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

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

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, 130403

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

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

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

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

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

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

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, 130403

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

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) 0.1 0.2 0.3 J/U

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)

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)