Colloque Microscope EQUIVALENCE PRINCIPLE AND MATTER WAVE INTERFEROMETRY. Luc Blanchet

Colloque Microscope 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 Bordé, Serge Reynaud, Christophe Salomon & Claude Cohen-Tannoudji 29 janvier 2013 Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 1 / 29

Foundations of metric theories of gravity FOUNDATIONS OF METRIC THEORIES OF GRAVITY Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 2 / 29

Foundations of metric theories of gravity Two fundamental principles METRIC THEORIES OF GRAVITY principle of relativity principle of equivalence EXPERIMENTS Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 3 / 29

Foundations of metric theories of gravity Weak version of the equivalence principle WEP [Philiponus V th century, Galileo 1610, Newton 1687, Laplace 1780, Bessel 1850, Eötvös 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, m i = m g where F = m i a (m i = inertial mass) F g = m g g (m g = passive gravitational mass) Precision is measured in terms of the Eötvös ratio ( ) ( ) η AB = mg mg m i A m i B Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 4 / 29

Foundations of metric theories of gravity Experimental limits on the weak equivalence principle Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 5 / 29

Foundations of metric theories of gravity Experimental limits on the weak equivalence principle -SCOPE 10-15 STE-Quest 10-15 2016 2022 Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 5 / 29

Foundations of metric theories of gravity Einstein equivalence principle trajectories of test particles 1 Weak equivalence principle (WEP). Guarantees existence of a family of trajectories followed by all test bodies 2 Local Lorentz invariance (LLI). Implies existence of fields Φ (A) reducing in local inertial frames to P local inertial frame Φ (A) αβ = Φ(A) (P) η αβ 3 Local position invariance (LPI). Implies independence of results of experiments on position Φ (A) (P) = }{{} c (A) Φ(P) set to one by change of units Einstein equivalence principle equivalent to coupling to a universal field [Will 1993] g µν = Φ 1 Φ µν reducing to Minkowski metric η αβ in local inertial frames Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 6 / 29

Foundations of metric theories of gravity Metric theories of gravity EEP Gravitation must be a curved-space-time phenomenon Postulates of metric theories [Thorne, Lee & Lightman 1973] Space-time is endowed with a metric g µν The world-lines of test bodies are geodesics of that metric The non-gravitational laws of physics are those of special relativity Some metric theories Nordström-Einstein-Fokker g µν = Ω 2 η µν General relativity g µν Jordan-Brans-Dicke g µν, φ Rosen g µν, η µν Ni g µν, ψ, t Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 7 / 29

Foundations of metric theories of gravity Gravitational redshift experiment z 2 z t 2 t 1 g g z 1 t 1 t Two identical clocks at different elevations in a static gravitational field δτ = g 00 (z 1 ) δt 1 = g 00 (z 2 ) δt 2 The clocks are compared by continuous exchanges of light signals ν ν = δt 1 1 = g z δt 2 c 2 With g and z known from independent measurements one can check the GR prediction for the redshift Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 8 / 29

Foundations of metric theories of gravity Experimental limits on the local position invariance Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 9 / 29

Foundations of metric theories of gravity Experimental limits on the local position invariance Pharao-ACES 2 10-6 2016 STE-Quest 10-7 2022 Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 9 / 29

Foundations of metric theories of gravity Testing the gravitational redshift with ACES ACES clock signal station signal ground station Prediction from GR in a two-way transfer [Blanchet, Salomon, Teyssandier & Wolf 1999] ν ν = 1 [ c 2 U 1 ] 2 v2 R a (1 + 1c ) ( ) 1 N v + O c 4 This will permit to test the redshift with an accuracy of 2 10 6 Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 10 / 29

Equivalence principle and atom interferometry EQUIVALENCE PRINCIPLE AND ATOM INTERFEROMETRY Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 11 / 29

Equivalence principle and atom interferometry Beam-slitting process in atom interferometry The beam-splitter is realized through the interaction of atoms with laser beams resonant with an hyperfine atomic transition The atoms undergo a Raman transition g g, resulting in a recoil velocity k where k = k 1 + k 2 is the effective wave vector transferred to the atoms by the Raman lasers e, p + h k > 1 z k = k + k 1 2 h k 1 h k 2 path I g' g g path II g' g, p > g', p + h (k +k ) > 1 2 t Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 12 / 29

Equivalence principle and atom interferometry Phase shift due to free propagation of atoms Computation performed for a quadratic Lagrangian [Storey & Cohen-Tannoudji 1999] L [z, ż] = a(t) ż 2 + b(t) żz + c(t) z 2 + d(t) ż + e(t) z + f(t) Theorem [Bordé 1989, Kasevitch & Chu 1991, Storey & Cohen-Tannoudji 1999, Wolf & Tourrenc 1999] The phase difference due to the free propagation of atoms is zero, ϕ S = S cl = 1 L [z cl (t), ż cl (t)] dt = 0 The proper time on the two paths I and II is the same The result is independent of the mass of the atom Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 13 / 29

Equivalence principle and atom interferometry Proof of theorem [Wolf, Blanchet, Bordé, Reynaud, Salomon & Cohen-Tannoudji 2011] 1 Difference of classical actions is S cl = T 0 ( ) L[z 1, ż 1 ] L[z 2, ż 2 ] dt + T +T T ( ) L[z 3, ż 3 ] L[z 4, ż 4 ] dt 2 For any quadratic Lagrangian the difference of on-shell Lagrangians is a total time derivative L[z 1, ż 1 ] L[z 2, ż 2 ] = d [ ( (z 1 z 2 ) a(t)(ż 1 + ż 2 ) + 1 )] dt 2 b(t)(z 1 + z 2 ) + d(t) 3 Taking into account the boundary conditions in positions S cl = a(t ) [ z 1 (T ) z 2 (T ) ][ ] ż 1 (T ) + ż 2 (T ) ż 3 (T ) ż 4 (T ) 4 Taking into account the boundary conditions in velocity S cl = 0 Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 14 / 29

Equivalence principle and atom interferometry Total phase shift in the atom interferometer z path I B D C path II A t The phase shift results from the interaction of the lasers with the atoms ϕ = ϕ laser = φ A + φ B + φ C φ D where φ is the phase of the laser light as seen by the atom at the interaction points Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 15 / 29

Equivalence principle and atom interferometry Atom interferometers are gravimeters In the case of general relativity the Lagrangian is L GR (z, ż) = mc 2 + GMm r mgz + 1 2 mż2 + O where g is the gravitational field. The result is [Bordé 1989] ϕ = k g T 2 ( ) 1 c 2 where k is the effective wave vector of the lasers and T is the time interval between pulses The atom interferometer is a gravimeter. It measures the local acceleration of gravity g. The phase shift arises from the interactions with the lasers and the fact that the atoms are falling with respect to the experimental platform There is no dependence on the mass (and Compton frequency) of the atom Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 16 / 29

Equivalence principle and atom interferometry Atom interferometers test the WEP 1 In a theory violating the EP the atom obeys L(z, ż) = m i c 2 + GMm g r 2 Since ϕ S = 0 we have m g gz + 1 ( ) 1 2 m iż 2 + O c 2 ϕ = ϕ laser = m g m i k g T 2 3 Comparing with g meas = gm g /M i measured with some classical mass ( ϕ 1 + m g M ) g k g meas T 2 m i M i Atom interferometers test of the WEP between atoms and macroscopic masses η Eötvos = m g M g m i 4 10 9 This is the most sensitive test comparing quantum objects with classical masses M i Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 17 / 29

GRAVITATIONAL REDSHIFT AND ATOM INTERFEROMETRY Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 18 / 29

A new proposal [Müller, Peters & Chu 2010] Vol 463 18 February 2010 doi:10.1038/nature08776 LETTERS A precision measurement of the gravitational redshift by the interference of matter waves Holger Müller 1,2, Achim Peters 3 & Steven Chu 1,2,4 One of the central predictions of metric theories of gravity, such as general relativity, is that a clock in a gravitational potential U will run more slowly by a factor of 1 1 U/c 2, where c is the velocity of light, as compared to a similar clock outside the potential 1. This effect, known as gravitational redshift, is important to the operation of the global positioning system 2, timekeeping 3,4 and future experiments with ultra-precise, space-based clocks 5 (such as searches for variations in fundamental constants). The gravitational redshift has been measured using clocks on a tower 6,an aircraft 7 and a rocket 8, currently reaching an accuracy of 7 3 10 25. Here we show that laboratory experiments based on quantum interference of atoms 9,10 enable a much more precise measurement, yielding an accuracy of 7 3 10 29. Our result supports the view that gravity is a manifestation of space-time curvature, an underlying principle of general relativity that has come verify that there is zero variation between different clocks that move together through space-time. Still, the verification of local position invariance may be called the weakest link in the experimental underpinning of the EEP. Our determination of the gravitational redshift is based on a reinterpretation of atom interferometry experiments that have been used to measure the acceleration of free fall 9,10,14. As shown in Fig. 1a, a laser-cooled atom launched vertically upwards in a vacuum chamber 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 " is h/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 is Luc Blanchet (GRεCO) Equivalence principle and matter adjusted wavesuch interferometry that this process happens with colloque a probability Microscope of 50%. As 19 / 29

Some important implications? General relativity tested on a tabletop Atomic-clock experiment pins down accuracy of fundamental gravity measurement. By measuring a spectacularly small difference in the ticks of two quantum clocks, physicists have proven a pillar of Albert Einstein s 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 Einstein s more profound predictions: that clocks in stronger gravitational fields run more slowly. For decades they have put clocks at higher elevations, where Earth s 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 Müller of the University of California, Berkeley, has measured the time-shifting effects of gravity 10,000 Holger Müller used laser-trap technology to test one of Einstein s predictions from general relativity. times more accurately than ever before. They show that gravity s effect on time is predictable to 7 parts per billion (H. Müller, A. Peters speed-up of time. In the other, the atom stuck to clocks as they are flown into orbit on Global experiencing a tiny weakening of gravity and had to worry about the accuracy of new atomic and S. Chu Nature 463, 926 929; 2010). And the lower path, where gravity was stronger and Positioning System (GPS) satellites. But Müller they did it using two laboratory clocks with a time moved slightly more slowly. A difference has demonstrated the effect to be extraordinarily consistent. Now we know that the physics height difference of just 0.1 millimetres a in phase in the atom s fundamental frequency, set-up that seems quaintly small in this day of measured by the interferometer, indicated a tiny is fine, he says. big physics. Precision experiments on a tabletop are not something of the past, says Müller, Clock Ensemble in Space (ACES), an experi- difference in time. The test also puts pressure on the Atomic whose research team consisted of Achim Peters Laser traps ment being run by the European Space Agency of the Humboldt University of Berlin and The experiment takes advantage of the laser that is due to be attached to the International Steven Chu, the US Secretary of Energy. atom trap, for which Chu won a Nobel prize Space Station in 2013. The current study Many atomic clocks use the extremely regular pulsations of atoms shifting obtained shortly after that, of gravity s time-shifting effect by almost three in 1997. The data for the current study were already betters ACES s planned measurement between excited energy states. Precision experiments when Chu was using the orders of magnitude. ACES s principal investigator Christophe Salomon says that the mission But Müller s apparatus relied on a tabletop are not set-up to measure a different on the fundamental quantum something of the past. constant, the acceleration of will cost about 100 million (US$136 million), frequency of a caesium atom gravity (A. Peters, K. Y. Chung plus the cost of a launch rocket. By comparison, Müller says that his tabletop apparatus cost associated with the atom s rest energy. This and S. Chu Nature 400, 849 852; 1999). frequency was so high that physicists never But Müller says that in October 2008, he much less than $1 million. Salomon says that thought to use it as a clock. But a special interferometer could measure the difference between used to show the constancy of gravity s effect two other fundamental physics tests, as well as had an epiphany that the same data could be ACES is still justified because it will perform two such clocks experiencing gravity s effect. on time. He e-mailed Chu, then the director of help researchers to improve the coordination of What s fascinating about their work is that the Lawrence Berkeley National Laboratory in ground-based atomic clocks. they were using the entire atom as a clock, says Berkeley, California, who responded three days Physicist Clifford Will of Washington University in St Louis, Missouri, says that Müller s atomic-clock expert Jun Ye of the Joint Institute for Laboratory Astrophysics in Boulder, Chu says in an e-mail that he found time result narrows the window for the alternative later saying it was a good idea. Colorado. to work on the current study during nights, theories of gravity that some theorists are Müller and his team shot caesium atoms, weekends and on planes after putting in exploring. Will was also impressed that Chu cooled nearly to absolute zero, in an arc across a 70 80-hour weeks as energy secretary. I like found time to contribute to the study. When gap. Mid-stream, photons from a laser bumped juggling a lot of balls, he says. was the last time that a sitting member of the the atoms into two, quantum-mechanical The result could one day have practical president s cabinet had a paper in Nature on alternate realities. In one, an atom absorbed applications. If gravity s time-shifting effect fundamental physics? he asks. a photon and arced on a slightly higher path, were not constant, then researchers might have Eric Hand D. ENGLISH A team led by Holger Müller of the University of California, Berkeley, has measured the time-shifting effects of gravity 10, 000 times more accurately than ever before The test puts pressure on the Atomic Ensemble in Space, an experiment being run by the European Space Agency When was the last time that a sitting member of the president s cabinet had a paper in Nature on fundamental physics? [Clifford Will] 862 2010 Macmillan Publishers Limited. All rights reserved Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 20 / 29

A raging controversy PROS Holger Müller, Achim Peters & Steven Chu, A precision measurement of the gravitational redshift by the interference of matter waves, Nature 463, 926 (2010) Holger Müller, Achim Peters & Steven Chu, Atom gravimeters and gravitational redshift, Nature 467, E2 (2010) Mike Hohensee, Achim Peters, Steven Chu, Holger Müller et al, Gravitational redshift, equivalence principle, and matter waves, J. Phys. Conf. Ser. 264, 012009 (2011) Mike Hohensee, Stephen Chu, Achim Peters & Holger Müller, Equivalence principle and gravitational redshift, Physical Review Letters 106, 151102 (2011) CONS Peter Wolf, Luc Blanchet, Christian Bordé, Serge Reynaud, Christophe Salomon & Claude Cohen-Tannoudji, Atom gravimeters and gravitational redshift, Nature 467, E1 (2010) Peter Wolf, Luc Blanchet, Christian Bordé, Serge Reynaud, Christophe Salomon & Claude Cohen-Tannoudji, Does an atom interferometer test the gravitational redshift at the Compton frequency?, Classical and Quantum Gravity 28, 145017 (2011) Supurna Sinha & Joseph Samuel, Atom interferometers and the gravitational redshift,classical and Quantum Gravity 28, 145018 (2011) Domenico Giulini, Equivalence principle, quantum mechanics, and atom-interferometric tests, arxiv:1105.0749 (2011) AND SO ON Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 21 / 29

Basic idea of this proposal g path I g path II The atoms are viewed as clocks ticking at the de Broglie-Compton frequency ω C = mc2 The atom-clocks propagate in the two paths I and II of the interferometer at different elevations in the gravitational field g. They experience a measurable phase shift due to the gravitational redshift Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 22 / 29

An analogy with clock experiments Two identical clocks are synchronized and moved at different elevations in a gravitational field. The total phase difference when the two clocks are brought back together is [ ϕ clock = ω dτ I II ] dτ ω where dτ is the proper time and ω the proper frequency The phase shift in an atom interferometer contains a contribution similar to the clock phase shift, ϕ = ω C dτ + ϕ laser dτ with the role of the clock s proper frequency played by the atom s Compton frequency Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 23 / 29

Arguments in details [Müller, Peters & Chu 2010] In general relativity the total shift in the atom interferometer is [ ϕ = ω C gz ] c 2 + ż2 2c 2 dt + ϕ laser Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 24 / 29

Arguments in details [Müller, Peters & Chu 2010] In general relativity the total shift in the atom interferometer is [ ϕ = ω C gz ] [ ] ż2 c 2 dt + ω C 2c }{{} 2 dt }{{} redshift Doppler shift + ϕ laser Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 24 / 29

Arguments in details [Müller, Peters & Chu 2010] In general relativity the total shift in the atom interferometer is [ ϕ = ω C gz ] c 2 dt }{{} k g T 2 [ ] ż2 + ω C 2c 2 dt }{{} k g T 2 2 + ϕ }{{ laser = k g T } k g T 2 Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 24 / 29

Arguments in details [Müller, Peters & Chu 2010] In general relativity the total shift in the atom interferometer is [ ϕ = ω C gz ] [ ] ż2 c 2 dt + ω C 2c }{{} 2 dt + ϕ }{{ laser }}{{} k g T 2 k g T 2 k g T 2 2 = k g T Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 24 / 29

Arguments in details [Müller, Peters & Chu 2010] In general relativity the total shift in the atom interferometer is [ ϕ = ω C gz ] c 2 dt = k g T 2 Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 24 / 29

Arguments in details [Müller, Peters & Chu 2010] In general relativity the total shift in the atom interferometer is ϕ = k g T 2 = ω C g z c 2 where z = k m T is the spatial separation between wave packets T Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 24 / 29

Arguments in details [Müller, Peters & Chu 2010] In an alternative theory the total shift in the atom interferometer is ϕ = ( 1 + β ) ω C g z c 2 The classical trajectory of the atom obeys g The quantum phase of the matter wave obeys g = (1 + β)g where β represents the redshift-violating parameter T Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 24 / 29

The arguments are incorrect [Wolf, Blanchet, Bordé, Reynaud, Salomon & Cohen-Tannoudji 2010, 2011] Rewriting the GR prediction is that form is misleading ϕ = ω C g z c 2 where z = k m T is the spatial separation between wave packets T 1 The mass of the atom cancels so the Compton frequency is irrelevant. Nowhere is there a measurable signal at ω C 2π 3.0 10 25 Hz 2 The separation between wave packets z is theoretically deduced but is not measured. Measuring it in an atom interferometer would destroy the interference fringe pattern Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 25 / 29

The arguments are incorrect [Wolf, Blanchet, Bordé, Reynaud, Salomon & Cohen-Tannoudji 2010, 2011] The statement that the Doppler and laser terms cancel out is not invariant [ ϕ = ω C gz ] [ ] ż2 c 2 dt + ω C 2c }{{} 2 dt + ϕ }{{ laser }}{{} k g T 2 k g T 2 k g T 2 2 = k g T Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 26 / 29

The arguments are incorrect [Wolf, Blanchet, Bordé, Reynaud, Salomon & Cohen-Tannoudji 2010, 2011] The invariant cancellation is between the redshift and Doppler terms [ ϕ = ω C gz ] [ ] ż2 2 c 2 dt + ω C 2c }{{} 2 dt + ϕ laser = k g T }{{}}{{} k g T 2 k g T 2 k g T 2 Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 26 / 29

The arguments are incorrect [Wolf, Blanchet, Bordé, Reynaud, Salomon & Cohen-Tannoudji 2010, 2011] z g t ϕ = k g T 2 k g T 2 + ϕ }{{} laser = k g T 2 0 Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 27 / 29

The arguments are incorrect [Wolf, Blanchet, Bordé, Reynaud, Salomon & Cohen-Tannoudji 2010, 2011] 1 2 z' = z + g t 2 a = - g t' = t ϕ = } 0 {{ 0 + ϕ } laser = k g T 2 0 Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 27 / 29

The arguments are incorrect [Wolf, Blanchet, Bordé, Reynaud, Salomon & Cohen-Tannoudji 2010, 2011] Such an alternative theory in non viable ϕ = ( 1 + β ) ω C g z c 2 The classical trajectory of the atom obeys g The quantum phase of the matter wave obeys g = (1 + β)g T 1 Schiff s conjecture that WEP implies EEP is not valid 2 No principle of least action for matter waves 3 Quantum Mechanics (e.g. Feynman s path integral formalism) is violated Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 28 / 29

Conclusions 1 Atom interferometers test the WEP or universality of free fall at the level 10 9 between atoms and macroscopic masses 10 7 between two different atoms 2 The satellite STE-Quest should test it by atom interferometry in space at the level 10 15 (comparable to Microscope) for different isotopes of the rubidium atom 3 Atom interferometers do not permit to test the gravitational redshift at the Compton frequency Luc Blanchet (GRεCO) Equivalence principle and matter wave interferometry colloque Microscope 29 / 29