József Fortágh, Universität TübingenOn-chip atom interferometry

On-chip atom interferometry

József FortághPhysikalisches Institut

Eberhard-Karls-Universität Tübingen

József Fortágh, Universität TübingenOn-chip atom interferometry

Separation between the condensates: d

Relative momentum of the condensates after an expansion time τ:

τdmp ≈Δ

mdhx τ≈Δ

The corresponding waveleght is the periodeof the fringe pattern:

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

Interference of two condensates

József Fortágh, Universität TübingenOn-chip atom interferometry

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

Interference of two condensates

József Fortágh, Universität TübingenOn-chip atom interferometry

Interference of condensates

1 1( , )1 1 1( , ) ( , ) −Ψ = − ⋅ i x x tx t f x x t e ϕ

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

Condensate wavefunctions:

2 2( , )2 2 2( , ) ( , ) −Ψ = − ⋅ i x x tx t f x x t e ϕ

Interference:2 2 2

1 2 1 2 1 2 1 22 cos( )Ψ + Ψ = + + −f f f f ϕ ϕ

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

xδ

József Fortágh, Universität TübingenOn-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

ξπ

healing lengthint. energyξ

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)

József Fortágh, Universität TübingenOn-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 xαϕ

( )( )( )

= ii

i

m b ttb t

α 1( ) =imt

tαfor

large t

Castin & Dum PRL 77, 5316 (1996)

József Fortágh, Universität TübingenOn-chip atom interferometry

Interference of two condensates in the TF regime

2 2 21 2 1 2 1 2 1 22 cos( )Ψ + Ψ = + + −f f f f ϕ ϕ

1 2

1 10,= == =x x x

α α αδ

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

α αϕ ϕ

α α δ

αα δ δ

δ

2

fringe

πλ

substitute

1( ) =imt

tαuse

xδ

József Fortágh, Universität TübingenOn-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 Eϕ

= −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)

József Fortágh, Universität TübingenOn-chip atom interferometry

Spatial phase

Δ= ⋅ π πλ

( 2 mod 2 )fringe

spatial phase

x

N(x)

Δreference measurement

József Fortágh, Universität TübingenOn-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

József Fortágh, Universität TübingenOn-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”

József Fortágh, Universität TübingenOn-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

1Φ

2Φ

Adiabatic limit: ω

2 2S L

1E2

= ± ω + Ω

Ω

( )( )

2

LZ 2 1dE / dt

ΩΓ = π >>

xB const.=

zB z∝

H B= −μ

BSgμ = μ

LωΩ

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

⎡ ⎤Φ = σ + σ + σ Φ⎢ ⎥⎣ ⎦

József Fortágh, Universität TübingenOn-chip atom interferometry

RF-induced adiabatic potentials

z

E

z

E

RFω

22S RF

1E ( )2

= ± ω − ω + Ω

RF1 e B2 m

Ω =

Energy eigenvalues

Static + RF magnetic field

Ω

x 0 RF RFB B B cos( t)= + ω

zB z∝

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

József Fortágh, Universität TübingenOn-chip atom interferometry

Long phase coherence due to number squeezing

Expected coherence time for coherent states: τc ~ 20 ms

Experiment: τc > 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)

József Fortágh, Universität TübingenOn-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

József Fortágh, Universität TübingenOn-chip atom interferometry

Adiabatic merging Sudden removal of a thin “phase membrane”

Excitation energy:

N ω

0

Phase sensitive recombination of BECs

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

József Fortágh, Universität TübingenOn-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)

József Fortágh, Universität TübingenOn-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

József Fortágh, Universität TübingenOn-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

Δ

Weidemüller, Zimmermann (Eds.) „Interaction in ultracold gases“Wiley-VCH 2003, Weinheim

József Fortágh, Universität TübingenOn-chip atom interferometry

Bragg diffraction

Weidemüller, Zimmermann (Eds.) „Interaction in ultracold gases“Wiley-VCH 2003, Weinheim

Oscillatory behavior with the two photon Rabi-frequency:

with the resonant single photon Rabi-frequencies

and Γ=1/τ

József Fortágh, Universität TübingenOn-chip atom interferometry

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

pulses2 2π π− π − −

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

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

József Fortágh, Universität TübingenOn-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

József Fortágh, Universität TübingenOn-chip atom interferometry

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

Atom Michelson interferometer based on Bragg pulses

József Fortágh, Universität TübingenOn-chip atom interferometry

Magnetic lattice

3 mm

1 μm wide gold conductors at 1 μm spacing

I= – 0.2mAI= 0.2mA

4 μm

lattice constant

100 μm

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

József Fortágh, Universität TübingenOn-chip atom interferometry

Diffraction from a lattice

1. BEC prepared 30µm below the lattice

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

d

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

József Fortágh, Universität TübingenOn-chip atom interferometry

Diffraction from the lattice

displacement d (μm)

phas

em

odul

atio

nin

dex

S 100μm

d

József Fortágh, Universität TübingenOn-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)

József Fortágh, Universität TübingenOn-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)

József Fortágh, Universität TübingenOn-chip atom interferometry

Diffraction and interference

without interference includes interference

20ms time of flight

S = 1.2

*( , ) ( , ) ( , ) ( , ) ( , )∞

∗

=−∞

= Ψ ⋅ Ψ − Ψ ⋅ Ψ∑ n nn

n S x S x S x S x S x

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

József Fortágh, Universität TübingenOn-chip atom interferometry

Reproducible phase

• New type of atom interferometer based on a single diffraction pulse

• Lattice constant defined by the fabricated conductor geometry

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

József Fortágh, Universität TübingenOn-chip atom interferometry

Single atom manipulation & detection

J. Fortágh et al., Opt. Comm. 243, 45 (2004)

single atom detectionby photoionization

quantum gate

Bose-Einstein condensate

outcoupled atom

laser

antenna

output coupler lattice