Dynamical model for the quantum-to-classical crossover of shot noise

Abstract

We use the open kicked rotator to model the chaotic scattering in a ballistic quantum dot coupled by two point contacts to electron reservoirs. By calculating the system-size-over-wave-length dependence of the shot noise power we study the crossover from wave to particle dynamics. Both a fully quantum mechanical and a semiclassical calculation are presented. We find numerically in both approaches that the noise power is reduced exponentially with the ratio of Ehrenfest time and dwell time, in agreement with analytical predictions.

Full text

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arXiv:cond-mat/0304327v2 [cond-mat.mes-hall] 7 Jul 2003
Dynamical model for the quantum-to-classical crossover of shot noise
J. Tworzyd? lo,1,2A. Tajic,1H. Schomerus,3and C.W.J. Beenakker1
1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
2Institute of Theoretical Physics, Warsaw University, Ho˙ za 69, 00–681 Warsaw, Poland
3Max Planck Institute for the Physics of Complex Systems, N¨ othnitzer Str. 38, 01187 Dresden, Germany
(Dated: April 2003)
We use the open kicked rotator to model the chaotic scattering in a ballistic quantum dot cou-
pled by two point contacts to electron reservoirs. By calculating the system-size-over-wave-length
dependence of the shot noise power we study the crossover from wave to particle dynamics. Both
a fully quantum mechanical and a semiclassical calculation are presented. We find numerically in
both approaches that the noise power is reduced exponentially with the ratio of Ehrenfest time and
dwell time, in agreement with analytical predictions.
PACS numbers: 05.45.Mt, 03.65.Sq, 72.70.+m, 73.23.-b
I.INTRODUCTION
Noise plays a uniquely informative role in connection
with the particle-wave duality.1This has been appreci-
ated for light since Einstein’s theory of photon noise.
Recent theoretical2,3,4,5,6and experimental7work has
used electronic shot noise in quantum dots to explore
the crossover from particle to wave dynamics. Particle
dynamics is deterministic and noiseless, while wave dy-
namics is stochastic and noisy.8
The crossover is governed by the ratio of two time
scales, one classical and one quantum. The classical time
is the mean dwell time τDof the electron in the quantum
dot. The quantum time is the Ehrenfest time τE, which is
the time it takes a wave packet of minimal size to spread
over the entire system. While τDis independent of ¯ h, the
time τEincreases ∝ ln(1/¯ h) for chaotic dynamics. An ex-
ponential suppression ∝ exp(−τE/τD) of the shot noise
power in the classical limit ¯ h → 0 (or equivalently, in the
limit system-size-over-wave-length to infinity) was pre-
dicted by Agam, Aleiner, and Larkin.2A recent exper-
iment by Oberholzer, Sukhorukov, and Sch¨ onenberger7
fits this exponential function.
and range of the experimental data is not sufficient to
distinguish this prediction from competing theories (no-
tably the rational function predicted by Sukhorukov9for
short-range impurity scattering).
Computer simulations would be an obvious way to test
the theory in a controlled model (where one can be cer-
tain that there is no weak impurity scattering to compli-
cate the interpretation). However, the exceedingly slow
(logarithmic) growth of τE with the ratio of system size
over wave length has so far prevented a numerical test.
Motivated by a recent successful computer simulation of
the Ehrenfest-time dependent excitation gap in the su-
perconducting proximity effect,10we use the same model
of the open kicked rotator to search for the Ehrenfest-
time dependence of the shot noise.
The reasoning behind this model is as follows. The
physical system we seek to describe is a ballistic (clean)
quantum dot in a two-dimensional electron gas, con-
However, the accuracy
nected by two ballistic leads to electron reservoirs. While
the phase space of this system is four-dimensional, it can
be reduced to two dimensions on a Poincar´ e surface of
section.11,12The open kicked rotator10,13,14,15is a stro-
boscopic model with a two-dimensional phase space that
is computationally more tractable, yet has the same phe-
nomenology as open ballistic quantum dots.
We study the model in two complementary ways. First
we present a fully numerical, quantum mechanical solu-
tion. Then we compare with a partially analytical, semi-
classical solution, which is an implementation for this
particular model of a general scheme presented recently
by Silvestrov, Goorden, and one of the authors.5
II. DESCRIPTION OF THE MODEL
We give a description of the open kicked rotator, both
in quantum mechanical and in classical terms.
A.Closed system
We begin with the closed system (without the leads).
In this subsection we follow Refs. 16,17. The quantum
kicked rotator has Hamiltonian
H = −¯ h2
2I0
∂2
∂θ2+KI0
τ0
cosθ
∞
?
k=−∞
δs(t − kτ0).(2.1)
The variable θ ∈ (0,2π) is the angular coordinate of
a particle moving along a circle (with moment of iner-
tia I0), kicked periodically at time intervals τ0 (with a
strength ∝ K cosθ).
time-reversal symmetry later on, when we open up the
system, we represent the kicking by a symmetrized delta
function: δs(t) =
mal ǫ. The ratio ¯ hτ0/2πI0≡ heffrepresents the effective
Planck constant, which governs the quantum-to-classical
crossover. The stroboscopic time τ0 is set to unity in
most of the equations.
To avoid a spurious breaking of
1
2δ(t − ǫ) +1
2δ(t + ǫ), with infinitesi-

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The stroboscopic time evolution of a wave function is
given by the Floquet operator F = T exp(−i?τ0
where T indicates time ordering of the exponential. For
1/heff≡ M an even integer, F can be represented by an
M × M unitary symmetric matrix. The angular coordi-
nate and momentum eigenvalues are θm= 2πm/M and
Jm = ¯ hℓ, with m,ℓ = 1,2,...M. We will use rescaled
variables x = θ/2π and p = J/¯ hM in the range (0,1).
The eigenvalues exp(−iεm) of F define the quasi-
energies εm ∈ (0,2π). The mean spacing 2π/M of the
quasi-energies plays the role of the mean level spacing
δ in the quantum dot. In coordinate representation the
matrix elements of F are given by
Fmm′ = (XU†ΠUX)mm′,
Umm′ = M−1/2e2πimm′/M,
Xmm′ = δmm′e−i(MK/4π) cos(2πm/M),
Πmm′ = δmm′e−iπm2/M.
0dtH/¯ h),
(2.2a)
(2.2b)
(2.2c)
(2.2d)
The matrix product U†ΠU can be evaluated in closed
form, resulting in the manifestly symmetric expression
(U†ΠU)mm′ = M−1/2e−iπ/4exp[i(π/M)(m′− m)2].
(2.3)
Classically, the stroboscopic time evolution of the
kicked rotator is described by a map on the torus
{x,p | modulo1}. The map relates xk+1,pk+1 at time
k + 1 to xk,pkat time k:
xk+1 = xk+ pk+K
4πsin2πxk,(2.4a)
pk+1 = pk+K
4π
?sin2πxk+ sin2πxk+1
?. (2.4b)
The classical mechanics becomes fully chaotic for K>
with Lyapunov exponent λ ≈ ln(K/2). For smaller K the
phase space is mixed, containing both regions of chaotic
and of regular motion. We will restrict ourselves to the
fully chaotic regime in this paper.
Forlaterusewegive
M(xk,pk), which describes the stretching by the map
of an infinitesimal displacement δxk, δpk:
∼7,
themonodromy matrix
?δxk+1
δpk+1
?
= M(xk,pk)
?δxk
δpk
?
.(2.5)
From Eq. (2.4) one finds
M(xk,pk) =
?
Λ(xk)1
Λ(xk)Λ(xk+1) − 1 Λ(xk+1)
2cos2πx.
?
,(2.6a)
Λ(x) = 1 +K
(2.6b)
B.Open system
We now turn to a description of the open kicked ro-
tator, following Refs. 10,15,18. To model a pair of N-
mode ballistic leads, we impose open boundary condi-
tions in a subspace of Hilbert space represented by the
indices m(α)
n = 1,2,...N labels the modes and the superscript
α = 1,2 labels the leads. A 2N × M projection ma-
trix P describes the coupling to the ballistic leads. Its
elements are
n
in coordinate representation. The subscript
Pnm=
?
1 if m = n ∈ {m(α)
0 otherwise.
n },
(2.7)
The matrices P and F together determine the quasi-
energy dependent scattering matrix
S(ε) = P[e−iε− F(1 − PTP)]−1FPT.
Using PPT= 1, Eq. (2.8) can be cast in the form
(2.8)
S =PAPT− 1
PAPT+ 1, A =1 + eiεF
which is manifestly unitary. The symmetry of F ensures
that S is also symmetric, as it should be in the presence
of time-reversal symmetry.
By grouping together the N indices belonging to the
same lead, the 2N × 2N matrix S can be decomposed
into 4 sub-blocks containing the N ×N transmission and
reflection matrices,
1 − eiεF= −A†,(2.9)
S =
?rt
t′r′
?
.(2.10)
The Fano factor F follows from19
F =Trtt†(1 − tt†)
Trtt†
.(2.11)
This concludes the description of the stroboscopic
model studied in this paper. For completeness, we briefly
mention how to extend the model to include a tunnel bar-
rier in the leads.
To this end we replace Eq. (2.8) by
S(ε) = −(1 − KKT)1/2
+ K[e−iε− F(1 − KTK)1/2]−1FKT. (2.12)
The N × M coupling matrix K has elements
?√Γn if m = n ∈ {m(α)
0
Knm=
n },
otherwise,
(2.13)
with Γn∈ (0,1) the tunnel probability in mode n. Ballis-
tic leads correspond to Γn= 1 for all n. The scattering
matrix (2.12) can equivalently be written in the form
used conventionally in quantum chaotic scattering:20,21
S(ε) = −1 + 2W(A−1+ WTW)−1WT,
with W = K(1 +√1 − KTK)−1and A defined in Eq.
(2.9).
(2.14)

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0
0.05
0.1
0.15
0.2
0.25
0.3
102
103
104
105
F
M
K = 7






102
103
104
105

M
K = 14






102
103
104
105

M
K = 21
τD = 5
10
30
FIG. 1: Dependence of the Fano factor F on the dimensionality of the Hilbert space M = 1/heff, at fixed dwell time τD = M/2N
and kicking strength K. The data points follow from the quantum mechanical simulation in the open kicked rotator. The solid
line at F =
4is the M-independent result of random-matrix theory. The dashed lines are the semiclassical calculation using
the theory of Ref. 5. There are no fit parameters in the comparison between theory and simulation.
1
III. QUANTUM MECHANICAL CALCULATION
To calculate the transmission matrix from Eq. (2.8) we
need to determine an N×N submatrix of the inverse of an
M ×M matrix. The ratio M/2N = τDis the mean dwell
time in the system in units of the kicking time τ0. This
should be a large number, to avoid spurious effects from
the stroboscopic description. For large M/N we have
found it efficient to do the partial inversion by iteration.
Each step of the iteration requires a multiplication by F,
which can be done efficiently with the help of the fast-
Fourier-transform algorithm.22,23We made sure that the
iteration was fully converged (error estimate 0.1%). In
comparison with a direct matrix inversion, the iterative
calculation is much quicker: the time required scales ∝
M2lnM rather than ∝ M3.
To study the quantum-to-classical crossover we re-
duce the quantum parameter heff = 1/M by two or-
ders of magnitude at fixed classical parameters τD =
M/2N = 5,10,30 and K = 7,14,21. (These three val-
ues of K correspond, respectively, to Lyapunov expo-
nents λ = 1.3,1.9,2.4.) The left edge of the leads is at
m/M = 0.1 and m/M = 0.8. Ensemble averages are
taken by sampling 10 random values of the quasi-energy
ε ∈ (0,2π). We are interested in the semiclassical, large-
N regime (typically N > 10). The average transmission
N−1?Trtt†? ≈ 1/2 is then insensitive to the value of heff,
since quantum corrections are of order 1/N and there-
fore relatively small.21The Fano factor (2.11), however,
is seen to depend strongly on heff, as shown in Fig. 1.
The line through the data points follows from the semi-
classical theory of Ref. 5, as explained in the next section.
In Fig. 2 we have plotted the numerical data on a
double-logarithmic scale, to demonstrate that the sup-
pression of shot noise observed in the simulation is in-
deed governed by the Ehrenfest time τE. The functional
dependence predicted for N >√M is5
F =1
4e−τE/τD, τE= λ−1ln(N2/M) + c,(3.1)
000
111
222
333
444
555
666
777
0001112223
ln(N2/M)ln(N2/M)ln(N2/M)
444555666777
− τD ln(4F)
K = 7
τD = 5
10
30
33
− τD ln(4F)
K = 14
− τD ln(4F)
K = 21
FIG. 2: Demonstration of the logarithmic scaling of the Fano
factor F with the parameter N2/M = M/(2τD)2. The data
points follow from the quantum mechanical simulation and
the lines are the analytical prediction (3.1), with c a fit pa-
rameter. The slope λ−1= 1/ln(K/2) of each line is not a fit
parameter.
with c a K-dependent coefficient of order unity.
shown in Fig. 2, the data follows quite nicely the log-
arithmic scaling with N2/M = M/(2τD)2predicted by
Eq. (3.1). This corresponds to a scaling with w2/LλF in
a two-dimensional quantum dot (with λFthe Fermi wave
length and w and L the width of the point contacts and
of the dot, respectively.) We note that the same para-
metric scaling governs the quantum-to-classical crossover
in the superconducting proximity effect.10,24
As

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4
10-3
10-1
101
103
105
107
10-5
10-5
10-4
10-4
10-3
10-3
10-2
10-2
AA
ρ(A)
K = 7 K = 7





ρ(A)






10-5
10-5
10-4
10-4
10-3
10-3
10-2
10-2

AA
K = 14 K = 14











10-5
10-5
10-4
10-4
10-3
10-3
10-2
10-2

AA
K = 21K = 21





FIG. 3:
represents an initial condition for the classical map (2.4), that is either transmitted through lead 2 (black/red) or reflected
back through lead 1 (gray/green). Only initial conditions with dwell times ≤ 3 are shown for clarity. Lower panels: histogram
of the area distribution of the transmission and reflection bands, calculated from the corresponding phase space portrait in the
upper panel. Areas greater than heff = 1/M correspond to noiseless scattering channels.
(Color online.) Upper panels: phase space portrait of lead 1, for τD = 10 and different values of K. Each point
IV.SEMICLASSICAL CALCULATION
To describe the data from our quantum mechanical
simulation we use the semiclassical approach of Ref. 5.
To that end we first identify which points in the x,p
phase space of lead 1 are transmitted to lead 2 and which
are reflected back to lead 1. By iteration of the classical
map (2.4) we arrive at phase space portraits as shown
in Fig. 3 (top panels). Points of different color (or gray
scale) identify the initial conditions that are transmitted
or reflected.
The transmitted and reflected points group together
in nearly parallel, narrow bands. Each transmission or
reflection band (labeled by an index j) supports noiseless
scattering channels provided its area Aj in phase space
is greater than heff = 1/M. The total number N0 of
noiseless scattering channels is estimated by
N0= M
?
j
Ajθ(Aj− 1/M),(4.1)
with θ(x) = 0 if x < 0 and θ(x) = 1 if x > 0. In the
classical limit M → ∞ one has N0= N, so all channels
are noiseless and the Fano factor vanishes.8
As argued in Ref. 5, the contribution to the Fano factor
from the N − N0 noisy channels can be estimated as
1/4N per channel. In the quantum limit N0 = 0 one
then has the result F = 1/4 of random-matrix theory.25
The prediction for the quantum-to-classical crossover of
the Fano factor is
M
4N
j
?1/M
0
F =
?
Ajθ(1/M − Aj)
=
M
4N
Aρ(A)dA,(4.2)

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5
0
0.1
1
0.15
K=14
7 modes
0
0.1
1
0.15
6 modes
K=21
0
0.1
1
0.15
K=7
10 modes
x
p
xx
FIG. 4: (Color online.) Contour plots of the Husimi function (5.3) in lead 1 for M = 2400, τD = 10, and K = 7,14,21. The
outer contour is at the value 0.15, inner contours increase with increments of 0.1. Yellow regions are the classical transmission
bands with area > 1/M, extracted from Fig. 3.
with band density ρ(A) =?
limit F = 1/4 follows from the total area?1
N/M.
We have approximated the areas of the bands from the
monodromy matrix (2.6), as detailed in the Appendix.
The lower panels of Fig. 3 show the band density in the
form of a histogram. The solid curves in Fig. 1 give the
resulting Fano factor, according to Eq. (4.2).
jδ(A − Aj). The quantum
0Aρ(A)dA =
V.SCATTERING STATES IN THE LEAD
To investigate further the correspondence between the
quantum mechanical and semiclassical descriptions we
compare the quantum mechanical eigenstates |Ui? of t′†t′
with the classical transmission bands.
Phase space portraits of eigenstates are given by the
Husimi function
Hi(mx,mp) = |?Ui|mx,mp?|2.
The state |mx,mp? is a Gaussian wave packet centered
at x = mx/M, p = mp/M. In position representation it
reads
∞
?
k=−∞
(5.1)
?m|mx,mp? ∝e−π(m−mx+kN)2/Ne2πimpm/N.
(5.2)
The summation over k ensures periodicity in m.
The transmission bands typically support several
modes, thus the eigenvalues Tiare nearly degenerate at
unity.We choose the group of eigenstates with Ti >
0.9995 and plot the Husimi function for the projection
onto the subspace spanned by these eigenstates:
H(mx,mp) =
?
Ti>0.9995
Hi(mx,mp).(5.3)
As shown in Fig. 4, this quantum mechanical function
indeed corresponds to a phase-space portrait of the clas-
sical transmission bands with area > 1/M.
VI.CONCLUSION
We have presented compelling numerical evidence for
the validity of the theory of the Ehrenfest-time dependent
suppression of shot noise in a ballistic chaotic system.2,5
The key prediction2of an exponential suppression of the
noise power with the ratio τE/τDof Ehrenfest time and
dwell time is observed over two orders of magnitude in the
simulation. We have also tested the semiclassical theory
proposed recently,5and find that it describes the fully
quantum mechanical data quite well. It would be of in-
terest to extend the simulations to mixed chaotic/regular
dynamics and to systems which exhibit localization.
Acknowledgments
We have benefitted from discussions with Ph. Jacquod
and P. G. Silvestrov. This work was supported by the
Dutch Science Foundation NWO/FOM. J.T. acknowl-
edges the financial support provided through the Euro-
pean Community’s Human Potential Programme under
contract HPRN–CT–2000-00144, Nanoscale Dynamics.
APPENDIX: CALCULATION OF THE BAND
AREA DISTRIBUTION
We approximate the bands in Fig. 3 by straight and
narrow strips in the shape of a parallelogram, disregard-
ing any curvature. This is a good approximation in par-
ticular for the narrowest bands, which are the ones that
determine the shot noise. Each band is characterized by
a mean dwell time n (in units of τ0). We disregard any
variations in the dwell time within a given band, assum-
ing that the entire band exits through one of the two leads
after n iterations. (We have found numerically that this
is true with rare exceptions.)