Tunable quantum switcher and router of cold atom matter waves using artificial magnetic fields

TL;DR

This paper demonstrates a tunable quantum switch and router for cold atom matter waves using artificial magnetic fields in a minimal two-rung ladder system, enabling control over atom transport and routing.

Contribution

It introduces a novel method to control atom transport and routing in a minimal two-rung ladder with artificial magnetic flux, with potential applications in quantum device interfaces.

Findings

Atom transport can be switched from blockade to transparency by tuning system parameters.

Atoms can be perfectly routed into different channels with wavevector modulation by magnetic flux.

The approach can be extended to other artificial quantum systems like superconducting circuits.

Abstract

We investigate the single-atom transport in a two-leg ladder with only two rungs, which together with the legs, enclose an artificial magnetic flux. Here, the atoms on the two legs possess opposite onsite energies that produce an energy offeset. We find that the atom incoming from the left leg can experience from blockade to tranparency via modifying the onsite energy, hopping strength, or magnetic flux, which can be potentially used for a quantum switcher. Furthermore, the atom incoming from the left leg can also be perfectly routed into the right leg, when, intriguingly, the outgoing atom in the R channel possesses a wavevector that can be modulated by the magnetic flux. The result may be potentially used for the interface that controls the communication between two individual quantum devices of cold atoms. The method can also be generalized to other artificial quantum systems, such as superconducting quantum circuit system, optomechanical system, etc.

Equations

$$H=$$

$$-\sum _l(J_{\text{R}}{b}_{l+1,\text{R}}^{†}{b}_{l,\text{R}}+J_{\text{L}}{b}_{l+1,\text{L}}^{†}{b}_{l,\text{L}})+\text{H.c.}$$

$$-K{b}_{0,\text{L}}^{†}{b}_{0,\text{R}}-K{e}^{-i\varphi }{b}_{1,\text{L}}^{†}{b}_{1,\text{R}}+\text{H.c.,}$$

$$E{u}_{l,\text{R}}=$$

$$-{\delta }_{l,0}K{u}_{l,\text{L}}-{\delta }_{l,1}K{e}^{i\varphi }{u}_{l,\text{L}},$$

$$E{u}_{l,\text{L}}=$$

$$-{\delta }_{l,0}K{u}_{l,\text{R}}-{\delta }_{l,1}K{e}^{-i\varphi }{u}_{l,\text{R}}.$$

$$u_{l,\sigma }=e^{i{k}_{\sigma }l},$$

$$E={\epsilon }_{\sigma }-2J_{\sigma }\cos k_{\sigma }.$$

$$u_{l,\text{L}}$$

$$u_{l,\text{R}}$$

$$t_{\text{LL}}$$

$$t_{\text{RL}}$$

$$r_{\text{LL}}$$

$$r_{\text{RL}}$$

$$D=$$

$$-\xi [1+e^{-i\left(k_{\text{L}}-k_{\text{R}}+\varphi \right)}][1+e^{i\left(k_{\text{L}}-k_{\text{R}}+\varphi \right)}].$$

$$E={\epsilon }_{\text{L}}-2J_{\text{L}}\cos k_{\text{L}}={\epsilon }_{\text{R}}-2J_{\text{R}}\cos k_{\text{R}}\text{,}$$

$$\varphi$$

$$e^{-i{k}_{\text{R}}}\mp \sqrt{\xi +1}$$

$$J_{\text{L}}\cos k_{\text{L}}\mp \frac{J_{\text{R}}}{2}\left(\sqrt{\xi +1}+\frac{1}{\sqrt{\xi +1}}\right)-\epsilon$$

$$J_{\text{L}}\cos k_{\text{L}}-\frac{J_{\text{R}}}{2}\left(\frac{2\cos k_{\text{L}}}{\gamma }+\frac{\gamma }{2\cos k_{\text{L}}}\right)-\epsilon$$

$$e^{i{k}_{\text{R}}}-\frac{2\cos k_{\text{L}}}{\gamma }$$

$${\tilde{T}}_{\text{LL}}+{\tilde{R}}_{\text{LL}}+{\tilde{T}}_{\text{RL}}+{\tilde{R}}_{\text{RL}}=1.$$

$$\xi$$

$$\epsilon$$

$$k_{\text{R}}$$

$$k_{\text{L}}$$

$$H_{\text{ld}}=$$

$$-\sum _{l\text{; }q=\text{L,R}}{J}_q{b}_{l+1,q}^{†}{b}_{l,q}-K\sum _l{b}_{l,\text{L}}^{†}{b}_{l,\text{R}}+\text{H.c.,}$$

$$H_{\text{e}}\left(t\right)=\hslash \Omega \sum _{l=0,1\text{; }q=\text{L,R}}\cos \left({\phi }_{l,q}+\omega t\right)b_{l,q}^{†}{b}_{l,q},$$

$$H_{\text{eff}}=$$

$$-\sum _{\begin{array}{c}l\ne -1,0,1\\ q=\text{L,R}\end{array}}{J}_y{b}_{l+1,q}^{†}{b}_{l,q}-\sum _{l\ne 0,1}{J}_x{b}_{l,\text{L}}^{†}{b}_{l,\text{R}}+\text{H.c.}$$

$$-\sum _{\begin{array}{c}l=-1,0,1\\ q=\text{L,R}\end{array}}{J}_{l,q}{b}_{l+1,q}^{†}{b}_{l,q}-\sum _{l=0,1}{K}_l{b}_{l,\text{L}}^{†}{b}_{l,\text{R}}+\text{H.c..}$$

$$H=$$

$$-\sum _l{J}_y{b}_{l+1,\text{L}}^{†}{b}_{l,\text{L}}+J_y{b}_{l+1,\text{R}}^{†}{b}_{l,\text{R}}+\text{H.c.}$$

$$-\sum _{l=0,1}K{b}_{0,\text{L}}^{†}{b}_{0,\text{R}}+K\exp \left(i\varphi \right)b_{1,\text{L}}^{†}{b}_{1,\text{R}}+\text{H.c..}$$

Full text

cold atoms, blockade, amplification, spin-orbit coupling, nonlinear interaction cold atoms, blockade, amplification, spin-orbit coupling, nonlinear interaction

LABEL:FirstPage1 LABEL:LastPage#1

I Introduction

The artificial tunability has made ultracold atoms an attractive platform for conducting reasearch on quantum information processing Brennen1999PRL ; Jaksch1999PRL ; Pachos2003PRL ; Kay2006PRA and high-precision instruments Hansel2001Nature ; Fortagh2005Science ; Colombe2007Nature ; Riedel2010Nature ; Zoest2010Science . Integrating different cold atom systems together, one can possibly realize the scalable quantum network, with each quantum node being a cold atom subsystem. However, reaching this destination demands the coherent manipulation of cold atom matter waves such that different nodes can be individually addressed for the effective communication of quantum signals. To this end, there have already been many researches on matter wave transport based on atom interactions Zhao2018arXiv ; Liu2000PRL ; Liang2005PRL ; Morsch2006RMP ; Miroshnichenko2010RMP ; Kartashov2011RMP ; Chien2015NP ; Poulsen2003PRA ; Smerzi2003PRA ; Vicencio2007PRL ; Zhang2008EPJD ; Arealo2009PLA ; Hennig2010PRA ; Bai2015AP ; Bai2016PRE ; Qi2009PRL . For example, matter wave switchers have been proposed for spinless Vicencio2007PRL and spinful Zhao2018arXiv plane waves. Besides, spinful plane waves with opposite spins can also be isolated and converted Zhao2018arXiv . However, both schemes Vicencio2007PRL ; Zhao2018arXiv rely on first preparation of a localized Bose-Einstein condensate (BEC), which increases the complexity in experiment. Moreover, to practically control the signal flow in quantum network, the ideal quantum router needs to have multiaccess channels, which has been studied in the circuit QED system Zhou2013PRL but rarely investigated in cold atom systems.

Meahwhile, we have noted that chiral currents can emerge in bosonic ladders due to the effect of the artificial magnetic field Atala2014NP . This motivates us to explore the quantum switcher and router controlled by the artificial magnetic field which can coherently couple two individual atomic channels together. In detail, we consider that the magnetic flux only penetrates one particular loop enclosed by four lattice sites, while elsewhere, the two legs of the ladders are decoupled but lifted by an energy detuning. We will study the transport properties in the single-atom scenario for the incident plane wave with varying energy detunings and magnetic fluxes. We will demonstrate that the one-channel switcher and two-channel router can be realized within our model.

Our paper is organized as follows. In Sec. II, we introduce the Hamiltonian and derive the single-atom scattering coeffecients between the two atomic channels. In Sec. III, we discuss the single-atom switcher effect for different parameters. In Sec. IV, we demonstrate that the model can also be used as single-atom router. In Sec. V, we make some discussions in experiment and conclude the main results.

II Single-atom transport

II.1 Two-channel interaction

We study the scattering problem for a single-atom plane wave from the L leg of the bosonic ladders incident on a loop enclosing an artificial magnetic flux $\phi$ (see Fig. 1). The L and R legs, detuned by onsite energies $\varepsilon_{\text{L}}=\varepsilon$ and $\varepsilon_{\text{R}}=-\varepsilon$, are decoupled except at the sites on the loop [see Appendix. A for details of the experimental realization]. Away from the loop, the reflection and transmission will be stimulated. The intraleg hopping strengths are respectively $J_{\text{L}}$ and $J_{\text{R}}$ for the L and R legs. The interleg hopping strengths are $K$ and $K\exp(i\phi)$ for the sites $l=0$ and $l=1$. The theoretical model can be characterized by the following Hamiltonian

[TABLE]

where $b_{l,\sigma}$ and $b_{l,\sigma}^{{\dagger}}$ ($\sigma=\mathrm{L},\mathrm{R}$) are the bosonic annihilation and creation operators for the $\sigma$ leg at site $l$.

The single-atom eigenstate of the full Hamiltonian can be given by $\left|E\right\rangle=\sum_{l}u_{l,\text{R}}b_{l,\text{R}}^{{\dagger}}\left|0\right\rangle+u_{l,\text{L}}b_{l,\text{L}}^{{\dagger}}\left|0\right\rangle,$ where the coefficients $u_{l,\mu}$ for each single-atom component are constrained by

[TABLE]

One can see that the interaction between both channels are determined by the coupling strength $K$ and magnetic flux $\phi$ at $l=0,1$.

II.2 Free modes

Now, assuming the interaction between both channels is zero, i.e., setting $K=0$ in Eqs. (2) and (3), we can obtain the free modes in both channels as

[TABLE]

where the energy $E$ depends on $k_{\sigma}$ according to the relation

[TABLE]

Here, we stress that $k_{\sigma}$ can be either real or complex depending on the regime of $E$. In detail, for either atomic channel $\sigma=\mathrm{L}$ or $\mathrm{R}$, the free modes can be classified into three cases:

(i) The energy $E$ belongs to the band $\left[\varepsilon_{\sigma}+2J_{\sigma},\infty\right]$, such that $k_{\sigma}$ must be complex and can be rewritten as $k_{\sigma}=\pi+i\kappa_{\sigma}$ for real $\kappa_{\sigma}$. In this case, the energy turns into a hyperbolic cosine, $E=\varepsilon_{\sigma}+2J_{\sigma}\cosh\kappa_{\sigma}$ [see Fig. 2(a)], and the eigenstate becomes $u_{l,\sigma}=\left(-1\right)^{l}e^{-\kappa_{\sigma}l}$, a staggered decaying mode [see Fig. 2(b)].

(ii) The energy $E$ is in the band $\left(\varepsilon_{\sigma}-2J_{\sigma},\varepsilon_{\sigma}+2J_{\sigma}\right)$, such that $k_{\sigma}$ is gauranteed to be real. In this case, the eigenstate and correpsponding energy will retain their original forms as in Eqs. (4) and (5). Dispersive as $E=\varepsilon_{\sigma}-2J_{\sigma}\cos k_{\sigma}$ [see Fig. 2(c)], the eigenstate $u_{l,\sigma}=e^{ik_{\sigma}l}$ is a transmission mode [see Fig. 2(d)] with the group veloctiy $v_{g}=2J_{\sigma}\sin k_{\sigma}$.

(iii) The energy $E$ belongs to the band $\left(-\infty,\varepsilon_{\sigma}-2J_{\sigma}\right)$, such that $k$ is purely imaginary and can be rewritten as $k_{\sigma}=i\kappa_{\sigma}$ for real $\kappa$. In this case, the energy also turns into a hyperbolic cosine, $E=\varepsilon_{\sigma}-2J_{\sigma}\cosh\kappa_{\sigma}$ [see Fig. 2(e)], and the eigenstate becomes $u_{l,\sigma}=e^{-\kappa_{\sigma}l}$, also a decaying mode [see Fig. 2(f)].

II.3 Single-atom transport

We consider the single-atom plane wave comes from the negative lattices along the L channel, which will be scattered into both L and R channels. In detail, we can express $u_{l,\text{R}}$ and $u_{l,\text{L}}$ as

[TABLE]

where $\theta_{l}$ is the Heaviside step function, $t_{\text{LL}}$ ($r_{\text{LL}}$) is the transmitted (reflected) amplitude, and $t_{\text{RL}}$ ($r_{\text{RL}}$) is the forward (backward) transfer amplitude.

Then, solving Eqs. (2) and (3) by assigning $l=0,1$, we finally obtain the scattering coefficients as

[TABLE]

where $\xi=\frac{K^{2}}{J_{\text{L}}J_{\text{R}}}$ represents the normalized square interleg coupling strength and the denominator $D$ reads

[TABLE]

For the lattice site $l\neq 0,1$ in Eqs. (2) and (3), we can obtain the energy matching condition

[TABLE]

which renders the constraint of $k_{\text{L}}$ and $k_{\text{R}}$ in Eqs. (8)-(11). Noting that $\varepsilon_{\text{L}}=\varepsilon$ and $\varepsilon_{\text{R}}=-\varepsilon$ have been previously hypothesized as treating Eq. (13), we can then determine that the transmission bands of both channels can have no overlap, partial overlap, or maximum overlap if $\left|\varepsilon\right|>J_{\text{L}}+J_{\text{R}},0<\left|\varepsilon\right|<J_{\text{L}}+J_{\text{R}},$ or $\varepsilon=0$, respectively. Figure 3 has shown the relative position of the transmission energy bands of both channels in the special case $J_{\text{L}}=J_{\text{R}}=1$ for different $\varepsilon$. In the case of no overlap [see Figs. (a) and (e)], the incoming atom will stimulate a localized profile [$k_{\text{R}}=i\kappa_{\text{R}}$ or $k_{\text{R}}=i\kappa_{\text{R}}+\pi$ with $\kappa_{\text{R}}>0$, see Eq. (2)], meaning no scattered atom in the R channel. In the case of maximum overlap [see Fig. (c)], the atom incoming from the L channel can be redirected into the R channel ($k_{\text{R}}$ is real). However, in the case of partial overlap [see Figs. (b) and (d)], the atom can either be redirected into the R channel or stimulate a localized profile there, just depending on the detailed energy $E$ for the incoming atom.

III Switching single atom

Now we explore the possibility to realize the atom switcher in the L channle, a device which can tune the blockade and transparency of the incident atom. The blocakde and transparency require that no scatterred atom occur in the R channel, where a localized mode should be stimulated instead [$k_{\text{R}}=i\kappa_{\text{R}}$ or $k_{\text{R}}=i\kappa_{\text{R}}+\pi$].

In detail, by investigating Eqs. (8) and (10), we can obtain the the condition for blockade ($t_{\text{LL}}=0$, $\left|r_{\text{LL}}\right|=1$) as

[TABLE]

Similarly, the condition for transparency ($\left|t_{LL}\right|=1$, $\left|r_{LL}\right|=0$) can also be obtained as

[TABLE]

where $\gamma=\xi-2\cos\phi\geq-2$ represents the resultant effect from both $\phi$ and $\xi$ and $\left|\frac{2\cos k_{\text{L}}}{\gamma}\right|<1$ must be satisfied to guarantee that $k_{\text{R}}$ corresponds to a localized mode near the loop: $\exp\left(ik_{R}\right)=\pm\exp\left(-\kappa_{R}\right)$ with $\kappa_{R}>0$.

In the special case with equal intraleg hopping strengths: $J_{\text{L}}=J_{\text{R}}=1$, which results in $\xi=K^{2}$, the relation in Eqs. (16) and (17) can be illustrated in Fig. 4. We find in Fig. 4(a) that, to blockade the incoming atom at a particular wavevector $k_{\text{L}}$, one can use a smaller $\varepsilon$ in combination with a bigger $\xi$, or vice versa. For given $\xi$, the blockade point $k_{\text{L}}$ monotonically decreases as $\varepsilon$ increases. In Fig. 4(b), we also see that, if $\gamma$ is given (e.g., $\gamma=-1.6$, etc), two transparency points may occur if $\varepsilon$ is assigned appropriately. The detailed conditions to obtain two transparency points can be obtained as $-2<\gamma<0$ and $\sqrt{1-\gamma}<\left|\varepsilon\right|\leq 1-\frac{\gamma}{2}$. Both Fig. 4(a) and Fig. 4(b) are centrally symmetic about $\left(\varepsilon=0,k_{\text{L}}=\frac{\pi}{2}\right)$. In addition, $\varepsilon\neq 0$ can be verified, indicating the matching relations of the transmission energy bands correspond to all the subfigures in Fig. 3 except Fig. 3(c). In a word, the transparency and blockade points can be tuned in principle trough modifying $\phi$, $\varepsilon$, or $\xi$.

To furthermore gain an intuitive picture, in Fig. 5, we have plotted the transmittance $T_{\text{LL}}=\left|t_{\text{LL}}\right|^{2}$, reflectance $R_{\text{LL}}=\left|r_{\text{LL}}\right|^{2}$, and forward (backward) transfer rate $T_{\text{RL}}=\left|t_{\text{RL}}\right|^{2}$ ($R_{\text{RL}}=\left|r_{\text{RL}}\right|^{2}$) as the function of $k_{\text{L}}$ for the detailed parameters $\phi=\pi$, $\xi=K^{2}=4$, and different $\varepsilon$. In such parameter setup, the matching relations of the transmission energy bands for both channels are respectively no overlap [$\varepsilon=-2.02$, see Fig. 3(a)], partial overlap [$\varepsilon=-0.7$, see Fig. 3(b)], maximum overlap [$\varepsilon=0$, see Fig. 3(c)], partial overlap [$\varepsilon=0.7$, see Fig. 3(d)], and no overlap [$\varepsilon=2.02$, see Fig. 3(e)] for each columb in Fig. 5 from left to right. From the curves of $T_{\text{LL}}$ and $R_{\text{LL}}$ in Figs. 5(a), 5(c), 5(g), and 5(i), the transparency-bockade transitions are observed as $k_{\text{L}}$ gradually increases.

In Figs. 5(b), 5(d), 5(h), and 5(j), we also note that the transfer rate $T_{\text{RL}}$ can exceed 1. On one hand, this is because T_{\text{RL}}\and $T_{\text{RL}}$ reveal the amplitudes of the localized mode stimulated in R channel when $k_{\text{R}}$ is complex. On the other, if $k_{\text{R}}$ is real, when transmission modes are stimualted in R channel, the atom flow conservation are subject to the relation

[TABLE]

Here, $\tilde{T}_{\text{LL}}=T_{\text{LL}}$ ($\tilde{T}_{\text{RL}}=T_{\text{RL}}J_{\text{R}}\sin k_{\text{R}}/J_{\text{L}}\sin k_{\text{L}}$) and $\tilde{R}_{\text{LL}}=R_{\text{LL}}$ ($\tilde{R}_{\text{RL}}=R_{\text{RL}}J_{\text{R}}\sin k_{\text{R}}/J_{\text{R}}\sin k_{\text{R}}$) are respectively the forward and backward atom flows scattered in the L (R) channel. In this case, one can conclude that $T_{\text{RL}}$ can be greater than 1 once $k_{\text{R}}$ is small enough, which features a sufficiently small group velocity ($=J_{\text{R}}\sin k_{\text{R}}$) in R channel. Similarly, $R_{\text{RL}}$ can in principle exceed 1 as well. When $\varepsilon=0$, the equal intraleg hopping strengths $J_{\text{L}}=J_{\text{R}}=1$ result in $k_{\text{R}}=k_{\text{L}}$, only transmission modes stimulated in R channel. Thus, we have $T_{\text{LL}}+R_{\text{LL}}+T_{\text{RL}}+R_{\text{RL}}=1$, in which case, $T_{\mathrm{LL,RL}}<1$ and $R_{\mathrm{LL,RL}}<1$ always hold [see Figs. 5(e) and 5(f)]. In particular, one finds that $T_{\text{RL}}=0$, which can be verified by Eq. (9) under the condition $k_{\text{R}}=k_{\text{L}}$ and $\phi=\pi$.

Now we investigate the transmittance $T_{\text{LL}}$ and reflectance $R_{\text{LL}}$ via independently modifying the values of $\varepsilon$, $\xi$, and $\phi$ [see Fig. 6]. We see two (one), one (one), and one (two) blockade (transparency) points in Fig. 6(a), 6(b), and 6(c), respectively as $\varepsilon$, $\xi$, and $\phi$ varies. The number of blockade or transparency points for $\varepsilon$, $\xi$, or $\phi$ can be exactly predicted using Eqs. (14)-(18) with other parameters determined. We also find that the transfer rates $T_{\text{RL}}$ and $R_{\text{RL}}$ can exceed 1, the reason for which is similar to Fig. 5. Thus, to tune the transport of the atom at a given wavevector, we can optionally select $\varepsilon$, $\xi$, or $\phi$ as the controllable parameter.

IV Routing single atom

Now we investigate the phenomenon of routing the atom from the L channel into the R one, in which case, the wavevectors $k_{\text{L}}$ and $k_{\text{R}}$ are both real and thus $\left|\varepsilon\right|\leq 2$ must hold [see Figs. 3(b)-3(d)]. Via examining Eqs. (8) and (11), we find that when the normalized square interleg coupling strength $\xi$ and onsite energy $\varepsilon$ satisfy the conditions

[TABLE]

the perfect routing defined by $\tilde{T}_{\text{LL}}=\tilde{R}_{\text{LL}}=0$ will occur at

[TABLE]

Here, to guarantee that both $\xi$ and the group velocity corresponding to $k_{\text{R}}$ (i.e., $J_{\text{R}}\sin k_{\text{R}}$) are positive, there should be the constraint $-\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$. Under the conditions in Eqs. (20)-(23), the transfer amplitudes are $t_{\text{RL}}=r_{\text{RL}}=\frac{K}{J_{\text{R}}}\frac{i}{2\cos\phi}$, and the corresponding atom flows are $\tilde{T}_{\text{RL}}=\tilde{R}_{\text{RL}}=\frac{1}{2}$.

Hereafter, we still consider equal intraleg hopping strengths, i.e., $J_{\text{L}}=J_{\text{R}}=1$. Figures 7(a)-7(c) have shown respectively how $\xi$, $\varepsilon$, and $k_{\text{R}}$ at perfect routing [see Eqs. (20)-(22)] will change as $\phi$ varies from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. The perfect routing requires that both $\xi$ and $\varepsilon$ be sine functions of the artificial magnetic flux $\phi$. Besides, the wavevector of the atom outgoing from the R channel can be linearly modulated by $\phi$, despite the fact that the atom must income at the fixed wavevector $k_{\text{L}}=\frac{\pi}{2}$ through the L channel.

If $\xi$ and $\varepsilon$ are specified according to Eqs. (20) and (21), how the atom flows vary with the wavevector $k_{\text{L}}$ for different magnetic flux $\phi$ can be shown in Fig. 8, where, to guarantee the reality of both $k_{\text{L}}$ and $k_{\text{R}}$, we have constrained $\arccos\left(\left|\varepsilon\right|-\varepsilon-2\right)\leq k_{\text{L}}\leq\arccos\left(-\left|\varepsilon\right|-\varepsilon+2\right)$. We can observe that despite the varying of $\phi$, the perfect routing always occurs at $k_{\text{L}}=\frac{\pi}{2}$. However, the wavevector $k_{\text{R}}$ for the outgoing atom in the R channel will accordingly change. It is also obvious that the two boundaries of the $k_{\text{L}}$ axis are changed as $\phi$ varies, which is due to the variation of $\varepsilon$.

V Discussion and conclusion

In experiment, the quasimomentum $k$ of the incident ${}^{\text{87}}$Rb atoms can be generated via phase imprinting method Denschlag2000Science , Bragg scattering, or simply acceleration of the matter-wave probe in an external potential. The localization magnetic flux is achieved by two laser beams whose intersection only covers four lattice sites that encloses the magnetic flux inside the ladder. The interleg hopping strength, magetic flux, and onsite energy can be tuned via modifying the lasers, e.g, the intensity, angles, and wavevectors. In measurement, the motion of the atoms can be recorded by absorption imaging Lin2011Nature for furthemore analysis.

In conclusion, we have investigated the single-atom transport in a two leg ladder with only two rungs, which together with the legs, enclose an artificial magnetic flux. Here, the atoms on the two legs possess opposite onsite energies that produce an energy offset. We find that the atom incoming from the left leg can experience from blockade to tranparency via modifying the onsite energy, hopping strength, or magnetic flux, which can be potentially used for a quantum switcher. Furthermore, the atom incoming from the left leg can also be perfectly routed into the right leg, when, intriguingly, the outgoing atom in the R channel possesses a wavevector that can be modulated by the magnetic flux. The result may be potentially used for the interface that controls the communication between two individual quantum devices of cold atoms. The method can also be generalized to other artificial quantum systems, such as superconducting quantum circuit system, optomechanical system, etc.

VI Acknowledgments

We are grateful to Ru-Quan Wang for helpful discussions. This work is supported by the National Key R&D Program of China under grants No. 2016YFA0301500, NSFC under grants No.s 11847165, 11434015, 61227902, 11611530676, 61775242, 61835013, SPRPCAS under grants No.s XDB01020300, XDB21030300.

Appendix A Experimental implementation

To create the two-leg ladder model Atala2014NP , we can apply a standing wave in the $y$ ($z$) direction with the wave length $\lambda_{\text{s}}$ ($\lambda_{z}$), but two standing waves in the $x$ direction, which creates a double-well potential: $V_{\text{s}}\left(x\right)=V_{\text{l}x}\sin^{2}\left(k_{\text{l}}x+\varphi/2\right)+V_{x}\sin^{2}\left(k_{\text{s}}x\right)$ with $k_{i}=2\pi/\lambda_{i}$ and $\lambda_{\text{l}}=2\lambda_{\text{s}}$ [see Fig. (9)]. The lattice depths $V_{\text{l}x/x}$ and phase $\varphi$ are properly chosen to create an array of isolated tilted double well potentials, where each double well realizes a single ladder. In the tight-binding limit, the ladder Hamiltonian can be described by

[TABLE]

where the tunneling is inhibited by the energy defference $\Delta$, suppose that $\Delta\gg K$ is satisfied.

The generation of the localized artificial magnetic flux is Inspired by Refs. Aidelsburger2011PRL ; Aidelsburger2013PRL ; Jaksch2003NJP ; Atala2014NP . We control the thickness of both beams such that the intersection of them only cover the very four lattice sites that will enclose the effective magnetive flux [also see Fig. (9)]. We assume two travelling laser beams with wavevectors $\mathbf{k}_{i}=k_{i}\left(\cos\phi_{i}\mathbf{e}_{x}+\sin\phi_{i}\mathbf{e}_{y}\right)$ and Rabi frequencies $\Omega_{i}$ ($i=1,2$) are employed to illuminate the atoms. Both lasers are incident in the $xy$-plane at angles $\phi_{1}$ and $\phi_{2}$, and couple two internal atomic energy levels $\left|g\right\rangle$ (ground state) and $\left|e\right\rangle$ (intermediate state) through large detunings $\delta_{i}$ ($\left|\delta_{i}\right|\gg\Omega_{i}$), which result in an external potential $V_{\text{e}}\left(\mathbf{r}\right)=\hbar\Omega\cos\left[\left(\mathbf{\mathbf{k}_{1}-\mathbf{k}_{2}}\right)\mathbf{r+}\omega t\right]$ with $\Omega=\frac{\Omega_{1}\Omega_{2}}{2\delta_{1}}$. Here, the frequency difference $\omega$ can be represented by $\omega=c\left(k_{2}-k_{1}\right)=\delta_{2}-\delta_{1}$ with $\left|\omega\right|\ll\left|\delta_{i}\right|$ such that $\delta_{1}\approx\delta_{2}$. In the tight-binding limit, the external potential corresponds to the Hamiltonian

[TABLE]

where $\varphi_{l,\text{L/R}}=\mp\frac{k_{x}\lambda_{\text{s}}}{4}+\frac{lk_{y}\lambda_{\text{s}}}{2}$. The wavevectors $k_{x}$ and $k_{y}$ can be represented as $k_{x}=k_{1}\cos\phi_{1}-k_{2}\cos\phi_{2}$ and $k_{y}=k_{1}\sin\phi_{1}-k_{2}\sin\phi_{2}$. To eliminate the time-dependent Hamiltonian $H_{\text{e}}\left(t\right)$, we perform a unitary transformation $U=\!\exp[-i\frac{\Omega}{\omega}\sum_{l=0,1\text{; }q=\text{L,R}}\sin\left(\varphi_{l,q}\mathbf{+}\omega t\right)b_{l,q}^{{\dagger}}b_{l,q}]$, which yields the following effective Hamiltonian

[TABLE]

Here, we assume $\omega$ is nearly resonant with $\Delta$: $\Delta-\omega=-2\varepsilon$ and $\left|\varepsilon\right|\ll\Delta$, which convinces us to discard the fast-oscillating terms. Furthermore, in the perturbative limit ($\frac{\Omega}{\omega}\ll 1$), the parameters take the forms $J_{l,q}\approx J_{y}$ ($l=-1,0,$and $1$), $K_{0}\approx K\exp\left(i\omega t\right)$, and $K_{1}\approx K\exp\left(i\phi\right)\exp\left(i\omega t\right)$, where the interleg coupling strength $K=\frac{J_{x}\Omega}{\omega}\cos\left(\frac{k_{x}\lambda_{\text{s}}}{4}\right)$, and the effective magnetic flux $\phi=\frac{1}{2}k_{y}\lambda_{\text{s}}=\frac{1}{2}\left(k_{1}\sin\phi_{1}-k_{2}\sin\phi_{2}\right)\lambda_{\text{s}}$ can be tuned via the Raman lasers, e.g., the angles $\phi_{i}$. Applying the unitary transformation $U^{\prime}=\exp[-\frac{1}{2}i\omega t\sum_{l}(b_{l,\text{R}}^{{\dagger}}b_{l,\text{R}}-b_{l,\text{L}}^{{\dagger}}b_{l,\text{L}})]$, we then obtain the Hamiltonian in Eq. (1) as

[TABLE]