Introduction 2 Short Resume of Work with the Mott · PDF file• State Selective Production...

Neutral Atoms in Optical Lattices Neutral Atoms in Optical Lattices -- from Quantum Simulators to Multiparticle Entanglement from Quantum Simulators to Multiparticle Entanglement --

Markus Greiner, Olaf Mandel, Tim Rom, Artur Widera, Simon Fölling

T. W. Hänsch & I.B.

LES HOUCHES 2004

Ludwig-Maximilians-University, Munich,Max-Planck-Institut for Quantum Optics, Garching

Johannes-Gutenberg University, Mainz

Introduction

• During the last 20 years a high degree of sophistication has been reached in the control of single quantum systems

• New challenges ahead: control, engineer and understand complex quantum systemquantum computers, quantum simulators, novel (states of) quantummatter, advanced materials, multi-particle entanglement

Atoms, Ions, Photons, Quantum Dots

Introduction 2

Control and engineer quantum

matter at the ultimate limit

search for novel quantum phases

solve longstanding open questions of condensed matter

physics

High precision spectroscopy Control interaction properties (mag. & opt. Feshbach resonances)

production of ground state moleculues

realize Quantumsimulators

create large scale entanglement

fast & robustqubits

single site addressing ?

Short Resume of Work with the Mott State

Quantum Information applications – The Mott state as a quantum register

• Spin dependent potentials for individual atomsO. Mandel et al., PRL 91, 010407 (2003)

• Collaps and revival of the matter wave field of a BECM. Greiner et al., Nature, 419, p. 51, 2002.

• Controlled collisions – massively parallel quantum gate array – controllable Ising interactionO. Mandel et al. Nature, 425, p. 937, (2003)

Atomic/Molecular Physics

• Entanglement Interferometry – Precision measurement of atomic scattering propertiesA. Widera et al., PRL 92, 160406 (2004)

• State Selective Production of Molecules in Optical LatticesT. Rom et al. (accepted for publication in PRL)

Novel Quantum Phases

• Tonks-Girardeau Gas in an Optical LatticeB. Paredes et al., Nature 429 (2004)

Neutral Atoms in Optical LatticesPart 1

•Introduction

•Experimental Setup

•Loading a BEC into an Optical Lattice

•From a Superfluid to a Mott Insulator

•Collapse and Revival of a Macroscopic Quantum Field

Part 2

•Spin dependent lattice potentials –a quantum atomic conveyor belt

•Neutral Atom Quantum Gate Arrays

Part 3

•Entanglement Interferometry

•Controlled Molecule Formation in Lattices

•Conclusion and Outlook

From a Classical Gas to a Bose-Einstein Condensate

cT T>>Classical Gas

cT T>1 2

dB h mv T −λ = ∝

cT T<

dB dλ ≈

0T =Coherent Matter WaveNobel prize 2001 !

Matter Waves at Work !

(MIT) Andrews et al., Science (1997)

How about the Particles beneath the Waves ?

How Can We Describe the Ground State of a Many-Body System in a Trap ?

Externalconfinement

N particle system

From a Bose Gas without Interactionsto a Strongly Correlated Bose System

No Interactions

intN⊗Ψ ∝ ψWeak Interactions

Strongly Correlated Systemint

N⊗Ψ ∝ ψ

N⊗Ψ ∝ ψ

Many-Body State


•Introduction





Part 2



Part 3




Experimental Setup for Lattice ExperimentsExperimental Setup for Lattice Experiments

Magnetic Transport of Cold Atoms

MOT

BEC

Laser Cooling in Action

Magnetic Transport of Cold Atoms

M. Greiner et al., PRA 63, 031401

Plus a lot of Optics and Electronics (Table 1) !!

Table 2How do we detect these quantum gases ?

Trapping Atoms in Light Field -Optical Dipole Potentials

Energy of a dipole in an electric field: dipU d E= − ⋅r r

An electric field induces a dipole moment: d E= αr r

( ) ( )dipU I r∝ −α ωr

Red detuning:

Atoms aretrapped in theintensity maxima

Blue detuning:

Atoms arerepelled from theintensity maxima

See R. Grimm et al., Adv. At. Mol. Opt. Phys. 42, 95-170 (2000).Pioneering work by Steven Chu

Optical Lattices

2 beams 4 beams

4 mm Polystyrol Particles in Water

see also work by: Hemmerich & Hänsch, Phillips, Grynberg,Cohen-Tannoudji, Salomon

Optical Lattices 2D Lattice Potential

• Resulting potential consists of an array of tightly confing potential tubes

• BEC is split into up to 10,000 coupled 1D quantum gases(radial motion confined to zero point oscillations)

V0 up to 40 Erecoil

ωr up to 2π × 45 kHz

n ≈ 10-50 atoms per tube

3D Lattice Potential

•Resulting potential consists of a simple cubic lattice

•BEC coherently populates more than 100,000lattice sites

V0 up to 40 Erecoil

ωr up to 2π × 45 kHz

n ≈ 1-5 atoms on average per site

General Lattice Parameters

All beams intensitystabilized

Intensity control

425 nmLattice spacing

87RbAtomic Species

Simple cubicLattice geometry

Orthogonal betweenstanding wave pairs

Polarization

125 µmWaist (1/e2)

850 nmWavelength


•Introduction





Part 2



Part 3




Loading the Atoms into the Lattice Potential

Turn on lattice potential adiabatically, so that the wavefunction remains in the manybody ground state of thesystem !

Here:

•Exponential ramp up in 80 ms witha timeconstant of 20 ms.

•Not very sensitive to exactwaveform, or ramp speed

Start with a pure condensate in a magnetic trap

Macroscopic Wave Function of a BEC in an Optical Lattice

( ))( () )( jj

i

j

xjw x xx x eA ⋅ −= ⋅Ψ ∑ φ

Number of atoms onjth lattice site

Phase of wavefunction on jth

lattice site

Localized wave function onJth lattice site

If there is a constant phase shift Df between lattice sites, the state is an eigenstate (Bloch wavefunction) of the lattice potential!

Quantum number characterizing these Bloch waves:2q = ∆φλh

Crystal (Quasi-) momentum

Detecting the Atoms in the Lattice

Spacing between neighboring lattice sites (≈425 nm) is too small to bedetectable by optical means !

Switch off the lattice light

Observe the multiple matter waveinterference pattern !

Localized wavefunctions expand

and interfere with each other

(simulation)

Momentum Distributions – 1D

( ))( () )( jj

i

j

xjw x xx x eA ⋅ −= ⋅Ψ ∑ φ

Momentum distribution canbe obtained by Fouriertransformation of themacroscopic wave function.

Matter Wave Interference Pattern of a BEC in an Optical Lattice

Individual condensates in the lattice expand and interfere witheach other, revealing the momentum distribution of the atoms in the lattice.

Time of flight measurement

time of flight

2 ms 6 ms 10 ms 14 ms 18 ms

Interference Pattern of a 3D Lattice s-Wave Scattering (Experiment)

experiment theory

s-Wave scattering sphere

2D Lattice

see also work from J. Walraven on d-wave scattering !

Preparing Arbitrary Phase DifferencesBetween Neighbouring Lattice Sites

0φ =∆ φ = π∆

( )' / 2 /j V tφ = λ ⋅∆ h

Phase difference betweenneighboring lattice sites

(cp. Bloch-Oscillations)

But: dephasing if gradientis left on for long times !

Mapping the Population of the Energy Bands onto the Brillouin Zones

Crystal momentum

Free particlemomentum

Population of nth band ismapped onto nth Brillouinzone !

vωh

Crystal momentum is conserved while lowering the lattice depth adiabatically !

A. Kastberg et al. PRL 74, 1542 (1995)M. Greiner et al. PRL 87, 160405 (2001)

Experimental Results

Piet

Mon

dria

n

Brillouin Zones in 2DMomentum distribution of a dephased condensateafter turning off the lattice potential adiabtically

2D

3D

Populating Higher Energy Bands

Stimulated Raman transitionsbetween vibrational levels areused to populate higher energybands.

Single lattice site Energy bands

Measured MomentumDistribution !


•Introduction





Part 2



Part 3




Ramping up the Lattice Potential

Lattice Potential is increased

Observe at time-of-flight images

Momentum Distribution for Different Potential Depths0 Erecoil

22 Erecoil

Can We Restore Coherence ?

Ramp down for different times t and monitormomemtum distribution !

Before rampingdown

0.1 ms 1.4 ms 4 ms 14 ms

How Long Does it Take to Restore Coherence ?

Dephased Bose-Einstein condensate

0.1 ms 1.4 ms 4 ms 14 ms

Tunneling time

Coherence is restored on theorder of a tunneling time !

Bose-Hubbard Hamiltonian

Expanding the field operator in the Wannier basis of localized wave functions on each lattice site, yields :

†

,

1 ˆ ˆ( 1)2

ˆ ˆˆ i ii

i jj

i iii

UJ a aH n n n++ ε−= −∑ ∑∑

Bose-Hubbard Hamiltonian

ˆ ˆ( ) ( )i ii

a wψ = −∑x x x

23 2( ) ( ) ( )

2i lat jJ d x w V wm

⎛ ⎞= − − − ∇ + −⎜ ⎟⎝ ⎠∫

hx x x x x2

434 ( )aU d x wm

π= ∫

h x

Tunnelmatrix element/Hopping element Onsite interaction matrix element

M.P.A. Fisher et al., PRB 40, 546 (1989); D. Jaksch et al., PRL 81, 3108 (1998)

Related experiments in 1D: C. Orzel et al., Science 291, 2386 (2001).

Superfluid Limit

†

,

ˆ ˆ 1 ˆ ˆ( 1)2i j

ii i

ij

J a a U n nH +−= −∑∑

†

1

ˆ 0NM

SF ii

a=

⎛ ⎞Ψ ∝ ⎜ ⎟⎝ ⎠∑

Atoms are delocalized over the entire lattice !Macroscopic wave function describes this state very well.

Poissonian atom number distribution per lattice site

n=1

ˆ 0iia ≠

Atom number distribution after a measurement

“Atomic Limit“ of a Mott-Insulator

†

,

ˆ ˆ 1 ˆ ˆ( 1)2i j

ii i

ij

J a a U n nH +−= −∑∑

n=1

( )†

1

ˆ 0i

M n

Motti

a=

Ψ ∝ ∏ ˆ 0iia =

Atoms are completely localized to lattice sites !

Fock states with a vanishing atomnumber fluctuation are formed.

Atom numberdistribution aftera measurement

++

The Simplest Possible “Lattice“:2 Atoms in a Double Well

0.5 x

Superfluid State MI State

0.25 x

0.25 x

Average atomnumber per site:

Average onsiteInteraction per site:

<n> = 1 <n> = 1

<Eint> = ½ U <Eint> = 0

( ) ( )1 12 2l r l rφ +φ ⊗ φ +φ

1 12 2l r r lφ ⊗φ + φ ⊗φ

+

+

Quantum Phase Transition (QPT) from a Superfluid to a Mott-Insulator

At the critical point gc thesystem will undergo a phasetransition from a superfluid to an insulator !

This phase transition occurs evenat T=0 and is driven by quantumfluctuations !

Characteristic for a QPT

•Excitation spectrum is dramatically modified at the critical point.

•U/J < gc (Superfluid regime)Excitation spectrum is gapless

•U/J > gc (Mott-Insulator regime)Excitation spectrum is gapped U/J = z 5.8

Critical ratio for:

see Subir Sachdev, Quantum Phase Transitions, Cambridge University Press

Superfluid – Mott-Insulator Phase Diagram

ilocµ = µ −εFor an inhomogeneous system an effective local chemical potential can be introduced

Jaksch et al. PRL 81, 3108 (1998)

Ground State of an Inhomogeneous System

From Jaksch et al. PRL 81, 3108 (1998)

From M. Niemeyer and H. Monien(private communication)

Creating Excitations in the MI Phase

Energy Scales:

Without gradient potential

ijE U=∆

With gradient potential

Special case:

Mott-Insulator with ni=1 atom per lattice site

hων ≈ 20 UU ≈10-300 J

S. Sachdev, K. Sengupta, S. M. Girvin, cond-mat/0205169

Excitation Probability vs. Gradient

10 Erecoil tperturb = 2 ms 13 Erecoil tperturb = 4 ms

16 Erecoil tperturb = 9 ms 20 Erecoil tperturb = 20 ms

Resonance Gradient vs. Potential Depth

Shaded green areadenotes possible experimental values due to systematic uncertainties.

Utheory


•Introduction





Part 2



Part 3




What Happens to the Relative Phase of two Quantum Liquids over Time ?

Start with a singleBose-Einstein condensate

Split it into two BECs with a constant relative phase φ

BEC 1 BEC 2

Fundamental question arises:

What happens to the relative phase between the two condensates over time ?

What happens to the individual wave functions of the two BECs over time ?

Relative phase φ

M. Greiner, O. Mandel, T. W. Hänsch and I. Bloch Nature, 419 (6901), 2002

Dynamical Evolution of a Many Atom State due to Cold Collision

How do collisions affect the many body state in time ?

Phase evolution of the quantum state of two interacting atoms:

/2 ( ) 2 Ui tt e−= ⋅ h

Collisional phase

•Phase shift is coherent !

•Can be used for quantum computation(see Jaksch, Briegel, Cirac, Zoller schemes)

•Leads to dramatic effects beyond mean-field theories !

Collisional phase of n-atoms in a trap:

1/ ( 1) /2nE t Un n t= −h h

Time Evolution of a Coherent State due to Cold Collisions

1

1

+

+

+

/iU te− h

3 /i U te− h

2| | / 2

!

n

in

e nn

− α αΨ = ∑

Coherent state in each lattice site !

=

1. Here α = amplitude of the coherent state

2. Here |α|2 = average number of atoms per lattice site

Freezing Out Atom Number Fluctuations

A

BRamp up lattice fast from the superfluid regime (A) to the MI regime (B), such that atoms do not have time to tunnel !

Atom number fluctuations at (A) are “frozen“ !

( ) 2| | /20!

n

in

nn

e− α αΨ = ∑

Collapse and Revival of the Matter Wave Field due to Cold Collisions

( ) 21 ( 1) /| | / 2 2

!

n i U n n t

in

t e e nn

− −− α αΨ = ∑

h

Quantum state in each lattice site (e.g. for a coherent state)

Matter wave field on the ith lattice site

( ) ˆ( ) ( )i ii it t a tΨ = Ψ Ψ

1. Matter wave field collapsesbut revives after times multiple times of h/U !

2. Collapse time depends on the variance σN of the atom number distribution !

Yurke & Stoler, 1986, F. Sols 1994; Wright et al. 1997; Imamoglu, Lewenstein & You et al. 1997, Castin & Dalibard 1997, E. Altman & A. Auerbach 2002

Similiar to Collapse and Revival of Rabi-Oscillations in Cavity QED !

Dynamical Evolution of a Coherent State due to Cold Collisions

The dynamical evolution can be visualized through the Q-function

2( )i tα ψ

=π

Q

Characterizes overlap of our input state with an arbitrary coherent state α

Re(a)

Im(a

)

Dynamical Evolution of a Coherent State due to Cold Collisions

time

Re(α)

Im(α

)

0

0

-4

4

-4 4

G.J. Milburn & C.A. Holmes PRL 56, 2237 (1986);B. Yurke & D. Stoler PRL 57, 13 (1986)

Dynamical Evolution of the Interference Pattern

t=600 mst=450 mst=400 ms

t=300 mst=200 mst=150 mst=50 ms

After a potential jump from VA=8Er to VB=22Er.

Collapse and Revival Ncoh/Ntot

Up to 5 revivals are visible !

Oscillations after lattice potential jump from 8 Erecoil to 22 Erecoil

Revival Frequency vs. Lattice Potential Depth

h/U from theory

Influence of the Atom Number Statistics on the Collapse Time

Final potential depth VB=22Er

VA=11Er

VA=4Er

tc/trev for Different Initial Potential Depths

Atom NumberStatistics

n=1, U/J≈0

Independent proof of sub-Poissonianatom number statistics for finite U/J !

Atom NumberStatistics

n=1, U/J=17

Controlled Collisions

D. Jaksch et al., PRL 82,1975(1999), G. Brennen et al., PRL 82, 1060 (1999)A. Sorensen & K. Molmer, PRL 83, 2274 (1999)

Moving the Lattice Potentials

0

20

2sin (

sin ( / )

/ 2)

2II I k

I kxx

−

+

= − θ

= + θ

State Selective Lattice Potentials

1( , ) ( , )Vx xV −θ = θ

0 (1 3( , )4

) ),4

( ,V x VV x x+− θθ = θ+

s- s+

87Rb Fine- structure

|1>

|0>

Hyperfine structure

Delocalization “by Hand”

Initial state |0>

π/2 microwave pulse

Shift

π/2 microwave pulse

TOF

Single Particle State after Delocalization

More general, the single particle state in |1> will be:

1i iie α

+Ψ + Ψ

single particle phase

Single particle phase depends on:

• Accumulated kinetic phase

• Accumulated potential energy phase shift

1) optical potentials not constant during transport

2) magnetic field fluctuations can be removed by photon echo π/2-π-π/2 sequence

2( )

2m x t dt∫ &h

If α is constant for all atoms, we can observe an interference pattern with absorption imaging after a time of flight period!

α=0

α=π

Shifting is Coherent !

Localized

2 sites

4 sites

3 sites

V=70%

V=50%

We have been able to delocalize atoms over up to 7 lattice sites !

Delocalized

O. Mandel, M. Greiner et al., Phys. Rev. Lett. 91, 010407 (2003)

Temporal Phase Control

Delocalization over two lattice sites (∆x = 400 nm), shift time 50 µsVarying phase α‘ of the last microwave pulse

α‘ = 0°

α‘ = 180°

( )12

1 1ii i n

eϕ+

+

Moving Atoms in Harmonic Potentials

How fast can we move, in order to avoid excitations ?

22 2

1 20

2( ) sin ( / 2)( )

vc t ta

= ωω

For a constant shift velocity v, first order perturbation theory yields:

ω=2π 30 kHz, ∆x = 200 nm

x [nm]

tshift [µs]0 t

200

Mapping the Population of the Energy Bands onto the Brillouin Zones

Crystal momentum

Free particlemomentum

Population of nth band ismapped onto nth Brillouinzone !

vωh

Crystal momentum is conserved while lowering the lattice depth adiabatically !

A. Kastberg et al. PRL 74, 1542 (1995)M. Greiner et al. PRL 87, 160405 (2001)

Populating Higher Energy Bands

Stimulated Raman transitionsbetween vibrational levels areused to populate higher energybands.

Single lattice site Energy bands

Measured MomentumDistribution !

Measuring the Excitation Probability vs. Shift Velocity

Population of higher vibrational states (energy bands) can be mapped onto the corresponding Brillouin zones by adiabatically decreasing the lattice potential !

A. Kastberg et al. PRL (1995)M. Greiner et al. PRL (2001)

Atoms moved over a distance of approx. 200 nm

Start with ground state atoms

Constant Velocity

Stop; measure remaining atoms in ground state

Complete Sequence used in the Experiment

Well suited for exploring

Quantum Random Walks !

First Trapped Atom Interferometer !

Measure Atoms in State |1>

Ramsey Fringe

( )112

0 ieϕ= +

See also work by W. Ketterle, PRL 2004

Controlled Collisions

In time tHold a phase factor of

is acquired.

/HoldtEi ie eϕ = h

22 2 3

0

10

1( )(4

)a

U d xx xm

ww

E nU n

π= ⋅

= ⋅ ⋅

∫h

D. Jaksch et al., PRL 82,1975 (1999)

Building a Quantum Gate

D. Jaksch et al., PRL 82,1975 (1999)

Input state Final state

1 11 1

0 0 0 00 0

0 01 1 1 1

ie φ⋅

Fundamental quantum gate forneutral atoms

Engineering Entanglement

10 0

i i+

i i+1


i i+1

( )( )12 11 0 1 10

i ii i+ ++ +


i i+1

( )1 12 1 1 212 1 10 0 0 10 1

i i i ii i i i++ + ++ ++ + +

( )( )12 11 0 1 10

i ii i+ ++ +

( )12 1 1 212 1

1 1 1 10 0 0 0i ii

ii ii ii

e ϕ+++ ++ +

+ + +

( )( )12 11 0 1 10

i ii i+ ++ +


i i+1

( )12 1 1 212 1

1 1 1 10 0 0 0i ii

ii ii ii

e ϕ+++ ++ +

+ + +

( )( )12 11 0 1 10

i ii i+ ++ +


( )112 11 1

10 0 0 101 1i i i

ii i i ii

e ϕ+ ++ +

+ + +

i i+1

( )12 1 1 212 1

1 1 1 10 0 0 0i ii

ii ii ii

e ϕ+++ ++ +

+ + +

( )( )12 11 0 1 10

i ii i+ ++ +


i i+1

1 12 2 1(1 ) ( 11 ) 1

ii i

ie eBellϕ ϕ

+− + +

More General Entanglement

p/2-pulse

Entangled State

p/2-pulse

Shift trap

Shift trap

D. Jaksch et al., PRL 82,1975 (1999), H.-J. Briegel et al., J.Mod.Opt. 47, 215 (2000)

H.-J. Briegel & R. Raussendorf PRL 86, 910 (2001) & PRL 86, 5188 (2001).

Collisional Phase Shift

With N atoms one obtains maximallyentangled cluster states from such a sequence !

( ) ( 1)1 1( ) exp2 2

j jz z

jU i σ σφ φ

+⎛ ⎞+ −= −⎜ ⎟

⎝ ⎠∑

Collapse and Revival of the Ramsey fringe

One atom per site

D. Jaksch, PhD-Thesis, Innsbruck


One atom per site


( ) ( ) 1 1 10 0 0

BELL

BELL c

φ π ψ= ⇒ =

= +−⊗ ⊗ +⋅


One atom per site


( ) ( ) 1 1 10 0 0

BELL

BELL c

φ π ψ= ⇒ =

= +−⊗ ⊗ +⋅

Measured

Entanglement Dynamics I

The gate operation does not cause a loss of visibility!

α

β

c

Conditional Double-Slit

Conditional Double-Slit Conditional Double-Slit


Before π/2-Puls: ( )1 1 1 1 212 2

10 0 10 01 1i i i

ii ii i i

e ϕ+− + + +−

+ + +


( ) ( ) 214 21 1

0 00 0 0 0ii i iii i

e ϕ− +++

⊗ + + ⊗ +After π/2-Puls,detection in :0

Entanglement Dynamics II

30 µs 60 µs 90 µs 120 µs 150 µs

180 µs 210 µs 240 µs 270 µs 300 µs

330 µs 360 µs 390 µs 420 µs 450 µs

Entanglement Dynamics II

35%

47%

36%

23%

16%

7.6 % 6.5 % 5.3 % 4.7 % 4.7 %

Visibility for φ=2n π decreases for increasing n due to dephasing / decoherence.Visibility for φ=2 (n+1) π never falls below 4 %; best data for first collapse andmaximum entanglement: V= 4.5 %.

Outlook

Experiments here: 3600 copies of one dimensional arraysof 60 atoms

Straightforward extension: 2D or 3D entangled clusterstates with up to 216.000 trapped atoms !!!

H.-J. Briegel & R. Raussendorf PRL 86, 910 (2001) & PRL 86, 5188 (2001).


•Introduction





Part 2



Part 3




Controlling the effective interaction

222, 3

,

4( ) ( )i j

i j i j

at d x w x w x

mπ

φ⋅ ⋅

= ⋅ ∫h

( )0 110100 1 1 1 112

00 0 10i i i ie e e eφ φ φ φ+ + +

Two atoms in a coherent superposition of the internal states:

Controlling the effective interaction

222, 3

,

4( ) ( )i j

i j i j

at d x w x w x

mπ

φ⋅ ⋅

= ⋅ ∫h

( )0 110100 1 1 1 112

00 0 10i i i ie e e eφ φ φ φ+ + +

Entanglement evolution for 00 11 012 0φ φ φ+ − ≠

( ) 0221

101100 ≠−+≡ UUUχ

Cf.: A. Sørensen et al., Nature 409, 63 (2001), see also further work of D. Jaksch and L. You

Two atoms in a coherent superposition of the internal states:

Hyperfine Feshbach resonance for 87Rb

M. Erhard et al., PRA 69, 032705 (2004)

1, 1 2, 1 :F FF m F m= = + + = = −

Extracted from: E.G.M. van Kempen et. al, PRL 88 #9, 093201 (2002)

State preparation

Starting from Mott-insulating phase with one and two atoms per site

State preparation

0

1

Cf. Work of E. Cornell (JILA)

π/2

Feshbach resonance I – Loss channel

B

timeπ/2

τ

Magnetic field values calibratedby microwave transitions

Measured position9.121(9) G

Theoretical prediction: E.G.M. van Kempen et. al, PRL 88 #9, 093201 (2002)Experimental detection of loss channel: M. Erhard et al., Phys. Rev. A 69, 032705 (2004)

Measurement in 2D lattice configuration

Elastic channel: Entanglement dynamics

B

time

τ

π/2 π/2

Variable Phase α

Ramsey fringe:State selective detection of relative atom numberversus microwave phase α

Use of spin echo techniques optional

Phase α

Two atoms per site

0 0⊗

Two atoms per site

0 0⊗ ( ) ( )12

0 01 1+ ⊗ +π/2

Two atoms per site

( )00 01 10 111 1 112

0 0 0 10i i i ie e e eφ φ φ φ+ + +

0 0⊗ ( ) ( )12

0 01 1+ ⊗ +π/2

Two atoms per site

( )00 01 10 111 1 112

0 0 0 10i i i ie e e eφ φ φ φ+ + +

0 0⊗ ( ) ( )12

0 01 1+ ⊗ +π/2

01 10φ φ=

00 11φ φ≈

00 10φ φ φ∆ ≡ −( )00 1 1 10

20 0 01 1ii iee eφφ φ∆ ∆+ + +Global phase

Two atoms per site

( )00 01 10 111 1 112

0 0 0 10i i i ie e e eφ φ φ φ+ + +

( )11 1 12

0 0 10 0i ie eφ φ∆ ∆+ + +

0 0⊗ ( ) ( )12

0 01 1+ ⊗ +π/2

π/2 (Phase α) ( ) ( )1 11 1 12

0 0 i ie eφ φ∆ ∆⎡ ⎤⋅ + + ⋅ −⎣ ⎦

Entanglement dynamics

( ) ( )1 11 1 12

0 0 i ie eφ φ∆ ∆⎡ ⎤⋅ + + ⋅ −⎣ ⎦

∆φ = 0 (2π,...):

∆φ = π (3π,...):

∆φ = π/2 (3π/2,...):

0 0

1 1

( ) ( )( )1 1 1 10 02

i i+ ⋅ + − ⋅

For one special α:


( ) ( )1 11 1 12

0 0 i ie eφ φ∆ ∆⎡ ⎤⋅ + + ⋅ −⎣ ⎦

∆φ = 0 (2π,...):

∆φ = π (3π,...):

∆φ = π/2 (3π/2,...):

0 0

1 1

( ) ( )( )1 1 1 10 02

i i+ ⋅ + − ⋅

Signal of Ramsey experimentwith two atoms per site

α

For one special α:

Entanglement evolution

Scetch of optical lattice:

0=∆φ



2/πφ =∆



πφ =∆

Lattice sites with one and two atoms separated without destruction


+ =N2 x Atom pairsN1 x Isolated atoms Total signal (rescaled)

Increasing interaction phase corresponds to longer interaction time

∆φ=0

∆φ=π/2

∆φ=π

Experimental results

B = 9.081 G

2π

Measured in experiment

Can be measured using methods of:Greiner et al., Nature 419, 51 (2002)

Can be extracted

A. Widera et al., PRL 92, 160406 (2004)

(a)

(a)

(b)

(b)

(c)

(c)

Measurement of elastic channel

Measured data; Not sensitive to sign of scattering length

Fitresults:Position: 9.128(9) GWidth: 15(4) mG

A. Widera et al., PRL 92, 160406 (2004)

Fit to dispersive profile;sign of scattering lengthincluded

( ), 00 11 011 22sa a a aχ∆ = − + −






Part 2



Part 3




Microarray of Identical Two-Particle Systems

Great environment to form molecules !

Ultracold molecules via Feshbach resonances:C. Wieman, D. Jin, Ch. Salomon, R. Grimm, G. Rempe, W. Ketterle

Starting from a Mott insulator phase with exactly two atoms per lattice site.

The motion is described by :

The motion can be separated in center-of-mass ( ) motion and relative coordinate ( ) motion:

PA in Optical Lattices

Rur

rr

2 22 2.

1 (| |)2 2rel rH r V rµω

µ= − ∇ + +r

r rh

2 22 2. .

12 2c m RH M RM

ω= − ∇ +ur

urh

2 2 22 2 21 2 2 11 2

1( ) ( ) (| |)2 2

H m r r V r rm

ω= − ∇ + ∇ + + + −r r r rh

Advantages of Photoassociation in Optical Lattices

• Possibility to create a M ott insulator with exactly two atoms per lattice site. A complete microscopic understanding of the two-atom dynamics and PA is possible.

• Isolated single molecules

• Higher Franck-Condon factor thanks to a bound-bound transition instead of a free-bound transition.

• High atomic densities without 2-and 3-body decay

• No inelastic molecule-molecule collisions

• Complete control of internal and external degrees of freedom

• No mean-field shifts and broadening of linewidth

3/ 41 trap

ωΩ ∝ Ω ∝

1. An atomic BEC is loaded into an optical lattice

2. The lattice depth is increased to create a MI with exactly two particles per site

3. A molecular MI is produced by two-color PA under tight trapping conditions

4. By decreasing the lattice depth the MI is “melted” and a molecular BEC is created in a quantum phase transition

Creating a Molecular BEC by Melting a Mott Insulator

Proposal: J. I. Cirac, P. Zolleret al. PRL Vol. 89, 040402-1 (2002)

Using a sequence of Raman lasers to reach the vibrational ground state with high efficiency

The Experimental Setup

Solution:

LA LB phase locked

The Experimental SetupMobile Raman laser system :

Measurement: 1-Color PA

Transition into the molecular excited electronic state. The moleculeproduction is detected via the atom loss.

Trap Loss as a Probe of the Occupation Numbers

3D lattice

1D lattice

Molecule Formation via Two-Photon Photoassociation- Controlling the Internal Degrees of Freedom -

A selection of groups performing PA experiments:D. Heinzen, P. Pillet, J. Weiner, M. Leduc & C. Cohen-Tannoudji, W.D. Phillips, R. Hulet

Ultracold molecules can also be created through Feshbach resonances:D. Jin, Ch. Salomon, R. Grimm, G. Rempe, W. Ketterle

center-of-mass motion

Resolving the External Quantum States

The Raman spectrum additionally resolves the quantized vibrational levels of the trapped molecules.

2 22 2.

1 (| |)2 2rel rH r V rµω

µ= − ∇ + +r

r rh 2 22 2. .

12 2c m RH M RM

ω= − ∇ +ur

urh

relative motion

Full Control over all Internal and External Quantum Degrees of a Chemical Reaction Should be Possible

By resolving the vibrational quantum state of the center of mass coordinate, one should be able to control the external degrees of freedomas well.

Ground State Molecules

Molecules in 1st excited state

Molecules in 2nd excited state

Final Quantum State is Now Completely Controllable !

Photoassociation to the last bound vibrational state No Mean Field Broadening !

High Resolution Spectroscopy at High Densities: Peak density 2*1015 cm-3

Compare e.g. Dan Heinzen experiments : Peak density 2*1014 cm-3

Linewidth 1.1 kHz

Autler-Townes Splitting

Autler-Townes splitting induced on bound-bound transitions allows us to measure coupling strengths!

see also work by C. Zimmermann

Linewidth of the Raman Transitionand Lifetime of the Ground State Molecules

Lifetime limiting effects:1. Spontaneous Raman scattering

2. Collisionaldecay

3. Resonant light (ASE, lattice lasers)

Line broadening: If the polarizabilityof the molecules is not twice theatomic one, the inhomogeneous external trapping potential leads to a differential energy shift.

1 2. 2eff

Ω ⋅ΩΩ =

∆

Conclusion & Outlook

• Control quantum matter at the ultimate limit• Engineer novel many particle quantum states with

fundamental and practical applications (e.g. precision metrology)• Understand and explore complex quantum matter

(use this enhanced knowledge to build new materials)• Quantum computers

(need to control approx. 10000 qubits, when ???)• Quantum simulators to solve tough problems in condensed

matter physics e.g. high-Tc-superconductivity (seem to be more within reach over the next few years)

• Engineer complex matter (molecules) in a bottom-up-approach in precisely defined quantum states

Introduction 2 Short Resume of Work with the Mott · PDF file• State Selective Production...

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Transcript of Introduction 2 Short Resume of Work with the Mott · PDF file• State Selective Production...