Quantum Goos-Hänchen shift and tunneling transmission at a curved step potential

We study the quantum Goos-Hänchen (GH) shift and the tunneling transmission at a curved step potential by investigating the time evolution of a wave packet. An initial wave packet is expanded in terms of the eigenmodes of a circular step potential. Its time evolution is then given by the interference of their simple eigenmode oscillations. We show that the GH shift along the step boundary can be explained by the energy-dependent phase loss upon reflection, which is defined by modifying the one-dimensional (1D) effective potential derived from the two-dimensional (2D) circular system. We also demonstrate that the tunneling transmission of the wave packet is characterized by a free-space image distant from the boundary. The tunneling transmission exhibits a rather wide angle divergence and the direction of maximum tunneling is slightly rotated from the tangent at the incident point, which is consistent with the time delay of the tunneling wave packet computed in the 1D modified effective potential.

Figure 1

(a) The effective potential $V_{\mathrm{eff}}\left(r\right)$ defined by Eq. (4) for $m=120$ and $V_0=5000$ is depicted by the blue line. The yellow and purple color correspond to the energy ranges where the bound and tunneling modes lie, respectively. (b) For the same $V_0$, the eigenmodes are represented in the $(m,\text{Re}(k\left)\right)$ plane. The central values of $m$ and $k$ of four wave packets (A–D) are marked by large solid dots, i.e., $(m_0,k_0)=(75,75)$ (red), $(120,90)$ (blue), $(140,122.065)$ (green), and $(140,140)$ (yellow), respectively. The tiny orange dots around B mark the relevant eigenmodes used to reconstruct the wave packet B (see Sec. 3).

Figure 2

(a) The initial Gaussian wave packet $|{\Psi }_I{(x,y)|}^2$ with $(m_0,k_0)=(120,90)$ is plotted with the 2D potential in the ($x,y$) plane. (b) Comparison between the original Gaussian wave packet defined by Eq. (18), depicted by the red line, and the reconstructed wave packet (dashed green line) obtained from Eq. (19). The horizontal and vertical profiles of $|{\Psi }_{\mathrm{I}}{(x,y)|}^2$ are shown. The center of $|{\Psi }_{\mathrm{I}}{(x,y)|}^2$ is located at $(x_0,y_0)$.

Figure 3

The time evolution of the wave packets is shown in the $(x,y)$ plane. The wave packets A and B are plotted in (a) and (b), respectively, at $t=0.5$ and $1.5$. The thick vertical yellow line corresponds to the $y$ axis, while the white curved line depicts the boundary of the circular cavity. In (c) and (d) the red rectangles are the expected positions of the wave packets A and B at various times obtained from Eq. (24) with 1% and 2.5% increased reflection angles, respectively, and show a very good agreement with the average positions ${\vec{x}}_{\mathrm{av}}\left(t\right)$ denoted by the solid blue dots.

Figure 4

The effective potential $V_{\mathrm{eff}}\left(r\right)$ [Eq. (4)] is modified according to (a), from left to right, in order to define the phase change ${\phi }_R$ upon reflection. The effective potential is plotted for $m=120$ and $V_0=5000$. (b) The phase change ${\phi }_R$ (red solid line) is plotted with respect to the energy and compared to the phase $\varphi$ of the 1D step potential case [see Eq. (22)] with ${\mathcal{V}}_0=5000$. The phases are nearly the same except for higher energy as shown in the inset.

Figure 5

Tunneling transmission of the wave packet B. (a) Schematic illustration of the coordinates $(D,h)$ and the rotation angle $\alpha$ for the emission Husimi function $H_E(D,h;\alpha )$. (b) The strength of tunneling transmission $f\left(\alpha \right)$ defined by Eq. (28) is evaluated at $t=7.5$ (red dotted line). The vertical green dotted line indicates the maximum of $f\left(\alpha \right)$. (c) The distance $\Delta$ of the normal tunneling wave component is plotted as a function of the angle $\alpha$ at two different times $t=7.5$ and $t=10$ (respectively, the red solid dots and the open blue circles). The vertical green dotted line marks the value ${\alpha }_T=-0.045$ where the curves intersect. (d) The normal tunneling wave component is plotted at $t=7.5$ and $t=10$ for the particular value ${\alpha }_T=-0.045$. (e) The average positions ${\vec{x}}_{\mathrm{av}}$ of the normal tunneling wave component are shown in the $(x,y)$ plane. The position at $t=1$ (solid red triangle) is extrapolated from the observed positions at $t=7.5$ and $t=10$ (solid red circles).

Figure 6

The resonance mode $(m,k)=(120,113-i1.57\times {10}^{-6})$ is shown in the upper part of the figure. In the lower part, the blue line is the corresponding effective potential with $V_0=5000$ and the red line represents the emission Husimi function $H_E(D_0,h)$ of the mode with $D_0=1.5R$, whose maximum gives the individual gap ${\Delta }_j$. The horizontal dashed line superimposed on the potential denotes the energy $E_j$ of the resonance mode.

Figure 7

Dynamics of tunneling wave packets in a rectangular barrier. (a) The potential shape depicted here is given by Eq. (32) with $V_{\max }=100$, $V_0=60$, $x_a=0$, and $x_b=1$. The red (blue) arrow approaching the barrier denotes the energy of the incident wave packet ${\Psi }_H$ (${\Psi }_L$). The average energy of tunneling wave packets are marked by the arrows beyond the barrier. (b) The upper panel describes the weighting distribution $W\left(E\right)\left|T\right(E\left)\right|$ of the plane wave components of the tunneling wave packets ${\Psi }_H$ and ${\Psi }_L$ (red and blue line, respectively) as a function of the energy. The lower panel depicts the phase ${\phi }_T$ of the transmission coefficient $T\left(E\right)$ (black solid line). The phase of the reflection coefficient (black dashed line) is also plotted for comparison. The time evolution of the tunneling wave packets $|{\Psi }_H{|}^2$ and $|{\Psi }_L{|}^2$ is shown in (c) and (d), respectively.

Figure 8

The dynamics of the tunneling wave packet with an average energy $E_0\simeq 118$ [horizontal orange arrow in (a)] is investigated in the modified effective potential shown in (a) and defined in Fig. 4 for the parameters $m=17$ and $V_0=100$. (b) The distribution of the plane wave components of the incident wave packet is plotted (red line) with respect to the energy and the maximum of the function is denoted by the vertical orange dotted line. The time evolution of the square modulus of the wave packet is depicted in (c). The time variation of ${\left|\Psi (x,t)\right|}^2$ at $x=2.7$ and $2.8$ is displayed in (d) and the time range delimited by the two maxima (light red window) corresponds to the time delay of the tunneling wave packet.

Figure 9

Time evolution of the high-energy wave packets. The wave packets C and D are respectively plotted in (a) and (b) in the $(x,y)$ plane at $t=0.6$ and $1.4$ and the average positions ${\vec{x}}_{\mathrm{av}}\left(t\right)$ at various times are denoted by red solid dots in (c) and (d) for the wave packets C and D, respectively.