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