bec-bcs crossover in a bose gas
DESCRIPTION
This presentation is about an earlier version of the work published in Phys. Rev. A 79, 063609 (2009) [arXiv:0809.4189]. Made in Powerpoint 2003 + TeX4ppt, source and high-resolution images available upon request.TRANSCRIPT
Rembert Duine
Arnaud Koetsier
Henk Stoof
Pietro Massignan
Bac
kgro
und
imag
e: ©
2000
Tor
Ola
v K
riste
nsen
2
Introduction
• Fermions must form pairs in order to undergo BEC. (Old news…)
• Bosons can undergo BEC solitarily. (Older news…)
• Bosons can also form pairsWhat about a paired-boson BEC?Is there a crossover between:a BEC condensate of pairs tightly bound by two-body effects,
anda BCS condensate of pairs loosely bound by many-body
effects?
Let’s do a NoziNozièèresres--SchmittSchmitt--RinkRink calculation for bosons and find out!
3
Single Channel Model
• Use a single-channel model of an interacting bose gas.
• Action:
• Local interaction:
S = S0 + Sint
V (x− x0) = V0δ(x− x0)
S0[φ∗,φ] =
Z ~β
0
dτ
Zdx φ∗(x, τ )
½~∂τ −
~2∇22m
− μ¾φ(x, τ)
Sint[φ∗,φ] =
1
2
Z ~β
0
dτdτ 0Zdxdx0 φ∗(x, τ )φ∗(x, τ)V (x− x0)φ(x0, τ 0)φ(x0, τ 0)
4
• Expand partition function as usual
• Nozières-Schmitt-Rink: neglect bubble diagrams.
Partition Function
Z =
ZDφ∗Dφ e− 1
hS0[φ∗,φ]e−
1hSint[φ
∗,φ]
=Z0he−1hSint[φ
∗,φ]i0
∼Z0µ1− 1
~hSint[φ∗,φ]i0 +
1
~21
2!hSint[φ∗,φ]2i0 + · · ·
¶∼Z0
µ1 +
1
22 +
1
2!
µ1
2
¶2 "4 + 4 + 16
#+ · · ·
¶'Z0 exp
∙+1
2+ · · ·
¸
5
T-matrix
• Lippmann-Schwinger equation:
• has a UV-divergence:
T = +T
+1
2T + · · ·=
V0Ξ(K, iΩn)
=−1~2βV
Xn,n0
Xk,k0
G(k, n)
∙V0 +
1
2V 20 Ξ(k+ k
0, n+ n0) + · · ·¸G(k0, n0)
=Xn,K
∞Xp=1
1
p
£V0Ξ(K, iΩn)
¤p=Xn,K
ln
∙1
1− V0Ξ(K, iΩn)
¸εk =
~2k22m
1
V
XK
1
iΩn − 2εK
After renormalization, this becomes the T-matrix
6
Renormalization to the two-body T-matrix
The divergence is related to the 2-body T-matrix:
Subtracting the divergence from then gives:
T 2B(z) =4π~2am
1
1− ap−zm/~2
2-body T-matrix:
Ξ(k, iωn) =1
V
Xq
∙N(ξk/2+q) +N(ξk/2−q)
i~ωn − ξk/2−q − ξk/2+q
¸Renormalized Correlation function:
T
T 2B(z) =1
V0− 1
V
XK
1
z − 2ξK Will not contribute to n, P
Ξ
=XΩn,K
µln
∙T 2B(iΩn)
1− T 2B(iΩn)Ξ(K, iΩn)
¸+ ln
1
V0
¶
ξk =~2k22m − μ
7
• Recall:
• Grand potential:
Grand thermodynamic potential
T
Many-Body T-matrix:
Ideal atomic gas Paired atoms
2-body T-matrix
T 2B(z) =4π~2am
1
1 − ap−zm/~2
Renormalized Correlation function
Ξ(k, iωn) =1
V
Xq
∙N(ξk/2+q) +N (ξk/2−q)
i~ωn − ξk/2−q − ξk/2+q
¸
Z =Z0 exp
∙+1
2+ · · ·
¸= Z0 exp
∙ ¸
TMB(k, iωn) =T 2B(z)
1 − T 2B(z)Ξ(k, iωn)
Ω ≡− 1
βVlnZ
=1
β
Xk
ln[1− exp(−βξk)] +1
β
Xn,k
lnT 2B(0)
TMB(k, iωn)
8
Finding Tc
1. Number equation
gives2.2. Thouless criterionThouless criterion
at we have:
− 1V
∂Ω
∂μ= n
T = Tc
μ(T )
1
T (0, 0)= 0
-1/n1/3a
T c / T a
-10 0 10 20 300
0.2
0.4
0.6
0.8
1
T = T 2B(k, iωn)
T = TMB(k, iωn)
Pair Condensation
reso
nanc
e
Ω =Ω0 + kBTXn,k
lnT 2B(0)
T (k, iωn)
9
Spectral function
• Why do many-body effects significantly lower Tc?
Many-body, finite-lifetime resonance
→ Many-body picture allows for pairs on the right of the resonance.
Two-body picture does not.
hω / k B
Ta
-Im
[ T( k
, ω+
i 0)]
(a
rbitr
ary
units
)
0 30 60 900
0.5
1
1.5
2
-1/n1/3a ε
m /
k B T
a
-1 0 1 2 3-5
-4
-3
-2
-1
0
Many-body
2-body
No resonance
10
Pressure of the gas – does the gas collapse?
• Pressure:• Compressibility positive: mechanically stable
n Λ(T)3
P /
n k B
T
0 5 10 150
1
2
3Near resonance
Deep BEC regime
Deep BCS regime
P = −∂Ω/∂V∝ ∂n/P
Ideal Molecular:P = nkBT/2
Ideal Atomic:P = nkBT
11
Below Tc: Formation of atomic condensate
• Below Tc atomic dispersion becomes:→ Atomic condensation criterion:
-1/n1/3a
T c / T a
AC+PCPC Tca
Tc
-10 -5 0 5 100
0.2
0.4
0.6
0.8
1
~ωk =qξ2k − |∆0|2
|∆0| = −μ
Pairing gap is related to the condensate fraction :nc
|∆0|2 = 2nc∙∂
∂μ
1
TMB(0, 0)
¸−1
nc = ~N(0)
V
∂TMB(0, 0)
∂μ
∙∂TMB(0, 0)
∂iωn
¸−1Condensate fraction related to zero-momentum divergence of the T-matrix:
12
10 20 30 40 5010
8
6
4
2
0
6g(6)
bindin
gene
rgyE b/h
(MHz
)
magnetic field (G)
2g4d6l(5)6s6l(4)4g(4)6l(3)4g(3)
Cesium: a good candidate for the observing the crossover
• Cs has many Feshbach resonances and bound states…• Need a molecule that is stable against inelastic losses.• [Ferlaino, et al. ’08], [Knoops, et al. ’08]: Look in this region
[Mark et.al. 07]
13
Close-up of region: avoided crossing
• Crossover: Populate upper branch only.• Need to take 6s state into account:
B - B 0 (G)
E (
kHz)
6s
4d
-2 -1 0 1-100
-80
-60
-40
-20
0
-0.1 0
-4
-2
0
BEC BCSCesium 2-body T-matrix:
Energy-dependent scattering length:
B-dependent background:
abg(B) = (1722 + 1.52B)
∙1 − 28.72
B + 11.74
¸a0
a(z) = abg(B)
∙1 +
∆μ∆B
z − δ
¸T 2BCs (z) =
4π~2a(z)m
∙1
a(z)−r−zm~2
¸−1
14
Tc for Cesium
• Solve for Thouless criterion with :[TMB(0, 0)]−1 = 0 T 2BCs (z)
ΔB
Tm/Ta
B - B 0 (G)
Tc /
T a
-0.2 -0.1 0 0.1
0.2
0.4
0.6
0.8
1
Tc line:
Atomic Tc lines
Tc line:n = 10−12cm−3
n = 10−13cm−3
BEC BCS
15
Discussion
• Experimental observation of pairs:atom shot noiseRF spectroscopySupression of atomic Tc (pairs reduce atomic density)
• Half-vortex unbinding transition present across atomic Tc line.
• Inelastic losses present at all temperatures can cause resonant pairs to decay into deeply bound states (Effimov states, atom-dimerrelaxation, dimer-dimer relaxation).
• Mechanical stability below Tc is uncertain. Need to explicity include condensate contribution to the grand potential: NSR calculation below Tc is very difficult!
• Minor corrections, e.g. non-analytic dependence of Tc on the scattering length above [Holzmann, et al. ’01]B −B0 > ∆B
16
Conclusion
• Tc calculated for bosonic atoms across the Feshbach a resonance
• Medium effects give rise to pair formation possible for negative scattering lengths where two atoms can not pair in the vacuum
• Pairs → strong supression of Tc
• Compressibility > 0 throughout the crossover: gas is mechanically stable
• Cs is a good candidate for experimental observation