# Hartman effect from layered $PT$-symmetric system

## TL;DR

This paper investigates the Hartman effect in layered $PT$-symmetric systems, showing tunneling time independence from system length and analyzing the limit where the number of layers becomes infinite.

## Contribution

It provides an analytical study of tunneling times in layered $PT$-symmetric systems, demonstrating the Hartman effect and its consistency with phase space methods.

## Key findings

- Tunneling time is independent of system length in the large layer width limit.
- In the infinite layer limit, tunneling time equals crossing time of empty space.
- The study confirms the Hartman effect in layered $PT$-symmetric systems.

## Abstract

The time taken by a wave packet to cross through a finite layered $PT$-symmetric system is calculated by stationary phase method. We consider the $PT$- symmetric system of fix spatial length $L$ consisting of $N$ units of the potential system `$+iV$' and `$-iV$' of equal width `$b$' such that $L=2Nb$. In the limit of large `$b$', the tunneling time is found to be independent of $L$ and therefore the layered $PT$-symmetric system display the Hartman effect. The interesting limit of $N \rightarrow \infty$ such that $L$ remains finite is investigated analytically. In this limit the tunneling time matches with the time taken to cross an empty space of length $L$. The result of this limiting case $N \rightarrow \infty$ also shows the consistency of phase space method of calculating the tunneling time despite the existence of controversial Hartman effect. The reason of Hartman effect is unknown to present day however the other definitions of tunneling time that indicate a delay which depends upon the length of traversing region have been effectively ruled out by recent attosecond measurements.