Coherent Optical Control of Electronic Excitations in Wide-Band-Gap Semiconductor Structures

Transcription

1 ARL-TR-785 MAY 015 US Army Rsarch Laboratory Cohrnt Optical Control of Elctronic Excitations in Wid-Band-Gap Smiconductor Structurs by Srgy Rudin, Grgory Garrtt, and Vladimir Malinovsy Approvd for public rlas; distribution is unlimitd.

2 NOTICES Disclaimrs Th findings in this rport ar not to b construd as an official Dpartmnt of th Army position unlss so dsignatd by othr authorizd documnts. Citation of manufacturr s or trad nams dos not constitut an official ndorsmnt or approval of th us throf. Dstroy this rport whn it is no longr ndd. Do not rturn it to th originator.

3 ARL-TR-785 MAY 015 US Army Rsarch Laboratory Cohrnt Optical Control of Elctronic Excitations in Wid-Band-Gap Smiconductor Structurs Srgy Rudin and Grgory Garrtt Snsors and Elctron Dvics Dirctorat, ARL Vladimir Malinovsy Oa Ridg Associatd Univrsitis Oa Ridg, TN Approvd for public rlas; distribution is unlimitd.

4 REPORT DOCUMENTATION PAGE Form Approvd OMB No Public rporting burdn for this collction of information is stimatd to avrag 1 hour pr rspons, including th tim for rviwing instructions, sarching xisting data sourcs, gathring and maintaining th data ndd, and complting and rviwing th collction information. Snd commnts rgarding this burdn stimat or any othr aspct of this collction of information, including suggstions for rducing th burdn, to Dpartmnt of Dfns, Washington Hadquartrs Srvics, Dirctorat for Information Oprations and Rports ( ), 115 Jffrson Davis Highway, Suit 104, Arlington, VA Rspondnts should b awar that notwithstanding any othr provision of law, no prson shall b subjct to any pnalty for failing to comply with a collction of information if it dos not display a currntly valid OMB control numbr. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY) May TITLE AND SUBTITLE. REPORT TYPE DRI Cohrnt Optical Control of Elctronic Excitations in Wid-Band-Gap Smiconductor Structurs 3. DATES COVERED (From - To) Octobr 011 Sptmbr 014 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) Srgy Rudin, Grgory Garrtt, and Vladimir Malinovsy 5d. PROJECT NUMBER 5. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) US Army Rsarch Laboratory ATTN: RDRL-SEE-I Abrdn Proving Ground, MD PERFORMING ORGANIZATION REPORT NUMBER ARL-TR SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) 11. SPONSOR/MONITOR'S REPORT NUMBER(S) 1. DISTRIBUTION/AVAILABILITY STATEMENT Approvd for public rlas; distribution is unlimitd. 13. SUPPLEMENTARY NOTES 14. ABSTRACT Th main objctiv of this rsarch is to study cohrnt quantum ffcts, such as Rabi oscillations in optical spctra of widband-gap matrials, and to dtrmin th fasibility of fast optical control of quantum stats in gallium nitrid and zinc oxid htrostructurs. Bcaus of much strongr xciton-photon intraction in ths matrials as compard to gallium arsnid, it is possibl to raliz th strong coupling rgim at room tmpratur. W prform an xprimntal and thortical study on th dphasing procsss in availabl nitrid and zinc oxid matrials. This is a ncssary wor toward cohrnt optical control of quantum stats at highr tmpraturs, with ultimatly room-tmpratur cohrnt control. W also propos a schm to prform arbitrary unitary oprations on a singl lctron-spin qubit in a quantum dot. Th dsign is solly basd on th gomtrical phas that th qubit stat acquirs aftr a cyclic volution in th paramtr spac. 15. SUBJECT TERMS wid-band-gap smiconductors, cohrnt control, dphasing procsss, quantum gats 16. SECURITY CLASSIFICATION OF: a. REPORT Unclassifid b. ABSTRACT Unclassifid c. THIS PAGE Unclassifid 17. LIMITATION OF ABSTRACT UU ii 18. NUMBER OF PAGES 68 19a. NAME OF RESPONSIBLE PERSON Srgy Rudin 19b. TELEPHONE NUMBER (Includ ara cod) Standard Form 98 (Rv. 8/98) Prscribd by ANSI Std. Z39.18

5 Contnts List of Figurs List of Tabls Acnowldgmnts v viii ix 1. Objctivs Approach to Achiving th Objctivs 1 1. Progrss Mad toward Achiving th Dirctor s Rsarch Initiativ (DRI) Rsarch Objctivs 1. Tmpratur Effcts in th Kintics of Photoxcitd Carrirs in Wid- Band-Gap Smiconductors.1 Thortical Modl 3. Numrical Procdur 10.3 Exprimnt 11.4 Rsults and Discussion for th Gallium Nitrid (GaN) Cas 1.5 Summary for Gallium Nitrid (GaN) Exprimnts and Modling 3.6 Rsults and Discussion for th Zinc Oxid (ZnO) Cas 3 3. Gomtric Singl-Qubit Quantum Gats for an Elctron Spin in a Quantum Dot Introduction 7 3. Gnral Equations of Motion Nonimpulsiv Cas Adiabatic Solution Bloch Vctor Rprsntation Evolution Oprator of th Bloch Vctor Ultrafast Qubit Rotations Using Gomtrical Phas Rotation in th Bloch Rprsntation Gnralization of th Singl-Qubit Opration Using Bright-Dar Basis Elctron Spin in a Quantum Dot as a Qubit Possibility to Raliz Gomtric Gats Using Nitrid Structurs 50 iii

6 4. Conclusions Rfrncs Publications and Prsntations for this DRI 53 Distribution List 55 iv

7 List of Figurs Fig. 1 Fig. Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8 Fig. 9 Fig. 10 Constant nrgy surfacs of (a) GaN conduction band at an nrgy of 750 mv and (b) GaN valnc band at an nrgy of 100 mv of GaN at 300 K...3 Calculatd polar optical scattring rat for th nonparabolic conduction band in GaN including th scrning ffct. Th solid lins show th phonon mission rat and th dashd lins show th phonon absorption rat. Th lctric fild of th light is Vm 1 and th xcss photon nrgy is 750 mv....9 A schmatic rprsntation for tim-rsolvd photoluminscnc through optical gating by frquncy down-convrsion. Excitation lasr provids pulss tunabl from 3.6 to 5.33 V, suitabl for studying GaN with gnratd lctron-hol pairs up to 10 0 cm Calculatd tim volution of th lctron (a) and hol (b) numbr dnsity functions at room tmpratur...13 (a) Calculatd ffct of scattring mchanisms on th intgratd dnsity of lctrons in GaN vs. lctric fild of th xtrnal light at 300 K whn th xcss photon nrgy is 50 mv. (b) Calculatd ffct of scattring mchanisms on th intgratd dnsity of lctrons in GaN vs. lctric fild of th puls at 300 K whn th xcss photon nrgy is 750 mv.15 Calculatd normalizd lctron dnsity vs. tim at room tmpratur in GaN. Th lctric fild of th light puls is 10 8 Vm 1 for valus of xcss photon nrgy. Ths curvs hav bn computd only considring th cohrnt ffcts as dscribd in Eqs., 3, and 4, and nglcting th dphasing du to scattring mchanisms. Th valus ar normalizd to th maximum of th curv corrsponding to 750 mv...16 Luminscnc spctra in GaN at room tmpratur (300 K) calculatd for diffrnt nrgy rgions...18 (a) Luminscnc spctra in GaN at 150 K collctd at diffrnt nrgy rgions. Th xcss photon nrgy is 750 mv. (b) Luminscnc spctra in GaN at 50 K collctd at diffrnt nrgy rgions. Th xcss photon nrgy is 750 mv. Th lctric fild of th xtrnal light puls is V m 1 in both cass Luminscnc spctra of GaN at diffrnt tmpraturs collctd at band dg nrgy rgion. Th xcss photon nrgy is 750 mv. (a) shows th xprimntal rsults and (b) prsnts th simulatd rsults for an lctric fild of th xtrnal light puls of V m Luminscnc spctra of GaN at diffrnt lasr powrs collctd at band dg nrgy rgion. Th xcss photon nrgy is 750 mv and th lattic tmpratur is 150 K. (a) shows th xprimntal rsults and (b) prsnts th simulatd rsults. In (a), E rprsnts th baslin powr usd in th xprimnt. In (b), E is th baslin valu of lctric fild mployd in th simulation....1 v

8 Fig. 11 Fig. 1 Fig. 13 Fig. 14 Fig. 15 Fig. 16 Fig. 17 Fig. 18 Fig. 19 Fig. 0 Fig. 1 Fig. Fig. 3 Calculatd luminscnc spctra of GaN at diffrnt tmpraturs collctd at band dg nrgy rgion. Th xcss photon nrgy is 50 mv and th lctric fild of th xtrnal light puls is V m 1. Th intgratd luminscnc spctra of GaN collctd up to 8 ps vs. normalizd lasr powr collctd at band dg nrgy rgion. Th xcss photon nrgy is 750 mv and th lattic tmpratur is 150 K.... Calculatd polar optical scattring rat for th nonparabolic conduction band in ZnO including th scrning ffct. Th solid lins show th phonon mission rat, and th dashd lins show th phonon absorption rat. Th lctric fild of th light is Vm 1, th xcss photon nrgy is 380 mv, and th lattic tmpratur is 300 K....4 Th rsult of th normalizd luminscnc intnsity from ZnO obtaind by xprimnt (a) and simulation (b). Th xcss photon nrgy is 380 mv, and th lattic tmpratur is 300 K....5 Th rsult of th normalizd luminscnc intnsity from ZnO obtaind by xprimnt (dashd lin) and simulation (solid lin). Th xcss photon nrgy is 40 mv, th lctric fild of th lasr puls is Vm 1, and th lattic tmpratur is 75 K....6 Th normalizd luminscnc intnsity from ZnO at diffrnt tmpratur valus calculatd at band dg nrgy rgion. Th xcss photon nrgy is 380 mv, and th lctric fild of th xtrnal light puls is V m Enrgy structur of th 3-lvl systm comprising th lctron spin stats and th trion stat...8 Th dnsity plot of th 1 stat population (a) and cohrnc (b) as a function of th ffctiv puls ara and frquncy chirp; P, 0. S Initially, only th 1 stat is populatd Th dnsity plot of th 1 stat population (a) and cohrnc (b) as a function of th ffctiv puls ara and frquncy chirp; P, S Initially, only th 1 stat is populatd Th Bloch vctor rprsntation of th qubit stat. Excitation of th qubit by an xtrnal fild corrsponds to th rotation of th B vctor about th psudo fild vctor, Ω, with componnts dtrmind by th ffctiv Rabi frquncy () t, dtuning, and th rlativ phas. 37 Th Bloch vctor trajctory for th qubit stat 0 in panl (a) and th qubit stat 1 in panl (b) gnratd by th squnc of -pulss with th rlativ phas...41 Th Bloch vctor trajctory for th qubit stat i in panl (a) and th qubit stat i in panl (b) gnratd by th squnc of /-pulss and on puls with th rlativ phas...4 Optical slction ruls in diffrnt bass: (a) th mixd basis is usd, whr th lctron spin stats ar in th x basis whil th trion stats ar in th z basis; (b) coupling schm is in th x basis vi

9 Fig. 4 Th population dynamics of th rsonant qubit stats with (dottd lins) and without (solid lins) adiabatic limination of th trion stat in th 3- lvl systm. Th xcitation is gnratd by th squnc of pairs of pulss (Gaussian puls nvlops) with th rlativ phas /...49 vii

10 List of Tabls Tabl 1 Paramtrs usd in th simulation for GaN...13 Tabl Paramtrs usd in th simulation...4 viii

11 Acnowldgmnts Parts of th wor wr prformd in collaboration with Enrico Bllotti and Sara Shishhchi (Boston Univrsity). W than Michal Wrabac for usful discussions. ix

12 INTENTIONALLY LEFT BLANK. x

13 1. Objctivs Th main objctiv of this study is to xamin cohrnt quantum ffcts, such as Rabi oscillations and quantum ntanglmnt in optical spctra of wid-band-gap matrials, and to dtrmin th fasibility of fast optical control of quantum stats in gallium nitrid (GaN) and zinc oxid (ZnO) htrostructurs. Bcaus of much strongr xciton-photon intraction in ths matrials as compard to gallium arsnid (GaAs), it is possibl to raliz th strong coupling rgim at room tmpratur. W prform an xprimntal and thortical study on th dphasing procsss in availabl nitrid and zinc oxid matrials. This will allow us to ta advantag of thir uniqu proprtis for obsrving cohrnt quantum ffcts at tmpraturs highr than th cryognic, at liquid nitrogn tmpratur and highr. This is a ncssary wor toward cohrnt optical control of quantum stats at highr tmpraturs, with ultimatly room-tmpratur cohrnt control. 1.1 Approach to Achiving th Objctivs W usd tim-rsolvd photoluminscnc (TRPL) to study th volution of carrir dnsity and intrband polarization in GaN and ZnO matrials. Our modling of ultrafast optical rspons from smiconductor structurs is basd on th dnsity matrix formalism. Th dtaild numrical solution rquirs a gnralizd Mont Carlo simulation. An important stp toward cohrnt control of lctronic xcitations in GaN htrostructurs will b an obsrvation of Rabi oscillations in th diffrntial transmission signal. In th wid-band-gap matrials li GaN and ZnO, th xciton transition oscillator strngth is at last an ordr of magnitud largr than in GaAs; thrfor, th strong coupling rgim can b achivd at significantly highr tmpraturs. 1. Progrss Mad toward Achiving th Dirctor s Rsarch Initiativ (DRI) Rsarch Objctivs W prformd thortical and xprimntal studis of th sub-picoscond intics of photoxcitd carrirs in GaN and ZnO. In th thortical modl, intraction with an xtrnal ultrafast lasr puls is tratd cohrntly, and to account for th scattring mchanisms and dphasing procsss, a gnralizd Mont Carlo simulation is usd. Th scattring mchanisms includd ar carrir intractions with polar optical phonons and acoustic phonons, and carrir-carrir Coulomb intractions. W studid th ffcts of various scattring mchanisms on th carrir dnsitis and also prsntd th tmpratur and puls-powr-dpndnt luminscnc spctra. Th rsults ar prsntd ovr a rang of tmpratur, 1

14 lctric fild, and xcitation nrgy of th lasr puls. Hr w rport th xprimntal tim-rsolvd photoluminscnc studis of GaN and ZnO sampls. W also xplain th intics of th photoxcitd carrirs by including only carrircarrir and carrir-phonon intractions and a rlativly simpl -band lctronic structur modl. Th analysis and modling of polarization volution and cohrnt oscillations in GaN quantum wlls will rquir an inclusion of carrir confinmnt ffcts in th quantum intic modl. Toward this goal, w hav includd polarization filds and strain in th.p modl of lctron and hol stats in GaN quantum wlls as part of our thory of radiativ rcombination rats in nitrid htrostructurs. Th quantum confinmnt of lctron stats is rquird for ffctiv optical control. Toward that goal, w modld th volution of th spin of quantum-confind lctrons, providing a clar gomtrical intrprtation of th qubit dynamics. Using th analytic form of th volution oprator, w proposd a st of singlqubit rotations that is solly basd on th gomtrical phas. W dmonstratd th adiabatic manipulation of a qubit using only th gomtric phas. This has som advantags, sinc it rducs th rquirmnts of prfct tuning of th controlld paramtrs and is significantly mor robust against nois.. Tmpratur Effcts in th Kintics of Photoxcitd Carrirs in Wid-Band-Gap Smiconductors Th intics of photoxcitd lctrons and hols in GaN and ZnO wr studid by sub-picoscond tim-rsolvd photoluminscnc. Th ris tim of th carrir rcombination was analyzd as a function of sampl tmpratur, from 14 K to room tmpratur, and th xcitation powr. In on st of masurmnts, th xcitation wavlngth was chosn such that th xcss photon nrgy was 750 mv abov th band gap, wll abov th polar optical phonon nrgy. In anothr st of masurmnts, th xcss photon nrgy was 50 mv, blow th polar optical phonon nrgy. Th goal was to gain a bttr undrstanding of th scattring and dphasing procsss that accompany th initial photoxcitation of carrirs. Rsults showd littl dpndnc in th ris tim on th initial carrir dnsity but strong dpndnc on th sampl tmpratur, with th ris tim incrasing by mor than 1 ordr of magnitud as tmpratur dcrass from room tmpratur down to 14 K. To lucidat th findings, simulations wr prformd basd on a -band modl and using a Mont Carlo tchniqu. In th modl, intraction with an xtrnal ultrafast lasr puls is tratd cohrntly, and to account for th scattring mchanisms and dphasing procsss, a gnralizd Mont Carlo simulation is usd. Th scattring mchanisms includd ar carrir intractions with polar optical phonons and

15 acoustic phonons, and carrir-carrir Coulomb intractions. W find rasonabl agrmnt btwn th simulation and xprimnt..1 Thortical Modl In this wor w considr a bul wid-band-gap smiconductor matrial dscribd by an isotropic parabolic havy hol band and an isotropic nonparabolic conduction band. 1 Considring th nonparabolicity factor for th conduction band incrass th accuracy of th band structur. For th valnc band, w just considr th havy hol band. Within th nrgy rgions considrd in this wor, hols will b confind to th havy hol band bcaus of its highr ffctiv mass, which rsults in a highr dnsity of stats compard to that of th light hol band. Sinc th xcitation nrgis that w considr in this wor ar lowr than th scondary minima, as shown in Fig. 1, th particls will not b acclratd or xcitd to nrgy stats outsid th Γ vally. Fig. 1 Constant nrgy surfacs of (a) GaN conduction band at an nrgy of 750 mv and (b) GaN valnc band at an nrgy of 100 mv of GaN at 300 K Figur 1a shows th nrgy surfacs for th conduction band in GaN at an nrgy of 750 mv abov th conduction band minima, and Fig. 1b shows th nrgy surfacs for th valnc band in GaN at 100 mv blow th valnc band maxima, calculatd using an mpirical psudopotntial modl. From ths figurs w can conclud that this simpl -band modl is a good approximation of th ralistic lctronic structur for th rang of nrgis considrd in this wor. Howvr, th mthod can b xtndd to multipl vallys or full band structurs. 3

16 Th Hamiltonian dscribing th systm contains parts: H 0, which includs th parts that can b tratd xactly using a singl particl modl, and H 1, which contains various intractions and dphasing procsss that will b tratd with som lvl of approximations. In this wor w considr a numbr of dphasing procsss that ar du to th intraction of carrirs and intrband polarization with polar optical and acoustic phonons, and th carrir carrir intractions via Coulomb potntial. W dscrib th dynamics of th systm using th dnsity matrix mthod.,3 Th singl particl Hamiltonian that dscribs th intraction of fr carrirs with th incoming light fild and fr phonons is givn by H 0 ( ) c c h ( ) d d ilt i t M E t c d M E t d c L 0 ( ) 0 ( ) ( q) b q b q, (1) whr ϵ (1 + αϵ ) = ħ /(m) and ϵh = ħ /(mh ) ar th non-parabolic and parabolic nrgy stats for lctrons and hols, and m and mh ar lctron and hol ffctiv masss, rspctivly. Oprators c +, d +,(c, d) ar cration (annihilation) oprators of lctrons, hols with Frmi anticommutation rlations. Oprators b and b + ar phonon oprators with Bos commutation rlations. ωq is th phonon frquncy, M is th intrband dipol matrix lmnt, and E0(t) is th amplitud of th Gaussian-shapd lasr puls with a frquncy cntrd at ω L and dfind as E0 (t) = E L xp( t /τ ), whr E L is th pa lctric fild and τ can b rlatd to th full width at half maximum of th masurd fild intnsity by τ = τ I /( ln). Th xtrnal light fild is tratd smi-classically within th rotating wav approximation. 4 Th optical dipol matrix lmnt in th parabolic bands approximation is givn by M = dcv(1+ ħ /mreg) -1, which can b approximatd by a constant, M dcv, for larg band gap valus Eg. Th rducd mass is dfind by mr = (m 1 + mh 1 ) 1. Th intrband lctric dipol dcv is rlatd to intrband momntum matrix lmnt dcv/ = ħpcv/egm0 and in th ffctiv mass approximation for -band modl dcv = [( 1+m0/m)/Egm0)] 1/, from which w obtain in GaN cas dcv/.1 Å. To dscrib th intics of th systm, w nd to obtain th distribution function of th carrirs and polarization as wll as th rat of chang of ths functions. d Using th Hisnbrg quations of motion, i Oˆ ( t) O ˆ, H for th lctron and dt hol oprators and incorporating appropriat commutation rlations, w obtain th quation of motion for particls and polarization that rprsnt th gnration rat for a nonintracting systm: 4

17 5 ) ( t g dt df dt df h, () h t i h f f t E M p i dt dp P 1 ) ( ) ( ) ( 0 0 1, (3) with th gnration rat p t E M p t E M i t g t i t i P P ) ( ) ( ) ( , (4) whr f, f h and p ar th distribution functions of lctrons, hols, and polarization, rspctivly: ) ( ) ( ) (, ) ( ) ( ) (, ) ( ) ( ) ( t c t d t p t d t d t f t c t c t f h, with bracts dnoting th xpctation valu. Various intractions among th particls ar introducd in th modl through a prturbing Hamiltonian. For ach intraction mchanism, w considr a Hamiltonian H 1 i, whr i rprsnts th spcific intraction mchanism. In this wor w considr polar optical and acoustic phonons, and carrir carrir intractions. A numbr of suitabl approximations ar also includd to ovrcom th difficultis inhrnt in th many-body natur of ths intractions. Th carrir carrir intraction Hamiltonian is givn by,p,q q q p p q q p p q q p p c d d c d d d d c c c c V H q cc 1 1 1, (5) whr th first and scond trms ar th rpulsiv lctron lctron and hol hol intractions, and th third trm is th attractiv intraction of lctrons and hols, and Vq is th scrnd Coulomb potntial. Th first-ordr approximation in th Hisnbrg quations of motion lads to a rnormalization of th carrir nrgis by a slf-nrgy factor and rnormalization of th light fild by an intrnal fild. Aftr obtaining th first-ordr corrction, using th Hisnbrg quation of motion again, on can driv th quations of th scond-ordr contribution of carrir-carrir intraction. Th quation for xpctation valus of 4 oprators involvs th strings of 6 oprators. W us factorizations to truncat ths chains of quations. 3 Th Marov approximation 3,4 is usd to avoid tim intgral quations and to nglct th rtardation ffcts. This approximation also implis that th lctric fild and th distribution functions vary slowly compard to th oscillation of th xponntial trms of th lasr puls; thrfor, it liminats th ffct of initial

18 corrlations and dynamic mmory. Furthrmor, th validity of this approach is limitd to th rang of carrir dnsitis considrd in this wor and may nd to b rvisd for highr dnsitis. In this wor w considr th scond-ordr contribution of th carrir carrir intraction. Th rsult of th scond-ordr carrir carrir intraction xcluding th Hartr Foc trms ar givn in Eqs. 6 and 7, whr α =,h: df cc() dt ( cc) i( cc) W -q, f W, -q f -q, (6) dp dt cc() p W -q, p W q p, -q p -q, (7) whr th carrir carrir transition matrics ar givn by α,cc W q, = π ħ V q α D (ε q α α ε α α f q ) [f α (1 f +q α + ε α ε +q ) (1 ) p +qp ] + c. c., (8) and th polarization transition matrics ar givn by W p,cc q, = π ħ V q α D(ε q α,α α ε α α + ε +q ε α ) [ p +q + f α (1 f ) (1 f +q q ) + f q α α α f +q p (1 f α )]. (9) Th function D (ϵ) is dfind as follows: whr P dnots principal valu. D(ω) 1 P iπ ω + δ(ω), (10) Bcaus of th prsnc of fr carrirs, th Coulomb potntial is scrnd. W us a static scrning potntial V q 1, q 0 s (11) whr ϵ0 and ϵs ar th vacuum prmittivity and static dilctric constant, rspctivly. Th scrning wav vctor is calculatd according to th following quation: 6

19 V s 0, ( ) 1 f, (1) whr ϵ() is th carrir nrgy stat, and V is th volum of th matrial. Bsids th intraction of carrirs with ach othr, in this wor w ta into account th intraction with polar optical and acoustic phonons as wll. Th structur of th Hamiltonian dscribing ths intractions is indpndnt of th typ of th phonon branch. Th only diffrnc is th intraction matrix lmnt that dscribs th typ of lattic dformation, in th cas of acoustic phonons, and th long-rang lctric fild whn polar optical mods ar considrd. Th Hamiltonian that dscribs th carrir-phonon intraction is H cp h h 1 qcqbqc q c bq cq q dqbqd q d bq dq, (13),q whr γ ;h q is th intraction matrix lmnt. Bcaus of th opposit charg of lctrons and hols, th matrix lmnts in th cas of a polar intraction ar γ q = γ h q = γq. Th procdur of obtaining th scond-ordr contribution of th carrir phonon intraction to th quations of motion is similar to that of carrir carrir intraction. W driv Hisnbrg quations of motion using th Marov approximation. Th scond-ordr contribution to th quations of motion, xcluding th intrnal fild trms, is th sam as Eqs. 6 and 7; howvr, instad of W ;h(cc) and W p(cc), th contribution of phonons W ;h(cp) and W p(cp) must b substitutd. Th carrir phonon scattring rats ar givn by α,(cp) = π γ ħ q δ(ε α α ε q ± ħω q ) (N q + 1 ± 1 ± ) (1 f q α ), (14) W q, and th polarization transition matrics ar givn by W p,(cp) q, = π ħ γ q D(ε α α ε q α=,h ± + (N q + 1 ± 1 ) (1 f q α )], ± ħω q ) [(N q ) f α q (15) whr Nq is th numbr of phonons with wav numbr q at a spcific tmpratur, which is calculatd using th Bos Einstin distribution and ωq is th phonon frquncy. To obtain th distribution functions, th quations ar sparatd into parts, a cohrnt part and a dphasing part. Th cohrnt contribution includs th gnration quations that ar calculatd according to Eqs. 3 and 4 for particls and carrirs, rspctivly. Th valu of th distribution functions in th cohrnt cas 7

20 can b obtaind by dirctly intgrating th gnration quations. A gnralizd Mont Carlo mthod, 3 which is vry similar to th traditional Mont Carlo mthod, is usd to account for th dphasing phnomna. Ths distribution functions ar thn usd to calculat th luminscnc spctrum. Th rat of th scattring mchanisms mployd in th Mont Carlo procdur is valuatd using th Frmi goldn rul. On of th most important scattring mchanisms is th carrir carrir intraction, spcially in GaN with carrir dnsitis as high as cm 3 as considrd in this wor. At this rang of carrir dnsitis, th scrning ffct is ssntial both for th carrir carrir scattring and carrir phonon scattring. In fact, ach carrir, whil intracting with othr carrirs through th scrnd Coulomb potntial, is also influncd by th scrning potntial in its intractions with phonons. Thrfor, th carrir phonon intraction is wand considrably by th scrning potntial. Th scrnd carrir carrir scattring is calculatd basd on th modl, 5,6 which tas into account th static scrning wav vctor as shown in Eq. 1. Th matrix lmnt of th intraction of th lctrons with th polar optical phonons via th Frohlich intraction is as follows 7 : q 0 LO 1 1 s q q I, q), (16) whr ϵ is th high-frquncy dilctric constant, q is th wav vctor of phonons, ω LO is th polar optical phonon nrgy, and I(, ) is th ovrlap intgral for th conduction band. 7 For th hols, w considr th ordinary polar optical scattring rat for parabolic band structurs. Th calculatd polar optical scattring rat at room tmpratur, including th ffct of scrning, is shown in Fig.. 8

21 Fig. Calculatd polar optical scattring rat for th nonparabolic conduction band in GaN including th scrning ffct. Th solid lins show th phonon mission rat and th dashd lins show th phonon absorption rat. Th lctric fild of th light is Vm 1 and th xcss photon nrgy is 750 mv. Th xcss photon nrgy th amount of photon nrgy aftr subtracting th band gap nrgy is 750 mv, and th lctric fild of th xtrnal light is Vm 1. Th solid lins show th phonon mission rats, and th dashd lins show th rat for th phonon absorption procsss. Th blac lins (topmost solid and dashd lins) show th calculatd scattring rat at 100 fs. At this tim, th gnration procss has just startd, and th dnsity of lctrons in th systm is cm 3. Th blu lins (cntr solid and dashd lins) show th scattring rats at 0 fs whn th maximum of th lasr puls occurs and th dnsity of lctrons is cm 3. Bcaus of th incrasing carrir dnsity rsulting from a highr gnration rat, th scrning wav vctor changs dynamically as th simulatd systm volvs. Thrfor, th polar optical scattring will b scrnd, and vntually th scattring rat will dcras. This ffct is mor pronouncd, as shown by th rd lins (bottom solid and dashd lins), whn th tim is 100 fs and mor carrirs ar gnratd. In this cas, th dnsity of lctrons is cm 3, and th lctron phonon scattring rat is at its lowst valu. Without th scrning ffct, th polar optical scattring rat will not chang during th simulation run. Th dformation potntial acoustic scattring rat for isotropic parabolic and nonparabolic bands ar calculatd basd on th mthod prsntd in Jacoboni and Lugli. 5 For th parabolic bands, only longitudinal mods contribut to th scattring. Th scattring rat is calculatd considring th acoustic dformation 9

22 potntials (D and Dh), longitudinal sound vlocity (V L), dnsity of th matrial (ρ), and th phonon numbr, nq, which is calculatd basd on Bos Einstin distribution. For th nonparabolic conduction band, th avrag valu of th longitudinal and transvrs sound vlocity (V T ) is usd. Th transfrrd nrgy is calculatd basd on nrgy and momntum consrvation. Th matrial paramtrs that hav bn usd in this wor ar listd in Shishhchi t al. 1,8. Numrical Procdur This sction dscribs th numrical approach 1 usd to valuat th distribution functions basd on th modl prsntd in Sction.1. Basd on this modl, th nsmbl of th carrirs is dscribd by mans of a numbr rprsntation. This mans that th numbr of supr particls rprsnting th carrirs is dfind at ach cll in spac; thrfor, th occupation of ach cll is dcrasd or incrasd by on, dpnding on th mchanism that tas plac. 9 Furthrmor, th maximum occupancy of ach cll is limitd by th dnsity of stats. Th distribution function is thn dfind by dividing th occupation numbr of ach cll by th dnsity of stat. Th distribution functions of lctrons and hols ar ral variabls, whras that of th polarization is complx. Consquntly, th magnitud and th phas of polarization ar assignd to a -vctor matrix. Th simulation procss initiats with th vacuum stat of th distribution of carrirs and polarization; in othr words, th distribution functions and polarization ar qual to zro. Subsquntly, th nsmbl of th carrirs and polarization volvs as tim lapss as a rsult of th lasr puls taing th systm out of quilibrium. Th simulation priod is dividd into a numbr of tim stps. At ach tim stp, first th cohrnt part is calculatd. Th gnration rat for th carrirs is valuatd by a dirct analytical intgration, and th tim volution of th polarization is obtaind by intgrating th coupld diffrntial Eqs. and 3 mploying a Rang-Kutta algorithm. Th obtaind distributions until this point ar thn transformd into a many-body particl distribution, which can b fficintly usd to implmnt th nsmbl Mont-Carlo calculation. Spcifically, w us N supr-particls for ach intic variabl. Ths particls ar gnratd randomly according to th distribution function calculatd during th cohrnt part of th tim stp. Sinc th polarization distribution function is complx, th phas is discrtizd according to th complx valus of p. Nxt, th scattring rats ar calculatd. Bcaus of th dpndnc of th scrning lngth on th distribution functions, which ar updatd at ach tim stp, scattring rats should b rcalculatd onc th nw distributions ar availabl. Thn, a traditional Mont Carlo simulation is prformd using th nwly calculatd scattring rats. Aftr 10

23 ach scattring vnt, th numbr of supr particls in th initial cll is dcrasd by on, and th numbr in th final cll is incrasd by on. In this wor, w also includ th ffct of th Pauli xclusion principl. Although th Mont Carlo mthod trats th lctrons and hols as smi-classical particls, thy should oby th Pauli xclusion principl, spcially at high carrir dnsitis whr th distribution of th particls is highly influncd by thir frmionic proprtis. 10 Aftr th scattring procss, onc th final stat is dfind, a random numbr btwn 0 and 1 is gnratd to valuat whthr th transition is accptd or rjctd. 11 In this mthod, th distribution function, fi, at ach cll, i, in th spac is compard to th random numbr, r. If r > fi, th transition is accptd and th occupancy of that cll is incrasd by on. Othrwis, th transition is dnid and th particl gos through a slf-scattring vnt. Evntually, th distribution function is updatd according to th xclusion principl aftr scattring by ach mchanism..3 Exprimnt A schmatic rprsntation of th xprimntal stup is shown in Fig. 3. A 50-Hz rgnrativly amplifid Ti:Sapphir lasr producs 150-fs pulss at 800-nm wavlngth. Part of this bam is split off, acting as a gating puls, and th othr part pumps an optical paramtric amplifir tunabl from 760 to 465 nm. Output from th amplifir is again doubld in a nonlinar optical crystal (Xtal) to produc pulss tunabl from 380 to 3.5 nm. Th pump puls, comprssd to a width of approximatly 80 fs, is focusd onto th sampl whr it photo-xcits lctron-hol pairs. Th luminscnc from th sampl is collctd using nondisprsiv optics and focusd on anothr nonlinar crystal (NLXtal), whr it is mixd with th gating puls to yild a down-convrtd signal that is spatially and spctrally sparatd from th othr bam. Th down-convrtd signal is passd through a spctromtr and dtctd with a photomultiplir tub in photon counting mod. Th sampl usd in this xprimnt was a -μm-thic htropitaxial GaN film dpositd by mtalorganic chmical vapour dposition (MOCVD) with a fully gradd aluminum gallium nitrid (AlGaN) layr on bul aluminum nitrid (AlN). Th arrival tim of th nar-ir gat puls rlativ to th UV photoluminscnc maps th TRPL with 350-fs rsolution as dtrmind by th group vlocity disprsion btwn th bams in th nonlinar crystal. Th nrgy rang of th carrir distribution monitord is a function of th accptanc angl of th phas matching condition in th nonlinar crystal. 11

24 Fig. 3 A schmatic rprsntation for tim-rsolvd photoluminscnc through optical gating by frquncy down-convrsion. Excitation lasr provids pulss tunabl from 3.6 to 5.33 V, suitabl for studying GaN with gnratd lctron-hol pairs up to 10 0 cm 3..4 Rsults and Discussion for th Gallium Nitrid (GaN) Cas In this sction w prsnt and compar th thortical and th xprimntal rsults for th cas of GaN. In particular, w will loo at th tim volution of th lctron and hol dnsitis as a function of th lctric fild strngth and intnsity of th xtrnal puls. Th matrial paramtrs ar listd in Tabl 1. W prsnt th ffct of various scattring mchanisms on ths rsults, with particular mphasis on th analysis of th phnomna that lad to th disruption of th cohrnc in th systm. In this contxt, to highlight th rol of polar optical phonon scattring, w considr xcitation nrgis: on with an xcss photon nrgy of 750 mv, which is highr than th polar optical phonon nrgy (91. mv), and th othr on with xcss photon nrgy of 50 mv, which is lowr than polar optical phonon nrgy. For th cas of xcitation with th lowr nrgy, longitudinal optical (LO) phonon mission procsss ar significantly supprssd and do not driv th nrgy rlaxation in th systm. Finally, w considr th calculatd luminscnc for both valus of th xcss photon nrgis. Morovr, w study th ffct of tmpratur on th luminscnc spctra and compar it with th masurd valus. 1

25 Tabl 1 Paramtrs usd in th simulation for GaN Figur 4a shows th calculatd tim volution of th lctron numbr dnsity function at room tmpratur whn th xcss photon nrgy is 750 mv and th lctric fild of th lasr puls is V m 1. Th numbr dnsity function n(e) is dfind as th product of th dnsity of stats and th carrir distribution: n(e) = g(e) f(e). Figur 4b provids th sam information for hols. Th nrgy in (a) is masurd from th bottom of th conduction band, and th nrgy in (b) is masurd from th top of th valnc band. Fig. 4 Calculatd tim volution of th lctron (a) and hol (b) numbr dnsity functions at room tmpratur 13

26 Th tim scal has bn chosn so that th pa of th Gaussian lasr puls occurs at 0 fs, and th calculatd lctric fild intraction xtnds from 00 to 00 fs. Th particl distribution rsults ar shown at th instants corrsponding to 50 fs, 0 fs, 50 fs, 100 fs, 150 fs, and 500 fs. Th dashd blac lin shows th normalizd lctron dnsity at 50 fs. At this tim stp, carrirs ar bing gnratd and, as th tim lapss, th numbr of th carrirs dnsity incrass. Sinc th xcss photon nrgy of 750 mv is dividd btwn th conduction and valnc bands (basd on th rspctiv dnsity of stats), w obsrv that th maximum of th lctron distribution is blow th xcss photon nrgy. For hols, th pa of th distribution occurs just blow 00 mv. As th tim lapss, th scattring procsss rlax th carrirs nrgy and chang th shap of th distribution. Th dot-dashd grn lin shows th rsults at 0 fs th tim at which th lasr puls intnsity is maximum. At this tim, anothr pa mrgs at an nrgy of about 91 mv, which corrsponds to th optical phonon nrgy. Th pa is du to th polar optical phonon scattring mission mchanism that cannot occur blow this nrgy. At latr tims, 100 and 150 fs, th shouldr at 91 mv bcoms mor and mor pronouncd. Evntually, at 500 fs whn th lasr puls is xtinguishd, th distribution is compltly rlaxd and carrirs hav accumulatd in a rgion clos to th band dg nrgy. From th tim volution of th carrir numbr dnsity functions, it is possibl to garnr additional information on th cohrnt procsss in th systm. Th cohrnt ffcts can b analyzd ithr as a function of th lctric fild at a givn tim or for a givn lctric fild intnsity as a function of tim. Figurs 5a and 5b show th calculatd normalizd total lctron dnsity as a function of th lctric fild of th xtrnal light for th cas that th xcss photon nrgy is 50 and 750 mv, rspctivly. In this cas, th lattic tmpratur is 300 K, and th dnsity valus ar obtaind aftr th systm has rachd stady stat, long aftr th lasr puls is finishd (1.5 ps). Furthrmor, th intgratd numbr of th lctrons in th whol spac has bn calculatd, thn th data hav bn normalizd to th maximum valu obtaind at ach fild valu. 14

27 Fig. 5 (a) Calculatd ffct of scattring mchanisms on th intgratd dnsity of lctrons in GaN vs. lctric fild of th xtrnal light at 300 K whn th xcss photon nrgy is 50 mv. (b) Calculatd ffct of scattring mchanisms on th intgratd dnsity of lctrons in GaN vs. lctric fild of th puls at 300 K whn th xcss photon nrgy is 750 mv. Th various curvs in Figs. 5a and 5b illustrat th ffct of various scattring mchanisms on th dnsity of th carrirs and dgr of cohrnc in th systm. Th dashd blac lin shows th cohrnt cas whr no dphasing procss is includd. In this cas, th Mont Carlo part of th numrical modl is xcludd, and th rsults ar obtaind by just taing into account th first-ordr solution of th quations of motion. Sinc no dphasing procss is prsnt, th distribution functions of lctrons and hols ar highly corrlatd. Furthrmor, Rabi oscillations 4 can b sn in th rsult. Th oscillation is mor pronouncd whn th xcitation nrgy is highr (750 mv in Fig. 5b). Figur 6 prsnts th Rabi oscillation in th tim domain for th xcss nrgy valus. 15

28 Fig. 6 Calculatd normalizd lctron dnsity vs. tim at room tmpratur in GaN. Th lctric fild of th light puls is 10 8 Vm 1 for valus of xcss photon nrgy. Ths curvs hav bn computd only considring th cohrnt ffcts as dscribd in Eqs., 3, and 4, and nglcting th dphasing du to scattring mchanisms. Th valus ar normalizd to th maximum of th curv corrsponding to 750 mv. Th normalizd lctron dnsity at room tmpratur as a function of tim for th prvious xcitation nrgis whn th lctric fild is qual to 10 8 Vm 1 is shown in Fig. 6. Th xcss photon nrgis for th solid and dashd lins ar 750 and 50 mv, rspctivly. It can b sn that th Rabi oscillation bcoms strongr as th xcitation nrgy incrass. Bsids th cas in which no dphasing is prsnt, Fig. 5 shows th ffcts of various scattring procsss on th cohrnc in th systm. Th prsnc of intractions with phonons and othr carrirs changs th pictur of th corrlation btwn lctrons and hols. Th dot-dashd grn lins in Figs. 5a and 5b show th cas that only phonon-scattring mchanisms ar includd. Th phonon-scattring procss includs both th polar optical and acoustic scattring mchanisms. In polar optical phonon intraction, th carrirs mit or absorb an intgr numbr of phonons, and th distribution rlaxs toward lowr nrgy stats. Consquntly, if th xcitation nrgy is lowr than th polar optical nrgy, th polar optical phonon mission procss is lss probabl. Howvr, acoustic phonon procsss, having a much lowr nrgy than polar optical phonons, can occur at any xcitation nrgy. Th dashd blu lins rprsnt th cas whn only th carrir carrir intraction is includd in th systm. Th rol of this procss is to basically rdistribut th nrgy in th systm; thrfor, as a rsult of this rlaxation mchanism, th pas of th carrir distribution function broadn. Evntually, th solid rd lin rprsnts th cas that all th abov mchanisms ar includd. Th broadning ffct du to th carrir-carrir scattring is dominant whn th xcss photon nrgy is 50 mv 16

29 sinc th polar optical is no longr th dominant nrgy loss mchanism. To summariz, according to Fig. 5a, for th cas that th xcss photon nrgy is lss than th polar optical phonon nrgy, sinc th polar optical phonon mission is no longr dominant, th total dnsity of lctrons, whr all th intractions ar prsnt, follows th shap of th carrir carrir dphasing curv. This suggsts that in this cas, th carrir carrir scattring mchanism is th dominant procss. Howvr, in Fig. 5b whr th photon xcss nrgy is highr than th polar optical phonon nrgy, polar optical phonon mission is a strong procss in rlaxing th carrirs; thrfor, th final dnsity of lctrons follows th phonon dphasing curv. In this figur, th lctron dnsity valus of th carrir carrir dphasing cas at ach lctric fild sm to b largr than th valus of th curv whr all th scattring rats ar includd. Howvr, car must b tan in comparison sinc ths ar th normalizd valus. Th actual valus of th carrir dnsity at ach lctric fild in th cas whr all th scattring mchanisms ar includd xcd th valus of all th othr cass. In fact, th prsnc of ach scattring mchanism affcts th distribution of polarization, which has a major rol in th carrir gnration rat. To compar th rsults obtaind from th numrical modl with th xprimntal data, w hav computd th nrgy-rsolvd tim-dpndnt luminscnc. To calculat th luminscnc at various nrgis of intrst, onc th systm rachs th stady stat condition, w slct spcific rangs of photon nrgis. For xampl, w considr narrow nrgy rgions clos to th band gap nrgy, th polar optical phonon nrgy, twic th polar optical phonon nrgy, and othr multipls. Subsquntly, w multiply th lctron and hols at ach nrgy within th narrow band and sum all ths partial products ovr th slctd nrgy rgion. Aftr that, w normaliz th luminscnc data to th rspctiv maximum valus. Figur 7 shows th calculatd nrgy-rsolvd tim-dpndnt photoluminscnc for th cas that th xcss photon nrgy is 750 mv. Th lctric fild of th xtrnal puls is Vm 1 and th lattic tmpratur is 300 K. Th pa of th lasr puls is locatd at th origin of th tim axis, and th tim volution of th carrir nsmbls has bn collctd up to 8 ps. 17

30 Fig. 7 Luminscnc spctra in GaN at room tmpratur (300 K) calculatd for diffrnt nrgy rgions Th inst of th figur shows th rsults up to 0.6 ps whr th dtails and th squnc of th curvs ar mor distinguishabl. Th solid blac lin shows th rsults whn th photoluminscnc nrgy is collctd at band dg nrgy, which is th slowst procss. Th dottd rd lin, th dot-dashd grn lin, and th dashd blu lin show th rsults whn th slctd nrgy is qual to 1,, and 3 polar optical phonon nrgis (1LO, LO, 3LO), rspctivly. Th 3LO cas is th fastst procss, thn th LO, and aftrwards th 1LO curvs rach th stady stat. Figurs 8a and 8b prsnt th luminscnc rsults in GaN whn th tmpratur is qual to 50 and 150 K, rspctivly. Th lctric fild and th xcss photon nrgy ar th sam as th ons in Fig. 7. Th rsults clarly show that as th tmpratur dcrass, th ris tim incrass. Mor spcifically, th ris tim for th band dg nrgy rgion changs significantly. This is du to th fact that at lowr tmpraturs, th polar optical scattring rat is lowr, and thrfor this procss is war in rlaxing th carrir distributions. Although w hav ndavord to dvlop our simulation modls to provid th bst dscription of th physical phnomna that mrg from th xprimnt, a quantitativ comparison is not always possibl. In fact, th simplifid band structur and nglcting th intrnal fild and th imaginary part of th slfnrgy arising from th scond-ordr corrction to th carrir carrir and carrir phonon intractions may lad to simulation rsults that in som cass can only b in qualitativ agrmnt with th xprimnt. 18

31 Fig. 8 (a) Luminscnc spctra in GaN at 150 K collctd at diffrnt nrgy rgions. Th xcss photon nrgy is 750 mv. (b) Luminscnc spctra in GaN at 50 K collctd at diffrnt nrgy rgions. Th xcss photon nrgy is 750 mv. Th lctric fild of th xtrnal light puls is V m 1 in both cass. Figurs 9a and 9b show th rsults of luminscnc for th tmpratur valus ranging from 150 to 300 K obtaind by xprimnt and simulation, rspctivly, whn th xcss photon nrgy is 750 mv. It clarly shows that as th lattic tmpratur incrass, th luminscnc ris tim dcrass. Bcaus of th spcific xprimntal conditions, it is not always possibl to dirctly corrlat th bam powr usd in th masurmnts with th lctric fild magnitud mployd in th simulations. In fact, for th modl w hav mployd, as opposd to th cas of a smiclassical gnration modl, th numbr of gnratd carrirs dpnds on th intrplay btwn th cohrnt and dphasing phnomna. Nvrthlss, th numrical modl prdicts th corrct trnd of th xprimntal rsults. It is important to apprciat this point, sinc th xprimntal conditions undr which th rsults in Fig. 9a wr obtaind ar not th sam as th ons considrd in computing th valus prsntd in Fig. 9b. 19

32 Fig. 9 Luminscnc spctra of GaN at diffrnt tmpraturs collctd at band dg nrgy rgion. Th xcss photon nrgy is 750 mv. (a) shows th xprimntal rsults and (b) prsnts th simulatd rsults for an lctric fild of th xtrnal light puls of V m 1. In ordr to show that this is still a valid comparison, vn if th powr of th bam in th xprimnt and th lctric fild magnitud in th numrical modl cannot b dirctly corrlatd, w hav computd th luminscnc as a function of th lctric fild strngth. According to th xprimntal rsults shown in Fig. 10a, th luminscnc ris tim is indpndnt of th powr of th lasr puls. In both Figs. 10a and 10b, th luminscnc rsults at GaN collctd at th band dg nrgy rgion ar shown ovr a rang of lasr puls powrs in which th ris tims ar approximatly th sam. 0

33 Fig. 10 Luminscnc spctra of GaN at diffrnt lasr powrs collctd at band dg nrgy rgion. Th xcss photon nrgy is 750 mv and th lattic tmpratur is 150 K. (a) shows th xprimntal rsults and (b) prsnts th simulatd rsults. In (a), E rprsnts th baslin powr usd in th xprimnt. In (b), E is th baslin valu of lctric fild mployd in th simulation. In Fig. 10, th lattic tmpratur is 150 K and th xcss photon nrgy is 750 mv. Fig. 10b shows that th numrical simulation rsults dlivr th sam trnd as th xprimnt. On th basis of this rsult, w can justify th comparison mad in Figs. 9a and 9b whr w prsntd th tmpratur-dpndnt xprimntal and simulatd luminscnc ris tim. Figur 11 shows th luminscnc spctra collctd at th band dg nrgy rgion whn th lctric fild of th light is Vm 1 and th xcss photon nrgy is 50 mv. Th rsults ar just for th band dg nrgy bcaus th highr nrgis ar lss important in this cas, th xcss photon nrgy is lowr than th polar optical phonon nrgy. 1

34 Fig. 11 Calculatd luminscnc spctra of GaN at diffrnt tmpraturs collctd at band dg nrgy rgion. Th xcss photon nrgy is 50 mv and th lctric fild of th xtrnal light puls is V m 1. Th solid blac lin shows th rsults whn th tmpratur is 50 K, th dashd rd lin shows th rsults whn th tmpratur is 150 K, and th dot-dashd grn lin shows th rsults whn th tmpratur is 300 K. As sn in th figur, whn th tmpratur dcrass, th ris tim of th luminscnc spctrum incrass. Figur 1 shows th rsults of th intgratd luminscnc spctrum in GaN collctd up to 8 ps at th band dg nrgy rgion. Th lattic tmpratur is 150 K, and th xcss photon nrgy is 750 mv. Th rsults ar shown vrsus th normalizd powr of th lasr puls. Th solid lin shows th rsults of th thory, and th marr points show thos of xprimnt. Th figur clarly shows a rasonabl agrmnt in th rsults. As th powr incrass, bcaus of th gnration of mor carrirs, th intnsity of th luminscnc spctrum incrass. Fig. 1 Th intgratd luminscnc spctra of GaN collctd up to 8 ps vs. normalizd lasr powr collctd at band dg nrgy rgion. Th xcss photon nrgy is 750 mv and th lattic tmpratur is 150 K.

35 .5 Summary for Gallium Nitrid (GaN) Exprimnts and Modling W prsntd a thortical and xprimntal study of th subpicoscond intics of photoxcitd carrirs in GaN. W considrd a bul GaN modl with an isotropic parabolic havy hol band and an isotropic nonparabolic conduction band. In th thortical modl, intraction with an xtrnal ultrafast lasr puls is tratd cohrntly, and to includ th scattring mchanisms and dphasing procsss, a gnralizd Mont Carlo simulation is usd. Th scattring mchanisms includd ar carrir intractions with polar optical phonons and acoustic phonons, and carrir carrir Coulomb intractions. Th rsults of th intgratd dnsity of lctrons ar shown for diffrnt xcss photon nrgis. In th cas whr th xcss photon nrgy (50 mv) is lss than th polar optical phonon nrgy, carrir carrir scattring is th dominant procss. Howvr, whn th xcss photon nrgy (750 mv) is gratr than optical phonon nrgy, th polar optical phonon scattring is th most significant rlaxing procss. Bsids th carrir dnsitis, tmpratur-dpndnt luminscnc spctra hav bn studid. W showd that for all valus of th xcss photon nrgis, as th tmpratur dcrass, th luminscnc ris tim incrass. This ffct is du to th lowr polar optical scattring rat, which rsults in th fact that this procss is no longr strong nough to rlax th carrirs. For comparison, w also rport th xprimntal tim-rsolvd photoluminscnc studis on GaN sampls. In conclusion, w hav shown that w can xplain th intics of th photoxcitd carrirs in GaN by including only carrir carrir and carrir phonon intractions and a rlativly simpl -band lctronic structur modl. Furthrmor, w hav prsntd a dtaild analysis of th ffct of a diffrnt dphasing machanism on th carrir intics. Finally, w hav xprimntally masurd and simulatd using our thortical modl th tmpratur and puls powr dpndnc of th photoluminscnc spctra..6 Rsults and Discussion for th Zinc Oxid (ZnO) Cas In this sction, w prsnt and compr th normalizd luminscnc intnsity rsults obtaind by thory and xprimnt in th cas of ZnO. Mor spcifically, w compar th luminscnc ris tims and th ffct of tmpratur on thm. Th rsults ar obtaind in a rang of xcitation nrgis and tmpratur valus. Th matrial paramtrs for ZnO ar givn in Tabl. Th calculatd polar optical scattring rats for lctrons ar shown in Fig. 13, whn th lctric fild of th light is Vm 1, th xcss photon nrgy is 380 mv, and th lattic tmpratur is 300 K. 3

36 Tabl Paramtrs usd in th simulation Fig. 13 Calculatd polar optical scattring rat for th nonparabolic conduction band in ZnO including th scrning ffct. Th solid lins show th phonon mission rat, and th dashd lins show th phonon absorption rat. Th lctric fild of th light is Vm 1, th xcss photon nrgy is 380 mv, and th lattic tmpratur is 300 K. Figurs 14a and 14b show th rsults of th normalizd luminscnc intnsity obtaind by xprimnt and simulation, rspctivly. Th xcss photon nrgy is 380 mv, and th lattic tmpratur is 300 K. Diffrnt curvs show th nrgy rgions whr th luminscnc has bn obtaind, and ach curv is normalizd to its maximum valu. Th solid blac lin shows th rsult of th luminscnc obtaind for photon nrgis nar th band dg transition. Th dashd rd lin, dot-dashd grn lin, and doubl-dot-dashd blu lin show th sam rsults obtaind at 0.5, 1, and 1.5 polar optical phonon nrgy in xcss of th band dg optical nrgy transition, rspctivly. 4

37 Fig. 14 Th rsult of th normalizd luminscnc intnsity from ZnO obtaind by xprimnt (a) and simulation (b). Th xcss photon nrgy is 380 mv, and th lattic tmpratur is 300 K. Figur 15 shows th rsult of th normalizd luminscnc intnsity from ZnO obtaind at th band dg nrgy rgion. Th dashd lin rprsnts th xprimntal rsult, and th simulatd rsult is shown by th solid lin. In this cas, th xcss photon nrgy is 40 mv, and th lattic tmpratur is 75 K. 5

38 Fig. 15 Th rsult of th normalizd luminscnc intnsity from ZnO obtaind by xprimnt (dashd lin) and simulation (solid lin). Th xcss photon nrgy is 40 mv, th lctric fild of th lasr puls is Vm 1, and th lattic tmpratur is 75 K. Bsids th rsults of th ris tim, w also invstigatd th ffct of tmpratur on th ris tim. Figur 16 shows th rsults of th normalizd luminscnc intnsity, whn th xcss photon nrgy is 380 mv, calculatd at th band dg nrgy rgion ovr a rang of lattic tmpratur valus from 150 to 300 K. It clarly shows that, as th lattic tmpratur incrass, th luminscnc ris tim dcrass. This is bcaus at lowr tmpraturs, th polar optical scattring rat is war than at room tmpratur and, thrfor, this procss is not as ffctiv in rlaxing th carrir distributions. Fig. 16 Th normalizd luminscnc intnsity from ZnO at diffrnt tmpratur valus calculatd at band dg nrgy rgion. Th xcss photon nrgy is 380 mv, and th lctric fild of th xtrnal light puls is V m 1. 6

39 In this study w considrd bul ZnO modld by an isotropic parabolic havy hol band and an isotropic nonparabolic conduction band. In th thortical modl, intraction with an xtrnal ultrafast lasr puls is tratd cohrntly, and th scattring mchanisms and th dphasing procsss ar tan into account through a gnralizd Mont Carlo simulation. Th scattring mchanisms includd ar carrir intractions with polar optical phonons and acoustic phonons, and carrir carrir Coulomb intractions. Th rsults of th calculatd normalizd luminscnc intnsity from ZnO at xcitation nrgis and lattic tmpratur valus ar in good agrmnt with thos of th xprimntal study obtaind by th tim-rsolvd photoluminscnc mthod. W also prsntd th calculatd rsults of th tmpratur-dpndnt luminscnc intnsity. As a conclusion, as th tmpratur dcrass, th luminscnc ris tim incrass. This ffct is du to th lowr polar optical scattring rat, which rsults bcaus this procss is no longr strong nough to rlax th carrirs. In conclusion, w hav shown that w can xplain th dynamics of th photoxcitd carrirs in ZnO by including only carrir carrir and carrir phonon intractions and a rlativly simpl -band lctronic structur modl. 3. Gomtric Singl-Qubit Quantum Gats for an Elctron Spin in a Quantum Dot W propos a schm to prform arbitrary unitary oprations on a singl lctronspin qubit in a quantum dot. Th dsign is solly basd on th gomtrical phas that th qubit stat acquirs aftr a cyclic volution in th paramtr spac. Th schm uss ultrafast linarly chirpd pulss, providing adiabatic xcitation of th qubit stats, and th gomtric phas is fully controlld by th rlativ phas btwn pulss. Th analytic xprssion of th volution oprator for th lctron spin in a quantum dot, which provids a clar gomtrical intrprtation of th qubit dynamics, is obtaind. Using paramtrs of InGaN/GaN and GaN/AlN quantum dots, w provid an stimat for th tim scal of th qubit rotations and paramtrs of th xtrnal filds. 3.1 Introduction Th lctron spin in a singl quantum dot is on of th prspctiv ralizations of a qubit for th implmntation of a quantum computr. During th last dcad, svral control schms to prform singl gat oprations on a singl quantum dot spin hav bn rportd Hr w propos a schm that allows prforming ultrafast arbitrary unitary oprations on a singl qubit rprsntd by th lctron spin. Th ida of gomtric manipulation of th qubit wav function has bn 7

40 rcntly dvlopd into a nw rsarch dirction calld gomtric quantum computing Th main motivation of this dvlopmnt is th robustnss of gomtric quantum gats against nois. 19,0 In this rport, w dmonstrat how to us th gomtric phas, which th Bloch vctor gains along th cyclic path, to prpar an arbitrary stat of a singl qubit. W show that th gomtrical phas is fully controllabl by th rlativ phas btwn th xtrnal filds. W also discuss th ralistic implmntation of th proposd dsign using th lctron spin in a chargd quantum dot as an xampl of a qubit. 3. Gnral Equations of Motion Lt us considr th cohrnt Raman xcitation in th 3-lvl -typ systm consisting of th lowst stats of lctron spin 0 X and 1 X coupld through an intrmdiat trion stat T consisting of lctrons and a havy hol 1 (Fig. 17). Fig. 17 Enrgy structur of th 3-lvl systm comprising th lctron spin stats and th trion stat W assum that th trion stat is far off-rsonanc with th xtrnal filds to nsur that dcohrnc on th trion-qubit transitions can b nglctd. Th lctron spin stats ar split by an xtrnal magntic fild; th sparation nrgy is. Th total wav function of th systm whr a0,1 t and ( t) a ( t) 0 a ( t) 1 b( t) T, (17) 0 1 btar th probability amplituds, is govrnd by th timdpndnt Schrödingr quation with th Hamiltonian 0 0 ( P( t) S( t)) H 0 ( P( t) S ( t)), ( P( t) S ( t)) ( P( t) S ( t)) T (18) whr 8

41 ( t) ( t)cos[ t ( t)], P, S P0, S 0 P, S P, S ( t) ( t)cos[ t ( t)], P, S P0, S 0 P, S P, S ( t) E ( t) /, and P0, S0 0 T, T1 P, S ( t) E ( t) / ar th Rabi frquncis; P0, S0 1 T, T 0 P, S 0 TT, 1 PS, ar th dipol momnts; E t ar th puls nvlops; PS, ar th cntr frquncis; PS, () t ar th tim-dpndnt phass; and T is th nrgy of th trion stat. W ar considring hr a cas of linarly chirpd pulss such that P, S ( t) P, S P, St /, (19) whr PS, ar th initial phass and PS, ar th chirps of th pulss. Using transformation ( t) U ( t), whr RWA i( SP) t U RWA 0 0, (0) ipt i( S P) t U RWA 0 0, (1) ip t 0 0 w ma th rotating wav approximation (RWA) by nglcting th rapidly oscillating trms with frquncy S, P and S P. In th RWA, th Hamiltonian has th following form: 9

42 H U H U i U U 1 1 RWA RWA RWA. RWA P i t P S it 0 P( t ) S * * it * it * P S P S T i 0 0 P S it 0 ( ) ( t ) * * it * i t * P S P S P t P S, () whr,, P T P P S S ( t), ( t), i P () t i () t P P0 S S 0 () i P () t P P0 t, and () i S () t S S0 t. Assuming larg dtunings of th pump and Stos fild frquncis from th transition frquncis to th trion stat, w apply th adiabatic limination of th trion stat. In that cas, b 0 and w find 1 * * it * it * b P S a0( t) P S a1( t). P (3) Substituting Eq. 3 into quations for a0,1 aftr som algbra, w obtain th following form of th Hamiltonian for th ffctiv -lvl systm: whr, ac0( t) ff ( t) H *, ff ( t) ac1( t) (4) 1 ( t) ( ( t) ( t) ( t) ( t)cos ( t) ( t) t, ac0 P0 S 0 P0 S 0 P S P 1 ( t) ( ( t) ( t) ( t) ( t)cos ( t) ( t) t, ac1 S0 P0 S 0 P0 P S P (5) (6) 30

43 i t ( t) ( t) ( t) ( t) ( t) ff P0 P0 S 0 S 0 P ( t) ( t) ( t) ( t). i[ P ( t ) S ( t ) t ] i[ P ( t ) S ( t ) t ] P0 S 0 P0 S 0 (7) Sinc P0( t) S0( t) P0( t) S0( t), w hav i t ( t) ( t) ( t) ( t) ( t) ff P0 P0 S 0 S 0 P ( t) ( t)cos[ ( t) ( t) t]. P0 S 0 P S (8) Maing th transformation ( t) U ( t), with t i t / i dt st ( t ) t 0 i t / i dt st ( t ) U, t I (9) i t / i dt st ( t ) 0 whr st ( t) ( ac0( t) ac1( t)) / 4, w can rwrit Eq. 4 in a mor symmtric form. In th nw basis, th Hamiltonian tas th form. 1 1 i st t H U HU U U H ( ( ) / ) I dif ( t) ff ( t) *, ff ( t) dif ( t) (30) whr dif ( t) ( ac0( t) ac1( t)) /. Taing into account th xact tim dpndnc of th phass in Eq. 19, w apply anothr transformation ( t) U ( t), with F i t /4 t 0 i σz 4 U F, (31) i t /4 0 whr P S. Thrfor, in th fild intraction rprsntation th Hamiltonian tas th following form: 31

44 H U HU i U U 1 1 F F F. i t /4 dif ( t) ff ( t) t 0 * i t /4 ( ) ( ) 0 t ff t dif t i t /4 t dif ( t) ff ( t) * i t /4 ff ( t) t dif ( t) * ff ( t) ( t) i ( t ) ( t) F ff i, (3) whr ( t) ( S P) t dif ( t), P S, and P0( t) S 0( t) P0( t) P0( t) S 0( t) S 0( t) ff ( t) 1 P P0( t) S 0( t) ( t) ( t). P0 S0 i[ t t /] P0( t) S0( t) i[ t t /] (33) In gnral, th diffrntial AC Star shift, () t, is not zro and has to b tan into account. Using th dfinition of th Rabi frquncy, w can rwrit Eqs. 5 and 6 in th form dif ( t) ( E ( t) E ( t) E ( t) E ( t)cos ( t) ( t) t, 0T ac0 P S P S P S P ( t) ( E ( t) E ( t) E ( t) E ( t)cos ( t) ( t) t. 1T ac1 P S P S P S P (34) (35) In th cas whn 0 T 1 T, th AC Star shifts ar th sam for both qubit stats, so that ( t) 0, and w hav dif i t ff () t H. * i ff () t t (36) Not that w do not rquir compltly ovrlappd pulss hr. 3.3 Nonimpulsiv Cas In som xcitation schms, bcaus of slction ruls that ta into considration th polarization of th xtrnal fild, th pump and Stos fild intract only with th corrsponding transitions, and th gnral Hamiltonian can b simplifid. For 3

45 that cas, w can obtain th corrct form of th Hamiltonian by putting ( t) ( t) 0 in Eqs. 5, 6, and 33. It rsults in P0 S0 1 ( t) ( t), ac0 P0 P 1 ( t) ( t), ac1 S 0 P 1 ( t) [ ( t) ( t)], dif P0 S0 4P ( t) ( t) P0 S0 ff ( t). P (37) (38) (39) (40) Thrfor, in th fild intraction rprsntation, th Hamiltonian has th form whr i ( t) ff ( t) H, * i (41) ff ( t) ( t) ( t) ( ) t [ ( t) ( t)] / 4. (4) S P P0 S0 P 3.4 Adiabatic Solution Th Hamiltonian in Eq. 41 controls th dynamics of th qubit wav function in th approximation of th adiabatic limination of th trion stat. Hr w considr th adiabatic xcitation of th qubit and find th adiabatic solution of th Schrödingr quation with th Hamiltonian in Eq. 41. i Sinc th phas factor,, of th coupling trm in Eq. 41 is tim indpndnt, it is convnint to us th following transformation ( t) A ( t), whr i i / i z / σ A, (43) so that th nw wav function is govrnd by th Hamiltonian ( t) ( ) 1 t H AHA ( t) z ( t) x. ( t) ( t) σ σ (44) To solv th Schrödingr quation in th adiabatic rprsntation, w apply anothr transformation, ( t) R ( t) ( t), whr cos ( t) sin ( t) i () t σy R ( t), sin ( t) cos ( t) (45) 33

46 and tan[ ( t)] ( t) / ( t). In th nw basis th Hamiltonian, Eq. 44, tas th form ( t) ( t) 0 1 ( t) ( t) ( t) ( t) ( t) z / 0 ( t) ( t) H R H R σ, (46) whr ( t) ( t) ( t). As w s, th Hamiltonian in Eq. 46 is diagonal in th adiabatic basis, and w can radily writ down th solution. Howvr, sinc th transformation R() t is tim dpndnt, an additional nonadiabatic coupling trm is prsnt in th gnral Schrödingr quation whr 1 i ( t) ( t) σz ( t) ( t) σ y ( t), (47) ( t) ( t) ( t) ( t) ( t). ( t) ( t) Nglcting th nonadiabatic coupling trm in Eq. 47, w radily obtain for th qubit wav function in th original basis (48) () t i σ z i( t) σ i y σz i(0) σ i y σz ( t) (0), (49) whr t ( t) dt ( t ). Th gnral form of th volution oprator in Eq. 49 is 0 3 wll justifid if th following condition ( t) ( t) ( t) ( t) ( t) is valid. In th cas of compltly ovrlappd pulss, P0( t) S0( t), with idntical chirp rats, P S, for th rsonant qubit, ( t) 0, w hav ( t) (0) / 4, and th transformation matrix bcoms i /4 σ y R( t) R (0). Thrfor, th unitary volution oprator for th wav function of th rsonant qubit tas th form U() t i cos S( t) / i sin S( t) / i i sin S( t) / cos S( t) / i i S t I i S t σ σ is ( t) n σ/ S t i S t cos ( ) / sin ( ) / ( ) cos ( ) / I sin ( ) / nσ, (50) 34

47 t whr S( t) dt ( ) 0 t is th ffctiv puls ara, n ( cos,sin,0), σ ( σ iσ x y) / ar th Pauli raising and lowring oprators, and σ x, y, z ar th Pauli oprators. Th nonadiabatic coupling trm, Eq. 48, is zro for th rsonant qubit, and th solution of th Schrödingr quation in th adiabatic approximation, Eq. 50, is th xact solution. * Th dnsity plots of th population and cohrnc ( a0( T) a1( T ) ) at th final tim (aftr th puls xcitation) as a function of th ffctiv puls ara, ST ( ), and th dimnsionlss frquncy chirp paramtr, /, dscribd by th unitary volution oprator in Eq. 50 is dpictd in Fig. 18. W us th Gaussian shap for th puls nvlops, assuming that linar chirp is obtaind by applying a linar optics tchniqu, maning that a transform-limitd puls of duration 0 is chirpd, consrving th nrgy of th puls. Th tmporal and spctral chirps ar rlatd as / (1 / ), whr 0 is th transform-limitd puls duration. W obsrv th Rabi oscillation rgim, whn th population of th qubit stats is changing btwn 0 and 1 whil th cohrnc is changing btwn 0 and 1/. This bhavior dos not dpnd on th chirp rat, sinc th ffctiv Rabi frquncy () is dtrmind by th product of th pump and Stos Rabi frquncis: t 4 1/4 P0, S0( t) 0 xp{ t / ( )}/[1 / 0 ] with th 4 1/ chirp-dpndnt puls duration [1 / ] and th amplitud.,3 0 0 Fig. 18 Th dnsity plot of th 1 stat population (a) and cohrnc (b) as a function of th ffctiv puls ara and frquncy chirp; P, 0. Initially, only th 1 stat is S populatd. In turn, for th off-rsonant qubits, 0, th volution oprator in th adiabatic approximation tas th form 35

48 i ( t) i ( t) i cos ( t) sin ( t) U ( t), i ( t) i i ( t) (51) sin ( t) cos ( t) whr 1 1 cos ( t) 1,sin ( t) 1 ( t) ( t) (5) 1 t and ( t) ( ) 0 t dt is th ffctiv puls ara. For th off-rsonant cas, th transformation matrixs ar cos ( t) sin ( t) R ( t), R (0). sin ( t) cos ( t) 0 1 (53) Figur 19 dmonstrats xcitation of th off-rsonant qubit. As xpctd, th population of th off-rsonant qubit at th final tim is not changd by th xtrnal filds as long as th puls xcitation paramtrs ar in th adiabatic rgim. This is th rgim of adiabatic rturn. Howvr, w still obsrv th Rabi oscillation for th valu of th chirp 5. This is th ara of nonadiabatic population transfr whr th nonadiabatic coupling trm cannot b nglctd in Eq Fig. 19 Th dnsity plot of th 1 stat population (a) and cohrnc (b) as a function of th ffctiv puls ara and frquncy chirp; P, S Initially, only th 1 stat is populatd. 36

49 3.5 Bloch Vctor Rprsntation Th dynamics of th qubit wav function can b dscribd qually wll using th Bloch vctor rprsntation. In addition, th Bloch vctor formalism allows a vry nic and clar gomtrical intrprtation of qubit dynamics. 4 In this sction, w giv a short ovrviw of th Bloch pictur. A gnral stat of a qubit can b dscribd as 0 1 i a ( t) 0 a ( t) 1 cos / 0 sin / 1, (54) whr and ar th phas paramtrs. Up to an insignificant global phas, th wav function can b mappd into a unitary Bloch vctor B ( u, v, w), as shown in Fig. 0. Fig. 0 Th Bloch vctor rprsntation of th qubit stat. Excitation of th qubit by an xtrnal fild corrsponds to th rotation of th B vctor about th psudo fild vctor, Ω, with componnts dtrmind by th ffctiv Rabi frquncy () t, dtuning, and th rlativ phas. To us Bloch vctor rprsntation, w construct th qubit dnsity matrix whr * ij ai ( t) a j ( t) i cos sin ρ, i (55) sin 1 cos, i, j 0,1. Using a Pauli matrix dcomposition u x v y w z 1 1 ρ I B σ I σ σ σ 1 I cos sin σx sin sin σ y cos σ z, (56) 37

50 w idntify rlation btwn th componnts of th qubit wav function, th qubit dnsity matrix lmnts, and th Bloch vctor componnts. Thrfor, w obtain th following xprssion: u cossin B v i( 01 10) sinsin. (57) w cos Taing into account th Hamiltonian in Eq. 41, th quation of motion for th dnsity matrix i ρ H ρ ρ H tas th following xplicit form i ( t) /, i i i i i i ( t) ( t) /, i i ( t) ( t) /, i ( t) /. Using th rlations in Eq. 57, w can also writ th dynamic quation in th Bloch vctor rprsntation as (58) u v( t) w ( t)sin, v u( t) w ( t)cos, w u ( t)sin v ( t)cos. (59) Introducing a psudo fild vctor, Ω, with componnts dtrmind by th ffctiv Rabi frquncy, -photon dtuning, and th rlativ phas btwn pump and Stos pulss, w can rwrit Eq. 59 in th compact form ( t)cos Ω ( t)sin, (60) () t B ΩB. (61) This is th Bloch quation, which dscribs a prcssion of th Bloch vctor, B, about th psudo fild vctor, Ω, and allows clar, intuitiv intrprtation of qubit dynamics. 38

51 3.6 Evolution Oprator of th Bloch Vctor Sinc w alrady now th xact form of th volution oprator U () t in Eq. 50 and ( t) U ( t) (0), (6) w can asily construct th volution oprator for th Bloch vctor. Using th dfinition of th dnsity matrix, w hav ρ( t) ( t) ( t) U( t) (0) (0) U ( t) U( t) ρ(0) U ( t), (63) whr 00(0) 01(0) 1 1w0 u0 iv0 ρ (0) (0) (0) (0) (0) u iv 1w is th initial condition and U () t i cos S( t) / i sin S( t) / i i sin S( t) / cos S( t) / i i S t I i S t σ σ is ( t ) n σ/ S t i S t cos ( ) / sin ( ) / ( ) cos ( ) / I sin ( ) / nσ. (64) Thrfor, i cos S( t) / i sin S( t) / ρ( t) U( t) ρ(0) U ( t) i i sin S( t) / cos S( t) / i S t i S t 00(0) 01(0) cos ( ) / sin ( ) /. 10(0) 11(0) i i sin S( t) / cos S( t) / (65) Taing into account Eq. 57, w obtain for th Bloch vctor: C S cos S sin C S sin B( t) S sin C S cos C S cos B (0), (66) C S sin C S cos C S whr C cos S( t) /, S sin S( t) /. 3.7 Ultrafast Qubit Rotations Using Gomtrical Phas At this point, w ar rady to discuss implmntation of th singl qubit gats sinc w hav obtaind th analytic solution for th qubit wav function and constructd th volution oprator in th Bloch vctor rprsntation. A univrsal 39

52 st of quantum gats has bn intnsivly discussd in th litratur rlatd to th univrsality in quantum computation. 5 To prform quantum computation, w must hav major building blocs at our disposal: arbitrary unitary oprations on a singl qubit and a controlld-not opration on qubits. Hr w addrss only singl qubit manipulation. To dmonstrat arbitrary gomtric oprations on a singl qubit, w us th Bloch vctor rprsntation discussd in th prvious sction. Sinc any unitary rotation of th Bloch vctor can b dcomposd as whr i i i σ R ( i, U R ( ) R ( ) R ( ), (67) i 0 z 1 y z 3 y z ) ar th rotation oprators, w nd to dmonstrat rotations of th qubit Bloch vctor about th z and y axs by applying various squncs of xtrnal pulss. Th dcomposition in Eq. 67 plays an important rol in circuit-basd quantum computing, as it shows xplicitly that singl-qubit oprations allow us to crat th arbitrary stat of th qubit. Hr w show how this can b accomplishd by controlling th paramtrs of th xtrnal pulss, which ar dfind by th xplicit form of th volution oprator (s Eq. 50). Thr ar distinct ways of th implmntation of rotation, dpnding on which part of th total qubit phas w mploy: dynamical or gomtrical. Quantum gats rlying on gomtrical quantum phass ar calld holonomic gats, and thy ar xpctd to b robust with rspct to nois. 16,17 To implmnt th rotation of th Bloch vctor about th z axis (th phas gat) basd on th gomtrical phas, w can us th volution oprator of th rsonant qubit, Eq. 50. Th product of volution oprators corrsponding to th squnc of π pulss with th rlativ phas givs i 0 R z( ) U ; U ;0, i (68) 0 whr th first subindx of U indicats th puls ara, ST ( ), and th scond on indicats th phas,. Figur 1 shows th Bloch vctor trajctoris of th qubit basis stats 0 and 1, which corrspond to th angls 0 and in Eq. 57 and th Bloch vctor initially pointing in z and z dirctions whil th vctor Ω 1 (,0,0) is pointing in x dirction. For simplicity, w chos 0 for th first -puls. Th first -puls flips th population to th stat 1 ( 0 ); corrspondingly, th Bloch vctor turns about th ffctiv fild vctor Ω 1 (about th x axis), and it stays in th yzplan, all th tim and points in th z ( z ) dirction at th nd of 40

53 th puls. Bcaus of th scond -puls, th population is transfrrd bac to th initial stat 0 ( 1 ); thrfor, th Bloch vctor rturns to its original position pointing along th z ( z ) axis. Howvr, sinc w chos for th scond -puls, th psudo-fild vctor is rotatd countrclocwis by th angl in th xyplan,, Ω ( cos, sin,0), and th Bloch vctor movs in th plan prpndicular to th xyplan, and has th angl / ( / ) with th xz, plan. Fig. 1 Th Bloch vctor trajctory for th qubit stat 0 in panl (a) and th qubit stat 1 in panl (b) gnratd by th squnc of -pulss with th rlativ phas Th Bloch vctors rprsnting a pair of orthogonal basis stats 0 and 1 follow a path nclosing corrspondingly solid angls of and. Th gomtrical phas is qual to on-half of th solid angl, which mans th basis stats 0 and 1 gain phass and, and th volution oprator tas th form of th phas gat, Eq. 68, with th rlativ phas controlling th phas of th gat. Th rotation oprator about th y axis can b constructd using 3 pulss. Th first and third puls is /-puls with 0, whil th scond puls is -puls with th rlativ phas. It is asy to show, using Eq. 50, that this 3-puls squnc rsults in ( i y y ; σ ;0 ;0 R ) U U U. (69) To dmonstrat th gomtrical natur of th R y ( ) opration, w us th fact that it crats th rlativ phas btwn th qubit basis stats i ( 0 i 1 ) /. In th Bloch rprsntation, ths stats hav th form 4 4 i i cos 0 sin 1, 41

54 which ar vctors dfind by th angls / and / and pointing in th y and y dirctions, as shown in Fig.. Th trajctory of th Bloch vctor rprsnting th stats i is shown in Fig.. Th psudo-fild vctors Ω 1 and Ω 3 ar dfind by th ffctiv Rabi frquncis of th first and third pulss and ar pointing in th x dirction sinc 0. Th scond psudofild vctor Ω is rotatd countrclocwis by th angl in th xyplan,, th sam as in th cas abov. Th initial Bloch vctor is pointing in th y ( y) dirction. Th first / -puls rotats th Bloch vctor about Ω 1 to th position of th stat 1 ( 0 ). Th scond puls flips th dirction of th Bloch vctor. Th third /-puls rturns th Bloch vctor to its original position. Th Bloch vctor and th psudo-fild vctor ar orthogonal during th whol volution. Similar to th prvious cas, w obsrv that th basis stats i and i follow a path nclosing corrspondingly solid angls of and. Thrfor, thy gain th rlativ phas, which is th gomtrical phas dfind by th rlativ phas btwn pulss. It is asy to show that th phas gat in th i basis is quivalnt to th R y ( ) gat in th 0, 1 basis. Fig. Th Bloch vctor trajctory for th qubit stat i in panl (a) and th qubit stat i in panl (b) gnratd by th squnc of /-pulss and on puls with th rlativ phas 3.8 Rotation in th Bloch Rprsntation Rotation oprations in th Schrödingr pictur ar whr i { x, y, z}, 0 is th initial wav function, i σ cos isin x R x, i sin cos R, (70) i 0 4