Excitation mechanisms for hydrogen atoms in strong microwave fields

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Excitation mechanisms for hydrogen atoms in strong microwave fields

H. Breuer, M. Holthaus

To cite this version:

H. Breuer, M. Holthaus. Excitation mechanisms for hydrogen atoms in strong microwave fields.

Journal de Physique II, EDP Sciences, 1991, 1 (4), pp.437-449. �10.1051/jp2:1991178�. �jpa-00247528�

Classification

Physics

Abstracts

42 50 32.80K 03 658

Excitation mechanisms for hydrogen atoms in strong

microwave fields

H. P Breuer and M. Holthaus

Physikahsches

Institut der Universitat Bonn, Nussallee 12, D-5300 Bonn I,

Gerrnany

(Received

5

September1990,

revised 27 December1990,

accepted

ii January

1991)

Abstract. We

analyse

the excitation mechanisms which determine the

dynamics

of

highly

excited

hydrogen

atoms traversing a

wavegulde.

Experimental measurements to

verify predictions

denved from Floquet

theory

_are

suggested.

1. Introduction.

Rydberg

atoms

exposed

to

short,

_{strong microwave}

pulses [1-4] provide

_a

particularly

attractive

possibility

to

study

short-time

phenomena

_{m quantum} mechanics. In such

expenments

^the

amplitude

of the _microwave field _can be

adjusted

to be of the _same order of

magnitude

_as the Coulomb field

binding

the

electron, therefore,

_{a major} theoretical task _is the

development

of _new,

non-perturbative

methods On the

expenmental

side essential data

required

for _a

thorough

_companson of theoretical

predictions

and

expenmental

results such

as

length

and

shape

of the

pulse

_can be determined with

great

precision.

Thus,

there is _a unique

opportunity

to unravel the mechanisms

goveming

the

quantum dynamics

of

systems

m

strong penodic

fields _;

insights gained

from _an

analysis

of the _microwave

problem

should also throw

light

_on the s1mllar

problem

of atoms

interacting

^with strong ^laser

pulses

In _a recent work

[5],

we have

applied

the

Floquet

picture to the

description

of _microwave

ionization

expenments performed

at

high

scaled

frequencies ^n~

_{w m} I_ using ^cavities

[2]

: an atom expenences, when

entering

the cavity, a

slowly nsing

field which leads _to diabatic-

adiabatic motion _on

coarse-grained

quasienergy surfaces such that the state that

finally

ends up m the cavity is a

superposition

of

Floquet

states. The results of the ionization

expenments

performed

in the

cavity

are determined

by

the

amplitudes

of these

Floquet

states

which,

m

turn, refldct both the structure of the

underlying

quasienergy spectrum ^{and the}

shape

of the

fringe

field

[5].

The concept ^{of adiabatic motion} is, of _course, of central

importance

m many areas of

physics.

For instance, within the framework of the

Born-Oppenheimer theory

_m molecular

physics

one encounters adiabatic evolution on energy surfaces at avoided

crossings

of energy levels Landau-Zener transitions _occur. For

penodically time-dependint

_{quantum systems}

such _as atoms in strong ^{laser fields} or

Rydberg

atoms m microwave

fields,

there _is adiabatic

evolution of

Floquet

states on

quasienergy

surfaces

[5, 6].

^Now an

important

new point appears : The quasienergy spectrum ^is

organized

m Bnllouin _zones This fact leads to avoided

JOURNAL DE PHYSIQUE II T I,M 4, AVRIL (W( 20

crossings

of

quasienergies corresponding

to «distant» states, Landau-Zener transitions among

Floquet

states,

therefore,

_open the

possibility

of

exciting high lying

states with

large probability. Thus,

in the _case of

Rydberg

atoms

interacting

^with a

pulsed

microwave field there _{is a}

vanety

^of processes which _are induced

by

the

pulse shape,

I-e-

by

the

parameter

motion and the

resulting

^adiabatic or diabatic evolution of the _wave function.

Obviously

such effects _can not be understood

by

_an

approach [7]

which

simply

_{assumes a}

temporally

constant

microwave

amplitude.

In the present paper we will discuss the

question

^which

experimental signatures

detectable

m

present-day experiments

_can be

employed

for _an

unambiguous

test of

predictions

denved from the

Floquet theory.

We

will, therefore, analyse

_a

paradigmatic _type

of

experiment

_{: we}

assume a beam of

fast,

excited

hydrogen

atoms to traverse _a

rectangular waveguide operated

m the

TEjo

mode

thus,

the

envelope

of the _microwave

pulse

_«seen»

by

the atoms _is assumed _to be half _a sine _wave, modified

by

the presence of the _entrance holes in the

waveguide.

Whereas _m

cavity expenments

^the

pulse shape

has _{a «} flat top », in the situation considered

by

_us the

amplitude

_vanes

during

^{the whole} interaction time and it should be clear

immediately

that

expenmental

results obtained with both types ^of

experimental

_{set-up can} differ

significantly

_even

though

the maximal

amplitude

and the

frequency employed

_may be the _same.

We will

proceed

in two

steps

: in the

following chapter

_we

_present

model calculations which illustrate the excitation mechanisms for the

type

of

experiment

under consideration. Seen in _a

more

general

context, these calculations _serve to establish the fact that adiabatic motion _on quasienergy surfaces _is of central importance for the

dynamics

of

penodically

dnven quantum systems. ^{We then turn} to the

question

of

expenmental

consequences and

suggest

measure-

ments which could _{serve as} a test of the

viability

of the

Floquet

_picture.

2. Excitation mechanisms for atoms in microwave fields.

For _our

niodel

calculations _{we use} the

(I

+ I

)-dimensional

Hamlltonlan H

=

Ho

_{+ H,~~}

(2.1)

describing

_{a «} I-dimensional

hydrogen

atom »

~

-~,

_x>0

Ho "jP

^{+ ~}

(2.2)

~~' ~'°

interacting

with _a

linearly polanzed

_microwave field H,~~ = A

(t)

_x sin _w t

,

(2.3)

where

(t) deiotes

the

time-dependent amplitude

^of the field and _w

= 2 _w

IT

its

frequency.

We denote

stationary eigenstates

^of

Ho by w~(x),

n

being

the

principal

_quantum

number,

and the

Floquet

states of the full Hamiltonian

(2.I) by u~(x, t)

with quasienergy e~.

Actual

experiments

done with

real,

three-dimensional

hydrogen

atoms _are often

performed

m the

jresence

of _a static electric field. The effect of such _a field is obvious _: the quasienergy Hamlltonlan of the three-dimensional

hydrogen

atom m a microwave field of constant

amplitude

_is invariant with

respect

to the

symmetry operatioq

lx-

^-x

S

=

T

(2.4)

~

~2'

this

symmetry

is broken

by

_an additional _static field. As _is well

known, _symmetry

_is essential

m the

theory

of level

crossing

_; the _static field thus not

only

leads to _a distortion of the

quasienergy

surfaces but also _to the appearance of further avoided _crossings of

quasienergles.

In the one-dimensional

model,

the situation is different _{: one}

could,

of _course, include _a term

simulating

_a static field of

strength

_{A~ in} the model

(2.I)

and _write

H~=Ho+Axsinwt-A~x. (2.5)

However,

the

(I

₊ I

)-dimensional

quasienergy Hamiltonian is

lacking

the symmetry ^S even

without the term A~ x.

Therefore,

this term does _now not

give

rise to _new

anticrossmgs

ⁱⁿ the

quasienergy

_spectrum but

only

to _a Stark-shift of the individual levels.

Furthermore,

the

basic mechanisms _we want to illustrate

effectively

adiabatic evolution _on smooth

quasienergy

surfaces,

Landau-Zener transitions at reactive avoided

crossings

remain unaltered. In order _to

keep

the model _as

simple

_as

possible

_we therefore

drop

the term

A~ x end

employ

the the Hamiltonian

(2,I)

for _our numerical

analysis

All calculations _are

performed

_{m a} basis of101 bound states of

Ho (n~~=40,

n~~~=140).

We first

compute

^the

quasienergy spectra

^for three different

frequencies w/2

_w =13,00 GHz

(Fig,1),

15.51GHz

(Fig. 2)

and 18.00 GHz

(Fig. 3).

When

displaying

quasienergles

graphically,

_a technical

difficulty

_{occurs : a}

quasienergy

_e~

actually

_{is a} class

(e~:

=

e~+mw)meZ)

E/W

1'

,

,

', , l

j ' '

'

fl

/ f

/

/

,'

, '

71 ?

,

~i

i i ii

I iiliii

Fig

I.- Part of the

quasienergy

_spectrum for

w/2

_w =1300GHz

Quasienergles

_are labelled

by

pnncipal quantum ^numbers

corresponding

to the static Hamlltonian Ho-

E/W

'

'

i i ₁₁

[(Iii]

Fig

2. Part of the quasienergy spectrum for w/2 _w

=

15 51GHz

of

physically equivalent representatives

_e~, the first Bnllouin _zone

e I

~i~w ~i'

hence,

contains _one

representative

^of every class. An

attempt

to

display

all levels contained in

one Brillouin _zone would

obviously

result in _a black

picture,

no structures would be

recognizable.

One therefore has to select from all the levels certain

dynamically

relevant

ones. We do this

by plotting only

those

quasienergies

for which the

corresponding Floquet

states are concentrated _{m a} certain

subspace

of the full Hilbert space. For

figures

I and

2,

this

subspace

is

spanned by eigenfunctions (w~s,.., wgs)

of the

time-independent

_system

(2.2),

for

figure

3

by (w~o,.., _wgo). Thus,

_a

vanishing

line in the

quasienergy

_spectra

simply

indicates that the

Floquet

state _is

mainly

contained in _a different

subspace

of the Hilbert space

The approximation

actually implicit

_{m our} calculation _is that _we

neglect

the

imaginary

_part of the

quasienergles

which determines the

decay _properties

of the

Floquet

states

[5].

However,

_since the _aim of the

present

paper is to

investigate

excitation

mechanisms,

_we will in the

following part only

consider

examples

with

parameters

^{well below the} ionization border.

For the

frequencies

considered in

figures

1, ² and

3,

the levels closest to the _{main resonance}

n~

w = I _{are n}

=

80,

75 and

-72, respectively.

In all three _{cases we} observe the _same

strong

e/w

i

i

I liii

ig

3

deformation of the _group of levels

adjacent

to the resonant one, the resonant level itself exhibits

large

avoided

crossings only

for

comparatively high

field

strengths

and thus enables

purely

adiabatic motion _even for

relatively large amplitudes.

This fact leads to a

quantum

mechanical

explanation [5]

of the local

stability

at

n~

w =

I observed in actual expenments 121.

In the

following

_we will _use these

spectra

to

explain

the behaviour of solutions of the time-

dependent Schr6dmger equation

Hw(t)

=

I

ojw(t) (2.6)

In order to model the

pulse shape

_seen

by

atoms traversing a

waveguiie

_we choose

(t)

= ~~~ sin

I

,

0 _{« t «} t

~,

(2.7)

t~

where t~ ^{denotes the} interaction time. As initial _states at t ₌ 0 _we take the

eigenstates

w~~ of

Ho.

As _our calculations

show, basically

three different mechanisms

goveming

the short-time behaviour have to be

distinguished

: adiabatic motion

(a),

Landau-Zener transitions

(b)

and

transitions due to _near

degeneracies

at

vanishing

field

strength (c).

a)

To give an

example

for adiabatic motion _we choose no =

72, w/2

_w

=13.00GHz,

t~ ⁼ 98

T, A~~

= 3 44

V/cm (in

atomic units, the scaled

parameters

are :

n(A~~~

=

0.018,

n(

w =

0.737).

We

decompose

the

Schr6dmger

_wave function

evolving

from the initial state w~~ with respect ^to the

eigenstates

_w~ of

Ho

#(X, t)

"

£ c~(t) _q~~(x) (2.8)

at time t ₌

t~/2,

^{when the maximal field}

amplitude

is reached

(Fig. 4),

and at t ₌ t~, ^{i-e- at the} end of the

pulse (Fig. 5). Although

several states w, are excited _m the middle of the

pulse,

after the

pulse practically

all

probability

_{is again} concentrated _on the initial state. The

explanation

for this non-diffusive behaviour is obvious from the quasienergy

spectrum

in

figure

I _{: since} the quasienergy surface _{e~~ is} smooth in the relevant interval of field

strength,

i-e- does not show any reactive avoided _crossing

[5],

we conclude that the time evolution _is

almost adiabatic. This _means that the _wave function #i^is descnbed

by

a

single Floquet

state u~~ ^at every instant of time.

Thus,

from the

point

of view of

Floquet theory only

_a

single

state is excited

during

the whole process ; this state «moves»

adiabatically

_on its quasienergy

surface _e~~ in _response to the

continuously changing amplitude. However,

the _naive

decomposition (2.8)

for _t

=

t~/2

^{obscures this}

simple

behaviour.

b)

We _now take w~~ as 1nltial state,

w/2w=15.51GHz, A~~~=4.0V/cm

[n( A~~

=

0.021, n(

w =

0.880

]. Again

we

decompose

the _wave function

according

to

(2.8)

after the

pulse,

this time,

however,

we

plot

the coefficients _{[c~ [~ as} functions of the

pulse

N N

ii ii i~ _~~

jrincqil _juinluR

nuRler

jriniijil _juinluR

nuRler

Fig

4

Fig

5

Fig.

4

Decomposition

with respect to static states (2

8)

of the _wave function which evolves from the initial state p~~ under the influence of the

pulse (2.3), (2.7)

for _w/2 _WI 13.00 GHz,

A~~

=

3 44

V/cm,

t~ = 98 T

,

the

decomposition

has been

performed

at i ₌ t~/2 ^{when the maximal}

amplitude

_was reached This _wave function _{is an} almost pure

Floquet

state

Fig.

5.

Decompos1tlon

of the wavefunction

evolving

from _p72 after the

pulse

at t ₌ t~, ^the parameters used _are the same as in

figure

4.

e/tv

~

r

i

.,_

~

_-.~,-.

r ,-,----.-~

j

II

_~ ~~~_

ii iii iii i i i 1

l~ii ^{i tin&i}

Fig

6

Fig.

7.

Fig.

6

Occupatlon probabilities

of the bound states _n

= 72 (upper full line), 78 (lower full

line),

79

(larger dashes)

and 84

(short dashes)

after the pulse _{as a} function of the

pulse length

_{t~ in} units of T= 2

grim

The initial state was no = 72; final _states other than the _ones

displayed

_are of minor

importance ^The parameters are

w/2

_{w =} 15 51 GHz, A~~ ₌ ^{4 00}

V/cm

Fig

7 Part of the quasienergy spectrum ^relevant for the

dynamics

of the initial state no =

72 at

w/2

_{w =} 15 51GHz

(cf. Fig. 6),

the

straight

lines indicate Landau-Zener transitions

length

_t~. The

resulting _pattem (Fig. 6)

has _a clear

explanation

from the

quasienergy _spectrum

in

figure

7: the initial state no = 72 _is shifted into the connected

Floquet

state u72 and

undergoes

two Landau-Zener transitions to u~g ^and u~~

during

the _nse of the

field,

when the

amplitude

decreases _again interferences _occur

leading

to

Stfickelberg

oscillations visible _m the excitation

probabilities

of the final _states. In order to observe these oscillations

expenmentally

one has to vary the interaction time t~, which _up to _now has not been done _{m a}

systematic

manner.

c)

As _an

example

for the third mechanism _we take no

=72, w/2w =18.00GHz,

A~~~ = 6.5

V/cm [n(

A

~~~ ⁼

0.034, n(

_w

= 1.021 and

display

in

figure 8a), o

the

excitation

probabilities _[c~[~

of the final bound states as functions of t~ ^{The intncate} oscillation

pattems

can be understood from the

underlying quasienergy _spectrum (Fig. 3).

Now the initial state _is very close to resonance, i-e- several quasienergles are

nearly degenerate

in the Brillouin-zone at A

=

0 This _near

degeneracy

is

rapidly

^removed for

increasing amplitude. Thus, right

at the

beginning

of the

pulse

the 1nltial _state is distnbuted

over several

Floquet

states which then

decouple

and evolve _on their _own

quasienergy

surface.

At the end of the

pulse

the

Floquet

states

couple

_again when A =0 and the

resulting

interference

pattem

_is determined

by

the

dynamical phases

each

Floquet

state has

accumulated.

,

N N

i

u u

i i

iii iii _i~~ iii

t~ii

ⁱⁱ

a)

)

Fig

8a

Occupatlon probability

of the final bound state n = 69 after the

pulse,

the initial state was

no = 72 Parameters w/2 w = 18.00 GHz, _A~~~

= 6 5 V/cm

Fig 8b As

figure

8a for the final state _n

=

70.

m N

- N

~ ~

U

iii iii iii iii

ii

t~li

~) d)

Fig.

8c As

figure

8a for the final state _n

=

71.

Fig

8d As

figure

8a for the final state _{n =} 72

N N

I /

u u

i i

iii iii iii iii

t~li i~li

e)

o

Fig

8e. As

figure

8a for the final _{state n}

= 73

Fig 8f As

figure

8a for the final state _{n =} 74

We finish this section with two remarks

conceming

the

important

differences between mechanism

b)

and

c)

The transitions _m

b)

_are due to avoided _crossings at A _» 0 which

couple

_states

differing by

several units in

pnncipal

_quantum number in _our

example

_we had transitions 72

=

78,

79

corresponding

to _an energy difference of 6.05 Am and 6.93

Am, respectively.

In contrast, the

near

degeneracy

at

= 0 _m

c) perturbatively couples

states which _are the nearest

neighbours

of the initial _one.

Secondly

_we note that mechanism

c) depends

_very

sensitively

_on the way the field is tumed

on and off. We have checked this

by smoothing

the sinusoidal

amplitude (2.7)

at the

edges

such that _an

exponential

function _is fitted

continuously

differentiable at times ti ^and t~

ti(ti

«

t~)

^{of the}

original pulse.

Two

examples

for the influence of this modification _are

given m

figures

9 and lo

(ti

=

3 T and ti ₌ ⁵

T).

This

dependence

_on details of the

pulse shape

is not

surpnsing

_since the

pulse

is modified

exactly

when the

Floquet

states

interact,

I-e-

near A

= 0. In contrast, _as we have checked the _same modification is of

negligible

influence for

a)

and

b).

3.

Experimental hnpficafions.

We _{saw m} the previous

chapter

that the

Floquet

representation offers _{a very} convenient and efficient tool to

analyse

the

dynamics

of atoms _m

pulsed

fields A process that appears ^to be

complicated

m the

eigenbasis

of

Ho

_can be very

simple

when

analysed

m the

Floquet basis,

_our mechanism

a) being

_a

convincing example

for this fact. As _{a measure} of the

complexity

of _a process

during

the interaction _we define the quantity

S(t)

=

£

_{p~ In p~}

(3.1)

N N

» »

u u

ii ii ii ii

jriniijil juiflluR

^fluRler

jriniijil juifltuR

nuRler

Fig

9

Fig

lo

Fig.

9.

Occupation probabilities

of final bound states after _a

pulse

which

slightly

differs from

(2 7)

the

edges

of the

pulse

have been smoothed such that at t ₌ tr ^and t

= t~ tr an

exponential

function has been fitted

continuously

differentiable _to the

onglnal amplitude

For this

figure,

_tr= 3T and t~ = 135 T, the other parameters are as in

figure

8.

Fig

lo As

figure

9, however tr = 5 T

where

p, m

p,(t)

=

jut

_(t)i

pi _12, (3 2)

#i is the

Schrbdinger

_wave function and _{the u~~~'~} denote the

Floquet

states

corresponding

to

the instantaneous field

strength A(t). Compared

to the «Shannon entropy»

recently

considered

by

Casati _et al.

[8]

to characterize localization

properties

^of

eigenfunctions,

the

complexity (31)

_{is a measure} of the dimension of the _space of effective

Floquet

states involved _m the

dynamics.

For _a

purely

adiabatic _process

S(t) stays

^{constant and increases} when non-adiabatic

phenomena

_{occur, e-g-} when the _wave function

splits

_» due to Landau- Zener transitions if

only

_a

single Floquet

state is excited in a

purely

adiabatic _process

(mechanism a) S(t)

remains

identically

_zero. This fact

again nicely

illustrates that the

stationary Floquet

states

provide

the appropnate basis for _a

description

of the

dynamics

:

if in the definition

(3.2)

of the

probabilities p~(t)

the

Floquet

states u~ ^were

replaced by

static

eigenstates

_{wi one} would obtain the

misleading

result that the

complexity

of _an adiabatic process grows

during

the rise of the

pulse

and then decreases to its

onginal

value. As _a

stnking example

for this _we show _m

figures I1,

12 _a static state

decomposition

of the _wave function for an

adiabatic

_process. Even

though

the initial static state is almost

completely depopulated

in the middle of the

pulse

this process is

adiabatic,

_i-e- the above defined

complexity

remains

zero for the whole

pulse.

The

observables

_m actual excitation expenments

performed

with

waveguides

_are the

occupation probabilities

of the final static bound states wi after the interaction with the

N N

/

i

/

ii 11 _~~ ii i~ ii

jrifliipil juifltuR

hunter

jriflcijil juifltun

fluRler

Fig,

I I

Fig

12

Fig I I.

Decomposition

of the _wave function which has evolved from the initial state no =

66 _in the middle of the

pulse

,

w/2

_v

=

18.00 GHz, A

~~ ⁼

60

V/cm,

_t~

=

136 T. Note that the initial static state seems to be

«

depopulated

Fig.

12

Decomposition

of the _wave function considered _in

figure

II after the

pulse

microwave field and the

challenge

_now is to detect clear evidence for the above descnbed

excitation mechanisms. Of _{course one} faces several difficulties

expenmental

results may be

modified

by

the presence of _noise

[4]

and _an accurate final bound state

analysis

is

complicated

because of the

high degeneracy

in real

hydrogen

atoms The latter

problem

^could

partially

be

circumvented

by working

with helium

[9]

_or alkali

[3, 4]

atoms m order to lift the

f- degeneracy

_{or, as a} different

possibility, by superimposing

_a static electnc field to achieve quasi

one-dimensionahty. Despite

these

complications,

_we believe that

skilfully performed experiments

_can

unambigously identify

the basic mechanisms. To this end _we

suggest

the

following

measurements :

I)

The interaction time t~ ^can be varied

by changing

the

velocity

of the atom beam

Landau-Zener transition

probabilities during

the

pulse

do

depend

_{on t~} and determine the final bound state distnbution.

Thus,

_measuring the [c~ [~ ^as functions of t~

yields important

information about transitions occurnng

during

the

pulse.

In

particular, waveguide

_exper-

iments _are

ideally

suited for _a detection of

Stfickelberg

oscillations _: if the

length

of the whole

pulse

is

adjusted

to be of the order of

only

loo

cycles

of the external

field,

the

dynamical

phases acquired by

the

interfering Floquet

states can be

prevented

from _{growing too}

large

_{; m} contrast, ^{the flat} top ^{of the}

pulse

_m

_cavity

expenments

[2]

lasts

longer

than 300

cycles

and thus induces

large phases

and _a

correspondingly sharper

interference pattern which _is harder to resolve. In order to resolve

St0ckelberg

oscillations _as

depicted

_m

figure

6 in such

waveguide

expenments,

^the necessary range of velocities translates into _a variation of the proton acceleration

voltage

_{over a range} of _some 10 kev.

448 JOURNAL DE

PHYSIQUE

II M 4

2)

The holes

by

which the atoms enter and leave the

waveguide modify

the

pulse shape

near t ₌ 0 and _t

= t~ such that the actual

envelope

obtains

long

tails. It is well known

[10]

that smoothness conditions at the

edges

of the

pulse

are essential for adiabatic behaviour _;

indeed,

as our calculations have

shown,

for _near resonant 1nltial conditions

(mechanism c)

the [c~ [~

(t~)

can be influenced

by shape

and

magnitude

of the

holes,

whereas _{m case}

a)

and

b)

properties ^{of the} holes have

only

_a

negligible

effect.

Hence, repeating

^the expenments

using waveguides

with

differently shaped

holes

will,

m

general, yield

identical

results, however,

for certain initial states the

expenmental signal

will be charactenstic for the

microscopic

structure of the holes of the

particular _apparatus employed.

3)

In order _to

investigate pulse-shape dependencies

it would be _very

interesting

to

operate

the

waveguide

_m

higher modes,

_e-g- m the

TE~O-mode

^{which leads} to _a double

pulse

the

Floquet

states involved thus interfere several times

during

the interaction which _may

drastically

alter the final bound state distribution.

Of

particular importance

_{is an}

expenmental

verification of mechanism

b),

_{i-e- an} excitation of bound states

considerably higher

than the initial _one induced

by

Landau-Zener transitions at broad avoided

crossings.

The _{very same}

phenomenon

_occurnng for atoms irradiated

by

short

strong

^laser

pulses explains

trans1tlons

requiring

_an excitation energy of several Am.

We would like to remark that _m actual

expenments performed

in the presence of a static electric field there _{is an} additional

possibility

of

controlling

the

system:

The «scaled

frequency (that

_is, the ratio of the microwave

frequency

and the

angular velocity

of the

corresponding

classical system ^for

vanishing

_microwave

amplitude)

is then _no

longer given by n(

w but rather becomes a function of the

strength

of the static field _;

furthermore,

the

structure of the Bnllouin _{zone at} A

=

0 will be determined

by

the

energies

of the Stark states.

Thus,

it is _an

interesting

_question whether it is

possible

to observe

dynamical

behavlour

govemed by

different mechanisms when

changing

the

strength

of the static field.

Summarizing,

_we have

theoretically

discussed microwave _excitation expenments on

highly

excited

hydrogen

atoms which _are characterized

by

_a continuous

change

of the

amplitude during

the interaction time. As we have

pointed

out, such _a

time-dependence

of the field

strength

makes the _use of adiabatic methods

indispensable

for _a

transparent descnption.

We conclude that

waveguide experiments

could _{serve as a}

stnngent

test for

Floquet dynamics

when the final bound state distribution is measured _{as a} function of

pulse-shape parameters.

In this _way, microwave expenments ^could

yield important insights

_into short-time

phenomena

m

strongly interacting

quantum systems.

Acknowledgment.

Ii

is a

pleasure

to thank Professor K. Dietz for fruitful

discujsions

and

clarifying

remarks _on the manuscript.

References

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