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Hartman effect from layered PT-symmetric system

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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.

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Acknowledgements

MH acknowledges supports from Director-SSPO for the encouragement of research activities. BPM acknowledges the support from MATRIX project (Grant No. MTR/2018/000611), SERB, DST Govt. of India.

Author information

Correspondence to Bhabani Prasad Mandal.

Appendix

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.

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Hasan, M., Mandal, B.P. Hartman effect from layered PT-symmetric system. Eur. Phys. J. Plus 135, 84 (2020). https://doi.org/10.1140/epjp/s13360-019-00084-1

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