1. Spectroscopy of ultracold bosons by periodic lattice modulations
A. Iucci, C.Kollath, T. Giamarchi, W. Hofstetter, and U. Schollwöck
Superfluid-Mott insulating transition
Bose-Fermi mixtures
Mainz/München
Low-dimensional systems
LENS 1D 2D
disordered systems
Challenge to combine BEC and cavity (UHV requirements)
BEC held in a magnetic trap
Weak output coupling and quasi cw (500ms)
Graph already seen in the talk by J. Dalibard
Table tilt
Cavity used for single atom detection
ETHZ

2. Probing cold atoms time-of-flight measurement
-> momentum distribution
periodic lattice modulation
-> energy spectrum
Mainz/München
& (in)commensurability
noise measurement:
-> density-density correlations
ETHZ
„Noch mal kurz erklären“
Mainz

3. Experimental results Response to a periodic modulation
periodic modulation of
optical lattice height
absorbed energy
experiment of Greiner et al:
Rb^87 atoms condensed into BEC,
subjected to optical lattice
strength can be tuned from zero to weak to stronger to very strong
in a typical run, they would turn on the optical lattice up to a certain strength Vo, then remove the entire trap
suddenly and take absorption images, imaging the momentum distribution (right fig.)
2. found that resulting image depends very strongly on strength of potential
correspond to different phases SF and MI phase:
explain fig: very sharp interference peaks -> long range order -> SF
side peaks arise
pattern is smeared out, incoherent background arises until peak vanishes,
corresponds to localized stated whith short range correlations
-> have driven a quantum phase transition (QPT) between SF and MI
3. comment on QPT in condensed matter systems??
advantage here better tunable...
4. investigate this system using a numerical method, the DMRG,
which is well suited to describe strongly correlated quasi-1-dim quantum systems
5. Aim is to describe the static properties as well as timedependent effects
6. Start presenting a theoretical description of the system by the Bose-Hubbard Model (BH) (all atoms are in the lowest vibrational level, only s-wave scattering)
T. Stöferle et al. PRL 92,

4. Bosonic atoms in an optical lattice
interaction energy
kinetic energy
experimental parameter -> J and U
periodic modulation of lattice height
-> time dependent J(t) and U(t)
explicitly time-dependent Hamiltonian
experiment of Greiner et al:
Rb^87 atoms condensed into BEC,
subjected to optical lattice
strength can be tuned from zero to weak to stronger to very strong
in a typical run, they would turn on the optical lattice up to a certain strength Vo, then remove the entire trap
suddenly and take absorption images, imaging the momentum distribution (right fig.)
2. found that resulting image depends very strongly on strength of potential
correspond to different phases SF and MI phase:
explain fig: very sharp interference peaks -> long range order -> SF
side peaks arise
pattern is smeared out, incoherent background arises until peak vanishes,
corresponds to localized stated whith short range correlations
-> have driven a quantum phase transition (QPT) between SF and MI
3. comment on QPT in condensed matter systems??
advantage here better tunable...
4. investigate this system using a numerical method, the DMRG,
which is well suited to describe strongly correlated quasi-1-dim quantum systems
5. Aim is to describe the static properties as well as timedependent effects
6. Start presenting a theoretical description of the system by the Bose-Hubbard Model (BH) (all atoms are in the lowest vibrational level, only s-wave scattering)
Methods: - adaptive t-DMRG
- linear response

5. Strong interaction
response to periodic modulation -> creation of excitations
single particle hole excitation -> energy U
two particle hole excitations -> energy 2U
-> expect peaks at multiples of U
U 2U 3U
experiment of Greiner et al:
Rb^87 atoms condensed into BEC,
subjected to optical lattice
strength can be tuned from zero to weak to stronger to very strong
in a typical run, they would turn on the optical lattice up to a certain strength Vo, then remove the entire trap
suddenly and take absorption images, imaging the momentum distribution (right fig.)
2. found that resulting image depends very strongly on strength of potential
correspond to different phases SF and MI phase:
explain fig: very sharp interference peaks -> long range order -> SF
side peaks arise
pattern is smeared out, incoherent background arises until peak vanishes,
corresponds to localized stated whith short range correlations
-> have driven a quantum phase transition (QPT) between SF and MI
3. comment on QPT in condensed matter systems??
advantage here better tunable...
4. investigate this system using a numerical method, the DMRG,
which is well suited to describe strongly correlated quasi-1-dim quantum systems
5. Aim is to describe the static properties as well as timedependent effects
6. Start presenting a theoretical description of the system by the Bose-Hubbard Model (BH) (all atoms are in the lowest vibrational level, only s-wave scattering)
absorbed energy ħw

6. C. Kollath et al. cond-mat 2006
Energy absorption
L=24
L=32
at resonance ħw=U
absorbed energy
experiment of Greiner et al:
Rb^87 atoms condensed into BEC,
subjected to optical lattice
strength can be tuned from zero to weak to stronger to very strong
in a typical run, they would turn on the optical lattice up to a certain strength Vo, then remove the entire trap
suddenly and take absorption images, imaging the momentum distribution (right fig.)
2. found that resulting image depends very strongly on strength of potential
correspond to different phases SF and MI phase:
explain fig: very sharp interference peaks -> long range order -> SF
side peaks arise
pattern is smeared out, incoherent background arises until peak vanishes,
corresponds to localized stated whith short range correlations
-> have driven a quantum phase transition (QPT) between SF and MI
3. comment on QPT in condensed matter systems??
advantage here better tunable...
4. investigate this system using a numerical method, the DMRG,
which is well suited to describe strongly correlated quasi-1-dim quantum systems
5. Aim is to describe the static properties as well as timedependent effects
6. Start presenting a theoretical description of the system by the Bose-Hubbard Model (BH) (all atoms are in the lowest vibrational level, only s-wave scattering)
away of resonance
C. Kollath et al. cond-mat 2006

7. Energy absorption at commensurate filling
experiment
U/J =72
U/J =95
absorption rate
U 2U ?
experiment of Greiner et al:
Rb^87 atoms condensed into BEC,
subjected to optical lattice
strength can be tuned from zero to weak to stronger to very strong
in a typical run, they would turn on the optical lattice up to a certain strength Vo, then remove the entire trap
suddenly and take absorption images, imaging the momentum distribution (right fig.)
2. found that resulting image depends very strongly on strength of potential
correspond to different phases SF and MI phase:
explain fig: very sharp interference peaks -> long range order -> SF
side peaks arise
pattern is smeared out, incoherent background arises until peak vanishes,
corresponds to localized stated whith short range correlations
-> have driven a quantum phase transition (QPT) between SF and MI
3. comment on QPT in condensed matter systems??
advantage here better tunable...
4. investigate this system using a numerical method, the DMRG,
which is well suited to describe strongly correlated quasi-1-dim quantum systems
5. Aim is to describe the static properties as well as timedependent effects
6. Start presenting a theoretical description of the system by the Bose-Hubbard Model (BH) (all atoms are in the lowest vibrational level, only s-wave scattering)
peak at U
small peak at U/2
no peak at higher frequencies
linear response:
A. Iucci et al. accepted by PRA

8. Energy absorption at incommensurate filling
experiment
experiment
U/J =95
n~1.2
U/J =72
absorption rate
experiment of Greiner et al:
Rb^87 atoms condensed into BEC,
subjected to optical lattice
strength can be tuned from zero to weak to stronger to very strong
in a typical run, they would turn on the optical lattice up to a certain strength Vo, then remove the entire trap
suddenly and take absorption images, imaging the momentum distribution (right fig.)
2. found that resulting image depends very strongly on strength of potential
correspond to different phases SF and MI phase:
explain fig: very sharp interference peaks -> long range order -> SF
side peaks arise
pattern is smeared out, incoherent background arises until peak vanishes,
corresponds to localized stated whith short range correlations
-> have driven a quantum phase transition (QPT) between SF and MI
3. comment on QPT in condensed matter systems??
advantage here better tunable...
4. investigate this system using a numerical method, the DMRG,
which is well suited to describe strongly correlated quasi-1-dim quantum systems
5. Aim is to describe the static properties as well as timedependent effects
6. Start presenting a theoretical description of the system by the Bose-Hubbard Model (BH) (all atoms are in the lowest vibrational level, only s-wave scattering)
peak at U
peak at 2U

9. 2U peak corresponding processes two particle-hole excitations
single particle-hole excitation
experiment of Greiner et al:
Rb^87 atoms condensed into BEC,
subjected to optical lattice
strength can be tuned from zero to weak to stronger to very strong
in a typical run, they would turn on the optical lattice up to a certain strength Vo, then remove the entire trap
suddenly and take absorption images, imaging the momentum distribution (right fig.)
2. found that resulting image depends very strongly on strength of potential
correspond to different phases SF and MI phase:
explain fig: very sharp interference peaks -> long range order -> SF
side peaks arise
pattern is smeared out, incoherent background arises until peak vanishes,
corresponds to localized stated whith short range correlations
-> have driven a quantum phase transition (QPT) between SF and MI
3. comment on QPT in condensed matter systems??
advantage here better tunable...
4. investigate this system using a numerical method, the DMRG,
which is well suited to describe strongly correlated quasi-1-dim quantum systems
5. Aim is to describe the static properties as well as timedependent effects
6. Start presenting a theoretical description of the system by the Bose-Hubbard Model (BH) (all atoms are in the lowest vibrational level, only s-wave scattering)
2U peak measure of incommensurability

10. Intermediate interaction strength
experimental results: U
1.9U
U/J =28
2.6U
absorbed energy
experiment of Greiner et al:
Rb^87 atoms condensed into BEC,
subjected to optical lattice
strength can be tuned from zero to weak to stronger to very strong
in a typical run, they would turn on the optical lattice up to a certain strength Vo, then remove the entire trap
suddenly and take absorption images, imaging the momentum distribution (right fig.)
2. found that resulting image depends very strongly on strength of potential
correspond to different phases SF and MI phase:
explain fig: very sharp interference peaks -> long range order -> SF
side peaks arise
pattern is smeared out, incoherent background arises until peak vanishes,
corresponds to localized stated whith short range correlations
-> have driven a quantum phase transition (QPT) between SF and MI
3. comment on QPT in condensed matter systems??
advantage here better tunable...
4. investigate this system using a numerical method, the DMRG,
which is well suited to describe strongly correlated quasi-1-dim quantum systems
5. Aim is to describe the static properties as well as timedependent effects
6. Start presenting a theoretical description of the system by the Bose-Hubbard Model (BH) (all atoms are in the lowest vibrational level, only s-wave scattering)
peaks at U, 1.9 U, and 2.6 U
peaks at approx. U, 1.9 U, and 2.6 U
T. Stöferle et al. PRL 92,

11. Energy absorption at incommensurate filling
U/J =9
n~1.2
absorption rate
experiment of Greiner et al:
Rb^87 atoms condensed into BEC,
subjected to optical lattice
strength can be tuned from zero to weak to stronger to very strong
in a typical run, they would turn on the optical lattice up to a certain strength Vo, then remove the entire trap
suddenly and take absorption images, imaging the momentum distribution (right fig.)
2. found that resulting image depends very strongly on strength of potential
correspond to different phases SF and MI phase:
explain fig: very sharp interference peaks -> long range order -> SF
side peaks arise
pattern is smeared out, incoherent background arises until peak vanishes,
corresponds to localized stated whith short range correlations
-> have driven a quantum phase transition (QPT) between SF and MI
3. comment on QPT in condensed matter systems??
advantage here better tunable...
4. investigate this system using a numerical method, the DMRG,
which is well suited to describe strongly correlated quasi-1-dim quantum systems
5. Aim is to describe the static properties as well as timedependent effects
6. Start presenting a theoretical description of the system by the Bose-Hubbard Model (BH) (all atoms are in the lowest vibrational level, only s-wave scattering)
peaks at U, 2.1 U, and 2.6 U
shift of position of energy eigenvalues
at incommensurate filling

12. Positions of peaks small system at incommensurate filling:
position shifts,
splitting of peaks
experiment of Greiner et al:
Rb^87 atoms condensed into BEC,
subjected to optical lattice
strength can be tuned from zero to weak to stronger to very strong
in a typical run, they would turn on the optical lattice up to a certain strength Vo, then remove the entire trap
suddenly and take absorption images, imaging the momentum distribution (right fig.)
2. found that resulting image depends very strongly on strength of potential
correspond to different phases SF and MI phase:
explain fig: very sharp interference peaks -> long range order -> SF
side peaks arise
pattern is smeared out, incoherent background arises until peak vanishes,
corresponds to localized stated whith short range correlations
-> have driven a quantum phase transition (QPT) between SF and MI
3. comment on QPT in condensed matter systems??
advantage here better tunable...
4. investigate this system using a numerical method, the DMRG,
which is well suited to describe strongly correlated quasi-1-dim quantum systems
5. Aim is to describe the static properties as well as timedependent effects
6. Start presenting a theoretical description of the system by the Bose-Hubbard Model (BH) (all atoms are in the lowest vibrational level, only s-wave scattering)
J/U

13. 20% perturbation -> beyond linear response
absorption rate, 1% modulation (scaled)
absorption rate, 20% modulation
integrated absorption, 20% modulation
saturation effects occur
in height and width
experiment of Greiner et al:
Rb^87 atoms condensed into BEC,
subjected to optical lattice
strength can be tuned from zero to weak to stronger to very strong
in a typical run, they would turn on the optical lattice up to a certain strength Vo, then remove the entire trap
suddenly and take absorption images, imaging the momentum distribution (right fig.)
2. found that resulting image depends very strongly on strength of potential
correspond to different phases SF and MI phase:
explain fig: very sharp interference peaks -> long range order -> SF
side peaks arise
pattern is smeared out, incoherent background arises until peak vanishes,
corresponds to localized stated whith short range correlations
-> have driven a quantum phase transition (QPT) between SF and MI
3. comment on QPT in condensed matter systems??
advantage here better tunable...
4. investigate this system using a numerical method, the DMRG,
which is well suited to describe strongly correlated quasi-1-dim quantum systems
5. Aim is to describe the static properties as well as timedependent effects
6. Start presenting a theoretical description of the system by the Bose-Hubbard Model (BH) (all atoms are in the lowest vibrational level, only s-wave scattering)

14. Summary
occurence of higher frequency peaks for incommensurate filling: confining potential, temperature
shift in peak position stem from shift in energy
20% perturbation -> saturation effects in width and height
not described by linear response
full time-dependent calculation: adaptive t-DMRG
many more applications possible
experiment of Greiner et al:
Rb^87 atoms condensed into BEC,
subjected to optical lattice
strength can be tuned from zero to weak to stronger to very strong
in a typical run, they would turn on the optical lattice up to a certain strength Vo, then remove the entire trap
suddenly and take absorption images, imaging the momentum distribution (right fig.)
2. found that resulting image depends very strongly on strength of potential
correspond to different phases SF and MI phase:
explain fig: very sharp interference peaks -> long range order -> SF
side peaks arise
pattern is smeared out, incoherent background arises until peak vanishes,
corresponds to localized stated whith short range correlations
-> have driven a quantum phase transition (QPT) between SF and MI
3. comment on QPT in condensed matter systems??
advantage here better tunable...
4. investigate this system using a numerical method, the DMRG,
which is well suited to describe strongly correlated quasi-1-dim quantum systems
5. Aim is to describe the static properties as well as timedependent effects
6. Start presenting a theoretical description of the system by the Bose-Hubbard Model (BH) (all atoms are in the lowest vibrational level, only s-wave scattering)