Feshbach resonance and

BCS-BEC crossover

Victor Gurarie

Windsor, August 2012

1

Atomic gases 2

Very dilute gas of Li, K or Rb

Interparticle spacingd = 103 nm

De Broglie wavelength�dB =

hpmT

Condition of quantum degeneracy

�dB =hpmT

⇠ d T ⇠ h2

md2⇠ 1µK

d

hot

d

cold

Degenerate weakly interacting gases 3

Bosons:

• Integer spin• Symmetric wave function

Bose condensate

Fermions:

• Half-integer spin• Antisymmetric wave function

EF= kBTF

(two iso-spin states)

Degenerate Fermi gas: “Fermi condensate”

Bose Einstein

Fermi Dirac

Bose-Einstein condensate 4

Macroscopic occupation of the single particle ground state

# of particles # of particles with nonzero momentum

# of particles “in the condensate”, with zero momentum

N =X

i

1

ep2i

2mT � µT � 1

=

Zd3p

(2⇡)31

ep2

2mT � µT � 1

+N0

N0

TTc

BEC detection 5

Time of flight measurement

BEC detection 5

` =p

mt

Time of flight measurement

6

2001

Cornell Wieman

June 1995 in 87Rb

Ketterle

September 1995 in 23Na

CU Boulder

MIT

First BEC

Atomic interactions 7

Atoms are neutral, no direct Coulomb interaction

Van der Waals attraction, hard core repulsion

U

r

Typical scattering lengthRe ⇡ 2.5 nm

a ⇠ Re

Re ⌧ d

Level structure of Lithium 8

6Li 3 protons + 3 electrons + 3 neutrons = fermionNuclear spin I=1, electronic spin S=1/2

Hyperfine interactions + magnetic fieldH = g ~S · ~I + µ ~B · ~S

Feshbach resonance 9

BSz =

1

2 Sz =1

2

Iz = 1 Iz = 0

Atoms in “open channel”

nucleus

electron

U

r

Feshbach resonance 9

BSz =

1

2 Sz =1

2

Iz = 1 Iz = 0

Atoms in “open channel”

nucleus

electron

U

r

Atoms in “closed channel”

Sz =1

2

Iz = 1 Iz = 1

U

r

Sz = �1

2

Feshbach resonance 9

BSz =

1

2 Sz =1

2

Iz = 1 Iz = 0

Atoms in “open channel”

nucleus

electron

U

rg ~S · ~I flips channels

Atoms in “closed channel”

Sz =1

2

Iz = 1 Iz = 1

U

r

Sz = �1

2

Notations 10

Sz =1

2

Iz = 1

Sz =1

2a"

Iz = 0

a#

Sz =1

2

Iz = 1 Iz = 1

U

r✏0

detuningmolecule

Sz = �1

2

b

Two channel model 11

Free motion of atoms Free motion of moleculesshifted by detuning Atomic-molecular interconversion

H =X

p,�=",#

p2

2ma†p,�ap,�+

X

q

✓✏0 +

q2

4m

◆b†q bq+

gpV

X

p,q

b†qa q2+p,"a q

2�p,# +X

p,q

bqa†q2�p,#a

†q2+p,"

!

Weak g limit 12

Free motion of atoms Free motion of moleculesshifted by detuning Atomic-molecular interconversion

H =X

p,�=",#

p2

2ma†p,�ap,�+

X

q

✓✏0 +

q2

4m

◆b†q bq+

gpV

X

p,q

b†qa q2+p,"a q

2�p,# +X

p,q

bqa†q2�p,#a

†q2+p,"

!

It is equivalent to noninteracting bosons with chemical potential 2μ and fermions, with chemical potential μ.

g->0 limit

H�µN =X

p,�=",#

p2

2ma†p,�ap,�+

X

q

✓✏0 +

q2

4m

◆b†q bq�µ

0

@X

p,�=",#a†p,�ap,� + 2

X

q

b†q bq

1

A

Weak g limit, zero temperature 13

H � µN =X

p,�=",#

✓p2

2m� µ

◆a†p,�ap,� +

X

q

✓✏0 +

q2

4m� 2µ

◆b†q bq

0!21

F

F

F

"

k

"

k(b)

"

k(a)

(c)

0!21

µ

"

"

µ

µ="

!2 01

✏0 > 2✏F No bosons, all particles are fermions. µ = ✏F

2✏F > ✏0 > 0 There are both fermions and bosons µ = ✏0/2

0 > ✏0 There are only bosons µ = ✏0/2 < 0

!0 ⌘ ✏0

Nb

✏02✏F

Scattering amplitude 14

H =X

p,�=",#

p2

2ma†p,�ap,�+

X

q

✓✏0 +

q2

4m

◆b†q bq+

gpV

X

p,q

b†qa q2+p,"a q

2�p,# +X

p,q

bqa†q2�p,#a

†q2+p,"

!

fp =1

� 1a + r0

p2

2 � ipScattering length a = � mg2

4⇡!0

Effective range r0 = � 8⇡

m2g2

!0 = ✏0 �g2m

2⇡2ReThe poles of the scattering amplitude with Re p=0, Im p > 0 correspond to the bound state of two atoms with the wave function

⇠ eip|r1�r2| exist only for a>0 or ω0<0

fp =1

g(p)� ip

g(p) = g0 + g2p2 + g4p

4 + . . .

generally

Small g behavior 15

H =X

p,�=",#

p2

2ma†p,�ap,�+

X

q

✓✏0 +

q2

4m

◆b†q bq+

gpV

X

p,q

b†qa q2+p,"a q

2�p,# +X

p,q

bqa†q2�p,#a

†q2+p,"

!

mean field theory bq ! �q,0B

H =X

p,�=",#

✓p2

2m� µ

◆a†p,�ap,�+V (✏0 � 2µ) BB+g

BX

p

ap,"a�p,# +BX

p

a†�p,#a†p,"

!

minimize the ground state energy@

@BEG.S.(B) = 0

!0 � 2µ =g2

2

Zd3p

(2⇡)3

2

6641r⇣

p2

2m � µ⌘2

+ g2B2

� 2m

p2

3

775

Zd3p

(2⇡)3

2

6641�p2

2m � µr⇣

p2

2m � µ⌘2

+ g2B2

3

775+ 2B2 = n

!0

Nb = B2Two important equations

zero g

nonzero g

BCS-BEC crossover 16

BCS (Bardeen-Cooper-Schrieffer)superconductor

BEC (Bose-Einstein condensate) of the diatomic molecules

!0

H =X

p,�=",#

p2

2ma†p,�ap,�+

X

q

✓✏0 +

q2

4m

◆b†q bq+

gpV

X

p,q

b†qa q2+p,"a q

2�p,# +X

p,q

bqa†q2�p,#a

†q2+p,"

!

One channel model limit 17

Interesting limit: ✏0 ! 1, g ! 1,g2

✏0= �

dbqdt

= i[H, bq] = �✏0bq �gpVa q

2+p,"a q2�p,# ⇡ 0

H =X

p,�=",#

p2

2ma†p,�ap,� � �

V

X

q,p,p0

a q2+p,"a q

2�p,#a†q2�p0,#a

†q2+p0,"

One channel model: interacting particles with variable attractive interactions

✏0

�

The physics of one channel model 18

H =X

p,�=",#

p2

2ma†p,�ap,� � �

V

X

q,p,p0

a q2+p,"a q

2�p,#a†q2�p0,#a

†q2+p0,"

Critical interaction strength for two fermions to form a bound state � > �c

fp =1

� 1a � ip

Scattering amplitude of two atoms

�

a

�c

no bound states

bound states

Ebinding = � 1

ma2

(~r1 � ~r2) ⇠ e�|~r1�~r2|

a

a�1 = � 4⇡

m�+

1

r0

One channel model: superconductor 19

H =X

p,�=",#

p2

2ma†p,�ap,� � �

V

X

q,p,p0

a q2+p,"a q

2�p,#a†q2�p0,#a

†q2+p0,"

n =

Zd3p

(2⇡)3

2

6641�p2

2m � µr⇣

p2

2m � µ⌘2

+�2

3

775 Δ - gap function

Unlike small g case, these equations are valid only for very small λ or very large λ

� m

4⇡a=

1

2

Zd3p

(2⇡)3

2

6641r⇣

p2

2m � µ⌘2

+�2

� 2m

p2

3

775

Small λ - BCS 20

Small λ - small and negative a - large integral - small Δ

Conventional weakly coupled superconductor

� m

4⇡a=

1

2

Zd3p

(2⇡)3

2

6641r⇣

p2

2m � µ⌘2

+�2

� 2m

p2

3

775

� ⇠ e⇡

4kF a

Large λ - BEC 21

Large λ - small and positive a.μ<0, |μ| >> Δ

� m

4⇡a=

1

2

Zd3p

(2⇡)3

"1

p2

2m + |µ|� 2m

p2

#

µ = � 1

2ma2

μ - binding energy of a diatomic moleculeThis is a Bose condensate of noninteracting (weakly interacting) diatomic molecules

� m

4⇡a=

1

2

Zd3p

(2⇡)3

2

6641r⇣

p2

2m � µ⌘2

+�2

� 2m

p2

3

775

It has a very high BEC transition temperature T ⇠ ~2m`2

If we arranged for this BEC among electrons in a solid, T would have been 10,000K

Chemical potential 22

µ

� 1

kFa

✏F

Unitary point 23

� ! �c a ! 1

�

a

�c

no bound states

bound states

fp = � 1

ip

The strongest possible scattering

Ground state energy is proportional to the Fermi energy in the absence of interactions

Eg = ⇠✏F

ξ is a constant which can be determined only numerically:

⇠ ⇡ 0.4

Transition temperature 24

Tc

� 1

kFa

Tc ⇠~2m`2

BCS: conventional superconductorBEC: weakly interacting diatomic molecules

Tc ⇠ ✏F e⇡

4kF a

T ⇤

µ =4⇡ab(n/2)

m

RG picture of the BCS-BEC crossover

25

µ

Interaction strength

VacuumUnitary critical point

Noninteracting Fermi gas fixed

point

Fermi liquidBCS Molecular BEC

Sachdev, ’06Radzihovsky, ‘06

0 �

µ =

�3⇡2n

�2/3

2m

µ =4⇡ab(n/2)

m

RG picture of the BCS-BEC crossover

25

µ

Interaction strength

VacuumUnitary critical point

Noninteracting Fermi gas fixed

point

Fermi liquidBCS Molecular BEC

Crossover

Sachdev, ’06Radzihovsky, ‘06

0 �

µ =

�3⇡2n

�2/3

2m

µ = ⇥

�3⇤2n

⇥ 23

2m

µ =4⇡ab(n/2)

m

RG picture of the BCS-BEC crossover

25

µ

Interaction strength

VacuumUnitary critical point

Noninteracting Fermi gas fixed

point

Fermi liquidBCS Molecular BEC

Crossover

Sachdev, ’06Radzihovsky, ‘06

At unitarity

Universal critical amplitude

� ⇥ 0.3� 0.45

0 �

µ =

�3⇡2n

�2/3

2m

Feshbach resonance for bosons 26

Molecular relaxation processes

Suppressed for fermions due to Pauli principle

Not suppressed for bosons

Results in the bosonic condensate lifetime close to resonance

⌧ ⇠ m`2

~That’s very short, less than a ms

These fly out of the condensate

27

The end