Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications

368

O.

Umut

x

-0.5

20 40 60 80

-1.0

-1.5

-2.0

-2.5

-3.0

y

100120tj:6l

'

~~

-I

2.5

2.0

-2

1.5

-3

1.0

0.5

-4

20

40 60 80 100120

t_5

20 40

t

80 100 120

Fig.

4.

The

time

variations

of x , y

and

z for

the

initial

conditions

outside

the

stability

region,

Xo

=

-3.3,

yo = 3.5,

Zo

=

-2

.

75

5

Conclusions

Using

Cartwright

method

, a

Lyapunov

function is

generated

in a system-

at

ic way for nuclear

spin

generator

syst

em

under

suitabl

e ass

umption

s

an

d a

three

dimensional region, in which

stability

is

guaranteed

by

the

Lyapunov

theorem, is

portrayed

graphically

and

the

results

ar

e verified

by

computer

solutions.

References

1.J

.P.LaSalle

and

S.Lefschetz, Stability

of

Lyapunov's direct method, Academic

Press, New York N.Y., 1961.

2.R

.

D.Ingw

erson, A modified

Lyapunov

m

et

hod

for nonlinear

stability

analysis,

IRE

Trans. voI.AC-6, 1961, pp.199-21O.

3.G.P.Szego, A co

ntribution

to

Lyapunov

second

method;

nonlinear

autonomous

systems,

ASME

paper No.61-WA-192, 1961.

4.Z.V.Rekasius

and

J.E.Gibson

,

Stability

analysis

of

nonlin

ear

control

systems

by

the

second m

et

hod

of

Lyapunov,

IRE

Trans. voI.AC-7, 1962, pp.3-15.

5

.M.L.Cartwri

ght,

On

the

stabi

li

ty

of

solutions

of

certain

differential

equations

of

fourth

order,

Quar.

J.

Mech. Appl. Math., 9(2), 1956, pp.185-194.

6.S.

Sherman,

A

third-order

nonlinear

syst

em arising from a nuclear

spin

generator,

Contrib.Diff. Eqn., 2, 1963, pp.197-227.

7.P.L.Sachdev a

nd

R.

Sarathy

, Periodic

and

chaotic

solution

s for a

nonlinear

syste

m

arising from a

nuclear

s

generator,

Chaos, Solitons and Fractal

s,

voI.4(1l),

1994, pp.2015-2041.

8.W.G.

Cunningham,

Introduction to

nonlin

ear analysis, M

cGra

w-Hill, 1958.

9.P.C.

Parks,

A

new

proof

of

the

Routh-Hurwit

z

stability

criterion

using

the

second

method

of

Lyapunov,

Pr

oc.

Cambridge

Phil. Soc.,

58

, 1962, pp.694-702.

Atomic

de

Broglie-Wave

Chaos

V. O. Vitkovsky, L. E. Konkov,

and

S.

V.

Prants

Laboratory

of

Nonlinear

Dynamical

Systems

Pacific Oceanological

In

stitute

of

the

Russian

Academy

of

Sciences

690041 Vladivostok,

Russia

(e-mail:

prants@poi.dvo.ru)

369

Abstract:

Co

he

rent

motion

of cold

atom

ic wave

packets

is

studied

in a one-

dimensional

optical

potential

created

by

a

stan

ding

las

er

wave.

Their

evo

lution

is

interpreted

in

the

dress

ed

-

state

basis

where

it

is

the

motion

in a

bipotential.

Man-

ifestations of

de

Broglie-wave chaos,

caused

by

nonadiab

atic

transitions

between

components

of

this

bipot

ent

ial,

are

found.

The

probability

of

thos

e

transitions

is large

in

that

range

of

the

atom

-field

detuning

and

recoil frequen

cy

where

the

classical

center

-of-mass

motion

is

shown

to

be

chaotic

in

the

sense

of

exponent

ial

sensitivity

to

sma

ll

variations

in

initial

conditions

or

parameters.

Keywords:

cold

atom,

matter

wave,

quantum

chaos.

1

Introduction

Cold

atoms

in a laser

standing

wave

are

ideal

objects

for

studying

fundamen-

tal

principles

of

quantum

physics, quantum-classical correspondence,

and

quantum

chaos.

The

proposal

[2]

to

st

ud

y

atomic

dynamics

in a

far-detuned

modulated

sta

nding

wave

made

atomic

opt

ics

[4

] a

testing

ground

for

quan-

tum

chaos. A

number

of impressive

expe

riment

s have

been

carried

out

in

accordance

with

this

proposal

[3].

New possibilities

are

opened

if one works

near

the

atom-field resonance where

the

interaction

between

the

internal

and

external

atomic

degrees

of

freedom is intense. A

number

of

Hamiltoni

an

non-

linear

dynamical

effects have been found in Refs.

[6

,1

]

with

point-like

a~oms

in a

near-resonant

rigid

optical

lattic

e:

chaotic

Rabi

oscillations,

chaotic

walking,

dynamical

fractals,

Levy

flights,

and

anomalous

diffusion.

Dynamic

al chaos

in

classical mechanics is a special kind

of

random-like

motion

without

any

noise

and/or

rand

om

parameters.

It

is

char

acterized by

exponent

ial sens

itivity

of

tra

jectories in

the

phase

space

to

small

variations

in initial conditions

and/or

control

parameters.

Such sensitivity does

not

exist in isolated

quantum

systems b

eca

use

their

evo

lution

is unitary,

and

there

is no well-defined

notion

of

a

quantum

traj

ectory.

Thus,

there

is a

fundamental

problem

of

emergence

of

classical

dynami

ca

l chaos from

more

profound

quantum

m

echan

ics which is known

as

quantum

chaos

problem

and

the

of

quantum-classical correspondence.

In

a

more

general

context

it

is a

problem

of

wave chaos.

It

is clear now

that

quantum

370

V.

O.

Vitkovsky,

L.

E.

KOllkov,

and

S.

V.

Prants

chaos, microwave,

optical,

and

acoustic

chaos

[5]

have

much

in

common.

The

common

practice

is

to

construct

an

analogue

for a given wave

object

in

a

semiclassical (ray)

approximation

and

study

its

chaotic

properties

(if any)

by

well-known

methods

of

dynamical

system

theory.

Then,

it

is

necessary

to

solve

the

corresponding

linear

wave

equation

in

order

to

find

manifestations

of

classical chaos

in

the

wave-field

evolution

in

the

same

range

of

the

control

parameters.

If

one

succeeds

in

that

the

quantum-classical

or

the

wave-ray

correspondence

are

announced

to

be

established.

In

this

paper

we

perform

this

program

with

cold two-level

atoms

in

a one-

dimensional

standing-wave

laser field

and

show

that

evolution

of

the

atomic

matter

waves

in

a

standing

wave is

really

complicated

in

that

range

of

the

control

parameters

where

the

corresponding

classical point-like

atomic

mo-

tion

can

be

strictly

characterized

as a

chaotic

one.

The

effect is

explained

by

non

adiabatic

transitions

in

the

optical

bipotential

whose

probability

increases

greatly

in

that

range.

2

Atomic

wave-packet

motion

in

a

laser

standing

wave

The

Hamiltonian

of

a two-level

atom,

moving

in

a

one-dimensional

classical

standing-wave

laser field,

can

be

written

in

the

frame

rotating

with

the

laser

frequency

W f

as

follows:

where

G-±,z

are

the

Pauli

operators

for

the

internal

atomic

degrees

offreedom,

X

and

P

are

the

atomic

position

and

momentum

operators,

Wa

and

[l

are

the

atomic

transition

and

Rabi

frequencies, respectively. We will

work

in

the

momentum

representation

and

expand

the

state

vector

as follows:

Itlf(t)) = J

[a(P,

t)

12)

+

b(P,

t)

11)

]IP)dP,

(2)

where

a(P, t)

and

b(P,

t)

are

the

probability

amplitudes

to

find

atom

at

time

t

with

the

momentum

P

in

the

excited

and

ground

states,

respectively.

Schri:idinger

equation

for

the

probability

amplitudes

is

written

as

.. 1 1

za(p)

= 2(w

r

P

2

-

L1)a(p)

- 2

[b(p

+

1)

+

b(p

- 1)],

. 1 2 1

ib(p) = 2(w

r

P +

L1)b(p)

- 2

[a(p

+

1)

+ a(p

-1)],

(3)

where

dot

denotes

differentiation

with

respect

to

dimensionless

time

T

==

[It,

and

the

atomic

momentum

p is

measured

in

units

of

the

photon

momentum

fik

f.

The

normalized

recoil frequency,

Wr

==

fikJ /

rna

[l,

and

the

atom-field

detuning,

L1

==

(w

f - w

a

) /

[l,

are

the

control

parameters.

We

will

interpret

Atomic de Broglie-

Wave

Chaos

371

Fig.

1.

The

resonant

E~±)

(.::1

=

0,

dotted

curve)

and

non-resonant

E1±)

(.::1

oF

0,

solid curve)

potentials

of

a two-level

atom

in

the

sbmding-,\vave field.

the wave-packet

motion

in

the

dressed-state

basis

1+

)

L1

=

sin812)

+ cos 811),

1-

)

L1

cos 812) - sin 811).

(4)

The

dressed

states

are

eigcnstates

of

an

atom

at

rest

in

a laser field

with

the

eigenvalues

of

the

quasienergy

and

the

mixing angle 8

E~±)

=

±J

LF

+ cos

2

x,

tan8

==~.

-

4 2 cos:c

(

Ll

)2

--

+1.

2 cos:r

(5)

The

resonant

Ef~~:)

(.:1

=

0)

and

nOll-resonant

E);:l

(Ll

1=

0)

potentials

of

the

atom

in

the

standing-wave field

are

drawn

in Fig.

1.

The

wave function in

the

position space is connected

to

the

wave function in

the

IIlOlnentum space

by a Fourier transform.

The

probability

amplitude

to

find

the

atom

nearby

a;

==

k f X

in

the

potential

E1-)

in

the

dressed

state

basis is

iXJ

c(;r;) = const / dpcipx[a(p)eos8 -

b(p)sin8].

(6)

-f:xJ

Let

the

atom

be

initially

prepared

in

the

ground

state

as a

Gaussian

wave

packet

in

the

momentum

space with

the

normalized recoil frequency

w,.

=

10-:

3

,

the

average initial

momentum

Po

= 55

and

the

initial

width

Llp =

1,

which corresponds

to

the

position initial

width

Lh:

=

1/2

or

X = )...t!41f.

The

centroid

of

the

wave packet is initially

at

the

antinode

Xo

=

O.

The

momentum

probability

at

the

time

moment

T is

computed

as

vV(p,

T)

=

la(p,

T)

12

+

Ihep,

T)

12.

The

atomic

wave packet, placed

at

the

alltinode

of

the

standing

wave, is simultaneously

at

the

top

of

the

potential

E(+)

and

at

the

bottom

of

the

potential

E(-)

(see Fig. 1).

Its

I ± ! components move

downwards

(upwards) in

the

potentials

E(±),

respectively.

The

evolution

of

the

wave packet depends, mainly,

on

the

relation

between

the

depth

of

the

optical

potential

and

the

atomic kinetic energy.

One

can

estimate

the

character

of

the

bipotential

motion as follows.

The

Hamiltonian

of

the

internal

degree

of

freedom in

the

basis I

±!

=

(11)

±

12))/

/2

has

the

form

(7)

372

V.

O.

Vitkovsky, L.

E.

Konkov,

and

S.

V.

Prants

We linearize

the

potential

in

the

vicinity

of

a

node

of

the

standing

wave

and

estimate

a

small

distance

the

atom

makes

when

crossing

the

node

as follows:

Ox

"':'

Wr

<

Pnode

>

T,

where <

Pnode

> is

the

mean

atomic

momentum

near

the

node. We

arrive

now

at

the

famous

Landau-Zener

problem

to

find a

transition

probability

when

the

energy

difference varies

linearly

in

time.

The

asymptotic

solution

is well

known

P

LZ

=

exp

(-If

L12 ) .

Wr

<

Pnode

>

(8)

The

quantity

WrP

is a

measure

of

the

atomic

velocity.

If

WrP

« 1,

then

an

atom

takes

for

the

period

of

the

Rabi

oscillations a

small

distance

as

compared

with

the

wavelength.

There

are

three

possibilities.

l.

L12 »

Wr

<

Pnode

>.

The

transition

probability

from one

potential

to

another

one

is

exponentially

small.

The

wave-packet

motion

is, gener-

ally speaking,

bipotential

but

without

transitions

between

the

nonresonant

potentials.

2.

L12 «

Wr

<

Pnode

>.

The

probability

of a

Landau-Zener

transition

is

close

to

unity,

and

an

atom

changes

the

potential

each

time

upon

crossing

any

node.

At

L1

= 0,

it

is a

bipotential

motion

in

the

resonant

potentials.

3. L12 "':'

Wr

<

Pnode

>.

The

probabilities

to

change

the

potential

or

to

remain

in

the

same

one

upon

crossing a

node

are

of

the

same

order.

In

this

regime one

may

expect

proliferation

of

components

of

the

atomic

wave

packet

and

strong

complexification

of

the

atomic

wave packets.

Let

the

average

initial

atomic

momentum

and

the

recoil frequency

be

fixed, P = 55

and

Wr

=

10-

3

,

respectively. So,

Wr

<

Pnode

>"':'

0.055

and

we will

change

the

detuning

in

order

to

investigate

all

the

three

cases.

At

L1 = 1,

one

gets L12 »

Wr

<

Pnode

>.

This

is

the

first case

with

negligibly

small

values

of

the

probability

of

nonadiabatic

transitions.

At

L1 = 0.2, we

have

the

relation

L12 "':'

Wr

<

Pnode

>,

corresponding

to

the

third

case

with

nonadiabatic

transitions

at

the

nodes

of

the

standing

wave. To satisfy

to

the

second condition, L12

«wr

<

Pnode

>, one

can

work

at

exact

resonance,

L1 = O.

In

the

first case

one

expects

a

comparatively

simple

evolution

of

the

atomic

wave packet. Fig.

2a

demonstrates

that

the

initial

wave

packet

splits

from

the

beginning

to

a few

components

because

the

initial

ground

state

is a su-

perposition

of

the

dressed

states

(4).

The

atomic

kinetic

energy

is

enough

to

perform

a ballistic

bipotential

motion.

The

momentum

changes in a

compar-

atively

small

range,

from 40

to

70

photon-momentum

units.

The

packet

does

not

split

at

the

nodes

but

it,

on

the

contrary

, recollects in

the

momentum

space

at

the

nodes

after

initial

spreading.

However,

this

recollection

smears

out

in course

of

time.

To

be

sure

that

the

wave-packet

evolution

is essen-

tially

nonadiabatic,

we go

to

the

position

space

and

compute

the

probability

Ic(x)12

to

find

the

atom

at

the

position

x in

the

potential

E~-)

in

the

frame

moving

with

the

mean

initial

atomic

velocity

Po

= 55. Fig.

2b

demonstrates

jitomie de Broglie-

Wave

Chaos 373

Fig.

2.

Adiabatic

wave-packet

motion

in

the

nonresonant

potentials

at

Ll.

"'~

1

in

the

momentum

and

position

spaces. (a)

Momentum

probability

tV(p,T)

and

(0)

position

probability

Ic_(;:r)1

2

to find

the

atom

in

the

potential

E~).

The

color

codes

the

corresponding

probability

values.

clearly

the

absence

of

nonadiabatic

transitions

in

the

position

space

at

the

llodes which

are

indicated

by

the

slope lines

in

the

moving frame.

Fig.

3.

'Nave-packet

motion

at

exact

resonance

L1

position

spaces.

o

in

t.he

momentum

and

Fig

. 4. Nonadiabat.ic

wave-packet

motion

and

transitions

between

the

nonresonant

potentials

a.t

L1

= 0.2 in

the

momentum

a.nd

position

spaces.

Figure

:3a

shows evolution

of

the

Gaussian

waVG

packet

in

the

mornentum

space

at

exact

resonance

(L1

=

0)

when

L12

«w,. <

Pnode

>.

The

color codes

374

V.

O.

Vitkovsk};

L.

E.

Konkov,

and

S.

V.

Prants

the

values

of

the

mornentmn probability.

It

follows from (5)

that

the

initial

ground

atomic

state

is now

an

equal superposition of

the

dressed

states.

The

initial wave packet splits into a mimber of components.

The

components

with

the

momentum

increasing from

the

beginning move ballistically in

the

potentia.!

E~

+).

The

components

with

the

momentum

decreasing from

the

beginning have

not

enough kinetic energy to overcome

the

barrier

of

the

first

potential

well

and

they

oscillate in

the

potential

E6-)

between

the

nodes

at

:r

=

'if

/2

and

:r

=

-II

/2.

Thus, we observe

the

quantnm

phenomenon:

atom

moves in

the

positive direction over

the

hills

of

the

optical

potential

and

oscillates simultaneously in

the

well

of

this

potential.

It

should

be

stressed

that

the

period

of

the

resonant

potentials

is twice

as

large as

the

period for

the

nonresonant

ones.

The

resonant potentia.ls cross each

other

at

the

nodes

(see Fig. 1).

In

the

position space we observe Landau-Zener

transitions

at

the

nodes

with

the

unit

probability as

it

is given by

the

formula (8).

The

color in Fig. :)b codes the values

ofthe

probability

(;r

W

to

find

the

atom

in

the

potential

E~-).

0.1

.--~--~---,

60.------------~

w

0.075

40

0.05

20

0.025

a)

o

......

_

.L..-'-.;...:...:~___'___>_'

o

30

60

P 90

200

400

1:

600

Fig.

5.

(a)

Momentum

probability

distributions

vV

at

the

fixed

time

moment

in

the

nonadiabatic

(bold

curve,

Li.

= 0.2),

adiabatic

(dashed

curve.

Li.

= 1),

and

resonant

(dotted

curve,

Li.

= 0)

c&ses.

(b)

Mean

atomic

momentum

VB

time

in

the

nonadiabatie

(bold lower curve,

Li.

= 0.2),

adiabatic

(bold

upper

curve,

Li.

= 1),

and

resonant

(dotted

curve,

Li.

= 0) eases.

At

Ll

= 0.2,

the

atomic

ground

state

is

the

following superposition of

the

dressed

states

:

11)

c::

0.741

+).c, +

0.661-

).c,.

The

part

of

the

initial wave

packet,

starting

from

the

top

of

the

potential

E~+)

, overcomes

the

barriers of

that

potential

and

moves in

the

positive direction of

the

axis :r proliferating

at

the

nodes.

The

larger

part

of

the

initial packet,

starting

from

the

bottom

of

the

potential

E~-),

will be

trapped

in

the

first well of

that

potential.

Only

its

small

part

is

capable

to

overcome

the

barrier. To illustrate

that

we

show in Fig. 4 evolution of

the

initial Gaussian wave packet. As compared

to

Fig. 3a,

we

see

an

interference-like picture in

the

InomentulYl space

and

dear

evidence

of

the

Ilonadiabatic transitions between

the

nonresonant poten[,ials

Atomic de Broglie-Wave Chaos 375

in

the

position

space.

This

results in a proliferation

of

the

components

of

the

atomic

wave

packet

at

the

nodes

of

the

standing

wave.

Figure

5a

shows

the

momentum

probability

distributions

W

at

the

fixed

time

moment

T = 30 in

the

nonadiabatic,

adiabatic,

and

resonant

cases.

The

distribution

is narrower in

the

adiabatic

case

than

in

the

other

ones.

The

peaks

of

Ware

more

pronounced

in

the

nonadiabatic

case

than

in

the

reso-

nant

one.

Figure

5b

shows

the

dependence

of

the

mean

atomic

momentum

< P >

on

time

for

the

three

cases.

This

quantity

oscillates

in

the

adiabatic

case in

much

nore

narrow

range

than

in

the

other

cases where < p > is a

wildly oscillating function.

3

Quantum-classical

correspondence

In

the

semiclassical

approximation,

one

treats

the

electronic degree

of

freedom

quantum

mechanically

and

the

translational

motion

classically.

The

problem

of

motion

of a point-like

atom

in a standing-wave laser field

has

been

studied

in

detail

in Refs. [6,1].

The

five-dimensional Hamilton-Schrodinger

equa-

tions

of

motion

have been deduced. Along

with

two integrals of motion,

they

constitute

a

Hamiltonian

autonomous

system

with

two degrees

of

freedom

moving

in

the

phase

space

on a three-dimensional hypersurface

with

a given

energy

value. A

number

of

nonlinear

dynamical

effects have

been

found

in

Refs. [6,1]: chaotic

Rabi

oscillations,

Hamiltonian

chaotic

atomic

transport

and

dynamical

fractals, Levy flights

and

anomalous

diffusion.

One

of

the

impressive effects is a random-like

motion

of

an

atom

in

a rigid

optical

lat-

tice

when

the

atom

can

fly

through

the

optical

potential

in

one

direction, be

trapped

in its wells for a while

and

then

continue

to

fly over

its

hills in

the

other

direction

[1].

This

and

the

other

effects

are

caused

by local

instability

of

the

center-of-mass

motion

in a laser field. A

set

of

atomic

trajectories

under

certain

conditions becomes

exponentially

sensitive

to

small

variations

in initial

quantum

internal

and

classical

external

states

or/and

in

the

control

parameters.

A

detailed

theory

of

Hamiltonian

chaotic

transport

of

atoms

in a

laser

standing

wave

has

been

developed in

[1].

The

maximal

Lyapunov

expo-

nent

,.\

characterizes

the

mean

rate

of

the

exponential

divergence of initially

close

trajectories

and

serves as a

quantitative

measure

of

dynamical

chaos.

In Fig.

6a

we show

the

dependence

of

,.\

on

the

detuning

Ll

for

an

initially

deexcited

atom

with

the

fixed value

of

the

recoil frequency

Wr

=

10-

3

and

initial conditions:

Xo

= 0

and

Po

= 55.

As a

measure

of complexity

of

the

wave packets, we

take

a

number

of

local

extrema

N

in

the

distribution

function

W(p)

on

a large

time

interval.

Figure

6b

plots

the

dependence

of

this

quantity

on

the

value

of

the

detuning

Ll.

Comparing

this

quantum

result

with

the

semiclassical

one

in Fig. 6a,

we see

rather

good correlation.

It

follows from

the

semiclassical

simulation

that

the

maximal

Lyapunov

exponent

is zero

nearby

the

resonance

and

when

the

detuning

is large

enough

providing a

regular

point-like

atomic

motion.

376

V.

O.

Vitkovsky,

L.

E.

Konkov,

and

S.

V.

Prants

0.04

A

"")\'

,

'(

\

O.Q3,

.

~

a)

0.02 ( \

""L",

"'

~~"

J

"O':----C

0

:-::.

2

---:

O

:-:.4---:0:-:.6---:0'"'.S-L\---'1

Fig.

6.

(a)

Maximal

Lyapunov

exponent

A

with

the

classical

atomic

motion

and

(b)

number

of local

extrema

in

the

momentum

distribution

with

th

e

quantiz

ed

atomic

motion

vs

the

detuning

.:1

.

Positive values

of

A

mean

chaotic

atomic

motion.

Quantum

evolution is

rather

regular

in

the

same

ranges of

the

detuning

and

it

is

complicated

in

that

range

of

the

detuning

where

the

corresponding

classical

atomic

motion

is

chaotic. We have shown

that,

varying

the

Rabi

frequency

or

the

detuning,

it

is possible

to

som

e

extent

to

control

propagation

of

de Broglie waves

creating

regular

and

chaotic wave packets.

This

work was

supported

by

the

RFBR

(project

no. 09-02-00358)

and

by

the

Program

"

Fundamental

Problems

of

Nonlinear

Dynamics

" .

References

l.V.Yu.

Argonov

and

S.V.

Prants.

Theory

of

chaotic

atomic

transport

in

an

optical

lattice. Physical Review

A,

75:a

rt.no.

063428, 2007.

2.R.

Graham,

M.

Schlautmann,

and

P. Zoller.

Dynamical

localization

of

atomic-

beam

deflection

by

a

modul

ate

d

standing

light wave. Physical Review A,

45:R19, 1992.

3.W

.K. Hensinger, N.R. Heckenberg,

G.J.

Milburn,

and

H.

Rubinszt

e

in

-

Dunlop.

Experimental

tests

of

quantum

nonlinear

dynamics

in

atom

optics. Journal

of

Optics B, 5:R83, 2003.

4.A. P.

Kazantsev,

G.

1.

Surdutovich,

and

V. P. Yakovlev.

World

Scientific, Singa-

pore

, 1990.

5.D. Makarov,

S.

Prants,

A. Virovlyansky,

and

G. Zaslavsky. World Scientific,

Singapore,

2009.

6.S. V.

Prants.

Chaos, fractals a

nd

flights

of

ato

ms in cavities.

JETP

Letters,

75:651, 2002.

377

Chaotic Behavior

of

Plasma Surface Interaction. A

Table

of

Plasma Treatment Parameters Useful to the

Restoration

of

Metallic Archaeological Objects

C. L. Xaplanteris,·2 and E. Filippaki2

, Hellenic Military Academy, Vari Attika, Greece

2 Plasma Physics Lab, IMS, NCSR

"Demokritos", Athens, Greece

(e-mail: lfilip@irns.dernokritos.gr)

Abstract:

In Plasma Physics laboratory

of

N.C.S.R. "Demokritos" the plasma chemistry method has

been used for the restoration and conservation

of

metallic archaeological objects during the last

decades. The obtained experience had led

us

to

conclude that plasma parameters and different status

of treated objects are so specific, so

as

to

become unique. In the present paper the theoretical and

experimental results

of

our laboratory are summarized. A treatment table of plasma parameters is

given, which claims

to

be useful for the conservators.

It

is obvious that this treatment table needs to

be completed and extended, so that

it

meets the uniqueness of each artifact. A theoretical study and

the treatment

of

a variety

of

iron objects are presented.

Keywords: plasma sheath potential, plasma restoration, corrosion, external potential, plasma

parameters.

1. Introduction

Three decades have passed since Daniels and coworkers used for first

time the mechanical and chemical action

of

the plasma [1] to reduce the tarnish

of

silver on daguerreotypes. During the 80's, Veprek and coworkers

systematized the plasma cleaning and restoring method [2-4], by creating the

homonymous plasma reactor, where a low-pressure hydrogen plasma acts upon

the artifacts. These procedures were very soon adopted by the

Plasma laboratory

of

NCSR "Demokritos" due

to

the many archaeological artifacts, which are

found at the Hellenic area. The plasma treatment processes are usually preferred,

for the following reasons: the plasma temperature is low and consequently, the

destruction risk

of

artifacts is avoided; the chemical reactions produce steady

compounds on the objects (e.g. magnetite for iron objects), which consist the

restorating behavior

of

plasma; finally, the plasma reactor volume allows the

simultaneously treatment

of

many objects.

Many experiments and studies have been carried out [5, 6], at the two

plasma reactors

of

"Demokritos" since the early 90's. Recently, the utilization

of

an

external d.c. electric current on the treated objects have showed very good

results

[7,

8].

The whole previous meditation leads to the conclusion that plasma

cleaning proceedings are chaotic due to multi-parametric behavior

of

plasma and

the complicated corrosion

of

metallic objects. Thus, the factors which affect the

cleaning and restoring can

be

listed as below.

Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications

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