G en eral Rela tiv ity an d G ravi tation , Vol. 3 1, No. 5 , 1 999

Atom ic Interferom etry in Grav itational Fields: In¯ uence of Grav itation on th e B eam Splitter Clau s L Èam m erzah l 1,2 an d Christian J. Bord Âe 1 ,3 Rece ived October 26 , 19 98 T he laser induced splitt ing of at om ic beam s in t he presenc e of a grav itat ional ® eld is an aly zed . In t he fram e of a quasiclassical ap proxim at ion, t he m ot ion of t he at om ic beam t hrough a laser region wit h rect angular pro® le is calculat ed . B eside t he usual beam splitt ing due t o t he at om ± laser int eract ion, an ad dit ional splitt ing occurs due to t he an om alous eŒect ive int eract ion w ith t he gravit at ional ® eld. In a ® rst order app rox im at ion in the grav it at ional acceler at ion, t he outcom e of an at om int erferom et ry ex perim ent is given , w hich includes t he variou s correct ions ow ing t o t he grav itat ional m odi® cat ion of t he b eam splitt ing process. KE Y W ORDS : Anom alous eŒect ive int eract ion

1. INTRODUCTION Interferometry using laser beam split ters [1± 6] has proved to be a very successful tool in probing the int eraction of quant um ob jects with gravit ational and inertial ® elds. T he accuracy of these devices makes it necessary to consider also the relat ivist ic corrections of their phase shift s due to accelerat ion and rotation as well as t o t reat t he in¯ uence of spin [7,10]. 1

Lab orat oire de P hysique des Lasers, Univers it Âe P aris 13, Av. J .-B . C l Âem ent , F-93430 V illetan eu se, France Faku lt Èat f Èur P hysik, Un iversit Èat Konst anz, P ost fach 5560 M674, D-78343 4 Konst anz, Germ any. E -m ail: clau s.laem m erzah l@uni-konstan z.de 3 Lab orat oire de Gravit at ion et C osm ologie Relat ivist e, Un iversit Âe P ierre et Marie C urie, C NRS / URA 679, F-7525 2 P aris Ced ex 05, Fran ce. E -m ail: chb [email protected] 2

635 0 0 0 1 -7 7 0 1 / 9 9 / 0 5 0 0 -0 6 3 5 $ 1 6 .0 0 / 0 ° c 1 9 9 9 P le n u m P u b lis h i n g C or p ora t ion

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L Èa m m e r z a h l a n d B o r d Âe

T he theoretical importance of quant um int erference experiments for the developm ent of a theory of gravity which is appropriat e t o t he quant um descript ion of matter has been st ressed by Hehl [8,9]. T he main part of these devices is the beam split ter, which consist s of a laser beam int eract ing wit h the atoms or the molecules. Since this int eract ion takes place in the presence of external gravit ational and inert ial ® elds, it is expected that there will be a modi® cat ion of the atom beam split ting process as compared to the split ting in the case of vanishing gravit ational and inertial ® elds. It is necessary t o give an appropriat e descript ion of the beam split ting process and to calculat e the corresponding corrections in order to give a correct int erpretation of the experim ental result s. T here are two kinds of beam split ters and corresponding ly two kinds of atom int erferometers: one uses a stationary int eraction geometry wit h time-indep endent laser beams while the other uses laser pulses. T he ® rst case is a time-indep endent quant um mechanical problem while the second is time-dependent. T he lat ter problem has an exact solut ion, even in the case of an addit ional gravit ational ® eld [11,12]. Here we want to give a quant um mechanical descript ion of the beam split t ing process in the st ationary case. We treat the beam split ting process as the scattering of an atomic beam by a periodic pot ential which is given by the laser beam. In our case, an addit ional gravit ational pot ential is present. We proceed in the same manner as in usual calculat ions of t he transm ission of a plane wave through a pot ential barrier. However, in our case the problem is more complicat ed owing to the two-level structure of the quant um st ate, the periodic structure of the scat tering pot ential, and the addit ional gravit ational pot ential. We assum e that the periodic pot ential has a rectangular pro® le (see Fig. 1) . T he laser region is bounded by two parallel planes S and S 9 possessing a common normal n . We calculat e the int ensity of the out going atomic beam as a funct ion of the incom ing beam. T here we can addit ionally dist inguish between the cases where the incoming atoms are in the ground state or in the excited st at e. In order to describe the beam split ters in t he spat ial int eraction geometry we restrict ourselves to stationary states which are described by the stationary two-level Schr Èodinger equat ion, 2

E C (x ) = Here C

= ( CC

b a

p ± 2m

1 hÅ x s 3 + H a + H d ip ( x ) + H int (x ) C (x ). 2

(1)

) is a two-component wave funct ion, s 3 is the third Pauli-

A t o m ic I n t e r fe r o m e t r y in G r a v i t a t io n a l F i e l d s

637

F i g u r e 1 . Up per: Geom et ry of a sim ple at om b eam int erferom et er; see [13]. An at om ic beam wit h at om s in the grou nd st at e wit h m om ent um p , | a, p i , is split during t he int eract ion w it h a ® rst laser b eam int o t he original st at e an d t he excit ed stat e w it h m om entum p + hÅ k . T he ot her two laser beam s serve as m irror and analyz er. T he int erferen ce pat t ern is observed by cou nting t he ex cit ed at om s leavin g t he an aly zer. Lower: Geom et ry of t he b eam splitt ers. Atom ic plane waves wit h m om entum p ent er t he laser region bet ween t he surfaces S an d S 9 . T hese surface s have t he x coordinat es d an d d9 = d + l . T he wave vect or of the laser beam k is t ran sferred w ith a cert ain probab ility to t he at om ic wave so that the transm itt ed at om ic wave is split int o two waves.

638

L Èa m m e r z a h l a n d B o r d Âe

matrix, and

Ha =

Eb 0

1 1 ( E a + E b ) + hÅ x 2 2 ik ¢ x 0 e , e - ik ¢ x 0

0 Ea

H d ip (x ) = ± hÅ V ba H int (x ) = ± mg . x ,

=

ba s

3

,

(2) (3) (4)

with x ba := ( E b ± E a )/ hÅ , where E b and E a are the upp er and lower energy levels of the atom, m is the mass of the atom and V ba = ( e/ 2hÅ ) j h aj m E j bi j is the Rabi frequency. x and k are the frequency and the wave vector of the laser beam, respectively. We neglect any relaxat ion eŒect. We have already removed any time-dependence from the dipole int eraction term Ð compare [11]. 2. W K B SOLUTION IN THE LA SER REGION In two-component not ation (1) is given by 2

E9 C

=

± hÅ D ± mg . x ¢ ± hÅ V ba e - ik x

± ( hÅ / 2m) Ñ

where 4 D = x

± x

ba

± hÅ V ba eik ¢ x

2

2

± ( hÅ / 2 m) Ñ

2

± mg . x

C ,

(5)

is the detuning. We ® rst de® ne

w (x ) =

eik ¢ x 0

0 1

C ( x ),

(6)

which means that the upper state acquires an extra momentum k. T hen we make the W KB-ansat z

w (x ) = e -

( i/ hÅ ) S ( x )

a( x )

with a real valued scalar phase S (x ). We assume that a( x ) and p = ± Ñ are only slowly varying. Insertion of the ansat z (7) gives 0=

( (p + hÅ k) 2 / 2 m) ± E 9 ± hÅ D ± mg . x ± hÅ V ba

± hÅ V ba (p 2 / 2 m) ± E 9 ± mg . x

(7)

S

a. (8)

4

T he Laplacian is den ot ed by

Ñ

2

.

A t o m ic I n t e r fe r o m e t r y in G r a v i t a t io n a l F i e l d s

639

T he Hamilton± J acobi equat ion is given by the solvability condit ion 2

hÅ V 2ba =

(p + hÅ k) 2 ± E 9 ± hÅ D ± mg . x 2m

p2 ± E 9 ± mg . x . 2m

(9)

For the momentum p , this is a fourt h-order equat ion, that is, if e.g. py and pz are given then there will be four solut ions for the component px . T herefore, in general the solut ion consist s of four waves and t he mass shell has two disconnect ed component s, called branches. (A thorough discussion of the dispersion surface will is in [14].) We solve this equat ion for constant energy E 9 and get two Hamilt on± J acobi equat ions or two dispersion surfaces with t he corresponding solutions p 1 and p 2 :

E9 =

p 21 ,2 2m

± mg . x + hÅ V ba ( y(p 1,2 ) ¨

with

y( p ) :=

(p + hÅ k) 2 ± 2m

1 2hÅ V ba

1 + y2 (p 1 ,2 ) )

p2 ± hÅ D 2m

.

(10)

(11)

Note t hat y does not depend explicit ly on the posit ion 5 . It depends on p through the combinat ion p . k. Consequent ly, it is most appropriat e to choose one of the momentum component s which have to be prescribed, to be that component in the direction of k. In this case we have y(p 1 ) = y(p 2 ) = y( p ). T his funct ion y is int roduced in analogy to the theory of dynamical neut ron diŒraction which has many features in common wit h our theory (see Ref. 10) . T he two Hamilton± J acobi equat ions can be rewritt en as (p t = ( py , pz ) )

e1 ,2 (p t ) = 5

p2x ± mg . x 2m

(13)

T his is no longer t rue for a cou pling t o a rot at ing fram e. In t his case t he coe cient y reads

y( x , p ) = =

1 2 hÅ V b a

H ( x , p + hÅ k ) -

1

1

2V ba

m

p- x £

V

H (x , p ) -

¢

k+

hÅ 2m

hÅ D 2

k - D

,

( 12)

so that t here is an ad dit ional x -dep en dence which has t o b e t aken int o accou nt an d lead s to a correct ion of t he Sagnac eŒect .

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L Èa m m e r z a h l a n d B o r d Âe

with

e1,2 (p t ) =

p> 2 ± hÅ V ba ( y( p ) ¨ 2m

E9 ±

1 + y2 (p ) ).

(14)

In the following we will neglect re¯ ected waves. We ® rst discuss some general feat ures of the classical equat ions of motion. T hese equat ions are given by p 1 ,2 d hÅ k x 1,2 = + A ¨ (p ), dt m m with

A ± (p ) :=

d p = mg, dt 1,2 y(p )

1 1± 2

.

1 + y2 (p )

(15)

(16)

T he solut ion of (15) is 0

p 1,2 ( t ) = p 1 ,2 + mg t

(17)

where t he subscript 1, 2 for p 01 ,2 indicat es that for two given component s of t he init ial momentum there are two solut ions for t he third component due to the two Hamilt on± J acobi equat ions. T he equat ion for x has to be diŒerentiat ed once more:

d2 hÅ k (g . k) x = g¨ . 2 1 ,2 dt 4 mV ba (1 + y2 (p )) 3 / 2

(18)

T his equat ion of mot ion describes that each branch is sub ject to anot her eŒective accelerat ion induced by the gravit ational force. T he second term can be varied by manipulat ing the Rabi frequency and the wave vector. If we int roduce the eŒective mass t ensor 1

m*1 ,2

ij

:=

¶ 2H ¶ pi ¶ pj

,

(19)

p= p1 , 2

then (18) has the form

d2 i x1,2 = dt 2

1

m*1 ,2

ij

gj .

(20)

T he eŒective mass tensor, which still depends on t he momentum p, is responsible for two diŒerent accelerat ions of the atom which leads to two diŒerent atomic trajectories inside the laser beam region.

A t o m ic I n t e r fe r o m e t r y in G r a v i t a t io n a l F i e l d s

641

F ig u r e 2 . Anom alous int eract ion of t he two-level sy stem w ith an ex t ern al force: aft er t he splitting of t he incom ing at om ic beam one part of t he beam acceler at es diŒerent ly from t he other, t hough bot h are sub ject to the sam e force.

With t he solut ion for the momentum, the equat ion for the pat h can be explicit ly int egrat ed. However, in order to display compact result s, we restrict ourselves to ® rst order in t he gravit ational accelerat ion and specialize to the case k = k e z , g = g e z , y0 = 0, and p0 y = 0. 1 mgz 2me1,2 ( p0 z ) 1 + t, m 2e1 ,2 ( p0 z ) 1 z1 ,2 ( t ) = z0 + p0 z + hkA Å ± ( p0 z ) t m 1 hk Å 2 1 + 1¨ gt 2 . 2 4 mV ba (1 + y2 ( p0 z )) 3 / 2

x1 ,2 ( t ) = x0 +

(21)

(22)

T he second term describes the usual beam split t ing of the atomic beam inside the laser region. T he last term is a gravit ational modi® cation of the beam split t ing process. We have to look for t he phase and the corresponding momenta depending on t he spat ial coordinat es only. We choose again x = x 0 as the boundary along which the phases S 1,2 ( x , z) are given. We get for the phases ( p0 z ( z) = pz ( x0 , z) ),

S 1,2 ( x, z) = ± p0 z ( z) z ± +

2 m( e1,2 ( p0 z ( z )) + mgz) ( x ± x 0 )

gmp0 z ( z) 2 ( x ± x0 ) 4e1 ,2 ( p0 z ( z))

(23)

and the corresponding momenta

px 1 ,2 ( x, z) =

2 m ( e1,2 ( p0 z ( z )) + mgz ) ±

gmp0 z ( z) ( x ± x 0 ) (24) 2 e1 ,2 ( p0 z ( z))

642

L Èa m m e r z a h l a n d B o r d Âe

pz 1 ,2 ( x, z) = p0 z ( z) + m2 g

x ± x0 2 me1,2 ( p0 z ( z ))

.

(25)

T hese momenta solve the Hamilton± J acobi equat ion to ® rst order in g. Note that there is a split ting of the momentum in the z -direction, which is not present in t he case of vanishing gravit ational int eraction. T he normalized amplit udes can be found from eq. (8) which gives the solut ion

w ( x ) = a1 ±

A+ (p) A - (p )

e - iS 1 ( x ) + a2

A - (p ) A + (p )

e - iS 2 ( x ) .

(26)

In t he case of vanishing laser int ensity (V ba = 0) we have A + (p ) = 1 and A - (p ) = 0 and therefore 6

w

0

( x ) = a01

1 0

e - iS 1 ( x ) + a02 0

0 1

e - iS 2 ( x ) . 0

(27)

We also have in this case from (10) E 9 = ( 1/ 2 m)( p2x 1 + p2z 1 ) ± mgz so that for z = 0 the E 9 is just t he kinet ic energy of a gound state at om. 3. THE B EAM SPLITTER IN A GR AVITATIONAL FIELD We turn t o the descript ion of t he beam split ter in t he presence of a gravit ational ® eld. For doing so, we ® rst take an atomic beam with a certain phase wit h boundary condit ion S 0 ( x = 0, z) = const. T his atomic beam with a given phase propagat es in a laser free region to the entrance surface at x = d. T here it enters the laser region, propagat es through the laser region, and leaves it. At the entrance and exit surfaces, jump condit ions have to be used, and during the propagat ion the calculat ed phase gives the solut ion. Since at each point we can assign a wave vect or to the wave funct ion, we can use the jump condit ions derived ab ove locally. We proceed in several steps. We consider how t he wave funct ion ent ers the laser region at the entrance surface at x = d, propagat es to the exit surface at x = d9 and leaves the laser region. At t he end we combine all these steps and get a relat ion between the wave funct ion entering the laser region and t he wave funct ion leaving the laser region. St e p 1 : T he incoming state in front of the beam split t er is given by (27) . St e p 2 : T he state behind the ent rance surface of the beam split ter is (26) . T hese two wave funct ions are connect ed by jump condit ions: (i) the 6

A sup erscrip t 0 refers t o quant it ies in t he laser-free region .

A t o m ic I n t e r fe r o m e t r y in G r a v i t a t io n a l F i e l d s

643

tangent ial part of the momentum is continuous along the boundary, and (ii) the state is continuous over the entrance surface. T he ® rst condit ion, together with t he constancy of the energy, gives px ( d, z) as a funct ion of p0z ( d, z):

px 1 ,2 ( d, z ) = p0x 1,2 ( d, z) +

mhÅ V ba ( y( p0z ( d, z) ) ¨ px 1,2 ( d, z) 0

1 + y2 ( p0z ( d, z) ) ) ,

(28)

where we have made anot her approximat ion hÅ V ba ¿ ( p0x 1,2 ( d, z)) 2 / 2 m, which means t hat the kinet ic energy of the atoms is large compared to the int eract ion energy of the laser beam. T he continuity of the wave funct ion, that is, the equality of (26) and (27) at x = d, relat es the amplit udes inside the laser region t o t he amplitudes out side:

a1 = a01 a2 =

a01

A + ( pz ( d, z)) eis 1 1 ± a02 A - ( pz ( d, z)) e

is 1 2

+

a02

A - ( pz ( d, z)) e is 2 1 , A + ( pz ( d, z)) e

is 2 2

(29)

,

(30)

and the corresponding inverse relat ions, where we have int roduced S i0 ( d, z ) ± S j ( d, z ) = : hs Å ij ( i, j = 1, 2). Consequent ly, the state behind the entrance boundary is given by (26) with a1 and a2 from (29) and (30) . St e p 3 : T he state at the exit surface at x = d9 inside the laser region is

w ( d9 , z) = w 1 ( d9

w

1 ( d9 0

, z) = ( a1

± a02

w

2 ( d9

, z) = ( a10 + a02

, z) + w

2 ( d9

, z),

A + ( pz ( d9 , z )) eis 1 1 A - ( pz ( d9 , z)) eis 2 1 ) ±

A + ( pz ( d9 , z)) A - ( pz ( d9 , z))

e iS 1 ( d

9 ,z )

, (32)

A - ( pz ( d9 , z)) e is 1 2 A + ( pz ( d9 , z)) e is 2 2 )

A - ( pz ( d9 , z)) A + ( pz ( d9 , z))

eiS 2 ( d

9 ,z)

.

(33)

It is now essential to not e that the two states no longer possess t he same pz ( d9 , z) as it is the case without gravity. T hese moment a are given by (24) ,(25)

px 1 ,2 ( d9 , z) =

2 m( e1 ,2 ( pz ( d) ) + mgz),

pz 1 ,2 ( d9 , z) = pz ( d, z) +

m2 g 2 me1,2 ( pz ( d) )

(34) ( x ± x0 ) .

(35)

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L Èa m m e r z a h l a n d B o r d Âe

St e p 4 : Now we calculat e the state behind the exit surface. To begin with, we have now two jump condit ions for the moment a at the exit surface 1 0

px 1 ,2 ( d9 , z) = px 1 ( d9 , z) +

mhÅ V ba ( ± y( pz 1 ( d9 , z)) px 1 ( d9 , z)

±

1 + y2 ( pz 1 ( d9 , z)) ), mhÅ V ba 2 0 px 1 ,2 ( d9 , z) = px 2 ( d9 , z) + ( y( pz 2 ( d9 , z)) px 2 ( d9 , z)

±

(36)

1 + y2 ( pz 2 ( d9 , z)) ).

(37)

T he component s pz 1 and pz 2 are continuous at the boundary. T he two diŒerent 1,2 p0z ( d9 , z) lead to two pairs of phases 1 ,2 S 10,2 ( x, z) behind the laser beam. Without gravity both st at es w 1 and w 2 are connect ed with the same pz ( d9 , z) so that they give bot h the same two st at es w 10 and w 20 . In the case with gravity the situat ion changes: Owing to the anom alous int eraction, we have two diŒerent component s pz 1,2 ( d9 , z ). T he momentum pz 1 ( d9 , z ) belongs to the state w 1 , and pz ,2 ( d9 , z) to the stat e w 2 . Both states now split int o two states: w 1 split s int o 1 w 10 and 1 w 20 , and w 2 int o 2 w 10 and 2 0 0 w 2 . Correspondin gly, the phase S 1 split s int o 1 S 1,2 and S 2 int o 2 S 10 ,2 . 1 0 2 0 1 0 2 0 However, the states w 1 and w 1 ( and w 2 and w 2 ) are diŒerent ; it is only for vanishing gravity that they coincide and simply add up. T his may be compared with the two diŒerent solut ions (20) for the z-coordinat e. T he int eraction with the gravit ational ® eld changes the velocity and t he moment um of the two branches in a diŒerent way, thus leading t o momenta at the exit surface which are not relat ed by jump condit ions corresponding to two energy eigenst ates behind t he laser beam. Consequent ly, for each w 1 ( d9 , z) and w 2 ( d9 , z ), we have to require the continuity of the wave funct ion separat ely. T herefore, we get from w 1 ( d9 , z ) the two states 1

0

w

( d9 , z) = 1 a01

2 ( d9

and from w 2

w

0

1 0

ei

0

+ 1 a02

0 1

ei

1

S 2 ( d9 , z )

0

+ 2 a02

0 1

ei

2

S 2 ( d9 , z )

1

S 1 ( d9 , z )

2

S 1 ( d9 , z )

0

,

(38)

,

(39)

, z) t he two states

( d9 , z) = 2 a01

1 0

ei

0

with 1

a01 = a1

A + ( pz 1 ( d9 , z)) e - i

1

s 91 1

,

(40)

A t o m ic I n t e r fe r o m e t r y in G r a v i t a t io n a l F i e l d s

1

a02 = ± a1

2

a01

2

a02

645

A - ( pz 1 ( d9 , z)) e - i

1

s 92 1

- i 2s 1 2

,

(41)

= a2

A - ( pz 2 ( d9 , z )) e

,

(42)

= a2

2 9 A + ( pz 2 ( d9 , z)) e - i s 2 2 ,

(43)

with i S j0 ( d9 , z) ± S i ( d9 , z) = : hÅ is9j i . T he total out going wave funct ion is

w

0

( d9 , z ) = 1 w

0

( d9 , z) + 2 w

0

( d9 , z) .

(44)

Su m m ar y o f s t e p s 1 t o 4 : If we insert the momenta and express t hem as funct ions of the init ial momentum p = p( d) at the entrance surface, and t he a1 ,2 as funct ions of the a01,2 of the wave funct ion in front of the laser region, then the total wave funct ion leaving the laser region has the st ructure

w

ou t ( x , z)

1 0

= 1 a01 + 2 a01

ei 1 0

1

S 10 ( x , z )

ei

2

0 1

+ 1 a20

S 10 ( x , z )

+ 2 a02

ei 0 1

1

S 20 ( x , z )

ei

2

S 20 ( x , z )

(45)

for x > d9 and where t he coe cients 1 ,2 a01,2 can be read oŒfrom (40) ± (43) with a1 ,2 from (29) ,(30) . T he ® rst two wave funct ions are the two energy eigenfunct ions belonging to the + branch inside the beam split ter, t he last two to the ± branch. T he ® rst and third wave funct ion describe atoms in the excited state, the second and fourt h atoms in t he ground state. From this we can int roduce two matrices 1 and 2 describing the split ting of the two branches 1 1

a01 a02

1

=

a01 a02

2

,

2

a01 a02

=

2

a01 a02

.

(46)

We express the momenta (36) ,(37) of the out going waves with the momenta of the ingoing wave to ® rst order in the gravit ational accelerat ion, 1 0

¼

p0x 1 ( d, z) 1 ±

2 mhÅ V ba A ¨ ( d) d y+ ( p0x 1 ( d, z)) 2

,

(47)

2 0

¼

p0x 1 ( d, z) 1 ±

2 mhÅ V ba A ¨ ( d) d y - , ( p0x 1 ( d, z)) 2

(48)

px 1,2 ( d9 , z ) px 1,2 ( d9 , z )

with

d ±y¼

hÅ V ba ( y ± 1 + y2 ) 2E 9 mg hk Å K g := . 2hÅ V ba Ö 2 mE 9

Kgl 1 + with

(49)

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L Èa m m e r z a h l a n d B o r d Âe

W ithout the gravit ational ® eld the out going momentum is the same as the incoming one. However, t he in¯ uence of gravity is modi® ed by the laser beam. For t he z component of t he momentum we get (see Fig. 3)

m2 g l = p0z ( d) + 2me1,2 ( d)

p0z 1 ,2 ( d9 , z) = pz ( d) +

m2 g l. 2 me1 ,2 ( d)

(50)

W ithout the anomalous int eraction of gravity with the atoms due to the addit ional periodic pot ential, bot h atomic beams will arrive at the exit surface in such a way that for bot h beams the created beams out side the beam split ter can sup erpose with the same wavelengt hs thus giving eŒectively two out going waves. However, due t o the anom alous gravit ational int eract ion we get four out going waves. 4. OMISSION OF THE GRAVITATIONALLY INDUCED SPLITTING In an int erference experiment with the simple geometry of Fig. 1, these four states propagat e t o the next beam split ter where they are split int o eight beams and a further beam split ter gives 16 beams. T he task is to calculat e these 16 beams. 8 of them describe atoms in t he ground state, the other eight beams atoms in the excited st ate. Since this is a huge amount of calculat ion, we make an approximat ion which consist s of omitt ing this addit ional split ting due to the gravit ational int eract ion. T hat is, we neglect an addit ional term of the order hÅ V ba / 2 E 9 in (49) . In this case there will be no addit ional split ting so that bot h branches give t he same out going waves after leaving the laser region. Consequent ly, it is no longer necessary to dist inguish between the matrices 1 and 2 . Instead, the coe cients of the out going wave are connect ed with the coe cient s of the ingoing wave by means of 0,ou t

a1 0,ou t a2

0 ,in

=

a1 0 ,in a2

(51)

with ( A 9± := A ± ( pz ( d9 ) , z))) := A + A 9+ e i ( s

11

± s 911 )

A ± A 9+ e i ( s

12

± s 922 )

-

+

A ± A 9 ± ei ( s

12

± s 92 1 )

A + A 9 ± ei ( s

22

± s 912 )

A + A 9 ± ei ( s

++

± s 921 )

A ± A 9 ± ei ( s

21

± s 921 )

+

A ± A 9+ e i ( s

21

± s 911 )

A + A 9+ e i ( s

22

± s 922 )

(52)

where we approximat ed s 9ij ¼ j s9ij . T his matrix represents the behavior of the beam split ter in the gravit ational ® eld. T he moduli j 1 2 j 2 and j 2 2 j 2

A t o m ic I n t e r fe r o m e t r y in G r a v i t a t io n a l F i e l d s

647

F ig u r e 3 . Top : Splitting of t he z-com p onent of t he m om ent um in t he presen ce of a grav it at ional ® eld. T he m ain diŒerence to t he case wit hout gravit at ion is t he splitt ing of pz inside t he laser region ( beside t he ad dit ional stim ulation of an add itional pz + hk Å m om entum ) . An incom ing m om entum pz 2 ex cit es the sam e m om ent a inside an d beh ind t he laser region. B ott om : For com parison: Sam e process in grav ity± free space.

648

L Èa m m e r z a h l a n d B o r d Âe

give the probability to ® nd the atom behind the beam split ter in an excit ed, or ground state, respectively, provided t he incoming at om was prepared in the ground state. Similarly, j 11 j 2 and j 1 2 j 2 is the probability to ® nd the atom in an excited or ground state, respectively, provided the incoming atom was in an excited state. With the usual Rabi wavelengt h k R := b 1 + y2 , b := mV ba / Ö 2 mE 9 ( b is a wavelengt h charact erist ic for the ratio of Rabi energy and kinet ic energy; in other words, it is the spat ial version of t he Rabi frequency V ba : since E 9 is the kinet ic energy of a ground stat e atom in the laser free region, E 9 = 12 mv2 , we have b = V ba / v ), we de® ne a gravit ationally modi® ed Rabi phase y g S R := k R l + (53) b K g l( d + l ) 1 + y2 and the corresponding local gravit ationally modi® ed Rabi wavelengt h

y

g

K R := k R +

1 + y2

2bK g ( d + l) .

(54)

T he evaluat ion of the matrix elements gives ( d := b y) 11

= e - i(

d + b K g ( d+ l ) ) l

y

+i

12

=

e - i( d

=

+

1

ei( d

2

= ei(

1+ y

± i 1 ±

1+ y

1

± i

g

2( 1 + y2 )

K g l sin S R

K g l cos S R ,

2

(56)

y 2( 1 + y2 )

g

K g l sin S R

g

1 + y2

d + b K g ( d+ l ) ) l

y

(55)

g

2

( 2 d+ l ) + b K g l ( d+ l ) )

2

1 g K g l sin S R , 2(1 + y2 ) 3 / 2 ± i 1 ±

1 + y2

+ 22

1+ y

2

( 2 d + l ) + b K g l ( d+ l ) )

±

21

g

cos S R

K g l cos S R ,

(57)

g

cos S R

y 1 + y2

+

1 K gl 2(1 + y2 ) 3 / 2

g

sin S R ,

(58)

where all quant ities have to be evaluat ed at the entrance surface x = d. We have used a ® rst order approximat ion in g and hÅ V ba / 2 E 9 in the amplit ude as well as in the exp onent .

A t o m ic I n t e r fe r o m e t r y in G r a v i t a t io n a l F i e l d s

649

We have always made t he calculat ion to ® rst order in mhÅ V ba / ( p0x ) 2 . 5. INTER FER ENCE EX PER IMENT TAK ING INTO ACCOUNT THE MODIFICATION OF B EA M SPLITTING A sequence of three beam split ters, which realizes the simple int erferometer (see Fig. 1), is given by the following procedure: We have at d = d1 = 0 a laser region with lengt h l 1 , then at d = d2 a laser region with lengt h l 2 and at d = d3 the third laser region with lengt h l 3 . T he assemblage of these t hree laser regions is described by

S=

3 ( d3 , l 3 )

0

( d3 , d2 )

2 ( d2 , l2 )

0

( d2 , d1 ) ;

1 ( d1 , l 1 ) ,

(59)

where 0 describes the propagat ion of the at oms in the region between the laser beams. For simplicity, we choose l1 = l2 = l 3 = l and we choose l in such a way that at the ® rst beam split ter is exact ly a p/ 2 beam in the absence of gravit at ion. T he condit ion for t hat is kR l = p/ 4. T he second beam split ter is assumed to have t he lengt h 2l and the third has the lengt h l . T he dist ance between t he laser regions is d. If we neglect t erms quadrat ic in the gravit ational accelerat ion in the amplit ude and in the phase separat ely, then we get as observed int ensity of the atoms leaving the int erferometer in the excited state

I2 =

1 1 ± cos 2 + Kg

mgk (2 d2 + 2 dl ± l 2 ) E 3 mgk d + l sin (2 d2 + 2 dl ± l 2 ) 2 E

,

(60)

provided t hat all incom ing atoms are in the ground stat e. In t he case l ® 0 and K g ® 0 we get for this phase shift the classical result for in® nit ely thin laser beams,

d w = kg

d2 , v2

(61)

which corresponds directly to the phase shift for t he time pulsed int erferometers d w = kgT 2 , where T is the time between the laser pulses. Note that because of m/ F 2 ~ 1/ N 2 for (61) and also for the more general case (60) there appears no mass in the phase shift . Consequent ly, the equivalence principle is ful® lled Ð see also [15± 17]. T here are two causes for a modi® cation of t he phase shift ( 61) : First, the eŒect that even if at the ® rst beam split ter the atoms are in resonance

650

L Èa m m e r z a h l a n d B o r d Âe

F ig . 4 ( a ) .

F ig . 4 ( b ) .

F ig. 4( c) . F ig u r e 4 . Upp er: T y pical interference pat t ern for an int erferen ce exp erim ent in the grav it at ional ® eld taking into accou nt t he m odi® cat ion of t he beam splitt ing process induced by the gravit at ional ® eld. Lower Left : T his ® gu re is for van ishing w idt h of t he beam splitter l = 0. Lower Right : For com parison: Int erferen ce pat t ern for the sam e speci® cat ions as in upp er ® gure but neglec t ing all grav it at ional m odi® cat ions of t he beam splitting process.

A t o m ic I n t e r fe r o m e t r y in G r a v i t a t io n a l F i e l d s

651

with the laser, the addit ional velocity gained during the ¯ ight to t he second beam split ter brings the atoms slight ly out of resonance. Second, the modi® ed accelerat ion inside t he beam split ter. T he ® rst modi® cation vanishes for K g ® 0, the second for l ® 0. Finally we present some diagrams of possible int erference pat terns. In addit ion, we specialize d to the case that t he atomic beam is at resonance y = 0 at the entrance surface of the ® rst beam split ter. We show the possible modi® cat ions for variat ions of the thickness of t he laser regions. T he corresponding diagrams for neut ron int erferometry in a gravit ational ® eld have been presented in [15]. A charact eristic feature of their result was that t he cont rast decreases for larger accelerat ions. T his is due to t he fact that in neut ron diŒraction there is absorpt ion of the neut rons if they do not hit the crystal surface with the Bragg angle. T he presence of a gravit ational accelerat ion acts in this way. In the case of atoms int eracting wit h a laser beam no such absorpt ion will occur. T herefore we do not expect to lose the contrast by varying the gravit ational accelerat ion. A speci® c feature of our diagram s is obviously the superposition of two waves. T his can be underst ood from the structure of the matrix elements describing the beam split ter. T his matrix contains g two wavelengt hs: ® rst the local Rabi wavelengt h K R and second the wavelengt h d in the phase of each mat rix element. T hese two wavelengt hs give the beat structure of the ® rst two pat t erns in Fig. 4. For comparison, the lower left diagram of Fig. 4 is the int erference pat tern wit hout gravitat ional modi® cat ion of the beam split ting process. It is calculat ed by neglect ing in the matrix elements all modi® cations of the amplit udes and of the beam split ting process, that is, by setting K g = 0. R EFER ENCES 1. B ord Âe C h. J . (1989) . P h ys . Lett. A 1 4 0 , 10. 2. Kasevi ch, M., an d C hu, S. (1991) . P h ys. Re v. Le tt. 6 7 , 181. 3. Riehle, F., Kist ers, T h., W it t e, A., Helm cke, J ., and B ord Âe, C h. J . (1991) . P hys . Rev . Lett. 6 7 , 177. 4. Sterr, U., Sen gst ock, K., M Èuller, J . H., B et t erm an n, D., an d E rtm er, W . ( 1992) . A p pl. P hys . B 5 4 , 341. 5. B ord Âe, C h. J ., Court ier, N., B urck, F. D., Gon ch arov , A. N., an d Gorlicki, M. ( 1994) . P hys. Le tt. A 1 8 8 , 187. 6. Chapm an , M. S., E kst rom , Ch. R., Ham m ond, T . D., Rubenst ein, R. A., Schm iedm ayer, J ., Weh inger, S., an d P rit ch ard , D. E . ( 1995) . P hys. Re v. Le tt. 7 4 , 4783. 7. Heh l, F. W ., and Ni, W .-T . ( 1990) . P hys. Rev . D4 2 , 2045. 8. Heh l, F. W ., Lem ke, J ., an d Mielke, E . W . ( 1991) . In G eom etr y a n d T h eoreti ca l P hysi c s , J . Debru s and A. C. Hirshfeld, eds. (Springer-Verlag, B erlin), p. 56. 9. Audret sch , J ., Hehl, F. W ., an d L Èam m erzah l, C. ( 1992) . In Re lativistic G ravi ty Re search. W ith E m p has is on E xper im e n ts an d Obse rvati on s , J . E hlers an d G. Sch Èafer, eds. (Lect ure Not es in P hy sics 410, Springer± Verlag, B erlin), p. 368.

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10. B ord Âe, Ch. J . ( 1996) . In A to m In ter ferom etr y , P. R. B erm an , ed. (Acad em ic P ress, B ost on) , p. 257. 11. L Èam m erzah l, C ., and B ord Âe, Ch. ( 1995) . P hys. Le tt. A 2 0 3 , 59. 12. Marzlin , P., an d Audret sch, J . ( 1996) . P hys. Rev . A 5 3 , 1004. 13. B ord Âe, Ch. J . ( 1992) . In Las er Spec tros co py X , G. Camy, M. Du cloy, E . Giacob ino, eds. (W orld Scienti® c, Singap ore) . 14. B ord Âe, C h. J ., an d L Èam m erzah l, C . ( 1999) . An n . P hysi k (Le ipzi g) 8 , 83.. 15. L Èam m erzah l, C . ( 1996) . G e n . Rel. G rav. 2 8 , 1043. 16. L Èam m erzah l, C . ( 1998) . Ac ta P hys . P olon . B 2 9 , 1057. 17. B ord Âe, C h., Karasiev icz, A., Tourrence, P h. ( 1994) . In t. J . Mod. P hys . D3 , 157. 18. B onse, U., and Wrob lew ski, T . ( 1984) . P hys . Rev. D3 0 , 1214.