Transport of quantum states of periodically driven systems

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Transport of quantum states of periodically driven

systems

H.P. Breuer, K. Dietz, M. Holthaus

To cite this version:

Transport

of

_quantum

states

of

_periodically

driven

_systems

H. P.

Breuer,

K. Dietz and M. Holthaus

Physikalisches

Institut, Universität Bonn, Nussallee 12, D-5300 Bonn 1, F.R.G.

(Reçu

le 17 août 1989,

accepté

sous

_{forme définitive}

le 23 novembre

₁₉₈₉₎

Résumé. 2014 On traite des

holonomies quantiques

sur des surfaces de

_{quasi-énergie}

de

systèmes

soumis à une excitation

_périodique,

et on établit leur

_topologie

_globale

non triviale. On montre

que cette dernière est _{causée par des transitions}

_diabatiques

entre niveaux au _passage

d’anti-croisements serrés. On donne brièvement

_quelques

_{conséquences expérimentales}

concernant le

transport

adiabatique

et les transitions de Landau-Zener entre états de

_Floquet.

Abstract. 2014 We discuss the

transport of _quantum states on

_quasi-energy

surfaces of

_periodically

driven _systems and establish their non-trivial structure. The latter is shown to be caused

_by

diabatic transitions at lines of narrow avoided

_crossings.

Some

_experimental

_consequences

pertaining

to adiabatic _transport and Landau-Zener transitions among

Floquet

states are

_briefly

sketched. Classification

Physics

Abstracts 03.65 - 32.80 - 42.50

1. Introduction.

Periodically

driven

_{quantum systems}

are of very broad

_physical

interest and are

experimen-tally

readily

realised

_by

_studying

laser/maser - matter interactions.

Among

the many facets of related

_phenomena

we shall address theoretical

_problems

and their

_experimental

repercussions

which

_{specifically pertain}

to the

_regime

of

_strong

laser/maser -

fields ;

strong

field - matter interaction cannot be described

by perturbation theory, non-perturbative

methods have to be

_applied.

The

_{particular problem}

we shall treat in this note is the

transport

of

_quantum

states

_[1]

_by

variation of external

_parameters,

i. e. their adiabatic - diabatic motion.

In order to

_get

a clear-cut

_{physical picture}

let us consider a

_strong

_single

- mode laser field

interacting

with a

_{quantum system}

(atom,

molecule or,

_{generally speaking,}

any

mesoscopic

system)

which is described

_by

the

_following

set of Hamiltonians

where

we

_explicitly

indicated the

_parameter

_dependence

of H :

À --+-

_strength

of the laser/maser field

to laser/maser

_frequency

e1

cp =

arg

-

- phase

controlling

the transition from

linearly

to

_circularly

_{polarised plane}

e2

wave

_{fields ;}

Ho

describes the static

_system,

the

_{dipole approximation}

is not assumed at this state of

argument,

D is the matter current

_density.

Introducing

now the

_région

A in which the external

_{parameters À,}

(J), cp can be varied

_(in

a

given

experiment)

and

we can formulate our

_problem

as follows : a time-variation in an interval of

_length

Ta

of external

_parameters

is

_represented

as a curve r in

_parameter

_space

the

_problem

of

_quantum

_transport

is now to

_study

the behaviour of states, i.e. of solutions of the

_{Schrôdinger equation}

under motions on the curve r.

A

_particularly

_transparent

_description

_{emerges if we} use

_Floquet

states

_{[2, 3]}

as a basis

[4].

We write the

_general

solution of the

_{Schrôdinger equation}

as a

_{superposition}

of

_Floquet

states

where the

_{« stationary »}

states

_{Ua (t)}

_obey

Of course, ua and the

_{quasi-energies Ea}

do

_depend

on the

parameters

R

adiabatic motion of the state

_gi

is characterised

_by

the

_requirement

that the

_{probabilities}

1 aa 12

in

_(1.5)

remain constant for a

_sufficiently

slow

_change

of

_parameters

_along

T and are thus

independent

of R.

Speaking

in terms of

_experiments,

parameter

variations on curves T are well known : the

curves

_necessarily

embedded in

_higher

dimensional

_parameter

_regions

_{appear if in addition}

to a

_varying

field

_strength

the

_{frequency cv}

and/or the

_phase

cp are modulated. Closed

curves T

are of

_particular

interest if at least two of the three

_parameters

_{vary in time such that} a

non-vanishing

surface is enclosed. The ad libitum realisation of such curves is

_doubtlessly

an

experimental challenge,

we

_believe,

_however,

and

_hope

to convince the reader that new

physics

can be uncovered in

_experiments

of this

_type.

The

_requirement

of

_periodicity

for the «

stationary

» states _{ua in the}

_{eigenvalue problem}

(1.6)

entails energy conservation modulo W , i. e.

and the

_quasi-energy

spectrum

is

_arranged

in Brillouin zones. We assume a discrete

_spectrum

of the static Hamiltonian

and

_anticipate

a choice

_{for {e j}}

such that

we then have

Nonetheless,

the

_quasi-energy

_{spectrum {Ea}}

_is,

in

_general,

dense. For our context this fact has as a _{consequence that the adiabatic}

_theorem

cannot _{be proven in its}

_original

form for

periodically

driven

_{quantum systems :}

an essential

_part

in its

_proof

is the

_requirement

that the state to be

_{transported adiabatically}

be

_separated

_{in energy from all other} states of the

_system

[5].

On the other

_hand,

_simple

model calculations

_{[6, 7]}

have

_clearly

demonstrated the

_validity

of adiabatic

_{concepts ;}

more realistic calculations

[8, 9]

pertaining

to the

_problem

of microwave ionisation of excited

_hydrogen

atoms

_[10]

have shown the

_importance

of adiabatic evolution for the

_{interpretation}

of

_{experimentally}

observed features. Numerical evidence of this kind and new

_experimental

_support

_[11]

for the

concept

of diabatic - adiabatic

transitions at avoided level

_crossings

_{may be taken} as sound

_guide

- lines in the

_following

argumentation.

We start

_{by dividing}

the

_quasi-energy

_spectrum

into sets of

_specific

symmetry

of states, sets

which we denote

_{by { ê a } sym}

The

_point

is here that levels

_belonging

to the same

_symmetry

class do not cross

_[12]

in the

_generic

case, avoided

_crossings

(AC’s)

appear as characteristic

patterns.

A

_typical

feature of

_periodically

driven

_systems

is that AC’s _{emerge, in}

_general,

densely

distributed : the

_{set (s/) sym 8>}

JL contains dense

_patterns

of AC’s.

Not all of these AC’s are

_dynamically

relevant for the

_description

of the

_parameter

motion

(1.3).

In a recent

_publication

_[13]

we have described a

_simple

selection

_{procedure : given}

an

with

5 e = level distance in _a Brillouin zone

OT = time which the

_system

spends

in the

vicinity

of the AC

(more

precisely speaking

OT

₌

_{15R 1/1 Ji}

₁

_where

_{1 5R}

₁

is the

_parameter

distance within which the level distance

_{changes by}

a factor of

J2).

The levels connected

_by

these AC’s

determine a finite dimensional

_subspace

of states

_{{ua }}

eff

which,

in a very

good

approximation,

effectively

suffices to describe the diabatic-adiabatic motion of

_quantum

states in

_experiments

performed

with the very

set-up

considered :

where the aa do

depend,

in

general,

on time

during

the motion

(1.3)

on the

_parameter

curve

r.

We are thus led to the discussiôn of a finite set of smooth effective

_quasi-energy

surfaces

It so

_happens

that

_quasi-energy

surfaces in this set

_get

_{very close :}

_they

almost touch

_along

curves

_of,

in

_general,

finite

_length.

A cross section of this

_approach

is a _very narrow

_AC,

_very narrow in the scale set

_{by Ta}

defined in

_(1.3)

and used for the construction of

{ Ua }

eff’

Dynamically,

these narrow AC’s are

ignored

while

moving

on T in the sense that the

system jumps

across from one

_quasi-energy

surface to the

_adjacent

one with

probability

_very

close to one.

_{Geometrically speaking,}

on the other

hand,

this line of narrow AC’s

_plays

the role of a

_twist,

the two surfaces should be visualised as manifolds cut

_along

the line of narrow

AC’s with cross-wise identified

borders,

the cross-wise structure

_being

due to the

_purely

diabatic nature of transitions at these narrow AC’s. To

_get

this

_{point unequivocally}

clear :

having

set the time-scale

_{Ta by specifying experimental}

conditions we are led to a

coarse-graining

[13]

of the

_quasi-energy

_spectrum

and thus to the construction of an effective set of

states

_{{ua} cff’}

narrow AC’s

_imply

diabatic transitions

and, hence,

cross-identification of

quasi-energy

surfaces the

_geometric

_purport

of which we _express

_by

taking

the two surfaces

diabatically

connected as two branches of one

_{geometrical object}

connected in the twisted

manner

_just

described. Needless to _{say, this is}

_again

an

_{effective, however,}

_consequent

construction

_translating

into

_geometry

the

_dynamic

notions of diabatic-adiabatic and

_purely

adiabatic transitions in

_periodically

driven

_{quantum systems.}

Clearly,

for a

_{sufficiently large}

_parameter

_region

M the set

_{ua}

cff

will,

in the

generic

case,

appear as one

_{geometrical entity}

in which all the states _ua are connected to all the other states

by

the above described cut ; the

_assignment

of

_quantum

numbers for its various sheets u,,,, is a matter of convention fixed

_by

the choice of

_parameters

R.

In the

_following,

we shall describe the

_precise

construction of such an « universal

_covering

»

in some detail and

_discuss,

in

_particular,

connections for adiabatic motion and their

implications

for

_generalised

_{Berry phases}

_[14].

Before

_going

into the details we should state

very

clearly

the

logic underlying

our construction :

_starting

with the

_{heuristically}

determined

effective set of

_dynamically

relevant

_{states {ua} cff}

which suffices to describe the

_dynamics

of

structures

_responsible

for diabatic - adiabatic motion. It should become clear to the reader that even basic

_experiments

on this

_type

of

_questions

need such a framework for an

_adequate

theoretical

_explanation.

The

_following

two _{sections will first introduce the necessary} mathematical

_concepts

and then discuss their

_{physical implications.}

2. Formal considérations on diabatic-adiabatic motion.

Motion on a curve r in _{parameter space}

of a

_system

driven

_by

a

_single

mode laser/maser field is

_{govemed by}

two time-scales : the

laser/maser

_period

which

is,

of course, in

_{general, T dependent,}

and the time

_Ta

_required

to cover the curve

T. We treat these scales

_{separately by}

first

_defining

the

_{eigenvalue problem}

_(1.6)

_explicitly

as

e.g. for a non-relativistic n-electron

_system,

3C R

acts on functions _ua

_(rl,

..., rn,

t).

The

« slow » motion on r is

_integrated

_[8]

_by

means of

w

_{(t ) :}

= e

_{(t, t )}

is then a solution of the

_{Schrôdinger equation}

_(1.4)

_including

the

_parameter

variation on r.

For every

R (T )

the set of effective

_{states {ua} cff}

_spans a finite dimensional

_complex

vector

space

z

The fundamental mathematical structure

_[15]

_underlying

the diabatic - adiabatic motion of

our

_system

is the N-dimensional

_(N

= dim

FReff)’

complex

_vector bundle B _over the base space

M

The

_projection

defines fibres

Because _{of the very construction}

_{of { ua} eff’}

to a

_{high degree}

of accuracy, motion on

r with the fixed scale

_Ta

never drives the state of the

_system

out

_{of 9 ;}

motion

from,

say,

We write a solution

_{et> a}

_{(T )}

of

_{(2.4) (we}

do not indicate the

_t-dependence

_explicitly)

_obeying

the « initial » condition

as

and

_stipulate

that

and

Inserting

into

_(2.4)

and

_demanding

that the

_projection

of the

_right-hand

side onto the fibre

_{F,,R4’) vanish}

we obtain

the

_equation

for the

_transport

matrix

_{Uf3a (T)}

thus

_defining

_{the gauge field}

A

on the bundle t which effectuates

_parallel

_transport

of

_quantum

states for

_parameter

motion on r.

_Explicitly

we find

where

is the scalar

_product

_{in the Hilbert space}

_underlying

the

_{eigenvalue problem (2.3),}

a(J)

and dots indicate differentiation with

_respect

to (J) and T

_respectively

which,

when

acting

on

_periodic

functions with

_period

T = 2 TI’ / (J) (T),

acts

_only

on the Fourier

_components,

the

shift of the Fourier basis

_{being already explicitly}

taken into account in

_(2.13).

Introducing

the scaled variable s =

T /Ta

we see that the first two terms in

_(2.13)

are

proportional

to

_Ta

whereas the last term is

_independent

of the scale

_Ta

_(if

the curve

r is

_kept

_fixed)

and defines a hermitian connection 1-form

hermiticity

is immediate because of

_{« uo 1 uy >}

=

S f3’Y ; .ae

can be taken as a

_Lie-algebra

valued 1-form :

A concurrent

_{gauge-structure}

now _emerges

_{[16]. Changing}

the basis in the fibre

Feff

induces a transformation of the connection 1-form

The

_{corresponding}

curvature 2-form

reads

_explicitly

We now consider closed curves

_T,

i.e.

and write the formal solution of

_(2.12)

as

where we

_decomposed

_(2.13)

and used the

_symbol

3’ to indicate the

_{T ordering}

to define the

_exponential

_integrals

in

_(2.21).

The holonomies

_{U(l’) ;}

i.e. the

_images

of all closed curves in

_parameter

_space

_mapped

into

the set of solutions of our

_transport

_{equation (2.12)}

or,

rather,

the

holonomy

group

’

closed curve

_{through :}

contain information on the

_topological

structure of the

bundle t ;

the

_{non-triviality}

of

g,

for

instance,

is related to its curvature.

From

_(2.21)

we have

Assuming

the existence of a surface S on which A is

_smoothly

defined such that j’ = aS _we find from Stokes’ theorem for the

geometric phase

factor

To recall a familiar

_example,

let r =

S

1 and

S±

denote the upper and lower

half-sphere

connected to it. We then have

and, hence,

The

_quantisation

_[15]

of the first Chern class

again

appears as a

_consistency

condition on the

_holonomy

U(T ).

We now come to the essential

_point

of our

_discussion,

_namely

to establish a relation

between the time-scale of the motion on T and the form of

energy-surfaces

in the

corresponding parametrisation

R =

_{(À, (J), cp) (1.2).}

The situation

_{qualitatively}

shown in

figure

1 is

_{relatively uninteresting.}

The

_quasi-energy

surfaces are

_{separated by}

finite,

approximately

constant _{gaps which} are bounded from below such that for

_{sufficiently large}

Ta

the adiabatic limit is attained. The

_non-diagonal

elements of the

_{gauge-field A}

are

exponentially

small such that the

_equations

for the

_transport

matrix

_{Ul3a (T)}

_{decouple :}

Fig.

1. - A sketch of

quasi-energy

surfaces _tat’ ..., taN

plotted

as functions of parameters

and

where

Initial states

_propagate

_{independently}

with their « own »

Berry phase

[14],

the

holonomy

group is

simply

the direct sum of N

_copies

of

_{U ( 1 )}

A richer case is

_{qualitatively}

shown in

_figure

2. The two

_quasi-energy

surfaces can be visualised as

_being

constructed

_{by moving}

a double cone

_along

some line of finite

length

and

then

_cutting

out a small

_{region containing}

this

line,

smoothing

the cut, of course. The

_typical

feature is now that our

_system

_e.g. moved

along

the closed

loop F depicted

in

figure

2

Fig.

2. -

Quasi-energy

surfaces _Eal,

_{Ea2 plotted}

as functions of k, co

showing

narrow

_{(S1 )}

and broad

(S2 )

avoided

_crossings

in cross-sections

_Si

and

_S2.

The line

_separating

the two « roofs » has to be

visualised as

_separating

them

_by

a line of narrow AC’s as indicated in the cross-section

Sl.

For time scales such that transitions induced

by

e.g.

moving

the _system in

_Si

are

_(almost)

_completely

_diabatic,

opposite

surfaces of these roofs are identified and a _square-root like Riemannian surface appears

giving

an effective

description

of our _system. A closed _parameter curve r

_{passing through}

a narrow and a

encounters two _{very different AC’s :} a _very narrow and a broad one, the

corresponding

cross-sections

_Si

and

_S2

are likewise shown in

_figure

2.

We now demonstrate the relation between the time

_Ta

needed to run

_along

r and the

geometry

of the bundle 6.

i)

Certainly,

there is a scale

_Ta

such that the

system

passes

Si

as well as

_S2

_{adiabatically}

and

the

_geometry

of 6 is characterised

_by

_(2.30),

the situation

_being

identical to the one shown in

figure

1.

ii)

If we now increase the

_speed

in

_traversing

_r,

i.e. decrease

_Ta,

we shall reach the

_point

where diabatic-adiabatic

_{(Landau-Zener)}

transitions occur at

_S1.

The

_transport

matrix obtains

non-diagonal

elements ;

the

_holonomy

group has a non-abelian structure.

iii)

A further decrease of

_Ta

leads to

_(almost)

_pure diabatic transitions at

_Si

and adiabatic transitions at

_S2.

In the latter case the

_system,

_starting

on one

_quasi

_energy

_surface,

_jumps

on

the other surface and remains there until the

_{starting point}

on the

_loop

T is reached

_{again :}

in

traversing

the

_loop

T the

_system

has

_changed

its

_quantum

state

The energy surfaces _{Eal and Ea2} are thus

melted,

under the

_premiss

that time-scales be such

that the line of narrow AC’s

_along

the double-rim

_displayed

in

figure

2 cannot be resolved and

purely

diabatic transitions take

_place,

into one surface which

has,

topologically

speaking,

the

structure of a

_square-root

Riemannian

surface,

the outermost

_point

in

_figure

2

_being

the

branch-point.

Hence,

the diabolical

_point

of a double-cone structure can turn into a square-root branch

_pÓÎ11t

when the

_system

under consideration is

_periodically

driven with a

sufficiently large

amplitude.

The

_integration

_{of e.g.} the form

_(2.15),

therefore,

does not

encounter _any

_singularity

and is well defined.

Geometrically

speaking,

iii)

means that states

_{corresponding}

to

_parameters

on the line in u1t

tracing

out the narrow AC’s have to be cross-wise identified in order to model the diabatic

jumps

thus

_introducing

non-trivial

_topological

structure into the bundle 6. To see this in some

_detail,

let us

_identify

the fibre

F,,R40)

with

C2

The cases

_(i)

and

_(üi)

with

_(2.32)

then

_correspond

to the holonomies

and

respectively.

Diabatic

_jumps

reflect themselves in the

_geometry

of 6

_locally

as

_topological

structures of

_square-root

_type

Riemannian surfaces.

In

_general

terms, the

_point

to be noted here is that it is the effective adiabatic theorem

which introduces non-trivial structures in the set of holonomies. We see that the bundle 6 in

our

_example

_has,

_roughly

_speaking,

the structure of a Môbius band : to construct the

_identity

element the

_appropriate

_loop

r has to be followed twice. The observation that the notion of

an effective adiabatic theorem contains a time

_scale,

either of intrinsic nature or

_{given by}

the

experimental

apparatus,

leads to the

_{important physically}

observable fact that the

_connectivity

of

_quasi-energy

_surfaces,

tested

_by

holonomies

_{U( ),}

is not an a

_priori

notion but rather

The

_physical

realisation and

_experimental

_implications

of such

_geometrical

features of 8 will

be discussed in the

_following

section.

3. Conclusions.

The

_{non-perturbative}

character of

strong

laser/maser - matter interaction is most

_effectively

taken into account

_{by treating}

its

_dynamics

in a _{space of}

_Floquet

states ; in order to illustrate their

_importance

for

_{quantum transport}

_phenomena

we discuss a

_{simple example}

in a

_separate

appendix.

Considered as a function of external

_parameters

the

quasi-energy

spectrum

has a

complicated

structure even if the static

_spectrum

is discrete : avoided

crossings

are

_densely

distributed over _any

_region

M of

_parameters

which we took as field

_{strength, frequency}

and

polarisation angle

in the case of

_single-mode

laser/maser interaction.

The observation that avoided

_crossings

_{appear in very distinct}

_scales,

extracted from model calculations

_(see,

_e.g.

_figure

1 in

_[13]),

led us to the construction of effective sets of

_Floquet

states { ua }

eff which describe the

dynamics

of an atom, molecule etc. for a

parameter

motion on curves F in it.

_{Physical examples}

are interactions with

_pulsed

laser/maser

_beams,

switch-on

processes,

atomic and

_molecular

beams in cavities etc. which introduce a time-scale

Ta.

This scale set, states

_{in {ua} eff}

are, at a fixed laser/maser

_frequency,

connected

_only

via

avoided

_{crossings producing}

Landau-Zener transitions. Seen in the 3-dimensional

_region

f1,

states in

_{ua}

eff are,

however,

also connected

by

lines of narrow avoided

crossings

where our

system

jumps diabatically

between

_adjacent

states on the time-scale

_Ta.

The set of states

{ua} eff’

hence,

seems to be connected at

_{square-root type}

_{branch-points}

the

_square-root

character

_resulting

from the cross-wise identification of states at narrow avoided

_crossings.

The

_typical

situation is

_depicted

in

_figure

2 where the diabatic transition in

_Si

leads to the

cross-identification ;

starting

at

_parameters

_R(0)

with a state

ua1R(0)

a modulated laser/maser

pulse

of

_{length Ta}

will lead us

_along

the curve r to the state

UR(T.)with

_{R ( Ta )}

=

R(0) :

_we do

not _{end up in the} same state, the _{Ua, a} E

_Xeff

are multivalued functions of the extemal

parameters,

it is

_only

an even number of

_pulses

that will lead us back into the same state.

We took care of this

_{non-uniqueness}

of

_Floquet

states

_by

_constructing

the bundle t defined in

_(2.6)

to

_(2.9)

where the

_assignment

of

_quantum

numbers to the states _{u a} is a matter of choice of

coordinates,

quantum

numbers

_being

a

_{path-dependent}

_concept.

The set

{ua}

eff of effective states is thus assembled into one structure, the bundle 6 which

represents

a

« universal

_{covering »}

of the

parameter

space M. On such a

_bundle,

adiabatic motion is a

well-defined

_procedure,

multi-valuedness is translated into non-trivial holonomies on 8. On the other

_hand,

this _{bundle construction is also necessary for the discussion of adiabatic} and Landau-Zener

_{dynamics : quasi-energy}

surfaces

_{separated by}

avoided

_{crossings leading}

to Landau-Zener transitions in one small

_parameter

_region

will be connected

_by

narrow

avoided

_crossings

in a

_parameter

_{region sufficiently enlarged,}

the latter

_leading

to

_purely

diabatic transitions

and, hence,

introducing

non-trivial

geometry.

There is a

_simple

line of

_reasoning

which

_helps

to visualise the occurence of structures like the one shown in

_figure

2 : choose w such that for

_{given integer n}

and static

_eigenvalues

E J O) ’ E k (0)

and, hence,

_{representatives £a.}

_J and

_£ak

_ion

a Brillouin zone are

_exactly

degenerate

at

À = 0. For

non-vanishing À

these levels will

repel

each other and the

resulting

AC will be

Ta

is the

_{corresponding dynamical}

mechanism.

_Increasing

the field

_{strength À}

leads to an

increase of

5 e,

for 6 E dT = h Landau-Zener transitions become

_important

and the «

branch-point »

in

_figure

2 is reached. We checked this scenario in model calculations similar to the

ones

reported

in

[6, 7].

In

particular,

we considered a

_{one-dimensional,}

_periodically

driven

electron in a box and

_{reproduced quantitatively}

the structure which we

_displayed

in

figure

2 as a

_qualitative,

_paradigmatic

_example

relevant for the construction of a non-trivial

holonomy

(see (2.33)).

We

_{repeatedly emphasised}

that the

_position

of this

_{« branch-point » depends}

on

Ta :

in the

_vicinity

of each such

_point

the wave function

_splits

into a

_{superposition}

of the two wave functions of the AC-levels.

_Thus,

_purely

diabatic and adiabatic motions are

_occurring

in

parameter

regions

M with holes around « branch

points

_», the holes

being

chosen such that

parameters

leading

to Landau-Zener transitions are removed from M.

An

_experimental

verification of elemental structures as the one

_{depicted schematically}

in

figure

2 has to _{cope with the}

_{following boundary}

conditions : first of

all,

the time scale

Ta

faces an _{upper limit which is either due} to

_spontaneous

emission or to the continuous

_part

in atomic and molecular

_spectra

_leading

to a finite life-time of

_Floquet

states. The

_question

at

stake, is,

of course, whether or not the time-scale

_Ta

can be chosen

_{large enough}

for effective adiabatic motion to be dominant.

_Furthermore,

one has to search for

_systems

where

technically

feasible

_loops

r in

_parameter

_{space do} not enclose too _{many branch}

_points

and AC’s in order not to obscure the basic

_{square-root type}

connectedness in

_{{ua} eff}

and the

corresponding

ramifications in the effective space of states caused

_by

Landau-Zener transitions.

This does not mean that the construct of a bundle 6 of

_physical

states is

_meaningless

for the

description

of,

say, microwave ionisation of

hydrogen

atoms

where,

in certain

parameter

regions

4t,

the reaction is dominated

_by

a

_large

number of

_close,

_overlapping

AC’s. On the

contrary,

it is our belief that even a statistical treatment of such

_{agglomerations}

of AC’s has to

take into account the non-trivial

_topological

structure of 8 : an

_{understanding}

of such

experiments

in more detail will

_require

a

_deciphering

of these

_{global geometrical properties}

of

quasi-energy

surfaces.

In

_preceeding

_publications

_{[8, 9]}

we have shown that a

_{simple purely}

_quantum

mechanical

frame for the

_description

of microwave ionisation of excited

hydrogen

atoms and related

phenomena

is constituted

_{by invoking}

Landau-Zener transitions in an effective set of

_Floquet

states :

_typical

scales for the

parameters

of the

_{corresponding}

AC’s and their distributions

over

_parameter

_regions

M are

_responsible

for

typical

features

of,

for

instance,

ionisation

curves. In this paper, we extended these notions

pointing

out non-trivial

_geometrical

structures caused

_by

diabatic transitions which

_prevail

in

_significant

sets of external

parameters.

It is

_doubtlessly

an

interesting challenge

to hunt for

_experimental

verification. On the theoretical

_side,

an

understanding

of such a rich but

_elementally

_quantum

mechanical

scenario in terms of a semi-classical

_approach

to

_classically

_complicated,

chaotic

_phenomena

is

an

_outstanding

_problem.

Remains the

_question

of the continuum. As we have remarked in

(1.10)

to

_(1.13),

a discrete

spectrum

of

_Ho

_leads,

in the

_generic

case, to a dense

_quasi-energy

_{spectrum ;}

a continuous

part

of the

_spectrum

of

_Ho

to a

_quasi-energy

spectrum

comprising

the whole real axis. The

provocative question

arises of how any

computer

simulation of the

_dynamics

of

_periodically

driven

_{quantum systems}

should ever

distinguish

the two cases ! It is

_again

a

_question

of

time-scales and

_{coarse-graining}

and

_leaving

out

_dynamically

irrelevant states which can be

_applied

Acknowledgment.

This work was done in

_part

at the Harvard-Smithsonian

Observatory,

one of us

(K. D.)

would

like to thank Alex

_Dalgarno,

Kate

_Kirby

and

_George

Victor for their kind

_hospitality.

Fruitful

discussions with B.

_Zygelman

are

_{equally acknowledged.}

Appendix.

To illustrate the fact that

_Floquet

states

_provide

the

_appropriate

basis for a

description

of

adiabatic

_transport

in

_periodically

driven

_systems

we

_give

an

_{explicit example}

for an abelian

phase

factor. Let us consider a

_{quantum system}

interacting

with a

linearily

polarized

radiation

mode described

_by

the

_Floquet

Hamiltonian

with variable

_phase

5. It is easy to see that the

_{quasi-energies}

_Sa are

_independent

of

8 and that the

_general

solution of

is

_{given by}

with

_{8-independent}

Fourier modes

_uam.

We now _vary 6 from 0 to 2 ir and obtain from

(2.29)

the

Berry-phase

Since

we

_finally

arrive at

As an observable

_quantity

this

_phase

factor

_does,

for £a -

_c

+ mw, not

_depend

on m, i.e. on the second index in

(1.13).

_Furthermore,

the role of

_Floquet

states for adiabatic

transport

is reflected in the

_explicit

occurence of the

_{Floquet index E a .}

It is instructive to _{compare the}

_expression

_(A6)

with the result of Bamett et al.

_[17].

Considering

a two level

_system

with level

_spacing

_Wo and

_using

the

_rotating

wave

approximation

these authors obtain the

_following

_expression

for the

_phase

where

To compare

(A7)

with out formula

_(A6)

we have to calculate the

_Floquet

indices for the same

system

and obtain

Therefore we have

and we see

(A7)

to be a

_special

case of the

_general

formula

_(A6)

which is neither restricted to a two level

_system

nor to the

_rotating

wave

_{approximation.}

We remark that

_(A6)

could also be of

_experimental

interest in order to

_distinguish

an

avoided

_crossing

in the E -

w-plane

from a real

_{crossing by}

means of a _pure

_phase

variation

since at an avoided

crossing

we have

_{2 ’TT dEa/dW = 0}

mod 2 ?r.

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