PACS number(s): v, q, Yz

Transcription

1 Transen nuaons deca n dlued paramagnec solds: a radaon dampng mechansm N. Ya. Asadullna T. Ya. Asadulln Deparmen of General Phscs Kaan naonal research echncal unvers KAI K.Marx r. Kaan 4 UIA PAC number(s): v 76..+q 3.65.Y Here a heor of he nens and concenraon dependen dampng of nuaon sgnals observed b Boscano e al. (Phs. ev B (993); Phs. ev. A (999)) and b ohers n varous wo-level spn ssems s proposed. I s shown ha n dlued paramagnec solds conrbuon of dpole-dpole neron o he nuaon deca s neglgbl small. We elaboraed a cav loss (radaon dampng) mechansm ha explans he nens- and concenraon dependence of he dampng. I s shown ha nsead of ordnar Bloch s ransverse T and longudnal T dampng parameers he deca of ransverse and longudnal spn componens n nuaon process are descrbed b one and he same nens- concenraon- frequenc- and me dependen dampng parameer. I. INTODUCTION Quanum nformaon processng ncludes excaon of a ssem of wo-level parcles (elecron or nuclear spns qubs) b a sequence of pulses of he resonan coheren elecromagnec feld of varous duraon and nens. The nsered o he ssem coheren nformaon decas wh some rae boh durng he pulse and afer (or beween he pulses n he case of a sequence of pulses). Ths deca durng he pulse wh lengh of he order of a charersc relaxaon me manfess self n he phenomenon of he ransen nuaons. Transen nuaons are he response of a ssem nall a hermal equlbrum o he abrup applcaon of a ver nense resonan feld n gong o a new saonar sae. In he case of he srong feld appled o he paramagnec ssem he response represens damped n me oscllaons (ab oscllaons) wh he ab frequenc B where B s he amplude of he resonan feld s gromagnec rao. Ths effec nall observed n he ssem of nuclear spns [] laer was seen also n he ssem of elecron spns [] n he opcal ransons of aomc and molecular ssems [3] ec. In dlue paramagnec solds o be consdered n hs paper here are several sources of dampng of nuaons. Here he resonance lne s srongl nhomogeneousl broadened. As a resul he sgnal decas due o decoherence of he precessng spns. Anoher source s a relaxaon assocaed wh he cav losses and losses n he spn ssem. The losses n he spn ssem nclude spn-spn (dpole-dpole) spn-lace hperfne and oher possble nerons. Under he assumpon ha hese nerons are descrbed b he Bloch equaons wh consan parameers T and T (n solds usuall T T amplude of oscllaons where J ( ) V s damped as [] V ~ J K ) he ( ) () s he eroh order Bessel funcon K exp( ) s he deca funcon and / T rae n he Bloch model. B s he deca In conras o hs heorecal predcon he deca observed n glass slca conanng E ceners and n quar crsals wh AlO 4 ceners [45] s faser han expeced and depends on he drvng-feld nens and on he concenraon of he B

2 ceners. The power dependence of he deca rae s well descrbed b a lnear dependence of on he nduced ab frequenc: where s equal o () B whn he expermenal unceranes and s a parameer of he order of. Furhermore parameer ncreases on ncreasng spn concenraon n. Laer he deca raes of he form () were found n he oher ssems of he elecron spns [6-]. In a recen paper Babekov [3] found a conrbuon of he man-parcle dpole-dpole nerons beween he paramagnec ceners o he deca rae conanng he dependence boh on he amplude of he mcrowave feld (.e. on frequenc ) and on he concenraon n of paramagnec ceners (see also ef. 4). However obaned numercal value of s occurred o be sgnfcanl less han expermenal one. B numercal solvng he chroednger equaon he conrbuons of dpole-dpole nerons and of some oher possble fors o he process of nuaon dampng were examned also n paper []. I s pernen o noe ha papers [4 5] are he sole publcaons where due o wo-quanum excaon he ransen nuaons are observed and measured drecl durng he excaon pulse whereas usuall such measuremens are performed ndrecl b use of free nducon deca and/or he echoes. Expermens on ransen nuaons shed lgh on he phscs of he neron of he resonan feld wh he ssem of he wo level parcles and he naure of he relaxaon processes n he ssem. In hs conex he resuls of he papers [4 5 6-] e reman largel unexplaned. The purpose of he presen arcle s o furher sud of he relevan relaxaon processes. Below wll be shown ha he conrbuon of he dpole-dpole nerons o he dampng of ransen processes (ncludng nuaons) s neglgbl small and ha under nense resonan feld he man conrbuon o hese processes has s orgn n he cav losses or more precsel n he radaon dampng [5]. The radaon dampng s a consequence of he Joule hea a he cav walls caused b precessng ve spns and s a funcon of he concenraon of he spns. The paper s organed as follows. In ubsecon II. A we oban expressons for specral and oal concenraons of spns exced durng he pulse solvng a smple rae equaon for a generc spn pke of he nhomogeneousl broadened spn ssem. nce he spns are exced from an equlbrum sae where he ransverse componens of spns are absen he excaon process s descrbed wh use of he low nens (equlbrum) relaxaon parameers T and T known from expermen. ubsecon II. B s devoed o soluon of he Hesenberg equaons for he exced spns; he equaons coun for boh he cav (radaon) losses and losses due o spn-spn nerons. As a resul general expressons are obaned for he nuaon response and for nens dependen relaxaon parameers. econ III s devoed o numercal analss of hese general expressons. The resuls of prevous secons and her relaon o some earl publshed resuls are brefl dscussed n econ IV. II. THEOY A. Knecs of excaon The elecron spn resonance lnes n solds are nhomogeneousl broadened.e. he resonance frequences are dsrbued around he mean frequenc cordng o a some law. Followng efs. 4 5 we approxmae he lne form of he ssem of spns ½ under consderaon b a Gaussan / exp / f (3) wh a sandard devaon. For comparson purposes n econ IV wo oher dsrbuon forms a Lorenan and a recangular wll be consdered.

3 The ssem s exced b resonan feld Hcos uned a he cenre of he nhomogeneousl broadened lne. The absorpon lne of a narrow band of spns (spn pkes) wh deunng s homogeneousl broadened and a low nenses of he resonan feld s descrbed b a Lorenan g L (4) wh he wdh / T and T s he low power value of he ransverse relaxaon me. Wh ncreasng nens he absorpon (and he concenraon of exced spn pkes) undergoes a power dependen sauraon (see below). The concenraon dependence of he response s of grea mporance and hs s deermned b wo reasons. Frsl he response s obvousl proporonal o he concenraon n of he ve spns exced durng he pulse. econdl he response depends on he concenraons of he ve and of he passve spns due o he concenraon dependence of he relaxaon mechansms under consderaon: he dpole nerons of he ve spns wh eh oher and wh he passve spns are dfferen because he ve spns have ransverse componens bu passve spns have he -componen onl. Hence her conrbuons o he deca process depend on he n and radaon dampng s a funcon of concenraon of he ve spns as well. n pas respecvel. As noed above and wll be shown below he nce n nuaon expermens he pulse lengh s of he order or shorer han he phase relaxaon me he excaon of spns s an nens and me dependen ransen process. Moreover due o he nhomogeneous broadenng he concenraon of exced spns s also frequenc dependen and as a resul we are n need of specral funcon n ( ) for generc spn pke x. To our knowledge he concenraon dependence of he nuaon response was aken no coun for he frs me n ef. 6. Here hs queson s consdered n somewha more deal. The ssem of he spns n equlbrum s charered b populaons n of he lower and n of he upper levels and wh n n n beng he full concenraon of he paramagnec ceners. Durng he pulse he me dependence of he specral concenraon of he spns a he upper level n s descrbed b he rae equaon ( n he spns a he lower level) s he concenraon of n w W n W n. (5) As follows from expermens [4 5] and wll be shown below he longudnal and ransverse relaxaon parameers of he spn ssem undergo power- concenraon- and me- dependen drasc changes. These changes are conneced wh he me-varng ransverse componens of he spns. nce he excaon process akes ple from equlbrum sae wh no ransverse componens he knecs n should be governed b he equlbrum values of he sponaneous ranson probabl w / T and of he probables of he nduced ransons W W. nce he absorpon profle of he homogeneousl broadened lne s Lorenan he nduced ranson probabl nroduce he frequenc-dependen specral sauraon parameer s s wh T and T beng he low-power (equlbrum) consan parameers [4 5]. W follows he Lorenan lne and we can s TT (6) Because of large nens of he resonan feld n expermens [4 5] s varaon due o absorpon b he spn ssem can be gnored and W ~ are consdered as consan parameers.

4 pecral concenraons of he ve n and of he passve n pas spns a he me are s Δn s n n n f exp n χ f n Δ n f n pas (7) E n n n n n n anh. kt Here n s he nal populaon dfference where E s he energ dfference beween he upper and he lower levels k he Bolmann consan T emperaure and The full concenraon of he ve and passve spns a me wll be respecvel n χ sδ. (8) T n n d (9) n Δn n. () pas B. Hesenberg equaons The nuaon response s proporonal o he ensemble average of he ransverse spn componen of he ssem M. I s convenen o perform calculaons of he response usng he Hesenberg equaons for spn operaors. For -h spn-pke of he nhomogeneousl broadened ssem we have d / H x. () In he appled magnec feld B and mcrowave feld Bcos x (seen b spns due o wo-quanum excaon) he Hamlonan H consss of H H H H H () where frs wo erms on he rgh hand sde descrbe nerons of he wh he appled and mcrowave magnec felds respecvel. To show ha conrbuon from magnec dpole nerons o he dampng s small we nclude n equaon () erm H descrbng neron wh he ve spns. Term H descrbes neron wh cav feedbk feld B. The feedbk feld s conneced wh curren nduced a he cav walls b precessng spns. Noe ha equaon () no ncludes he dpole neron wh he passve spns (he fnal resul for neron wh he passve spns wll be gven below whou calculaons). In he reference frame roang wh frequenc around axs of he laboraor reference frame one has H ; H x ; H u [ ( )]. (3) 4 ħ 3cos 3 r Here u s he coordnae funcon n he dpole neron and n H we have omed oscllang me-dependen erms. In descrbng he cav feedbk feld B we proceed from he Krchhoff law [5]. In resonance

5 condons of expermens [4 5] he onl conrbuon o curren s due o he precessng spns so B ~ ~ dm d M and as a consequence he radaon dampng Hamlonan for -h spn s H B a. (4). d x x The feedbk feld componen B a he se of -h spn due o -h spn s a coordnae funcon ha we specf b a a r 3 where he roang reference frame are a s a cav geomercal for. Hamlonan H s smplfed as follows. The Hesenberg equaons n d / H. (5) If we rean n he Hamlonan H onl H and H he equaon for wll be d x. (6) ubsuon no equaon (4) for d gves fnal expresson for he radaon dampng Hamlonan x x H a. (7) Usng Hamlonan ndcae ) H () gven b expressons (3) and (7) he Hesenberg equaons for are (he prmed sum d x ' u d x u x x a x ' ' (8) d ' ' u a. x x x ummand wh a n he radaon dampng Hamlonan gves neglgbl small conrbuons o he drvng force and o frequenc shf. I s easer o solve ssem (8) n he reference frame ransformed as follows: where / x x x x x x s he effecve ab frequenc. The new ssem of equaons s (9) d x ' u x ' u x ' u ' ' d x u x x u x ()

6 d x ' u. Here s conrbuon o he effecve ab frequenc from he nonlnear mechansms: ' u x ' u x () ' a ' a x If we coun for also neron wh he passve spns (denoed as I ) s conrbuon o s I. ui. () I follows from expermens [4 5] ha nuaon response represens well defned oscllaons (he ab oscllaons). Furhermore wll be shown b numercal calculaons below ha conrbuons of he dpole nerons o he deca are neglgbl small. Hence equaons () are smplfed as Frequenc parameers and d x d d. (3) are slowl varng n me charerscs of he local magnec feld a -h se due o he dpolar neron and he feedbk feld respecvel and are consdered n equaons (3) as me-ndependen. Eh -h componen of o equaons for x I however has he oscllaor dependence on me. Parameer s reall me-ndependen. The soluons and are ha eners no parameer resul wh he use of equaons (9) one has cos sn (4) cos sn. can be obaned from he frs of equaons (3) whou he rgh hand sde here. As a x cos sn (5) Δ χ cos n he roang reference frame where we rean erms proporonal o onl. Noe ha for equaons (5) represen he relaxaon-free soluon of he Bloch equaons ( T T ). I s seen from equaons () and (5) ha are nerrelaed. ubsuon of from equaons (5) no defnons () of and gves and

7 u χ cos ' (6) Le us consder he average value of cos '. a : Tr Tr sn (7) where he equlbrum dens marx s ħ ħi ħ ħ n n n n exp B / kt / Z I pas pas kt kt kt kt and where concenraons of he ve spns n and he passve spns Thus n he roang reference frame one has n pas are gven b equaons (9) and (). (8) where m ' l and I k ħ I I I m sn k ml m l mk n npas kt ' I ml mk k l l k m denoe he confguraons of n ve and n pas passve spns respecvel wh m. Performng summaon over all possble spn confguraons we oban (9) ħ sn cos α cos ; 4kT (3) u cos ; u ; (3) Fors cos and cos a cos. 3 r should be averaged over posons r and resonance frequences he sample [7 3]. Le us suppose ha here s one relaxaon mechansm onl sa dpole neron spns. Then one has [3] 3 cos = exp n d f d r cos exp r of spns n H beween ve where C 4 g B / 9 3 spns Cn d f (3) (here g s g-for B Bohr magneon). mlar calculaons for he neron wh passve H I gve

8 I cos exp r and for radaon dampng Cn I pas 3 r cos exp n d f d r cos exp (33) where ~ a n d f (34) a a s anoher numercal parameer. The same averagng process appled o cos equaons (3)) leads o cos exp cos respecvel. Below wll be shown b numercal calculaons ha deca parameers and ha man conrbuon o he nuaons deca s due o he radaon dampng (3) gves The response of he ssem s gven b sum over ve spns n and I (n are neglgbl small and. eanng hs las one n expresson ħ sn exp 4kT. (35) g sn Bħ M n exp d 4kT where specral concenraon of he ve spns n s gven b expresson (7). (36) III. NUMEICAL CALCULATION In expermens [4 5] power of he excaon pulses expressed n ab frequenc s vared n range / 5 kh 3 kh. Fgure shows he me evoluon of he specral dsrbuon of he ve spns n gven b formula (7) for / kh. Oher parameers nvolved are relaed o sample #3 of ef. 5: n cm / MH wh use of hese parameers. n.4 cm 7 3 T 75 μs T. s. All oher fgures below excep Fgures 4 and 5 are obaned

9 FIG.. Tme evoluon of he specral dsrbuon of ve spns n 7 3 5: n.4 cm T 75 μs T. s / MH T 4. K. (formula (7)) for / kh for sample #3 of ef. FIG.. The excaon knecs of spns dsrbuon s Gaussan Lorenan or recangular wh one and he same lne wdh. n durng he nuaon process wh parameers gven n Fg. when nhomogeneous Tme dependence of he concenraon of ve spns respecvel and n and passve spns pas n are gven b formulae (9) and () n s depced n Fg. wh parameers relaed o sample #3 of ef. 5. I s obvous from expressons (7) and (9) ha he maxmall possble value of n s equal o n /. Fgure shows ha even wh he larger values of he ab frequenc / kh onl a lle par of n s parcpae n he excaon process. As s seen from expressons (3) (33) and (34) respecvel nsead of ordnar consan dampng parameer / T he nuaon response s descrbed b nens- concenraon- frequenc- and me-dependen parameers and I parameer. respecvel. Tha s an -h spn pke s charered b s proper me-varng dampng

10 FIG. 3. pecral dsrbuon and me dependence of he conrbuons o he nuaon dampng parameer nerons beween ve spns ; (b) from dpolar nerons of ve spns wh passve ones I 5 3 dampng. The used parameers are as n Fg. geomercal for a. cm axs. Fgures 3(a) 3(b) and 3(c) depc he : (a) from dpolar ; (c) from radaon. Noe he scale change along vercal - dependen conrbuons o he dampng from dpolar nerons beween ve spns of ve spns wh passve ones and conrbuon due o he radaon dampng I respecvel (agan for / kh and wh he use of parameers relaed o sample #3 of ef. 5). Fgure 3(c) and wha follows are obaned wh he value of he geomercal for resonanl exced expresson (36) nsead of ve spns conrbuon and a leads o he neglgble dampng of nuaons cm. A comparson wh Fg. shows ha for. Tha s subsuon of hose n I

11 FIG. 4. Nuaonal response and deca funcon K() (op) calculaed for sample # n ef. 5 wh / 3 kh (for oher parameers see ex). Fgure 4 shows pcal nuaon response and he deca funcon K (obaned b comparng maxma of he sgnal and of he Bessel funcon J ( ) see [4 6]) calculaed wh he use of parameers n.6 cm μs T T.7 s relaed o sample # of ef. 5 for / 3 kh (see Fg. n ef. 5). Noe ha Hesenberg equaons () and her soluons are wren down whou use of he low-power conrbuons o he dampng and hence all he resuls of he presen work are correc for. From comparson of he nuaon response of Fg. 4 wh he expermenal one n Fg. of ef. 5 we see ha he presen heor s correc a he mddle par of he nuaonal process bu fals a he begnnng (frs maxmum n Fg. 4) where curved up) where s small due o small concenraon s small due o decrease of. n and a he al (where he deca funcon s lne s somewha FIG. 5. Calculaed nuaon deca rae versus ab frequenc for hree samples of ef. 5 wh concenraons n n n3. Nex Fg. 5 shows he calculaed dependence of he nuaon deca rae on he ab frequenc for hree samples of ef. 5 obaned b use of K resuls. Comparson wh Fg. 3 of ef. 5 reveals qualave concdence beween heor and expermen n he concenraon dependence alhough he - dependence s no so sragh lne as n he expermen and as

12 should be cordng o formula (). If one reans n he fgure onl daa pons wh 75 kh / kh (ha s ) lnear f gves for n formula () values.5.6 and β 3 =.95 whereas expermen (ef. 5) gves 6..5 and for he hree samples respecvel. As should be he agreemen beween heor and expermen s beer for sample wh larger concenraon snce ~ n ~ n and he condon s fulflled easer n hs case. IV. DICUION AND CONCLUION Obaned n precedng secons resuls can be commened as follows. If he wavelengh of he drvng feld s large compared wh he sample dmensons or wh he dsances beween spns all he spns wll ner collecvel wh a sngle mode of he feld formng he cooperave superradan Dcke saes [8]. As a resul nsead of raher weak couplng beween a sngle spn and he elecromagnec feld wh he couplng srengh g he srong collecve couplng of he ensemble of N spns o he cav mode wh he couplng srengh gcol ~ N g s realed [9]. For large values of phoon number nph N n he cav he srong collecve couplng srengh g converges o he classcal lm gcol ~ nph g ~ [9]. In our col classcal descrpon of he cav mode he spn-cav couplng s gven b Hamlonan H x (3). nce n he whole range of Boscano expermens [4 5] condon s fulflled he srong couplng lm s realed here. The srong couplng leads o he enhancemen of he sponaneous emsson rae n he resonan cav [] and as a consequence o he radaon dampng. I s remarkable ha due o he wo-phoon excaon procedure n expermens [4 5] one has a pure sponaneous emsson n he cav. The nhomogeneous lne broadenng deermnes he specral dsrbuon of he ve spns concenraon (see Fg. ) and has drasc nfluence on he frequenc dependence of he deca raes due o he dephasng of he spns durng he me evoluon (Fg. 3). As s seen from expressons (5) all he hree deca mechansms I affec onl he oscllang pars of he spn componens. Furher he mechansms affec he longudnal and ransverse spn componens equall servng as phase and energ relaxaon mechansms smulaneousl. I was shown [3] ha he deca rae of he ab oscllaons depends no onl on he wdh of he specral dsrbuon bu on s form as well. In parcular he deca rae s sgnfcanl slower n he case of he Gauss dsrbuon (he so-called cav proecon effec ef. ) han for he Lorenan one. To es hs effec n our case we compare he nuaon deca raes for Gaussan Lorenan and recangular forms of he nhomogeneous broadenng wh one and he same value of he lne wdh. I s no so eas o observe he dfferences however beween 3D pcures of hese hree forms. Hence n Fg. 6 we compare he resonance componens (smlar o Fg. 3(c)) for (.e. he hump lne n Fg. 3(c)). Noe ha he smlar relaonshp s reaned for as well. As s seen from formula (34) concenraon n and negral I a d f Lorenan dsrbuon funcons ha n r n G n L s gven b produc of he. I s clear from defnons of he recangular Gaussan and excepng far wngs (wh nsuffcen conrbuon o n due o small s here see formulas (6) and (7)) and he same nequal s rue for n as s shown n Fg.. On he oher hand he negral pcured for n Fg. 7 s nall less n he Lorenan case bu graduall becomes larger han ha for he recangular and Gaussan dsrbuons.

13 FIG. 6. Tme dependence of deca rae / π of resonanl exced funcon f s of a Gaussan Lorenan or recangular pe. spns when nhomogeneous dsrbuon f s of a FIG. 7. Tme dependence of he negral I a d f when nhomogeneous dsrbuon funcon Gaussan Lorenan or recangular pe. Agan a such behavor s reaned for funcon K exp funcon. As a resul n ransen nuaon regme he deca rae and deca are more nensve n he case of he Lorenan dsrbuon as s seen from Fg. 8 for he deca K. Here afer removal of frs hree daa pons a lnear f gves for he overall deca rae numercal values /.75 (kh) /.96 (kh) and / 5.8 (kh). r G L

14 FIG. 8. Deca funcon K exp when nhomogeneous dsrbuon funcon recangular pe. f s of a Gaussan Lorenan or In concluson we have consdered drvng-feld nens and concenraon dependen dampng of nuaon sgnals observed n varous dlued paramagnec solds. We have shown ha n nhomogeneousl broadened spn ssem conrbuon of he dpolar neron o he nuaons deca s neglgbl small. A radaon dampng mechansm s elaboraed well descrbng he expermenall observed properes of nuaons n dlued paramagnec solds. I s shown ha nsead of usual Bloch deca parameers T and T he dampng caused b hs mechansm s descrbed b he nens- concenraon- frequenc- and me-dependen dsrbuon of parameer and he dependences are dsplaed. The consdered mechansms affec he longudnal and ransverse spn componens equall servng as phase and energ relaxaon mechansms smulaneousl. Fnall snce repea expermens [4 5] wh superconducng caves. ~ a ~ where s he ve ressance of he cav s desrable o [] H. C. Torre Phs. ev (949). [] P. W. Akns A. J. Dobbs and K. A. McLauchlan Chem. Phs. Le. 5 5 (974). [3] G. B. Hocker and C. L. Tang Phs. ev. Le. 59 (968). [4]. Boscano F. M. Gelard and J. P. Korb Phs. ev. B (993). [5]. Agnello. Boscano e al. Phs. ev. A (999). [6]. M. akhmaulln I. N. Kurkn e al. Phs. ev. B (9). [7] E. I. Babekov I. N. Kurkn e al. J. Magn. es 9 69 (). [8]. Berana. Gambarell e al. Naure Nanoech. 39 (7). [9]. Nellula K. Y. Cho e al. Phs. ev. Le (7). [] J. H. hm. Berana e al. Phs. ev. Le (). [] H. De aed B. Barbara e al. Phs. ev. B (). []. Berana C.-E. Duo e al. arxv: (4). [3] E. I. Babekov JETP Le.93 9 (). [4] E. I. Babekov M.. Gafurov e al. arxv (cond-ma): (4). [5] N. Bloembergen and. V. Pound Phs. ev (954). [6] N. Ya. Asadullna T. Ya. Asadulln and Ya. Ya. Asadulln J. Phs. Condens. Maer (). [7] W. B. Mms Phs. ev (968). [8]. H. Dcke Phs. ev (954). [9] I. Chorescu N. Groll e al. Phs. ev. B (). [] E. M. Purcell Phs. ev (946). [] Z. Kuruc J. H. Wesenberg and K. Molmer Phs ev. A (). [] I. Dn. Porolan e al. Phs ev. A (). [3] D. O. Krmer. Pu e al. Phs ev. A (4).