The Feynman Path Integral Approach to Atomic Interferometry. a Tutorial Pippa Storey, Claude Cohen-Tannoudji

The Feynman path integral approach to atomic interferometry. A tutorial Pippa Storey, Claude Cohen-Tannoudji

To cite this version:

Pippa Storey, Claude Cohen-Tannoudji. The Feynman path integral approach to atomic interferometry. A tutorial. Journal de Physique II, EDP Sciences, 1994, 4 (11), pp.1999-2027. 10.1051/jp2:1994103. jpa-00248106

HAL Id: jpa-00248106 https://hal.archives-ouvertes.fr/jpa-00248106 Submitted on 1 Jan 1994

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés.

(1994) Phys.

II trance

1999-2027 4 1994, 1999 NOVEMBER PAGE

Classification

Physics Abstracts

32.80 42.50

integral path approach Feynman interferometry. atomic The to

A tutorial

Cohen-Tannoudji Pippa Storey Claude and

Collbge

de France and Sup4rieure Kastler l'Ecole Normale Laboratoire Brossel de (*

Lhomond, Cedex

05, Paris France 75231 24 rue

(Received September1994) September accepted 1994, 22 26

problems interferometry Many Abstract. lend themselves in of atomic interest current to

guide integral practical path taking solving problems, We

such

present treatment. to

a a as

examples gravitational Chu, experiments Kasevich of the equivalents and atomic and the of the

Sagnac and Aharonov-Bohm effects.

Introduction.

interferometry rapidly-developing physical research,

and field of concerned Atomic with is

a new

ill.

plays phenomena

which of neutral role important wide the The in wave-nature atoms an

possibilities of

degrees investigation of of internal freedom for

variety

atom opens up new an

interferometry

using photons, do traditional of which the electrons and exist in types

not more

neutrons.

interferometry development by advances, been aided The of has technical atomic recent

deflecting, slowing, particularly manipulation cooling of the New mechanisms for in atoms.

and Also position important control of allow both their trapping and atoms momentum.

providing equivalent of of

optics", mechanisms the has been the birth of "atomic

range a

Recently pointed beamsplitters for been that has and certain it mirrors, lenses out atoms.

realizing

Doppler

the effect high-resolution techniques which avoid amount to spectroscopy an

adapted inertial fields methods have been

interferometer These since atomic to [2]. measure

gravitation) interferometry. (due by atomic and rotation to

often interferometry experiments close the classical is The encountered atomic to situation in

analysis

integral approach appropriate the since it path is

limit. When this is the to

very case a

simplifications

paths. Further made be integrals along

of classical calculation

reduces

to can a

gravitational

field

rotating particle for

quadratic, in Lagrangian is if the

true or a a as a

CNRS with the and the Universit4 Pierre Marie (*) Kastler Brossel associated is Laboratoire The et

Curie.

PHhSIQUE hOVEMBER li Vii JOURN~L DE T 4 t994

PHYSIQUE JOUIINAL II N°t1 DE 2000

perturbative integra1method possible. path simple reference frame. A also The is treatment

physical insight,

and traditional formulation of links the mechanics allows with quantum new

Huygens-Fresnel principle the of mechanics. wave

being body of first

article of the of the the presentation The main consists parts, two a

integral path applications method, of consisting examples and the second in atomic various to

interferometry. first begins Lagrangian dynamics.

The brief of The review with part quantum a

introduced,

quadratic expression simple for for derived

the of is and it propagator

a case

Lagrangians. applied the wavefunctions, then of The propagation is quantum propagator to

practical perturbative approach phase difference obtained. used

calculate the and This is is to a

applied interferometer. the of

theory between In the second the is the part two

to arms an

following free

particle particle,

two-level laser crossing in systems:

atom

wave, a a a a a

(including gravitational Chu), analysis by particle field Kasevich

of and

experiment the

an a

frame, equivalents the effects. of rotating

and the atomic Aharonov-Bohm in a

integral Path method. 1.

this

el- LAGRANGIAN review

REVIEW In section 1.I

cLAssicAL DYNAMICS. OF some we

beginning classica1Lagrangian dynamics principle of the of with least and action ements [3],

Hamiltonian, Lagrange define the equations. their the We and the and rela- momentum state

partial Using tionship derivatives of

classical this formalism the the

obtain action. to

an we

following about initial which of expansion position, the classical will be used the action in

sections.

travelling particle space-time. classical from Paths

We consider point the I.I.I in zata a

figure infinity

possible point paths the There shown of in spacetime exists I. zbtb to

as an

jr,

z(t)

linking r'.. being by path points,

described function such these each that two a

z(ta) z(tb)

paths, actually by possible Of the there which the and all taken

is is

zb. za

one " =

Lagrangian particle which determined from of and be Lagrangian the the The L system. can

velocity function

of particle position I,

and

potential

certain which for of is M in

a a mass z a

Viz) is

Liz, I) Viz). ii)

)Ml~

Principle path

least of The by particle taken classical for actual action. the I.1.2 is

one a

Liz, I) the

defined which extremal. The Lagrangian integral

action is action the of the is as

z(t) given path the over

/ tb

[z(t),

lit)] (2) Sr dt L

by rci the will denote path, actual classical for We which by extremal, the Sci and the action is

corresponding value the of

This only endpoints function "classical" action. of the action is a

Scj(zbtb, zata).

path, of the

SCI +

Lagrange Lagrange equations. differential

The principle equations

form I.1.3 of the

are a

(see Appendix of least A-I action dynamics ), describe the of and They the be

system.

can

written

~ ~

~~~

equivalent the and well-known equation Newton

to are

F Ma (4)

the the force acceleration F relating to a.

INTEGRAL A PATH ATOMIC APPROACH TO N°11 INTERFEROMETRY 2001

la t 16

Fig. paths spacetime connecting the Possible in point initial the final rci point is 1. zata zbtb, to

path, solution equations the actual classical of the of motion.

partial Partial derivatives classical The of of the derivatives the classical action IA action.

Defining Appendix

calculated A.2. in momentum

are as

IS)

+ P

partial and positions derivatives with final the initial be the written respect

to can as

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

)~d

(Zblb, Zala)

(~)

Pbi

(pb) along particle path rci

point classical where of the the the the is zata at momentum pa

(zbtb).

Similarly, by Hamiltonian defined the is

H+Pi~L, 18)

initial and final partial with the and the derivatives of classical times action respect to are

by given

(Hb) along path particle rci of the the classical point Ha Hamiltonian the the where zata is at (zbtb).

PHYSIQUE JOURNAL II DE N°11 2002

partial derivatives, Using of differential of

obtain the the classical the values the

we can

fixed,

Keeping spacetime point the classical the due the initial action variation in action. to

(z, t) change point the final spacetime in is

(~'

ill) pdz dsci

dz dt dt. H

~j~'

= =

expression This for the classical gives alternative action an

dt)

(p (12) H Sci

r~,

(holding

for

about expansion position the classical the initial We also obtain action

za an

constant). final coordinates the initial and the time zbtb la

define

THE this the The

In 1.2 section propagator. quantum PROPAGATOR. QUANTUM we

intuitively obeys explain equivalence composition used the between this which is it property to

Feynman's simple definition formulation. expression usual and A for the then is propagator

quadratic Lagrangians. obtained for the of

case

Definition of

The the of final 1.2.I time quantum propagator. system quantum state at

a a

through by determined the evolution U earlier its time is ta operator tb at state an

(la)) la)

(~~) (~b)) (~b,

i~k i~k "

wavefunction, by basis, final projection position The the final given of the the is state onto

therefore

i~biL~(tb,ta) (~b,tb)

i~k(ta)) (16)) i~bi~k

~k " "

(la)) (tb,ta)

~~a iZbiU

iZa) i~ai~k "

(Zaila) (IS) Zala) dZa (Zblbi K

16 "

where the function defined has been K as

jz~ju (z~i~, ji~, j16) i~)

z~i~) K jz~) e

particle amplitude the and denotes for

known the the arrive K propagator, is quantum to as

point from point given that zata. it zbtb starts at

wavefunction

analogy between and the propagation Fresnel- reveals

Equation quantum IS an

point wavefunction the the of principle: superposition of the all is the value Huygens zbtb at

z('ta... (see z[ta, 2). Fig. by "point sources" radiated the zata, wavelets

N°11 INTEGRAL APPROACH TO A PATH INTERFEROMETRY ATOMIC 2003

, la

,, la

la 16 t

Fig.

point

superposition contributions The wavefunction is of from all the 2.

zbtb zata, at

sources a

z[tar z$ta...

The of of composition composition the The 1.2.2 property propagator. quantum property

following

for using be derived the relation evolution the the propagator quantum operator can

(lc,la) (tbt (Ii) ta) lc)

U U U

(16,

" t

simply equation intermediate between and This that the where time tc is la 16- states any

arbitrary from calculated

final

of initial be in evolution time time

to state two

may an an a

by time, of from the init1al followed

the evolution that time intermediate state to stages: any

final resulting of the from intermediate the evolution intermediate the time. the state to

(16) gives definition of the Substituting equation the the above into propagator quantum

(tb, la) (zbtb, (tc,

zata) (zb(U tc) U K (za)

(Zc(U(lc,la)

(Zb(U(16,lc)

dZc

(Za) (Zc) "

(zeta, zata) (18)

(zbtb, zeta) dzc K K

of and of the

composition the expression propagator, The last line property is quantum an

possible paths connecting

the

All

interpreted intermediate be summation states.

may as a over

through

position point final spacetime the spacetime point initial zbtb

zata must to pass some

3). amplitude through partic- for this (see

passing Fig.

The the intermediate time tc at zc

(zeta,zata). amplitude K(zbtb,zeta) The total product K just the intermediate is ular state

of equals zata) the point final (zbtb,

from initial

point the arriving for the zata K zbtb at sum

The fact possible through

positions time amplitudes all intermediate passing for tc. at the zc

distinguishing probabilities

amplitudes applies

rather than is composition rule this that to a

of mechanics. feature quantum

The Feynman's composition for the is the expression property 1.2.3 quantum propagator.

Feynman's formulation, which defines of of the starting point propagator quantum

as sum a

PHYSIQUE JOURNAL II N°11 DE 2004

la 16 lc

paths through points point and the connecting Fig. Possible which intermediate

3. zata zbtb pass

zeta.

paths possible from all and modulus final connecting points contributions initial The the 5]. [4,

/lt, independent path phase equals Sr of of contribution However the factor the r. where each is

Feynman's along path for be Sr the the expression

the r. is action propagator quantum may

written

(zbtb, fit zata) e~~r/~, (19) K

fit £~

integral normalization denotes of

functional the where is and constant,

space a a over

possible paths connecting all and An alternative r is notation zata zbtb.

~ /

l7z(t)e~~r/~, K(zbtb,zata) K(b,a) (20)

l7z(t) of

integration. Feynman's denotes element

the shown that where be It

[4, 5] expres- can

(16). completely equivalent the usual definition sion is to

Sr/lt

generally Sr lt, rapidly phase classical limit where the between In the varies » very

Along neighbouring paths r, path

destructive interference and the classical however occurs.

extremal, paths. interference neighbouring the action and constructive the is

among occurs

(20).

only paths integral path classical the Therefore the close contribute the This will to to

reduces explains

mechanics classical the how lt mechanics limit in 0.

quantum to -

simple quadratic for A Lagrangians.

The for form be found quantum propagator 1.2.4 can

Lagrangian quadratic I, function of and form if if is that is it has the the the propagator

a z

a(t)l~

c(t)z~ e(t)z (21) b(t)lz

d(t)I f(t).

+ + + + + =

particle,

those free

of quadratic Lagrangians examples of Three

a are

(22)

L )Ml~, =

INTEGRAL PATH N°11 A APPROACH TO ATOMIC INTERFEROMETRY 2005

gravitational field,

particle in

a a

jml~ Mgz, L (23)

particle velocity reference frame and angular rotating

in fl with Galilean

respect to a a a

frame [3]

(n ir r)~

r) )M

Mn L (24)

)Mr~

+ +

x x =

only paths path expected (20), close Since the

classical contribute integral the

to to to very are

z(t) path 2(t) each of convenient it is deviation from in path the classical its terms to express

z(t) 2(t) f(t)1 (25)

+ =

boundary where the conditions are

z(t~)

t(t~)

t(t~) z(i~) i(i~) i(i~) (26)

z~ z~

======

, ,

fit) of deviation from (20) path, classical this the In for expression the

terms quantum propa-

be written gator

can as

zata) Dflt) lzbtb,

(27) K

)S12(t) +

exP

f(t)1) =

I, (25) (21)

Substituting Lagrangian gives expression

( quadratic for the which into is in 2, an

((t)]

[2(t) (. consider the the and will from of We contribution S each action the in to terms +

independent off

[2(t)], which The contribute and value just the S which is turn. terms are

(zbtb, z~ta).

classical from Sci the

contribution that The action ( the linear and in terms are

2(t). z(t) ( path the first-order difference path the the between the and is in classical action

along

the path. from extremal classical This is contribution the action the The since is zero,

quadratic however Writing explicitly, this is have

term out terms nonzero. we

((t)] zata)

b(t)(( (zbtb,

[2(t) dt S (28) Scj

+ + + +

c(t)(~j (a(t)(~

(27) Substituting expression equation the above for

the gives into propagator

zata)

(zbtb, (zbtb,

Sci K

zata

exp x = h

b(t)((

(29) dt

+ + /~17((t) c(t)(~j (a(t)j~ exp

~ t

integral independent

denoted equation functional above of The the and and be is

in zb, may za

F(tb,

following by simplified obtained expression Hence for the the is ). propagator la

(zbtb, (zbtb, F(tb,ta) zata) (30) zata)

Scj K

exp

= lt

(30)

result calculate the We the

PROPAGATION

1.3 to

OF THE WAVEFUNCTION. use now

simple Lagrangian quadratic. A

wavefunction the of the the where is propagation in case

by assuming

final wavefunction then obtained the that initial for the is expression is state a

phase Lagrangian perturbation evaluated,

the plane shift is and due the

The

on a wave.

interferometry applied calculations. result to

PHYSIQUE II N°11 JOUIINAL DE 2006

Lagrangian. Lagrangian quadratic quadratic

the For Wavefunction calculation for 1.3.I

a a

(30),

dependence entirely contained the the by which is in expression in given is propagator za

(IS) Scj(zbtb,zata). gives of Substituting expression equation the evolution this into action

wavefunction the

/ 16

(~l) (Za,la) Zala)

)~d l~(tb,la) (Zb,tb) (Zblb, dZa

eXP 16

(zbtb, (isci

phases

of neighbourhood position of where the integral

the the In the

exp za, over

/lt) la) of (za, predominant. point

position zata)

and1b will be This is known the cancel out as

phase" "stationary

plane wavefunction

the where the initial is Plane incident. In

1.3.2

wave a case wave

final appealing intuitively

result for wavefunction. phase the of gives stationary method the an

(13) expanded where initial position about the the

the Using equation action momentum is zo

of This the the incident path the classical is the zeta

zbtb momentum

po wave. same as on -

quadratic

Lagrangian phase.

Since stationary of the point

be is the will position out to turn zo

quadratic velocity, that the classical function be shown is action position and

the it in a can

verified that consider: the the for all positions. final This be of initial and systems the

we can

(73), (55),

particle reference particle gravitational rotating

field and particle

in in free

a a a a

(97). after therefore the expansion

for the classical action about terminates The frame zo

order second term,

C(tb,ta)(~,

z0ta) (33) (>taj

P0( (Zbtb, lid lid

(Zb>tbi +

20 "

where

)~d

(Zblb, (~~)

Z01a)

PO "

~~~~'~~~ ~~~~~' ~°~~~ ~~~~ ~~~

Expanding point

wavefunction the independent position. of the initial about and C

is zo same

gives

~ ~°~~°

~°~~~

la) (

+l~ (zo

exP

~fi -

~ ~i~

~~~~~

~~~~ ~~~ ~~~

(31), (33) (36)

( the

equation and substituted linear in expansions into into When terms are

exponential the and cancel in gets out one

d( )~d laj (, (,

la (Zb,16 l~(~b,

la (Zb,16 (20

+ +

~XP

16 16 "

-Eota) In

~i(pozo j

~~ ~~~~

~~~~'

~~~~~' ~~~ ~~~ ~°~~~ ~~~

fi lt

~~~~'~~~

~~~

~~~~~' ~~~~ ~°~~~ ~~~ ~~~ ~~~

PATH INTEGRAL APPROACH N°11 A INTERFEROMETRY TO ATOMIC 2007

simple

This interpretation. wavefunction final result calculate the has To particular

at very a a

trajectory particle position of

consider

classical the whose initial

and is

momentum a we po

through

phase final The of final which that wavefunction point. the by determined is passes

along path, phase this and the of the action classical trajectory's the wavefunction the initial at

amplitude final independent point. The of the wavefunction position, of depends only and is

final the initial and

times. on

formalism extended The provided non-quadratic potentials be of higher the to terms may

sufficiently

second than order small. are

analysis Generalizations. the

wavefunction above calculated the value 1.3.3 In of the we

considering by final

point trajectory particle

of classical the which zbtb that arrives at

at a a

possible assumed particle point. determine from We that the position which the it to zo was

classically left, given happen

that of have that had initial this It

momentum must po may an

phase point fact stationary unique. positions

if then all In unique, is initial it is

not not zo are

points phase. of This quadratic stationary follows. Since classical

be the action is

may as seen

position linear function the initial the

of in is

momentum

za, a za

~'~

2C(tb,ta)(za (38) zo).

pa PO

= " 8za

C(tb, la) the If becomes then condition satisfied for all initial

is time

any zero, pa po =

focussing

collapses

wavefunction this and positions function. the delta In

occurs, za. case a

C(tb, (37).

vanishing divergence Mathematically, of equation

the in la

causes a

happens One which this harmonic oscillator. Consider of classical the is the in system set

possible starting from After trajectories positions. all initial with la time 0

at momentum

po "

T/4

Quantum period the

of all trajectories point the will focus 0. quarter

onto

a zb a "

mechanically,

period and of the eigenstate, in later system quarter momentum starts

a a a

C[ta

position finishes be shown for that

the eigenstate. It harmonic oscillator in +

a can

m/2), T(1/4

integers for all and all 0 ta ta]

m. =

complicated

quadratic general both Lagrangian the If the situation is since in is not more

sufficiently dependent. slowly

long position

position, As and with initial F C F varies

are as

Firstly, phase used. should noted. points the method still be However be stationary two may

multiple phase. the final wavefunction stationary there discrete of In this points exist

may case

(37)

replaced by solutions, equation be

superposition of of and consists quantum must

a sum a

phase Secondly, C But C stationary point of from vanish. since is contributions each may zo.

globally

quadratic

locally, the for

dependent rather than position this will

as case was occur

only position. values this general of the initial In Lagrangians. In C vanish discrete will at

single focal described family point

of trajectories the classical will all

not to

converge as a case

neighbourhood stationary of such the above, points the

instead form will In but caustic. a

(33)

valid,

for be extended longer phase expansion and the the action method is to must no

general phenomenon physics,

higher of in include order The is

caustics

terms. very occurrence a

be by catastrophe theory semi-classical calculations More elaborate and treated is [6]. may

iii. performed approximation uniform the using

perturbation

the the effect of

limit. discuss Perturbative this section In 1.3A

on a we

perturbation calculated by the be phase introduced show shift Lagrangian. We that the may

path. unperturbed along perturbation the integrating by the

which initial the 1.3.2, discussed section in We consider the situation in

wave- same as was

final calculated wavefunc-

the plane

with In that function section

momentum

po. we was a wave

(37) following particle had which

classical the of initial point trajectory the tion zbtb at a

through perturbative passed preparation final for and point. which the the In momentum po

PHYSIQUE JOURNAL DE II N°11 2008

~ 16

--T------

Moi

16 la t

points

Fresnel paths in the Fig. Classical from

4. two zone. same

neighbouring replaced by be follows, trajectory this

show first that that

treatment can a we

possible

replacement through final This

point. the is which trajectory classical

passes same

sufficiently

close. trajectories initial the of provided positions the two are

through

figure the trajectories Both paths classical sketched in 4. Consider the two pass

departs Mo from the point labeled The which has ini- trajectory N. point final zbtb,

same

used phase

of which

in corresponds point stationary the and tial

zeta to momentum was po,

neighbouring point final

with point calculate the wavefunction. The M is

1.3.2 section to a

coordinates zta.

(up factor)

(37) amplitude

obtain for final wavefunction

the Using equation to

we an

Mo)/hl lisci(N, (39)

1b(N) 1b(Mo)

exP

= ,

(N,

along Mo the and N. connecting trajectory points where the the classical Sci action is MO

neighbouring Along

have

connecting point the M. the M N trajectory We consider to

we now

M)/lt) (isci(N, 1b(M) exp

~i(poz-Eota) In

~~~

~°~~

~~~~~'~°~~~

~°~ ~

~~~

~°~~~ ~~~~ /~ lt

~~~

~~~~~' ~ ~°~~~ ~~~

~°

~~~~

fi lt

0)~)

liscl(N> M0)/hl (40) lfi(M0)

)C(~

~~p ~~p

approximation

the

make Hence

can we

M)/lip Mo)/hj jisci(N, (41) jisciiN,

~ljM)

~ljmo)

exp m exp

(40)

exponential equation final factor of

sufficiently Mo the close that M is point if the to can

N°11 INTEGRAL PATH APPROACH A

ATOMIC TO INTERFEROMETRY 2009

' l~j

~Cl

~'°~

Cl '

la 16

Fig. paths unperturbed Perturbed perturbative phase calculate and used the 5. shift. to

Quantitatively, replaced

by unity. be expressed this condition be

may as

)

(@~

(42) l, <

the positions and When this satisfied where distance between the M. points the is is MO

and M be said

points the lie "Fresnel zone". All this

radiate in in to

can same zone

approximately perturbative phase with approximation calculation will this the We

in in MO use

wavefunction which of the follows.

perturbation Lagrangian the We consider

a on

eLi (43) Lo L

+ =

phase by perturbation shift

calculate wish introduced this We

where the the I.

to < on e

wavefunction, plane final wavefunction given that the init1al with is momentum

po. wave a

r))~ figure path corresponding denotes the classical In with initial the 5, momentum to po

o). unperturbed path

rcj Lagrangian point Mo. labeled the classical Its initial is (e is

perturbed linking Lagrangian corresponding N, Mo points the initial final and and but to same

r[j

po.)

0). (The general equal

# path

will be the initial this is in (e to momentum not on

o).

perturbed

corresponding Lagrangian # the path (e with initial classical momentum to po

M[.

follows summarized initial labeled The be point Its is situation

as can

r')~ p(ta)

Mo N 0

e - = =

r)j

(~~) p(ta) # #

N 0

po e

rjj Ml p(ta) #

N o

Po e - =

phase unperturbed 1b'°)(N) perturbation wavefunction shift

introduces between the The a

perturbed Up amplitude wavefunction factors wavefunctions the these given 1b(N). and to are

PHYSIQUE JOURNAL II DE N°11 2010

~l(N)

~li

M[ If and are MO

#(Mo)

(47)

ELI (Lo dt 1b(N)

exp m

endpoints r))~

first rcj the differ order Since the rcj have and

However and is

to same e.

# for path which the 0, maximizes action e

IL /

(48)

dt L dt

O(e~ ).

r~, r~(~

(47)

Substituting find first equation this that order into in

to we e

~ /

mlb(Mo)

(49) ELI dt (Lo 1b(N)

exp

r(o)

(45) Expression gives then

~l'"(N)e~~', ~l'°)(N)

~l(N) eLi

(So) dt

exp m

~ ri~

where

~~ ~~ ~~' ~~~~ /(j)

ii phase This result shows first introduced final wavefunc- the shift the that order in into to e,

simply by perturbation by determined the integral perturbation of the tion is along the the

unperturbed path.

interferometry Application

phase calculate Here 1.3.5 the shift introduced between to we

perturbation the interferometer due of Langrangian. the

two

an arms a on

(labeled Figure unperturbed possible I) the A'CB 6 paths classical ADB and represents two

(labeled II)

by particle followed

which be through the interferometer, of the the

two can arms

the final and which general point

arrive B We have considered here the tb.

at at t

most same =

paths situation, which the classical points initial from the in different time two start two at

example A' for

lengths

and the Michelson A. This interferometer where is in the type

case a

(between mirrors) reflecting beamsplitter

unequal. Very of the and the the two

two are arms

9). A' (see Fig. points example A for often however the coincide and

(51) phase According along equation by paths shifts induced the perturbation the the to

interferometer I and II the marked in

are

Li dt b#1

~'CB = li

~~ ~~ ~~~~ ~~~~ ~DB

PATH INTEGRAL APPROACH N°11 A ATOMIC TO INTERFEROMETRY 2011

~fl

A ~_ 11

paths through Fig. spacetime Two interferometer. 6. an

AA') (segment

path. of the diagram However shift Section classical III phase the is

not a

A' independent points the A known. It of perturbation, equals is between and the and is

/lt,

displacement

and Ax where the initial

points. between the Ax the is is momentum two po po

phase by perturbation shift introduced the The total the therefore between is two arms

(53) ii b#I bill dt, Li

AA'CBDA

j~~,~~~~ f~,~~~~ f~, replaced

by have Li where dt since

phase perturbation result

Lagrangian by This that introduced final shift shows the the

on a

integral simply by perturbation unperturbed path of the determined around closed the the is

the of the interferometer. comprising

two arms

Examples interferometry. in 2.

developed

apply above this formalism several of the In interest. systems part current to we

2,I) (Sect.

free of particle modification consider and the of propagation We first the the a

2.2), (Sect.

crossing of laser interfer-

wavefunction 2-level several since atomic

atom a a wave

separated

propagation free The results consist of laser-atom interactions. ometers zones

analysis Kasevich of used of of Chu be the the experiment sections and will and in in 2.2 2.I

section 2.3.

Lagrangian particle

just

The for free

contains the kinetic THE

2.I PARTICLE. FREE energy a

term,

Mi~.

(54) L

straight velocity paths simply space-time. classical Since the lines the

remains in constant, are

therefore The classical action is

~~~ ~~~~

Ml~ (zbtb, zata) (55)

dt Sci

= =

2 2 tb ta

directly of

using expansion be calculated The in

quantum

propagator terms

momen- can an

which of eigenstates the evolution operator, states, tum are

/ -i~~ (

lzblU(tb,ta)lza)

lPlzal dP lzblP)

exP [~~~~

PHYSIQUE JOURNAL DE 2012 II N°11

(tb ta) zb) (za

~~~

2irlt 2Mlt lt

~~)~

)~~~

~~~

~~~~

ta)

(30), Comparing exp(isci/lt),

equation with

exponential the indeed factor that above is note

independent factor and the F position that of is

~'~~b'~a~ ~57) =

Evaluating the (6) using gives expression expected the result momentum

-)Sci

zata) (zbtb (58) M~a. Mll

= = =

(9), Similarly

Hamiltonian using yields which the be calculated expression can

(35) Definition for C gives

(t~

~~~~'~~~

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

lal' 2

plane

propagation The with initial calculated using be

momentum

a wave equa- po can

(37). of the the wavefunction initial The value point tion is at

~~~~ ~~~~

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

~~~~'~~~

Pita p(

2Mpozo la (16

2Mlt 2Mlt

~°~~ ~°~~

(62)

~ ,

/M. la) Po(tb (57)

A(tb) Using since (60) equations amplitude obtain the and

zo zb we "

of wavefunction the final the point at

expected, Hence wavefunction the point is, the final at as

PATH APPROACH A INTEGRAL N°11 INTERFEROMETRY TO ATOMIC 2013

~l~l Zih

kLzi-~Lti-v~

~-'

-L~~ti-v)

u«~ (kLzi u~«

U««

U~~

involving travelling and Fig.

possible two-level laser

Four interaction 7.

atom

processes wave: a a

la)

photon, gaining making absorbs and transition from

&kL, momentum state state to atom

a a o

emits and from 16) &kL, state atom momentum state atom to

o, a

(d) fl,

remains in in remains atom state state n.

freely-

A consider

2-LEVEL

2.2 ATOM cROssiNG TRAVELLING wAvE. LASER A a

travelling Initially propagating two-level which laser the with interacts

atom atom wave. a

of the

have well-defined and internal assumed be in is momentum, two states

to to one a a

by fl.

accompanied of the change The with laser field the the interaction atom,

state may or

longitudinal

figure

the shown We that change

in in 7

momentum,

transverse [2]. assume as a

(along sufficiently propagation that of the velocity the of the the great is atom atom transverse

beam) neglected.

during

be Hence associate with the interaction crossing laser the

we a can

Depending effect of of single the final the spacetime point initial and the atom, ziti. states on

multiplying

change by of four factors: the wavefunction the laser interaction atomic is to one

(a #)) (kLzi fl)

wLti Upn

exp -

j-I i)j (kLzi (p

U«p wLti

exp -

CY)

~65~

(" ") uaa

(fl fl)

Upp -

phase respectively of wavenumber, frequency and kL, and quantities #

The denote the wL

jth internal

amplitude from the the ith defined the the laser transition is

wave. as U~~

phase # and be by taking origin calculated coordinate be the the

ii atomic state, to to zero. zi

exp(+ikLzi) change

+ltkL of

the in factors associated with Note that the

transverse

a are

emitted. photon

absorbed of the when is the

momentum atom or cases a

the assuming interaction, in after

is example

of the the it calculate the As

state atom

we an

that the We

position propagates the initial

internal to time atom and

state at

at suppose zo a

makes position and freely with the laser

which interacts until time it at

ii, moment at a zi wave

freely the final when reaches fl. again until time it the It then t2, propagates transition state to

t2)

by product of ~lp(z2, given wavefunction is

At final the atomic position the this time

a z2.

wavefunction, free before phase propagation

shift due the four contributions: the initial to a

phase itself, free

due the another shift the interaction, due interaction and the factor to to a

PHYSIQUE DE JOURNAL II N°1t 2014

~L"L~

Ii t~ t~

photon absorption final

propagation and which determine the Fig. The of free 8.

processes wave-

(z2,t2).

function ~bp

figure

indicated result The after the These is in 8. propagation interaction.

processes are

~pjz2,t2) jiScijz2t2,ziti)/&j

exp =

(I #)) (kLzi

uJLti Upn exp x

jis~ijziti, zeta)/&I (zo,to) (66)

~fin exp x

to)

of the and where and calculated the wavefunction initial

is where atom,

~fin(zo, zi z2 are

change knowing and the due the initial

and t2, from ti &kL momentum to zeta, momentum po

laser interaction. the

apply previous

We results of

A will the the

2.3 GRAVITATIONAL FIELD. PARTICLE IN A now

particle field, of and gravitational experiment

and discuss Kasevich the sections in

two a a

Chu [8].

gravitational

Lagrangian particle field for Classical The in is action.

2.3.1

a a

Md~

Liz, (67) d) Mgz.

the applied.

quadratic directly previous I, this and be of

Since results the sections is in

may z

paths,

Lagrange be derived which from the equations, The classical

can are

(68) ~(t) 9(t ta)

Ua "

g(t ta) ta)~. (69)

Ua(t

Z(~)

Za "

gives final velocity position Evaluating the time the and tb at

ta) i~°1 9(tb

Ua Ub "

ta)~. ta)

Uajtb 171)

)gl~b Za Zb "

INTEGRAL APPROACH TO PATH ATOMIC INTERFEROMETRY N°11 A 2015

Using

endpoints equation, this last the coordinates of the of in

terms

express ua we can

~~ ~~

ta).

(72)

+ (g(tb

ua =

tb ta

along path by given the classical The is action

~~~~

~~~

~~~~~ ~~~ ~~~~~~ ~~~~~~g

~(~ (tb tat

l~~ ~~~~

~ 2 tb ta

~~~~

expected (6) yields expression for the that the result We momentum note

Mu(

(zbtb, zata) (75) Mgza. Ha

S~i

= =

recently observed Chu. Kasevich and of Chu have experiment Kasevich and The 2.3.2 [8]

They

gravitational effects the field. stimulated

interference which sensitive atomic to

use are

hyperfine ground of the of sodium

between levels and Raman the

transitions state two g2 gi

subsequently recombine wavepacket

coherent

and atomic into components separate two to an

vertically-oriented

pulses applied counterpropagating laser using

Three Raman them. two are

ki,uJi, Ii k2,uJ2,42 phases by

wavenumbers, frequencies and denoted and beams, whose are

9). (see respectively Fig. of insert

wavepacket /2 (at

pulse, which the into pulse

components 0) first The separates is two t

a 7r =

exchanges pulse applied,

which the differing by &(ki At T k2 is momenta ). t

momentum

a 7r =

spatially

2T, At when the of the internal components and t two components.

states two =

coherently. pulse recombine them 7r/2 used overlap, another is to

paths the figure The classical

shown in paths by followed the 9. spacetime in The atoms are

AoDoBo. AocoBo

the and In depicted by trajectories the straight

lines, gravity of absence are

represented by ADB. parabolic ACB and

the trajectories of the gravity

curves presence are

Note that

-jgT~

= =

-2gT~.

@ (76)

first interfering

beams. We shift between the phase gravitational

field introduces The a

perturbed trajectories.

using shift the phase

calculation of this

present exact an

path and AC along (73) the the actions the difference between segments equation two From

AD is

PHYSIQUE JOURNAL DE II N°11 2016

~u~ ~ji

~12

ii I-

2T t

experiment Spacetime by and of Kasevich Chu. Raman Fig. paths followed the in the 9. atoms

level scheme of

the pulses and The shows the atomic and the directions 0, times T 2T. insert

at occur

beams. laser

Similarly along the CB the difference between the actions and DB is segments

$(zc

S~i(CB) S~i(DB) (78) (zc

zD) gT~) 2zB

zD =

phase

difference due the the the Hence contribution the total between to to

two propa- arms

gation is

(AD) (AC)

S~i(CB) S~i(DB)] b#P"P

S~i

+ + [S~i

)

(zc (zc (79)

gT~ zD)

zD zB zA =

(76), taking However, relations factor second the the expression above into in is account

gT~ (80)

+ +

zA zDo zc zB zco zAo zBo zD

= =

parallelogram. AoBocoDo

since Hence is a

6#P"P (81)

phase

difference evolution, Neither there the from internal contribution the is since to 6#~~~ any

wavepacket spend

both of the of internal the time the in components amount two states. same

phase contribution difference from consider the the the laser The

We interactions. to now

directly applied pulses replacements after of

the Raman the results be section 2.2 to can

~#ki~k2, ~"~i~~2. (~~)

~J"~Ji~~J2

from path contribution Along the laser interactions is ACB the the

iIj ~ zco T~)

(83)

UJT

Uj)(~,

Uj~( (I Uj~(~ exp

2 ~ ~ ~

INTEGRAL APPROACH TO N°11 A PATH ATOMIC INTERFEROMETRY 2017

U(~) U(~)

U(~), amplitude

and where the matrices transition 0, and times T

t at t

are

= =

respectively, phase III #III and of # 2T and the values the 42 11

these times.

WI,

are = =

Similarly along path the the contribution from ADB the laser interactions is

2uJT #IIIj 2gT~)

Uj)(~ [~ (zBo exp x

gT~

#II

-I UJT

Uj)(~

exp zDo x ~

(84)

[~zAo Uj)(~ exp x uJ WI]

phase the total difference the of the interferometer between Hence is two arms

6jt°t ~gT2

(85)

6jiaSer 21II 4III iI

+ +

= =

(We gravitational neglected

which contribution the acceleration have the is sensitive to to g.

phases amplitudes they independent

U, of from the the of and of transition since 6#'~~~~

g are

lasers.) phases of the the

perturbative phase difference,

treating this result with calculation of the We

compare now a

gravitational potential perturbation. phase

the shift the As described the in section 1.3.5

as can

unperturbed paths. by along perturbation The integrating calculated contribution the the be

(85) from

phase

equation difference from the laser this obtained the interactions in is to case

by setting 0,

g =

(86) 2jII IIII

61[~~~~

WI =

(53) using equation from calculated propagation contribution the is The

~~ 6#("P ~~

(87) z(t) A,

= =

AocoBoDoAo

by Using

of interferometer. fact the the that distance A enclosed the the

where the is

area arms

Do Co point from the is to

~T,

(88) Doco

of interferometer find that the the is

we area

x2Tx@=-~T~,

(89) A=

phase propagation shift the and the due is to

6#["P ~gT~. (90)

phase difference evolution, total the from the internal

contribution there Since is

6#(~~ no

of interferometer the is between the two arms

(gi)

~gT2

i~~~,

21~~ + 611~°~ + 641°~ +

611~~~~

= =

(85).

with the calculation which exact agrees

velocity, independent of and hence phase difference the initial remains the Note that is

spread velocity experiment of the beam. The unchanged the atomic averaging

after over

10~~

gravitational

field,

sensitivity Ag of of the provides precise 3

measurement

x m a very a

and Chu. by the authors Kasevich being estimated

PHYSIQUE JOURNAL DE II N°11 2018

X' z'

rotating Fig. Galilean and reference frames. 10.

perturbative path

this

A In section

2.4

present

ROTATING FRAME. PARTICLE IN A a we

apply interferometry. integral particle frame,

and rotating of

the results in treatment to

a a

compared optical of interferometers that of sensitivity The with is rotation to matter-wave

(the effect). Sagnac interferometers

x'y'z' R' by We define Galilean frame described

the coordinates Classical 2.4.1 action. a

figure

and frame shown The the with coordinates of rotating R 10. in

two

xyz, axes a as z

R' angular velocity coincide, the Galilean frame and frame with R frames respect rotates to at

Qez, along being fl fl the direction. the where unit

vector

ez z =

Lagrangian particle, expressed of

of The for free the coordinates the Galilean in terms a

R', frame is

Since

the

ively

= v' + x fl r,

v (93)

L'(r', jM v') (r, (v v) r)~

fl L

= =

)M (r (fl v)

(94) r)~

Ma jmv~

+ +

x x =

quadratic

Lagrangian

and this is in Note that

r v.

by given The is momentum

(95)

Mfl

x r, p

= =

by Hamiltonian and the

H=p.v-L=j-Q.(rxp). (96)

INTERFEROMETRY PATH APPROACH ATOMIC INTEGRAL TO A N°11 2019

8 j

r+dr

Unperturbed path perpendicular plane fl. rotation AB in the the axis Fig. 11. to

straightforward for classical the gives calculation action A

(xayb

(97) xbya)

xayata) xa)~

(xbybtb, MQ (yb

S~i

+ +

((xb ya)~j

(tb ta) , 2

quadratic of and

which function

Ya yb. xa, xb, a

the

which phase of shift. consider in the We situation Perturbative calculation 2.4.2 now

sufficiently

small velocity that intervals At of the angular and Q time interest the are

(98)

flat 1. <

perturbation, and the order in considered

second the

be terms this

In rotation

as a can case

Lagrangian neglected the in

Mv~

(99) v). (r

Mfl L

+ x m

path,

unperturbed which along calculated the perturbation shift is due the phase The to

previous trajectory unlike the figure

that Note

shown straight in

line 11. segment is

as a

depicts the figure and real representations, this is in

spacetime which diagrams,

space, were

and particle position

fl. leaves time The perpendicular ta rotation axis plane the at

to ra

by along given

this is phase trajectory shift accumulated

The position time tb. arrives at at rb

(100)

jr(t)

V(t)j 6j dt

But since

(101) dtv(t)

dr(t),

shift phase be rewritten the

as can

/ ~

2~~

~~ ~

r(t) dr(t) (102)

Ao, 6#

= =

r(t)

by origin. path the the Ao subtended the DAB where is at area

PHYSIQUE JOURNAL DE N°11 2020 II

I o

(in space) paths describing Fig. Unperturbed

real the

atomic interferometer of 12.

two

an arms

sensitive rotation. to

phase

interferometer. the shift Since the We this result calculate in rotation

to now use an

perturbation

just applied, results obtain motion, the of section be and the is 1.3.5

a on can we

2~~

(103) 6# A

figure by A ACBDA enclosed

interferometer, where the the of the shown is in 12.

area arms as

figure diagram plane Note that like this real spacetime, 11, rather and is in than the space

perpendicular shown the is rotation axis. to

Sagnac Sagnac Conlparison effect. phase

effect the The with the shift observed 2.4.3 is in an

optical optical interferometer due Here sensitivity the of

rotation rotation.

to to compare we

interferometers. and matter-wave

figure rotating optical circular interferometer Consider Two emitted shown

in 13.

as a rays

circulate interferometer point opposite directions from the A time around and the in o

at t =

beamsplitter, interfere which assumed the point be of B is

the absence In o. at to at at t

the travel of beamsplitter the the equal.

rotation times However when the

two to rays are

travelling rotated,

interferometer the

longer

is the takes in fl

time

to ray sense same as a

according beamsplitter

reach the travelling observer Galilean frame.

Likewise

an ray a a

opposite the The travel

takes shorter determined time in and times be t~

t~ sense a can

following by equations the

7rp+pot+ ct+

(104) pot~

ct~

7rp =

difference Hence the time is

~rp ~rp

/~~

pn pn

+ c c

l~'~

?~[~

(105)

~ =

Sagnac phase therefore shift interferometer. The is of

the

the A is where

7rp~

area =

(~~

(106)

6§iphoton

~J0

# " ,

INTEGRAL APPROACH PATH ATOMIC N°11 TO A INTERFEROMETRY 2021

/ l'

/ '

t=t~

t=t-

Fig. Rotating optical beamsplitter circular interferometer. The point from B the time 13. starts at

angular frequency and point leaving Q. The

o, A circulate in the o t rotates at two

at t

rays

= =

beamsplitter different

and the opposite reach times and at t~ senses

(angular) light. frequency optical of sensitivity between the relative the the The

where is uJo

for interferometer enclosed and the is

matter-wave

area same

MAR

2 ~ ~ j~ §

~0 2~

~~~~

6§i~~n

might by

result, sensitivity From the of this factor much

last increase expect

a as one as

10~°

by photons. of instead be remembered that the using However it

matter must area waves

by considerably

of optical enclosed interferometer be increased that

matter

a wave over may an

signal-to-noise by optical also A achieved

interferometer of fibres. better ratio be the

use can

optical higher through because interferometers of flux of the the in quanta apparatus.

scalar will and

this the AHARONOV-BOHM In section review

2.5 vector EFFECTS. we

analogous particles. effects using Aharonov-Bohm and describe neutral effects [9, 10]

charged predicts

Quantum that Aharonov-Bohm effect. mechanics scalar The

2.5.1 par- a

potential, absence of electrical force. The

the in ticle be will sensitive electric

to any even an

potentials subjecting different electric wavepackets by them phase shift induced between to two

Aharonov-Bohm effect.

the scalar is known as

charged electromagnetic particle

field Lagrangian for is The in

an a

qU(r), (108) A(r)

L jmv~

qv =

A(r) U(r) potential,

charge electric and

the particle, the the is the is where is vector

q on

effect and U potential. Aharonov-Bohm A scalar the 0 0. In

figure effect The hypothetical Aharonov-Bohm indicated 14. experiment detect the is in A to

by Wavepacket wavepacket split the slits and I particle's I and II

into is components two si s2

through cylinder wavepack- wavepacket cylinder the The through

Cl, C2 the and II two passes

subsequently point interfere the M. at ets

PHYSIQUE JOURNAL DE II N°11 2022

experimental Fig. Sketch of scalar Aharonov-Bohm effect. the the setup 14. to test

longer length cylinders

wavepackets, assumed than coherence the be the

The of to

are so

wavepackets

that, interval, localised close the well the middle of time will be certain

to over a

cylinders. During potential applied Cl C2 the this time, and for duration the between

U is a

spatially-uniform potential, particle wavepacket Each

then

the T. and experiences

a sees no

potential phase wavepackets the introduces between the force. However shift of two a

f -~, ~ ~~

(109)

6# dt U

= = =

displacement fringes. of results the interference which in

effect.

charged The Aharonov-Bohm effect To the Aharonov-Bohm 2.5.2

vector test vector a

subjected potential effect, particle magnetic Aharonov-Bohm As the scalar is A. in to vector a

particle of region field, the travels electric and magnetic and hence in experiences

a zero no

force.

possible figure design experimental A shown charged particle wavepacket of is

The in is 15. a

split which travel recombining. opposite of around into

solenoid before sides components, two a

particle

experiences force, To that field the

the magnetic entirely confined be

must ensure no

of this, achieve the the solenoid. order interior principle solenoid In the should be in to to

long infinitely

shape. toroidal long alternatively practice be In could solenoid in

or very a

used.

phase by wavepackets The introduced shift potential the between the is two

f f

~~ ~~~

i ~

A(r)

6# A(r) dt dr

v =

= =

/ /

14lB,

IV A(r)] (110)

= =

by of the

A surface flux enclosed the interferometer of the the and is the 4lB where is arms

through field the solenoid. magnetic

effects

the scalar Aharonov-Bohm absence of both and the features important vector The are:

effect,

topological phase of the and the fact that the shift particle,

the is the of force nature on

fringes possible

observe that is non-dispersive. global This last it and property to

means even

length wavepacket. of 6# larger coherence the the than length associated with is the if

INTEGRAL PATH APPROACH TO ATOMIC N°11 A INTERFEROMETRY 2023

solenoid

tony

experimental Fig. Sketch the Aharonov-Bohm effect. of the 15. setup vector to test

particles. effect,

The Extension neutral scalar Aharonov-Bohm such

2.5.3 it to

was as

above, described has due small been observed. The difficult the experiment is yet to not very

beams, interfering high speeds particles, the of separation the the and the technical between

(of realizing voltages sufficiently high microcylinders frequencies applying problems and in at

GHz). order of the

(see therein), Aharonov-Bohm effect been references and al- observed has The vector [11]

non-dispersive phase-shift though large the of the has been tested due the character not to

lengths wavepackets. of the coherence

recently proposed problems,

have been

experiments effort these In to to

overcome new an

generalize particles. Aharonov-Bohm effects Particles both the scalar and neutral vector to

they

coupled fields which magnetic electric in magnetic static

with

moment to

are or a ~t

undergo phase experience force. shift but

a no

Zeilinger proposed by effect for

Aharonov-Bohm equivalent of the An scalar neutrons was

through of

separated which A beam of into components, is

[12]. neutrons two passes a one

wavepacket

applied the while well-localised the solenoid. A solenoid the is is current to near

although shift, undergoes potential phase wavefunction

giving The

B. rise

centre,

to -p. a a

by Allman

proposal experimentally realized has particle experiences force. This been the no

theory,

good large but phase shift observed with

the

in agreement al. The not

[13]. et was was

verify effect. last this enough non-dispersive character of the order In property, the to to test

performed by Their using al. Badurek slow experiments have been et neutrons. [14] very

single al., of differed from of that beam those Allman in experiments however neutrons et a

prepared with their spatially separated

than beams. The used rather

neutrons two were was

applied magnetic field

the perpendicular magnetic field. When the polarisation the to was

phase

shift from

parallel underwent different the

B component spin whose to component

a was

enough

enough long dispersion high antiparallel The and the transit time spin.

with

energy was

much than possible phase shifts fields achieve magnetic

greater with small that it to

was even

phase shift length. non-dispersive of corresponding The character the coherence the those to

fields. magnetic which and by comparing experiments transitory used

tested permanent was

performed by Chormaic using Similar have been al. experiments et [15]. atoms

equivalent of Aharonov-Anandan-Casher effect the the Aharonov- The is [16, vector 17]

split particles. wavepacket particle of effect neutral for The the Bohm is into components two

long charged recombining. particle before opposite around of rod The which sides propagate a

[E(r) /c~,

acquired vi

although phase potential The shift force. experiences

x -p a sees no

PHYSIQUE JOURNAL II DE N°11 2024

of charge the the interferometer A length between where the is

of is A~tleo&c~ unit two arms per

Aharonov-Anandan-Casher experimentally the effect been tested The has by rod. Cimmino et

charged condensing by replaced using pairs al. the rod of with plates. has It neutrons, two [18]

by San using also been molecules. their

tested al. single experiment In beam gster [19], et

was a

condensing

plates. prepared passed between The molecules

been of had superposition in two a

experienced phase of different

which spin electric field. each shift the in states, a

analogy effects, the Aharonov-Bohm particles experiments these

In with using neutral are

topological effect, by force, characterized absence global the the the of of

the and nature non-

they dispersive phase shift. of the However differ effects Aharonov-Bohm from the property in

particle evolves regions fields where electric that the magnetic the and in

are non-zero.

Conclusion.

solving path

described integral problems approach interfer-

this have In

in atomic to

paper a we

Simple physical insight, solutions, offer useful for which have been obtained situations ometry.

wavefunction

classical limit the

plane when treated close the incident be

as can a wave.

particular possible Lagrangian quadratic

results when the of In function

is exact

are some a

velocity, particle for

and gravitational position field refer- the is in rotating

case as a a or a

simple

perturbative A has presented. have frame. also been We shown that treatment ence

by phase perturbation

introduced wavefunction

shift Lagrangian

into the the be in

can a a

by perturbation integrating along unperturbed calculated the path. the

applicable

The methods wide of examples variety research have

current

are a areas. we

discussed gravitational the interferometric of the by

acceleration Kasevich and measurements g

Chu, equivalents Sagnac of the and the Aharonov-Bohm and effects. atomic

Acknowledgements.

Dalibard thank Yvan Castin and grateful Storey Pippa for fruitful We Jean

discussions. is very

for from Hugot Collbge the Fondation France. de du support

Appendix A.

6z(t)

LAGRANGE Consider DERIVATION A.I

from clas- variation

EQUATIONS.- OF THE a a

z(t) path boundary which satisfies sical conditions the

6z(t~) (iii) 6z(t~)

= =

path changes values of The Lagrangian the the variation by and the action amounts

)6z(t)

(112)

~)6i(t) 6L

+ =

/~~()6z(t)+)6i(t)jdt.

(113) 65

z z t~

Making the substitution

t~~~~~

~~~~~ ~~~~~

by integrating gives and parts

() )j

(

~~ ~~

)6z(t)

6z(t) (115) dt. 65

Z Z Z

t~ t~

INTEGRAL PATH APPROACH ATOMIC N°11 A TO INTERFEROMETRY 2025

~b~~~b

tb t ta

paths Fig. spacetime point A, different classical from

Two actual leave the but arrive 16.

at same

final positions the time tb. at

z(t) along

path equal extremal,

classical bS Since the the actual The first action is must zero.

(111). boundary equation requirement the conditions The above vanishes due the in to term

6z(t)

arbitrary

Lagrange for describe gives equations, the second be the which that term zero

particle dynamics of classical the a

d3L_~

(116)

$~&j~'

The classical

THE

A.2 action is

CLASSICAL ACTION.- OF PARTIAL DERIVATIVES THE a

partial function and final and calculate derivatives positions. of the initial times Here

its we

these of with each respect parameters. to

(I.e. paths Consider actual classical position.- A.2.1 Partial derivatives with respect two to

motion) figure 6zb, by final differ shown

positions 16. equations of the of whose in solutions as

along paths equation calculated using difference the these be The between actions two can

(115).

identically

The first paths second is Since the classical the is

term term zero. are

boundary calculated the conditions using

(117) 6z(ta)

(i18) 6z(t~) 6z~,

gives and

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

(5)

Using definition obtain the we

(120)

)~~ "Pb>

path. final the classical

position the where the

is momentum at on pb

PHYSIQUE

N°11 JOURNAL DE II 2026

tb+6tb I ta tb

final A, arrive but

spactime point the paths leave from the actual classical Fig.

Two at 17. same

different pOSltlOl1 at times. Zb

partial of classical with the derivative the action respect that similar calculation shows to A

initial position the is za

~~~'

(121)

-pa

= 3za

path. the classical

position the initial the

where

is at momentum on pa

paths classical whose actual Consider with Partial derivatives time. A.2.2 respect to two

AB'

path figure

6tb with the by point 17, let C be the the differ and in final

times

same on as

the between B and C

Then B. distance is abcissa as

(122) 6tb. 6tb 6tb

-iB< -iB -16

" m "

(120) along paths Using find that difference between the classical the the equation actions

AB AC and is

PbM

(123) 6tb -pblb

SAC SAB

* m

AB' along paths difference AC know the between the the and is We also that actions

(l~~)

SCB' 6tb, Lb SAB' SAC

" "

path

Lagrangian the AB. B point value of the Lb where the the is at on

change the due the in equations gives the the Adding in action the above variation to two

final time

(125)

6tb (Pbib

Lb 65 SAB

SAB>

= =

equation gives the above (8) of Hamiltonian definition the into Substituting the

3S~i

(126)

$ ~'

N°11

A PATH INTEGRAL APPROACH ATOMIC TO INTERFEROMETRY 2027

calculation A similar partial shows that

derivative the of classical the with action respect to

initial the time is

~~'

~~~~~ ~)~

References

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