Feshbach resonance and BCS-BEC crossover · 2012-08-20 · RG picture of the BCS-BEC crossover 25...
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Feshbach resonance and BCS-BEC crossover Victor Gurarie Windsor, August 2012 1
Transcript of Feshbach resonance and BCS-BEC crossover · 2012-08-20 · RG picture of the BCS-BEC crossover 25...
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
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The end
