Zitterbewegung and Klein-tunneling phenomena for transient quantum waves

Fernando Nieto-Guadarrama and Jorge Villavicencio

Phys. Rev. A 101, 042104 – Published 6 April 2020

Abstract

We explore the dynamics of relativistic quantum waves in a potential step by using an exact solution to the Klein-Gordon equation with a point-source initial condition. We show that in both the propagation and Klein-tunneling regimes, the zitterbewegung effect manifests itself as a series of quantum beats of the particle density in the long-time limit. We demonstrate that the beating phenomenon is characterized by the zitterbewegung frequency and that the amplitude of these oscillations decays as $t^{- 3 / 2}$ . We show that the beating effect also manifests itself in the free Klein-Gordon and Dirac equations within a quantum shutter setup, which involve the dynamics of cutoff quantum states. We also find a time domain where the particle density of the point source is governed by the propagation of a main wavefront, exhibiting an oscillating pattern similar to the diffraction-in-time phenomenon observed in nonrelativistic systems. The relative positions of these wavefronts are used to investigate the time delay of quantum waves in the Klein-tunneling regime. We show that depending on the energy difference $E$ between the source and the potential step, the time delay can be positive, negative, or zero. The latter case corresponds to a super-Klein-tunneling configuration, where $E$ equals half the energy of the potential step.

Received 16 November 2019
Revised 20 February 2020
Accepted 6 March 2020

DOI:https://doi.org/10.1103/PhysRevA.101.042104

Physics Subject Headings (PhySH)

Relativistic wave equations Tunneling & traversal time

General Physics

Authors & Affiliations

Fernando Nieto-Guadarrama and Jorge Villavicencio ^*

Facultad de Ciencias, Universidad Autónoma de Baja California, 22800 Ensenada, Baja California, México

^*villavics@uabc.edu.mx

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Vol. 101, Iss. 4 — April 2020

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Images

Figure 1
Step potential of intensity $V$ , and the energy range of the different regimes corresponding to the (i) propagation regime ( $E > V + μ$ ), (ii) evanescent case ( $V - μ < E < V + μ$ ), and (iii) Klein-tunneling regime ( $E < V - μ$ ). Here and in all of our numerical calculations, we use $ℏ = m = c = 1$ .
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Figure 2
(a) Time evolution of the particle density $ρ$ [Eq. (12)] at the position $x = 10.0$ for $V = 0.2$ , and different values of the source energy $E$ . The stationary density $ρ = E$ (gray dotted line) is included for comparison in all cases. The density exhibits an oscillating pattern similar to the diffraction-in-time phenomenon. The velocity of the main wavefronts increases as the energy of the source $E$ (or the free-type energy $E$ ) increases. (b) Dynamical delay $Δ t$ of the free-type density (orange solid line) with respect to the free point-source case ( $V = 0$ ) (blue dashed line) for the case $E = 1.6$ . (c) Time evolution of $ρ$ at the position $x = 10.0$ for high energy, $E = 5$ and $V = 0.5$ , where we can see a distortion with respect to the time evolution shown in case (a). The stationary solution (gray solid line) is also included.
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Figure 3
Time evolution of the particle density $| ρ (x, t) |$ [Eq. (12)] in the Klein-tunneling regime for $V = 3.0$ at a fixed value of position $x = 10.0$ , for different values of the source energy $E$ .
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Figure 4
Time evolution in the Klein-tunneling regime of the particle density $| ρ (x, t) |$ [Eq. (12)] at a fixed position $x = 10.0$ for a potential step with $V = 3.0$ (orange solid line) for different values of the source energy $E$ , and compared with the free case ( $V = 0$ ) density (blue dashed line). (a) For an energy $E = 1.8$ , we observe a positive delay (time delay), while in case (b) with $E = 1.2$ , a negative delay (time advance) is observed. In case (c), a source energy $E = 1.5$ yields $Δ t = 0$ . See the inset for the comparison of the positive and negative densities, $ρ (x, t)$ .
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Figure 5
(a) The ZBW effect manifests in the solution of the point-source model a series of quantum beats in the long-time behavior of particle density calculated from Eq. (8) (blue solid line) with an energy $E = 10.0$ , and $V = 0.5$ for a fixed value of position $x = 10.0$ . The asymptotic density $ρ_{a}$ (orange dashed line) reproduces the transient behavior. The exact period of the beatings, $T = π$ , is indicated in the density plot. The contributions to the asymptotic $ρ_{a}$ given by (b) $ρ_{μ}$ (blue solid line) and (c) $ρ_{E}$ (orange solid line) exhibit a beating effect governed by the ZBW frequency, $Ω$ .
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Figure 6
The ZBW effect in relativistic plane-wave shutter models involving (a) free Klein-Gordon particles of spin 0 and (b) Dirac particles of spin $1 / 2$ . In both cases, the ZBW effect appears as a series of quantum beats in the long-time behavior of particle density. In case (a), we plot the exact Klein-Gordon density using Eq. (20) (orange solid line) and the corresponding asymptotic density $ρ_{K G}$ . In case (b), we plot the exact Dirac probability density $ρ = | ψ_{1} |^{2} + {| ψ_{2} |}^{2}$ (orange solid line) using Eq. (25) and the corresponding asymptotic probability density $ρ_{D}$ . In both cases, we consider an energy $E = 10.0$ and a fixed value of position $x = 10.0$ .
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