Quantum dynamics of tight-binding networks coherently controlled by external fields

Abstract

With some reviews on the investigations on the schemes for quantum state transfer based on spin systems, we discuss the quantum dynamics of magnetically-controlled networks for Bloch electrons. The networks are constructed by connecting several tight-binding chains with uniform nearest-neighbor hopping integrals. The external magnetic field and the connecting hopping integrals can be used to control the intrinsic properties of the networks. For several typical networks, rigorous results are shown for some specific values of external magnetic field and the connecting hopping integrals: a complicated network can be reduced into a virtual network, which is a direct sum of some independent chains with uniform nearest-neighbor hopping integrals. These reductions are due to the fermionic statistics and the Aharonov-Bohm effects. In application, we study the quantum dynamics of wave packet motion of Bloch electrons in such networks. For various geometrical configurations, these networks can function as some optical devices, such as beam splitters, switches and interferometers. When the Bloch electrons as Gaussian wave packets input these devices, various quantum coherence phenomena can be observed, e.g., the perfect quantum state transfer without reflection in a Y-shaped beam, the multi-mode entanglers of electron wave by star-shaped network, magnetically controlled switches, and Bloch electron interferometer with the lattice Aharonov-Bohm effects. With these quantum coherent features, the networks are expected to be used as quantum information processors for the fermion system based on the possible engineered solid state systems, such as the array of quantum dots that can be implemented experimentally.

Graphical abstract

Keywords

quantum information / quantum state transfer / entanglement / A-B effect

1 Introduction

Quantum information processing (QIP) has been a very active area of research in the past few years [1, 2]. The current challenge for QIP is to coherently integrate a sufficiently large and complex controllable system, and then require the ability to transfer quantum information between spatially separated quantum bits. Since then, numerous approaches for this purpose have been proposed, ranging from linear and nonlinear quantum optical devices to various interacting quantum systems. Among them, many studies proposed using the internal dynamics of coupled spins for the transfer of quantum information [3–13]. However, the basic and necessary “optical devices” (for the electron wave or the spin wave) in a solid system are scarce due to the technology at hand. Therefore, it is significant to probe the possibilities to construct the artificial “optical devices” and then build the corresponding electronic networks for the matter wave of electron within a solid state system [14–23]. Here, we notice that, for the boson system, Plenio, Hartley and Eisert [18, 19] have studied the quantum network dynamics of a system consisting of a large number of coupled harmonic oscillators, while Kay and Ericsson [20] have studied the coupled spin systems of various geometric configurations for the similar purpose.

This paper will pay attention to the fermion systems where Bloch electrons move along the quantum lattice network. We consider various geometrical configurations of tight-binding networks that are analogous to quantum optical devices, such as beam splitters, switches and interferometers. As an application, we then consider the functions of these tight-binding networks in detail when the initial Gaussian wave packets are entering these devices. Analytical and numerical investigations show that these devices are controllable by the magnetic flux through the network. Characteristic parameters of devices can be adjusted by changing the flux or the interactions on the nodes. The relevant quantum phenomena, such as generation of entanglement and the Aharonov-Bohm (A-B) effects in the solid state-based devices are also discussed systematically.

This paper is organized as follows. In Section 2, we present the Hamiltonians of the simplest tight-binding lattice systems with and without magnetic field as building blocks to construct various networks, which can be formed topologically by the linear and the various connections among their ends. In Section 3, we theoretically design and analytically study the basic properties of a star-shaped TBN, then also explore the further dynamic property of the Y-shaped network. Surprisingly, for appropriate joint hopping integrals, the complicated network can be reduced into an imaginary linear chain with homogeneous NN hopping terms plus a smaller complicated network. It is known that such TBNs are analogous to quantum optical devices such as beam-splitters, entanglers and interferometers. In Section 4, we investigate the dynamic properties of a Bloch electron model on a Q-shaped lattice, which consists of a terminal chain and a ring threaded by a magnetic flux. The appropriate flux through the network can reduce the network to the linear virtual chain, which indicates that the flux can control the propagation of GWP in the network. In Section 5, the interferometer network, a mimic of the AB effect experiment, is also studied in a similar way. In addition, in the whole paper, a moving Gaussian wave packet (GWP) localized in a linear dispersion regime is a good example to illustrate the properties of the above Bloch electron networks. In Section 6, we extend the results of the TBNs to the spin network (SN) for the dynamics of the single magnon. In Section 7, we summarize the results of this paper and suggest the possible applications of these TBNs.

2 Basic setup

In this section, we introduce the systems under consideration, namely the tight-binding Bloch electron systems and the Hamiltonians of the building blocks to construct various networks. Without loss of generality, we concentrate our attention on the simplest tight-binding system, in which only the hopping term or kinetic energy is considered.

A general tight-binding network (TBN) is constructed topologically by the linear tight-binding chains and the various connections between their ends. An important element in the system is the Aharonov-Bohm flux through some loops of the TBN. Here, we consider the simplest tight-binding model, in which only the nearest neighbor (NN) hopping terms are taken into account. The Hamiltonian of a tight-binding linear chain with N_l sites reads as

\begin{array}{}\text{(1)}& H_l=H_l(N_l)\equiv {\displaystyle \sum _{j=1}^{N_l-1}(t_j^{[l]}{a}_{l,j}^{†}{a}_{l,j+1}+H.c.)}\end{array}

Here, the label l denotes the chain with the distribution of the hopping integrals \{t_1^{[l]},t_2^{[l]},\cdots t_{N_{l-1}}^{[l]}\} and a_{l,j}^{†} is the fermion creation operator at the j th site of the chain l. Here, the spin degree of freedom is omitted for notational brevity. However, all the discussion and conclusion throughout the paper are available for the spinful system with a_{l,j}^{†}{a}_{l,j+1}\to{\displaystyle {\sum }_{\sigma =\pm }{a}_{l,j,\sigma }^{†}}{a}_{l,j+1,\sigma }, since the model we concerned in this paper does not contain the interaction involving spin. The hopping integral could be the complex number due to the presence of the external magnetic field. In this paper, we restrict our study to the simplest case described as follows: when the field is absent, the hopping integral for a homogeneous chain

\begin{array}{}\text{(2)}& t_j^{[l]}=t{\mathrm{e}}^{\mathrm{i}{\Phi }_{l,j+1}}\end{array}

is real and identical (i.e., independent of sites) while in the presence of a vector potential the hopping integral is modified by a phase factor. Here, we defined the link phase

\begin{array}{}\text{(3)}& {\Phi }_{l,j+1}=\frac{2\mathrm{\pi }}{{\varphi }_0}{\displaystyle {\int }_j^{j+1}A\cdot \mathrm{d}l}\end{array}

with flux quanta {\varphi }_0=hc/e as an integral of the vector potential A along the link between the sites j and j+1 in the lth chain. The above observation about the phase modification of hopping integral can be found in many modern literatures [24–27] but the proof can be cast back to Peierls [28].

Another important portion of the quantum networks is the joints or nodes between two linear chains, of which the Hamiltonian can be presented in the form

\begin{array}{}\text{(4)}& H_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}\equiv -(t_{ji}^{[lm]}{a}_{l,j}^{†}{a}_{m,i}+H.c.)\end{array}

where t_{ji}^{[lm]} denotes the hopping integral over the j th site of chain l and the ith site of chain m. Here, only one joint term connecting two chains is listed in H_joint as an illustration. In a general TBN, the joint Hamiltonian H_joint should contain many connection terms.

In the remainder of this paper, we will show that, under certain conditions, complex TBNs can be decomposed as a simple sum of several independent imaginary chains. In order to avoid confusion, we describe one of the imaginary chains with N_l sites by a Hamiltonian

\begin{array}{}\text{(5)}& {\tilde{H}}_l={\tilde{H}}_l(N_l)\equiv -t{\displaystyle \sum _{j=1}^{N_l-1}({\tilde{a}}_{l,j}^{†}{\tilde{a}}_{l,j+1}+H.c.)}\end{array}

Here, {\tilde{a}}_{l,j}^{†} denoted by tilde are the fermion creation operators for j th site of the imaginary chain l, which are linear combinations of a_m,i. Namely, there exists a mapping R between the two sets of fermion operators, \left\{a_{l,j}^{†}\right\}\stackrel{R}{\to }\left\{{\tilde{a}}_{m,i}^{†}\right\} or by a transformation R:

\begin{array}{}\text{(6)}& {\tilde{a}}_{m,i}^{†}={\displaystyle \sum _{l,j}{R}_{l,j,m,i}{a}_{l,j}^{†}}\end{array}

We will investigate several types of networks based on the notations introduced above. As an example to demonstrate the application of the notation, we can express the main conclusion of this paper by using the above well-defined notations as

\begin{array}{}\text{(7)}& {\displaystyle \sum _l{H}_l}+H_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}\stackrel{R}{\to }{\displaystyle \sum _m{\tilde{H}}_m}\end{array}

i.e., a network can be equivalent to the simple sum of several independent imaginary chains with the aid of the transformation R.

There is a remark to be made here: Usually, the role of the potential A is shown as the A-B effect of Bloch electron in a close chain (or called a tight-binding ring). Here, the local magnetic field strength for Bloch electron may vanish, but the loop integral of A-the magnetic flux does not. Due to the A-B effect, the magnetic flux ϕ can be used to control the single-particle spectrum of a homogeneous chain, ε_k = –2t \cos (k+2\mathrm{\pi }\varphi /N). When the flux \varphi \sim {\varphi }_n\equiv (n/2+1/4)N, the lower spectrum becomes a linear dispersion approximately, i.e., ε_k ~ k. For the wave packet as a superposition of those eigenstates with small k, the linearization of Hamiltonian lead to the transfer of the wave packet without spreading [29].

3 The basic blocks of quantum network : star- and Y-shaped beams

In this section, we use analytical method and numerical simulation to study the basic blocks of quantum network, the star- and Y-shaped beams, which are constructed by connecting one end of several chains to the end (a node) of a single chain. Beam splitters are the elementary optical devices frequently used in classical and quantum optics [30, 31], which can even work well in the level of single photon quanta [32–34] and are applied to generate quantum entanglement [35, 36]. For matter wave, an early beam splitter can be referred to the experiments of neutron interference based on a perfect crystal interferometer with wavefront and amplitude division [37]. Moreover, for cold atoms, a beam splitter has been experimentally implemented on the atom chip [38–40] The theoretical method has been suggested to realize the beam splitter for the Bose-Einstein condensate [41, 42].

In the following, we begin with the basic properties of the corresponding networks for fermions. We will show that, for appropriate joint hopping integrals, the complicated network can be reduced into an imaginary linear chain with homogeneous NN hopping terms plus a smaller complicated network. We also further study the dynamic property of such kind of network by taking the Y-shaped network as an example of star shape beams, in which only three chains are involved. We investigate various aspects that are analogous to quantum optical devices, such as beam-splitters, entanglers, and interferometers.

3.1 Star-shaped beam splitter and its reduction

We consider a simple TBN, a star shape (we also call the star-shaped tight-binding network (STBN)) as shown in Fig. 1 (a). The STBN is constructed by linking the m output chains B_p (p = 1, 2, \cdots, m) to the one end of the input chain A by the hopping integrals t_{N_{A}1}^{[A{B}_p]}. The Hamiltonian of an STBN consists of the linear chain part (1) and the joint part (4) around the last N_A th site of the input chain A. Obviously, since there is no vector potential acting on the network, all the hopping integrals are real.

We will show that, under some restriction for the intrachain hopping constants t and interchain hopping constants t_{N_{A}1}^{[A{B}_p]}, an STBN can be reduced into an imaginary linear tight-binding chain with homogeneous hopping constants plus a smaller complicated network. The fact that the input chain A is a part of this virtual linear chain implies that the Bloch electron can perfectly propagate in this virtual linear chain without the reflection by the node. This also indicates that there is a coherent split of the input electronic wave because the wave function in this virtual chain actually is just a superposition of wave functions in the m chains.

To sketch our central idea, we first consider a special STBN, which consists of m identical “output” chains B₁, B₂, \cdots, B_m with homogeneous intrachain hopping constants t, interchain hopping constant t_{N_{A}1}^{[A{B}_p]}=t_n and the same length N, while the length of chain A is M. The Hamiltonian of the network

\begin{array}{}\text{(8)}& H={\displaystyle \sum _{p=1}^m{H}_{B_p}}+H_A+H_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}\end{array}

is now explicitly written in terms of the chain Hamiltonians H_{B_p} and H_A defined by Eq. (1). Here, the basic parameters for the network are t_j^{[B_p]}=t_j^{[A]}=t,N_{B_p}=N, and N_A = M, where p = 1, 2, \cdots, m. The joint Hamiltonian is

\begin{array}{}\text{(9)}& H_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}=-t_n\left(a_{A,\text{}M}^{†}{\displaystyle \sum _{p=1}^m{a}_{B_p,1}}+H.c.\right)\end{array}

Note that we only consider the case that all the hopping integrals over the joints are identical for the convenience of illustration. In the next section, the different joint hopping integrals will be taken into account for a simplest case of m = 2.

Now we construct the new fermion operators denoted by the tilde notation, {\tilde{a}}_{a,j}^{†} of virtual tight-binding chain a of length M + N as:

\begin{array}{}\text{(10)}& \begin{array}{l}{\tilde{a}}_{a,j}^{†}=a_{A,j}^{†}\hfill \\ {\tilde{a}}_{a,M+l}^{†}=\frac{1}{\sqrt{m}}{\displaystyle \sum _{p=1}^m{a}_{B_p,l}^{†}}\hfill \end{array}\end{array}

where j\in [1,M] and l\in [1,N]. There exist m–1 complementary tight-binding chains with the collective operators:

\begin{array}{}\text{(11)}& {\tilde{a}}_{b_q,j}^{†}=\frac{1}{\sqrt{m}}{\displaystyle \sum _{p=1}^m\exp (-\mathrm{i}2\mathrm{\pi }pq/m)a_{B_p,j}^{†}}\end{array}

where j\in [1,N],p=1,2,\cdots ,m, and q=1,2,\cdots ,m-1.It can be checked that, all the tilde operators are also the standard fermion operators, which satisfy the anticommutation relation:

\begin{array}{}\text{(12)}& \left\{{\tilde{a}}_{\alpha ,i},{\tilde{a}}_{\beta ,j}^{†}\right\}={\mathrm{\delta }}_{\alpha \beta }{\mathrm{\delta }}_{ij}\end{array}

where \alpha ,\beta \in (a,b_1,b_2,\cdots ,b_{m-1}) denote the labels of the virtual chains. By inverting Eqs. (10) and (11) we have

\begin{array}{}\text{(12)}& \begin{array}{l}{a}_{A,j}^{†}={\tilde{a}}_{a,j}^{†},j=1,2,\cdots ,M\\ a_{B_p,j}^{†}=\frac{1}{\sqrt{m}}{\displaystyle \sum _{q=1}^{m-1}{e}^{\mathrm{i}\frac{2\mathrm{\pi }pq}{m}}}{\tilde{a}}_{b_q,j}^{†}+\frac{1}{\sqrt{m}}{\tilde{a}}_{a,M+j}^{†}\end{array}\end{array}

where p\in [1,m] and j\in [1,N]. These establish the mapping R between the two sets of fermion operators, \left\{a_{l,j}^{†}\right\}\stackrel{R}{\to }\left\{{\tilde{a}}_{m,i}^{†}\right\}.Therefore, we have

\begin{array}{}\text{(13)}& \begin{array}{l}H=-t{\displaystyle \sum _{j=1}^{M-1}{\tilde{a}}_{a,j}^{†}}{\tilde{a}}_{a,j+1}-t{\displaystyle \sum _{j=1}^{N-1}({\displaystyle \sum _{q=1}^{m-1}{\tilde{a}}_{b_q,j}^{†}{\tilde{a}}_{b_q,j+1}}}\\ +{\tilde{a}}_{a,M+j}^{†}{\tilde{a}}_{a,M+j+1})-\sqrt{m}{t}_n{\tilde{a}}_{a,M}^{†}{\tilde{a}}_{a,M+1}+H.c.\end{array}\end{array}

The above Hamiltonian depicts a TBN with different geometry. It is easy to observe that only when the matching condition of the joint hopping constants

\begin{array}{}\text{(14)}& t_{N_{A}1}^{[A{B}_p]}=t_n=\frac{t}{\sqrt{m}}\end{array}

is satisfied, we have H={\tilde{H}}_a+{\displaystyle {\sum }_{q=1}^{m-1}\tilde{H}}{b}_q

\begin{array}{}\text{(15)}& \begin{array}{l}{\tilde{H}}_a=-t{\displaystyle \sum _{j=1}^{M+N-1}{\tilde{a}}_{a,j}^{†}}{\tilde{a}}_{a,j+1}+H.c.\\ {\tilde{H}}_{b_q}=-t{\displaystyle \sum _{j=1}^{N-1}{\tilde{a}}_{b_q,j}^{†}}{\tilde{a}}_{b_q,j+1}+H.c.\end{array}\end{array}

where N_a = M + N and N_{b_q}=N. The tilde Hamiltonians are also illustrated in Fig. 1 (b). Interestingly, all the sub-Hamiltonians {\tilde{H}}_a,{\tilde{H}}_{b_q} commutate with each other, i.e.,

\begin{array}{}\text{(16)}& \left[{\tilde{H}}_{\alpha },{\tilde{H}}_{\beta }\right]=0\end{array}

where \alpha ,\beta \in (a,b_1,b_2,\cdots ,b_{m-1}). This fact means that the virtual chain described by {\tilde{H}}_a is just a standard tight-binding chain of length N_a with uniform NN couplings. For an arbitrary initial state localized within the chain H_a, it will evolve driven by the virtual chain of length M+N. If the local state moves out of chain H_a, an ideal beam splitter can be realized since there is no reflection at the node.

So far we have shown the decomposition of the Hamiltonian exactly, which are valid for many-particle system. In the following numerical simulation, only single-particle case is considered for simplicity. To demonstrate the decomposition and its application to quantum information processing (QIP), we take an example with the initial state as the Gaussian wave packet (GWP), which was first well-studied for quantum state transfer in spin system [9] and then tight-binding system [29]. The GWP with momentum \mathrm{\pi }/2 has the form:

\begin{array}{}\text{(17)}& |\psi (N_0)\rangle =\frac{1}{\sqrt{\Omega }}{\displaystyle \sum _{j=1}^M{\mathrm{e}}^{-\frac{{\alpha }^2}{2}{(j-N_0)}^2}}{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{2}j}|j\rangle \end{array}

here, \Omega ={\displaystyle {\sum }_{j=1}^{N_A}\exp [-{\alpha }^2{(j-N_0)}^2]} is the normalization factor and N_0\in [1,N_A] is the initial central position of the GWP at the input chain A, while the factor \alpha is large enough to guarantee the locality of the state in the chain A. Accordingly, the basis |j\rangle is defined as |j\rangle ={\tilde{a}}_{a,j}^{†}|0\rangle for j\in [1,M+N]. In the previous work [29], it has been shown that such GWP can approximately propagate along a homogenous chain without spreading. Actually, at a certain time τ such GWP evolves into

\begin{array}{}\text{(18)}& \begin{array}{l}|\Psi (\tau )\rangle ={\mathrm{e}}^{-\mathrm{i}{\tilde{H}}_a\tau }|\psi (N_0)\rangle \approx |\psi (N_0+2t\tau )\rangle \\ =\frac{1}{\sqrt{\Omega }}{\displaystyle \sum _{j=M+1}^{M+N}{\mathrm{e}}^{-\frac{{\alpha }^2}{2}{(j-N_0-2t\tau )}^2}}{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{2}j}|j\rangle \end{array}\end{array}

in the virtual space. From the mapping of the operators (10), we have the final state as:

\begin{array}{}\text{(19)}& |\Psi (\tau )\rangle =\frac{1}{\sqrt{m}}{\displaystyle \sum _{p=1}^m|{\varphi }_p(N_{\tau })\rangle }\end{array}

here, the state

\begin{array}{}\text{(20)}& |{\varphi }_p(N_{\tau })\rangle =\frac{1}{\sqrt{\Omega }}{\displaystyle \sum _{j=1}^N{\mathrm{e}}^{-\frac{a^2}{2}{(j-N_{\tau })}^2}{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{2}j}{a}_{B_{p,}j}^{†}|0\rangle }\end{array}

is the clone of the initial GWP with the center at N_τ = N₀ + 2tτ–M in the chain B_p. Then we conclude that the beam splitter split the single-particle GWP into m cloned GWPs without any reflection. Furthermore, it is obvious that the function of the splitter originates from the reduction (15) of the Hamiltonian, which is available for every invariant subspaces of fixed particle number. Therefore, such splitter can be applied to the many-particle system.

The above discussion is limited to the simplest case of identical joint hopping integrals. We would like to say that the marching condition (14) is not unique for constructing an independent virtual chain. We will demonstrate this for the case m = 2 in the next section.

3.2 Y-shaped beam splitter

The reduction for star-shaped network was demonstrated under the restriction (14). This kind of reduction can also be performed with different joint hopping integrals. If the splitter is applied to the local GWP, the node interactions of the two output legs are not necessary to be identical. In the following, we will investigate this issue by considering the simplest configuration with m = 2, which is called Y-beam. The asymmetric Y-beam consists of three legs A, B and C with the intrachain hopping integrals t for F = A, B, C and the joint ones t_nF for F = B, C [see Fig. 2 (a)]. The total Hamiltonian reads

\begin{array}{}\text{(21)}& H={\displaystyle \sum _{F=A,B,C}{H}_F-{\displaystyle \sum _{F=B,C}(t_{nF}{a}_{A,M}^{†}{a}_{F,1}+H.c.)}}\end{array}

where t_j^{[A]}=t_j^{[B]}=t_j^{[C]}=t,N_A=M,\text{\hspace{0.33em}and\hspace{0.33em}}{N}_B=N_C=N.

In order to decouple this Y-beam as two virtual linear tight-binding chains, we need to optimize the asymmetric couplings so that the perfect transmission can occur in the decoupled linear tight-binding chains. For this purpose, we introduce the tilde operators of fermion:

\begin{array}{}\text{(22)}& \begin{array}{l}{\tilde{a}}_{a,j}^{†}=a_{A,j}^{†}\\ {\tilde{a}}_{a,M+l}^{†}=\cos \theta a_{B,l}^{†}+\sin \theta a_{C,l}^{†}\\ {\tilde{a}}_{b,m}^{†}=\sin \theta a_{B,m}^{†}-\cos \theta a_{C,m}^{†}\end{array}\end{array}

where j\in [1,M],l\in [1,N], and m\in [1,N]. Here, the mixing angle θ is to be determined as follows by the optimization for quantum state transmission. In comparison with the optical beam splitter, the above equations (22) can be regarded as a fundamental issue for the electronic wave beam splitter.

Together with the original creation operator {\tilde{a}}_{a,j}^{†}=a_{A,j}^{†} for the input leg, the set \left\{{\tilde{a}}_{a,M+j}^{†}\right|j\in [1,N]\} defines a new linear chain a with the effective hopping integrals:

\begin{array}{}\text{(23)}& \begin{array}{l}{t}_{aj}=t_{a,M+l}=t\\ t_{aM}=t_{nB}\cos \theta +t_{nC}\sin \theta \end{array}\end{array}

where j\in [1,M-1] and l\in [1,N-1]. Another virtual linear chain b is defined by {\tilde{a}}_{b,j}^{†} with homogeneous hopping integral t_bj = t, j\in [1,N-1].

In general, these two linear chains are dependent, since there is a connection interaction around the node

\begin{array}{}\text{(24)}& H_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}=-t_{AB}({\tilde{a}}_{a,M}^{†}{\tilde{a}}_{b,1}+H.c.)\end{array}

where

\begin{array}{}\text{(25)}& t_{AB}=t_{nB}\sin \theta -t_{nC}\cos \theta \end{array}

Fortunately, the two virtual chains decouple with each other when the mixing angle θ and the intrachain connections are optimized by setting tan θ = t_nC/t_nB. And if we take t_AB = t, the virtual chain a becomes a completely homogeneous chain of length M + N as illustrated in Fig. 2 (b). Thus, these lead to the matching condition

\begin{array}{}\text{(26)}& \sqrt{t_{nC}^2+t_{nB}^2}=t\end{array}

for Y-beam network. It can be employed to transfer the quantum state without reflection on the node in the transformed picture. Transforming back to the original picture, we can see that such network behaves as a perfect beam splitter.

In the point of view of linear optics, such beam splitting process can generate the mode entanglement between the separated waves in chains B and C, and the measure of such mode entanglement is determined by the values of t_nB and t_nC. We will show that the strength of t_nB and t_nC can be used to control the amplitudes of the evolving Bloch electron wave packets on legs B and C.

Now we apply the beam splitter to a special Bloch electron wave packet, a GWP with momentum π/2, which has the form (17) at τ = 0. It is known from the previous work [29] that such GWP can approximately propagate along a homogenous chain without spreading. Then at a certain time τ, such GWP evolves into

\begin{array}{}\text{(27)}& |\Psi (\tau )\rangle =\cos \theta |{\psi }_B(N_{\tau })\rangle +\sin \theta |{\psi }_C(N_{\tau })\rangle \end{array}

where

\begin{array}{}\text{(28)}& N_{\tau }=N_0+2t\tau -M\end{array}

i.e., the beam splitter can split the GWP into two cloned GWPs completely. The possibility of GWPs in the arms B and C is determined by the mixed angle θ.

In order to verify the above analysis, a numerical simulation is performed for a GWP with α = 0.3 in a finite system with N_A = N_B = N_C = 50. Let |\Phi (0)\rangle be a normalized initial state. Then the reflection factor at time τ can be defined as

\begin{array}{}\text{(29)}& R(t_{nC},t_{nB},\tau )={\displaystyle \sum _{j_A=1}^{M-1}{|\langle j_A\left|{\mathrm{e}}^{-\mathrm{i}H\tau }\right|\Phi (0)\rangle |}^2}\end{array}

to depict the reflection at the node. At an appropriate instant τ₀,

\begin{array}{}\text{(30)}& R(t_{nC},t_{nB})=R(t_{nC},t_{nB},{\tau }_0)\end{array}

as a function of t_nC and t_nB is plotted in Fig. 3. It shows that around the matching condition (26), there flection factor vanishes, which is just in agreement with our analytical result.

The conclusion for such GWP comes from the reduction of the Y-shaped network, which the equal length of two output arms is crucial. However, for the local wave packet (17), the local environment around the wave packet only results in its behavior at the next instant. This can be seen from the speed of the GWP (17). According to the study in Ref. [29], the speed of the GWP is independent of the size and the boundary condition (ring or open chain). Therefore, for a splitter to GWP, the equality of two output arms is not necessary. This argument will be demonstrated in the following content about quantum interferometer.

3.3 Entangler of Bloch electron

Now we consider how the STBN can behave as an entangler to produce entanglement with the Y-beam as an illustration. Let the input state |\varphi (0)\rangle, represent a single-particle state located in the arm A. It can propagate into the arms B and C through the node with some reflection. On the other hand, the electronic wave can be regarded as being transferred along the virtual legs a and b. Once we manipulate the joint hopping integrals to satisfy the matching condition, the electronic wave can only enter the virtual chain a rather than b without any reflection. Then the final state is in the subspace of the virtual chain a. Similar to optical splitter, such Y-beam splitting can also be regarded as an entangler of fermion. For instance, consider a state |D(j)\rangle ={\tilde{a}}_{a,j}^{†}|0\rangle for j\in [M,M+N], which is a local state in the viewpoint of the virtual chain a. However, in the real space, this state is nonlocal and possesses mode entanglement, while state |D(j)\rangle for j\in [1,M] is still a non-entangled state. Obviously, the Y-beam acts as an entangler similar to that in quantum optical systems.

To quantitatively characterize mode entanglement generated by the splitter on the joint hopping integrals t_nC, t_nB, we consider the GWP (17) as an initial state. Through the splitter, two separated GWPs are obtained. The total concurrence with respect to the two waves located at the arms B and C can be calculated as

\begin{array}{}\text{(31)}& C(\tau )={\displaystyle \sum _{j=1}^N|\langle \Psi (\tau )|(a_{B,j}^{†}{a}_{C,j}+a_{B,j}{a}_{C,j}^{†})|\Psi (\tau )\rangle |}\end{array}

according to Refs. [43–45]. When the interchain connections are optimized by setting t_nB = tcosθ, t_nc = tsinθ, the above mode concurrence can be calculated as

\begin{array}{}\text{(32)}& C(\tau )=\mathrm{s}\mathrm{i}\mathrm{n}(2\theta )\end{array}

from the Eq. (27).

It is obvious that if cos θ = sin θ = 1/\sqrt{2}, i.e.,

\begin{array}{}\text{(33)}& t_{nB}=t_{nC}=\frac{t}{\sqrt{2}}\end{array}

C (τ) reaches its maximum value 1. Numerical simulation for t_{nB,}{t}_{nC}\in [0,2t] is performed for a GWP with α= 0.3 and momentum π/2 in a finite system with N_A = 50, N_B = 50, and N_C = 50. The concurrence is also the function of time τ due to the dynamics of the system. We choose maximal concurrence

\begin{array}{}\text{(34)}& C_{\max }(t_{nC},t_{nB})=\max \{C(\tau )\}\end{array}

as a function of t_nC and t_nB to depict the property of the splitter. Numerical result is plotted in Fig. 4. It shows that two split wave packets yield the maximal entanglement just at the matching point (33).

3.4 Quantum interferometer for Bloch electron

Now we consider in detail a more complicated TBN than the Y-beam, the quantum interferometer for Bloch electron wave. This setup consists of two Y-beams, which is illustrated schematically in Fig. 5 (a). It is similar to the optical interferometer, where state |a\rangle of a single photon is split into two parts |b\rangle and |c\rangle by the splitter and then a new state |d\rangle =U_B|b\rangle +U_C|c\rangle can be achieved by the unitary transformations U_B and U_C. In the tight-binding Bloch electron interferometer, the analogue of the import state |a\rangle is the moving GWP (17).

Firstly, we consider the simplest case with the path difference [defined in Fig. 5 (a)] Δ = 0. It is shown that such network is equivalent to two independent virtual chains with lengths N_A + N_B + N_D and N_B, respectively, when the coupling matching condition is satisfied. Then the initial GWP will be transmitted into the arm D without any reflection. This fact can be understood according to the interference of two split GWPs. Actually, from the above analysis about the GWP propagating in the Y-beam, we note that the conclusion can be extended to the Y-beam consisting of two different length arms N_B ≠ N_C. It is due to the locality of the GWP and the fact that the speed of the GWP only depends on the hopping integral. Then the arrival time of the two split GWPs at arm D depends on the lengths N_B and N_C. It means that the nonzero Δ should affect the shape of the pattern of output wave.

To verify the analysis above, we investigate this problem again numerically. According to quantum mechanics, the interference pattern at site r₀ and time τ₀ in arm D can be presented as

\begin{array}{}\text{(35)}& I(r_0,{\tau }_0,\Delta )={|\langle r_0|\exp (-\mathrm{i}H{\tau }_0)|\Phi (0)\rangle |}^2\end{array}

Numerical simulation of I (r₀, τ₀, Δ) for the input GWP in the interferometric network with N_A = N_B = N_D = 50, N_C = N_B + Δ is performed. For r₀ = 50, τ₀ = 100/t, a perfect interference phenomenon by I (r₀,τ₀,Δ) is observed for the range \Delta \in[–25, 25] in Fig. 5 (b). This observation shows that the quantum interferometer can be realized by the TBN.

4 Q - shaped TBN controlled by flux

From the above discussion, it can be found that the essence of the reduction for the TBN lies on the interference of the matter wave. On the other hand, the presence of vector potential can induce a phase factor to the wave function and then the magnetic flux can control the coherent reduction to some extent. In this section, we investigate how to control the motion of the Bloch electron along this TBN by an external magnetic field. We will show that the appropriate flux through the network can reduce the network to the linear virtual chain, which indicate that the flux can control the propagation of GWP in the network.

4.1 Model and Hamiltonian

Consider a quantum network constructed by connecting the two free ends of the two identical chains in Y-beam. Such a network is called Q-shaped TBN, or QTBN labeled by {A, B, C}. As illustrated schematically in Fig. 6(a), QTBN is placed in an external magnetic field. The ring of the model is threaded by a magnetic flux ϕ in the unit of flux quanta. Here, we only consider the effect of the vector potential A without the Zeeman term for simplicity. The Hamiltonian of our Q-shaped lattice model reads

\begin{array}{}\text{(36)}& H=H_A+H_B+H_C+H_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}\end{array}

where

\begin{array}{}\text{(37)}& \begin{array}{ll}{H}_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}=\hfill & -(t_{nB}{a}_{A,M}^{†}{a}_{B,1}{\mathrm{e}}^{\mathrm{i}{\Phi }_{B,1}}+t_{nC}{a}_{A,M}^{†}{a}_{C,1}{\mathrm{e}}^{-\mathrm{i}\Phi {}_{C,1}})\hfill \\ \hfill & -t{a}_{B,N}^{†}{a}_{C,N}{\mathrm{e}}^{\mathrm{i}{\Phi }_{BC}}+H.c.\hfill \end{array}\end{array}

and the parameters N_A=M,N_B=N_C=N,t_j^{[A]}=t,t_j^{[B]}=t \exp (\mathrm{i}{\Phi }_{B,j}{}_{+1}),t_j^{[C]}=t\exp (-\mathrm{i}{\Phi }_{C,j+1}). Here, Φ_B,l, Φ_C,l, l\in[1, N] denote the phase differences between the neighboring sites l and l + 1 in the chains B and C, while Φ_BC with respect to the connection between the two chains. The values of the phase difference is defined as (3) and Φ_B,l, Φ_C,l, and Φ_BC are not required to be identical in order to avoid losing generality.

In the following discussion, what we concern is only the sum of the phase difference along the loop

\begin{array}{}\text{(38)}& \Phi ={\displaystyle \sum _{l=1}^{{\rm N}}({\Phi }_{B,l}+{\Phi }_{C,l})+{\Phi }_{BC}}\end{array}

corresponding to the flux ϕ= Φ/2π through the loop. We will show that the flux ϕ can control the dynamics of the Q-shaped lattice system.

4.2 Reduction of the Q-shaped lattice model

The QTBN with the flux ϕ, and the joint hopping strengths t_nB, t_nC, exhibits a rich variety of dynamic behaviors. Fortunately, we find that there exist analytical results in some range of parameters. Together with the numerical simulation, these analytical results are helpful to get a comprehensive understanding of the mechanism. We start with the cases of fixed Φ but various t_nB, t_nC and then investigate the cases vice-versa.

4.2.1 Case: \varphi =\frac{n}{2}

Our aim is try to decouple this Q-shaped model as two virtual linear chains. We first introduce two anticommutative sets of fermion operator \left\{{\tilde{a}}_{a,M+j}^{†}\right\},\left\{{\tilde{a}}_{b,j}^{†}\right\} defined by

\begin{array}{}\text{(39)}& \begin{array}{l}{\tilde{a}}_{a,M+j}^{†}=\cos \theta {\mathrm{e}}^{-\mathrm{i}{\phi }_B^j}{a}_{B,j}^{†}+\sin \theta {\mathrm{e}}^{\mathrm{i}{\phi }_C^j}{a}_{C,j}^{†}\hfill \\ {\tilde{a}}_{b,j}^{†}=\sin \theta {\mathrm{e}}^{-\mathrm{i}{\phi }_B^j}{a}_{B,j}^{†}-\cos \theta {\mathrm{e}}^{\mathrm{i}{\phi }_C^j}{a}_{C,j}^{†}\hfill \end{array}\end{array}

for j\in [1,N], where {\phi }_{\alpha }^j={\displaystyle {\sum }_{l=1}^j{\Phi }_{\alpha ,l}(\alpha =B,C).}

We can check that they still satisfy the anticommutation relations \{{\tilde{a}}_{\alpha ,i},{\tilde{a}}_{\beta ,j}^{†}\}={\mathrm{\delta }}_{\alpha \beta }{\mathrm{\delta }}_{ij}, where \alpha ,\beta \in (a,b). The inverse transformations of Eq. (39), together with the original fermion operator {\tilde{a}}_{a,j}^{†}=a_{A,j}^{†},j\in [1,M], define a new linear chain a, while another virtual linear chain b is only constructed by {\tilde{a}}_{b,j}^{†},j\in [1,N]. Therefore, the parameters are taken as:

\begin{array}{}\text{(40)}& \varphi =\frac{n}{2};t_{nB}=t_{nC}=\frac{t}{\sqrt{2}}\end{array}

the Hamiltonian can be reduced as:

\begin{array}{}\text{(41)}& \begin{array}{l}H={\tilde{H}}_a+{\tilde{H}}_b+{\tilde{H}}_{\mu }\hfill \\ {\tilde{H}}_{\mu }=t{(-1)}^n{\tilde{n}}_{a,M+N}-t{(-1)}^n{\tilde{n}}_{b,}{}_N\hfill \end{array}\end{array}

with N_a = M + N and N_b = N. Here, {\tilde{n}}_{a,M+N}={\tilde{a}}_{a,M+N}^{†} {\tilde{a}}_{a,M+N} and {\tilde{n}}_{b,N}={\tilde{a}}_{b,N}^{†}{\tilde{a}}_{b,N} are the particle number operators. The {\tilde{H}}_{\mu } represents the chemical potential at the ends of chains a and b. For a large N system, the effect of the end potentials can be ignored. Thus, the Q-type lattice can be reduced into two independent virtual linear chains a and b with homogeneous NN hopping integrals, and length M + N and N respectively as illustrated in Fig. 6 (b).

4.2.2 Case: \varphi =\frac{n}{2}+\frac{1}{4}

In this case, we will show that the model can be reduced to a virtual chain with M + 2N sites if the joint hopping integrals satisfy \sqrt{t_{nC}^2+t_{nB}^2}=t. We introduce the fermion operators

\begin{array}{}\text{(42)}& \begin{array}{l}{\tilde{a}}_{a,j}^{†}=a_{A,j}^{†}\hfill \\ {\tilde{a}}_{a,\text{}M+l}^{†}=\cos \theta {\mathrm{e}}^{-\mathrm{i}{\phi }_B^l}{a}_{B,l}^{†}+\sin \theta {\mathrm{e}}^{\mathrm{i}{\phi }_C^l}{a}_{C,l}^{†}\hfill \\ {\tilde{a}}_{a,\text{}M+N+l}^{†}=\mathrm{i}{(-1)}^n[\sin \theta {\mathrm{e}}^{-\mathrm{i}{\phi }_B^l}{a}_{B,l}^{†}-\cos \theta {\mathrm{e}}^{\mathrm{i}{\phi }_C^l}{a}_{C,l}^{†}]\hfill \end{array}\end{array}

where j\in [1,M],l\in [1,N]. Similarly, when we set

\begin{array}{}\text{(43)}& \varphi =\frac{n}{2}+\frac{1}{4};t_{nB}=t\cos \theta ,t_{nC}=t\sin \theta \end{array}

the Hamiltonian becomes

\begin{array}{}\text{(44)}& H={\tilde{H}}_a\end{array}

with N_a = M + 2N, which is illustrated schematically in Fig. 6 (c). Then we conclude that, when the flux is ϕ = n/2+1/4 and the joint hopping integrals satisfy \sqrt{t_{nC}^2+t_{nB}^2}=t, the QTBN is equivalent to a single virtual open chain a with length M + 2N.

4.2.3 Case: arbitrary \varphi ,t_{nB}=t_{nC}=\frac{t}{\sqrt{2}}

When we take the interchain hopping integrals as t_{nB}=t_{nC}=t/\sqrt{2}, the mapping (39) reduces the network Hamiltonian as

\begin{array}{}\text{(45)}& \begin{array}{l}H={\tilde{H}}_a+{\tilde{H}}_b+{\tilde{H}}_a{}_b\hfill \\ {\tilde{H}}_a{}_b=-\mu ({\tilde{a}}_{a,M+N}^{†}{\tilde{a}}_{a,M+N}-{\tilde{a}}_{b,N}^{†}{\tilde{a}}_{b,N}^{})\hfill \\ -\tilde{t}({\tilde{a}}_{b,N}^{†}{\tilde{a}}_{a,M+N}+H.c.)\end{array}\end{array}

with N_a = M + N, N_b = N. We have replaced \mathrm{i}{\tilde{a}}_{b,N}^{†} by {\tilde{a}}_{b,N}^{†} for j\in [1,N] without influence on the physics of dynamical process. Here, {\tilde{H}}_a and {\tilde{H}}_b stand for two virtual chains with length M + N and N, respectively, {\tilde{H}}_{ab} represents the chemical potential at the ends of chains a and b and the connection between the two sites. Note that the end-site chemical potentials possess the same magnitude μ = t cos Φ, but of opposite sign and the hopping integral between the two end sites is \tilde{t}=t\sin \Phi . The reduced model is also illustrated in Fig. 6 (d).

Physically, the chemical potentials μ and the hopping integral \tilde{t} have the complementary relation u^2+{\tilde{t}}^2=t^2. When Φ = (n + 1/2)π, the network is equivalent to a linear chain with length M + 2N; while for Φ = nπ it corresponds to two independent chains with lengths M + N and N. In the next section, we will focus on such system for arbitrary Φ. We will show that such system behaves as an optical device, a transmission-reflection film, while the flux determines the coefficient. In conclusion, the magnetic flux ϕ can influence the “effective length” or the connective status of the virtual chains and then can be used to control the dynamics of the network.

4.3 Transmission-reflection film

To make the above observation more clear, we consider two identical tight-binding chains {A, B}, which consist of N sites respectively. There exists a connection interaction between the two terminal sites of the two virtual chains. The hopping constant is t sin Φ and each terminal site also has a chemical potential, ± t cos Φ. The Hamiltonian reads

\begin{array}{}\text{(46)}& H=H_A+H_B+H_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}\end{array}

where N_A = N_B = N, t_j^{[A]}=t_j^{[B]}=t, and

\begin{array}{}\text{(47)}& \begin{array}{cc}{H}_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}=& -t\sin \Phi (a_{A,N}^{†}{a}_{B,N}+H.c.)\hfill \\ \hfill & -t\cos \Phi a_{A,N}^{†}{a}_{A,N}+t\cos \Phi a_{B,N}^{†}{a}_{B,N}\hfill \end{array}\end{array}

Obviously, it is the simplest case of the system described by Eq. (45).

In order to study the properties of this Bloch electron model more clearly, we introduce two anticommutative sets of fermion operators

\begin{array}{}\text{(48)}& \begin{array}{l}{\tilde{a}}_{a,j}^{†}=\frac{\sqrt{2}}{2}\left(f+a_{A,j}^{†}-f_{-}{a}_{B,j}^{†}\right)\\ {\tilde{a}}_{b,j}^{†}=\frac{\sqrt{2}}{2}\left(f-a_{A,j}^{†}+f_{+}{a}_{B,j}^{†}\right)\end{array}\end{array}

where j\in [1,N] and

\begin{array}{}\text{(49)}& f_{\pm }=\cos \frac{\Phi }{2}\pm \sin \frac{\Phi }{2}\end{array}

The inverse transformation of Eq. (48) results in the reduction of the network described by

\begin{array}{}\text{(50)}& H={\tilde{H}}_a+{\tilde{H}}_b+{\tilde{H}}_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}\end{array}

where N_a = N_b = N, and

\begin{array}{}\text{(51)}& {\tilde{H}}_{joint}=-t({\tilde{a}}_{a,N}^{†}{\tilde{a}}_b{}_{,N}+H.c.)\end{array}

Obviously, the Hamiltonian (50) depicts an imaginary linear chain with homogeneous couplings t no matter how much the magnitude of the flux Φ is taken. Then, the Hamiltonian describing transmission-reflection is mapped into a chain in virtual space. Interestingly, such mapping is irrelevant to the state concerned.

In order to demonstrate the function of such network, we study the propagation of a moving GWP. To this end, we consider a GWP defined as (17) at chain A, i.e.,

\begin{array}{}\text{(52)}& |{\psi }_A(N_0)\rangle =\frac{1}{\sqrt{{\Omega }_1}}{\displaystyle \sum _{j=1}^N{\mathrm{e}}^{-\frac{{\alpha }^2}{2}{(j-N_0)}^2}}{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{2}j}{a}_{A,j}^{†}|0\rangle \end{array}

We require this wave packet to satisfy \langle 0\left|a_{B,j}\right|{\psi }_A(N_0)\rangle\approx 0, so that it ensures the initial GWP being located in chain A. The transformation (48) means that such GWP corresponds to the combination of two GWPs in virtual space with the centers at N₀ respectively,

\begin{array}{}\text{(53)}& |{\psi }_{a(b)}(N_0)\rangle =\frac{f_{+(-)}}{\sqrt{2{\Omega }_1}}{\displaystyle \sum _{j=1}^N{\mathrm{e}}^{-\frac{{\alpha }^2}{2}{(j-N_0)}^2}}{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{2}j}{a}_{a(b),j}^{†}|0\rangle \end{array}

Based on the analytical result in Ref. [30, 31], the two GWPs in the virtual chain should travel along the chain defined by Eq. (50) without spreading as time evolution. Then, at a certain time τ, the evolved state |{\varphi }_{a(b)}(\tau )\rangle =\exp (-\mathrm{i}\tilde{H}\tau ) |{\psi }_{a(b)}(N_0)\rangle , or

\begin{array}{}\text{(54)}& \begin{array}{ll}|{\varphi }_{a(b)}(\tau )\rangle =& \frac{1}{\sqrt{2{\Omega }_1}}[f_{+(-)}{\displaystyle \sum _{j=1}^N{\mathrm{e}}^{-\frac{{\alpha }^2}{2}{(j-N_{\tau })}^2}}{\mathrm{e}}^{\mathrm{i}\frac{\pi }{2}j}\hfill \\ \hfill & +f{}_{-(+)}{\displaystyle \sum _{j=1}^N{\mathrm{e}}^{-\frac{{\alpha }^2}{2}{(Pj-N_{\tau })}^2}}{\mathrm{e}}^{\mathrm{i}\frac{\pi }{2}Pj}]a_{a(b),j}^{†}|0\rangle \hfill \end{array}\end{array}

describes the superposition of two GWPs. Here, Pj = 2N + 1 – j, N_τ= N₀ + 2tτ.

Transforming back to the real space, we rewrite the time evolution by the state

\begin{array}{}\text{(55)}& |\Psi (\tau )\rangle =\cos \Phi |{\varphi }_A(\tau )\rangle +\sin \Phi |{\varphi }_B(\tau )\rangle \end{array}

in terms of the two components of the wave function

\begin{array}{}\text{(56)}& \begin{array}{ll}|{\varphi }_A(\tau )\rangle =& \frac{1}{\sqrt{\Omega {}_1}}[{\displaystyle \sum _{j=1}^N{e}^{-\frac{{\alpha }^2}{2}{(Pj-N_{\tau })}^2}}{\mathrm{e}}^{\mathrm{i}\frac{\pi }{2}Pj}\hfill \\ \hfill & +\frac{1}{\cos \Phi }{\mathrm{e}}^{-\frac{{\alpha }^2}{2}{(j-N_{\tau })}^2}{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{2}j}]a_{A,j}^{†}|0\rangle \\ \hfill & \approx \frac{1}{\sqrt{{\Omega }_1}}{\displaystyle \sum _{j=1}^N{\mathrm{e}}^{-\frac{{\alpha }^2}{2}{(Pj-N_{\tau })}^2}}{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{2}Pj}{a}_{A,j}^{†}|0\rangle \hfill \end{array}\end{array}

and

\begin{array}{}\text{(57)}& |{\varphi }_B(\tau )\rangle =\frac{1}{\sqrt{{\Omega }_1}}{\displaystyle \sum _{j=1}^N{\mathrm{e}}^{-\frac{{\alpha }^2}{2}{(Pj-N_{\tau })}^2}{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{2}Pj}}{a}_{B,j}^{†}|0\rangle \end{array}

here, in Eq. (56), the second term is ignored in the case | j – N_τ|\gg1.

Obviously, the central positions of the final sub-GWPs 2N + 1–N_τdecrease with time τ. This observation indicates that, the beam splitter can split the GWP into two cloned GWPs with opposite moving directions along with AB chain. Therefore, state |{\varphi }_A(\tau )\rangle represents the reflection component with probability cos²Φ, while state |{\varphi }_B(\tau )\rangle is the transmission component through the connection of AB with probability sin²Φ. So this Bloch electron network for a moving GWP behaves like an optical transmission-reflection film for photons. Interestingly, transmission and reflection coefficients are governed by the parameter Φ, the flux through the network. This feature is illustrated in Fig. 7 in detail.

4.4 The dynamic properties of the Q-shaped Bloch electron model

Now we take the propagation of the GWP |{\psi }_A(N_0)\rangle as an example. Its advantage is that the GWP we often used can move along a homogeneous chain without spreading approximately. So we can easily see the various characteristics of the model through the propagation of the GWP.

4.4.1 Case: \varphi =\frac{n}{2},t_{nB}=t_{nC}=\frac{t}{\sqrt{2}}

The initial GWP is moving with speed 2 t along chain A. Before it reaches the node, it can also be regarded as moving along the virtual chain a. From the above discussion, the virtual chain a is homogeneous with length M + N and decoupled with another virtual chain b. Thus, in the virtual space, we can see that the GWP moves toward the end site of virtual chain a and then reflects at the boundaries with “π-phase shift ”. It never appears on virtual chain b. Notice that, in this case, t_nB = t_nC = t/\sqrt{2} must be satisfied, and then the GWP in virtual space can be mapped into two identical GWPs with half amplitude of the initial one in the real space. Therefore, the whole propagation process in the real space is as follows: When the initial GWP reaches the node, it is divided into the two identical GWPs which also move with speed 2 t along the legs B and C respectively without spreading. Then the two GWPs reflect completely at the opposite site of the node and come back along the original paths. When they reach the node again, they merge as a big GWP and move out of the ring.

4.4.2 Case: \varphi =\frac{n}{2}+\frac{1}{4},t_{nB}^2+t_{nC}^2=t^2

As shown in subsection (2), the reduction of the Q-shaped Bloch electron model has two main characteristics. First and foremost, it is decoupled as a long virtual chain a with length M + 2N. Secondly, t_nB = t cosθ, t_nC = t sinθ. Thus, when the initial GWP reaches the node for the first time, it is divided into two GWPs with (t_nB /t)² and (t_nC /t)² amplitude of the initial one. They move along the ring for one circle and reach the node again. This time, instead of going out of the ring to the real chain A, they reflect back and continue moving along the ring for another circle until they meet at the node for the third time. After circumambulating two circles, they finally merge into a big one and move out of the ring towards the input leg.

4.4.3 Case: arbitrary \varphi ,t_{nB}=t_{nC}=\frac{t}{\sqrt{2}}

For other values of ϕ and t_nB, t_nC, when the two GWPs reach the node for the second time, parts of them move out while the other parts remain moving on the ring. Especially, when t_nB = t_nC = t\sqrt{2}, the coupling constants and the chemical potentials satisfy the relation discussed in the section “Transmission-reflection film”. Therefore, when the initial GWP reaches the end of the virtual chain a, some novel phenomena occur. Part of it can move onto the virtual chain b and forms a new GWP with sin²ϕ amplitude of the initial one. At the same time, the other part is reflected by the joint and forms a GWP with cos²ϕ amplitude of the initial one. On mapping them to the real space, we can image that when the two sub-GWPs reach the node again. Parts of them are merged as a GWP with cos²Φ amplitude getting out of the ring. However, the other parts move along the ring for another circle before going out.

Therefore, the magnetic flux ϕ can control the amplitude of the outgoing GWP. Such a Q-shaped Bloch electron model can also be used to test the flux ϕ by measuring the probability of the outgoing GWP at some certain instance.

5 Flux-controlled interferometer and its reduction

5.1 Model and Hamiltonian

In this subsection, we consider the interferometer model {A, B, C, D} for Bloch electron, illustrated schematically in Fig. 8 (a). This quantum interferometer consists of two chains A, D and one ring {B, C} with one end of each chain connecting to two opposite points of the ring. The ring is threaded by a magnetic flux ϕ in the unit of flux quanta. The Hamiltonian reads

\begin{array}{}\text{(58)}& H=H_A+H_D+H_B+H_C+H_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}\end{array}

where N_A = M, N_D = L, N_B = N_C = N, t_j^{[A]}=t_j^{[D]}=t,t_j^{[B]}= t\exp (\mathrm{i}{\Phi }_{B,j+1}), and t_j^{[C]}=t\exp (-\mathrm{i}{\Phi }_{C,j+1}). The connection Hamiltonian reads

\begin{array}{}\text{(59)}& \begin{array}{ll}{H}_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}=& -(t_{nAB}{a}_{A,M}^{†}{a}_{B,1}{\mathrm{e}}^{\mathrm{i}{\Phi }_{B,1}}+t_{nAC}{a}_{A,M}^{†}{a}_{C,1}{\mathrm{e}}^{-\mathrm{i}{\Phi }_{C,1}}\hfill \\ \hfill & +t_{nBD}{a}_{B,N}^{†}{a}_{D,1}{\mathrm{e}}^{\mathrm{i}{\Phi }_{B,N+1}}\hfill \\ \hfill & +t_{nCD}{a}_{C,N}^{†}{a}_{D,1}{\mathrm{e}}^{-\mathrm{i}{\Phi }_{C,N+}{}_1}+H.c.)\end{array}\end{array}

here, \Phi ={\displaystyle {\sum }_{l=1}^{N+1}{\Phi }_{B,l}}+{\displaystyle {\sum }_{l=1}^{N+1}{\Phi }_{C,l}}=2\mathrm{\pi }\varphi is the sum of Φ along the ring.

In this section, we investigate the basic properties of the flux-controlled interferometer. Similarly, we will find out that there still exist some analytical results, which reveal the dynamics of such network for the appropriate parameters.

5.2 Reduction of the interferometer network

To reduce the network of interferometers, the four sets of new fermion operator

\begin{array}{}\text{(60)}& \begin{array}{l}{\tilde{a}}_{a,j}^{†}={\tilde{a}}_{A,j}^{†}\\ {\tilde{a}}_{a,M+l}^{†}=\cos \theta {\mathrm{e}}^{-\mathrm{i}{\phi }_B^l}{a}_{B,l}^{†}+\sin \theta {\mathrm{e}}^{\mathrm{i}{\phi }_C^l}{a}_{C,l}^{†}\hfill \\ {\tilde{a}}_{b,l}^{†}=\sin \theta {\mathrm{e}}^{-\mathrm{i}{\phi }_B^l}{a}_{B,l}^{†}-\cos \theta {\mathrm{e}}^{\mathrm{i}{\phi }_C^l}{a}_{C,l}^{†}\hfill \\ {\tilde{a}}_{c,s}^{†}={\mathrm{e}}^{\mathrm{i}{\phi }_C^{N+1}}{a}_{D,s}^{†}\hfill \end{array}\end{array}

for j\in [1,M],l\in [1,N], and s\in [1,L], are introduced to satisfy

\begin{array}{}\text{(61)}& \left\{{\tilde{a}}_{a,M+j},{\tilde{a}}_{b,j}^{†}\right\}=0\end{array}

here,

\begin{array}{}\text{(62)}& {\phi }_{\alpha }^j={\displaystyle \sum _{l=1}^j{\Phi }_{\alpha ,l},\alpha =B,C}\end{array}

is the sum of the phase.

The inverse transformation of the above Eqs. (60) reduces the Hamiltonian (58) into

\begin{array}{}\text{(63)}& \begin{array}{ll}H=& {\tilde{H}}_a+{\tilde{H}}_b+{\tilde{H}}_d\hfill \\ \hfill & -t{\displaystyle \sum _{j=1}^{N-1}({\tilde{a}}_{a,M+j}^{†}{\tilde{a}}_{a,M+j+1}+H.c.)}\hfill \\ \hfill & -[(t_{nAB}\cos \theta +t_{nAC}\sin \theta ){\tilde{a}}_{a,M}^{†}{\tilde{a}}_{a,M+1}\hfill \\ \hfill & +(-t_{nAB}\sin \theta +t_{nAC}\cos \theta ){\tilde{a}}_{a,M}^{†}{\tilde{a}}_{b,1}\hfill \\ \hfill & +(t_{nBD}\cos \theta {\mathrm{e}}^{\mathrm{i}\Phi }+t_{nCD}\sin \theta ){\tilde{a}}_{a,M+N}^{†}{\tilde{a}}_{c,1}\hfill \\ \hfill & +(-t_{nBD}\sin \theta {\mathrm{e}}^{\mathrm{i}\Phi }+t_{nCD}\cos \theta ){\tilde{a}}_{b,N}^{†}{\tilde{a}}_{c,1}+H.c.]\hfill \end{array}\end{array}

where N_a = M, N_b = N, and N_c = L. Now we concentrate on two special cases with different ϕ and other parameters.

5.2.1 Case: \varphi =n+\frac{1}{2}, t_nAB = t_nCD = t cosθ, t_nAC = t_nBD = t sinθ

It is obvious that exp (iΦ) =–1. The Hamiltonian can be rewritten as

\begin{array}{}\text{(64)}& \begin{array}{ll}H=& -t({\displaystyle \sum _{j=1}^{M+N-1}{\tilde{a}}_{a,j}^{†}+{\tilde{a}}_{a,j+1}+{\displaystyle \sum _{j=1}^{L-1}{\tilde{a}}_{c,j}^{†}{\tilde{a}}_{c,j+1}}}\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^{N-1}{\tilde{a}}_{b,j}^{†}{\tilde{a}}_{b,j+1}+{\tilde{a}}_{b,N}^{†}{\tilde{a}}_{c,1}+H.c.})\hfill \end{array}\end{array}

We define the new fermion operator

\begin{array}{}\text{(65)}& {\tilde{a}}_{b,N+j}^{†}={\tilde{a}}_{c,j}^{†},j\in [1,L]\end{array}

to extend the virtual chain b. Its Hamiltonian

\begin{array}{}\text{(66)}& H={\tilde{H}}_a+{\tilde{H}}_b\end{array}

is given by the parameters N_a = M + N and N_b = N + L.

From the reduced Hamiltonian (66), we can see that the interferometer network is decoupled into two imaginary linear chains with homogeneous couplings t. The set of \left\{{\tilde{a}}_{a,j}^{†}\right|j\in [1,N+M]\} defines one of them with length M + N sites, and the set \left\{{\tilde{a}}_{a,j}^{†}\right|j\in [1,N+L]\} defines the other one with length N + L. As illustrated schematically in Fig. 8 (b).

5.2.2 Case: ϕ = n, t_nAB = t_nBD = t cos θ, t_nAC = t_nCD =t sin θ

In this case, the Hamiltonian becomes

\begin{array}{}\text{(67)}& \begin{array}{ll}H=\hfill & -t({\displaystyle \sum _{j=1}^{M+N-1}{\tilde{a}}_{a,j}^{†}{\tilde{a}}_{a,j+1}^{†}}+{\displaystyle \sum _{j=1}^{L-1}{\tilde{a}}_{c,j}^{†}{\tilde{a}}_{c,j+1}}\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^{N-1}{\tilde{a}}_{b,j}^{†}{\tilde{a}}_{b,j+1}+{\tilde{a}}_{a,M+N}^{†}{\tilde{a}}_{c,1}+H.c.})\hfill \end{array}\end{array}

with the newly defined operators

\begin{array}{}\text{(68)}& {\tilde{a}}_{a,M+N+j}^{†}={\tilde{a}}_{c,j}^{†},j\in [1,L]\end{array}

the reduced Hamiltonian

\begin{array}{}\text{(69)}& H={\tilde{H}}_a+{\tilde{H}}_b\end{array}

describes an extended virtual chain of length N_a = M + N + L and another of N_b = N.

It is clear that, when the conditions

\begin{array}{}\text{(70)}& \begin{array}{l}{t}_{nAB}=t_{nBD}=t\mathrm{c}\mathrm{o}\mathrm{s}\theta \hfill \\ t_{nAB}=t_{nCD}=t\mathrm{s}\mathrm{i}\mathrm{n}\theta \hfill \\ \varphi =n\end{array}\end{array}

are satisfied, the interferometer network is decoupled into two imaginary linear chains with length M + N + L sites and N sites, respectively. See also Fig. 8 (c).

5.2.3 Case: arbitrary ϕ, t_nAB = t_nAC = t_nBD = t_nCD =\frac{t}{\sqrt{2}}

Under this condition, the Hamiltonian is reduced as

\begin{array}{}\text{(71)}& H={\tilde{H}}_a+{\tilde{H}}_b+{\tilde{H}}_c+{\tilde{H}}_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}\end{array}

where N_a = M + N, N_b = N, and N_c = L.

Here, the joint Hamiltonian is

\begin{array}{}\text{(72)}& \begin{array}{ll}{\tilde{H}}_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}=& -t[{\mathrm{e}}^{\mathrm{i}\frac{\Phi }{2}}\cos (\frac{\Phi }{2}){\tilde{a}}_{a,M+N}^{†}{\tilde{a}}_{c,1}\hfill \\ \hfill & -\mathrm{i}{\mathrm{e}}^{\mathrm{i}\frac{\Phi }{2}}\sin \left(\frac{\Phi }{2}\right){\tilde{a}}_{b,N}^{†}{\tilde{a}}_{c,1}+H.c.]\end{array}\end{array}

while the sub-Hamiltonians {\tilde{H}}_a,{\tilde{H}}_b and {\tilde{H}}_c present three homogeneous virtual linear chains {a, b, c} with length M + N, N and L, respectively. In {\tilde{H}}_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}, there exists a connection interaction exp (iΦ/2) cos (Φ/2) between the two end sites of virtual chain a and c. Meanwhile, there exists another connection interaction –i exp (iΦ /2) sin (Φ /2) between the two end sites of virtual chain b and c. The geometry of such network is illustrated in Fig. 8 (d). Obviously in the virtual space, such network is equivalent to the Y-shaped beam splitter with different lengths of output arms and complex joint hopping constants controlled by the flux Φ. From the discussion about Y-shaped network, we have known that the lengths of output arms do not affect the feature as beam splitter for local input wave packet. In the following we will investigate the property of such Y-shaped network by considering equi-length case for simplicity.

5.3 Y-shaped Beam splitter controlled by Φ

Now we consider a Y-shaped network {A, B, C} with complex joint hopping constants. The model Hamiltonian reads

\begin{array}{}\text{(73)}& H=H_A+H_B+H_C+H_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}\end{array}

where N_A = L, N_B = N_C = N, and t_j^{[A]}=t_j^{[B]}=t_j^{[C]}=t. The joint Hamiltonian

\begin{array}{}\text{(73)}& H_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}=-(t_{AB}{a}_{A,L}^{†}{a}_{B,1}+t_{AC}{a}_{A,L}^{†}{a}_{C,1}+H.c.)\end{array}

describes the connections with the complex hopping integrals

\begin{array}{}\text{(74)}& t_{AB}=t{\mathrm{e}}^{-\mathrm{i}\frac{\Phi }{2}}\cos \left(\frac{\Phi }{2}\right),t_{AC}=\mathrm{i}t{\mathrm{e}}^{-\mathrm{i}\frac{\Phi }{2}}\sin \left(\frac{\Phi }{2}\right)\end{array}

Interestingly, if we get rid of the exponential terms in the hopping integrals, i.e., t cos (Φ /2), t sin (Φ/2), we recover the matching condition (26) in the original Y-shaped beam splitter t_{AB}^2+t_{AC}^2=t^2. In order to decouple this network, we introduce three commutative sets of fermion operators

\begin{array}{}\text{(75)}& \begin{array}{l}{\tilde{a}}_{a,l}^{†}={\tilde{a}}_{A,l}^{†}\\ {\tilde{a}}_{a,L+j}^{†}={\mathrm{e}}^{\mathrm{i}\frac{\Phi }{2}}[\cos \left(\frac{\Phi }{2}\right){\tilde{a}}_{B,j}^{†}-\mathrm{i}\sin \left(\frac{\Phi }{2}\right){\tilde{a}}_{C,j}^{†}]\hfill \\ {\tilde{a}}_{b,j}^{†}={\mathrm{e}}^{-\mathrm{i}\frac{\Phi }{2}}\left[\mathrm{i}\sin \left(\frac{\Phi }{2}\right)a_{B,j}^{†}+\cos \left(\frac{\Phi }{2}\right)a_{C,j}^{†}\right]\hfill \end{array}\end{array}

for l\in [1,L] and j\in [1,N].

The inverse transformations of Eqs. (75) result in the reduction of Hamiltonian in terms of {\tilde{a}}_{a,j}^{†},{\tilde{a}}_{a,L+j}^{†}, and {\tilde{a}}_{b,j}^{†}:

\begin{array}{}\text{(76)}& H={\tilde{H}}_a+{\tilde{H}}_b\end{array}

where N_a = L + N and N_b = N.

Thus, this kind of Y-shaped Bloch electron network is also decoupled into two imaginary chains. Similarly, we apply the beam splitter to the special Bloch electron GWP |{\psi }_A(N{}_0)\rangle. At a certain time τ, such GWP evolves into

\begin{array}{}\text{(77)}& \begin{array}{ll}|\Psi (\tau )\rangle ~& \cos \left(\frac{\Phi }{2}\right)|{\psi }_B(N_{\tau })\rangle \hfill \\ \hfill & -\mathrm{i}\sin \left(\frac{\Phi }{2}\right)|{\psi }_C(N_{\tau })\rangle \hfill \end{array}\end{array}

where N_τ = N₀ + 2tτ – L. This means that the beam splitter can split the GWP into two cloned GWPs completely with the probabilities

\begin{array}{}\text{(78)}& \begin{array}{l}{|\langle j_B|{\psi }_B(N_{\tau })\rangle |}^2={\cos }^2\frac{\Phi }{2}\\ {|\langle j_C|{\psi }_C(N_{\tau })\rangle |}^2={\sin }^2\frac{\Phi }{2}\end{array}\end{array}

which can be controlled by the external flux ϕ.

5.4 A-B effect in a solid system

This virtual model of an interferometer network is very similar to the second type of Y-shaped beam splitter we discussed before. The only difference between them is that in this virtual model the lengths of the two legs are unequal. Fortunately, by appropriately choosing α, the width of the wave packet, the GWP can be regarded as a classical electron. It not only propagates along a homogeneous chain without spreading approximately, but also does not regard the length of the chain. Now we prepare such a moving GWP at the input leg. When it reaches the node, it will be split into two cloned GWPs with the amplitudes of cos² (Φ/2) and sin² (Φ /2), respectively. According to our discussion above, the sub-GWP with the amplitude of sin² (Φ /2) will be reflected by the opposite node of the ring, but the other sub-GWP with the amplitude of cos² (Φ /2) will move onto the output leg directly. Therefore, some time later, we will receive a cloned GWP with the probability cos² (Φ /2) at the output leg.

The interferometer based on the Bloch electron network can be regarded as a mimic of the A-B effect [46] experimental device in a solid system illustrated in Fig. 9 (a). Here, the initial GWP

\begin{array}{}\text{(79)}& |{\psi }_A(N_0)\rangle =\frac{1}{\sqrt{{\Omega }_1}}{\displaystyle \sum _{j=1}^M{\mathrm{e}}^{-\frac{{\alpha }^2}{2}{(j-N_0)}^2}}{\mathrm{e}}^{\mathrm{i}\frac{\mathrm{\pi }}{2}j}|j\rangle \end{array}

is taken as a good example to demonstrate the physical mechanism of such setup.

We first focus on the GWP at the input site N₀ = A and detect it later on a distant site D₁ or D₂. The maximal probability of the GWP in some certain site j is

\begin{array}{}\text{(80)}& {\left|\psi (j,\alpha )\right|}_{\max }^2=\max \left\{{|\langle j\left|{\mathrm{e}}^{-\mathrm{i}H\tau }\right|{\psi }_A(N_0)\rangle |}^2\right\}\end{array}

Thus, we can define the relative probability Q as a function of α, the magnetic flux ϕ, and the site of the detector j,

\begin{array}{}\text{(81)}& Q(j,\varphi ,\alpha )=\frac{{\left|\psi (j,\alpha )\right|}_{\max }^2}{{\left|\psi (A,\alpha )\right|}_{\max }^2}\end{array}

Obviously, Q is an observable physical quantity, which describes the AB effect and the influence of lattice scattering. Numerical simulation of Q (D₁, ϕ, α) and Q (D₂, ϕ, α) for different initial GWP with half-width Δ = 16.65 (α = 0.1), Δ = 5.55 (α = 0.3), and Δ =1 (\alpha =\infty) are plotted in Fig. 9 (b). Here, the optical paths between the input site and the detector-sites are L₁ = 200, L₂ =400. The ring of the system is threaded by a magnetic flux \varphi \in [-2,2].The numerical results show that the relative probabilities Q are periodic in the magnetic flux ϕ with a period of unit flux quantum Φ₀ = h/e. This is the so called A-B effect in a solid system. At ϕ = integer, our previous discussion shows that the interferometer network of Bloch electron model is decoupled into one long chain and one short chain. The initial GWP localized in the input arm can be transmitted to the detector-arm without any reflection. Consequently, the relative probability Q reaches its maximum of the curve. On the other hand, when ϕ is a half-integer, the initial GWP cannot be transmitted to the detector-arm. Thus, the corresponding Q equals to its minimum zero.

Then we consider the GWPs with different half-width Δ = 2\sqrt{\text{ln}2}/\alpha . If Δ is larger, the GWP is localized in thelinear dispersion regime more exactly. In this case, it can be well transferred without spreading [29]; or in another point of view, it is a free particle which will not be scattered by the lattice. We can see from the numerical results, Q = (1 + cos Φ) /2, the maximum Q of Δ = 16.65 (α = 0.1) is approximately equal to 1. Otherwise, a GWP with smaller Δ, i.e., Δ = 5.55 (α = 0.3) is scattered by the lattice severely, so the relative probability Q of which are smaller than Q of wider GWP. It is reasonable that when the optical path is longer, the influence of lattice scattering is larger, the relative probability Q is smaller. The black dot-dash line also shows that in large α limit, Q is approximately equal to zero. Therefore, a GWP localized beyond the linear dispersion regime is not suitable for demonstrating the A-B effect experiment in a solid system.

To sum up, when the half-width of the initial wave packet is narrower or the detect-length is longer, the relative probability Q is smaller, the A-B effect is weaker to be observed. From these results, we get two possible reasons why we cannot observe the A-B effect in a macroscopically solid system. Firstly, we do not choose an appropriate wave packet. Secondly, the total site of the macroscopically solid system is so large that the influence of lattice scattering cannot be ignored. To solve these problems and to realize AB effect in a solid system, we should choose a wider GWP mentioned in our previous work [29]. At the same time, we should decrease the optical paths between the input site and the detector-sites.

6 Applications for spin network

In the above discussion, we studied the fermion systems where the Bloch electrons move along the quantum lattice network. We consider various geometrical configurations of TBN that are analogous to quantum optical devices, such as beam splitters and interferometers. In this section, we will apply the results obtained for TBN to another analogue system, spin network (SN) where the spin wave acts as the Bloch electron.

The basic setup of a SN is constructed topologically by the linear spin chains and the various connections between their ends. Here, we consider the spin-1/2 XY model, in which only the nearest neighbor (NN) coupling term is taken into account. The Hamiltonian of a SN reads

\begin{array}{}\text{(82)}& H^s={\displaystyle \sum _l{H}_l^s+H_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}^s}\end{array}

where the Hamiltonians of leg l consisting of N_l spins and the joints are

\begin{array}{}\text{(83)}& \begin{array}{l}{H}_l^s=H_l^s(J_l,N_l)\approx {\displaystyle \sum _{j=1}^{N_{l-1}}{J}_j^{[l]}(S_{l,j+1}^{+}+H.c.)}\hfill \\ H_{\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}^s\equiv J_{ji}^{[lm]}{S}_{l,j}^{+}{S}_{m,i}^{-}+H.c.\hfill \end{array}\end{array}

Here, S_{l,j}^{\pm } are the Pauli spin operators acting on the internal space of electron on the j th site of the l th leg. Although the SNs and TBNs have the same structure, the physics should be different due to the difference of the intrinsic statistical properties. Then the analytical conclusions for TBN are not available to SN. However, in the context of state transfer, only the dynamics of the single-magnon is concerned. Notice that for Hamiltonian (82), the z-component of the total spin S^z={\displaystyle {\sum }_{l,i}{S}_{l,i}^z} is conserved, i.e., [S ^z, H ^s] = 0. Thus, in the invariant subspace with S^z=({\displaystyle {\sum }_l{N}_l-1})/2, this model can be mapped into single-particle TBN.

7 Conclusions and remarks

In summary, we studied various geometrical configurations of tight-binding networks for the fermion systems. It is found that the lattice networks for moving GWPs are analogous to quantum optical devices, such as beam splitters, switches and interferometers. The networks are constructed by connecting several tight-binding chains with uniform nearest-neighbor hopping integrals. The external magnetic field and the connecting hopping integrals can be used to control the intrinsic properties of the networks. The results presented in this paper are hopefully to be realized in experiments. First, the model we considered is the tight-binding model, which is more practical in both theory and experiment. Second, the conditions to realize the various devices in the solid system are presented in quite general cases. The networks are not as artificially engineered as the ones shown in [20]. Third, we used the magnetic field to control the devices, which is more feasible in experiments. The tight-binding networks will enable an elementary quantum device for scalable quantum computation, which can coherently transfer quantum information among the integrated qubits. The observable effects for electronic wave interference may be discovered in the dynamics of magnetic domain in some artificial quantum material.

In the above studies, we only consider the spinless Bloch electron. Actually, since the model concerned in this paper does not contain the interaction involving spin, the spin degree of freedom can be omitted for notational brevity. Therefore, the conclusion for the spinless fermion is available for spinful fermion. Moreover, all the conclusions we obtained can be extended to the networks of spin-1/2 electrons, if the external magnetic field does not exert any forces or torques on the magnetic moment of spin, but only a phase on the wave function of electron. In addition, the Hamiltonian of such system has a similar form with (1) and (4) under the transformation a_{l,j}^{†}{a}_{l,i}\to {\displaystyle {\sum }_{\sigma =\pm 1}{a}_{l,j,\sigma }^{†}{a}_{l,i,\sigma }.}Note that, for the new Hamiltonian, the spin of the electron is a conservative quantity that cannot be influenced during the propagation [29]. The electronic wave packet with spin polarization is an analogue of photon “flying qubit”, where the quantum information was encoded in its two polarization states. Thus, these networks can function as some optical devices, such as beam splitters and interferometers. These are expected to be used as quantum information processors for the fermion system based on the possible engineered solid state systems, such as the array of quantum dots, Josephson junctions or other artificial atoms that can be implemented experimentally.

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