Lecture 2:Matter. Wave Interferometry

Size: px

Start display at page:

Download "Lecture 2:Matter. Wave Interferometry"

Kory Reed
5 years ago
Views:

1 Lecture :Matter Wave Interferometry Matter wave interferometry: as old as Quantum Mechanics Neutron diffraction and electron diffraction are standard investigation tools in solid state physics Cold atoms: new possibilities h λ DB = T 1μ K Mv = λ 1μm Rapid advances in precision measurements DB

2 Interferometry Beam splitter LASER DETECTOR Beam recombiner Change of optical path, Length, pressure, temperature, Atom interferometry Change of phase of interference pattern Atomic source Beam splitters, mirrors Detectors

3 Matter wave diffraction Huygens-Fresnel principle Monokinetic beam of particles described by ψ (,) rt Δ E m ψ = ψ Analogy with Δ + k = me Δ ψ + k ψ = 0 with k = E E 0 for electric field E(,) rt of light beam with λ = π k with T = D/ v k = mv 0 0 ik( X + Y ) im( X + Y ) ψ ( X, Y, D) = ψ0 exp = ψ0 exp D T

4 Neutron Diffraction from a slit Experiment with slow neutrons: λ = 1.96 nm v0 = 06 m/ s ILL Grenoble Neutrons produced at km/s slowed in successive steps to T=0K Detection by nuclear reaction with 100% efficiency producing alpha particles Slit width: 93 microns

5 Atom holography Generalization to arbitrary image patterns Principle of a transmission hologram We start from the momentum distribution I( X, Y) that we want to produce on the screen. Its amplitude is axy (, ) = IXY (, ) and is the amplitude diffracted in direction with k, k k = kx / f, k = ky / f x y a( X, Y ) = Fourier Transform of t( x, y) the transmission function of the diffracting hologram x y

Tokyo experiment i( kxx+ kyy) ak ( x, ky) = e txydxdy (, ) By inverse

) Solution: code: This gives two images: I( XY, ) andi( X, Y) txy (, ) +

PRL, 77, 80 (1996) One also adds a constant t 0 to the function to be

6 Tokyo experiment i( kxx+ kyy) ak ( x, ky) = e txydxdy (, ) By inverse Fourier transform we deduce the required t(x,y) But! In general t(x,y) is complex! In fact, we can only code real values and only 0 or 1 (block or transmit!) Solution: code: This gives two images: I( XY, ) andi( X, Y) txy (, ) + t( xy, )/ M. Morinaga et al. PRL, 77, 80 (1996) One also adds a constant t 0 to the function to be coded Such that t0 + t( x, y) + t ( x, y)/ is a positive number between 0 and The transmission 0 or 1 of each pixel is then obtained by comparing tx ( i, yi) to η where is a constant t max η t max

$Matter wave diffraction by periodic$ standing wave 100% transmission

7 Matter wave diffraction by periodic structures Material grating Laser standing wave 100% transmission Standing wave MIT P. Gould et al. PRL 56,87 (1986)

8 Thin phase grating approximation Standing wave Period:λ / k = π / λ Simple treatment: 1D along z, quasi monochromatic wave packet: mean momentum p 0, Δpz k pz pz U0 U0 Hamiltonian H = + U0 sin kz = + cos kz m m pz U0 U0 ikz ikz = + ( e + e ) m 4 iht / ψ ( T) = e ψ(0 with ψ(0 = p 0 First we neglect motion along z during T n=+ iut ( 0 / )cos( kz) n UT 0 inkz ψ ( T) = e p0 = ( i) Jn e p0 n= n=+ n UT 0 = () i Jn p0 n k n= UT 0 Intensity of diffracted peaks prop. to J n

9 3D case and validity of thin grating approx. w 3D motion T = v p p n k Validity: m m x 0 ( 0 + c ) TUE 0 R / 1 U Introducing the classical oscillation period in potential well Validity: 0 where n c is a typical diff. order n U0 T / ω = UE 0 R / ω T 1 If condition is violated: thick grating and Bragg diffraction c w Bragg diffraction

10 Energy conservation and Bragg regime Stationary problem: total energy is conserved For small momenta along z, and n c ~1 p 0 k The kinetic energy along z changes by p ( p ± k) Δ Ez = m m z z It must be compensated by a change of kinetic energy along x E Δ E R z = ΔE x How does a standing wave along z changes the kinetic energy along x? Finite transit time: standing wave (diameter w ) has angular divergence This enables momentum changes along x of and of kinetic energy As long as h T E Otherwise, for long times, choose R p x kθ x ± x x h/ w ( p h/ w) p p h h Δ Ex = = m m mw T z θ = λ / w total energy can be conserved and exchanged between x and z p = k Diffraction leads to pz = + k momentum along z is conserved. This is the Bragg regime. Rabi oscillation between the two states with pulsation U / 0 Adjustable beamsplitter

11 MIT, Konstanz Stanford,Vienna Yale,Toulouse, Paris, Hannover Three grating Interferometers Beam with Velocity v 0 x Phase shift calculated on classical path (See below) φ( k ) = k ( x) dx k ( x) dx 0 1 Γ1 Γ where k 0 = me / and kx ( ) = me ( Vx ( )/ xb V E 0 ( ) 1 tb For k V x and stationary: δϕ ( k0) = ( ) dx= V t dt xa E v ta 0 Example: Na beam with v 0 =1000m/s High precision measurement of x b -x a =0.1m, V= ev, δφ = 1 rad electric polarizability, A- B effect, index of refraction for matter waves,

12 Path integral formalism r, t r, t A particle travels from to with a probability amplitude given by: a a b b / b b a a Γ 3 b tb K b tb a ta ψ a ta d ra is K( r, t ; r, t ) = e Γ ψ ( r, ) = ( r, ; r, ) ( r, ) t b S = Γ L((),),) r t r t dt t a The sum is over all paths Γ connecting r, t r, t Remarkable result: For a Lagrangien which is linear or quadratic in position and momentum we have: isclass. K( r, t ; r, t ) = F( t, t ) e / b b a a b a b F( t, t ) b b a S class. is the action calculated along the path Γ where is independent of initial and final positions r a a a to r b, and where is the action evaluated on the classical path connecting to r, t Examples Particle in gravity field Particle in harmonic trap Particle in rotating frame L = mr / mgz L= mr / mω r / L= mr / + mr.( Ω r) + m( Ω r) / r t a a b b

13 Young slit experiment with free falling atoms F. Shimizu et al., Phys. Rev A, 46 R17 (199) rt v a, a 0 a H rt b, b T = v + gh + v 0 0 g ( rb, tb) cos ( mxba/ T) ψ δ x = a=6 μm, H= 0.85 m Fringe period b π T ma

14 Atom interferometers, Spectroscopy,, and Clocks Atoms have internal states Two level atom: g, e Laser resonant on g e transition Neglect spontaneous emission Use long lived upper states Mg, Ca, Sr, or Raman Transition between hyperfine ground states in alkalis for instance Effective two-level system e +k 1 -k g g 1

15 Optical clocks Mach-Zehnder interferometer with light beams π π π e, p+ k Atom beam g, p g, p t 0 t 1 t Laser beams Time δφ = φ ( t ) φ ( t ) + φ ( t ) Sensitive to rotation and accelerations: gyrometers and gravimeters

16 M. Kasevich and S. Chu, PRL 67, 1991 δφ = k at = k gt = k gt a Cold atom gravimeter eff eff L λ Tcycle = δφ 4πT τ L min noise min 1/ δφnoise = 0.01 rad, T = 0.4s a = 9 10 / 1 m s at s Detection of tides effects at ~10-7 g

17 Sagnac interferometer Interferometer area: A Rotation with angular speed Ω around z Path length increase (decrease) for counter clockwise (clockwise) δ l = ΩRT Travel time from A to B: Phase difference T = π R/ v δφ = kδl = kω RT = kω π R / v= kωa/ v Photons: v = c k = ω / c δφ photons = ΩAω / c Matter: k = mv/ δφ = ΩAm / matter Ratio of sensitivity: δφ δφ matter photons = mc 100GeV 10 ω 1eV 11

18 Stanford Gyroscope T. Gustavson, P. Bouyer and M. Kasevich PRL 78, 046, 1997 δω Current sensitivity 610 rad s Hz / Measurement of Earth rotation rate : 43 μrad/s

19 Optical clocks Ramsey-Bordé interferometer π π π π C. Bordé, Phys. Lett. A,140 (1989) Atom beam g, p g, p e, p- k T T with t 1 t t 3 P cos T( ω ω + δ) + φ t 4 ( ) e L 0 L δ = k /m and L Time φ = φ φ + φ φ 1 4 3

$increases Lasers locked to Ca with fractional instability of 5$

20 Recoil doublet in optical clocks U. Sterr et al., atom interferometry, P. Berman ed D D Cold atoms frequency standards Calcium: PTB, NIST T increases Lasers locked to Ca with fractional instability of at 1 s and at 000 s Accuracy: ~10-14

21 h/m and fine structure constant α Recoil splitting is: Δ f = hν / m c recoil atom Measuring Δ f recoil is a measure of h/ matom R h R h m m atom p = = c me c m atom m p me α All other quantities can be measured at 10-9 or better thus a photon recoil measurement at 10-9 can give α at 10-9 Cesium atom interferometry: α at Wicht et al. Phys. Scripta (00) Bloch oscillations of Rb atoms : α at Cladé et al., PRL 96, (006) g- of electrons with QED calculations: α at : Gabrielse et al, PRL, 97, (006)

Cold atoms and precision measurements Interferometers and clocks T : interaction time with ELM field Slow atoms: T large; atomic fountain or microgravity of space Interferometers on chips

22 Cold atoms and precision measurements Interferometers and clocks T : interaction time with ELM field Slow atoms: T large; atomic fountain or microgravity of space Interferometers on chips Clocks: gain prop. to T L Inertial sensors: Accelerometers: gain as T Sagnac gyroscopes : gain as L T Current sensitivity: Acceleration: δg/g= in 1min Rotation: Ω= rad s -1 in 1 s

23 Summary Atom interferometry has entered into high precision measurement phase Fine structure constant and h/m Towards a redefinition of the kilogram based on atomic masses Earth rotation, g, g gradients, inertial base (GOM, CASI) G: Magia (Firenze) Tests of Newton law at short distances Test of Equivalence principle Prospects for ultra-high sensitivity inertial sensors in space with long interrogation times: HYPER Quantum gases sources and atom lasers with atom chips (ICE, Quantus) See several talks at the workshop for most recent developments

24 Further reading Atom Interferometry ed. Paul Berman, Academic Press, 1997 Atoms quanta and relativity, special Issue of J Phys B: Atomic, Molecular and Optical physics, IoP (005) ed., T. Haensch, H. Schmidt-Boecking and H. Walther C. Bordé, metrologia, 39, 435 (00) C. Cohen-Tannoudji, lectures at Collège de France J. Dalibard, DEA lectures on cold atoms

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