We show that the delay time distribution for wave reflection from a
one-dimensional random potential is related directly to that of the reflection
coefficient, derived with an arbitrarily small but uniform imaginary part added
to the random potential. Physically, the reflection coefficient, being
exponential in the time dwelt in the presence of the imaginary part, provides a
natural counter for it. The delay time distribution then follows
straightforwardly from our earlier results for the reflection coefficient, and
coincides with the distribution obtained recently by Texier and Comtet
[C.Texier and A. Comtet, Phys.Rev.Lett. {\bf 82}, 4220 (1999)],with all moments
infinite. Delay time distribution for a random amplifying medium is then
derived . In this case, however, all moments work out to be finite.Comment: 4 pages, RevTeX, replaced with added proof, figure and references. To
  appear in Phys. Rev. B Jan01 200

A. Comtet

A.M. Jayannavar

B. Doucot

B. Raghavendra Prasad

C.W.J. Beenakker

Christophe Texier

D.S. Wiersma

E.P. Wigner

F.T. Smith

J. Heinrichs

N. Kumar

N.M. Lawandy

P.W. Brouwer

Prabhakar Pradhan

R. Landauer

R. Rammal

S. Anantha Ramakrishna

V.A. Gopar

Xunya Jiang

Y.V. Fyodorov

Z.Q. Zhang

English

arXiv

Crossref

Imaginary potential as a counter of delay time for wave reflection from a one-dimensional random potential

We show that the delay time distribution for wave reflection from a one-dimensional (one-channel) random potential is related directly to that of the reflection coefficient, derived with an arbitrarily small but uniform imaginary part added to the random potential. Physically, the reflection coefficient, being exponential in the time dwelt in the presence of the imaginary part, provides a natural counter for it. The delay time distribution then follows straightforwardly from our earlier results for the reflection coefficient, and coincides with the distribution obtained recently by Texier and Comtet [C. Texier and A. Comtet, Phys. Rev. Lett. 82, 4220 (1999)], with all moments infinite. The delay time distribution for a random amplifying medium is then derived. In this case, however, all moments work out to be finite

Ramakrishna, S. Anantha

Kumar, N.

Publications of the IAS Fellows

PHYSICAL REVIEW B 1 FEBRUARY 2000-IVOLUME 61, NUMBER 5Imaginary potential as a counter of delay time for wave reflection from a one-dimensional
random potential
S. Anantha Ramakrishna* and N. Kumar†
Raman Research Institute, CV Raman Avenue, Bangalore 560 080, India
~Received 17 June 1999!
We show that the delay time distribution for wave reflection from a one-dimensional ~one-channel! random
potential is related directly to that of the reflection coefficient, derived with an arbitrarily small but uniform
imaginary part added to the random potential. Physically, the reflection coefficient, being exponential in the
time dwelt in the presence of the imaginary part, provides a natural counter for it. The delay time distribution
then follows straightforwardly from our earlier results for the reflection coefficient, and coincides with the
distribution obtained recently by Texier and Comtet @C. Texier and A. Comtet, Phys. Rev. Lett. 82, 4220
~1999!#, with all moments infinite. The delay time distribution for a random amplifying medium is then
derived. In this case, however, all moments work out to be finite.When a wave packet centered at an energy E is scattered
elastically from a scattering potential, it suffers a time delay
before spreading out dispersively. This delay is related to the
time for which the wave dwells in the interaction region. For
the general case of a scatterer coupled to N open channels
leading to the continuum, one defines the phase-shift time
delays through the Hermitian energy derivative of the S ma-
trix, 2i\S21]S/]E , whose eigenvalues give the proper de-
lay times. These delay times then averaged over the N chan-
nels give the Wigner-Smith delay times introduced by
Wigner1 for the one-channel case, and generalized later by
Smith2 to the case of N open channels. Thus the scattering
delay time is the single most important quantity describing
the time-dependent aspect, i.e., physically, the reactive as-
pect of the scattering in open quantum systems, e.g., the
chaotic microwave cavity and the quantum billiard ~whose
classical motion is chaotic! and the solid-state mesoscopic
dots coupled capacitively to open leads terminated in the
reservoir. The delay time is, however, not self-averaging and
one must have its full probability distribution over a statisti-
cal ensemble of random samples. The latter may be related
ergodically to the ensembles generated parametrically, e.g.,
by energy E variation over a sufficient interval. Thus we
have the random matrix theory ~RMT! for circular ensembles
of the S matrix giving delay times for all three Dyson uni-
versality classes for the case of a chaotic cavity connected to
a single open channel.3 Generalization to the case of N chan-
nels corresponded to the Laguarre ensemble4 of RMT. The
RMT approach has been treated earlier through the super-
symmetric technique for the case of a quantum chaotic cavity
having a few equivalent open channels.5 However, it has
been suspected for quite sometime that the RMT-based re-
sults and the universality claimed thereby may not extend to
a strictly one-dimensional ~1D! random system where
Anderson localization dominates, and that the 1D random
system may constitute after all a different universality class.6
This important problem has been reexamined recently by
Texier and Comtet7 who have derived the delay time distri-
bution for a 1D conductor with the Frish-Lloyd model ran-
domness in the limit of high energy and weak disorder and
the sample length @ the localization length. The universalityPRB 610163-1829/2000/61~5!/3163~3!/$15.00of the distribution is amply supported by numerical simula-
tions for different models of disorder.7,8
In this work we reexamine this question of the universal-
ity of the delay time distribution for a 1D random system and
relate it to the universality of the distribution of the reflection
coefficient, a quantity that we have direct access to from our
earlier work.9 To this end we introduce a counter that liter-
ally clocks the time dwelt by the wave in the scattering re-
gion, obviating the need for calculating the energy derivative
of the phase shift.10 This involves adding formally an arbi-
trarily small but uniform imaginary part iVi to the 1D ran-
dom potential Vr . Now, the reflection coefficient, being ex-
ponential in the time dwelt in the scattering region in the
presence of iVi , provides a literal ‘‘counter’’ for this time.
The distribution derived by us agrees exactly with the uni-
versal time-delay distribution of Texier and Comtet.7 Be-
sides, our technique allows us to treat the time-delay distri-
bution for the important case of light reflected from a
random amplifying medium equally well. In this case, how-
ever, unlike the case for the passive random medium, all
moments of the delay time are finite for long samples.
Consider first the electronic case for a 1D disordered
sample of length L having a random potential Vr , 0<x<L ,
and connected to infinitely long perfect leads at the two ends.
Let the electron wave of energy E5\2k2/2m be incident
from the right at x5L , and be partially reflected with
a complex amplitude reflection coefficient R(L)
5uR(L)uexp@iu(L)# and uR(L)u25r(L), the real reflection
coefficient. Inside the sample we have the Schro¨dinger equa-
tion
d2c~x !
dx2
1k2@11hr~x !#c~x !50, ~1!
with hr(x)52Vr(x)/E .
As we will be interested in the reflection coefficient, it is
apt to follow the invariant imbedding technique9–12 and re-
duce the Schro¨dinger equation ~1! to an equation for the
emergent quantity R(L):3163 ©2000 The American Physical Society
3164 PRB 61BRIEF REPORTSdR~L !
dL 52ikR~L !1
ik
2 hr~L !@11R~L !#
2
. ~2!
We now introduce a uniform imaginary part iVi , with Vi
.0, and accordingly define h(L)5hr1ih i , with h i
52Vi /E . For an analytical treatment, we take for Vr(x) a
Gaussian d-correlated random potential ~the Halperin model!
with ^hr(L)&50 and ^hr(L)hr(L8)&5D2d(L2L8). The
Fokker-Planck equation corresponding to the stochastic
equation ~2! can be solved analytically in the limit L→‘ ,
giving9
P‘~r !5
D expS 2 D
r21 D
~r21 !2
, r>1,
50, r,1, ~3!
with D5(4Vi)/(ED2k). This result is obtained in the high-
energy and weak-disorder limit. Now, clearly for a passive
medium, i.e., with Vi50, the distribution P‘(r) must col-
lapse to a d function d(r21) as L→‘ . However, with Vi
Þ0, for a short dwell time T in the sample, the reflection
coefficient r5uRu25exp(2ViT/\), giving r2152ViT/\ to
first order in Vi as Vi is taken to be arbitrarily small. Thus,
P‘(r) can at once be translated into the dwell time distribu-
tion P‘
0 (t):
P‘
0 ~t!5
a
t2
expS 2at D , ~4!
where a52(D2k)21 and the dimensionless time t5ET/\ .
This is precisely the result of Texier and Comtet.7 Note that
Vi , the counter, drops out in the limit Vi→0, as it should. It
should also be noted that the invariant imbedding equation
for the energy derivative of the phase shift10 also yields the
same result for the delay time distibution when the high-
energy limit (k→‘ while keeping Vr /k constant! is explic-
itly taken. This again reconfirms our delay time distribution
given above.
At this point, it is perhaps apt to demystify our time delay
counter, viz., the introduction of an imaginary potential (Vi)
in the limit Vi→0 as a mathematical artifice for the elec-
tronic case, in terms of the well-known analytic property of
the S matrix, corresponding to wave reflection from the 1D
infinitely long disordered system. The S matrix in this case is
simply the complex amplitude reflection coefficient R(E)
5exp@iu(E)# with uRu251 for real E. Now from the analyt-
icity of the S matrix in the complex energy plane, we have
](Re u)/](Re E)5](Im u)/](Im E), where ‘‘Re’’ and
‘‘Im’’ denote the real and imaginary parts, respectively. As
we approach the real axis, i.e., in the limit Im E→0, we have
](Re u)/](Re E)5T/\ ~Wigner time delay!, while
](Im u)/](Im E)→Im u/Vi as Vi→0 ~along with Im u).
Thus we have uRu25exp@2ViT/\#, giving uRu22152ViT/\
in the limit Vi→0 ~the latter corresponds to treating our elec-
tronic problem as a limit of vanishing imaginary part of the
scattering potential!. This is what has been used above to
obtain the delay time distribution from the reflection coeffi-
cient distribution given by Eq. ~3! in the limit Vi→0.Encouraged by this result for for the electronic case, we
now turn to the case of a light wave reflected from a random
amplifying medium. The latter has received much attention
in recent years in the context of random lasers.13–15 To fix
ideas, consider the case of a single-mode optical fiber doped
with Er31, say, optically pumped and intentionally disor-
dered refractively. All we have to do now is to keep Vi finite,
a measure of medium gain, and use T5(\/2)(] ln r/]Vi) for
the dwell time, and translate P‘(r) into P‘(t):
P‘~t!5~Dj!
expS 2 D
ejt21 D
~ejt21 !2
ejt, ~5!
where j52Vi /E . Again, P‘ vanishes in the limit t→‘ as
also for t→0. Also, P‘(t)→P‘0 as Vi→0. All moments
^tn& are, however, finite in this case. An explicit expression
can be obtained for the first moment as
^t&5
1
j
@ ln D1C2eD Ei~2D !# , ~6!
where C is the Euler’s constant16 and ‘‘Ei’’ is the exponen-
tial integral.16 This expression diverges as Vi→0. In Fig. 1,
we show the delay time distributions given by Eq. ~5! for
different values of the parameter j , keeping a fixed corre-
sponding to different values of the imaginary potential Vi
while keeping the disorder fixed.
Several interesting points are to be noted here. The
counter introduced by us literally counts the dwell time in
the interaction region for total reflection in the 1D, i.e., the
one-channel case. A large delay time is dominated by the
dwell time when the wave penetrates deeper into the sample,
which is true at high energy and low disorder. It is this
‘‘equilibrated’’ part of the reflected wave, and not the
prompt part, that is expected to give universality. Hence the
universal 1/t2 tail in Eq. ~4!. Indeed, the universality of the
delay time distribution directly reflects that of the reflection
coefficient given by Eq. ~3!.17–19 Indeed, we have verified
that Eq. ~3! is obtained for telegraph disorder also. It is to be
FIG. 1. The delay time distribution from an amplifying medium.
PRB 61 3165BRIEF REPORTSremarked here that this universal delay time distribution as in
Eq. ~4! is not obtained for a chaotic cavity connected to a
reservoir by a single open channel.3 Here the localization
picture may not hold. As for the finiteness of all the moments
^tn& for the case of the random amplifying medium, it is
quite consistent with the known fact that amplification en-
hances localization and thus prevents deep penetration in the
random sample. Of course, there is also an enhanced prompt
part of the reflection resulting from the increased refractive
index mismatch with respect to its imaginary part at the
sample-lead interface.In conclusion, we have introduced a ‘‘counter’’ that mea-
sures the dwell time in the scattering medium. We have used
it successfully to derive the delay time distribution in terms
of that of the reflection time. Both passive and amplifying
media have been treated. Our counter can be used equally
well in principle to calculate the traversal time for the prob-
lem of tunneling across a potential barrier.20,21
One of us ~N.K.! was partially supported by the Max-
Planck Institut fu¨r Komplexer Systeme, Dresden 01187, Ger-
many. We thank Y. V. Fyodorov for helpful comments.*Electronic address: sar@rri.ernet.in
†Electronic address: nkumar@rri.ernet.in
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Anantha Ramakrishna, S.

Raman Research Institute Digital Repository

Physical Review B

Imaginary potential as a counter of delay time for wave reflection  from a one-dimensional random potential

http://link.aps.org/abstract/PRB/v61/p3163