Complex-barrier tunnelling times

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Complex-barrier tunnelling times

Fabio Raciti, Giovanni Salesi

To cite this version:

Fabio Raciti, Giovanni Salesi. Complex-barrier tunnelling times. Journal de Physique I, EDP Sciences,

1994, 4 (12), pp.1783-1789. �10.1051/jp1:1994220�. �jpa-00247032�

Classification Physics Abstracts

73.40G _{03.80 03.658}

Complex-barrier tunnelling times(*)

Fabio Raciti and Giovanni Salesi

Dipartimento di Fisicaj Università Statale di Catania, Catania, Italy I.N F-N-j Sezione dl Catania, Catania, Italy

(Received 24 August 1994, received in final form 9 September 1994, accepted 20 September 1994)

Abstract. In this paper we calculate trie analytic expression of trie phase lime for trie

scattering of _an electron off

complex _square barrer As _is well known trie (negative) imagmary part of trie potential takes into account, phenornenologically, trie absorption. We mvestigate trie Hartman-Fletcher elfect, and find that _it is suppressed by trie _presence of _a (non negligible)

imagmary potential. In factj when _a sufàciently large absorption _{is present,} trie asyrnptotical

transmission speed _is finite. Actually, trie tunnelhng time does _increase lmearly with trie barrier width. A recent optical experirnent seerns to be

m agreernent with _our theoretical expectation.

l. Introduction.

In recent titres trie longstanding question of trie tunnelling titres bas acquired new urgency because of the recent experimental results claiming for superluminal tunnelling speeds. The problem of defining tunnelhng times has _a long history, since _{it arose} in the foities and fifties [1-3] simultaneously with the fundamental problem of introducing lime _{as a} quantum me-

chamcal observable and, _m particularj of _a definition (in Quantum Mechamcs) of the collision durations. Furthermorej _a corpuscular picture ^of tunnelling is _very hard to be realized be-

cause of the lack of _a direct classical limit for the sub-barrer particle paths and velocities.

Nevertheless~ _vanous typical definitions for the time spent by _a particle in the classically for- bidden _regions have been proposed _{[4, Si; we} underline that the differences _among them _are not only merely formol but, on the contrary~ they _{are a} consequence of different _views, physical interpretations ^and experimental expectations. Let _us quote, mcidentally, the most important definitions of the tunnelling times:

a) the dwell time [4-6]. i-e, the time spent ^{inside the} barrier, averaged _over all the incoming particles,with no distinction between transmission and reflection channelsj

(* ^Work partially supported by INFN

© Les Editions de Physique 1994

1784 JOURNAL DE PHYSIQUE I N°12

b) the local "Larmor times" [6]: 1e., the traversal time _as measured by the spin _precession of the tunnelling particle _in a uniform infinitesimal magnetic fieldj

c) the "complex time approach" _{[Sj 7]: a} quantum extension _via path integral _averages

over the classical paths of the classical complex time spent by the particle in the scattering

processj

d) the "Buttiker-Landauer" times là, _{6~ 8]} namely interaction times of the particle with _a time modulated barrierj

e) the _so called "spatial approaches" là, 9], ^based on the probabilistic quantum standard interpretation ^of the flux densities J[~, _ii involved during transmission and reflectionj

f) the ordinary "phase times"~ il, 2, 4, Sj _11~] or group delays: _1-e-, the times taken by quasi-

monochromatic _wave packets to appear on the other side of the barner~ as given by the sta-

tionary phase approximation.

Before going _onj let _us stress the important fact that many of those theoretical pictures do imply the so-called Hartman-Fletcher effect _[11~]j that is to say the surprising occurrence, for sufliciently _opaque barriers, of tunnelhng delays mdependent of the barr~er width. Namelyj

those delays imply traversal _mean group velocities larger than the light speed in _vacuum.

Recent optical _{expenment seem} to confirm the existence of superluminal tunnelling speeds [11]. They _can be grouped in 3 main classes:

1) evanescent wave propagation in _a low dielectric _constant region, separating two regions of higher dielectric constant or~ similarly, in optical devices allowing for frustrated total internal

reflection;

2) propagation of _a Gaussian light pulse through an anomalous dispersion mediumj 3) evanescent microwaves in a wave guide below cut-off.

We _are particularly interested in evanescent microwaves since the recent optical experiments by Nimtz et al. [12] agree with _our quantum mechanical calculation. The analogy between quantum tunnelhng and propagation of _microwaves in _wave guides is based _on the fact that the _group velocity is obtained from the derivative of the transmission amplitude _in both _cases.

Moreover the Schroedinger equation _is formally identical to the Helmholtz equation for the

propagation of _a scalar field, electric _or magnetic component ^of the _wave:

d~~fildz~ + k~~fi

l~

where k is the _wave number. Thus is possible to simulate quantum tunnelhng studying the

propagation of _{e.m. waves in wave} guide, where the region below the cut off is the equivalent

of the quantum ^barrier [11]. ^{The connection} between the Schroedinger equation and the Helmholtz equation _is also vahd for the Dirac equationj which is _a relativistic equation. Using

the Dirac equation instead of the Schroedinger equation _we obtain again the Hartman effect,

so that this effect is not a "wrong" quantum mechanical effect due to the fact that _one is not using a relativistic equation (this has been showed by C.R- Leavens).

Dur proposal in this paper is studying the tunnelhng time and the Hartman-Fletcher effect

in the _presence of absorption. To that purpose we introduce _a complex _square potential (with

a negative immaginary part) such non-real potentials _are customary in scattering theory and

in nuclear physics (where they _are named optical potentials). We shall show that the _non

reality of the effective Hamiltonian (related to the existence of other interaction channels) does

in general destroy the Hartman-Fletcher effect. Superluminal speeds _can be achieved only if the imaginary part ^{of the} Hamiltonian _is chosen sufliciently small. Let _us notice that these

theoretical predictions _are in agreement ^{with the} experimental results obtained by Nimtz et al. at Cologne _[12] employing evanescent microwaves _in absorptive _wave guides below cut-off.

In this context, ^the most suitable and natural theoretical approach for the evaluation of the

global tunnelling times and of the _mean tunnelling speeds _is perhaps trie _most "direct" _one~

trie phase time approach already mentioned in f): thus, _m trie following _{we are} going to adopt such _a theoretical approach.

2. Analytic calculation of trie phase time.

Following the ordinary procedures employed in references _[11~], the _group delay for quasi-

monochromatic packets _m the stationary phase approximation~ is given by:

ôT(E)

=

&((arg AT) (la)

where E is the mcoming particle _energy, AT is the (complex) transmission amplitude and is the reduced Planck constant. The global tunnelling time is given, _as usualj by the _sum of the

semiclassical traversal and delay time:

T

T~ + ôT(E) (16)

In order to find out the analytic expression for AT _we have to solve the (stationary) Schroedinger equation with a potential different from _zero only in the interval (l~,a)j namely:

V(~)

Vo ivi

j

z E (l~j a) (2)

Let _us observe that:

lfiI

lfi>n + lfiR

e~~~ + BIe ^~~~

and

film + lfiT

Àme~~~

as in the real potential _case (with k~

=

2mE/&~)j while <II _is obtained by solving ^the Schroedinger equation in the barrier region:

ÎÎ~ ^~ Î~~ _{~ ~ ~~~~ ~ ~~~}

Thusj _{we get} for _~fiiI the following expression:

4II

Alie~~°~ ₊ BIIe~~~II~ (4)

'~~~~~

~ii + ~/2m(E vo ₊ ivi)/à (Sa)

Let _{us notice} that equation (Sa) implies that kII is _a complex quantity:

kil + f + i~ (Sb)

where (, _{~ are} real numbers.

In the E < Vo case we get:

f

lw/&)~Î/(E Vo)~ + Vi~ IA E) 16a)

and:

~

(W/&) /(E _vo)2

+ K~ + (Vo El (6bl

1786 JOURNAL DE PHYSIQUE I N°12

Imposing the continuity boundary conditions for # and its derivative, 1e.:

lfiI10)

i~II(oli _IfiII(tl)

4III(tl)1 lfil(o)

lfill(o)1 lfill(tl)

lfilII(tl)

we get, ^after some algebra, the expression for AT + AIII 4kkIIe~~iI~e~~~~

(7a)

~~ k2 ₊ k)1(1 e2~~II~) + 2kkii(1 _{+ e~~~~~~)}

Since _{we are} interested in determining # _{e arg} AT let _{us express} Aj~ in agebraic formj _e-g-

~~~ ^~~ Îkli ^~~

~ ~~~ 2kÎi ^~~ _~~~~

~~~~~'

A + ((k~ ₊ kiikli) _~~~~

B ₊ ~(kiikii k~) (8b)

C _a 2kkiik(1 (8c)

z % sin fa cosh _~a (8d)

y + cos fa sinh _~a (8e)

w e cos fa cosh _~a (8f)

r + -sm fa sinh _~a (8g)

Thus _we may write:

~ _~~~ ~~ _~~~~~~~ Î~ÎÎ+ _~~'

~~~~

After _some manipulations, quantity # _can be written _as follows:

~ _~~~~~~~ ÎÎÎÎÎÎÎ

A Î~II~ÎÎÎ Î Î _~~'

~~~~

According to equation (16) the tunnelhng time _is:

T(E)

=

Î + &£(arg AT

&£(arg ^AT + ka) _(11~)

where _~

hk/m.

By inserting equation (9b) into equation _{(11~) we} get~ ^after some elaborations (see trie Ap- pendix), the rather comphcate expression:

T

(11)

where

n e sin2(a[(-am~t/&~p~)(C~ ^B~ A~ ₊ (A'C AC')]

+ cos 2(a[2BCam~/(&~p~ _{)] +} sinh 2~a[(am( _/&~p~ )(A~ + B~ ₊ C~

+(BC' B'C)] ₊ (2am(AC/h~p~ _{cosh 2~a + 2(A'B} AB')(sin~ _{fa +} ^sinh~ _~a)

(1la)

an d

d ₊ 2(A~ + B~ (sin~ fa + sinh~ ~a) + 2AC sinh 2~a

(11b) +2BC sin 2(a + 2C~ (cos~ fa + smh~ ~a)

Since _{we want to} check the _occurrence of the Hartman-Fletcher effect~ _{we are} interested in

studying the opaque barrier limit. That is~ we are interested in the asymptotic condition:

akiik(1 » 1 (12)

(13)

~~~~~ ~~~~~~

T~~~

fi(/~~2)

that _is~ T~~Y is directly proportional to the barrier width. In other words the _mean tunnel speed _~i is asymptotically

~i +

~

"

~~~~ ^~ ~~~

(14)

T~~Y In~

Thus, _we do not obtain the Hartman-Fletcher effect and the saturation of the transmission times anylonger, but _a limiting speed _as it is shown in equation (14). Nevertheless, _{as we can} deduce from (14), for sufliciently small values of (~ ^{i-e- in} the _case of low absorption, _we get:

~l

oQ

so that superluminal tunnelling velocities _are not a priori forbidden.

3. Conclusion.

Most of the theoretical models proposed for the tunnelling time give rise to the Hartman effect~

1-e- to an anomalously short tunnelling time. In the _very last years the phase time approach, _as

well _as the Hartman effect, has received experimental confirmation by several _groups which _ex-

pIoited the analogy ^between tunnelling particles and evanescent electromagnetic _waves. Thus,

we propose ^to extend this approach to the _case of _a complex potential~ _m order to introduce

an absorption charnel. Dur calculations shows that _a strong absorption tends to destroy the Hartman effect and agree with the experimental results of Nimtz _et al. However, although

aII those experiments do confirm the Hartman effect, there _are controversial opinions ^about

the mterpretation of this effect and the possibility of superluminal signal transmission. In _our

opinion~ for _a better physical interpretation of this phenomenon _one should also study the distributions of tunnelling times and the distortion of the _wave packet. Work is _m progress m

this direction.

4. Appendix.

Dur starting point will be the analytical _expression for #, already written in (9b). Thus, _we

can calculate the global tunnelling time (ll~) and obtain that in the general _case:

((#+ka)=j

1788 JOURNAL DE PHYSIQUE I N°12

~~~~~

n + (i / ^cosh~ fa) (sin fa _cos ~a[a~'(C~ B~ A~)

+(A'C AC' _{)] +} a~'BC(sin~ fa cos~ fa)

+ tanh ~ta[al'(A~ ₊ ^B~ ₊ C~) ₊ (BC' B'C)]

+al'RC il ₊ tanh~ ~a) + (A'B AB')(sin~ fa + tanh~

~a cos~ ~a)

and

~ ~ ~ ~ ~

de (A +B )(sin fa +tanh _~acos fa) ₊ 2ACtanh~a +C~ (cos~ fa + sin~ fa tanh~ ~a) ₊ ^~~ _~~( ~~~,

cosh _~a

symbol i meaning derivation with respect ^to the _energy.

At this point~ _{we can} check _our comphcated formula in the special _case l§

l~. Let _us observe that the assumption Vi

"

l~ implies:

i

o, i'

=

o~ ~'

=

-m/à2(1/~) k'

=

m/à2(1/k)

and also:

A

l~ A'

=

l~ B

=

~~ ~k~ B'

=

3~~~J' /k~ 2~kk'

C

=

2k~~ ^C'

=

2~~k' ₊ 4~~'k

After _some manipulations, _we get ^the expression for the phase time in the particular _case

Vi

l~:

~~~ _~ ~~~~"° ~~~~~~~~~ÎÎmÎÎ~)

ÎÎÎÎÎÎÎ~Î~Î~~~~ ^{~~~~ ~~~}

If _we define (2mVo/&~)~ e k( and D _e [(2m/&~)V/ sinh~a~ _{+ 4k~~~]~ we} realize that _our expression is identical with equation (12a) of reference [Si. After this check, _{we can now go} back to the general _expression (II) of the tunnelhng time which after further algebra

can be written:

r

~

d where now

n + s1n2(a[(-am~J/&~p~)(C~ B~ A~) + (A'C AC' ii

+ cos 2(a[2BCam~/(&~p~)] + sinh 2~a[(am(/&~p~)(A~ + B~ ₊ C~) ₊ (BC' B'C)]

+(2am(AC/h~p~) cosh 2~a + 2(A'B AB')(sin~ fa + sinh~ ~a)

and

~ ~ ~ ~

d _a 2(A ₊ B )(sin fa ₊ sinh pa) + 2AC sinh 2~a + 2BC sin 2fa

+ 2C~ (cos~ fa + sinh~ ~a) Acknowledgments.

Special thanks _are due to Prof. V.S Olkhovsky for having suggested this work~ ^and to Profs.

E. Recami and V.S. Olkhovsky for stimulating discussions and their continuous help. We also

thank Profs. S. Sambataroj R. Mignani and Drs. G. Andronico, G. Angilella, P. Falsaperlaj

A.Lamagna, G-D- Maccarrone, R. Maltese. At lastj the collaboration of G. Navaj S. Parietti, Dr. P. Saurgnani and T. Venasco is also acknowledged.

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