Fundamental Issues of Quantum Gases

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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
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Where we stand Whats new Fundamental
Issues Challenges
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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
(MURIDARPA) Puzzling phenomena being to emerge
in fermion expts Worldwide experimental effort
to simulate strongly correlated CM systems using
cold atoms
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J1/2
alkali atoms
J
Bosons and Fermions with large spins
e
I
Spin F1, F2 bosons
FIJ
Hyperfine spin
Spin F1/2, 3/2, 5/2, 7/2, 9/2 fermions
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Magnetic trap
B
Spinless bosons and fermions
Atoms lose their spins!
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Magnetic trap
B
Mixture of quantum gases
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)
Pseudo-spin 1/2 bosons
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Focused laser
Optical trapping
BEC or cold fermions
All spin states are trapped,
Spin F1, F2 bosons
T.L.Ho, PRL 1998
Spin F1/2, 3/2, 5/2, 7/2, 9/2 fermions
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Atomic Physics
Condensed Matter Physics
BEC
Quantum Optics
Nuclear Physics
Quantum Gases
Quantum Information
High Energy Physics
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Quantum Gases
system
environments
symmetry
interaction
3D 2D 1D 0D
B BB BF FF F
U(1) Magnetic trap, spins frozen S0(3)Optical
trap, spins released
stationary fast rotating
single trap lattice
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• 1996 Discovery of BEC!
• 1997 Mixture of BEC and pseudo spin-1/2
• Condensate interference collective modes
  solitons
• Spin-1 Bose gas (Super-radiance)
• Bosanova Bragg difffration, super-radience,
  Superfluid-Mott oscillation
• 1999 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 S1/2 BEC, Coreless vortex in
  S1 BEC,
• evidence of universality near
  resonance)
• 2003 Molecular BEC, (Spin dynamics of S1
  BEC, noise measurements)
  
• Fermion pair condensation! (pairing gap,
  collective mode)
• BEC-BCS crossover,
• Vortices in fermion superfluids, discovery
  of S3 Cr Bose condensate, observation of
  skymerion in S1 Bose gas.
• Effect of spin asymmetry and rotation on
  strongly interacting Fermi gas.

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New Bose systems spin-1/2, spin-1, spin-2
Bose gas,
Molecular Bose gas. (BEC at T0) 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
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Often described as experimental driven, but in
fact theoretical ideas are crucial.
Bose and Einstein, Laser cooling, Evaporative
cooling
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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
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Question
How do Bosons find their ground state?
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Question
How do Bosons find their ground state?
Conventional Bose condensate all Bosons
condenses into a single state.
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What happens when there are several degenerate
state for the Bosons to condensed in?
G Number of degenerate states
N Number of Bosons
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What happens when there are several degenerate
state for the Bosons to condense in?
G Number of degenerate states
N Number of Bosons
Pseudo-spin 1/2 Bose gas G 2
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G Number of degenerate states
N Number of Bosons
Spin-1 Bose gas G3, GltltN
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G Number of degenerate states
N Number of Bosons
Spin-1 Bose gas G3, GltltN
Bose gas in optical lattice G N
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G Number of degenerate states
N Number of Bosons
Spin-1 Bose gas G3, GltltN
Bose gas in optical lattice G N
Fast Rotating Bose gas GgtgtN
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Effect of spin degeneracy on BEC
Effect of spin degeneracy on BEC
Spin-1 Bose Gas
A deep harmonic trap
Only the lowest harmonic state is occupied
gt
a zero dimensional problem
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Spin dynamics of spin-1 Bose gas
Spin-1 Bose Gas
A deep harmonic trap
Hilbert space
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Effect of spin degeneracy on BEC
Effect of spin degeneracy on BEC
Spin-1 Bose Gas
A deep harmonic trap
Under spin rotation,
rotates like a 3D Cartesean vector
.
3D rotation
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Cgt0
Conventional condensate
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Cgt0

Exact ground state
Ho and Yip, PRL, 2004
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Relation between singlet state and coherent state
z
Average the coherrent state over all directions
Because
y
The system is easily damaged
x
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Transformation of singlet into coherent states as
a function of External field and field gradient
If the total spin is non-zero
Bosonic enhancement
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Transformation of singlet into coherent states as
a function of External field and field gradient
If the total spin is non-zero
Bosonic enhancement
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Transformation of singlet into coherent states as
a function of External field and field gradient
If the total spin is non-zero
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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
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S2 Cyclic state S3 Spin biaxial Nematics
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A geometric representation Generalization of
Barnett et.al. PRL 06
T.L.Ho, to be published
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Cycle Tetrahedron S2
Octegonal S3
Cubic S4
Icosahedral S6
T.L. Ho, to be published
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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)
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The fate of a fast rotating quautum gas
Superfluidity ----gt Strong Correlation
Boson
Quantum Hall
Vortex lattice
Overlap gt Melting
Normal
Quantum Hall
Fermion
In superconductors
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A remarkable equivalence
Rotating quantum gases in harmonic traps
Electrons in Magnetic field
external rotation
trap
as
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No Rotation Two dimensional harmonic
oscillator
, ngt0, mgt0.
E
m
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, ngt0, mgt0.
E
As
,
Angular momentum states organize into Landau
levels !
m
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E
m
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Mean field quantum Hall regime
in Lowest Landau level
E
condensate
m
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E
Strongly correlated case interaction dominated
m
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E. Mueller and T.L. Ho, Physical Rev. Lett. 88,
180403 (2002)
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Simulate EM field by rotation
Eric Cornells latest experiment cond-mat/0607697
TL Ho, PRL 87, 060403(2001)
V. Schweikhard, et.al. PRL 92, 040404 (2004)
(JILA group, reaching LLL)
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Fermion quantum Hall
Boson Fermion
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Strongly interacting
Fermi gases
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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)
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Weak coupling
Dilute Fermi Gas
S-wave scattering length
Normal Fermi liquid
Weak coupling BCS superfluid
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What Happens?
Dilute Fermi Gas
S-wave scattering length
Normal Fermi liquid
Weak coupling BCS superfluid
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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

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BCS
Molecular BEC
Universality A statement about the energetics
at resonance
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How Resonance Model acquire universality
has to hybridize with many
pairs.
If is large -- strong hybridization, then
has relatively little weight in the pair!
Small effect of means universality !
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Two channel Model
Single Channel model
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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
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In two channel model
Small closed channel contribution ltgt pair size
are given by interparticle spacing ltgt ltgt
single channel description ok ltgt universal
energy density
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• Current Development
• Unequal spin population
• Rotation

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Single vortex
Melting of vortex lattice
c
To quantum Hall regime
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Other possible Fermion superfluids P-wave
Fermion superfluids.
Optiuum phase
B
B
agt0
alt0
o
Molecular condensate
Fermion Superfluid
Ho and Diener, to appear in PRL
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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
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• 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

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0
1
2
3
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0
1
2
3
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0
1
2
3
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0
1
2
3
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0
1
2
3
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Superfluid
Mott
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Superfluid State


ODLRO
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Superfluid State


ODLRO
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Mott State
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Mott State
Resists addition of boson require energy
U, hence insulating
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Nature, 419, 51-54 (2002)
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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)
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Phase diagram of Boson-Hubbard Model
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Part IC Current experiments
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Expts involving superfluid-insulator transitions
Fermions in optical lattice, 2 fermions per site
I. Bloch, et.al, PRA72, 053606 (2005)
Ketterle et.al, cond-mat/0607004
Sengstock et.al. PRL 96, 180403 (2006)
Esslinger, PRL 96, 180402 (2006)
F-B mixture
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ETH Experiment very deep lattice, less than two
toms per site
Band insulator
2 to 3 bands occupied
2 atoms per site
0
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2 fermions Per site
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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
• The current method of detecting superfluidity in
  lattices is misleading
• B. A precise determination of superfluidity gt
  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
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Part II Necessary conditions for studying
strongly correlated physics Quantum
Degeneracy Method of Detection
Quantum Degeneracy
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Condition for quantum degeneracy
Condition for BEC
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Free space
Lattice
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Lowest temperature attainable
Quantum degeneracy
Free space
Optical lattice
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Current method of identifying superfluidity
sharpness of n(k)
Fermions in optical lattice, 2 fermions per site
I. Bloch, et.al, PRA72, 053606 (2005)
Ketterle et.al, cond-mat/0607004
Esslinger, PRL 96, 180402 (2006)
Sengstock et.al. PRL 96, 180403 (2006)
F-B mixture
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However, a normal gas above Tc can also have
sharp peak!
Diener, Zhao, Zhai, Ho, to be published.
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Current method of identifying superfluidity
sharpness of n(k)
Fermions in optical lattice, 2 fermions per site
I. Bloch, et.al, PRA72, 053606 (2005)
Ketterle et.al, cond-mat/0607004
Esslinger, PRL 96, 180402 (2006)
Sengstock et.al. PRL 96, 180403 (2006)
F-B mixture
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Part II Necessary conditions for studying
strongly correlated physics Quantum
Degeneracy Method of Detection
Method of Detection
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An accurate method for detecting superfluidity
Visibility
Reciprocal lattice vector
Not a reciprocal lattice vector
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DZZH, to be published
T0 visibility
2nd Mott shell
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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
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Finite temperature effect becomes important