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  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    On-chip atom interferometry

    Jzsef FortghPhysikalisches Institut

    Eberhard-Karls-Universitt Tbingen

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Separation between the condensates: d

    Relative momentum of the condensates after an expansion time :

    dmp

    mdhx

    The corresponding waveleght is the periodeof the fringe pattern:

    M.R. Andrews et al., Science 275, 637-641 (1997)

    Interference of two condensates

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    M.R. Andrews et al., Science 275, 637-641 (1997)

    Interference of two condensates

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Interference of condensates

    1 1( , )1 1 1( , ) ( , )

    = i x x tx t f x x t e

    Time evolutions of the amplitude fi(x-xi,t) and phase i(x-xi,t) have to be taken into account !

    Condensate wavefunctions:

    2 2( , )2 2 2( , ) ( , )

    = i x x tx t f x x t e

    Interference:2 2 2

    1 2 1 2 1 2 1 22 cos( ) + = + + f f f f

    For a BEC in the Thomas-Fermi regime: Castin & Dum PRL 77, 5316 (1996).

    x

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Thomas-Fermi approximation

    We approximate the GP equation for repulsive interaction (a>0) in the limit

    interaction energy >> kinetic energy

    The density the shape of the potential (TF limit):

    18

    =ngna

    healing lengthint. energy

    single particle energy in the condensate

    density n(x)

    potential V(x)

    TF radiusx

    x

    Dalfovo et al., Rev. Mod. Phys. 71, 463-512 (1999)

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Expansion of condensates in the TF regime

    Density profile

    Phase distribution

    In the harmonic trap free expansion

    parabolic, with TF radius parabolic

    scaling parameterdescribing the size of the condensate at any time in units of the Thomas-Fermi radius

    uniform parabolic 2

    0 0( ) ( )2 = ii x x x x

    ( )( )( )

    = iii

    m b ttb t

    1( ) =imt

    tfor

    large t

    Castin & Dum PRL 77, 5316 (1996)

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Interference of two condensates in the TF regime

    2 2 21 2 1 2 1 2 1 22 cos( ) + = + + f f f f

    1 2

    1 10,= == =x x x

    Phase difference:

    fringe spacing

    2 21 21 2 1 2

    2 2

    2

    ( ) ( )2 2

    ( )2 2

    ( )2

    = =

    =

    = +

    x x x x

    x x x

    x x x

    m x xt

    2

    fringe

    substitute

    1( ) =imt

    tuse

    x

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Force measurement

    In general, the phase develops as the time integral of the Lagrangian.

    01 1 ( )

    = = +

    f

    i

    t

    kin pott

    S S dt E E

    = kin potL E E

    0 ( )= + f

    i

    t

    kin pott

    S S dt E E

    Lagrange function:

    Action:

    Phase:

    In case of a condensate, S0 contains the parabolic phase profile, any initial centre of mass motion, and initial phase.

    Principle of an atom interferometer:

    Kasevich & Chu, PRL 67, 181 (1991)

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Spatial phase

    =

    ( 2 mod 2 )fringe

    spatial phase

    x

    N(x)

    reference measurement

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Y. Shin et al. PRA 72, 21604 (2005)

    Coherence is lost after breaking the tunnel coupling due to excitation of the condensate at the splitting point.

    Spatial interferometer: magnetic double well

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    T. Schumm et al., Nature Physics 1, 57 (2005)

    Coherence is preserved after the splitting for ~4 oscillation periods (2ms). Dephasing for longer observation times.

    Splitting a condensate in an rf-double well

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Magnetic trap as an avoided crossing

    z

    E

    Hamiltonian for a spin S=1/2 system in a magnetic field

    Static magnetic field

    Energy eigenvalues

    1

    2

    Adiabatic limit:

    2 2S L

    1E2

    = +

    ( )( )

    2

    LZ 2 1dE / dt

    = >>

    xB const.=

    zB z

    H B=

    BSg =

    L

    x y zS x y z1 e e eE B B B2 m m m

    = + +

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    RF-induced adiabatic potentials

    z

    E

    z

    E

    RF

    22S RF

    1E ( )2

    = +

    RF1 e B2 m

    =

    Energy eigenvalues

    Static + RF magnetic field

    x 0 RF RFB B B cos( t)= +

    zB z

    Courteille et al., J. Phys. B 39 1055 (2006)Lesanovski et al., Phys. Rev. A 73, 033619 (2006)

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Long phase coherence due to number squeezing

    Expected coherence time for coherent states: c ~ 20 ms

    Experiment: c > 200 ms

    Splitting results in number squeezed states: lower phase diffusion rate for number squeezed states.

    Jo et al., Phys. Rev. Lett. 98, 030407 (2007)

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Phase diffusion rate in a condensate:

    Derivative of the chemical potential

    Standard deviation of the relative atom number

    Coherent state:

    Squeezed state:

    s: squeezing parameter

    Jo et al., Phys. Rev. Lett. 98, 030407 (2007)

    Number squeezing

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Adiabatic merging Sudden removal of a thin phase membrane

    Excitation energy:

    N

    0

    Phase sensitive recombination of BECs

    Jo et al., PRL 98, 180401 (2007)

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    The excitation due to the phase difference decays and heats the cloud:loss of condensate fraction

    Phase sensitive recombination of BECs

    Jo et al., PRL 98, 180401 (2007)

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Jo et al., PRL 98, 180401 (2007)

    Recombination time

    slow

    sudden

    Hold time in the trap determines the relative phase

    Oscillation of the condensed fraction as a function of the relative phase and the recombination time

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Bragg diffraction

    Two photon transition between two momentum states of the ground state:

    initialp1k 2k

    final initial 1 2p p (k k )= +

    With k1 k2 ( k) and

    Weidemller, Zimmermann (Eds.) Interaction in ultracold gasesWiley-VCH 2003, Weinheim

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Bragg diffraction

    Weidemller, Zimmermann (Eds.) Interaction in ultracold gasesWiley-VCH 2003, Weinheim

    Oscillatory behavior with the two photon Rabi-frequency:

    with the resonant single photon Rabi-frequencies

    and =1/

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Output ports:1. P1 population in p=02. P2 population in p=2kIf there is no phase shift, the populations P1 an P2 are equal.

    pulses2 2

    First pulse split the condensate into equal populations with p=0 and p=2k.

    Second pulse inverts the states.

    Third pulse recombines the wavepackets,

    This is a single particle interferometer scheme. BEC just makes the detection easier.

    Kasevich & Chu, PRL 67, 181 (1991)

    Bragg pulse interferometer

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    J. E. Simsarian et al., PRL 85, 2040-2043 (2000)

    measuring the phase of a Bose-Einstein condensate wave function

    Bragg pulse interferometer

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Wang et al., PRL 94, 090405 (2005)

    Atom Michelson interferometer based on Bragg pulses

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Magnetic lattice

    3 mm

    1 m wide gold conductors at 1 m spacing

    I= 0.2mAI= 0.2mA

    4 m

    lattice constant

    100 m

    Produced by: thin film evaporation (300nm thick gold), and dry etching. Substrate: Si, covered by SiO2 isolation layer.

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Diffraction from a lattice

    1. BEC prepared 30m below the lattice

    2. BEC forced to oscillate towards the lattice3. Observation after 20ms TOF: diffraction & interference

    d

    Gnther et al., PRL 95, 170405 (2005)Gnther et al., PRL 98, 140403 (2007)

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Diffraction from the lattice

    displacement d (m)

    phas

    em

    odul

    atio

    nin

    dex

    S 100m

    d

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Phase modulation

    Expansion in momentum eigenfunctions of the axial motion (Bessel functions of first kind):

    Number of atoms in the nth diffraction order proportional to:

    xy

    z

    Periodic potential of the meander:

    Phase modulation index

    Wave function directly after phase imprinting

    Ref: C. Henkel, J.-Y. Courtois, and A. AspectJ. Phys. II. France 4, 1955 (1994)

  • Jzsef Fortgh, Universitt TbingenOn-chip atom interferometry

    Phase modulation

    Expansion in momentum eigenfunctions of the axial motion (Be