Relativistic treatment of Raman light-pulse atom beam interferometer with applications in gravity theory

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Relativistic treatment of Raman light-pulse atom beam interferometer with applications in gravity theory

Claus Lämmerzahl

To cite this version:

Claus Lämmerzahl. Relativistic treatment of Raman light-pulse atom beam interferometer with ap- plications in gravity theory. Journal de Physique II, EDP Sciences, 1994, 4 (11), pp.2089-2097.

�10.1051/jp2:1994111�. �jpa-00248111�

Classification

Physics

Abstracts

04.80 42.50

Relativistic treatment of Raman light-pulse atom beam interferometer with applications in gravity theory

Claus Limmerzahl

(*)

Fakultit fir

Physik

der Universitit

Konstan2,

Pf 5560 M 674, 78434

Konstan2, Germany

(Received

16 March 1994, revised 23 June 1994,

accepted

I

August 1994)

Abstract. The first relativistic correction in the

phase

shift of _a Raman

light-pulse

atom beam interferometer in _a

gravitational

field is

being

calculated. The

starting point

is _a three level system

coupled

to

gravity

which is treated to order

c~~. Gravity

is described

using

the PPN formalism. It will be

proposed

that _{a new} measurement with better estimate of the PPN

parameter p could be

possible.

1. Introduction.

Atom beam interferometers

provide

_{a new}

highly

accurate tool for

detecting

influences of external fields _on

quantum

matter. The atom beam interferometer based _on two

counterprop- agating pairs

of

copropagating

laser _waves

corresponding

to a

Ramsey

excitation scheme _was first

proposed by

Bordd

iii

based _{on [2]} and

subsequently

realized

by

Riehle et al.

[3],

^who demonstrated the

Sagnac

effect with _atoms. A second type ^of atom beam interferometers is the Raman

light-pulse

interferometer built

by

Kasevich and Chu

[4, 5].

These _new devices created _new interest in

testing

the interaction of

quantum

matter with

gravity,

which is essential for the construction of General

Relativity

_[6]. In

addition,

it has been

proposed

_[7] that _a

spin-rotation coupling

_may be

directly

measurable with atomic beam

interferometry.

In [8] ^{it is} shown that atom beam interferometric tests of Lorentz invariance may

give

better estimates than other tests. In

addition,

^Audretsch and Marzlin [9]

proposed

that the curvature

(second

derivative of Newtonian

potential)

_may lie within the limits of

detectability.

In

[10]

^{neutron and} atom beam interferometric tests of local

position

invariance

are

analysed.

It is clear that for

larger

interaction times of the _atoms with the external

gravitational

field the

phase

shifts increase. This may be stated

loosely

in the form that atoms _can be

prepared

with _very slow

velocity.

The

large

interaction time is the

general

_reason

why

atomic beam interferometers achieve such _a

high

_accuracy. We want to show that the accuracy is

high

(*)

e,mad:

Claus@spock.physik.uni-konstanz.de.

2090 JOURNAL DE

PHYSIQUE

II N°11

enough

^for atomic beam

interferometry

to be sensible for relativistic effects. Beside the usual acceleration induced

phase

shift

rw

kVU,

where U is the Newtonian

gravitational potential

and k the _wave vector transferred from the laser _wave to the atom

beam,

_one also

expects

terms of the form

kVU(U/c~)

and

kVU(u~/c~)

where _u is the

velocity

of the atoms and _c the

velocity

of

light.

Since

u~/c~

for the atoms _are in

general

at last of the order

10-~~ only

the first term may lie within the _range of

detectability.

We want to show that relativistic effects in the interaction with external

potentials

^{of the} structure mentioned above _may well be within the _range of

measurability.

However,

to

give

_a correct form and a correct estimate of the

phase

shift due to relativistic terms,

consistency requires

that the atom beam interferometer should also be treated in _a

relativistic _manner. This is the purpose of this letter. We first treat the Raman

light-pulse

atom beam interferometer of Kasevich and Chu _{[4] to} first order in the relativistic correction.

Thereby

we base _our calculations _{on a} Hamiltonian which is derived from _a relativistic

expansion

of the Klein-Gordon

equation

to the order

c-~.

Relativistic corrections also refer to the used laser

beams,

and to the

Doppler

shift of the atoms in motion. The _use of the

equation

^{of motion} for the center of _mass of the atom to first relativistic order

finally

leads _to the

relativistically

corrected

phase

shift.

Gravity

is described

by

_means of the PPN formalism. This _means that _{we are}

treating

_{a very}

general

class of

gravitational

theories which enables _us to calculate matter _wave interferometric

tests of

gravitational

theories. It turns out that the current estimate of the PPN

parameter p

_may be

improvable

for the Raman

light-pulse

atom beam interferometer. Beside

this,

the

experiment

described below will

provide

the first quantum test of

p.

All other tests use bulk matter.

2. The Hamiltonian.

2.I GENERAL FORM. We start with the

description

of the three level atoms used

by

the

Raman

light-pulse

interferometer. We shall treat this three-level

system

to order

c-~

The three levels _are denoted

by

_{o =} 1,

2, I, whereby

I stands for "intermediate state". The

corresponding

Hamiltonian

(§

is the momentum

operator)

is

H

"

) ~(3~2

^~

~j"t _°11° +filsl

₊

fi©~ _(i)

~~ ~

a

uJj

is the energy of the ath level of the atom.

fit'/

_is the usual

dipole

interaction -d E in the rest frame of the atom and

Hf/~

describes the interaction with the

gravitational

field.

The latter is

given by

_a relativistic

expansion

in terms of

c~~

_{and consists in terms of the form}

£,

U~'...~>

(R)§a, §a,,

that

is,

in _a

polynomial

of finite order in the momentum

operators.

In the

following

we

only

need

polynomials

of second order in the momentum operator;

however, generalization

to

higher

order is

straightforward.

We

neglect spontaneous

emission.

Any

state of the three level

system

may be

decomposed according

to

plane

_waves

whereby

we

split

the free evolution

(bold symbols

_are

3-vectors):

2 4

1fi) =

~j / _aa,pit)e ^{'("~~ ' d~)~}

a,

pjd3p _j2)

~

with

lo, p)

₌

[a)@[ p)

where

§ la, p)

=

pi

_o,

p)

and where p is the 3-momentum of the

plane

wave in the rest frame of the

observer,

that

is,

of the

interferometer,

_or

equivalently,

of the

generators

^{of the laser beams.}

We do _{not treat}

gravitational

modifications of the

dipole

momentum _or of the _energy levels

of the atoms. These modifications do not enter our formalism because the Rabi

frequency

_or

the transferred momentum

belong

to the

experimental input

and need _no further

analysis

in

our setup. Even

gravitational

corrected Rabi

frequencies

and transferred momenta

(see below)

are

parameters

^which are read off from the

apparatus

^{and which} are the entities

entering

the

phase

shift.

In the

following

entities defined in the rest frame of the atom _are underlined. All other

symbols

are understood to be defined in the rest frame of the interferometer.

2.2 THE DIPOLE INTERACTION. In the rest frame of the atom the

dipole

interaction is the

same as

usual, namely

-d E. This _can be founded

by

use of _a covariant introduction of this

coupling ii Ii -dPfpau"

where _u is the

4-velocity

^{of the}

atom, F~v

₌

b~Av bvA~

the Maxwell field

strength

with the

4-potential A~

and dP _a covariant formulation of the

dipole

momentum

with the

property u~d"

= 0. In _a 3+1

decomposition

this

4-potential

has to be identified with

A~

-

(-#, A).

In the rest frame of the _{atom we} have u"

=

6(

due to the normalization

goo

condition

g~vu"u"

₌ -I where g~v ^is the

space-time

metric. In this _{case we}

get

for the

dipole coupling -dPfpau"

= -dP

Fpo

"

-d'E, ii

=

1, 2, 3)

^{with the} electric field measured in goo

the rest frame of the atom

E,

=

F,o

"

(-b~# ~btA,)

- E.

goo goo c

E

again

consists in two

counterpropagating electromagnetic

_waves. In the atom's rest frame

we have E

=

E~ cos(ki

_.x

_-uJ~t

+

ii

+

E~ cos(k~

x

uJ~t+ #2

^with

ki

_m

-k~.

We _assume that

d.E~

j~

only

induces transitions between

11

/2)

and

ii)

and that the

rotating

wave

approximation

is valid. Both

assumptions

_are in _no way correlated with

relativity problems

_so that

they

can

also be taken _as valid in _our first relativistic

approximation. Consequently,

in the rest frame of the atom _we have

HIT

"

~j II _(li,

P

d'El~~'~~'~~~~~~

_I>

_P~)

_i>

P)~~'~~

_{Ii> P~}

~(~>P ~'~2~

~~~~ ~ ~~~~

~>

P')

~>

P)~

^'~~ _(~,P~

~~'~')~~P~~P' ^(~)

The

phases

of the

electromagnetic

_{waves are}

easily

formulated with

respect

to the interferom- eter's rest frame: _as the

phase

is a scalar _we have

k~

_x uJ~t =

ki

_{x wit.} Because the states

la, pi

_as well _as the

phases ^e+'k"X

_{are now}

given

in the interferometer frame of reference _we

can

apply ^e+'k'X1

₌

f pi (p

_~ hk

d~p.

The fact

why

we want to describe the momentum transfer in the rest frame of the interferometer

(that is,

the 1

-operator

is _an

operator

^{in the} Hilbert _space attached to the

interferometer)

is the _reason for

expressing

the

phase

of the elec-

tromagnetic

wave with

respect

to the interferometer frame.

By using

the above relations _we

get

^the

dipole

interaction operator ^{which is}

represented

in the rest frame of the interferometer

fit'/

=

(Hi, I,p)e'"'~ ii,

_{p +}

ltki +Q(, 2,p)e'"2~(I,p

₊

ltk2

+

c-c-) d~p (4)

2

where _we defined the

complex

Rabi

frequencies Qi~

_."

)(I[d E[I)e"i

and _Q2>

."

Ii

d

lt

E[2)e"2

in the rest frame of the atom. Of _course, the electric field E is the Lorentz transform of the

electromagnetic

field

(E,B) produced by

the laser beams in their rest frame: E

=

~E

^~

~

(E v)v

_{~v x} B where _v is the

velocity

of the _center of _mass of the _atom with

v

respect

^{to the rest frame of the} interferometer

(~

=

ii v~)-~'~). However,

because the atoms

2092 JOURNAL DE

PHYSIQUE

II N°11

in the Raman

light pulse

interferometer have _a

velocity

of around 6 _cm

Is

modifications which

are of the order

u~/c~

_are

negligible (see

also

below).

2.3 THE GRAVITATIONAL INTERACTION. An

expression

for the

gravitational

interaction to

first relativistic order is achieved

by taking

the Klein-Gordon

equation minimally coupled

to Riemann space

ix

_are

4-coordinates,

_{x =}

it, x))

fib~ ig""bv~7iz)) m~~7iT)

₌ 0

15)

where _g

:=

det(g~v)

is the determinant of the metric tensor.

By using

this

equation

we are

modelling

the center-of-mass motion of the atom

by

_means of _a scalar field

equation

thus

ignoring (in

this

simplified model)

the influence of _any internal

degrees

of freedom like

spin

_on

the

trajectory

^{of the} atom. Since these influences _are

generally

of the order h this will

lead,

in

our case, ^to no measurable errors. In the

following

we use the metric in the form

[12, 13]

go~ = -1 +

2j

+

P)~

901 "

°,

_{9v 61J}

~

~

~~~

~~~

with two PPN

parameters

~ and

p

which indicate the

strength

of the

gravitational

field _pro- duced

by

_a unit rest mass and the

nonlinearity

of the

gravitational

^field. U is the Newtonian

potential,

and _{we assume}

empty

^{space /hU = 0 and}

^stationary

^matter distribution:

bU/dt

= 0.

U is

given by U(x)

=

/ ^~~~ _d~z'.

_{There is}

no gauge freedom _to transform U. Due to

ix x'[

boundary

conditions the Newtonian

potential

in the PPN formalism _is

uniquely

fixed.

We insert the ansatz

q7(z)

= exp

(c~so(z)

+

Si lx)

₊

c~~S2(z)

+ into

is)

and set

equal

powers of _c

equal

to _zero. The zeroth order will

give

the usual

Schr6dinger equation coupled

to the Newtonian

potential.

The first relativistic

approximation

is

given by

a

Schr6dinger equation

of the form

ih~)=Hq7

=

-~/hq7-mUq7

-~

~ 3~~§7 () _#) ~§7

C m C

~ ~~ ^~

~~~ ~~~~'

~

~~ ^~~i~~''

_~~~

This Hamiltonian is hermitean if and

only

if _one chooses _as scalar

product

1k7 lfi) .-

/

k7*ifi

~lI ^{d~T 18)}

with

(~)g

as the determinant of the 3-metric g,j ^{in the t} = Cte.

hypersurfaces. Furthermore, by using (7)

_one can derive a conservation law

d

j

~

/§

h

^~'~'

goo~~

~ ^{~ ~~~}

(For deriving

the

hermiticity

of H and for

(9)

_we used /hU ₌ 0 and

at

U ₌

0.)

The first line in

(7) obviously

is the non-relativistic

Schr6dinger operator

and the next two lines describe relativistic corrections in the kinetic energy as well _as in the Newtonian

potential.

Alternatively

_{we can} transform the matter fields and the

operators

to _a "flat" scalar

product

1~7

lift)

=

/ @+ii-goo)~~/~d~T (lo)

by

_means of

q7-@=(1+~~~)q7

^and

fi:=(1+~~~)H(1-(~~), ill)

C C C

With

respect

to the 'flat" scalar

product

H is hermitean.

Again

_we

_get

_a conservation law of the form

(9):

^~

@*@(-goo)~~~~d~x

₌ 0. With

respect

to this "flat" scalar

product

_we

get

dt

/

as interaction Hamiltonian

Hf[~~@= -mU@- p ~@+ _II

+

2~)

^~

lip

₊

2~

^~ VU

V@ (12)

2 _C

~

C

2~ 2~

C

which is calculated to first order in

c~~

_{and second order in the}

gravitational potential

U.

Completed

with the kinetic and

dipole parts

the

resulting operator

is hermitean with

respect

to the "flat" scalar

product (10).

The first

part

^{is the usual} non-relativistic Newtonian

potential.

The next terms describe rela- tivistic corrections of this Newtonian

part.

^These corrections consist in kinematical and in grav- itational modifications of the non-relativistic terms. We do not absorb the factor

(-goo)~~~~

into _a redefinition of the scalar

product

because this term

represents

the fact that the expec- tation value of the _energy is

given

with respect ^{to the}

eigentime

_goo dt of the observer.

The

problem

_now is to take the relativistic

parts

of the

Doppler

shift into account and to deal with the

x-dependent

interactions in

fit[~~.

We treat the second

problem

in _an

approximative

manner

using

_a kind of _a classical limit to obtain the main contributions to the effects in

question.

3. The

dynamical equations.

We

perform

the interference

experiment

with atoms

obeying

the

Schr6dinger equation ii)

with the

dipole

interaction

(4)

and with the

relativistically

corrected

gravitational

interaction

(12).

Our derivation of the

relativistically

corrected

phase

shift is _a

generalization

^of the _case

given by

Kasevich and Chu [5] based _on

[14].

^Our

strategy

^{is to solve the free} relativistic evolution

of states at first and then to calculate the

phase

shift of the interference

pattern.

^Then we use

the

equation

of motion for the atoms which lead via the

Doppler

shift to interaction induced

phase

shifts.

The interference

experiment

starts with the

preparation

^of the state

ii, p). Projecting

_succes-

sively

the above

Schr6dinger equation

on this state and _on

I,

_{p +}

ltki

and

2,

_{p +}

lt(ki k2)) gives

the time evolution of the

respective

coefficients defined

by (2):

~Q*

^iA>t

j~~)

ai,p "

j

^ii~ ~~,P+&k>

'~ ^-~ ,

~ '~'~~~'

~~

^~

~~~~~'~'

~

~~~~~ ~~~~~2,p+&(k,-k2) (14)

Q2,p+&(k,-k~j "

)Q(~e~~~~ai,p+&k> (IS)

2094 JOURNAL DE

PHYSIQUE

II N°11

where _we defined

~~ ~~

_{~ ~~ ~}

2~h ^{i~~~2h "~ ~~ ~~~~~}

^~

~~~j$j/~ ⁽¹⁶⁾

~~

₌

w)

_{+ w~ +}

^~~

^~

^{~~~l k2))~ (P}

⁺

^{h(kl k2))~}

2mlt 8m3~2&

A

(p

₊

ltki

_)~

(p

₊

ltki

₎₄

-uJ +

j~~)

' 2mlt 8m3c21t

and

()*

means

complex conjugation. Equations (13-15)

describe the

dynamics

of _a set of

coupled

states.

These

equations

have the _same structure _as in the non-relativistic _case. The differences lie

in the _occurrence of

p4-terms

in

/hi

and

/h2.

We solve them

approximately by eliminating

adiabatically

the intermediate level

~~~~'P

"

(Q)~ai,p+~o~we~~~a~

,

~~

2 ^,P+ (ki-k2)

(18)

~~'~~'~~~ ~~~~

§~~W~ ~~~ai,p

+

)Q)Ca~

~~,~~ ~

' i- 2)

(19)

where _we introduced the definitions

~~~ 2/h~" ~~~

2/h~" ^~~"'~ (12

^~'

2/h~

^~~~~

6 =

~~~/h2 ^~~'~~

^~~

^~~' ~~~~~

lp p2 ltp

k

lt~k~ p21tk~

"~ "~ "~~~ ~

m

~ 2m2c2 2m2c2 2m2c2 4m3c2

fi~2 fi2~2

~ 2m

4m~c~

^~~~~

where _{uJhfs ."}

uJ) WI

is the

frequency

of the

hyperfine

transition and k _:=

ki k2

The p

/m-

term describes the

Doppler

shift with the first relativistic corrections. The last term in

(21)

is the relativistic

generalization

of the recoil shift. The _error due to the adiabatic elimination is of the order

Q~tl1h2.

In

addition,

the _error due to the introduction of the effective

Rabi-frequency

Q~w is

Q~t611h2.

In the _same way as shown for the atomic fountain

geometry

_[5] the

phase

shift which is related to the

probability

to detect the state

2,

_{p +}

lt(ki k2))

after the three

7r/2

7r

7r/2 light pulses (whereby

the duration of the _7r and the

7r/2 pulses

_{are T} and

T/2, respectively)

is

given by

/~1= ijti) 21jt~)

₊

ijt~),

with

wit,)

=

j~' ^{6jt)dt. j22)}

to

^{is the time of}

launching

the atoms from the

optical

molasses. ti, t2 ^and t3 are the times of the three

light pulses

and T is the

flight

time between the

light-pulses.

To

get

this result _one has to

assume 6 < Q~w and that the AC-Stark shifts of the energy levels _are

time-independent. (21)

generalizes

the result _[5] to include first relativistic corrections. It is clear from the definition

(21)

that _an accelerated atom leads to _a

time-dependent Doppler

shift which _enters the

phase

shift

(22).

Therefore _we have _to calculate and to

integrate

the

equation

of motion for _{an atom} in _a

gravitational

field.

4. The

equation

of motion for the center of _mass of the atom.

There _{are many ways} to derive the

equation

of motion for _an

object

in _a PPN framework. We

can use

Ii)

the

geodesic equation

and insert the PPN

metric, (it)

_{we can use} the Hamiltonian

II)

and derive in _a canonical _way the

equation

of motion for the

momentum,

_or

(iii)

_{we can}

use the

vanishing

^{of the}

divergence

of the _{energy momentum tensor}

[12].

We choose the second

possibility.

We start with

Heisenberg's equation

of motion for the

momentum

operator §

=

[H,§].

Here the dot denotes the derivative with

respect

to the coordinate time. The

resul(is

t

=

mvujR)

+ 11

2p) @mvuj~)

11 ⁺

2~)vuj~)£ ~[ja~vuj~))pa _j23)

To

get

the

equation

of motion for the momentum of the _center of _mass of the

atom,

_we

perform

a classical

approximation

in the _sense that the average of _a

product

of

operators

_can be

replaced by

the

product

^{of the} _averages. For

consistency,

_we also take h

- 0

explicitely. (The

fact that the WEB

approximation gives

the main contribution to interference effects has been shown in

[15].)

^In

addition,

for the classical limit _we correct the above time-derivative to be _a time derivative with respect ^{to the} _proper ^time

(measured

_e.g. from _an atomic

clock).

For

doing

so

we have to take

~~

=

fi.

The result is

da _goo

~~

=

mVU(x)

+

2(1 p) _~~~~ mVU(x)

₁₁ +

2~)VU(x) ^~~

~

(24)

The most

interesting

term is the second _one which vanishes for Einstein's General

Relativity.

We _assume that the

inhomogeneities

of the

gravitational potential

_are small

compared

to

the dimension of the interferometer

(it

is of _course

possible

to treat this _case

also)

_so that _we

can

replace

VU

by

_a constant g. Since the atoms in the interferometer _are in _a free fall

only

for _{a very} short time (rw ^o.I

s)

and since

they

_are

staying

within _{a very} small

region

where the difference of the

potential

and of the

gradient

of the

potential

_{are very}

small,

_we

approximate

u

~2

p(ao

+

a)

= p~ +

mVU

+

2(1 p)~mVU

₁₁ +

2~)VUfi)

a

(25)

C 2mC

whereby

_po

"

P(ao)

is the initial momentum of the atoms

entering

the interferometer.

5. The

phase

shift and numerical estimates.

The above

equation

for the momentum is _now used to calculate the

phase

shift

according

to

(22)

with

(21).

After _some calculation _we

get

/h#

= -g k

T~

II ^{)~~~ ~(~§ (~}

⁺

²⁾ £

+

2(1 p) po

+

~) ~~j

2cm 2cm cm c 6 _mc

~

~~~

~

~

i~

~°

~

~~

^{~ ~}

^~

^{~ ~~~~}

whereby

_we

only

took terms of the order

c-2

_{into account} and where _we chose po ⁱ k

ii g. In addition _we assumed short

pulse durations,

that is _T < T. We aIso defined

6p

_:= hk _as the

2096 JOURNAL DE

PHYSIQUE

II N°11

transferred momentum.

(26)

_are the domimant terms in _a relativistic

expansion

^{of the}

phase

shift in _a Raman

light-pulse

interferometer. We

neglected post-WEB

quantum corrections.

Note that the

explicit

_appearance of the

potential

U is not

problematic

because U is the Newtonian

potential

which is

uniquely

fixed

by boundary

conditions.

The first term in

(26)

is the Newtonian non-relativistic part ^{of the total}

phase

shift and is of the order 2 x 10~

[4,

5]

land

which has also been tested

by

Neutron

interferometry [16]).

The next three terms _are relativistic corrections to the kinetic energy as well _as to the recoil momentum.

They

_are of the order

10~~~

and therefore far below the _accuracy of this inter- ferometer. In the nonrelativistic _case the

phase

shift does not

depend

_on the initial momenta of the atom so that all initial velocities _can

coherently

contribute to the interference. This is _no

longer

the _case in the relativistic treatment,

which,

because of the

negligible order,

has

fortunately

_no

practical importance.

The fifth _term

rw

ii p)U/c~

_{is a} relativistic correction of the

potential

at the

position

of the interferometer. This term is also the most

interesting

_one because it allows to test the

combination

p

I which vanishes in Einstein's General

Relativity.

A null test of this term may

give stronger

constraints _on alternative

gravitational

theories. This term modifies the

Newtonian term

by

_a factor

U/c~

which is the

gravitational potential

at the

position

of the interferometer. It is

generated by

the

earth,

the _sun and _our

galaxy

and turns out to be

U/c2

rw 5 x

10~~

_This

means that the

corresponding phase

shift is of the order

/h#

m 2 and

is therefore within the _accuracy of the Raman

light-pulse

interferometer. Their accuracy is about

/h#/#

_m

^10~~ (and they _expect

to reach _{an accuracy} of about

lo~~°).

Note that this term also does not

depend

_on the initial momentum.

~

The

parameter p

is determined

by p

-1 ₌

~~

~

~(~

^~~

Thereby

_{g is} the Newtonian

gkT

2U

acceleration of the earth which has to be determined

by

satellites _or

by

other

gravimeters.

If _{we assume} that the measurement of the term in

question gives

a null

effect,

then the

accuracy of the interferometer may

give

the estimate

p _$ ^10~~ whereby

the estimate

is limited

by

the accuracy of the

knowledge

of the

position

in

height

of the

interferometer,

that is

essentially,

the radius of the earth.

Thereby

_we assumed for the

geocentric gravita-

tional constant

/h(GMq~)/(GMq~)

_m

10~~,

for the radius of the earth

/hRq~/Rq~

m

10~~,

and

/h(kT~)/(kT~)

_m

10~~ (The

current estimate for

p

with uncertainties in the

angular

_momen-

tun of the _sun, is

p

-1 <

10~~,

_compare

[12]).

To _{sum up, a} test of the influence of the

p

-term

by

_a Raman

light-pulse

interferometer _may put a limit _on this

parameter

^{which is}

stronger

than the _{current one.}

The last two terms in the first line _are of the order

lo~~~

and _are therefore

uninteresting

in

comparison

to the

previous

_ones.

The second line consists in terms

depending

_on the

launching

^time ti These terms _are

again

of the order u~

/c~

which

again

is

negligible.

In the

experiment

of Kasevich and Chu ti ^is chosen

to be 20 _ms which _can be made smaller.

6. Discussion.

To sum up,

despite

of the fact that the Raman

light-pulse

atom beam interferometer works with very slow atoms _{u rw} 10 _cm

Is

it may be able to _measure effects due to relativistic corrections in the

gravitational potential.

These relativistic corrections _are

given

within the PPN framework of

gravitational

theories. We have shown that the relativistic correction due to the

parameter

~ which enters the

phase

shift

only

via the relativistic correction of the kinetic _energy, is not measurable. On the other

hand,

the term

U/c~

_may be within the range of

measurability.

Since

this interaction term is

multiplied

with

p

I which vanishes for Einsteinian General

Relativity,

a measurement of this _term with _a null result _may

give

constraints _on the

parameter fl.

Acknowledgements.

thank Prof. J. Audretsch and F.

Burgbacher

for discussions and the Deutsche

Forschungsge-

meinschaft for financial

support.

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