decay of quantum accelerator modes

DECAY OF QUANTUM

ACCELERATOR MODES

MICHAEL SHEINMAN

DECAY OF QUANTUM ACCELERATOR

MODES

RESEARCH THESIS

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS

FOR THE DEGREE OF MASTER OF SCIENCE IN PHYSICS

MICHAEL SHEINMAN

SUBMITTED TO THE SENATE OF THE TECHNION — ISRAEL INSTITUTE OF TECHNOLOGY

ADAR I, 5765 HAIFA MARCH, 2005

THIS RESEARCH THESIS WAS SUPERVISED BY PROF. SHMUEL FISHMAN

UNDER THE AUSPICES OF THE PHYSICS DEPARTMENT

ACKNOWLEDGMENT

I am grateful to prof. Shmuel Fishman for his guidance and I wish to

express gratitude to Peter Schlagheck, Denis Ullmo, Italo Guarneri and

Yevgeny Krivolapov.

THE GENEROUS FINANCIAL HELP OF THE US-ISRAEL BINATIONAL

SCIENCE FOUNDATION (BSF), THE MINERVA CENTER OF NONLINEAR

PHYSICS OF COMPLEX SYSTEMS AND THE ISRAEL SCIENCE

FOUNDATION (ISF) AND TECHNION IS GRATEFULLY ACKNOWLEDGED

Dedicated to Lena

Contents

List of Symbols 1

Abstract 3

1 Introduction 4

1.1 Quantum dynamics in the presence of classical resonance . . . . . . . 5

1.1.1 Effective Hamiltonian in the vicinity of a resonance . . . . . . 6

1.1.2 Calculation of resonance parameters . . . . . . . . . . . . . . 9

1.1.3 Resonant coupling . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Kicked accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.1 Atom-Field interaction . . . . . . . . . . . . . . . . . . . . . 11

1.2.2 Realization of kicked accelerator . . . . . . . . . . . . . . . . 16

1.2.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.4 Transformation to dimensionless parameters . . . . . . . . . . 18

1.3 Quantum Accelerator Modes . . . . . . . . . . . . . . . . . . . . . . 18

2 Resonance-Assisted Tunneling 22

2.1 The case without degeneracy of the regular states . . . . . . . . . . . 22

2.2 Degeneracy of the regular states . . . . . . . . . . . . . . . . . . . . 28

iv

CONTENTS v

3 Direct Tunneling 34

4 Summary and discussion 39

A Derivation of formula (3.11) 41

B Numerical calculations 44

B.1 Dynamic simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

B.2 Truncation of the one-kick-evolution operator . . . . . . . . . . . . . 46

References 48

Hebrew Abstract g

List of Figures

1.1 Comparison between the real phase space and the integrable approximation 6

1.2 Temporal typical profile of the pulse strength . . . . . . . . . . . . . 16

1.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1 Phase space of the J map . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Comparison between the numerical simulation and the analytical for-

mula for the ground state . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Comparison between the numerical simulation and the analytical for-

mula for an excited state . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 Eigenfunctions of the evolution operator for various values of Planck’s

constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Escape rate against quasienergy for all regular states and various values

of Planck’s constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1 Schematic phase space of the system after transformation to the new

coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Schematic phase space obtained by by shifts of the circle . . . . . . . 36

3.3 Comparison between the numerical simulation and the analytic formula 38

3.4 Extrapolation of the analytical formula . . . . . . . . . . . . . . . . . 38

vi

LIST OF FIGURES vii

B.1 Wave functions after the different number of kicks . . . . . . . . . . . 45

B.2 Husimi plots of the wave functions after different number of kicks . . 46

B.3 Decay of population in the regular island . . . . . . . . . . . . . . . . 47

B.4 Same as previous graph but for different Planck’s constant . . . . . . 48

B.5 The eigenvector of the truncated one-kick-evolution operator . . . . . 49

B.6 Escape rate against the cutoff . . . . . . . . . . . . . . . . . . . . . . 49

B.7 The comparison between two numerical methods . . . . . . . . . . . . 50

List of Symbols

I action

J shifted action

p momentum

q x coordinate

θ angle

n l m integer

ψ χ ϕ wave function

H Hamiltonian

~ Planck’s constant

k strength of kicks

τ kick period

η dimensionless gravitational acceleration

r order of resonance

U evolution operator

L number of excited regular states with quantum number fissionable by r

t time

mr:s Ir:s Vr:s resonance parameters

Nc number of chaotic states

1

2 LIST OF SYMBOLS

Γ escape rate

Veff effective coupling

E regular energy

W chaotic energy

Lm m-th Legendere polynomial

C dimensionless constant

ε pseudo-Planck’s constant

ε expansion parameter

Abstract

Cooling and trapping of atoms enables the exploration of quantum mechanical effects

of their center of mass motion. One of the most surprising phenomena, that was

discovered, was the existence of ”Quantum accelerator modes”. These modes were

found in the experiments with kicked (by a standing electromagnetic wave) cesium

atoms. It was found that there is a fraction of the atoms that accelerate, on average,

relative to the gravity. This discovery was explained using a pseudo-classical limit.

The quantum dynamics far from the classical limit turn out to be described by a

fictitious classical map. The accelerator modes result of wave packets trapped in a

stable island of this classical map. The decay of these modes results of tunneling

out of the island and is the subject of the thesis. Recently it was demonstrated

that in spite of the fact that tunneling is a pure quantum phenomenon it is strongly

influenced by classical phase space structures. Results that were obtained recently in

the studies of tunneling between regular islands, embedded in the chaotic sea, were

used to calculate the tunneling rate from a regular island to the chaotic sea. Of

particular importance are the classical resonant island chains. The decay rates of the

population of the islands, and the related accelerator modes were calculated in the

semiclassical approximation and were tested numerically. A nontrivial dependence

on Planck’s constant is found in the semiclassical limit.

3

Chapter 1

Introduction

Most of the systems which are studied experimentally and numerically are too com-

plicated to be described analytically in detail. As a result simple models are chosen

to represent the physical phenomena of interest while maintaining minimal level of

complexity [1–7]. Two dimensional, symplectic maps are an example of dynamical

systems which are very simple yet complicated enough to exhibit many interesting

properties, which are characteristic of general Hamiltonian systems. Such maps can

be generated by a one dimensional Hamiltonian with explicit time dependence. The

dynamics that is generated by such maps can be described by the three dimensional

phase space (two variables and time), but it is more appropriate to describe such dy-

namics by the stroboscopic map. An example of such a phase space is shown in Fig.

1.1(a). The phase space of such type can be separated into three important parts :

chaotic part, integrable part and the resonant island chain. Dynamics in the chaotic

part can be approximated as ergodic motion and dynamics in the regular part can be

approximated by some integrable (time independent) Hamiltonian. Such integrable

4

CHAPTER 1. INTRODUCTION 5

Hamiltonian can be calculated as some series that provide a good description of mo-

tion in the regular part of the phase space and rapidly diverges in the chaotic domain

and near the resonance. For example, in Fig. 1.1 one can see the comparison between

the phase space of the map and its integrable approximation. However, near the

resonance the effective integrable Hamiltonian is constructed by means of resonant

perturbation theory. This procedure is illustrated hereafter for the particular case of

a periodically driven one-degree-of-freedom system. The main problem that we will

consider is the decay of the wave function localized at the regular part of the phase

space as a result of transitions to the chaotic part of the phase space.

1.1 Quantum dynamics in the presence of classical

resonance

Influence of the classical resonances on the tunneling in the phase space was described

in detail by Ullmo and co-workers in the case where the tunneling occurs between

two symmetric regular islands in the phase space [8–10]. In our problem there is no

symmetry between regular islands in the phase space but the mechanism that was

introduced by Ullmo and co-workers still play first fiddle in the tunneling of wave

from the regular part of the phase space to the chaotic part. In this section this

mechanism resulting in so-called resonant coupling between different regular states

will be described.

6 CHAPTER 1. INTRODUCTION

Figure 1.1: (a) Phase space of the system of (1.81) with the negative sign, projectedon the [−π, π) × [−π, π) space . (b) The same phase space as in (a) but insteadof the regular island there is an integrable approximation of the second order. Theparametes are: k = 2.50 , τη = 1.00 , l = 1.

1.1.1 Effective Hamiltonian in the vicinity of a resonance

We write the Hamiltonian of a weakly perturbed system, after the canonical transfor-

mation to the action-angle variables of the unperturbed Hamiltonian, in the standard

form

H(I , θ, t) = H0(I) + εV (I , θ, t) (1.1)

where H0 represents the unperturbed Hamiltonian or a suitable approximation of H

obtained e.g. by standard classical perturbation theory as some series and (I , θ) are

the action-angle variables associated with H0. The perturbative term εV contains


the rest of the Hamiltonian H and has periodic property

V (I , θ, t + τ) = V (I , θ, t). (1.2)

A resonance condition arises when the frequency w = 2π/τ of the external driving V

equals a rational multiple of the internal oscillation frequency of the system - i.e.

w = Ω(Ir:s) · r

s(1.3)

with

Ωr:s ≡ dH0

dI(Ir:s) (1.4)

where r and s are coprime positive integers and Ir:s is the action at the resonance.

In Fig. 1.1 this resonance results in an island chain that is marked by an arrow. In

this case r = 4 and s = 1. In the vicinity of such r : s resonance, standard classical

perturbation theory diverges rather quickly due to small denominators. To avoid this

problem and, to get the effective integrable Hamiltonian it is convenient to perform

a canonical transformation to the frame that co-rotates with the angle variable θ on

the resonance by the canonical transformation (Eq. 17, 18) of [8, 9]:

Θ = θ − Ωr:st (1.5)

H(r:s)(I , Θ, t) = H0(I) + εV (I , θ(Θ), t)− Ωr:sI ≡ H0(I) + V (r:s)(I , Θ, t)− Ωr:sI

after which the new angle variable remains constant, under the evolution by H0 on

the r : s resonance and varies slowly in its vicinity. So, now it is convenient to apply

the adiabatic perturbation theory and to eliminate the explicit time dependence by

a canonical transformation to new slightly shifted phase space variables (I , ϑ)

I = I +∂G

∂Θ(I , Θ, t) (1.6)

ϑ = Θ +∂G

∂I(I , Θ, t) (1.7)


by the generating transformation

G(I , Θ, t) = −∫ t

0

[εV (r:s)(I , Θ, t′)− V (r:s)(I , Θ)

]dt′ (1.8)

with

V (r:s)(I , ϑ) =1

rτ

∫ rτ

0

εV (r:s)(I , ϑ, t)dt. (1.9)

Expanding V in the double Fourier series in θ and t

εV (I , θ(Θ), t) =∞∑

k,l=−∞Vk,l(I) exp[i(kθ − lwt)] (1.10)

and H0 to the second order of the Taylor series around the action Ir:s, we obtain (Eq.

30 of [8, 9])

H(r:s)(I , ϑ) = H(r:s)0 (I) + V (r:s)(I , ϑ)

= H0(Ir:s) + Vr.0(Ir:s) +(I − Ir:s)

2

2mr:s

+∞

2∑

n=1

Vr.n(Ir:s) cos(nrϑ + λn)

(1.11)

where

Vr.neiλn = Vnr,−ns. (1.12)

and

mr:s =

[d2H0

dI2(Ir:s)

]−1

. (1.13)

If the perturbation V (I , θ, t) of the integrable Hamiltonian is some analytical and

complicated function of θ (as it will be in the generic case), the magnitude of the

Fourier coefficients Vr.n decreases very rapidly (exponentially) with n. So, in the

leading order the effective time independent Hamiltonian near the resonance (omitting

the constant terms and high harmonics) reduces to (Eq. 1 of [10])

H(r:s)(I , ϑ) =(I − Ir:s)

2

2mr:s

+ 2Vr:s cos(rϑ + λ1) (1.14)


with

Vr:s ≡ Vr.1(Ir:s). (1.15)

1.1.2 Calculation of resonance parameters

The resonance parameters Ir:s, mr:s and Vr:s can be calculated from the numerically

obtained classical phase space of the system due to invariance of the phase space areas

under the canonical transformation. This fact allows us to calculate some area in the

phase space of the action-angle variables of the unperturbed Hamiltonian(I , θ

)by

calculating appropriate area in the original variables that are the variables of the

physical system before the canonical transformation to (1.1) was performed. Also,

the invariance property of the trace of monodromy matrix was used. To this end,

we calculated approximately the monodromy matrix Mr:s of a stable periodic point

and the phase space areas S+r:s and S−r:s that are enclosed by the outer and inner

separatrices of the resonance, respectively. Then we use [10]

Ir:s =1

4π

(S+

r:s + S−r:s)

(1.16)

√2mr:sVr:s =

1

16

(S+

r:s − S−r:s)

(1.17)

and √2Vr:s

mr:s

=1

r2τcos−1(Mr:s/2) (1.18)

to derive the required parameters.


1.1.3 Resonant coupling

Quantization of the Hamiltonian (1.14) is done by introducing the Hamiltonian op-

erator

H =(I − Ir:s)

2

2mr:s

+ 2Vr:s cos(rϑ + λ1), (1.19)

where I and ϑ are canonically conjugate variables i.e. the action operator is defined

by

I = −i~∂

∂ϑ. (1.20)

The system that is generated by a Hamiltonian (1.14), is naturally described in the

basis of the unperturbed states i.e. eigenstates of the Hamiltonian

H0 =(I − Ir:s)

2

2mr:s

. (1.21)

These eigenstates are the angular momentum eigenstates

〈ϑ|ψn〉 =1√2π

ei(n+1/2)ϑ (1.22)

where the quantized action [3]

In = ~(n + 1/2) (1.23)

and the appropriate (unperturbed) energy is

En =~2

2mr:s

(n− nr:s)2 (1.24)

where

nr:s ≡ Ir:s

~− 1/2 (1.25)

is not integer in general. In the basis of such states the Hamiltonian (1.14) takes the

form

H =Nr−1∑n=0

En |ψn〉〈ψn|+ Vr:s |ψn〉〈ψn+r|+ Vr:s |ψn+r〉〈ψn| (1.26)


that is a tridiagonal matrix. Of course, the size of this Hamiltonian is not an infinite

but limited by Nr - the number of the regular states localized in the regular part of

the phase space, like the island of Fig. 1.1.

1.2 Kicked accelerator

The map that was taken in this work as a demonstration has also an experimental

realization in the field of Atom Optics [11–17]. In this subsection the experimental

system that will be studied in the present work will be reviewed.

1.2.1 Atom-Field interaction

Consider the one-dimensional problem of a two-level atom moving in a standing wave

of light. The standing wave is described by the sum of two traveling waves:

E(x, t) = zE0 [cos (kLx− wLt) + cos (kLx + wLt)] (1.27)

= zE0 cos (kLx)(e−iwLt + eiwLt

)= E+ + E−

where E+ and E− are the positive- and negative-rotating components of the field,

respectively, E0 is the amplitude of either one of the two constituent travelling waves,

and wL is the laser frequency. The atomic free-evolution Hamiltonian is then given

by

HA = Ip2

2m+ ~w0 |e〉〈e| (1.28)

where the excited and ground internal atomic states are |e〉 and |g〉, respectively and

~w0 is the energy difference between the ground state and the excited state, while

I ≡ |e〉〈e|+ |g〉〈g| . (1.29)


Defining the slowly varying excited state

|e〉 ≡ eiwLt |e〉 (1.30)

the atomic free evolution Hamiltonian can be written as

HA =p2

2m− ~∆L |e〉〈e| (1.31)

where

∆L = wL − w0 (1.32)

is the detuning of the laser from the atomic resonance and the unity operator was sup-

pressed for the sake of simplicity. The atom-field interaction part of the Hamiltonian

is given (in the dipole approximation) by

HAF = −d · E (1.33)

where d is the atomic dipole operator

d= deg

(a + a†

)(1.34)

deg = 〈e|d |g〉 (1.35)

a = |g〉〈e| . (1.36)

Assuming that |wL − w0| ¿ wL +w0, we can make the rotating-wave approximation,

neglecting the ”antiresonant” terms e−iwLt |g〉〈e| and eiwLt |e〉〈g|, resulting in

HAF =~ΩR

2

(a + a†

)cos (kLx) (1.37)

where

ΩR ≡ −2 〈e| dz |g〉~

E0 (1.38)


is the maximal Rabi frequency.

Since, we assume that the detuning from the atomic resonance is large compared

to natural width of the excited state (i.e. ∆L À Γ where 1/Γ is the natural lifetime

of |e〉), we will neglect spontaneous emission and use the Schrodinger equation,

i~∂t |Ψ〉 = (HA + HAF ) |Ψ〉 (1.39)

to describe the atomic evolution. It is convenient to decompose the state vector |Ψ〉into a tensor product of internal and external (center of mass) states,

|Ψ〉 = |ψe(t)〉 |e〉+ |ψg(t)〉 |g〉 (1.40)

where the |ψi(t)〉 (i = e, g) are states in the center-of-mass space of the atom. In

the following, we will associate all time dependence of the atomic state with the

center-of-mass component of the state vector. Defining the coefficients:

ψi(x, t) = 〈x |ψi(t)〉 (1.41)

the equations of motion for these wave functions are

i~∂tψe =p2

2mψe + (

~ΩR

2cos kLx)ψg − ~∆Lψe (1.42)

i~∂tψg =p2

2mψg + (

~ΩR

2cos kLx)ψe. (1.43)

For the slow center-of-mass atomic motion we can assume

⟨p2

2m

⟩¿ ∆L~. (1.44)

We assume also that the occupation of the excited state is small:

|〈ψe |ψe〉|2 ¿ |〈ψg |ψg〉|2 . (1.45)


This enables to use so-called adiabatic approximation i.e. to set ∂tψe = p2

2mψe = 0.

This approximation then enables to eliminate ψg leading to

i~∂tψg =p2

2mψg + V0 cos (2kLx) ψg (1.46)

where we shifted the zero of the potential energy and denote

V0 =~Ω2

R

8∆L

=|〈e| dz |g〉|2 E2

0

2~∆L

. (1.47)

For detuning that is sufficiently large, nearly all the population is indeed in |g〉, so the

excited state could be eliminated. Hence, the atom obeys the Schrodinger equation

with the effective center-of-mass Hamiltonian

Heff =p2

2m+ V0 cos (2kLx) . (1.48)

We now generalize these results to the case where the two traveling waves have dif-

ferent amplitudes and frequencies. In this case, the electric field is

E(x, t) = z [E01 cos (kL1x− wL1t) + E02 cos (kL2x + wL2t)] (1.49)

= z(E01e

i(kL1x−δLt/2) + E02e−i(kL2x−δLt/2)

)e−iwLt + c.c.

where

wL = (wL1 + wL2) /2 (1.50)

is the mean laser frequency and

δL = wL1 − wL2 (1.51)

is the frequency splitting between the two travelling waves, and we assume that |δL| ¿|∆L|, with ∆L = wL−w0. Then, in the rotating-wave and the dipole approximations,

the interaction Hamiltonian becomes [11]

HAF =~2

(ΩR(x, t)a† + Ω∗

R(x, t)a)

(1.52)


where a and a† are defined as before and the time- and space-dependent Rabi fre-

quency is defined by

ΩR(x, t) ≡ −2 〈e| dz |g〉~

(E01e

i(kL1x−δLt/2) + E02e−i(kL2x−δLt/2)

)(1.53)

Writing the Schrodinger equation

i~∂tψe =p2

2mψe + (

~ΩR(x, t)

2cos kLx)ψg − ~∆Lψe (1.54)

i~∂tψg =p2

2mψg + (

~Ω∗R(x, t)

2cos kLx)ψe. (1.55)

and making adiabatic approximation we obtain the equation of motion for the ground-

state amplitude

i~∂tψg =p2

2mψg +

~2∆L

|Ω(x, t)|2 ψg (1.56)

From (1.53), we find that

|Ω(x, t)|2 =|〈e| dz |g〉|2

~2

[|E01|2 + |E02|2 + 2E01E02 cos (2kLx− δLt)]

(1.57)

where

kL = (kL1 + kL2) /2. (1.58)

Hence, we see that the atom obeys the center-of-mass Schrodinger equation, with the

effective Hamiltonian (after energy shift)

Heff =p2

2m+ V0 cos (2kLx− δLt) (1.59)

and

V0 =|〈e| dz |g〉|2 E01E02

2~∆L

. (1.60)

Thus, the motion is the same as before, except that the standing wave moves with

velocity δL/2kL.


Figure 1.2: Temporal typical profile of the pulse strength used at the experiment[12].

1.2.2 Realization of kicked accelerator

The atom-optics realization of the kicked accelerator is described by the Hamiltonian

(the positive direction of x refers to the gravitation direction)

H(p, x, t) =p2

2m−mgx + V0 cos (2kLx)

∞∑j=−∞

δ(t− jT ) (1.61)

Then, the realization of the kicked accelerator can be done experimentally by putting

the atoms in the gravitation field and setting the strength of the oscillating perturba-

tion as V0

∞∑j=−∞

δ(t− jT ) with an Acousto-Optic Modulator. Of course it is impossible

to create such temporal dependence and the real strength always has a finite width.

For example, the typical strength can be seen in Fig. 1.2 [12].

1.2.3 Experimental results

In the experiments of the Oxford group [14, 15] the experimental realization of the

Hamiltonian (1.61) was implemented and its results one can see on Fig. 1.3. In this

picture we see that a large number of atoms falls more or less with the gravitational


Figure 1.3: Experimental data that shows the number of atoms with specified mo-mentum relative to the free falling frame as the number of kicks varies (each linerepresents a specified number of kicks).

acceleration (in the laboratory frame) but there is a finite fraction of atoms that

accelerate with constant acceleration relative to gravity. This phenomenon was called

”Quantum accelerator mode”. Its theoretical explanation was given by Fishman,

Guarneri and Rebuzzini [18,19] and is briefly summarized below. In the present work

the decay rate of the such accelerator modes, will be described.


1.2.4 Transformation to dimensionless parameters

Comfortable rescaling of Hamiltonian (1.61) can be done by introducing dimensionless

variables:

x′ = 2kLx (1.62)

p′ =p

2~kL

(1.63)

t′ =4k2

L~m

t (1.64)

H ′ =m

4(kL~)2H, (1.65)

and setting Planck’s constant to one. Then the rescaled Hamiltonian takes the fol-

lowing form (dropping the primes):

H(p, x, t) =p2

2− η

τx + k cos x

∞∑j=−∞

δ(t− jτ) (1.66)

where

k =V0

~τ =

4~Tk2L

mη =

mgT

2kL~. (1.67)

The dynamics is, then, fully characterized by the dimensionless parameters k, τ , η.

1.3 Quantum Accelerator Modes

Due to the gravity, the Hamiltonian (1.66) has no spatial symmetry and then there is

no conservation of the quasimomentum. So, it is expedient to measure the momentum

in the free falling frame, namely to replace p− ητt by p, resulting in the spatial periodic

Hamiltonian

H(p, x, t) =

(p + η

τt)2

2+ k cos x

∞∑j=−∞

δ(t− jτ). (1.68)


Dynamics, generated by this Hamiltonian conserve quasimomentum that in these

units is simply the fractional part of the momentum (this Hamiltonian couples only

such eigenstates of momentum, that whose momenta differ by an integer). The quan-

tum mechanical study of the system described by (1.68) starts with decomposing the

total momentum

p = n + β (1.69)

where n and β are defined as the integer and the fractional part of p respectively.

Then, the evolution is equivalent to that of a superposition of independent kicked

rotors, each characterized by a different value of β. Such a rotor will be called the β

rotor, and the one step evolution operator is

Uβ(t) = e−ik cos(θ)e−i(τ/2)(N+β+ηt+η/2)2 (1.70)

where

θ = x mod 2π (1.71)

N = −id

dθ. (1.72)

For β = η = 0 this is the usual kicked rotor. However, in the presence of gravity

(η 6= 0), the most surprising experimental result was that an appreciable fraction of

atoms were found to accelerate (in the free falling frame) for various values of the

experimental parameters, for values of τ in intervals of appreciable size around integer

multiples of 2π. It is a quantum resonance that is robust, in contrast to the usual

quantum resonance and it has no counterpart in the classical limit of (1.68). Below

we will describe the theoretical explanation of this effect [18,19]. Assuming that value

of τ is close to the resonant value 2πl, where l is a positive integer and the kicking


strength is large, we write

τ = 2πl + ε (1.73)

k = k/ε (1.74)

with small ε. Noticing that e−ilπn2= e−ilπn, the evolution operator (1.70) takes the

form

Uβ(t) = e−(i/|ε|)k cos(θ)e−(i/|ε|)Hβ(I,t) (1.75)

where

I = |ε| N = −i |ε| d

dθ(1.76)

Hβ

(I , t

)= ± I2

2+ I

[πl + τ

(β + tη +

η

2

)](1.77)

± ≡ ε

|ε| . (1.78)

An irrelevant phase factor of the evolution operator was omitted. Treating |ε| as

the pseudo-Planck’s constant we see that (1.75) is the so-called ε-quantization of the

following classical map

It+1 = It + k sin θt+1, θt+1 = θt ± It + πl + τ(β + tη + η/2) (1.79)

Consequently ,the small |ε| asymptotics of the quantum β rotor is thus equivalent to a

quasiclassical approximation based on evolution determined by the map (1.79), called

the ε-classical dynamics . We stress that the ε-classical limit ε −→ 0 has nothing in

common with the real classical limit ~ −→ 0. Making the transformation

Jt = It ± πl ± τ(β + tη + η/2) (1.80)

we can remove the explicit time dependence to get

Jt+1 = Jt + k sin θt+1 ± τη, θt+1 = θt ± Jt mod 2π. (1.81)


Then, examining (1.80) one can see that a periodic (mod 2π) point of (1.81) (that

will sometimes be referred to as the J-frame), namely

Jt+µ = Jt + 2πν, (1.82)

for some integer µ and ν, gives rise to the linear (in time) growth of I with acceleration

(see [18])

a =2πν

µ∓ τη (1.83)

and, consequently, constant acceleration of the atoms. This motion was called the

accelerator mode.

However, this ε-classical dynamics of I and J is just an approximation to the

real (ε-quantum) dynamics. Consequently, the decay of quantum accelerator modes,

that was observed experimentally and numerically, results of tunneling by ε-quantum

dynamics from the accelerator mode (that in the ε-classical phase space looks like

the regular island in Fig. 1.1, with µ = 1 and ν = 0). To obtain the decay rate of

the quantum accelerator mode we need to solve a problem of the tunneling from the

regular part of the phase space to the chaotic sea. In the present work the tunneling

rate is estimated analytically for small |ε|. For this purpose the ε-classical dynamics

are described by a Hamiltonian of the form (1.19). The decay rate is calculated

analytically, in the semiclassical approximation where |ε| plays the role of ~. This

parameter can be changed in experiments, because this is the difference between the

kicking period τ and 2πl, with some integer l.

Chapter 2

Resonance-Assisted Tunneling

2.1 The case without degeneracy of the regular

states

In this chapter resonance-assisted tunneling from a regular island to the chaotic sea

will be studied following the method that was introduced in [8–10]. Consider the

island presented in Fig. 2.1. There are many trajectories on regular tori around the

center of the island. There is also an island chain of large area. Actually there are

other island chains that are not visible in the simulation. The assumption of our

calculation, that is discussed in detail in [8–10] is that the largest island chain domi-

nates the escape from the center of the island to the chaotic sea. Since the motion in

the island is approximately regular, one can introduce a canonical transformation to

angle action variables(I , θ

)where each torus is characterized by an action I while

θ is the conjugate angle variable. For the tori one can introduce a canonical trans-

formation to a time independent Hamiltonian that can be computed approximately.

Then one has to analyze the effect of the resonant island chain. Assuming in r time

22

CHAPTER 2. RESONANCE-ASSISTED TUNNELING 23

Figure 2.1: Phase space of the system of (1.81) with the negative sign, describedin terms of the (J, θ) variables projected on the [−π, π) × [−π, π) space. With theparametes: k = 2.50 , τη = 1.00 , l = 1.

periods a point in the chain rotates s times around the center, leading to the r : s

resonance (in the case of Fig. 2.1 r = 4, s = 1) and the corresponding chain consists

of r islands. Introducing a canonical transformation to the frame rotating with the

resonance, θ −→ θ − sr2πt = Θ and making use of the fact that Θ varies slowly, one

obtains the approximate Hamiltonian [10]

H1(I, ϑ) =(I − Ir:s)

2

2mr:s

+ 2Vr:s cos rϑ. (2.1)

The parameters Ir:s, mr:s and Vr:s are computed numerically from the phase portrait

like Fig. 2.1 of the relevant system, exploiting invariance properties of canonical

transformations using (1.16-1.18) [10].

The quantum mechanical behavior of (2.1) will be studied in the semiclassical

24 CHAPTER 2. RESONANCE-ASSISTED TUNNELING

limit. First we will ignore the second term in (2.1). In this case the quantization

condition of the action is

In =

(n +

1

2

)~ (2.2)

and the corresponding eigenfunctions are

ψn =einϑ

√2π

, (2.3)

while the eigenenergies are

En =(n− nr:s)

2

2mr:s

~2 , (2.4)

where nr:s is defined by Ir:s =(nr:s + 1

2

)~ and generally it is not integer. If we

consider the second term in (2.1) as a perturbation, it couples, in leading order, only

energies En1 and En2 such that n1 − n2 is an integer multiple of r. Assume there are

L + 1 such states that can be approximated by quantization of tori (n = 0, r, ..., Lr),

and therefore

L = Int

[(A

2π~− 1

)/r

], (2.5)

where A is the area of the regular part of the phase space and Int(x) the integer part

of x. The location of what would be the (L + 2)-th torus is in the chaotic sea. If

n = 0 is one of the states we are interested in (actually we will start often from that

state), the relevant Hamiltonian is [10]

H1 =

0 :

: :

:

:

:

0

0

r s

r s r r s

r s

Lr r s

r s

E V

V E V

V

E V

V

CHAOS

(2.6)


The block ”chaos” stands for a matrix taken from an ensemble of random matrix

theory (RMT). We will be interested in tunneling from the most internal state in the

island to the chaotic sea. For this purpose we first calculate the coupling of this state

to the state n = (L + 1)r, in the leading nonvanishing order of perturbation theory.

It is given by the Hamiltonian

H2 =

0 0 0

0

0

eff

eff

E V

V

C H A O S

(2.7)

with

Veff = Vr:s

L∏

l=1

Vr:s

E0 − Elr

(2.8)

This result can be generalized to the case where E0 is an excited state with quantum

number n0. Then L looks like:

L = Int

[(A

2π~− 1− n0

)/r

], (2.9)

and

Veff = Vr:s

L∏

l=1

Vr:s

En0 − En0+lr

. (2.10)

The assumption that we made is the standard assumption of perturbation theory,

Vr:s ¿ En0 − En0+lr (2.11)

for all l ≤ L (case without the degeneracy of the regular states). In the next section

the situation where this assumption doesn’t hold, will be described.

If Nc is the number of states in the chaotic component then H2 is an N = Nc + 1

dimensional matrix. Let χj, with j = 1..Nc be the eigenstates of the chaotic block


and let their corresponding eigenvalues be Wj. The wave function ψ(L+1)r is located

in the chaotic region in the vicinity of the regular island and does not approximate

any eigenstate of the system. Within the RMT assumption ψ(L+1)r can be expanded

in terms of the chaotic block with equal weights, namely

ψ(L+1)r =1√Nc

Nc∑j=1

e−iφjχj (2.12)

Where φj are random phases. In the basis ψ0, χ1, χ2, ..., χNc the Hamiltonian matrix

takes the form

H3 = (2.13)

where

V =Veff√

Nc

(2.14)

In the first order perturbation theory the eigenstates of H3 are:

ϕ0 = ψ0 +Nc∑j=1

V e−iφj

E0 −Wj

χj (2.15)

ϕj = χj +V eiφj

Wj − E0

ψ0

The transition probability from the most inner state of the regular island and chaotic

component assisted by the r : s resonance is :

Γr =Nc∑j=1

∣∣∣〈χj| U |ψ0〉∣∣∣2

(2.16)


Within first order perturbation theory:

Γr =Nc∑j=1

∣∣∣∣∣Nc∑

l=0

〈χj|U |ϕl〉〈ϕl|ψ0〉∣∣∣∣∣

2

= (2.17)

Nc∑j=1

∣∣∣∣V

E0 −Wj

(e−i

E0~ − e−i

Ej~

)∣∣∣∣2

=

= 2Nc∑j=1

V 2

(E0 −Wj)2

(1− cos

E0 −Wj

~

)

In the semiclassical limit where ~ −→ 0 and Nc −→ ∞ the sum can be replaced by

an integral

Γr =2V 2

~2Nc

∫dwρ(w)

1− cos (w0 − w)

(w0 − w)2 (2.18)

where

w =W

~(2.19)

w0 =E0

~

while ρ(w) is the density of states (in the normalized variable w). Therefore

Γr =2V 2

eff

~2C (2.20)

where the numerical factor C is the integral in (2.18). It depends on w0 and on the

density of states. If the wj =Wj

~ are considered as the quasienergies of a random

evolution operator, they are uniformly distributed in the interval [0, 2π). If we con-

sider eigenenergies of a GOE random matrix the probability level density satisfies

semicircle law [20]:

ρ(w) =2

π2

√1− (w − π)2

π(2.21)

For

w0 = E0/~mod 2π =n2

r:s

2mr:s

~mod 2π (2.22)


one finds

C ≡ C1 =1

2π

∫ 2π

0

dw1− cos (w0 − w)

(w0 − w)2 (2.23)

for uniform density (this integral has a long analytical expression) while

C ≡ C2 =2

π2

∫ 2π

0

dw

√1− (w − π)2

π2

1− cos (w0 − w)

(w0 − w)2 (2.24)

for the semicircle law (2.21). The constants C1 and C2 are both of order unity for

all possible values of w0 and are insignificant for the evaluation of Γr, that changes

by several orders of magnitude. Moreover, the theory that is presented here is not

accurate enough for the computation of constants of order unity. In Fig. 2.2 the

results of (2.20) are compared with numerical calculation for the map (1.81). The

relevant numerical method is described in App. B Reasonable agreement is found for

sufficiently small ~. In particular, the ”step structure” is found. A step is found when

L changes by one, resulting in an additional factor of Veff in (2.8). On the scale of the

figure, curves that corresponding to the C1 and C2 are indistinguishable (for Fig.2.2

C1 varies between 0.2257 and 0.2630 and C2 varies between 0.2521 and 0.2643), so

for the following graphs we will take one of the values of C or their average. One can

see, for some values of ~, narrow peaks in the analytic approximation for the first

excited state in the Fig. 2.3. The cause for this is that for such values of ~ regular

perturbation theory diverges due to the degeneracy of corresponding regular states.

This subject will be discussed in the next section.

2.2 Degeneracy of the regular states

At the end of the previous section, it was mentioned that the narrow peaks that can

be seen on the Fig. 2.3, are consequence of the degeneracy of two states located in the


Figure 2.2: Comparison between the numerical simulation (circles) of the system,parameters and phase space of which are presented on Fig. 2.1 and the analyticformula (2.20) (solid line).

regular domain of the phase space, resulting in large decay. Such a situation, cannot

be described by standard perturbation theory. From (2.1) we can evaluate the values

of ~ where such peaks for a state n0:

1

~n0(l)=

2n0 + rl + 1

2Ir:s

. (2.25)

However from (2.9) follows that maximal l is L so,

l ≤ Int

[(A

2π~− 1− n0

)/r

](2.26)

Then, for such values of ~ and near such values, we cannot use standard perturbation

theory and if we use it, anyway, we will get such non physical divergencies as shown

on Fig. 2.3. As can be seen from Fig. 2.3, for n0 = 0, there is no value of ~ that

satisfies both (2.25) and (2.26) for the values of A and Ir:s, that are used in Fig. 2.2

(that are 5.2 and 0.43, relatively), for 1~ < 22.


Figure 2.3: Comparison between the numerical calculation of the system, parametersand phase space of which are shown on Fig. 2.1 and the analytic formula (2.20) forvarious states with quantum number n0. Bold line is the analytic approximation forthe n0 = 0 (as on Fig. 2.2), thin line is the analytical approximation for n0 = 1.Squares (dots) are the numerical values for the lowest (before lowest) escape rate,and crosses are for the bigger escape rates.

In order to obtain the analytical approximation for the height of the peaks and

behavior near the peak, we need to diagonalize the regular part of the Hamiltonian

(2.6), namely:

HR =

0 :

:

:

:

0

0

r s

r s r

r s

r s Lr

E V

V E

V

V E

. (2.27)

The quantity, that we need to calculate after the diagonalization, is the L-th entry

of the eigenvector that has the maximal n0-th entry. If we do it numerically we get

the approximation for the escape rate also for the systems where there is degeneracy


between two regular states, and results of such calculations are shown on Fig. 2.4, for

the first excited state (n0 = 1). The way to check that there is a peak at the point

where the theory predicts is described below. At the point where the peak is located

for some n0, its escape rate can be higher than the one of the higher excited state.

Then it is impossible to find the escape rate of n0 based on the monotonic dependence

between the quantum number of the state and its escape rate. In other words, for

the peak one can find contributions with all escape rates, but one cannot know which

escape rate is related to which state. Due to this, to identity the peak resulting from

the degeneracy, we will plot the escape rates as a function of the quasienergy for each

regular state in the system. We will see that at the point where the theory predicts,

the two states will have the same quasienergy and very close escape rates. These plots

can be compared with the plots of the eigenstate that corresponds to the appropriate

escape rate and check visually that this is a correct state. For example for the first

ten plots in Fig. 2.4 we can see that the eigenstate that corresponds to the escape

rate that is marked in Fig. 2.5 by ellipse really is the first excited eigenstate (the

identification can be done by the evaluating the number of maxima inside the island,

since the states that are approximately the eigenstates of a harmonic oscillator). From

Fig. 2.4 one can see that in the place predicted by the approximation (1/~ ' 8.2) the

wave function suddenly changes the structure and gets sizeable values on the more

external torus (and consequently sizeable projection on the higher excited state).From

Fig. 2.5 one can see that in the place predicted by the approximation (2.25) (with

Ir:s = 0.43, n0 = 1, r = 4, l = 1, resulting in 1/~ ' 8.2) the first excited state of

the unperturbed Hamiltonian is really degenerate with the fifth excited state. It has

almost the same escape rate as of the fifth excited state and also the real parts of the

eigenvalues are degenerate.


Figure 2.4: First ten plots in this figure describe the part that is localized on theregular island of first excited (n0 = 1 ) eigenstate with the different values of 1/~ thatare shown in the center of each graph. The axes of these plots are Log10

(|ψ(J)|2)and J as the y and x axes respectively. The last graph describes the escape rate ofthis state as a function of 1/~. The line with the points is the numerical data andthe solid line is the approximation obtained from the diagonalization of (2.27).


Figure 2.5: The escape rates (log10 (Γ)) of all the states that are localized on theregular island against the real part of the appropriate eigenvalue of the truncatedevolution operator (if the eigenvalues of the evolution operator are eiw−Γ, w is thehorizontal axis and log10 (Γ) is the vertical). On top of each graph one can see thevalue of 1/~. The ellipse and the rectangle on each graph mark the first excited stateand the fifth one respectively.

Chapter 3

Direct Tunneling

In this chapter the contribution of the direct tunneling from the most inner state in

the regular island to the chaotic sea will be calculated. It is expected to dominate

when the resonance assisted tunneling is not effective. This will happen if ~ is larger

than the size of an island in the chain that forms the resonance, and the resonance

cannot be resolved by the quantum mechanical system. In this situation we approx-

imate eigenstates of the regular region by the ones of a harmonic oscillator. This

approximation is expected to be best for states with large amplitude in the center

of the island, since there the harmonic approximation is best. The overlap of these

states with the states of the chaotic region, that are expressed as superpositions of

eigenstates of harmonic oscillators is calculated. This approximation was tested for

systems with similar structure of phase space and good statistical agreement was

found [21]. It should be noted, however, that this is not a controlled approximation

(the errors are not controlled by a small parameter).

The calculation follows Podolski and Narimanov [21] that applied a perturbative

34

CHAPTER 3. DIRECT TUNNELING 35

method that was introduced by Bardeen [22]. First, the effective coupling Veff be-

tween the most inner state of the regular island and the chaotic sea will calculated.

For this purpose we apply a canonical transformation that transforms the island that

is studied to a circular one that is presented in 3.1. The dynamics in the island is

described by the Hamiltonian of the unit Harmonic oscillator:

H(p.q) =p2

2+

q2

2(3.1)

The area of the island in Fig. 3.1 is equal to the one of the original island since the

transformation is canonical. Outside the island (grey region in Fig. 3.1) the motion

is chaotic.

Figure 3.1: Schematic phase space of the system after transformation to the co-ordinates for which the regular island looks like the phase space of unit Harmonicoscillator (circles).

The eigenstates in the island are the eigenstates of the Harmonic oscillator and

the corresponding energies are

E0,m = ~(

m +1

2

)(3.2)

If r is the radius of the island, its area is A = πr2 and the maximal m for states in

the island is:

M ' A

2π~− 1 (3.3)

36 CHAPTER 3. DIRECT TUNNELING

The main idea of Podolski and Narimanov [21] was to cover the chaotic region with

islands like the one shown in Fig. 3.1, transformed by rα in phase space from the

center of the original island, so that they do not overlap, as shown in Fig. 3.2.

0α =

Figure 3.2: Schematic phase space obtained by by shifts of the circle regular islandof Fig. 3.1.

On an island α, centered at (pα, qα) one defines wave functions Ψα,m that are just

Ψ0,m with p and q replaced with p + pα and q + qα. For α 6= 0 these wave functions

do not approximate any of the wave functions of the original system. Since

〈Ψα,m1|Ψα,m2〉 = δm1,m2 (3.4)

and in the semiclassical limit the phase space overlap of functions with different values

of α is small, to a good approximation, a chaotic state χj can be expanded in terms

of the Ψα,m with α 6= 0 :

|χj〉 ' 1√Nc

∑α>0

M∑m=0

eiβjα,m |Ψα,m〉 (3.5)

where Nc is the number of states in the chaotic component, the phases βjα,m are

assumed to be random. The coupling matrix element between a state m0 in the

regular island α = 0 and a chaotic state is:

Vm0,j = 〈Ψ0,m0|H |χj〉 =E0,m0√

Nc

∑α>0

M∑m=0

eiβjα,m 〈Ψ0,m0|Ψα,m〉 (3.6)

Let us assume for simplicity that m0 = 0. Then, the variance of the coupling is

⟨|V0,j|2⟩

j'

∑α>0

M∑m=0

∣∣∣Vα,m

∣∣∣2

(3.7)

CHAPTER 3. DIRECT TUNNELING 37

with [23]

∣∣∣Vα,m

∣∣∣2

=

∣∣∣∣E0,0√

Nc

∣∣∣∣2

|〈Ψ0,0 |Ψα,m〉|2 = (3.8)

π

2Nc~

∫ ∞

−∞dp

∫ ∞

−∞dqWm,α(p, q)W0,0(p, q)

where 〈...〉j denotes average over the various chaotic states |χj〉. As a result of the ran-

domness of the phases βjα,m only terms with the same α and m survive the averaging

resulting in (3.7). The Wigner function of the unit Harmonic oscillator is [23]

Wm,α(p, q) =(−1)m

π~m!Lm(

2

~[(p− pα)2 + (q − qα)2])e−

1~ [(p−pα)2+(q−qα)2] (3.9)

and Lm is the m-th Laguerre polynomial [24]:

Lm(x) = (−1)m

m∑s=0

(−1)m−s xm−s

(m− s)!

(m!

s!

)2

(3.10)

The sum (3.7) is calculated in the Appendix A. The result is given by

⟨|V0,j|2⟩

j' 3~2

2Nc

Γ(M + 1, a)

Γ(M + 1)(3.11)

where

a =2A

~π(3.12)

and M is given by (3.3).Here this variance plays the role of V 2 in the previous chapter.

Consequently

V 2eff = Nc

⟨|V0,j|2⟩

j(3.13)

and the transition rate is given by the relation (2.20) of the previous chapter, leading

to

Γd = 3CΓ(M + 1, a)

Γ(M + 1)(3.14)

with C of the previous chapter. In Fig. 3.3 this result is tested numerically.

As we can see from this figure, for large values of ~ (larger than 1/3) this expression

gives the right order of magnitude of the escape rate and comparing with Fig. 2.2

we see that this mechanism is dominant for large values of ~ . But for small values

of ~ this expression is not justified, but as shown in Fig. 3.4, in this, specific case it

apparently gives approximately the right order of magnitude.

38 CHAPTER 3. DIRECT TUNNELING

Figure 3.3: The comparison between the numerical simulation of the tunnelingpresented in Fig. 2.2 (circles) with the analytical formula (3.14) (solid line).

Figure 3.4: Extrapolation of the formula (3.14) to the domain where the resonantassisted mechanism is dominant.

Chapter 4

Summary and discussion

In this work an analytical approximation of the escape rate from a regular island in

the phase space to the chaotic sea was developed. This problem is motivated by the

problem of decay of the quantum accelerator modes in atom optics. Statistical ap-

proach to the problem of related tunneling was applied by Narimanov and Podolsky

[21]. A more detailed approach that was proposed by Ullmo et al. [8–10] takes into ac-

count also the influence of the classical resonance in this system and the consequences

of both approaches were described in the present work.

We have presented a semiclassical picture of tunneling from the regular island to

the chaotic sea. Classical, nonlinear resonances are identified to provide a key coupling

mechanism between the invariant tori (quantum number of them differs by the order

of resonance) due to the modulation of the phase space structure in an analogous

way as in the generalized pendulum in the vicinity of every such classical resonance.

Via the succession of such couplings each regular state is connected to the chaotic

states and tunneling to the chaotic sea takes place. An analytical approximation of

the escape rate, based on such a mechanism reveals the step structure of the escape

39

40 CHAPTER 4. SUMMARY AND DISCUSSION

rate (see Fig 2.2) and the extremely large escape rates for some parameters (obtained

due to the degeneracy for these parameters as demonstrated on Fig 2.4).

Also, to test the validity of such analytical approximations, numerical simulations

were performed and discussed. The comparison between the exact numerical results

and the analytical approximation demonstrated that the analytical approximation

based on the mechanism of resonance-assisted tunneling is very good and reproduces

the most important features. The method was applied to the exploration of the decay

of accelerator modes that were found for kicked cold atoms. The relevant phase space

is the ε-classical one.

In the present work the case with only one significant classical resonance was

studied and the case with two or more significant resonances in the regular island was

not explored. It may produce some interesting phenomena.

In the present work the evolution of the atomic state with definite quasimomentum

was explored. What happens when the initial state is some coherent superposition

with various quasimomenta? This question is relevant since such states can be pre-

pared experimentally.

How can one describe the case where interactions between the atoms are impor-

tant?

Another interesting question that can be asked is: For which parameters the step

structure and/or the effect of the degeneracy can be tested experimentally? The

main problem is that in experiments one cannot kick the atoms ∼ 1/Γ times if Γ is

very small because of the finite width of the kicks. Then, to solve this problem it is

necessary to find the parameters for which at least the first step (and the first peak

because of degeneracy) occurs when the escape rate is as large as possible.

Appendix A

Derivation of formula (3.11)

In this Appendix we will derive (3.11). But, firstly we will evaluate∣∣∣Vα,m

∣∣∣2

. Substi-

tuting (3.9) in (3.8), changing variables p − pα −→ p, q − qα −→ q and substituting

E0,0 = ~2, we get:

∣∣∣Vα,m

∣∣∣2

=

∣∣∣∣~

2√

Nc

∣∣∣∣2

2(−1)m

π~m!× (A.1)

×∫ ∞

−∞dp

∫ ∞

−∞dqLm(

2

~[p2 + q2])e−

1~ (p2+q2)e−

1~ [(p+pα)2+(q+qα)2]

It is convenient to introduce polar coordinates :

p = r sin ϕ (A.2)

q = r cos ϕ

In terms of these we get

∣∣∣Vα,m

∣∣∣2

=~2

2Nc

(−1)m

π~m!× (A.3)

×∫ ∞

0

rdr

∫ 2π

0

dϕLm(2

~r2)e−

1~ r2

e−1~ r2

e−1~ rαe−

1~2rrα cos(ϕ−ϕα)

41

42 APPENDIX A. DERIVATION OF FORMULA (3.11)

First we perform the integral over ϕ by expansion of the last exponent in the (A.3)

in a Taylor series leading to:

∣∣∣Vα,m

∣∣∣2

=~2

4Nc

(−1)m

π~m!e−

1~ r2

α

∫ ∞

0

dr2Lm(2

~r2)e−

2~ r2× (A.4)

×∞∑

k=0

(2rrα/~)2k

(2k)!

2√

πΓ(12

+ k)

k!

and after substitution

u =2

~r2 (A.5)

uα =2

~r2α

we get:

∣∣∣Vα,m

∣∣∣2

=(−1)m~2

4m!Nc

√π

e−uα2

∞∑

k=m

ukα

(2k)!

Γ(12

+ k)

k!

∫ ∞

0

duLm(u)e−uuk (A.6)

where we used the identity [24]:

∫ ∞

0

xpe−xLm(x)dx = 0 for p < m. (A.7)

Now we substitute the expansion of the Laguerre polynomial (3.10) and get:

∣∣∣Vα,m

∣∣∣2

=~2

4m!Nc

√π

e−uα2

∞∑

k=m

ukα

(2k)!

Γ(12

+ k)

k!× (A.8)

×∫ ∞

0

e−uukdu

m∑s=0

(−1)m−s us

(m− s)!

(m!

s!

)2

At this stage we can perform the integration over u, to get:

∣∣∣Vα,m

∣∣∣2

=~2

4m!Nc

√π

e−uα2

∞∑

k=m

ukα

(2k)!

Γ(12

+ k)

k!× (A.9)

×m∑

s=0

(−1)m−s

(m− s)!

(m!

s!

)2

Γ(k + s + 1)

APPENDIX A. DERIVATION OF FORMULA (3.11) 43

The second sum is performed with the help of Mathematica 5.2 [25], using the fact

that k ≥ m and noticing that :

limε−→0

Γ(m− k + ε)

Γ(−k + ε)=

(−1)mk!

(k −m)!(A.10)

for natural m ≤ k :

m∑s=0

(−1)m−s

(m− s)!

(m!

s!

)2

Γ(k + s + 1) =(−1)mk!Γ(m− k)

Γ(−k)=

(k!)2

(k −m)!(A.11)

Substituting this in (A.9) and using again Mathematica 5.2 one finds :

∣∣∣Vα,m

∣∣∣2

=~2

4m!Nc

√π

e−uα2

∞∑

k=m

ukα

(2k)!Γ(

1

2+ k)

k!

(k −m)!=~2

4Nc

e−uα4

(uα

4)m

m!(A.12)

Also the sum over m in (3.7) is performed with help of Mathematica 5.2 and gives :

⟨|V0,j|2⟩

j' ~2

4Nc

∑α>0

M∑m=0

e−uα4

(uα

4)m

m!=~2

4Nc

∑α>0

Γ(M + 1, uα

4)

Γ(M + 1). (A.13)

If r is the radius of regular island, we will neglect all the terms in this sum at which

rα > 2r. From Fig. 3.2 we can see that the number of contributing terms is approxi-

mately 6 and their contributions are similar. The area of an island is A = πr2. Hence

we consider term with rα = 2r. For such terms - uα = 2~r

2α = 8A

π~ ≡ 4a and

⟨|V0,j|2⟩

j' 3~2

2Nc

Γ(M + 1, a)

Γ(M + 1)(A.14)

and we get the formula (3.11). This value is the variance of the matrix element that

connects the most inner quasimode with the chaotic sea.

Appendix B

Numerical calculations

In this appendix we will describe two ways to obtain the numerical values of the

escape rate for different values of ~. Similar methods were implemented by Fishman,

Guarneri and Rebuzzini [18,26].

B.1 Dynamic simulation

Conceptually, this is the simplest method but its implementation is more difficult

for technical reasons like computer memory and time that the simulation runs. The

initial state is a Gaussian state centered in the center of the regular island as shown

in the most left graph in Fig. B.1 and B.2. Then we evolve this state (the evaluated

state can be seen in the rest graphs on Fig. B.1 and B.2) with the one-kick-evolution

operator (as will be described later) and after each kick we compute the projection of

the resulting state on the eigenstates of I of (1.79) that intersect the regular island

(forming the observation window).

In the ”I frame” (namely the map (1.79)) the island is ”falling”, but we know

44

APPENDIX B. NUMERICAL CALCULATIONS 45

Figure B.1: Wave functions after the different number of kicks (t). From left to rigth- t = 0, 10, 20, 30. The evolution was performed for ~ = 0.1 and classical parametersas on Fig. 2.1.

its velocity at each kick, so at each kick we know the eigenstates of I that ”cover”

the island. So, if our initial state is some superposition of the regular states in the

island, its projection on the window will decay with time as some sum of decaying

exponentials with different rates and coefficients. Then, to find the slowest decay

rate we have to wait a sufficiently long time at which only the state with the lowest

rate will survive and this is the escape rate of the most inner state of the regular

island. An example of such a graph we can see in Fig. B.3 and Fig. B.4. But, when

~ becomes very small, there are many states that ”live” in the regular island (the

number of the regular states and their escape rates can be seen in the Fig. 2.3, that

was obtained by the second method, as the function of ~) and consequently there are

many exponents with decay rates that are very close.

46 APPENDIX B. NUMERICAL CALCULATIONS

Figure B.2: Husimi plots of the wave functions presented in Fig. B.1.

For the map (1.79) the one-kick-evolution operator in the ”I frame” looks like:

U(t) = exp

(− i

~k cos θ

)exp

(− i

~

± I2

2+ I [πl + τ(β + tη + η/2)]

)≡ U2U1

(B.1)

where t is the number of the kick (a discrete variable). So, this operator is the

multiplication of the two operators, one of them depends only on I and the second

only on θ. Then, the more natural way to perform the evolution of the wave packet

by the direct and inverse (Fast) Fourier Transform (FFT,iFFT), i.e. to start with

the function in the I representation, to act in the trivial way with U1, to turn the

state by iFFT to the θ representation, to act in the trivial way with U2, to get back

I representation with FFT and so on.

B.2 Truncation of the one-kick-evolution operator

In this method we will use the fact that the real eigenvectors of our system in the ”J

frame” (quantization of the map (1.81)) are not normalizable and they even diverge

at infinity due to the ”gravity” (an eigenvector of the ”truncated” evolution operator


Figure B.3: (a) The survival probability of the wave function projected on thewindow that covers the regular island for the parameters of Fig. B.1. The initialwave function is a Gaussian state shown on the left plots in Fig. B.1 and Fig. B.2.(b) Local slope of the curve in (a) (blue) and the smooth (by averaging on 100 points)of this slope (black). In this graph, the decay rate (slope of the curve that is shown in(b)) doesn’t approach the value obtained by the truncation method (log10 Γ0 = −7.5)even after 10000 kicks, due to the other decay rates, the slowest of them is expectedto be log10 Γ1 = −6.6. Consequently, the required time for resolution is ∼ 107 steps.

is shown on Fig. B.5.

Schematically we can consider our system as one described by a Wannier-Stark

potential [27] in which each local minimum corresponds to the regular island and

the distance between two local minima in the J - representation is 2π (see (1.81)).

The states that are localized on the regular island are metastable states with complex

eigenvalues of the Hamiltonian and consequently result in non unitary evolution. The

imaginary part of the eigenvalue (and hence the escape rate of the metastable state)

can be found numerically by truncation the matrix of the evolution operator in the

”J - frame” (that originally is infinite and is discrete if τη/~ takes a rational value).

In Fig. B.6 it is demonstrated that the escape rate that is found in this way is

independent of the cutoff.


Figure B.4: Same as Fig. B.3 but with ~ = 0.2. Here, the decay rate (slope ofthe curve that is shown on (b)) approaches the value obtained in the second way(log10 Γ0 = −5.184).

In Fig. B.7 we can see the consistency of these two methods. From this graph,

one can see that the values obtained by the dynamic simulation approach the values,

obtained by the truncation of the evolution operator, as the time (and consequently

the basis size) of the simulation increases. The reason for this is, that at large time,

only the state with the minimal escape rate survives. For very small values of ~,

the time required for evaluating the lowest escape rate is huge (if our initial state

is superposition of all regular eigenstates in the regular island), therefore, for small

values of ~ it is more practical to use the second method.


Figure B.5: The eigenvector of the truncated one-kick-evolution operator of themap (1.81) with ~ = 0.1 and the classical parameters are as on Fig. 2.1. The size ofthe basis (cutoff) is (a) 504 and (b) 756.

Figure B.6: Evaluated by the second method, the escape rate of the ground state,as a function of the cutoff. The parameters are the same as in Fig. B.3. One can seethat the escape rate is independent of the cutoff.


Figure B.7: The comparison between the values of Γ obtained by the two methodsas a function of ~. Crosses are the values that were obtained by the second method.Circles, are the values that were obtained by the first method with different numberof kicks - 2000, 3000, 4000, 8000 corresponding to the size of the each circle (fromsmall to large).

References

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[4] A.J. Lichtenberg and M.A. Liberman, Regular and Stochastic Motion, hard-

cover ed. (Springer-Verlag, New York, 1982).

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[6] S. Fishman, in Proceedings of the International School of Physics Enrico

Fermi: Varenna Course CXIX, edited by G. Casati, I. Guarneri, and U.

Smilansky (North-Holland, Amsterdam, 1993), p. 187.

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bridge, 2002).

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51

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REFERENCES 53

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onpiiy l`kin

miihpeew dv`d ipte` ly dkirc

xwgn lr xeaig

x`ez zlawl zeyixcd ly iwlg ielin myl

dwifita mircnl xhqibn

onpiiy l`kin

l`xyil ibelepkh oekn — oeipkhd hpql ybed

2005 uxn dtig d"qyz '` xc`

onyit lèny 'text zkxcda dyrp xwgn lr xeaig

dwifitl dhlewt

dcez zxkd

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.aetleaixw

dwifitl daxpin fkxnl ,l`xyi-a"dx` rcnl zine`l ecd oxwl dcen ip`

dkinzd lr rcnl zine`ld oxwle zeakxen zekxrn ly zix`pil `l

izenlzyda daicpd zitqkd

dpll zycwen dceard

mipipr okez

1 milnq zniyx

3 zilbpà xivwz

4 èan 1

5 . . . . . . . . . . . . . . . . . . ziqlw dcedz zekgepa zihpeewd winpic 1.1

6 . . . . . . . . . . . . . . . ziqlw dcedz cil iaihwt` oìpehlind 1.1.1

9 . . . . . . . . . . . . . . . . . . . . dcedzd ly mixhnxt ly aeyig 1.1.2

10 . . . . . . . . . . . . . . . . . . . . . . ziqlw dcedz zxfra ceniv 1.1.3

11 . . . . . . . . . . . . . . . . . . . . . . . . . . . caek dcya oaxecn wiwlg 1.2

11 . . . . . . . . . . . . . izpbnexhwl` dcyl meh` oia divw`xhpi` 1.2.1

16 . . . . . . . . . . . . . . . . . caek dcya oaxecn wiwlg ly yenin 1.2.2

16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zeipeiqip ze`vez 1.2.3

18 . . . . . . . . . . . . . . . . . . . zecigi ixqg zehpicxeèwl xarn 1.2.4

18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . miihpeew dve`z ipte` 1.3

22 ziqlw dcedz zxfra xedpn 2

22 . . . . . . . . . . . . . . . . . . . . miixlebxd miavnd ly oeeip `ll dxwn 2.1

28 . . . . . . . . . . . . . . . . . . . . . . . . . . . miixlebxd miavnd ly oeeip 2.2

c

d mipipr okez

34 xiyi xedpn 3

39 oeice mekiq 4

41 (3.11) dgqep ly gezt `

44 mixnep miaeyig a

44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zinpic divleniq a.1

46 . . . . . . . . . . . . . . . . . . . . . . . . . divelea`d xehxte` ly jezig a.2

48 zexewn zniyx

g xivwz

mixei` zniyx

6 . . . . . . . iliaxbhpi`d aexiwd oial izin`d df`td agxn oia dèeyd 1.1

16 . . . . . . . . . . . . . . . . . . . oeiqipa oeaxicd znver ly iqetih litext 1.2

17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . zeipeiqip ze`vez 1.3

23 . . . . . . . . . . . . . . . . . . . . onfa ielz `ld ietind ly df`td agxn 2.1

29 . . . . . ceqid avnl zihilp`d dgqepde zeixnepd ze`vezd oia dèeyd 2.2

30 . . . . . xxern avnl zihilp`d dgqepd oial ixnepd aeyigd oia dèeyd 2.3

32 wplt reaw ly mipey mikxrl divelea`d xehxte` ly zeinvr zeivwpet 2.4

mikxrl mixlebxd miavnd lk ly dibxpìfeew ly divwpetk dkircd mcwn 2.5

33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . wplt reaw ly mipey

35 zeycgd zehpicxeèw divnxetqpxh xg`l df`td agxn ly iznkq xeiv 3.1

36 . . . . milebird ly ly zeffdd xg`l zlawznd df`td agxn ly dnkq 3.2

38 . . . . . . . . . . . . . zihilp`d dgqepde zeixnepd ze`vezd oia dèeyd 3.3

38 . . . . . . . . . . . . . . . . . . . . . zizilp`d dgqepd ly divletxhqw` 3.4

45 . . . . . . . . . . . . . . . . . . . . . . . dpey oeaxic onf xg`l lb zivwpet B.1

46 . . . . . . . . . . . . dpey oeaxic onf xg`l lbd zeivwpet ly iniqeg bevi B.2

47 . . . . . . . . . . . . . . . . . . . . . . . . . ixlebxd i`d ly qelk` zkirc B.3

48 . . . . . . . . . . . . . . . . . . . . dpey wplt reawl j` ,mcewd sxbd enk B.4

e

f mixei` zniyx

49 . . . . . . . . . . . . . . . . . . . . . . divelea`d xehxte` ly invr xehwe B.5

49 . . . . jezgd divelea` xehxte` ly lcebd ly divwpetk dkircd mcwn B.6

50 . . . . . . . . . . . . . . . . . . . . . . . . zeixnep zehiy izy oia d`eeyd B.7

xivwz

ixyt` didiy ickn zeakxen opid dcarna zeccnp xy` zeilwiqitd zertezd ziaxn

.zepey zertez x`zl ick miheyt milcena miynzyn ,okl .zihilp` dxeva ox`zl

mre mdizepekz z` aygl ozip didiy ick witqn miheyt zeidl mikixv dl` milcen

-iqlw miietin .zxwgpd zilwiqitd drtezd z` x`zl zpn lr witqn mikaeqn z`f

zeakxen j` ,zeheyt ody el` zekxrn beql dnbec mpid micnin cge miihwltniq ,mi

ozip dl`k miietin .zeizin` zekxrn ly zeaeyg zeilwiqit zepekz x`zl ick witqn

mietin ly dwinpic .onfa zyxetn zelze zg` yteg zbxc mr oìpehlind ici-lr x`zl

ici-lr dze` bviil xyt` j` ,icnin zlz df`t agxn ici-lr ,oexwira ,zxèzn dl`k

itewqeaexhq ietin ly df`td agxn .(Poincare) dxwpèt jzg e` ,itewqeaexhq ietin

-inqd) ihpeewd xeìzd .mihe`k mixef`ne miixlebx mixef`n akxen llk jxca df beqn

dwipknd ly miipey`xd minidn .mixwind ipya dpey epid efd dwinpicd ly (iqlw

dwifitd oia izeki` oteà zelicany zehlead zertezdn zg`k xked xedpn zihpeewd

-eà e`) dibxp`d meqgn z` xearl wiwlg ly zexyt` .zihpeewd dwifitl ziqlwd

,bviin (ziqlw xeq`d megz zxne` z`f ,df`td agxna in`pic meqgn illk xzei ot

-ifita miax mineyiil dliaene zihpeewd dxezd ly zeniyxnd zepwqnd zg` ,oaenk

drtez èd xedpndy zexnl ,la` .itewqefnd rcna mb enk zixlewlene zineh` dw

jildz .mi`znd iqlwd df`td agxn ly dxeva ce`n ielz ely izenk xeìz ,zihpeew

jk ,zihxwqic dixhniq oda yiy zekxrna ,lynl .zepey zekxrna ygxzn xedpnd

g

h xivwz

-xna miixhniq mixefà minwenny miavnd oia xveeidl zeleki (Rabi) iax zecepzy

ihpè ixhniqd mipte`d oia levitl ziqgi ìd xedpnd ly zeiaihwt`d .df`td ag

svxl ceniv mr iliahqhn avn zlleky zkxrn ef zxg` zexyt` .mini`znd ixhniq

iteq miig onf ly dxeva f` dlbzn xedpnd jildz .`tl` zpixw ly dxwna lynl enk

ef dceara .mi`znd il`xhwtqd qppefxd ly iteq agexa lewy oteà e` avnd ly

ipyl wlegn eply iqlwd df`td agxn eay dxwnd ,ipyd dxwnl epnvr z` liabp epgp`

oiiprnd xedpnd jildze .(ihe`k mi) ihe`kd xef`de (ixlebxd i`d) ixlebxd xef`d :miwlg

oteà ixlebxd i`d z` x`zl ick .ihe`kd mil ixlebxd i`dn xedpn zexyt` èd epze`

x`zl ickae (EBK) xlwe oèlixa oiihypi` ly ziqlwinq divfihpeewa ynzyp ihpeew

xeyrd zligzn .zei`xw`d zevixhnd zxeza ynzyp ihpeew oteà ihe`kd xef`d z`

dziid xzeia zeriztnd ze`vezd zg` .mixw mineh` mr miax mieqip eyrp mcewd

meifv ineh` mday miieqipa elbzd dl`d mipte`d .mihpeew dv`d ipte` ly meiwd

zivwpet enk onfa zbdpzny dcehiltn` mr cner ihpbnexhwl` lb ici-lr epaxec mixw

ly mieqn wlg yi minrtly dlbzd dl`k miieqipa .zixefgn (Dirac) w`xic ly dzlc

z`fd drtezd .dciakd zve`z mr iyteg zltepy zkxrnl ziqgi mivi`nd mineh`d

(Fishman, Guarneri and ipifeaxe ixpxèb ,onyit ly dceara ihxeìz oteà dxaqed

zecewp ly meiwl xeyw dl`d mipte`d ly meiwy e`xd md ef dceara . Rebuzzini)

dv`d ote`n mineh`d ly dgixad avwe mieqn iqlw ietina zeaivi (mod 2π) zay

iqlw ietinn xvepy) df`td agxna ixlebxd xef`n wiwlg ly dgixa avwl deey dfk

iaihwt` (Planck) wplt reaw mr ixlebxd xef`d z` siwny ihe`kd mil (lirl xkfend

-xa ,w`b`ly ,enle` ly zeceara zepexg`d mipya .izin` wplt reawl xyw el oi`y

dpany dcaerd dzdef (Ullmo, Schlagheck, Brodier and Eltschka) dwyhlè xìce

ly cenivd lr dwfg dxeva ritydl leki zkxrnd ly df`td agxn ly ilìaixh `l

ixlebxd xef`dn xedpnd lr okle df`td agxn ly ixlebxd xefà minwenny miavnd

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xivwz i

xef`d ly dxevdn dwfg dxeva rtyen zeidl leki ely izenkd xeìzd ,ihpeew oihelgl

zeiqlwd (miqppefx) zecedzd ly meiw .zkxrnd ly iqlwd df`td agxn ly ixlebxd

-iy zbven zigkepd dceara . dcedz zxfra xedpn `xwpd xedpnd ly sqep mfipkn xxern

mfipknd lr zqqeand ihe`kd mil ixlebxd i`dn xedpnd mcwn ly ihilp` aexwl dh

lecibl znxeb miixlebxd miavnd ly oeeipd ly drtezd ik `vnp ok-enk .lirl xaqedy

-ibdn d`vezk mixhnxtd ly minieqn mikxr xear xedpnd mcwn ly izernyn ce`n

bvedy mfipknd xèzi ,dfd xeìzl sqepa .mipeepnd miavnd oia ceniva izernynd lec

lk `ly dcaerd lr qqeany ,(Narimanov and Podolskiy) iwqlecete aepnixp ici-lr

mb mieqn "apf" dl yi dl` ,ixlebxd xefà znwenn ixlebxd avnd ly lbd zivwpet

znerl ,dfd mfipknd lr qqeand ihilp`d aexiwd .df`td agxn ly ihe`kd xefà

xy`ke i`d fkxna minwenn miavn xear xedpnd mcwnl ieaip wx wtql leki ,z`f

ik ,miihilp` ixnbl aygidl mileki `l dl`d miaexiwd ipy .ziqgi lecb wplt reaw

-wzd dfd rcind .zkxrnd ly iqlwd df`td agxn lr mieqn rcin lr miqqean md

zihilp` ayegn xedpnd .dni`znd ziqlwd dwinpicd ly zixnep divleniq ici-lr la

jxevl miixnep miaeyig mb eyrp ef dceara .zeaxewnd zeiqlwd ze`vezd jnq lr

mcwn ly ixnep aeyigl dgzety dhiyd .zeaxewnd zeihilp`d ze`vezd mr dèeyd

dhiy ly zihpeew dwipknl oey`xd yeniyd ìd ,dvixhnd lceb zrihw ici-lr dkircd

ici-lr dyrp yeniyd) zeiqlw divlxew zeivwpet ly dkircd aeyig jxevl dgzety ef

elawzdy ze`vezd .(eply xwgnl mitzey mde dnqxet `ly dceara ezveawe ixpxèb

-ny zeiqlw zecedz ly meiwa dax dcina ielz xedpnd mcwny ,ok` ,ze`xn ef dceara

(dcedzd ly xcqa lcaip odly ihpeewd xtqndy) zeihpìxapi`d zerahd z` zecnv

-na enk dxev dzeà dcedzd zaxwa df`td agxnd ly (divlecen) oepeekd ly d`vezk

mixxern xzei miixlebx miavnl cnvip ixlebx avn dl`k miceniv jxc .zllkend zlheh

-ilp`d aexiwd .ygxzn ihe`kd xef`l xedpnd jke miihe`kd miavnl cnvip èd mkxce

xedpnd mcwn ly zebxcnd zxev z` syeg dfd mfipknd lr qqeany xedpnd mcwnl ih

k xivwz

minwennd miixlebxd miavnd xtqn ly zebxcnd zxevn raep) wplt reaw ly divwpetk

mikxr lawn xedpnd mcwn mday mixef` elbzp .(wplt reaw ly divwpetk ixlebxd ià

ly dèeyd .(wfgd cenivl mxebd) miixlebxd miavnd ly oeeipn d`vezk cgeina milecb

ixnep oteà elawzdy mikxrd mr ihilp` oteà elawzdy xedpnd mcwn ly mikxrd

dpekpd zebdpzdd z` d`xn df aexw ,z`fn dxzie ,aeh èd ihilp`d aexiwdy d`xn

xedpnd dpey`xl ayeg ef dceara ,mekiql .wplt reaw ly divwpetk xedpnd mcwn ly

.zihe`k xwira ìd drepzd ea megzl zixlebx dxwira ìd drepzd ea i`n xiyid

ly reawd (ly iktedd) ly divwpetk xedpnd mcwn ik ìd zixwird dycgd d`vezd

.miiqlwinq milewiyn aygl ozipy dpan d`xn `l` ilìvppetqw` heyt epi` wplt

dl` zepeirxa yeniyd jez dbyedy dn`zddn xzei daeh ef dceara dbyedy dn`zdd

iqlwd leabd .iqlw ienc ietin ly ìd dhiyd dlrted dilr znieqnd zkxrnd .xara

qt`l sèy (dirad ly zirahd dlerta wlegn) wplt ly reawd ea leabd epi` mi`znd

ietind .zipeiqip dhilyl ozip okle qt`l sèy (qppefx) dcedzdn wgxnd ea leabd `l`

bvend aeyigd .dciakd zrtyda mirpd ,xfiil ici-lr mipaxecnd mineh` xen`k x`zn

deedn èd .df beqn iqlw ienc agxna xedpn ly oey`xd ihilp`d aeyigd èd ef dceara

zekxrna divfihpeewd enk zbdpzn df iqlw ienc ietin ly divfihpeewd ik jkl zecr

-peew zeira xwgl dpecpd dhiya yeniy ly ycg wte` gzet df .zelibxd zeihpeewd

.xedpnn d`vezk dv`dd ipte` zkirc dayeg ef dceara .zetqep (milb zeiral mbe) zeih

èd lcendy jkn raep xacd .mitqep zexewnn reapl dleki dkircd dcarn zepeiqipa

,o`k zxèznd ,dixeìzdn dtevnd dkircd oia lcadd .zipeiqipd zkxrnl aexw wx

geztle zipeiqipd zkxrnd ly zepiiprn zepekz lr riavdl dleki zipeiqipd d`vezde

.miycg miihxeìz zepeirxl miwte`

decay of quantum accelerator modes

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