Hartman effect from layered PT-symmetric system

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 displays 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.

Appendix

Consider that the transfer matrix M, of a ‘unit cell’ potential is known,

$$\begin{aligned} M(k)= \begin{pmatrix} M_{11}(k) &{} \quad M_{12}(k) \\ M_{21}(k) &{} \quad M_{22}(k) \end{pmatrix}, \end{aligned}$$

(49)

such that,

$$\begin{aligned} \begin{pmatrix} A_{+}(k) \\ B_{+}(k) \end{pmatrix}= M(k) \begin{pmatrix} A_{-}(k) \\ B_{-}(k) \end{pmatrix}, \end{aligned}$$

(50)

where $A_{+}, B_{+}$ are the coefficients of the asymptotic solution of the scattering wave to the right of the potential (Eq. 8) and $A_{-}, B_{-}$ are the coefficients of the asymptotic solution of the scattering wave to the left of the potential (Eq. 9). Then, the transmission coefficient (for incidence from left) of a periodic system made by n repetitions of the ‘unit cell’ is given by

$$\begin{aligned} t_{n}= \frac{\hbox {e}^{-i k n s}}{\left[ M_{22}(k) \hbox {e}^{-iks} U_{n-1}(\zeta ) - U_{n-2} (\zeta )\right] }, \end{aligned}$$

(51)

where

$$\begin{aligned} \zeta = \frac{1}{2} \left( M_{11}\hbox {e}^{iks} +M_{22} \hbox {e}^{-iks}\right) . \end{aligned}$$

(52)

Here $s=w+g$, where w is width of the ‘unit cell’ potential and g is the gap between consecutive ‘unit cell’ potentials. For the present problem, $w=2b$ and $g=0$, thus $s=2b$. The procedure to derive Eq. 51 is outlined in [56]. A caution note for the reader is, in [56], the transfer matrix considered is $M^{-1}$; however, the procedure to arrive at Eq. 51 is the same as detailed in [56]. From Eq. 18, the $M_{11}$ and $M_{22}$ elements of our ‘unit cell’ PT-symmetric system are

$$\begin{aligned} M_{11}= & {} \frac{1}{4} \hbox {e}^{-2ikb} \left( P_{+}^{1} P_{+}^{2}-S^{1}S^{2}\right) , \end{aligned}$$

(53)

$$\begin{aligned} M_{22}= & {} \frac{1}{4}\hbox {e}^{2ikb} \left( P_{-}^{1} P_{-}^{2}-S^{1}S^{2}\right) . \end{aligned}$$

(54)

Using Eqs. 15 and 16 in Eq. 53, the $M_{11}$ element can be simplified to

$$\begin{aligned} M_{11}= (\xi +i\chi ) \hbox {e}^{-2ikb}. \end{aligned}$$

(55)

In the above, we have essentially separated the term $P_{+}^{1} P_{+}^{2}-S^{1}S^{2}$ in real and imaginary parts to arrive at Eq. 55. The expression for $\xi$ and $\chi$ is given by Eqs. 21 and 22, respectively. Next, we simplify $M_{22}$ element using Eqs. 15 and 16 in Eq. 54. It is easy to show that

$$\begin{aligned} M_{22}= (\xi -i\chi ) \hbox {e}^{2ikb}. \end{aligned}$$

(56)

We observe $M_{22}=M_{11}^{*}$ which implies that the argument, $\zeta$ of Chebyshev polynomial is real. We substitute Eqs. 55 and 56 in Eq. 52. This gives $\zeta = \xi$. Substituting Eq. 56 and $\zeta =\xi$ in Eq. 51, our final expression of transmission coefficient $t_{n}=t$ is given by

$$\begin{aligned} t=\frac{\hbox {e}^{-ikL}}{G(k)} , \end{aligned}$$

(57)

where $G(k)=(\xi -i\chi ) U_{N-1}(\xi )-U_{N-2}(\xi )$ (Eq. 20). In Eq. 57, we have identified $ns=2Nb=L$, where L is the net spatial extent of our layered PT-symmetric system.