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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 138358, 13 pages
http://dx.doi.org/10.1155/2011/138358
Research Article

Dissipative Effect and Tunneling Time

Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata 700 108, India

Received 15 March 2011; Revised 7 June 2011; Accepted 10 June 2011

Academic Editor: Yao-Zhong Zhang

Copyright © 2011 Samyadeb Bhattacharya and Sisir Roy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The quantum Langevin equation has been studied for dissipative system using the approach of Ford et al. Here, we have considered the inverted harmonic oscillator potential and calculated the effect of dissipation on tunneling time, group delay, and the self-interference term. A critical value of the friction coefficient has been determined for which the self-interference term vanishes. This approach sheds new light on understanding the ion transport at nanoscale.

1. Introduction

Caldeira and Laggett [1] started a systematic investigation of the quantum dissipative system and quantum tunneling in dissipative media. After that influential work, many authors have discussed the dissipative tunneling in numerous papers but not with profound illustration in the aspect of tunneling time. In fact, the proper definition of tunneling time has been debated for decades and it is yet to have definite answer [2]. Hauge and Støveng [3] mentioned seven different definitions of tunneling time of which the dwell time and the phase time or group delay are well accepted by the community. Winful [4] studied a general relation between the group delay and the dwell time. In case of quantum dissipative system Caldeira and Laggett used the path integral technique to study the dissipative quantum tunneling. Brouard et al. [5] made an important clarification of the existence of many tunneling times and the relations among them in a comprehensive framework. Ford et al. [6] investigated the dissipative quantum tunneling using quantum Langevin equation. The quantum Langevin equation is nothing but the Heisenberg equation of motion for the coordinate operator of a particle with certain mass, under a particular potential. This is the macroscopic description of a quantum particle interacting with a bath. The interaction with the bath corresponds to energy loss; in other words, it is the signature of dissipation. The memory function present in the equation describes the interaction with the bath. The nature of the dissipation is contained in the memory function. In a recent work [7] one of the present authors (S. Roy) investigated the transport of ions in biological system and constructed a nonlinear Schrödinger equation where the transport of ion occurs at nanoscale. Here, the authors proposed a particular type of memory kernel associated to the non-Markovian behaviour of ions so as to understand the observational findings. The mechanism of ion transport at nanoscale is becoming an important area of research. Considering the memory kernel for ion transport at nanoscale, we will discuss the tunneling phenomena within quantum Langevin framework. Here we consider a parabolic potential barrier of the form 𝑉(𝑥)=(1/2)𝑚Ω2(𝑑2𝑥2). The transmittance is calculated using the method devised by Ford et al. [6]. The misty aspect of tunneling time and various approaches towards it are critically analyzed. We will also try to figure out a process to calculate the tunneling time for a dissipative medium. In Section 2 we will briefly review quantum Langevin equation for convenience. Then, we will discuss various concepts related to tunneling time, that is, phase time, dwell time, and so forth, and the effect of dissipation in Section 3. The self-interference effect is discussed within this framework for dissipative systems. This method has been applied to understand the transport of K+ ion in the biological domain which is much relevant at the nanoscale in Section 4. Possible implications are indicated in Section 5.

2. Quantum Langevin Equation

We begin with the discussion on the tunneling of the ions through the dissipative potential barrier. The theory of dissipative quantum tunneling is pioneered by Caldeira and Laggett [1], where they treated the problem through the technique of path integral. But we will address how quantum Langevin equation can be used to discuss dissipative quantum tunneling using the approach developed by Ford et al. [6, 8, 9]. Here it is easy to incorporate non-Markovian and strong coupling effects using suitable memory function. The memory function present in the quantum Langevin equation describes the interaction with the bath. At first we will briefly discuss the general theory with a general memory function. Then, we will deal with a specific memory function which we have at our hands [7], in our problem of potassium ion transfer through ion channels. We consider an inverted harmonic oscillator potential barrier and see how the transmission coefficient is modified for inclusion of the memory function. The quantum Langevin equation has the form 𝑚̈𝑥+𝑡𝑑𝑡1𝜇𝑡𝑡1𝑡̈𝑥1+𝑈(𝑥)=𝐹(𝑡),(2.1) where the dot and the prime, respectively, describ the derivative with respect to 𝑡 and 𝑥. This is nothing but the Heisenberg equation of motion for the coordinate operator 𝑥 of a particle of mass 𝑚, under a potential 𝑈(𝑥). Here the coupling with the bath is described by two terms, the random force 𝐹(𝑡) with mean zero and a mean force characterized by the memory function 𝜇(𝑡𝑡1). The autocorrelation of 𝐹(𝑡) is given by 12𝑡𝐹(𝑡)𝐹1𝑡+𝐹1=1𝐹(𝑡)𝜋0𝑑𝜔Re𝜇𝜔+𝑖0+𝜔coth𝜔𝜔2𝐾𝑇cos𝑡𝑡1.(2.2) In this expression 𝜇(𝑧)=0𝑑𝑡𝑒𝑖𝑧𝑡𝜇(𝑡);Im(𝑧)>0(2.3) is nothing but the Fourier transformation of 𝜇(𝑡). The coupling to the bath is given by the function 𝜇(𝑧).

Now this function has three important mathematical properties corresponding to some very important physical principles [5, 7]. (1)The Im(𝑧)>0 condition states that the function is analytic in the upper half plane. This is a consequence of causality.(2)The second condition is the “positivity condition” stated as Re𝜇𝜔+𝑖0+0;<𝜔<.(2.4)This is a consequence of the 2nd law of thermodynamics.(3)The third condition is the reality condition stated as 𝜇𝜔+𝑖0+=𝜇𝜔+𝑖0+.(2.5)

This follows from the fact that 𝑥 is a Hermitian operator. These properties are very elaborately explained by Ford and his coauthors. Based on these three properties, the function can be specified to belong in a restricted class of functions, having a general representation in the upper half plane 𝜇(𝑧)=𝑖𝑐𝑧+2𝑖𝑧𝜋0𝑑𝜔Re𝜇𝜔+𝑖0+𝑧2𝜔2,(2.6) where 𝑐 is a positive constant, which can be absorbed in the particle mass.

Ford and his collaborators have considered harmonic oscillator potential 𝑈(𝑥)=(1/2)𝑚𝜔2𝑥2 as a simple example. Under this potential, the quantum Langevin equation takes the form 𝑚̈𝑥+0𝑑𝑡𝜇𝑡𝑡̇𝑥+𝑚𝜔20𝑥=𝐹(𝑡).(2.7) This equation can be solved by the method of Fourier transformation. We get the Fourier transformation of the coordinate operator 𝑥 as 𝐹̃𝑥(𝜔)=𝜂(𝜔)(𝜔),(2.8) where 𝜂(𝜔) is called the susceptibility and expressed as 𝜂(𝜔)=𝑚𝜔2+𝑚𝜔20𝑖𝜔𝜇(𝜔)1.(2.9) We consider an inverted harmonic oscillator potential of the form 𝑈(𝑥)=(1/2)𝑚Ω20(𝑑2𝑥2), where 2𝑑 is the width of the barrier. In the absence of dissipation we have an exact expression for the transmittance by the WKB approximation method 𝐷0=exp𝜋𝑚Ω0𝑑222𝐸𝑚Ω20.(2.10) If dissipation is included, the tunneling frequency will be changed. Then the expression will be modified by replacing the frequency in the nondissipative case by that in the dissipative case.

In case of the inverted harmonic oscillator potential the susceptibility takes the form 𝜂(𝜔)=𝑚𝜔2𝑚Ω20𝑖𝜔𝜇(𝜔)1.(2.11) The normal mode frequencies of this coupled system are the poles of the susceptibility. However, there is an isolated imaginary normal mode frequency corresponding to a pole of the susceptibility, which is classically forbidden and can be interpreted as the tunneling frequency 𝜔=𝑖Ω(Ω). The determining equation for Ω is []𝜂(𝑖Ω)1=𝑚Ω2𝑚Ω20+Ω𝜇(𝑖Ω)=0.(2.12) Putting the expression of 𝜇(𝑖Ω) from (2.6), we getΩ2+2Ω2𝑚𝜋0𝑑𝜔Re𝜇𝜔+𝑖0+Ω2+𝜔2=Ω20.(2.13) Since the left-hand side of (2.13) is a monotonically increasing function, the value of Ω will always be less than Ω0.

Let us consider a simple frictional coefficient “𝛾”. Under which the frequency determining equation becomes Ω2+𝛾ΩΩ20=0.(2.14) Since Ω must be real and positive, we take the positive solution of this quadratic equation 𝛾Ω=2+Ω0𝛾1+24Ω201.(2.15) Considering 𝛾Ω0, we get ΩΩ0(𝛾/2).

Then by replacing Ω0 by Ω in the expression of transmittance, we get the transmittance for dissipative medium: 𝐷=exp𝜋𝑚Ω𝑑222𝐸𝑚Ω2,(2.16)thatis,𝐷=𝐷0exp𝜋𝑚𝛾𝑑2+2𝜋𝐸𝛾Ω20,(2.17) where 𝐷 and 𝐷0 are the transmittance with and without dissipation, respectively.

Now the presence of dissipation can be incorporated in the potential function. The potential barrier without dissipation is 𝑉0(𝑥)=(1/2)𝑚Ω20(𝑑2𝑥2).

The potential barrier with dissipation can be expressed as 𝑉(𝑥)=(1/2)𝑚Ω2(𝑑2𝑥2). Relating these two, we get 𝑉(𝑥)𝑉0𝛾(𝑥)1Ω0.(2.18) So we can say that dissipation reduces the potential function. The dissipative contribution is included in the tunneling frequency Ω. We will use this fact to calculate the tunneling time and incorporate effect of dissipation in it.

3. Tunneling Time

In case of quantum mechanical tunneling through a barrier, it is well known how to calculate the probability of tunneling, the escape rate, and the lifetime in initial well. But the question is if there is a time analogous to classical time spent in the barrier region how long does it take a particle to tunnell through a barrier? The subject of this so-called tunneling time or traversal time has been covered by many authors in numerous independent approaches [3, 1015]. Brouard et al. [5] have discussed various aspects of tunneling time in a very systematic approach. The very elegant review of Hauge and Støveng [3] lists at least seven different types of tunneling time of which the phase time (group delay) and dwell time are considered well established.

3.1. Relation between Phase Time and Dwell Time

The group delay or phase time measures the delay between appearance of a wave packet at the beginning and the end of the barrier. By the method of stationary phase, it is given by the energy derivative of the transmission phase shift: 𝑡𝑝𝑇=𝑑𝜙𝑇.𝑑𝐸(3.1) Here 𝜙𝑇=𝜙𝑡+𝑘𝐿, where 𝐿 is the length of the barrier. Similarly the group delay for reflection is given by 𝑡𝑝𝑅=𝑑𝜙𝑅,𝑑𝐸(3.2) where 𝜙𝑅 is the reflection phase shift. The total group delay is defined as the total group delay:𝑡𝑝=||𝑇||2𝑡𝑝𝑇+||𝑅||2𝑡𝑝𝑅,(3.3) where 𝑇 and 𝑅 are the transmission and reflection coefficients, respectively. In case of symmetric barriers 𝑡𝑝=𝑡𝑝𝑇=𝑡𝑝𝑅.

Regardless of transmission or reflection, the dwell time is a measure of the time spent by a particle in the barrier region 𝐴<𝑥<𝐵. It is given by the expression 𝑡𝑑=𝐵𝐴||||𝜓(𝑥)2𝑑𝑥𝑗in,(3.4) where 𝜓(𝑥) is the wave function corresponding to energy 𝐸 and 𝑗in=𝑘/𝑚 is the flux of the incident particles. This equation gives us the time that the incident flux has to be turned on, to provide the accumulated particle storage in the barrier. Winful [4] has discussed that delay time and dwell time are related by a linear relation: 𝑡𝑝=𝑡𝑑+𝑡𝑖,(3.5) where 𝑡𝑖 is called the self-interference term given by the expression 𝑡𝑖=Im(𝑅)𝑘𝜕𝑘.𝜕𝐸(3.6) The self-interference term comes from the overlap of incident and reflected waves in front of the barrier. This term is of considerable importance at low energies, when the particle spends most of its time dwelling in front of the barrier, interfering with itself. In the relation given by (3.5), the self-interference term is disentangled and given by a separate expression in (3.6).

3.2. Calculation of Dwell Time and Self-Interference Term in Dissipative Case

The delay time can be calculated considering the effect of dissipation. In order to do that, first we will calculate the dwell time for the parabolic barrier we have taken as a model potential. Following Er-Juan and Qi-Qing [16] we begin with the parabolic potential barrier and then subdivide the potential into infinitesimal rectangular barrier elements and then summing up the individual dwell times spent by the particles inside the barrier elements, and the dwell time of the parabolic barrier is calculated.

Let us take a rectangular potential 𝑉(𝑥)=𝑉𝑖for𝑥𝑖1<𝑥<𝑥𝑖,𝑉(𝑥)=0for𝑥>𝑥𝑖1or𝑥<𝑥𝑖.(3.7) The solutions for the three regions can be written as 𝜓1(𝑥)=𝑒𝑖𝑘𝑥+Re𝑖𝑘𝑥for𝑥<𝑥𝑖,𝜓2(𝑥)=𝐶𝑒𝜅𝑥+𝐷𝑒𝜅𝑥for𝑥𝑖1<𝑥<𝑥𝑖,𝜓3(𝑥)=𝑇𝑒𝑖𝑘𝑥for𝑥>𝑥𝑖.(3.8) After some manipulations 1𝐶=𝑇2𝑘1+𝑖𝜅𝑒𝑖𝑘Δ𝜅Δ,1𝐷=𝑇2𝑘1𝑖𝜅𝑒𝑖𝑘Δ+𝜅Δ.(3.9) Now the transmission coefficient can be found as 𝑇=4𝑖𝑘𝜅𝑒𝜅Δ𝑒𝑖𝑘Δ(𝜅𝑖𝑘)2(𝜅+𝑖𝑘)2𝑒2𝜅Δ,(3.10) and the probability of transmission is ||𝑇||𝑃=2=16𝑘2𝜅2||(𝜅𝑖𝑘)2(𝜅+𝑖𝑘)2𝑒2𝜅Δ||2.(3.11) For thin barrier we have 𝑒2𝜅Δ1. Then the transmission coefficient and probability of transmission, respectively, become 𝑇𝑒𝜅Δ𝑒𝑖𝑘Δ and 𝑃𝑒2𝜅Δ. The dwell time 𝑡𝑑 is defined as 𝑡𝑑=(1/j)𝐵|𝜓2|2𝑑𝑥. Substituting these values, we get 𝑡𝑑=1𝑗𝐵𝑒2𝜅Δ1𝑑𝑥=𝑗𝐵||𝑇||21𝑑𝑥=𝑗𝐵𝑃𝑑𝑥.(3.12) Considering such barrier as a succession of the adjacent thin rectangular barrier elements in width Δ𝑖=𝑥𝑖𝑥𝑖1,

The corresponding dwell time for the 𝑖th barrier 𝑡𝑖𝑑=1𝑗Δ𝑖𝑃𝑖𝑒𝑑𝑥=2𝜅𝑖Δ𝑖𝑗Δ𝑖=𝑃𝑖Δ𝑖𝑗.(3.13) The dwell time for the entire barrier becomes 𝑡𝑑=𝑃1Δ1𝑗+𝑃1𝑃2Δ2𝑗+𝑃1𝑃2𝑃3Δ3𝑗++𝑖𝑛=1𝑃𝑛Δ𝑖𝑗+𝑡𝑑=𝑖=1𝑖𝑛=1𝑃𝑛Δ𝑖𝑗=1𝑗𝑖=1𝑒2𝑖𝜅𝑖(𝑥)Δ𝑖Δ𝑖.(3.14) Since the length of each barrier is very small, the summation can be replaced by integral: 𝑡𝑑=1𝑗𝑥2𝑥1𝑒2𝑥𝑥1𝜅(𝑥)𝑑𝑥𝑑𝑥,(3.15) where 𝜅(𝑥)=2𝑚(𝑉𝐸)/2, 𝑉=(1/(𝑥2𝑥1))𝑥2𝑥1𝑉(𝑥)𝑑𝑥 is a kind of average height approximation for the potential barrier 𝑉(𝑥). Our model potential is the inverse harmonic oscillator potential of the form 𝑉(𝑥)=(1/2)𝑚Ω2(𝑑2𝑥2). Therefore 𝑉=(1/3)𝑉𝐷0+(2/3)𝐸, where 𝑉𝐷0=(1/2)𝑚Ω2𝑑2.

Therefore, the integrant becomes𝑃𝑥=𝑒2𝑥𝑥1[2𝑚(𝑉𝐸)/2]𝑑𝑥,thatis,𝑃𝑥=𝑒2[2𝑚(𝑉𝐸)/2][𝑥𝑑2(2𝐸/𝑚Ω2)].(3.16)

Now 𝑡𝑑=(1/𝑗)𝑥2𝑥1𝑃𝑥𝑑𝑥. Finally putting the value of 𝑃𝑥, we get 𝑡𝑑=𝑚𝑒2𝐸2𝜅𝐴𝜅sinh2𝜅𝐴,(3.17) where 𝜅𝐴=2𝑚(𝑉𝐸)/2𝑑2(2𝐸/𝑚Ω2)=𝑑2𝑚/32𝑉𝐷0(𝑉𝐷0𝐸).

The effect of dissipation is included in 𝑉𝐷0, where 𝑉𝐷0=(1/2)𝑚Ω2𝑑2=(1/2)𝑚Ω20𝑑2(1(𝛾/2Ω0))2𝑉0(1(𝛾/Ω0)), 𝛾 is the coefficient of friction.

Now we have to calculate the self-interference term 𝑡𝑖=(Im(𝑅)/𝑘)(𝜕𝑘/𝜕𝐸). To calculate this term, we have to know the reflection coefficient 𝑅 for the concerned potential.

By the method of WKB approximation the reflection coefficient 𝑅 can be easily shown as 𝑅=((𝜃/4)(1/𝜃))𝑒((𝜃/4)+(1/𝜃))𝑖(𝜋/2).(3.18) Therefore, Im(𝑅)=((𝜃/4)(1/𝜃)),((𝜃/4)+(1/𝜃))(3.19) where 𝜃=exp𝑥2𝑥1𝜅(𝑥)𝑑𝑥=exp𝜋𝑚Ω𝑑2412𝐸𝑚Ω2𝑑2=𝑒𝛼.(3.20) For small 𝜃, Im(𝑅)(3+5𝛼).(5+3𝛼)(3.21) Therefore, the self-interference term is found to be 𝑡𝑖=2𝐸(3+5𝛼).(5+3𝛼)(3.22) So from (3.17) and (3.22), we find the complete expression of the group delay for the case of a particle tunneling through a barrier of inverse harmonic oscillator potential: 𝑡𝑝=𝑚𝑒2𝐸2𝜅𝐴𝜅sinh2𝜅𝐴+2𝐸(3+5𝛼).(5+3𝛼)(3.23)

3.3. Effect of Dissipation in Self-Interference Term

Now we will estimate the effect of dissipation in the self-interference term. The effect of dissipation is included in the frequency term Ω=(Ω0(𝛾/2)). From (3.20) we can write 𝛼=𝜋𝑚Ω𝑑24𝜋𝐸2Ω𝜋𝑚Ω0𝑑24𝜋𝐸2Ω0𝑚Ω0𝜋𝑑2+8𝜋𝐸4Ω0𝛾Ω0.(3.24) Hence, we can write 𝛼=𝛼0𝛼𝛾Ω0,(3.25) where 𝛼0=((𝜋𝑚Ω0𝑑2/4)(𝜋𝐸/2Ω0)) and 𝛼=((𝑚Ω0𝜋𝑑2/8)+(𝜋𝐸/4Ω0)).

Therefore, 𝑡𝑖=56𝐸11.0671.67+𝛼0𝛼𝛾/Ω0,(3.26)𝛼 and 𝛾 are both positive terms. Therefore, the denominator of the 2nd term on the right-hand side will reduce due to them and the 2nd term will increase. It will reduce the whole term. So the presence of dissipative term reduces the self-interference effect.

Now if we take Im(𝑅)=0, then𝛼𝛾=0+0.603𝛼Ω0.(3.27) Therefore, it is evident that, for a critical value of dissipation coefficient given by (3.27), the self-interference term vanishes. Let us now apply the approach to a biological phenomena, that is, transport of potassium ion through ion channels.

4. Potassium Ion Transfer through Ion Channel

Here we will consider a special case for potassium ion transport through ion channel. From the above discussion we can emphasize on the fact that the memory function 𝜇 is the all important function in this theory. It signifies the nature of the dissipative medium. By choosing this memory function properly, we can determine the tunneling coefficients and tunneling times for various dissipative media. Now at this very moment, our memory function representing the potassium ion transfer through ion channel comes into play. Ion channels are transmembrane protein structures that selectively allow given ion species to travel across the cell membrane. Zhou et al. [17] demonstrate that the channel protein transiently stabilizes three K+ states, two within the selectivity filter and one within the water basket towards the intracellular side of the selectivity filter. Experimental evidence indicates that the selectivity filter is devoid of water molecules other than single water molecule between K ions [18]. The memory kernel of our specific problem [7] can be written as 𝜇𝑡𝑡=𝑎0𝛿𝑡𝑡+𝑎1𝜏1𝑒|𝑡𝑡|/𝜏1𝑎2𝜏2𝑒|𝑡𝑡|/𝜏2.(4.1)

The oscillatory ionic dynamics in K+ ion channels is proposed to occur at the limit of the weak non-Markovian approximation associated with a time reversible Markov process, at the selectivity filter. This reversible stochastic process belongs to a different time scale to that governing diffusion across the rest of the channel, which is determined by the glue-like properties of water at the water basket. The framework of stochastic mechanics provides a model for such dissipative force in terms of quantum theory. That channel ionic permeation can be associated with nonlinear Schrödinger equation which addresses the issue of de-coherence and time scale considerations. Now the memory kernel contains both Markovian and non-Markovian contributions that allows a continuous change from Markovian to non-Markovian dynamics and enables identification of both the terms. The non-Markovian process has two time scales 𝜏1 and 𝜏2 whose contributions are dominated by the parameters 𝑎1 and 𝑎2. The first term contains the Markovian contribution. It is also clear that 𝑎1,𝑎2𝑎0 is the weak non-Markovian limit.

Averaging over 𝑡 and taking the Fourier transformation over the memory kernel, we get =𝑎Re𝜇(𝜔)0+𝑎1𝑎2𝑎1𝜏11+𝜔2𝜏21+𝑎2𝜏21+𝜔2𝜏22.(4.2) The determining equation of the tunneling frequency (Ω) is given by (2.13). Under the present circumstances, the equation becomes Ω2+1𝑚𝑎0+𝑎1𝑎2𝑎1𝜏11+Ω𝜏1𝑎2𝜏21+Ω𝜏2ΩΩ20=0.(4.3) Here we consider an approximation Ω𝜏1,Ω𝜏21. That is, Ω1/𝜏1,1/𝜏2. That is, the time scales 𝜏1 and 𝜏2 are very small. Since 𝜔 is a finite positive quantity, it is very small compared to the inverse of 𝜏1 and 𝜏2.

Taking up to the first order of the binomial terms, we get Ω2+1𝑚𝑎0+𝑎1𝑎2𝑎1𝜏11Ω𝜏1+𝑎2𝜏21Ω𝜏2ΩΩ20=0.(4.4) Neglecting the second-order terms of 𝜏1 and 𝜏2, we get Ω2+1𝑚𝑎0+𝑎1𝑎2𝑎1𝜏1𝑎2𝜏2ΩΩ20=0.(4.5) Let 𝛾=(1/𝑚)[(𝑎0+𝑎1𝑎2)(𝑎1𝜏1𝑎2𝜏2)]. So (4.5) may be written in the same form of (2.14),Ω2+𝛾ΩΩ20=0.(4.6) The tunneling coefficient is found to be𝐷=𝐷0exp𝜋𝑚𝑑2+2𝜋𝐸Ω201𝑚𝑎0+𝑎1𝑎2𝑎1𝜏1𝑎2𝜏2.(4.7)

In case of weak non-Markovian limit (𝑎0𝑎1,𝑎2), we neglect the 𝑎1,𝑎2 part, and 𝐷𝑤𝑛𝑚=𝐷0exp𝜋𝑚𝑑2+2𝜋𝐸Ω20𝑎0𝑚.(4.8) This is similar to (2.17).

For the strong non-Markovian case, we get 𝐷𝑠𝑛𝑚=𝐷𝑤𝑛𝑚exp𝜋𝑚𝑑2+2𝜋𝐸Ω201𝑚𝑎1𝑎2𝑎1𝜏1𝑎2𝜏2.(4.9) If we put this 𝛾 in the tunneling time expression, we get the group delay for this specific case of potassium ion transfer too.

The expression of delay time is given by (3.23), where 𝜅𝐴=𝑑2𝑚/32𝑉𝐷0(𝑉𝐷0𝐸).

The effect of dissipation is included in 𝑉𝐷0𝑉0(1(𝛾/Ω0)). So we get 𝜅𝐴=𝑑2𝑚32𝑉0𝛾12Ω01𝑉0𝑉𝐸0𝛾Ω0𝑑2𝑚32𝑉0𝑉0𝐸𝑑2𝑚32𝑉0𝛾2Ω0𝑉0.+𝐸(4.10)

In this case of potassium ion channel with the memory kernel as given in (4.1), the expression of the delay time will be 𝑡𝑝=𝑚𝑒2𝐸2𝜅𝐴𝜅sinh2𝜅𝐴+2𝐸(3+5𝛼),(5+3𝛼)(4.11) with 𝜅𝐴=𝑑2𝑚32𝑉0𝑉0𝐸𝑑2𝑚32𝑉0𝑎(1/𝑚)0+𝑎1𝑎2𝑎1𝜏1𝑎2𝜏22Ω0𝑉0,+𝐸𝛼=𝛼0𝛼1𝑚Ω0𝑎0+𝑎1𝑎2𝑎1𝜏1𝑎2𝜏2.(4.12)

So the delay time or phase time will depend on the parameter values 𝑎0, 𝑎1, 𝑎2, 𝜏1, and 𝜏2.

5. Possible Implications

It is evident from the above analysis that the effect of dissipation on group delay can be estimated directly in terms of the frictional coefficient. It is also possible to express the self-interference term in terms of the friction coefficient (𝛾), and we can estimate the critical value of 𝛾 for which the interference term vanishes. The chosen biological example indicates that the present approach may play an important role in understanding the ion transport at nanoscale which will be considered in subsequent papers. We are also interested in the numerical estimation of tunneling time and the effect of dissipation on it. For that purpose, currently we have the required data for electron tunneling through water. But when one considers electron tunneling through water, the electron-phonon interaction must also play an important role. This interaction will contribute in the potential in a considerable manner. But at nanoscale water behaves more like frozen ice [19]. In that frozen water configuration, the electron transfer through water is only weakly affected by electron-phonon interaction [20]. But currently we do not have sufficient data for that numerical calculation. We hope to present this thorough numerical estimation of tunneling times in subsequent papers.

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