5D relativistic atom optics and interferometry�������������������

Christian J. BordéSYRTE, Observatoire de Paris, F-75014 Paris, France

and Laboratoire de Physique des Lasers, F-93430 Villetaneuse, Francehttp://christian.j.borde.free.fr

April 26, 2013

Abstract

This contribution is an update of a previous presentation of 5D matter-wave optics and interferometry with a correction of some algebraic errors.Electromagnetic interactions are explicitly added in the 5D metric tensorin complete analogy with Kaluza�s work. The 5D Lagrangian is rederivedand an expression for the Hamiltonian suitable for the parabolic approx-imation is presented. The corresponding equations of motion are alsogiven. The 5D action is shown to cancel for the actual trajectory which isa null geodesics of the 5D metric. This presentation is mainly devoted tothe classical aspects of the theory and only general consequences for thequantum phase of matter-waves are outlined. The application to Bordé-Ramsey interferometers is given as an illustration.

1 Introduction

The foundations of relativistic 5D-optics for matter waves have been presentedin an earlier publication [1]. This is a natural framework to unify and comparephoton and atom optics thanks to formulas valid for arbitrary mass. The conceptof mass and its relationship with proper time in terms of associated dynamicalvariables and conjugate quantum observables are presented again here. Gravito-inertial �elds and electromagnetic �elds are included in the 5D metric tensor asin Kaluza�s theory. A corrected expression is given for the 5D Lagrangian andcorresponding equations of motion are derived. As in 4D, a superaction makesthe link with the quantum mechanical phase in 5D. The 5D generalization ofthe ABCD theorem [2, 3, 4, 5] for matter-wave packets leads to a single formulafor the quantum phase in presence of external �elds taking into account theinternal degrees of freedom of the particle.

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2 The status of mass in classical relativistic me-chanics: from 4 to 5 dimensions

In special relativity, the total energy E and the momentum components p1; p2; p3

of a particle, transform as the contravariant components of a four-vector

p� = (p0; p1; p2; p3) = (E=c;�!p ) (1)

and the covariant components are given by :

p� = g��p� (2)

where g�� is the metric tensor. In Minkowski space of signature (+;�;�;�):

p� = (p0; p1; p2; p3) = (E=c;�p1;�p2;�p3) (3)

These components are conserved quantities when the system considered is in-variant under corresponding space-time translations. They will become thegenerators of space-time translations in the quantum theory. For massive par-ticles of rest mass m, they are connected by the following energy-momentumrelation (see �gure 1):

E2 = p2c2 +m2c4 (4)

or, in manifestly covariant form,

p�p� �m2c2 = 0 (5)

This equation cannot be considered as a de�nition of mass since the originof mass is not in the external motion but rather in an internal motion (seeAppendix A). It simply relates two relativistic invariants and gives a relativisticexpression for the total energy. Thus mass appears as an additional momentumcomponent mc corresponding to internal degrees of freedom of the object andwhich adds up quadratically with external components of the momentum toyield the total energy squared (Pythagoras�theorem). In the reference frame inwhich p = 0 the mass squared is responsible for the total energy and can thusbe seen as stored internal energy just like kinetic energy is a form of externalenergy. Even when this internal energy is purely kinetic e.g. in the case of aphoton in a box, it appears as pure mass m� for the global system (i.e. thebox). This new mass is the relativistic mass of the stored particle:

m�c2 =pp2c2 +m2c4 (6)

The concept of relativistic mass has been criticized in the past but, as we shallsee, it becomes relevant for embedded systems. We may have a hierarchy ofcomposed objects (e.g. nuclei, atoms, molecules, atomic clock ...) and at eachlevel the mass m� of the larger object is given by the sum of energies p0 of theinner particles. It transforms as p0 with the internal coordinates and is a scalarwith respect to the upper level coordinates.

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Mass is conserved when the system under consideration is invariant in aproper time translation and will become the generator of such translations inthe quantum theory. In the case of atoms, the internal degrees of freedom giverise to a mass which varies with the internal excitation. For example, in thepresence of an electromagnetic �eld inducing transitions between internal energylevels, the mass of atoms becomes time-dependent (Rabi oscillations). It is thusnecessary to enlarge the usual framework of dynamics to introduce this newdynamical variable as a �fth component of the energy-momentum vector.

Figure 1: 5D energy-momentum picture

Equation (5) can be written with a �ve dimensional notation :

G�̂�̂bp�̂bp�̂ = 0 with �̂; �̂ = 0; 1; 2; 3; 4 (7)

where bp�̂ = (p�; p4 = �mc) ; G�� = g�� ; G�̂4 = G4�̂ = 0 ; G44 = G44 = �1

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This leads us to consider also the picture in the coordinate space and itsextension to �ve dimensions. As in the previous case, we have a four-vectorrepresenting the space-time position of a particle:

x� = (ct; x; y; z)

and in view of the extension to general relativity:

dx� = (cdt; dx; dy; dz) = (dx0; dx1; dx2; dx3) (8)

The relativistic invariant is, in this case, the elementary interval ds, alsoexpressed with the proper time � of the particle:

ds2 = dx�dx� = c2dt2 � d�!x 2 = c2d�2 (9)

which is, as that was already the case for mass, equal to zero for light

ds2 = 0 (10)

and this de�nes the usual light cone in space-time.For massive particles proper time and interval are non-zero and equation

(9) de�nes again an hyperboloid. As in the energy-momentum picture we mayenlarge our space-time with the additional dimension s = c�

dbxb� = (cdt; dx; dy; dz; cd�) = (dx0; dx1; dx2; dx3; dx4) (11)

and introduce a generalized light cone for massive particles1

d�2 = G�̂�̂dbx�̂dbxb� = c2dt2 � d�!x 2 � c2d�2 = 0 (12)

As pointed out in the case of mass, proper time is not de�ned by this equationfrom other coordinates but is rather a true evolution parameter representativeof the internal evolution of the object. It coincides numerically with the timecoordinate in the frame of the object through the relation:

cd� =pG00dx

0 (13)

Finally, if we combine momenta and coordinates to form a mixed scalarproduct, we obtain a new relativistic invariant which is the di¤erential of theaction. In 4D:

dS = �p�dx� (14)

1 In this picture, anti-particles have a negative mass and propagate backwards on the �fthaxis as �rst pointed out by Feynman. Still, their relavistic mass m� is positive and hencethey follow the same trajectories as particles in gravitational �elds as we shall see from theequations of motion.

4

Figure 2: 5D coordinates

5

and in 5D we shall therefore introduce the superaction:

bS = �Z bp�̂dbx�̂ (15)

equivalent to

bp�̂ = � @ bS@bx�̂ with �̂ = 0; 1; 2; 3; 4 (16)

If this is substituted inG�̂�̂bp�̂bp�̂ = 0 (17)

we obtain the Hamilton-Jacobi equation in 5D

G�̂�̂@�̂ bS@�̂ bS = 0 (18)

which has the same form as the eikonal equation for light in 4D. It is alreadythis striking analogy which pushed Louis de Broglie to identify action and thephase of a matter wave in the 4D case. We shall follow the same track for aquantum approach in our 5D case.What is the link between the three previous invariants given above? As in

optics, the direction of propagation of a particle is determined by the momentumvector tangent to the trajectory. The 5D momentum can therefore be writtenin the form: bp�̂ = dbx�̂=d� (19)

where � is an a¢ ne parameter varying along the ray. This is consistent withthe invariance of these quantities for uniform motion.In 4D the canonical 4-momentum is:

p� = mcg��dx

�pg��dx�dx�

= mcg��u� (20)

where u� = dx�=d� is the normalized 4-velocity with d� =pg��dx�dx� given

by (9).We observe that d� can always be written as the ratio of a time to a mass:

d� =d�

m=dt

m� =d�

M= ::: (21)

where � is the proper time of individual particles (e.g. atoms in a clock or in amolecule), t is the time coordinate of the composed object (clock, interferometeror molecule) and � its proper time; m;m�;M are respectively the mass, therelativistic mass of individual particles and their contribution to the scalar massof the device or composed object.In the usual paradigm of relativity, the time t is a coordinate variable and

the proper time � is taken as the evolution parameter to describe the motionof particles in space-time. In this presentation however, proper time is an inde-pendent coordinate describing the internal motion of massive particles, so thatwe shall rather chose the coordinate time as the evolution parameter. Another

6

good reason for this choice is that, in order to describe an ensemble of atomsor of atom waves within a clock or an atom interferometer, it cannot be a goodchoice to use the proper time of a speci�c atom to describe the full device. Weshall therefore write in 5D:

bp�̂ = m�G�̂�̂ _bxb� = m� _bx�̂ (22)

expressed with the "relativistic mass" :

m� = mdt

d�=

mcpg�� _x� _x�

(23)

and where the dot refers to derivation with respect to a "laboratory time"(identical to the proper time � of the apparatus only in the absence of gravitation

or inertial e¤ects). With this choice _bx0 = c and bp0 = m�c: An alternate choicecould be to take the proper time � of the full device as the evolution parameter.In which case:

cd� =pG00dx

0 and M = m�pG00 (24)

From :d�2 = G�̂�̂dbx�̂dbxb� = 0 (25)

we infer in 5DdbS = 0 (26)

and in 4DdS = �p�dx� = �mc2d� (27)

In Appendix A, we generalize these relations to an object, such as a clock, amolecule.., composed of a number of subparticles and illustrate the origin ofproper time as coming from the inner structure of the object.

3 Generalization in the presence of gravitational

and electromagnetic interactions

The previous 5D scheme can be extended to general relativity with a 4D metrictensor g�� and an electromagnetic 4-potential A�

g�� (p� � qA�) (p� � qA�) = m2c2 (28)

(q = �e for the electron).We shall search for a metric tensor G�̂�̂ for 5D such that the generalized

interval given by:d�2 = G�̂�̂dbx�̂dbxb�

is an invariant.

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Let us recall that, from the equivalence principle, the metric tensor g�� canbe obtained from the Minkovski �at space-time tensor ��� using in�nitesimalframe transformations from a locally inertial frame. Quite generally any in�ni-tesimal coordinate transformation considered as a gauge transformation can beused to introduce a component of the gravito-inertial �eld. As an example, in4D, the transformation (case of a rotation):

dx0i = dxi + �i0dx0

dx00 = dx0 (29)

transforms the interval

ds2 = g000(dx00)2 + g0ijdx

0idx0j (30)

intods2 = g00(dx

0)2 + 2g0idx0dxi + gijdx

idxj (31)

with

g00 = g000 + �i0�

j0g0ij (32)

g0i = �i0g0ij (33)

gij = g0ij (34)

g00 = g000 = 1=g000 (35)

g0ij = 1=g0ij (36)

Using :

gijgi0 = �g00gj0 (37)

we �nd

�i0 = � gi0

g00(38)

�i0�j0g0ij = �gi0g

i0

g00(39)

In the case of rotation we recover the usual metric tensor in the rotating frame.The action S becomes

S = �Zp0�dx

0� = �Zp00dx

00 �Zp0idx

0i (40)

= �Zp00dx

0 �Zp0i(dx

i + �i0dx0) (41)

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S = �Z �

p00 + p0i�i0

�dx0 �

Zp0idx

i = �Zp�dx

�

which gives the Sagnac phase asR �pig

i0=g00�dx0.

The same approach can be used with the �fth dimension by introducing thegauge transformation

dx04 = dx4 + �4�dbx�dbx0� = dbx� (42)

to generate the o¤-diagonal elements G�4

d�2 = G44�dx4�2+ 2G44�

4�dx

4dbx� + �g�� + �4��4�G44� dbx�dbx�G44 = G044 (43)

G�4 = �4�G44 (44)

G�� = g�� + �4��

4�G44 (45)

The superaction bS given by (15) becomesbS = �

Z bp�̂dbx0�̂ = �Z p�dbx0� � Z bp4dx04 (46)

= �Zp�dbx� + Z mc(dx4 + �4�dbx�) (47)

bS = �Z �p� �mc�4�� dbx� + Z mc2d� (48)

which yields the Aharonov-Bohm phase if mc�4� = qA�.The metric tensor in �ve dimensions G�� is thus written as in Kaluza�s

theory to include the electromagnetic gauge �eld potential A�

G�̂�̂ =

�G�� G�4G4� G44

�=

�g�� + �

2G44A�A� �G44A��G44A� G44

�G�̂�̂ =

�G�� G�4

G4� G44

�=

�g�� ��A���A� G44

�(49)

where � is given by the gyromagnetic ratio of the object. This metric tensor issuch that

G�̂b�Gb��̂ =

�G�� G�4

G4� G44

��G�� G�4G4� G44

�= ��̂�̂ (50)

=

�G�� ��A���A� G44

��g�� + �

2G44A�A� +�G44A�+�G44A� G44

�=

�G��g�� �G44G

��A� � �G44A� = 0��A�(g�� + �2G44A�A�) + �G44G44A� ��2G44A�A� +G44G44

�= ��̂�̂

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which implies

G��g�� = ���

G44 = 1=G44 + �2A�A� (51)

The equation :

G�̂�̂bp�̂bp�̂ = 0 (52)

with bp�̂ = (p�;�mc) (53)

and G44 = �1 is therefore equivalent to equation (28)

g�� (p� � qA�) (p� � qA�) = m2c2 (54)

Higher order electromagnetic interactions are introduced via the multipolarexpansion p� � qA� + Q�F��, where dipole moments will become operatorsin the quantum description.

4 Hamiltonian and Lagrangian : parabolic ap-

proximation

In some cases it is convenient to assume that the energy E is close to a knownvalue E0 either because energy is conserved and remains equal to its initialvalue or because of a slow variation of parameters. This means that the usualhyperbolic dispersion curve is locally approximated by the parabola tangent tothe hyperbola for the energy E0. This approximation scheme applies to massiveas well as to massless particles. We can then make use of the identity: E =E02 +

E2

2E0+O("2) valid to second-order in " = E�E0 (parabolic approximation).

Let us start with the exact formula:

bp0 = bp02+(bp0)22bp0 (55)

in which (bp0)2 is obtained from:0 = G�̂�̂bp�̂bp�̂ = G00(bp0)2 + 2G0bibp0bpbi +Gbibjbpbibpbj (56)

= G00(bp0 + G0bi

G00bpbi)2 +

Gbibj � G0biG0bj

G00

! bpbibpbj (57)

=1

G00(bp0)2 + f̂bibjbpbibpbj (58)

i.e.:(bp0)2 = �G00f̂bibjbpbibpbj (59)

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where

f̂bibj = Gbibj � G0biG0bj

G00(60)

is the 4D metric tensor, inverse of Gbibj . Hence

bp0 = bp02�G00f̂

bibjbpbibpbj2bp0 = G00bp0 +G0bibpbi (61)

and

bp0 =bp02G00

�f̂bibjbpbibpbj2bp0 �

G0bjbpbjcG00

(62)

bi;bj = 1; 2; 3; 4

With the choice of time coordinate such that _bx0 = c the Hamiltonian can be�nally written:

H =m�c2

2G00�f̂bibjbpbibpbj2m� �

G0bjbpbjcG00

(63)bi;bj = 1; 2; 3; 4

This expression is exact but requires the knowledge of the relativistic mass m�.In the parabolic approximation this quantity will �nally be approximated byits central value. From the previous exact expression of the Hamiltonian, theLagrangian is recovered as:

bL = �bp�̂ _bx�̂ = �12m�G�̂�̂ _bx�̂ _bx�̂ (64)

5 Equations of motion

From this Lagrangian we may infer the following equations of motion:

bp�̂ = � @bL@ _bx�̂ = m�G�̂�̂ _bx�̂ (65)

i.e.

_bxbi = f̂bibjbpbjm� +

G0bjc

G00(66)

and_bpb� = 1

2m� �@b�G�̂�̂� _bx�̂ _bxb� (67)

or_bp�̂ = 1

2m�G�̂

b� �@b�Gb��̂ � 2@b�Gb�b�� _bxb� _bxb� (68)

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These equations can be compared to those obtained either from the equationfor geodesic lines in 5D obtained from �d�2 = 0 with

d�2 = G�̂�̂dbx�̂dbx�̂ (69)

or from the condition:D _bx�̂ = 0 (70)

We proceed as in 4D and �nd:

�bx�̂ + (5)��̂b�b� _bxb� _bxb� = 0with

(5)��̂b�b� _bxb� _bxb� = 1

2G�̂b� �2@b�Gb�b� � @b�Gb�b�� _bxb� _bxb� (71)

We wish now to check that we recover the usual equations of motion in 4D whenthe metric is independent of the 5th coordinate:

�bx� +(5) ���� _bx� _bx� +(5) ��4� _bx4 _bx� +(5) ���4 _bx4 _bx� +(5) ��44 _bx4 _bx4 = 0 (72)

with

(5)��4� =G442�F�� (73)

(5)���4 =G442�F�� (74)

(5)��44 = G��@4G�4 = 0 (75)

The Christo¤el symbols in 4D and 5D are connected by:

(5)���� �(4)���� =

�2

2(A�F

�� +A�F

�� ) (76)

Hence

�bx� + (4)����_bx� _bx� + �2

2(A�F

�� +A�F

�� )_bx� _bx� = �G44 _bx4�F�� _bx� (77)

using

�G44 _bx4 = � _bx4 +G4� _bxb� (78)

we recover the usual 4D equation of motion:

m�(�bx� + (4)����_bx� _bx�) = qF�� _bx� (79)

sincem� _bx4 = bp4 = �mc (80)

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If we reintroduce the dependence in the �fth coordinate, we obtain the rateof mass change associated with the change of internal motion induced by anelectromagnetic �eld:

_bp4 =1

2m� (@4G�̂�̂) _bx�̂ _bxb�

= m� (@4G�4) _bx� _bx4 (81)

and similar expressions for the rate of energy-momentum changes induced byinternal transitions. In the case of electric dipole transitions, the photon energy-momentum is exchanged at the Rabi frequency rate. However in this approx-imation we do not obtain the Rabi oscillations (pendellösung) which requireto introduce two coupled modes and therefore a quantum treatment of theiramplitudes.

6 5D expression of the phase shift

The total phase di¤erence between both arms of an interferometer is usuallycalculated as the sum of three terms: the di¤erence in the action integral alongeach path, the di¤erence in the phases imprinted on the atom waves by thebeam splitters and a contribution coming from the splitting of the wave packetsat the exit of the interferometer [3, 5]. If � and � are the two branches of theinterferometer:

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��(q) =NXj=1

[S� (tj+1; tj)� S� (tj+1; tj)] =~

+NXj=1

�~k�jq�j � ~k�jq�j

�� (!�j � !�j) tj +

�'�j � '�j

�+ [~p�;D (q � q�;D)� ~p�;D (q � q�;D)] =~ (82)

where S�j = S� (tj+1; tj) and S�j = S� (tj+1; tj) are the action integrals along� (�) paths; ~k�j(~k�j) are the momenta transferred to the atoms by the j-thbeam splitter along the � (�) arm; q�j and q�j are the classical coordinates ofthe centers of the beam splitter/atom interactions; !�j(!�j) are the angularfrequencies of the e.m. waves; '�j('�j) are the �xed phases of the j-th beamsplitters; D is the detection port.With our new approach in 5D the action terms should be replaced by the

phase jumps induced by the beam splitters along the fourth space coordinatec� :

NXj=1

c2 [�m�j��j � �m�j��j ] =~ (83)

in which �m�j (�m�j) are the mass changes introduced by each splitter. Toobtain this result we write the action terms as:

NXj=1

S� (tj+1; tj) =NXj=1

�c2 [m�j+1��j+1 � (m�j + �m�j)��j ] (84)

with m�N+1 = m�D and ��N+1 = ��D. We shift j by one unit for the �rstterm:

NXj=1

S� (tj+1; tj) = c2m�1��1+

NXj=1

�c2 [m�j��j � (m�j + �m�j)��j ]�c2m�D��D

(85)We suppress the �rst term c2m�1��1 and add a current term c2m�D� followingthe logic of a phase term analogous to the spatial terms (these terms are gen-erally eliminated between both arms of the interferometer but they are indeednew phases arising in the 5D approach):

NXj=1

S� (tj+1; tj) is replaced byNXj=1

�c2 [m�j��j � (m�j + �m�j)��j ] + c2m�D (� � ��D)

=NXj=1

c2�m�j��j + c2m�D (� � ��D) (86)

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We have therefore in 5D:

��(q) =

NXj=1

�~k�jq�j � ~k�jq�j

�� (!�j � !�j) tj +

�'�j � '�j

�+ [~p�;D (q � q�;D)� ~p�;D (q � q�;D)] =~ (87)

where p�j ; ~k�j and q�j have now an additional 4-component equal respectivelyto m�jc; �m�jc and c��j . For Hermite-Gauss wave packets, this phase shouldbe evaluated at the mid-point (mid-point theorem [4]) q = (q�;D + q�;D) =2 .This mid-point phase shift is:

��((q�;D + q�;D) =2) =

NXj=1

�~k�jq�j � ~k�jq�j

�� (!�j � !�j) tj +

�'�j � '�j

�+ [(~p�;D + ~p�;D) (q�;D � q�;D) =2] =~ (88)

If energy is conserved, we may use the conservation of the Lagrange invariant(derived from Stokes theorem):

(~p�j+1 + ~p�j+1) (q�j+1 � q�j+1)�h(~p�j + ~p�j) + ~

�~k�j + ~k�j

�i(q�j � q�j) = 0

(89)and obtain the 5D scalar product:

��((q�N+1 + q�N+1) =2) =NXj=1

"~k�j � ~k�j

2(q�j + q�j)

#�(!�j � !�j) tj+

�'�j � '�j

�(90)

(we have also assumed q�1 = q�1). If energy is not conserved, we may use instead

the symplectic Lagrange-Helmholtz invariant in the quadratic approximation(Hamiltonian of degree 2 at most in position and momentum):

~p�j+1m��

(q�j+1 � q�j+1)�~p�jm��

(q�j � q�j) =~p�j+1m��

(q�j+1 � q�j+1)�~p�jm��

(q�j � q�j)

(91)which reduces to the previous Lagrange invariant with a good approximationfor small relative energy changes. This explains the cancellation of the actionand of the mid-point phase shift in the usual 4D approach as emphasized inreference [1].For the illustration, let us apply the previous formulas to the Bordé-Ramsey

interferometer [6, 7] represented on the �gure.

15

16

Formula (88) gives for the mid-point 5D phase:

��((bq�4 + bq�4) =2) = ~k:~q1 + (mb �ma)c2�1=~� !t1

�~k:~q�2 + (�mb +ma)c2��2=~+ !t2

�~k:~q�3 + (mb �ma)c2��3=~� !t3

+~k:~q�4 + (�mb +ma)c2��4=~+ !t4

+4Xj=1

�'�j � '�j

�+h( ~p�b4 + ~p�a4 + ~~k): (~q�4 � ~q�4) =2

i=~

+ [(mb +ma +ma �mb) (��4 � ��4) =2] c2=~ (92)

In the absence of gravito-inertial �elds (e.g. in the inertial frame of the atoms):

t2 = t1 + T ; ~q�2 = ~q1 +

�~p1 + ~~k

�T

m�b1

; ��2 = �1 +mb

m�b1

T ; ~p�2 = ~p1 + ~~k

t3 = t2 + T0; ~q�3 = ~q1 +

�~p1 + ~~k

�T

m�b1

+~p1T

0

m�a

; ��3 = �1 +mb

m�b1

T +ma

m�a

T 0; ~p�b3 = ~p1

t4 = t3 + T ; ~q�4 = ~q1 +

�~p1 + ~~k

�T

m�b1

+

�~p1 � ~~k

�T

m�b2

+~p1T

0

m�a

;

��4 = �1 +mb

m�b1

T +mb

m�b2

T +ma

m�a

T 0; ~p�b4 = ~p1 � ~~k (93)

and for the lower branch:

~q�2 = ~q1 +~p1T

m�a

; ��2 = �1 +ma

m�a

T ; ~p�2 = ~p1

~q�3 = ~q1 +~p1 (T + T

0)

m�a

; ��3 = �1 +ma

m�a

(T + T 0) ; ~p�3 = ~p1

~q�4 = ~q1 +~p1 (2T + T

0)

m�a

; ��4 = �1 +ma

m�a

(2T + T 0) ; ~p�4 = ~p1

We see that the �nal positions and proper times di¤er on both arms by smallquantities, owing to the relativistic di¤erences of velocities on both arms:

~q�4 � ~q�4 =

�1

m�b1

+1

m�b2

� 2

m�a

�~p1T +

�1

m�b1

� 1

m�b2

�~~kT

��4 � ��4 =

�mb

m�b1

+mb

m�b2

� 2ma

m�a

�T

Finally�� =

�2! � (m�

b1 +m�b2 � 2m�

a) c2=~�T (94)

17

For each segment, we check the conservation of the symplectic invariant (Lagrange-Helmoltz), e.g.:�

~p�2m�b1

+~p�2m�a

�: (~q�2 � ~q�2) =2 +

�mb

m�b1

+ma

m�a

�(��2 � ��2) c2=2 = 0 (95)

~p�3 + ~p�3m�a

: (~q�3 � ~q�3)�~p�2 + ~p�2

m�a

: (~q�2 � ~q�2)+2ma

m�a

(��3 � ��3) c2=2�2ma

m�a

(��2 � ��2) c2=2 = 0

(96)which correspond to a conserved Lagrange invariant only in the approximationof conserved energy along the arms of the interferometer. Within this approxi-mation, we may use the approximate expression

�� = ~k: [~q1 � (~q�2 + ~q�2) =2� (~q�3 + ~q�3) =2 + (~q�4 + ~q�4) =2]+!ba [�1 � (��2 + ��2) =2� (��3 + ��3) =2 + (��4 + ��4) =2]�! (t1 � t2 + t3 � t4)

= 2!T � !baT2

�2ma

m�a

+mb

m�b1

+mb

m�b2

��~k

2T

2

�1

m�b1

+1

m�b2

�+~k:~p1T

2

�1

m�b2

� 1

m�b1

�(97)

The same recoil and second-order Doppler corrections are obtained from thisapproximate formula to �rst-order but only expression (94) is exact. The second-term corresponds to the di¤erence in proper times for the clock term (that wehave called a quantum Langevin twin paradox in reference [1]). Note that thisclock term implies that the mass di¤ers on both arms and does not correspondto a di¤erent clock on each arm as in the classical Langevin twin paradox. Thecoherent quantum superposition of both arms is essential to generate a clock.The remaining piece of the recoil shift comes from �rst-order Doppler shiftscontained in the third term and originating from the spatial ~q part of the shift.

7 Conclusion

As a conclusion, the motion of massive or massless particles in 5Dfollows a null geodesic just as it is the case for photons in 4D. TheLagrangian is proportional to the interval squared and both vanishfor the real motion. This has the consequence that the phase, whichis proportional to the 5D superaction, will also vanish between twopoints of the real trajectory of the particle. As a consequence thephase shift in atom interferometers results only from the phase jumpsintroduced by the beam splitters.

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8 Appendix A

If we consider an object (such as a clock, a molecule...) composed of a numberof subparticles, the 5D superaction di¤erential is given by the sum:

dbS =XA

��pA�dx�A +mAc

2d�A�

(98)

where mA is the mass of particle A. With the following change of coordinates:

dx�A = dX� + d��A (99)

dbS = �P�dX� +XA

��pA�d��A +mAc

2d�A�

(100)

withP� =

XA

pA� and pA� = (m�A=m

�)P� + �A� (101)

The coordinates X� and ��A are such thatXA

m�Ad�

�A = 0 and �

0A = 0 (102)

(common time coordinate for all the particles of the composed object). Oneobtains for the full object:

dbS = �P�dX� +Mc2d� = �P�̂dX �̂ = 0 (103)

provided that:

Mc2d� =XA

��pA�d��A +mAc

2d�A�

(104)

=XA

���Ajd�jA +mAc

2d�A

�(105)

The source of the proper time � for the object lies in the internal degrees offreedom and its mass Mc2 is given by its internal Hamiltonian. A well-de�nedquantum phase for the composed object requires that it should be in an eigen-state of this internal Hamiltonian.

9 Acknowledgements

Special thanks to Dr Luc Blanchet and to Dr Peter Wolf for many fruitful andstimulating discussions.

19

References

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[2] Ch. J. Bordé, Propagation of Laser beams and of atomic systems, LesHouches Lectures, Session LIII, 1990, Fundamental Systems in QuantumOptics, J. Dalibard, J.-M. Raimond and J. Zinn-Justin eds, Elsevier SciencePublishers (1991) p.287-380.

[3] Ch.J. Bordé, Theoretical tools for atom optics and interferometry, C. R.Acad. Sci. Paris, t.2, Série IV, 509-530 (2001).

[4] Ch.J. Bordé, Atomic clocks and inertial sensors, Metrologia, 39 (5) 435-463(2002).

[5] Ch.J. Bordé, Quantum Theory of Atom-Wave Beam Splitters and Appli-cation to multidimensional Atomic Gravito-Inertial Sensors, General Rel-ativity and Gravitation, 36, 475-502 (2004).

[6] Ch.J. Bordé, Atomic interferometry with internal state labelling, Phys.Lett., A140, 10-12 (1989).

[7] Ch.J. Bordé, Atomic interferometry and laser spectroscopy, in: Laser Spec-troscopy X, World Scienti�c (1991) pp. 239-245.

[8] Ch.J. Bordé, A. Karasiewicz and Ph. Tourrenc, General relativistic frame-work for atomic interferometry, Int. J. of Mod. Phys. D 3, 157-161 (1994).

[9] Ch.J. Bordé, N. Courtier, F. du Burck, A.N. Goncharov and M. Gorlicki,Molecular interferometry experiments, Phys. Lett. A 188, 187-197 (1994).

[10] Ch.J. Bordé, Matter-wave interferometers: a synthetic approach, in AtomInterferometry, ed. P. Berman, Academic Press (1997) pp. 257-292

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