Forced synchronization of a quantum dissipative dynamics

AIP Con e ence P oceedings 922, 507 (2007); h ps://doi.o g/10.1063/1.2759730 922, 507 © 2007 Ame ican Ins i u e o Physics. Fo ced synch oniza ion o a quan um dissipa i e dynamics Ci e as: AIP Con e ence P oceedings 922, 507 (2007); h ps:// doi.o g/10.1063/1.2759730 Published Online: 20 July 2007 Igo Goychuk, Jesús Casado-Pascual, Manuel Mo illo, Jö g Lehmann, and Pe e Hänggi Fo ced synch oniza ion o a quan um dissipa i e dynamics Igo Goychuk*, Jesus Casado-Pascual^ Manuel Mo ilk^, Jo g Lehmann** and Pe e Hanggi* *Ins i u ii Physik, Uni e si a Augsbu g, Uni e si d ss . 1, D-86135 Augsbu g, Ge many ^Fisica Ted ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, Se illa 41080, Spain **Depa emen u Physik und As onomie, Uni e si a Basel, Klingelbe gs asse 82, CH-4056 Basel, Swi ze land Abs ac . We gene alize he phenomenon o o ced s ochas ic synch oniza ion in o he quan um domain wi hin he amewo k o a pa adigma ic spin-boson model ( unneling cha ge, o lipping spin 1/2 coupled o an en i onmen ) which is d i en by an ex e nal pe iodic ec angula ield. The o e damped egime o dissipa i e quan um unneling is s udied. The mal noise assis ed synch o- niza ion o a e y high quali y is shown o occu in a b oad ange o empe a u es, d i ing s eng hs and equencies, i he ex e nal d i ing equency exceeds he ze o- empe a u e limi o dissipa i e unneling a e, he dissipa ion s eng h exceeds a c i ical alue, and he d i ing is su icien ly s ong. A simple c i e ion o such s ochas ic synch oniza ion is es ablished. Bo h he simila i ies and he p o ound di e ences wi h he akin phenomenon o quan um s ochas ic esonance a e ou lined. Keywo ds: o ced s ochas ic synch oniza ion, wo-s a e quan um dynamics, dissipa ion PACS: 05.60.Gg, 05.40-a, 05.45.X , 82.20.Gk INTRODUCTION Synch oniza ion is a e y uni e sal phenomenon in classical physics [1]. In pa icula , o ced synch oniza ion e e s o he locking o he phase o a nonlinea d i en oscilla o o ha o a pe iodic d i ing ield. In he p esence o noise, i is also possible o de ine a s ochas ic clock wi h some mean phase equency which depends on he noise s eng h. The o de ing and locking o his s ochas ic clock o he pacemake d i ing clock wi hin some noise ange is called s ochas ic synch oniza ion [2]. The less p obable phase slips be ween he d i e and he d i en sys em a e, he be e is he quali y o synch oniza ion. Can a quan um s ochas ic clock p o ided by a unneling cha ge jumping a andom imes be ween wo di e en si es o localiza ion in a dissipa i e en i onmen , o a quan um spin 1/2 lipping a andom be ween wo o ien a ions be synch onized wi h a pe iodic ield and unde wha condi ions? We answe his in iguing ques ion wi hin he amewo k o he pa adigma ic spin-boson model in he p esence o an ex e nal ec angula d i ing. CP922, Noise and Fluc ua ions, I1?11 In e na ional Con e ence, edi ed by M. Tacano, Y. Yamamo o, and M. Nakao © 2007 Ame ican Ins i u e o Physics 978-0-7354-0432-8/07/S23.00 507 MODEL, THEORY, AND RESULTS The model is desc ibed by he ollowing Hamil onian A( ) = -e( )dz + -Mdx + -dz^Kj(h) + hj) + ^n<oj(h)hj + -), (l) whe ein, he ope a o s az and ax deno e he s anda d Pauli ope a o s, e( ) is a ime- dependen ene gy bias, and HA is he unnel ma ix elemen . The ba h Hamil onian [las e m in Eq. (1)] is exp essed in e ms o he ope a o s c+ and bj associa ed o he j' h ba h no mal mode wi h equency COj. The s ochas ic in luence o he quan um he mal ba h is cap u ed by an ope a o andom o ce |( ) = £/ Kj(P-e"°^ + bje~l<°' ). I can be cha ac e ized by he spec al densi y /( o) = (n/Ti) Ly K?5( o — o,). We assume ha J((o) acqui es he Ohmic o m, /( o) = 27zhoc(De~m>mc, wi h ic ion s eng h a and an exponen ial cu o . We conside he o e damped limi a > 1/2 and weak unneling limi A <C oc. Then, he dynamics o he diagonal elemen s o he educed densi y ma ix, Pp (0 = P±( )> is gi en by he Pauli mas e equa ion [3] Pp( ) =W_p( )p_p( )-Wp( )pp( ) , (2) whe e he ime-dependen elaxa ion a es Wp ( ) wi hin he Golden Rule app oxima ion and he app oxima ion o adiaba ically slow a ying bias e( ) ead: 1 °° W±( ) = -A2 dTexp[-g'(T)]cos 2 Jo The unc ions Q'( ) and Q"( ) deno e he eal and imagina y pa s o he dissipa ion ke nel 2(0 = % + i /V *2 (l( e)l(0)) , (4) n Jo Jo whe ein X = J0°° dcoJ( o) / (%co) is he ba h eo ganiza ion ene gy [3]. Fo he conside ed model one inds ha X = 2ahcoc and m = 2«iB{^T^| (1^^0|2} (5) g"( ) = 2aa c an( oc ) . (6) In Eq. (5), T{£) deno es he Gamma unc ion, COT = ksT/h, and K = (OT/(OC. The quan um a es sa is y W_( ) = exp(—e( )/( cB ))W+( ) o each ins an o ime . A ze o empe a u e, .2 / i /. 2a—l W±,M ) = 0^(0]^^ (^J exp[T£(0/(. oc)], whe e 0( ) is he Hea iside s ep unc ion [4], i.e. one o he a es is non- ze o, while ano he one is exac ly ze o. In he limi o high- empe a u es (clas- sical en i onmen ) and quasi-s a ic en i onmen al noise limi , he a es a e e"(T)T^(o* (3) 508 well app oxima ed by he celeb a ed Ma cus-Le ich-Dogonadze a e exp ession W±{ ) = {n/2)hA2/V4nnBTexpl-(±e( ) - I /{AXkBT) . In a ully quan um egime o kgT < hcoc, he a es ha e o be e alua ed nume ically. Ou hough -expe imen al se up mimics he quan um analogue o a classical (phase)- synch oniza ion beha io elabo a ed in Re s. [5, 6]. A quan um pa icle wi h he cha ge q (elec on, o p o on) makes unneling (ins an ) jumps o h and back be ween wo si es o localiza ion sepa a ed by he dis ance 0. The ene gy bias is modula ed in ime by he applied elec ic ield <?( ) yielding e( ) = q 0S'( ) (al e na i ely, i could be a spin 1/2 and a magne ic ield). The ex e nal ield al e na es also dicho omously, changing i s di ec ion, howe e , pe iodically in ime. The unneling jump p ocess can be desc ibed ma hema ically as a classical eleg aph noise de ined by he mas e equa ion (2) wi h a es which a e ully quan um-mechanical and ime-dependen . We a e in e es ed in he synch oniza ion o unneling e en s sepa a ed by andom ime in e als wi h he ex e nal d i e al e na ions. One hen coun s he numbe o jumps n( ) wi hin a ime window [ o, ). Following Re . [5], we in oduce he an- dom phase (j>( , o) = nn{ ), which inc eases by % a each swi ching e en ( wo sub- sequen swi ches co espond o a 27 -cycle o andom du a ion), and de ine he a e - age equency and di usion coe icien s associa ed o he s ochas ic phase-p ocess as Qph := lim^oo(<KMo))/(?-?o) and2Dph := lim,^, [(<j>2( , 0)) - (<p{ , 0))2] /( - 0), espec i ely. Using a s ochas ic pa h-in eg al desc ip ion o he d i en eleg aph p o- cess [7], he ollowing exac esul s we e ob ained o a pe iodic ec angula d i ing e( ) = ±8o wi h ampli ude EQ and equency Q. = 2%j 2? [6, 8] — nW i , Qph = — ^ 1 - 8piq and 2n2 4 anh(W^/4) (7) 2Dph = 7 Qph-^5^q anh3(W^/4) -^8p2eq(l-8p2eq){mmHW^/4) -W^[l+2sech2(W^/4)] . (8) He e, W deno es he sum o he o wa d and backwa d a es in Eq. (3) o a ixed alue o he ield <?o, i.e., o e( ) = eo = q oSb, and 8peq = anh(eo/(2 c£ )) is he absolu e alue o he di e ence o he equilib ium popula ions. The in e se Fano ac o o he coun ing p ocess R := 7 Qph/(2Dph) p o ides a eliable quali y measu e o o ced syn- ch oniza ion [2, 5]. I co esponds o he numbe o unneling jumps synch onized o he d i ing al e na ions. Desynch oniza ion occu s, when he a iance o he synch onized coun ing p ocess becomes abou uni y, (8n2(Synch o)} = 1 (a phase slip occu s). F om his condi ion, he (de)synch oniza ion ime ollows as 'synch o ^ K--J /£ • 509 C i e ion o s ochas ic synch oniza ion Fo s ochas ic synch oniza ion o occu he pa icle mus ha e su icien ime o make a ansi ion o he lowe ene gy s a e du ing he il ing hal -pe iod. This equi es ha he mean esidence ime (T+(e)) = l/W+(6o) in he highe ene gy s a e should be much less han ST jl. On he o he hand, he backwa d ansi ions du ing such a o wa d il mus be p ohibi ed. The synch oniza ion c i e ion hus eads, l/W+{eo) < 572 < l/W-{eo) = exp{e0/kBT)/W+{e0) . (9) I should be con as ed wi h he c i e ion o S ochas ic Resonance (SR) eading [9] l/W+(eo) = STjl o 7 W+(e0) = Q. (10) Clea ly, o a weak d i ing, wi hin he linea esponse (LR) app oxima ion, eo <C ksT, he c i e ion (9) can ne e be jus i ied, while SR can occu [9]. Mo eo e , SR is e- quen ly no ela ed o synch oniza ion a all. In pa icula , he ea lie wo ks on quan- um SR (QSR) [10] wi hin he same spin-boson model unco e ed, wi hin he pa ame e egime a < 1, ha QSR in he conside ed symme ic (in he absence o d i ing) sys em is impossible, and an addi ional s a ic bias is equi ed [9]. This e en was hough o be a main ea u e o QSR, as compa ed wi h he classical SR [9]. Such QSR in a biased spin- boson sys em has, howe e , no ela ion o synch oniza ion, as i does no co espond o any ma ching o he ime scale o d i ing and ha o he dissipa i e unneling dynamics. In [11] i was shown, howe e , ha QSR is also possible wi hin he LR heo y o he s udied symme ic model p o ided ha quan um ic ion is su icien ly s ong, a > 1. This pa ame e egime is ele an , e.g. o nonadiaba ic elec on unneling in condensed molecula sys ems [3], whe e a can be as la ge as a ~ 5 — 10 and e en la ge . We we e guided by his ea lie QSR esea ch o ind he pa ame e egime, whe e he he mal noise-assis ed Quan um S ochas ic Synch oniza ion (QSS) can occu . A undamen al di e ence o QSS, as compa ed wi h i s classical coun e pa , is ha QSS is expec ed o occu always a T = 0, i he d i ing is su icien ly slow, Q. <C W =o(£o). This is so because one o he a es is exac ly ze o, as i ollows om he de ailed balance condi ion. Thus, a desynch oniza ion ansi ion occu s wi h enhancing he empe a u e abo e some h eshold alue [8]. The ques ion is, howe e , whe he he he mal noise can help o synch onize when Q. > W =o(so) and he pa icle canno ollow he bias al e na ions a T = 0. As in he case o QSR, his ques ion is highly non i ial because o he non-A henius, powe law empe a u e dependence o he unneling a es. Ou nume ical s udy based on he ou lined analy ical heo y e ealed ha such a quan um he mal noise assis ed QSS is only possible when a > 1, in ag eemen wi h he QSR esea ch. Mo eo e , la ge a ~ 5 — 10 a e p e e able. Nume ical esul s Le us i s illus a e a undamen al di e ence be ween QSR and QSS o a la ge ic ion a = 10, some ixed unnel ma ix elemen A = 4 • 10~4, equency Q = 10~n and 510 o a mode a e d i ing s eng h eo = 0.5 which is al eady beyond he LR condi ion. All he dimensional quan i ies a e scaled (h = I,kg = 1) in e ms o he quan um c osso e empe a u e Tc = Hcoc/kB o he mal ba h oscilla o s. The pa ame e s a e chosen o mimic nonadiaba ic elec on unneling in he molecula dime s o azu in [8]. i-H o o «J Pi 1 3 1.2 1.1 1 II u.y 0.8 0.7 1 1 _(b) / ~ , , - - V — X. '0.3 0.4 0.5 0.6 K 0.7 0.8 FIGURE 1. (a) Mean s ochas ic equency 2ph, mean phase di usion coe icien Dph and (b) he in e se Fano ac o R e sus he scaled empe a u e K o he d i ing s eng h £o = 0.5. O he pa ame e s a e gi en in he ex . Fig. 1(a) showsjha synch oniza ion o he chosen pa ame e s is absen . The mean phase equency Qph as a unc ion o he scaled empe a u e K c osses he line Q. = 10~n a a ce ain poin . Howe e , any equency locking supplemen ed by a minimum o he phase di usion coe icien is absen . The e o e, his is no a synch oniza ion, pe de ini ion, as any synch oniza ion equi es a equency locking. The c ossing poin a ound K « 0.5 co esponds, howe e , o QSR as i can be deduced om Fig. 1(b). Indeed, he quali y, o cohe ence ac o R displays a smoo h maximum when quan um s ochas ic esonance occu s. The ampli ude o his maximum is, howe e , so small ha desynch onizing phase slips occu pe manen ly, e en whe e a d i ing-induced phase cohe ence, R > 1, exis s. Such QSR can be conside ed also as a kind o cohe ence esonance [12] induced by he ex e nal d i ing. I cons i u es a p ecu so o quan um s ochas ic synch oniza ion ha occu s wi h a u he inc ease o he d i ing s eng h abo e some h eshold eo « 2.5, o o he pa ame e s ixed. Fo eo = 5 he quali y o synch oniza ion is al eady e y high as e idenced by Fig. 2. 10 * io-10 IQ ^ io"12 OH IG .14 10 (a)"1 iiph -- 2Dph x" y V,; 10 10 10 K FIGURE 2. The same as Fig. 1 o he d i ing s eng h £Q = 5. 511 In his igu e, he mean phase equency Qph is clea ly locked o he ex e nal equency Q in a b oad ange o empe a u es, while he mean phase di usion coe icien Dpb displays a p onounced minimum a some empe a u e. A his minimum, he quali y o he synch oniza ion is as onishingly high [see Fig. 2(b)] eaching he maximum abou 105 phase-locked unneling jumps be o e a phase slip occu s. Ou es ima ions o elec on unneling in molecula dime s like azu in show [8] ha o unneling dis ances TQ ~ 15 A, c osso e empe a u es Tc ~ 150 K, unnel couplings TiA ~ 5 • 10~3 meV, he equi ed d i ing equencies a e in he ange om se e al Hz o se e al hund eds Hz, while he elec ical ield s eng h should be abou 5 • 104 V/cm o a i e a a high quali y synch oniza ion. Such expe imen al s udies should be easible in labo a o ies. ACKNOWLEDGMENTS J. C.-P. and M. M. acknowledge he suppo o he Minis e io de Educacion y Ciencia o Spain (FIS2005-02884) and he Jun a de Andalucia. I. G. and P. H. acknowledge suppo by he DFG h ough SFB 486 and by he Ge man Excellence Ini ia i e ia he Nanosys ems Ini ia i e Munich (NIM). REFERENCES 1. A. Piko sky, M. Rosenblum, and J. Ku hs, Synch oniza ion: A Uni e sal Concep in Nonlinea Science, Camb idge Uni e si y P ess, Camb idge, 2001. 2. V. Anishchenko, A. Neiman, A. As akho , T. Vadia aso a, and L. Schimansky-Geie , Chao ic and S ochas ic P ocesses in Dynamic Sys ems, Sp inge , Be lin, 2002. 3. I. Goychukand P. Hanggi, Ad . Phys. 54, 525-584 (2005). 4. U. Weiss, Quan um Dissipa i e Sys ems, 2nd ed., Wo ld Scien i ic, Singapo e, 1999. 5. J. A. F eund, A. B. Neiman, and L. Schimansky-Geie , Eu ophys. Le . 50, 8 (2000); J. A. F eund, L. Schimansky-Geie , and P. Hanggi, Chaos 13, 225 (2003). 6. J. Casado-Pascual, J. Gomez-O donez, M. Mo illo, J. Lehmann, I. Goychuk, and P. Hanggi, Phys. Re . E71, 011101 (2005); J. Casado-Pascual, J. Gomez-O donez,andM. Mo illo, PhysicaA35l, 111 (2005). 7. I. Goychuk and P. Hanggi, Phys. Re . £ 61, 4272 (2000). 8. I. Goychuk, J. Casado-Pascual, M. Mo illo, J. Lehmann, and P. Hanggi, Phys. Re . Le . 97, 210601 (2006). 9. L. Gammai oni, F. Ma chesoni, P. Jung and P. Hanggi, Re . Mod. Phys. 70, 223 (1998). 10. R. Lo s ed and S. N. Coppe smi h, Phys. Re . Le . 72, 1947 (1994); M. G i oni and P. Hanggi, Phys. Re . Le . 76, 1611 (1996). 11. I. Goychuk and P. Hanggi, Phys. Re . E 59, 5137 (1999); I. Goychuk and P. Hanggi, New J. Phys. 1, 14 (1999). 12. A. S. Piko sky and J. Ku hs, Phys. Re . Le . 78, 775 (1997). 512