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  mr r 2
2
2
n 
 (r , t )   ann (r )e  E t / 
n
t 
n 0
 (r, t )  a00 (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(x1)
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
2R
-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
2R
-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 nq  j, n q  j 
j  1 n  q  j , n q  j  1
a  q n q  j , n q  j 
j nq  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 kq 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  8na vk  dq 2 Akq
2k
2
1.0
Ek  Eq
2
1  Ek  Eq )
2
Excitations
2
k [8a ]
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