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  • Hot Topics in General Relativity and Gravitation

    EQUIVALENCE PRINCIPLE ANDMATTER WAVE INTERFEROMETRY

    Luc Blanchet

    Gravitation et Cosmologie (GRCO)Institut dAstrophysique de Paris

    Based on a collaboration with

    Peter Wolf, Christian Borde, Serge Reynaud,Christophe Salomon & Claude Cohen-Tannoudji

    30 juillet 2013

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 1 / 24

    http://www.iap.fr

  • Weak version of the equivalence principle

    WEP [Philiponus Vth century, Galileo 1610, Newton 1687, Laplace 1780, Bessel 1850, Eotvos 1898]

    All test bodies follow the same universal trajectory in a gravitational field,independently of their mass, detailed internal structure and composition

    For all test bodies, mi = mg where

    F = mi a (mi = inertial mass)

    Fg = mg g (mg = passive gravitational mass)

    Precision is measured in terms of the Eotvos ratio

    Eotvos =

    (mgmi)A

    (mgmi

    )B

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 2 / 24

  • Experimental limits on the weak equivalence principle

    -SCOPE10-15

    2016

    STE-Quest

    10-15

    2022

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 3 / 24

  • -SCOPE experiment (2016)

    Expected accuracy 1015

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 4 / 24

  • STE-Quest experiment (2022?)

    Expected accuracy 1015

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 5 / 24

  • Einstein equivalence principle (EEP)

    1 Weak equivalence principle (WEP). Guarantees existence of a universal classof frames freely falling with all test bodies (local inertial frames)

    2 Local Lorentz invariance (LLI). Results of non-gravitational experiments inthe local inertial frame are independent of the velocity of the frame

    3 Local position invariance (LPI). Results of non-gravitational experiments inthe local inertial frame are independent of the location and space and time

    EEP is equivalent to a universal coupling to the space-time metric [Will 1993]

    g

    reducing to Minkowskis metric in local inertial frames

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 6 / 24

  • Gravitational redshift experiment

    z

    t

    z

    z

    t

    1

    2

    1

    t 1t 2

    g g

    Two identical clocks at different elevations in a static gravitational field

    =g00(z1) t1 =

    g00(z2) t2

    The clocks are compared by continuous exchanges of light signals

    =t1t2 1 = gz

    c2

    With g and z known from independent measurements one can check theGR prediction for the redshift

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 7 / 24

  • Atomic Clock Ensemble in Space [Cacciapuoti & Salomon 2009]

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 8 / 24

  • Testing the gravitational redshift with Pharao/ACES

    signalclock

    ACES

    ground

    station

    station

    signal

    Prediction from GR in a two-way transfer [Blanchet, Salomon, Teyssandier & Wolf 1999]

    =

    1

    c2

    [U 1

    2v2 R a

    ](1 +

    1

    cN v

    )+O

    (1

    c4

    )This will permit to test the redshift with an accuracy of 2 106

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 9 / 24

  • A new proposal [Muller, Peters & Chu 2010]

    LETTERS

    A precision measurement of the gravitational redshiftby the interference of matter wavesHolger Muller1,2, Achim Peters3 & Steven Chu1,2,4

    One of the central predictions of metric theories of gravity, such asgeneral relativity, is that a clock in a gravitational potential U willrun more slowly by a factor of 1 1 U/c2, where c is the velocity oflight, as compared to a similar clock outside the potential1. Thiseffect, known as gravitational redshift, is important to the opera-tion of the global positioning system2, timekeeping3,4 and futureexperiments with ultra-precise, space-based clocks5 (such assearches for variations in fundamental constants). The gravita-tional redshift has been measured using clocks on a tower6, anaircraft7 and a rocket8, currently reaching an accuracy of7 3 1025. Here we show that laboratory experiments based onquantum interference of atoms9,10 enable a much more precisemeasurement, yielding an accuracy of 7 3 1029. Our result sup-ports the view that gravity is a manifestation of space-time cur-vature, an underlying principle of general relativity that has comeunder scrutiny in connection with the search for a theory ofquantum gravity11. Improving the redshift measurement is par-ticularly important because this test has been the least accurateamong the experiments that are required to support curved space-time theories1.

    Metric theories of gravity are based on the Einstein equivalenceprinciple (EEP), which states that the local effects of gravity are thesame as those of being in an accelerated reference frame. The EEP isderived from three separate experimental observations1: the weakequivalence principle (that is, the universality of free fall), localLorentz invariance, and local position invariance. The first two havebeen verified experimentally to accuracies of 10213 or better(although some loopholes have not been closed)1,11. Local positioninvariance requires the outcome of a non-gravitational experiment tobe independent of where and when it is performed. In practice, thehighest-precision tests of local position invariance are measurementsof the gravitational redshift: the frequency of an oscillating system (aclock) is measured as a function of location. If the EEP holds, therewill be no variations other than those caused by gravity, that is, thegravitational redshift.

    The basic concept of redshift measurements like ours is to synchron-ize a pair of clocks when they are located closely to one another, andmove them to different elevations. The gravitational redshift willdecrease the oscillation frequency of the lower clock relative to thehigher one. When we bring the clocks together afterwards and comparethe number of elapsed oscillations, there will be a measurable phaseshift between them. A famous version of such a measurement was thecomparison7 of atomic clocks in aircraft against ground-based clocks,which confirmed Einsteins prediction with an accuracy of roughly10%. An accuracy of 7 3 1025 was obtained by using a hydrogen maserclock in a rocket8; 30 years later, this remains the most precise absolutemeasurement of the gravitational redshift. A higher accuracy of3.5 3 1026 is reached by relative redshift measurements12,13, which

    verify that there is zero variation between different clocks that movetogether through space-time. Still, the verification of local positioninvariance may be called the weakest link in the experimental under-pinning of the EEP.

    Our determination of the gravitational redshift is based on a re-interpretation of atom interferometry experiments that have beenused to measure the acceleration of free fall9,10,14. As shown inFig. 1a, a laser-cooled atom launched vertically upwards in a vacuumchamber is subjected to three pulses from a pair of anti-parallel,vertical laser beams having respective wavenumbers of k1 and k2.Each laser pulse transfers the momentum "(k1 1 k2) (where " ish/2p, h being the Planck constant) of two photons to the atom(Fig. 1b). The recoil gives a combined momentum impulse of "k,where k ; k1 1 k2. The intensity and duration of the first laser pulse isadjusted such that this process happens with a probability of 50%. Asa result, the first laser pulse places the atom into a coherent super-position of two quantum states, which physically separate owing totheir relative momentum "k. The second pulse redirects the atom

    1Department of Physics, 366 Le Conte Hall MS 7300, University of California, Berkeley, California 94720, USA. 2Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley,California 94720, USA. 3Institut fur Physik, Humboldt-Universitat zu Berlin, Hausvogteiplatz 5-7, 10117 Berlin, Germany. 4US Department of Energy, 1000 Independence Avenue SW,Washington, District of Columbia 20585, USA.

    Hei

    ght

    Time

    Laser pulse

    A

    B

    Path 2

    Path 1

    t0 t0 + T t0 + 2T

    g1, pz

    g2, pz + (k1 + k2)

    e, pz + k1

    k1 k2

    a b

    Figure 1 | Atom interferometer and Raman beam splitter. a, Atominterferometer (schematic). The trajectories of the atom are plotted asfunction of time in the laboratory frame of reference. They are acceleratingowing to gravity. The oscillatory lines depict the phase accumulation of thematter waves. Arrows indicate laser pulses applied at times t0, t0 1 T andt0 1 2T that change the trajectories. At time t0, the atom is put into asuperposition of two trajectories. At time t0 1 T, a laser pulse is used to alterthe trajectory of the atoms, and at time t0 1 2T, the phase differenceDQ 5DQ2 2DQ1 is recorded. b, Two-photon Raman beam splitter. An atomin a quantum state g1,pzj i, moving upwards with momentum pz, interactswith photons of two counter-propagating laser beams. The first one transfersthe momentum "k1 and brings the atom into a virtual excited statee,pzzBk1j i. The second laser beam stimulates the atom to emit a photon of

    momentum "k2, which transfers the atom to another hyperfine ground stateg2,pzzB(k1zk2)j i. With appropriate duration and intensity of the laser

    pulses, the process can have 50% or 100% probability, creating beamsplitters or mirrors for atomic matter waves.

    Vol 463 | 18 February 2010 | doi:10.1038/nature08776

    926Macmillan Publishers Limited. All rights reserved2010

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 10 / 24

  • Some important implications ?

    Atomic-clock experiment pins down accuracy of fundamental gravity measurement.

    General relativity tested on a tabletopBy measuring a spectacularly small difference in the ticks of two quantum clocks, physicists have proven a pillar of Albert Einsteins theory of gravity to be on firmer footing than ever before.

    The experiment is the latest in a series of tests in which scientists have scrutinized one of Ein-steins more profound predictions: that clocks in stronger gravitational fields run more slowly. For decades they have put clocks at higher elevations, where Earths gravity is slightly weaker, and measured the ensuing changes. From a clock in a tower at Harvard University in Cambridge, Massachusetts, in the 1960s, to others flown on planes in the 1970s, to a clock that flew thousands of kilometres into space on a rocket in 1980, physicists have not been able to show that Einstein was wrong.

    Now, a team led by Holger Mller of the University of California, Berkeley, has meas-ured the time-shifting effects of gravity 10,000 times more accurately than ever before. They show that gravitys effect on time is predict-able to 7 parts per billion (H. Mller, A. Peters and S. Chu Nature 463, 926929; 2010). And they did it using two laboratory clocks with a height difference of just 0.1 millimetres a set-up that seems quaintly small in this day of big physics. Precision experiments on a table-top are not something of the past, says Mller, whose research team consisted of Achim Peters of the Humboldt University of Berlin andSteven Chu, the US Secretary of Energy.

    Many atomic clocks use the extremely regu-lar pulsations of atoms shifting between excited energy states. But Mllers apparatus relied on the fundamental quantum frequency of a caesium atom associated with the atoms rest energy. Thisfrequency was so high that physicists never thought to use it as a clock. But a special inter-ferometer could measure the difference between two such clocks experiencing gravitys effect.

    Whats fascinating about their work is that they were using the entire atom as a clock, says atomic-clock expert Jun Ye of the Joint Insti-tute for Laboratory Astrophysics in Boulder, Colorado.

    Mller and his team shot caesium atoms, cooled nearly to absolute zero, in an arc across a gap. Mid-stream, photons from a laser bumped the atoms into two, quantum-mechanical alternate realities. In one, an atom absorbed a photon and arced on a slightly higher path,

    experiencing a tiny weakening of gravity and speed-up of time. In the other, the atom stuck to the lower path, where gravity was stronger and time moved slightly more slowly. A difference in phase in the atoms fundamental frequency, measured by the interferometer, indicated a tiny difference in time.

    Laser trapsThe experiment takes advantage of the laser atom trap, for which Chu won a Nobel prize in 1997. The data for the current study were

    obtained shortly after that, when Chu was using the set-up to measure a different constant, the acceleration of gravity (A. Peters, K. Y. Chung

    and S. Chu Nature 400, 849852; 1999).But Mller says that in October 2008, he

    had an epiphany that the same data could be used to show the constancy of gravitys effect on time. He e-mailed Chu, then the director of the Lawrence Berkeley National Laboratory in Berkeley, California, who responded three days later saying it was a good idea.

    Chu says in an e-mail that he found time to work on the current study during nights, weekends and on planes after putting in7080-hour weeks as energy secretary. I like juggling a lot of balls, he says.

    The result could one day have practical applications. If gravitys time-shifting effect were not constant, then researchers might have

    had to worry about the accuracy of new atomic clocks as they are flown into orbit on Global Positioning System (GPS) satellites. But Mller has demonstrated the effect to be extraordinar-ily consistent. Now we know that the physics is fine, he says.

    The test also puts pressure on the Atomic Clock Ensemble in Space (ACES), an experi-ment being run by the European Space Agency that is due to be attached to the International Space Station in 2013. The current study already betters ACESs planned measurement of gravitys time-shifting effect by almost three orders of magnitude. ACESs principal investi-gator Christophe Salomon says that the mission will cost about 100 million (US$136 million), plus the cost of a launch rocket. By compari-son, Mller says that his tabletop apparatus cost much less than $1 million. Salomon says that ACES is still justified because it will perform two other fundamental physics tests, as well as help researchers to improve the coordination of ground-based atomic clocks.

    Physicist Clifford Will of Washington Uni-versity in St Louis, Missouri, says that Mllers result narrows the window for the alternative theories of gravity that some theorists are exploring. Will was also impressed that Chu found time to contribute to the study. When was the last time that a sitting member of the presidents cabinet had a paper in Nature on fundamental physics? he asks. Eric Hand

    Holger Mller used laser-trap technology to test one of Einsteins predictions from general relativity.

    Precision experiments on a tabletop are not something of the past.

    D. E

    NG

    LISH

    862

    Vol 463|18 February 2010

    862

    NATURE|Vol 463|18 February 2010NEWS

    862 News MH AB.indd 862862 News MH AB.indd 862 16/2/10 11:54:2816/2/10 11:54:28

    20 Macmillan Publishers Limited. All rights reserved10

    A team led by Holger Mullerof the University of California,Berkeley, has measured thetime-shifting effects of gravity10, 000 times more accuratelythan ever before

    The test puts pressure on theAtomic Ensemble in Space, anexperiment being run by theEuropean Space Agency

    When was the last time thata sitting member of thepresidents cabinet had a paperin Nature on fundamentalphysics? [Clifford Will]

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 11 / 24

  • A raging controversy

    PROSHolger Muller, Achim Peters & StevenChu, A precision measurement of thegravitational redshift by the interferenceof matter waves, Nature 463, 926(2010)

    Holger Muller, Achim Peters & StevenChu, Atom gravimeters andgravitational redshift, Nature 467, E2(2010)

    Mike Hohensee, Achim Peters, StevenChu, Holger Muller et al, Gravitationalredshift, equivalence principle, andmatter waves, J. Phys. Conf. Ser. 264,012009 (2011)

    Mike Hohensee, Stephen Chu, AchimPeters & Holger Muller, Equivalenceprinciple and gravitational redshift,Physical Review Letters 106, 151102(2011)

    CONSPeter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Atom gravimetersand gravitational redshift, Nature 467, E1(2010)

    Peter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Does an atominterferometer test the gravitational redshiftat the Compton frequency?, Classical andQuantum Gravity 28, 145017 (2011)

    Supurna Sinha & Joseph Samuel, Atominterferometers and the gravitationalredshift,Classical and Quantum Gravity 28,145018 (2011)

    Domenico Giulini, Equivalence principle,quantum mechanics, andatom-interferometric tests, arXiv:1105.0749(2011)

    AND SO ON Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 12 / 24

  • A raging controversy

    PROSHolger Muller, Achim Peters & StevenChu, A precision measurement of thegravitational redshift by the interferenceof matter waves, Nature 463, 926(2010)

    Holger Muller, Achim Peters & StevenChu, Atom gravimeters andgravitational redshift, Nature 467, E2(2010)

    Mike Hohensee, Achim Peters, StevenChu, Holger Muller et al, Gravitationalredshift, equivalence principle, andmatter waves, J. Phys. Conf. Ser. 264,012009 (2011)

    Mike Hohensee, Stephen Chu, AchimPeters & Holger Muller, Equivalenceprinciple and gravitational redshift,Physical Review Letters 106, 151102(2011)

    CONSPeter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Atom gravimetersand gravitational redshift, Nature 467, E1(2010)

    Peter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Does an atominterferometer test the gravitational redshiftat the Compton frequency?, Classical andQuantum Gravity 28, 145017 (2011)

    Supurna Sinha & Joseph Samuel, Atominterferometers and the gravitationalredshift,Classical and Quantum Gravity 28,145018 (2011)

    Domenico Giulini, Equivalence principle,quantum mechanics, andatom-interferometric tests, arXiv:1105.0749(2011)

    AND SO ON Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 12 / 24

  • A raging controversy

    PROSHolger Muller, Achim Peters & StevenChu, A precision measurement of thegravitational redshift by the interferenceof matter waves, Nature 463, 926(2010)

    Holger Muller, Achim Peters & StevenChu, Atom gravimeters andgravitational redshift, Nature 467, E2(2010)

    Mike Hohensee, Achim Peters, StevenChu, Holger Muller et al, Gravitationalredshift, equivalence principle, andmatter waves, J. Phys. Conf. Ser. 264,012009 (2011)

    Mike Hohensee, Stephen Chu, AchimPeters & Holger Muller, Equivalenceprinciple and gravitational redshift,Physical Review Letters 106, 151102(2011)

    CONSPeter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Atom gravimetersand gravitational redshift, Nature 467, E1(2010)

    Peter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Does an atominterferometer test the gravitational redshiftat the Compton frequency?, Classical andQuantum Gravity 28, 145017 (2011)

    Supurna Sinha & Joseph Samuel, Atominterferometers and the gravitationalredshift,Classical and Quantum Gravity 28,145018 (2011)

    Domenico Giulini, Equivalence principle,quantum mechanics, andatom-interferometric tests, arXiv:1105.0749(2011)

    AND SO ON Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 12 / 24

  • A raging controversy

    PROSHolger Muller, Achim Peters & StevenChu, A precision measurement of thegravitational redshift by the interferenceof matter waves, Nature 463, 926(2010)

    Holger Muller, Achim Peters & StevenChu, Atom gravimeters andgravitational redshift, Nature 467, E2(2010)

    Mike Hohensee, Achim Peters, StevenChu, Holger Muller et al, Gravitationalredshift, equivalence principle, andmatter waves, J. Phys. Conf. Ser. 264,012009 (2011)

    Mike Hohensee, Stephen Chu, AchimPeters & Holger Muller, Equivalenceprinciple and gravitational redshift,Physical Review Letters 106, 151102(2011)

    CONSPeter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Atom gravimetersand gravitational redshift, Nature 467, E1(2010)

    Peter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Does an atominterferometer test the gravitational redshiftat the Compton frequency?, Classical andQuantum Gravity 28, 145017 (2011)

    Supurna Sinha & Joseph Samuel, Atominterferometers and the gravitationalredshift,Classical and Quantum Gravity 28,145018 (2011)

    Domenico Giulini, Equivalence principle,quantum mechanics, andatom-interferometric tests, arXiv:1105.0749(2011)

    AND SO ON Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 12 / 24

  • A raging controversy

    PROSHolger Muller, Achim Peters & StevenChu, A precision measurement of thegravitational redshift by the interferenceof matter waves, Nature 463, 926(2010)

    Holger Muller, Achim Peters & StevenChu, Atom gravimeters andgravitational redshift, Nature 467, E2(2010)

    Mike Hohensee, Achim Peters, StevenChu, Holger Muller et al, Gravitationalredshift, equivalence principle, andmatter waves, J. Phys. Conf. Ser. 264,012009 (2011)

    Mike Hohensee, Stephen Chu, AchimPeters & Holger Muller, Equivalenceprinciple and gravitational redshift,Physical Review Letters 106, 151102(2011)

    CONSPeter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Atom gravimetersand gravitational redshift, Nature 467, E1(2010)

    Peter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Does an atominterferometer test the gravitational redshiftat the Compton frequency?, Classical andQuantum Gravity 28, 145017 (2011)

    Supurna Sinha & Joseph Samuel, Atominterferometers and the gravitationalredshift,Classical and Quantum Gravity 28,145018 (2011)

    Domenico Giulini, Equivalence principle,quantum mechanics, andatom-interferometric tests, arXiv:1105.0749(2011)

    AND SO ON Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 12 / 24

  • A raging controversy

    PROSHolger Muller, Achim Peters & StevenChu, A precision measurement of thegravitational redshift by the interferenceof matter waves, Nature 463, 926(2010)

    Holger Muller, Achim Peters & StevenChu, Atom gravimeters andgravitational redshift, Nature 467, E2(2010)

    Mike Hohensee, Achim Peters, StevenChu, Holger Muller et al, Gravitationalredshift, equivalence principle, andmatter waves, J. Phys. Conf. Ser. 264,012009 (2011)

    Mike Hohensee, Stephen Chu, AchimPeters & Holger Muller, Equivalenceprinciple and gravitational redshift,Physical Review Letters 106, 151102(2011)

    CONSPeter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Atom gravimetersand gravitational redshift, Nature 467, E1(2010)

    Peter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Does an atominterferometer test the gravitational redshiftat the Compton frequency?, Classical andQuantum Gravity 28, 145017 (2011)

    Supurna Sinha & Joseph Samuel, Atominterferometers and the gravitationalredshift,Classical and Quantum Gravity 28,145018 (2011)

    Domenico Giulini, Equivalence principle,quantum mechanics, andatom-interferometric tests, arXiv:1105.0749(2011)

    AND SO ON Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 12 / 24

  • A raging controversy

    PROSHolger Muller, Achim Peters & StevenChu, A precision measurement of thegravitational redshift by the interferenceof matter waves, Nature 463, 926(2010)

    Holger Muller, Achim Peters & StevenChu, Atom gravimeters andgravitational redshift, Nature 467, E2(2010)

    Mike Hohensee, Achim Peters, StevenChu, Holger Muller et al, Gravitationalredshift, equivalence principle, andmatter waves, J. Phys. Conf. Ser. 264,012009 (2011)

    Mike Hohensee, Stephen Chu, AchimPeters & Holger Muller, Equivalenceprinciple and gravitational redshift,Physical Review Letters 106, 151102(2011)

    CONSPeter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Atom gravimetersand gravitational redshift, Nature 467, E1(2010)

    Peter Wolf, Luc Blanchet, Christian Borde,Serge Reynaud, Christophe Salomon &Claude Cohen-Tannoudji, Does an atominterferometer test the gravitational redshiftat the Compton frequency?, Classical andQuantum Gravity 28, 145017 (2011)

    Supurna Sinha & Joseph Samuel, Atominterferometers and the gravitationalredshift,Classical and Quantum Gravity 28,145018 (2011)

    Domenico Giulini, Equivalence principle,quantum mechanics, andatom-interferometric tests, arXiv:1105.0749(2011)

    AND SO ON Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 12 / 24

  • Basic idea of this proposal

    path I

    path II

    g g

    The atoms are viewed as clocks ticking at the de Broglie-Compton frequency

    C =mc2

    ~

    The atom-clocks propagate in the two paths I and II of the interferometer atdifferent elevations in the gravitational field g. They experience a measurablephase shift due to the gravitational redshift

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 13 / 24

  • An analogy with clock experiments

    Two identical clocks are synchronized and moved at different elevations in agravitational field. The total phase difference when the two clocks arebrought back together is

    clock =

    [I

    d

    II

    d

    ]

    d

    where d is the proper time and the proper frequency

    The phase shift in an atom interferometer contains a contribution similar tothe clock phase shift,

    = C

    d + laser

    with the role of the clocks proper frequency played by the atoms Comptonfrequency

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 14 / 24

  • Beam-slitting process in atom interferometry

    The beam-splitter is realized through the interaction of atoms with laserbeams resonant with an hyperfine atomic transitionThe atoms undergo a Raman transition g g, resulting in a recoil velocity~k where k = k1 + k2 is the effective wave vector transferred to the atoms bythe Raman lasers

    g

    g'

    g

    g'

    path I

    path II

    t

    z

    | g , p >

    | g', p + h (k +k ) >

    | e , p + h k >

    h k h k 2

    1

    1 2

    1

    k = k + k1 2

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 15 / 24

  • Phase shift due to free propagation of atoms

    Theorem [Borde 1989, Kasevitch & Chu 1991, Storey & Cohen-Tannoudji 1999, Wolf & Tourrenc 1999]

    The phase difference due to the free propagation of atoms is zero,

    S =Scl~

    =1

    ~

    L [zcl(t), zcl(t)] dt = 0

    The proper time on the two paths I and II is the same

    The result is independent of the mass (and Compton frequency) of the atom

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 16 / 24

  • Total phase shift in the atom interferometer

    path I

    path II

    t

    z

    A

    B

    C

    D

    The phase shift results from the interaction of the lasers with the atoms

    = laser = A + B + C Dwhere is the phase of the laser light as seen by the atom at the interaction points

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 17 / 24

  • Atom interferometers are gravimeters

    In the case of general relativity the Lagrangian is

    LGR(z, z) = mc2 +GMm

    rmgz + 1

    2mz2 +O

    (1

    c2

    )where g is the gravitational field. The result is [Borde 1989]

    = k g T 2

    where k is the effective wave vector of the lasers and T is the time intervalbetween pulses

    The atom interferometer measures the local acceleration of gravity g. Thephase shift arises from the interactions with the lasers and the fact that theatoms are falling with respect to the experimental platform

    The atom interferometer permits to test the WEP between atoms andmacroscopic bodies or between different species of atoms

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 18 / 24

  • Arguments in more details [Muller, Peters & Chu 2010]

    In general relativity the total shift in the atom interferometer is

    = C

    [gzc2

    +z2

    2c2

    ]dt+ laser

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 19 / 24

  • Arguments in more details [Muller, Peters & Chu 2010]

    In general relativity the total shift in the atom interferometer is

    = C

    [gzc2

    ]dt

    redshift

    +C

    [z2

    2c2

    ]dt

    Doppler shift

    +laser

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 19 / 24

  • Arguments in more details [Muller, Peters & Chu 2010]

    In general relativity the total shift in the atom interferometer is

    = C

    [gzc2

    ]dt

    k g T 2

    +C

    [z2

    2c2

    ]dt

    k g T 2

    + laser k g T 2

    = k g T 2

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 19 / 24

  • Arguments in more details [Muller, Peters & Chu 2010]

    In general relativity the total shift in the atom interferometer is

    = C

    [gzc2

    ]dt

    k g T 2

    +C

    [z2

    2c2

    ]dt

    k g T 2

    + laser k g T 2

    = k g T 2

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 19 / 24

  • Arguments in more details [Muller, Peters & Chu 2010]

    In general relativity the total shift in the atom interferometer is

    = C

    [gzc2

    ]dt = k g T 2

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 19 / 24

  • Arguments in more details [Muller, Peters & Chu 2010]

    In general relativity the total shift in the atom interferometer is

    = k g T 2 = Cgz

    c2T

    where z = ~ km T is the spatial separation between wave packets

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 19 / 24

  • Arguments in more details [Muller, Peters & Chu 2010]

    In an alternative theory the total shift in the atom interferometer is

    =(1 +

    )C

    gz

    c2T

    The classical trajectory of the atom obeys g

    The quantum phase of the matter wave obeys g = (1 + )g where represents the redshift-violating parameter

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 19 / 24

  • The arguments are incorrect[Wolf, Blanchet, Borde, Reynaud, Salomon & Cohen-Tannoudji 2010, 2011]

    Rewriting the GR prediction is that form is misleading

    = Cgz

    c2T

    where z = ~ km T is the spatial separation between wave packets

    1 The mass of the atom cancels so the Compton frequency is irrelevant.Nowhere is there a measurable signal at C 2 3.0 1025 Hz

    2 The separation between wave packets z is theoretically deduced but is notmeasured. Measuring it in an atom interferometer would destroy theinterference fringe pattern

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 20 / 24

  • The arguments are incorrect[Wolf, Blanchet, Borde, Reynaud, Salomon & Cohen-Tannoudji 2010, 2011]

    The statement that the Doppler and laser terms cancel out is not invariant

    = C

    [gzc2

    ]dt

    k g T 2

    +C

    [z2

    2c2

    ]dt

    k g T 2

    + laser k g T 2

    = k g T 2

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 21 / 24

  • The arguments are incorrect[Wolf, Blanchet, Borde, Reynaud, Salomon & Cohen-Tannoudji 2010, 2011]

    The invariant cancellation is between the redshift and Doppler terms

    = C

    [gzc2

    ]dt

    k g T 2

    +C

    [z2

    2c2

    ]dt

    k g T 2

    + laser k g T 2

    = k g T 2

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 21 / 24

  • The arguments are incorrect[Wolf, Blanchet, Borde, Reynaud, Salomon & Cohen-Tannoudji 2010, 2011]

    t

    z

    g

    = k g T 2 k g T 2 0

    + laser k g T 2

    = k g T 2

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 22 / 24

  • The arguments are incorrect[Wolf, Blanchet, Borde, Reynaud, Salomon & Cohen-Tannoudji 2010, 2011]

    t' = t

    z' = z + g t

    a = - g

    12

    2

    = 0 0 0

    + laser k g T 2

    = k g T 2

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 22 / 24

  • The arguments are incorrect[Wolf, Blanchet, Borde, Reynaud, Salomon & Cohen-Tannoudji 2010, 2011]

    Such an alternative theory in non viable

    =(1 +

    )C

    gz

    c2T

    The classical trajectory of the atom obeys g

    The quantum phase of the matter wave obeys g = (1 + )g

    1 Schiffs conjecture that WEP implies EEP is not valid

    2 No principle of least action for matter waves

    3 Quantum Mechanics (e.g. Feynmans path integral formalism) is violated

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 23 / 24

  • Conclusions

    1 Atom interferometers test the WEP or universality of free fall at the level

    109 between atoms and macroscopic masses 107 between two different atoms

    2 The satellite STE-Quest should test it by atom interferometry in space at thelevel 1015 (comparable to Microscope) for different isotopes of therubidium atom

    3 Atom interferometers do not permit to test the gravitational redshift at theCompton frequency

    Luc Blanchet (GRCO) Equivalence principle and matter wave interferometry HTGRG 24 / 24

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Hot Topics in General Relativity and Gravitation EQUIVALENCE PRINCIPLE AND MATTER WAVE INTERFEROMETRY Luc Blanchet Gravitation et Cosmologie (GRεCO) Institut d’Astrophysique de Paris Based on a collaboration with Peter Wolf, Christian Bord´ e, Serge Reynaud, Christophe Salomon & Claude Cohen-Tannoudji 30 juillet 2013 Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry HTGRG 1 / 24
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