THULIUM IONS IN A YTTRIUM ALUMINUM GARNET HOST FOR QUANTUM COMPUTING APPLICATIONS: MATERIAL ANALYSIS AND SINGLE QUBIT OPERATIONS.

Transcription

1 THULIUM IONS IN A YTTRIUM ALUMINUM GARNET HOST FOR QUANTUM COMPUTING APPLICATIONS: MATERIAL ANALYSIS AND SINGLE QUIT OPERATIONS by Ijaz Zafarullah A dissraion submid in parial fulfillmn of h rquirmns for h dgr of Docor of Philosophy in Physics MONTANA STATE UNIVERSITY ozman, Monana May 8

2 COPYRIGHT by Ijaz Zafarullah 8 All Righs Rsrvd

3 ii APPROVAL of a dissraion submid by Ijaz Zafarullah This dissraion has bn rad by ach mmbr of h dissraion commi and has bn found o b saisfacory rgarding conn, English usag, forma, ciaion, bibliographic syl, and consisncy, and is rady for submission o h Division of Gradua Educaion. Dr. Wm. Randall abbi Approvd for h Dparmn of Physics Dr. William Hiscock Approvd for h Division of Gradua Educaion Dr. Carl A. Fox

4 iii STATEMENT OF PERMISSIONTO USE In prsning his dissraion in parial fulfillmn of h rquirmns for a docoral dgr a Monana Sa Univrsiy, I agr ha h Library shall mak i availabl o borrowrs undr ruls of h Library. I furhr agr ha copying of his dissraion is allowabl only for scholarly purposs, consisn wih fair us as prscribd in h U.S. Copyrigh Law. Rquss for xnsiv copying or rproducion of his dissraion should b rfrrd o ProQus Informaion and Larning, 3 Norh Zb Road, Ann Arbor, Michigan 486, o whom I hav grand h xclusiv righ o rproduc and disribu my dissraion in and from microform along wih h nonxclusiv righ o rproduc and disribu my absrac in any forma in whol or in par. Ijaz Zafarullah May 8

5 iv To my wif Samina, Thanks for prayrs, painc, and suppor

6 v ACKNOWLEDGEMENTS Firs of all I would lik o hank Physics Dparmn for supporing m as a gradua sudn. I spcially hank Profssor Wm. Randall abbi, my advisor, whos suppor, ncouragmn, and hlpful discussions, allowd m o compl my rsarch work. This rsarch work would no b possibl wihou his consan suppor. I also lik o hank Profssor William Hiscock, had of h Physics Dparmn, for providing a wondrful nvironmn in h dparmn for gradua sudns. I lik o hank Dr. Mingzhn Tian for hr ddicaion and consan hlp. Hr xpris and ddicaion wr invaluabl for h succssful complion of his work. I also lik o hank Dr. Charls Thil for hlpful discussions and his hlp in undrsanding and inrpring h xprimnal rsuls. I also wan o hank him for his hlpful commns on dissraion wri up. I also hank Dr. Krishna Rupavaharam for his hlp in rfining h marial in h dissraion. H wn byond h duis of a commi mmbr and hlpd m o improv h conn of h dissraion. I wan o hank Profssor Rufus Con for loaning h ky quipmns and his hlpful commns. I also wan o hank all my commi mmbrs for hir hlp. I also hank Ariana Paliobagis for proofrading h dissraion. I would also lik o hank Margar Jarr, Janni Gundrson, Jrmy Gay, Ros Waldon, Sarah aruha, and Norm Williams for aking car many adminisraiv and chnical mars. I also wan o hank my family mmbrs for hir suppor and prayrs hroughou my lif.

7 vi TALE OF CONTENTS. INTRODUCTION... Classical Compuing... Quanum Compuing...3 Quanum Compuing and DiVincnzo Rquirmns for a Physical Sysm...4 Dissraion Oulin...6. COHERENT INTERACTION OF LIGHT WITH MULTI-LEVEL SYSTEMS...9 Inroducion...9 Inracion of a Singl Puls wih a -Lvl Sysm... Rabi Oscillaions... 4 Fr Inducion Dcay (FID)... 6 FID afr h Inracion wih a Shor Opical Puls... 6 FID afr h Inracion wih a Long Opical Puls... 8 Inracion of Two rif Pulss wih a -Lvl Sysm... 8 Inracion of a Singl Opical Puls wih Thr Lvls in a 4-Lvl Sysm... Opical Rphasing in a 4-Lvl Sysm wih Mulipl Opical Pulss... 4 Inracion of Tmporally Ovrlappd Pulss wih a 4-Lvl Sysm... 3 Summary SPECTRAL HOLE URNING AND ENERGY LEVEL STRUCTURE OF THULIUM IONS IN A YTTRIUM ALUMINUM GARNET HOST (YAG) Inroducion Spcral Hol urning Enrgy Lvl of h Tm 3+ :YAG wihou h Exrnal Magnic Fild Enrgy Lvl of h Tm 3+ :YAG wih h Exrnal Magnic Fild CHARACTERIZATION AND OPTIMIZATION OF THULIUM IONS IN A YTTRIUM ALUMINUM GARNET HOST (YAG) FOR QUANTUM COMPUTING APPLICATIONS Inroducion Sampl and Exprimnal S-up... 5 Magnic Fild and h Dsign of h Hlmholz Coil... 5 Th Sabilizd Lasr Sourc, RF Elcronics, and h Dcor Th Alignmn of h Opical am and h Magnic Fild Th Crysal Srucur of YAG... 6 Th Elcronic and h Magnic Dipol Dircions in h Crysal Inracion of Thulium Ions wih h Opical Fild Inracion of Thulium Ions wih h Magnic Fild... 67

8 vii TALE OF CONTENTS CONTINUED Masurmn of h Magnic Anisoropy in h Mdium... 7 Inroducion... 7 Thory... 7 Exprimn Th Magnic Fild Orinaions for Opimal Marial Prformanc Th Hyprfin Spliings and h Inracing Thulium Ions Exprimn... 9 Th Hol urning Spcrum and h Inracing Thulium Ions Inroducion Exprimn... Populaion Lifim of h Hyprfin Lvls a 4K... 3 Inroducion... 3 Exprimn... 4 Populaion Lifim of h Hyprfin Lvls a 5K... 9 Modling h Rlaxaion Dynamics in h Mdium... 3 Inroducion... 3 Thorical Modl... 5 Exciaion wih a Narrowband Puls... 5 Rlaxaion Dynamics of h Excid Populaion... 8 Scanning h Mdium wih a Wak Chirp Puls... Analysis of h Thorical Rsuls... 4 Exprimnal Rsuls for Exciaion and Rlaxaion Paramrs a 4K... 5 Exprimnal Rsuls for Exciaion and Rlaxaion Paramrs a 5K... 9 Exprimns o Explor h Hyprfin Cohrnc Tim... 3 Exciaion wih Tmporally Ovrlappd i-frquncy Pulss Obsrvaion of a Cohrn Raman a Signal Exprimn Conclusions QUANTUM COMPUTING: THEORY FOR SINGLE QUIT OPERATIONS Inroducion loch Sphr and Pur Sa loch Sphr and Mixd Sa... 5 Dnsiy Marix Formalism... 5 Evoluion of h loch Vcor of a -Lvl Sysm Inracion wih h Opical Fild and Evoluion of h loch Vcor Cohrnc in h Mdium and Componns of h loch Vcor Singl Qubi Opraions Using h Gomric Phas Roaions Th loch Vcor Roaion abou h Z-axis Dcion of h loch Vcor Roaion abou h Z-axis... 6 Th loch Vcor Roaion abou h Y-axis... 6 Th loch Vcor Roaion abou h Y-axis wih a Singl Opical Puls... 63

9 viii TALE OF CONTENTS CONTINUED Dcion of h loch Vcor Roaion abou h Y-axis Th loch Vcor Roaion abou h Y-axis using h Gomric Phas Dcion of h loch Vcor Roaion abou h Y-axis using h Gomric Phas Exciaion wih h Gaussian Spaial am for Singl Qubi Opraions Th loch Vcor Roaion abou h Z-axis Th loch Vcor Roaion abou h Y-axis... 7 Th loch Vcor Roaion wih a Singl Opical Puls... 7 Th loch Vcor Roaion abou h Y-axis Using Gomric Phas... 7 Qubi Sa Masurmn and h Opraion Fidliy Dgradaion of h Opraion Fidliy for Singl Qubi Opraions Opraion Fidliy for h loch Vcor Roaion abou h Z-axis Opraion Fidliy for h loch Vcor Roaion abou h Y-axis Conclusions QUANTUM COMPUTING: EXPERIMENTAL DEMONSTRATION... 8 Inroducion... 8 Ion Slcion for a Singl Qubi Prparaion Inroducion Thorical and Simulaion Rsuls Marial and h Frquncy Sabilizd Lasr Sourc Exprimnal S-up Exprimn Rsuls for h Ensmbl Slcion Characrizaion of h Slcd Ensmbl of Ions... 3 Uniform Exciaion of h Slcd Ensmbl of Ions... 3 Opical Cohrnc Tim of h Slcd Ensmbl of Ions... 5 Singl Qubi Opraions Using an Ensmbl of Ions... 9 Calibraion of h Dcion Sysm... 9 Dcing h Phas of a Lasr Puls... 9 Dcing h Phas of h -Puls Echo... Dmonsraion of h loch Vcor Roaion abou h Z-axis... 4 Dmonsraion of h loch Vcor Roaion abou h Y-axis... loch Vcor Roaion abou h Y-axis wih a Singl Opical Puls... loch Vcor Roaion abou h Y-axis Using h Gomric Phas... 3 Ion-Ion Inracion... 8 Th Slcion of Two Ensmbls... 3 Inroducion... 3 Exprimn... 3 Inracion wn Two Slcd Ensmbls Inroducion Exprimn Conclusions... 4

10 ix TALE OF CONTENTS CONTINUED 7. SUMMARY Fuur Dircions REFERENCE CITED... 5 APPENDICES.58 APPENDIX A: Fiing Rouin Usd To Obain Linar Fi To Th Daa APPENDIX : Malab Cod For Roaion Around Z-axis Wih Th Gaussian Spaial ams... 6 APPENDIX C: Malab Cod For Roaion Around Y-axis Wih Th Gaussian Spaial ams... 63

11 x LIST OF TALES Tabl Pag. Th componns of h gyromagnic nsors for h xcid sa and for h ground sa manifolds o. Th fixd orinaion, 5, of h applid magnic fild and h hyprfin spliing valus for hulium ions in YAG o 3. Th fixd orinaion, 5, of h applid magnic fild and h hyprfin spliing valus for hulium ions in YAG o 4. Th fixd orinaion, 5, of h applid magnic fild and h hyprfin spliing valus for hulium ions in YAG

12 xi LIST OF FIGURES Figur Pag 5. Growh of componns on a compur chip pr yar A -Lvl Sysm FID afr xciaion wih a shor lasr puls wih dphasing, duning, and populaion dcay nglcd during h puls A V-yp sysm in a 4-lvl sysm and fw markd opical ransiions. All opical ransiions ar no markd for simpliciy A -yp sysm in a 4-lvl sysm and fw markd opical ransiions. All opical ransiions ar no markd for simpliciy A six puls squnc o achiv a rphasing in a 4-lvl sysm o masur h cohrnc bwn non-opical lvls A 4-lvl sysm wih markd opical ransiions Enrgy lvl srucur of Tm 3+ [(39)] wihou an xrnal magnic fild. Hr ransiion a 793nm and dcay from xcid sa o h inrmdia lvl ar markd Simulaion of h hol burning spcrum for a -lvl sysm. Th horizonal axis is rfrncd wih rspc o h burning frquncy Slcd nrgy lvls of Tm:YAG wih h xrnal magnic fild applid o h sampl Th hol burning spcrum from a 4-lvl sysm. Again h horizonal axis is rfrncd wih rspc o h burning frquncy Two diffrn 3-lvl sysms for quanum compuing applicaions. Hr h cross-ransiion is givn by R +-, and h dirc ransiions ar givn by R ++ and R Exprimnal valus of h magnic fild along h common axis of wo coils a h midpoin bwn wo coils Magnic fild as masurd wih h magnic prob a diffrn posiions away from h midpoin a h common axis bwn coils Exprimnal s-up for spcral hol burning, fr inducion dcay, and phoon cho xprimns Hlmholz coil as arrangd around h cryosa for applicaion of uniform magnic fild o h sampl in h cryosa On ocan of h uni cll of YAG laic is shown [(6)].... 6

13 xii LIST OF FIGURES CONTINUED Figur Pag. Trivaln hulium ions a diffrn sis in YAG crysal. Each si is assignd a uniqu local axis which is rlad o h crysal axis. Th axs ar idnifid using Millr indics, whr squar bracks rfr o dircions and angular bracks o axs, as is dfind on pag 3 of Rf. [(66)]. Picur is rproducd from Sun,. al.[(67)] Thorical plo for dirc ransiion probabiliy for ions in h Tm 3+ :YAG a diffrn sis in h crysal. In his plo h horical valus, [(55)], of h componns of h gyromagnic nsor for h ground sa, g 8.9,559.6,. MHz, and h xcid sa, Tsla.3,75.,6.3 MHz Tsla, manifolds ar usd in quaion (4.3) Thorical rsuls for cross-ransiion raios for ions a various sis in Tm 3+ :YAG and hir dpndnc on magnic fild orinaion. Th valus,g chosn for ar h sam as in Figur Thorical rsuls for cross-ransiion raios for ions a various sis in Tm 3+ :YAG and hir dpndnc on magnic fild orinaion. Th componns of gyromagnic nsors ar h sam as in Figur 9. Th magnic fild is ild from Thorical rsuls for h hyprfin spliings for ions a all six sis in h crysal whn an xrnal magnic fild of hypohical Tsla is applid o h sampl. Th plo uss h horical valus (55) of h componns of h gyromagnic nsor for h ground sa, g 8.9,559.6,. MHz, and h xcid sa, Tsla.3,75.,6.3 MHz Tsla, manifolds in quaion (4.3) Thorical rsuls for hyprfin spliings for ions a all six sis in h crysal whn an xrnal magnic fild of Tsla is applid o h sampl. Hr, h magnic fild is no rsricd o (-) plan and is ild from o h plan by il angl,, fixd a 3. Th valus of h componns of h gyromagnic nsor ar h sam as in Simulaion of h hol burning spcrum of h ions a sis 4 and 6 for h o orinaion, 34, of magnic fild giving maximum hyprfin spliings wih an applid magnic fild fixd a 567 Gauss. Th spcrum shows ha h widhs of absorpion faurs ohr han a, ar incrasd, as doubl humps appar for hs absorpion faurs which signify conribuions from ions a wo diffrn sis in h crysal

14 xiii LIST OF FIGURES CONTINUED Figur Pag 9. Exprimnal rsuls for h hol burning spcrum obaind for h Tm 3+ :YAG sampl a 4.5K. A fixd magnic fild of 75 Gauss applid o h sampl. Th orinaion of h sampl wih rspc o h magnic fild was varid in ( - ) plan by roaing h sampl in ha plan. Th opical fild orinaion was fixd along h [] dircion Exprimnal daa (circls) and horical rsuls (lins). Th componns of h gyromagnic nsor for h ground and h xcid sa manifolds ar obaind by fiing xprimnal daa using quaions (4.-4.) for diffrn valus of h componns of h gyromagnic nsor Thorical cross-ransiion raios for ions a various sis in Tm 3+ :YAG and hir dpndnc on magnic fild orinaion. In his plo, our xprimnally-obaind valus of gyromagnic nsors ar usd in quaion (4.5) Th gomry of h Tm 3+ :YAG crysal and h hom-mad moun for h crysal Posiions of h crysal moun for various orinaions Exprimnal daa (squars, diamonds, sars, circls, riangls, and crosss) showing hypfin spliings for 5. Lins ar h horical rsuls for hyprfin spliings for ions a ach si wih 5 obaind using quaions (4.-4.) and xprimnally calculad componns of h gyromagnic nsor Exprimnal daa (circls and sars) showing hypfin spliing for h orinaion giving 5. Lins ar plod using horical rsuls in quaions (4.-4.) for Exprimnal daa (riangls, diamonds, sars, and circls) showing hypfin spliing for h orinaion of h magnic fild along 65. Lins ar h horical rsuls obaind using 5 in quaions (4.-4.) Thorical plo for absorpion srngh of ions a diffrn sis conribuing o h absorpion faurs in h hol burning spcrum. Th magnic fild is along h dircion ha is -5 dgr away from <>. Th rsuls ar obaind wih h xprimnal valus of h gyromagnic nsor....

15 xiv LIST OF FIGURES CONTINUED Figur Pag 38. Exprimnal rsuls showing h hol burning spcrum from ions a diffrn sis in h crysal. Hr h 56 Gauss magnic fild was o o alignd along 5 rlaiv o <> and h orinaion of h lcric fild was varid in h (-) plan Exprimnally-masurd populaion rlaxaion bwn hyprfin lvls for ions a sis 4 and 6 in h crysal. Th crysal was orind along o o 5. Las squar fis, shown as sraigh lins o h xprimnal daa ss (circls and squars), gav us h hyprfin populaion lifims Exprimnally-masurd populaion rlaxaion bwn hyprfin lvls for ions a si in h crysal for a fixd magnic fild orinaion a o o 5. Th las squar fis rprsnd by lins o h daa ss (circls and squars) gav us h hyprfin populaion lifims Exprimnally-masurd populaion rlaxaion bwn hyprfin lvls o o a 5.5K for ions a sis 4 and 6 wih crysal orind a 5. Th xprimnal daa (circls and squars) and h las squar fis o hs daa ss ar lins giving us h hyprfin populaion lifims Exprimnally-masurd populaion rlaxaion bwn hyprfin lvls a 5K for ions a si wih crysal orind along h dircion o o 5. Each xprimnal daa is shown as a poin (circl and squar). Th las squar fi o h xprimnal daa ss ar rprsnd wih lins. Ths lins wr usd o obain h hyprfin populaion lifims Enrgy lvl diagram of Tm 3+ :YAG wih an xrnal magnic fild applid o h sampl a 4K showing h hyprfin lvls Sub-group of ions considrd for h analysis. Hr,,,, is h hyprfin spliing for h xcid sa, and 3 g is h hyprfin spliing for h ground sa for ach ion. In his figur, for simpliciy, ohr possibl opical ransiions ar no shown Exprimnal masurmns of absorpion chang a a sampl mpraur of 4.5K. Thorical rsuls (lins) ar obaind using quaions (4.37 and ) g

16 xv LIST OF FIGURES CONTINUED Figur Pag 46. High sampl mpraur, 5.5K, im-dpndn masurmns for spcral hol burning xprimn, poins, and h horical rsuls obaind using quaions (4.37, 4.4, 4.4, 4.4, and 4.43). Th magnic o o fild orinaion was 5, ha gav h highs cross-ransiion raio for ions a sis 4 and Thr puls squnc for hyprfin cohrnc im masurmns Squnc wih wo bi-frquncy pulss usd o sudy hyprfin cohrnc Fas Fourir ransform (FFT) of an xprimnally-obaind ba signal. Th signal was obaind wih wo bi-frquncy pulss using h puls squnc givn in Figur Dcay of a Raman cho obaind from a Raman cho using wo bifrquncy opical pulss as givn in Figur 44. Th las squar fi (solid lin) o h xprimnal daa (circls) was usd o obain h hyprfin cohrnc im Puls squnc applid o h mdium for obsrving h cohrn Raman bas in Tm:YAG Exprimnal daa for h cohrn Raman signal obsrvd a diffrn magnic filds from ions a sis 4 and 6. Each rac rprsns h Fourir ransform of a ba signal a a diffrn magnic fild Exprimnal daa (circls) for hyprfin spliing as obaind using o o cohrn Raman ba wih 5. Thorical rsuls ar o rprsnd by lins and ar obaind using 5 and il angl o 5 in quaions (4.-4.). Exprimnal daa giv h hyprfin spliings for h xcid sa as 5.7. MHz Tsla for ions a si 6 and MHz for ions a si Tsla 54. Exprimnal daa (circls) and horical rsuls (solid lin) for h ampliud of h FFT signal in Figur 48. Th plo shows h dpndnc of h signal ampliud on h sparaion of hyprfin nrgy lvls in a 3- lvl sysm bing xcid wih a s long -puls. Th prob puls was s long and abou 5 ims wakr han h xciaion puls A Lvl sysm Th loch sphr wih diffrn sas of a -lvl sysm mappd on h sphr.... 5

17 xvi LIST OF FIGURES CONTINUED Figur Pag 57. A puls squnc o roa and dc h roaion of h loch vcor around h z-axis A puls squnc usd o roa and masur h roaion of h loch vcor abou h y-axis for a singl puls cas A puls squnc o roa and dc h conrol roaion abou h y-axis of h loch vcor wih rlaiv phas conrol of h opical pulss Thorical rsuls for h loch vcor roaion around h z-axis using opical pulss having Gaussian spaial bam profils Thorical rsuls show h componns of h loch vcor for roaion around h z-axis. Hr, aras a h cnr of h uniform innsiy and h Gaussian spaial bam ar assumd o hav idal valus of h puls aras ndd o impar h corrc loch vcor roaion abou h z-axis Thorical rsuls show h dpndnc of h cho innsiy on h puls ara a h cnr of h bam of h conrol puls whn h mdium is xcid wih a puls having a uniform spaial profil (solid lin) and Gaussian spaial profil (dod lin) for h loch vcor roaion around h y-axis Thorical rsuls for h dcd cho powr for h loch vcor roaion around h y-axis. Th cho powr srongly dpnds on h spaial profils of h conrol pulss Thorical rsuls show h componns of h loch vcor for roaion around h y-axis. Th aras a h cnr of ach conrol puls ar () as rquird by h squnc for idal loch vcor roaion around h y-axis (Figur 55) Thorical rsuls [(5.5), (5.5)] for h fidliy dgradaion of h z-axis roaion of h loch vcor wih conrol pulss having Gaussian spaial profils Thorical rsuls [(5.5), (5.5)] for h fidliy dgradaion of h y-axis roaion of h loch vcor wih conrol pulss having Gaussian spaial profils Th nrgy lvl of ions bing addrssd wih a scan pumping chirp puls o mpy a frquncy spcrum around in Tm 3+ :YAG. Th hyprfin srucur is obaind wih an xrnal magnic fild of 44 Gauss which givs hyprfin spliing in h ground sa of MHz and hyprfin spliing in h xcid sa of MHz... 87

18 xvii LIST OF FIGURES CONTINUED Figur Pag 68. A puls squnc o mpy a spcral rnch in h inhomognous absorpion profil of Tm 3+ :YAG Thorical rsuls for h populaion in ach nrgy lvl for ion A afr a ms dcay from h Nh scan pumping chirp puls as a funcion of h numbr of rpad scan pumping chirp pulss Thorical rsuls for h populaion in ach lvl for ion afr a ms dcay from h Nh scan pumping chirp puls as a funcion of h numbr of rpad scan pumping chirp pulss Thorical rsuls for h populaion in ach lvl for ion C afr a ms dcay from h Nh scan pumping chirp puls as a funcion of h numbr of rpad scan pumping chirp pulss Thorical rsuls for h populaion in ach lvl for ion D afr a ms dcay as from h Nh scan pumping chirp puls as a funcion of h numbr of rpad scan pumping chirp pulss Exprimnal s-up for qubi prparaion and dmonsraion of singl qubi opraions Exprimnal rsuls for h spcrum of h mdium a 4K. Th spcrum shows h mpid rnch of 5MHz widh in h inhomognously broadnd absorpiv mdium. Th iniial absorpion in h mdium was L A puls squnc o slc an nsmbl of ions. Hr g is h hyprfin spliing for h ground sa manifold. An nsmbl of ions is slcd in h mpid rnch using a burn-back puls afr N rpiions of h broadband pump chirp puls Th spcrum of h mdium afr h applicaion of h puls squnc in Figur 7. Th spcrum shows burnd back ions a MHz in h mpid rnch wih a burn-back chirp applid a MHz... g 77. A puls squnc usd o iniializ h slcd nsmbl of ions o h lows nrgy Eign-sa for h dmonsraion of singl qubi opraions Th ransmission spcrum from h mdium shows h burnd back ions in h lows nrgy Eign-sa in h mpid frquncy. Sinc h numbr dnsiy of.% hulium-dopd in YAG is 9 ions/cm 3, h lasr puls of bandwidh >.5MHz wih a spo siz of abou m will b inracing wih abou 3 ions ovr a mm long sampl....

19 xviii LIST OF FIGURES CONTINUED Figur Pag 79. A puls squnc usd o characriz h slcd nsmbl of ions for is inracion wih h opical fild Th slcd nsmbl, in Figur 74, afr i was addrssd wih a singl -puls (lf) and wo -pulss (righ). Each -puls has a Rabi frquncy.3mhz a h cnr of h Gaussian spaial bam, and is puls widh was 4 ns Th dod lin is h -puls cho signal from h background ions in h rnch in Figur 7. Th solid lin is h -puls cho signal from h slcd nsmbl of ions in Figur 74. Th vrical scal in h plo is shifd o show wo signals Exprimnal rsuls for h masurd cho innsiy as a funcion of h dlay bwn h wo pulss in PE squnc. Th las squar fi o h xprimnal daa givs a cohrnc im of 35.7s a 4K Th calibraion rsuls of h phas dcion s-up usd o masur h phas of an opical signal. Th lowr plo shows h rsidual rror in ach masurmn wih a maximum rror of abou Exprimnal rsuls (circls) of phas dcion sysm for h PE squnc. Hr, h phas of h scond puls in h PE squnc was fixd, and h phas of h firs puls was varid. Th circls rprsn six singl xprimnal acquisiions; h solid lin is h xpcd horical rsul. Th boom plo shows h rror in h xprimnal daa compar o h horical rsul Th avrag (circls) of h xprimnal daa in Figur 8 and h xpcd horical rsul (solid lin). Th boom plo is h comparaiv rror Th puls squnc o dmonsra h gomric roaion abou h z-axis. Hr C and C ar h conrol pulss of ara and phas diffrnc. P and P ar par of h dcion pulss, wih aras / and and zro phass Th xprimnal dmonsraion of h gomric roaion abou h z-axis. Th roaion angl was masurd hrough phoon chos. Th solid lin is h xpcd horical rsuls, and h xprimnal daa poins ar six singl acquisiions of h xprimn akn on diffrn days wih no avraging prformd on ach daa poin.... 6

20 xix LIST OF FIGURES CONTINUED Figur Pag 88. Th avragd xprimnal daa (circls) showing h phas of h dcd cho for h roaion abou h z-axis. Th xpcd roaion angl is rprsnd as a solid lin. Th rror bars in h lowr plo show dviaion from h xpcd valu of h roaion Esima of h fidliy of singl qubi opraion for roaions around h z- axis using quaion (5.5) A puls squnc o raliz and dc h roaion around h y-axis wih a singl conrol puls, C Innsiy of h xprimnally-dcd cho afr h loch vcor around h y-axis wih a singl opical puls. Th xprimnal (circls) and horical rsuls (solid and do-dashd lins) ar normalizd o h maximum cho powr A puls squnc usd in h xprimn o dmonsra h loch vcor roaion around h y-axis. Hr, h conrol puls squnc has pulss C, C, and C 3, whil P is h dcion puls Rsuls for h roaion abou y-axis wih h conrol puls squnc whn h slcd nsmbl was iniializd o h xcid sa. Rsuls show h masurd innsiis of h dcd cho and hir dpndnc on h roaion angls for h loch vcor roaion around h y-axis. Each puls in h squnc has idal innsiis () a h cnr of h Gaussian spaial bam Rsuls for h roaion abou y-axis wih h conrol puls squnc whn h slcd nsmbl was iniializd o h ground sa. Rsuls show h masurd innsiis of h dcd cho and hir dpndnc on h roaion angls for h loch vcor roaion around h y-axis. Each puls in h squnc has idal innsiis () a h cnr of h Gaussian spaial bam A puls squnc o slc wo nsmbls of ions in h inhomognously broadnd absorpion profil in Tm:YAG Th xprimnally-masurd absorpion spcrum of h marial afr xciaion wih h puls squnc in Figur 9. Th xprimnal rsul dmonsrad h isolaion of h wo nsmbls a 4MHz apar in h inhomognous absorpion profil of h mdium wih h ions in ach nsmbl in h xcid sa

21 xx LIST OF FIGURES CONTINUED Figur Pag 97. Th xprimnally-masurd absorpion spcrum of h marial. Th rsuls dmonsra h iniializaion of h wo slcd nsmbls (nsmbl and nsmbl ) of ions o h ground sa A puls squnc usd o masur ion-ion inracion. Each puls has a MHz Rabi frquncy. Th -puls has 5ns duraion, and h -puls was wic as long in duraion. Th prurbing puls was always kp in h cnr of h PE squnc, and cho dcay was masurd as a funcion of h dlay bwn h wo pulss in h PE squnc Th xprimnally-masurd dcay of h -puls cho as a funcion of h dlay bwn wo pulss in h PE squnc

22 xxi ASTRACT Rar-arh-dopd crysals hav bn usd for opical signal procssing and sorag applicaions. In his dissraion, hir ponial for quanum compuing applicaions is xplord. In on quanum compuing schm, informaion is sord in nuclar spin sas and his informaion is hn procssd by using opical pulss hrough h coupling of hs nuclar spin sas o a common lcronic lvl. To implmn his schm, nuclar spin sas and coupling of hs nuclar spin sas o a common lcronic lvl is rquird. Prliminary work in rar-arh marials lik Pr 3+ and Eu 3+ has shown promising rsuls rgarding hir suiabiliy for quanum compuing applicaions. On paricular problm wih hs marials is ha hir ransiion wavlnghs ar only accssibl wih dy lasrs. Ths lasrs ar inhrnly unsabl, and currnly fw availabl sysms xhibi h sabiliy rquird for quanum compuing applicaions. An alrnaiv choic was o invsiga ohr rar-arh ions lik hulium. Thulium has a ransiion wavlngh ha can b accssd wih diod lasrs, which ar commrcially availabl, asy o sabiliz, and compac. This dissraion is basd on our invsigaions of Tm 3+ :YAG for quanum compuing applicaions. Invsigaions involvd a daild characrizaion of h marial. Nuclar spin sas, in Tm 3+ :YAG, wr obaind by applying an xrnal magnic fild o h sampl. Firs, inracion of an xrnal magnic fild wih h hulium ions a various sis in h crysal was analyzd. This analysis was usd o masur h magnic anisoropy in h marial. Ths rsuls show ha i is possibl, wih h suiabl choic of h magnic orinaion and h si in h crysal, o build a working 3-lvl quanum sysm. In h dmonsraion of singl qubi opraions in Tm 3+ :YAG, w firs horically sudid h ffc of Gaussian spaial bam on h singl qubi opraions. Lar on, w xprimnally prpard a singl isolad nsmbl of ions in h inhomognously broadnd absorpion profil of h mdium. This singl isolad nsmbl of ions was usd as a s-bd o implmn h singl qubi opraions. W also isolad wo nsmbls of ions in h inhomognous absorpion profil of h mdium. Th inracion bwn hs wo isolad nsmbls of ions was also sudid.

23 CHAPTER ONE INTRODUCTION Th main purpos of his work is o invsiga h rar-arh-dopd marial, Tm 3+ :YAG, for is suiabiliy in quanum compuing applicaions. Ths invsigaions includ characrizing h hulium-dopd crysal and ailoring h opraing condiions for quanum compuing applicaions. This work provids nhancd insigh ino h marial propris and suiabiliy of h marial as a s-bd for implmnaion of basic quanum compuing schms. In his chapr, an inroducion o classical and quanum compuing and h dissraion oulin ar prsnd. Classical Compuing Classical compuing and is implmnaion in modrn compurs sard wih h groundbraking work by A. M. Turing in 936 [ ()] whn h dvlopd an absrac modl of a programmabl compur. This programmabl compur is known as a Turing machin in honor of his pionr of modrn compur scinc. H also dvlopd h noion of a univrsal compuing machin ha can simula any ohr compuing machin. Turing showd ha an algorihm can b implmnd in his univrsal Turing machin o prform a ask. In gnral, any algorihmic procss can b implmnd using a univrsal Turing machin. This assrion is known as h Church-Turing hsis, in honor of Turing and Alonzo Church, anohr pionr of modrn compuing [ ()].

24 In h modrn ra, progrss achivd in compur hardwar has bn phnomnal. Th ra of compur hardwar growh was prdicd by Gordon Moor in 956 and is known as Moor s law. Moor simad ha compuing powr will doubl for consan cos roughly onc vry wo yars. Figur : Growh of componns on a compur chip pr yar. Currn growh in hardwar has followd Moor s law (Figur ), bu i is blivd ha hardwar growh will cas o follow Moor s law as convnional fabricaion chniqus ncounr h problm of sacking incrasd numbr of componns on a compur chip. On way o incras h compuing powr wihou using addiional rsourcs on h compur chip is o us a diffrn compuing chniqu: on ha is mor fficin in using h availabl compuing rsourcs. In an fficin compuing chniqu, h rsourcs ndd for compuing grow polynomially wih h siz of h problm. On h ohr hand, in h cas of an infficin

25 3 programming chniqu, compuing rsourcs grow xponnially wih h siz of h problm. Th convnional compuaion modl dfind in h Church-Turing hsis can sill b applid wih a sligh modificaion. In mid-97s, h Church-Turing hsis facd a nw challng. A ha im i was discovrd ha probabilisic algorihms ar mor fficin han h drminisic algorihms. In probabilisic algorihms, h soluion is no a dfini answr; rahr i drmins h probabiliy of a crain oucom. This ld o anohr modificaion in h Church-Turing hsis: Any algorihm procss can b simulad fficinly using a probabilisic Turing machin. This prompd many o hink ha hr may b som ohr compuaion modl ha can b usd o simula any physical sysm. This lin of hinking prompd using a compuaional modl basd on h laws of physics [ (3)]. Sinc hs laws of physics ar bs dscribd in quanum mchanics, his ld David Dusch in 985 o propos h Univrsal Quanum Compur ha was basd upon h principls of quanum mchanics [ (4)]. Quanum Compuing Th building block in quanum compuing is calld a qubi, and any -lvl sysm can srv as a qubi. Th lowr nrgy lvl,, of h wo lvls can b hough of as h sa of h qubi, and h highr nrgy lvl,, of h wo lvls can b hough of as h sa of h qubi. This dfiniion of a qubi is analogous o h dfiniion of a classical bi whr informaion is sord as s and s.

26 4 Howvr, a qubi, in addiion o hs wo sas, can also xis in a suprposiion of hs wo sas. Thus, in gnral, a qubi sa will b a b, whr a and b ar complx numbrs. A qubi is bs dscribd by a cohrn suprposiion of wo sas. This propry of h qubi lis a h har of quanum compuaion. This mans ha a qubi is prforming wo paralll compuaions a any givn im. If hr ar wo inracing qubis, hn ach qubi can xis in h suprposiion of four sas. Thus, i can prform four paralll compuaions a any givn im. Th compuaion powr of a quanum compur hus incrass xponnially wih h incras in h numbr of qubis. To raliz h full ponial of h suprposiion sa of h qubis, on has o us spcial algorihms, somims calld quanum algorihms, which incorpora quanum paralllism. Currnly fw quanum algorihms xis, hus limiing h us of quanum compuaion [ (5), (6), (7), (8), (9)]. Apar from h scarciy of fficin quanum algorihms, hr is also h challnging ask of building a pracical quanum compur. A physical sysm has o fulfill crain condiions in ordr for i o b considrd a candida for quanum compur hardwar. Ths condiions ar dscribd in h nx scion. Quanum Compuing and DiVincnzo Rquirmns for a Physical Sysm David DiVincnzo pu forward fiv basic rquirmns for a physical sysm. Th physical sysm has o fulfill hs condiions o b considrd for quanum compuing applicaions [ ()]. Ths condiions ar. A scalabl physical sysm wih wll-characrizd qubis.

27 5. Th abiliy o iniializ h sa of h qubis o a simpl iniial sa, such as A cohrnc im much longr han h ga opraion im. 4. A univrsal s of quanum gas. 5. A qubi-spcific masurmn capabiliy Th firs condiion rquirs h prparaion of a qubi ha has wo wll characrizd sas, for xampl, wo spin lvls of a spin ½ paricl, h ground and xcid sas of an aom/molcul/ion, or wo disinc polarizaion sas of a singl phoon. Th scond condiion rquirs ha h sa of a qubi can b conrolld, so ha i can b iniializd o a known valu bfor h sar of compuaion. Th hird condiion dals wih h abiliy of h qubi o sor informaion wihou much loss. This rquirs h qubi o hav minimal conac wih h nvironmn, bu i should also b abl o inrac wih ohr qubis, as pr condiion 4, and wih h masurmn apparaus, as pr condiion 5. Condiion 3, coupld wih condiions 4 and 5; highlighs h fundamnal difficuly in building a quanum compur. A quanum ga can b hough of as a uniary ransformaion ha acs on a qubi and ransforms is currn sa o a nw sa. A s of quanum gas will b univrsal if any uniary ransformaion of a qubi sa can b ralizd by combinaions of hs quanum gas. A numbr of physical sysms ar bing invsigad by ohrs in ordr o assss how wll hy fulfill hs rquirmns. Thy includ Nuclar Magnic Rsonanc (NMR) [ ()], Ion raps [ ()], Suprconducing circuis mad wih Josphson juncions [ ()], Linar opics quanum compuing [ ()], Solid sa (quanum do and spin

28 6 qubis) [ ()], Caviy quanum lcrodynamics (Caviy QED) [ ()], and Nural aoms [ ()]. Th Roadmap documn, [ ()], dscribs ach approach and compars progrss for implmnaion of quanum compuing in ach physical sysm for ach of h DiVincnzo criria [ ()]. Th radr is rfrrd o his documn and furhr rfrncs in i for grar dail on hs sysms. Rfrnc () also includs nw approachs for quanum compuing applicaions, on of which is calld h spcral hol-burning approach in rar-arhdopd crysals. Rar-arh-dopd crysals ra favorably agains h fiv rquirmns oulind by DiVincnzo. In h rar-arh-dopd crysals, dopan ions ar randomly disribud, wih ach dopan rsiding in a uniqu, shildd nvironmn. This inhomognously broadnd absorbr, a low mpraur, can provid a significan numbr of ponial candidas for qubis. In h currn sudy, h hulium-dopd marial, Tm 3+ :YAG, is invsigad for is applicaion in quanum compuing as oulind in h nx scion. Dissraion Oulin Th dissraion is dividd ino wo main pars wih h firs par daling wih h marial characrizaion of Tm 3+ :YAG and h scond par giving rsuls for nsmbl slcion for qubi prparaion and implmnaion of singl qubi opraions on his slcd nsmbl in Tm 3+ :YAG. Th dissraion sars wih h basic hory ncssary for characrizing h marial. No ha, wo main chniqus ar mployd o characriz h marial: on

29 7 uss h cohrnc ffcs in h mdium whil h ohr rlis on absorpion spcroscopy. In Chapr, rsuls for cohrn ransin ffcs arising from coupling of h opical fild wih h wo and hr lvls ar summarizd. Ths rsuls provid h framwork ndd o invsiga h marial s cohrnc propris. In Chapr 3, horical rsuls for hyprfin spliing in a spin ½ sysm ar prsnd and ransiion probabiliis for various ransiions ar dfind. Chapr 4 includs h xprimnal rsuls for characrizaion of h marial. Th characrizaion of h marial for quanum compuing applicaions ssnially rquird us o opimiz h condiions for cross-ransiion probabiliis, h populaion lifim of hyprfin lvls, and h hyprfin cohrnc im. Th horical rsuls in Chapr 3 ar usd xnsivly o characriz h marial for hs hr paramrs. Firs, w characriz h hyprfin spliing for Tm 3+ ions a diffrn sis in h hulium-dopd crysal. Characrizing h hyprfin lvls includs a sudy of h populaion lifim of hs lvls and variaion in h populaion lifim wih h orinaion of h applid magnic fild. W also invsigad h spliing of hyprfin lvls for ions a diffrn sis. Rsuls from Chapr 3 ar also usd o sudy h ransiion probabiliis for ions a hs sis for diffrn magnic fild orinaions. Ths rsuls mappd h hyprfin spliing and ransiion probabiliis for ions a diffrn sis in h crysal. W also sudid h ffc of h misalignmn of h applid magnic fild on h hyprfin spliings and h ransiion probabiliis. Various dcay ras for h marial wr modld horically, and his modl was usd o

30 8 xprimnally sudy hs rlaxaion mchanisms in h marial. Th ffc of h sampl mpraur on hs rlaxaion paramrs was also sudid. Lar, w xprimnally invsigad h hyprfin cohrnc ims for ions a diffrn sis in h crysal and for various orinaions of h magnic fild. Inroducory work dscribing h rlvan hory for a singl qubi opraion is givn in Chapr 5. I dals wih h singl qubi opraion undr idal condiions and also xnds h opraion o non-idal condiions. Ths non-idal condiions includ nonuniformiy in h opical bam innsiy. Prdicions for opraions on a singl qubi for a Gaussian spaial opical bam ar also prsnd. In Chapr 6, a mhod o slc an nsmbl of ions for dmonsraing a singl qubi opraion is dscribd. This was followd by a dscripion of his mhod for implmnaion in h xprimn. This mhod providd us a slcd nsmbl of slcd ions. Th propris of his nsmbl wr sudid using cohrnc and absorpion spcroscopy. Th slcd nsmbl of ions was usd as a s-bd o implmn h basic qubi opraions. Exprimnal rsuls from hs singl qubi roaions wr compard wih h horical prdicions for h uniform and non-uniform (Gaussian) xciaions. Finally, wo nsmbls a wo diffrn locaions in h absorpion spcrum of h mdium wr slcd. Th inracion bwn hs wo slcd nsmbls was invsigad using phoon cho xprimns. In Chapr 7, a summary of all h rsuls is prsnd. This chapr also conains h fuur dircions, on should pursu o implmn h muli-qubi opraions in Tm 3+ :YAG.

31 9 CHAPTER TWO COHERENT INTERACTION OF LIGHT WITH MULTI-LEVEL SYSTEMS Inroducion A quanum compuing schm rlis on h muli-lvl sysm. In h muli-lvl sysm som lvls ar usd o sor and manipula h populaion and h ohr lvls ar usd o sor and manipula h cohrncs in h sysm. Th daild informaion rgarding h sysm will b prsnd in Chapr 6. In his chapr, w considr h inracion of such a muli-lvl sysm wih an opical fild. Th primary purpos hr is o dscrib h inracion of an opical fild wih wo and hr opical lvls in - and 4-lvl aomic sysms. Th dscripion is carrid ou using h dnsiy marix formalism wih h inroducory hory prsnd in h nx scion. Whn an opical fild, E xe cos kz, inracs wih hs sysms, i no only rdisribus populaion bu i also cras cohrncs in h sysm [ ()]. In dnsiy marix formalism, h populaion dnsiy of ach of hs nrgy lvls is givn by h diagonal lmns of h dnsiy marix, and h opical cohrnc in h sysm is givn by Trac (.), whr is h ransiion dipol marix and is h dnsiy marix. In an inhomognously broadnd mdium, h macroscopic polarizaion consiss of conribuions from aoms/ions a diffrn dunings. Thrfor, h macroscopic polarizaion for such a sysm will b

32 P z g whr is h frquncy duning, and such ha g d., d (.), g is h lin shap of inhomognous profil Th oupu opical fild from h mdium can b obaind from h Maxwll quaion E c c rrad, rrad, z E, z P z (.3), whr c is h spd of ligh in a vacuum and is h prmiiviy of fr spac. Equaion (.3) is qui complicad and has no analyical soluion in gnral, bu i is possibl o g an approxima rsul for h spcial cas considrd hr. Hr, w considr a cas for which macroscopic polarizaion, P, is solly crad by an incidn fild, E. If h ampliuds of h incidn filds, h polarizaion, and h oupu fild ar slowly varying in im, hn h oal opical fild mid from h sysm is givn by [ ()] E T, L E cos kl sin kl o c s (.4), L L c c c, c whr, L P L,, L P L P s c s c, L xp cos kl P sin kl c s, and L is h lngh of mdium. For his paricular cas, if h macroscopic polarizaion, P, is givn, hn h rradiad fild can b asily calculad using quaion (.4). Thus, h focus in h rs of h chapr will b o find h macroscopic polarizaion for diffrn sysms. This in urn

33 involvs finding dnsiy marics for hs sysms. No ha h populaion rdisribuion manifss islf as a chang in h absorpion profil of h mdium. Inracion of a Singl Puls wih a -Lvl Sysm Firs considr a - lvl sysm as shown in Figur. Th wav funcion in h and basis is wrin as. Figur : A -Lvl Sysm Th voluion of his sa of h sysm is govrnd by h Schrodingr quaion as i H (.5), whr H is h Hamilonian of h sysm and, for a -lvl sysm, is givn as, H whr is h nrgy diffrnc bwn h wo lvls. For his siuaion i is sraighforward o solv h Schrodingr quaion (.5) and hus find h voluion of h sysm. In his cas, ach lvl will oscilla in im wih an oscillaion frquncy.

34 Now, if his sysm inracs wih h xrnal lcric fild via dipol inracion, hn h Schrodingr quaion will also hav an inracion rm, and h sysm voluion in im will b givn by h Schrodingr quaion as i H H I (.6), whr H I is h inracion Hamilonian and is givn by H I E cos L. Hr h xrnal fild, E cos for and E for E L, is assumd paralll wih h dipol momn and E is consan in im; is h coupling of h wo sas and is assumd ral, i.., [No, if is assumd complx, hn i jus inroducs a phas facor]; L is h lasr frquncy; and is h phas of h lasr puls. In his discussion, h opical cohrnc dcay, h populaion dcay, and h propagaion ffcs ar ignord. In ordr o solv h Schrodingr quaion (quaion (.6)), h wav funcion is ia ransfrrd o a nw fram ha is roaing wih h lasr frquncy as, whr A. Thrfor, h Schrodingr quaion in h nw fram will b L i H (.7), ia ia whr H H H A. I

35 3 Using h roaing wav approximaion and ignoring rms oscillaing a wic h lasr frquncy in quaion (.7), w g, i i H whr L and E. In h nw fram, H is indpndn of im. Thus quaion (.7) can b solvd and h wav funcion would b. i H Hr i H u can b wrin as sin cos sin sin sin cos,,, i i i i i i i i S S u S H S (.8), whr h marix S diagonalizs H and. Thrfor, h dnsiy marix is (.9), whr sin cos, sin sin cos i i i, sin sin cos i i i, sin, h iniial populaion is in h ground sa, and no iniial cohrncs ar prsn in h sysm.

36 4 Nx, w us h rsuls givn in his scion and prsn h rsuls for inracion of a singl opical puls and mulipl opical pulss wih a -lvl sysm. Rabi Oscillaions W follow h drivaion givn in [ ()] and considr h inracion of a singl puls wih a -lvl sysm. W us quaions (.9) and (.) o obain h cohrnc crad in a -lvl sysm wih a singl opical puls as i L i L sin i sin sin L cos L (.), Firs, considr h cas whn an opical fild is inracing wih aoms/ions on rsonanc i..,. In his cas, h opical fild cohrnly drivs h populaion bwn ground sa and xcid sa wih frquncy. This frquncy is ofn rfrrd o as h rsonan Rabi frquncy, owing is nam o I. I. Rabi who firs calculad h rsuls o dscrib h magnic fild inracions wih h nuclar spin sysms [ (3)]. For h cas whr ach ion/aom in h sysm is occupying a uniqu nvironmn and ions/aoms in h marial ar randomly disribud wih h nrgy lvl of ach ion prurbd by h rsidual srain of h crysal, h absorpion profil of h sysm is ypically Gaussian. For his cas, h macroscopic polarizaion can b obaind from quaion (.) by using h xprssions for h absorpion profil of h

37 5 sysm, g, and h cohrnc in h sysm, is givn by quaion (.). Rsulan ingral in quaion (.) is asy o valua if low bandwidh xciaion, i.., h bandwidh of h puls is smallr han h inhomognous broadning of h sysm, is considrd. For his cas, h absorpion profil of h sysm is assumd o b fla, and h scond rm in (.) will ingra o zro as i is an odd funcion of h duning,. Thn h surviving rm in quaion (.) will b h in-quadraur componn, P s, of macroscopic polarizaion and is givn by [ ()] J P T L s (.), whr J is h firs ordr ssl funcion and T dpnds on h bandwidh of h xcid ions and on h inhomognous dphasing im of h marial. Th oupu innsiy from h sysm will b J L E c I T L (.), whr c is h spd of ligh in a vacuum and is h prmiiviy of fr spac. Th rsul in quaion (.) is valid for an opical fild wih spaially uniform innsiy and i can b gnralizd o an opical bam wih a Gaussian innsiy profil as [ ()] J L I w I T L (.3),

38 6 whr w is h full widh of / maximum of innsiy, I = c (E ) /, and J is h firs ordr ssl funcion. Fr Inducion Dcay (FID) Now w follow drivaions in rfrncs [ (4), ()] and prsn h rspons of h sysm afr i was xcid wih an opical puls. In h fr inducion dcay sudy, h sysm is firs xcid wih an opical fild. Th voluion of h nsmbl of ions/aoms in h sysm is sudid a a lar im. Thrfor, in h FID sudy, h voluion of an nsmbl of aoms/ions is sudid afr h fild is urnd off. Th opical fr inducion is vry similar o h NMR fr inducion dcay ha was firs obsrvd by Hahn in 949 [ (5)]. Following h drivaions in [ (4), ()] w prsn h rsuls for wo cass. FID afr h Inracion wih a Shor Opical Puls: In his cas, a shor opical puls of duraion and Rabi frquncy such ha and is applid o h sysm saring a =. Thrfor, in his cas, h inhomognous broadning of h sysm is assumd o b smallr han h bandwidh of h puls. Hnc h duning is nglcd during h puls and h dnsiy marix, using quaion (.8), jus afr h puls will b * u,, u,,,, (.4a). Th sysm is now allowd o volv frly, and h voluion of h sysm a a lar im will b givn by

39 * u,,, u,,, 7 (.4), whr * u is h Hrmiian conjuga of u. Th macroscopic polarizaion can b calculad using quaions (.,.) and h xprssion for h Gaussian absorpion profil of h sysm. In his cas, h in-phas componn, P c, will vanish and h in-quadraur componn, P s, will b givn by T 4 P sin (.5). s In his cas, h dcay of h FID signal will dpnd on h combinaion of h bandwidh,, of h sampl bing xcid, and h marial cohrnc im, T, as shown in Figur 3. Figur 3: FID afr xciaion wih a shor lasr puls wih dphasing, duning, and populaion dcay nglcd during h puls.

40 8 Th plos in Figur 3 show ha h FID signal from h sysm having a narrowband absorpion profil, givn by dod lin, will las longr han h FID signal from h sysm having a broadband absorpion profil wih h sam marial dphasing im, T. Thrfor, h FID signal provids a ool o characriz h inhomognous widh of h nsmbl of ions. FID afr h Inracion wih a Long Opical Puls: Now considr h inracion of a -lvl sysm wih a long opical puls such ha h sysm is in a sady sa a h nd of h puls. In h cas of sady sa, h lmns of h dnsiy marix rach a consan valu and do no chang wih im. In his cas, again following h drivaion in [ ()], h in-phas macroscopic polarizaion will vanish and h in-quadraur componn will b P s T T (.6), whr TT, T is h cohrnc im of h sysm, T is h populaion dcay im of h sysm, and h inhomognous profil of h sysm is assumd o b fla. Equaion (.6) shows ha h dcay of h cohrn mission is xponnial and i dpnds on h combinaion of h cohrnc im and h populaion dcay im. Inracion of Two rif Pulss wih a -Lvl Sysm In h prvious scion, rsuls for h inracion of on puls wih a -lvl sysm wr prsnd and rlvan phnomna arising from h inracion wr brifly

41 9 discussd. Hr, h inracion of wo brif pulss wih a -lvl sysm is prsnd. In his discussion, h pulss ar considrd o b sufficinly brif so ha h duning and any ohr dcay procsss (cohrnc dcay and populaion dcay) during h puls ar ignord. In his sudy w again follow h drivaion givn in rfrncs [ (4), ()]. Th firs brif puls is assumd o b applid around im, from o. Assuming ha h iniial populaion is in h ground sa and hr ar no cohrncs in h sysm, h dnsiy marix jus afr h puls can b wrin from quaion (.9) as i i.5 sin sin sin cos i (.7), i.5 whr, is h duraion, is h phas of h puls, and i is assumd ha, T, and. Th cohrnc crad in h sysm wih puls is givn by h off diagonal lmns of h dnsiy marix in quaion (.7). Afr h applicaion of puls, h sysm will volv in im and h dnsiy marix a im will b i i T i.5 i cos sin i.5 sin i sin (.8). Now apply a scond puls of puls ara and phas from o. Thn h dnsiy marix jus afr h scond puls will b * u,, u, (.9),,

42 whr h duning and h cohrn dphasing during h puls ar nglcd bcaus h puls duraion,., is assumd o b smallr han h cohrnc im and Th inrs hr is o find h oupu fild crad by h inracion of wo opical pulss applid o h sysm a diffrn ims. As nod arlir, h rradiad fild from h sysm can b obaind using off diagonal lmns of h dnsiy marix. Thrfor, jus afr h scond puls, h off diagonal lmn of h dnsiy marix will b T cos sin sin sin i i sin i i cos i (.), whr, is h duraion, and is h phas of h scond puls. Th cohrnc s-up by hs wo opical pulss in h sysm will dphas and h cohrnc a a lar im will b T cos sin sin i i sin i i sin i i cos (.). Th analysis of h voluion of cohrnc, quaion (.), in h sysm a a lar im shows ha h scond rm in will prfcly rphas a indpndn of h duning of h individual aom/ion in h sysm. Hnc, h cohrn

43 mission from h sysm will yild an opical cho a his rphasing im, which is usually calld h -puls cho [ ()]. Inracion of a Singl Opical Puls wih Thr Lvls in a 4-Lvl Sysm Considr a 4-lvl sysm as shown in Figur 4. Th hory of h lcric fild inracion wih a 4-lvl sysm is wll-known, and for a daild dscripion h radr is rfrrd o h liraur in rfrncs [ (6), (7)]. Hr w will jus prsn rlvan rsuls from hs rfrncs. Considr Figur 4, whr lvls 3 and 4 ar opically coupld o lvl or lvl, whil hr is no dirc opical coupling bwn lvls 3 and 4 or lvls and. Considr an opical puls givn by E cos, for and E L and zro lswhr such ha. Thus, anohr cas, whr, is discussd in h nx scion. No in his cas h bandwidh of h xciaion puls is smallr han h nrgy diffrnc bwn lvls and. This singl opical puls will xci hr lvls in h 4-lvl sysm. Again, dcay procsss in h marial and h dphasing im during h puls ar ignord and his problm is solvd using similar chniqus, as in h cas of a -lvl sysm [ (6)].

44 + - = 4 = 3 g + = g - = Figur 4: A V-yp sysm in a 4-lvl sysm and fw markd opical ransiions. All opical ransiions ar no markd for simpliciy. This opical puls will cra cohrnc no only bwn h opical ransiions bu also bwn lvls 3 and 4. Assum ha h iniial populaion is in lvl. Thn h cohrnc crad bwn lvls 3 and 4 wih his singl opical puls is givn by h marix lmn of dnsiy marix in h rfrnc [ (8)] as 34 sin (.), E whr 3 4, L, 3 is ransiion dipol for ransiion from lvl o lvl 3, and 4 is h ransiion dipol for ransiion from lvl o lvl 4. No ha h cohrncs crad bwn opical ransiions ar similar o hos givn in quaion (.9) and hos cass ar no discussd hr. Hr h aim is o discuss h cohrnc crad bwn non-opical ransiions.

45 3 No ha a singl puls can only cra a cohrnc bwn lvls 3 and 4, as givn in quaion (.), if boh of h opical ransiion dipols ar non-zro. In h laboraory fram, h cohrnc bwn lvls 3 and 4 afr h puls will b [ (8)] sin T i (.3), 4 3 whr T is h cohrnc im for cohrnc bwn lvls 3 and 4. In his cas, dphasing and duning during h puls ar ignord as T and. Th cohrnc givn in quaion (.3) is h maximum if h applid puls has puls ara. No ha lvls 3 and 4 ar no opically coupld. Thrfor, w canno opically dc h cohrnc givn in quaion (.3). In ordr o prob his cohrnc and dc an opical signal, h Raman forward scaring mhod is usd [ (6), (9)]. In his mhod anohr narrowband opical puls wih h bandwidh smallr han h sparaion bwn lvls 3 and 4 is applid bwn any of h opically allowd lvls. Firs assum for our calculaions ha h opical puls is applid bwn lvls and 4. Thn his opical puls will convr h non-opical (RF) cohrnc o h opical cohrnc. Using his mhod, h macroscopic polarizaion crad in h sysm for opimum,, firs puls will b

46 P 3 c. c. 4 3, 4 3 T 4 sin E i i (.4), whr is h duning dfind arlir, E is h ampliud, and is h Rabi frquncy of h scond opical puls. Hr, h scond puls is assumd o sar jus afr h nd of h firs puls. Th rradiad fild obaind from quaion (.4) will hav frquncy whil h frquncy of h scond puls is. Thus, h oal fild will b h sum of hs wo filds, and hnc, a ba no of frquncy will b obsrvd. This ba signal is ofn rfrrd o as a cohrn Raman ba or quanum ba signal and is ofn usd o masur Sark spliing, hyprfin spliing, c. [ (), ()]. Opical Rphasing in a 4-Lvl Sysm wih Mulipl Opical Pulss In h prvious scion, h FID signal from h inracion wih a singl opical puls was usd o xplor h cohrncs in a 4-lvl sysm. In h currn scion mulipl pulss ar usd o sudy h cohrncs in h 4-lvl sysm. Hr w follow h drivaion givn in h rfrnc [ ()]. Th inracion of mulipl opical pulss wih a 4-lvl sysm can rsul in h rphasing phnomnon similar o h cas in h -lvl sysm. In currn scion w prsn h rsuls and condiions o obsrv a rphasd