Properties of dilute Bose-Einstein
condensates
Thorsten Köhler
Department of Physics and Astronomy, University College London,
Gower Street, London, WC1E 6BT, United Kingdom
Cold gases 2009 – p.1
Outline
Ground-state Bose-Einstein condensates
Cold gases 2009 – p.2
Outline
Ground-state Bose-Einstein condensates
Thomas-Fermi approximation
Cold gases 2009 – p.2
Outline
Ground-state Bose-Einstein condensates
Thomas-Fermi approximation
Breathing mode in a spherical trap
Cold gases 2009 – p.2
Outline
Ground-state Bose-Einstein condensates
Thomas-Fermi approximation
Breathing mode in a spherical trap
Collective excitations in a cylindrical trap
Cold gases 2009 – p.2
Outline
Ground-state Bose-Einstein condensates
Thomas-Fermi approximation
Breathing mode in a spherical trap
Collective excitations in a cylindrical trap
Literature
Cold gases 2009 – p.2
Ground-state Bose-Einstein condensates
Stationary condensate wave functions
Time-dependent Gross-Pitaevskii equation:
h
~2
∂
Ψ(r, t) = − 2m ∇2 + Vext (r) + g |Ψ(r, t)|
i~ ∂t
2
i
Ψ(r, t)
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
Stationary condensate wave functions
Time-dependent Gross-Pitaevskii equation:
h
~2
∂
Ψ(r, t) = − 2m ∇2 + Vext (r) + g |Ψ(r, t)|
i~ ∂t
Product ansatz:
2
i
Ψ(r, t)
Ψ(r, t) = e−iµt/~ Ψ(r)
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
Stationary condensate wave functions
Time-dependent Gross-Pitaevskii equation:
h
~2
∂
Ψ(r, t) = − 2m ∇2 + Vext (r) + g |Ψ(r, t)|
i~ ∂t
Product ansatz:
2
i
Ψ(r, t)
Ψ(r, t) = e−iµt/~ Ψ(r)
Stationary Gross-Pitaevskii equation:
h
~2
i
µΨ(r) = − 2m ∇2 + Vext (r) + g |Ψ(r)|2 Ψ(r)
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
Stationary condensate wave functions
Time-dependent Gross-Pitaevskii equation:
h
~2
∂
Ψ(r, t) = − 2m ∇2 + Vext (r) + g |Ψ(r, t)|
i~ ∂t
Product ansatz:
2
i
Ψ(r, t)
Ψ(r, t) = e−iµt/~ Ψ(r)
Stationary Gross-Pitaevskii equation:
h
~2
i
µΨ(r) = − 2m ∇2 + Vext (r) + g |Ψ(r)|2 Ψ(r)
Chemical potential:
µN =
R
h
~2
i
Ψ∗ (r) − 2m ∇2 + Vext (r) + g |Ψ(r)|2 Ψ(r) dr
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
Stationary condensate wave functions
Time-dependent Gross-Pitaevskii equation:
h
~2
∂
Ψ(r, t) = − 2m ∇2 + Vext (r) + g |Ψ(r, t)|
i~ ∂t
Product ansatz:
2
i
Ψ(r, t)
Ψ(r, t) = e−iµt/~ Ψ(r)
Stationary Gross-Pitaevskii equation:
h
~2
i
µΨ(r) = − 2m ∇2 + Vext (r) + g |Ψ(r)|2 Ψ(r)
Chemical potential:
µN = K + Eho + 2Emf
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
Energy balance
Virial relation:
d2
2i
hr
t
2
dt
=
6E
Nm
−
6Eho (t)
Nm
−
2
Nm
R
|Ψ(r, t)|2 r · ∇Vext (r) dr −
2K(t)
Nm
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
Energy balance
Virial relation:
0=
6E
Nm
−
6Eho
Nm
−
2
Nm
R
|Ψ(r)|2 r · ∇Vext (r) dr −
2K
Nm
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
Energy balance
Virial relation:
0=
6E
Nm
−
6Eho
Nm
−
4Eho
Nm
−
2K
Nm
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
Energy balance
Virial relation:
0 = 2K − 2Eho + 3Emf
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
0 = 2K − 2Eho + 3Emf
Rotational invariance:
Ψ(r) = Ψ(r)
1/2 -3/2
Virial relation:
condensate wave function [N aho ]
Examples: spherical trap and attractive interactions; g ≤ 0
3
Ng/(4πaho )=-0.5 h- ωho
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
r [aho]
4
5
6
K/N = 1.105 ~ωho
Eho /N = 0.522 ~ωho
Emf /N = −0.389 ~ωho
E/N = 1.238 ~ωho
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
0 = 2K − 2Eho + 3Emf
Rotational invariance:
Ψ(r) = Ψ(r)
1/2 -3/2
Virial relation:
condensate wave function [N aho ]
Examples: spherical trap and attractive interactions; g ≤ 0
3
Ng/(4πaho )=-0.4 h- ωho
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
r [aho]
4
5
6
K/N = 0.961 ~ωho
Eho /N = 0.592 ~ωho
Emf /N = −0.246 ~ωho
E/N = 1.307 ~ωho
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
0 = 2K − 2Eho + 3Emf
Rotational invariance:
Ψ(r) = Ψ(r)
1/2 -3/2
Virial relation:
condensate wave function [N aho ]
Examples: spherical trap and attractive interactions; g ≤ 0
3
Ng/(4πaho )=-0.3 h- ωho
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
r [aho]
4
5
6
K/N = 0.880 ~ωho
Eho /N = 0.642 ~ωho
Emf /N = −0.159 ~ωho
E/N = 1.364 ~ωho
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
0 = 2K − 2Eho + 3Emf
Rotational invariance:
Ψ(r) = Ψ(r)
1/2 -3/2
Virial relation:
condensate wave function [N aho ]
Examples: ideal gas in a spherical trap; g = 0
3
Ng/(4πaho )=0
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
r [aho]
K/N =
3
4
~ωho
Eho /N =
3
4
~ωho
4
5
6
Emf /N = 0
E/N =
3
2
~ωho
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
0 = 2K − 2Eho + 3Emf
Rotational invariance:
Ψ(r) = Ψ(r)
1/2 -3/2
Virial relation:
condensate wave function [N aho ]
Examples: spherical trap and repulsive interactions; g ≥ 0
3
Ng/(4πaho )=1 h- ωho
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
r [aho]
4
5
6
K/N = 0.588 ~ωho
Eho /N = 0.969 ~ωho
Emf /N = 0.254 ~ωho
E/N = 1.811 ~ωho
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
0 = 2K − 2Eho + 3Emf
Rotational invariance:
Ψ(r) = Ψ(r)
1/2 -3/2
Virial relation:
condensate wave function [N aho ]
Examples: spherical trap and repulsive interactions; g ≥ 0
3
Ng/(4πaho )=10 h- ωho
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
r [aho]
4
5
6
K/N = 0.356 ~ωho
Eho /N = 1.772 ~ωho
Emf /N = 0.944 ~ωho
E/N = 3.072 ~ωho
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
0 = 2K − 2Eho + 3Emf
Rotational invariance:
Ψ(r) = Ψ(r)
1/2 -3/2
Virial relation:
condensate wave function [N aho ]
Examples: spherical trap and repulsive interactions; g ≥ 0
3
Ng/(4πaho )=100 h- ωho
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
r [aho]
4
5
6
K/N = 0.190 ~ωho
Eho /N = 4.087 ~ωho
Emf /N = 2.598 ~ωho
E/N = 6.875 ~ωho
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
0 = 2K − 2Eho + 3Emf
Rotational invariance:
Ψ(r) = Ψ(r)
1/2 -3/2
Virial relation:
condensate wave function [N aho ]
The ratio K/E decreases as N g/
4πa3ho ~ωho
increases!
3
Ng/(4πaho )=100 h- ωho
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
r [aho]
4
5
6
K/N = 0.190 ~ωho
Eho /N = 4.087 ~ωho
Emf /N = 2.598 ~ωho
E/N = 6.875 ~ωho
Cold gases 2009 – p.3
Ground-state Bose-Einstein condensates
0 = 2K − 2Eho + 3Emf
Rotational invariance:
Ψ(r) = Ψ(r)
Stationary Gross-Pitaevskiiequation
in the limit N g/ 4πa3ho ~ωho → ∞:
µ ≈ Vext (r) + g |Ψ(r)|2
1/2 -3/2
Virial relation:
condensate wave function [N aho ]
The ratio K/E decreases as N g/
4πa3ho ~ωho
increases!
3
Ng/(4πaho )=100 h- ωho
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
r [aho]
4
5
6
K/N = 0.190 ~ωho
Eho /N = 4.087 ~ωho
Emf /N = 2.598 ~ωho
E/N = 6.875 ~ωho
Cold gases 2009 – p.3
Thomas-Fermi approximation
Ground-state condensate density
in the limit
N g/ 4πa3ho ~ωho → ∞
Parabolic density profile:
Thomas-Fermi
0
[µ − Vext (r)]
if µ ≥ Vext (r)
elsewhere
-3
1
g
condensate density [Naho]
ρ(r) =
(
0.008
0.006
0.004
0.002
0
0
1
Ng
4πa3
ho
2
3
r [aho]
4
5
= 100 ~ωho
Cold gases 2009 – p.4
6
Thomas-Fermi approximation
Ground-state condensate density
in the limit
N g/ 4πa3ho ~ωho → ∞
Parabolic density profile:
Thomas-Fermi
0
[µ − Vext (r)]
if µ ≥ Vext (r)
elsewhere
Number of atoms:
N=
R
ρ(r) dr
-3
1
g
condensate density [Naho]
ρ(r) =
(
0.008
0.006
0.004
0.002
0
0
1
Ng
4πa3
ho
2
3
r [aho]
4
5
= 100 ~ωho
Cold gases 2009 – p.4
6
Thomas-Fermi approximation
Ground-state condensate density
in the limit
N g/ 4πa3ho ~ωho → ∞
Parabolic density profile:
Thomas-Fermi
0
[µ − Vext (r)]
if µ ≥ Vext (r)
elsewhere
Number of atoms:
N=
µ
g
2µ
mωho
3/2
4π
R1
0
u2
1
− u2
du
-3
1
g
condensate density [Naho]
ρ(r) =
(
0.008
0.006
0.004
0.002
0
0
1
Ng
4πa3
ho
2
3
r [aho]
4
5
= 100 ~ωho
Cold gases 2009 – p.4
6
Thomas-Fermi approximation
Ground-state condensate density
in the limit
N g/ 4πa3ho ~ωho → ∞
Parabolic density profile:
Thomas-Fermi
0
[µ − Vext (r)]
if µ ≥ Vext (r)
elsewhere
Number of atoms:
N=
4πa3ho
25/2
15 g(~ωho )3/2
µ5/2
-3
1
g
condensate density [Naho]
ρ(r) =
(
0.008
0.006
0.004
0.002
0
0
1
Ng
4πa3
ho
2
3
r [aho]
4
5
= 100 ~ωho
Cold gases 2009 – p.4
6
Thomas-Fermi approximation
Ground-state condensate density
in the limit
N g/ 4πa3ho ~ωho → ∞
Parabolic density profile:
Thomas-Fermi
0
[µ − Vext (r)]
if µ ≥ Vext (r)
elsewhere
Number of atoms:
N=
4πa3ho
25/2
15 g(~ωho )3/2
µ5/2
-3
1
g
condensate density [Naho]
ρ(r) =
(
0.008
0.006
0.004
0.002
0
0
1
Chemical potential:
µ=
~ωho
2
15N g
4πa3ho ~ωho
2/5
Ng
4πa3
ho
2
3
r [aho]
4
5
= 100 ~ωho
Cold gases 2009 – p.4
6
Thomas-Fermi approximation
Example: ground-state condensate density
in a spherical
trap in the limit N g/ 4πa3ho ~ωho → ∞
Parabolic density profile:
0
Thomas-Fermi
R2 − r
2
if r ≤ R
elsewhere
Condensate radius:
R = aho
15N g
4πa3ho ~ωho
Chemical potential:
µ=
~ωho
2
1/5
15N g
4πa3ho ~ωho
2/5
-3
2
mωho
2g
condensate density [Naho]
ρ(r) =
(
0.008
0.006
0.004
0.002
R
0
0
1
Ng
4πa3
ho
2
3
r [aho]
4
5
= 100 ~ωho
Cold gases 2009 – p.4
6
Thomas-Fermi approximation
Exact versus approximate ground-state condensate
densities in a spherical trap
Parabolic density profile:
0.12
ρ(r) ≈
2g
0
R2 − r
2
if r ≤ R
elsewhere
Condensate radius:
R = aho
15N g
4πa3ho ~ωho
Chemical potential:
1/5
-3
2
mωho
condensate density [Naho]
3
(
Ng/(4πaho )=1 h- ωho
Thomas-Fermi
0.1
0.08
0.06
0.04
0.02
0
0
1
2
3
4
6
5
r [aho]
7
8
9
10
K/N = 0.588 ~ωho
µ≈
~ωho
2
15N g
4πa3ho ~ωho
2/5
E/N = 1.811 ~ωho
Cold gases 2009 – p.4
Thomas-Fermi approximation
Exact versus approximate ground-state condensate
densities in a spherical trap
Parabolic density profile:
0.03
ρ(r) ≈
2g
0
R2 − r
2
if r ≤ R
elsewhere
Condensate radius:
R = aho
15N g
4πa3ho ~ωho
Chemical potential:
1/5
Ng/(4πaho )=10 h- ωho
Thomas-Fermi
-3
2
mωho
condensate density [Naho]
3
(
0.02
0.01
0
0
1
2
3
4
6
5
r [aho]
7
8
9
10
K/N = 0.356 ~ωho
µ≈
~ωho
2
15N g
4πa3ho ~ωho
2/5
E/N = 3.072 ~ωho
Cold gases 2009 – p.4
Thomas-Fermi approximation
Exact versus approximate ground-state condensate
densities in a spherical trap
Parabolic density profile:
2g
0
R2 − r
2
if r ≤ R
elsewhere
Condensate radius:
R = aho
15N g
4πa3ho ~ωho
Chemical potential:
1/5
3
Ng/(4πaho )=100 h- ωho
Thomas-Fermi
-3
2
mωho
condensate density [Naho]
ρ(r) ≈
(
0.008
0.006
0.004
0.002
0
0
1
2
3
4
7
6
5
r [aho]
8
9
10
K/N = 0.190 ~ωho
µ≈
~ωho
2
15N g
4πa3ho ~ωho
2/5
E/N = 6.875 ~ωho
Cold gases 2009 – p.4
Thomas-Fermi approximation
Exact versus approximate ground-state condensate
densities in a spherical trap
Parabolic density profile:
2g
0
R2 − r
2
if r ≤ R
elsewhere
Condensate radius:
R = aho
15N g
4πa3ho ~ωho
Chemical potential:
1/5
-3
2
mωho
condensate density [Naho]
ρ(r) ≈
(
0.003
3
Ng/(4πaho )=500 h- ωho
Thomas-Fermi
0.0025
0.002
0.0015
0.001
0.0005
0
0
1
2
3
4
7
6
5
r [aho]
8
9
10
K/N = 0.118 ~ωho
µ≈
~ωho
2
15N g
4πa3ho ~ωho
2/5
E/N = 12.807 ~ωho
Cold gases 2009 – p.4
Thomas-Fermi approximation
Exact versus approximate ground-state condensate
densities in a spherical trap
Parabolic density profile:
2g
0
R2 − r
2
if r ≤ R
elsewhere
Condensate radius:
R = aho
15N g
4πa3ho ~ωho
Chemical potential:
1/5
3
Ng/(4πaho )=1000 h- ωho
Thomas-Fermi
-3
2
mωho
condensate density [Naho]
ρ(r) ≈
(
0.002
0.0015
0.001
0.0005
0
0
1
2
3
4
7
6
5
r [aho]
8
9
10
K/N = 0.096 ~ωho
µ≈
~ωho
2
15N g
4πa3ho ~ωho
2/5
E/N = 16.83 ~ωho
Cold gases 2009 – p.4
Thomas-Fermi approximation
Example: Bose-Einstein condensate in a cigar-shaped trap
Trapping potential:
r⊥ =
Hau et al.
where
800
-2
+ ωz2 z 2
x2 + y 2
2 2
r⊥
ω⊥
p
column density [µm ]
Vext (r) =
m
2
1000
600
400
200
0
-60
-40
Ng
4πa3
ho
-20
0
z [µm]
20
40
60
= 125 ~ωho
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.4
Thomas-Fermi approximation
Example: Bose-Einstein condensate in a cigar-shaped trap
Trapping potential:
r⊥ =
Hau et al.
where
Angular trap frequencies:
ω⊥ = ωx = ωy = 2050 s−1 and
ωz = 170 s−1
800
-2
+ ωz2 z 2
x2 + y 2
2 2
r⊥
ω⊥
p
column density [µm ]
Vext (r) =
m
2
1000
600
400
200
0
-60
-40
Ng
4πa3
ho
-20
0
z [µm]
20
40
60
= 125 ~ωho
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.4
Thomas-Fermi approximation
Example: Bose-Einstein condensate in a cigar-shaped trap
Trapping potential:
r⊥ =
where
Angular trap frequencies:
ω⊥ = ωx = ωy = 2050 s−1 and
ωz = 170 s−1
Column density:
ρ(z) =
R
ρ(x, 0, z) dx
ideal gas
Hau et al.
800
-2
+ ωz2 z 2
x2 + y 2
2 2
r⊥
ω⊥
p
column density [µm ]
Vext (r) =
m
2
1000
600
400
200
0
-60
-40
Ng
4πa3
ho
-20
0
z [µm]
20
40
60
= 125 ~ωho
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.4
Thomas-Fermi approximation
Example: Bose-Einstein condensate in a cigar-shaped trap
Trapping potential:
r⊥ =
where
Angular trap frequencies:
ω⊥ = ωx = ωy = 2050 s−1 and
ωz = 170 s−1
Column density in the Thomas-Fermi
approximation:
ρ(z) =
4
3
q
2 1
mωx2 g
µ−
mωz2
2
z2
ideal gas
Thomas-Fermi
Hau et al.
800
-2
+ ωz2 z 2
x2 + y 2
2 2
r⊥
ω⊥
p
column density [µm ]
Vext (r) =
m
2
1000
600
400
200
0
-60
-40
Ng
4πa3
ho
-20
0
z [µm]
20
40
60
= 125 ~ωho
3/2
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.4
Breathing mode in a spherical trap
Time-dependent mean-square
radius in the limit
N g/ 4πa3ho ~ωho → ∞
Assumption of rotational invariance:
Ψ(r, t) = Ψ(r, t)
Cold gases 2009 – p.5
Breathing mode in a spherical trap
Time-dependent mean-square
radius in the limit
N g/ 4πa3ho ~ωho → ∞
Assumption of rotational invariance:
Ψ(r, t) = Ψ(r, t)
Virial relation for a Bose-Einstein condensate in a general atom trap:
d2
2i
hr
t
2
dt
=
6E
Nm
−
6Eho (t)
Nm
−
2
Nm
R
|Ψ(r, t)|2 r · ∇Vext (r) dr −
2K(t)
Nm
Cold gases 2009 – p.5
Breathing mode in a spherical trap
Time-dependent mean-square
radius in the limit
N g/ 4πa3ho ~ωho → ∞
Assumption of rotational invariance:
Ψ(r, t) = Ψ(r, t)
Virial relation for a Bose-Einstein condensate in a general atom trap:
d2
2i
hr
t
2
dt
=
6E
Nm
−
6Eho (t)
Nm
−
2
Nm
R
|Ψ(r, t)|2 r · ∇Vext (r) dr −
2K(t)
Nm
Relation between mean-square radius and trap energy:
Eho (t) =
m 2
2 ωho
R
r2 |Ψ(r, t)|2 dr =
Nm 2
2i
hr
ω
t
ho
2
Cold gases 2009 – p.5
Breathing mode in a spherical trap
Time-dependent mean-square
radius in the limit
N g/ 4πa3ho ~ωho → ∞
Assumption of rotational invariance:
Ψ(r, t) = Ψ(r, t)
Virial relation for a Bose-Einstein condensate in a spherical trap:
d2
2i
hr
t
2
dt
=
6E
Nm
2 hr 2 i −
− 5ωho
t
2K(t)
Nm
Cold gases 2009 – p.5
Breathing mode in a spherical trap
Time-dependent mean-square
radius in the limit
N g/ 4πa3ho ~ωho → ∞
Assumption of rotational invariance:
Ψ(r, t) = Ψ(r, t)
Approximate virial relation neglecting the kinetic energy:
d2
2
hr
it
2
dt
≈
6E
Nm
2
hr2 it
− 5ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
Time-dependent mean-square
radius in the limit
N g/ 4πa3ho ~ωho → ∞
Assumption of rotational invariance:
Ψ(r, t) = Ψ(r, t)
Approximate virial relation neglecting the kinetic energy:
hr2 it
Phase shift:
√
≈ A cos 5ωho t + ϕ +
tan ϕ = − √5ω
ho
6E
2
5N mωho
d
2i
hr
t=0
dt
2 )
]
[hr2 it=0 −6E/(5N mωho
Amplitude:
A=
2 )
hr2 it=0 −6E/(5N mωho
cos ϕ
Cold gases 2009 – p.5
Breathing mode in a spherical trap
Example: excitation of the breathing mode by a sudden
change of the interactions
Initial non-linearity parameter:
Ng
4πa3ho
= 500 ~ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
Example: excitation of the breathing mode by a sudden
change of the interactions
Initial non-linearity parameter:
Ng
4πa3ho
= 500 ~ωho
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
Example: excitation of the breathing mode by a sudden
change of the interactions
Initial non-linearity parameter:
Ng
4πa3ho
= 500 ~ωho
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
Cold gases 2009 – p.5
Breathing mode in a spherical trap
Example: excitation of the breathing mode by a sudden
change of the interactions
Initial non-linearity parameter:
= 500 ~ωho
Initial mean-square radius:
2
hr it=0 =
15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
2
Ng
4πa3ho
mean-square radius [aho ]
25
20
15
10
estimate
5
0
0
10
20
30
40
t [1/ωho]
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
50
Breathing mode in a spherical trap
Example: excitation of the breathing mode by a sudden
change of the interactions
Initial non-linearity parameter:
= 500 ~ωho
Initial mean-square radius:
2
hr it=0 =
15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
2
Ng
4πa3ho
mean-square radius [aho ]
25
20
15
10
estimate
simulation
5
0
0
10
20
30
40
t [1/ωho]
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
50
Breathing mode in a spherical trap
The mean-square radius oscillates with an angular
√
frequency . 5ωho !
Initial non-linearity parameter:
= 500 ~ωho
Initial mean-square radius:
hr2 it=0
=
15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2
hr
it=0
dt
=
2
N
R
r · j(r, 0) dr = 0
2
Ng
4πa3ho
mean-square radius [aho ]
25
20
15
10
estimate
simulation
5
0
0
10
20
30
40
t [1/ωho]
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
50
Breathing mode in a spherical trap
The trapped Bose-Einstein condensate undergoes a
breathing motion!
Initial non-linearity parameter:
0.025
50
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
0.02
40
0.015
30
0.01
20
0.005
10
potential energy [h- ωho]
= 500 ~ωho
-2
Ng
4πa3ho
column density [Naho]
t=0
0
-10
-5
0
z [aho]
5
10
0
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
The trapped Bose-Einstein condensate undergoes a
breathing motion!
Initial non-linearity parameter:
0.025
50
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
0.02
40
potential energy [h- ωho]
= 500 ~ωho
-2
Ng
4πa3ho
column density [Naho]
-1
t=0.281 ωho
0.015
30
0.01
20
0.005
10
0
-10
-5
0
z [aho]
5
10
0
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
The trapped Bose-Einstein condensate undergoes a
breathing motion!
Initial non-linearity parameter:
0.025
50
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
0.02
40
potential energy [h- ωho]
= 500 ~ωho
-2
Ng
4πa3ho
column density [Naho]
-1
t=0.562 ωho
0.015
30
0.01
20
0.005
10
0
-10
-5
0
z [aho]
5
10
0
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
The trapped Bose-Einstein condensate undergoes a
breathing motion!
Initial non-linearity parameter:
0.025
50
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
0.02
40
potential energy [h- ωho]
= 500 ~ωho
-2
Ng
4πa3ho
column density [Naho]
-1
t=0.843 ωho
0.015
30
0.01
20
0.005
10
0
-10
-5
0
z [aho]
5
10
0
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
The trapped Bose-Einstein condensate undergoes a
breathing motion!
Initial non-linearity parameter:
0.025
50
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
0.02
40
potential energy [h- ωho]
= 500 ~ωho
-2
Ng
4πa3ho
column density [Naho]
-1
t=1.124 ωho
0.015
30
0.01
20
0.005
10
0
-10
-5
0
z [aho]
5
10
0
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
The trapped Bose-Einstein condensate undergoes a
breathing motion!
Initial non-linearity parameter:
0.025
50
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
0.02
40
potential energy [h- ωho]
= 500 ~ωho
-2
Ng
4πa3ho
column density [Naho]
-1
t=1.405 ωho
0.015
30
0.01
20
0.005
10
0
-10
-5
0
z [aho]
5
10
0
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
The trapped Bose-Einstein condensate undergoes a
breathing motion!
Initial non-linearity parameter:
0.025
50
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
0.02
40
potential energy [h- ωho]
= 500 ~ωho
-2
Ng
4πa3ho
column density [Naho]
-1
t=1.686 ωho
0.015
30
0.01
20
0.005
10
0
-10
-5
0
z [aho]
5
10
0
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
The trapped Bose-Einstein condensate undergoes a
breathing motion!
Initial non-linearity parameter:
0.025
50
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
0.02
40
potential energy [h- ωho]
= 500 ~ωho
-2
Ng
4πa3ho
column density [Naho]
-1
t=1.967 ωho
0.015
30
0.01
20
0.005
10
0
-10
-5
0
z [aho]
5
10
0
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
The trapped Bose-Einstein condensate undergoes a
breathing motion!
Initial non-linearity parameter:
0.025
50
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
0.02
40
potential energy [h- ωho]
= 500 ~ωho
-2
Ng
4πa3ho
column density [Naho]
-1
t=2.248 ωho
0.015
30
0.01
20
0.005
10
0
-10
-5
0
z [aho]
5
10
0
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
The trapped Bose-Einstein condensate undergoes a
breathing motion!
Initial non-linearity parameter:
0.025
50
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
0.02
40
potential energy [h- ωho]
= 500 ~ωho
-2
Ng
4πa3ho
column density [Naho]
-1
t=2.529 ωho
0.015
30
0.01
20
0.005
10
0
-10
-5
0
z [aho]
5
10
0
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
The trapped Bose-Einstein condensate undergoes a
breathing motion!
Initial non-linearity parameter:
0.025
50
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
0.02
40
potential energy [h- ωho]
= 500 ~ωho
-2
Ng
4πa3ho
column density [Naho]
-1
t=2.81 ωho
0.015
30
0.01
20
0.005
10
0
-10
-5
0
z [aho]
5
10
0
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
Breathing mode in a spherical trap
Due to the rotational invariance, the breathing mode is also
referred to as a monopole excitation!
Initial non-linearity parameter:
0.025
50
Initial mean-square radius:
hr2 it=0 = 15.321 a2ho
Initial first time derivative of the mean-square
radius:
d
2i
hr
t=0
dt
=
2
N
R
r · j(r, 0) dr = 0
0.02
40
potential energy [h- ωho]
= 500 ~ωho
-2
Ng
4πa3ho
column density [Naho]
-1
t=2.81 ωho
0.015
30
0.01
20
0.005
10
0
-10
-5
0
z [aho]
5
10
0
E/N = 17.834 ~ωho
Non-linearity parameter during the evolution:
Ng
4πa3ho
= 1000 ~ωho
Cold gases 2009 – p.5
Collective excitations in a cylindrical trap
Dynamics of the condensate wave function in an infinitely
elongated cigar-shaped trap
Time-dependent Gross-Pitaevskii equation (g > 0):
h
~2
i
∂
i~ ∂t
Ψ(r⊥ , t) = − 2m ∇2⊥ + Vext (r⊥ ) + g |Ψ(r⊥ , t)|2 Ψ(r⊥ , t)
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Dynamics of the condensate wave function in an infinitely
elongated cigar-shaped trap
Time-dependent Gross-Pitaevskii equation (g > 0):
h
i
~2
∂
i~ ∂t
Ψ(r⊥ , t) = − 2m ∇2⊥ + Vext (r⊥ ) + g |Ψ(r⊥ , t)|2 Ψ(r⊥ , t)
Two-dimensional Laplacian and trapping potential:
∇2⊥
=
∂2
∂x2
+
Vext (r⊥ ) = Vext (x, y) =
∂2
∂y 2
m 2
2 ω⊥
x2
+ y2
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Dynamics of the condensate wave function in an infinitely
elongated cigar-shaped trap
Time-dependent Gross-Pitaevskii equation (g > 0):
h
i
~2
∂
i~ ∂t
Ψ(r⊥ , t) = − 2m ∇2⊥ + Vext (r⊥ ) + g |Ψ(r⊥ , t)|2 Ψ(r⊥ , t)
Two-dimensional Laplacian and trapping potential:
∇2⊥
=
∂2
∂x2
+
Vext (r⊥ ) = Vext (x, y) =
∂2
∂y 2
m 2
2 ω⊥
x2
+ y2
Number of atoms integrated over a distance L in z-direction:
N = L |Ψ(r⊥ , t)|2 dr⊥
R
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Dynamics of the mean-square radius
Time-dependent mean-square radius:
2
it
hr⊥
=
L
N
R
2
x +y
2
|Ψ(r⊥ , t)|2 dr⊥
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Dynamics of the mean-square radius
Time-dependent mean-square radius:
2
it
hr⊥
=
L
N
2
R
x +y
=
2L
N
2
|Ψ(r⊥ , t)|2 dr⊥
First time derivative of the mean-square radius:
d
2i
hr
t
⊥
dt
R
r⊥ · j(r⊥ , t) dr⊥
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Dynamics of the mean-square radius
Time-dependent mean-square radius:
2
it
hr⊥
=
L
N
2
R
x +y
=
2L
N
2
|Ψ(r⊥ , t)|2 dr⊥
First time derivative of the mean-square radius:
d
2i
hr
t
⊥
dt
Virial relation in two dimensions:
d2
2i
hr
2
⊥ t
dt
=
4E
Nm
−
4Eho (t)
Nm
−
R
2L
Nm
r⊥ · j(r⊥ , t) dr⊥
R
|Ψ(r⊥ , t)|2 r⊥ · ∇⊥ Vext (r⊥ ) dr⊥
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Dynamics of the mean-square radius
Time-dependent mean-square radius:
2
it
hr⊥
=
L
N
2
R
x +y
=
2L
N
2
|Ψ(r⊥ , t)|2 dr⊥
First time derivative of the mean-square radius:
d
2i
hr
t
⊥
dt
Virial relation in two dimensions:
d2
2i
hr
2
⊥ t
dt
=
4E
Nm
−
4Eho (t)
Nm
−
R
2L
Nm
r⊥ · j(r⊥ , t) dr⊥
R
|Ψ(r⊥ , t)|2 r⊥ · ∇⊥ Vext (r⊥ ) dr⊥
Main difference in the derivations between the two-dimensional and
three-dimensional cases:
∇⊥ · r⊥ = 2 whereas ∇ · r = 3
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Dynamics of the mean-square radius
Time-dependent mean-square radius:
2
it
hr⊥
=
L
N
2
R
x +y
=
2L
N
2
|Ψ(r⊥ , t)|2 dr⊥
First time derivative of the mean-square radius:
d
2i
hr
t
⊥
dt
R
r⊥ · j(r⊥ , t) dr⊥
Virial relation for a harmonic trap in two dimensions:
d2
2
hr
2
⊥ it
dt
=
4E
Nm
−
4Eho (t)
Nm
−
4Eho (t)
Nm
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Dynamics of the mean-square radius
Time-dependent mean-square radius:
2
it
hr⊥
=
L
N
R
2
x +y
2
|Ψ(r⊥ , t)|2 dr⊥ =
2
2 Eho (t)
N mω⊥
First time derivative of the mean-square radius:
d
2i
hr
dt ⊥ t
=
2L
N
R
r⊥ · j(r⊥ , t) dr⊥
Virial relation for a harmonic trap in two dimensions:
d2
2i
hr
2
⊥ t
dt
=
4E
Nm
2 hr 2 i
− 4ω⊥
⊥ t
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Dynamics of the mean-square radius
Time-dependent mean-square radius:
2
it
hr⊥
=
L
N
R
2
x +y
2
|Ψ(r⊥ , t)|2 dr⊥ =
2
2 Eho (t)
N mω⊥
First time derivative of the mean-square radius:
d
2i
hr
dt ⊥ t
=
2L
N
R
r⊥ · j(r⊥ , t) dr⊥
Exact time dependence of the mean-square radius in two dimensions:
2 i = A cos (2ω t + ϕ) +
hr⊥
t
⊥
E
2
N mω⊥
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Dynamics of the mean-square radius
Time-dependent mean-square radius:
2
it
hr⊥
=
L
N
R
2
x +y
2
|Ψ(r⊥ , t)|2 dr⊥ =
2
2 Eho (t)
N mω⊥
First time derivative of the mean-square radius:
d
2i
hr
dt ⊥ t
=
2L
N
R
r⊥ · j(r⊥ , t) dr⊥
Exact time dependence of the mean-square radius in two dimensions:
2 i = A cos (2ω t + ϕ) +
hr⊥
t
⊥
E
2
N mω⊥
Phase shift and amplitude:
tan ϕ = − 2ω
⊥
d
hr2 i
dt ⊥ t=0
2
2i
[hr⊥
t=0 −E/(N mω⊥ )]
and A =
2
2i
hr⊥
t=0 −E/(N mω⊥ )
cos ϕ
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
The mean-square radius oscillates with the frequency ω⊥ /π!
Time-dependent mean-square radius:
2
it
hr⊥
=
L
N
R
2
x +y
2
|Ψ(r⊥ , t)|2 dr⊥ =
2
2 Eho (t)
N mω⊥
First time derivative of the mean-square radius:
d
2i
hr
t
⊥
dt
=
2L
N
R
r⊥ · j(r⊥ , t) dr⊥
Exact time dependence of the mean-square radius in two dimensions:
2 i = A cos (2ω t + ϕ) +
hr⊥
t
⊥
E
2
N mω⊥
Phase shift and amplitude:
tan ϕ = − 2ω
d
hr2 i
dt ⊥ t=0
2
2
⊥ [hr⊥ it=0 −E/(N mω⊥ )]
and A =
2
2i
hr⊥
t=0 −E/(N mω⊥ )
cos ϕ
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
The results are applicable in the Thomas-Fermi limit!
Time-dependent mean-square radius:
2
it
hr⊥
=
L
N
R
2
x +y
2
|Ψ(r⊥ , t)|2 dr⊥ =
2
2 Eho (t)
N mω⊥
First time derivative of the mean-square radius:
d
2i
hr
dt ⊥ t
=
2L
N
R
r⊥ · j(r⊥ , t) dr⊥
Exact time dependence of the mean-square radius in two dimensions:
2 i = A cos (2ω t + ϕ) +
hr⊥
t
⊥
E
2
N mω⊥
Phase shift and amplitude:
tan ϕ = − 2ω
⊥
d
hr2 i
dt ⊥ t=0
2
2i
[hr⊥
t=0 −E/(N mω⊥ )]
and A =
2
2i
hr⊥
t=0 −E/(N mω⊥ )
cos ϕ
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Some other collective excitations: dipole modes excited by
shaking the trap about a symmetry axis
Centre of mass of a Bose-Einstein condensate:
hrit =
1
N
R
r |Ψ(r, t)|2 dr
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Some other collective excitations: dipole modes excited by
shaking the trap about a symmetry axis
Centre of mass of a Bose-Einstein condensate:
hrit =
Centre of mass motion:
d
dt hrit
1
N
R
=
r |Ψ(r, t)|2 dr
1
N
R
j(r, t) dr
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Some other collective excitations: dipole modes excited by
shaking the trap about a symmetry axis
Centre of mass of a Bose-Einstein condensate:
hrit =
1
N
Centre of mass motion:
d
dt hrit
d2
hrit
dt2
=
R
=
− N1m
R
r |Ψ(r, t)|2 dr
1
N
R
j(r, t) dr
|Ψ(r, t)|2 ∇Vext (r) dr
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Some other collective excitations: dipole modes excited by
shaking the trap about a symmetry axis
Centre of mass of a Bose-Einstein condensate:
1
N
hrit =
Centre of mass motion:
d
dt hrit
d2
hrit
dt2
R
=
r |Ψ(r, t)|2 dr
1
N
R
j(r, t) dr
1
h∇Vext (r)it
= −m
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Some other collective excitations: dipole modes excited by
shaking the trap about a symmetry axis
Centre of mass of a Bose-Einstein condensate:
hrit =
1
N
R
r |Ψ(r, t)|2 dr
Centre of mass motion in a harmonic trap:
d
dt hrit
=
1
N
R
j(r, t) dr
d2
hxit
dt2
= −ωx2 hxit
d2
hyit
dt2
= −ωy2 hyit
d2
hzit
dt2
= −ωz2 hzit
Cold gases 2009 – p.6
Collective excitations in a cylindrical trap
Some other collective excitations: dipole modes excited by
shaking the trap about a symmetry axis
Centre of mass of a Bose-Einstein condensate:
hrit =
1
N
R
r |Ψ(r, t)|2 dr
Centre of mass motion in a harmonic trap:
d
dt hrit
=
1
N
R
j(r, t) dr
d2
hxit
dt2
= −ωx2 hxit
d2
hyit
dt2
= −ωy2 hyit
d2
hzit
dt2
= −ωz2 hzit
Courtesy W. Ketterle
Cold gases 2009 – p.6
Literature
Some classic experimental references
Bose-Einstein condensation of atomic gases
M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269,
198 (1995); K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,
D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995)
Cold gases 2009 – p.7
Literature
Some classic experimental references
Bose-Einstein condensation of atomic gases
M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269,
198 (1995); K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,
D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995)
Bose-Einstein condensates in the Thomas-Fermi limit
M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle,
Phys. Rev. Lett. 77, 416 (1996)
Cold gases 2009 – p.7
Literature
Some classic experimental references
Bose-Einstein condensation of atomic gases
M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269,
198 (1995); K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,
D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995)
Bose-Einstein condensates in the Thomas-Fermi limit
M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle,
Phys. Rev. Lett. 77, 416 (1996)
Collective excitations in Bose-Einstein condensates
D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 77,
420 (1996); M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee,
C. G. Townsend, and W. Ketterle, Phys. Rev. Lett. 77, 988 (1996)
Cold gases 2009 – p.7
Literature
Some classic theory references
Gross-Pitaevskii equation
E. P. Gross, Nuovo Cimento 20, 454 (1961); L. P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961)
Cold gases 2009 – p.7
Literature
Some classic theory references
Gross-Pitaevskii equation
E. P. Gross, Nuovo Cimento 20, 454 (1961); L. P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961)
Thomas-Fermi approximation
M. Edwards and K. Burnett, Phys. Rev. A 51, 1382 (1995); G. Baym and C. J. Pethick,
Phys. Rev. Lett. 76, 6 (1996)
Cold gases 2009 – p.7
Literature
Some classic theory references
Gross-Pitaevskii equation
E. P. Gross, Nuovo Cimento 20, 454 (1961); L. P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961)
Thomas-Fermi approximation
M. Edwards and K. Burnett, Phys. Rev. A 51, 1382 (1995); G. Baym and C. J. Pethick,
Phys. Rev. Lett. 76, 6 (1996)
Collective excitations in Bose-Einstein condensates
P. A. Ruprecht, M. Edwards, K. Burnett, and C. W. Clark, Phys. Rev. A 54, 4178 (1996);
M. Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and C. W. Clark, Phys. Rev. Lett. 77, 1671
(1996); K. G. Singh and D. S. Rokhsar, Phys. Rev. Lett. 77, 1667 (1996); S. Stringari,
Phys. Rev. Lett. 77, 2360 (1996); L. P. Pitaevskii, Phys. Lett. A 221, 14 (1996)
Cold gases 2009 – p.7
Literature
Some review articles and books
Review articles
F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999);
A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001); K. Bongs and K. Sengstock,
Rep. Prog. Phys. 67, 907 (2004)
Cold gases 2009 – p.7
Literature
Some review articles and books
Review articles
F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999);
A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001); K. Bongs and K. Sengstock,
Rep. Prog. Phys. 67, 907 (2004)
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
Literature
Some review articles and books
Books on many-particle physics
A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinskii, Methods of Quantum Field Theory in
Statistical Physics (Prentice Hall, 1963)
A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle systems (McGraw-Hill, 1971)
Cold gases 2009 – p.7