2009 1202 NTHU BEC的熱力特性 Kao.pdf · 2016. 2. 19. · Tisza-Landau two-fluid hydrodynamics (1938~1941) Laszlo Tisza (July 7, 1907 – April 15, 2009 Lev Landau (1908~1968)

Dr. 柯宜謀

The Thermodynamic Properties of Bose-Einstein Condensation

彰化師大物理系

專長 : 統計力學超冷原子

玻色-愛因斯坦凝聚態的統計特性

2009_1201 於清華物理系

Outline :

- Bose-Einstein Condensate 簡介

- Thermodynamic Limit

- Weakly Interacting

- Phase Transition

(1901) Black Body Radiation

Max Planck (1858-1947)

/ 1h kThE

e νν

=−

, 0,1,2,3,quantaE nh nν= = L

(1905)The Photoelectric Effect

photonE hν=

Albert Einstein(1879~1955)

(1917) stimulated emission3

212 3 /

112 21

2

1( ) 1T kT

AWc eN B B

N

ω

ωωπ

= =−⎛ ⎞

−⎜ ⎟⎝ ⎠

h

h

The Bose-Einstein Distribution

Satyendra N. Bose, (1894~1974)

For photon (Bose,1924) :

** Identical particle **

For atoms (Einstein,1925) :

( ) /1

1E kTf E

e=

−

( ) ( ) /1

1E kTf E

e μ−=

−μ is a Lagrangian multiplier for the conservation of particle number

Albert Einstein(1879~1955)

Einstein considered N non-interacting bosonic and non-relativistic particles in a cubic box of volume L3 with periodic boundary conditions. In the thermodynamic limit, defined as

N,L→∞ with N / L3 = ρ = constant,

22

31

3/ 2, , ( )( ) , j

j B

zz e g zj mV T

Nk

g z βμη η

πλ

λ∞

=

== = =∑ h

a phase transition occurs at a temperature Tc defined by:

Bose–Einstein condensate

0

3/ 2

0

0

1

c

cc

NT TN

N TT TN T

> →

⎛ ⎞< = − ⎜ ⎟

⎝ ⎠

( )31

3 12.612, 2dB j jη

ρλ ζ ζ η∞

=

⎛ ⎞= ≅ =⎜ ⎟⎝ ⎠

∑

For T < Tc the system has formed a Bose–Einstein condensate in p = 0.

The number N0 of particles in the condensate is on the order of N, that is macroscopic.

As we will see, the macroscopic population of a single quantum state is the key feature of a Bose–Einstein condensate, and gives rise to interesting properties, e.g. coherence (as for the laser).

Wolfgang Ketterle

李政道 1979 統計力學p.57 凡異出版社

李政道(1926~) (1957)

Family tree of atomic physicists.People with names in italics are Nobel laureates.

Rev. Mod. Phys., Vol. 74,p.1131 , 2002

Spin-polarized Hydrogen

Hecht (1959)Stwalley and Nosanow (1976)

Kleppner at MIT, since 1985,by magnetic trapping andevaporative cooling.(1999)Hydrogen BEC

Daniel Kleppner awarded Wolf Prize in Physics

The Nobel Prize in Physics 2001

Eric A. Cornell Wolfgang Ketterle Carl E. Wieman

"for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates"

Magneto-optical TrapLaser Cooling

Magnetic Field Gradient

Why interesting?I. Simple systems for the theory

An important theoretical frame for Bose–Einstein condensation in interacting systems was developed in the 50’s by Beliaev, Bogoliubov, Gross,Pitaevskii in the context of superfluid helium.

II. New features : (respect to superfluid He 4)Spatial inhomogeneity

Finite size effects

Tunability

This theory however is supposed to work better if applied to Bose condensed gases where the interactions are much weaker.

Gross, E. P., 1961, Nuovo Cimento 20, 454.Pitaevskii, L. P., 1961, Zh. Eksp. Teor. Fiz. 40, 646 [Sov. Phys. JETP 13, 451 (1961)].

2 2 22

04 ( ) | | ,

2V r g g a

m mπμ

⎛ ⎞∇Φ = − + Φ + Φ Φ =⎜ ⎟

⎝ ⎠

h h

2 2 2 2 2 21 1 12 2 2x y z

V x y zω ω ω= + +

Grand Canonical Ensemble for non-interacting particles

I. Energy Level

II. Grand Partition Function

III. Thermodynamic Functions

ˆ i i iE Hψ ψ=

0

3 3( , )

3, , r! Tn

N NEH p qN

NN

N Nn

d pd qz e Q e eN

D z Qh

ββμ β ρ−∞

=

−= = == =∑∫∑

0

ln , ln , ln

, ,

( , , ) , ( , , )

B

T VV

V

A N V T U TS G N P T A PV

PV D N z D U Dk T z

CUC S dTT T

β∂ ∂

= = = −∂ ∂

∂⎛ ⎞= =⎜ ⎟

= −∂

=⎠

+⎝ ∫

Thermodynamic limitStatistical Mechanics Kerson Huang 2nd p.289

Grand Canonical Ensemble for non-interacting Bose gas

5/ 23

3/ 23

1 1 ( ) ln(1 )

1 1 ( ) 1

B

P g z zk T V

N zg zV V z

λ

λ

= − −

= −−

Free Gas, no trap

Exact to B

Ek T

⎛ ⎞Δ⎜ ⎟⎝ ⎠

Trapped Gas

11ln ( ) ln(1 )

1 ( ) 1

char

B

char

B

ED g z zk T V

E zN g zk T V z

η

η

η

η

+

⎛ ⎞= − −⎜ ⎟

⎝ ⎠

⎛ ⎞= −⎜ ⎟ −⎝ ⎠

Euler's Theorem for homogenous functions

, ,,

dU TdS PdV dN U TS PV NA U TS PV N G A PV N

μ μμ μ

= − + = − += − = − + = + =

Expansion of Bose-Einstein function F. London

( )

1/ 22 3

1/ 2

1/ 2 2 33/ 2

1/ 23 2 35/ 2

( ) 1.460 (0.208) (0.0128) ( )

( ) 2.612 2( ) (1.460) (0.104) ( )4( ) 1.342 (2.612) (0.730) ( )3

g e O

g e O

g e O

α

α

α

π α α αα

πα α α α

α πα α α

−

−

−

⎛ ⎞= − + − +⎜ ⎟⎝ ⎠

= − + − +

= − + − +

Chebyshev Polynomial ExpansionIf η≠ integer, and – π < α < 0 , then ( )

10

( ) ( 1 )! ( 1)sin( )

j

j

g e jj

ηα

η

π αα ζ ηη ηπ

∞

+=

−= + − −

Γ +∑

If η≠ integer, and 0 < α < π , then ( )1

0

( ) ( 1 )! ( 1) tan( )

j

j

g e jj

ηα

η

π αα ζ ηη ηπ

∞

+=

= + − −Γ +∑

If η = integer, and -2π < α < 2π then

( ) [ ]10

( ) ( 1 ) ln | | ( 1) (1)! ( 1)

j

j

g e jj

ηα

ηη

αα ζ η α ηη

∞

+= ≠

= + − − −Ψ + +ΨΓ +∑ where ( ) ln ( )

dd

η ηη

Ψ = Γ

Note on the Bose-Einstein Integral Functions (John E. Robinson,1951)

Two-fluid model in BECTisza-Landau two-fluid hydrodynamics (1938~1941)

Laszlo Tisza (July 7, 1907 – April 15, 2009

Lev Landau (1908~1968)

Superfluid : component of liquid which is associated with macroscopic occupation(BEC) of a single particle state .Carries zero entropy, flows without dissipation with irrotational velocity.

Normal fluid : comprised of incoherentthermal excitations , behaves like any fluid at finite temperatures in local thermodynamic equilibrium. This requires strong collisions.

A. Griffin, Phys. Rev. B, Vol 53, p.9341 (1996)

The condensate part

and the thermal part

Diverge ?

Projected Gross-Pitaevskii equationGardiner-Zoller Quantum Kinetic Theory

Projectors :

Condensate band and non-condensate band

In weakly interacting,

Mean Field Theorem 是否足以決定相變溫度 ?

PHYSICAL REVIEW A ,R4633 (1996)

Condensate fraction and critical temperature of a trapped interacting Bose gasS. Giorgini, L. P. Pitaevskii and S. Stringari

1/ 600 1.33c

c ho

T a NT aΔ

− where is the harmonic oscillator length.hoa mω=h

Phys. Rev. Lett. 92, 030405,(2004)

δTc/Tc0 = αN1/6,Theo.α = - 0.07Exp. α = - 0.09

Phys. Rev. Lett. 96, 060406,(2006)

A1: This is the first-order analytic estimate of Giorgini et al.A2: This is the full second-order result

PHYSICAL REVIEW A 73, 043604 (2006)

Transition temperature of a weakly interacting Bose gas

2 2 ,

2 ,

idealT T TT cT

idealc c Tc cc

E E gn gn

E E gn gn

⎧ = + +⎪⎨

= + +⎪⎩

1/ 600 (1.33 0.45)c

c ho

T a NT aΔ

− +

1900年初，當時在英國皇家學會的新年致辭中，發表了題為「籠罩在熱和光的動力理論上的十九世紀之雲」的著名演講。他說

“物理的大廈已經落成，所剩的只是一些修飾的工作 …”

Yee-Mou Kao and T. F. Jiang

PRL 98, 080407 (2007)

Critical Temperature of Weakly Interacting Dipolar Condensates

Dipolar BEC

Konstantin Glaum, Axel Pelster, Hagen Kleinert, and Tilman Pfau

PHYSICAL REVIEW A 75, 033607 2007

Transition temperature of the interacting dipolar Bose gas

1/ 600 (2.78 5.46) ( )

c d

c ho

T a NT a

χ κΔ − +


In three-dimensional space, we show that the shift of Tc changes its sign from a negative value for n < 3 to a positive one for n > 3, where n is the exponent of the power-low potential.

International Journal of Modern Physics BVol. 17, No. 12 (2003) 2439-2446© World Scientific Publishing Company

CRITICAL TEMPERATURE OF A WEAKLY INTERACTING BOSE GAS IN A POWER-LAW POTENTIAL

HIDENORI SUZUKI and MASUO SUZUKI

Department of Physics, Graduate School of Science,Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan [email protected]

Received 20 December 2002

Mean field approximation ( ) nextV r Ar=r

Mean-field analysis of Bose–Einstein condensation in general power-law potentials

J. Phys. B: At. Mol. Opt. Phys. 37 (2004) 2593–2605

O ZobayInstitut f¨ur Angewandte Physik, Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany

1 2 31 2 3

1 1 1 1( ) , 2

sp l

extx y zV r E E EL L L p l s

η⎛ ⎞⎛ ⎞ ⎛ ⎞

= + + = + + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

r

0 , c

c T

T at qT λΔ

= =

Mean-field calculation of critical temperature

Mean-field theory is applied to describe the condensation of dilute interacting Bose gases in general power-law potentials.


Phase transition of interacting Bose gases in general power-law potentials

Institut für Angewandte Physik, Technische Universität Darmstadt, 64289 Darmstadt, Germany

O. Zobay, G. Metikas, and G. Alber

Using energy-shell renormalization and the ε expansion

Bold curves: RG results with ε expansion. Dashed curves: mean-field LDA. Dotted curves: linear mean-field approximation


Nonperturbative effects on Tc of interacting Bose gases in power-law traps

O. Zobay, G. Metikas, and H. Kleinert

calculated with the help of variational perturbation theory.

Schematic diagram

Phase Transition of Trapped Interacting Bose Gases

O. Zobay

Laser Physics, 2009, Vol. 19, No. 4, pp. 700–724.

A first calculation by Lee and Yang in 1957 [P.R.105,p.1119] obtained an increase in Tc compared to the ideal gas that is proportional to the square root of the scattering length.0a

It is only since about 1999 that a consistent description of the critical Bose gas has emerged and has led to a generally accepted result for the critical temperature.

1/300

c

c

T ca nTΔ

=

Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom

2

int 2

,

, c

c

k k kE

k k k

η−⎧ ≤⎪⎨

≥⎪⎩

In weakly interacting,

Mean Field Theorem 是足以決定相變溫度 !

Eur. Phys. J. D /e2006-00157-4 (2006)

Transition temperatures of the trapped ideal spinor Bose gas


0particle number magnetization

N N N NM N N

+ −

+ −

= + +

= −

For a gas of 87Rb atoms.

PHYSICAL REVIEW A 70, 043611 (2004)Interacting Spinor BECWenxian Zhang, Su Yi, and L. You

Several Phase Transition ?

Paul Erdös(1913~1996)

Dr. Erdös defined the word"mathematician"

as"a machine for turning

coffee into theorems."

The End

2009 1202 NTHU BEC的熱力特性 Kao.pdf · 2016. 2. 19. · Tisza-Landau two-fluid hydrodynamics (1938~1941) Laszlo Tisza (July 7, 1907 – April 15, 2009 Lev Landau (1908~1968)

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