Tunneling Flight Time, Chemistry, and Special Relativity - The Journal

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Tunneling Flight Time, Chemistry and Special Relativity Jakob Petersen, and Eli Pollak J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b02018 • Publication Date (Web): 09 Aug 2017 Downloaded from http://pubs.acs.org on August 10, 2017

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The Journal of Physical Chemistry Letters

Tunneling Flight Time, Chemistry and Special Relativity Jakob Petersen and Eli Pollak∗ Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel E-mail: [email protected]

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Abstract Attosecond ionization experiments have not resolved the question: “What is the tunneling time?”. Different definitions of tunneling time lead to different results. Secondly, a zero tunneling time for a material particle suggests that the non-relativistic theory includes speeds greater than the speed of light. Chemical reactions, occurring via tunneling, should then not be considered in terms of a non-relativistic quantum theory calling into question quantum dynamics computations on tunneling reactions. To answer these questions, we define a new experimentally measurable paradigm - the tunneling flight time - and show that it vanishes for scattering through an Eckart or a square barrier, irrespective of barrier length or height, generalizing the Hartman effect. We explain why this result does not lead to experimental measurement of speeds greater than the speed of light. We show that this tunneling is an incoherent process by comparing a classical Wigner theory with exact quantum mechanical computations.

Keywords: tunneling; quantum tunneling time; quantum scattering; Hartman effect.

Table of contents graphic |hq|Ψt i|2 [10−2 ] 0

0.5

1

1.5

2

12 10 t [102 ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

8 6 4 −4

−3

−2

−1

0

1

2

3

4

2

q [10 ]


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The quantum phenomenon of material tunneling through a potential barrier was predicted 90 years ago. 1 One of the intriguing and not fully understood aspects is the tunneling time. 2–11 The simple question of how long does it take a particle to tunnel through a barrier has met with a spectrum of responses, ranging from a vanishing tunneling time 9,12,13 to assertions that the question is invalid and there is no such tunneling time. 14 Especially the observation by Hartman that a tunneling time for transition through a symmetric square barrier is independent of the thickness of the barrier 15 has intrigued many. 16 It seemingly implies that material particles can travel at speeds greater than the speed of light (c). 17 Some of the confusion is due to differing definitions 18–20 of quantum mechanical time which also give different answers. 20 Is there then a unique answer to the tunneling time question? Recently we showed 21–23 that the tunneling time may be related to what is known in Chemistry and Biology as the transition path time distribution. 24 It is defined as the probability that a transition between two points in space of a molecular system will take a time t. An analytic expression for the classical distribution has been worked out for a parabolic barrier in the presence of dissipation 25–27 and applied successfully to the experimentally measured transition path time distribution of proteins moving from an unfolded to a folded state and vice versa. 28 A thermal quantum transition path time distribution was then used to study tunneling transitions. 21 For a parabolic barrier, when above barrier transitions are excluded, as is the case at the crossover temperature between tunneling and thermal activation, 22 the mean tunneling time vanishes. In this paper we suggest a related but new paradigm which we name the Tunneling Flight Time (TFT). Consider a wavepacket incident on a barrier, and due to tunneling is reflected and transmitted. One may define separately the transition path time distribution for the transmitted and reflected parts. This may be obtained in principle from a time of flight experiment, by which particles are released at t = 0 from a trap 29,30 and their subsequent time of arrival at a predetermined point is measured. The TFT is determined by considering the difference between the mean transition path time of transmitted and reflected particles.



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We will show that for symmetric Eckart and square barriers the TFT vanishes. The specific Eckart barrier we will employ is characteristic of the hydrogen exchange reaction. 31 The fact that the TFT vanishes seemingly invalidates the usage of non-relativistic quantum mechanics for the hydrogen exchange reaction and perhaps all light atom transfer tunneling reactions since it implies speeds faster than c. We will argue that a vanishing TFT does not violate special relativity. The Hartman effect states that the time it takes to tunnel through a square barrier, although finite, is independent of the barrier width. Here we show that the TFT vanishes and is independent not only of the width but also the height of the barrier. This generalizes the Hartman effect. We also show though that for a square barrier a vanishing TFT does not lead to experimental measurement of speeds greater than c. The TFT formalism is presented for one dimensional systems, generalization to many ˆ = dimensions is straightforward. The Hamiltonian operator is H

pˆ2 2M

+ V (ˆ q ). qˆ and pˆ are

the coordinate and momentum operators, V (ˆ q ) is the potential with barrier height V ‡ . It is assumed to be symmetric about q = 0 and vanishes when q → ±∞. Within a time dependent scattering formalism it is convenient to consider an initial Gaussian minimum uncertainty wavepacket Ψ localized about the coordinate position x and momentum px ( )1/4 ( ) Γ Γ i 2 ⟨q|Ψ⟩ = exp − (q − x) + px (q − x) , π 2 ~

(1)

where Γ is the width parameter. Since tunneling is the phenomenon to be studied, one must assure that the transmission of the particle indeed occurs only via tunneling through the barrier. 12,37 For example, for the parabolic barrier studied in Refs. 21,22 this occurs only at the crossover temperature. At higher temperature the mean tunneling time is finite due to above barrier crossing. To assure that only tunneling contributes to the transmission implies that the initial kinetic energy (Ei = p2x / (2M )) is smaller than the barrier height V ‡ and that the variance about


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this initial mean

√

~2 Γ/M is sufficiently smaller than V ‡ − Ei so that the probability for

initially having an incident momentum whose energy is greater than or equal to the barrier height is much smaller than the transmission probability at the mean incident momentum. A third condition is that the initial wavepacket does not “leak” into the region to the right of the barrier. This implies that the initial location of the wavepacket x, chosen to the left of the barrier is such that the distance to the barrier is much larger than the spatial width √ of the incident wavepacket (1/ Γ). The transition path time distribution is defined in terms of a (positive) density correlation function at time t about the final position y: [ Ct (y; Ψ) = T r |Ψ⟩⟨Ψ| exp

(

ˆ iHt ~

)

(

ˆ iHt δ (ˆ q − y) exp − ~

)]

⟨ ( ) ⟩ 2 ˆ iHt = y exp − Ψ . (2) ~

We are assuming the existence of a clock, which ticks when the particle exits an orifice preparing it in the state |Ψ⟩. It ticks again if at time t the particle crosses the point y. The transition path time probability distribution is then Ct (y; Ψ) Pt (y; Ψ) = ∫ ∞ , dtCt (y; Ψ) 0

(3)

and we have tacitly assumed that the time integral of Ct does not diverge. This probability distribution is related to the so-called presence time distribution Ref. 32–35 but is not the same. The latter is defined on the time interval [−∞, ∞] leading to easier analytic manipulation than for the transition path distribution which is defined on the time interval [0, −∞]. ∫∞ ˆ ˆ The time integrated density operator defined as −∞ dt exp(−iHt/~)|Ψ⟩⟨Ψ| exp(iHt/~) comˆ but if one takes only the interval [0, ∞] it does not. Since mutes with the Hamiltonian H a time of flight experiment occurs in the interval [0, ∞] it is necessary to consider the more complex transition path time distribution. An analysis of the presence time distribution may be found in Ref. 36. As the wavepacket is incident upon the barrier it will split up into transmitted and 5 ACS Paragon Plus Environment


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reflected parts with corresponding probabilities |TΨ |2 and |RΨ |2 . If the final point y is to the left of the barrier and sufficiently to the left of the center of the incident wavepacket √ (y < x < 0, x − y > 1/ Γ), then as a function of time one will only see the reflected wavepacket at y. If the final point is to the right of the barrier (y > 0) one will find a single contribution at y resulting from the transmitted part of the wavefunction. This allows us to decompose the transition path time distribution into two independent parts, the reflected and transmitted transition path time distributions

Pt (y; Ψ) = |RΨ |2 PR,t (−y; Ψ) + |TΨ |2 PT,t (y; Ψ)

(4)

and by definition the two distributions PR,t (−y; Ψ) and PT,t (y; Ψ) are normalized to unity (y > 0) when time integrated. The mean transition path time for the reflected and trans∫∞ mitted distributions is ⟨t⟩(R,T ) = 0 dt tP(R,T ),t (∓y; Ψ) . In the following we will study the mean time difference ∆t (Γ) ≡ ⟨t⟩R − ⟨t⟩T

(5)

as a function of the width parameter Γ. For any finite width of the initial wavepacket one expects that the transmitted mean time will be shorter than the reflected mean time, as shown in Fig. 1 for the Eckart barrier. The transmission through the barrier is sensitive to the incident momentum and will prefer the higher initial momentum components, since they have higher transmission probabilities. The opposite works for the reflected part. For the Eckart and square barriers, for which the transmission probability is known analytically, one finds that for sufficiently small Γ this mean time difference is positive and scales linearly: ⟨t⟩R − ⟨t⟩T ∝ Γ. The tunneling flight time (TFT) is then defined as:

tTFT ≡ lim [−∆t(Γ)]. Γ→0


(6)

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35 PR,t (−1000;Ψ) PT,t (1000;Ψ)

30 25 Pt [10−5 ]

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20 15 10 5 0 10 11 12 13 14 15 16 17 18 19 20 21 t [103 ]

Figure 1: Reflected and transmitted transition path time distributions at ∓y = ∓1000 for an initial Gaussian wavepacket Ψ with width parameter Γ = 7 · 10−5 and center position and momentum at (x, px ) = (−500, 0.1) incident upon a unit height Eckart barrier. The transmitted particle arrives slightly earlier than the reflected part, due to the preferential filtering of higher momentum components. The units are the dimensionless units defined in the text. where the minus sign is used to indicate that the TFT is a measure of the transmission time. This definition strips away the filtering of the incident momentum components, since in this limit there is only one component. The critical reader might note that when the width ( ) vanishes one is left with the simple free wave exp ~i px (x − q) and this wave has amplitude over all of space including the region to the right of the barrier. In fact, the limit has to be taken carefully. The incident point x and the final points −y and y must always be chosen √ such that their magnitude is larger than 1/ Γ as discussed above. As Γ becomes smaller these points also move out to infinity. More specifically, for any finite value of the final location y, no matter how large, there exists an incident point x whose magnitude is smaller than that of y such that initially the wavefunction is centered about x and is sufficiently far from the barrier so that it does not overlap with it. The mean time difference is therefore well defined for any finite value of x, y and Γ. As the momentum width Γ becomes smaller, the momentum filtering becomes smaller and ultimately one is left with the contribution of tunneling only.



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If the potential is not symmetric, the momenta in the two asymptotic regions are different, affecting the mean time difference ∆t. To generalize the definition of the TFT we consider separately a mean time difference for particles incident upon the barrier from the left ∆tL (Γ) and from the right ∆tR (Γ). The overall mean time difference is taken as the average

∆t(Γ) =

1 [∆tL (Γ) + ∆tR (Γ)] . 2

(7)

The TFT is then defined as in the symmetric case (Eq. 6). In this asymmetric case, experimental determination of the flight time implies measuring the transition path time distributions in the forward and backward directions of the same reaction. This is more involved than in the symmetric case, but conceptually straightforward and in principle measurable. First we consider the TFT for a model of the hydrogen exchange reaction. The mass is the relative translational mass (M = 2mH /3), the barrier height and frequency are taken to be V ‡ = ~ω ‡ = 0.01 a.u.. The symmetric Eckart potential is V (ˆ q ) = V ‡ / cosh2 (ˆ q /d) where the characteristic length scale is d. The barrier frequency is ω ‡2 =

2V ‡ . M d2

Using the

¯ = ~ω ‡ /V ‡ , the Eckart barrier Hamiltonian dimensionless variables q¯ = q/d, E¯ = E/V ‡ and ~ ¯ ≡ reduces to the simple form: H

ˆ 2H V‡

¯2 d2 d¯ q2

= − ~2

+

2 . cosh2 q¯

For the model hydrogen exchange

¯ = 1. We will also use the dimensionless values reaction with parameters as given above, ~ t¯ = ω ‡ t, p¯ =

p M ωd

¯ = Γd2 . Henceforth the notation will be without the bars. and Γ

The numerical propagation of the wavepacket Ψ has been carried out using the split operator technique with the grid range [−2800; 2800] (absorbing boundaries are applied), 216 equidistant grid points, and the time step 5 · 10−3 . The mean time difference ∆t (Γ) between the reflected and transmitted particle is shown in Fig. 2 as a function of the reduced width parameter for the incident central position x = −500, incident central momenta px = 1 (panel a) and 0.1 (panel b), final points ∓y = ¯2 = 1. The mean time difference is positive and may be fitted to the linear form ∓1000, and ~ ∆t (Γ) = αΓ. The quality of such a fit is high, as shown explicitly in Table 1 which gives the


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2

6

(b)

(a) 5

∆t [102 ]

4 ∆t [10−1 ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3

1

2 1 0

¯2 = 1 QM: px = 1, ~ ¯2 = 1 W: px = 1, ~

0

1 2

3

4 5

6 7 Γ [10

−5

8

¯2 = 1 QM: px = 0.1, ~ ¯2 = 1 W: px = 0.1, ~

0

9 10 11 12 13

0

1

2 3

4 5

]

6 Γ [10

7 8 −5

9 10 11 12 13

]

Figure 2: Mean time difference ∆t as a function of the width parameter Γ for the Eckart barrier. QM refers to exact quantum mechanical results, W to results obtained using the classical Wigner prescription as described in the text, which explicitly assumes a vanishing tunneling time. Panels (a) and (b) are for incident momenta 1 and 0.1, respectively. The units are dimensionless as in Fig. 1. linear fit parameter and its estimated standard error for the numerically exact quantum data (αQM ) and a classical Wigner approximation (αW ) described below. The average relative dif∑ 2 2 ference between the two estimates is calculated as: Ω = N1 N j=1 (∆tQM,j − ∆tW,j ) /∆tQM,j . As discussed above, the parameters were chosen such that the probability of above barrier transmission or leakage of the initial wavefunction into the transmitted region is at least two orders of magnitude smaller than the transmission coefficient at the average incident energy. Table 1: Numerical data for the linear dependence of the mean time difference on the width parameter for the Eckart barrier. For further details, see the text. ¯2 ~

px

|TΨ |2

αQM

αW

Ω

1

1

2.8 · 10−3

4.71959 · 103 ± 4 · 10−2

4.71799 · 103 ± 1 · 10−3

1.2 · 10−7

1

0.1

2.1 · 10−6

1.589 · 106 ± 4 · 103

1.567 · 106 ± 3 · 103

1.4 · 10−4

0.4

1

1.6 · 10−6

4.71774 · 103 ± 3 · 10−2

4.71593 · 103 ± 3 · 10−3

1.5 · 10−7

10

0.1

2.9 · 10−2

1.590 · 106 ± 3 · 103

1.583 · 106 ± 3 · 103

2.0 · 10−5

Two competing effects contribute to ∆t. Filtering of the initial momenta favors higher momenta for the transmitted part and lower momenta for the reflected part leading to a 9 ACS Paragon Plus Environment


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positive time difference. The other would have to do with the tunneling time. If it is finite, it would tend to make the transmitted part slower than the reflected part, reducing the mean time difference. This reduction should be insensitive to the magnitude of the width parameter Γ and so cannot be the source for the linear dependence shown in Fig. 2. The characteristic barrier width (depending on the incident momentum) as well as the mass and incident momentum are of the order of unity. Hence, for a free particle, the traversal time of the barrier region would be of the order of unity. This is much larger than the magnitude of the time differences shown in Fig. 2(a) and would lead to a negative time difference and not a positive one as shown in the figure. In this context it is also worthwhile to note that the imaginary time 38 associated with the transmission (tim =

dlnP (E) ) 2dE

at an incident momentum

of unity is 3.14, much larger than the mean time differences shown in Fig. 2(a). For each of the incident central momenta, a corresponding wave packet calculation has ¯2 , which effectively changes the height of the Eckart been carried out using a non-unit ~ barrier. Changing the barrier height but maintaining the central momentum of the incident wavepacket obviously affects the transmission probability, but the linear fit parameter α is virtually invariant, see Table 1. The linear extrapolation shown in Fig. 2 and detailed in Table 1 is compelling. Within numerical error, the TFT vanishes irrespective of the magnitude of the incident momentum or the barrier height (see Table 1). This conclusion is based on a numerical computation and an extrapolation. To further verify it we consider a Wigner dynamics approximation which incorporates also the filtering property as well as an explicit zero TFT. The correlation function of Eq. (2) is a trace of two operators. It can be rewritten as a phase space trace of two Wigner densities. One is the Wigner representation of the ) ( 2 x) incident Gaussian wavepacket ρΨ (p, q) = π1~¯ exp −Γ (q − x)2 − (p−p . The other is the 2 ¯ Γ ~ Wigner representation of the Heisenberg time evolved density ρˆ (t) = eiHt δ (ˆ q − y) e−iHt . We ˆ

ˆ

then replace the exact Wigner representation of the time evolved quantum density with its classical Wigner approximation δ (qt − y) where qt is the classical trajectory that is evolved


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(analytically) to time t from the initial condition (q, p), but with a caveat. When the trajectory reaches the (left) turning point of the potential it is reflected with probability |R (p)|2 and transmitted (with probability |T (p)|2 ) to the right turning point at the same energy instantaneously and then continued in the transmitted direction. In this way, the initial wavepacket is separated into a reflected and transmitted part and one can as before determine the reflected and transmitted transition path time probability distributions and their time averages. As may be seen in Table 1, the Wigner dynamics estimate for the mean time difference is practically identical to the quantum one, but with two differences. Since an instantaneous transition from the right to the left turning point has been imposed, the classical Wigner estimate for the mean time difference must extrapolate to a zero TFT in the limit that Γ → 0. As shown in Table 1, the slopes for the Wigner dynamics and the numerically exact result are almost identical, supporting the contention that the TFT vanishes. Moreover, the Wigner dynamics time difference is consistently a bit smaller than the exact quantum time difference. This is another indication that the TFT vanishes, since a nonvanishing tunneling flight time would increase the mean transmission time within the Wigner approximation and the Wigner based time difference would become smaller, thus increasing the discrepancy with the numerically exact results. The comparison with the Wigner based results also illustrates that the source of the positive time difference shown in Fig. 2 is related to the filtering of momenta due to the small transmission probability. This filtering is included in the classical Wigner computation. Finally, the excellent agreement between the numerically exact quantum and approximate Wigner results indicates that quantum coherence is not very important in the tunneling process, as the Wigner approximation is an incoherent approximation having to do with probabilities but not with amplitudes. The vanishing TFT seems to imply that the tunneling particle may have speeds greater than c and this would invalidate using a non-relativistic theory for such a system. To answer, one should reconsider what one means when talking about a measured speed. The particle



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is assumed to be non-relativistic, its incident velocity is small compared to c. In the time of flight experiment, each particle arrives at the desired final point at some time t. Repeating the experiment many times will give the associated mean transition time. The mean of the distance between the source and the detector, divided by this mean time is the measured speed of the particle. As noted though, to assure that above barrier transmission is negligible, √ the characteristic distance between the source and the detector is of the order of 1/ Γ which is much larger than the distance traveled under the barrier. The mean measured time will therefore be characteristic of free particle motion at the incident momentum which is much much smaller than c. In different words, assuring that the flight time is associated only with tunneling, implies that the tunneling path must be much shorter than the path between source and detector and one will essentially measure only the initial free particle speed. The instantaneous TFT contribution cannot lead to a measurement of speeds of the order of c. We also undertook a study of the TFT for a symmetric square barrier potential V (ˆ q) = V ‡ [Θ(ˆ q + a) − Θ(ˆ q − a)], where Θ is the step function and 2a is the width of the barrier. As in the Eckart barrier the initial wavepacket is a coherent state (Eq. (1)). The transmission √ 4E(V ‡ −E) 1 2 2M V ‡ − p2 . probability is |T (E)|2 = 4E(V ‡ −E)+V ‡2 sinh2 (2K(E)a) with K (E = p /(2M )) = ~ The quantum propagation of the wavepacket is carried out as described in Ref. 39. For the square barrier atomic units are used throughout the calculations (i.e., ~ = M = 1). Results are presented for the mean time difference as a function of Γ in Fig. 3. Numerical results for the linear fits using the numerically exact quantum mechanical and the approximate Wigner propagations are given in Table 2. It is evident that in all cases the resulting TFT is zero, as in the case of the Eckart barrier. This zero TFT for all barrier configurations is consistent with the Hartman effect 15 but goes beyond it. The TFT is not only independent of the width of the barrier it is also independent of the barrier height. Since the width of the barrier can become arbitrarily long, this would seem to support the conclusion that the speed of the tunneling particle is unbounded and thus greater than c. 17 We now proceed to show that this is not the case.


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6

6 QM: a = 1 W: a = 1 QM: a = 5 W: a = 5 QM: a = 10 W: a = 10

QM: V ‡ = 1

5

W: V

‡

5

=1

QM: V ‡ = 5 W: V ‡ = 5

4

4

QM: V ‡ = 10 W: V ‡ = 10

3

∆t

∆t [10−1 ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3

(a)

(b)

2

2

1

1

0

0

1

2

3

4

5

6

7

8

9

0

10

0

1

2

3

Γ [10−5 ]

4

5

6

7

8

9

10

Γ [10−5 ]

Figure 3: Mean time difference ∆t between reflected and transmitted wavepackets is plotted as a function of the width Γ of the initial wavepacket Ψ for the square barrier. In (a) the barrier width is constant a = 1, and in (b) the barrier height is constant V ‡ = 1. QM (W) refers to exact quantum mechanical results (classical Wigner approximation). Here a.u. are used throughout.

Table 2: Numerical data for the linear dependence of the mean time difference on the width parameter for the square barrier. The notation is as in Table 1 except that the units are a.u.. Constant barrier width: a = 1 V‡

|TΨ |2

αQM

αW

Ω

1

7.1 · 10−2

6.22148 · 103 ± 5 · 10−2

6.22035 · 103 ± 5 · 10−2

3.3 · 10−8

5

8.8 · 10−6

4.66507 · 103 ± 5 · 10−2

4.66418 · 103 ± 5 · 10−2

3.6 · 10−8

10

2.0 · 10−8

4.21695 · 103 ± 5 · 10−2

4.21637 · 103 ± 5 · 10−2

1.9 · 10−8

Constant barrier height: V ‡ = 1 a

|TΨ |2

αQM

αW

Ω

1

7.1 · 10−2

6.22148 · 103 ± 5 · 10−2

6.22035 · 103 ± 5 · 10−2

3.3 · 10−8

5

8.3 · 10−9

2.99189 · 104 ± 1.8 · 100

2.99192 · 104 ± 1.7 · 100

1.0 · 10−10

10

1.7 · 10−17

5.96815 · 104 ± 7.6 · 100

5.96822 · 104 ± 7.6 · 100

1.6 · 10−10



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The experimentally measured speed of the particle is given by the distance between the incident wavepacket and the point at which one measures the arriving particle. This distance √ may be written as L = 2a+m/ Γ with m a number greater than 2. Assuming that tunneling occurs instantaneously, and noting that in the free particle region the speed is px /M , implies that the time it takes to cross the distance L on the average is at least pM√mΓ and the velocity x √ px n+m is given by (with 2a Γ ≡ n) M m . Velocities greater than c are possible only if n ≫ m. As in the Eckart barrier it is necessary to consider pure tunneling. The condition that there is no leakage into the transmitted zone implies that the initial density |⟨a|Ψ⟩|2 is much smaller than the transmission probability, which for deep tunneling goes as exp [−4K(Epx )a]. This implies that (n + m)2 ≫ 4K(Epx )a. The condition that the momentum spread is suffi[ ] ciently small to prevent above barrier transmission implies that exp −(p‡ − px )2 /(~2 Γ) ≪ √ 1−ϵ √ exp (−4K(Epx )a). But this means that n ≪ π(1−ϵ) , where λ‡ = 2π~/p‡ is the de1+ϵ λ‡ Γ Broglie wavelength associated with the barrier height and ϵ2 ≡ Epx /V ‡ . Combining the two conditions one finds that necessarily (

√ 1+ϵ m )2 2π √ 1−ϵ≫4 1+ ≫ , ‡ n aΓλ (1 − ϵ)

(8)

which taking into consideration that ϵ ≤ 1 implies that m ≫ n. In other words, the conditions that the initial momentum is lower than the barrier, that momenta cannot leak over the barrier, and that the incident wavepacket cannot leak through the barrier assure that in any measurement, the velocity will be of the same order as the incident velocity and it is impossible to obtain velocities of the order of c. The TFT has been studied in this letter for one dimensional symmetric Eckart and square barriers under a variety of incident conditions. We found that the TFT vanishes for the square barrier irrespective of its width and height, generalizing the Hartman effect. We have also shown why the vanishing TFT cannot lead to experimentally measured velocities greater than c. The observation that the TFT vanishes is to be expected for any one dimensional


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symmetric barrier. Evidence for this was presented for thermal tunneling through a parabolic barrier in Ref. 22. There is no obvious reason why different barrier shapes should give different results, however this needs to be further verified. Similarly, one should study the TFT for scattering through asymmetric barriers as well as in multidimensional systems. The excellent agreement between the numerically exact quantum mechanical results with the more approximate classical Wigner propagation indicates that coherences are not very important in the parameter range studied. What does this imply for the attosecond tunneling ionization experiments? The TFT has to be considered separately for these experiments, since they involve a time dependent Hamiltonian.

Acknowledgements We thank Professors Ilya Averbukh, Julius Jellinek and Gershon Kurizki for stimulating discussions. This work was supported by grants from the Israel Science Foundation, the Minerva Foundation and the German Israel Foundation for Basic Research.

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Tunneling Flight Time, Chemistry, and Special Relativity - The Journal

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