Irregular Bloch-Zener oscillations in $\alpha$-$\mathcal{T}_{3}$ lattices

The Landau-Zener transition (LZT) [1, 2] is a fundamental phenomenon in time-dependent quantum systems. The paradigmatic setting for LZT is a two-level system in which the two energy levels do not cross each other and vary adiabatically with time. When the energy gap between the two levels is sufficiently small, a non-adiabatic transition from one energy level to another can occur - leading to an LZT. The phenomenon of LZT is relevant to quantum information science and technology, because qubits are essentially two-level systems [3, 4, 5]. In addition to quantum systems, LZTs can arise in other physical situations such as optical lattices [6, 7, 8] and electromechanical systems [9, 10]. When a two-level system is periodically driven by an electric field, the transition probability will depend on the phase accumulated by the two energy bands between subsequent crossings, leading to the so-called Landau-Zener-Stückelberg interferometry [11]. In general, interference among the quantum states in different energy bands is determined by two phases: geometric and dynamic, which correspond to the adiabatic and Stokes phases, respectively, in the adiabatic-impulse model [11, 12] underlying the Landau-Zener-Stückelberg interferometry, where the sum of the adiabatic and Stokes phases gives the Stückelberg phase - a concept originated from strongly periodically driven two-level systems. A generalization from the two-level setting is the three-level LZT model [13] with an additional flat band [14, 15].

In solid state physics, Bloch oscillations [16, 17] are a fundamental phenomenon closely related to LZT, which occur when a static electric field is applied to a periodic lattice, leading to a linear increase with time in the electron momentum and generating a time-dependent quantum system. The basic periodicity of the momentum space stipulates that the electron must execute oscillatory motions in the physical space at a frequency determined by the lattice constant and the electric field strength (typically in the terahertz regime). In fact, insofar as the electron moves in a periodic potential, Bloch oscillations can occur, rendering them a common quantum phenomenon beyond a solid-state lattice. In the past, the oscillations have been observed in diverse systems such as semiconductor superlattices [18], photonic structures [19, 20, 21, 22] and plasmonic waveguide arrays [23]. The phenomenon provides a viable way to convert a direct current to a high-frequency signal [24, 25].

Bloch oscillations arise from the time evolution of the electron in a single energy band. When there are multiple energy bands, LZTs can occur at the avoided crossing points between the bands. Driven by the static electric field, a quantum state initialized in the lower energy band evolves with time. At certain time, the state will reach an avoided crossing point between distinct energy bands and possibly experience an LZT. Thus, in systems with multiple energy bands, a combination of LZTs and Bloch oscillations can occur, leading to the so-called Bloch-Zener oscillations [26, 27, 15, 28], which have applications in, e.g., matter-wave beam splitters and Mach-Zender interferometry [29, 30]. In the past, Bloch-Zener oscillations were extensively studied in one-dimensional (1D) gapped periodic lattices [31, 29, 30, 32] and were demonstrated to be sensitive to the size of the energy gap. In particular, if the gap is relatively large, LZTs are inhibited so the Bloch-Zener oscillations are restricted to within a single energy band. For a smaller gap, interband LZTs can occur, which destroys the periodicity of the Bloch-Zener oscillations [31, 29, 30, 33, 34]. In a 2D lattice with multiple energy bands, in the first Brillouin zone, different subregions can arise that can inhibit or excite LZTs, resulting in irregular Bloch-Zener oscillations [35, 36]. For example, in pure graphene, when a constant electric field is applied in a lattice-rational direction, the oscillation amplitude can decay with time rapidly in a non-adiabatic fashion [37]. When the electric field is in an irrational direction with respect to the lattice structure, Bloch oscillations of complicated patterns can arise [38].

In this paper, we study the effects of a constant and uniform electric field on a broad class of 2D Dirac materials, the $\alpha$-$\mathcal{T}_{3}$ lattice [39, 40], which has an additional atom at the center of each unit cell of the honeycomb graphene lattice. The interaction between the central atom and any of its nearest neighbors is characterized by the parameter $0\leq\alpha\leq 1$ - effectively the strength relative to that between two neighboring atoms at the vertices of the graphene cell. For $\alpha=0$, the lattice reduces to that of graphene with quasiparticles being pseudospin-1/2 Dirac fermions. As $\alpha$ increases from zero, a flat band through the conic interaction of the two Dirac cones emerges [39, 40]. The maximal value $\alpha=1$ gives a pseudospin-1 lattice where, because of the extra atom, the low energy excitations need to be described by the pseudospin-1 Dirac-Weyl equation with a three-component spinor [41]. A distinct feature of the entire spectrum of $\alpha$-$\mathcal{T}_{3}$ lattices is the existence of two distinct valleys centered about the two non-equivalent Dirac points of the backbone hexagonal lattice, denoted as $+\mathbf{K}$ and $-\mathbf{K}$.

Depending on the direction and magnitude of the electric field, electrons initiated from distinct valleys can exhibit characteristically different LZTs. As the dynamic phases associated with different valleys cancel each other exactly, the distinct LZTs are due to the different adiabatic phases of the quantum states in the energy bands between consecutive crossings. This can be understood by considering the reciprocal periodic momentum space with the hexagonal Brillouin zone, as shown in Fig. 1. Now apply an electric field in the $x$ direction. The $x$ component of the momentum will then increase linearly with time under the premise of the same energy value. As a result, the Dirac points $+\mathbf{K}$ and $-\mathbf{K}$ will shift towards the right. At an original Dirac point (+$\mathbf{K}$ or $-\mathbf{K}$), the energy will increase from zero, reach a maximum, and then decrease to zero when the next Dirac point arrives, generating a time-periodic behavior. Because of the hexagonal structure of the momentum space, the energy variations associated with $+\mathbf{K}$ and $-\mathbf{K}$ are distinct, as indicated in Fig. 1(b). The LZT probability depends on the accumulated phase between subsequent crossings, where the adiabatic phase is the integral of the energy variation over time. In the specific setting of Figs. 1(a) and 1(b), the integral associated with the Dirac point $+\mathbf{K}$ will have a much larger value than that associated with the other Dirac point $-\mathbf{K}$. As a result, if an electron initiates with a momentum value near $+\mathbf{K}$, the adiabatic phase will be nearly constant for a large range of energy gaps determined by the momentum deviation from the trajectory of Dirac points $\pm K$ in the $p_{y}$ direction. However, if an electron starts with a momentum value near $-\mathbf{K}$, the adiabatic phase will depend sensitively on the energy gap. Based on this property, it is possible to generate specific destructive or constructive interference for a large range of momentum deviation from the trace of the Dirac point $+K$, whereas there is mixed interference associated with all possible phases for electrons with initial momentum near $-K$.

Our first finding is the emergence of complicated LZT morphological patterns in the vicinity of distinct Dirac points, which is associated with mixed quantum interference among the quantum states in multiple bands. Say we apply an electric field in the $x$ direction, initialize electrons in the lower Dirac cone (the lower band), and calculate the differential LZT probability, defined as the difference between the probability that an electron is in the upper band and that in the lower band. Different momentum values about a Dirac point and the magnitudes of the electric field give distinct interference phases. As a result, in the momentum plane, the differential LZT probability displays different values, giving rise to some morphological pattern that can be complex. During its temporal evolution, the pattern can be maintained for some time but it can change from time to time due to the switching of two two distinct Dirac points $\pm\mathbf{K}$ in the Brillouin zone as caused by the external electric field and the periodic structure of the momentum space. Specifically, during one period of the Bloch-Zener oscillation, the two valleys go through a complete cycle in the sense that they are switched and then returned to their respective original positions.

The second finding is that changes in the LZT morphology can lead to irregularities in Bloch-Zener oscillations in $\alpha$-$\mathcal{T}_{3}$ lattice. To explain this, we recall the two typical cases where periodic Bloch oscillations are generated. One case is a single-band material, such as a normal conductor, where the Bloch oscillations are characterized by a perfect temporally periodic behavior in the electrical current density. Another case is where an LZT causes all electrons initialized in one band to transition completely to another band, i.e., the transition probability is one - the so-called ideal LZT. In this case, the resulting Bloch-Zener oscillations behave as if the electrons were in a single band. In the three-band $\alpha$-$\mathcal{T}_{3}$ lattice, LZTs are typically not ideal. The mutual switchings of the two Dirac points $\pm\mathbf{K}$ in the Brillouin zone changes the LZT morphology, which can produce an abrupt, nonsmooth change in the current density, thereby leading to aperiodic, irregular Bloch-Zener oscillations. More specifically, the coexistence of a variety of LZT possibilities in the momentum space generates complex, mixed quantum interference between the states in the upper, lower, and flat bands, disrupting the originally periodic Bloch-Zener oscillation rhythm before the Dirac point switch. While aperiodic Bloch oscillations [31, 29, 30, 33, 34] and irregular Bloch-Zener oscillations [35, 36] have been noted before, to our knowledge, the physical mechanisms underlying these irregular behaviors were not clear. Especially, it has not been reported previously that LZTs can form a complicated morphology in $\alpha$-$\mathcal{T}_{3}$ lattice and a change in the morphology can lead to irregular Bloch-Zener oscillations.

In Sec. II, we describe the $\alpha$-$\mathcal{T}_{3}$ lattice model and derive the current density associated with the Bloch-Zener oscillations from the adiabatic basis in the hexagonal Brillouin zone. In Sec. III, we present a general treatment of LZTs in the $\alpha$-$\mathcal{T}_{3}$ lattice, display the LZTs morphology in the long time, and analyze the relationship between morphology and irregular Bloch-Zener oscillations. In particular, in Sec. III.1, we linearize the Hamiltonian about the Dirac points to obtain the effective Landau-Zener Hamiltonian for the two limiting cases: $\alpha=0$ and $\alpha=1$. For $0<\alpha<1$, we numerically demonstrate the occurrence of LZT. In Sec. III.2, we elucidate the interplay between LZT morphological changes and irregular Bloch-Zener oscillations. In Sec. IV, we focus on the Landau-Zener-Stückelberg interferometry in $\alpha$-$\mathcal{T}_{3}$ lattice, where in Sec. IV.1, we establish the equivalence of the $\alpha$-$\mathcal{T}_{3}$ lattice to two- or three-level time-dependent systems and provide an understanding of the LZT based on the Stückelberg phase. In Sec. IV.2, we address the problem of harnessing irregular Bloch-Zener oscillations through selection of the LZT morphology and discuss the experimental feasibility of this scheme. A discussion is offered in Sec. V. Our main code is uploaded to GitHub: https://github.com/liliyequantum/Irregular-Bloch-Zener-oscillations-in-two-dimensional-flat-band-Dirac-materials.

The $\alpha$-$\mathcal{T}_{3}$ lattice interpolates between the graphene honeycomb lattice ($\alpha=0$) and the dice lattice ($\alpha=1$) with the parametrization $\tan\varphi=\alpha\in[0,1]$ with the duality [42] $\alpha\rightarrow 1/\alpha$. The tight-binding Hamiltonian is given by

where

$\boldsymbol{k}=(k_{x},k_{y})$, and $t_{\epsilon}$ is the nearest-neighbor hopping energy between A and B sites, as shown in Fig. 1(a). The primitive translation vectors are $\boldsymbol{a}_{1}=a\left(\sqrt{3}/2,3/2\right)$ and $\boldsymbol{a}_{2}=a\left(-\sqrt{3}/2,3/2\right)$ with $a$ being the lattice constant. The corresponding primitive translation vectors in the hexagonal Brillouin zone of the reciprocal lattice are $\boldsymbol{b}_{1}=\left(\sqrt{3}/3,1/3\right)2\pi/a$ and $\boldsymbol{b}_{2}=\left(-\sqrt{3}/3,1/3\right)2\pi/a$. The eigenenergy spectrum of the $\alpha$-$\mathcal{T}_{3}$ lattice is independent of $\alpha$, which consists of two conic dispersive bands $\epsilon_{\lambda}=\lambda\left|f_{\boldsymbol{k}}\right|$ distinguished by the band index $\lambda=\pm$ and a zero energy flat band $\epsilon_{0}=0$, as shown in Fig. 1(b). The eigenstates of the $\alpha$-$\mathcal{T}_{3}$ lattice in the whole hexagonal Brillouin zone can be obtained through the following effective Hamiltonian about the Dirac points:

where $\theta_{\boldsymbol{k}}$ is the angle of $f_{\boldsymbol{k}}$ associated with the specific momentum.

We describe the Hamiltonian underlying Bloch-Zener oscillations. Apply a uniform and constant electric field to the $\alpha$-$\mathcal{T}_{3}$ lattice in the $+x$ direction, which is switched on at $t=0$. With the time-dependent vector potential [44, 45] $\boldsymbol{A}\left(t\right)=\left[A\left(t\right),0,0\right]$, where $A\left(t\right)=Et\Theta\left(t\right)/\hbar$, the Hamiltonian becomes

where $f_{\boldsymbol{k}}(t)$ is given by

with $k_{x}\left(t\right)\equiv k_{x}-eEt/\hbar$. For convenience, we define the dimensionless quantities

so that $f_{k}(t)$ has the same dimension as the hopping energy $t_{\epsilon}$.

In the presence of the electric field, the eigenenergy spectrum of the positive dispersion band in the whole hexagonal Brillouin zone is determined by

where $X_{\boldsymbol{k}}\left(t\right)=\widetilde{k}_{x}\left(t\right)\sqrt{3}/2$ and $Y_{\boldsymbol{k}}=\widetilde{k}_{y}3/2$. Because the function $f_{\boldsymbol{k}}(t)$ includes $\cos(L(t))$, where $L(t)$ is a linear function of time, the Hamiltonian becomes time-periodic. Consider the special case of $\alpha=0$ (graphene). Expanding the Hamiltonian around the Dirac points yields the standard Landau-Zener Hamiltonian [46]:

That is, when the electrons are near the Dirac points, LZTs between distinct energy bands can arise.

The quantum dynamics are governed by

On the adiabatic basis, the evolution of a quantum state is under an infinitesimal electric field [44, 47]:

where $S_{z}$ is the $z$-component of the vector of spin-1 matrices. The transformed quantum dynamics are governed by

through the time-dependent unitary transformation $U(t)$ given by

where $\Phi_{\boldsymbol{k}}(t)=U^{\dagger}(t)\psi_{\boldsymbol{k}}(t)$ and the term incorporating $C_{0}(t)$ contributes to the time dependence of $U(t)$ through $-i\hbar U^{\dagger}\partial_{t}U$. Specifically, we have

where

For $\alpha=1$, $\widetilde{S}_{x}$ reduces to $S_{x}$, the $x$ component of spin-1 matrix and Eq. (21) becomes the quantum evolution equation for a dice lattice. This form of Eq. (21) is consistent with that reported in a previous work [45] except for a periodic factor in $C_{0}(t)$ due to the intrinsic lattice structure. For $\alpha=0$, $\widetilde{S}_{x}$ become $-S_{L}$ and Eq. (21) describes the graphene lattice, which is consistent with a previous work [44] except for the identity term $I\exp(-i\theta(t)/2)\partial_{t}\exp(i\theta(t)/2)/2$ due to the different form of the basis. Overall, $\widetilde{S}_{x}$ reflects the different coupling strength among the three bands and the periodic term in $C_{0}(t)$ originates from the intrinsic property of the $\alpha$-$\mathcal{T}_{3}$ lattice.

We set the initial state as one corresponding fully occupied lower band:

The average current density associated with the momentum $\langle J_{x}\rangle_{\boldsymbol{k}}\left(t\right)$ in the hexagonal Brillouin zone is given by

where the current density matrix with the definite momentum is

and $\Phi_{\boldsymbol{k}}\left(t\right)$ in the adiabatic basis can be written as

Due to the periodic structure of the energy band (referred to as the Bloch band), the average current density will exhibit Bloch oscillations.

The average current density $\langle J_{x}\rangle_{\boldsymbol{k}}\left(t\right)$ can be decomposed into two components, the intraband and interband currents, respectively [44]:

which can be written as

The matrix $J_{x,\boldsymbol{k}}(t)$ provides insights into the current density $\langle J_{x}\rangle_{\boldsymbol{k}}(t)$. In particular, the intraband component consists of both electrons and holes, corresponding to

respectively, where the minus sign comes from the opposite sign of the equivalent charge in the electron-hole pair. The zero group velocity of the flat band results in zero intraband contribution. The interband contribution arises from the interference between the transition from the lower to the flat band or the upper band and that from the flat to the upper band, corresponding to $J_{x,\;\boldsymbol{k}}^{23}\left(t\right),\;J_{x,\;\boldsymbol{k}}^{13}\left(t\right)\;$ and $J_{x,\;\boldsymbol{k}}^{12}\left(t\right)$, respectively, which are given by

where

and $J_{x,\;\boldsymbol{k}}^{0}(t)$ is the common factor with the dimension of the current density:

To facilitate numerical calculations, we define the dimensionless quantities:

where $t_{0}\equiv\hbar/t_{\epsilon}$, $E_{0}\equiv t_{\epsilon}/(ea)$, $\epsilon_{0}=t_{\epsilon}$ and $J_{0}=eat_{\epsilon}$. We use the fourth-order Runge–Kutta method to calculate the adiabatic evolution of the particle. The discrete step sizes in time and momentum are chosen according to Eq. (21) with the error tolerance $10^{-2}$ and normalized wavefunction error within $10^{-4}$. Figure 1(b) indicates that the $\pm\mathbf{K}$ correspond to zero energy, which results in numerical divergence of Eq. (21), so we set a minimum energy cut-off to be $\widetilde{\epsilon}_{k}\geq 10^{-5}$.

The dimensionless current density $\widetilde{J}$ is the result of integrating $\langle J_{x}\rangle_{\boldsymbol{k}}(t)/J_{0}$ over the rectangular reference area in the hexagonal Brillouin zone, as shown in Fig. 1(b), which contains three Dirac points and is $3/2$ times larger than the first Brillouin zone. In addition, we divide the current density $\widetilde{J}$ by the electric field $\widetilde{E}$ and a constant $3\pi^{2}/4$, which normalizes the current density in the weak field regime [37] for $\alpha=0$.

Bloch or Bloch-Zener oscillations, in addition to being a fundamental phenomenon in solid state physics, practically provides the foundation to convert a direct current into an oscillating current in the terahertz frequency regime [24]. Research in the past decade or so has suggested that, in 2D multiband materials, Bloch-Zener oscillations can be vulnerable to Landau-Zener transition or tunneling [29, 30, 34, 33, 35, 36]. Is this generally true? The question is important to physics and our present work has provided two-fold answers: yes LZTs can indeed affect the Bloch oscillations but what is destroyed is not the oscillations themselves but just the perfect time periodicity of the oscillations; no because the irregular oscillations can persist even with frequent occurrences of LZTs. In fact, approximately periodic Bloch-Zener oscillations can be maintained if the LZTs are near ideal such that they result in a near-one probability for the electrons to switch into a different band or when near destructive interference arises between the quantum states in different energy bands. Nonsmooth or irregular behaviors in the current density arise when the LZTs are not ideal and the interference is partially destructive.

Our study encompasses the entire spectrum of a class of 2D Dirac materials modeled by $\alpha$-$\mathcal{T}_{3}$ lattices. We have found that the set of points in the 2D momentum space near a Dirac point at which LZTs occur can possess a complex morphology, and it is the change in the morphology that results in irregular Bloch-Zener oscillations. Theoretically, when driven by a static electric field, an $\alpha$-$\mathcal{T}_{3}$ lattice is equivalent the Landau-Zener-Stückelberg interferometry. Specifically, the $\alpha=0$ lattice (graphene) is effectively a two-level periodically driven quantum system while the general $\alpha$-$\mathcal{T}_{3}$ lattice for $0<\alpha\leq 1$ is equivalent to a three-level periodically driven system. For the three-level system, we have exploited the concept of Stückelberg phase from the adiabatic impulse theory to understand the LZTs that occur in the neighborhoods of the Dirac points in distinct valleys.

In the $\alpha$-$\mathcal{T}_{3}$ lattice, a non-zero coupling parameter $\alpha$ induces a flat band in between the positive and negative energy bands, effectively resulting a three-level system. After the first LZT in Fig. 2, the quantum state is a superposition of the three energy bands and the LZT is enhanced by the flat band (compared with the two-level case). For subsequent LZTs (e.g., as shown in Figs. 8 and 9), the flat band modifies the Stückelberg phase $\Phi_{st}$ for destructive interference in the three-band interferometry (compared with the two-band one). More specifically, for $\alpha=0$, the destructive interference corresponds to $\Phi_{st}=0,\leavevmode\nobreak\ \pi$ from Eq. (66). For $\alpha=1$, the destructive interference means $\Phi_{st}=0$ only and the case $\Phi_{st}=\pi$ is excluded from Eqs. (75-77). For $0<\alpha<1$, the behavior of destructive interference is obtained numerically, as shown in Figs. 8 and 9, which is slightly different from the $\alpha=1$ case, especially in Figs. 9(e,f). Taken together, a nonzero $\alpha$ modifies the physical picture from two- to three-band quantum interference: it generates a flat band in the original two-level system, creates three-band interferometry, enhances LZT in the momentum space around the Dirac points, and modifies destructive interference.

Theoretically, the asymmetrical morphology pattern around the $\pm K$ points results from the different characteristics of the Stückelberg phases $\Phi_{st}$. As shown in Fig. 6(a), $\Phi_{st}$ is nearly constant with $\Delta\widetilde{k}_{y}$ measured from the $+K$ Dirac point after two LZTs and $\Phi_{st}$ changes greatly with $\Delta\widetilde{k}_{y}$ from $-K$. Different values of the interference phase $\Phi_{st}$ give distinct interference patterns, such as constructive, destructive, and mixed quantum interference (neither constructive nor destructive) in the momentum space, as exemplified in Fig. 4(c).

For LZTs with a near unity transition probability, which can occur for an infinitesimal energy gap, the $\pm K$ points make the same contribution to the Bloch oscillations. However, slightly away from the Dirac points where the energy gap is no longer infinitesimal, the contributions differ. The reason why the Bloch oscillations around $K$ are more prominent compared with those around the $-K$ point lies in the interference phase $\Phi_{st}$. In particular, the LZTs from the $+K$ point with different momentum deviation $\Delta\widetilde{k}_{y}$ are in phase with each other, as shown in Fig. 6(a). However, the LZTs measured from the $-K$ point are out of phase, so the contributions to the Bloch oscillations cancel each other to some degree.

The irregular Bloch-Zener oscillations in the average current density integrated over the Brillouin zone arise from the LZTs induced by different energy gaps associated with the momentum deviation $\Delta\widetilde{k}_{y}$ measured from Dirac points. In the $\alpha$-$\mathcal{T}_{3}$ lattice, there is an alternative way to induce an energy gap between the valence and conduction bands by adding the positive (negative) onsite energy on the A (B) sublattice  [49, 50]. Because of the sensitive dependence of the LZT on the size of the momentum-dependent energy gap, irregularities in the LZTs are anticipated, so are the irregular Bloch-Zener oscillations.

Experimentally, it may be feasible to observe at least the first peak of the Bloch-Zener oscillation in ballistic time [37]. To probe into the momentum-space morphology associated with LZTs and to directly observe the irregular Bloch-Zener oscillations in a longer time interval in Dirac material systems remain to be difficult at the present. Alternatively, it may be possible to use quantum simulators [51, 52] by exploring equivalent optical systems [20]. In the past, Bloch oscillations have been experimentally studied in photonic systems [21, 53, 23, 35, 54]. The observation of Bloch-Zener oscillations and Landau-Zener tunneling in photonic graphene [35] is particularly relevant to possible experimental checks of our findings. In this artificial Dirac system, the wave packet of light is driven by an index gradient on a non-adiabatic basis and the two sublattices are subjected to a potential imbalance. When the momentum deviation is zero from a Dirac point and the index gradient is applied in a specific direction, perfect (ideal) LZT with transition probability one can occur due to the zero energy gap. However, when the index gradient is applied in the orthogonal direction, the LZT becomes imperfect (non-ideal) due to the non-zero gap. Our work suggests that, if the time evolution of the wavefunction in photonic graphene can be approximated as constituting two processes: adiabatic evolution and non-adiabatic LZT, in principle the Stückelberg phase can be calculated to choose the appropriate index gradient value to create destructive interference between the quantum states in the upper and lower energy bands. It may thus be possible to generate destructive interference to obtain a near-ideal LZT. Our work predicts that, in this case, the resulting Bloch-Zener oscillations will become more periodic. Photonic graphene may be a feasible experimental testbed for these phenomena.

We thank Drs. Chen-Di Han and Cheng-Zhen Wang as well as Prof. Liang Huang for valuable discussions. This work was supported by AFOSR under Grant No. FA9550-21-1-0186.