Atom Interferometry I. F. Pereira Dos Santos

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1 Atom Interferometry I F. Pereira Dos Santos

2 SYRTE SYRTE is one of the 4 Departments of Paris Obs. SYRTE research activities : - Time and Frequency Metrology (LNE-SYRTE) - Fundamental Astronomy - History of sciences (astronomy) LNE-SYRTE research activities : - Atomic time scales (French Atomic Time, TAI) - Atomic clocks (µwave+optical) - Links (µwave, optical links) - Atom Interferometry and Inertial Sensors

3 SYRTE Atomic Interferometry and Inertial Sensors Team 5 permanent researchers (3 CNRS, 1 LNE, 1 MDC) ~15 students and post-docs Our activities : Development of laser cooled atom interferometers, inertial sensors

4 Organization of the lecture 1 : Matter wave optics 2 : Tools to manipulate atomic wavepackets 3 : Interferometers and calculation of the phase shift 4 : Experimental demonstrations 5 : The sensitivity function

5 Organization of the lecture 1 : Matter wave optics 2 : Tools to manipulate atomic wavepackets 3 : Interferometers and calculation of the phase shift 4 : Experimental demonstrations 5 : The sensitivity function

6 De Broglie : Wave/particle duality Louis De Broglie (1923) : to a particule of momentum p is associated a matter wave with de Broglie wavelength : Λ db =h/p Matter-wave optics is as old as quantum mechanics Neutron and electron diffraction have been demonstrated in the 1920 s Standard tools to investigate solid state physics Atom optics/interferometry exploits the wave nature of atoms Analogous to light wave phenomena, but for de Broglie waves

7 De Broglie : duality wave/particule Light Light wave (l,c) Photon (E, p) Velocity c = m.s -1 Electric field Intensity Matter Matter wave (L db,v) with L db =h/p Particle (E, p) Velocity of the atoms 0 < V < a few km.s -1 Wave function Probability of presence Thermal beam : v ~ m/s => L db ~ 10 pm Laser cooled atoms : v ~ 1cm/s In fact, p ~ hk => L db ~ l ~ 1 µm

8 Matter wave propagation Analogy between light and matter Wave equation for light Wave equation for matter (Schrodinger equation) Let s compare relations of dispersion in vacuum (V=0) i t ( kx t) E, e E, i e d vg v Wavepacket spreads dk k Light : velocity is independent of wavelength Matter : velocity depends on wavelength => Vacuum is a dispersive medium for matter waves

9 Matter wave propagation Index of refraction for matter waves (V 0) Wave equation for matter (Schrödinger equation) Look for harmonic solutions e it with where k 0 is the wavevector in vacuum Equation for light in a medium : By analogy, the index of refraction for matter waves is given by

10 Matter wave propagation Wave number is thus given by which can also be written as By analogy with optics, the phase accumulated can be obtained by integrating the wave number along the distance The phase shift due to the potential is thus given by To first order, V

11 Matter wave propagation Orders of magnitude : Consider Na atom thermal velocity (Pritchard, 91) : v kbt m m/s Energy : ev De Broglie wavelength: L db h 2mk b T 30 pm Propagation along 10 cm increases the phase by rad Increasing its height by 1 mm increases its potential energy Index of refraction V 1 1/ mgz ev Phase shift : 1000 rad!!!!

12 Matter wave propagation Influence of other potentials which affect atoms Sensitivity of matter waves to external potentials can be exploited to measure - atomic polarizability and/or electric fields - magnetic fields - light shifts - short range forces (Van der Waals, Casimir Polder) - gravity

13 Matter wave propagation Let s go back to the Schrodinger equation The solution is an approximation It is valid in the limit where the potential varies slowly over the scale of Analog to the WKB approximation. L db It is equivalent to the result obtained following Feynam s path integral formalism

14 Matter wave propagation Propagation of a wave function K is the quantum propagator : the wave function in B is the sum of all waves radiated from all point source As Feynman : Sum over all possible paths connecting A and B is the action calculated along the path Γ is the Lagrangian : L = Ec - V

15 Matter wave propagation If L is a quadratic function of position and velocities, the sum over all Γ reduces to a single contribution, corresponding to the classical path for which the action is extremal (principle of least action, principle of Fermat in optics) is the classical action calculated along the classical path Γ In the perturbative limit, the phase shift due to the potential is then given by

16 Organization of the lecture 1 : Matter wave optics 2 : Tools to manipulate atomic wavepackets 3 : Interferometers and calculation of the phase shift 4 : Experimental demonstrations 5 : The sensitivity function

17 Atomic diffraction Estermann and Stern (1930) He atoms diffracted from the surface of a LiF crystal Momentum transferred d : grating period ~ 4A => large diffraction angles Coherent illumination across several grating periods requires the tranverse coherence length to be larger than the grating period => small transverse velocity dispersion => Collimation of the beam to increase transverse coherence length

$Wave nature of atoms : diffraction on a grating With the$ nanostructured materials with characteristic pattern size

18 Wave nature of atoms : diffraction on a grating With the progress of nanotechnology, possibility to realize nanostructured materials with characteristic pattern size below 1µm Material grating : amplitude modulation = loss D. Keith et al (1988)

19 Young double slit experiment

20 Young double slit experiment F.O. Carnal, J. Mlynek, Phys. Rev. Lett. 66, p 2689 (1991) S 1 = 2 µm, S 2 = 1 µm, d = 8 µm L = L = 68 cm Diffraction with a first slit => Transverse coherence Angle : Λ db /s 1 = rad Transverse coherence length > separation of secondary slits Fringe separation : Λ db L /d ~ 8 µm

21 Young double slit experiment F. Shimizu et al., Phys. Rev A, 46 R17 (1992) Laser cooled Ne* atoms Point like initial source

22 Atomic holography Arbitrary atomic pattern are in principle possible The atomic pattern is the Fourier transform of the mask

23 Young double slit experiment F. Shimizu (1996) Increase the separation between slits and screen => larger separation a=6 mm

24 Young double slit experiment Calculation of the fringe pattern Two slits at x=-d and x=+d in the plane z=0 A screen in the plane z=-h Atoms with initial velocity v 0 z Langragian 1 ( 2 2 x z ) L m v v mgz 2 Let s consider the trajectory A(x A,y A,t=0) -> B(x B,y B,t=T) x() t x v t A vx ( x x ) / T B z( t) z v t 1/ 2gt A x z0 v ( z z ) / T 1/ 2gT A z0 B A 2 A x B Action T cl ( ) ( )... ( A B) 2T S T L t dt m r mg z z T mg T

25 Initial state : Ponctual sources at the slits T S ( T ) L( t) dt... mr mg( z z ) T mg T Action cl A B T 0 imv0 z / ( x, z) ( x d) ( x d) ( z) e i Χ(z) is an envelope function It is a wavepacket centered on z=0 with central velocity vz(0)=-v0 The phase term corresponds to the central velocity Final state : i( Scl ( T ) mv 0z a )/ ( xb, zb, T ) e ( x d) ( x d) ( z ) dx dz f a a a a a We can notice 1 2 cl cl a a cl a cl a S ( T ) S ( x, z, T ) S ( x, T ) S ( z, T ) => the integral separates

26 1 is cl ( x, )/ (,, ) ( ) ( ) a T f xb zb T e xa d xa d dxa 2 i( S cl ( za, T ) Mv0za)/ z e dz ( ) a a 2 2T (,, ) ( ) ( ) f xb zb T e xa d xa d dxa 2 i S cl zat Mv0za ( ) 1 m( xbxa) / ( (, ) )/ za e dza 2 i S cl zat Mv0za ( ) ( (, ) )/ za e dza i i m( xbd ) / i m( xbd ) / 2T 2T f ( xb, zb, T ) e e Replacing S 1 par its expression, we get mx 2 bd i S cl zat Mv0za ( x, z, T ) cos ( z ) e dz T ( (, ) )/ f b b a a

27 Evaluate the expression at zb=-h mxbd f ( xb, zb, T ) cos T i( m( H za) mgzat mv0za)/ 2T 2 z e dz ( ) a a Fringes!! Fringe spacing 2 T md What is T? 2 m za 2HzA i ( gzat 2 v0za) 2 T z e dz ( ) a a X is peaked around z a =0. The integral is zero unless the argument is stationary around z a =0. This means that 2 2 v0 2gH v0 2H 1 gt 2v 0 H gt v T T T We recover the classical expression for the center of mass g

28 Fringe spacing h D v 2gH v 2mdg Let s consider the case where v 02 <<gh D h 2 h H L gh 2mdg m 2gH d d db H Similar to optics The wavelength is Λ db at the detector

29 Diffraction by light grating Light near resonance makes index grating for the atomic waves Physical quantities to conserve during the diffraction process: Total Energy (atom + light) Momentum (atom + light) Consider 3 cases: Thin grating Thick grating: Bragg diffraction Raman transition

$Diffraction by thin grating 1D along z, quasi monochromatic wave$ Hamiltonian describing the evolution in grating : x First we

30 Diffraction by thin grating 1D along z, quasi monochromatic wave packet: z Standing wave Period:λ /2 mean momentum p 0, Hamiltonian describing the evolution in grating : x First we neglect motion along z during T Thin grating approximation Jn: Bessel functions

31 Diffraction by thin grating z x

32 Diffraction by thin grating Stationary problem: total energy is conserved For small momenta along z, p z << hk and n~1 The kinetic energy along z changes by z x It must be compensated by a change along x : Finite size of the laser beam w (the beam diverges) enables momentum changes along x of ~ h/w Bessel functions Change in kinetic energy As long as h T E R, the total energy can be conserved and exchanged between x and z

33 Diffraction by thick grating Bragg diffraction Diffraction occurs only in the direction of constructive interferences Long pulse regime Bragg diffraction couples only two states

34 Diffraction by thick grating Bragg diffraction Couples only two states : conservation Energy and momentum Energy Diffraction in a standing wave Different initial momentum : Use of a running wave (difference of frequency between the two lasers) Absorption of photon from one laser and stimulated emission in the retroreflected beam E0 P (momentum)

35 Diffraction by thick grating First demonstration of Bragg diffraction Martin et al, Phys Rev Lett 60, 515 (1988) Highly collimated sodium beam Large beam waist for the standing wave ~ 5 mm Kapitza Dirac Bragg First order Second order

$Diffraction of matter waves Raman transitions Two photon transition : Absorption of photon from one laser, stimulated emission into the second one Rabi$

36 Diffraction of matter waves Raman transitions Two photon transition : Absorption of photon from one laser, stimulated emission into the second one Rabi oscillation between 2 states of different momenta Alkali atoms (Rb, Cs) Transition between 2 momentum states ~ 1 GHz k 1, 1 k 2, 2 b a a and b Hyperfine states

37 Diffraction of matter waves Raman transitions Two photon transition : Absorption of photon from one laser, stimulated emission into the second one Rabi oscillation between 2 states of different momenta Alkali atoms (Rb, Cs) Energy ~ 1 GHz k 1, 1 k 2, 2 b Ee Eg a P (momentum) a and b Hyperfine states

p/2 pulse Atomic beam splitter Laser phase printed on the atomic

38 Transition probability Wave packet manipulation Rabi oscillation Rabi oscillations between and p pulse Atomic mirror p/2 p W Rabi t p/2 pulse Atomic beam splitter Laser phase printed on the atomic wave during a transition e +φ eff e -φ eff 12 f, p e, p k eff e i g g

39 Transition probability Wave packet manipulation : Rabi oscillation 1.0 Rabi oscillations Effective parameters Resonance conditions Conservation of momentum p' p k Conservation of energy t (µs) Doppler term Recoil term

40 Wave packet manipulation : Rabi oscillation Wave function evolution where is the effective Rabi frequency depends on the laser parameters

41 Wave packet manipulation : Rabi oscillation Laser phase imprinted on the atomic wave during a transition +φ eff -φ eff e g e g

$state Creates the coherent superposition 12 f, p e, p k eff e i adds a phase term onto the diffracted wave$

42 Wave packet manipulation : Rabi oscillation Useful cases : π/2 pulse : beamsplitter when starts from a pure state Creates the coherent superposition 12 f, p e, p k eff e i adds a phase term onto the diffracted wave packet

43 Organization of the lecture 1 : Matter wave optics 2 : Tools to manipulate atomic wavepackets 3 : Interferometers and calculation of the phase shift 4 : Experimental demonstrations 5 : The sensitivity function

44 How to built an interferometer An atom interferometer will use a series (at least two) coherent splitting processes to create multiple paths that will interfere.

45 Mach-Zehnder interferometers Let us exploit once more the analogy with light MZ Light interferometer mirror Exit port 2 Exit port 1 light beam splitter mirror Exit port 2 1,0 0,5 MZ type atomic interferometer atoms Exit port 1 0,0 1,0 0,5 lasers 0, Phase shift These interferometers are both two wave interferometers : Δφ: difference of the phase shifts accumulated along the two arms

46 Realization of an interferometer 3 Raman pulse interferometer Exit port 2 1,0 T T 0,5 0,0 1,0 Exit port 1 0,5 0, Phase shift Similar to a Mach-Zehnder in optics ΔΦ is the difference of atomic phase shift along the two paths

47 Contributions to the interferometer phase shift ΔΦ is the difference of atomic phase shift accumulated along the two paths Calculated along the classical trajectory of the center of the wave packet the laser phase shift during interaction Φi=ki.ri at the position taking into account the real trajectories between pulses free propagation phase (action during free fall) phase shift due to displacement of the wave packet Path B Path A Wavepacket separation

48 Contributions to the interferometer phase shift ΔΦ is the difference of atomic phase shift accumulated along the two paths Calculated along the classical trajectory of the center of the wave packet the laser phase shift during interaction Φi=ki.ri at the position taking into account the real trajectories between pulses free propagation phase (action during free fall) phase shift due to displacement of the wave packet Canceled in general case of an Hamiltonian at most quadratic in r and P : acceleration, rotation, gradient of acceleration Ch.J. Bordé, Metrologia 39, (2002) -Φ eff,2 +Φeff,1 Path B +Φeff,2 -Φeff,3 Path A

49 Interferometer Phase Shift Laser phase gets imprinted b a + b a - p 2 p A 2 p 2 3 A A B B B 2 A B A 2 2 B 2

50 Acceleration phase shift ( t) k. r( t) eff 1 at 2 2 T 2 T 3 a 1 1 ( t 1 ) ( t2) keff. at 2. at 1 (t 1 ) 2 2 (t 2 ) + 3 (t 3 ) = k eff ( t3) keff. a (2T )

51 Rotation phase shift k 2 VT VT W k 1 ( t1 keff 1VT 1 ) k eff )VT ( ( t 2 ) 0 3 t3) eff 3 ( k VT 2. W V T 2 k eff

52 Inertial forces sensors summary The phase shift can be calculated taking into account only the laser phase shift: measures the displacement of the reference frame of the laser (lab) compared to the reference frame of the atoms in free fall (defines an inertial reference frame) The measurement is done with a ruler materialized by the laser equi-phases (keff) : allows for an accurate measurement The sensitivity scales as T 2 : use of cold atoms/ultra-cold The systematic errors come from the interaction laser/atom

53 Organization of the lecture 1 : Matter wave optics 2 : Tools to manipulate atomic wavepackets 3 : Interferometers and calculation of the phase shift 4 : Experimental demonstrations 5 : The sensitivity function

54 Key date for atom interferometry 1991: demonstration of atomic interferometry First atom interferometer with double-slit: O. Carnal, J. Mlynek, "Young s double-slit experiment with atoms : A simple atom interferometer ", Phys. Rev. Lett., 66, 2689 (1991) First atomic interferometer: gyroscope with atomic beam and light beam splitter F. Riehle, Th. Kister, A. Witte, J. Helmcke, Ch. Bordé, Phys. Rev. Lett., 67, p 177 (1991) First atom interferometer with mechanical gratings : gyroscope with atomic beam and mechanical gratings D.W. Keith, C.R. Ekstrom, Q.A. Turchette, D.E. Pritchard, Phys. Rev. Lett., 67, p 2693 (1991) First cold atom interferometer : accelerometer with cold atoms and light beam splitter (Raman transitions) M; Kasevich, S. Chu, Phys. Rev. Lett., 67, p 177 (1991)

55 First atomic beam interferometers Keith et al. (1991) - Highly collimated supersonic beam of Na atoms - Nanofabricated gratings 400 nm period Interference signal PZT position

56 First atomic beam interferometers F. Riehle, Th. Kister, A. Witte, J. Helmcke, Ch. Bordé, Phys. Rev. Lett., 67, p 177 (1991) - Ca beam - Beamsplitters : lasers (one photon transition) - Geometry of the interferometer : Ramsey-Bordé (4 p/2 pulses)

57 First cold atom interferometer M. Kasevich, S. Chu, Phys. Rev. Lett., 67, p 181 (1991) - Cold atom interferometer - Raman transitions - Vertical beams : sensitivity to g => gravimeter

58 Key dates Test of many different configurations : mechanical gratings, one or two photon transitions, Bragg regime, Raman transition, multi-path interferometers... Paul Berman, Atom Interferometry (Academic Press, San Diego, 1997) End of the 90 s: high performances for atom interferometer High sensitivity gyroscope : double atomic beam in M. Kasevich group (Stanford and Yale University) : atomic beams and light beam splitter (Raman transitions) High accuracy gravimeter and h/m measurement: S. Chu group at Stanford University: cold atoms and light beam splitter (Raman transitions) => High stability and accuracy : cold atoms and Raman transitions Since 2000 : many experiments based on cold atoms/ Raman transitions have been developed for practical applications More recently : interferometers with guided ultra-cold atoms, demonstration of high momentum splitting (multi-hk)

59 Mechanical grating interferometers Measurement of atom polarizability (Schmiedmayer, et al. 1997) Index of refraction of gaz for matter waves Atome-surface interactions (VdW, CP)

$, Phys Rev Lett 75, 2633 (1995) Highly collimated Ar* beam Kapitza Dirac diffraction Signals from complementary ports$

60 Standing wave interferometers Rasel et al., Phys Rev Lett 75, 2633 (1995) Highly collimated Ar* beam Kapitza Dirac diffraction Signals from complementary ports Giltner et al., Phys Rev Lett 75, 2638 (1995) Highly collimated Ne* beam Bragg diffraction

61 Signal Atomic beam gyroscope Laser collimation 2 atomic beams Sensitivity : rad.s -1 / Hz Cs oven (Yale University) State preparation Detection Raman pulses Magnetic shields Interferometer fringes Rotation velocity (x10-5 ) rad/s Parameters : Longitudinal velocities : 290 m.s -1 Interferometer length : 2 m (duration 2T = 6,9 ms) => area ~26 mm 2 Flux (Mf=0) ~ at.s -1

62 Cold atom gravimeter Parameters Cs atoms T=1µK Atomic fountain T = 160 ms Performances : Sensitivity g à 1 s (2001) g at 1 s (2008) Accuracy : ~ g?

Atom interferometry. Quantum metrology and fundamental constants. Laboratoire de physique des lasers, CNRS-Université Paris Nord

Atom interferometry. Quantum metrology and fundamental constants. Laboratoire de physique des lasers, CNRS-Université Paris Nord Diffraction Interferometry Conclusion Laboratoire de physique des lasers, CNRS-Université Paris Nord Quantum metrology and fundamental constants Diffraction Interferometry Conclusion Introduction Why using