Table Of ContentSuperluminal tunneling through two successive barriers
Vladislav S. Olkhovsky
Institute for Nuclear Research, Kiev-03028; Research Center ”Vidhuk”, Kiev, Ukraine;
e-mail: [email protected]
2
0
0 Erasmo Recami
2
Facolt`a di Ingegneria, Universit`a statale di Bergamo, 24044–Dalmine (BG), Italy;
n
a
I.N.F.N.–Sezione di Milano, Milan, Italy; and
J
3 C.C.S., State University at Campinas, Campinas, S.P., Brazil;
2
e-mail: [email protected]
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v and
2
2 Giovanni Salesi
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2 I.N.F.N.–Sezione di Milano, Milan, Italy; and
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0 Facolt`a di Ingegneria, Universit`a statale di Bergamo, 24044–Dalmine (BG), Italy;
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/ e-mail: [email protected]
h
p
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t Abstract — We study the phenomenon of one-dimensional non-resonant tunneling
n
a
u through two successive (opaque) potential barriers, separated by an intermediate free
q
: region R, by analyzing the relevant solutions to the Schroedinger equation. We find that
v
i the total traversal time does not depend not only on the barrier widths (the so-called
X
r “Hartman effect”), but also on the R width: so that the effective velocity in the region
a
R, between the two barriers, can be regarded as practically infinite. This agrees with
the results known from the corresponding waveguide experiments, which simulated the
tunneling experiment herein considered due to the known formal identity between the
Schroedinger and the Helmholtz equation.
PACS numbers: 73.40.Gk ; 03.65.-w ; 03.30.+p ; 41.20.Jb ; 84.40.Az
(†) WorkpartiallysupportedbyMURST,INFNandCNR(Italy),byCAPES(Brazil),andbyI.N.R.
(Kiev)andS.K.S.T.ofUkraine. [AppearedinpreliminaryversionasLanlArchives#quant-ph/0002022].
1
1. – Introduction — It is known within quantum mechanics, with regard to the
tunnelingprocesses, thatthetunnelingtime—eitherevaluatedasasimple“phasetime”[1]
or calculated through the analysis of the wavepacket behaviour[2]— does not depend on
the barrier width in the case of opaque barriers. Such a phenomenon, sometimes called
“Hartman effect”[3], implies Superluminal and arbitrarily large (group) velocities v inside
long enough barriers[2]. Experiments that may verify this prediction by, say, electrons are
difficult. Luckilyenough,however, theSchroedingerequationinthepresenceofapotential
barrier is mathematically identical[4] to the Helmholtz equation for an electromagnetic
wavepropagating,e.g.,downametallicwaveguide alongthex-axis: andabarrierheightV
bigger thanthe electronenergy E corresponds (foragiven wave frequency) toa waveguide
transverse size smaller than a cut-off value. A segment of undersized guide does therefore
behave as a barrier for the wave (photonic barrier): So that the wave assumes therein
—like an electron inside a quantum barrier— an imaginary momentum or wave-number
and gets exponentially damped along x, as a consequence. In other words, it becomes
an evanescent wave (going back to normal propagation, even if with reduced amplitude,
when the narrowing ends and the guide returns to its initial transverse size). Thus, a
tunneling experiment can be simulated by having recourse to evanescent waves (for which
the concept of group velocity can be properly extended[5]).
And the fact that evanescent waves travel with Superluminal speeds has been actually
verified in a series of famous experiments. Namely, various experiments —performed
since 1992 onwards by R.Chiao’s and A.Steinberg’s group at Berkeley[6], by G.Nimtz at
Cologne[7], by A.Ranfagni and colleagues at Florence[7], and by others at Vienna, Orsay,
Rennes[7]— verified that “tunneling photons” travel with Superluminal group velocities;
in other words, they confirmed, directly or indirectly, the occurrence of the Hartman
effect.
Letusemphasizethatthemostinterestingexperimentalsetup,dealingwithevanescent
waves, seemstobe—however—theonecomprehending twosuccessive evanescenceregions
(“classical barriers”), separated by a segment of normal region. For suitable frequency
bands —i.e., far from resonances—, it was found that the total crossing time does not
depend on the length of the intermediate (normal) region: namely, that the beam speed
2
along it is infinite. The related experimental results[8] have been already confirmed by
numerical simulations, based onMaxwell equations only[9]. But they are so amazing that
we want to check whether they agree also with what is predicted by quantum mechanics
in the analogous case of two successive potential barriers.
In this note we are actually going to show that, for non-resonant tunneling trough two
successive, rectangular (opaque) potential barriers (Fig.1), the (total) phase time does
depend neither on the barrier widths nor on the distance between the barriers. In other
words, far from resonances the tunneling phase time, which does depend on the entering
energy, can be shown to be independent of the distance between the two barriers.
2. – Phase time evaluation — Let us consider the (quantum-mechanical) sta-
tionary solution for the one-dimensional (1D) tunneling of a non-relativistic particle, with
mass m and kinetic energy E = h¯2k2/2m = mv2/2, through two equal rectangular
barriers with height V (V > E) and width a, quantity L − a ≥ 0 being the distance
0 0
between them. The Schr¨odinger equation is
h¯2 ∂2
− ψ(x) + V(x)ψ(x) = Eψ(x), (1)
2m ∂x2
where V(x) is zero outside the barriers, while V(x) = V inside the potential barriers. In
0
the various regions I (x ≤ 0), II (0 ≤ x ≤ a), III (a ≤ x ≤ L), IV (L ≤ x ≤ L+a) and V
(x ≥ L+a), the stationary solutions to eq.(1) are the following
ψ = e+ikx +A e−ikx
I 1R (2a)
ψ = α e−χx +β e+χx
II 1 1 (2b)
ψψIII == AA1T hαeikxe−+χ(Ax−2RL)e+−ikβxie+χ(x−L) ((22dc))
IV 1T 2 2
ψV = A1TAh2Teikx , i (2e)
where χ ≡ 2m(V −E)/h¯, and quantities A , A , A , A , α , α , β and β are the
0 1R 2R 1T 2T 1 2 1 2
q
reflection amplitudes, the transmission amplitudes, and the coefficients of the “evanes-
cent” (decreasing) and “anti-evanescent” (increasing) waves for barriers 1 and 2, respec-
tively. Such quantities can be easily obtained from the matching (continuity) conditions:
3
ψ (0) = ψ (0)
I II
(3a)
∂ψ ∂ψ
I = II (3b)
∂x (cid:12) ∂x (cid:12)
(cid:12)x=0 (cid:12)x=0
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
ψ (a) = ψ (a)
II III
(4a)
∂ψ ∂ψ
II = III (4b)
∂x (cid:12) ∂x (cid:12)
(cid:12)x=a (cid:12)x=a
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
ψ (L) = ψ (L)
III IV
(5a)
∂ψ ∂ψ
III = IV (5b)
∂x (cid:12) ∂x (cid:12)
(cid:12)x=L (cid:12)x=L
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
ψ (L+a) = ψ (L+a)
IV V
(6a)
∂ψ ∂ψ
IV = V (6b)
∂x (cid:12) ∂x (cid:12)
(cid:12)x=L+a (cid:12)x=L+a
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Equations (3-6) are eight equations for our eight unknowns (A , A , A , A , α ,
1R 2R 1T 2T 1
α , β and β ). First, let us obtain the four unknowns A , A , α , β from eqs.(5) and
2 1 2 2R 2T 2 2
(6) in the case of opaque barriers, i.e., when a is large enough (and χ not too small) so
that one can assume that χa → ∞:
2ik
α −→ eikL (7a)
2 ik −χ
Aβ22R−→−→eikeL2i−k2Lχaiikk−−+2(iiχχkk(−ikχ+)2χ) ((77bc))
Then, we may obtainthAe 2oTth−e→r foeu−rχauen−kiknao(wi−kns4−iAkχχ)2, A , α , β from eqs.(3) and(7(d4)).
1R 1T 1 1
Aagain in the case of large enough barriers (and χa → ∞), one gets:
4
V V
0 0
1 2
I II III IV V
0 a L L+a
Figure 1: The non-resonant tunneling process, through two successive (opaque) potential
barriers, considered in this paper. We show that, far from resonances, the (total) phase
time for tunneling through the two barriers does depend neither on the barrier widths
nor on the distance between the barriers.
5
2ik
α −→ (8a)
1 ik −χ
βA1 −→−→e−i2kχa+(kχ−iχ)sink(χL−a)A ((88bc))
1R ik −χ
where A1T −→ e−χae−ikLA, (8d)
2χk
A ≡ (9)
2χk cosk(L−a) + (χ2 −k2) sink(L−a)
results, incidentally, to be real.
At this point, by applying the well-known definition of phase-time (see, for instance,
refs.[1-3]), we can derive that the tunneling time
∂arg A1TA2Teik(L+a) ∂ −4ikχ
τph ≡ h¯ = h¯ arg =
tun h ∂E i ∂E "(ik −χ)2#
∂ k2 −χ2 1 2m
= h¯ arctan = , (10)
∂E " k χ # h¯χ k
while depending on the energy of the tunneling particle, does not depend on L + a (it
being actually independent both of a and of L).
Thisresultdoesnot onlyconfirmtheso-called“Hartmaneffect”[2,3]forthetwoopaque
barriers —i.e., the independence of the tunneling time from the opaque barrier widths,—
but it does also extend such an effect by implying the total tunneling time to be inde-
pendent even of L (see Fig.1). This might be regarded as a further evidence of the fact
that quantum systems seem to behave as non-local; but is has a more general meaning,
it being associated with the properties of any waves (and, in fact, something very similar
happens also, e.g., with electromagnetic waves: see below). It is important to stress once
more that the previous result holds, however, for non-resonant (nr) tunneling: i.e., for
energies far from the resonances that arise in region III due to interference between for-
ward and backward traveling waves (a phenomemon quite analogous to the Fabry-P´erot
one in the case of classical waves). Otherwise it is known that the expression for the time
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delay τ near a resonance is rather larger: for example, for a gaussian resonance at E with
r
half-width Γ, it would be τ = h¯Γ[(E −E )2 +Γ2]−1 + τ .
r nr
3. – Discussion — The tunneling-time independence from the width (a) of each
one of the two opaque barriers is itself a generalization of the Hartman effect, and might
be a priori understood —following refs.[4,6]— on the basis of the reshaping phenomenon
which takes place inside a barrier.
With regard to the even more interesting tunneling-time independence from the dis-
tance L−a between the two barriers, it may beunderstood on the basis ofthe interference
between the waves coming out of the first barrier (region II) and traveling in region III
and the waves reflected from the second barrier (region IV) back into the same region III.
Such an interference has been shown[2] to cause an “advancement”, i.e., an effective
acceleration of the forward-traveling waves, even in region I: Namely, going on to the
wavepacket language, we noticed in refs.[2] that the arriving wavepacket does interfere
with the reflected waves that start to be generated as soon as the packet forward tail
reaches the first barrier edge: In such a way that, already before the barrier, the backward
tail of the initial wavepacket decreases —because of destructive interference with those
reflected waves— at a larger degree than the forward one. This simulates an increase of
the average speed of the entering packet; hence, the effective (average) flight-time of the
approaching packet from the source to the barrier does decrease.
So, a reshaping and “advancement” of the same kind (inside the barriers, as well as to
the left of the barriers) may qualitatively explain why the tunneling-time is independent
of the barrier widths and of the distance between the two barriers. Phenomena of this
kind, actually, do notseem tobeatvariance withSpecial Relativity, asit hasalreadybeen
discussed in a number of papers (cf., e.g., refs.[9] and [5], and refs. therein). It remains
impressive, nevertheless, that in region III —where no potential barrier is present, the
current is non-zero and the wavefunction is oscillatory,— the effective speed (or group-
velocity) is practically infinite.∗ After some straightforward but rather bulky calculations,
one can moreover see that the same effects (i.e., the independence from the barrier widths
∗Loosely speaking, one might say that the considerd two-barriers setup can behave as a “space de-
stroyer” with reference to its intermediate region.
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and from the distances between the barriers) are still valid for any number of barriers,
with different widths and different distances between them.
Finally, let us recall that the known similarity between photon and (nonrelativistic)
particle tunneling[2,4,10,11] implies our previous results to hold also for photon tunneling
through successive “barriers”: For example, for photons in presence of two successive
band gap filters, like two suitable gratings or two photonic crystals. Experiments should
be easily realizable; while indirect experimental evidence seems to come from papers such
as [12].
Let us also repeat that the classical, relativistic (stationary) Helmholtz equation for an
electromagnetic wavepacket in a waveguide is known to be formally identical to the quan-
tum, non-relativistic (stationary) Schroedinger equation for a potential barrier;† so that,
forinstance, thetunnelingofaparticleunderandalongabarrierhasbeensimulated[2,4,7-
11,13] by the traveling of evanescent waves along an undersized waveguide. Therefore,
the results of this paper are to be valid also for electromagnetic wave propagation along
waveguides with a succession of undersized segments (the “barriers”) and of normal-sized
segments. This confirms the results obtained, within the classical realm, directly from
Maxwell equations[9,13], as well as by the known series of “tunneling” experiments per-
formed —till now— with microwaves (see refs.[7] and particularly [8]).
Acknowledgements
The authors are grateful to S.Esposito, as well as to A.Agresti, G.G.N.Angilella,
P.Bandyopadhyay, F.Bassani, G.Benedek, R.Bonifacio, L.Bosi, C.Becchi, G.C.Cavalleri,
R.Y.Chiao, R.Colombi, G.C.Costa, G.Degli Antoni, P.Falsaperla, F.Fontana,
R.Garavaglia, A.Gigli Berzolari, H.E.Hern´andez F., J.Jakiel, E.Kapuschik, L.C.Kretly,
G.Kurizki, G.Marchesini, R.Mignani, D.Mugnai, G.Privitera, A.Ranfagni, A.Salanti,
A.Shaarawi, A.Steinberg, J.W.Swart, M.Zamboni-Rached and B.Zappa for scientific col-
laboration or stimulating discussions; as well as to the Referees for very useful remarks.
†These equations are however different (due to the different order of the time derivative) in the time-
dependent case. Nevertheless, it can be shown that they still have in common classes of analogous
solutions, differing only in their spreading properties[2,11].
8
At last, for the active cooperation special thanks are due to G.Giuffrida, R.Salesi and
M.T.Vasconselos.
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