Tunneling time and stochastic-mechanical trajectories for the double-barrier potential

Physics Letters A 377 (2013) 357–361

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Tunneling time and stochastic-mechanical trajectories for the double-barrier potential Tomoshige Kudo, Hideo Nitta ∗ Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184-8501, Japan

a r t i c l e

i n f o

Article history: Received 9 May 2012 Received in revised form 21 November 2012 Accepted 3 December 2012 Available online 14 December 2012 Communicated by P.R. Holland Keywords: Tunneling time Resonant tunneling Stochastic mechanics Dwell time Presence time Double-barrier potential

a b s t r a c t Quantum motion of particles tunneling a double barrier potential is considered by using stochastic mechanics. Stochastic-mechanical trajectories give us information about complex motion of tunneling particles that is not obtained within the framework of ordinary quantum mechanics. Using such information, we calculate the tunneling times within each of the barriers which depend on the distance between them. It is found that the stochastic-mechanical tunneling time shows better asymptotic behavior than the quantum-mechanical dwell time and presence time. © 2012 Elsevier B.V. All rights reserved.

1. Introduction In the standard framework of quantum mechanics physical quantities are expressed by the self-adjoint operators called observables. There is an exception, however. Time is not an observable [1,2]. Indeed, as proved by Pauli, there is no self-adjoint time operator conjugate to a Hamiltonian bounded from below [3,4]. This means that expectation values related to the time variable cannot be calculated within the standard framework of quantum mechanics. As a result, there are some ambiguities in the calculation of tunneling times [2,5–8]. Although several definitions such as the phase time [9], the dwell time [10,11], the local Larmor time [12–14,11] and the complex time [15] have been proposed, tunneling times have remained to be controversial until now. Recently, the use of stochastic mechanics [16] has been proposed for the calculations of the tunneling time [17,18], arrival time [19], and presence time [18,19]. These works have shown that stochastic mechanics may provide a unique numerical method of analyzing the time development of wave packets in spite of the fact that Nelson himself repudiated his original theory of stochastic mechanics [20]. Indeed, in stochastic mechanics the quantum motion of a particle is described as a set of sample paths or “stochastic-mechanical trajectories” [21]. Therefore, it becomes possible to define tunneling time in the similar way of classical mechanics [4]. This advantage of stochastic mechanics may give us

*

Corresponding author. E-mail address: [email protected] (H. Nitta).

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information about tunneling time, which is not obtainable within ordinary quantum mechanics. In this Letter we study numerically the resonance tunneling phenomena under a symmetric double barrier potential and compare the tunneling times between two barriers using stochastic mechanics as well as quantum mechanics. In the following sections, we show that the tunneling times for two barriers are different despite the symmetry of the double barrier potential. The ratio of these tunneling times changes as a function of the distance, height and width of two barriers. It is also shown that, in contrast to tunneling times based on the definition of the dwell time and presence time in quantum mechanics, the numerical result based on stochastic mechanics has a correct asymptotic behavior. 2. Theory 2.1. Model We consider the simple symmetric double rectangular potential,





V = V 0 Rect[−l/2 − d, −l/2; x] + Rect[l/2, l/2 + d; x] , where

Rect[x1 , x2 ; x] =



0 (x < x1 ), 1 (x1  x  x2 ), 0 (x2 < x),

(1)

(2)

d is the width of a barrier, and l the distance between barriers. In this case the wave function for the stationary states becomes

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T. Kudo, H. Nitta / Physics Letters A 377 (2013) 357–361

Fig. 1. (Color online.) Examples of (a) tunneled trajectories and (b) reflected trajectories calculated by stochastic mechanics (V 0 = 1.5E 0 ). The grey-color areas represent the region of the double-barrier potential.

⎧ ikx Ae + Be −ikx (x < −l/2 − d), ⎪ ⎪ ⎪ ⎨ C e κ x + De −κ x (−l/2 − d  x < −l/2), ϕk (x) = F e ikx + Ge−ikx (−l/2  x < l/2), ⎪ κx −κ x ⎪ ⎪ (l/2  x < l/2 + d), ⎩ He + Ie K e ikx (l/2 + d  x)

(3)

where the amplitudes A , B , . . . , K are determined by the boundary conditions √ as usual, k is the wave number of the incoming plane wave, κ = 2m( V 0 − E )/¯h , and E = h¯ 2 k2 /(2m). For the calculation of tunneling times, we use the Gaussian-like wave packet ψ(x, t ) given by [17]

∞ ψ(x, t ) =

2 2 Ne −(k−k0 ) /(2σ ) e −ikx0 ϕk (x)e −i Et /¯h dk,

(4)

−∞

where N is the normalization factor, k0 the mean value of the wave number for the superposition, σ −1 the width of the initial wave packet, and x0 the central position of the wave packet at t = 0. In stochastic mechanics [16], the forward time evolution of a stochastic-mechanical trajectory x(t ) is given by Ito’s stochastic differential equation,





dx(t ) = b x(t ), t dt + dw (t ),

(5)

Fig. 2. (Color online.) The motion of a wave packet tunneling a double-barrier potential. The red solid lines represent |ψ(x, t )|2 . The blue crosses represent distributions derived from stochastic-mechanical trajectories.

use a finite time t instead of infinitesimal time dt. We determine t so that the stochastic-mechanical calculations reproduce quantum-mechanical probability density |ψ(x, t )|2 . Examples of such agreements are shown in Fig. 2. 2.2. Tunneling times Following the definition of classical time spent in a certain region [4,18], the tunneling time τ (i ) for the ith stochasticmechanical trajectory xi (t ) (i = 1, 2, . . . , n) passing through a barrier region x1  x  x2 is defined by

t f

τ (i ) =

∂ b(x, t ) = (Re + Im) ln ψ(x, t ), m ∂x and dw (t ) is the Wiener process satisfying





dw (t ) = 0,





dw (t )2 =

h¯ m

dt ,

(7)

where · · · represents the statistical average over all possible trajectories. In Fig. 1 we show examples of the stochastic-mechanical trajectories calculated by using Eqs. (1)–(7) with the Monte Carlo method. We note that stochastic mechanics makes it possible to separate clearly the transmitted trajectories (Fig. 1(a)) from the reflected trajectories (Fig. 1(b)). For numerical calculations one must

(8)

0

τs =

n  τ (i )

n

i =1

(6)

We introduce the stochastic-mechanical tunneling time

where b(x, t ) is the drift velocity,

h¯



Rect x1 , x2 ; xi (t ) dt .

.

τ s [18] as (9)

It is noted that τ s has been calculated by choosing only the tunneled trajectories passing through the barrier at t → ∞, as in Fig. 1(a): the trajectories passing through barriers once but finally go back to the initial region have not been included. For comparison, we have calculated four types of dwell times [5,10,11,22] and the presence time [4] within the framework of quantum mechanics. It is worthwhile to mention that the phase time is related to the dwell time for stationary wave functions [23–25] and hence we will not consider further the phase time in this work. For a stationary wave function ϕ (x) in the region x1  x  x2 the dwell time is given by [10,11]

τ

d0

=

1 j

x2   ϕ (x)2 dx, x1

(10)

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where j = h¯ k/m is the incoming current density. It should be noted that τ d0 includes both the transmitted wave and the refracted wave. To distinguish these contributions, transmission and reflection dwell times for stationary wave function are introduced [26,27]. The transmission dwell time is given by

τ

dt0

=

1 jt

x2   ϕ (x)2 dx,

(11)

x1

where jt = h¯ k| K |2 /m is the transmitted current density, | K |2 being the transmission coefficient (see Eq. (3)). The reflection dwell time is given by

τ

dr0

=

x2   ϕ (x)2 dx,

1 jr

(12)

x1

where j r = h¯ k| B |2 /m is the reflection current density, | B |2 being the reflection coefficient. Due to the conservation of the probability current density, j = jt + | j r |, these dwell times, τ d0 , τ dt0 , and τ dr0 , satisfy the relation [26],

1

τ d0

=

1

τ dt0

+

1

τ dr0

.

(13)

For comparison with the stochastic-mechanical tunneling time we also introduce the dwell time for the time-dependent wave packet ψ(x, t ). The dwell time for ψ(x, t ) in the region x1  x  x2 is given by [5,22]

∞ x2   ψ(x, t )2 dx dt .

τd =

(14)

0 x1

As is pointed out in Ref. [5], the dwell time should be considered as the time spent in the barrier rather than the tunneling time. An alternative method to obtain a kind of tunneling time is to use the presence time. The presence time distribution of the wave packet arriving at a fixed point X is defined by

|ψ( X , T )|2 dT . |ψ( X , T )|2 dT 0

ρ X ( T ) dT =  ∞ Using p T  X

(15)

ρ X ( T ) the presence time [4,18,28] is given by ∞

=

T ρ X ( T ) dT .

(16)

τ1d0 , τ2d0 , τ1d and τ2d , respectively, τ1p p s d and τ2 , respectively. In (b) and (c), τ1s /τ2s and τtot (blue crosses), τ1d /τ2d and τtot p p p d0 (black dash–dot lines), τ1d0 /τ2d0 and τtot (gray dashed lines), and τ1 /τ2 and τtot and gray dashed line represent the dwell times of

0

Then, the presence time for the region x1  X  x2 is calculated by

τ p =  T  pX =x2 −  T  pX =x1 .

Fig. 3. (Color online.) (a) The tunneling times of the first barrier and the second barrier, (b) their ratio, (c) the total time tunneling through the double-barrier potential as a whole, and (d) the transmission probability for the tunneling case, V 0 = 2E 0 . In (a), blue circles and gray crosses represent stochastic-mechanical tunneling times of τ1s and τ2s , respectively, green solid line, cyan broken line, black dash–dot line

(17)

3. Numerical results The numerical results of the tunneling times as a function of the distance between two barriers, l, are shown in Fig. 3 for a tunneling case, i.e. V 0 > E 0 , and in Fig. 4 for a traverse case, i.e. V 0 < E 0 , where E 0 = h¯ 2 k20 /(2m). In these figures τ1s , τ2s , s and τtot represent stochastic-mechanical tunneling times for the first (or left) barrier, the second (or right) barrier, and the doublebarrier potential as a whole, respectively. In the numerical calculations, we have set k0 = 5, d = 1/k0 , σ = 0.5, N = 0.18 and x0 = −10 in the atomic units. The values of t have been chosen in such a manner that the stochastic-mechanical calculations of the transmission probability agree with the corresponding quantummechanical calculations. Such agreements are demonstrated in Fig. 3(d) and Fig. 4(c).

and magenta thin solid line and red dotted line represent presence times of

(magenta solid lines) are shown. In (c), green dotted line and cyan dashed line repdt0 dr0 resent τtot and τtot , respectively. In (d), blue crosses represents normalized number of tunneling trajectories calculated by stochastic mechanics, and red solid line and gray dashed line represent transmissivities of the wave packet and of the stationary wave function, respectively.

Fig. 3(a) shows the difference between τ1s (blue circles) and τ2s (gray crosses). For small l, i.e. when the second barrier is close to the first one, τ1s becomes about two times larger than τ2s . Fig. 3(b) shows the tunneling time ratio τ1s /τ2s (blue crosses). As l increases, the amplitude of oscillation of τ1s /τ2s decreases. For large l, the ratio τ1s /τ2s approaches to unity. This is because the scattering of tunneling particles by the first barrier and the second barrier becomes almost incoherent when the distance between barriers becomes much larger than the size of the wave packet, i.e. l  σ −1 . For comparison, the tunneling times calculated by quantum mechanics are also shown. In Fig. 3(a) the green solid line, cyan

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Fig. 5. (Color online.) (a) Ratio of tunneling (traversal) times and (b) resonant tunneling (traversal) times as a function of V 0 / E 0 for k0 = 5, d = l = 1/k0 , x0 = −10.

Fig. 4. (Color online.) (a) Ratio of traversal times, resonant traversal times, (b) resonant traversal time in total barrier, and (c) normalized number of traversed particles. The potential height is V 0 = 0.5E 0 . The other parameter values and conventions are the same as in Fig. 3.

broken line, black dash–dot line and gray dashed line represent the dwell times of τ1d0 , τ2d0 , τ1d and τ2d , respectively, and the magenta thin solid line and red dotted line represent presence times τ1p and τ2p , respectively. All of these values oscillate as l varies. p The oscillation of τ1 is so large that the value multiplied by 0.3 is shown. According to our numerical data these large oscillations are p due to the contribution of reflected waves in  T  X =−l/2 . In contrast, p p since  T  X =l/2+d includes only the transmitted waves, τ2 shows small oscillations. The black dash–dot lines and gray dashed lines in Fig. 3(b) and (c) represent dwell times calculated by using τ d and τ d0 , respectively, and the magenta solid lines represent the presence time τ p . The ratio τ1d0 /τ2d0 oscillates without damping as a result

of the use of stationary waves. On the other hand τ1d /τ2d shows a damped oscillation, although its amplitude is much larger than that of τ1s /τ2s . In addition, for large l the ratio becomes constant

value, τ1d /τ2d = 2.2, as shown in the inset. These results of τ1d /τ2d may be caused by the inclusion of reflected waves in the dwell p p time [5]. The ratio based on the presence time, τ1 /τ2 , has been also shown as the value multiplied by 0.05 (magenta solid line): the amplitude of oscillation is very large. This is caused by the rep p markable difference of oscillation between τ1 and τ2 mentioned p p above. It should be noted that τ1 /τ2 keeps oscillating even at large l. Fig. 3(c) shows the resonant tunneling time for the doubles barrier potential as a whole. In Fig. 3(c), τtot (the blue crosses), p d0 d τtot (the magenta solid line), τtot (the black dash–dot line), τtot (the dt0 (the green dotted line) oscillate roughly gray dashed line), and τtot

dr0 d0 in phase while τtot (the cyan dashed line) out of phase. At τtot = dt0 dr0 τtot in Fig. 3(c), the values of τtot become infinite. This is cond0 dt0 dr0 d0 , τtot , and τtot , only τtot firmed by using Eq. (13). Among τtot shows a moderate behavior. In Fig. 3(d) we show the tunneling (transmission) probability of the double-barrier potential as a whole. The agreement between stochastic-mechanical calculation and quantum-mechanical calculation is almost perfect. In fact, as mentioned before, such an agreement of the total transmission probability as shown in Fig. 3(d) or Fig. 4(c) has been used as the criterion for choosing the suitable values of t. In Fig. 4 we show the traverse case, i.e. the potential height is smaller than the average energy of the incoming wave packet (V 0 = 0.5E 0 ). In this case, as one can see from Fig. 4(c), the transmission probability is large and hence the dwell time is not affected very much by reflected waves. Therefore, in the traverse case we may expect better agreement between τ1s /τ2s and τ1d /τ2d than that of the tunneling case. However, as seen in Fig. 4(a), the asymptotic value of τ1d /τ2d is again beyond unity, τ1d /τ2d = 1.1. In stochastic mechanics, on the other hand, τ1s /τ2s approaches to unity asymptotically. Comparing these asymptotic behaviors, one may consider that the tunneling time/traverse time calculated by using stochastic mechanics provides more physically valid values than that of quantum-mechanical dwell time. In Fig. 5(a) we show τ1s /τ2s as a function of the normalized po-

tential height, V 0 / E 0 , for the case of l = d. Again, neither τ1d /τ2d p p nor τ1 /τ2 agrees with the stochastic-mechanical result. Fig. 5(b)

d represents τtot as a function of V 0 / E 0 . At V 0 < E 0 , τtot agrees well s with τtot . However, at V 0  E 0 where tunneling becomes substantial, they begin to separate. This is, as we have already pointed out, due to the influence of reflected waves in the dwell times. s It should be emphasized that τtot includes only transmitted trajectories. This kind of discrimination is impossible in the calculations of the dwell time or presence time. Fig. 6(a) shows τ1s /τ2s as a function of the barrier width, d, for V 0 = 0.5E 0 , i.e. the traverse case. It is expected that at d  l, where the resonant effect ceases, the traversal time of the first barrier coincides with that of the second barrier. At d  l we find that τ2s /τ1s only approaches to unity while τ2d /τ1d and τ2p /τ1p do not represent such a proper asymptotic behavior. Fig. 6(b) shows τ1s /τ2s

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between two barriers becomes much larger than the size of the wave packet. In our calculations only stochastic mechanics shows this asymptotic behavior. References

Fig. 6. (Color online.) (a) Ratio of traversal times as a function of the barrier width d for V 0 = 0.5E 0 with k0 = 5, l = 0.2, and x0 = −10. (b) Ratio of tunneling times for V 0 = 1.5E 0 .

for a tunneling case (V 0 = 1.5E 0 ). Again, we have not found any agreements among three calculated methods of tunneling time. 4. Concluding remarks We have shown that, for a double barrier potential, the tunneling time of the first barrier is different from that of the second barrier. Their ratio oscillates due to the resonance effect. Stochastic mechanics makes it possible to use only the tunneled trajectories in the calculation of tunneling time. We have pointed out that the tunneling time ratio should approach to unity when the distance

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