System and definitions In harmonic trap (ideal): er.
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Transcript of System and definitions In harmonic trap (ideal): er.
System and definitions
In harmonic trap (ideal):
er
Dilute interacting Bosons
Single particle field operators:
Macroscopic occupation assumption:
Homogeneous result:
Dilute interacting BosonsInhomogeneous (time and space):
Single particle density matrix formalism:
Scattering theory (see ahead):
Time evolution of operator in Heisenberg Rep.
Time-Dependent Gross-Pitaevskii equation (TDGPE)
Mean-field assumption – discard fluctuating part
Not an operator! an operator!
A short review of scat. theory
Fourier Trans.
Born Approx.
Indistinguishable particles…Low k limit
“)s-wave(”
Eigenvalue scattering problem:
Effectivepotential!
GPE – ground state properties
Interaction energy: 3/ hoaNn
Variational derivation+ Energy functional
Smallness parameter:
intE
Kinetic energy: kinE
1,1
Weak interactions ≠ ideal gas behavior!
1
)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 TDGPEImaginary time evolution:
Interacting ground-state Non-interacting ground-state
n
n
tEnn
neratr0
/)(),(
/)(),( tEeratr 000
t
22
2
1
2
1rmzmV rzext
)Mean-field repulsioncauses increase inSize(
Thomas-Fermi approx.
Neglect kinetic term:
Relaxed
T.F.
Excitations – Bogoliubov equationsAnsatz (plugInto TDGPE):
Bogoliubovequations(“linearized GPE”):
Homogeneous system (u(r) and v(r) are plane waves):
Neglect terms of orderu2, v2 and uv
Homogeneous Bogoliubov spectrum
“healing length”
Interaction vs. Quantum Pressure
k(
E(
m
k
2
2
1
Bragg Spectroscopy
M. Kozuma, et. al., PRL 82, 871 (1999). J. Stenger, et. al., PRL 82, 4569 (1999).
o o
pk pk
2sin2
pkk
0.0 0.5 1.0 1.5 2.0 2.5 3.00
2
4
6
8
10
(
/)
k (-1)
2/35P
kkkR bbSNH
02
int
The Measured Excitation Spectrum
0 2 4 6 8 10 12 140
2
4
6
8
10
12
14
2R-1 -1
/(2)
(kH
z)
k (m-1)
)using Bragg spectroscopy(
Liquid Helium)scaled for
comparison(
Phonon Region
0 1 2 30.0
0.5
1.02R-1
/(2)
(kH
z)
k (m-1)
Superfluidity!
Landau criteria:
k
kc
)(
Superfluid velocity
A few mm/sec in experimentalsystems!
Interactions – lead tosuperfluidity!
Many body theory (homogeneous)
Assume macroscopic occupation of S.P.Ground state:
Put in assumption + keep terms of order 20N and 0N
The number operator is conserved – can be placed in H
Many body theory (homogeneous)
Neglected :
Bogoliubov Transform:
Atomic commutation relations give:
Many body theory (homogeneous)
Eliminate off-diagonal thirdline:
Convenient representation:
Solution of quasi-particleamplitudes:
Diagonalized Hamiltonian
Energy spectrum: (again)
Ground state is a highly non-trivialSuperposition of all momentum states:
Ground state energy:
Quasi-particle physicsInverse transformation:
Particle creation
Particle Annihilation
2
2
v
u
b
a
Quasi-particle factors for repulsivecondensates
Low k limit
High k limit
Quasi-particle physics
s!
s
kα ???
0 0
,1
q
j
jqq
j
q
q
q
jnjnu
v
u
jnjnjjnjna
jnjnjjnjna
qqqqq
qqqqq
,1,
1,1,Don’t Forget BosonicEnhancement!
22kk vsu 21 kvs
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 inLow k’s(
Experimental values:A few thousand atoms!
Structure factor and Feynman relation
Static structure factor (Fourier transformof the density-density correlation function(
T=0
Static Structure FactorMeasure of:
• Response at k
• Fluctuations with wave-number k
0.0 0.5 1.0 1.5 2.0 2.5 3.00
2
4
6
8
10
(
/)
k (-1)
k ceff
m
k
2
2
2largemc
1
)(
)()(
k
kkS o
Feynman Relation
0.0 0.5 1.0 1.5 2.0 2.5 3.00.0
0.2
0.4
0.6
0.8
1.0
S(k
)
k (-1)
1
Excitation Spectrum of Superfluid 4He
D. G. Henshaw and A. D. B. Woods, Phys. Rev. 121, 1266 (1961).
)(
)()(
k
kkS o
Feynman Relation
0 5 10 15 20 250
200
400
600
800
1000
1200
(k)
/ 2
(G
Hz)
k (nm-1)
1
D. G. Henshaw, Phys. Rev. 119, 9 (1960).
Higher order – Beliaev and Landau damping
00
0
2 kqkkqkqqkqkkq
,int
ˆ AV
NgH
Beliaev
k
k-q
q
Landau
k
k-q
q
qkqqkqqkqqkqqkqqkqkq uvvuvvvvuuvuuuA kk 22
Akq The many-body suppression factor:
Damping rate
Fermi golden rule: q
qkk EEEAV
Ngqkkq
2
20
2
2
0 1 2 3 4-2
-1
0
1
2
q [-1
]q
|| [-1]
222 11
2
1)cos( qk EEqk
kq
The function can be turned into a geometrical condition:
Damping rate
0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
vc
k [8a
2 ]
k [-1]
22
22
128
qk
qkkkkk
EE
EEA
k
qdqvnavn
kq
Impurities
Excitations
Points not covered -Inhomogeneous Bogoliubov theory
-Beyond T=0 -Coherent collisions of excitations (FWM)
-Hydrodynamic representation of GPE -Na3 ~ 1 – theory and experiment
