Bad structure cacuatos group semar 00-0 Georg Wrth Isttut für Laser Physk Jauary 00
Motvato attce ad stab. aser are outcouped from SM-PM-fber Mcheso terferometer braches overap uder 90 attce forms overappg rego pezo-drve mrrors aow to adust reatve phase dfferece θ pae waves vertca poarzato photodode cod atoms 50% pezomouted mrror Georg Wrth Isttut für Laser-Physk
Motvato dpoe poteta at has terferece term attce structure depeds o phase dfferece θ bad structure ad wavefuctos chage wth phase at y 0 cos k cos ky cos k cos ky cos θ 444 4443 4444 4444 3 sum of pae waves terferece term Georg Wrth Isttut für Laser-Physk 3
Oute How to cacuate a bad structure Boch Theorem ad Fourer epaso Schrödger euato matr form Oe- ad two-dmesoa bad structure Boch- ad Waer-fuctos Topc Iterferece chages depth of eghbour stes Effect o oe-dmesoa bad structure Modfed Boch ad Waer?-fuctos Epermeta sgature & Summary Georg Wrth Isttut für Laser-Physk 4
Georg Wrth Isttut für Laser-Physk 5 Boch theorem ad Fourer epaso ˆ E H Φ Φ wth Schrödger euato wth perodc poteta cos ˆ ˆ k m p H at / ep u Φ h Boch theorem ˆ u E u H B Schrödger euato for u-fuctos wth cos ˆ ˆ k m p H at B k ep Epress u-fuctos ad poteta as dscrete Fourer sums k c u ep ad ep / ˆ k c E k u m p rec h ketc eergy of H B becomes
Georg Wrth Isttut für Laser-Physk 6 Schrödger euato matr form Fourer coeffcets at at at at k k k 4 0 4 ep ep cos ep k c u poteta eergy of H B becomes put t together ± ± at at k c k c u ep : 0 ep : 4 wrte Schrödger euato matr form 0 : ese : / : ˆ ˆ ep ep ˆ ˆ 4 at at rec B E k H c E c H k c E k c H u H h M M M L
-dm bad structure sove egevaue probem umercay Mathematca: Egevaues[] wth fte umber of Fourer coeffcets good appromato f hgher bads are ot popuated Hamto-operator s matr oy bads are take to accout! free partce harm. oscator k π / λ s NOT recproca attce vector Georg Wrth Isttut für Laser-Physk 7
-dm bad structure reca attce poteta at z 0 cos k cos kz cos k cos kz cos θ 444 4443 4444 4444 3 sum of pae waves terferece term for θ π/ phase dfferece Hamto separabe two or three dmesos wavefuctos ca be cacuated separatey ad supermposed eergy s sum of dfferet drectos Georg Wrth Isttut für Laser-Physk 8
Boch-fuctos for gve bad ad uas-mpuse use egevector c - c to cacuate u- fuctos ad Boch-wavefucto Φ ep / h c ep k Georg Wrth Isttut für Laser-Physk 9
Waer-fuctos Boch-fuctos are the atura bass the mode of uas-free partce but deocazed Waer-fuctos are ocazed o partcuar stes ad form a compete set of orthogoa bass states w N / BZ d ep / h Φ appromate Itegrato over Brou-zoe by dscrete sum makes Waer-fucto perodca rgg! Georg Wrth Isttut für Laser-Physk 0
Waer-fuctos dscrete sum Georg Wrth Isttut für Laser-Physk
Waer-fuctos Georg Wrth Isttut für Laser-Physk
Iterferece chages depth of eghbour stes at z 0 cos k cos kz cos k cos kz cos θ 444 4443 4444 4444 3 sum of pae waves terferece term Georg Wrth Isttut für Laser-Physk 3
Iterferece chages depth of eghbour stes ook at vertca cut at z0 sma devatos from θ π/ ead to sgfcat chages we depth Georg Wrth Isttut für Laser-Physk 4
Effect o -dm. bad structure -dm. attce poteta s at 0 at at [ cos k cos θ cos k ] [ 4 ep k 4 ep k cos θ ep k cos θ ep k 3 ] Fourer sums ow wth perodcty k stead of k 3 4 at cos θ at 0 at Schrödger euato s ow Hˆ Hˆ c E c / hk : E : cos θ at : 4 at ese : 0 rec 3 at Georg Wrth Isttut für Laser-Physk 5
Effect o -dm. bad structure Georg Wrth Isttut für Laser-Physk 6
Effect o -dm. bad structure f phase devates from θ π/ addtoa Bragg-refecto from superattce sets coupg betwee mpuse casses ħk/ ad -ħk/ eads to spttg eergy gap Georg Wrth Isttut für Laser-Physk 7
Effect o -dm. bad structure the Bose-Hubbard-mode the tueg matr eemet s gve by J 4 0 0 ma m E E owest bad s dramatcay fatteed partces are ped to deep attce stes tueg s suppressed euay spaced eves ke harm. osc. Georg Wrth Isttut für Laser-Physk 8
Modfed Boch-fuctos cacuate Boch-fuctos wth ew k/ perodcty Φ ep / h c ep k Georg Wrth Isttut für Laser-Physk 9
Modfed Waer-fuctos cacuate waer-states by summato over smaer Brou-zoe w N / BZ d ep / h Φ Georg Wrth Isttut für Laser-Physk 0
Epermeta sgatures & Summary TOF-mages show mpuse-spectra of atoms the attce whe attce poteta s swtched of fasty sma devatos from 90 strogy popuate the 0 peaks comparabe to theoretca resuts Georg Wrth Isttut für Laser-Physk