Atom interferometry with arbitrary laser configurations : exact phase shift for potentials including inertia and gravitation

(1)

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Atom interferometry with arbitrary laser configurations : exact phase shift for potentials including inertia and

gravitation

Jürgen Audretsch, Karl-Peter Marzlin

To cite this version:

Jürgen Audretsch, Karl-Peter Marzlin. Atom interferometry with arbitrary laser configurations : exact

phase shift for potentials including inertia and gravitation. Journal de Physique II, EDP Sciences,

1994, 4 (11), pp.2073-2087. �10.1051/jp2:1994248�. �jpa-00248110�

(2)

Classification Phi'.ii<..I Ab.itia< _t-I

42.50 04.90

Atom interferometry with arbitrary laser configurations

_:

exact

phase shift for potentials including inertia and gravitation

Jiirgen

Audretsch and Karl-Peter Marzlin

FaLultfit for PhysiL der Universitit Kon~tanz, Po~tfach 5560 M 674. 78434 Kon~tanz. Germany

(Ret eiied /4 Malt h /994, a«eptet/ 6./iih /994)

Abstract. We ,tudy _in _a general treaimeni the influence oi _a cla,, of perturbation~ acting on the

atom, ot _an atomic interferometer. An _exact

expre,,ion

for the re,ulting ;hift of the interference

pattern ^I, given for _an

arbitrary

number of la;er _zone; with

travelling

_or ~tanding _wave~. The

a;;umption

made i~ that the two-level model _is

appropriate

for the description of _atom~ and that the perturbation~ _can be neglected _in the la~er _zone~. As _an important application _we calculate the influence of acceleration, rotation, and ~pace-time curvature on the Ramsey interferometer and the

atomic fountain. The me»urability of the re~pective pha~e ~hift~ i~ di~cu~sed.

1. Introduction.

The idea of

using

lasers _a~ beam

splitters

for atomic interferometers _was

developed by

Bordd in 1989. Due to momentum conservation atoms can

gain

or loose a momentum of k

it

they

absorb _or emit _a

photon

with wavevector k. This effect _can be used to

split

and

recombine _an atomic beam. The interaction with a laser beam which is tuned to be

nearly

in

resonance with _a certain atomic transition _can be described within the two-level model of

atoms. This _was used to calculate the

complete fringe

pattern ⁱⁿ ^the

Ramsey

spectrometer

which consists of _two

pairs

of

counterpropagating

laser _waves

[2).

The atomic fountain

device

[3)

_may be treated _as _an effective two level system.

There _are extended efforts _to

;tudy

the influence of

electromagnetic

fields of laser beams _on

moving

atoms with the intention to

design

mirrors and beam

splitters

in _an atomic

interferometer. In the

following

_we

study

the

complementary problem

_:

using

a black box

approach

^for the laser _zones _we treat

exactly

the influence of

a class of

perturbations acting

_on

the atoms when these _are

moving through

the dark _zones between the lasers in the

interferometer. Our intention

thereby

is to treat the induced shift of _a

fringe

_pattem for _a set-up with _an

arbitrary

number of la;er _zones with

standing

or

running

waves. We restrict _to _two level _atoms and strive for _a result terms of _a recursion formula which allows _us _to work _out the

influences of the different part~ ^of ^the

perturbation

to any desired order of

magnitude.

^This i~ of

(3)

particular importance

if these _terms differ

largely

in

magnitude.

In thi~ _case _one _must be able to control all the mixed terms which may occur in order to obtain the

phase

^shift ^for a

particular experimental specification

_up to a

given

_accuracy. In thi~ _sense _we

generalize

_our calculations in reference

[4].

There _are at the moment two

potential applications

of _our

general

scheme. The

perturbations

may be unwanted, for

example

^when ^the ^atomic interferometer is used _as _a

frequency

standard.

In this _case _one has to know the _structure of their influence in order to eliminate it. On the other

hand, the

perturbation

_may _go back _to the influence of

acceleration,

rotation, and

space-time

curvature. In this connection the intention i~ to examine whether such _an inertial _or

gravitational

influence _on quantum matter can be detected in the

laboratory using

atomic

interferometry. Interferoinetry

with neutrons

[5. 6],

electrons

[7],

and _atoms

[8, 3]

is at present the

only possibility

to examine this effect. Its relevance for

gravitational theory

has been

discussed in

[9].

For

space-time

curvature we have ~tudied the

measurability by

a

Ramsey

interferometer in reference

II 0].

For the influence of acceleration _on the

phase

^shifts there is _a

difference between neutron and atomic

interferometry.

For neutrons the beam

~plitters

_are

realized with _a

single crystal,

and _a series

expansion

of the WKB

phase

is needed 1-13

].

We will show below that for atom interferometers where lasers _are used _as beam

splitters

the

phase

shift due to acceleration alone has

only

_one part the first order contribution.

The paper is

organized

as follows. In section 2 _we will

explain

the

general

scheme for the

algebraic

calculation of the shift. The idealization of the laser beams is addressed in section 3.

The elaboration of the

complete phase

shift is contained in section 4 and will be

applied

to the

Ramsey

interferometer and the atomic fountain in section 5. On this basis the

measurability

of

space-time

curvature is studied in section 6. A

quick

direct calculation of the

phase

shift if

only

acceleration is present ^is

given

in ~ection 7. We will _use natural units

(h

= c ₌ unless

otherwise stated.

2. General scheme.

In the two-level model the Hamiltonian of the _atoms in the atomic interferometer is

given by

H<oi ~ H~m + H~~~~ d E ix, t

(1)

~~~~

~Lm 6

~

~p

~~~

Hj<om =

HA )

_r ₍₃₁

where _x, _p, and M are the center of _mass

position

and _momentum, and the _mass of the _atom,

respectively. H~

= E~

la) jai

₊

E~ 16) (b

is the internal energy operator ^restricted to the

two atomic energy

eigenstates la)

and

16)

^of ^interest. ^We ^will assume

E~<E~.

r y~

la) jai

₊

y~16) (b[

contains the

decay

factor~ y~, y~ of the two state~. The

coupling

^of the atoms to the laser iield in the laser _zones i~

given

in the

dipole approximation by

d E (x ), where E i~ the electric field of the laser and d is the

dipole

^to ^be ^of ^the ^form

H~

=

Ma _x ₊

R,>

_I,i,,< _"f ."«< "' (~ ~

P).

~~~

(4)

Here and in the remainder _we use the convention that any index of

R,ji,j,,,

which appears a

second time _in the _same formula ha~ to be summed from _to 3. The first _two _terms may ^be

regarded

as the first _terms of _a

Taylor expansion

of _some scalar

potential.

The third _term represents an

angular

momentum

coupling.

In the

applications

discussed below

H~

^will represent ^the ^influence ^of

gravitation

and of the rotation of the reference system. ^In ^this case

a ₌ const. is the acceleration, _w

= const. the

angular velocity,

and

Rot

o~, ⁼ const. are the

components ^of ^the ^Riemann tensor with re~pect to the Fermi coordinates of the observer

[10(.

It is the aim of this paper to

give

_a

general

treatment of the

resulting

shift of the interference pattern ^if

H~

^is ^switched on.

The

general

arrangement ^for an atom interferometer

using

lasers _as beaui

.iplitteis

consists of

an atom beam

passing

_a _sequence of N lasers with

standing

or

travelling

waves and

moving

free of la~er influences in the dark _zone~ between the lasers for time intervals

T,.

The free

evolution is

governed by

the Hamiltonian

li=H~~n+H,~,,~~

of

equation

(I). Because

h

_i~ _time

independent

the

corresponding

time evolution operator is

simply

_exp

[- iliT,

_in _the

I-th interval of free evolution.

Denoting

the evolution operator ^which de~cribes the re~ult of the interaction with the I-th laser

by U~ (I

the

complete

evolution of the _sate of the atoms is given

by

~ (t~

))

=

U~ (N )exp [- iliT~ _]U~ _(N _U~

₍₂

_{)exp [-} iliT, _]U~

_{)exp [-}

iliTo] _~o)

(5)

where

~j,)

is the initial state of the _atoms and _t~ is the final time.

During

its evolution the

ingoing

atomic beam is

split

into _many

partial

path

in

,.

,' ,'

,~,~ i~)j

Tj

T~ T3

""",,~

l 2 3 4

Fig. I. All partial beam~ of _an atomic beam passing the Ramsey device _as they _are predicted in the rotating _wave approximation. The vertical _arrows denote the running la~er _wave~ and their direction.

Dihed lines correspond to excited _atoms. The atoms are

initially

deexcited and travel freely for _time~

T, ^between ^the ^lasers. ^Beam Î ând ÎI are able _to interfere.

(5)

configuration

_space in the quantum ^evolution. ^In ^the

following

_we

single

_out two

partial

beams I and II

resulting

at the final time _t~ in

~,)

and

~,j)

and

;tudy

the

corresponding

interference pattern.

A formal

description

of the

partial

beam~ can be made if _we

decompose

the state vector into its deexcited part ând îts êxcited part. The Hilbert space of _our

problem

is the direct

product

of the two dimensional internal Hilbert space and the one of the _center of _mass motion. We _can

therefore write

)

~~)

~~~

with

[~~)

=

(a) (a (~)

and similar for

(b). Independent

of the details of the atom's interaction with the laser~ in the laser _zones the transition from the

ingoing

matter state to the

outgoing

one can be described

phenomenologically by

~

Y< ~>

where I

= I,

,

N denotes the number of the laser _zone. The component~ ^{of thi~} ^matrix are in

general

operators

containing

_x and p. A more

speciiic

form will be

given

for idealized laser

zone; in the next section. The

physical meaning

of the components ^is ^clear cv, ^occurs if _an

excited atom remain~ excited,

6,

it it continues _to be in the state

a).

If

p,

arise _we know that the I-th la;er ha~ the _atom excited, and _y, describes the proces; of deexcitation. For _a

particular partial

beam

pa,;ing

_a

splitter only

one oi the component; ^of

U~

(I will become effective. It I, therefore obviou; that each total

partial

beam i~

;pecified

it _we write down the

corresponding

;equence oi components. Consider for in;tance the two

interfering

beam; I and II oi

figure

I.

Beam I

correspond~

to the sequence

pi

_y, p~ _cv~ ^and ^beam II I; described

by

_&j _&~

6~ p~.

To concentrate in the calculation _on the relevant term, the

general

scheme in thi; paper will

be the iollow _{in g} for _~ome

configuration

of la;ers and two

given particular partial

beams I and II

resulting

in

~j)

and

~,j)

there is

already

_an interference pattern ^if ^the ^external

potentials

vanish

(H~

= 0 ). This pattern ^is ^assumed to be known. We do not addre;; the

question

which

pairs

of

partial

beam; lead _to _a detectable pattern.

They

_may for instance be

;pecified

_as tho~e with the ;mallest

vanishing

out

by Doppler broadening.

If _we _now switch _on the di~turbance

H~

^these

fringe

_patterns _are shifted. The

corresponding

terms in the calculation

depend

_on _a, _w,

or

Rj,

_it,,,,. We are

only

interested in thi~

pha;e

shift. In

~,)

and

~,,)

the contributions which

are

already

there in the undisturbed _case _can be isolated _a~

complex

iactors. If not otherwise stated _our <.r)iiientir)n when

working

out

~,)

and

~,j

will therefore be _to omit contributions to these factors whenever

possible. Thereby

we mud

keep

certain contributions in order _not _to

loose terms oi

H~. Finally

the

resulting phase

shift caused

by switching

on

H~

i~ obtained in

restricting

in

(~, ~,,)

to those parts ^which ^contain ^the constituents of

H~.

This convention has consequences. ^Because _H,~~~~~~ commutes with _H~~~ it contributes

only

to

the omitted

complex

factor. Therefore

(~,)

and

(~,,)

_go

solely

back to the action of

H~~~ in the dark _zones and to

respective

_components of

U~(I)

of

equation

(7).

For

example,

for the _two beams in the

Ram~ey

device of

figure

we iind

>fi i <ii r >ii i <n

~,)

= cv~ ^e ^~"'

p

~ e ^~"' y~ e ^~"' ^'

p,

_e ^"" ^~'

~,>)

(8)

->ii i~ -<ii i, -<ii i, -<ii in

4'III

_~ P

4 ~ ^""

61

e ^~"'

62

e ^~"' 61 e ^~"'

4'n)

(6)

In thi; _ca~e both

partial

beams _are excited aiter the passage of the iourth laser. In many devices such _as the

Ramsey

interierometer the number of excited atoms after the evolution is

measured. The observed

quantity

is therefore

( )

and it contains beside other

contributions _an interierence term of the form 2 Re

(~j ~,,).

In the

Ramsey

device almost all the other interierence _term~, I-e- those between other

partial

beam~, are washed out due to the

Doppler

effect

(2].

This entails that the

signal

con~ists of _some

background

and some

sinusoidal variation

coming

from the interference term. The

phase

^of this variation _can be calculated

by evaluating

_(4~j

~,,).

The

phase

shift

originating

from

H~

^of

equation

_can be read off

directly.

Thi~ i~ the

general

scheme of the paper.

Our aim is to

give

a

general expression

for

~, ~,,

for any ^two

given partial

beam~ for _an

arbitrary

la~er

coniiguration provided

the lasers _can be idealized _as follow~.

3. Idealization of the laser beams.

We need the

knowledge

oi the evolution operators

U~(I

^which refer _to laser _zone~ with

travelling

or

;tanding

laser _waves

serving

as beam

splitters.

Ii the

complete

Hamiltonian H,,~, of

equation

Ii

including _H~

is taken into _account the related microstructure of the time

evolution will lead to

U~(I

) which _are in the _exact version

complicated

functions of the

operators ^of

position

and momentum, of the

ingoing

matter state, and of the duration

T of the la;er influence. Because _we follow two

given partial

beams I and II we have in fact

U[

_ii and

U)

_(I_). Our main idealization is then that the influence of the

perturbing

Hamiltonian

H~

over the interaction time

T in the laser _zone is _so small that it may be

neglected

when

elaborating

the final interierence pattern. ^Thi~ ^is to be

expected

^it T is very short

compared

with the times of

flight

_T, between the lasers. We _assume it therefore _to be

justified

to de;cribe the influence of the

travelling

or

standing

laser _waves in

~,)

_as _a momentum tran;fer _at _one

instant oi time oi the form

cv,' = exp

[in) k, xi

(9)

where

k,

is the _wave _vector of the

respective

laser _wave and ¹⁾

= 0, ± 1, ± 2, The

position

operator is the

only cl-number

which appears here. We have

corresponding expressions

for

p,', y,', 6,',

and for the beam II.

According

to our convention any

amplitude

is omitted because it does _not contribute to the

pha~e

shift which goe~ back to

H~.

^The same is the _case ior the

laser

phase

omitted in

equation

(9). The restriction of the laser influence to the momentum transfer which will be

ju~tified theoretically

below is reasonable. It is this feature which is used for the construction oi

optical

beam

splitters

_as

proposed,

for instance,

by

Pfau, Adams, and

Mlynek

[14] in the context of _a three-level atom in _a constant

magnetic

field.

The time evolution of _a

partial

beam _i~ now

given by

,«~k,, , <H~,,,I(, <I(,,,Tj <,i( ^k , <H~,,, T,,

$ij)

= e e e e e

$ij,)

( (0)

and

similarly

for

~jj)

with H~~~ of

equation

(2). For

~,)

in

figure

_we have. for

example, al

=

I,

ii(

=

I,

n(

=

I, ₁₁₍

=

0, and

k,

₌

k~

₌ k~ ₌

k~.

The

approximation

made above which leads to the idealization of the laser _zones

by

beam

splitters acting instantaneously

can

formally

be obtained

by taking

the limit _to very small

T and very

large

Rabi's

frequency

_J2~~

=

E~

_{d~~ in} ^such a way that _x

:=

TJ2~~

^remains ^fixed

characterizing

the

respective experimental

set up. To

justify

the structure

(9)

assumed above

we

give

an exact calculation of the output ^of a laser _zone for the extreme limit

T - 0 IX

= const. ). The electric field E may

thereby

be

given by E,,cos [wt

k _x ₊ _~

(running

wave) or

Ej,

_cos [wt cos [~ k _x

(~tanding

wave). _~ is _~ome constant

pha~e.

For

(7)

T <

I/(k(

there is _one _common result for

running

and

standing

waves. The

Schr6dinger equation

with the full Hamiltonian

given by equation

(I) ha~ the formal solution

~ T ',,

~ jr

))

=

jj

(- _I16'~

dt, dt,, Hit, Hit,,

_j ~

(0 ))

' '

,, =,1

i >

whereby

the

simplifying

convention of section ? has not been used. We _now

reparametrize

the variable~ of

integration by

t, ₌

A,

T and

perform

the de~cribed limit.

Exploiting

lim

TH(t,

₌

TA,

)

= X cos

[p

k

xi (12)

~ _~,

l

which is valid for both

running

and

standing

waves it i~ easy ^to

perform

the _sum in

equation

I).

Remembering

that the square of the Pauli matrix «, occurring in

equation (12)

is the

identity

matrix _we arrive at

~

j0~ ))

co~ Ly cos (~ k _x )] ₊ I ~in Ly cos (~ k _x

l~

4'lo )) (13)

Thi~ form of the evolution operator ^is not very

illuminating

(what is the ~ine of the cosine of the

position operator?)

but it _can be written in _a _more

appropriate

form if _one represents

cos

[~

k _x as

(exp ii

_~ ik _x ₊ exp [- I _p ₊ ik

xi

if? and writes down the

Taylor

~erie~

of the

remaining

^sine and cosine. In this way we find, _e.g.,

cos

lx

_CDS (w k _x

)i

=

I )j ((' (

^~

_i _~~/[ _exP12 i~

_(k

x ~

Jl 1'4)

,)

u i

~'n,

A resummation of the r-h-s- leads _us _to

~

co~ Ly cos q7 k x)

=

jj

_exp

[2 ir(k

x ~

II

(-

l'J~

j,

(X (15)

, ~

where

~~ _''' ~~

2

~j~ I! (I

₊ ₂

jr

~!

~~~~

are Be~~el's functions. A similar calculation _can be done for the sine in

equation (13).

The time evolution in its final forms is then

given by

~ (0_

))

₌

( (

exp

[2

iii (k _x _p (- )"

Jj

_j,, ix +

,, ~

+ i i ^'

f ie<i~«+ii<k ,~>+ e-<'~«+"'k ~~>i

i-

J«/~,,+>(x)l'~f(°)i

^1'?~

,,~~

This

expression

makes the

physical

effect of the laser _waves rather obvious. Since

exp(ink

xi

corresponds

to a

change

of nk in the momentum of the _atoms the atomic beam i~

split

into _many

partial

beams which _are

weighted

with Bes~el functions. For _even

ii there is _a momentum tran~fer without excitation and deexcitation. It

depends

on the apparatu~

(8)

parameter _x which Bessel function and

accordingly

which _momentum transier _n is

predominant

in

equation (17).

Note that in the _extreme limit of

equation (17)

also for

running

waves excitation with _n <0 and deexcitation with _n ~0 is

possible.

Such processe~ are

explicitly

removed in the

rotating

wave

approximation

because

they

are far from resonance

with the atomic transition. In the limit of very short interaction time between lasers and _atoms

the _re~onance condition has _not to be fulfilled. Seen in this way the result

(17)

makes it

plausible

that for very short

(running)

laser

pulses

atoms could get ^the _« wrong » momentum transfer.

Having justified

the _structure of

equation

(9) we ~tress that the calculation should _not be mi~understood _as _an

adequate

treatment of the influence of

extremely

short laser

pulses

_on

atom~. This is due to the fact that for ~hort laser

pul~es

^the contribution of non-resonant

electromagnetic

waves cannot be

neglected.

An

important implication

^is that for too short

pulses

also atomic states which _are not described in the two-level

approach

_can be excited

by

the laser _waves. Since _a very ~hort

pulse

_may not extend _over _one

wavelength

also the

shape

oi the

pulse

becomes

important.

Nevertheless the result

(17)

is _a

good approximation

if the

pulses

are not too short and it the

flight

times T, ⁱⁿ ^the ^dark zone~ are much

longer

^than ^the interaction times _T, between the atoms and the lasers. Its range oi

validity

_may be constrained _to situation~

where the

rotating

wave

approximation

and oi _cour;e the two-level

approximation

_are

applicable.

4. Phase shift for

arbitrary

interferometer

configurations.

In thi~ section _we will be concerned with the calculation of the

pha,e

shiit between two

partial

beams I and II caused

by H~

^of

equation

(4) in an atomic interferometer with _an

aibin.ai=>.

miuibei

t)j

la.iei beuui.I

serving

as beam

~plitters.

In _a fimt ;tep ^the

expre;sion

(10) can be

brought

into _a more convenient form if _we in~ert the

identity

operator ⁱⁿ the form

exp(-

_{iH~,~~ ii}

exp(iH~,~~

t) for certain times t and make _use of the

equality

exp (- B ) _exp

IA

) _exp

(B

)

= exp

[e

^~ A

e~] (18)

which is valid for any operator~ ^A ^and ^B

provided

the

exponentials

make _sense. We get

~,)

=

_means of the well-known formula

exp [itH~~~~ ^k x exp [-

itH~,~]

^~ ^~' ^~

K,

(?

~j,

.~

with

K,,

= k _x and

K,~

=

[H~,~. K,I.

It i~ not difficult to prove

by

induction that each

K,

_can be written in the form

K,=i'~U;x+ V;p+W;a) (2?)

M

(9)

where

U,, V,,

^and

W,

are c -numbers and dr) _not

depend

_I)ii the a<.<.eleiutir)n _a. This will _turn _out

to be the _reason for the

independence

of the accelerational

phase

shift of the initial state of the

atoms. The <.-number vectors

obey

the recursion law

(~

i +

£<

~ ~£,l),

(~,

_)b (" _), ₊ ~U

£<0 h

(~,

_)b

IV,

+

I,

= F'ai>,

IV,

_Ii> lW

)~

Ill,

)<,

(w (v j (23)

, +1t< _ii'

with the initial condition Uj, = k and

V,,

=

Wj,

=

0. In

general

the solution of'this scheme

cannot be

given

in _a closed

form,

but it is

possible

to calculate the operators _I, to any wanted

accuracy. When this has been done the operator~ _I, ^of

equation

(20) are

given by

'., =~Lix+»;P+P;a (24)

with well defined _~L,, v,, and p,. This is the first step ⁱⁿ ^the ^evaluation ^of

equation (20).

The

specification

of _~L,, v,, and p, follows in

equations (27), (28),

and (30).

The rest of the calculation i~

only

a

repeated

utilization of the famou~

Baker-Campbell-

Haussdori formula which

gives again

exact results becau~e the operators _I, ^have ^the

simple

iorm (24). In _a fir~t ;tep one can prove

by

induction that

l~

^~

e"'" e""'~

= exp

jj

_n, _I, _exp

jj jj

_ii, _{ii~ (~L,} _v~ _v,

~L~j (25)

2

-1 _, =, ,»>

hold~. This _can be u;ed _to rewrite the

expression (19)

in _a _more convenient form. After the

~ame is done ior

4~j,)

the interference _term

(~, ~,,)

_i~

quickly

calculated to be

IA

(~, ~j,)

= exp

jj (ii)'

ii) p, a x

=1

x exp

~~ jj~

jj (nl' nl'

_{ii) ii)} _(~L, _v~ _v,.~L~ ₊ ^~

(

~L, v~

(n)

_ii) ₊

nl'nl'

?

iil'n)

_x

~

,

~

, j =1

l,;

^~

x

~,,

_exp

~ (n)'

ii) _~L, x exp I

~ (ii)'

ii) _v, _p

~,, (26)

,

where we have reintroduced h and _<. ior convenience. In realistic ~ituation, we will have k =

E~ E,,. Doppler

shift and recoil _are still contained. To read off the

phase

shift induced

by H~

^of

equation

(4j one has _to work out ~L,, v,, and p,

explicitly

for the interferometer

~ituation in

que~tion

and to substract from the

pha~e given

in

equation (26j

the

corresponding expression

for _a

=

0, _w

=

0, and

R,,

_I,,,,,

=

0.

Equation

(26) i~ the main re;ult of this paper. It may look ,omewhat

complicated

but thi, I;

only

due to it~ great

generality.

In

particular,

thi;

form I; well-suited for the

application

of

algebraic

computer programs. After

specification

of the

magnitude,

^oi _a, _w, and

R,,

_I,,,,, it I; _now in the

expression

(?6) where _one ha; to make

sure that the iteration

(23)

above has led to the

respective

contributions of the desired order. An

example

will be

given

in the _next section.

We remark that in _two

physically important ;pecial

cases closed

expression;

for

I, ^can be

given

from which the

respective

_~L,, _v,, and _p, _can be read off. If the rotation vani;he;

we find

I, =

(k,

Ii (CDS

It,

N )i,,, I,,, +

(k,

Ji (N Sin

It,

N )I,,,P,,, ⁺

+

(k,

ii IN

~(cos _It,

_N

_))I,,,

a,,, (271

(10)

where the matrix N is defined

by (N~ _)I~,.= Rot

j,,,,.

Surely

this is _a formal

expression

defined

solely

iia its

expansion

which _contains

only

even powers of N. In this _sense _we need not to care ior the exi;tence of N.

In absence of _curvature the result for the operator~ ^is

t,

", =

(k, )I (exP It,

° )I,,,-I,,, +

~ ^(k,

^)I

^(exp it,

O )I,,, p,,, +

(k, ii

^O- ~

(exp it,

^O _)I,,, _u,,,

(k,

)I (°~ _exP

It, °I

_Ji,,, _an, _t,

(281

where the matrix O is deiined

by Of,,,

₌ _ti,,,~ _w~.

A short discussion oi _our result (?6) will close this section. Fir~t, it is obvious that the accelerational induced

ph»e

shift is

completely independent

of the initial state

~,,)

since the acceleration _a _occurs

only

in the iir;t _row of

equation (?6).

The

independence

oi this shift irom the initial momentum oi the atom; ior the atomic fountain _was noted

by

Kasevich and Chu [3).

Second, there _are ;tate

dependent

;hiit; _as the la;t _row indicates. In order to get an

impres;ion

oi their iorm _we _a;sume that the initial ;tate is

(a)

g

(G)

where

(G)

denotes _a Gaussian

wavepacket

(in momentum

;pace)

with _mean _momentum

(p)

and half width

« in all

directions.

Setting

A

:=

~

_(ii)~ _ii) _~L, and B

:=

jj inl'

_ii) _v, _we

quickly

iind that the state

dependent

part ^oi

equation

(?6) contain; _a

pha,e

iactor and _a iactor

modiiying

the

amplitude

exp

iB (p) ^~

A

B)

^exp

~- ^fl

^B~

~~ (29)

2

~

8 _« ^~

We will _see in the next section that ior the

Ramsey

interferometer

only

the term

B

(p)

contributes _to the

phase

^shiit to the desired order in _our

applications

since the term A B will be

entirely

oi

higher

order.

5.

Application

to the

Ramsey

interferometer and to the atomic fountain.

It i~ _now

~traightforward

to

apply equation (26)

to two

~pecial

devices, the

Ramsey

interferometer and the atomic fountain. As

~pecification

of the

experimental

set up we may

assume

flight

times T of about _one ~econd. The order of the initial

velocity

v, the laser _wave

vector k, and the atomic _mass M _are about _v _m _s~

(k ^10? ^m~',

and M 10~ ^~~

kg,

respectively.

If _one

applies

such _an atomic interferometer ior the mea~urement oi the influence of the earth's acceleration _a, the earth's rotation _w, and

space-time

curvature

Rot

o~, of the

earth _or of _some lead block _we have the

magnitudes

_a 10 _m _s~

w 7 _x

10~~

_s~

',

and

R,>I,,,,,

5 _x

10~ m respectively.

These

specifications

make it reasonable _to work out

~L,, v,, and p, of

equation

(24)

by

_means of the recursion law

(23)

_up to second order in the rotation and iirst order in the _curvature. In the exponents ^oi

equation

(26) we will take into

account

only phase

shifts which _are at least of the order of 0.I rad. In detail _we obtain

~P,If

'f~~<

_~f +

'/

~~< ~ " ~f

'f

_l~< _~ _"~ _~

" If +

?~

^~~^~'~

^~"

f"«< ~~<_~«<.

(30)

(11)

The first apparatus we will consider is the

Ram.ie)>

inter

fieiometer

which is described in _some detail in reference

[2].

Four _our purpose the

following

facts _are sufficient. First, in the

Ratnsey

interferometer

essentially only

two atomic states _are involved and the laser

frequency

is

closi

to the transition

frequency

between these _states. It is therefore allowed to use the _two level model for _atoms _as _we have done in the

preceding

sections. Four lasers _serve _as beam

splitters,

and their _wave _vectors _are

given by

k

:=

k,

₌ k~ ₌

k~

=

k~.

The times of

flight

_as

they

appear in

equation (lo)

_are

given by

Tj, =0,

Tj =T~=:T,

and

T':=T~.

We introduced

T and T'

just

to be in concordance with the usual notation in the literature. We will examine the

phase

shift between the

partial

beams I and II of

figure

_as it is induced

by _H~. According

to

section 2 these beams are characterized

by

^the sequences

p,

_y~

p~

_n~ and

6j 6~

6~

p~,

respectively.

From this _we _can infer _to the sequence~

(n))

=

(I,-I,1,0)

and

(nl')

=

in,

0,

0, Ii. Inserting

these values in

equation (26)

and

using equation (30)

leads after _some

algebra

to the

expression

(~i ~,j)

= exp

it ^k

^aT

^(T

⁺

^T')

^a ^(k ^x ^w ⁾

^T(T

⁺

^{T') (2}

^T ⁺

^T')

+

( _{(c~ Rj,}

I,,,,,

ki

k,,, + 3 (k _x

w)~) T~(2

T ₊ 3 T')

+( (((kxw)xw).a-)Roij>~,lia,,~j

^~

T(T+T')(7T~+7TT'+2T'~)j~

~

(~u _(eXP (i

[C'~

Rut

_ii,,,,'f k,,, ((k x ml x w ).

xi

T(T +

T'))

exp

)

_i(c.~

_R,,

~,~,~~

i~

_p,,, 3 ((k _x _m i x ml _p

T(T

₊ T~) 12 T ₊ T~)

+ 4 p (k x w

T(T

₊

T')j~ ^~o) ⁽³¹⁾

which contains

only

the

phase

shift due _to

H~

^# ^0.

Again

_we have reintroduced h and

c. The first order contribution due to rotation _was measured with the

Ramsey

interferometer

by

Riehle _et al.

[8].

For the measurement of

space-time

curvature the device of Sterr _et al.

[16]

may be better suited because of the

longer

lifetime of the

respective

excited state in

,' ,I+Il ,' ,, 3

,

beam I,,'

, T

,

2 ^'

,,'

,'

beam

II T

,' i

Fig.

2. The

interiering

beam; in the atomic fountain device under~tood _ah _an effective two level system ^w>th running la~er pul~es. The horizontal line~ represent ^the ^three ^laser pul~e~. The atom~ are

initially

in the lower

hyperfine

level. The

higher hyperfine

level is denoted

by

dashed I>nes.

T i~ the flight time between the laser

pulse~.

(12)

magnesium.

As announced the

product

of the two vectors A and B

entering equation (29)

is of

higher

order in this _case. A discussion of the

leading

_terms of the

phase

shift _was

given

elsewhere

[10].

The order of

magnitude,

however, is the _same _as for the atomic fountain which will be estimated below.

Next _we turn out to the atr~uii(.

fr~ifiitain

_a~ it _was

developed by

Kasevich and Chu

[3].

k~.

The

splitting

of the atomic beam

is

given

in

figure

2.

It now seems that _our scheme is _not

applicable

to this

experiment

since it is

only

valid for

two level _atoms. however, _as

pointed

out in reference

[3]

the

particular tuning

of the laser

pulses

makes it

possible

to treat the

hyperfine

_states _as _an effective _two level system ^with an

effective momentum transfer of k~

k~.

This observation allows _us to use the results of this paper for the calculation of the

phase

^shift in the atomic fountain apparatus.

We will model the situation

by

three

copropagating running

laser beams with _wave _vector

equal

to the effective _momentum transfer k

k,, k~.

From

figure

2 _we can read off the

defining

sequences of operators ^for ^the

interfering

beams. For beam I, it is

6, p~

_cv~ and

(n))

=

(0,1. 0).

Beam II is described

by p, y~p~

and

(n)')

=

(I,

I,

I). Setting T,

₌ ^T~ ₌ ^T ^and Tt, ₌ ⁰ ^the

application

of

eqution (26)

leads _us to the

phase

shift

fi~,2

(4r,14rtt)

_=exp

(I(-k.aT~+3a. (kxw)T~-mRoion,kikmT~

~~P

( _[(3

_(k

~ ") ~ "

l'

_P ~'~

~010

n<

ki

_{Pn<) T~} 2 _P

(k

~ "

~l) ~°) (3~)

The

state-independent

part ^of ^the

phase

shift _can be

directly

read off from this

expression.

Assuming

that the center of _mass _state of the _atoms i~ _a Gaussian

wavepacket

_we can

apply equation

(291 to calculate the

state-dependent phase

factor. For the atomic fountain it is

simply given by

the

exponential

in the last line of

equation

(32) if _one

replaces

the operator p

by

the

initial _mean momentum

(p)

of the

wavepacket.

In the _case of pure acceleration

ii.e.,

_R,> _I,,,,~

= w I =

o)

the

complete

shift reduces _to the first term in

equation (32)

and agrees with the result of reference

[31.

The last term in

equation (32)

describe~ the well-known

Sagnac phase

^shift. ^Becau~e ^the ^earth's acceleration alter~ the

motion of the atoms

substantially

it is modified

by

the ~econd term in

equation (32).

The

structure of this modification is

easily

understood if _one considers _a classical free

particle

with

initial momentum p,,. In _an accelerated frame of reference its _momentum is

changed according

to

pit

_pi, Mat. The

replacement

of the _mean initial _momentum

(p)

in the

Sagnac phase

shift

by

MaT reflects this fact

by leading

to correct structure of the

modifying

term. Corrections to the

Sagnac phase

which _are of

higher

^order ⁱⁿ ^the ^earth's

angular velocity

are the first term in the last line and the first term in the second line. The latter can be understood in the _same way

as the modification to the first order

Sagnac phase

as a consequence of the modified atomic motion _an the accelerated reference frame. The influence of

space-time

curvature on atoms,

(13)

I,e., of the variation of the acceleration _over the interferometer, _can be

spl>tted

^into two parts.

The last term in the second line and the second term n the last line of

equation (32)

are

connected _to the motion oi the _atoms whereas the last term in the iirst line is

only

due _to the

non-vanishing

relative acceleration

experienced by

two

partial

beams which _are

separated along

the direction of the laser beams. Note

again

that the third line does _not contribute _to the

phase

shift to the order under con~ideration if the atomic

wavepacket

^is

sufficiently

localized in

momentum space.

6.

Measurability

of the influence of

space-time

curvature _on quantum systems.

A detailed discussion of the

measurability

oi _curvature eifects with the

Ramsey

device _was

given

in reference

[10).

It is necewary to restate here _some remarks _on the

phy~ical

value of the

measurement of the

curvature-dependent phase

shift. In

general relativity

the acceleration _a i~

tied _to the

particular

observer who is accelerated with _a.

Loosely speaking everything

seems to be accelerated with acceleration _a in its frame of reference. This

implies

that in the frame of _a

freely falling

ob~erver there is _no acceleration at all. Acceleration is _a non-inertial efiect, tied to the observer, and is not connected with the structure of

space-time.

The latter is described

by

space-time

curvature which is present ⁱⁿ any frame oi reference. A curved

space-time implies

_a

relative acceleration between

nearby

observers, I.e. the distance between _two

freely falling

ob~ervers

changes

and the acceleration of two

nearby

observers which _are held in _a iixed

distance is different.

It was

suggested

that _one _can _use the atomic fountain _as _a

gravimeter [3).

This is true in _a twofold _sen~e. If _one varies the

position

of the whole device in order to detect the acceleration of the earth at two diiierent

points

this i~

indirectly

also _a detection of

~pace-time

curvature since the variation of the acceleration _over space is measured. This is, however,

by

no means

the demonstration of how

space-time

curvature acts on atoms. Rather, the classical apparatus

lead block

/

atoms

"2 mi

lead block

Raman laser Ram» laser

Fig.

3. A ~cheme of the

proposed experiment

to mewure the influence of the

space-time

curvature

caused

by

two leads blocks _on the atomic fountain. The

big

arrow~ repre~ent ^the ^three ^Raman ^lawr

pulses performed

^with two

counterpropagating

I»er beam~ with frequencies _w and _w

~.

The point O denote~ the

origin

of the coordinate sy~iem.