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”
Low temperature T:
De Broglie wavelength
lDB=h/mv~T-1/2
“Wave packets”
T=Tcrit : Bose Einstein Condensation
De Broglie wavelength
lDB=d
“Matter wave overlap”
T=0 : Pure Bose Condensate
“Giant matter wave ”
Ketterle
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
C. Wieman
Using Rb and
W. Ketterle
Na atoms
• A BEC opened the possibility of studying quantum phenomena
on a macroscopic scale.
a : Scattering Length
s
• Ultra cold gases are dilute
n: Density
Eint .
a n
* n1 / 3  1
 2 s 2 / 3  ma
s
 n
Ekin
2m*
Cold gases have almost 100% condensate fraction: allow
for mean field description
How can increase interactions in cold atom systems?
1. increase as: Using Feshbach resonances
2. Increase the effective mass m m*
One way to achieve 2. is with an optical lattice
Periodic light shift potentials for atoms created by the
interference of multiple laser beams.
|e
|g

hn
Two counter-propagating beams
 
 ~  d
Standing wave
 2 ~Intensity
4
o
V ( x) 
Sin2 (kx)
4
2
a=l/2
Perfect Crystals
AMO Physics
• Precision Spectroscopy
• Polar Molecules
• Scattering Physics e.g.
Feshbach resonances
Quantum
Simulators
• Bose Hubbard and
Hubbard models
Quantum
Information
• Quantum gates
• Quantum magnetism
• Robust entanglement
generation
• Many-body dynamics
• Reduce Decoherence
Single particle in an Optical lattice
Solved by Bloch Waves
q: Quasi-momentum
–k/2≤ |q| ≤ k/2
n: Band Index
k=2 p/a Reciprocal lattice vector
Recoil Energy:
ћ2k2/(2m)

k
2
k
k

2
2
 k k

2
2
Effective mass
d E

m   
2 
 dq 
2
*
2
1
m* grows with lattice depth
k
2
Single particle in an Optical lattice
Bloch Functions
V=0
V=4 Er
Wannier Functions
localized wave functions:
V=0.5 Er
V=20 Er
We start with the Schrodinger Equation
 2 2


2
 
 Vo sin [kx]  V ( x)  ( x)  i  ( x)
2
t
 2m x

And expand Y in lowest band Wannier states
Y   i w0 ( x  xi )
i
Assuming: Lowest band, Nearest neighbor hopping
i j  J ( j 1  j 1 )  V ( x j ) j
2

p
J    dx3 w0 ( x  xi ) H sp w0 ( x  xi 1 )  exp 
 4
Vo
ER
Cosine spectrum
If V=0
 j  Ae
iqja
E (q)  2 J cos[qa]
Band width = 4 J




Vortices
Quantum
Simulation
Coherence
Weakly
interacting
Bose Gas
Strongly
interacting
Bose Gas
Quantum
Information
Superflow
Non
Linear
optics
M. Greiner
Quantum
Phase
transitions
New states
of matter
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
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 models
Quantum phase transitions
• Fermi Hubbard models
Cuprates, high temperature
superconductors,
• Quantum magnetism
•…
Richard Feynman
We start with the full many-body Hamiltonian and expand the field
operator Y in Wannier states
ˆ   aˆ w ( x  x )
Y
j 0
j
j
Assuming:
Lowest band, Short -range interactions, Nearest neighbor hopping
H=-J<i,j> âi† âj
+ U/2 j âj† â†j âj âj + j (Vj –m)âj† âj
Hopping Energy
External
potential
Interaction Energy
J

w0(x)
U
j
  kT ,U , J
j+1
V
4pas  2
U
2m

dx3 | w0 ( x) |4
J   dx3w( xi ) H sp w( xi1 )
D. Jacksh et al, PRL, 81, 3108 (1998)
m U
M.P.A. Fisher et al.,
PRB40:546 (1989)
n=3 Mott
4
n 1
n=2
Mott
Superfluid
n=1
2
n=1
0
Mott
J U
U  n J
U  nJ
Superfluid phase
Weak interactions
Mott insulator phase
Strong interactions
Superfluid
–
Mott Insulator
Quantum phase transition: Competition between kinetic and interaction
Deep potential: U>>J
energy
Shallow potential: U<<J
• Weakly interacting gas
Superfluid
• Strongly interacting gas
Mott insulator
Superfluid
 | SF
1 ˆt N
1

(b0 ) | 0 
( aˆ ti ) N | 0
N!
N! i
Mott Insulator
 |  MI  
(aˆ tj )
j
n
n!
|0
• Poissonian Statistics
• Atom number Statistics
• Condensate order parameter
• No condensate order parameter
i
ˆ
ˆ
b0  b0  N e
• Off diagonal long Range Order
lim|i  j|  aˆ tj aˆi  n
• Gapless excitations
bˆ0  0
• Short Range correlations
aˆit aˆ j   ij
• Energy gap ~ U
• 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 .
SF: E(2) < 0
E(2) = 0
Critical
point
Energy
Energy
Mott: E(2) >0


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
after time of flight
->
Momentum distribution
at t=0
In the lattice at t=0
 ( x,0)   nei w0 ( x  Ri ,0)
so
| |
2
x
j
a
After time of flight
σ(t)= tħ/(mσo)
| |
 ( x, t )   ne w0 ( x, t )e
2
x
n( x )
|G|=
t
th
am
m
~
n(Q)
t
i
j
0
  Q  nG , 0
iQR j
Superfluid
Mott insulator
Quantum Phase transition
Lattice depth :
Laser Intensity
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
m U
4
2
0
mi  m0  i 2
nn o=3
Mott
1
no=2 Mott
no=1
Superfluid
Mott
J U
ultracold.uchicago.edu
Observing the Shell structure
S. Foelling et al., PRL 97:060403 (2006)
Spatially selective microwave transitions
and spin changing collisions
G. Campbell et al,
Science 313,649 2006
J. Sherson et al : Nature 467, 68
(2010).
S. Waseems et al Science, 2010
Also N. Gemelke et al Nature 460,
995 (2009)
Why are some materials ferro or anti- ferromagnetic
A 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 applications
Ferromagnetic RAM
Magnetic Heads
High Tc
Superconductivity
Exchange interactions
Effective spin-spin interactions can arise due to the interplay between
the SPIN-INDEPENDENT forces and EXCHANGE SYMMETRY
f2
• Exchange
Direct overlap
Basic Idea
Energy
Triplet
Singlet
f1
Experimental Control of Exchange Interactions
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
 
Hex  Vex (S1  S2 )
8pa 2
Vex 
2m

dx3 | w0 ( x) |2 | w1 ( x) |2
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
Prepare a
superposition of
singlet and triplet
Measured spin
exchange:
using band-mapping
techniques and SternGerlach filtering
Experimental Control of Exchange Interactions
Spin
dynamics
Super-Exchange Interactions
• Spin order can arise even though the wave function overlap
is practically zero.
Virtual processes
Super- Exchange
f1
f2
Triplet
Singlet
Energy
E.g. Two electrons in a hydrogen molecule, MnO
Mn
O
P.W. Anderson, Phys. Rev. 79, 350 (1950)
Super-exchange in optical lattices
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 lifts the degeneracy: An effective Hamiltonian can be derived
using second order perturbation theory via virtual particle hole
excitations
J
J
0, 
2J J
,   
U 
, 
 ,0
Super-exchange in optical lattices
For spin independent parameters
H eff
 
 2 J ex S R  S L
- Bosons , + Fermions
J ex 
2J 2
U
Reversing the sign of super-exchange
Add a bias:
Jex 
J2
U  2

J2
U  2

J2U
U2  42
2>U implies Jex<0
S. Trotzky et. al , Science, 319,295(2008)
Two bosons in a Double Well with Sz=0
Only 4 states:
Vibrational spacing o>>U,J
 2 singly occupied configurations:
(1,1)
(1,1)|s , (1,1)|t
s 
1
(    )
2
Singlet
t 
1
(    )
2
Triplet
2 doubly occupied configurations: (2,0)|t , (0,2)| t
(0,2)| S
(2,0)
(0,2)
o
Energy levels in symmetric DW: o»U
Good basis:
( 1,1) t ,s 
  
2
| , 
20  02
2
t
|s, |- are not coupled by J. They have E=0,U for any J.
 |t, |+ are coupled by J: Form a 2 level system
|++ a|t
|-
ħ2
ħ1
U
|s
|t + a’|+
In the U>>J limit
ħ1~ 4J2/U: Super-exchange
ħ2~ U
Magnetic field gradient
In the limit U>>J, only the singly occupied states are populated
and they form a two level system: |s= | and |t=|
A Magnetic field gradient couples | and |
B ( BR  BL )zˆ
  J ex
H  
 B
B 

0 
• If |B|« Jex then | s and | t 
• If | B |» Jex
are the eigenstates
then | ↓↑  and |↑↓ are the eigenstates
Experimental Observation
S. Trotzky et. al , Science, 319,295(2008)
Prepare |↑↓
|B|>0
Turn of B
M
t
Evolve
Measure spin imbalance
Nz:
|s
# atoms |↑↓ - # atoms |↓↑
N z( t )  f( 1,2 )
In the limit J<<U,
|↑↓
|tz
Nz (t )  cos(2Jex t )
Simple Rabi oscillations
Measuring Super-exchange
Two frequencies
V=6Er
V=11 Er
V=17 Er
Almost one frequency
Comparisons with B. H. Model
2Jex
Shadow regions: 2%
experimental lattice
uncertainty
Extended B. H. Model
Real Materials:
• Complicated
• Disorder
Direct experimental
test of condensed
matter models:
Great success and a
lot of new challenges
?
Condensed
matter models:
Difficult to
calculate
Cold atoms in
optical lattice:
Clean realization
of CM models