Dissipative Effect and Tunneling Time
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.
Caldeira and Laggett  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 . Hauge and Støveng  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  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.  made an important clarification of the existence of many tunneling times and the relations among them in a comprehensive framework. Ford et al.  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  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 . The transmittance is calculated using the method devised by Ford et al. . 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 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 , 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 , 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 where the dot and the prime, respectively, describ the derivative with respect to and . This is nothing but the Heisenberg equation of motion for the coordinate operator of a particle of mass , under a potential . Here the coupling with the bath is described by two terms, the random force with mean zero and a mean force characterized by the memory function . The autocorrelation of is given by In this expression is nothing but the Fourier transformation of . The coupling to the bath is given by the function .
Now this function has three important mathematical properties corresponding to some very important physical principles [5, 7]. (1)The 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 This is a consequence of the 2nd law of thermodynamics.(3)The third condition is the reality condition stated as
This follows from the fact that 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 where is a positive constant, which can be absorbed in the particle mass.
Ford and his collaborators have considered harmonic oscillator potential as a simple example. Under this potential, the quantum Langevin equation takes the form This equation can be solved by the method of Fourier transformation. We get the Fourier transformation of the coordinate operator as where is called the susceptibility and expressed as We consider an inverted harmonic oscillator potential of the form , where is the width of the barrier. In the absence of dissipation we have an exact expression for the transmittance by the WKB approximation method 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 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 . The determining equation for is Putting the expression of from (2.6), we get Since the left-hand side of (2.13) is a monotonically increasing function, the value of will always be less than .
Let us consider a simple frictional coefficient “”. Under which the frequency determining equation becomes Since must be real and positive, we take the positive solution of this quadratic equation Considering , we get ).
Then by replacing by in the expression of transmittance, we get the transmittance for dissipative medium: where and 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 .
The potential barrier with dissipation can be expressed as . Relating these two, we get So we can say that dissipation reduces the potential function. The dissipative contribution is included in the tunneling frequency . 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, 10–15]. Brouard et al.  have discussed various aspects of tunneling time in a very systematic approach. The very elegant review of Hauge and Støveng  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: Here , where is the length of the barrier. Similarly the group delay for reflection is given by where is the reflection phase shift. The total group delay is defined as the total group delay: where and are the transmission and reflection coefficients, respectively. In case of symmetric barriers .
Regardless of transmission or reflection, the dwell time is a measure of the time spent by a particle in the barrier region . It is given by the expression where is the wave function corresponding to energy and 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  has discussed that delay time and dwell time are related by a linear relation: where is called the self-interference term given by the expression 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  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 The solutions for the three regions can be written as After some manipulations Now the transmission coefficient can be found as and the probability of transmission is For thin barrier we have . Then the transmission coefficient and probability of transmission, respectively, become and . The dwell time is defined as . Substituting these values, we get Considering such barrier as a succession of the adjacent thin rectangular barrier elements in width ,
The corresponding dwell time for the th barrier The dwell time for the entire barrier becomes Since the length of each barrier is very small, the summation can be replaced by integral: where , is a kind of average height approximation for the potential barrier . Our model potential is the inverse harmonic oscillator potential of the form . Therefore , where .
Therefore, the integrant becomes
Now . Finally putting the value of , we get where .
The effect of dissipation is included in , where , is the coefficient of friction.
Now we have to calculate the self-interference term ). To calculate this term, we have to know the reflection coefficient for the concerned potential.
By the method of WKB approximation the reflection coefficient can be easily shown as Therefore, where For small , Therefore, the self-interference term is found to be 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:
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 . From (3.20) we can write Hence, we can write where and .
Therefore, and 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 , then 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 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.  demonstrate that the channel protein transiently stabilizes three 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 . The memory kernel of our specific problem  can be written as
The oscillatory ionic dynamics in 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 and whose contributions are dominated by the parameters and . The first term contains the Markovian contribution. It is also clear that is the weak non-Markovian limit.
Averaging over and taking the Fourier transformation over the memory kernel, we get The determining equation of the tunneling frequency is given by (2.13). Under the present circumstances, the equation becomes Here we consider an approximation . That is, . That is, the time scales and are very small. Since is a finite positive quantity, it is very small compared to the inverse of and .
Taking up to the first order of the binomial terms, we get Neglecting the second-order terms of and , we get Let . So (4.5) may be written in the same form of (2.14), The tunneling coefficient is found to be
In case of weak non-Markovian limit (), we neglect the part, and This is similar to (2.17).
For the strong non-Markovian case, we get If we put this 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 .
The effect of dissipation is included in . So we get
In this case of potassium ion channel with the memory kernel as given in (4.1), the expression of the delay time will be with
So the delay time or phase time will depend on the parameter values , , , , and .
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 (), and we can estimate the critical value of 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 . In that frozen water configuration, the electron transfer through water is only weakly affected by electron-phonon interaction . 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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