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Quantum-Classical Dynamics of Wave Fields Alessandro Sergi ∗ Dipartimento di Fisica, Universitá degli Studi di Messina, Contrada Papardo 98166 Messina, Italy An approach to thequantum-classical mechanics of phase space dependent operators, which has beenproposedrecently,isremodeledasaformalism forwavefields. Suchwavefieldsobeyasystem ofcouplednon-linearequationsthatcanbewrittenbymeansofasuitablenon-Hamiltonianbracket. As an example, the theory is applied to the relaxation dynamics of the spin-boson model. In the adiabaticlimit,agoodagreementwithcalculationsperformedbytheoperatorapproachisobtained. Moreover, the theory proposed in this paper can take nonadiabatic effects into account without resorting to surface-hopping approximations. Hence,theresults obtained follow qualitatively those of previous surface-hopping calculations and increase by a factor of (at least) two the time length over which nonadiabatic dynamics can be propagated with small statistical errors. Moreover, it is 7 worthtonotethatthedynamicsofquantum-classicalwavefieldshereproposedisastraightforward 0 non-Hamiltonian generalization of the formalism for non-linear quantum mechanics that Weinberg 0 introduced recently. 2 n a I. INTRODUCTION condensed phases. Therefore, it is worth to search for a J reformulation of the theory of Refs. [6, 7, 8] that, while 0 There are many instances where a quantum-classical maintaining such features, could be used to integrate re- 1 liably long-time nonadiabatic dynamics. description can be a useful approximation to full quantum dynamics. Typically, a quantum-classical picture To this end, one can note that, within standard quan- 5 often allows one to implement calculable algorithms on tum mechanics, some problems that are formidable to v 9 computers wheneverchargetransferis consideredwithin solve by means of the dynamics of operators become 2 complex environments, such as those provided by pro- much simpler to handle when, instead, the time evolu- 2 teins or nano-systems in general [1]. With respect tion of wave functions is considered [9]. Hence, for anal- 1 to this, an algebraic approach has been recently pro- ogy,it might also happen that, within quantum-classical 1 posed [2, 3] in order to formulate the dynamics and mechanics, the correspondence between operators and 5 the statistical mechanics [4] of quantum-classical sys- quantum-classical wave functions could open new possi- 0 tems. Generalquestionsregardingthequantum-classical bilities for useful approximations in order to carry long- / h correspondence have also been addressed within a sim- timecalculationsefficiently. Indeed,findingandapplying p ilar framework [5]. The approach of Refs. [2, 3] rep- thecorrespondencebetweenoperatorandwaveschemeof t- resents quantum-classical dynamics by means of suit- motioninquantum-classicalmechanicsisthescopeofthe n able brackets of phase space dependent operators and present paper. A wave picture for quantum-classical dy- a u describes consistently the back-reaction between quan- namics can be found by direct algebraic manipulation of q tum and classical degrees of freedom. Notably, a partic- the equation of motion for the density matrix. In prac- : ular implementation of this formalism has been used to tice,thesingleequationobeyedbythequantum-classical v calculate nonadiabatic rate constants in systems model- density matrix is mapped onto two coupled non-linear i X ing chemical reactions in the condensed phase [6]. How- equations for quantum-classical wave fields. Despite its r ever,suchschemeshaveonlypermittedthesimulationof non-linear character,such a quantum-classical dynamics a short-time nonadiabatic dynamics because of the time- ofphasespacedependentwavefieldscorrespondsexactly growing statistical error of the algorithm. Neverthe- to the dynamics ofphase space dependent operatorsdis- less, the algebraic approach [2, 3], underlying the algo- cussed in Refs. [2, 3, 6, 7, 8] and can be used to devise rithms of Refs. [6], has some very nice features, such as novel algorithms and approximation schemes. the (above mentioned) proper description of the back- The abstract algebraic equations here presented are reaction between degrees of freedom, that one should readily expressed in the adiabatic basis and applied, in not give up when addressing quantum-classical statis- order to provide an illustrative example, to the spin- tical mechanics. Moreover, quantum-classical brackets boson model and its relaxation dynamics both in the define a non-Hamiltonian algebra [7] so that their ma- adiabatic and nonadiabatic limit. By making a suit- trix structure allows one to introduce quantum-classical able equilibrium approximation to the non-linear wave Nosé-Hoover dynamics [7] and to define the statistical equations,itisfoundthatnonadiabaticdynamicscanbe mechanics of quantum-classical systems with holonomic propagated, within the wave picture, for time intervals constraints[8]. Alloftheabovefeaturesoftheformalism that are a factor of two-three longer than those which are highly desirable when studying complex systems in have been spanned in Ref. [10] by means of the operator theory [2, 3, 6, 7, 8]. Such a result is very encouraging forpursuingthelong-timeintegrationofthenonadiabatic dynamics of complex systems in condensed phases. ∗E-mail: [email protected] Following a line of research that investigates the rela- 2 tions between classical and quantum theories [11], it is is the commutator and worthtonotethatthe wavepictureofquantum-classical 2N ∂Hˆ ∂χˆ mechanics,whichis introducedinthis paper,generalizes Hˆ,χˆ = (3) withinanon-Hamiltonianframeworktheelegantformal- { }B ∂XiBij∂Xj i,j=1 X ism that Weinberg [12] proposed for describing possible isthePoissonbracket[14]. Boththecommutatorandthe non-linear effects in quantum mechanics [13]. Poissonbracketaredefinedintermsoftheantisymmetric This paper is organized as follows. In Section II the matrix non-Hamiltonianalgebraofphasespacedependentoper- ators is briefly summarized. In Section III the quantum- B= 0 1 . (4) classicaldynamicsofoperatorsistransformedintoathe- 1 0 (cid:20) − (cid:21) ory for phase space dependent wave fields evolving in ThelastequalityinEq.(1)definesthequantum-classical time. Such a theory for wave fields is also expressed bracket. Following Refs. [7, 8, 15], the quantum-classical by means of suitable non-Hamiltonian brackets: in this law of motion can be easily casted in matrix form as wayalinkisfoundwiththegeneralizationofWeinberg’s non-linear formalism given in Appendix A. More specif- d χˆ = i Hˆ χˆ D Hˆ ically, in Appendix A, Weinberg’s formalism is briefly dt ¯h · · χˆ (cid:20) (cid:21) reviewed and its symplectic structure is unveiled. Then, i (cid:2) (cid:3) thisstructureisgeneralizedbymeansofnon-Hamiltonian = ¯h[Hˆ,χˆ]D , (5) brackets. Therefore, one can appreciate how the gen- where eralized Weinberg’s formalism establishes a more com- prehensive mathematicalframeworkfor non-linearequa- 0 1 h¯ ...,... D = − 2i{ }B . (6) tions of motion, comprising phase space dependent wave 1+ h¯ ...,... 0 fields as a special case. In Section IV the abstract non- (cid:20) − 2i{ }B (cid:21) The structure of Eq. (5) is that of a non-Hamiltonian linear equations of motion for quantum-classical fields commutator, which will be defined below in Eq. (10), arerepresentedintheadiabaticbasisandsomeconsider- andassuchgeneralizesthestandardquantumlawofmo- ations, which pertain to the numerical implementation, aremade. Bymakinganequilibriumansatz,inSectionV tion [7]. The antisymmetric super-operatorD in Eq. (6) introduces a novel mathematical structure that charac- the non-linear equations of motion are put into a linear terizes the time evolution of quantum-classical systems. form and the theory is applied to the spin-boson model. The Jacobi relation in quantum-classical dynamics is Section VI is devoted to conclusions and perspectives. = χˆ, ξˆ,ηˆ + ηˆ, χˆ,ξˆ + ξˆ,[ηˆ,χˆ] . (7) J D D D D D D h h i i h h i i h i The explicit expression of has been given in Ref. [7] II. NON-HAMILTONIAN MECHANICS OF J where it was shown that it may be different from zero QUANTUM-CLASSICAL OPERATORS at least in some point X of phase space: for this reason the quantum-classical theory of Refs. [2, 3, 7, 8] can be Aquantum-classicalsystemiscomposedofbothquan- classified as a non-Hamiltonian theory. tum χˆ and classical X degrees of freedom, where X = It is worth to note that the quantum-classical law of (R,P) is the phase space point, with R and P coordi- motion in Eq. (5) is a particular example of a more gen- nates and momenta, respectively. Within the operator eral form of quantum mechanics where time evolution is formalism of Refs. [2, 3, 7, 8], the quantum variables definedbymeansofnon-Hamiltoniancommutators. The depend from the classical point, X, of phase space. The non-Hamiltoniancommutator betweentwoarbitraryop- energyofthesystemisdefinedintermsofaHamiltonian erators χˆ and ξîs defined by operator Hˆ = Hˆ(X), which couples quantum and clas- χˆ sical variables, by E = Tr′ dXHˆ(X). The dynamical [χˆ,ξˆ] = χˆ ξˆ Ω , (8) Ω · · ξˆ evolution of a quantum-classical operator χˆ(X) is given (cid:20) (cid:21) R (cid:2) (cid:3) by [2, 3] whereΩisanantisymmetricmatrixoperatoroftheform 0 f[ηˆ] d i 1 Ω= , (9) χˆ(X,t) = Hˆ,χˆ(X,t) Hˆ,χˆ(X,t) f[ηˆ] 0 dt ¯h B− 2 B (cid:20) − (cid:21) 1 h i n o where f[ηˆ] can be another arbitrary operator or func- + χˆ(X,t),Hˆ = Hˆ,χˆ(X,t) , (1) tional of operators. Then, generalized equations of mo- 2 B n o (cid:16) (cid:17) tion can be defined as where dχˆ i Hˆ = Hˆ χˆ Ω dt ¯h · · χˆ (cid:20) (cid:21) Hˆ,χˆ = Hˆ χˆ B Hˆ (2) = i[(cid:2)Hˆ,χˆ] (cid:3). (10) B · · χˆ ¯h Ω h i (cid:2) (cid:3) (cid:20) (cid:21) 3 The non-Hamiltonian commutator of Eq. (8) defines a where the two operators generalizedformofquantummechanicswhere,neverthe- ¯h less, the Hamiltonian operator Hˆ is still a constant of −→ = Hˆ Hˆ,... (15) motion because of the antisymmetry of Ω. H − 2i B ¯h n o ←− = Hˆ ...,Hˆ (16) H − 2i B n o III. QUANTUM-CLASSICAL WAVE havebeenintroducedand isanorderingoperatorwhich DYNAMICS S is chosen so that the left and the right hand side of Eq.(14)coincide by construction[4], whenthe exponen- In Refs. [2, 3], quantum-classical evolution has been tialoperatorsaresubstitutedwiththeirseriesexpansion. formulatedin terms ofphasespace dependentoperators. The existence of such an ordering problem, and of the In this scheme of motion operators evolve according to ordering operator , in Eq. (14) is caused by the Pois- S son bracket parts of the operators in Eqs. (15) and (16). χˆ(X,t) = exp t Hˆ,... χˆ(X) Hence,onecanimaginethatthesolutiontothisproblem D can be found by dealing properly with these parts of the = expnith χˆ(Xi),o (11) brackets. To this end, one can consider the quantum- { L} classical equation of motion for the density matrix where the last equality defines the quantum-classical Li- ouville propagator. Quantum-classical averages are cal- ∂ρˆ i = Hˆ ρˆ culated as ∂t −¯h (cid:2) 0(cid:3) 1 h¯ ...,... Hˆ χˆ (t) = Tr′ dXρˆ(X)χˆ(X,t) · 1+ h¯ ...,... − 2i{0 }B · ρˆ . h i (cid:20)− 2i{ }B (cid:21) (cid:20) (cid:21) Z (17) = Tr′ dXρˆ(X,t)χˆ(X), (12) As above discussed, in Eq. (17) the ordering problem Z arisesfromthetermsintherighthandsidecontainingthe where ρˆ(X) is the quantum-classical density matrix and Poisson bracket operator ...,... . Then, considering ρˆ(X,t) = exp it ρˆ(X). Either evolving the dynam- theidentity1=ρˆρˆ−1 =ρˆ{−1 ρˆ,Eq}.B(17)canberewritten {− L} ical variables or the density matrix, one is still dealing · · as with phase space dependent operators: viz., one deals i with a formof generalizedquantum-classicalmatrix me- ∂ ρˆ = Hˆ ˆ1 t chanics. As it has been discussed in the Introduction, −¯h this theory has interesting formal features and a cer- (cid:2) 0(cid:3) 1 h¯ ...,ρˆ Hˆ − 2i{ }B tain number of numerical schemes have been proposed · 1+ h¯ ρˆ,... 0 · ˆ1 (cid:20) − 2i{ }B (cid:21) (cid:20) (cid:21) tointegratethe dynamicsandcalculatecorrelationfunc- i tions[6,10,16]. However,thesealgorithmshavebeenap- = Hˆ ρˆρˆ−1 −¯h pliedwithsuccessonlytoshort-timedynamicsbecauseof (cid:2) 0 (cid:3) 1 h¯ ...,ρˆ Hˆ statistical uncertainties that grow with time beyond nu- − 2i{ }B merical tolerance. With this in mind, it is interesting to · (cid:20) −1+ 2h¯i{ρˆ,...}B 0 (cid:21)·(cid:20)ρˆ−1ρˆ(cid:21) see which featuresare found whenthe quantum-classical = i Hˆ ρˆ D Hˆ , (18) theory of Refs. [2, 3] is mapped onto a scheme of mo- −¯h · B,[ρˆ]· ρˆ (cid:20) (cid:21) tion where phase space dependent wavefields, insteadof (cid:2) (cid:3) operators, are used to represent the dynamics. where As it is well known [9], in standard quantum mechan- D = B,[ρˆ] ics,the correspondencebetweendynamics inthe Heisen- 0 1 h¯ ...,ln(ρˆ) berg and in the Schrödinger picture rests ultimately on 1+ h¯ ln(ρˆ),... − 2i{ 0 }B the following operator identity: (cid:20)− 2i{ }B (cid:21) (19) eYˆXê−Yˆ =e[Yˆ,...]Xˆ , (13) The operator D in Eq. (19) depends from the B,[ρˆ] quantum-classical density matrix, ρˆ, itself. However, if where [Yˆ,...]Xˆ [Yˆ,Xˆ]. Thus, in quantum-classical one momentarily disregards this non-linear dependence, ≡ theory,onewouldliketoderiveanoperatoridentityanal- Eq.(18)canbemanipulatedalgebraicallyinordertode- ogous to that in Eq. (13). However, as already shown in velopa wavepicture of quantum-classicalmechanics. To Ref.[4],becauseofthenonassociativityofthequantum- thisend,onecanintroducequantum-classicalwavefields, classical bracket in Eq. (5), the identity that can be de- ψ(X) and ψ(X),andmakethefollowingansatzforthe rived is d|ensityi mathrix | eih¯t [Hˆ,...]Dχˆ = S eih¯t−→Hχê−ih¯t←H− , (14) ρˆ(X) = wι|ψι(X)ihψι(X)|, (20) (cid:18) (cid:19) Xι 4 where one has assumed that, because of thermal disor- A. Non-linear wave dynamics by means of der, there can be many microscopic states ψι(X) (ι = non-Hamiltonian brackets | i 1,...,l) which correspond to the same value of the macroscopic relevant observables [17]. In terms of the The waveequationsin (22)were derivedstarting from quantum-classicalwave fields, ψι(X) and ψι(X), and thenon-Hamiltoniancommutatorexpressingthedynam- | i h | consideringthesinglestatelabeledbyι,Eq.(18)becomes ics ofphase space dependent operators[7]. It is interesting to recast quantum-classical wave dynamics itself by ψ˙ι(X) ψι(X) + ψι(X) ψ˙ι(X) = means of non-Hamiltonian brackets. It turns out that | ih | | ih | i this form of the wave equations generalizes the mathe- Hˆ ψι(X) ψι(X) matical formalism first proposed by Weinberg [12] in or- − ¯h | ih | (cid:16) der to study possible non-linear effects in quantum me- + ψι(X) ψι(X)Hˆ | ih | chanics (see Appendix A). 1 (cid:17) Consider a case in which a single state is present, i.e. + Hˆ,ln(ρˆ) ψι(X) ψι(X) ι = 1. Then, consider the wave fields ψ and ψ as 2 B| ih | | i h | (cid:16)n o coordinatesofanabstractspace,anddenotethepointof ψι(X) ψι(X) ln(ρˆ),Hˆ . such a space as − | ih | B n o (cid:17)(21) ψ ζ = | i . (28) ψ Equation(21)canbe written as a systemoftwo coupled (cid:20) h |(cid:21) equations for the wave fields [18]: Introduce the function d ¯h = ψ Hˆ ψ , (29) i¯h ψι = Hˆ Hˆ,ln(ρˆ ) ψι H h | | i dt| (X,t)i (cid:18) − 2in (X,t) oB(cid:19)| (X,t)i and the antisymmetric matrix operator ←d− ¯h −i¯hhψ(ιX,t)|dt = hψ(ιX,t)| Hˆ − 2i ln(ρˆ(X,t)),Hˆ B . 0 1 h¯ {Hˆ,ln(ρˆ)}B|ψi (cid:18) n o (cid:19) Ω= − 2i Hˆ|ψi (22)  1+ h¯ {ln(ρˆ),Hˆ}B|ψi 0  − 2i hψ|Hˆ   Equations (22), which are obeyed by the wave fields, are  (30) non-linear since their solution depends self-consistently Equations (22) can be written in compact form as from the density matrix defined in Eq. (20). These equations are also non-Hermitian since the operators ∂ζ i ∂ζ Hˆ,ln(ρˆ) and ln(ρˆ),Hˆ are not Hermitian. How- ∂t = −¯h ∂∂|Hψi ∂∂hHψ| ·Ω· ∂∂|ψζi  B B h i ∂hψ| enver, thisodoes nont cause poroblems for the conservation i   of probability. The wave fields ψι and ψι evolve ac- = ,ζ . (31) | i h | −¯h{H }Ω;ζ cording to the different propagators Equations (22), or their compact “Weinberg-like” form −→UB,[ρˆ](t) = exp −i¯ht Hˆ − 2¯hi Hˆ,ln(ρˆ) B , icnlasEsqic.a(l3d1y)n,aemxpicrsesosfpthheasweasvpeacpeicdteupreendfoerntthqeuaqnutaunmtudme-- (cid:20) (cid:18) n o (cid:19)(cid:21) (23) grees of freedom [2, 3]. Such a wave picture makes one recognizethe intrinsic non-linearity of quantum-classical it ¯h ←U−B,[ρˆ](t) = exp(cid:20)−¯h (cid:18)Hˆ − 2inln(ρˆ),HôB(cid:19)(cid:21) , dotyhnearmisicsus.esT,hinistshpeecnifiexctfesaetcutiroens.willbediscussed,among (24) so that time-propagating wave fields are defined by IV. ADIABATIC BASIS REPRESENTATION AND SURFACE-HOPPING SCHEMES |ψι(X,t)i = −→UB,[ρˆ](t)|ψι(X)i (25) hψι(X,t)| = hψι(X,)|←U−B,[ρˆ](t). (26) ordEeqrutaotidoenvsis(e2a2)nuarmeerwicraitlteanlgoinritahnmatbosstoralvcettfhoermm,.onIne has to obtain a representation in some basis. Of course, Quantum classical averages can be written as any basis can be used but, since one would like to find a comparison with surface-hopping schemes, the adiabatic χˆ (t) = dX wι ψι(X,t)χˆψι(X,t) . (27) basisisagoodchoice. Tothisend,considerthefollowing h i h | | i Z Xι form of the quantum-classicalHamiltonian operator: One can always transform back to the operator picture P2 to show that the probability is conserved. Hˆ = +hˆ(R), (32) 2M 5 where the first term provides the kinetic energy of the Thewavefields ψι(X) and ψι(X) canbe expandedin classical degrees of freedom with mass M, while hˆ(R) the adiabatic ba|sis as i h | describes the quantum sub-system and its coupling with ψι(X) = α;R α;Rψι(X) = Cι α;R the classical coordinates R. The adiabatic basis is then | i | ih | i α| i α α defined by the following eigenvalue equation: X X ψι(X) = ψι α;R α;R = α;RCι∗(X), hˆ α;R =E (R)α;R . (33) h | h | ih | h | α | i α | i Xα Xα Sincethenon-linearwaveequationsin(22)havebeende- (41) rivedfromthebracketequationforthequantum-classical and the density matrix in Eq. (20) becomes densitymatrix(17),bydealinginasuitablemannerwith the Poisson bracket terms, the most simple way to find ραα′(X,t) = wιCαι(X,t)Cαι∗′(X,t). (42) the representationof the wave equations (22) in the adi- ι X abatic basis is to first represent Eq. (17) in such a basis In order to find two separate equations for Cι and Cι∗, and then deal with the terms arising from the Poisson α α′ one cannot insertEq.(42) directly into Eq.(40) because brackets. The adiabatic representationof Eq. (17) is [3] of the presence of the derivatives with respect to the ∂tραα′(X,t) = i αα′,ββ′ρββ′(X,t), (34) phase space coordinates R ad P. One must set Eq. (40) −ββ′ L intotheformofamultiplicativeoperatoractingonραα′. X To this end, for example, consider where ∂ ∂ ∂ iLαα′,ββ′ = iL(α0α)′,ββ′δαβδα′β′ −Jαα′,ββ′ ∂Pρβα′ = ∂Pρβγ δγα′ = ∂Pρβγ ρ−γµ1ρµα′ γ (cid:18) (cid:19) γµ (cid:18) (cid:19) = (iωαα′ +iLαα′)δαβδα′β′ Jαα′,ββ′ .(35) X X − ∂(lnρˆ) βµ Here, ωαα′ =(Eα(R)−Eα′(R))/¯h≡Eαα′/¯h and = ∂P ρµα′ . (43) µ P ∂ 1 ∂ X iLαα′ = M · ∂R + 2(Fα+Fα′)∂P , (36) Equation(43)showshowtotransformformallyaderiva- tive operator acting on ρˆ into a multiplicative operator where which, however,depends on ρîtself. Therefore, Eq. (40) ∂ˆh(R) becomes F = α;R α;R (37) α −h | ∂R | i i i istheHellmann-Feynmanforceforstateα. Theoperator ∂tραα′ = −¯hEαραα′ + ¯hEα′ραα′ J that describes nonadiabatic effects is P P P 1 ∂ − M ·dαβρβα′ − M ·d∗α′β′ραβ′ Jαα′,ββ′ = −M ·dαβ 1+ 2Sαβ · ∂P δα′β′ Xβ Xβ′ (cid:18) (cid:19) 1 P ∂(lnρ) αµ P 1 ∂ ρµα′ −M ·d∗α′β′ 1+ 2Sα∗′β′ · ∂P δαβ , − 2Xµ M · ∂R (cid:18) (cid:19) (38) 1 P ∂(lnρ)µα′ρ αµ − 2 M · ∂R wheredαβ = α;R(∂/∂R)β;R isthenonadiabaticcou- Xµ h | | i pling vector and 1 ∂(lnρ)αµ Fα ρµα′ P −1 − 2 µ ∂P S =E d d . (39) X αβ αβ αβ M · αβ 1 ∂(lnρ)µα′ (cid:18) (cid:19) Fα′ ραµ − 2 · ∂P Using Eqs. (36) and (38), the equation of motion for µ X the density matrix in the adiabatic basis can be written 1 P ∂(lnρˆ) βµ explicitly as dαβSαβ ρµα′ − 2 M · · ∂P β,µ P ∂ X ∂tραα′ = −i1ωαα′ραα′ − M ∂· ∂Rραα′ − 12 MP ·d∗α′β′Sα∗′β′ · ∂(ln∂ρPˆ)µβ′ραµ . (Fα+Fα′) ραα′ βX′,µ −2 · ∂P (44) P 1 ∂ dαβ 1+ Sαβ ρβα′ Insertingtheadiabaticexpressionforthedensitymatrix, − M · 2 · ∂P β (cid:18) (cid:19) given in Eq. (42), into Eq. (44), one obtains, for each X P 1 ∂ quantum state ι, the following two coupled equations − M ·d∗α′β′ 1+ 2Sα∗′β′ · ∂P ραβ′ . i P Xβ′ (cid:18) (cid:19) C˙αι(X,t) = −¯hEαCαι(X,t)− M ·dαβCβι(X,t) (40) β X 6 1 P d S ∂(lnρˆ)βµCι(X,t) Eq. (35) can build and destroy coherence in the system − 2 M · αβ αβ · ∂P µ by creating and destroying superposition of states. As β,µ X explained above, this is a feature of a non-linear theory. 1 P ∂(lnρ) αµCι(X,t) Such a non-linear character is simply hidden in the op- − 2 M · ∂R µ eratorversionofquantum-classicaldynamicsandclearly µ X manifested by the wave picture of the quantum-classical 1 ∂(lnρ) F αµCι(X,t) (45) evolution, which has been introduced in this paper. − 2 α ∂P µ µ Since Eqs. (45) and (46) are non-linear, their numeri- X i P cal integration requires either to adopt an iterative self- C˙αι∗′(X,t) = +¯hEα′Cαι∗′(X,t)− M ·d∗α′β′Cβι∗′(X,t) consistent procedure (according to which one makes a Xβ′ first guess of ραα′, as dictated by Eq. (42), calculates − 12 MP ·d∗α′β′Sα∗′β′ · ∂(ln∂ρPˆ)µβ′Cµι∗(X,t) rtheceuervsiovlevepdrCocαιe(dXu,rte)uanntdilCnαιu∗′(mXe,ritc)a,lancdontvheerngegnoceesiisntooba- βX′,µ tained) or to choose a definite form for ρGαα′, follow- 1 P ∂(lnρ)µα′Cι∗(X,t) ing physical intuition, and then calculating the time − 2 M · ∂R µ evolution, according to the form of Eqs. (45) and (46) µ X which is obtained by using ρG . This last method is − 12 Fα′ · ∂(ln∂ρP)µα′Cµι∗(X,t). (46) atulrmeadmyeckhnaonwicnsw[2it4h]inasththeeWmiegαtnαhe′ordfoorfmWulaigtnioenr otrfaqjeucatno-- µ X ries [25]. It is also important to find some importance Quantum-classical averages of arbitrary observables can samplingschemeforthephasespaceintegralinEq.(47). be calculated in the adiabatic as Such sampling scheme may depend on the specific form hχî(t)= wι dXCαι(X,t)Cαι∗′(X,t)χα′α(X), Eχαqαs.′ (o4f5)t,he(4o6b),searnvadb(le4.7)Ictanisbineteurseesdtintog atoddnreostse bthotaht Xι Xαα′Z equilibrium and non-equilibrium problems on the same (47) footing. However, the dynamical picture provided by wherethecoefficientsCι(X,t)andCι∗(X,t)areevolved α α′ Eqs. (45) and (46) is very different both from that of according to Eqs. (45) and (46), respectively. Equa- the usual surface-hopping schemes [26] and from that of tions(45)and(46)arenon-linearequationswhichcouple the nonadiabatic evolution of quantum-classical opera- all the adiabatic states used to analyze the system. tors [6]. In order to appreciate this, for simplicity, one At this stage, a general discussion about such a non- canconsiderasituationinwhichthereis nothermaldis- linear character is required. With a wide consensus, order in the quantum degrees of freedom so that ι = 1: quantum mechanics is considered a linear theory. This viz., the density matrix becomes that of a pure state leads,for example, to the visualizationof quantumtran- sitions as instantaneous quantum jumps. The linearity ραα′(X,t) → Cα(X,t)Cα∗′(X,t). Then, equations (45) and(46)remainunalteredandonehasjusttoremovethe of the theory also determines the need of considering in- index ι from the coefficients. Therefore, it can be real- finite perturbative series which must be re-summed in izedthatnoclassicaltrajectorypropagation,andnostate some way in order to extract meaningful predictions. switching areinvolvedby Eqs.(45) and(46). Instead, in Density Functional Theory is an example of a non-linear ordertocalculateaveragesaccordingtoEq.(47),onehas theory [19] but it is usually considered just as a com- to sample phase space points and integrate the matrix putational tool. However, there are other approaches to equations. quantumtheory thatrepresentinteractionsby anintrin- In the next section, an equilibrium approximation of sic non-linear scheme [20]. It is not difficult to see how Eqs.(45)and(46),alongthelinesfollowedbythemethod this is possible. Matter is represented by waves, these of Wigner trajectories [25], is given and applied, with very same waves enter into the definition of the fields goodnumericalresults,totheadiabaticandnonadiabatic defining their interaction [21]. This point of view has dynamics of the spin-boson model. been pursued by Jaynes [22] and Barut [23], among oth- ers. These non-linearapproachesdepictquantumtransi- tionsasabruptbutcontinuousevents[20]inwhich,togo from state 1 to state 2 , the system is first brought by V. WAVE DYNAMICS OF THE SPIN-BOSON | i | i MODEL theinteractioninasuperpositionα1 +β 2 ,andthen,as | i | i the interaction ends, it finally goes to state 2 . It is un- | i derstoodthatthisismadepossiblebythenon-linearityof The theory developed in the previous sections can be suchtheoriesbecause,instead,alineartheorywouldpre- applied to simulate the relaxation dynamics of the spin- servethesuperpositionindefinitely. Incidentally,thepic- boson system [27]. This system has already been stud- ture of the transition process just depicted also emerges ied within the framework of quantum-classicaldynamics from the numerical implementation [6] of the nonadia- of operators in Ref. [10] and “exact” numerical results batic quantum-classical dynamics of phase space depen- were obtained at short-time by means of an iterative dent operators [2, 3]: The action of the operator J in path integral procedure developed by Nancy Makri and 7 co-worker[28]. Theshort-timeresultsofRef.[27]numer- Theprocessofrelaxationfromtheinitialstatecanbefol- ically coincide with those obtained by the path integral lowedbymonitoringthesubsystemobservablesσˆ,which z calculation of Ref. [28]. However, as it is shown later in the adiabatic basis reads by Fig. 2, the quantum-classicalresults of Ref. [27] have 1 2G 1 G2 somelimitationsconcerningthenumericalstabilityofthe σz = 1+G2 1 G2 −2G . (56) algorithm beyong a certain time length. Using the di- (cid:18) − − (cid:19) mensionless variables of Ref. [10], the quantum-classical The adiabatic basis is real so that the initial density Hamiltonian operator of the spin-boson system reads matrix of the system can be written as N P2 1 2 Hˆ(X) = −Ωσˆx+ 2j + 2ωj2Rj2−cjσˆzRj! ραα′(X,0)= ψα(X,0)φα′(X,0), (57) j=1 α=1 X X = hˆ +H +Vˆ (R), (48) where s b c whNerePhˆ2s/=2 +−Ω1/σˆ2xωi2sRt2he=subsyNstemP2H/2am+iltVon(iRan),iHs bth=e ψ1(X,0)=φ1(X,0) = ρb(X) 21(1++GG2) , j=1 j j j j=1 j b p Hamiltonian of a classical bath of N harmonic oscilla- (58) p atPocrtsio,nanbdetVˆwce(eRn)t=he−subNjs=y1stcejmPσˆzaRnjd=thγe(Rba)σˆthz.isAtnheOihntmeric- ψ2(X,0)=φ2(X,0) = ρb(X) 1−G . P 2(1+G2) spectral density is assumed for the bath. Hence, denot- p ingtheKondoparameterasξK andthecut-offfrequency p (59) as ω , the frequencies of the oscillators are defined by max Suchcoefficients enter into the calculationofthe observ- ω = ln(1 jω ), where ω = N−1(1 exp( ω )), j − − 0 0 − − max able and the constants entering the coupling by and c = j r√eξspKeωc0tivωejl.y,Tahree adiabatic eigenvalues and eigenvectors, hσz(t)i= dXφα′(X,t)σzα′α(X)ψα(X,t). (60) αα′Z X E =V Ω2+γ2(R), (49) The coefficients evolve in time according to Eqs. (45) 1,2 b ∓ p and (46), where one must set Cαι ≡ ψα and Cαι∗′ ≡ φα′. 1 1+G In order to devise an effective computational scheme for 1;R = | i 2(1+G2 1 G such equations, one could assume that the density ma- (cid:18) − (cid:19) trix entering Eqs. (45) and (46) is taken to be that at p 1 1+G t= , whenthe totalsystem(subsystemplus bath) has |2;Ri = 2(1+G2 −1+G , (50) reac∞hedthermal equilibrium. The equilibrium quantum- (cid:18) (cid:19) classicaldensitymatrixisknownasaseriesexpansionin where p ¯h[4]. Ifonemakestheadditionalassumptionofcomplete G(R)=γ−1(R) Ω+ Ω2+γ2(R) . (51) decoherenceatt= ,onlythe (h¯0)termcanbe taken − ∞ O The coupling vector dαα′h= α;Rp−→∂/∂Rα′;Ri is ρ(e0)αα′(X)=Z0−1e−β( jPj2/2+Eα(R))δαα′ , (61) h | | i d = d =(1+G2)−1∂G/∂R. (52) where Z = dXρ(0)Pαα′(X). Then 12 − 21 0 αα′ e Assuminganinitiallyuncorrelateddensitymatrix,where ∂lnρ(0P)αα′ R ∂E e α the bath is in thermal equilibrium and the subsystem is = β δαα′ βFα(R)δαα′ , (62) ∂R − ∂R ≡ in state , the initial quantum-classicaldensity matrix in the ad|i↑aibatic basis takes the form ∂lnρ(e0)αα′ = βPδαα′ . (63) ∂P − ρ(0)=ρ (0)ρ (X), (53) s b Equations (45) and (46) become where d 1 (1+G)2 1 G2 ψ (X,t) = iE ψ (X,t) ρ (0)= − , (54) dt α − α α s 2(1+G2) 1 G2 (1 G)2 (cid:18) − − (cid:19) β P d 1 E ψ (X,t) (64) and − · αβ − 2 αβ β β (cid:18) (cid:19) N X tanh(βωi/2) d ρb(X) = ωi dtφα′(X,t) = iEα′φα′(X,t) I=1 Y β exp 2tanh(βωi/2) Pi2 + ωi2Ri2 . − P ·dα′β′ 1− 2Eα′β′ φβ′(X,t). × (cid:20)− ωi (cid:18) 2 2 (cid:19)(cid:21) Xβ′ (cid:18) (cid:19) (55) (65) 8 In Eqs. (46) and (65) the terms (β/2)P Fαψα (and 1 ± · the analogous terms with ξα′) cancel each other. In the adiabatic basis d11(R) = d22(R) = 0. Hence, defining 0.5 the matrix > σz 0 iE P d 1 βE < Σ = − 1 − · 12 − 2 12 , -0.5 P d 1+ βE (cid:16)iE (cid:17) · 12 2 12 − 2 -1  (cid:16) (cid:17) (66) 0 2 4 6 8 10 12 14 t Equations (64) and (65) can be written as FIG.1: Adiabaticdynamicsofthespin-bosonmodel. β =0.3, d ψ ψ d φ φ 1 = Σ 1 , 1 =Σ∗ 1 , Ω= 1/3, ωmax =3, ξK =0.007, N = 200. The black circles dt ψ2 · ψ2 dt φ2 · φ2 show the results of the calculation with the theory proposed (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (67) in this paper while the light dashed line with squares shows, for comparison, theresults of thecalculation in Ref. [10]. which can be integrated by means of the simple algorithm Ψ(X,dτ) = Ψ(X,0)+dτΘ(X,0) η(X,0), where Ψ = (ψ,φ) and Θ = (Σ,Σ∗). The phas·e space part of 1 the initial values of ψ and φ can be used as the weight 0.5 forsamplingthecoordinatesX enteringtheclassicalinte- gralinEq.(60). Then,foreachinitialvalueX,Eqs.(67) σ>z 0 must be integrated in time so that averages can be cal- < culated. It is worth to note that in such a wave scheme -0.5 the Eulerian point of view of quantum-classical dynamics [6, 10] is preserved. This is different from what hap- -1 0 5 10 15 20 25 pens in the original operator approach [6, 10], where in t order to devise an effective time integration scheme by means of the Dyson expansion, one is forced to change FIG. 2: Nonadiabatic dynamics of the spin-boson model. from the Eulerian point of view (according to which the β=0.3,Ω=1/3,ωmax=3,ξK =0.007,N =200. Theblack phase space point is fixed and the quantum degrees of circlesshowtheresultsofthecalculationwiththetheorypro- freedom evolve in time at this fixed phase space point) posed in this paper while the line with error bars shows, for to the Lagrangian point of view, where phase space tra- comparison, the results of the calculation in Ref. [10]. These latter are indistinguishable, at short time, from the “exact” jectories are generated. Moreover,it must be noted that resultsof Ref. [28], which wereobtained bymeansof an iter- the numerical integration of Eqs. (67) provides directly ative path integral procedure. thenonadiabaticdynamicswithouttheneedtointroduce surface-hopping approximations. Inordertobeableofcomparingtheresultswiththose thecalculationofnonadiabaticrateconstantsofcomplex presentedinRef.[10],thenumericalvaluesoftheparam- systems in the condensed phase [1]. eters specifying the spin-boson system have been chosen to be β = 0.3, Ω = 1/3, ω = 3, ξ = 0.007, and max K N = 200. Figure 1 shows the results in the adiabatic case, obtained by setting d = 0 in Eqs. (67). One can VI. CONCLUSIONS 12 seethat,inspiteofthesimpleapproximationoftheform of the density matrix made in the equations of motion, In this paper the approach to the quantum-classical thewavetheoryprovidesresultswhichareingoodagree- mechanics of phase space dependent operators has been ment with those obtained with the operator approachof remodeled as a non-linear formalism for wave fields. It Ref.[10]. Instead,Fig.2showstheresultsofthenonadia- hasbeenshownthattwocouplednon-linearequationsfor baticcalculation. Thisistobecomparedwiththeresults phasespacedependentwavefieldscorrespondto the sin- of the operator theory [10] (which are identical with the gle equation for the quantum-classical density matrix in exact” ones of Ref. [28]). Of course, since different ways the operatorschemeofmotion. The equationsofmotion of dealing with the nonadiabatic effects are used in the for the wave fields have been re-expressedby means of a two approaches the results do not need to be the same. suitablebracketandithasbeenshownthattheemerging However, the results of the wave theory follow qualita- formalism generalizes within a non-Hamiltonian frame- tivelythoseofRef.[10]whileimprovingsubstantiallythe workthe non-linear quantummechanicalformalismthat statistical convergence and increasing the length of the has been proposed recently by Weinberg. Finally, the timeintervalspannedbyafactorof2 3. Suchresultsare non-linearwaveequationshavebeenrepresentedintothe − particularly encorauging and suggest the possible appli- adiabatic basis and have been applied, after a suitable cationof the wavetheory here proposed,for example, to equilibriumapproximation,tothenumericalstudyofthe 9 adiabatic and nonadiabatic dynamics of the spin-boson ∂ i ∂ Ψ = H . (A6) 12 model. Good results have been obtained. In particular, ∂th | ¯h∂ Ψ B | i thetimeintervalthatcanbespannedbythenonadiabatic It is easy to see that, when the Hamiltonian function is calculation within the wave scheme of motion turns out chosenasinEq.(A3), Eq.(A4),oritsexplicitform(A5- to be a factor of two-three longer than that accessible A6), gives the usual formalism of quantum mechanics. within the operator scheme of motion. This encourages It is worth to remark that in order not to alter gauge one to pursue the application of the wave scheme of mo- invariance, the Hamiltonian and the other observables tiontothecalculationofcorrelationfunctionsforsystems must obey the homogeneity condition: inthe condensedphase. Futureworkswillbe specifically devoted to such an issue. = Ψ(∂ /∂ζ ) = (∂ /∂ζ )Ψ . (A7) 2 1 H h | H i h H | i Weinberg showedhow the formalismabove sketched can Acknowledgment be generalized in order to describe non-linear effects in I acknowledge ProfessorKapralfor suggesting the possi- quantummechanics[12]. Tothisend,onemustmaintain bility of mapping the quantum-classicaldynamics of op- thehomogeneitycondition,Eq.(A7),ontheHamiltonian erators into a wave scheme of motion. I am also very butrelaxtheconstraintwhichassumesthattheHamilto- grateful to Professor P. V. Giaquinta for continuous en- nianmustbeabilinearfunctionofthewavefields. Thus, couragement and suggestions. Finally, discussions with the Hamiltonian can be a general function given by DrGiuseppe Pellicaneduringthe finalstageofthis work n are gratefully acknowledged. ˜ = ρ−i , (A8) i H H i=1 X APPENDIX A: WEINBERG’S FORMALISM where n is arbitrary integer that fixes the order of the correction, =h, and 0 H Consider a quantum system in a state described by the wave fields Ψ and Ψ, where Dirac’s bra-ket nota- = ρ−1 drdr′dr′′dr′′′Ψ∗(r)Ψ∗(r′) tion is used to d|eniote Ψh(r)| rΨ and Ψ∗(r) Ψr . H1 Z Observables are defined by f≡unhct|ionis of the type≡ h | i G(r,r′,r′′,r′′′)Ψ(r′′)Ψ(r′′′), (A9) × a= ΨAˆΨ , (A1) withanalogousexpressionsforhigherorderterms. Appli- h | | i cations and thoroughdiscussions of the above formalism where the operators are Hermitian, Aˆ=Aˆ†. Weinberg’s can be found in Ref. [12]. formalismcanbeintroducedbydefiningPoissonbrackets Once Weinberg’s formalism is expressed by means of in terms of the wavefields Ψ and Ψ. To this end, one | i h | the symplectic form in Eq. (A4), it can be generalized considers the wave fields as “phase space” coordinates very easily in order to obtain a non-Hamiltonian quan- ζ (Ψ , Ψ), so that ζ = Ψ and ζ = Ψ, and then ≡ | i h | 1 | i 2 h | tum algebra. To this end, one can substitute the an- introduce brackets of observables as tisymmetric matrix B with another antisymmetric ma- 2 ∂a ∂b trix Ω = Ω[ζ], whose elements might be functionals a,b = . (A2) { }B ∂ζ Bαβ∂ζ of ζ (Ψ , Ψ) obeying the homogeneity condition in α β ≡ | i h | αX=1 Eq. (A7). By means of Ω a non-Hamiltonian bracket The bracket in Eq. (A2) defines a Lie algebra and a ...,... can be defined as { }Ω Hamiltonian systems. Typically, the Jacobi relation is satisfied, i.e. = a, b,c + c a,b + 2 ∂a ∂b J { { }B}B { { }B}B a,b = Ω [ζ] . (A10) {fobr,m{ca,lais}mB,}Bon=ec0a.nInintorroddeurcteotohbetHaianmtihlteonuisaunalfuqnucatniotunmal { }Ω αX=1∂ζα αβ ∂ζβ in the form Ingeneral,thebracketinEq.(A10)doesnolongersatisfy [ψ , ψ ] [ζ]= ψ Hˆ ψ , (A3) the Jacobi relation H| i h | ≡H h | | i where Hˆ is the Hamiltonian operator of the system. = a, b,c + c a,b + b, c,a =0. J { { }Ω}Ω { { }Ω}Ω { { }Ω}Ω 6 Equationsofmotionforthewavefieldscanbe writtenin (A11) compact form as Thus,non-Hamiltonianequationsofmotioncanbe written as ∂ζ i = [ζ],ζ . (A4) ∂t ¯h{H }B ∂ζ i = ,ζ . (A12) ThecompactformofEq.(A4)canbesetintoanexplicit ∂t ¯h{H }Ω form as In principle, the non-Hamiltonian theory, specified by ∂ i ∂ Eqs.(A10),(A11),and(A12),canbeusedtoaddressthe Ψ = H (A5) ∂t| i ¯h∂ Ψ B21 problem of non-linear correction to quantum mechanics, h | 10 as it was done in Refs. [12]. In the present paper, it has bymeansofsuitablebrackets. 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