Time Evolution of the Position Operators in a Bilayer Graphene

In this work, the researchers mainly focus on the trembling motion which is known as Zitterbewegung in a bilayer grapheme. This is effectively achieved by means of the long-wave approximation. That is, the Heisenberg representation is ultimately employed in order to derive the analytical expression concerning the expectation value related to the position operator along the longitudinal and transversal orientation, which describes the motion concerning the electronic wave packet inside the bilayer graphene. Parameters’ numbers are considered to explicate the packet of Gaussian wave, including the polarization of initial pseudo-spin as well as the wave number of the initial carrier number along with the localized wave packet’s width along the longitudinal as well as transversal orientation. Consequently, the researchers show that the obvious oscillation in position operator can be effectively controlled not only by what is known as the initial parameters concerning the wave packet. Rather, it can mainly be controlled by selecting the localized quantum state’s components. Furthermore, the interference’s analysis between the conduction as well as valence bands concerning quantum states is really emphasized as the ability of what can be described as the transient’s emergence, or in a sense, aperiodic temporal oscillations concerning the average value of position operator in the bilayer graphene.

Convergence theorems on multi-dimensional homogeneous quantum walks

We propose a general framework for quantum walks on d-dimensional spaces. We investigate asymptotic behavior of these walks. We prove that every homogeneous walks with finite degree of freedom have limit distribution. This theorem can also be applied to every crystal lattice. In this theorem, it is not necessary to assume that the support of the initial unit vector is finite. We also pay attention on 1-cocycles, which is related to Heisenberg representation of time evolution of observables. For homogeneous walks with finite degree of freedom, convergence of averages of 1-cocycles associated with the position observable is also proved.

Trembling motion of the wave packet in armchair graphene nanoribbons (AGNRs)

A treatment of the trembling motion or Zitterbewegung (ZB) phenomena in the armchair graphene nanoribbons (AGNRs) by using long-wave approximation is presented theoretically. We are first interested to study the time dependence of the average values for the current density in AGNR. The longitudinal and transversal components of the current density operator are derived analytically in the Heisenberg representation. The wave packet in a Gaussian distribution is considered with half-width d and a carrier wave vector in the longitudinal orientation k[Formula: see text]. The average values of the current density which represent the current induced by the motion of the electrons along the nanoribbon are calculated numerically. The interference between two energy branches, or the corresponding upper and lower energy states, leads to the trembling motion in the armchair graphene nanoribbon, and hence, our results emphasized that the phenomena of ZB has an aperiodic and nontransient oscillation in the longitudinal and transversal direction, respectively. The average values for the current density for AGNRs are calculated with extremely large value of N and compared with infinite pristine graphene.

An FPGA-based quantum circuit emulation framework using heisenberg representation

While physical realization of practical large-scale quantum computers is still ongoing, theoretical research of quantum computing applications is facilitated on classical computing platforms through simulation and emulation methods. Nevertheless, the exponential increase in resource requirement with the increase in the number of qubits is an inherent issue in classical modeling of quantum systems. In the effort to alleviate the critical scalability issue in existing FPGA emulation works, a novel FPGA-based quantum circuit emulation framework based on Heisenberg representation is proposed in this paper. Unlike previous works that are restricted to the emulations of quantum circuits of small qubit sizes, the proposed FPGA emulation framework can scale-up to 120-qubit on Altera Stratix IV FPGA for the stabilizer circuit case study while providing notable speed-up over the equivalent simulation model.

Invariant formula of the determinant of a Heisenberg representation

In this paper, we give an explicit formula of the determinant of a Heisenberg representation [Formula: see text] of a finite group [Formula: see text]. Heisenberg representations are induced by 1-dimensional characters in multiple ways, but our formula will be independent of any particular choice of induction.