Download Bose-Einstein condensation of an ideal gas Thorsten K ¨ohler

Bose-Einstein condensation of an
ideal gas
Thorsten Köhler
Department of Physics and Astronomy, University College London,
Gower Street, London, WC1E 6BT, United Kingdom
Cold gases 2009 – p.1
Outline
Planck’s radiation formula
Cold gases 2009 – p.2
Outline
Planck’s radiation formula
Modern interpretation of Planck’s formula
Cold gases 2009 – p.2
Outline
Planck’s radiation formula
Modern interpretation of Planck’s formula
Bose-Einstein condensation of an ideal gas
Cold gases 2009 – p.2
Outline
Planck’s radiation formula
Modern interpretation of Planck’s formula
Bose-Einstein condensation of an ideal gas
Bose-Einstein condensation in atom traps
Cold gases 2009 – p.2
Outline
Planck’s radiation formula
Modern interpretation of Planck’s formula
Bose-Einstein condensation of an ideal gas
Bose-Einstein condensation in atom traps
Literature
Cold gases 2009 – p.2
Planck’s radiation formula
Blackbody radiation
Electric field inside a perfectly reflecting
cavity:
1 ∂2
c2 ∂t2
z
L
− ∇2 E(r, t) = 0
L
y
L
x
Cold gases 2009 – p.3
Planck’s radiation formula
Blackbody radiation
Electric field inside a perfectly reflecting
cavity:
1 ∂2
c2 ∂t2
z
L
− ∇2 E(r, t) = 0
Transversality:
∇ · E(r, t) = 0
L
y
L
x
Cold gases 2009 – p.3
Planck’s radiation formula
Blackbody radiation
Electric field inside a perfectly reflecting
cavity:
1 ∂2
c2 ∂t2
z
L
− ∇2 E(r, t) = 0
Transversality:
∇ · E(r, t) = 0
L
y
Boundary conditions:
E(r, t) = 0 when x, y or z are 0 or L
L
x
Cold gases 2009 – p.3
Planck’s radiation formula
Blackbody radiation
Electric field inside a perfectly reflecting
cavity:
1 ∂2
c2 ∂t2
z
L
− ∇2 E(r, t) = 0
Transversality:
∇ · E(r, t) = 0
L
y
Boundary conditions:
E(r, t) = 0 when x, y or z are 0 or L
L
x
Cold gases 2009 – p.3
Planck’s radiation formula
Blackbody radiation
Electric field inside a perfectly reflecting
cavity:
1 ∂2
c2 ∂t2
z
L
− ∇2 E(r, t) = 0
Transversality:
∇ · E(r, t) = 0
L
y
Boundary conditions:
E(r, t) = 0 when x, y or z are 0 or L
L
x
Cold gases 2009 – p.3
Planck’s radiation formula
Cavity modes
Stationary wave functions (s = 1, 2):
n
y
Ek,s (r) ∝ sin(kx x) sin(ky y) sin(kz z)
0
n
x
Cold gases 2009 – p.3
Planck’s radiation formula
Cavity modes
Stationary wave functions (s = 1, 2):
n
y
Ek,s (r) ∝ sin(kx x) sin(ky y) sin(kz z)
Discrete wave vectors (nj = 1, 2, 3, . . .):
k=
π
L (nx , ny , nz )
0
n
x
Cold gases 2009 – p.3
Planck’s radiation formula
Cavity modes
Stationary wave functions (s = 1, 2):
n
y
Ek,s (r) ∝ sin(kx x) sin(ky y) sin(kz z)
Discrete wave vectors (nj = 1, 2, 3, . . .):
k=
π
L (nx , ny , nz )
Relation between frequency and wave
number:
q
π
2πν = c|k| = c L
n2x + n2y + n2z
0
n
x
Cold gases 2009 – p.3
Planck’s radiation formula
Cavity modes
Stationary wave functions (s = 1, 2):
n
y
Ek,s (r) ∝ sin(kx x) sin(ky y) sin(kz z)
Discrete wave vectors (nj = 1, 2, 3, . . .):
k=
π
L (nx , ny , nz )
Relation between frequency and wave
number:
π
n
2πν = c|k| = c L
0
n
x
Cold gases 2009 – p.3
Planck’s radiation formula
Density of cavity modes
Stationary wave functions (s = 1, 2):
Ek,s (r) ∝ sin(kx x) sin(ky y) sin(kz z)
ny
dn
Discrete wave vectors (nj = 1, 2, 3, . . .):
k=
π
L (nx , ny , nz )
Relation between frequency and wave
number:
π
n
2πν = c|k| = c L
Frequency interval ν . . . ν + dν:
g(ν) dν = 2
1
8
2
4πn dn =
0
dn
nx
8πL3 2
ν dν
c3
Cold gases 2009 – p.3
Planck’s radiation formula
Postulate of discrete energies
Allowed energies of radiation per cavity mode
(n = 0, 1, 2, . . .):
ǫ = ǫn = nhν
Cold gases 2009 – p.3
Planck’s radiation formula
Canonical description of the thermal equilibrium
Allowed energies of radiation per cavity mode
(n = 0, 1, 2, . . .):
ǫ = ǫn = nhν
Probability for a cavity mode of frequency ν to
carry the energy ǫn :
p(ǫn ) =
e−βǫn
∞
P
e−βǫj
=
e−βnhν
−1
(1−e−βhν )
j=0
Cold gases 2009 – p.3
Planck’s radiation formula
Canonical description of the thermal equilibrium
Allowed energies of radiation per cavity mode
(n = 0, 1, 2, . . .):
ǫ = ǫn = nhν
Probability for a cavity mode of frequency ν to
carry the energy ǫn :
p(ǫn ) =
e−βǫn
∞
P
e−βǫj
=
e−βnhν
−1
(1−e−βhν )
j=0
Average energy per cavity mode:
hǫi =
∞
P
n=0
ǫn p(ǫn ) =
hν
eβhν −1
Cold gases 2009 – p.3
Planck’s radiation formula
Cavity radiation in thermal equilibrium
Energy per cavity volume V = L3 in the
frequency interval ν . . . ν + dν:
1
V hǫig(ν) dν
=
8πh ν 3 dν
c3 eβhν −1
u(ν) dν
u(ν) dν =
1200 K
visible regime
0
0
2×10
14
14
4×10
frequency ν [Hz]
6×10
14
Cold gases 2009 – p.3
Planck’s radiation formula
Cavity radiation in thermal equilibrium
Energy per cavity volume V = L3 in the
frequency interval ν . . . ν + dν:
1
V hǫig(ν) dν
=
8πh ν 3 dν
c3 eβhν −1
u(ν) dν
u(ν) dν =
1800 K
1200 K
visible regime
0
0
2×10
14
14
4×10
frequency ν [Hz]
6×10
14
Cold gases 2009 – p.3
Planck’s radiation formula
Cavity radiation in thermal equilibrium
Energy per cavity volume V = L3 in the
frequency interval ν . . . ν + dν:
1
V hǫig(ν) dν
=
8πh ν 3 dν
c3 eβhν −1
Relation between β and temperature:
u(ν) dν
u(ν) dν =
2400 K
1800 K
1200 K
visible regime
β = 1/(kB T )
0
0
Boltzmann constant: kB = 1.381 × 10−23 J/K
2×10
14
14
4×10
frequency ν [Hz]
6×10
14
Planck constant: h = 6.626 × 10−34 Js
Cold gases 2009 – p.3
Modern interpretation of Planck’s formula
Occupation numbers
Probability for a cavity mode of frequency ν to
carry n photons of energy nhν (n = 0, 1, 2, . . .):
(n)
pν
=
e−βnhν
−1
(1−e−βhν )
Cold gases 2009 – p.4
Modern interpretation of Planck’s formula
Occupation numbers
Probability for a cavity mode of frequency ν to
carry n photons of energy nhν (n = 0, 1, 2, . . .):
(n)
pν
=
e−βnhν
−1
(1−e−βhν )
Average occupation number per cavity mode:
hNν i =
∞
P
n=0
(n)
npν
=
1
eβhν −1
Cold gases 2009 – p.4
Modern interpretation of Planck’s formula
Occupation numbers
Probability for a cavity mode of frequency ν to
carry n photons of energy nhν (n = 0, 1, 2, . . .):
(n)
pν
=
e−βnhν
−1
(1−e−βhν )
Average occupation number per cavity mode:
hNν i =
∞
P
(n)
npν
n=0
=
1
eβhν −1
Energy per cavity volume V = L3 in the
frequency interval ν . . . ν + dν:
u(ν) dν =
1
V hNν i hν
g(ν) dν =
8πh ν 3 dν
c3 eβhν −1
Cold gases 2009 – p.4
Bose-Einstein condensation of an ideal gas
Ideal gas in a periodic box
Single-particle Hamiltonian:
2
~
∇2
Hsp = − 2m
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Ideal gas in a periodic box
Single-particle Hamiltonian:
2
~
∇2
Hsp = − 2m
Single-particle energy states:
φk (r) =
ik·r
e√
V
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Ideal gas in a periodic box
Single-particle Hamiltonian:
2
~
∇2
Hsp = − 2m
Single-particle energy states:
φk (r) =
ik·r
e√
V
Discrete wave vectors (nj = . . . , −1, 0, 1, . . .):
k=
2π
L (nx , ny , nz )
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Ideal gas in a periodic box
Single-particle Hamiltonian:
2
~
∇2
Hsp = − 2m
Single-particle energy states:
φk (r) =
ik·r
e√
V
Discrete wave vectors (nj = . . . , −1, 0, 1, . . .):
2π
L (nx , ny , nz )
k=
Discrete energies:
ǫk =
~2 k2
2m
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Thermal equilibrium in the grand-canonical description
Many-particle Hamiltonian:
H=
N
P
(n)
Hsp
n=1
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Thermal equilibrium in the grand-canonical description
Many-particle Hamiltonian:
H=
N
P
(n)
Hsp
n=1
Occupation probabilities (n = 0, 1, 2, . . .):
(n)
pk
=
e−β(ǫk −µ)n
−1
[1−e−β(ǫk −µ) ]
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Thermal equilibrium in the grand-canonical description
Many-particle Hamiltonian:
H=
N
P
(n)
Hsp
n=1
Occupation probabilities (n = 0, 1, 2, . . .):
(n)
pk
=
e−β(ǫk −µ)n
−1
[1−e−β(ǫk −µ) ]
Average occupation numbers:
Nk =
∞
P
n=0
(n)
npk =
1
eβ(ǫk −µ) −1
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Determination of the chemical potential
Total number of particles:
N=
P
k
Nk =
P
k
1
eβ(ǫk −µ) −1
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Determination of the chemical potential
Total number of particles:
N=
P
k
Nk =
P
k
1
eβ(ǫk −µ) −1
Thermodynamic limit:
N=
V
(2π)3
R∞ R∞ R∞
−∞ −∞ −∞
dkx dky dkz
eβ(ǫk −µ) −1
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Determination of the chemical potential
Total number of particles:
N=
P
Nk =
k
P
k
1
eβ(ǫk −µ) −1
Thermodynamic limit:
N=
V
(2π)3
R∞ R∞ R∞
−∞ −∞ −∞
dkx dky dkz
eβ(ǫk −µ) −1
Change of variable to energy:
N/V =
1
4π 2
∞
R
3/2
2m
~2
0
√
ǫ dǫ
eβ(ǫ−µ) −1
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Phase-space density vs. chemical potential
Thermal de Broglie wavelength:
λT =
q
2π~2 β
m
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Phase-space density vs. chemical potential
Thermal de Broglie wavelength:
λT =
q
2π~2 β
m
Phase-space density:
ρph =
N 3
V λT
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Phase-space density vs. chemical potential
Thermal de Broglie wavelength:
λT =
q
2π~2 β
m
Phase-space density:
ρph =
N 3
V λT
Phase-space density vs. βµ:
ρph =
√2
π
R∞
0
√
x dx
e(x−βµ) −1
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Chemical potential in the classical limit
Power-series expansion:
√2
π
0
-1
n=0 0
βµ
ρph =
∞√
∞ R
P
xe−(n+1)(x−βµ) dx
-2
-3
classical
-4
-5
0
0.5
1
2
1.5
phase-space density
2.5
3
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Chemical potential in the classical limit
Power-series expansion:
√2
π
-1
n=0 0
Classical limit ρph → 0:
ρph =
0
eβµ √2π
R∞ √ −x
xe dx = eβµ
0
βµ
ρph =
∞√
∞ R
P
xe−(n+1)(x−βµ) dx
-2
-3
classical
-4
-5
0
0.5
1
2
1.5
phase-space density
2.5
3
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Chemical potential in the classical limit
Power-series expansion:
√2
π
-1
n=0 0
Classical limit ρph → 0:
ρph =
0
eβµ √2π
R∞ √ −x
xe dx = eβµ
0
βµ
ρph =
∞√
∞ R
P
xe−(n+1)(x−βµ) dx
-2
-3
classical
-4
-5
0
0.5
1
2
1.5
phase-space density
2.5
3
Classical chemical potential:
βµ = ln ρph
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Chemical potential of a Bose gas
Phase-space density vs. βµ:
√2
π
0
√
0
x dx
e(x−βµ) −1
-1
βµ
ρph =
R∞
-2
-3
classical
Bose
-4
-5
0
0.5
1
2
1.5
phase-space density
2.5
3
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Chemical potential of a Bose gas
Phase-space density vs. βµ:
√2
π
0
√
0
x dx
e(x−βµ) −1
Constraint of positive occupation
numbers:
-1
βµ
ρph =
R∞
-2
-3
classical
Bose
-4
βµ < 0
-5
0
0.5
1
2
1.5
phase-space density
2.5
3
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Chemical potential of a Bose gas
Phase-space density vs. βµ:
√2
π
0
√
0
x dx
e(x−βµ) −1
Constraint of positive occupation
numbers:
ζ(3/2)
-1
βµ
ρph =
R∞
-2
-3
classical
Bose
-4
βµ < 0
-5
0
Phase-space density in the limit
βµ ր 0:
ρph =
√2
π
R∞ √x dx
0
ex −1
0.5
1
2
1.5
phase-space density
2.5
3
= ζ(3/2)
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
What happens if the phase-space density increases?
Phase-space density vs. βµ:
√2
π
0
√
0
x dx
e(x−βµ) −1
Constraint of positive occupation
numbers:
?
-1
βµ
ρph =
R∞
-2
-3
classical
Bose
-4
βµ < 0
-5
0
Phase-space density in the limit
βµ ր 0:
ρph =
√2
π
R∞ √x dx
0
ex −1
0.5
1
2
1.5
phase-space density
2.5
3
= ζ(3/2)
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
-0.2
βµ
k
1
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
-0.2
βµ
k
1
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
-0.2
βµ
k
1
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
-0.2
βµ
k
1
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
-0.2
βµ
k
1
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
= N0 +
k6=0
1
eβ(ǫk −µ) −1
-0.2
βµ
k
1
P
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
= N0 +
k6=0
1
eβ(ǫk −µ) −1
-0.2
βµ
k
1
P
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
= N0 +
k6=0
1
eβ(ǫk −µ) −1
-0.2
βµ
k
1
P
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
= N0 +
k6=0
1
eβ(ǫk −µ) −1
-0.2
βµ
k
1
P
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
= N0 +
k6=0
1
eβ(ǫk −µ) −1
-0.2
βµ
k
1
P
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
= N0 +
k6=0
1
eβ(ǫk −µ) −1
-0.2
βµ
k
1
P
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
= N0 +
k6=0
1
eβ(ǫk −µ) −1
-0.2
βµ
k
1
P
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Occupation of the zero-energy level
Total number of particles:
0
eβ(ǫk −µ) −1
= N0 +
k6=0
1
eβ(ǫk −µ) −1
-0.2
βµ
k
1
P
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N=
P
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Thermodynamic limit
Chemical potential for ρph > ζ(3/2):
0
βµ = −0
βµ
-0.2
ζ(3/2)
-0.4
-0.6
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
N0
60
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Thermodynamic limit
Chemical potential for ρph > ζ(3/2):
0
βµ = −0
Number of particles for ρph > ζ(3/2):
βµ
-0.2
ζ(3/2)
-0.4
-0.6
R∞ √x dx
0
uniform
3
1µm box
-0.8
ex −1
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N = N0 +
√2V 3
πλT
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Thermodynamic limit
Chemical potential for ρph > ζ(3/2):
0
βµ = −0
βµ
Number of particles for ρph > ζ(3/2):
-0.2
ζ(3/2)
-0.4
-0.6
V
λ3T
ζ(3/2)
uniform
3
1µm box
-0.8
-1
0
2
4
8
6
phase-space density
10
100
80
60
N0
N = N0 +
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Thermodynamic limit
Chemical potential for ρph > ζ(3/2):
0
βµ = −0
Number of particles for ρph > ζ(3/2):
βµ
-0.2
ζ(3/2)
-0.4
-0.6
N = N0 +
V
λ3T
ζ(3/2)
-1
0
Number of zero-energy particles for
ρph > ζ(3/2):
mkB T
2π~2
3/2
ζ(3/2)
2
4
8
6
phase-space density
10
100
80
60
N0
N0 = N − V
uniform
3
1µm box
-0.8
40
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Critical temperature for condensation
Chemical potential for ρph > ζ(3/2):
0
βµ = −0
Number of particles for ρph > ζ(3/2):
βµ
-0.2
ζ(3/2)
-0.4
-0.6
N = N0 +
V
λ3T
ζ(3/2)
-1
0
Number of zero-energy particles for
ρph > ζ(3/2):
mkB T
2π~2
3/2
ζ(3/2)
Transition from N0 = 0 to N0 > 0:
Tc =
2π~2
mkB
h
N
Vζ(3/2)
i2/3
2
4
8
6
phase-space density
10
100
80
60
N0
N0 = N − V
uniform
3
1µm box
-0.8
40
Tc
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Critical temperature for condensation
Chemical potential for ρph > ζ(3/2):
0
βµ = −0
Number of particles for ρph > ζ(3/2):
βµ
-0.2
ζ(3/2)
-0.4
-0.6
N = N0 +
V
λ3T
ζ(3/2)
-1
0
Number of zero-energy particles for
ρph > ζ(3/2):
N0 = N 1 − (T /Tc )3/2
Transition from N0 = 0 to N0 > 0:
Tc =
2π~2
mkB
h
N
Vζ(3/2)
i2/3
i
2
4
8
6
phase-space density
10
100
80
60
N0
h
uniform
3
1µm box
-0.8
40
Tc
20
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Critical temperature for condensation
Chemical potential for ρph > ζ(3/2):
0
βµ = −0
Number of particles for ρph > ζ(3/2):
βµ
-0.2
ζ(3/2)
-0.4
-0.6
N = N0 +
V
λ3T
ζ(3/2)
-1
0
Number of zero-energy particles for
ρph > ζ(3/2):
N0 = N 1 − (T /Tc )3/2
Transition from N0 = 0 to N0 > 0:
Tc =
2π~2
mkB
h
N
Vζ(3/2)
i2/3
i
2
4
8
6
phase-space density
10
1000
800
600
N0
h
uniform
3
10µm box
-0.8
400
Tc
200
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation of an ideal gas
Critical temperature for condensation
Chemical potential for ρph > ζ(3/2):
0
βµ = −0
Number of particles for ρph > ζ(3/2):
βµ
-0.2
ζ(3/2)
-0.4
-0.6
N = N0 +
V
λ3T
ζ(3/2)
-1
0
Number of zero-energy particles for
ρph > ζ(3/2):
N0 = N 1 − (T /Tc )3/2
Transition from N0 = 0 to N0 > 0:
Tc =
2π~2
mkB
h
N
Vζ(3/2)
i2/3
i
2
4
8
6
phase-space density
10
10000
8000
6000
N0
h
uniform
3
100µm box
-0.8
4000
Tc
2000
0
0
0.2
0.4 0.6 0.8
1
temperature [µK]
1.2
1.4
Cold gases 2009 – p.5
Bose-Einstein condensation in atom traps
Trapping potential
Single-particle Hamiltonian:
100
2
80
energy/kB [nK]
~
∇2 + Vext (r)
Hsp = − 2m
60
40
20
0
-4
-3
-2
-1
0
z [µm]
1
2
3
4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Trapping potential
Single-particle Hamiltonian:
100
2
~
∇2 + Vext (r)
Hsp = − 2m
Harmonic-oscillator potential:
Vext (r) =
m
2
ωx2 x2
+ ωy2 y 2
+
ωz2 z 2
energy/kB [nK]
80
60
40
20
0
-4
-3
-2
-1
0
z [µm]
1
2
3
4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Trapping potential
Single-particle Hamiltonian:
100
2
~
∇2 + Vext (r)
Hsp = − 2m
Harmonic-oscillator potential:
Vext (r) =
m
2
ωx2 x2
+ ωy2 y 2
+
ωz2 z 2
Discrete energy levels (nj = 0, 1, 2, . . .):
ǫn =
P
j=x,y,z
~ωj nj +
1
energy/kB [nK]
80
60
40
20
0
-4
-3
-2
-1
0
z [µm]
1
2
3
4
2
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Example: spherical trap
Single-particle Hamiltonian:
100
2
Harmonic-oscillator potential:
Vext (r) =
m 2 2
2 ωho r
80
energy/kB [nK]
~
∇2 + Vext (r)
Hsp = − 2m
60
40
20
Discrete energy levels (n = 0, 1, 2, . . .):
0
-4
ǫn = ~ωho n +
3
-3
-2
-1
0
z [µm]
1
2
3
4
2
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Example: spherical trap
Single-particle Hamiltonian:
100
2
Harmonic-oscillator potential:
Vext (r) =
m 2 2
2 ωho r
80
energy/kB [nK]
~
∇2 + Vext (r)
Hsp = − 2m
60
40
20
Discrete energy levels (n = 0, 1, 2, . . .):
0
-4
ǫn = ~ωho n +
3
-3
-2
-1
0
z [µm]
1
2
3
4
2
Degeneracy:
gn = (n + 1)(n + 2)/2
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Single-particle ground state
Average angular frequency:
0.4
60
0.3
40
0.2
20
0
-4
0.1
-3
-2
-1
0
z [µm]
1
2
3
0
4
Cold gases 2009 – p.6
]
-3/2
0.5
80
wave function [µm
√
3 ω ω ω
x y z
energy/kB [nK]
ωho =
0.6
100
Bose-Einstein condensation in atom traps
Single-particle ground state
Average angular frequency:
Harmonic-oscillator length:
aho =
q
~
mωho
0.4
60
0.3
40
0.2
20
0
-4
0.1
-3
-2
-1
0
z [µm]
1
2
3
0
4
Cold gases 2009 – p.6
]
-3/2
0.5
80
wave function [µm
√
3 ω ω ω
x y z
energy/kB [nK]
ωho =
0.6
100
Bose-Einstein condensation in atom traps
Single-particle ground state
Average angular frequency:
Harmonic-oscillator length:
aho =
q
~
mωho
Ground-state wave function:
φ0 (r) =
e
2 x2 +ω 2 y 2 +ω 2 z 2 )
− m (ωx
y
z
2~
(πa2ho )3/4
0.4
60
0.3
40
0.2
20
0
-4
0.1
-3
-2
-1
0
z [µm]
1
2
3
0
4
Cold gases 2009 – p.6
]
-3/2
0.5
80
wave function [µm
√
3 ω ω ω
x y z
energy/kB [nK]
ωho =
0.6
100
Bose-Einstein condensation in atom traps
Spherical trap
Average angular frequency:
Harmonic-oscillator length:
aho =
q
~
mωho
1
φ0 (r) =
0.4
60
0.3
40
0.2
20
Ground-state wave function:
e− 2 (r/aho )
(πa2ho )3/4
0
-4
2
0.1
-3
-2
-1
0
z [µm]
1
2
3
0
4
Cold gases 2009 – p.6
]
-3/2
0.5
80
wave function [µm
√
3 ω ω ω
x y z
energy/kB [nK]
ωho =
0.6
100
Bose-Einstein condensation in atom traps
Thermal equilibrium in the grand-canonical description
Many-particle Hamiltonian:
H=
N
P
(n)
Hsp
n=1
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Thermal equilibrium in the grand-canonical description
Many-particle Hamiltonian:
H=
N
P
(n)
Hsp
n=1
Occupation probabilities (k = 0, 1, 2, . . .):
(k)
pn
=
e−β(ǫn −µ)k
−1
[1−e−β(ǫn −µ) ]
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Thermal equilibrium in the grand-canonical description
Many-particle Hamiltonian:
H=
N
P
(n)
Hsp
n=1
Occupation probabilities (k = 0, 1, 2, . . .):
(k)
pn
=
e−β(ǫn −µ)k
−1
[1−e−β(ǫn −µ) ]
Average occupation numbers:
Nn =
∞
P
k=0
(k)
kpn =
1
eβ(ǫn −µ) −1
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Determination of the chemical potential
Total number of particles:
N=
∞
P
nx ,ny ,nz =0
Nn =
∞
P
nx ,ny ,nz =0
1
eβ(ǫn −µ) −1
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Determination of the chemical potential
Total number of particles:
N=
∞
P
Nn =
∞
P
nx ,ny ,nz =0
nx ,ny ,nz =0
1
eβ(ǫn −µ) −1
Thermodynamic limit:
N=
R∞ R∞ R∞ dnx dny dnz
0 0 0
eβ(ǫn −µ) −1
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Determination of the chemical potential
Total number of particles:
N=
∞
P
∞
P
Nn =
nx ,ny ,nz =0
nx ,ny ,nz =0
1
eβ(ǫn −µ) −1
Thermodynamic limit:
N=
R∞ R∞ R∞ dnx dny dnz
eβ(ǫn −µ) −1
0 0 0
Partial integrations:
N=
1
(β~ωho )3
R∞
0
1 2
n dn
2
en−β(µ−ǫ0 ) −1
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Chemical potential in the classical limit
Classical approximation:
=
0
1 2
n dn
2
en−β(µ−ǫ0 ) −1
-1
β(µ-ε0)
N (β~ωho
)3
R∞
0
-2
-3
Boltzmann
-4
-5
0
0.2
0.4
0.8
0.6
3
N(βh- ωho)
1
1.2
1.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Chemical potential in the classical limit
Classical approximation:
≈
0
2
-1
n2 e−n+β(µ−ǫ0 ) dn
β(µ-ε0)
N (β~ωho
)3
R∞ 1
0
-2
-3
Boltzmann
-4
-5
0
0.2
0.4
0.8
0.6
3
N(βh- ωho)
1
1.2
1.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Chemical potential in the classical limit
Classical approximation:
≈
0
2
-1
n2 e−n+β(µ−ǫ0 ) dn
Limit of Boltzmann distribution:
β(µ-ε0)
N (β~ωho
)3
R∞ 1
0
-2
-3
Boltzmann
N (β~ωho )3 = eβ(µ−ǫ0 )
-4
-5
0
0.2
0.4
0.8
0.6
3
N(βh- ωho)
1
1.2
1.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Chemical potential in the classical limit
Classical approximation:
≈
0
2
-1
n2 e−n+β(µ−ǫ0 ) dn
β(µ-ε0)
N (β~ωho
)3
R∞ 1
0
Limit of Boltzmann distribution:
-2
-3
Boltzmann
N (β~ωho )3 = eβ(µ−ǫ0 )
-4
Classical chemical potential:
β(µ − ǫ0 ) = ln N (β~ωho
-5
0
)3
0.2
0.4
0.8
0.6
3
N(βh- ωho)
1
1.2
1.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Chemical potential of an uncondensed ideal Bose gas
N (β~ωho )3 vs. βµ:
=
R∞
0
1 2
n dn
2
en−β(µ−ǫ0 ) −1
-1
β(µ-ε0)
N (β~ωho
)3
0
-2
-3
Boltzmann
Bose
-4
-5
0
0.2
0.4
0.8
0.6
3
N(βh- ωho)
1
1.2
1.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Chemical potential of an uncondensed ideal Bose gas
N (β~ωho )3 vs. βµ:
=
R∞
0
1 2
n dn
2
en−β(µ−ǫ0 ) −1
Constraint of positive occupation
numbers:
β(µ − ǫ0 ) < 0
-1
β(µ-ε0)
N (β~ωho
)3
0
-2
-3
Boltzmann
Bose
-4
-5
0
0.2
0.4
0.8
0.6
3
N(βh- ωho)
1
1.2
1.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Chemical potential of an uncondensed ideal Bose gas
N (β~ωho )3 vs. βµ:
=
R∞
0
1 2
n dn
2
en−β(µ−ǫ0 ) −1
Constraint of positive occupation
numbers:
β(µ − ǫ0 ) < 0
3
N (β~ωho ) =
R∞ 12 n2 dn
0
en −1
-2
-3
Boltzmann
Bose
-4
-5
0
N (β~ωho )3 in the limit β(µ − ǫ0 ) ր 0:
ζ(3)
-1
β(µ-ε0)
N (β~ωho
)3
0
0.2
0.4
0.8
0.6
3
N(βh- ωho)
1
1.2
1.4
= ζ(3)
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Macroscopic occupation of the single-particle ground state
Total number of particles:
0
eβ(ǫn −µ) −1
-0.2
β(µ-ε0)
n
1
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N=
P
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Macroscopic occupation of the single-particle ground state
Total number of particles:
0
eβ(ǫn −µ) −1
-0.2
β(µ-ε0)
n
1
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N=
P
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Macroscopic occupation of the single-particle ground state
Total number of particles:
0
eβ(ǫn −µ) −1
-0.2
β(µ-ε0)
n
1
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N=
P
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Macroscopic occupation of the single-particle ground state
Total number of particles:
0
eβ(ǫn −µ) −1
-0.2
β(µ-ε0)
n
1
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N=
P
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Macroscopic occupation of the single-particle ground state
Total number of particles:
0
eβ(ǫn −µ) −1
-0.2
β(µ-ε0)
n6=0
1
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N = N0 +
P
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Macroscopic occupation of the single-particle ground state
Total number of particles:
0
eβ(ǫn −µ) −1
-0.2
β(µ-ε0)
n6=0
1
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N = N0 +
P
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Macroscopic occupation of the single-particle ground state
Total number of particles:
0
eβ(ǫn −µ) −1
-0.2
β(µ-ε0)
n6=0
1
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N = N0 +
P
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Macroscopic occupation of the single-particle ground state
Total number of particles:
0
eβ(ǫn −µ) −1
-0.2
β(µ-ε0)
n6=0
1
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N = N0 +
P
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Macroscopic occupation of the single-particle ground state
Total number of particles:
0
eβ(ǫn −µ) −1
-0.2
β(µ-ε0)
n6=0
1
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N = N0 +
P
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Macroscopic occupation of the single-particle ground state
Total number of particles:
0
eβ(ǫn −µ) −1
-0.2
β(µ-ε0)
n6=0
1
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N = N0 +
P
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Macroscopic occupation of the single-particle ground state
Total number of particles:
0
eβ(ǫn −µ) −1
-0.2
β(µ-ε0)
n6=0
1
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N = N0 +
P
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Macroscopic occupation of the single-particle ground state
Total number of particles:
0
eβ(ǫn −µ) −1
-0.2
β(µ-ε0)
n6=0
1
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N = N0 +
P
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Thermodynamic limit
Chemical potential for N (β~ωho )3 > ζ(3):
β(µ-ε0)
-0.2
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
β(µ − ǫ0 ) = −0
0
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Thermodynamic limit
Chemical potential for N (β~ωho )3 > ζ(3):
-0.2
3
Number of particles for N (β~ωho ) > ζ(3):
1
(β~ωho )3
R∞ 21 n2 dn
0
ζ(3)
-0.4
-0.6
thermodynamic limit
finite
-0.8
en −1
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N = N0 +
β(µ-ε0)
β(µ − ǫ0 ) = −0
0
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Thermodynamic limit
Chemical potential for N (β~ωho )3 > ζ(3):
-0.2
3
Number of particles for N (β~ωho ) > ζ(3):
ζ(3)
-0.4
-0.6
ζ(3)
(β~ωho )3
thermodynamic limit
finite
-0.8
-1
0
2
4
6
3
N(βh- ωho)
8
10
40000
30000
N0
N = N0 +
β(µ-ε0)
β(µ − ǫ0 ) = −0
0
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Thermodynamic limit
Chemical potential for N (β~ωho )3 > ζ(3):
-0.2
3
Number of particles for N (β~ωho ) > ζ(3):
N = N0 +
β(µ-ε0)
β(µ − ǫ0 ) = −0
0
ζ(3)
-0.4
-0.6
ζ(3)
(β~ωho )3
thermodynamic limit
finite
-0.8
-1
0
Number of condensed particles for
N (β~ωho )3 > ζ(3):
2
4
6
3
N(βh- ωho)
8
10
40000
N0 = N −
ζ(3)(kB T )3
(~ωho )3
N0
30000
20000
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Critical temperature for condensation
Chemical potential for N (β~ωho )3 > ζ(3):
-0.2
3
Number of particles for N (β~ωho ) > ζ(3):
N = N0 +
β(µ-ε0)
β(µ − ǫ0 ) = −0
0
ζ(3)
-0.4
-0.6
ζ(3)
(β~ωho )3
thermodynamic limit
finite
-0.8
-1
0
Number of condensed particles for
N (β~ωho )3 > ζ(3):
2
4
6
3
N(βh- ωho)
8
10
40000
N0 = N −
ζ(3)(kB T )3
(~ωho )3
Transition from N0 = 0 to N0 > 0:
Tc =
~ωho
kB
h
N
ζ(3)
i1/3
N0
30000
20000
Tc
10000
0
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Critical temperature for condensation
Chemical potential for N (β~ωho )3 > ζ(3):
β(µ − ǫ0 ) = −0
3
β(µ-ε0)
-0.2
Number of particles for N (β~ωho ) > ζ(3):
N = N0 +
0
ζ(3)
-0.4
-0.6
ζ(3)
(β~ωho )3
thermodynamic limit
finite
-0.8
-1
0
Number of condensed particles for
N (β~ωho )3 > ζ(3):
2
4
6
3
N(βh- ωho)
8
10
40000
N0 = N 1 − (T /Tc )3
i
Transition from N0 = 0 to N0 > 0:
N0
30000
h
20000
Tc
10000
0
Tc =
~ωho
kB
h
N
ζ(3)
i1/3
0
0.1
0.2
0.3
temperature [µK]
0.4
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Comparison with a dilute gas of interacting Bose atoms
Assumption on the fraction of
Bose-Einstein condensed atoms:
1
0.8
Nc/N
3
Nc /N ≈ N0 /N ≈ 1 − (T /Tc )
ideal gas
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
T/Tc
1.2
1.4
1.6
1.8
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Comparison with a dilute gas of interacting Bose atoms
Assumption on the fraction of
Bose-Einstein condensed atoms:
1
0.8
Critical temperature for condensation of an
ideal Bose gas in the thermodynamic limit:
Tc =
~ωho
kB
h
N
ζ(3)
i1/3
Nc/N
3
Nc /N ≈ N0 /N ≈ 1 − (T /Tc )
ideal gas
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
T/Tc
1.2
1.4
1.6
1.8
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Comparison with a dilute gas of interacting Bose atoms
Assumption on the fraction of
Bose-Einstein condensed atoms:
1
0.8
Critical temperature for condensation of an
ideal Bose gas in the thermodynamic limit:
Tc =
~ωho
kB
h
N
ζ(3)
i1/3
Nc/N
3
Nc /N ≈ N0 /N ≈ 1 − (T /Tc )
ideal gas
Ensher et al.
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
T/Tc
1.2
1.4
1.6
1.8
J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 77, 4984 (1996)
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Density of atoms
Condensate density of an ideal Bose gas:
ρc (r) = N0 |φ0 (r)|2
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Density of atoms
Condensate density of an ideal Bose gas:
Column density of a condensate:
ρc (z) =
R∞
−∞
condensate
2
ρc (x, 0, z) dx
400
-2
column density [aho]
ρc (r) = N0 |φ0 (r)|
500
300
200
100
0
-8
-6
-4
-2
0
z [aho]
2
4
6
8
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Density of atoms
Condensate density of an ideal Bose gas:
condensate
2
400
-2
Column density of a condensate:
ρc (z) =
R∞
ρc (x, 0, z) dx
−∞
ρth (r) ≈ (N − N0 )
β~ωho
2πa2ho
300
200
100
0
-8
Density of thermal atoms:
column density [aho]
ρc (r) = N0 |φ0 (r)|
500
3/2
-6
-4
-2
0
z [aho]
2
4
6
8
e−βVext (r)
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Density of atoms
Condensate density of an ideal Bose gas:
2
-2
Column density of a condensate:
ρc (z) =
R∞
ρc (x, 0, z) dx
−∞
ρth (r) ≈ (N − N0 )
β~ωho
2πa2ho
300
200
100
0
-8
Density of thermal atoms:
3/2
condensate
thermal (T=0.9 Tc)
400
column density [aho]
ρc (r) = N0 |φ0 (r)|
500
-6
-4
-2
0
z [aho]
2
4
6
8
e−βVext (r)
Column density of thermal atoms:
ρth (z) =
R∞
ρth (x, 0, z) dx
−∞
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Effect of interactions
Condensate density of an ideal Bose gas:
Column density of a condensate:
ρc (z) =
R∞
−∞
ρc (x, 0, z) dx
Hau et al.
800
-2
column density [µm ]
ρc (r) = N0 |φ0 (r)|
2
1000
600
400
200
0
-60
-40
-20
0
z [µm]
20
40
60
L. V. Hau, B. D. Busch, C. Liu, Z. Dutton, M. M. Burns, and J. A. Golovchenko, Phys. Rev. A 58, R54 (1998)
Cold gases 2009 – p.6
Bose-Einstein condensation in atom traps
Effect of interactions
Condensate density of an ideal Bose gas:
Column density of a condensate:
ρc (z) =
R∞
−∞
ρc (x, 0, z) dx
ideal gas
Hau et al.
800
-2
column density [µm ]
ρc (r) = N0 |φ0 (r)|
2
1000
600
400
200
0
-60
-40
-20
0
z [µm]
20
40
60
L. V. Hau, B. D. Busch, C. Liu, Z. Dutton, M. M. Burns, and J. A. Golovchenko, Phys. Rev. A 58, R54 (1998)
Cold gases 2009 – p.6
Literature
Some classic theory references
Planck’s radiation formula
M. Planck, Annalen der Physik 4, 553 (1901)
Cold gases 2009 – p.7
Literature
Some classic theory references
Planck’s radiation formula
M. Planck, Annalen der Physik 4, 553 (1901)
Modern interpretation of Planck’s formula
S. N. Bose, Z. Phys. 26, 178 (1924)
Cold gases 2009 – p.7
Literature
Some classic theory references
Planck’s radiation formula
M. Planck, Annalen der Physik 4, 553 (1901)
Modern interpretation of Planck’s formula
S. N. Bose, Z. Phys. 26, 178 (1924)
Bose-Einstein condensation of an ideal gas
A. Einstein, S.-B. preuß. Akad. Wiss., physik.-math. Kl. 22, 261 (1924); 23, 3 (1925)
Cold gases 2009 – p.7
Literature
Some classic theory references
Planck’s radiation formula
M. Planck, Annalen der Physik 4, 553 (1901)
Modern interpretation of Planck’s formula
S. N. Bose, Z. Phys. 26, 178 (1924)
Bose-Einstein condensation of an ideal gas
A. Einstein, S.-B. preuß. Akad. Wiss., physik.-math. Kl. 22, 261 (1924); 23, 3 (1925)
Bose-Einstein condensation in atom traps
S. R. de Groot, G. J. Hooman, and C. A. Ten Seldam, Proc. R. Soc. London, Ser. A 203, 266
(1950); V. Bagnato, D. E. Pritchard, and D. Kleppner, Phys. Rev. A 35, 4354 (1987)
Cold gases 2009 – p.7
Literature
Some review articles and books
Review article
F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999)
Cold gases 2009 – p.7
Literature
Some review articles and books
Review article
F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999)
Books
Bose-Einstein Condensation, edited by A. Griffin, D. W. Snoke, and S. Stringari (Cambridge
University Press, 1996)
C. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University
Press, 2002)
L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Clarendon Press, 2003)
Cold gases 2009 – p.7