Tunneling time in non-Hermitian space fractional quantum mechanics

TL;DR

This work develops non-Hermitian space-fractional quantum mechanics (NHSFQM) and uses the stationary-phase method to derive a closed-form expression for the tunneling time of a wave packet through a complex barrier. It shows that, in general, the Hartman effect is absent in SFQM with non-Hermitian potentials, but a Hartman-like restoration can occur for specific combinations of the absorption strength Vi and Levy index α. The analysis reveals competing influences: absorption tends to increase the tunneling time with barrier width, while fractional-order transport (smaller α) tends to reduce it, enabling nontrivial cancellations in certain parameter regimes. Overall, the work bridges NHQM and SFQM, providing analytic expressions for the tunneling time and offering insights for transport in nonlocal, lossy quantum media with potential applications in optics and quantum devices.

Abstract

We investigate the tunneling time of a wave packet propagating through a non-Hermitian potential $V_{r} - iV_{i}$ in space-fractional quantum mechanics. By applying the stationary phase method, we derive a closed-form expression for the tunneling time for this system. This study presents the first investigation of tunneling time at the interplay of non-Hermitian quantum mechanics and space-fractional quantum mechanics. The variation in tunneling time as the system transitions from a real to a complex potential is analyzed. We demonstrate that the tunneling time exhibits a dependence on the barrier width $d$ in the limit $d\rightarrow \infty$, showing the absence of the Hartman effect. A particularly striking feature of our findings is the potential manifestation of the Hartman effect for a specific combination of the absorption component $V_{i}$ and the Levy index $α$. This behavior arises from the fact that the presence of the absorption component $V_{i}$ leads to a monotonic increase in tunneling time with barrier thickness, whereas the Levy index $α$ reduces the tunneling time. The interplay of these contrasting influences facilitates the emergence of the Hartman effect under a specific combination of $V_{i}$ and the fractional parameter $α$.

Figures (6)

• Figure 1: (Color online). Figure presents the variation of tunneling time $\Gamma_{\alpha}$ as a function of barrier width $d$ for a complex potential in standard QM ($\alpha = 2$), with $V_r = 5$ and different values of $V_i$, for an incident energy of $E = 4$ ($< V_r$). The system exhibits the well-known Hartman effect for the real potential and this effect disappears when $V_{i}> 0$.
• Figure 2: (Color online). Figure presents the variation of tunneling time $\Gamma_{\alpha}$ with the absorption component $V_i$ for different values of Levy index $\alpha$ and for fixed barrier thicknesses (a) $d = 1.5$ and (b) $d = 5$. The incident wave energy is fixed at $E = 4$. For high value of $V_{i}$, the tunneling time decreases, and as $\alpha$ decreases, the tunneling time also decreases.
• Figure 3: (Color online). Contour plots illustrating the variation of tunneling time in the $\alpha-d$-plane for (a) a real potential ($V_{i}=0$) and for (b) a complex potential ($V_{i}=20$). The incident wave energy is fixed at $E=4$. The color scale represents the tunneling time.
• Figure 4: (Color online). Variation of tunneling time $\Gamma_{\alpha}$ with barrier thickness $d$ for different values of $\alpha$. For $\alpha<2$, tunneling time initially rises, reaching a peak at a specific threshold $d$ and then begins to decrease with increasing $d$. For $\alpha=2$, the well-known Hartman effect is recovered. The incident wave energy is fixed at $E = 4$.
• Figure 5: (Color online). Contour plots illustrating the variation of tunneling time in the $V_{i}-d$ plane, in (a) standard QM ($\alpha = 2$) and in (b) SFQM ($\alpha = 1.96$). The incident wave energy is fixed at $E = 4$. The color variation represents the tunneling time.