29 April 2002

Physics Letters A 296 (2002) 265–271

www.elsevier.com/locate/pla

Manifestation of Zeno and chaotic-like effectson a one-dimensional multibarrier potential of finite range

D. Bara,∗, L.P. Horwitza,b

a Department of Physics, Bar Ilan University, Ramat Gan, Israelb Raymond and Beverly Sackler Faculty of Exact Science, School of Physics, Tel Aviv University, Ramat Aviv, Israel

Received 13 February 2002; accepted 12 March 2002

Communicated by P.R. Holland

Abstract

We study the problem of a large numberN of one-dimensional (bounded) potential barriers in a finite interval. We show thatits various physical parameters depend sensitively upon the ratio of total spacing to total barriers width, exhibiting, for criticalparameter values, Zeno-type transmission and chaotic-like effects. 2002 Elsevier Science B.V. All rights reserved.

PACS: 05.45.Pq; 02.10.Yn; 03.65.Nk

Keywords: Transmission coefficient; Chaos; Scattering cross-section

Quantum systems with chaotic-like properties in the presence of tunneling through asingle barrier have beenstudied recently [1,2]. It was emphasized in the works of [2] that the complexity of the wave function at the barrierand its time dependence in the neighbourhood of it, where Zaslavsky’s criterion for the decrease of the Ehrenfesttime [3] is expected to be satisfied, indicate the existence of chaotic-like behaviour in the presence of the barrier.There are many physically important problems of conduction through more than one barrier. In particular, one mayhave a system with a large number of barriers, such as may occur in successively evaporated layers, or a largenumber of impurities or crystal defects. The result we obtain by studying a large numberN of barriers, includingthe limit of N → ∞, appear to be new and of theoretical interest (for example, the limit does not converge to givethe effect of a single barrier, but contains a much reacher structure).

We study here a one-dimensional set ofN rectangular barriers on a finite interval. For largeN we find thephenomena of anomalously large transmission for a critical range of parameter values as well as signaturesof chaotic-like behaviour (such as spectral distributions and complexity of the wave function). In this case thecomplex behaviour appears to be due to the large number of repeated interactions represented by the potential. Theemergence of Zeno-type anomalous transmission appears to correspond to the onset of decoherence for criticalparameter values. Although we obtain, in the limit of largeN , analytical expression for the scattering, our numerical

* Corresponding author.E-mail address: bardan@mail.biu.ac.il (D. Bar).

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9601(02)00272-4

266 D. Bar, L.P. Horwitz / Physics Letters A 296 (2002) 265–271

results for the form of the wavepacket forN = 4 andN = 150 for the case〈e〉packet> v indicate that the sourceof complexity is the repetitive structure of the potential which could be expected, as for the case of tunneling, todiminish the Ehrenfest time [1].

The array that we study starts atx = − a+b2 and ends atx = a+b

2 , so the finite length of this array isL = a + b.The parametera here is taken to be the sum of the widths of all theN potential barriers, that is, the length of allthe places in our entire array where the potentialv(x) = 0, so that the width of each barrier isa

N; b is taken to be

the sum of the separating spaces between all the neighboring barriers (the length of each such interval isbN−1). We

use the transfer matrix method [5,6] to calculate the transmission probability and theS-matrix (the Hamiltonian

for our problem is taken to beH = p2

2m + v(x)).We discuss first the case ofe > v. Using the terminology of [5] we can write the following transfer matrix

equation which governs the behaviour of our bounded system

(1)

[A2n+1B2n+1

]= T (n)T (n−1) · · ·T (2)T (1)

[A0B0

],

A2n+1 andB2n+1 are the amplitudes of the transmitted and reflected parts, respectively, of the wave function fromthenth potential barrier.A0 is the coefficient of the initial wave that approaches the potential barrier system, andB0 is the coefficient of the reflected wave to the left of the first barrier.T (n) is the product of three two-dimensionalmatrices

(2)

T (n) =[e−ik(

(n−1)bN−1 + (2n−1)a

2N ) 0

0 eik((n−1)bN−1 + (2n−1)a

2N )

][T11 T12T21 T22

][eik(

(n−1)bN−1 + (2n−3)a

2N ) 0

0 e−ik((n−1)bN−1 + (2n−3)a

2N )

],

where

T11 = cos

(aq

N

)+ i

ξ

2sin

(aq

N

), T12 = i

η

2sin

(aq

N

), T21 = −i

η

2sin

(aq

N

),

(3)T22 = cos

(aq

N

)− i

ξ

2sin

(aq

N

),

k =√

2me

h̄2 , q =√

2m(e−v)

h̄2 , andξ andη are given byξ = qk

+ kq, η = q

k− k

q. Taking the limit of very largeN we

obtain for the coefficients at the pointx = a + b wheren = N .[A2N+1B2N+1

]=(

1+ i

N

((kb + aq

ξ

2

)σ3 + iaq

η

2σ2

))N

(4)=[e−iz

(cosφ + if

sin(φ)φ

)ie−izd

sin(φ)φ

−eizdsin(φ)

φeiz(cosφ − if

sin(φ)φ

)][A0B0

],

whereσ2, σ3 are the Pauli matrices, and we have definedz = k(a + b), f = kb + aqξ2 , d = aq

η2 and φ2 =

(f σ3 + idσ2)2 = f 2 − d2. One sees that this result is very different from that of the single barrier of the same

total width and location [5] (they coincide only ifb → 0). The determinant of the matrix on the right-hand side ofEq. (4) is unity. The transmission probability at the pointx = a + b is found by settingB2N+1 = 0. We obtain forthis probability

(5)

∣∣∣∣A2N+1

A0

∣∣∣∣2

=∣∣∣∣ 1

eiz(cos(φ)− if

sin(φ)φ

)∣∣∣∣2

= 1

1+ d2(sin(φ))2

φ2

.

We will see that the presence of a finiteb in Eq. (5) results in a new possibility for the transmission probability toapproach unity without having to increase the energy as in the one potential barrier system. We may writea and

D. Bar, L.P. Horwitz / Physics Letters A 296 (2002) 265–271 267

b asa = L1+c

, b = Lc1+c

, wherec = ba. Fig. 1 shows a three-dimensional graph of the transmission probability from

Eq. (5) as a function of the energye andc. The total length isL = 70,v = 60, the range ofe is 61� e � 120, andthat of c is 0.01� c � 5. As expected, the transmission probability tends to 1 as the energye grows, but as seenfrom the graph it tends faster to unity, even for smalle, asc increases, that is, as the total spacing becomes larger.

Concerning thev > e case we can proceed in a similar manner to thee > v case and obtain equations similar toEqs. (1)–(5), but nowf , d andφ are given byf̀ = kb − aqη

2 , d̀ = aqξ2 , φ̀2 = f̀ 2 − d̀2. As for thee > v case, the

presence of a finiteb yields a new possibility for the transmission probability to be unity even whenv e. Fig. 2shows a three-dimensional surface of the transmission probability as a function of the energye andc, wherec hasthe same range as in Fig. 1. The potentialv is 200, ande has the range of 150� e � 192. We see that also here thetransmission probability tends to unity asc grows.

TheS matrix connects the outgoing wavesA2N+1 andB0 at the two sides of the system to the ingoing wavesB2N+1 andA0 [5]. The components of the matrixS can be determined from those of the matrix from Eq. (4) bypassing from the transfer matrix description to theS-matrix one by comparing ingoing and outgoing waves [5]. We

Fig. 1. A three-dimensional graph of the transmission probability from Eq. (5) fore > v as a function ofc and the energye. Note that ascgrows the transmission probability tends to unity even for smalle.

Fig. 2. A three-dimensional graph of the transmission coefficient for thev > e case as a function ofc and the energye. As in Fig. 1 whencgrows the transmission probability tends to unity even for smalle.

268 D. Bar, L.P. Horwitz / Physics Letters A 296 (2002) 265–271

can write the cross-sectionσ for the two eigenphases

σ± = 4 sin2 δ± = 2(1− cos(2δ±)

)

(6)= 2

(1−

cos(z)cos(φ) + f sin(z) sin(φ)φ

± d( sin(z) sin(φ) cos(φ)

φ− f cos(z)sin2(φ)

φ2

)cos2(φ)+ f 2 sin2(φ)

φ2

).

We see from Eq. (6) thatσ± → 0 for e → ∞, since d ≈ O( 1√e) → 0 and f

φ→ 1. It is further clear that

for large e the period of oscillation grows; the oscillations with respect toe go as√e, the period, for which√

e + ∆ = √e + 2π , is determined by∆ = 4π2 + 4π

√e, will therefore grows. Fig. 3 shows a graph ofσ+ as a

function of the energye (the same graph is obtained also forσ−). The total lengthL is 70, anda = 40, sob = 30.v is 70, and the range ofe is 71� e � 1000. One sees the increase in period on this graph, but the decrease inamplitude would not become visible untile a2v2/4, i.e; for our casee 2 × 106. An interesting property ofσ± emerges when we release the constraint of constantL = a + b. In this case it is shown that the dependence ofthe cross-section uponb is totally different from that upona. That is, for the same value ofa the cross-sectionsσ±, as functions of the energye, depend also onb only in a finite specific range that depends upon the value ofa. For example, forv = 70 anda = 10, the cross-sectionσ+ changes withb in the range of 0� b � 3.5a, and forb > 3.5a the change inσ+ is so small that its form, as a function ofe, can be considered constant. The same thingcan be said also forσ−. As for the dependence upona, the cross-section depends on the total widtha in such amanner thata and the period ofσ± are inversely proportional to each other. That is, as the total widtha of thepotential barrier system grows the growth rate of the period ofσ± becomes small.

For thev > e case the period ofσ± also becomes larger ase increases. As for thee > v case, if we ignore thecondition of a constantL = a + b, we find that the dependence of the cross-sectionsσ±, as functions of the energye, upon the total intervalb is non-trivial only for a specific finite range which depends upona. For example, forv = 140,e in the range 1� e � 120, anda = 10 the cross-sectionσ+, as a function ofe, changes withb only in therange 0< b � 8.7a. For any other value ofb > 8.7a we obtainσ+ = 0. The range of dependence uponb becomessmaller asa grows. For example, whena grows from the former value of 10 to 15 the former range ofb becomessmaller by almost a factor of 3, so that the new range in whichσ+ changes withb is 0< b � 3.13a. That is, forany other value ofb > 3.13a we obtainσ+ = 0.

We discuss now the energy level statistics [7] of the bounded dense array. For this we use theS-matrix and the

Fig. 3. The graph of the cross-sectionσ from Eq. (6) as a function of the energye. Note that the period of the graph becomes larger as theenergye grows. The graph is drawn forL = 70,a = 40,b = 30,v = 70, and 71� e � 1000.

D. Bar, L.P. Horwitz / Physics Letters A 296 (2002) 265–271 269

Fig. 4. The histogram is the level spacing distribution as a function ofs/D wheres is the spacing between neighbouring levels andD is themean spacing. The histogram is constructed from 102 energy levels found in the range 1� e � 600,v is v = 120, andc = 19. The continuouscurve is the chaotic Wigner distribution, and the dashed one is the Poisson distribution.

boundary conditions at the two sides of the system. We shall assume periodic boundary conditions so that the valuesof the wave function and its derivative at the end points of the spatial interval (say at pointsx = ±C far from theboundaries of the potential region at the pointsx = ±L) are equal. The energy spectrum is defined by the energiesfor which these constraints are satisfied. Obtaining these energies we use the unfolding procedure [7] to transformto the more appropriate energies from which we can try to discover an indication of chaos in the dense system. Thedistribution of the energy level spacings depends sensitively upon the value ofc. For smallc (small total intervalb and large total widtha) the relevant distribution is more of the Poisson distribution type [7] than of the chaoticWigner one, whereas whenc increases the corresponding distribution is more of the chaotic Wigner-type than ofthe Poisson one. Fig. 4 shows a histogram form of the level spacings distribution [7] of 102 energy levels obtainednumerically for both cases ofe > v andv > e. The potential height isv = 120,C = 90,L = 20, andc = 19. Thecontinuous curve is the chaotic Wigner distribution [7] (as obtained from a random matrix model), and the dashedone is the Poisson distribution [7]. One can see that the histogram-form curve resembles the chaotic Wigner one.We remark that including levels only fore > v results in a distribution which is not clearly of Wigner-type; ifv > e

the distribution appears more clearly of Wigner-type. The graph we have shown in Fig. 4, which is of Wigner-type,includes both regions of the spectrum with correspondingly better statistics.

In order to investigate further the properties of the bounded dense system we have passed a Gaussian wavepacket through it and study how it behaves in the bounded potential region. We have used the complex packet

(7)φ(x, t, x0,p0,w0) =√w0π

1/4 exp(− p2

04w2

0

)exp

(w20(i(x0−x)− p0

2w20

)21−2itw2

0

)√

1− 2itw20

,

wherex0 is the initial mean position of the packet in coordinate space, andp0 and w0 are the initial meanmomentum, and initial width (uncertainty) of the momentum, respectively, inp space. For our numericalsimulations we have discretized space and time with a resolution ofdx = 1

7 anddt = 150. This resolution ensures

the conditiondt < dx2 which is necessary for a stable and steady performance of the numerical method used here[8]. For the other parameters we choosem = 1

2, w0 = 12, x0 = −10, c = 2.333,v = 2, andp0 = 3. The last two

chosen values ensure the condition ofe > v. The dense system is arrayed between the pointsx = −10, andx = 10.The units we are using for length and time are therefore;x = xcm

h̄and t = tsec

mh̄(we takep to be momentum in

270 D. Bar, L.P. Horwitz / Physics Letters A 296 (2002) 265–271

Fig. 5. The density of the wave packet from Eq. (7) as a function ofx. The potential barriers between the pointx = −10 andx = 10 are notshown. The continuous curve is the form of the wave packet forN = 150, and the point-type curve is the form forN = 4. Both curves are attime t = 5.8.

unitsmv andw0 the dispersion inp). During and after the passage of the wave packet through the potential regionits initial Gaussian form is totally deformed. The point-type curve in Fig. 5 shows the form of the density of thewave packet from Eq. (7) when the number of potential barriers isN = 4, and the continuous graph is the formof this density forN = 150. The potential barriers arrayed betweenx = −10 andx = +10 are not shown. Bothcurves in Fig. 5 are for the same time oft = 5.8. Comparing these curves we see that for largeN the wave packetexpands across the whole potential region more rapidly than the expansion forN = 4. This behavior of the wavepacket (rapid expansion through the potential region) characterizes chaotic behaviour [3]. Moreover, we see thatthe structure of the function forN = 150 is of a much higher degree of complexity. For larger values oft (notshown) one sees that the forms of the transmitted and reflected waves are also of higher complexity. These resultsdepend sensitively on the value ofc. The effect is most pronounced whenc is in the neighbourhood of 4, suchthat if c is somewhat smaller, we have very complex structure, whereas, asc grows this effect diminishes until itcompletely disappears for very largec. Moreover, we obtain a significant transmission of the wave packet throughthe potential region even for the case ofv > e, and the part of the wave packet that passes this region increases asthe numberN or c or both of them grow. Note that we have found that the transmission of a plane wave growsrapidly withc, whereas it may be poorly transmitted whenc is very small; this is in complete accord with the wavepacket behavior just described.

The dependence uponc is manifested also in the resonances of the cross-sectionσ from Eq. (6). The real andimaginary components of the complex poles of thisσ are obtained from two simultaneous trigonometric-hyperbolicequations. Solving numerically these equations we obtain the result that as long asc is relatively small there is avery large number of solutions to these equations, but asc increases the available solutions diminish in numberuntil they disappear entirely for largec. This absence of resonances for largec is in accord with what we havefound earlier for the complete transmission of the plane wave and the wave packet for largec.

D. Bar, L.P. Horwitz / Physics Letters A 296 (2002) 265–271 271

One may understand this effect in terms of an increase in effective decoherence [10]. This preservation of theinitial wave function is typical of the space analog [9] of the Zeno effect [10]. As discussed by Bauer [11] andWillox et al. [11], in the context of a billiard model, there appears to be a classical analog to the polynomialapproach to the exponential behaviour of transitions, which is characteristic of the quantum Zeno phenomenon. Ithas been recently shown [4] that the multiple trap configuration for classical diffusion also exhibits a spatial Zeno-type effect in analogy to the quantum result we have discussed here. That is, even classical particles are passed [4]through a very large number of traps arranged along a finite section of thex axis.

We have shown in this Letter that a potential constructed of a large number of barriers can induce the sametype of chaotic-like behaviour observed in the neighbourhood of tunneling barriers [1]. One may consider, asfor the parallel drawn [1] between an unstable fixed point of the classical problem and the quantum chaotic-likebehaviour for the single barrier tunneling problem, a classical analog to the problem studied here. The repetitivepotential in the bounded region, approached from above (fore > v), appears as an accumulation of unstable fixedpoints. The single wide barrier, on the other hand, is quasi-stable when approached from above; it is only thetunneling configuration in this case that has a strong analogy to the effect of a separatrix. Forc large, when thepotential barriers are relatively well separated, we see the chaotic-like effect most strongly through the Wigner-type level distribution, and whenc is very small the distribution moves toward Poisson-type. The Zeno-typetransmission associated with decoherence, on the other hand, may be high forc small (but not very small) andN large. The repetitive barrier structure also appears to induce, for critical parameter values, Zeno-type anomaloustransition even for thev e case (overcoming the naive application of WKB approximation), indicating the onsetof decoherence (the Zeno effect usually follows from a sequence of projections rather than as a consequence of aHamiltonian model with potential). This type of decoherence has been discussed in [9]. In this very rich example wehave demonstrated both the onset of quantum chaotic-like behaviour and the Zeno-type anomalous transmission.

Acknowledgements

We wish to thank J. Schiff, D. Pearson and W. Amrein for discussions at an early stage of this work.

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