Reflecting slow atoms from a damped resonator

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Reflecting slow atoms from a damped resonator

Marco Battocletti, Berthold-Georg Englert

To cite this version:

Marco Battocletti, Berthold-Georg Englert. Reflecting slow atoms from a damped resonator. Journal

de Physique II, EDP Sciences, 1994, 4 (11), pp.1939-1953. �10.1051/jp2:1994241�. �jpa-00248097�

Classification

Physics

Abstracts

42.50 03.65N

Reflecting slow atoms from

_a

damped resonator

Marco Battocletti

(*)

and

Berthold-Georg Englert _{(**, ***)}

Sektion

Physik,

Universitit Mfinchen, Theresienstrasse

37/III,

D-80333 Mfinchen,

Germany

(Received

17

May

1994, revised 25

July1994, accepted

7

September1994)

Abstract. Previous studies of the reflection of slow excited atoms

entering

_an empty mi-

crowave resonator _are extended

by including

the dissipation of the

photon

_energy into the de-

scription.

For

typical

parameter values, this leads to _a reduction of the reflection

probability by

a factor of ten. Somewhat

unexpectedly,

_we also find that the transmitted atoms may

gain

in kinetic _{energy as a} result of the

photon damping.

1. Introduction.

The center-of-mass

(CM)

motion of _very slow two-Ievel atoms, ^{which interact}

predominantly

with _one mode of the

quantized electromagnetic

field in _a microwave resonator of

ultrahigh quality,

_was the

subject

of two recent

publications.

Since the

strength

of the interaction

changes enormously

when the atom enters the

cavity,

_a

position dependent

force arises which affects the atomic CM motion. One finds

that,

if the interaction with the

privileged photon

mode is

resonant

[I],

the

cavity

_may act like _a

semitransparent

mirror for the atoms and reflect

50%

of

them; if, however,

the interaction is well off _resonance

[2],

^{either all} atoms could be reflected

or none,

depending

on the

sign

of the

detuning.

These

previous investigations

_were facilitated

by

_a number of

simplifying assumptions.

In

particular,

the

damping

of the

photon

field _was

ignored,

which _can

hardly

be

justified. For,

the

velocity

of the atoms must be about 5

mm/s

_or less

(I],

_so that it would take them _a few seconds at least to traverse the resonator which is _a

couple

of centimeters in

length.

This is _a rather

long

time _on the scale set

by

the

photon lifetime,

which may be _as

long

_as half _a second in _{a resonator} of _very

high quality,

but _more

typical

lifetimes _are

considerably

shorter.

Clearly,

in _a realistic calculation the

photon damping

must be taken into account

(and possibly

other

(*)

Now at Institut for

Physikalische Chemie,

Universit£t

Mfinchen,

Theresienstrasse 41, D-80333

Mfinchen, Germany.

(**)

Mailing address: Sektion

Physik,

Universitit Mfinchen, Am Coulombwall I, D-85748

Garching,

Germany.

(***)

Also at Max-Planck-Institut ffir

Quantenoptik, Hans-Kopfermann-Strasse

I, D-85748

Garching,

Germany.

additional

dynamical

contributions _as

well).

It is the

objective

of the

present

_paper to

report

such _{a more} realistic treatment.

Another

simplification adopted

in references

[I]

^and [2] is the

disregarding

of other atomic transitions than those between the two

masing

levels that _are

strongly coupled

to the

privileged

mode of the resonator.

Clearly,

if such transitions

happen

at rates that _are

large

_on the scale set

by

the interaction

time,

their

dynamical

effect would be

quite significant.

In the

present

paper,

however,

_we shall

pretend

in the tradition of references

[I]

^and _[2] ^{that the} atom is well modeled

by

two relevant internal states. More about this will be said in the _summary.

2.

Equations

of motion.

We

adopt

the notational conventions of reference

[Ii.

The excited _state of the two-level atom is called _a, the deexcited _one is

fl;

the

operators

_{a =}

(fl) (al

and

at

=

la) (fl(

^{effect the} atomic

transition;

the

energetic

level

spacing

is

h(Q

₊

A),

where Q is

(27r times)

the

photon frequency

and A denotes the

detuning;

^the

photon

ladder

operators

are a and

at;

_{the CM}

_position

and momentum operators are x and _p. The state of the

system

is

specified by

_a

density

operator P(t;

_{z, p; a,}

at

a,

at

_{which is}

a function of all

dynamical

variables. The

explicit

time

dependence

of P is determined

by

the master

equation

~ P

=

I(H, _Pi

_{+ CP}

_(2.1)

The

unitary

evolution involves the Hamilton

operator

H =

(

+

hoaia

+

h(Q

₊

A)aid hg(x)(ala

₊

aai), _(2.2)

m

where _m is the _mass of the atom and the

spatially dependent

Rabi

frequency g(z)

_measures

the

coupling strength.

Inside the resonator, that is: in the range ⁰ < z <

L, g(x)

is

relatively large,

outside it vanishes. The

nonunitary

evolution

generated by

the Liouville

operator £,

£P

=

(aiaP ^2aPai

₊

Paia), _(2.3)

models the

photon damping

in the usual _way,

whereby

the

symbol

A denotes the

decay

rate of the

photon

number. For

simplicity,

_we

opt

^for a

photon

reservoir with temperature zero, so

that the resonator does not contain thermal

photons.

We shall restrict the discussion to situations in which the atom is

initially

excited and

approaches

the resonator from the

negative

_{x range,} and there _{are no}

photons

in the

cavity.

Thus at t ₌ 0 _we have the

density operator

Pot so(x,p)ala

:

exp(-aia)

:,

(2.4)

where _so

(x, p)

is the initial

(reduced) density _operator

of the CM

motion,

and the

pair

of colons indicates normal

ordering

of the

photon

variables _as usual. This initial

density operator

is _a

product

of three

factors,

_one for each

degree

of

freedom,

because there is _no

entanglement

at t ₌ 0. Both the

probability

for

finding

the atom in the _x > 0 range and the

probability

^for

finding negative

momentum values have to vanish

initially,

which

imposes

the

requirements

tram

(6(x z')so(z, p))

= o for

x'

_>

0,

tram j6(p p')so(x, p)j

₌ 0 for

p'

_< 0.

j2.5)

Here and below the

primed symbols

x' and

p'

denote numbers in

particular, eigenvalues

of the

operators

z and _p and trcm

signifies

the

injunction

to trace _over the CM variables.

A convenient ansatz for the

density operator

is

p ₌

~p(o)

~

l

~(-) ~(++)

_~ ^l

~(+) ~(--)

^l

~i p(+-) /~p(-+) _{(2 6)}

2 2 2 2 ^'

where

~(t,

_z,

p), ~(+)(t,

_z,

p),

and

~t(t,

_z,

p)

account for the

dependence

_on the time and _on the CM

variables,

and

p~°~ =

aai _exp(-a ia)

,

pl++~

=

(at

₊

^at _{)p(°~ (a}

₊

_{a) (2.7)}

are time

independent

_operators of the

photon

variables and the internal atomic variables.

(The

first

/second

+ _on the left-hand side

corresponds

to the first

/second

_{one on} the

right-hand

side.

This ansatz is motivated

by

the

identity

(ala

₊

aai)~

=

ala

+

ala, _(2.8)

which

implies

that the

coupling

term in the Hamilton

operator (2.2)

commutes with the _con- tribution

proportional

to Q. One is therefore invited _to

expand

P into

dyadic products

of the

eigenstates

^of

at,

+

aai,

_«dressed states" of _one kind.

Only

the

eigenstates

to the

eigenvalues

0 and +1 _are needed for the purposes of this paper, and of the nine

possible dyadic products

thereof

only

the five of

(2.7)

show up. In

particular, p(°) projects

to the state in which the atom is deexcited and _no

photons

are in the resonator. For the initial

density operator (2.4),

this state _can

get populated solely

_{as a} result of the

photon damping.

By construction,

the operators

(2.7)

commute with the _sum

ala

+

ala;

_other

_properties

_that

are relevant here _are listed in table I.

Upon tracing

_over the two-level

degree

of freedom

(tr,)

and the

photon

variables

(tra)

_we obtain the

(reduced) density operator

s,

s(t,

_z,

p)

= tr,tra

P(t;

_{z, p; a,}

at;

a,

at)

,

(2.9)

so that _s is _a

weighted

sum of _~,

A(+),

and

A(~),

s = ~ +

(>(+)

+

>(-)). (2.10)

This CM

density operator

^is the

quantity

that _{we are}

mostly

interested

in,

because it contains all information about the CM motion. For

instance,

the

probability

R that the atom is reflected

as a result of the

interaction,

^is

given by

the

negative-momentum

content of _s,

R " trcm

(Q(-P)s(T,

_z,

p))

,

(2.ii)

where q is Heaviside's unit

step function,

and T is any instant after the interaction.

After

inserting

the ansatz

(2.6)

into the master

equation (2.

I _we find the

equations

of motion

~

~

'~

~~~~ ~ ~~~~~~ ~~~~~ ~

(ll

₊

~ti

_~ ^{i ~} _/~

~i)

~~

~D

_A

~ 2 ^'

~

^~ _~ ^{~ ~ ~~~~~~ ~}

~~~~~

~ ~~ ^~ _~ ~~~~~

~~~~), (2.12)

Table I, Fundamental

properties

of the

operators (2.7).

Note in

particular

the

eigenvalues

exhibited in the fourth and fifth

columns;

the

superscripts (++)

_or

(0)

reflect these

eigenvalues.

p tratr«p («'«. PI ta'« +a«')p pta'« +a«') _£p

pt0) ₁ o o o o

~(++) _i

)~~j+-)

~t-+)) +pt++) +pt++)

fjpt++)

pro)~ _~ A(pt+-) ₊ pt-+))

p(--) _i

+)(p(+-)

-p(-+)) -pi--) -p(--)

-§(p(--)

p(0)) ₊

f(p(+-)

+p(-+))

pt+-) o

)(p(++)

pi--)) +pt+-) -pt+-)

f(pi+-)

₊ p(o)~ _~

f~~j++)

_~ ~j--)~

~(-+) _o

+)(p(++)

p(--)) -pt-+) +p(-+I

-~~p(-+)

₊p(o)) ₊ A(p(++) ₊ p(--))

as well _as

~'

~ =

~

j(+)

₊

j(-)

_{+ ~ +}

~i) _{(2 13)}

Dt 4 ^'

where the low dot indicates _a

symmetrized product,

9.ll "

(911

+

~tg), (2.14)

and

~

is defined

by

Dt

~~ _~t~~ ^'m'~~

'

~~'~~~

for _a free

particle

this would be the

comoving

time derivative.

Equations (2.12)

determine the

auxiliary

^functions

A(+)

and ~t, which _are needed in

(2.13)

for the

computation

of

~(t,x,p).

Then

s(t,

_x,

p)

results from

(2.10).

The initial conditions at t

= 0 _are

~(0,z,P)

"

0, A~~~(0,X,P)

"

~lL(°>X>P)

_"

80(X,P) (2.16)

as found

by _comparing (2.6)

with

(2A).

In

passing

_we note that the _reverse

situation,

in which the atom is

initially

deexcited and there is _one

photon

in the resonator, is treated

analogously

with the initial conditions _~

=

0, A(+)

_{= ~t = so} instead of

(2.16).

On _resonance

(A

=

0)

and without

photon damping (A

=

0), equations (2,12)

_are

decoupled.

This is the _case treated in reference

iii.

The

equation obeyed by ^A(+)

is then _a

unitary

master

equation

^with a Hamilton operator

p~ /(2m)

₊

hg(x).

If _we take

g(x)

to be

positive,

_{as we} shall do

throughout,

then this

component

encounters _a

repulsive potential

and is

reflected, provided

that the initial kinetic energy is not sufficient to cross the barrier. The component

A(~), by

contrast, ^evolves

unitarily

with _a Hamilton

operator p~/(2m) hg(x).

It _moves

through

_an

attractive

potential

and is transmitted. As mentioned

above,

the _~ channel

corresponds

to

having

the atom deexcited and _no

photons present. Consequently,

there is _no

potential

_energy

in this

channel,

and in the absence of

photon damping

this channel cannot be

populated.

indeed,

in this A ₌ 0

situation, equations (2.13), (2.16),

and

(2.10) imply

that _~

= 0 and

s =

j(+)

₊

j(~)

_{at all}

_times,

so that _we

immediately

arrive at the conclusion of reference

[1],

tram)jy

that

50%

each of the atoms are reflected and transmitted. We note further that the

nonhermitean _~t and

~ti components

are measures for the

quantum

^{coherence between the}

A(+)

and

A(~)

channels and evolve

accordingly.

Similarly,

with

sufficiently large detuning (usually (A(

» A _must

hold,

but

(A(

»

(g(x)(

is not

required)

and without

photon damping (A

₌

0), equations (2.12), (2.13),

and

(2,16)

_convey

the message of reference

[2].

^{This is} more

easily

_seen after

rewriting

the

equations suitably,

and _we shall return to this _matter in section 4.

For the numerical

treatment,

we represent ^the CM operator functions _{s, ~,}

A(+),

and _~t

by

their

Wigner

functions _{s~, ~~,}

A~f~,

and _~t~. A factor of

e~"~

_{is absorbed in the definition of} A~f~ and _{~t~, so} that

sit,z,p)

_-

sw(t,x',v'),

~(t, Z,P)

_-

~w(t, x', p') j(+J(t,

,

z,

p)

_-

A[+)jt, z', p'~~-At/2

ll(t, z,P)

_-

llw(t,z~, p')e~~"~ (2,17)

is the actual

correspondence

between the

operators

and their

Wigner

functions. Since _{s, ~,} and

A(+)

are hermitean

operators,

the

Wigner

functions _s~,

~~, and A)+~ _are real. The

Wigner

function of

~ti is,

of _course, the

complex conjugate ~t[

of _~t~.

The action of the differential operator

(2.15)

is then

given by

&~&~h~iw ^~~'~~~

For the commutators and

symmetrized products

that involve

g(x),

the WKB

approximations I[g(z), j(+Jj

_{_}

_&~9(~')

^~

_j(+le-At/2

ax'

dp'

^~

9(X)

_{11 -}

9(X')llwe~~~/~ (2.19)

are very well

justified,

because the de

Broglie wavelength

_(+~

10~~cm)

associated with the CM motion is _very small

compared

to the

macroscopic

distance (+~

lcm)

_over which

g(x') changes considerably.

The relative size of the

leading

corrections to these WKB

approximations

is

given by

the

squared

ratio of both

length

scales (+~

lo~~); consequently,

there is no need to go

beyond

the semiclassical relations

(2.19).

The

equations

of motion

(2.12)

and

(2.13)

_are thus turned into

()

+

f(x') ()

Al+~ = (/1w +

/1S)

~ i

)

(/1w

/1S),

()

+

ig(x'))

~t~ =

~

(A[+)

+

Al

I ~

(A(+~ Al

,

(2.20)

t 4 2

and

~

~~ =

~e~~~/~(A(+~

+

Al

_{+ ~tw +}

~t$), (2.21)

where

~

~~~~~ ~

~~$~~~'~ ^~~'~~~

is the force that _a classical

particle

would

experience

ⁱⁿ a

potential hg(z').

The initial conditions

(2.16)

_are

simply

translated into

~w(o,z',p')

=

o, >1+~(o,z',p')

=

-~w(o,z',p')

=

sow(x',v'), (2.23)

and

~~ = ~~ +

-e-At/2(j[+)

₊

j[-)) _(2.24)

2

replaces equation (2.10).

The

requirements (2.5)

_now read

m

dp'sow(x', p')

= 0 for

x'

_>

0,

-m

dx'sow(x',p')

= 0 for

p'

_<

_{0, (2.25)}

or, if _sow is

non-negative,

_more

simply sow(x', p')

= 0 when

x'

_{> 0}

or

p'

_< 0.

(2.26)

The reflection

probability (2.ll)

is available _as

m

, ,

R =

/ _dx' / _{dv' ~~~(jj'~'}

,

(2.27)

-m -m

which exhibits the convention that _{we are}

employing

for the normalization of the

Wigner

functions.

Equations (2.20)

and

(2.21) provide

the basis for _our numerical calculations. In the resonant

case,

they

_can be used _as

they stand;

results _are

presented

in section 3. When the

detuning

is

large enough,

an adiabatic

approximation

_as in reference _{[2] is}

fitting,

which

requires

that the

equations

^{of motion} are rewritten in _a form _more suitable for this purpose. This is discussed in section 4.

To be

specific,

_we shall consider the

63p~/~

-

61p5/~

^{transition in}

^8~Rb

^{that is used in the}

Garching

one-atom-maser

experiments

[3]. The _mass of the atom is then _m

= IA _x

10~~~

_g.

We take _a

cavity length

of L

= 2 _cm and _a

photon decay

rate of A

=

2.5s~~

The initial average ^momentum of the atom is

(p)o

_" 7 _x

10~~~ gcm/s,

which

corresponds

to a

velocity

of 5 _mm

Is,

to _a kinetic _energy of I.I x

10~~~ eV,

and to a de

Broglie wavelength

of 9.5 _x

10~~

cm.

The effective Rabi

frequency _g,

which is related to the

(positive) g(x')

function

by

§L

₌

/dx' ^g(x')

,

(2.28)

is chosen to

equal

g = 44 kHz.

Indeed,

the energy

fig

₌ 2.9 x

10~~~

_{eV associated with the} interaction is

markedly larger

than the kinetic energy of the

arriving

atom.

A realistic

shape

for

g(x)

would be

~jsin(7rx/L)

_{for 0 <}

x <

L,

g(x)

= 2

(2.29)

0 elsewhere

,

if the

coupling

is to _a TE mode _as is the usual

experimental

situation. None of the conclusions of this _paper

depends crucially

_on the

specific shape

of

g(x).

We shall therefore facilitate the

numerical

integration

of the

equations

of motion

by using

the _more convenient function

e/2

~ ~ "

~cosJ~2 (e(x/~

i

/~~~

(2,30~

because the

resulting

force

(2.22)

is such that the classical

equations

of motion possess an

analytical

solution. These classical

trajectories

_are the characteristics of the

partial

differential

equations obeyed by

A(+~ and

Al

~, and for this _reason

(2.30)

is

advantageous,

_as will be

discussed below. We fix the

parameter

e

by requiring

that

g(x

=

0)

₌

g(x

₌

L)

is of the

27 maximal value

g(x

=

L/2)

₌

ge/2;

thus

cosh(e/2)

₌

/fl.

_Then

g(x)

is

practically

_zero outside the range 0 _{< x <} L. Of _course, similar _e values _are

equally good.

In reference

[I]

^the

simple

_mesa function

has

been

_used.

It

has

its

approximate experimental ^ealization in TM modes.

The

the electromagnetic energy hrough the openings, through which the atom _enters and _leaves,

is rather

large

_in this _geometry and

does

_not allow for the very long photon

we eed. evertheless, we

shall briefly consider (2.31)

in

order to

makedirect

contact with

3. Resonant interaction

IA

=

0).

In this section _we shall deal with the _case of resonant

interaction,

_so that _we set A

= 0 in

equations (2.20).

We

begin

with _a reconsideration of the situation of reference

iii,

where the

mesa function

(2.31)

has been used. For this

g(x),

_an

analytical

solution of the

equations

of motion

(in

WKB

approximation)

is

available, provided

that the atom is

initially

well localized and also has _a rather well defined

velocity,

viz. [4]

sw

(T, x', p')

= sow

x'

₊

p'T/m, -p') (3.la)

2

+

(1

⁺

_g2 ~ll14)21~~~ Ill1

~°~

(~' ^l~ Al

_~

_{?'~'/~ ?'l}

(3.lb)

+1

g~

1/~~~

exP

li~ll

_Sow

_(ix' P'T/m) _/P~)~ ~l

(3.lc)

(A/8)~ mL)

~§~ (A/4)~

^~~~

2(P)o

x sow

lx'

⁺

~,i'-

~~

ii

L

P'T/ml ^IF')

^~

,

P~)

,

(3. id)

where jl ^is a momentum _measure for the interaction energy:

fig

=

jl~/(2m).

The first term

(3. la)

is the reflected

intensity.

It _accounts for

50%

of the total flux. Since the A(+~

component

is here reflected _at x'

= 0 without _ever

entering

the resonator, the reflection

probability

is _not affected

by

the

photon damping.

The three other terms in

(3.I)

_are transmitted intensities of various momentum contents. The

component (3.lb)

has the _same momentum distribution _as

the initial CM

density operator

_sow

(x', p');

this is

just

like the

A(~) component

in the

undamped

case, which

speeds

_up when

entering

and slows down when

leaving

the interaction

region.

The

component (3.lc)

has

gained

the amount jl~

/(2m)

=

fig

in kinetic energy

during

the interaction.

This _occurs because the

damping couples

the channels in

(2.20)

and

(2.21),

_so that the

A(~)

component,

^which is accelerated at x'

=

0,

feeds the _~

component,

which

experiences

_no force at x'

= L when it leaves the resonator and is therefore not decelerated. The

A(+)

channel is also fed

by A(~J,

and

A(+)

_is accelerated at x'

=

L,

_so that _a second

gain

of

fig

is

possible

and this

produces

the last contribution

(3. Id).

For the parameter values

specified

at the end of the

preceding section,

the

exponential

factor

exp(- ~AmL/(p)o)

_Qi 0.01 is rather small and the

2

ratio

(A/8)~ /[j~ (A/4)~]

ci 5 _x

10~~~

is

tiny,

and _so the dominant transmitted

component

in

(3.I)

is

(3,lc),

which represents

99%

of the transmitted atoms. The

remaining

I$lo _are in the elastic channel

(3. lb). Thus, owing

to the

photon damping

almost all atoms

gain fig

in kinetic energy when

traversing

the resonator.

This

example

demonstrates _an

intriguing

constructive effect of the

photon damping:

_a mech- anism for

converting

the

atom-photon

interaction energy into kinetic _energy is

provided.

In

principle,

this effect is also present for _{a more} realistic

g(x'),

such _as

(2.30).

Since the

entering

and

leaving

is _a much less dramatic event for such _a smooth

g(x'), however,

the

periods during

which the forces of

differing signs

and

strengths

_are

acting

_are not well

defined,

and there- fore momentum transfers of various amounts are

possible.

As _a consequence, the _momentum

distribution is broadened rather than

split.

For

g(x')

of

(2.30)

_we have

integrated (2.20)

and

(2.21) numerically.

The classical

phase

space

trajectories

_{T -}

(X(+)(T; x', p'), P(+)(T; x', p'))

that _are determined

by

those solutions of the classical

equations

of

motion,

~

x(+)

=

p(+)

^~

p(+)

= +

f( x(+)) _{j3 2)}

aT _m ^' aT ^'

which _run

through ix', p')

at time _T

=

o,

(X(+~, P(+~)

₌

ix', p')

_for

T =

0, (3.3)

are used to turn the

partial

differential

equations

for A)~~ into

ordinary

differential

equations,

~

A[+~

(t

_{+ T,}

X(+~,P(+~)

=

~

~tw

+

~t$) (t

_{+ T,}

X(+), P(+~). (3.4)

These _are

supplemented by

the

equations

for _~tw and _~~,

~ r

exp

(2i /dT'g(x'

⁺

^{p'T'/m) /Jw(t}

+ T,

x'

₊

p'T/m, p')

dT

o

r

=

~

exp 2i

/dT'

g

(x'

+

p'T'/m) _(A(+)

₊

Al ⁾ (t

_{+ T,}

x'

₊

p'T /m, p')

,

(3.5)

4

and

~

~wjt

_{+ T,}

^x'

+

p'T/m, p')

dT

~

A

~-A(t+r)/2 (j(+)

⁺

jj-)

₊

~~ +

~j) ^(t

_{+ T,}

^x'

+

v'T/m, v') _13.6)

4 ^~

16

,~",

b

~ ,.,~' ^",

i a@@=i;[

E _".,

~

o

..:...().

~~

)

-f~~~c

lo

10 0 20 30

x' (mmj

Fig.

I. Evolution of

sw(t, x', p')

for

g(x')

of

equation (2.30)

in the absence of

photon damping

and

for _{resonant atoms.} The plot shows

(a)

the initial distribution

(3.7)

at t

= 0 before the atom enters the resonator; the distributions

during

the interaction at

(b)

t

= 1.8s and

(c)

t ₌ 3s; and

(d)

the

final distribution after the interaction at t ₌ T

= 4.5s. The contour lines enclose

20%, 50il,

and 80il of the total

probability.

The dashed _curves mark the classical

phase

_space

trajectories

that start in

((x)o, (p)o)

at t ₌ 0,

computed according

to

equations (3.2)

and

(3.3).

The x' _{range covered is}

-10 _mm < x' < 30 _mm, and the

p'

_range is -1.4

x10~~~ gcm/s

<

p'

< 2.24 _x 10~~~

gcm/s,

which

corresponds

to _a velocity _range of

-10mm/s

<

p'/m

< 16

mm/s.

For 0 < x' _{< L}

= 20 _mm the atom

is inside the resonator.

Upon integrating

from _T

= -dt _{to T}

= 0 with the aid of the

trapezoidal rule,

_{one can express}

A(~~,

_~tw, and _{~~ at} time t in terms of their values _at time t dt.

This,

in essence, is the

relatively simple

_manner in which _we have

computed

the numerical solutions. The

g(x)

of

(2.30)

is

advantageous

for this purpose because it allows for

analytical

solutions of

(3.2).

In

particular,

_a numerical

integration

is not needed at all for A

= 0.

This

analytical

solution for A

= 0 is

reported

in

figure

I for reference. The

plot

shows

s~(t,x',p')

before

(t

=

0), during it

= 1.8s and _t

=

3s),

and after

it

= T

=

4.5s)

the interaction. One

clearly

_sees that _s~ consists of _two

components:

^the

A(+)

branch that is reflected and the

A(~)

_{branch that is} transmitted. Here and in the

following figures

the initial

distribution is

always

the _same

Gaussian,

~°~~~" ?'~ /~6p

~~~

l~

^~'

^~~~~

^~

^{~' j~~~~ ~}

_~~'~~

with the

expectation

^values

(x)o

_" -3 _mm and

(p)o

" 7 x

10~~~gcm/s

_as well _as their

spreads

fix

= I _mm and

6p

₌ 7 _x

10~~~ gcm/s.

This

corresponds

to an initial

velocity

of

(p)o/m

=

smm/s

with _a

spread

of

6p/m

₌

0.smm/s.

Of _course, this _sow does _not vanish

exactly

when x' _> 0 _or

p'

<

0,

but the conditions

(2.25)

_are

obeyed

with

sufficiently high

accuracy.

In

figure

2 _we present

sw(t, x', p')

when the

photon damping

is taken into account. In contrast to the A

= 0 _case, the reflected

intensity

is

strongly

reduced. Rather than R

= 50$lo, we obtain

16

(~)

10 0 20 30

X'

(mmj (b)

10 0 ?0 30

x'

(mni]

16

(c)

~

I i

^°

10

10 0 20 30

x'(mmj

Fig.

2, Evolution of

sw(t, x', p')

with photon

damping

for resonant atoms. The initial distribution is the _{same as} in

figure

I. We show the distributions at the instants

(a)

t ₌ 1.8 _s,

(b)

t ₌ 3_s, and

(c)

t = T

= 4.5 _s.

Along

^the contour

lines,

_sw

equals 20il, 13il, Gil,

and 0.4il of the maximal value at t ₌ 0. The x' and

p'

_{ranges are} the _{same as} in

figure

1.

a reflection

probability

^{of R} =

4.8$l.

The situation is here different from that of

eqaution (3.1)

because the _atom has to enter the resonator before it _can

get exposed

to the force

generated by (2.30)

in its full

strength.

^But _as soon as the

atom-photon coupling produces

_a

nonvanishing photon

number

(ala),

_the

_{photon damping}

_becomes effective and the force-free _~ channel is fed _at the _expense of the

A(+)

and

A(~)

channels. As _a consequence, the

repulsive

forces do not act

long enough

to _reverse the atom's

velocity

with _a

large probability.

At the final time t ₌ T ₌ 4.5 _s, there is still _a considerable

probability

for

finding

the atom inside the resonator.

Nevertheless,

this instant is "after the interaction" in the _sense of

equation (2.11)

because

only

~~ is

nonvanishing

^{inside the}

cavity,

_so that the forces have

already

ceased to act. Please note

also how the momentum distribution is smeared out. This is the smoothened version of the

splitting

that _we have observed in

(3.1).

4. Nonresonant interaction

IA

<

(A(

<

j).

The terms

proportional

to A in

(2.I)

_can be included _on the

right-hand

sides of

(3.4)

and

(3.5).

But when the

detuning

A is

markedly larger

than the

photon decay

rate

A,

these terms _are troublesome in _a numerical

integration,

^because

they

lead to rather

rapid

oscillations. In other

words,

the "dressed-states basis"

(2.7)

is inconvenient under these circumstances.

Therefore,

we switch from A(~~ and _~t~ to another set of

phase

_space functions

I)+~, Al

_{~, and}

_ji~

in accordance with

jj+)

₊

jj-)

=

jj+)

₊

jj-)

,

>[+~ Al

=

(i[+~ i[ ₎₎

cos ~2 +

(jiw

⁺

ji$)

sin _~2,

~tw +

~t$

=

(jiw

⁺

ji$)

_{cos ~a}

(I[+~ i(

~) sin ~a,

vw

AS

=

fiw AS

,

14.1)

where the

angle ~a(x')

is defined

by

g(x')sin(~(x'))

₌

( _cosl~(x'))

_with

<

~(x')

<

(4.2)

The

corresponding

linear combinations of the

p(++) _operators

in

(2.7)

_are

dyadic products

of the _x

dependent eigenstates

of

g(x) (ala

+

ala) ^hat,

₌

g(z) [ala

⁺

ala (aid aai)

tan

(~a(x))j

~

(4.3)

2

the relevant

eigenvalues

_are

~

+

g(x)/I

₊

^tan~ _(~a(x))

=

~

+

~/(g(x)]2

+

(A/2)2 (4.4)

These

eigenstates

are the "dressed states"

employed

in reference _[2] for _a

study

of the _nonres- onant situation in _an adiabatic

approximation.

The substitution

(4.I)

turns

(2.20), (2.21),

and

(2.24)

into

(D

⁺

^/(zi)

^~

^j(+)

=

+~j(+)si~~

Dt

dp'

~ 2 ^~

(~

^{cos ~a +}

^f(x')

^{sin ~a}

~ +

~ ~~ (fi~

+

#*

2 2

~P'

_m dX' ^~ ,

()

+

i§(x'))

fiw

=

A

cos ~a +

f(x')

sin

~a

~

(I(+~

⁺

i( ⁾

t 2

dP'

+

P' ~l~ j(+) j(-)) (4 5)

2 _m ax' ^{~ ~ '}

and

D

~

A

~-At/2 jj+)

~

jj-)

_~

_~

_~

_~* (4 6)

Dt ^~ 4 ^{~ ~ ~ ~}

as well _as

sw = ~w +

-e~~"~

(i(+~

⁺

i[ ⁾

,

(4.7)

where,

consistent with

(4A), 1(x')

=

g(x') cos(~(x'))

+

)

_sin

_(~(x'))

=

~/lg(x')12

₊

(A/2)2 (4.8)

is the effective local Rabi

frequency,

and

f(x')

=

f(x') C°S(~2(x'))

₌

-h£§(x') (4.9)

is the effective force associated with it.

The adiabatic

approximation

of reference [2] ^{amounts to}

discarding

those terms on the

right-hand

sides of

(4.5)

that involve

multiplications

with

p'

_or differentiations with respect to

p'.

This is well

justified

for the

typical experimental parameters

^that we have in mind.

For _a

detuning

of A

=

+looHz,

_say, which is

large compared

with A

= 2.5 Hz and small

compared

with

g

₌

44kHz,

_we

get

_~a ci

lo~~

inside the

resonator,

so that the relative _size of

f(x')

sin _~a

d/dp' compared

to A _cos

~a is 2

where _we have used that the

product

of

typical

x' and

p'

_ranges is _+~

lo~h

for atoms in _a beam. Likewise _one finds that the

(p'/m)d~a/dx'

terms _are

equally

small. Once these small

contributions _are

neglected

in

(4.5),

the numerical

procedure

of

equations (3A)

and

(3.5)

_can

be

employed

with the necessary

changes.

Inasmuch _as the

limiting

values of

~a(x')

_are

~

/2

for A > 0

~~~~~

g(z')

# 0

~~'~~~

~g(X')

_"

_~

/2

for A < 0

the initial conditions

(2.23)

_appear here _as

~w(o,z',p') =Jw(0,x',v') =o,

I(+~(0, x', p')

=

~~°~~~"?'~ )~ _~'

°~

,

~~ ~~~'~"?'~ sow~x',

p') ~~ ^~~'~~~

16

(a)

~

E E

I

do

~

10

10 0 20 30

x'

[mm]

16

16)

10

10 0 20 30

x' (mm]

ic)

~

E E

~ I

^~

lo

10 0 20 30

z'

(mmj

Fig.

3. Evolution of

sw(t, x',p')

with

photon damping

for nonresonant atoms with

a

positive

de-

tuning

of 6

= 100 Hz. The initial distribution is the _{same as} in

figure

1. We show the distributions at the instants

(a)

t

= 1.8s,

(b)

t

= 3s, and

(c)

t

= T

= 4.5s. The contour lines have the _same

significance

_as in

figure

2. The x' and

p'

_{ranges are} the _{same as} in

figure

1.

The close

analogy

between

(4.5)

and

(2.20)

tells _us that the

I(+~ _component

is reflected and the

Al _component

_is transmitted for A

=

o,

and thus _{we see} that all _{atoms are} reflected for A _> o and all transmitted for A < o if there is _no

photon damping.

This

is,

of _course, what

has been found in reference [2].

In

figure

3 _we show the results of _a numerical

integration

of

(4.5), (4.6),

and

(4.7)

in the adiabatic

approximation,

for _a

positive detuning

of A

= +100 Hz.

[For

the sake of

simplicity,

we have set

§(x) equal

to the

g(x)

function of

(2.30),

_so that _{we can} make _use of the _same classical

trajectories

_as in the resonant situation. The

g(x) corresponding

to this

§(x)

differs but little from the _one in

(2.30).]

^The

comparison

with

figure

2 identifies two main effects of the

detuning: ii)

it increases the reflected

intensity

to R

=

9.1$l,

that

is, roughly by

_a factor of

two; iii)

it suppresses the

large p' contribution,

which would

signify

_a

gain

in kinetic _energy, in the transmitted

intensity.

This is _as

expected.

We thus find _a reduction of the reflection

probability by

_a factor of

ten,

^both in the resonant _case

(4.8~

rather than

Soil)

and in the nonresonant _case

(9.1%

rather than

loo%).

5.

Summary.

We have extended

previous

studies

[1,

2] ^{of the reflection of slow atoms from} a resonator

(that

does not contain

photons initially) by including photon damping

into the

description.

This is necessary

because,

for

typical experimental parameters,

the

atom-photon

interaction lasts for several

photon lifetimes,

_so that the

dissipation

of the

electromagnetic

_energy cannot be

neglected

in _a realistic treatment.

We have found that the

photon damping

leads to _a reduction of the reflection

probability, roughly by

_a factor of ten. Rather

unexpectedly,

_we have further observed that transmitted

atoms may

gain

in kinetic energy. This

phenomenon

is well understood _{as a}

dynamical

_conse-

quence of the

coupling

between otherwise elastic

channels, brought

about

by

the

photon damping.

We have

consistently

referred to the parameters of the

Garching

micromaser

experiments

_[3]

in order to stay in close touch with reference [1].

Now,

since that

requires

atomic CM velocities of about 5 _mm

Is,

_a reflection

experiment

of the discussed kind cannot be

performed

within the

gravitational

field _on the surface of the earth. As mentioned also in reference _{[2] a}

microgravity

environment would be

mandatory.

But

perhaps

_progress in

increasing

^the

quality

of

optical

resonators will _one

day

enable the

experimenters

to realize _an

analogous experiment

with _an

optical

transition. Then the much

larger

Rabi

frequencies

would allow for the _use of faster atoms and the

gravitational pull

would _no

longer

be

disturbing.

The _use of _an

optical

transition rather than _a microwave transition between

Rydberg

states would offer additional benefits. The deexcited state of the

optical

transition _can be identical with the true

ground

state of the atom. Then transitions that lead out of the

masing

two-level

system briefly

mentioned in the Introduction would be of _{no concern.} Of _course, the

lifetime of the excited state, which results from spontaneous ^{emissions into other modes than} that of the

optical resonator,

must not be short

compared

with the interaction time needed for the reflection. In contrast to the situation dealt with

explicitly above,

ⁱⁿ such _an

optical experiment

_one would

presumably prefer ground-state

atoms

entering

_a resonator

containing

one

photon (or

_a few

photons perhaps).

In view of the remark after

equations (2.16)

this would not constitute _a difference in

principle.

Acknowledgments.

We would like to thank G.

Rempe

^{for valuable} discussions.

References

iii Englert

B.-G.,

Schwinger

J., Barut A-O- and

Scully

M-O-,

Europhys.

Lett. 14

(1991)

25.

[2] ^Haroche S., ^Brune M. and Raimond J-M-,

Europhys.

Lett. 14

(1991)

19.

[3] ^{A recent review is}

given by

Raithel

G., Wagner

C., Walther H., Narducci L-M- and Scully M-O-,

Cavity Quantum Electrodynamics,

P-R- Berman Ed.

(Academic Press, 1994)

_pp. 57-121.

[4] Battocletti M.,

Diplomarbeit,

Universitit Mfinchen, 1993

(unpublished).