11Cozumel 2004

Nonlinear optics of atoms and molecules

Chris Search (→ Stevens Inst. Of Technology )Markku JääskeläinenTakahiko Miyakawa

Omjyoti DuttaDominic MeiserHermann Uys

Pierre Meystre

ARO NASA NSF ONROptical Sciences Center

The University of Arizona

Cozumel 2004 22

Matter-wave field

(Courtesy E. Cornell and C. Wieman)

( )Matter-wave field ,ˆ r tψ(Courtesy W. Ketterle)

( ) ( ) ( )

( ) ( ) ( )

†

†

ˆ ˆ, , ', - '

ˆ ˆ, , ', - '

r t r t r r

r t r t r r

ψ ψ δ

ψ ψ δ−

+

⎡ ⎤ =⎣ ⎦

⎡ ⎤ =⎣ ⎦

bosons

fermions

Cozumel 2004 33

Collisions

Single-particle

(ˆ

)(ˆ

†2

31

3

0†3

rrdrd

HrrdH

ψ

ψψ

+

=

∫∫

)(ˆ)(ˆ)()(ˆ)

)(ˆ

12212†

1 rrrrVr

r

ψψψ −

Two-body collisions

• s-wave approximation : ( ) ( )2

2 1 2 14V r r r r

Maπ δ− = −

• Heisenberg equations of motion:

),(ˆ),(ˆ),(ˆ4),(ˆ),(ˆ †2

0 trtrtrM

atrHdt

trdi ψψψπψψ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

Cozumel 2004 44

Mean-field theory (bosons)

( ) ( ) ( )ˆ ˆ, ,r t r t r tψ ψ→ ≡ΦBosons : , “condensate wave function”

0

22( , ) ( , ) 4 ( , ) ( , )d r ti H r t r t

Mr at

dtπ⎛ ⎞

Φ Φ⎜ ⎟⎝

+⎠

Φ= ΦGross-Pitaevskii equation :

Two-body collisions

a : scattering length

(From S. L. Cornish et al, PRL 85, 1795 (2000))

Cozumel 2004 55

Examples

Four-wave mixing of matter waves Matter-wave phase conjugationAtom holographyAtomic solitonsSecond-harmonic generationAtom lasersAtom amplifiersMatter-wave superradianceMixing of optical and matter wavesNonclassical and entangled statesCoherence controlQuantum informationSensors

Fermionic matter wavesFermi-Bose wave mixingCoherent molecular fields…

Cozumel 2004 66

Recent trends

Beyond mean field- Fluctuations- Strongly correlated systems (Mott insulator transition, …)- Quantum atom optics (number states, entanglement, …)

Fermions

- BCS pairing- BEC-BCS cross-over- Fermionic atom optics

Molecules

- Feshbach resonances- Photoassociation- Fermions vs. bosons- Molecular condensates- Molecular optics

Cavity atom optics

- Quantum control of matter-wave field

Cozumel 2004 77

Three-wave mixing

• Particle number conservation ?

• Molecules !

Atoms Molecules Property

identical bosons bosonic short-lived

identical fermions(different internal state)

bosonic long-lived

different bosonic or fermionic atoms bosonic

heteronuclear,polar

one boson and one fermion fermionic heteronuclear,

polar

Cozumel 2004 88

Coherent molecule formation

• Feshbach resonances • Photoassociation

(Picture from R. Hulet, http://atomcool.rice.edu)(Picture from N. R.Claussen, PhD Thesis, U. Coloradohttp://jilawww.colorado.edu/www/sro/thesis/claussen/)

Cozumel 2004 99

Adiabatic sweep

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( )

2 2 2†

† †'

23

,

3, '

†

, '

2 2ˆ ˆ ˆ

ˆˆ .ˆ1 .2

ˆH d r V r V r Brm m

d r H

r r

r

r

r cr

σ σσ

σ

σ

σ

σ

σ σσσ

ψ ψ

ψ ψ

ϕ ϕε

ϕχ

=↑ ↓

⎡ ⎤⎛ ⎞ ⎛ ⎞∇ ∇= − + + − + +⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎣ ⎦

+ +

∑ ∫

∑ ∫

Atomic pair

Molecule

B

Energy

Cozumel 2004 1010

Molecular BEC

Feshbach resonance

“A molecular condensate emerges from the Fermi sea”,

M. Greiner, C. Regal, & D. JinNature 426, 537 (2003)

Also:

• R. Grimm et al., Science Express, Nov. 13, 2003• W. Ketterle et al., PRL 91, 250401 (2003)

Cozumel 2004 1111

Molecule statistics

Feshbach resonance

• Atomic BEC

Feshbach resonance

• Normal Fermi system

Feshbach resonance

• BCS state

Cozumel 2004 1212

Atomic BEC

( )2BEC

† †ˆ ˆ .ˆ ˆ .H ga ca c Haδ= + +Two-mode model:

010

2030

0

0.2

0.4

0

0.5

1.0

nt [g−1]

Pn(t)

( ) ( ) ( )

( )

2†

(2)1 2

ˆ ˆ 2 2 1

2, ...1

a a

a

n t a a t

g t tN

Ng N= = −

= − +

Short times:

Coherent state !

Cozumel 2004 1313

Normal Fermi Gas

( ) †† †1, 1, 2, 2, 1NFG , ,

†2ˆ ˆ ˆ ˆ ˆ ˆ .ˆˆ ˆ .k k k

kk

kk kkH E gc c c c c ca a a Hcδ− − −

⎛ ⎞= + + + +⎜ ⎟⎝ ⎠

∑ ∑

• Anderson mapping

1, 2,1 1k k ke −↔

1, 2,0 0k k kg −↔

,kσ−σ+,k

, 1, 2,

† †, 2, 1,

† †, 1, 1, 2, 2,

ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ ˆ 1

k k k

k k k

z k k k k k

c c

c c

c c c c

σ

σ

σ

− −

+ −

− −

=

=

= + −

,NFG ,† †ˆ . .ˆ ˆ ˆ ˆk z kk

k k

H E g Ha a a cσδσ −⎛ ⎞= + + +⎜ ⎟⎝ ⎠

∑ ∑

Cozumel 2004 1414

Normal Fermi Gas (2)

,NFG ,† †ˆ . .ˆ ˆ ˆ ˆk z kk

k kH E g Ha a a cσδσ −

⎛ ⎞= + + +⎜ ⎟

⎝ ⎠∑ ∑

† †,

,

ˆ ˆ ˆ ˆ ˆ 1ˆ ˆ ˆ

k z k k k k

k k k

c c c cc c

σ

σ− −

− −

⎧ = + −⎪⎨

=⎪⎩

( ) ( )

( )

2†

(2)1 2

ˆ ˆ 2

1, 1 . .2

2 .a

an t a a gt

g t t

N

N

= =

⎛ ⎞= − +⎜ ⎟

⎝ ⎠

Short times:

Chaotic state !

Cozumel 2004 1515

BCS state

( ) ( )2†2

ˆ ˆ an t a a gt NV∆⎛ ⎞

⎜⎡ ⎤

= = +⎢ ⎥⎢ ⎥⎣ ⎠ ⎦

⎟⎝

Short times:

(Number of Cooper pairs)2

, ',ˆ ˆkBCS NF kGH H Vσ σ+ −−=

0

5

10

0

1

2

0

0.5

1.0

nt [g−1]

Pn(t)

Π

0 0.1 0.2

1

1.2

1.4

1.6

1.8

2

∆/µ

g(2) (0

+,0

+)

<Na>≈ 100

<Na>≈ 1000

<Na>≈ 10000

Cozumel 2004 1616

Delay time statistics (NFG)

† †NFG , ,ˆ ˆ ˆ ˆ ˆ . .k k z k

k kH E a a g a H cσ δ σ −

⎛ ⎞= + + +⎜ ⎟⎝ ⎠

∑ ∑Normal Fermi gas:

analogous to superradiance problem

Mea

n m

ole

cule

num

ber

time

Del

ay t

ime

dis

trib

ution

time

Cozumel 2004 1717

Effective potential

( )

( )†

ˆˆ ˆ ˆ ˆˆ ˆ2 2

ˆ ˆˆ ˆˆ / 2

yby z y

y

dJdn J S J S Sdt dt

J bS S b i

χ χ + −

+ −

= − = − +

≡ −

• Heisenberg equations of motion(Exact resonance)

• “Semiclassical” description

0 Nbn

( )eff bV n

moleculesatoms

Cozumel 2004 1818

Outlook

• Quantum control of molecular field

• Heteronuclear molecules Ketterle et al., PRL 93, 160406 (2004);D. Jin et al., PRL 93, 112002 (2004)

• Detection

• Molecular matter-wave amplifier C. P. Search & PM, PRL 93, 140405 (2004)

• Molecular micromasersC. P. Search, W. Zhang, & PM, PRL 91, 190401 (2003)

For more details:

• D. Meiser & PM, cond-mat/0410349 • H. Uys, T. Miyakawa, D. Meiser, & PM, cond-mat/0412105• D. Meiser, PM, & C. P. Search, in preparation