expanded as $\Psi (t)\rangle ={\sum }_{j = 1}^{N}{c}_{j}(t)\vert {a}_{j},\{{0}_{k}\}\rangle +{\sum }_{k}{d}_{k}(t)\vert b,{1}_{k}\rangle$. From the Schrödinger equation, we derive

$$i{\dot{c}}_{j}(t)={\omega }_{0}{c}_{j}(t)+{\sum}_{k}{g}_{jk}{d}_{k}(t),$$

(10)

$$i{\dot{d}}_{k}(t)={\omega }_{k}{d}_{k}(t)+{\sum }_{l=1}^{N}{g}_{lk}^{* }{c}_{l}(t).$$

(11)

Substituting the solution of Eq. (11) ${d}_{k}(t)=-i{\sum }_{l}\int_{0}^{t}d\tau {g}_{lk}^{* }{e}^{-i{\omega }_{k}(t-\tau )}{c}_{l}(\tau )$ under the initial condition d_k(0) = 0 into Eq. (10), we have

$${\dot{c}}_{j}(t)+i{\omega }_{0}{c}_{j}(t)+\mathop{\sum}_{l=1}^{N}\int_{ 0}^{t}d\tau {f}_{jl}(t-\tau ){c}_{l}(\tau )=0,$$

(12)

where ${f}_{jl}(t-\tau )=\int_{0}^{\infty }d\omega {J}_{jl}(\omega ){e}^{-i\omega (t-\tau )}$ is the correlation function of the matter wave and ${J}_{jl}(\omega )={\sum }_{k}{g}_{jk}{g}_{lk}^{* }\delta (\omega -{\omega }_{k})$ is the correlated spectral density of the atom between the j th and l th lattice sites. Remembering the form of g_jk and γ_jk and making the continuum limit of k under $L\gg \bar{z}$, we obtain

$${rcl}{J}_{jl}(\omega ) =\frac{{\Omega }^{2}\sqrt{\pi }\bar{z}}{2L}{\sum}_{k}{e}^{-{\bar{z}}^{2}{k}^{2}-ik({z}_{l}-{z}_{j})}\delta (\omega -{\omega }_{k})\\ =\frac{{\Omega }^{2}{e}^{\frac{-2\omega }{\tilde{\omega }}}}{\sqrt{8\pi \tilde{\omega }\omega }}\cos \left[{\left(\frac{2\omega }{\tilde{\omega }}\right)}^{\frac{1}{2}}\frac{{z}_{j}-{z}_{l}}{\bar{z}}\right].\hfill$$

(13)

We see that J₁₁(ω) = ⋯ = J_NN(ω) and J_jl(ω) = J_lj(ω). Thus, J_jl satisfies the property J_jl(ω) = J_∣j−l∣(ω). Equation (12) is rewritten as a column-vector form

$$\dot{{{{\bf{c}}}}}(t)+i{\omega }_{0}{{{\bf{c}}}}(t)+\int_{ 0}^{t}d\tau {{{\bf{f}}}}(t-\tau ){{{\bf{c}}}}(\tau )=0,$$

(14)

where ${{{\bf{c}}}}(t)={({c}_{1}(t)\cdots {c}_{N}(t))}^{{{{\rm{T}}}}}$, ${{{\bf{f}}}}(t-\tau )=\int_{0}^{\infty }d\omega {e}^{-i\omega (t-\tau )}{{{\bf{J}}}}(\omega )$ is an N-by-N matrix with its elements being the correlation function f_jl(t − τ), and J(ω) is an N-by-N matrix with its elements being the correlated spectral density J_jl(ω). Via a Laplace transform $\tilde{{{{\bf{c}}}}}(s)\equiv \int_{0}^{\infty }{e}^{-st}{{{\bf{c}}}}(t)dt$, Eq. (14) is converted into $[s+i{\omega }_{0}+\tilde{{{{\bf{f}}}}}(s)]\tilde{{{{\bf{c}}}}}(s)={{{\bf{c}}}}(0)$, where $\tilde{{{{\bf{f}}}}}(s)=\int_{0}^{\infty }d\omega \frac{{{{\bf{J}}}}(\omega )}{s+i\omega }$ is the Laplace transform of f(t − τ). Using the Jordan decomposition of $\tilde{{{{\bf{f}}}}}(s)$ as

$${{{\bf{D}}}}(s)={{{{\bf{V}}}}}_{s}^{-1}\tilde{{{{\bf{f}}}}}(s){{{{\bf{V}}}}}_{s}=\,{\mbox{diag}}\,[{D}_{1}(s),\cdots \,,{D}_{N}(s)],$$

(15)

we have

$$\begin{array}{rcl}\tilde{{{{\bf{c}}}}}(s)&=&{{{{\bf{V}}}}}_{s}{[s+i{\omega }_{0}+{{{\bf{D}}}}(s)]}^{-1}{{{{\bf{V}}}}}_{s}^{-1}{{{\bf{c}}}}(0)\hfill\\ &=&{{{\bf{M}}}}{[s+i{\omega }_{0}+{{{\bf{D}}}}(s)]}^{-1}{\left(\begin{array}{ccc}1&\cdots &1\end{array}\right)}^{{{{\rm{T}}}}}.\end{array}$$

(16)

The inverse Laplace transform reads ${{{\bf{c}}}}(t)=\int_{\gamma -i\infty }^{\gamma +i\infty }\frac{{e}^{st}}{2\pi i}\tilde{{{{\bf{c}}}}}(s)$, where γ is chosen such that all the poles are included. According to the contour integration, see Fig. 5, the evaluation of the inverse Laplace transform needs finding the poles of  $\tilde{{{{\bf{c}}}}}(s)$ from

$$s+i{\omega }_{0}+{D}_{j}(s)=0.$$

(17)

In general, Eq. (17) has three types of poles determined by D_j(s), see Fig. 5. The first one is a continuum of poles along the negative imaginary axis of the complex plane formed by the real and imaginary parts of s, which forms a branch cut. The second one is several discrete roots formed by the removable singularities of D_j(s) along the negative imaginary axis, which is called the BIC. The third one is the isolated poles along the positive imaginary axis, which is called the BOC.

Fig. 5: Contour integration to evaluate the inverse Laplace transform.

Path of the contour integration in the complex plane Re(s) + iIm(s) for the calculation of the inverse Laplace transform $\tilde{{{{\bf{c}}}}}(s)$. The radii C_R tends to infinity and C_ϵ tends to zero. The continuous poles form a branch cut wrapped by two inverse paths B_±. The isolated poles form the bound states out of the continuum with eigenenergies ${E}_{j}^{{{{\rm{boc}}}}}$ in the positive-Im(s) axis and the bound states in the continuum with eigenenergies ${E}_{j}^{\,{\mbox{bic}}\,}$ in the negative-Im(s) axis.

According to the Cauchy’s residue theorem and contour integration⁶⁹, we have

$${{{\bf{c}}}}(t)={{{\bf{Z}}}}(t)+\left[{\int}_{{\! \! \! \! \! B}_{+}}\frac{d\varpi }{2\pi }+{\int}_{{\! \! \! \! \! B}_{-}}\frac{d\varpi }{2\pi }\right]\tilde{{{{\bf{c}}}}}(-i\varpi ){e}^{-i\varpi t},$$

(18)

where ${{{\bf{Z}}}}(t)={\sum }_{\alpha = {{\mathrm{bic,boc}}}}{\sum }_{j = 1}^{N}$Res$[\tilde{{{{\bf{c}}}}}(-i{\varpi }_{j}^{\alpha })]{e}^{-i{\varpi }_{j}^{\alpha }t}$, with $\,{\mbox{Res}}\,[\tilde{{{{\bf{c}}}}}(-i{\varpi }_{j}^{\alpha })]$ being the residue contributed by the j th bound state obtained from Eq. (17) with eigenenergy ${E}_{j}^{\alpha }=\hslash {\varpi }_{j}^{\alpha }$, and the second term comes from the branch cut of the contour. Oscillating with time in continuously changing frequencies, the second term tends to zero in the long-time limit due to the out-of-phase interference. Thus, if the bound state is absent, then ${\lim }_{t\to \infty }{{{\bf{c}}}}(t)={{{\bf{0}}}}$ characterizes a complete relaxation of the atom in the isolated tube, while if the bound states are formed, then ${\lim }_{t\to \infty }{{{\bf{c}}}}(t)={{{\bf{Z}}}}(t)$ implies either a stable distribution or a lossless oscillation of the atom among the optical lattice. Both of the results are absent in the Born-Markovian approximation.

Derivation of the energy spectrum

The eigenstate of Eq. (1) is expanded as $\left\vert \Phi \right\rangle ={\sum }_{j = 1}^{N}{x}_{j}\left\vert {a}_{j},\{{0}_{k}\}\right\rangle +{\sum }_{k}{y}_{k}\left\vert b,{1}_{k}\right\rangle$. From the eigen equation $\hat{H}\left\vert \Phi \right\rangle =E\left\vert \Phi \right\rangle$, we obtain

$$(E/\hslash -\omega ){x}_{j}=\mathop{\sum}_{k}{g}_{jk}{y}_{k},$$

(19)

$$(E/\hslash -\omega ){y}_{k}=\mathop{\sum}_{l=1}^{N}{g}_{lk}^{* }{x}_{l}.$$

(20)

The substitution of the solution of Eq. (20), i.e., ${y}_{k}={\sum }_{l = 1}^{N}{g}_{lk}^{* }{x}_{l}/(E/\hslash -\omega )$, into Eq. (19) leads to $(E/\hslash -\omega ){x}_{j}=-i{\sum }_{l = 1}^{N}{\tilde{f}}_{jl}(-iE/\hslash ){x}_{l}$, which can be tightly rewritten in a matrix form as

$$[E/\hslash -{\omega }_{0}+i\tilde{{{{\bf{f}}}}}(-iE/\hslash )]{{{\bf{x}}}}=0,$$

(21)

where ${{{\bf{x}}}}={({x}_{1}\cdots {x}_{N})}^{{{{\rm{T}}}}}$. The Jordan decomposition in Eq. (15) readily converts Eq. (21) into

$$E/\hslash -{\omega }_{0}+i{D}_{j}(-iE/\hslash )=0,$$

(22)

which determines the eigenenergies E of the system.

It is interesting to find that the pole equation (17) determining the tunneling dynamics is just the eigenenergy equation (22) after making the replacement of E/ℏ = is. Just for this reason, we call the eigenstates corresponding to the removable singularities of D_j(s) the BICs and the ones corresponding to the isolated poles the BOCs. This result reveals that the tunneling dynamics of the atom is essentially governed by the feature of the energy spectrum. If neither BOC nor BIC is formed, then ∣c_j(t)∣ tends to zero with time and thus the atom eventually relaxes into the tube as a matter wave. If either the BOCs or BICs are formed, then the atom tends to a lossless coherent tunneling among the N lattice sites in multiple frequencies determined by the difference of any pairs of the eigenenergies of the formed BOCs and BICs.

The case of N = 2

The spectral density matrix for the two lattice sites is ${{{\bf{J}}}}(\omega )=\left(\begin{array}{rc}{J}_{0}(\omega )&{J}_{1}(\omega )\\ {J}_{1}(\omega )&{J}_{0}(\omega )\\ \end{array}\right)$. It contributes to the Laplace transform of f(t − τ) as $\tilde{{{{\bf{f}}}}}(s)=\left(\begin{array}{rc}{\tilde{f}}_{0}(s)&{\tilde{f}}_{1}(s)\\ {\tilde{f}}_{1}(s)&{\tilde{f}}_{0}(s)\\ \end{array}\right)$, where ${\tilde{f}}_{j}(s)=\int_{0}^{\infty }d\omega \frac{{J}_{j}(\omega )}{s+i\omega }$. The Jordan decomposition of $\tilde{{{{\bf{f}}}}}(s)$ reads

$${{{\bf{D}}}}(s)=\,{{\mbox{diag}}}\,[{\tilde{f}}_{0}(s)+{\tilde{f}}_{1}(s),{\tilde{f}}_{0}(s)-{\tilde{f}}_{1}(s)]$$

(23)

under ${{{{\bf{V}}}}}_{s}=\left(\begin{array}{cc}1&1\\ 1&-1\end{array}\right)$. Thus, we have

$$\tilde{{{{\bf{c}}}}}(s)=\left(\begin{array}{rc}1/2&1/2\\ 1/2&-1/2\end{array}\right)\left(\begin{array}{l}{[s+i{\omega }_{0}+{D}_{1}(s)]}^{-1}\\ {[s+i{\omega }_{0}+{D}_{2}(s)]}^{-1}\end{array}\right).$$

(24)

We need to make the inverse Laplace transform to obtain c(t). The equations determining the poles of the inverse Laplace transform are

$${Y}_{1}(\varpi )\equiv {\omega }_{0}-\int_{0}^{\infty }d\omega \frac{{J}_{0}(\omega )+{J}_{1}(\omega )}{\omega -\varpi }=\varpi ,$$

(25)

$${Y}_{2}(\varpi )\equiv {\omega }_{0}-\int_{0}^{\infty }d\omega \frac{{J}_{0}(\omega )-{J}_{1}(\omega )}{\omega -\varpi }=\varpi ,$$

(26)

where ϖ = E/ℏ = is. Their solutions can be classified into the following types.

• 1. In the regime ϖ > 0, the integration in both Y₁(ϖ) and Y₂(ϖ) is divergent. Thus, they jump rapidly between  ± ∞ and Eqs. (25) and (26) have infinite continuous roots, which form an energy band after multiplying ℏ as well as the branch cut.

• 2. In the regime ϖ < 0, both Y₁(ϖ) and Y₂(ϖ) are a monotonically decreasing function with ϖ. Thus, one and only one root ${\varpi }_{j}^{{{{\rm{boc}}}}}$ exists for either Eq. (25) or Eq. (26) as long as Y_j(0) < 0. Since this type of root ${\varpi }_{j}^{{{{\rm{boc}}}}}$ resides in a position out of the continuous energy band, we call it eigenstate BOC.

• 3. In the regime ϖ > 0, the integration in Y₁(ϖ) and Y₂(ϖ) has removable singularities ${\varpi }_{j}^{\,{{\mathrm{bic}}}\,}$ determined by ${J}_{0}({\varpi }_{1}^{\,{{\mathrm{bic}}}})+{J}_{1}({\varpi }_{1}^{{{\mathrm{bic}}}\,})=0$ and ${J}_{0}({\varpi }_{2}^{\,{{\mathrm{bic}}}})-{J}_{1}({\varpi }_{1}^{{{\mathrm{bic}}}\,})=0$ such that Eqs. (25) or (26) is respectively satisfied when ${\omega }_{0}={\varpi }_{1}^{\,{{\mathrm{bic}}}}+i{D}_{1}(-i{\varpi }_{1}^{{{\mathrm{bic}}}\,})$ or ${\omega }_{0}={\varpi }_{2}^{\,{{\mathrm{bic}}}}+i{D}_{2}(-i{\varpi }_{2}^{{{\mathrm{bic}}}\,})$. We readily evaluate that the removable singularities are ${\varpi }_{1/2}^{\,{{\mathrm{bic}}}\,}=\tilde{\omega }{(\bar{z}{n}_{1/2}\pi /d)}^{2}/2$, with d = ∣z₁ − z₂∣, n₁, and n₂ being odd and even numbers. Since this type of root ${\varpi }_{j}^{\,{{\mathrm{bic}}}\,}$ resides in a position in the continuous energy band, we call its corresponding eigenstate BIC.

The residue contributed by the l th bound state with frequency ${\varpi }_{l}^{\alpha }$ evaluated from Y_l(ϖ) = ϖ to the inverse Laplace transform ${[s+i{\omega }_{0}+{D}_{j}(s)]}^{-1}$ in Eq. (24) is

$$\begin{array}{rcl}&&\mathop{\lim }_{s\to -i{\varpi }_{l}^{\alpha }}(s+i{\varpi }_{l}^{\alpha }){e}^{st}{[s+i{\omega }_{0}+{D}_{j}(s)]}^{-1}\\ &=&{Z}_{l}^{\alpha }{e}^{-i{\varpi }_{l}^{\alpha }t}{\delta }_{l,j},\hfill\end{array}$$

(27)

where ${Z}_{l}^{\alpha }={[1+{\partial }_{s}{D}_{l}(s)]}^{-1}{| }_{s\to -i{\varpi }_{l}^{\alpha }}$. Based on the fact that the steady-state solution of c(t) obtained via the inverse Laplace transform $\tilde{{{{\bf{c}}}}}(s)$ is governed by the residues contributed by the eigenenergies of the BOCs and BICs, we readily have

$${{{\bf{c}}}}(\infty )=\mathop{\sum}_{\alpha =\,{{\mathrm{bic,boc}}}\,}\left(\begin{array}{cc}1/2&1/2\\ 1/2&-1/2\end{array}\right)\left(\begin{array}{c}{Z}_{1}^{\alpha }{e}^{-i{\varpi }_{1}^{\alpha }t}\\ {Z}_{2}^{\alpha }{e}^{-i{\varpi }_{2}^{\alpha }t}\end{array}\right)$$

(28)

in the case of two bound states are formed.

The case of N = 3

The spectral density matrix in the case of N = 3 reads ${{{\bf{J}}}}(\omega )=\left(\begin{array}{rcl}{J}_{0}(\omega )&{J}_{1}(\omega )&{J}_{2}(\omega )\\ {J}_{1}(\omega )&{J}_{0}(\omega )&{J}_{1}(\omega )\\ {J}_{2}(\omega )&{J}_{1}(\omega )&{J}_{0}(\omega )\\ \end{array}\right)$. It contributes to the Laplace transform of f(t − τ) as  $\tilde{{{{\bf{f}}}}}(s)=\left(\begin{array}{rcl}{\tilde{f}}_{0}&{\tilde{f}}_{1}&{\tilde{f}}_{2}\\ {\tilde{f}}_{1}&{\tilde{f}}_{0}&{\tilde{f}}_{1}\\ {\tilde{f}}_{2}&{\tilde{f}}_{1}&{\tilde{f}}_{0}\end{array}\right)$, where the argument “s” of ${\tilde{f}}_{j}(s)$ has been omitted for brevity. The Jordan decomposition of $\tilde{{{{\bf{f}}}}}(s)$ is

$$\begin{array}{r}{{{\bf{D}}}}(s)=\,{{\mbox{diag}}}\,[{\tilde{f}}_{0}-{\tilde{f}}_{2},{\tilde{f}}_{0}+\frac{1}{2}({\tilde{f}}_{2}-\tilde{e}),{\tilde{f}}_{0}+\frac{1}{2}({\tilde{f}}_{2}+\tilde{e})]\,\,\,\end{array}$$

(29)

under ${{{{\bf{V}}}}}_{s}=\left(\begin{array}{ccc}-1&1&1\\ 0&\frac{{\tilde{f}}_{1}(\tilde{e}-3{\tilde{f}}_{2})}{{\tilde{f}}_{2}\tilde{e}-2{\tilde{f}}_{1}^{2}-{\tilde{f}}_{2}^{2}}&\frac{{\tilde{f}}_{1}(\tilde{e}+3{\tilde{f}}_{2})}{{\tilde{f}}_{2}\tilde{e}+2{\tilde{f}}_{1}^{2}+{\tilde{f}}_{2}^{2}}\\ 1&1&1\end{array}\right)$, where $\tilde{e}=\sqrt{8{\tilde{f}}_{1}^{2}+{\tilde{f}}_{2}^{2}}$. We thus have $\tilde{{{{\bf{c}}}}}(s)$ in the case of N = 3 as $\tilde{{{{\bf{c}}}}}(s)={{{\bf{M}}}}{[s+i{\omega }_{0}+{{{\bf{D}}}}(s)]}^{-1}{\left(\begin{array}{ccc}1&1&1\end{array}\right)}^{{{{\rm{T}}}}}$, where ${{{\bf{M}}}}=\left(\begin{array}{ccc}1/2&\frac{1-{\tilde{f}}_{2}/\tilde{e}}{4}&\frac{1+{\tilde{f}}_{2}/\tilde{e}}{4}\\ 0&-{\tilde{f}}_{1}/\tilde{e}&{\tilde{f}}_{1}/\tilde{e}\\ -1/2&\frac{1-{\tilde{f}}_{2}/\tilde{e}}{4}&\frac{1+{\tilde{f}}_{2}/\tilde{e}}{4}\end{array}\right)$. The poles and the eigenenergies are determined by

$${Y}_{j}(\varpi )\equiv {\omega }_{0}-i{D}_{j}(-i\varpi )=\varpi .$$

(30)

Being similar to the N = 2 case, Eq. (30) has three types of roots, i.e., continuous energy band when ϖ > 0, isolated poles ${\varpi }_{j}^{{{{\rm{boc}}}}} < 0$ provided Y_j(0) < 0, and discrete removable singularities ${\varpi }_{j}^{\,{{\mathrm{bic}}}\,} > 0$. We can analytically determine that the roots of the BICs for D₁(s) read ${\varpi }_{1}^{\,{{\mathrm{bic}}}\,}=\tilde{\omega }{(\bar{z}n\pi /d)}^{2}/8$, with n being an even number, which is formed when ${\omega }_{0}={\varpi }_{1}^{\,{{\mathrm{bic}}}}+i{D}_{1}(-i{\varpi }_{1}^{{{\mathrm{bic}}}\,})$. Only the BOCs and BICs survive in the long-term dynamics. Thus, according to the residue theorem, we finally obtain

$${c}_{j}(\infty )={\sum}_{\alpha =\,{{{\mathrm{bic,boc}}}}\,}{\sum}_{l=1}^{3}\left.\frac{{M}_{jl}{e}^{st}}{1+{\partial }_{s}{D}_{l}(s)}\right| _{s\to -i{\varpi }_{l}^{\alpha }}.$$

(31)

Equation (31) reveals that, in contrast to c₁(∞) and c₃(∞), c₂(∞) does not depend on ${\tilde{f}}_{0}(s)$. Therefore, the bound state formed by the ${\tilde{f}}_{0}(s)$-branch does not have an impact on c₂(∞).

The energy spectrum in Fig. 6a reveals that three BOCs and one BIC are formed at most. The evolution of ∣c_j(t)∣ in Fig. 6b–d exhibits a one-to-one correspondence to the feature of the energy spectrum. When one BOC is present, the atom tends to a stable probability distribution in the three lattice sites. When two BOCs are present, ∣c_1/3(t)∣ tends to a periodic oscillation in a frequency equal to the difference of the eigenenergies of the two BOCs divided by ℏ, while ∣c₂(t)∣ still tends to be a stable value. In this case, the emerged BOC is from ${\tilde{f}}_{0}$ and thus has no effect on ∣c₂(t)∣. When three BOCs are present, ∣c₁(t)∣ and ∣c₃(t)∣ tend to an oscillation in three frequencies equal to the differences of the eigenenergies of any pairs of BOCs divided by ℏ, while ∣c₂(t)∣ tends to a periodic oscillation in one frequency. The analytical results as Eq. (31) are plotted in Fig. 7. The matching of our analytical results with the numerical ones verifies that the long-range tunneling is realized among the three lattice sites as long as multiple BOCs or BICs are present in the energy spectrum.

Fig. 6: Energy spectrum and dynamics for different values of the site separation in the case of three lattice sites.

a Energy spectrum for different values of the site separations d. The red point marks the position of the BIC. Evolution of the absolute values of the probability amplitudes (b) ∣c₁(t)∣, (c) ∣c₂(t)∣, and (d) ∣c₃(t)∣. The parameters are N = 3, $\Omega =0.13\tilde{\omega }$, and ${\omega }_{0}=0.05\tilde{\omega }$.

Fig. 7: Steady-state distribution of the atomic probability amplitudes in the three sites.

In the one-BOC regime, (a) ∣c₁(∞)∣, (b) ∣c₂(∞)∣, and (c) ∣c₃(∞)∣ are equal to constants. In the two-BOC regime, (b) ∣c₂(∞)∣ keeps being a constant, while ∣c_1/3(∞)∣ in a and c exhibit periodic oscillations in a single frequency, with the maxima and the minima marked by red dots. In the three-BOC regime, (b) ∣c₂(∞)∣ exhibits a single-frequency periodic oscillation, with the maxima and the minima marked by red dots, while ∣c_1/3(∞)∣ in a, c exhibit a lossless oscillation in three frequencies, with the local maxima and the minima joined by red lines. The parameters are the same as in Fig. 6.