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

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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 in- terferometry. 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 J. 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 is 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 an 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 " t~ 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 as ~ ~ t ~~ ~~~ 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 z I i 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. 1. 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 " zb (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 a (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 dz = 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, L~ 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 " f (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 Z~ , la ~b ,, 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.
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