Transcript Slide 1
System and definitions
er
In harmonic trap (ideal):
Dilute interacting Bosons
Single particle field operators:
Macroscopic occupation assumption:
Homogeneous result:
Dilute interacting Bosons
Not an operator!
an operator!
Inhomogeneous (time and space):
Single particle density matrix formalism:
Time evolution of operator in Heisenberg Rep.
Scattering theory
(see ahead):
Mean-field assumption – discard fluctuating part
Time-Dependent GrossPitaevskii equation
(TDGPE)
A short review of scat. theory
Eigenvalue scattering problem:
Fourier Trans.
Born Approx.
Low k limit
(“s-wave”)
Indistinguishable particles…
Effective
potential!
GPE – ground state properties
Variational derivation +
Energy functional
1
Smallness parameter:
Interaction energy:
Kinetic energy:
Eint
3
n N / aho
Ekin
1, 1
Weak interactions ≠ ideal gas behavior!
(still small depletion, but strongly non-ideal)
GPE – ground state properties
TDGPE:
Ansatz
+ normalization:
TIGPE:
Note: energy is not a good quantum number (nonlinear problem!)
Numerical solution of TDGPE
Imaginary time evolution:
Vext
1
1
m z z 2 mr r 2
2
2
n
(r , t ) ann (r )e E t /
n
t
n 0
(r, t ) a00 (r)e E t /
0
Interacting ground-state
Non-interacting ground-state
(Mean-field repulsion
causes increase in
Size)
Thomas-Fermi approx.
Neglect kinetic term:
Relaxed
T.F.
Excitations – Bogoliubov equations
Ansatz (plug
Into TDGPE):
Neglect terms of order
u2, v2 and uv
Bogoliubov
equations
(“linearized GPE”):
Homogeneous system (u(r) and v(r) are plane waves):
Homogeneous Bogoliubov
spectrum
E(m)
Interaction vs. Quantum
Pressure
m
x 1
k 2
2m
“healing length”
k(x1)
Bragg Spectroscopy
k 2k p sin
2
5P3 / 2
10
o
o
kp
(m/)
8
6
4
2
kp
0
0.0
0.5
1.0
1.5
-1
k (x )
H int
M. Kozuma, et. al., PRL 82, 871 (1999).
R
2
N 0 S k bk bk
)
J. Stenger, et. al., PRL 82, 4569 (1999).
2.0
2.5
3.0
The Measured Excitation Spectrum
(using Bragg spectroscopy)
14
/(2) (kHz)
12
10
2R
-1
-1
x
8
6
4
Liquid Helium
(scaled for
comparison)
2
0
0
2
4
6
8
10
-1
k (mm )
12
14
/(2) (kHz)
Phonon Region
1.0
2R
-1
0.5
0.0
0
1
2
-1
k (mm )
3
Superfluidity!
Landau criteria:
c
(k )
k
Interactions – lead to
superfluidity!
Superfluid velocity
A few mm/sec in experimental
systems!
Many body theory (homogeneous)
Assume macroscopic occupation of S.P.
Ground state:
Put in assumption + keep terms of order N 02 and N 0
The number operator is conserved – can be placed in
Hˆ
Many body theory (homogeneous)
Neglected:
Bogoliubov
Transform:
Atomic commutation
relations give:
Many body theory (homogeneous)
Eliminate off-diagonal third
line:
Convenient representation:
Solution of
quasi-particle
amplitudes:
Diagonalized Hamiltonian
Energy spectrum:
(again)
Ground state is a highly non-trivial
Superposition of all momentum states:
Ground state energy:
Quasi-particle physics
Inverse transformation:
Particle creation
Particle
Annihilation
Low k limit
Quasi-particle factors for repulsive
condensates
a u2
b v2
High k limit
Quasi-particle physics
αk
s!
s
???
1
q 0 u q
Don’t
Forget Bosonic
Enhancement!
j
vq
n q j , n q j
j 0 u q
j
aq nq j, n q j
j 1 n q j , n q j 1
a q n q j , n q j
j nq j 1, n q j
s 1)vk2
suk2 vk2
Quantum depletion of S.P. ground
state
Evaluate the non-single-particle component of the ground state at T=0
About 1% for “standard”
experiments
Attractive collapse!
Complex energy –
unstable to excitation!
Finite size can save us (cutoff in
Low k’s)
Experimental values:
A few thousand atoms!
Structure factor and Feynman
relation
Static structure factor (Fourier transform
of the density-density correlation function)
T=0
Static Structure Factor
Measure of:
Feynman Relation
(k )
S (k ) o
(k )
• Response at k
• Fluctuations with wave-number k
1.0
10
x 1
0.8
8
(m/)
S(k)
0.6
0.4
0.2
2
m clarge
6
x 1
k 2
2m
4
2
ceff k
0.0
0.0
0.5
1.0
1.5
-1
k (x )
2.0
2.5
3.0
0
0.0
0.5
1.0
1.5
-1
k (x )
2.0
2.5
3.0
Excitation Spectrum of Superfluid 4He
Feynman Relation
(k )
S (k ) o
(k )
1200
x 1
(k) / 2 (GHz)
1000
800
600
400
200
0
0
5
10
15
20
25
-1
k (nm )
D. G. Henshaw, Phys. Rev. 119, 9 (1960).
D. G. Henshaw and A. D. B. Woods, Phys.
Rev. 121, 1266 (1961).
Higher order – Beliaev and Landau
damping
g N0
ˆ
H int
2V
A
kq
k 0 ,q k 0
q k q q kq k )
k
Akq The many-body suppression factor:
Akq 2uk uquk q vquk q uqvk q ) 2vk vqvk q uqvk q vquk q )
Landau
Beliaev
k-q
k-q
k
k
q
q
Damping rate
Fermi golden rule:
2
g 2 N0
k
A Ek Eq E k q
2 kq
2V q
)
The function can be turned into a geometrical condition:
cos( )
1
-1
1 2
2
k q 2 1 1 Ek Eq )
2kq
q [x ]
2
0
-1
-2
0
1
2
3
-1
q|| [x ]
4
Damping rate
q
n k vk k 8na vk dq 2 Akq
2k
2
1.0
Ek Eq
2
1 Ek Eq )
2
Excitations
2
k [8a ]
0.8
0.6
Impurities
vc
0.4
0.2
0.0
0
2
4
6
-1
k [x ]
8
10
Points not covered
- Inhomogeneous Bogoliubov theory
- Beyond T=0
- Coherent collisions of excitations (FWM)
- Hydrodynamic representation of GPE
- Na3 ~ 1 – theory and experiment