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Physical Review B

Time dependence of the position momentum and position velocity uncertainties in gapped graphene

Eylee Jung, Kwang S. Kim, DaeKil Park

Published Apr 9, 2012• 2 citations
Publisher

Abstract

We examine the time dependence of the position-momentum and position-velocity uncertainties in monolayer gapped graphene. The effect of the energy gap to the uncertainties is shown to appear via the Compton-like wavelength ${\ensuremath{\lambda}}_{c}$. The uncertainties in the graphene are mainly contributed by two phenomena, spreading and zitterbewegung. While the former determines the uncertainties in the long range of time, the latter gives high oscillation to the uncertainties in the short range of time. The uncertainties in the graphene are compared with the corresponding values for the usual free Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\mathrm{free}}=({p}_{1}^{2}+{p}_{2}^{2})/2M$. It is shown that the uncertainties can be under control within the quantum mechanical laws if one can choose the gap parameter ${\ensuremath{\lambda}}_{c}$ freely.

DOI
10.1103/physrevb.85.165418

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We examine the time dependence of the position-momentum and position-velocity uncertainties in monolayer gapped graphene. The effect of the energy gap to the uncertainties is shown to appear via the Compton-like wavelength \ensuremathλc{\ensuremath{\lambda}}_{c}\ensuremathλc​. The uncertainties in the graphene are mainly contributed by two phenomena, spreading and zitterbewegung. While the former determines the uncertainties in the long range of time, the latter gives high oscillation to the uncertainties in the short range of time. The uncertainties in the graphene are compared with the corresponding values for the usual free Hamiltonian H\ifmmode^\elseˆ\fifree=(p12+p22)/2M{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{\mathrm{free}}=({p}_{1}^{2}+{p}_{2}^{2})/2MH\ifmmode^\elseˆ\fi​free​=(p12​+p22​)/2M. It is shown that the uncertainties can be under control within the quantum mechanical laws if one can choose the gap parameter \ensuremathλc{\ensuremath{\lambda}}_{c}\ensuremathλc​ freely.
Motivated by graphene-based quantum computer we examine the time-dependence of the position-momentum and position-velocity uncertainties in the monolayer gapped graphene. The effect of the energy gap to the uncertainties is shown to appear via the Compton-like wavelength λcλ_cλc​. The uncertainties in the graphene are mainly contributed by two phenomena, spreading and zitterbewegung. While the former determines the uncertainties in the long-range of time, the latter gives the highly oscillation to the uncertainties in the short-range of time. The uncertainties in the graphene are compared with the corresponding values for the usual free Hamiltonian H^free=(p12+p22)/2M\hat{H}_{free} = (p_1^2 + p_2^2) / 2 MH^free​=(p12​+p22​)/2M. It is shown that the uncertainties can be under control within the quantum mechanical law if one can choose the gap parameter λcλ_cλc​ freely.
We investigate the impact of an induced mass term ΔΔΔ on the current density in graphene subjected to a space- and time-dependent periodic potential U(x,t)U(x,t)U(x,t). By solving the Dirac equation and deriving both the quasi-energy spectrum and the corresponding eigenspinors, we obtain explicit analytical expressions for the current density, which exhibits a clear dependence on ΔΔΔ. We show that ΔΔΔ acts as a tunable control parameter that governs the amplitude, sign, and resonance structure of Josephson-like current oscillations. For normal incidence and a purely time-periodic potential, our results reveal that the oscillations within the energy gap gradually diminish as the mass term ΔΔΔ increases. This suppression leads to a weakening of the Josephson-like effect typically observed in such systems. When the potential U(x,t)U(x,t)U(x,t) is periodic in both space and time, the behavior becomes more complex. The current density can take either positive or negative values depending on the magnitude of the induced gap, and it generally decreases over time. As a result, the resonance phenomena--prominent at lower gap values--become progressively less significant as ΔΔΔ increases. These findings underscore the tunable nature of light-matter interactions and quantum transport in gapped graphene, suggesting potential applications in terahertz (THz) nanoelectronic devices and optically controlled quantum switches.
In this paper we consider the interaction of electrons in bilayer graphene with a constant homogeneous magnetic field which is orthogonal to the bilayer surface. Departing from the energy eigenstates of the effective Hamiltonian, the corresponding coherent states will be constructed. For doing this, first we will determine appropriate creation and annihilation operators in order to subsequently derive the coherent states as eigenstates of the annihilation operator with complex eigenvalue. Then, we will calculate some physical quantities, as the Heisenberg uncertainty relation, the probabilities and current density as well as the mean energy value. Finally, we will explore the time evolution for these states and we will compare it with the corresponding evolution for monolayer graphene coherent states.
In this paper we consider the interaction of electrons in bilayer graphene with a constant homogeneous magnetic field which is orthogonal to the bilayer surface. Departing from the energy eigenstates of the effective Hamiltonian, the corresponding coherent states will be constructed. For doing this, first we will determine appropriate creation and annihilation operators in order to subsequently derive the coherent states as eigenstates of the annihilation operator with complex eigenvalue. Then, we will calculate some physical quantities, as the Heisenberg uncertainty relation, the probabilities and current density as well as the mean energy value. Finally, we will explore the time evolution for these states and we will compare it with the corresponding evolution for monolayer graphene coherent states.
In the Dirac theory for the motion of free relativistic electrons, highly oscillatory components appear in the time evolution of physical observables such as position, velocity, and spin angular momentum. This effect is known as zitterbewegung. We present a theoretical analysis of rather different Hamiltonians with gapped and/or spin-split energy spectrum (including the Rashba, Luttinger, and Kane Hamiltonians) that exhibit analogs of zitterbewegung as a common feature. We find that the amplitude of oscillations of the Heisenberg velocity operator v(t)\mathbf{v}(t)v(t) generally equals the uncertainty for a simultaneous measurement of two linearly independent components of v\mathbf{v}v. It is also shown that many features of zitterbewegung are shared by the simple and well-known Landau Hamiltonian, describing the dynamics of two-dimensional (2D) electron systems in the presence of a magnetic field perpendicular to the plane. Finally, we also discuss the oscillatory dynamics of 2D electrons arising from the interplay of Rashba spin splitting and a perpendicular magnetic field.