Conference-ISSS-5-Scar-Like States in Dynamical Electron-Wavepackets in Chaotic Billiard ∗

The time-evolution of the wavepacket inside chaotic and integrable two-dimensional (2D) nanostructures is numerically studied. We have found the enhancement around the classical periodic orbits during the time-evolution in the stadium billiard. It is similar to the scars in the standing wave of the chaotic billiards. The initial position and velocity, and the shape of the wavepacket are crucial for the enhancement, but we can observe that the remnant of the initial wavepacket travels along the unstable periodic orbit. Then the wavepacket gradually diffuses around the structure. This behavior has close relation to the dynamical properties of electrons in the structure, e.g., the conductivity, the magneto-resistance etc. The quantum fidelity, which can measure the robustness of dynamical states inside the nanostructures, is also discussed. [DOI: 10.1380/ejssnt.2009.721]


I. INTRODUCTION
It will be soon to become possible to fabricate about 10nm scale sturctures on two-dimensional (2D) GaAs/AlGaAs hetero-junction.In such nanostructures we surely have to treat only one or a few electrons inside and their dynamical behavior should be purely quantum mechanical phenomena, due to the Coulomb blockade and/or the spin blockade, etc. Numerical approaches to closed and open quantum systems have been done, but only stable (not necessary stationary or standing wave) states are usually considered [1].On the other hand, to clarify the dynamical property inside the devices, the study of the time-evolution of electron wave function is absolutely necessary.The propagation of the wavepacket is a fundamental subject in quantum dynamical properties.
About a quarter of a century ago the numerical simulation of the quantum wave function had been rising and the unexpected discovery of the scar states in chaotic systems surprised us [2].The scar state has the clear concentration of the wave function on a certain period orbit, which is apparently classical mechanical subjects.In chaotic systems almost all periodic orbits are unstable.Therefore this concentration seems unrealistic.However, the semi-classical approximation clarifies that the enhancement along a single periodic orbit can be significant in some eigenstates and it is also quantum effect [3,4].It has also turned out to be very important to understand the dynamical properties of electrons in the structure, which should have close relation to the conductivity, the magneto-resistance etc., when the quantum dot has chaotic shape.
Stadium billiard is just the typical 2D chaotic system (Fig. 1).It is a finite flat 2D region, its boundary is a wall of infinite height and its shape is just a stadium.Of course it was proven that the stadium billiard is chaotic [5].Unstable periodic orbits in the stadium are well searched and listed [3].Only an exception, which is periodic, but stable, is the bouncing ball mode.A particle bounces infinitely between the straight parts of the wall of the stadium.It can be considered as just a kind of the periodic orbit in a rectangle billiard and it is relatively easy to handle and, if necessary, to eliminate its contribution from the physical properties theoretically.Thus the shape of the stadium has been applied to various studies in both simulation [1,2,6] and experiment [7].Then we accumulate the absolute square of the time-evolving wave function on each mesh point and find scar-like feature as the stationary state.The quantum fidelity between the stadium and a slightly distorted stadium are also examined.It is a important tool to see the possibility of nano-sized device for practical application, e.g., the quantum computing.

II. TIME-EVOLUTION OF WAVEPACKET
In this work, to study quantum dynamical motion of electrons inside the nanostructure, the time-evolution of the quantum wavepacket is numerically calculated.We choose the stadium billiard as the shape of the nanostructure, because it is one of the typical chaotic examples.The nanostructure device of stadium billiard is molded in the 400 × 200(=80,000) mesh points inside a rectangle (Fig. 1).Outside the device model, we put V = 10 300 that is almost as large as possible for the double precision variable numerically instead of the infinity.If we put the lattice constant a=1 a.u.(=0.0529nm), where a.u.represents the Hartree atomic units, i.e. m e = e 2 = = 1, and
our stadium billiard model is molded in the scale 21.2 nm × 10.6 nm.
In the time-dependent Schrödinger equation the Hamiltonian is actually made in a very sparse matrix for our numerical simulation.
The time-evolution of the wave function is calculated by the Crank-Nicholson method [8] that is For our simulation, we put ∆t = t n+1 − t n = 0.025a.u. as one step (1 a.u.= 2.419 × 10 −17 sec) and calculation goes up to 500,000 steps.If the Hartree atomic units are adopted to our calculation, its total time is 12.1 psec.
To study the quantum dynamical properties of the stadium, we initially put a Gaussian wavepacket.The initial Gaussian wavepacket is given as where r 0 = (x 0 , y 0 ) represents the original location of the center of the wavepacket, σ is the standard deviation of the Gaussian shape and is the momentum of the wavepacket.Thus, at first, it goes with velocity v and gradually spreads out with the time-dependent deviation σ(t) = σ 1 + 2 t 2 m 2 e σ 4 .If there were not any boundaries, the wavepacket was always the Gaussian with σ(t).
An example of the time-evolution of the wavepacket with the initial condition: r 0 = (0, 0) in Fig. 2, σ = 5 and intial velocity p 0 = ( 2 ) are shown.Thus its launching direction is θ = 30 • and its absolute value of the momentum is |p 0 | = 1.The launching direction is measured as angle θ from the coordinate x.Of course, the wavepacket itself has the tendency of spreading out.The reflection against the wall of the billiard also causes the strong dispersion.The wavepacket seems to fly as a bunch up to about 5000 or 6000 steps, then gradually spread out and after about 20000 steps the wavepacket diffuses all over the billiard.

III. PERIODIC ORBITS SCARS IN TIME-EVOLVING WAVEPACKETS
It has been already found that we can often see scar states in stationary wave functions of chaotic system, especially in 2D billiard systems [2].The scar states have scars that are the magnificent concentration of the nodal pattern closely around some particular classically unstable orbit.It is also predicted that the wavepacket should have the similar concentration in their nodal patterns of the time-evolutions [9] and the concentration might also suppress the decay of the wavepacket itself.
Here we calculate the accumulation of the absolute square of the time evolving wave functions |Ψ| 2 at each mesh point r i = (x i , y i ) where N represents the number of the time steps.
The results for the Gaussian wavepackets of the same shape σ = 5, the same initial location of the center r 0 = (0, 0) and the same absolute value of the momentum |p 0 | = 1 with different launching angles θ are shown in Fig. 3.In the cases of θ = 0 • , 30 • , 45 • , and 90 • , the center of the wavepacket is launched along the period orbits, which have relatively short periods.Especially at θ = 0 • , 30 • , and 45 • , it is on the unstable periodic orbit.On the other hand, at θ = 90 • , it is just on the one of the one-parameter family of the stable orbits, socalled bouncing ball mode, which is the family of periodic orbits and similar to integrable systems.On all these orbits the scar states are already found in stationary wave states [2].Then we can also see the "scar" in the timeevolving wavepacket, if it is launched along the periodic orbit.We shall call these scar-like states as dynamical scar states.

IV. QUANTUM FIDELITY
To know whether the decay rate of the wavepacket is suppressed or not, when it is launched along the periodic orbit of chaotic systems, the quantum fidelity is also evaluated [10,11].The fidelity is defined as where Ψ(t) = exp(iHt)Ψ(0) is the time-evolution of the original Gaussian wavepacket Ψ(0), and Ψ (t) = exp(i(H + V )t)Ψ(0) is the time-evolution of the same wavepacket Ψ(0) by the slightly different Hamiltonian H = H + V .The additional potential term V is necessary to see the robustness of the dynamical scar state.
Here we take Increasing , it is known that the quantum fidelity has the transition from the pertubation theory(PT) region to the Fermi-golden-rule(FGR) region, and to the Lyapnov(L) region in the relatively simple system, i.e., the quantum standard map [12,13]

When
is small in our system: 0 < 0.006 (Fig. 4), it is found that the fidelity clearly shows the Gaussian decay; m(t) ≈ exp(−Γt 2 / 2 ) as the function of time t before it approaches the ergotic value.Therefore it is the PT regime.The perturbation theory tells us Γ V 2 ∝ 2 [13], and our result also agrees with it up to ≈ 0.006 (Fig. 4(a-2)).Then eventually the fidelity deviate from the Gaussian decay and approach to the exponential decay.Note that, at least from the numerical calculation, after arriving at the ergodic value, we cannot find its meaningful behavior.Of course, theoretically m(t) can approach to zero.Though, numerically there apparently exists the smallest value, which is estimated from the finite element approximation for the simulation and is called the ergodic value.
Then, it is also confirmed the exponential decay of the fidelity m(t) ≈ exp(−Λt/ ) in the range 0.01 0.03 (Fig. 4), the FGR region.The fidelity should show the exponential decay; m(t) ≈ exp(−Λt/ ).It is also predicted that its decay constant Λ should be proportional to 2 as Γ, however, it shows the linear dependence with , instead of 2 .
Getting still larger, it should enter the L regime, however, the simulated value of the slope of ln m(t) starts to fluctuate gradually.Then, it soon become pretty hard to evaluate the slope of m(t) even before it goes to the ergodic value.Here it is assumed that the slope is to be estimated from the origin and the point where m(t) arrives the ergodic value for the first time, and they are plotted in Fig. 4(b-2).The slopes seem to be almost independent in the range 0.05 0.15 as the L regime.However, their value also seem much smaller than the Lyapunov exponent of our system 8.6 × 10 −3 [11].The value is almost the half of the Lyapunov exponent, in the contrast to the Lyapunov regime of simpler quantum systems [12,13].

V. DISCUSSION
The dynamical scar in the time-evolution of the wavepacket can be understood by the expansion with respect to the stationary eigenstates.Using the stationary eigenstates of the system and their eigenvalues E n , generally a timedependent wavefunction Ψ(r, t) can be expanded as and we can have the coeffient Thus, putting Ψ(r, 0) = Ψ 0 (r), (11) and determining c n by Eq. ( 10), then, Ψ(r, t) is the timeevolution of the initial Gaussian wavepacket Ψ 0 (r).
In the case of the launching from the center of the stadium(0,0) with angle 30 • , the result of the expansion is presented in Fig. 5.The eigenstates which have larger value of |c n | 2 (n = 936, 879, ...) show the scars (Fig. 5).All of them also correspond to a specific unstable periodic orbit (see the 30 • case in Fig. 3).It is also clear that the initial position of the center of the wavepacket is on the periodic orbit, and the initial velocity is also along the periodic orbit.Thus, it strongly implies that the contribution of the scar states dominate in the eigenfunction expansion of the time-evolving wavepacket, if the wavepacket is launched along the periodic orbit.Calculating the time-evolution of various launching angles, this is numerically confirmed.Launching along the periodic orbit, consequently, the accumulation Eq. ( 5) has the scarlike concentration.
Of course each launching angle result has its own characteristics on the quantum fidelity, however, we cannot find any clear difference in the dependency on , particularly on the regime change (Figs. 4 and 6).The PT and the FGR regimes exist about the same ranges of in each launching angle.The slope of the L regime is always much smaller than the Lyapunov exponent.It means that the robustness of the fidelity has no specific difference whether the wavepacket is launch on the periodic orbit or not.
On the contrary to Γ of the PT regime, in the FGR regime the slope Λ of ln m(t) is just proportional to , not 2 .At least, naive physical inspection tells that Λ should be proportional to V 2 ∝ 2 as PT.In FGR the density of states also change rapidly against and it might affect its -dependency.

VI. CONCLUSION
The quantum time-evolution of the Gaussian wavepacket in the stadium billiard is calculated.The dynamical scar is found in the accumulation of the absolute square of the time-evolving wave function.It is the concentration of the nodal patterns near around the unstable periodic orbits, if the wavepacket is launched along the periodic orbit.Therefore they are very similar to the scars in the quantum stationary states.
The quantum fidelity is also evaluated to investigate the robustness of the dynamical scar states, making the flat bottom of the billiard bent a bit.At least, our system has the PT and the FGR regime as expected.It would have the L regime, however, its properties seems different from the previous expectation.The fidelity indicates no particular difference between its behaviors of the dynamical scar states and ordinary dynamical states.

FIG. 1 :
FIG. 1: Schematic illustration of the stadium billiard adopted in this work.It is a kind of 2 × 4 stadium.It is composed of a 100 × 100 square and two semicircles on the opposite sides of the square.The radii of the semicircles are 50.

FIG. 4 :
FIG.4: Logarithm of the quantum fidelity is plotted as a function of time t, varying the perturbation strength , at = 0.001 s 0.008(a-1) and = 0.01 s 0.23(a-2), with the initial condition r0 = (0, 0), σ = 5 and |p0| = 1 and launching angle 45 • .The broken line represents the ergodic limit, caused by the finite element approximation.The double broken line shows the decay, when the slope is just the Lyapnov exponent.In PT regime(0 < .0.006); ln m(t) ≈ −Γt 2 /~2, its Gaussian dependence on is clear(upper right).In FGR regime(0.01 . .0.03); ln m(t) ≈ −Λt/~, the slope Λ is found to be linearly dependent on .Then, in the region 0.05 . .0.15, the estimated slopes seem likely to be independent on (lower right), and almost the half of the Lyapnov exponent(double broken line).In the range of far larger , suddenly Λ exceeds the Lyapnov exponent.

50 ) 2 − 7 )
at |x| ≤ 50 and |y| ≤ 50 {( |x|−50 50 ) 2 + ( y 50 ) 2 } − at |x| > 50 and (|x| − 50) 2 + y 2 ≤ 50 (Thus represents the strength of the potential.If = 0, then m(t) never decay, it is just a constant m(t) ≡ 1 instead.Here we put the origin at the center of the stadium and the coordinates x and y are introduced as in Fig. 1.The bottom of the billiard is caved a bit to see the robustness of the states.Of course, m(0) ≡ 1 by definition.