images of excited condensates: diagonal dynamical bogoliubov vacuum
DESCRIPTION
Images of Excited Condensates: Diagonal Dynamical Bogoliubov Vacuum. Jacek Dziarmaga & Krzysztof Sacha. Kraków, Poland. PRA 67, 033608 (2003) cond-mat/0503328. Interference of BE condesates. Bogoliubov theory. Diagonal dynamical Bogoliubov vacuum. Density measurements - PowerPoint PPT PresentationTRANSCRIPT
Images of Excited Condensates:
Diagonal Dynamical Bogoliubov Vacuum
Jacek Dziarmaga
& Krzysztof Sacha
Kraków, Poland
PRA 67, 033608 (2003)cond-mat/0503328
Interference of BE condesates
Bogoliubov theory
Diagonal dynamical Bogoliubov vacuum
Density measurementson Bogoliubov vacuum
Example: dark soliton
1 )()( )(
and in , 22
xxx
ee xixiNN
Fock state
Condensate interferenceJavanainen & Yoo, PRL 76, 161 (1996)
Single density measurement
Condensate interference
22122122
21
, )()()....()()....()( ,
),....,,(NN
NNNN
N
xxxxxx
xxxP
In real experiment
22111122
11
2211221
,)()..( )()( )()..(,
)..|(
, )()( , )(
NNkkkk
NN
kk
NNNN
xxxxxx
xxxP
xxxP
In computer experiment
annihilation of atoms
k
xx
xdxx
xd
eeeed
k
NNk
xiixiiNN
1
),....,(
) cos( :N ),( , )()....(
) cos( :N
:N ,
10
20
2
0221
221
2
0
21
2
022
22
Condensate interference
Phase ``collapse’’ by density measurement
gxVH xx 2
121 )(dx
0GP00*0
221
0
0
)(
)()()(
HgxV
xxNx
x
Hamiltonian
Small fluctuations
Quantum Bogoliubov theory
Fig.Jav
0*0GP
*0
*0
000*0GP
2
with
,
gHg
ggHL
LdxH
Expansion
NdN
mmm
m
m
m
mm
mmmm
m
vub
v
uL
v
u
xvbxubx
)( )( )( *
1
0 0 bmb
Bogoliubov transformation
mmm
m bbH
12
particles <-> quasiparticles
Bogoliubov Hamiltonian Bogoliubov vacuum
),(
),(),(),(
00*0
221
0
0
txVgi
xtxtNxt
xt
Time-dependent Bogoliubov theory
Condensate in the ground state
Condensate in an excited state
),( txV
)0(0 b
)(0 tb
0 0 )(
)()()(
)(
)(
tbm
mmm
m
m
m
mt
tb
tvtutb
v
utL
v
ui
Solution
0 0
)( )(
t
*
1
bb
mmmm
m tvbtub
Ansatz
Excited state =t-dependent vacuum
Time-dependent Bogoliubov theory
Girardeau & Arnowitt, PR 113, 755 (1959)Castin & Dum, PRA 57, 3008 (1998)
0 0 )( )( tbm tb
N-conserving theory
N
amN
amm
mmm
vub
vub
00
N-conserving operator
Vacuum is N-particle state
Diagonal dynamical vacuum
0, 0 0
basis condensate-non lorthonorma )(
[0,1) )(
function wavecondensate )(
100)(
0
aaaabb
t
t
t
mtbm
0 02
100)(
N
aaaatb
Exact solution is
[0,1)
1 )(
),( ),( )1( ),( ),(
0 )( )( 0
),( ),( ),( ),(
0 )( )( 0
1
*
1
)(ˆˆ
)(
*
1
*
1
)()(
00
dN
dNt
ytxtdNdNytvxtu
yx
ytxtdNytvxtv
yx
mm
m
tbN
a
N
atb
mm
m
tbtb
)( )( )( )( tttvtu mm
Recipe:
qNqdq
aaaa
N
NdNN
10
22
1
00
: exp
0 2
Quantum superposition
Real coordinates q
Born rule (?)
)(
)'( :':
then, 1When
: 0
2
212
21
21
2
2
221
2
2)'(
1
10)(
qPee
qqqNqN
dN
qNedq
dN
N
q
tb
e
2
10
21
)()( )|(
exp )(2
xqxNqx
qP
N
dN
q
Density measurement
00
2
002
21
02
21
0 ),( xtVgxi xt
Dark soliton
Fig:imprinting
Burger et al., PRL 83, 5198 (1999)
Phase imprinting in Bogoliubov theoryCondesate Densityafter 2ms
Total densityafter 2ms
In focus Mode 1
250 105.1
00
15
111002
dNN
aaaaN
b
48.14with
:line Solid
1
2
101
q
NN
q
Density measurement
CONCLUSION
0 0 2
00)(
N
aaaatb
)(
2
21
dN
q
eqP
Exact diagonal vacuum
Distribution of images
Condensate in thermal state
Condensate in excited state
),( txV
RLbebbbbb
RL bNbNebdbd RLRRLL :: **
21*
2122
orthogonal-non are ),(
),(),( N )|( )()( Tr
)()( Tr
10
xt
xtqxtqxyx
yx