QMPT 540

Bosons• Symmetric N-boson states • Symmetrizer

• Symmetrized states

• with • Completeness (unrestricted)

• Ordered sp quantum numbers • Normalization

• Not ordered

S =1

N !

�

p

P

|�1�2...�N ⇥ =�

N !n�!n�� !...

⇥1/2

S |�1�2...�N )�

�

n� = N�

�1�2...�N

n�!n�� !...N !

|�1�2...�N ⇥ ��1�2...�N | = 1

ordered�

�1�2...�N

|�1�2...�N ⇥ ��1�2...�N | = 1

��1�2...�N |��1�

�2...�

�N ⇥ = ��1|��

1⇥��2|��2⇥...��N |��

N ⇥= ⇥�1,��

1⇥�2,��

2...⇥�N ,��

N

��1�2..�N |��1�

�2..�

�N ⇥ =

1[n�!..n�� !..]1/2

�

P

��1|��p1

⇥��2|��p2

⇥..��N |��pN

⇥

QMPT 540

Boson addition and removal operator• Notation

• As for fermions

• Two particles when • Otherwise

• General case

• Contrast with fermions: for bosons commutation relations

• from the requirement to generate symmetric states

|�1�2...�N ⇤ =�

N !n�!n�� !...

⇥1/2

S |�1�2...�N ) � |n�n�� ...⇤

|�� = a†� |0�

|�⇥� = a†�a†⇥ |0� � �= ⇥

|��� = |n� = 2� =1⇤2a†�a†� |0�

|n�n⇥ ...n⇤� =1

[n�!n⇥ !...n⇤!]1/2

�a†�

⇥n�⇤a†⇥

⌅n⇥

...�a†⇤

⇥n⇤ |0�

[a�, a†⇥ ] = a�a†⇥ � a†⇥a� = ��,⇥

[a�, a⇥ ] = [a†�, a†⇥ ] = 0

QMPT 540

Properties of boson operators• As for oscillators

• and

• As for fermions

• and

a†� |n�n⇥ ...n⇤� =⇤

n� + 1 |n� + 1 n⇥ ...n⇤�

a� |n�n⇥ ...n⇤⇥ =⌅

n� |n� � 1 n⇥ ...n⇤⇥

F =�

�⇥

��| F |⇥⇥ a†�a⇥

V =12

�

�⇥⇤⌅

(�⇥|V |⇤⌅)a†�a†⇥a⌅a⇤

QMPT 540

Statistical mechanics• Large systems • Finite temperature

• Grand canonical ensemble

• Statistical operator • with

• and chemical potential

• Grand partition function

�G =e��(H�µN)

ZG

� = (kBT )�1

µ

ZG = Tr�e��(H�µN)

⇥

=⇤

N

⇤

n

��Nn | e��(H�µN) |�N

n ⇥ =⇤

N

⇤

n

e��(ENn �µN)

QMPT 540

Thermodynamic potential• Standard result • and therefore

• Ensemble averages

• Noninteracting systems replace • Summing over complete sets of states in Fock space can also be

accomplished by summing over all possible occupations of sp states in occupation number representation!

• Replace relevant operators by eigenvalues:

• Apply

�(T, V, µ) = �kBT lnZG

�G = e�(��H+µN)

�O⇥ = Tr��GO

⇥=

Tr�e��(H�µN)O

⇥

Tr�e��(H�µN)

⇥

H � H0

H0 |n1...n�� =�

i

ni�i |n1...n��

N |n1...n�� =�

i

ni |n1...n��Z0 =

�

n1...n�

⌅n1...n⇥| e��(H0�µN) |n1...n⇥⇧

=�

n1...n�

⌅n1...n⇥| e��(P

i ⇤ini�µP

i ni) |n1...n⇥⇧

=�

n1...n�

exp {�(µn1 � ⇤1n1)}... exp {�(µn⇥ � ⇤⇥n⇥)}

=⇥⇥

i=1

Tr (exp {��(⇤i � µ)ni})

QMPT 540

Noninteracting grand partition function• Yields

• with Tr including a summation over possible occupation numbers • Bosons: all occupation numbers possible

• Thermodynamic potential

• Average particle number

Z0 =�

n1...n�

exp {�(µn1 � ⇤1n1)}... exp {�(µn� � ⇤�n�)}

=�⇥

i=1

Tr (exp {��(⇤i � µ)ni})

ZB0 =

⇥⇥

i=1

⇥�

n=0

[exp {�(µ� ⇤i)}]n =⇥⇥

i=1

[1� exp {�(µ� ⇤i)}]�1

�B0 (T, V, µ) = �kBT ln

⇥⇥

i=1

[1� exp {�(µ� ⇤i)}]�1

= kBT⇥�

i=1

ln [1� exp {�(µ� ⇤i)}]

N = ⇥N⇤ = ��

⇥�B0

⇥µ

⇥

TV

⇧N⌃ ⇥��

i=1

n0i =

��

i=1

1exp {�(⇤i � µ)}� 1

QMPT 540

Noninteracting fermions at finite T• Restriction to 0 and 1 for occupation numbers

• Thermodynamic potential

• Particle number

ZF0 =

�⇥

i=1

1�

n=0

[exp {�(µ� ⇤i)}]n =�⇥

i=1

[1 + exp {�(µ� ⇤i)}]

�F0 (T, V, µ) = �kBT

��

i=1

ln [1 + exp {�(µ� ⇤i)}]

⇧N⌃ ⇥��

i=1

n0i =

��

i=1

1exp {�(⇤i � µ)} + 1

n0i =

1

exp {�("i � µ)}+ 1

QMPT 540

BEC in infinite systems• Ground state of noninteracting bosons: all in lowest sp level • This limit is approached when T → 0

• boson spectrum

• As before

• Transform to energy integral

• Thermodynamic potential

�(k) =�2k2

2m�

i

� �V

(2⇥)3

⇥dk

�V

(2⇥)34⇥k2 dk =

�V

2⇥2

�2m

�2

⇥3/2 ⇤d⇤

2⇤1/2=

�V

4⇥2

�2m

�2

⇥3/2

⇤1/2d⇤

�B0 = kBT

⇤V

4⌅2

�2m

�2

⇥3/2 ⇤ �

0d⇧ ⇧1/2 ln [1� exp {�(µ� ⇧)}]

= � ⇤V

4⌅2

�2m

�2

⇥3/2 23

⇤ �

0d⇧

⇧3/2

exp {�(⇧� µ)}� 1

QMPT 540

BEC• Energy

• Particle number

• Note so one confirms ideal gas

• Denominator represents occupation so may not become negative so chemical potential such that so here

• Fix density and lower temperature: should decrease

• The limit for

E =⇤

i

n0i ⇧i =

⇤V

4⌅2

�2m

�2

⇥3/2 ⌅ �

0d⇧

⇧3/2

exp {�(⇧� µ)}� 1

N =⇤

i

n0i =

⇤V

4⌅2

�2m

�2

⇥3/2 ⌅ �

0d⇧

⇧1/2

exp {�(⇧� µ)}� 1

� = �PV PV =23E

⇥� µ ⇥ 0 µ � 0|µ|

µ = 0 N =⇥V

4⇤2

�2m

�2

⇥3/2 ⇤ �

0d⌅

⌅1/2

exp {⌅/kBT0}� 1

=⇥V

4⇤2

�2mkBT0

�2

⇥3/2 ⇤ �

0dx

x1/2

exp (x)� 1

=⇥V

4⇤2

�2mkBT0

�2

⇥3/2

�(32)12⌅

⇤

QMPT 540

BEC• Rewrite

• with Riemann -function

• For temperatures below

integral only yields particles with

so those particles represented by • Remaining particles must all have

• Macroscopic occupation (~ ) of single state → BEC

T0 =3.31�2/3

�2

mkB

�N

V

⇥2/3

�(32) = 2.612 �

T0

� > 0

N�>0(T ) = N

�T

T0

⇥3/2

� = 0

N�=0(T ) = N

⇤1�

�T

T0

⇥3/2⌅

N

QMPT 540

BEC for 4He• Check that at there is a discontinuity in the slope of the

specific heat (see Fetter and Walecka) • For 4He with

• Experimental transition at and ...

• called λ transition • N0 ~ 10%

• Superfluidity ≠ideal gas but transition still related to BEC!?

T0

� = 0.145 g cm�3 T0 = 3.14 KT0 = 2.2 K

Tilley & Tilley Superfluidity & Superconductivity

QMPT 540

BEC in traps• Laser cooling & magneto-optical

trapping techniques • Evaporative cooling

• Atomic gases are metastable

• Why? • Temperatures

• Densities

• Scales 10s to 100s of

• Magnetic traps for alkali atoms look like harmonic oscillators with different oscillator lengths

500 nK to 2µKfew �1014atoms/cm3

µm

Vext(r) =12

m��2

xx2 + �2yy2 + �2

zz2⇥

QMPT 540

Oscillators• Eigenvalues • Ground state: all atoms in state with

• Wave function

• with • bosons in this sp state

• Calculate density grows with

• shape does not and is determined by trap potential: • Actual scale

• Finite temperature: atoms occupy excited states

• For use classical Boltzmann distribution

• spherical (Landau&Lifshitz) • If width

⇥nxnynz = (nx + 12 ) ��x + (ny + 1

2 ) ��y + (nz + 12 ) ��z

nx = ny = nz = 0

⇥000(r) =�m⇤HO

��

⇥3/4exp

⇤�m

2� (⇤xx2 + ⇤yy2 + ⇤zz2)

⌅

�HO = (�x�y�z)1/3

|�N0 � =

1⇤N !

(a†000)N |0�

�(r) = N |⇥000(r)|2N

N

aHO =�

�m�HO

⇥1/2

aHO � 1 µm

kBT � ��HO

�cl(r) ⇥ exp��Vext(r)

kBT

⇥

Vext(r) =12m�2

HOr2 RT = aHO

�kBT

��HO

⇥1/2

QMPT 540

BEC observation• BEC observed in the form of sharp peak in the center • Wave function in momentum space also Gaussian: width

• So both in coordinate space and momentum space

• Infinite system all particles zero momentum but no signature in coordinate space

• Observe velocity distribution/ density distribution

Anderson et al.

Science 269, 198 (1995)

Rubidium atoms (velocity)

left: just above condensation

middle: just after

right: further cooled

asymmetric trap

� a�1HO

QMPT 540

Trapped bosons at finite temperature• Interaction between bosons important for the actual shapes • Still useful considerations for noninteracting bosons

• Number of particles

• Energy

• Usual thermodynamic limit not possible • Still separate lowest state from the sum with

• As for infinite system: can be of order when

• with

• This limit is reached for a critical temperature

N =⇤

nxnynz

1exp

��(⇤nxnynz � µ)

⇥� 1

E =⇤

nxnynz

⇤nxnynz

exp��(⇤nxnynz � µ)

⇥� 1

N0

N µ� µc =32

�⇥

� = (�x + �y + �z)/3

T = Tc

QMPT 540

Convert to integrals• Rewrite particle number

• do numerically or for

• semiclassical description: excitation energies level spacing

• valid for large and

• Integrate: • Imposing at

• yields

• or

• Evaluate energy similarly etc.

• Interaction important as are finite size corrections

N �N0 =�

nx �=0,ny �=0,nz �=0

1exp {��(⇥xnx + ⇥yny + ⇥znz)}� 1

N �⇥N �N0 =

� �

0dnxdnydnz

1exp {��(⇥xnx + ⇥yny + ⇥znz)}� 1

�N kBT � ��HO

N �N0 = �(3)�

kBT

�⇥HO

⇥3

�(3) � 1.202

N0 � 0 Tc

kBTc = �⇥HO

�N

�(3)

⇥1/3

= 0.94 �⇥HO N1/3

N0

N= 1�

�T

Tc

⇥3