[PPT] - Ultracold Atoms and Quantum Simulators Marc Cheneau Igor PowerPoint Presentation

SLIDE 1

Ultracold Atoms and Quantum Simulators

Marc Cheneau Igor Ferrier-Barbut

Laboratoire Charles Fabry Institut d’Optique, CNRS, Université Paris-Saclay

SLIDE 2

3

The Saga of Ultracold Atoms at a Glance

l a s e r c

l

i n g ( ~ 1 μ K* ) B

s

e

E

i n s t e i n c

n

d e n s a t i

n

( ~ 1 n K* ) t

w

a r d s u l t r a

l
w

t e m p e r a t u r e s w e a k l y

i

n t e r a c t i n g q u a n t u m g a s e s s t r

n

g l y

c
r

r e l a t e d q u a n t u m m a t t e r t

w

a r d s q u a n t u m s i m u l a t i

n
p

t i c a l l a t t i c e s a n d c

n

t r

l
f

i n t e r a c t i

n

s d y n a m i c a l c

n

t r

l

1 9 8 5 1 9 9 5 2 5 2 1 5 Historical perspective

From classical gases to strongly-correlated quantum systems

Several periods (~10 years each)

Milestone achievements and new perspectives

SLIDE 3

4

The Saga of Ultracold Atoms at a Glance

l a s e r c

l

i n g ( ~ 1 μ K* ) B

s

e

E

i n s t e i n c

n

d e n s a t i

n

( ~ 1 n K* ) t

w

a r d s u l t r a

l
w

t e m p e r a t u r e s w e a k l y

i

n t e r a c t i n g q u a n t u m g a s e s s t r

n

g l y

c
r

r e l a t e d q u a n t u m m a t t e r t

w

a r d s q u a n t u m s i m u l a t i

n
p

t i c a l l a t t i c e s a n d c

n

t r

l
f

i n t e r a c t i

n

s d y n a m i c a l c

n

t r

l

1 9 8 5 1 9 9 5 2 5 2 1 5 Towards ultra-low temperatures

Doppler and Sisyphus cooling schemes (friction force and momentum kicks; temperature limited to the mK range; classical gas)

Sub-recoil cooling (eg VSCPT, Raman cooling, side band cooling, … ; quantum gas)

Moreover, trapping (counteracts Brownian motion; magneto-optical and dipole traps)

SLIDE 4

5

The Saga of Ultracold Atoms at a Glance

l a s e r c

l

i n g ( ~ 1 μ K* ) B

s

e

E

i n s t e i n c

n

d e n s a t i

n

( ~ 1 n K* ) t

w

a r d s u l t r a

l
w

t e m p e r a t u r e s w e a k l y

i

n t e r a c t i n g q u a n t u m g a s e s s t r

n

g l y

c
r

r e l a t e d q u a n t u m m a t t e r t

w

a r d s q u a n t u m s i m u l a t i

n
p

t i c a l l a t t i c e s a n d c

n

t r

l
f

i n t e r a c t i

n

s d y n a m i c a l c

n

t r

l

1 9 8 5 1 9 9 5 2 5 2 1 5 Weakly-interacting quantum gases

Evaporative cooling (quantum gases, weak interactions)

Bose-Einstein condensation and degenerate Fermi gases

(eg coherence and supefluidity) Moreover, first simulations (Brownian motion, dissipative optical lattices, …)

SLIDE 5

6

The Saga of Ultracold Atoms at a Glance

l a s e r c

l

i n g ( ~ 1 μ K* ) B

s

e

E

i n s t e i n c

n

d e n s a t i

n

( ~ 1 n K* ) t

w

a r d s u l t r a

l
w

t e m p e r a t u r e s w e a k l y

i

n t e r a c t i n g q u a n t u m g a s e s s t r

n

g l y

c
r

r e l a t e d q u a n t u m m a t t e r t

w

a r d s q u a n t u m s i m u l a t i

n
p

t i c a l l a t t i c e s a n d c

n

t r

l
f

i n t e r a c t i

n

s d y n a m i c a l c

n

t r

l

1 9 8 5 1 9 9 5 2 5 2 1 5 Strongly-correlated quantum matter

Control of interactions

Optical lattices (non-dissipative ; realization

f tight-binding models for solids)

Fano-Feshbach resonances Low-dimensional systems

SLIDE 6

7

The Saga of Ultracold Atoms at a Glance

l a s e r c

l

i n g ( ~ 1 μ K* ) B

s

e

E

i n s t e i n c

n

d e n s a t i

n

( ~ 1 n K* ) t

w

a r d s u l t r a

l
w

t e m p e r a t u r e s w e a k l y

i

n t e r a c t i n g q u a n t u m g a s e s s t r

n

g l y

c
r

r e l a t e d q u a n t u m m a t t e r t

w

a r d s q u a n t u m s i m u l a t i

n
p

t i c a l l a t t i c e s a n d c

n

t r

l
f

i n t e r a c t i

n

s d y n a m i c a l c

n

t r

l

1 9 8 5 1 9 9 5 2 5 2 1 5 Towards quantum simulation

Simulating the dynamics of quantum matter in true experiments

Thermodynamic equilibrium

Out-of-equilibrium physics

SLIDE 7

B a s i c m

d

e l s p l a y a c e n t r a l r

l

e I n t e r e s t i n g p h y s i c a l s y s t e m s a r e u s u a l l y c

m

p l e x

Ma n y b

d

i e s , p

s

s i b l y n

t

a l l i d e n t i c a l

S i m u l a t i n g I n t e r e s t i n g P h e n

m

e n a i n P h y s i c s

C

m

p l i c a t e d m i c r

s

c

p

i c i n t e r a c t i

n

s S t r u c t u r e , f r u s t r a t i

n

, q u a n t u m e n t a n g l e m e n t , … ⇒! C a n n

t

b e s

l

v e d e x a c t l y a t t h e m i c r

s

c

p

i c l e v e l S i m p l e r a n d

f

t e n r e p r

d

u c e i n t e r e s t i n g p h y s i c s Ma y b e e a s i e r t

s
l

v e U n i v e r s a l b e h a v i

u

r ⇒! S i m p l e H a m i l t

n

i a n s d

n
t

g u a r a n t e e s i m p l e s

l

u t i

n

s

N u m e r i c a l s i m u l a t i

n

s

Ma n y

b
d

y a p p r

a

c h e s e x i s t ( Q MC , D MR G , D F T , D MF T , …) Me a n

fj

e l d a p p r

a

c h e s ( n

n

l i n e a r i t i e s , …) ⇒! C a n n

t

s

l

v e a n y p r

b

l e m , i n p a r t i c u l a r i n t h e q u a n t u m w

r

l d ( e x p

n

e n t i a l l y

l

a r g e H i l b e r t s p a c e , e n t a n g l e m e n t , s i g n p r

b

l e m f

r

f e r m i

n

s , …)

H=J ∑

⟨R , R'⟩

S R

x⋅SR' x −h∑ R

SR

z

SLIDE 8

R e q u i r e m e n t s a n d c h a l l e n g e s

B u i l d u p : C r e a t e a q u a n t u m s y s t e m ( b

s
n

s , f e r m i

n

s , s p i n s , …) t h a t c a n b e m a n i p u l a t e d b y e x t e r n a l fj e l d s

Wh y d

n

’ t w e l e t N a t u r e w

r

k f

r

u s ?

D e s i g n a q u a n t u m s y s t e m e x a c t l y g

v

e r n e d b y a p r e

d

e fj n e d H a m i l t

n

i a n Ĥ

m

d

e l

S i m u l a t i

n

: F r

m

N u m e r i c s t

Q

u a n t u m S y s t e m s

R.P. Feynman, Int. J. Theor. Phys. 21, 467 (1982) ; S. Lloyd, Science 273, 1073 (1996)

L e t t h e s y s t e m e v

l

v e u n d e r Ĥ

m

d

e l

t

w

a r d s i t s g r

u

n d s t a t e ( c

l

i n g )

r

a t h e r m a l s t a t e ( c

u

p l i n g t

a

b a t h ) ,

r

s t u d y i t s t i m e

d

e p e n d e n t d y n a m i c s Me a s u r e r e l e v a n t q u a n t i t i e s s

a

s t

c
n

s i d e r Ĥ

m

d

e l

i s s

l

v e d Q u a n t u m e n g i n e e r i n g : D e s i g n t h e d e s i r e d H a m i l t

n

i a n w i t h a t l e a s t

n

e c

n

t r

l

p a r a m e t e r ( e g b e n c h m a r k i n g ) I n i t i a l i z a t i

n

: P r e p a r e t h e s y s t e m i n a w e l l

k

n

w

n i n i t i a l s t a t e ( p u r e

r

m i x e d ) D e t e c t i

n

: S u ffj c i e n t l y a c c u r a t e a n d v a r i

u

s m e a s u r e m e n t s

SLIDE 9

N e w p r

m

i s i n g p l a t f

r

m s

S u p e r c

n

d u c t i n g c i r c u i t s

T

w

a r d s Q u a n t u m S i m u l a t i

n

Q u a n t u m

p

t i c s Ma g n e t i c i n s u l a t

r

s U l t r a c

l

d a t

m

s a n d i

n

s …

U l t r a c

l

d a t

m

s

[Aspuru-Guzik & Walther, Nat. Phys. 8, 285 (2012)] [Houck et al., Nat. Phys. 8, 292 (2012)] [Bloch et al., Nat. Phys. 8, 267 (2012); Blatt & Roos, Nat. Phys. 8, 277 (2012)] [Ward et al., J. Phys : Condens. Matter 25, 014004 (2013)]

A m a j

r

p l a y g r

u

n d A l m

s

t a n y p a r a m e t e r c a n b e c

n

t r

l

l e d e x p e r i m e n t a l l y

SLIDE 10

T

w

a r d s Q u a n t u m S i m u l a t i

n

w i t h U l t r a c

l

d A t

m

s

▲ Quantum gases in arbitrary dimensions ▲ Lattice spin Hamiltonians ▲ Disordered quantum systems ▲ Artificial gauge fields ▲ Bose- and Fermi-Hubbard models in optical lattices ▲ Out-of-equilibrium dynamics ▲ BEC-BCS crossover in strongly-correlated Fermi gases

SLIDE 11

1 3

Quantum mechanics and statistical physics

[1] J.-L. Basdevant, J. Dalibard, and M. Joffre, Mécanique Quantique (Presse de l’Ecole Polytechnique; available also in English at Springer, 2006). [2] C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique Quantique, parts 1, 2 & 3. [3] L. D. Landau and E. M. Lifshitz, Statistical Physics, parts 1 & 2 (Elsevier, Oxford, 1980). [4] B. Diu, D. Lederer, and B. Roulet, Physique Statistique (Hermann, Paris, 1996).

Ultracold atoms

[1] C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, 2008). [2] L.P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Clarendon press, Oxford, 2004).

N

n
e

x h a u s t i v e L i t e r a t u r e

SLIDE 12

1 4

Statistical Physics : A Reminder

SLIDE 13

C l a s s i c a l g a s e s

Mi c r

s

c

p

i c D e s c r i p t i

n

O n e

b
d

y s t a t e

f

p a r t i c l e j :

r p l1 l2 l3 l4

l j=(⃗ r j, ⃗ p j)

N

b
d

y s t a t e

f

t h e g a s :

Λ={l1,...,lN }

D y n a m i c s g

v

e r n e d b y t h e N e w t

n

e q u a t i

n

s

d⃗ r j dt =⃗ pj m d ⃗ p j dt =⃗ F j ({⃗ r j,⃗ p j},t)

a n d

(

r

, e q u i v a l e n t l y , b y t h e H a m i l t

n

e q u a t i

n

s )

Q u a n t u m g a s e s

O n e

b
d

y s t a t e

f

p a r t i c l e : ∣l =∑α cα(l)∣α j N

b
d

y s t a t e

f

t h e g a s :

∣Λ ≠{∣l1,...,∣lN }

D y n a m i c s g

v

e r n e d b y t h e S c h r ö d i n g e r e q u a t i

n

i ℏ d∣Λ  dt = ^ H (t )∣Λ 

∣Λ〉 =

c

1

( Λ) ∣1 〉 +

c

2

( Λ) ∣2 〉 ∣1 〉 ∣2 〉 c

1

( Λ) c

2

( Λ) measurement

( a n d q u a n t u m j u m p s i n d u c e d b y m e a s u r e m e n t s ) ⇒ I n b

t

h c a s e , t h e d y n a m i c s i s e s s e n t i a l l y d e t e r m i n i s t i c ( u p t

q

u a n t u m m e a s u r e m e n t s )

SLIDE 14

T h e m i c r

s

c

p

i c d y n a m i c s i s u n t r a c t a b l e

Ma c r

s

c

p

i c D e s c r i p t i

n

T

m

a n y p a r t i c l e s , O n e c a n n

t

p e r f

r

m c a l c u l a t i

n

s , n

r

s t

r

e i n f

r

m a t i

n

N∼N A≃6×10

23

Mi c r

s

c

p

i c d y n a m i c s m u c h f a s t e r t h a n t h e m a c r

s

c

p

i c d y n a m i c s ( s c a l e s e p a r a t i

n

) ⇒ U s e f u l i n f

r

m a t i

n

l i m i t e d t

a

s m a l l n u m b e r

f

v a r i a b l e s ( P , T , N , M , …) T h e s e v a r i a b l e s a r e r e l a t e d b y h e u r i s t i c e q u a t i

n

s ( e q u a t i

n
f

s t a t e ) , e g

PΩ=N k BT

T h e r m

d

y n a m i c a p p r

a

c h

dE=δW +δQ dS=δQ T +δ Screa δ Screa⩾0

w i t h

F i r s t l a w S e c

n

d l a w T h i r d l a w

Sclass(T=0)=0 Squant(T=0)=kB ln(g)

b u t

⇒ T h e r m

d

y n a m i c s i s e s s e n t i a l l y i r r e v e r s i b l e ( n

n

d e t e r m i n i s t i c ) H e u r i s t i c b u t r e m a r k a b l y e ffj c i e n t a n d u n i v e r s a l

SLIDE 15

Mi c r

s

c

p

i c v e r s u s Ma c r

s

c

p

i c D e s c r i p t i

n

s

e1 2e1 3e1 4e1 5e1 6e1 1 macrostate of energy E=5e1 Λ1 Λ2 Λ3 Λ4 Λ5

T h e r e a r e m a n y m

r

e m i c r

s

t a t e s t h a n m a c r

s

t a t e s

H e r e a f t e r , w e u s e t h e m a c r

s

c

p

i c v a r i a b l e s E ( a n d N ) A s i m p l e e x a m p l e : 3 n

n
i

n t e r a c t i n g p a r t i c l e s ; l a d d e r

n

e

b
d

y s p e c t r u m

5 corresponding microstates (realizations)

I n g e n e r a l , t h e n u m b e r

f

m i c r

s

t a t e s g r

w

s e x p

n

e n t i a l l y w i t h N ( e n e r g y g r

w

s a l g e b r a i c , e g ; H i l b e r t s p a c e g r

w

s e x p

n

e n t i a l l y , i e )

H N∼⊗j H j EN∼Σ j E j

SLIDE 16

B

l

t z m a n n ’ s S t a t i s t i c a l P h y s i c s

E s t a b l i s h t h e l i n k b e t w e e n t h e m i c r

s

c

p

i c a n d m a c r

s

c

p

i c d e s c r i p t i

n

s

F

c

u s

n

t h e s l

w

d y n a m i c s (

f

t h e m a c r

s

t a t e ) a n d g e t r i d

f

t h e r a p i d d y n a m i c s (

f

t h e m i c r

s

t a t e s ) T h e m i c r

s

t a t e i s r a n d

m

a n d t h e s y s t e m p e r f

r

m s f a s t e r r a t i c j u m p s b e t w e e n t h e m i c r

s

t a t e s . I t i s t h u s r e l e v a n t t

a

t t r i b u t e t h e m a p r

b

a b i l i t y d i s t r i b u t i

n

P

Λ

. N . B . : P r

b

a b i l i t i e s

f

m a c r

a

n d m i c r

s

t a t e s :P(E)= ∑

Λ, E Λ=E

PΛ

T h e m a c r

s

t a t e s w e

b

s e r v e a r e t h

s

e w i t h a d

m

i n a n t n u m b e r

f

m i c r

s

t a t e r e a l i z a t i

n

s

B a s i c p r i n c i p l e s

B

l

t z m a n n ' s p r i n c i p l e : T h e b e s t s t a t i s t i c a l d e s c r i p t i

n
f

a c

m

p l e x s y s t e m i s t h e

n

e t h a t a t t r i b u t e s t h e s a m e p r

b

a b i l i t y t

a

l l t h e a c c e s s i b l e m i c r

s

t a t e s i f n

p

a r t i c u l a r c

n

s t r a i n t f a v

r

s s

m

e m i c r

s

t a t e s w i t h r e s p e c t t

t

h e r

n

e s . E r g

d

i c i t y p r i n c i p l e : T i m e a n d s t a t i s t i c a l a v e r a g e s a r e e q u a l , .

⟨O⟩t=⟨O ⟩stat ⟨O⟩t= lim

tmeas→∞

1 tmeas ∫

tmeas

dt O(t ) ⟨O⟩stat=∑

Λ PΛ〈 Λ∣ ^

O∣Λ

SLIDE 17

Mi c r

c

a n

n

i c a l e n s e m b l e

T h e G i b b s C a n

n

i c a l E n s e m b l e s

I s

l

a t e d s y s t e m s ( n

e

x c h a n g e

f

e n e r g y ; n

e

x c h a n g e

f

m a t t e r )

B U U : u n i v e r s e B : b a t h S : s y s t e m S

PΛ= 1 W (E,Δ E; N , Δ N )

P r

b

a b i l i t y

f

a m a n y

b
d

y m i c r

s

t a t e :

S=k Bln [W (E ,Δ E ;N ,Δ N )]

S t a t i s t i c a l e n t r

p

y :

SLIDE 18

C a n

n

i c a l e n s e m b l e

T h e G i b b s C a n

n

i c a l E n s e m b l e s

C l

s

e d s y s t e m s ( e x c h a n g e

f

e n e r g y ; n

e

x c h a n g e

f

m a t t e r )

B U U : u n i v e r s e B : b a t h S : s y s t e m S

P r

b

a b i l i t y

f

a m a n y

b
d

y m i c r

s

t a t e : C a n

n

i c a l p a r t i t i

n

f u n c t i

n

:

S

PΛ= exp(−β EΛ ) ZC ZC=∑

Λ exp (−β EΛ)

SLIDE 19

G r a n d c a n

n

i c a l e n s e m b l e

T h e G i b b s C a n

n

i c a l E n s e m b l e s

O p e n s y s t e m s ( e x c h a n g e

f

e n e r g y ; e x c h a n g e

f

m a t t e r )

B U U : u n i v e r s e B : b a t h S : s y s t e m S

P r

b

a b i l i t y

f

a m a n y

b
d

y m i c r

s

t a t e : C a n

n

i c a l p a r t i t i

n

f u n c t i

n

:

S

PΛ= exp(−β EΛ+α N Λ) ZGC ZGC=∑

Λ exp (−β EΛ+α N Λ )

SLIDE 20

D i s c e r n a b l e v e r s u s I n d i s c e r n a b l e C

u

n t i n g

SLIDE 21

2 3

1/2 1/2 1

fermions bosons

A simple and instructive example

2 n

n
i

n t e r a c t i n g p a r t i c l e s 3 p

s

s i b l e

n

e

b
d

y s t a t e s Probability that the two particles are in the same state ?

C

u

n t i n g N

b
d

y Q u a n t u m S t a t e s

SLIDE 22

2 4

1/3 2/3 1/2 1/2 1

fermions bosons

discernible

A simple and instructive example

2 n

n
i

n t e r a c t i n g p a r t i c l e s 3 p

s

s i b l e

n

e

b
d

y s t a t e s Probability that the two particles are in the same state ?

C

u

n t i n g N

b
d

y Q u a n t u m S t a t e s

SLIDE 23

2 5

1/3 2/3 1/2 1/2 1

fermions

discernible bosons

A simple and instructive example

2 n

n
i

n t e r a c t i n g p a r t i c l e s 3 p

s

s i b l e

n

e

b
d

y s t a t e s Probability that the two particles are in the same state ?

C

u

n t i n g N

b
d

y Q u a n t u m S t a t e s

SLIDE 24

2 6

1/3 2/3 1/2 1/2 1

discernible fermions bosons

A simple and instructive example

2 n

n
i

n t e r a c t i n g p a r t i c l e s 3 p

s

s i b l e

n

e

b
d

y s t a t e s Probability that the two particles are in the same state ?

C

u

n t i n g N

b
d

y Q u a n t u m S t a t e s

SLIDE 25

2 7

1/3 2/3 1/2 1/2 1

discernible fermions bosons

A simple and instructive example

2 n

n
i

n t e r a c t i n g p a r t i c l e s 3 p

s

s i b l e

n

e

b
d

y s t a t e s Probability that the two particles are in the same state ?

C

u

n t i n g N

b
d

y Q u a n t u m S t a t e s

Bose amplification Pauli exclusion

SLIDE 26

2 8

classical gas quantum gas lT

Quantum degenerate regime :

n lT

d≿1

lT a

C l a s s i c a l v e r s u s Q u a n t u m d e B r

g