\documentstyle[preprint,revtex]{aps}
\tightenlines
\begin{document}
\preprint{Universit\`a di Roma ``La Sapienza'', preprint n. 830,
 October 15, 1991}
\draft

\begin{title}
Quantum Chaos Without Classical Counterpart
\end{title}

\author{Giovanni Jona-Lasinio and Carlo Presilla}
\begin{instit}
Dipartimento di Fisica dell'Universit\`a ``La Sapienza'',\\
Piazzale A. Moro 2, Roma, Italy 00185
\end{instit}

\author{Federico Capasso}
\begin{instit}
AT\&T Bell Laboratories,\\
600 Mountain Avenue, Murray Hill, New Jersey 07974
\end{instit}

\begin{abstract}
We describe a quantum many-body system undergoing multiple resonant
tunneling which exhibits chaotic behavior in numerical simulations
of a mean field approximation.
This phenomenon, which has no counterpart in the classical limit,
is due to effective nonlinearities in the tunneling process and
can be observed in principle within a heterostructure.
\end{abstract}

\pacs{03.65.-w, 05.45.+b, 73.40.Gk}

\narrowtext

%\section {Introduction}

In current studies
the problem of quantum chaos is usually posed in the following terms.
We are given a system which exhibits chaotic behavior classically:
does this property survive quantization?
Typically one studies the spectrum of the quantum mechanical Hamiltonian
operator.
However spectral differences at the quantum level between classically
integrable and chaotic systems do not seem to imply dramatic differences
in the time evolution of the quantum mechanical observables.
Relics of classical chaos exist in low dimensional systems in short
wavelength or large quantum numbers limits where the discrete quantum
spectrum approximates a continuous spectrum.
Persistent chaotic behavior in the evolution of the observables
however does not seem possible \cite{CHIR}.

In this paper we propose to explore the opposite road by asking
the following question: is it possible to have quantum chaos,
transient or persistent, independently of what happens in the classical
limit?
To tackle this question we observe that
another way of approximating a continuous spectrum is by increasing
the number of degrees of freedom of the system, that is by considering
a many body system.
It is well known that under appropriate conditions a many body
system can be approximately described in terms of a single particle
wave function obeying a nonlinear and nonlocal Schr\"odinger equation
(Hartree or Hartree-Fock equations).
The importance of effective nonlinearities due to the particle
interaction in concrete cases has been
emphasized in our previous paper \cite{PJLC} where we showed that
nonlinear quantum oscillations can be present in resonant tunneling of
electrons through a double barrier heterostructure \cite{FC}.
The situation studied in Ref. \cite{PJLC} represented a transient
phenomenon because we dealt with an open system.
In the present paper we consider the same system enclosed between
two potential barriers.
In other words we deal now with a cloud of electrons moving in
a three well heterostructure (see later).
In this way our Hartree-like equation describing the motion of the cloud
represents a confined non dissipative system which may be ergodic or even
mixing at least in some regions of its phase space and which therefore may
exhibit irregular behavior during its evolution.
Two points should be emphasized:
{\sl i)} chaos is found numerically in a mesoscopic variable,
the electric charge in a well,
which is in principle experimentally observable \cite{LEO},
{\sl ii)} the phenomenon we describe has no
classical analogue being based on resonant tunneling.

%\section{Description of the system}

As in our previous work we consider the quantum transport of
electrons through a double barrier via resonant tunneling.
However in the present case the heterostructure is shaped in such a way
that the system is closed.
The situation is described in Fig.\ \ref{FIG1}.
An electron cloud is initially created in one of the large wells
with a kinetic energy peaked around the resonance energy of the double
barrier.
As discussed in Ref. \cite{PJLC} we may reduce ourselves to one dimensional
propagation along the $x$ axis orthogonal to the
heterostructure junction planes.
We describe the state of the electrons with a one
particle wave function obeying a nonlinear Hartree equation \cite{PJLC}.
The effect of the nonlinearity depends strongly on the materials in the
heterostructure regions.
We assume that the wells $w_1$ and $w_3$ consist of a heavily doped
semiconductor so that in these regions the nonlinearity can be
neglected during the time evolution of the system,
while the barriers $b_1$ and $b_2$ and the well $w_2$ are undoped
semiconductors \cite{NOTA1}.
When a fraction of the charge penetrates inside the double barrier, the
resonance is shifted and the nonlinear charge oscillations described in
Ref. \cite{PJLC} appear.
The reflected and transmitted charges in regions $w_1$ and $w_3$ after some
time will return to the double barrier and a fraction of them will
penetrate inside the well $w_2$
depending on the height of the resonance at that moment.
We expect that
after a few cycles the charge in $w_2$ will show a very complicated
time dependence losing memory of how the process initiated.

Let us now turn to the mathematical model of the system.
The external potential, as depicted in Fig.\ \ref{FIG1}, is assumed to be a step
function (no electric field is applied):
\FL\begin{equation}
V(x)= V_0 \biggl [ \chi_{b_1}(x) + \chi_{b_2}(x) \biggr ] +
      V_1 \biggl [ \chi_{B_1}(x) + \chi_{B_2}(x) \biggr ]
\label{EXTPOT}
\end{equation}
where $V_0$ and $V_1$ are positive constants
({\sl i.e.} the height of the barriers $b_1$, $b_2$ and $B_1$, $B_2$
respectively) with $V_1>V_0$
and $\chi_S$ is the characteristic function of the set $S$
({\sl i.e.} $\chi_S(x) = 1$ if $x \in S$, 0 otherwise).

As in Ref. \cite{PJLC} we simplify the shape of the nonlinear term by
considering a rigid displacement of the well $w_2$.
This looks legitimate because the essential effect of the nonlinearity is
the displacement of the resonance energy and in addition this choice
simplifies enormously the numerical calculation.
On the other hand we have verified that a more precise calculation
using a screened Coulomb potential does not modify in any important
feature the dynamics of the system.
The nonlinear Schr\"odinger equation describing the evolution
of the electron cloud wave function is therefore:
\FL\begin{equation}
i\hbar {\partial \psi(x,t) \over \partial t} =
-{\hbar^2 \over 2 m} {\partial^2 \psi(x,t) \over \partial x^2}
+ \biggl [ V(x) + \alpha Q(t) \chi_{w_2}(x) \biggr ] \psi(x,t)
\label{SCHEQ}
\end{equation}
where $Q(t)$ is the charge inside the well $w_2$ at time $t$
\begin{equation}
Q(t) \equiv \int_{w_2} |\psi(x,t)|^2~dx
\label{QDEF}
\end{equation}
In these notations $\psi(x,t)$ is normalized to $1$, $Q(t)$ is dimensionless
and $\alpha$ has the dimension of an energy and measures the strength of
the mean field acting on each electron.
The parameter
$\alpha$ is proportional to the transversal areal density of the electrons
and depends on the electric capacitance of the double barrier.

The 1-particle state which is the initial condition in the mean field
equation  has been chosen to be a gaussian shaped
superposition of plane waves with mean momentum $\hbar k_0$:
\begin{equation}
\psi (x,0) = {1 \over \sqrt{\sigma \sqrt{\pi} } }
\exp{\biggl [- {1 \over 2} \biggl ( {x-x_0 \over \sigma} \biggr )^2
+ i k_0 x \biggr ]}
\label{PSISTART}
\end{equation}
$x_0$ is chosen to coincide with the middle point of the well $w_1$
which has a width so large with respect to $\sigma$ that
at the initial time no appreciable charge  sits in $w_2$,
i.e. $Q(0) \simeq 0$.

The choice of the various parameters has been made on the basis of the results
obtained in Ref. \cite{PJLC}.
Their values correspond to a situation
in which nonlinear oscillations in the transient regime are enhanced.
We consider $|w_1|=|w_3|=1100~a_0$, $|w_2|=15~a_0$, $|b_1|=|b_2|=20~a_0$,
$\sigma=110~a_0$ ($a_0\simeq 0.529$ \AA, being the Bohr radius),
$V_0=0.3~eV$.
The mean kinetic energy of the initial state is equal to the resonance
energy ($E_R \simeq 0.15~eV$) of the double barrier $b_1$-$b_2$.
The width of the two external barriers $|B_1|=|B_2|=440~a_0$ and their
height $V_1=0.9~eV$ assure the complete confinement of the electron cloud
between them.

The solution of the differential equation (\ref{SCHEQ}) with the initial
condition (\ref{PSISTART}) has been achieved by a numerical integration on
a suitable two-dimensional lattice, taking into account the remarks
of Ref. \cite{BEN1}.

%\section{Numerical Results}

Now we report the results of the numerical simulations
and we provide an analysis which shows that in our system a chaotic
behavior develops during its evolution.
We concentrate on the charge $Q(t)$ and we treat it as if it were an
experimental signal.
A mathematical study of the system is deferred to another publication.

In Fig.\ \ref{FIG2} we show the behavior of $Q(t)$ as a function of time
in two different widely separated time intervals.
The first structure appearing between $t=0$ and $t=10$
(everywhere we use as time unit
$10^3$ atomic units of time  $\simeq 4.83~10^{-14}$ s),
reproduces exactly the transient behavior explored in Ref. \cite{PJLC}.
Between $t=10$ and $t=40$ we observe a qualitative repetition of
almost the same structure due to the multiple reflections of the wave
packet inside the large wells $w_1$ and $w_3$.
The number of oscillations per structure increases progressively
until, after $t=40$ the isolated structures tend to disappear
and an apparently irregular motion sets in.
The irregularity increases with time as it is evident from the behavior
of $Q(t)$ between $t=500$ and $t=600$.
This suggests an approach to a stationary state as also evidenced
in Fig.\ \ref{FIG3} where the evolution of the mean charge
$\langle Q \rangle_t \equiv t^{-1} \int_0^t Q(t')~dt'$ is reported.
It is clear that a true stationary state
is not yet reached at $t=600$ but the approach at this stage is
already very slow so that one can safely speak of quasi equilibrium.

Concerning the attainment of quasi equilibrium a crucial question is
what happens if the initial conditions are varied.
In the case of an energetically equivalent initial condition the
same quasi equilibrium state is reached.
An example is shown in Fig.\ \ref{FIG3} where the dashed curve refers
to an initial condition in which
the single electron cloud of Fig.\ \ref{FIG1}\ is replaced by
two equal half-density electron clouds moving from the wells $w_1$
and $w_3$.
When the initial conditions are energetically nonequivalent
the asymptotic behavior changes.
For instance, we have examined the case of an initial cloud with
mean energy below the resonance energy.
In this case the mean charge $\langle Q \rangle_t$ tends
to a lower asymptotic limit and the irregular variation of the charge
$Q(t)$ in the time interval considered is less pronounced.
This is due to a reduced charge accumulation in the well $w_2$
and therefore to a weakening of the nonlinearity.
A more systematic study of the dependence of the phenomenon on the
initial conditions will be presented in a forthcoming paper.

We now show that the irregular behavior we have found has all the features of
chaotic behavior by estimating commonly used indicators like
correlation functions, power spectra, information dimension and
entropy \cite{ECKRUE}.

We examine first the autocorrelation function
$C(t)\equiv\int_{T}q(t')q(t'+t)~dt'$
$ / \int_{T}q(t')q(t')~dt'$ of the zero-mean charge
$q(t) \equiv Q(t)- |T|^{-1} \int_{T} Q(t')~dt'$.
In Fig.\ \ref{FIG4} we give for comparison the autocorrelation function
calculated in the time interval $T=[100,600]$ for the linear and nonlinear
case respectively.
The difference is striking.
In the nonlinear case we have a short correlation time followed by small
oscillations around zero.
The interpretation of these oscillations is not immediate.
In part they reflect the complicated dynamics of the system and
in part are a noise effect which tends to disappear when the interval
$T$ is enlarged.
Furthermore a preliminary analysis shows that
when the parameter $\alpha$ measuring the strength of the nonlinearity
decreases these oscillations increase in amplitude and their characteristic
times become comparable to those of linear case.
A rather sharp transition seems to take place in the region of values
of $\alpha$ in which the transient oscillations observed in our previous
work \cite{PJLC} disappear.

In order to study the approach of the system to the true equilibrium state
we chose to calculate the chaotic indicators in two different time
intervals $T_1=[180,260]$ and $T_2=[480,560]$
where $\langle Q \rangle_t$ is almost constant and has a slightly
different value.
Between these two intervals we observe a reduction of the correlation
time $\tau$.
We have $\tau =0.28$ in $T_1$ and $\tau =0.25$ in $T_2$.
This fact is also reflected in the behavior of the power spectrum
$P(\omega)$ reported in Fig.\ \ref{FIG5}.
Going from $T_1$ to $T_2$, in fact  we see an evident increase of the
spectrum flatness.

We have next calculated the information dimension $\dim_H \rho(T_{1,2})$ where
$\rho(T_{1,2})$ is the quasi equilibrium measure associated  with the evolution
of our system during the intervals $T_1$ and $T_2$.
We have used the Grassberger-Procaccia algorithm \cite{GP}.
As expected from the results of Ref. \cite{BEN2} $\dim_H\rho$,
{\sl i.e.} the slope of the straight lines in Fig.\ \ref{FIG6},
has a non integer value increasing slowly with time from $2.20 \pm 0.05$
in the first interval to $2.50 \pm 0.05$ in the second interval.
The straight line behavior of $\log N(r)$ versus $\log(r)$ is absent in
the linear case.
>From the same data of Fig.\ \ref{FIG6} we can compute the order-2 Renyi
entropy $K_2(\rho)$,
a lower bound for the Kolmogorov-Sinai entropy,
by an extrapolation process to infinite embedding dimension $d$ \cite{GP}.
For both time intervals  we obtain for $K_2(\rho)$ a
positive value of the order of magnitude of the inverse correlation time
$\tau^{-1}$ in agreement with the remarks of Ref. \cite{ZASLA}.

The example we have discussed raises several problems.
The relationship of the one dimensional Hartree approximation
to an exact treatment of the three dimensional many electron system
was briefly discussed in our previous paper \cite{PJLC}, but has
to be analysed further.
This is crucial for the following reasons.
Let us assume that the chaotic behavior found numerically is persistent
in time and is a feature of our nonlinear Schr\"odinger equation
that can be proved rigorously.
In general this does not imply that the many body system of which the
nonlinear Schr\"odinger equation (\ref{SCHEQ}) is an approximate description
exhibits persistent chaos.
In fact, if we view the equation (\ref{SCHEQ}) as an approximation
to a one dimensional many body system, the latter will have discrete
spectrum due to the fact that the number of particles must remain finite
(the system is confined) in order to have a finite density.
According to the accepted wisdom this is compatible with transient
chaos, possibly persisting over a long time interval.
However, our real heterostructure system is {\sl three dimensional} and
can be considered infinite in the transversal directions.
This means that the physical system should be characterized in this
limit by a continuous spectrum due to the electron-electron interaction
which couples the longitudinal and transversal variables.
In these conditions persistent chaos is not forbidden.


\acknowledgements

It is a pleasure to thank G. C. Benettin, Ya. G. Sinai  and
S. Tenenbaum for very useful discussions and J. Palis for critical
remarks and for his interest in our work.


\begin{references}

\bibitem{CHIR}
M. V. Berry in {\sl Chaotic Behavior of Deterministic Systems},
Les Houches Lectures {\sl XXXVI}, G. Iooss, R. H. G. Helleman
and R. Stora, eds. (North-Holland, Amsterdam, 1983);
G. Casati, B. V. Chirikov, I. Guarnieri and D. L. Shepelyansky,
Phys. Rep. {\bf 154}, 77 (1987);
B. V. Chirikov, F. M. Izrailev and D. L. Shepelyansky,
Physica D {\bf 33}, 77 (1988);
Ya. G. Sinai, Physica A {\bf 163}, 197 (1990).

\bibitem{PJLC} C. Presilla, G. Jona-Lasinio and F. Capasso,
Phys. Rev. B {\bf 43}, 5200 (1991).

\bibitem{FC} For a general discussion of heterostructures see
F. Capasso and S. Datta, Phys. Today {\bf 43} (2), 74 (1990).

\bibitem{LEO} K. Leo, J. Shah, E. O. G\"obel, T. C. Damen, S. Schmitt-Rink,
W. Sch\"afer and K. K\"ohler, Phys. Rev. Lett. {\bf 66}, 201 (1991).

\bibitem{NOTA1}The carrier concentration (doping density) in regions
$w_1$ and $w_3$ should be approximately equal to $10^{19}~cm^{-3}$
for a material such as $GaAs$.
In these conditions the dielectric relaxation time is
approximately $10^{-15}$ s so that the electric field in $w_1$ and $w_3$,
and therefore the nonlinearity, remains essentially negligible
during the time evolution of the system.
For the barriers and thin well materials a possible choice is
$AlGaAs$ and $GaAs$ (undoped) respectively.
The chaotic behavior predicted for this system can be investigated
experimentally using the subpicosecond optical technique described
in Ref. \cite{LEO}.

\bibitem{BEN1} G. Benettin, M. Casartelli, L. Galgani, A. Giorgilli
and J.-M. Strelcyn, Nuovo Cimento {\bf 44} B, 183 (1978).

\bibitem{ECKRUE} J. P. Eckmann and D. Ruelle, Rev. Mod. Phys.
{\bf 57}, 617 (1985).

\bibitem{GP} P. Grassberger and I. Procaccia, Phys. Rev. Lett.
{\bf 50}, 346 (1983); Phys. Rev. A {\bf 28}, 2591 (1983).

\bibitem{BEN2} G. Benettin, D. Casati, L. Galgani, A. Giorgilli
and L. Sironi, Phys. Lett. A {\bf 118}, 325 (1986).

\bibitem{ZASLA} G. M. Zaslavsky, {\sl Chaos in Dynamic Systems}
(Harwood Academic Publishers, Chur 1985).
\end{references}

\figure{Energy diagram of the three-well two-barrier heterostructure
considered in the present paper. The barriers $b_1$ and $b_2$ separated
by the well $w_2$ produce a resonance in the currents between wells
$w_1$ and $w_3$ at the energy indicated by the dashed line.
An electron cloud is initially localized in the well $w_1$ and moves
toward the well $w_3$ with a mean kinetic energy close to the resonance.
\label{FIG1}}

\figure{Time development of the charge $Q(t)$ in the case
$\alpha=3$ Ryd.
An atomic unit of time corresponds to $4.83~10^{-17}$ s.
\label{FIG2}}

\figure{Time development of the mean charge
$\langle Q \rangle_t$ from the data of Fig.\ \ref{FIG2} (solid line).
The dashed line represents the case where
the starting single electron cloud of Fig.\ \ref{FIG1}\ is replaced
by two equal half-density electron clouds moving from the wells $w_1$
and $w_3$.
\label{FIG3}}

\figure{Normalized time autocorrelation function $C(t)$ of
the zero-mean charge from the data of Fig.\ \ref{FIG2} (solid line).
The dot-dashed line represents the corresponding linear case $\alpha=0$.
\label{FIG4}}

\figure{Power spectrum $P(\omega)$ of the zero-mean charge
from the data of Fig.\ \ref{FIG2}\
in the two intervals $T_1=[180,260]$ and $T_2=[480,560]$.
\label{FIG5}}

\figure{Information dimension $\dim_H(\rho)$
in the two intervals $T_1$ and $T_2$ calculated by the slope
of the straight line portion of the curves $\ln N(r)$ versus $\ln r$.
$N(r)$ is the number of pairs of points
$(Q(t),Q(t+\Delta t), ... ,Q(t+(d-1)\Delta t))$
with distance less
than $r$ in a $d$-dimensional embedding space with $d=6,8,10,$...$,22$
(see Ref. \cite{GP}).
\label{FIG6}}

\end{document}