ana maria rey march meeting tutorial may 1, 2014
TRANSCRIPT
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