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02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
RPA Studies of the Dynamic
Response of Ultracold Atoms
in 1D Optical Lattices
Markus Hild
Panagiota PapakonstantinouFelix SchmittRobert Roth
1
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Outline
2
• Bose-Hubbard Model & Spectroscopy
• Equations of Motion (EOM)
• Random Phase Approximation (RPA)...
• ... and the Bose-Hubbard Model
• Results
H = −JI�
i=1
(a†iai+1 + a
†i+1ai ) +
U
2
I�
i=1
ni(ni − 1)
I N U J
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Bose-Hubbard Model
optical
lattice
tunneling on-site interaction
3
sites, particles, interaction strength , tunneling strength
T = 0
H = −JI�
i=1
(a†iai+1 + a
†i+1ai ) +
U
2
I�
i=1
ni(ni − 1)
I N U J
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Bose-Hubbard Model
optical
lattice
tunneling on-site interaction
3
sites, particles, interaction strength , tunneling strength
T = 0
H = −JI�
i=1
(a†iai+1 + a
†i+1ai ) +
U
2
I�
i=1
ni(ni − 1)
I N U J
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Bose-Hubbard Model
optical
lattice
tunneling on-site interaction
3
sites, particles, interaction strength , tunneling strength
T = 0
H = −JI�
i=1
(a†iai+1 + a
†i+1ai ) +
U
2
I�
i=1
ni(ni − 1)
I N U J
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Bose-Hubbard Model
optical
lattice
tunneling on-site interaction
3
sites, particles, interaction strength , tunneling strength
T = 0
H = −JI�
i=1
(a†iai+1 + a
†i+1ai ) +
U
2
I�
i=1
ni(ni − 1)
I N U J
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Bose-Hubbard Model
optical
lattice
tunneling on-site interaction
3
sites, particles, interaction strength , tunneling strength
T = 0
number-state representation
|φ� =�
l
cφl |n1, ..., nI�l,
�
i
ni = N
V (x) = V0 sin2(kx)
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Spectroscopy
via Lattice Modulation
static lattice potential
4
J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 4547
V (x) = V0 sin2(kx)
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Spectroscopy
via Lattice Modulation
static lattice potential
4
J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 4547
amplitude modulated lattice
V (x, t) = V0[1 + F sin(ωt)] sin2(kx)
V (x) = V0 sin2(kx)
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Spectroscopy
via Lattice Modulation
static lattice potential
4
I = N = 10U/J = 20
0 1 10 100 1000
ν
0
20
40
60
.
(Eν−
E0)/
J0
1
30 20 10
∆E/J0
0
20
40
60
ω/J
0
1
(Eν−
E0)/
J
ν∆E/J
ω/J
J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 4547
amplitude modulated lattice
V (x, t) = V0[1 + F sin(ωt)] sin2(kx)
exact time-evolution
V (x) = V0 sin2(kx)
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Spectroscopy
via Lattice Modulation
static lattice potential
4
I = N = 10U/J = 20
0 1 10 100 1000
ν
0
20
40
60
.
(Eν−
E0)/
J0
1
30 20 10
∆E/J0
0
20
40
60
ω/J
0
1
(Eν−
E0)/
J
ν∆E/J
ω/J
linearized Hamiltonian
Hlin(t) = H0 + sin(ωt)(λH0 − κHJ)
J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 4547
amplitude modulated lattice
V (x, t) = V0[1 + F sin(ωt)] sin2(kx)
V (x) = V0 sin2(kx)
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Spectroscopy
via Lattice Modulation
static lattice potential
4
I = N = 10U/J = 20
0 1 10 100 1000
ν
0
20
40
60
.
(Eν−
E0)/
J0
1
30 20 10
∆E/J0
0
20
40
60
ω/J
0
1
(Eν−
E0)/
J
ν∆E/J
ω/J
linearized Hamiltonian
Hlin(t) = H0 + sin(ωt)(λH0 − κHJ)
transition amplitudes
|�0|κHJ |ν�|2
J. Phys. B: At. Mol. Opt. Phys. 39 (2006) 4547
amplitude modulated lattice
V (x, t) = V0[1 + F sin(ωt)] sin2(kx)
H|ν� = Eν |ν�
�0|[δQ, [H,Q†ν ]]|0� = (Eν − E0)�0|[δQ,Q†
ν ]|0�
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
From the Schrödinger Equation
to the Equations of Motion (EOM)
5
phonon operators
Q†ν |0� = |ν�
Qν |0� = 0
Schrödinger equation
variations δQexhaust the whole
Hilbert space
!“complicated” reformulation of the Schrödinger equation
! requires the groundstate
!offers “handles” for approximations: phonon operators and restriction
of the variations
Q†ν =
�
k
X(ν)k c†k
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Approximation of the
Phonon Operators
6
simplest expansions of phonon operators
1 particle - 1 holeexcitations
Tamm-Dancoff Approximation (TDA)
Random PhaseApproximation (RPA)
Q†ν =
�
k
X(ν)k c†k −
�
k
Y (ν)k ck
1 particle - 1 hole excitations +de-excitations
c†k (ck)
systematic approach
expand phonon operators in terms of particle-hole (de-) excitations
higherRPAs
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Particle-Hole Operators and the
Bose-Hubbard Model
7
system properties
• filling factor
• strongly interacting
ground-state approximation
|RPA, 0� ≈ |1, 1, · · · , 1�N/I = 1U � J
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Particle-Hole Operators and the
Bose-Hubbard Model
7
c†ij = a†iaj
cij = a†jai
naive ansatz
system properties
• filling factor
• strongly interacting
ground-state approximation
|RPA, 0� ≈ |1, 1, · · · , 1�N/I = 1U � J
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Particle-Hole Operators and the
Bose-Hubbard Model
7
c†ij = a†iaj
cij = a†jai
naive ansatz just hopping in opposite directions
system properties
• filling factor
• strongly interacting
ground-state approximation
|RPA, 0� ≈ |1, 1, · · · , 1�N/I = 1U � J
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Particle-Hole Operators and the
Bose-Hubbard Model
7
c†ij = a†iaj
cij = a†jai
naive ansatz just hopping in opposite directions
F a i l
system properties
• filling factor
• strongly interacting
ground-state approximation
|RPA, 0� ≈ |1, 1, · · · , 1�N/I = 1U � J
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Particle-Hole Operators and the
Bose-Hubbard Model
7
c†ij = a†iaj
cij = a†jai
naive ansatz just hopping in opposite directions
F a i l
c†ij =1√2
a†ia
†iaiaj
cij =1√2
a†ja
†iaiai
creates ph-excitation
destroys ph-excitation
system properties
• filling factor
• strongly interacting
ground-state approximation
|RPA, 0� ≈ |1, 1, · · · , 1�N/I = 1U � J
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Solving RPA
8
�A B−B −A
� �Xν
Yν
�= Eν0
�S −T−T S
� �Xν
Yν
�
EOM formulated as generalized eigenproblem
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Solving RPA
8
�A B−B −A
� �Xν
Yν
�= Eν0
�S −T−T S
� �Xν
Yν
�
EOM formulated as generalized eigenproblem
[A,H,B] :=12
([A, [H,B]] + [[A,H],B])
Akl = �0|[ck,H, c†l ]|0� = Alk,
Bkl = −�0|[ck,H, cl]|0� = Blk,
Skl = �0|[ck, c†l ]|0� = Slk,
Tkl = �0|[ck, cl]|0� = −Tlk
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Solving RPA
8
�A B−B −A
� �Xν
Yν
�= Eν0
�S −T−T S
� �Xν
Yν
�
EOM formulated as generalized eigenproblem
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Solving RPA
8
�A B−B −A
� �Xν
Yν
�= Eν0
�S −T−T S
� �Xν
Yν
�
EOM formulated as generalized eigenproblem
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Solving RPA
8
�A B−B −A
� �Xν
Yν
�= Eν0
�S −T−T S
� �Xν
Yν
�
EOM formulated as generalized eigenproblem
provides
!excitation energies
!excited states (Xν , Yν)Eν0
|Eν0� via
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Solving RPA
8
�A B−B −A
� �Xν
Yν
�= Eν0
�S −T−T S
� �Xν
Yν
�
EOM formulated as generalized eigenproblem
strength function
R(ω) =�
(ν)
δ(ω − Eν0) |�0|κHJ |Eν0�|2
provides
!excitation energies
!excited states (Xν , Yν)Eν0
|Eν0� via
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Solving RPA
8
�A B−B −A
� �Xν
Yν
�= Eν0
�S −T−T S
� �Xν
Yν
�
EOM formulated as generalized eigenproblem
strength function
R(ω) =�
(ν)
δ(ω − Eν0) |�0|κHJ |Eν0�|2
provides
!excitation energies
!excited states (Xν , Yν)Eν0
|Eν0� via
transition amplitudes
R(ω
)in
a.u.
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Results I: 1U Resonance
Varying the System Size
Dynamic Response of Ultracold Bose Gases in 1D Optical LatticesMarkus Hild, Panagiota Papakonstantinou, Felix Schmitt, and Robert Roth
Institut fur Kernphysik, TU Darmstadt, Schloßgartenstraße 9, 64289 Darmstadt
Summary & Motivation
we investigate dynamic properties of strongly corre-lated ultracold Bose gases in 1D optical lattices withinthe Bose-Hubbard model (BHM) [1,2]
we use a weak periodic modulation of the lattice am-plitude to excite the system (Bragg spectroscopy)
exact time evolution yields precise description of theresonance structure [3,4] but is feasible for moderatesystem sizes only
the results are in excellent agreement with the predic-tion from a linear response analysis [4]
random phase approximation (RPA) allows for studiesof the resonance structure at the level of particle-holeexcitations and beyond [5]
transition amplitudes derived from the linear responseanalysis lead to response functions
Bose-Hubbard Model (BHM) & Lattice Modulation
N bosons on I lattice sites described by the Bose-Hubbard Hamiltonian
latticeamplitude V0
H =
kinetic term (”tunneling”)! "# $
!JI
%
i=1
&
a†iai+1 + a†
i+1ai
'
+
on-site interaction! "# $
U
2
I%
i=1ni(ni ! 1) = !JHJ + UHU
eigenstates in number basis representation(((!
)
=*D
j=1 c(!)j
((({n1 · · ·nI}j
)
modifications due to the amplitude modulation
potential modified by a time-dependent factor : V0 !" V0(t) = V0[1 +F sin("t)]
parameters J , U become time dependent
Linear Response Analysis
which excited states couple to ground state?
linearization of H based on the Taylor expansion in the modulation amplitude F :
Hlin(t) = H0 + F sin("t) [#H0 ! $HJ ]
in first order only HJ couples ground and excited states!
look for finite matrix ele-ments
+
0((( HJ
((( !
)
right panel: lower eigenspec-trum with matrix elements;left panel: resonance struc-ture (time evolution)
30 20 10"/J
0
20
40
60
!E
/J
I = N = 10U/J = 20
0 1 10 100 1000!
0
20
40
60
(E!!
E0)/
J
Generalized Random Phase Approximation (RPA) and Response Functions
RPA ground state approximated by(((RPA
)
#(((1, · · · , 1
)
for N/I=1
excitations generated by phonon operator Q†! (obeying Q!
(((RPA
)
= 0) :
Q†! =
%
ijX!
ij a†ia
†iaiaj !
%
ijY !
ij a†ja
†iaiai ,
generalized eigenproblem to solve,
-A B
!B !A
.
/
,
-X!
Y !
.
/ = "!
,
-S !T
!T S
.
/
,
-X!
Y !
.
/
matrix elementsAiji$j $=
0
1, · · ·, 1(( [a†ja
†iaiai ,H,a†i$a
†i$ai$aj $]
(( 1, · · · , 1
1
, Siji"$=0
1, · · ·, 1(( [a†ja
†iaiai ,a
†i$a
†i$ai$aj $]
(( 1, · · ·, 1
1
,
Biji$j $=0
1, · · ·, 1(( [a†ja
†iaiai ,H,a†j $a
†i$ai$ai$]
(( 1, · · · , 1
1
, Tiji$j $=0
1, · · ·, 1(( [a†ja
†iaiai ,a
†j $a
†i$ai$ai$]
(( 1, · · ·, 1
1
weighting the positive RPA spectrum {"!} by the transition amplitudes(((
+
1, · · · , 1((( HJ
((( "!
)(((
2yields the response function
R(") =%
!%(" ! "!)
(((
+
1, · · · , 1((( HJ
((( "!
)(((
2
Response Functions : RPA vs. Diagonalisation
I=N=10
0
100
200
300
400
500
"R
("),
a.u.
U/J=15
0 0.5 1 1.5 2"/U
0
200
400
600
"R
("),
a.u.
U/J=20
I=N=20
0
100
200
300
400
500 U/J=15
0 0.5 1 1.5 2"/U
0
200
400
600U/J=20
I=N=30
0
250
500
750
1000
1250 U/J=15
0 0.5 1 1.5 2"/U
0
500
1000
1500U/J=20
I=N=40
0
500
1000
1500
2000
2500 U/J=20
0 0.5 1 1.5 2"/U
0
1000
2000
3000U/J=30
( ) Random Phase Approximation ( ) Diagonalisation in ph-space ( ) Diagonalisation in 3p3h-space
plots show the " = 1U resonances for several interac-tion strengths and systems sizes in the strongly corre-lated regime excited by weak lattice modulations
diagonalization in 1p1h-space fails to describe reso-nance position & shape
RPA predicts correct centroid energy and resonanceshape
role of ground-state correlations on fine-structure &width to be investigated
RPA allows for the computation of response functionsfor large systems with minimal numerical e!ort
higher resonances and more complex lattice topologiesare possible by extension of the phonon operator
[1] D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998)[2] M. Greiner et al., Nature 415, 39 (2002)
[3] S.R. Clark and D. Jaksch, New J. Phys. 8, 160 (2006)[4] M. Hild et. al, J. Phys. B.: At. Mol. Opt. Phys. 39, 4547 (2006)
[5] D.J. Rowe, Rev. Mod. Phys. 40, 1 (1968)
9
R(ω
)in
a.u.
Dynamic Response of Ultracold Bose Gases in 1D Optical LatticesMarkus Hild, Panagiota Papakonstantinou, Felix Schmitt, and Robert Roth
Institut fur Kernphysik, TU Darmstadt, Schloßgartenstraße 9, 64289 Darmstadt
Summary & Motivation
we investigate dynamic properties of strongly corre-lated ultracold Bose gases in 1D optical lattices withinthe Bose-Hubbard model (BHM) [1,2]
we use a weak periodic modulation of the lattice am-plitude to excite the system (Bragg spectroscopy)
exact time evolution yields precise description of theresonance structure [3,4] but is feasible for moderatesystem sizes only
the results are in excellent agreement with the predic-tion from a linear response analysis [4]
random phase approximation (RPA) allows for studiesof the resonance structure at the level of particle-holeexcitations and beyond [5]
transition amplitudes derived from the linear responseanalysis lead to response functions
Bose-Hubbard Model (BHM) & Lattice Modulation
N bosons on I lattice sites described by the Bose-Hubbard Hamiltonian
latticeamplitude V0
H =
kinetic term (”tunneling”)! "# $
!JI
%
i=1
&
a†iai+1 + a†
i+1ai
'
+
on-site interaction! "# $
U
2
I%
i=1ni(ni ! 1) = !JHJ + UHU
eigenstates in number basis representation(((!
)
=*D
j=1 c(!)j
((({n1 · · ·nI}j
)
modifications due to the amplitude modulation
potential modified by a time-dependent factor : V0 !" V0(t) = V0[1 +F sin("t)]
parameters J , U become time dependent
Linear Response Analysis
which excited states couple to ground state?
linearization of H based on the Taylor expansion in the modulation amplitude F :
Hlin(t) = H0 + F sin("t) [#H0 ! $HJ ]
in first order only HJ couples ground and excited states!
look for finite matrix ele-ments
+
0((( HJ
((( !
)
right panel: lower eigenspec-trum with matrix elements;left panel: resonance struc-ture (time evolution)
30 20 10"/J
0
20
40
60
!E
/J
I = N = 10U/J = 20
0 1 10 100 1000!
0
20
40
60
(E!!
E0)/
J
Generalized Random Phase Approximation (RPA) and Response Functions
RPA ground state approximated by(((RPA
)
#(((1, · · · , 1
)
for N/I=1
excitations generated by phonon operator Q†! (obeying Q!
(((RPA
)
= 0) :
Q†! =
%
ijX!
ij a†ia
†iaiaj !
%
ijY !
ij a†ja
†iaiai ,
generalized eigenproblem to solve,
-A B
!B !A
.
/
,
-X!
Y !
.
/ = "!
,
-S !T
!T S
.
/
,
-X!
Y !
.
/
matrix elementsAiji$j $=
0
1, · · ·, 1(( [a†ja
†iaiai ,H,a†i$a
†i$ai$aj $]
(( 1, · · · , 1
1
, Siji"$=0
1, · · ·, 1(( [a†ja
†iaiai ,a
†i$a
†i$ai$aj $]
(( 1, · · ·, 1
1
,
Biji$j $=0
1, · · ·, 1(( [a†ja
†iaiai ,H,a†j $a
†i$ai$ai$]
(( 1, · · · , 1
1
, Tiji$j $=0
1, · · ·, 1(( [a†ja
†iaiai ,a
†j $a
†i$ai$ai$]
(( 1, · · ·, 1
1
weighting the positive RPA spectrum {"!} by the transition amplitudes(((
+
1, · · · , 1((( HJ
((( "!
)(((
2yields the response function
R(") =%
!%(" ! "!)
(((
+
1, · · · , 1((( HJ
((( "!
)(((
2
Response Functions : RPA vs. Diagonalisation
I=N=10
0
100
200
300
400
500
"R
("),
a.u.
U/J=15
0 0.5 1 1.5 2"/U
0
200
400
600
"R
("),
a.u.
U/J=20
I=N=20
0
100
200
300
400
500 U/J=15
0 0.5 1 1.5 2"/U
0
200
400
600U/J=20
I=N=30
0
250
500
750
1000
1250 U/J=15
0 0.5 1 1.5 2"/U
0
500
1000
1500U/J=20
I=N=40
0
500
1000
1500
2000
2500 U/J=20
0 0.5 1 1.5 2"/U
0
1000
2000
3000U/J=30
( ) Random Phase Approximation ( ) Diagonalisation in ph-space ( ) Diagonalisation in 3p3h-space
plots show the " = 1U resonances for several interac-tion strengths and systems sizes in the strongly corre-lated regime excited by weak lattice modulations
diagonalization in 1p1h-space fails to describe reso-nance position & shape
RPA predicts correct centroid energy and resonanceshape
role of ground-state correlations on fine-structure &width to be investigated
RPA allows for the computation of response functionsfor large systems with minimal numerical e!ort
higher resonances and more complex lattice topologiesare possible by extension of the phonon operator
[1] D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998)[2] M. Greiner et al., Nature 415, 39 (2002)
[3] S.R. Clark and D. Jaksch, New J. Phys. 8, 160 (2006)[4] M. Hild et. al, J. Phys. B.: At. Mol. Opt. Phys. 39, 4547 (2006)
[5] D.J. Rowe, Rev. Mod. Phys. 40, 1 (1968)
R(ω
)in
a.u.
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Results I: 1U Resonance
Varying the System Size
Dynamic Response of Ultracold Bose Gases in 1D Optical LatticesMarkus Hild, Panagiota Papakonstantinou, Felix Schmitt, and Robert Roth
Institut fur Kernphysik, TU Darmstadt, Schloßgartenstraße 9, 64289 Darmstadt
Summary & Motivation
we investigate dynamic properties of strongly corre-lated ultracold Bose gases in 1D optical lattices withinthe Bose-Hubbard model (BHM) [1,2]
we use a weak periodic modulation of the lattice am-plitude to excite the system (Bragg spectroscopy)
exact time evolution yields precise description of theresonance structure [3,4] but is feasible for moderatesystem sizes only
the results are in excellent agreement with the predic-tion from a linear response analysis [4]
random phase approximation (RPA) allows for studiesof the resonance structure at the level of particle-holeexcitations and beyond [5]
transition amplitudes derived from the linear responseanalysis lead to response functions
Bose-Hubbard Model (BHM) & Lattice Modulation
N bosons on I lattice sites described by the Bose-Hubbard Hamiltonian
latticeamplitude V0
H =
kinetic term (”tunneling”)! "# $
!JI
%
i=1
&
a†iai+1 + a†
i+1ai
'
+
on-site interaction! "# $
U
2
I%
i=1ni(ni ! 1) = !JHJ + UHU
eigenstates in number basis representation(((!
)
=*D
j=1 c(!)j
((({n1 · · ·nI}j
)
modifications due to the amplitude modulation
potential modified by a time-dependent factor : V0 !" V0(t) = V0[1 +F sin("t)]
parameters J , U become time dependent
Linear Response Analysis
which excited states couple to ground state?
linearization of H based on the Taylor expansion in the modulation amplitude F :
Hlin(t) = H0 + F sin("t) [#H0 ! $HJ ]
in first order only HJ couples ground and excited states!
look for finite matrix ele-ments
+
0((( HJ
((( !
)
right panel: lower eigenspec-trum with matrix elements;left panel: resonance struc-ture (time evolution)
30 20 10"/J
0
20
40
60
!E
/J
I = N = 10U/J = 20
0 1 10 100 1000!
0
20
40
60
(E!!
E0)/
J
Generalized Random Phase Approximation (RPA) and Response Functions
RPA ground state approximated by(((RPA
)
#(((1, · · · , 1
)
for N/I=1
excitations generated by phonon operator Q†! (obeying Q!
(((RPA
)
= 0) :
Q†! =
%
ijX!
ij a†ia
†iaiaj !
%
ijY !
ij a†ja
†iaiai ,
generalized eigenproblem to solve,
-A B
!B !A
.
/
,
-X!
Y !
.
/ = "!
,
-S !T
!T S
.
/
,
-X!
Y !
.
/
matrix elementsAiji$j $=
0
1, · · ·, 1(( [a†ja
†iaiai ,H,a†i$a
†i$ai$aj $]
(( 1, · · · , 1
1
, Siji"$=0
1, · · ·, 1(( [a†ja
†iaiai ,a
†i$a
†i$ai$aj $]
(( 1, · · ·, 1
1
,
Biji$j $=0
1, · · ·, 1(( [a†ja
†iaiai ,H,a†j $a
†i$ai$ai$]
(( 1, · · · , 1
1
, Tiji$j $=0
1, · · ·, 1(( [a†ja
†iaiai ,a
†j $a
†i$ai$ai$]
(( 1, · · ·, 1
1
weighting the positive RPA spectrum {"!} by the transition amplitudes(((
+
1, · · · , 1((( HJ
((( "!
)(((
2yields the response function
R(") =%
!%(" ! "!)
(((
+
1, · · · , 1((( HJ
((( "!
)(((
2
Response Functions : RPA vs. Diagonalisation
I=N=10
0
100
200
300
400
500
"R
("),
a.u.
U/J=15
0 0.5 1 1.5 2"/U
0
200
400
600
"R
("),
a.u.
U/J=20
I=N=20
0
100
200
300
400
500 U/J=15
0 0.5 1 1.5 2"/U
0
200
400
600U/J=20
I=N=30
0
250
500
750
1000
1250 U/J=15
0 0.5 1 1.5 2"/U
0
500
1000
1500U/J=20
I=N=40
0
500
1000
1500
2000
2500 U/J=20
0 0.5 1 1.5 2"/U
0
1000
2000
3000U/J=30
( ) Random Phase Approximation ( ) Diagonalisation in ph-space ( ) Diagonalisation in 3p3h-space
plots show the " = 1U resonances for several interac-tion strengths and systems sizes in the strongly corre-lated regime excited by weak lattice modulations
diagonalization in 1p1h-space fails to describe reso-nance position & shape
RPA predicts correct centroid energy and resonanceshape
role of ground-state correlations on fine-structure &width to be investigated
RPA allows for the computation of response functionsfor large systems with minimal numerical e!ort
higher resonances and more complex lattice topologiesare possible by extension of the phonon operator
[1] D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998)[2] M. Greiner et al., Nature 415, 39 (2002)
[3] S.R. Clark and D. Jaksch, New J. Phys. 8, 160 (2006)[4] M. Hild et. al, J. Phys. B.: At. Mol. Opt. Phys. 39, 4547 (2006)
[5] D.J. Rowe, Rev. Mod. Phys. 40, 1 (1968)
9
R(ω
)in
a.u.
Dynamic Response of Ultracold Bose Gases in 1D Optical LatticesMarkus Hild, Panagiota Papakonstantinou, Felix Schmitt, and Robert Roth
Institut fur Kernphysik, TU Darmstadt, Schloßgartenstraße 9, 64289 Darmstadt
Summary & Motivation
we investigate dynamic properties of strongly corre-lated ultracold Bose gases in 1D optical lattices withinthe Bose-Hubbard model (BHM) [1,2]
we use a weak periodic modulation of the lattice am-plitude to excite the system (Bragg spectroscopy)
exact time evolution yields precise description of theresonance structure [3,4] but is feasible for moderatesystem sizes only
the results are in excellent agreement with the predic-tion from a linear response analysis [4]
random phase approximation (RPA) allows for studiesof the resonance structure at the level of particle-holeexcitations and beyond [5]
transition amplitudes derived from the linear responseanalysis lead to response functions
Bose-Hubbard Model (BHM) & Lattice Modulation
N bosons on I lattice sites described by the Bose-Hubbard Hamiltonian
latticeamplitude V0
H =
kinetic term (”tunneling”)! "# $
!JI
%
i=1
&
a†iai+1 + a†
i+1ai
'
+
on-site interaction! "# $
U
2
I%
i=1ni(ni ! 1) = !JHJ + UHU
eigenstates in number basis representation(((!
)
=*D
j=1 c(!)j
((({n1 · · ·nI}j
)
modifications due to the amplitude modulation
potential modified by a time-dependent factor : V0 !" V0(t) = V0[1 +F sin("t)]
parameters J , U become time dependent
Linear Response Analysis
which excited states couple to ground state?
linearization of H based on the Taylor expansion in the modulation amplitude F :
Hlin(t) = H0 + F sin("t) [#H0 ! $HJ ]
in first order only HJ couples ground and excited states!
look for finite matrix ele-ments
+
0((( HJ
((( !
)
right panel: lower eigenspec-trum with matrix elements;left panel: resonance struc-ture (time evolution)
30 20 10"/J
0
20
40
60
!E
/J
I = N = 10U/J = 20
0 1 10 100 1000!
0
20
40
60
(E!!
E0)/
J
Generalized Random Phase Approximation (RPA) and Response Functions
RPA ground state approximated by(((RPA
)
#(((1, · · · , 1
)
for N/I=1
excitations generated by phonon operator Q†! (obeying Q!
(((RPA
)
= 0) :
Q†! =
%
ijX!
ij a†ia
†iaiaj !
%
ijY !
ij a†ja
†iaiai ,
generalized eigenproblem to solve,
-A B
!B !A
.
/
,
-X!
Y !
.
/ = "!
,
-S !T
!T S
.
/
,
-X!
Y !
.
/
matrix elementsAiji$j $=
0
1, · · ·, 1(( [a†ja
†iaiai ,H,a†i$a
†i$ai$aj $]
(( 1, · · · , 1
1
, Siji"$=0
1, · · ·, 1(( [a†ja
†iaiai ,a
†i$a
†i$ai$aj $]
(( 1, · · ·, 1
1
,
Biji$j $=0
1, · · ·, 1(( [a†ja
†iaiai ,H,a†j $a
†i$ai$ai$]
(( 1, · · · , 1
1
, Tiji$j $=0
1, · · ·, 1(( [a†ja
†iaiai ,a
†j $a
†i$ai$ai$]
(( 1, · · ·, 1
1
weighting the positive RPA spectrum {"!} by the transition amplitudes(((
+
1, · · · , 1((( HJ
((( "!
)(((
2yields the response function
R(") =%
!%(" ! "!)
(((
+
1, · · · , 1((( HJ
((( "!
)(((
2
Response Functions : RPA vs. Diagonalisation
I=N=10
0
100
200
300
400
500
"R
("),
a.u.
U/J=15
0 0.5 1 1.5 2"/U
0
200
400
600
"R
("),
a.u.
U/J=20
I=N=20
0
100
200
300
400
500 U/J=15
0 0.5 1 1.5 2"/U
0
200
400
600U/J=20
I=N=30
0
250
500
750
1000
1250 U/J=15
0 0.5 1 1.5 2"/U
0
500
1000
1500U/J=20
I=N=40
0
500
1000
1500
2000
2500 U/J=20
0 0.5 1 1.5 2"/U
0
1000
2000
3000U/J=30
( ) Random Phase Approximation ( ) Diagonalisation in ph-space ( ) Diagonalisation in 3p3h-space
plots show the " = 1U resonances for several interac-tion strengths and systems sizes in the strongly corre-lated regime excited by weak lattice modulations
diagonalization in 1p1h-space fails to describe reso-nance position & shape
RPA predicts correct centroid energy and resonanceshape
role of ground-state correlations on fine-structure &width to be investigated
RPA allows for the computation of response functionsfor large systems with minimal numerical e!ort
higher resonances and more complex lattice topologiesare possible by extension of the phonon operator
[1] D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998)[2] M. Greiner et al., Nature 415, 39 (2002)
[3] S.R. Clark and D. Jaksch, New J. Phys. 8, 160 (2006)[4] M. Hild et. al, J. Phys. B.: At. Mol. Opt. Phys. 39, 4547 (2006)
[5] D.J. Rowe, Rev. Mod. Phys. 40, 1 (1968)
Dynamic Response of Ultracold Bose Gases in 1D Optical LatticesMarkus Hild, Panagiota Papakonstantinou, Felix Schmitt, and Robert Roth
Institut fur Kernphysik, TU Darmstadt, Schloßgartenstraße 9, 64289 Darmstadt
Summary & Motivation
we investigate dynamic properties of strongly corre-lated ultracold Bose gases in 1D optical lattices withinthe Bose-Hubbard model (BHM) [1,2]
we use a weak periodic modulation of the lattice am-plitude to excite the system (Bragg spectroscopy)
exact time evolution yields precise description of theresonance structure [3,4] but is feasible for moderatesystem sizes only
the results are in excellent agreement with the predic-tion from a linear response analysis [4]
random phase approximation (RPA) allows for studiesof the resonance structure at the level of particle-holeexcitations and beyond [5]
transition amplitudes derived from the linear responseanalysis lead to response functions
Bose-Hubbard Model (BHM) & Lattice Modulation
N bosons on I lattice sites described by the Bose-Hubbard Hamiltonian
latticeamplitude V0
H =
kinetic term (”tunneling”)! "# $
!JI
%
i=1
&
a†iai+1 + a†
i+1ai
'
+
on-site interaction! "# $
U
2
I%
i=1ni(ni ! 1) = !JHJ + UHU
eigenstates in number basis representation(((!
)
=*D
j=1 c(!)j
((({n1 · · ·nI}j
)
modifications due to the amplitude modulation
potential modified by a time-dependent factor : V0 !" V0(t) = V0[1 +F sin("t)]
parameters J , U become time dependent
Linear Response Analysis
which excited states couple to ground state?
linearization of H based on the Taylor expansion in the modulation amplitude F :
Hlin(t) = H0 + F sin("t) [#H0 ! $HJ ]
in first order only HJ couples ground and excited states!
look for finite matrix ele-ments
+
0((( HJ
((( !
)
right panel: lower eigenspec-trum with matrix elements;left panel: resonance struc-ture (time evolution)
30 20 10"/J
0
20
40
60
!E
/J
I = N = 10U/J = 20
0 1 10 100 1000!
0
20
40
60
(E!!
E0)/
J
Generalized Random Phase Approximation (RPA) and Response Functions
RPA ground state approximated by(((RPA
)
#(((1, · · · , 1
)
for N/I=1
excitations generated by phonon operator Q†! (obeying Q!
(((RPA
)
= 0) :
Q†! =
%
ijX!
ij a†ia
†iaiaj !
%
ijY !
ij a†ja
†iaiai ,
generalized eigenproblem to solve,
-A B
!B !A
.
/
,
-X!
Y !
.
/ = "!
,
-S !T
!T S
.
/
,
-X!
Y !
.
/
matrix elementsAiji$j $=
0
1, · · ·, 1(( [a†ja
†iaiai ,H,a†i$a
†i$ai$aj $]
(( 1, · · · , 1
1
, Siji"$=0
1, · · ·, 1(( [a†ja
†iaiai ,a
†i$a
†i$ai$aj $]
(( 1, · · ·, 1
1
,
Biji$j $=0
1, · · ·, 1(( [a†ja
†iaiai ,H,a†j $a
†i$ai$ai$]
(( 1, · · · , 1
1
, Tiji$j $=0
1, · · ·, 1(( [a†ja
†iaiai ,a
†j $a
†i$ai$ai$]
(( 1, · · ·, 1
1
weighting the positive RPA spectrum {"!} by the transition amplitudes(((
+
1, · · · , 1((( HJ
((( "!
)(((
2yields the response function
R(") =%
!%(" ! "!)
(((
+
1, · · · , 1((( HJ
((( "!
)(((
2
Response Functions : RPA vs. Diagonalisation
I=N=10
0
100
200
300
400
500
"R
("),
a.u.
U/J=15
0 0.5 1 1.5 2"/U
0
200
400
600
"R
("),
a.u.
U/J=20
I=N=20
0
100
200
300
400
500 U/J=15
0 0.5 1 1.5 2"/U
0
200
400
600U/J=20
I=N=30
0
250
500
750
1000
1250 U/J=15
0 0.5 1 1.5 2"/U
0
500
1000
1500U/J=20
I=N=40
0
500
1000
1500
2000
2500 U/J=20
0 0.5 1 1.5 2"/U
0
1000
2000
3000U/J=30
( ) Random Phase Approximation ( ) Diagonalisation in ph-space ( ) Diagonalisation in 3p3h-space
plots show the " = 1U resonances for several interac-tion strengths and systems sizes in the strongly corre-lated regime excited by weak lattice modulations
diagonalization in 1p1h-space fails to describe reso-nance position & shape
RPA predicts correct centroid energy and resonanceshape
role of ground-state correlations on fine-structure &width to be investigated
RPA allows for the computation of response functionsfor large systems with minimal numerical e!ort
higher resonances and more complex lattice topologiesare possible by extension of the phonon operator
[1] D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998)[2] M. Greiner et al., Nature 415, 39 (2002)
[3] S.R. Clark and D. Jaksch, New J. Phys. 8, 160 (2006)[4] M. Hild et. al, J. Phys. B.: At. Mol. Opt. Phys. 39, 4547 (2006)
[5] D.J. Rowe, Rev. Mod. Phys. 40, 1 (1968)
ωR
(ω)
U/J
ω/U
30
20
10
0
0.5
1.0
1.5
0
50
100
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Results II: 1U Resonance
Varying U/J
10
ω = U!centroid slipping-off towards superfluid regime
centroid
width
30 particles 30 sites
02.03.2009 | TU Darmstadt | DPG 2009 Hamburg | Markus Hild |
Conclusion & Outlook
11
!RPA offers numerically efficient access to the strength function (solving a single eigenproblem)
!can be easily extended via generalization of the phonon operators
!adapt to other lattice topologies via additional matrix elements (e.g., superlattice potential)
! improve on the resonance’s width via correlated ground-state