Проект: Коллапс и квантовые ... - MIPT fileBose- Einstein condensation (BEC) ﬁrst realized in 1995 Atom’s de Broglie wavelength λdB = ~/(2πMkBT)1/2 is

Проект: "Коллапс и квантовые фазы вихрей в Бозе-Ферми смеси с

притяжением между бозонами и фермионами"

аналитической ведомственной целевой программы "Развитие научного потенциала

высшей школы (2006-2008 годы)

Рук. А.М. Белемук

НТС (МФТИ, 19 Декабря 2007) – p. 1/28

Introduction

Bose- Einstein condensation (BEC) first realized in 1995


Introduction


Atom’s de Broglie wavelength λdB = ~/(2πMkBT )1/2 is

small compared to the spacing between atoms ==>

quantum degeneracy


Introduction




quantum degeneracy

In most cases, quantum degeneracy would simply

be-preempted by the more familiar transitions to a liquid

or solid


Introduction




quantum degeneracy



or solid

Temperatures 500 nK - 2 µK


Introduction




quantum degeneracy



or solid


Densities 1014 - 1015 cm−3


Introduction




quantum degeneracy



or solid


Densities 1014 - 1015 cm−3

The full cooling cycle may take from a few seconds to as

long as several minutes


Introduction

The problem is interesting, because:


Introduction


– possible Fermi superfluidity, where the interaction

between fermions is mediated by bosons;


Introduction




– sympathetic cooling of fermions via a buffer gas of

bosons;


Introduction





bosons;

– wealth of novel quantum phases in these mixtures;


Introduction





bosons;

– wealth of novel quantum phases in these mixtures;

Example: mixture of 87Rb (bosons) and 40K (fermions).


Effective Hamiltonian

Grand-canonical partition function of the Bose-Fermi mixture:

Z =

∫

D[φ∗]D[φ]D[ψ∗]D[ψ] exp

{

−1

~(SB(φ∗, φ)+

+ SF (ψ∗, ψ) + Sint(φ∗, φ, ψ∗, ψ))} .

Here Bose field φ(τ, r) is periodic on the imaginary-time interval

[0, ~β], and Fermi (Grassmann) field ψ(τ, r) is antiperiodic on

this interval.

SB(φ∗, φ) =

∫

~β

0

dτ

∫

dr

{

φ∗(τ, r)

(

~∂

∂τ−

~2∇2

2mB

+ VB(r)−

− µB)φ(τ, r) +gB

2|φ(τ, r)|4

}

.



SF (ψ∗, ψ) =

∫

~β

0

dτ

∫

dr

{

ψ∗(τ, r)

(

~∂

∂τ−

~2∇2

2mF

+

+ VF (r) − µF )ψ(τ, r)} .

Sint(φ∗, φ, ψ∗, ψ) = gBF

∫

~β

0

dτ

∫

dr|ψ(τ, r)|2|φ(τ, r)|2,

where gB = 4π~2aB/mB, gBF = 2π~

2aBF/mI ,

mI = mBmF/(mB +mF ), aB and aBF - boson-boson and

boson-fermion s wave scattering lengths respectively.



Integral over Fermi fields is Gaussian, we can calculate this

integral and obtain the partition function of the Fermi system

as a functional of Bose field φ(τ, r).

ZF =

∫

D[ψ∗]D[ψ] exp

(

−1

~(SF (ψ∗, ψ) + Sint(φ

∗, φ, ψ∗, ψ))

)

=

=

∫

D[ψ∗]D[ψ] exp

{∫

~β

0

dτ

∫

dr

∫

~β

0

dτ ′∫

dr′×

× ψ∗(τ, r)G−1(τ, r, τ ′, r′)ψ(τ ′, r′)}

,

where

G−1 = G−10 − Σ

is the Dyson equation, and Σ(τ, r, τ ′, r′) is a selfenergy.



G−10 (τ, r, τ ′, r′) = −

1

~

(

~∂

∂τ−

~2∇2

2mF

+ VF (r)−

− µF ) δ(r − r′)δ(τ − τ ′).

Σ(τ, r, τ ′, r′) =gBF

~|φ(τ, r)|2δ(r − r′)δ(τ − τ ′).

Gaussian integral over the Grassmann variables:

∫

∏

n

dψ∗ndψn exp

{

−∑

n,n′

ψ∗nAn,n′ψn′

}

= detA = eSp[ln A]

ZF = exp(

Sp ln(

−G−1))

= exp

(

−1

~Seff

)



G−1 = G−10 − Σ = G−1

0 (I − G0Σ),

Sp(ln(−G−1)) = Sp(ln(−G−10 )) −

∞∑

n=1

1

nSp [(G0Σ)n] .

G0(τ, r, τ′, r′) =

∑

ω,n

−~

−i~ω + ǫn − µF

ξn(r)ξ∗n(r′)e−iω(τ−τ ′)

~β,

where ω = π(2s+ 1)/~β; s = 0,±1, ... and(

−~

2∇2

2mF

+ VF (r)

)

ξn(r) = ǫnξn(r).



In the semiclassical Thomas-Fermi approximation one has:

∑

n

ξn(r)ξ∗n(r)F (ǫn) =1

(2π~)3

∫

dpF (H0(p, r)),

where H0(p, r) = p2/(2mF ) + VF (r) and F (x) is an arbitrary

function.

We suppose that all |φ(τi, ri)|2 have one and the same argument

(τ1, r1). In this approximation we have:

Seff =

∫

~β

0

dτdrfeff (|φ(τ, r)|),

feff (|φ(τ, r)|) = −β−1

(2π~)3

∫

dp ln(

1 + eβ(µF−gBF |φ(τ,r)|2−H0(p,r)))

.



The effective boson Hamiltonian:

Heff =

∫

dr

{

~2

2mB

|∇φ|2 + (VB(r) − µB)|φ|2+

+gB

2|φ|4 + feff (|φ|)

.



The effective boson Hamiltonian:

Heff =

∫

dr

{

~2

2mB

|∇φ|2 + (VB(r) − µB)|φ|2+

+gB

2|φ|4 + feff (|φ|)

.

In low temperature limit µ/(kBT ) ≫ 1:

feff (|φ|) = −2

5κµ5/2 −

π2

4κ(kBT )2µ1/2,

where µ = µF −VF (r)− gBF |φ(τ, r)|2 and κ = 21/2m3/2F /(3π2

~3).



The effective boson Hamiltonian for gBF < 0 at T = 0:

Heff =

∫

dr

{

~2

2mB

|∇φ|2 + (Veff (r) − µB)|φ|2+

+geff

2|φ|4 +

κ

8µ1/2F

g3BF |φ|

6

}

.

Veff (r) =

(

1 −3

2κµ

1/2F gBF

)

1

2mBω

2B

(

x2 + y2 + λ2z2)

,

geff = gB −3

2κµ

1/2F g2

BF ,

µF = ~ωF (6λNF )1/3.


Variational approach

Variational boson wave function:

φ(r) =

√

NBλ

ω3a3π3/2exp

(

−(x2 + y2 + λ2z2)

2w2a2

)

.

a =

√

~

mBωB

.

Variational energy:

EB

NB~ωB

=2 + λ

4

1

w2+ bw2 +

c1NB

w3+c2N

2B

w6,

b =3

4

(

1 −3

2κµ

1/2F gBF

)

,



c1 =1

2

(

gB −3

2κµ

1/2F g2

BF

)

λ

(2π)3/2~ωBa3,

c2 =κ

8µ1/2F

g3BF

λ2

33/2π3~ωBa6.

Experimental system: Mixture of fermionic 40K and bosonic87Rb (Mogundo et al, Science 297, 2240 (2002)):

aB = 5.25 nm, aBF = −21.7+4.3−4.8 nm

Critical values: NBc ≈ 105;NK ≈ 2 × 104


Vortex state (variational approach)

Variational function

φ(r) =

√

λNB

(ωa)5π3/2ρ exp

(

−ρ2 + λ2z2

2ω2a2

)

eiϕl

Variational energy

Eb

NB~ωB

=2 + λ2 + 2l2

4

1

ω2+Bω2 +

C1NB

ω3+C2N

2B

ω6,

B =5

3b,

C1 =1

2c1,

C2 =2

9c2.



Fig. 1. Variational energy EB

NB~ωB

as a function of w for various numbers of

bosons.

Fig. 2. Critical number of bosons NBc as a function of the number of

fermions NF .



4,3 4,4 4,5 4,6 4,7 4,8 4,9 5,00,54

0,56

0,58

0,60

0,62

0,64

0,66

0,68

0,70

0,72

0,74 NF=20000 NF=200000 NF=500000 NF=100000

c

log10(NB)4,2 4,4 4,6 4,8 5,0 5,2 5,4 5,6 5,8 6,0

0,54

0,56

0,58

0,60

0,62

0,64

0,66

0,68

0,70

0,72

NB=50000 NB=70000 NB=40000 NB=80000

c

log10(NF )

Fig. 3. Critical angular vortex velocity as a function of a number of bosons

NB .


NF .НТС (МФТИ, 19 Декабря 2007) – p. 16/28


Fig. 5. Density distribution of Fermions.



Fig. 5. Critical number of bosons as a function of boson-fermion scattering

length aBF .НТС (МФТИ, 19 Декабря 2007) – p. 18/28

Thomas-Fermi Approximation

The Gross-Pitaevski equation that follows from the effective

hamiltonian:(

−~

2

2mB

△φ+ (Veff − µB) + geff |φ|2 +

3κ

8µ1/2F

g3BF |φ|

4

)

φ = 0.

At high enough densities one can neglect the kinetic energy.

|φ|2 = θ(µB − Veff)geff

−1 +√

1 + (µB − Veff) 3κ

g2

eff2µ

1/2

F

g3BF

3κ

4µ1/2

F

g3BF

→

→ θ(µB − Veff)(µB − Veff)

geff

, if gBF → 0.



Density of Bosons:

n(r) = n(0)

(

1 −

√

1 +x2 + y2 + λ2z2 −R2

R2max

)

.

Here

ri = ri/ah; R2 =

2µB

c0mBω2Ba

2h

;

n(0) = −4

3

geffµ1/2F

κg3BF

; R2max = −

4

3

g2effµ

1/2F

κg3BF c0mBω2

Ba2h

;

c0 =

(

1 −3

2κµ

1/2F gBF

)

.



Vortex state in TFA: φ(r) =√

n(r)eiϕl,

where n(r) = n(0)

(

1 −

√

1 +ρ2+λ2z2+ l2

c0ρ2−R2

R2max

)

.


Расчет

−1.5−1

−0.50

0.51

1.5

−1.5

−1

−0.5

0

0.5

1

1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

−1.5−1

−0.50

0.51

1.5

−1.5

−1

−0.5

0

0.5

1

1.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Распределение бозонной плотности в магнитной ловушке для l = 0 и

l = 1.




NB in TFA.

Fig. 10. Critical angular vortex velocity as a function of a number of

fermions NF in TFA.НТС (МФТИ, 19 Декабря 2007) – p. 23/28

Numerical solution of GGPE

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

r

φ(r)

φho

n′c= 1

n′c= 5

n′c= 10

n′c= 15

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

r

φ(r)

φho

n′c= 20

n′c= 40

n′c= 60

n′c= 90

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

r

φ(r)

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

r

φ(r)

n′c= 7 n′c= 20

Fig. 11. Evolution of the profile of the condensate wave function with

increasing central density n′

c. φho = π−3/4exp(−r2/2) corresponds to the

ground state of the ideal Bose.

Fig. 12 . The profile of the condensate wave function φ(r), found from the

numeric solution of GGPE (solid line), and the TF approximation

(dashed-dotted line).


Experiment

Fig. 6. Experimental stability diagram for the 40K-87Rb mixture(C.

Ospelkaus, S. Ospelkaus, K. Sengstock, and K. Bongs, Phys. Rev. Lett. 96,

020401 (2006)). The solid line is based on the present theory and

aBF = −284a0.


Nature of the collapse transition.

A strong rise of density of bosons and fermions in the collapsing

condensate enhances intrinsic inelastic processes, in particular,

the recombination in 3-body interatomic collisions, as is the

case for the well-known 7Li condensates. In the presence of a

vortex there appears a hole in the middle of the condensate.

This reduces the maximum density of the condensate and

increases the critical number of bosons. However, for the

Bose-Fermi mixtures with attraction between components the

formation of the boson-fermion bound states is also possible

(M. Yu. Kagan et al, cond-mat/0209481). It seems that the

description of the evolution of the collapsing condensate should

include both these mechanisms.


Summary

Using the effective Hamiltonian for the Bose system, the

instability and collapses of the trapped boson-fermion mixture

due to the boson-fermion attractive interaction in the presence

of the quantized vortices was investigated.


Summary





We analyze quantitatively properties of the 87Rb and 40K

mixture. The calculated instability boson number NBc for the

collapse transition is in agreement with experiment.


Summary





We analyze quantitatively properties of the 87Rb and 40K

mixture. The calculated instability boson number NBc for the

collapse transition is in agreement with experiment.

References: A.M. Belemuk, N.M. Chtchelkatchev, V.N. Ryzhov, S.-T. Chui,

"Vortex state in a Bose-Fermi mixture with attraction between bosons and

fermions", Phys. Rev. A 73, 053608 (2006); А.М. Белемук, В.Н. Рыжов, С.Т.

Чуи, "Механизм коллапса конденсатной волновой функции в

бозе-ферми-смеси с притяжением между компонентами", Письма в ЖЭТФ,

2006, т. 84, вып. 6, стр.354-359; A.M. Belemuk, V.N. Ryzhov, S.-T. Chui, "Stable

and unstable regimes in Bose-Fermi mixtures with attraction between

components", Phys. Rev. A 76, 013609 (2007); А.М. Белемук, В.Н. Рыжов, С.Т.


Заключение

"Сильно коррелированные электронные системы и квантовые критические

явления"(Троицк, 2007); "23rd International Conference on Statistical

Physics"(Genova, Italy, 2007).


Заключение

"Сильно коррелированные электронные системы и квантовые критические

явления"(Троицк, 2007); "23rd International Conference on Statistical

Physics"(Genova, Italy, 2007).

Thank you!


Documents

Проект: Коллапс и квантовые ... - MIPT fileBose- Einstein condensation (BEC) ﬁrst realized in 1995 Atom’s de Broglie wavelength λdB = ~/(2πMkBT)1/2 is