INTRO LIB RESEARCH
INTRO LIB RESEARCH INT 1
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we UCIrVine Univcrzily qualifumia1wim Multidimensional Spectroscopy with Classical Fields vs Entangled Photons Selecting Liouville Space Pathways Shaul Mukamel ad Oleksiy Rosyak Department of Chemistry UCI California Kavli Institute for Theoretical Physics Santa Barbara CA May 18 2008 2D NMR Spectroscopy ANGEWANDTE EUEMUE International Edition in English N at Mngncfic Risaname Fn winr Trunsfann swamquot Norm mm a mama n um 54m unn Wm M A HeterodyneDetected Four Wave Er7 24 ZEJVQ fjeXpikjr iajz39 27 ipjvz39 fj cc j1 v ksziklikzik3 Coherent non linear response to classical fields n 1 wave mixing n incoming fields generate 1 signal field ks k n incoming fields induce an nth 1 order polarization PM in the material system Semiclassical theory material lt gt quantum field lt gt classical k3 fourwave mixing n3 The Nonlinear Response Functions Pr t P1r t P2r t PBr t Nonlinear polarization Plt r t E ltltVp tgtgt E T39I Vpquot39t PynI qlLgt CH 11L71 Zt15 z nt1 ZL1 I 0 D xEr t t1Er t tn 2311Er t tn 2311 tl 3 Sam t2 t1 ltltl9t3 lgt2 it11 p oogtgt i I Twodimensional correlation plots Double Fourier transform 54915523 6173fdileiglt1ig3t35tl1213 Sm 0192523 fdrgfdrzem2f2m3rsltrprzJ3 Particularly useful for displaying structural information in analogy with 2D NMR Ultrafast 50 fs time resolution Probe intra and intermolecular interactions Spreading transitions in multiple dimensions Lineshapes give environment uctuations Semiclassical prescription for calculating the signal Two steps 1 microscopic step calculate the polarization PW induced by the classical incoming fields Pquotrt fdtnffdt1 Ert tnErt tn t15quottnt1 SW nth order response function 2 Macroscopic step solve Maxwell s equations with PM as a source Multi wavemixing amp the phasematching condition Suppose we know Plquot gt step 2 solve Maxwell eqn for the signal eld Plquot serves as a source N incoming elds E t 3 E i kjlrwtr 00 Solution depends on geometry boundary conditions and pulse con guration of the sample n2 82 47f 82 n 2 V Ema 3W g EW yields a directed signal Within the slowly varying amplitude approximation fiE 392 jam S 3L39Epcrquot hz 39 dz 6 iii 2 kc k5 4amp2 k k1 l k I 1 kn ItsSignal 17 v Esih r 5 39 LPLirjgm lkLBJTE39 rii mL I L j Phase matching condition ak Detection modes Homodyne detection measure intensity ism IESILr Ill Heterodyne holographic detection extract amplitude amp phase of the field by interference of E5 with external field ELo local oscillator Propagating along ks SHEr IE5 Em ELQT2 s2 2REEEDE5 Since LIE 5quot2 IELDI39E 9v l SHET 1111E Psii1le l How to calculate the polarization A Wave function Hilbert space approach PW r g lt rWW9 Zfn1 3 f Ntmu di2 E rt7fxrirkr17k1 Lx r171 my m xv w w my m w m r1 1 m 39kVY dI d formu t1 Diagrammatic representation Forward amp SchwmgerKeldysh loo backward In time B Density operator Liouville space approach PW t 7 TH WW 7 f c1th f it Er t 7 inErt 7 tn 7 7 n5lt Jtn t1 5 VlgthVIquottnilv VEt2V 7t1JV 7 X In 171 p 1 bath bra and ket gm Htexp in 1 1V 1 5 1 propagate forward in time Diagrammatic representation double sided Feynman diagram Liouville space pathways for third order response of excitons 3963 g kIII W4 XI 62 gt 3 leugt mg k k1k2 k3 kH k1 k2 k3 km 2 k1k2 k3 kW k1k2 k3 Merits of Liouville space picture We work in real physical time purely forward propagation The double sided Feynman diagrams connect directly to timedomain experiments One explicitly visualizes the real time intervals tj the k1 k1k2k3 Signal four wave mixing phasematching direction k1k1k2k3 ideal time domain experiment with temporally well separated pulses Uhr39 1 Hal 7 a Three level model system Double sided Feynman diagrams for the k1signal otating Wave approximation photon absorption arrow pointing inwardsquot aterial emission arrow pointing outwardsquot deexcites R excites the m rial the mate 3meiiesp1 epathw2 as 1mm me R undidUmqu PM minimum Rm llab beihbltba abt lui1tr 39ot39 In ItyJIAtvJlmitz LMhliml397utlt 5 R m Ra A EE2ESk st Quantum description of Field Hamiltonian of the joint matterfield system the bare field iii 2 Ej ug j j is eliminated by going to the interaction picture Hindi VF HM V Marr bare material matterfield coupling Replace classical field by field operator iii rt t 2 Sir t Silk t A quotgemsW 13quot fcir r11 Z n J i 3equot quotquot 39I photon field annihilation operator represents destruction of a photon at position r and time t 51 E1quot Photons are bosons Fiji 39 quotiquot Ma tter fleld co uplmg Within the dipole and Rotating Wave approximation Emir fir 1134 Em t matteratcited photon destroyed matter tieexcited phuten created Decompose dipole operator fig according to 5 W Z Zia Li RI raising operator creates excitation in i I39 14 the material if annihilates an excitation The optical signal The detector registers the number of photons per unit time in mode a The signal is given by the time averaged photon flux le T Sazllm ltaaaagtdt TewZTJdt The Heisenberg equation of motion for the photon number of i aTa H aTa a a h mt a a and the interaction Hamiltonian 1 give the following signal 1 00 I I Sa 2 E s 5E0 ora rdr Ea D a aa The expectation value lt gtais over the initial state of the entire system W lightmatter UCIrVine quvusity qualiIuruia llViJlt Heterodyne detection revisited a quantum eld perspective Semiclassical picture Local oscillator LO does not interact with the material but only interferes with the generated signal field We measure the change in intensity of the local oscillator Is this assumption of a spatially separated LO necessary In a quantum description we consider the whole process as a single n 1 photon eventwith n 1fields The L0 is singled out only by the detection process Calculate the change in intensity STdirectly from our microscopic definition of the signal ET I39 Sli39TJIjT ti me integrated signal 1 hf 39 539 3 dt fl T 5M Heterodyne detection is a stimulated process by the local oscillator Using Heisenberg s equation Sm 2 E 111 ii it ri T gt this is the starting point for a perturbative expansion in all the modes incoming amp L0 This looks like exactly like the semiclassical expression for the heterodyne detected Signal SHEr ImrEL39 Psrrl If we replace the field operatory is by a classical field Leading order contribution each mode enters to first order Replacing bril ls exact The perturbative expansion H int Perturbative expansion in the interaction Liouvillian superoperator m SO 2 SO m1 m wave mixing signals are given by products of material and corresponding optical field SNGF39s of mth order SOquot 2 Z T dt dt dt 0 vm o o m o o o 1 rmav ziw 29WzagtEs3yma rm1rmr1 In representation VJj j 17 7 J In LR representationzv AR 17 LR The optical field SNGF s The material SNGF39s m m EVm117m171tm19tm 9 911 V mjvmuyl tm1tm t1 ltTELM tm1E m rm E91 23 TVV39Mtm1WmtmKt1gt Conclusions Recover the classical result but now LO does not have to be spatially separated anymore One shot microscopic calculation of the signal as an n 1 photon event no need to create polarization and use Maxwell equations w UCIrVine University of California Iwine Manipulating Quantum Pathways of Matter by Coherent Multidimensional Spectroscopy with Entangled Photons The ks k1k2k3 signal Frequency domain no control over time l k1 k2 k3 ks Ell ordering different than k Construction of the diagrams 1 2 and 3 are represented by arrows pointing to the right absorption and 1 and s by arrows pointing to the left emission 2 Consider all possible ways to distribute these arrows around the loop Constraints The interaction with the detected mode s is fixed to the top left branch The material system must start amp end in the same state its ground state lagt 8 loop diagrams 7k 1 7 WW EVWW ikl 193 Vm wwwy V k3 7 k1 A WW 4W 19 19 1 Vgt a b 7 k 5 WW f 7 k L k2 k3 V ks 4V 1 4 39 k k3 V lt f ks 7kg 4 7 A r kg k 1 VVW ki k3 195 k2 myxm d e 7kg 7 MN f k3 k2 N 7 kl iki fv 193 Twin entangled photons generated by PDC Vgt HVgt4quot5 VH PDC crystal The idler A k and signal B k2 are populated from the vacuum state from the pump photon P km by means of the interactions mediated by the nonlinear 7 crystal 2 f f f Hm 1 aazauauaza The idler and signal are entangled G relayquotanti MHUiliWUalaHWMW H39ktsw39r525a39nlaa39amplagt 0 Twin photons Types of entanglement HVgtr VH PDC crystal Polanzatjon entanglement photons having nonzontal H signal k2 and vertical V idler k linear polanzatjons ofthe two cones is entangled Twin photons Types of entanglement HVgt4quot5 VH PDC crystal position space entanglement frequency energy entanglement wave vector momentum entanglement Iv Ifm mm 9 Halal kl MOM k2vr2dmld4edrrdra where f 39 39 39 q kn k k2 Twin photons Properties W x vabtvm A PDC crystal TR A t area1n ofeach other This quantities represent the width ofthe fourthorder temporal and spacial coherence functions s respectively Spectroscopy with Twin photons 4TPA 1 Several spectroscopic techniques had been reported by several groups Two photon absorption Dayan 2004 Saleh 1998epv TPA Sum frequency generation Silberberg 2007 Peer 2007 SFG Two photon induced uorescence Goodgon2007 Lee 2007 Teich1998 TPIF Siqnal linear dependence on pump intensity as a siqnature of entanqlement 2 9 SCL D IE12E2 SEN D lEplZ Theoretical prediction for homodynedetected SFG B Saleh B Jostm M Teich EntangledPhotons VirtualState spectroscopy PRL80 34833486 1998 A Sergienko M Teich etc Quantum theory of entangled photon photoemission PRB 69 165317 2004 Experimental verification for homodynedetected SFG and TPlF M Teich B Saleh Entangledphotons microscopy spectroscopy and display US Patent 5796 477 APe er B Dayan YSilberberg Temporal shaping of entangled photons PRL 94073601 2005 Nonlinearnteractions with an ultrahigh ux of Broadband entangled photons PRL 94 043602 2005 DLee TGoodson Entangled Photon Absorption in an organic porphyrine dendrimer Journal of physical chemistry 110 2558225585 2006 we UCIrvine UniversirynfCali xmin Thine over l 77 gTPA Bk 1 k PD vlmrr J pump w SFG The absorption no longer depends on an accidentally simultaneous arrival of two photons longer quadratically depends on photon ux density but appears to be linear Javanainen 1990 Absorption rate Power levels required for TP excitation can be dramatically reduced Spectral resolution can be improved since absorption only occur in a region where correlated photon pairs overlap in space Entanqled photons PDCMZI in nonlinear spectroscopy inch Zehendei interferomet637 f ff a39l2 3 quot k x i 2 35 ES a Ez391 V 5 V W 5557 1 2 f h Experiments are conducted with the nonorthogonal modes 31 32 with waveveotors k1 k2 w UCIrvine University of California Irvine Pumpprobe PP technique carried out with two optical modes Interacting with N three level molecules lii fjtol The signal is the timeaveraged photon flux in one of the modes S m1 wz D 2 material SNGF gtlt optical SNGF w UCIrVine University of California Irvine PumpProbe with classical beams All optical SNGF s are the same lt01gtlt612 Xaf gtlta gt NI E1 39239 E2 392 The classical pumpprobe signal is TPA Ground state bleaching SCal9a231 IZIEZ 392 321413a19a2i lg3a13a2 ems UCIrvine I Iniversiry nfCali 1mia Irvine TPA with maximally entanqled PDCMZI beams A L 11 it 2 r r v x F r l39i lfl k r l J i All ll lr i kl iii kl l rLHkl I Iquot r l kill lyl ill lyl Optical SNGF s of group A optical SNGF s of group B aTaTacD Ep 2 Ep 4 Jada Ep 4 At low pump intensity limit the signal is solely given by group A E 2 3 W S wIDwZNIEp SZA 1202 UCI C Univemrynrcaiifnmia Irvine Group A TPA pathways contributiormiEp 2 Cross peaks due to pathway i are A given by double resonance condition If 2 z A 01 z 0 1 Crosspeaks due to pathway iii are doubly resonant meg z wef Cross peaks due to pathways ii and iv are given by triply resonant ml z meg a1a2 zwfg Group B Raman pathways contributicmEp I4 The pathways of group B induce crosspeaks at wlzweg wzza l4 39 z 2 fl 7 gt gt R2 gt k2 k UCIrvme UniversirynfCali ymia Irvine Total siqnal group A B pathways large dephasing rate and close transition frequencies ywegzOl Spectral overlap meg N mfg All dipole moments are the same 01 arb units The pathways of group A TPA and B single photon transition may not be separated by using classical optical fields This is possible by the entangled PDCMZI photon signal w UCIrvine I Iniversiry of California Irvine The diaqonal section of the 2D spectra a1a2agf1 1 A Entangled photon PDCMZI signal Group A U 1 I B Group B contributions 392 Bf 390 C Classical signal AB quot C f A gtlt 47 Resonances are very sensitive to 1 v V a the overlap between pathways quotquot meg i 6f due to destructive interference between pathway iii and pathways of group B 300 Pathways ii and iv augment the central peak The resonances are given by pathway iand interfere W constructively with group B They disappear slowly with UCIrVine Increasing dephasing lJniversirynfCali xmin Irvine Classical vs entanqled PDCMZI siqnals Classical signal 1 Scales as intensity square C 2 2 5 0102 qu lEzl 2 Pathway selectivity NO 8 mg 3m 25 Entangled photons signal 1 Scales linearly with intensity 5Cd1d2 Ep 2 2 Pathway selectivity YES SCw1w2 31 w UCIrVine University nFCalifnrnia Irvine TPA on a closedtime path loop Detector d measures change in intensity of the H polarized mode k with and without place V polarizer before the sample mode k2 A r gr The two signals are blended together is for symmetric TPA is for asymmetric TPA Detector d2 measures change in intensity of the V polarized mode k2 uuml I El PP on a closedtime path loop yr m PMP Pump modulated Probe TPA Two photon absorption PumpProbe symmekric PMP TPA d1d2 PP on a closedtime path loop m 11 Ma 1 ah 7 k akl 7k 77L 7k h 7k l L 7L r 1 i 1 k k 7k 7L k Using the loop diagrams the PP signal can be Written as m 2sz 1 T N i3 m m sf QM F dmiiidzzdzqu 90109021090311ltV11V13V 12V 14gtltEJ 4192T 1031 23204 914 z399tz 1911 3ltV 3V11V 2V 14gtltE 233 1031 232 4gt 9012914133991311ltV11 V13VT GOVT ENET 11ngT 1032 1031 10gt 914129141191113ltV 13V11VT t4VTt2gtltE 10191T t1E2t4E1t2gt k1 lt gt k2 v vi vii viii PP with classical fields B Wm f u All optical eld correlation functions become the product of the elds intensities ltE39E39EEgt 4 E W F1 l2 At this point let us introduce the TP operator in the frequency domain T 44592 VV39Gtq mgGta2 mgVV39 is the Fourier transform of the retarded Green39s function operator 6a 7i T dt6r exp7iHnrequot PP with classical fields In the molecular eigenstates basis the only nonzero matrix elements of the TP operators TP amplitudes are ltf T 011012 ggt ng 01012 yilegmlwfg jzlegmzw ltf T WW I ggt T 01012 21 way 21 may ltg IT 011012 fgt Tgfwlwz gelgewlef gelgxwsz ltg I T Topaz I fgt Tgimp a geIea1Jef 31 wsz Here we introduce the Fourier transform of the single excited state Green39s function leg 0 i I Geerexpia wgrdr 700 a meg 17ge yge is the dephasing rate expiagr describes the free propagation of the molecule ground state TPA pathways group B Accompanied by emmision from state I f T1 zq all absorption on the left branch forward time with retarded Green39s functions k T tq all emission on the right branch backward time with advanced Green39s functions PMP pathways group A Accompanied by emmision from state le T1 tq 4 absorption on the right branch backward time with advanced Green39s f ctions 1 7 7k w w mm Tqa12 emission on the right branch backward time with advanced Green39s funcu39ons Siimaiwz PP classical signal 271N 2 2 g g s s 51w1wzrfgwlwzTgwpwzTgaawzTgaican TL I l x A by the double excited state Green39s func on 1 Igm mewEH For offresonant single excited state T1 111 02K lq 02 T1 tq JET 114710 and the signal is 5tq 102 imfgiTgmnm2iz Twin state as initial state of the field Twin state is defined as 11 CZZtI exp isincjexp ijsmcj k1k2gt k1 k2 Here A0 2 mp a1 a2Ak kp k1 k2 fl is the interaction time Within the PDC crystal of Width L2 The normalization constant C MDEP E is proportional to the nonlinearity of the PDC crystal 252 the pump electric eld amplitude EP and the entanglement area A12 The Fock state k1k2gt contains one photon in each mode k1 and k2 Twin state no delay correlation functions For the twin states the optical eld correlation functions ETETEEgtare factorized into the products of the field transition amplitudes 00 E2f4E1t2 VI 904570412112 9f2t4Ff2f4aT12 ltWE1Tf3E2Tt100gt 90301 11112 9t1t3F t1139T12 2 Zlt2EP 102 Q A12T12 Where rectt is the rectangular function equal to 1 for 0 S t S 1 and 0 otherwise t t Ft4t2T12 exp ia1t2 wzt4rect 4T 2 12 Twin state no delay correlation functions The signal becomes 339erLthSLLLdt zdt automatoeagagtltVlttgtVltr3gtV39tor39rnwrprhnnntmtpmw eltr4t2gteltt2agteltrt3gtltVltagtVlttgtV39WtM tvrpznnrwtpmw eltr4r2gteltr4t3gtelttgagtltwrm20Mwtowt3tnzgtFltr2t4mgt Margauxgteltrt3gtltV2ltr1gtV ltrgtV39tor39rnw mr3zzgtFltt2r4zzgt ltk 9km Twin state no delay TP transition amplitudes At this point let us introduce the following transformation of the material Green39s function Geg 139 due to the entanglement between the photons oo iw7w iy T 2g 2g 12 1 JegwT12 I Geeltrgtrectltigtexplttmgtdr foo TlZ w Qeg rlyeg That is the role of the entanglement is the modification of the Fourier transformation of the material Green39s function J 6 g 01 T12 can be alternatively Viewed as the Fourier transformation of the joined matterentangled photon Green39s function Modified TP transition amplitude is ng 01 2 Tiz geuef39leg 01 Tu geuef39leg 02 Tiz TP induced transparency WM 2 Buzz V aim c s 53 ivwszun Mm dqu 02Tf qv T Lw HTgldvmzv ziz CTPL correction Glauber twophoton counting For the offresonant single states T2 qm2T2 T1 q 10272 and the symmetric TPA signal does not dissapear What is the role ofT iv zy z T 1147102732 ten PP entanglement time spectra Tm szpmp ma N I dnz expowmnzmzpmp mu l8 16 15 r W 1 ply2 um LID 0 vdldzSymAsym W Group a of resonances at 7w e 0 03 Group b of resonances qg imp 25113 Group c of resonances 4226 PP entanglement time spectra zualtrn1lt515 mamms i ll l Smr 1 All pathways interfere constructively 2 b resonances can be detected by a conventional pumpprobe with short Well separated pulses 3 Increasing the dephasing rate le column quenches the resonances in regions a and c PP entanglement time spectra zuaaltsza unJltT1ltnns My 1 W 5 i1 a a 355 0 y a 55 4 PMP pathways contribute to spectml regions b and c 5 TPA pathways show resonances in regions a and b 6 Squot Sd SW 7 The assymer c signal vanishes PP with mutually delayed twin photons gm Now let us consider a different TPA setup with twin photons close to those proposed by Saleh The frequencies of the beams are xed to be tq to2 my 2 and we introduce a relative time delay I between the tween photons M 12 Et 1 exp7impt 7 r 2 2 3 12 9 ajexpkimp tr 22 E2t PP with mutually delayed twin photons Twin correlation functions Tr um dia lt0 0l E 130E032 l III 0gt 9t4t2Ft4t2Tu 9t2t4Ft2t4Tu W 0 l Elam rm 0 0gt WM mm 6ltr1t3F lttt3Tu 2 III E a t it Ft t T exp 71m 2t t rectE A 2 A 2 12 gm p 4 2 7 I For the diagrams VViiione has to interchange t4 lt gt t2 t3 gt t PP with mutually delayed twin photons Matter transition amplitudes Jag zap 27 m r describe the Fourier transform of the systemtwinmatter Green39sfunction by incorporating the relativedelay r between the entangled photons This delay breaks the symmetry of the Green39s function and makes it dependent on the order in which the twins are absorbed or emitted Isz mode precedes k the system evolves by Jag 1072 7 r Ifk mode precedes k2 the system evolves by Jag 1072 7 r TPA with mutually delayed twin photons Using the transition amplitude above the symmetrized signal can be rewritten as 27I3N 12 2 By 2 012 SigaP2Tmr39 WI T 93 P 31 gaprfwp2zzr39ngap2z2r39ngapzzr39V 12 12 So far we have considered the symmetrized TPA signal but as an alternative one can antisymmetrize it by monitoring the difference between the detector d1 and d2 27z3N lt2 2 E 202 lz llil pygmy 3AIZTIZQ g P 2th 2T12I39ng wp 2T12r39 Tgwp 2T12I39Tfwp 2T12r39 S rmkwp mm Here we introduced the symmetric and antisymmetric TP transition amplitudes Tg mp 2112 2J geJefleg mp 27 Tl2 ZJ gell tefleg mp 27l2 2J Tg mp 2quot112D 2J ge eflegap29112 15 ge eflegap29112 2J The asymmetric TPA signal vanishes in absense of the delay I39 0 PP delay time spectra 7 Slip 10 2 T127114 N Id 6XpiwrS p 11 27713927 r r n r 1 opz rap2 W 0 Group a of resonances fp Group b of resonances qg imp 25113 Group c of resonances quf4u PP delay time spectra In numerical simulations we assume that we can increase the time delay with a step A r S 2r maxmg a flap 0025 up to the Value ofthe entanglement time T2 2 A r Ifthe unit ofenergy is lEv then the step is A r3913fs PP delay time For PMP TPA pathways region a c resonances are produced by the pathways iviiivii L 39 fnllnw L L 39 order the region c a 39i iii L order for the emission is opposite to those ofthe absorption This peculiar feature is a result of broken symmetry of the eld transition amplitude all PP delay time spectra 1 For small r 39 2 At 1t 1 16 4 15 W 1 m2 W2 rgt 0 39 PMP and TPA quot t rr vviii aand c PP delay time spectra r 3 The spectra obtained from the two detectors are different 4 Only the S 2 signal reveals the resonances in the nonclassical regions a and c 5 Both SSym and SAym signals contain all resonances 6TheTPA39 39 39 39 39 an region a thus one can separate PMP from TPA iffocuses on the regions a and c only PP correlation spectra Detector 1 and 2 M m STan 2agt42w N I dz many I drexpawmsgrwp mm 7 n The correlation spectra from the two detectors are different Unlike the delay spectra detector d1 reveals all spectral regions while resonances aa and cc from detector d2 are suppressed PP correlation spectra TPA and Sym spectra 7 mV 1 The offdiagonal as well as bb resonances are sensiu39ve to the dephasing rate and Vanish for large dephasing The diagonal aa and cc resonances overlap as the dephasing rate increases PP correlation spectra TPA and Sym spectra m m 2 For both Values ofthe dephasing rate the TPA and PMP signals miss the cc and 2121 resonances correspondingly TPA with mutually delayed rectangular shaped elds E t E exp7imp t 7 r39 2 2rectH 77 vz2 12 E2 t E2 expeimp t r39 2 2rect 12 31004420 TPA with mutually delayed rectangular shaped elds E t E exp7imp t 7 r 2 2rectH 7722 12 E2 t E2 exp7imp t r 2 2rect 12 Taking into account the following identity reaper 27T22rectt2r 27722rectlt47t2r 12 12 Tu we obtain the eld transition amplitudes 0303 0 6t4t2Et47t239TlZ 90210302714732 t it t it 1 exp 7sz 2t4 t26t4t2rect 2 7 6t2t4rect7 ltE1ltt3gtE4lttgtgt eltt3r1E lttztTugt 6m than Wl39l fill H l E E expimp 203 t 6t3trectt tt3rect21t l t W W H SymmetricTPA signalbecomes 27mm 3K sgmmp 2T2r 7 512 7122002T2r39lTgmPT2r39l2 r independent prefactor Conclusions We used mutually delayed twin photons to study offresonant single excited states in a three level model system Two detectors measured the change in one of the beams intensity with and without other beam The symmetric asymmetric twophoton absorption was defined as sum difference of the detectors readings Fully microscopic entangled photon TPA formalism using close time path loop CTPL diagrams was developed Its predictions was compared with proposed before virtual state spectroscopy based on conventional twophoton counting Glauber theory For general quantum optical fields CTPL theory yields the signal in terms of Liouville pathways each scaled with corresponding four point optical correlation function There are two groups of pathways TPA and PM P The TPA pathways correspond to emission from the double excited states and may be taken into account by the twophoton counting Glauber theory The PMP pathways are related to emission from the single excited states where the one of the photons creates just an intermediate coherence between the ground and double excited states For the degenerate offresonant case of classical uncorrelated photons the signal PMP TPA Asymetric signal vanishes CS 130A BiTreesl mway Search TreesI m way Search Tree Empty or if not empty then 0 Each internal node has q children and q 1 elements for 2 3 q 3 m o Nodes With 19 elements have exactly 19 l 1 children 0 Suppose a node has 19 elements Let k1 k2 kip be the keys of these elements Then lt31 lt lt32 lt lt kip Let 6061 cp be the p 1 children of the node The elements in the subtree With root co have keys smaller than M The elements in the subtree rooted at Ci have keys larger than kri and smaller than k3 1y 1 S i lt P The elements in the subtree rooted at cp k have keys larger than kip CS 130A BiTrees2 2b10c80d o b 1e50 6020030f400500600700 0 Q o d 8 0840 86088g 40 2 30203040 0 CD 0 f 20320360 o g 50900920940960980 0 Searching Inserting 3165 0 Deleting 2084510 0 Format 7160 6161 6262 671671 CS 130A BiTrees BTrees of Order m gt 239 Different from textbook I A B Tree of order m is an m way search tree If the B tree is not empty the corresponding extended tree satis es the following properties 0 The root has at least two children 0 All internal nodes other than the root have at least Zl children 0 All external nodes are at the same level A B tree of order 7 CS 130A BiTreesl o m gt 2 because they cannot represent all possible sets 0 B Tree of order 3 is a 2 3 tree 0 B Tree of order 4 is a 2 3 4 tree Same as RB Tree Lemma 113 Let T be a B tree of order m and height ii Let d and let n be the number of elements in T 1MM4 1gngn 1 2 logmm 1 S h S logdquotT1 1 Proof 1 gt 1 follows from the fact that the minimum number of nodes on levels 1 2 3 4 uh 1is122d2dau2dh2andthe maximum num is 1 m m2 mh1 The ember of null pointers n 1 CS 130A BiTrees A B tree of order 200 and height 3 has at least 19999 elements and therefore can represent all UCSB students 0 A B tree of order 200 and height 5 has at least 199999999 and therefore can represent all US voters o The order of a B Tree is determined by the disk block size and size of individual elements 0 For obvious reasons all the B tree examples have small order 0 Searching is like in an m way search tree CS 130A BiTrees Insert Example BTree of Order 3 I CS 130A BiTreesY CS 130A BiTrees CS 130A BiTreesB Nodes are of the form nc0e1c1encn where the e s are the values or keys and the C s are the pointers Procedure Insertte t points to root and e will be inserted c NULL ec is to be inserted in leaf node SearchtePfound returns foundtrue if e in the tree and P will point to the node in main memory that has e Returns false if e is not in the Btree and P will point to the last node visited leaf node during the search Done false if not found while P NULL amp not Done do Insert ce into appropriate position in node P gt Let the resulting node be P gt nc0e1c1encn CS 13OA BiTreesJO quot if P gtn lt m1 Output P to Disk Done true else e P gteceilm2 d ceilm2 Split P into two nodes in main memory P d1COe1c1ed1cd1 Q mdcded1cd1emcm Output P and Q to Disk CQ P ParentP Parent may be obtained from stack that is built by the Search proce if not Done Create new node Q in memory Q 1tec tQ Output t to Disk a dure CS 130A BiTreesll Delete Example I CS 130A BiTreesJQ 50 75 10 30 60 70 a CS 130A BiTreesl Nodes are of the form nc0e1c1encn where the e s are the values or keys and the C s are the pointers Procedure Deletete t points to root and e will be deleted SearchtXPfound returns foundtrue if e in the tree and P will point to the node in main memory that has e Returns false if e is not in the Btree and P will point to the last node visited leaf node during the search k CS 130A BiTrees14 if found Let P point to node nc0e1c1encn and ei has value e if P gtc0 O P is not a leaf node Q P gtci Reads from Disk P gtci and stores it in memory node Q While Q is not a leaf node do Q Q gtc0 Reads from Disk P gtci and stores it in memory node Q P gtei Q gte1 Write P on Disk PQ i 1 CS 130A BiTrees15 delete P gtei P gtci from P nc0e1c1encn and replace P gtn by P gtn1 while Pgtn lt Ceilm21 ampamp P t do if P has a nearest right sibling Y Let Z point to the parent of P and Y Let j be such that Zgtcj1 P ampamp Z gtcj if Y gtn gt ceilm2 can borrow from right sibling P gteP gtn1 Z gtej move from Z to P PgtCPgtn1 Y gtc0 Pgtn Pgtn1 Z gtej Y gte1 move e1 from Y to Z Ygtnc0e1c1 gt Y gtn1c1e2c2 e1 is deleted Output nodes P Z amp Y on Disk return CS 13OA BiTrees16 quot k e if Pgtn Has a right child but cannot borrow from it I 2 Ceilm22 Borrow from parent and combine P and Y into one node Output r P gtc0P gte1P gtc1 P gteP gtnP gtcP gtn Z gtejY gtc0 Y gte1Y gtc1 Y gteYgtnYgtcYgtn as new node P Node P is now node Z except that Z gtejZ gtcj is del lse do the nearest left sibling instead 0 Output P onto Disk else t PgtC0 eted CS 13OA ees17 Threebody Interactions in Cold Polar Molecules HP Biichler Theoretlsche Physlk Unlversltat lnnsbruck Ausma lnsmut fur Ouantenoptlk und Ouantemnformalon der Osterrelchlschen Akademle derWlSSenScha en lnnsbruck Ausma In collaboration with A Micheli G Pupillo P Zoller Threebody in i39emc i39ions Manybody interaction potential V Hamiitonians c den ed matter are effective Hamiitonains atter integrating out nign energy excitations Vt an gvtrz 7 r7 gkw r1 rm twoparticie threeebody interaction interaction Application 7 ft an Wave function of fractionai quantum i iaii state More and Read Bi V xcnange interactions in spin systems microscopic modeis exotic onases Muessner and Sundi Eli Eaients eiai EIZ Muutncn and Semnii EI2 Route towards exotic and topoiogicai pnases7 7 string nets degenerate Hiioertspace tor he ioop gases FidDWSKi et ai Threebody in l39emc l39ions Extended BoseH ubbard models 7 hardcore bosons l 1 H 712b bj 5 Uijnmj 6 Wijkninjnkl 27gt 2 k hopplng energy Woebody rmeracuon threeebody lnteractlon Goal 7 large lnteracuon strengths 7 lndependent control of we and threeebody rmeracuon Realizable with polar molecules Polor39 molecules Hetronuclear Molecules r electromc excltatlons N lUlaHz r vlbrauonal excltatlons N 10 Hz r rotatlonal excltatlons I dlpole N lUIUHz d moment mu m 2 5Debye r electron 5pm 7 nuclear 5le Polar molecules in the electronic vibrational and rotational ground state Strong leOISdIP0le Interactions tunable With 7 permanent dlpole external elds moment d N 179Debye didQ 11100120 W0 T 375 v v r polanzable Wm stath electnc eld and mmwave elds Interaction energies Particles in an optical lattice e iattice spacing a A2 N 500mm 7 recoil energy ET 7 25 m Pseu dopotential e dominant interaction in atomic gases Magnetic dipole moment 7 SiZe of Wannier function aMNU2a nearestneignoor r Cnromium atoms Witn Electric dipole moment r LiCS netronuclear molecule a N 6 5Debye 7 increased wractor 1o2 N 1372 interaction interaction present but U N 0 5E srnaii U N 0 SET U1 N 10 5Em N777 U V V V U1 N SUET aai 93111 Polar39 molecules molecular ensembles AMO SOlld state Interface quantum memory solid state quantum processor molecular quantum memory P Rabl D DeMiIIe J Doyle M Lukin R Schoelkopf and P Zoller PRL 2006 Cooper Pair Box superconducting qubit Spin toolbox polar molecules with spin realization of Kitaev model A Micheli G Brennen P Zoller Nature Physics 2006 Polar molecules Raman asEr spumaneuus Emwssmn Experimental status 7 POW mo ecu es m the rotauond and vwbrauona ground State 7 coohng and trappmg techmquea beemg deve op ment 7 coohng of p0 ar mo ecmes e g stark dece erator Mme a e 2 Harvard G RempE Mumch G Meuer Eemn J VE JWLA DDE JD r p o o assooauon seeJ Ye sta k aH BUM alum abs 8 mo ecu es thh do ed rmatmna and muratmnax gmund state 5 e ectromc SheH e g SrO RDCS L CS Polar molecules Laser Crystalline phases long range dipoledipole interaction 4 x 3 a 3 3 interaction energy exceeds 39 kinetic energy Bipolar Ciystal Threebody interaction extended Hubbard models tunable threebody interaction Polor39 molecule tat f Low energy description ml mg igc e N r rigid rotor m an electric field i 2 H502 BN dEt N angular momentum di oie d dipole operator d moEnent BNN N 2 Accessible via microwave anharmonic spectrum electric dipole transition N1 ANi1 Am7101 20GIIz microwave transition frequencies 39 n N 0 no spontaneous emisslo In i39er39ac i39ion be i39ween polar39 molecules Hamiltonian 2 2 7 HM z Vn EN dds d d imxdzquot i1 kinetic trapping rigid electric interaction energy potential rotor field potential Without external drive 1 4153 7 van derWaais V r i vdW attraction 6 EnB Static electric eld 7 internal Hamilton 2 0 H503 BN dE r finite averaged dipole moment D gdlt1a S a2 Dipoledipole in l39emc on Dipoledipole interaction 7 coHap5 of me system for mcreasmg dwpo e mte r roton so emng r supersohds msotropwc mteracuon ongrrange Vr D 3 BornrOppenhewmer vahd for r gt Rm 03 r gt EdD1I3 nstabmty m the anracuon aW39DOdV syste Dm racuon ran 2 1 Gura a at nzL sanmsetaw EIB SmyameUv Elm attracuon repu s on Z un mng uterma Stability p 7 strong nteracUOHS uscmatur Wavemnctmn r con mng mto 2D by an optwca amce Sfabili fy via Transverse confining Effective interaction 7 interaction potentiai Witn transverse trapping potentiai 7 1 2 my 2 Vr7D 737475 72 r cnaracteristic I Dm 1 5 7 a iengtn scaie 4 h 4 r potentiai barrier iargertnan kinetic energy attempt frequency Tunneling rate serniciasmcai rate r Aexp isEh instanton tecnniques 25 7 Euciidean action ortne BM 35 7 n C instanton traiec ory numericai factur 01558 Cr39ysfulline phase mteracuon Hamiltonian 7 peter motecutes con ned mto a Wordtmenstona ptane E Dm y m H P7B a Em 52a 21 2jR Rj 0117qu KosterhtZrThomess Ftrst order memng ran mon Kama 81 2 tl quot 5 2 Quantum meltln m H P EUENEV E DEN ER M LUKW A M EHEH G Pupm P ZuHEr PRL 2UU7 annual r ndtcauon of a rst order ransmon r cnttcat mteracuon Strength m 20 Single polor39 molecule Static electric eld 7 along tne Zraxes r splitting tne degeneracy or tne nrst excited states degenerac r rnducesnnrte dipole moments 9 aldzw d5 ltE1ldzle1gt e Mircowave eld 7 coupirngtnestatelg and le1 A uetunrng 2 ram frequency 7 restnct to two states 7 rgnore influence or le 71 r rotatrng Wave approximation shifted away by External DCAC fields anharmonic spectrum electric dipole transition ANi1 Ann 7101 microwave transition frequencies no spontaneous emission Manybody Hamil l39onicm Manybody H amiltonian 2 H Vuagm nggt H35 Hf W W V externa potenuam r dwpo spro e mteracuon resmcuo w gg f a ag g a ntema states m e11 Twolevel System V rotaung Wave approwmauon M0 e gengtateg nggtlt 2 gt hsi gtzaggtz5e1gt1 42 g ME 1 r twoJeve System w an effecuve and energ es magneuc he d Ei i402 A22 Dipoledipole in l39emc on Microwave photon exchange 7 D 5 HOW H d23 ex 1 D H 7 Z uriirjSZSj 5952 2 7170056 dwpo edwpo e 111 3 mteracuon Induced dipole moments r We damV5 P1 mm 1 HE 5 Z D 1239 i 1139 779339 ngQi 77ng nEQj 1 Q Q BornOppenheimer potentials Effective interaction i diagonalizing the internal Hamiltonian for fixed interparticle distance Z H5 H3 H2 iiThe eigenenergies describe the BornOppenheimer potential a given state manifold iii Adiabatically connected to the groundstate iGgt Hiigti weak dipole interaction interpanicle istance d kc BornOppenheimer po i39en i39ictl First order perturbation r E1rzlt0ii i fot im 39 i0 H ai9gtz i D1 E1ri A1 2 Bum 7 rj 2 dipoierdipoie 17 31059 Eb ValtrgtA1T Dimensionless coupling parameter 2 A1 12779 32775 7 0252 rforamagicrabi frequency tne r tunabie bythe externai eiectncrieid dEB andtne ratio QA A1 U dipoierdipoie interaction vanishes BornOppenheimer po l39en l39ial Second order perturbation MV 2 7 2 7 7 threerbody E M k W0 m mm m z 7 2 N 2 Dwrzil y repu swetwobodv 7 W 7 mteracuon Matrix elements 39 M 7 M3 WW 32m U2 7 w 52 7 MW N 7 352 na 7 n02 1 Effec ve Hamil l39onicm Effective interaction dE n ltrzgt vltrrr7gt gkwrurm BS 39 39 r Morbody mteracuon Vltrgt MD Mr MERE v lt01 r threerbody mteracuon W 3931 mm 72R3D VF12VF13 VF12Vr23 VF13VF2s r vahduy 5 resmcted to D mterpamde 3 a dwstance R0 lt a n transverse con mng mo 2D u vamsmng dwpo erdwpde mteracuon