Ana Maria Rey

March Meeting Tutorial May 1, 2014

• Brief overview of Bose Einstein condensation in dilute ultracold gases

• What do we mean by quantum simulations and why are ultra cold gases useful

• The Bose Hubbard model and the superfluid to Mott insulator quantum phase transition

• Exploring quantum magnetisms with ultra-cold bosons

High temperature T: Thermal velocity v Density d-3

“billiard balls”

Ketterle

Low temperature T: De Broglie wavelength

DB=h/mv~T-1/2

“Wave packets”T=Tcrit : Bose Einstein

Condensation De Broglie wavelength DB=d “Matter

wave overlap”T=0 : Pure Bose Condensate

“Giant matter wave ”

In 1995 (70 years after Einstein’s prediction) teams in Colorado and Massachusetts achieved BEC in super-cold gas.This feat earned those scientists the 2001 Nobel Prize in physics.

S. Bose, 1924

Light

A. Einstein, 1925

Atoms

E. Cornell

W. Ketterle C. Wieman

Using Rb and Na atoms

• A BEC opened the possibility of studying quantum phenomena on a macroscopic scale.

• Ultra cold gases are dilute

1

2

3/13/22

.int nma

mn

na

E

Es

s

kin

How can increase interactions in cold atom systems? 1. increase as: Using Feshbach resonances

2. Increase the effective mass m m*

as: Scattering Length

n: Density

*

*

Cold gases have almost 100% condensate fraction: allow for mean field description

Periodic light shift potentials for atoms created by the interference of multiple laser beams.

Two counter-propagating beams

Standing wave

)(4

)( 22

kxSinxV o

|e

h|g

2

4

d ~

a=/2

~Intensity

Perfect Crystals

Quantum Simulators Quantum

Information

• Precision Spectroscopy

• Polar Molecules

• Scattering Physics e.g. Feshbach resonances

• Bose Hubbard and Hubbard models

• Quantum magnetism

• Many-body dynamics

• Quantum gates

• Robust entanglement generation

• Reduce Decoherence

AMO Physics

Single particle in an Optical latticeSingle particle in an Optical lattice

q: Quasi-momentum –k/2≤ |q| ≤ k/2

n: Band Index

Solved by Bloch Waves

Effective mass1

2

22*

dq

Edm

m* grows with lattice depth

k=2 /a Reciprocal lattice vector

2

k

2

k

2

k

2

k2

k2

kRecoil Energy: ћ2k2/(2m)

Single particle in an Optical latticeSingle particle in an Optical lattice

Wannier Functions

localized wave functions:

Bloch FunctionsV=0 V=0.5 Er

V=4 Er V=20 Er

R

oispi E

VxxwHxxwdxJ

4exp)()(

2

1003

And expand in lowest band Wannier states Assuming: Lowest band, Nearest neighbor hopping

)()()(][sin2

22

22

xt

ixxVkxVxm o

We start with the Schrodinger Equation

jjjjj xVJi )()( 11

iqjaj Ae ]cos[2)( qaJqE

If V=0 Cosine spectrum

)(0 ii

i xxw

Band width = 4 J

M. Greiner

Idea: Use one physical system to model the behavior of another with nearly identical mathematical description.

Important: Establish the connection between the physical properties of the systems

Richard Feynman

We want to design artificial fully controllable quantum systems and use them to simulate complex quantum, many-body behavior

What can we simulate with cold atoms?• Bose Hubbard modelsQuantum phase transitions• Fermi Hubbard modelsCuprates, high temperature superconductors,• Quantum magnetism• …

We start with the full many-body Hamiltonian and expand the field operator in Wannier states

)(ˆˆ

0 jj

j xxwa Assuming: Lowest band, Short -range interactions, Nearest neighbor hopping

40

32

|)(|2

4xwdx

m

aU s

H=-J<i,j> âi† âj

External potential

Interaction EnergyHopping Energy

+ U/2 j âj† â†

j âj âj + j (Vj –âj† âj

J

j+1

j

Uw0(x)

V

JUkT ,, )()( 1

3ispi xwHxwdxJ

D. Jacksh et al, PRL, 81, 3108 (1998)

M.P.A. Fisher et al.,PRB40:546 (1989)

Superfluid phase

Mott insulator phase

Weak interactions

Strong interactions

41n

U

2

0

Mottn=1

n=2

n=3

Superfluid

Mott

Mott

UJ

JnU

nJU

n=1

Superfluid – Mott Insulator Superfluid Mott InsulatorQuantum phase transition: Competition between kinetic and

interaction energyShallow potential: U<<J Deep potential: U>>J

• Weakly interacting gas • Strongly interacting gas

SuperfluidMott insulator

Superfluid Mott Insulator

• Poissonian Statistics

0|)ˆ(!

10|)ˆ(

!

1|

0

N

i

NSF i

aN

bN

tt

• Condensate order parameter

• Off diagonal long Range Order

ieNbb 00ˆˆ

j

nj

MIn

a0|

!

)ˆ(|

t

• Atom number Statistics

• No condensate order parameter

• Short Range correlations

0ˆ 0b

ijaa ji ˆˆ t

• Gapless excitations • Energy gap ~ U

naa iji j ˆˆlim ||

t

• Step 1: Use the decoupling approximation

• Step 2: Replace it in the Hamiltonian

z: # of nearest neighbor sites• Step 3: Compute the energy using as a perturbation parameter and minimize respect to

Mott: E(2) >0

En

erg

y

SF: E(2) < 0

En

erg

y

E(2) = 0Critical point

Van Oosten et al, PRA 63, 053601 (2001)

t=0 Turn off trapping potentials

Imaging the expanding atom cloud gives important information about the properties of the cloud at t=0:

Spatial distribution -> Momentum distribution after time of flight at t=0

0,0)(~)( nGQt

Qnt

mxn

In the lattice at t=0

After time of flight

2||x

am

th

σ(t)= tħ/(mσo)

2||

x a

|G|=

j

ii Rxwenx )0,()0,( 0

j

iQRi jetxwentx ),(),( 0

Lattice depth : Laser Intensity

Superfluid Mott insulatorQuantum Phase transition

Markus Greiner et al. Nature 415, (2002);

shallow deep shallow

The loss of the interference pattern demonstrates the loss of quantum phase

coherence.

Optical lattice and parabolic potential

41n

U

2

0

Mottno=1

no=2

no=3

Superfluid

Mott

Mott

UJ

20 ii

ultracold.uchicago.edu

Observing the Shell structure

Spatially selective microwave transitions and spin changing collisions

S. Foelling et al., PRL 97:060403 (2006) G. Campbell et al, Science 313,649 2006

S. Waseems et al Science, 2010

J. Sherson et al : Nature 467, 68 (2010).

Also N. Gemelke et al Nature 460, 995 (2009)

Why are some materials ferro or anti- ferromagneticA fundamental question is whether spin-independent interactions e.g. Coulomb fources, can be the origin of the magnetic ordering observed in some materials. • Study role of many-body interactions in quantum systems:

Non-interacting electron systems universally exhibit paramagnetism

• Useful applicationsFerromagnetic RAM Magnetic Heads High Tc

Superconductivity

Exchange interactions

Basic Idea

Singlet

Triplet

Effective spin-spin interactions can arise due to the interplay between the SPIN-INDEPENDENT forces and EXCHANGE SYMMETRY

En

erg

y

• Exchange Direct overlap

Experimental Control of Exchange Interactions

)( 21 SSVH exex

21

20

32

|)(||)(|2

8xwxwdx

m

aVex

M. Anderlini et al. Nature 448, 452 (2007)

Spin : |0=|F=1,mF=0

|1=|F=1,mF=-1 Singlet < Triplet

Orbitals: Two bands g and e

w0 w1

Superimpose two lattices: one with twice the periodicity of the other

Adjustable bias and barrier depth by changing laser intensity and phase

Experimental Control of Exchange Interactions

Measured spin exchange: using band-mapping techniques and Stern-Gerlach filtering

Prepare a superposition of singlet and triplet

Spin dynamics

Experimental Control of Exchange Interactions

Super-Exchange Interactions

Super- Exchange

Virtual processes

E.g. Two electrons in a hydrogen molecule, MnO

Singlet

Triplet

En

erg

y

P.W. Anderson, Phys. Rev. 79, 350 (1950)

Mn O

J lifts the degeneracy: An effective Hamiltonian can be derived using second order perturbation theory via virtual particle hole excitations

Consider a double well with two atoms

At zero order in J , the ground state is Mot insulator with one atom per site and all spin configurations are degenerated

J

Super-exchange in optical lattices

J

U

JJ2 , ,

,0

0,

Super-exchange in optical lattices

For spin independent parameters

LRexeff SSJH

2 UJ

J ex

22

- Bosons , + Fermions

Reversing the sign of super-exchange

Add a bias:

22

222

422

U

UJ

U

J

U

JJex

2>U implies Jex<0

S. Trotzky et. al , Science, 319,295(2008)

Two bosons in a Double Well with STwo bosons in a Double Well with Szz=0=0

2 singly occupied configurations:

2 doubly occupied configurations: (2,0)|t , (0,2)| t

(1,1)|s , (1,1)|t

(0,2)(2,0)

(1,1))(s

2

1

)(t 2

1

Singlet

Triplet

Only 4 states:

(0,2)| S

Vibrational spacing o>>U,J

Energy levels in symmetric DW: »UGood basis:

|s, |- are not coupled by J. They have E=0,U for any J.

|t, |+ are coupled by J: Form a 2 level system

In the U>>J limit

ħ1~ 4J2/U: Super-exchange

ħ2~ U

2,)1,1(

st t

2

0220,|

ħ

ħ

|-

|t + ’|+

U

|++ |t

|s

Magnetic field gradientIn the limit U>>J, only the singly occupied states are populated and they form a two level system: |s= | and |t=|

• If |B|« Jex then | s and | t are the eigenstates

• If |B |» Jex then | ↓↑ and |↑↓ are the eigenstates

0B

BJH ex

A Magnetic field gradient couples | and |

zBBB LR ˆ)(

Experimental Observation

t M

Measure spin imbalance

Prepare|↑↓

|B|>0

Turn of B

Evolve

Nz: # atoms |↑↓ - # atoms |↓↑

S. Trotzky et. al , Science, 319,295(2008)

|s

|↑↓

|tz

In the limit J<<U,

),()( 21 ftN z

)tJcos()t(N exz 2

Simple Rabi oscillations

Measuring Super-exchange

V=6Er

V=11 Er

V=17 Er

Two frequencies

Almost one frequency

Comparisons with B. H. Model

2Jex

Shadow regions: 2% experimental lattice uncertainty

Extended B. H. Model

?

Direct experimentaltest of condensedmatter models:Great success and a lot of new challenges