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 techniquekeep 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

Bosons and Fermions with large spins

F=I+J

alkali atoms

Hyperfine spin

J=1/2

I

J

e

Spin F=1, F=2 bosons:

Spin F=1/2, 3/2, 5/2, 7/2, 9/2 fermions

BMagnetic trap

Spinless bosons and fermions

Atoms “lose” their spins!

BMagnetic trap

Mixture of quantum gases:

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:

Ho and Shenoy, PRL 96

Optical trapping:Focused laser

BEC or cold fermions

All spin states are trapped,

Spin F=1, F=2 bosons:

Spin F=1/2, 3/2, 5/2, 7/2, 9/2 fermions

T.L.Ho, PRL 1998

Quantum Gases

Atomic PhysicsCondensed Matter Physics

Quantum Optics

Nuclear Physics

Quantum Information

BEC

High Energy Physics

Quantum Gases

BBBBFFF F

3D

2D

1D

0D

single trap

lattice

stationary

fast rotating

U(1)Magnetictrap, spins frozen

S0(3)Optical trap, spins released

€

Ω=0

€

Ω→ ωtrap

€

na3 <<1

€

na3 >>1

system environmentssymmetry interaction

1996 Discovery of BEC!1997 Mixture of BEC and pseudo spin-1/2 Condensate interference collective modes solitons1998 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 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?

Conventional Bose condensate : all Bosons condenses into a single state.

How do Bosons find their ground state?

Question:

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 N: Number of Bosons

Pseudo-spin 1/2 Bose gas: G =2

Spin-1 Bose gas : G=3, G<<N

G: Number of degenerate states N: Number of Bosons

Spin-1 Bose gas : G=3, G<<N

Bose gas in optical lattice: G ~N

G: Number of degenerate states N: Number of Bosons

Spin-1 Bose gas : G=3, G<<N

Bose gas in optical lattice: G ~N

Fast Rotating Bose gas: G>>N

G: Number of degenerate states N: Number of Bosons

Effect of spin degeneracy on BEC

Only the lowest harmonic state is occupied

=> a zero dimensional problem

Spin-1 Bose Gas

Effect of spin degeneracy on BEC

A deep harmonic trap

€

aμ

+

€

μ=1,0,−1

Spin-1 Bose Gas

Spin dynamics of spin-1 Bose gas

A deep harmonic trap

€

H = cr S

2

Hilbert space

Effect of spin degeneracy on BEC

Spin-1 Bose Gas

Effect of spin degeneracy on BEC

A deep harmonic trap

€

Ax = (−a1 + a−1) / 2

€

Ay = (a1 + a−1) / 2i

€

Az = a0

Under spin rotation, rotates like a 3D Cartesean vector .

€

aμ → (e−ir θ ⋅

r S a)μ

€

rA i → R(

r θ )ij

r A j

€

R(r θ ) : 3D rotation

€

aμ

+

€

μ=1,0,−1

Conventional condensate :

€

N0 = 0,

€

N±1 = N /2

€

H = cr S 2 C>0

€

< r

S >= 0

€

ΔN12 ~ N

€

Ax = (−a1 + a−1) / 2

€

Ay = (a1 + a−1) / 2i

€

Az = a0

Exact ground state :

€

| S = 0 >= ΘN / 2 | 0 >

€

< aμ

+aν >=N3

1 0 0

0 1 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

N0 = N1 = N−1 = N /3

€

H = cr S 2 C>0

€

Θ=2a1

+a−1

+ − a0

+2

€

ΔN12 ~ N 2

=

Ho and Yip, PRL, 2004

Average the coherrent state over all directions

Relation between singlet state and coherent state

x

y

z

Because

€

ΔN12 ~ N 2

The system is easily damaged

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

CycleTetrahedron S=2

Cubic S=4

Octegonal S=3

Icosahedral S=6

T.L. Ho, to be published

Rotating the Bose condensate

Generating a rotating quadrupolar field using a pair of rotating off-centered lasers

condensate

K. W. Madison, F. Chevy, W. Wohlleben, J. Dalibard PRL. 84, 806 (2000)

The fate of a fast rotating quautum gas : Superfluidity ----> Strong Correlation

Vortex lattice Overlap => Melting

Quantum Hall Boson

Fermion

Normal Quantum Hall

In superconductors

€

h =(r p − M

r Ω ×

r r )2

2M

€

Ω→ ω as

Rotating quantum gases

in harmonic traps

Electrons in

Magnetic field

€

h =r p 2

2m+

12

Mω2r2 −r Ω •

r r ×

r p

€

h =(r p − M

r Ω ×

r r )2

2M+

12

M (ω2 −Ω2 )r2

trap

external rotation

A remarkable equivalence

, n>0, m>0.

m

€

Ω=0, E = hω(n + m)

E

No Rotation : Two dimensional harmonic oscillator

€

Enm = h(ω +Ω)n + h(ω −Ω)m , n>0, m>0.

€

Ω→ ωAs

Angular momentum states organize into Landau levels !

,

m

E

m

E

€

μ

m

E

€

μ

condensate

€

<ψ >

Mean field quantum Hall regime: in Lowest Landau level

€

<ψ >

m

E

€

μ

Strongly correlated case: interaction dominated

€

<ψ >=0

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)

Boson + Fermion

Fermion quantum Hall

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)

Dilute Fermi Gas

Normal Fermi liquid

Weak coupling BCS superfluid

: S-wave scattering length

Weakcoupling

Dilute Fermi Gas

Normal Fermi liquid

Weak coupling BCS superfluid

: S-wave scattering length

What Happens?

Key Properties: Universality (Duke, ENS)

Evidence for superfluid phase: Projection expt: Fermion pair condensataion -- JILA, MIT Specific heat -- DukeEvidence 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 pairs.

If is large -- strong hybridization, then has relatively little weight in the pair!

Small effect of means universality !

Two channel Model

Single Channel model:

Origin of universality

Scattering amplitude: (from both single and two channel model)

r = effective range

Question: what happen to scattering on Fermi surface

Wide resonance

Narrowresonance

Bruun & Pethick PRL 03Petrov 04Diener and Ho 04Strinati et.al 04Eric Cornell, email

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

c

To quantum

Hall regime

Melting of vortex lattice

Single vortex

Other possible Fermion superfluids: P-wave Fermion superfluids.

BBo a>0 a<0

Molecular condensateFermion Superfluid

Ho and Diener, to appear in PRL

Optiuum phase

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)

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)

Observation of Superfluid-insulator transition

Phase diagram of Boson-Hubbard Model

Part IC

Current experiments

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)

Expts involving superfluid-insulator transitions:

F-B mixture

Fermions in optical lattice, 2 fermions per site

Band insulator

2 atoms per site 2 to 3 bands occupied

0

ETH Experiment: very deep lattice, less than two toms per site

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:

* Quantum Degeneracy

Condition for quantum degeneracy

Condition for BEC :

Free space

Lattice

Free space Quantum degeneracyLowest temperature attainable:

Optical lattice

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)

Current method of identifying superfluidity: sharpness of n(k)

F-B mixture

Fermions in optical lattice, 2 fermions per site

However, a normal gas above Tc can also have sharp peak!

Diener, Zhao, Zhai, Ho, to be published.

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)

Current method of identifying superfluidity: sharpness of n(k)

F-B mixture

Fermions in optical lattice, 2 fermions per site

Part II Necessary conditions for studying

strongly correlated physics:

* Quantum Degeneracy

* Method of Detection: * Method of Detection

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 More intriguing physics of quantum physics of quantum critical behavior can critical behavior can

be expected be expected