Transcript Fundamental Issues of Quantum Gases

Quantum Gases:
Past, Present, and
Future
Jason Ho
The Ohio State University
Hong Kong Forum in Condensed Matter
Physics: Past, Present, and Future
HKU and HKUST, Dec 18-20
Where we stand
What’s new
Fundamental Issues
Challenges
A decade since discovery of BEC :
Still expanding rapidly
Discoveries of new systems, new phenomena, and new technique
keep being reported in quick succession.
Highly interdisciplinary -- (CM, AMO, QOP, QI, NP)
New Centers and New Programs formed all over the world.
England, Japan, Australia, CIAR, US (MURI&DARPA)
Puzzling phenomena being to emerge in fermion expts
Worldwide experimental effort to simulate strongly
correlated CM systems using cold atoms
J=1/2
alkali atoms
Bosons and Fermions with large spins
J
I
e
Spin F=1, F=2 bosons:
F=I+J
Hyperfine spin
Spin F=1/2, 3/2, 5/2, 7/2, 9/2 fermions
Magnetic trap
Atoms “lose” their spins!
B
Spinless bosons and
fermions
Magnetic trap
Mixture of quantum gases:
Pseudo-spin 1/2 bosons:
B
Ho and Shenoy, PRL 96
D.S. Hall, M.R. Matthews, J. R. Ensher, C.E. Wieman,
and E.A. Cornell PRL 81, 1539 (1998)
Optical trapping:
Focused
laser
BEC or cold fermions
All spin states are trapped,
Spin F=1, F=2 bosons:
T.L.Ho, PRL 1998
Spin F=1/2, 3/2, 5/2, 7/2, 9/2 fermions
Condensed Matter Physics
Quantum
Optics
BEC
Quantum
Gases
Quantum Information
Atomic Physics
Nuclear
Physics
High Energy Physics
Quantum Gases
system
B
BB
BF
FF
F
symmetry
U(1)
Magnetic
trap, spins
frozen
environments
single
trap
stationary
 0
na 3  1
2D

S0(3)
Optical
trap, spins
released
3D
interaction
1D

fast
rotating
  trap
lattice
0D


na 3  1
1996 Discovery of BEC!
1997 Mixture of BEC and pseudo spin-1/2
Condensate interference collective modes
1998 Spin-1 Bose gas (Super-radiance)
Bosanova Bragg difffration, super-radience,
1999
solitons
Superfluid-Mott oscillation
Low dimensional Bose gas
(Vortices in 2-component BEC)
2000 (Vortices in BEC, Slow light in BEC)
2001 Fast Rotating BEC, Optical lattice,
BEC on Chips
2002 Quantum degenerate fermions (Spin dynamics
of S=1/2 BEC, Coreless vortex in S=1 BEC,
evidence of universality near resonance)
2003 Molecular BEC, (Spin dynamics of S=1 BEC, noise measurements)
2004 Fermion pair condensation! (pairing gap, collective mode)
BEC-BCS crossover,
2005 Vortices in fermion superfluids, discovery of S=3 Cr Bose condensate,
observation of skymerion in S=1 Bose gas.
2006 Effect of spin asymmetry and rotation on strongly interacting Fermi gas.
Boson-Fermion mixture in optical lattices.
New Bose systems: “spin”-1/2, spin-1, spin-2 Bose gas,
Molecular Bose gas. (BEC at T=0)
Un-condensed Bose gas: Low dimensional Bose gas,
Mott phase in optical lattice
Strongly Interacting quantum gases:
Atom-molecule mixtures of Bosons near Feshbach resonance
Fermion superfluid in strongly interacting region
Strongly interacting Fermions in optical lattices
Possible novel states:
Bosonic quantum Hall states,
Singlet state of spin-S Bose gas,
Dimerized state of spin-1 Bose gas on a lattice.
Fermion superfluids with non-zero angular momentum
Often described as experimental driven,
but in fact theoretical ideas are crucial.
Bose and Einstein, Laser cooling, Evaporative cooling
What is new ?
A partial list:
Bosons and Fermions with large spins
Fast Rotating Bose gases
Superfluid Insulator Transition in optical lattices
Strongly Interacting Fermi Gases
Question:
How do Bosons find their ground state?
Question:
How do Bosons find their ground state?
Conventional Bose condensate :
all Bosons condenses into a single state.
What happens when there are several degenerate
state for the Bosons to condensed in?
G: Number of degenerate states
N: Number of Bosons
What happens when there are several degenerate
state for the Bosons to condense in?
G: Number of degenerate states
Pseudo-spin 1/2 Bose gas: G =2
N: Number of Bosons
G: Number of degenerate states
Spin-1 Bose gas : G=3, G<<N
N: Number of Bosons
G: Number of degenerate states
Spin-1 Bose gas : G=3, G<<N
Bose gas in optical lattice: G ~N
N: Number of Bosons
G: Number of degenerate states
Spin-1 Bose gas : G=3, G<<N
Bose gas in optical lattice: G ~N
Fast Rotating Bose gas: G>>N
N: Number of Bosons
Effectofofspin
spin degeneracy
degeneracy onon
BEC
Effect
BEC
Spin-1 Bose Gas
A deep
harmonic
trap


a
1,0,1

Only the lowest harmonic state is occupied
=> a zero dimensional problem
Spin dynamics of spin-1 Bose gas
Spin-1 Bose Gas
A deep
harmonic
trap
H  cS
2
Hilbert space


Effectofofspin
spin degeneracy
degeneracy onon
BEC
Effect
BEC
Spin-1 Bose Gas
A deep
harmonic
trap

a
1,0,1
Ax  (a1  a1 )/ 2 Ay  (a1  a1 ) / 2i Az  a0

 Under spin rotation, a  (eiS a) rotates
like a 3D Cartesean vector Ai  R()ij A j .


R() : 3D rotation

H  cS
2
C>0
Conventional
condensate :
 S  0
N 0  0, N1  N /2

2
N1
~N
Ax  (a1  a1 )/ 2 Ay  (a1  a1 ) / 2i Az  a0


H  cS
C>0
2
Exact ground state :
  2a1 a1  a02 =
| S  0   N / 2 | 0 
1 0  0 
N 


 a a  0 1 0 
3 

0 0 1 
N0  N1  N1  N /3
2
N1
~N
2
Ho and Yip, PRL, 2004

Relation between singlet state and coherent state
z
Average the coherrent state over
all directions
Because
2
N1
The system is
easily damaged
~N
2
y
x
Transformation of singlet into coherent states as a function of
External field and field gradient:
If the total spin is non-zero
Bosonic enhancement
Transformation of singlet into coherent states as a function of
External field and field gradient:
If the total spin is non-zero
Bosonic enhancement
Transformation of singlet into coherent states as a function of
External field and field gradient:
If the total spin is non-zero
Transformation of singlet into coherent states as a function of
External field and field gradient:
If the total spin is non-zero
With field gradient
S=2
Cyclic state
S=3
Spin biaxial Nematics
A geometric representation : Generalization of Barnett et.al. PRL 06
& T.L.Ho, to be published
Cycle
Tetrahedron S=2
Octegonal S=3
Cubic S=4
Icosahedral S=6
T.L. Ho, to be published
Rotating the Bose condensate
condensate
Generating a rotating quadrupolar field
using a pair of rotating off-centered lasers
K. W. Madison, F. Chevy, W. Wohlleben, J. Dalibard PRL. 84, 806 (2000)
The fate of a fast rotating quautum gas :
Superfluidity ----> Strong Correlation
Quantum Hall
Vortex lattice
Overlap =>
Melting
Normal
In superconductors
Boson
Quantum Hall
Fermion
A remarkable equivalence
Rotating quantum gases
in harmonic traps
Electrons in
Magnetic field
2
p 1
2 2
h
 M r   r  p
2m 2
external rotation
trap
( p  M r ) 1
2
2
2
h
 M (   )r
2M
2
2
( p  M r )
h
2M
2
as
 
No Rotation : Two dimensional harmonic oscillator
  0, E   (n  m)
, n>0, m>0.
E
m
Enm  (  )n  ( )m
E

, n>0, m>0.
As    ,
Angular momentum
states organize into
Landau levels !
m
E


m

Mean field quantum Hall regime:
   in Lowest Landau level
E


condensate
 

m
E

Strongly correlated case:
 interaction dominated
   0

m
E. Mueller and T.L. Ho,
Physical Rev. Lett. 88,
180403 (2002)
Simulate EM field by rotation:
Eric Cornell’s latest experiment
cond-mat/0607697
TL Ho, PRL 87, 060403(2001)
V. Schweikhard, et.al.
PRL 92, 040404 (2004)
(JILA group, reaching LLL)
Fermion quantum Hall
Boson + Fermion
Strongly interacting
Fermi gases
Cooling of fermions
Pioneered by Debbie Jin
Motivation: To reach the superfluid phase
Depends only on density
For weakly interacting Fermi gas
To increase Tc, use Feshbach resonance, since
Holland et.al. (2001)
Weak
coupling
Dilute Fermi Gas
: S-wave scattering length
Normal Fermi liquid
Weak coupling BCS superfluid
What
Happens?
Dilute Fermi Gas
: S-wave scattering length
Normal Fermi liquid
Weak coupling BCS superfluid
Key Properties: Universality (Duke, ENS)
Evidence for superfluid phase:
Projection expt: Fermion pair condensataion -- JILA, MIT
Specific heat -- Duke
Evidence for a gap -- Innsbruck
Evidence for phase coherence -- MIT
BEC -- BCS crossover is the correct description
Largest
Origin of universality now understood
BCS
Molecular
BEC
Universality : A statement about the energetics at resonance
How Resonance Model acquire universality
has to hybridize with many
If
is large -- strong hybridization, then
has relatively little weight in the pair!
Small effect of
means universality !
pairs.
Two channel Model
Single Channel model:
Origin of universality
Scattering amplitude: (from both single and two channel model)
r = effective range
Bruun & Pethick PRL 03
Petrov 04
Diener and Ho 04
Strinati et.al 04
Eric Cornell, email
Question: what happen to scattering on Fermi surface
Wide resonance
Narrow
resonance
In two channel model:
Small closed channel contribution
<=> pair size are given by interparticle spacing
<=>
<=> single channel description ok
<=> universal energy density
Current Development:
•Unequal spin population
•Rotation
Single vortex
Melting of
vortex lattice
c
To quantum
Hall regime
Other possible Fermion superfluids: P-wave Fermion superfluids.
Optiuum phase
a>0
Molecular condensate
Bo
a<0
B
Fermion Superfluid
Ho and Diener, to appear in PRL
Many quantum phenomenon observed:
Condensate interference collective modes
solitons
Bosanova
Bragg difffration,
super-radience,
Superfluid-Mott oscillation
Engineering quantum states in optical lattices,
vortices and spin-dynamics of spin-1/2 Bose gas,
phase fluctuation in low dimensional Bose gas,
spatial fragmention of BEC on chips,
slow light in Bose gases,
large vortex lattice, Skymerion vortices in spin-1 Bose gas,
spin dynamics of spin-1 and spin-2 Bose gas,
dynamics in optical lattices
Unique Capability for Lattice Quantum Gases
•Solid State environment without disorder
•Simulate electro-magnetic field by rotation
•Great Ease to change dimensionality
•Great Ease to change interactions
Major Incentive:
•Observation of Superfluid-insulator transition
-- a QPT in a strongly correlated system
•Realization of Fermion Superfluid using
Feshbach resonance
Exciting Prospects:
•Novel States due to unique degrees of freedom of cold atoms
Bose and Fermion superfluids with large spin
Quantum Hall state with large spin
Lattice gases in resonance regime
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
Superfluid :
Mott :
Superfluid State :
+
+
ODLRO
Superfluid State :
+
+
ODLRO
Mott State
Mott State
Resists addition of boson
require energy U,
hence insulating
Nature, 419, 51-54 (2002)
Observation of Superfluid-insulator transition
Figure 2 Absorption images of multiple matter wave interference patterns.
These were obtained after suddenly releasing the atoms from an optical lattice potential
with different potential depths V0 after a time of flight of 15 ms. Values of V0 were:
a, 0 Er; b, 3 Er; c, 7 Er; d, 10 Er; e, 13 Er; f, 14 Er; g, 16 Er; and h, 20 Er.
M. Greiner et.al, Nature 415, 39 (2002)
M. Greiner, O. Mandel. Theodor, W. Hansch & I. Bloch,Nature (2002)
Phase diagram of Boson-Hubbard Model
Part IC
Current experiments
Expts involving superfluid-insulator transitions:
I. Bloch, et.al, PRA72,
053606 (2005)
Esslinger, PRL 96, 180402 (2006)
Fermions in optical lattice, 2 fermions per site
Ketterle et.al, cond-mat/0607004
Sengstock et.al. PRL 96, 180403 (2006)
F-B mixture
ETH Experiment:
very deep lattice, less than two toms per site
Band insulator
2 to 3 bands occupied
2 atoms per site
0
2 fermions
Per site
Part I: Why cold atoms for condensed matter?
A. Major developments in CM and Long Standing Problems
B. The Promise of cold atoms
C. Current experimental situation
Part II: Necessary conditions to do strongly correlated physics: Quantum
Degeneracy and method of detection:
A. The current method of detecting superfluidity in lattices is misleading
B. B. A precise determination of superfluidity => illustration of far from
quantum degeneracy in the current systems.
Part III: Solid state physics with ultra-cold fermions:
A. Metallic and semi-conductor physics with cold fermions
B. Studying semiclassical electron motions with cold fermions
Part II
Necessary conditions for studying
strongly correlated physics:
* Quantum Degeneracy
* Method of Detection:
Condition for quantum degeneracy
Condition for BEC :
Free space
Lattice
Free space
Optical lattice
Quantum degeneracy
Lowest temperature
attainable:
Current method of identifying superfluidity: sharpness of n(k)
I. Bloch, et.al, PRA72,
053606 (2005)
Esslinger, PRL 96, 180402 (2006)
Fermions in optical lattice, 2 fermions per site
Ketterle et.al, cond-mat/0607004
Sengstock et.al. PRL 96, 180403 (2006)
F-B mixture
However, a normal gas above Tc can also have sharp peak!
Diener, Zhao, Zhai, Ho, to be published.
Current method of identifying superfluidity: sharpness of n(k)
I. Bloch, et.al, PRA72,
053606 (2005)
Esslinger, PRL 96, 180402 (2006)
Fermions in optical lattice, 2 fermions per site
Ketterle et.al, cond-mat/0607004
Sengstock et.al. PRL 96, 180403 (2006)
F-B mixture
Part II
Necessary conditions for studying
strongly correlated physics:
* Quantum Degeneracy
Method of
of Detection:
Detection
** Method
An accurate method for detecting superfluidity:
Visibility
Reciprocal
lattice vector
Not a reciprocal
lattice vector
DZZH, to be published
T=0 visibility
2nd Mott shell
Main message:
Current Experiments in optical lattice are far from
quantum degeneracy
Need new ways to cool down to lower temperature
Need reliable temperature scale
Finite temperature effect becomes
important
More intriguing physics
of quantum critical
behavior can be
expected