Density matrix equation for a bathed small system and its application to molecular magnets

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1 City University of New York CUNY CUNY Acaemic Works Publications an Research Lehman College 013 Density matrix equation for a bathe small system an its application to molecular magnets Dmitry A. Garanin CUNY Lehman College How oes access to this work benefit you? Let us know! Follow this an aitional works at: Recommene Citation Garanin, D. A Density matrix equation for a bathe small system an its application to molecular magnets in Avances in Chemical Physics, volume 147, 013, pages or Avances in Chemical Physics, eite by S. A. Rice an A. R. Dinner John Wiley & Sons, Inc., Hoboken, NJ, USA, 011, vol. 147 DOI : / ch4 This Book Chapter or Section is brought to you for free an open access by the Lehman College at CUNY Acaemic Works. It has been accepte for inclusion in Publications an Research by an authorize aministrator of CUNY Acaemic Works. For more information, please contact AcaemicWorks@cuny.eu.

2 Density Matrix Equation for a Bathe Small System an its Application to Molecular Magnets D. A. Garanin Department of Physics an Astronomy, Lehman College, City University of New York, 50 Befor Park Boulevar West, Bronx, New York , U.S.A. Date: February 1, 013 arxiv: v1 [con-mat.stat-mech] 5 May 008 The technique of ensity matrix equation DME for a small system interacting with a bath is explaine in etail. Special attention is given to the nonsecular DME that is neee in the vicinity of overampe tunnelling resonances in molecular magnets MM. The relaxation terms of the DME for MM are represente in the universal form that oes not employ any unknown spin-lattice coupling constants an absorbs the information about the spin Hamiltonian in the exact basis states an transition frequencies. This makes aing new types of anisotropy easy an error-free. The Mathematica coe is available from the author. Contents I. General theory A. From the wave function to the ensity matrix B. Entangle states an quantum statistics 3 C. Density matrix as an operator 4 D. Temporal evolution an interaction representation 5 II. The ensity matrix equation 7 A. From the full to reuce DOE 7 B. From the DOE to DME 8 C. DME in the iagonal basis 9 D. Secular approximation an Fermi golen rule 10 E. Analysis of the non-secular DME 1 F. Semi-secular approximation 1 G. Transformation to the natural basis 1 III. Time-epenent problems 14 A. Free evolution Non-secular DME 14. Secular DME Semi-secular DME 16 B. Resonant perturbation 17 C. Linear response 0 IV. Application to molecular magnets 5 A. The moel 5 B. Spin-phonon interaction 5 C. DME for molecular magnets 7 1. Secular vs non-secular 7. Initial conition for free relaxation 7 3. Direct processes 8 4. Raman processes 30 D. The realistic phonon spectrum 3 E. Groun-state tunneling an relaxation The two-level moel 33. Groun-groun state resonance Dynamics of the groun-groun state resonance via effective classical spin Coherence in the groun-groun state resonance Relaxation rate between two tunnel-split states Groun-excite state resonance 38 F. Numerical implementation an illustrations 41

3 G. Discussion 46 Acknowlegments 46 References 47 I. GENERAL THEORY Density matrix is use to escribe properties of a system that is a part of a larger system with which it interacts. Whereas isolate systems such as the above mentione larger system can be escribe by a Schröinger equation, systems which interact with their environments cannot. Starting from the Schröinger equation for the isolate whole system, small system + environment or bath, an eliminating the environmental variables, one can, in principle, construct an object that can be use to calculate observables of the small system in a short way. This object is the ensity matrix of the small system. Of course, integrating or taking matrix elements in two steps, at first over the environment an then over the small system is not a big simplification. This approach becomes really useful if one obtains a close equation of motion for the ensity matrix of the small system, the ensity matrix equation DME. This is possible if the interaction between the small system an its environment is small an can be consiere as a perturbation, an the small system oes not strongly perturb the state of the environment. The erivation of the DME for a bathe small system will be presente in Sec. II. Here the necessary components of the formalism will be introuce. A. From the wave function to the ensity matrix The general wave function or state ψ of an isolate quantum system can be expane over a set of complete basis states Ψ m as Ψ = m c m Ψ m. 1 The coefficients c m completely characterize the state Ψ an can be use to calculate physical quantities A escribe by corresponing operators Â: A Â Ψ Â Ψ = mn c m c n Ψ n Â Ψ m = mn c m c na nm. One can efine the ensity matrix ρ corresponing to quantum-mechanical states pure states of our small system by its matrix elements as then Eq. becomes {ρ} mn = ρ mn = c m c n, 3 A = mn The ensity matrix satisfies the normalization conition ρ mn A nm. 4 Tr {ρ} m ρ mm = 1. 5 This conition follows from Eq. 3 for the pure states but it hols in general, too, since the average of the unity operator A mn = δ mn shoul be 1. Aitionally, the conition ρ mn = 1 6 mn is satisfie for the ensity partix of pure states, Eq. 3. For systems interacting with their environment, observables A still are given by Eq. 4, although the coefficients ρ nm in general o not reuce to proucts as in Eq. 3. For the whole system incluing the environment, one can use basis states that are irect proucts of those of the small system ψ m an those of the environment φ : Ψ m = ψ m φ ψ m φ. 7

4 The quantum mechanical states of the whole system consiere as isolate can be written, similarly to Eq. 1, in the form Ψ = m C m Ψ m. 8 3 The expression for the observable A of the small system becomes A = Ψ Â Ψ = Cm Cn Ψm  Ψn = m,n m,n C m Cn ψ m  ψ n φ φ. 9 For the orthonormal set of φ this can be rewritten in the form of Eq. 4 with ρ mn ρ s mn, where ρ s mn is the reuce ensity matrix of the subsystem given by ρ s mn = C m C n ρ m,n. 10 Since, in general, C m o not split into the factors epening on m an, the reuce DM ρ s mn is not a prouct of its wave-function coefficients, ρ s mn c mc n. Obviously ρs mn in Eq. 10 epens on the state of the whole system. One can check that it satisfies the normalization, Eq. 5. On the other han, Eq. 6 is not satisfie, in general. B. Entangle states an quantum statistics It woul be wrong to think that when the interaction between the small system an the environment becomes very weak, the ensity matrix of the small system, Eq. 10, simplifies to Eq. 3. If the coupling is vanishingly weak, quantum states of both the small system an the environment are well efine. Thus both the small system an the environment coul be in their pure quantum mechanical states. However, there are pure states of the whole system that o not consist of pure states of the two subsystems. Such states can be calle entangle. For instance, if both subsystems are two-level systems with the states ψ 1, ψ an φ 1, φ, respectively, an the whole system is in the state Ψ = C 11 ψ 1 φ 1 + C ψ φ 11 with C 11 + C = 1, Eq. 10 with C m = C mm δ m yiels the iagonal ensity matrix ρ s mn = C mm δ mn. 1 This ensity matrix satisfies Eq. 5. On the other han, iagonal ρ s mn oes not correspon to any pure state, cf. Eq. 3. Thus Eq. 6 is not satisfie, ρ s mn = C mm mn m In particular, for C 11 = C = 1/ the result is 1/ instea of 1. This is unrelate to the interaction between the two subsystems an it alone manates using the ensity matrix rather than the wave function for a subsystem in the general case. In a non-entangle state, in the absence of interaction, the wave function of the whole system factorizes: Ψ = ψ φ = m c m ψ m φ. 14 If the whole system an both subsystems are escribe by ensity matrices, the total ensity matrix uner the above conitions factorizes such as ρ m,n = ρ s mnρ b 15 for a small system an the bath. This factorization means that both subsystems are completely inepenent. If they are prepare at the initial moment inepenently from each other, one cannot expect their entanglement that requires a special care. As the time goes, interaction between the subsystems can cause some entanglement that, however, shoul remain small if the interaction is weak. The measure of entanglement is the mismatch in Eq. 15, where ρ s mn is efine by Eq. 10 an ρ b is efine by similar tracing out the variables of the small system. Note that both

5 rhs an lhs of Eq. 15 are properly normalize accoring to Eq. 5 an its variants. As a number measuring the entanglement, one can use, e.g., [ mn Ent = n n 1 ρs mn ρ b ] 1 16 ρm,n m,n that is relate to Eq. 6. Here n is the number of states of each subsystem that is assume to be common. If the whole system is in a pure state, the enominator is equal to 1. In particular, for Ψ given by Eq. 11, n =, one has ρ s mn given by Eq. 1 an similarly ρb = C δ, so that Ent = 4 [ 1 C C 4 ] In the case C 11 = C = 1/ entanglement reaches its maximal value 1. The coupling between subsystems also results in impossibility to escribe them in terms of wave functions. The coupling causes slow transitions between energy levels, so that the total energy of the whole system is conserve. For the small system this means that it spens time in its ifferent quantum mechanical states in the interaction is weak an they are well efine, so that over a large time its state is a mixture of ifferent states rather than a pure quantum mechanical state. Alternatively one can think about an ensemble of bathe small systems being in ifferent pure states. An example of a mixture state is the thermal-equilibrium state ρ mn = 1 Z s exp E m k B T δ mn, 18 where Z s = m e Em/kBT is the partition function. This formula hols if the basis states ψ m are eigenfunctions of the small-system s Hamiltonian Ĥs, that is a natural choice. We will see below that in the course of temporal evolution escribe by the DME, the initial ensity matrix approaches the form above. In fact, the environment of the small system also can be weakly couple to some super-bath, so that the small system + environment is not an isolate system an it also shoul be escribe by the ensity matrix instea of the wave function. The role of the super-bath is just is to ensure that the whole system is not in a pure quantum state but in a mixture state, whereas the coupling between the bath an super-bath is assume to be very weak an is never consiere explicitly. In most cases the state of the environment bath is the thermal equilibrium that is escribe by the ensity matrix similar to Eq. 18. Since the bath is much larger than the small system, the weak coupling to the latter won t rive it out of the equilibrium. C. Density matrix as an operator The formalism of quantum mechanics allows one to consier the ensity matrix as an operator. More precisely, one can introuce the ensity operator ˆρ = mn ρ mn ψ m ψ n, 19 where ψ m is a complete orthogonal set of states. The ensity matrix consists of matrix elements of the ensity operator: ρ mn = ψ m ˆρ ψ n, 0 so that both DM an DO contain the same information an are equivalent. Using the ensity operator allows one to put formulas in a more compact form without subscripts. In particular, the expectation value of an operator  can be obtaine as a trace of ˆρ over any complete orthogonal set of states ψ n. The calculation is especially simple if one uses the set of states in the efinition of ˆρ: } {ˆρ  = Tr = ψ ˆρ ψ n n = ψ  ψ n m ψ m ˆρ ψ n = A nm ρ mn, 1 n mn mn an the result coincies with Eq. 4. It can be proven that the trace of an operator is inepenent of } the choice { } of the basis in Tr {...} an operators can be cyclically permute uner the trace symbol, so that Tr {ˆρ = Tr ˆρÂ.

6 If the system interacts with its environment escribe by the basis φ, the total DO has the form ˆρ = ρ m,n ψ m φ ψ n φ ρ m,n ψ m ψ n φ φ. m,n m,n Calculating any observable A that correspons to the small system can be one in two steps: First calculating the trace over the variables of the bath an then calculating the trace over the basis states of the small system using Eq. 1. The first step yiels the reuce ensity operator for the small system that has the form of Eq. 19 with ˆρ s = Tr bˆρ Tr ˆρ 3 5 ρ s mn = ρ m,n. 4 If the whole system is in a pure state, this formula coincies with Eq. 10. The ensity operator corresponing to the thermal equilibrium of the small subsystem in contact with the bath has the beautiful form ˆρ s,eq = 1 Z s exp Ĥs k B T. 5 Because of applications in quantum statistics ensity operator is also calle statistical operator. Defining the ensity matrix ρ s mn with respect to the basis of eigenstates of Ĥ in Eq. 0, one arrives at Eq. 18. The eigenstate basis is the most convenient for calculating thermal averages of physical quantities. However, as pointe out above, this coul be one with the help of any other basis. If the whole system consisting of the small subsystem an the bath is in contact with a super-bath, the equilibrium statistical operator of the whole system has the form similar to Eq. 5. For a system in a pure state, the ensity operator efine by Eq. 19 can be with the help of Eqs. 3 an 1 rewritten in the form ˆρ = ψ ψ. 6 D. Temporal evolution an interaction representation Temporal evolution of the DM or DO of an isolate system such as the small system + bath obeys the equation that follows from the Schröinger equation. If, for instance, the whole system is in a pure state Ψ, its ensity operator is given by c.f. Eq. 6. Then with the help of the Schröinger equation an its conjugate one obtains the quantum Liouville equation ˆρ = Ψ Ψ 7 i t Ψ = Ĥ Ψ, i Ψ = Ψ Ĥ 8 t i ˆρ t = [Ĥ, ˆρ ]. 9 It shoul be note that this equation is not an equation of motion for an operator in the Heisenberg representation that has another sign. The DO consists of states that have their own time epenence, unlike Heisenberg operators whose time epenence is borrowe from the states. In the presence of the super-bath the system is not in a pure state but rather in a mixture of ifferent states Ψ, ˆρ = Ψ A Ψ Ψ Ψ. 30 Fortunately, Eq. 9 is vali also in this case. Remember that Eq. 9 escribes the whole system, small system + bath, although we roppe the inex tot for brevity. Our final aim is, however, to obtain a ensity matrix equation DME for the ensity matrix of the small system alone with the bath egrees of freeom eliminate.

7 Let us break up the Hamiltonian of the system into a simple Hamiltonian Ĥ0 an perturbation or interaction ˆV : Ĥ = Ĥ0 + ˆV. 31 In the absence of ˆV, evolution of quantum states Ψ is escribe by the unitary evolution operator Û0t: Ψ t = Û0t Ψ 0, Ψ t = Ψ 0 Û 0 t, 3 where Ψ 0 correspons to the starting moment t = 0. We call Û0t bare evolution operator. From the Schröinger equation 8 the equations of motion for Û0 an its conjugate follow : i tû0 = Ĥ0Û0, i tû 0 = Û 0Ĥ0, Û 0 If Ĥ0 is time inepenent, the solution of these equations is 1 t = Û0 t. 33 Û 0 t = e iĥ0t/, Û 0 t = eiĥ0t/. 34 Of course, one can write similar formulas with the total Hamiltonian Ĥ as well, Ût being the full evolution operator. The iea of using Û0t is to split the nontrivial part of the evolution ue to the perturbation ˆV from the trivial evolution ue to Ĥ0. To effectuate this, one can introuce the ensity matrix in the interaction representation The equation of motion for ˆρt I follows from Eqs. 9 an 33: ˆρt I = Û 0 tˆρtû0t i ˆρ I t = i Û 0 t ˆρtÛ0t + t Û ti ˆρ 0 t Û0t + Û 0 tˆρti Û0t t = Û 0Ĥ0ˆρtÛ0t + Û 0 t [Ĥ, ˆρ ]Û0 t + Û 0 tˆρtĥ0û0 = Û 0 t [ˆV, ˆρ ] Û0 t. 36 Inserting Û0tÛ 0 t = 1 [see Eq. 33] between the operators an efining this equation can be brought into the form ˆV I t Û 0 tˆv Û0t, 37 i ˆρ I t = [ˆVI, ˆρ I ]. 38 One can see that the temporal evolution of the ensity matrix in the interaction representation is governe by the interaction only. This facilitates constructing the perturbation theory in ˆV.

8 7 II. THE DENSITY MATRIX EQUATION A. From the full to reuce DOE In this section the equation of motion for the reuces ensity operator of a small system s weakly interacting with a bath will be obtaine. We will be following the metho of Karl Blum [1] that is most practical, although not rigorous. Rigorous methos using the projection operator technique [, 3] lea to the same result with much greater efforts. The Hamiltonian can be written in the form Ĥ = Ĥ0 + ˆV, Ĥ 0 Ĥs + Ĥb, ˆV Ĥ s b. 39 Eq. 38 for the DO of the whole system can be integrate over the time resulting in the integral equation ˆρt I = ˆρ0 I i Inserting it back into Eq. 38 one obtains the integro-ifferential equation i tˆρt I = t [ˆV ti, ˆρ0 I ] i 0 t [ˆV t I, ˆρt I ]. 40 t 0 [ˆV ]] t ti, [ˆV t I, ˆρt I which is still an exact relation. Although this equation seems to be more complicate than the initial equation 38, it is convenient for perturbative treatment of the interaction ˆV. If the small system is not entangle with the bath in the initial state, that is a natural assumption, a small interaction cannot cause a significant entanglement as well. Thus the ensity operator of the whole system nearly factorizes, see Eq. 15. Here we aitionally use that the bath is at thermal equilibrium that cannot be noticeably istorte by a weak interaction with the small subsystem: 41 ˆρt I = ˆρs t Iˆρ eq b = ˆρ s t 1 exp Ĥb Z b k B T 4 Note that this approximation cannot be one in Eq. 40 since it leas to isappearance of the effect of interaction. To properly escribe this effect, one has then to account for the corrections to Eq. 4 in the first orer in ˆV. This way is inconvenient. To the contrary, Eq. 41 alreay contains quaratic ˆV terms that capture the main effect of interaction. Here taking into account small corrections to Eq. 4 can bring only irrelevant small terms. Now using Eq. 4 one can transform Eq. 41 into the ensity operator equation for the small system by making a trace over the bath variables accoring to Eq. 3: tˆρ s t I = i Tr b [ˆV ti, ˆρ s 0 Iˆρ eq b ] 1 t t Tr b [ˆV ti, 0 ]] [ˆV t I, ˆρ s t Iˆρ eq b. 43 For the couplings ˆV we consier here, the first linear-ˆv term isappears. The equation above is an integro-ifferential equation with integration over preceing times, t t, in the rhs. Such equations are calle equations with memory an they are ifficult to solve irectly. In our case, however, the problem simplifies. The time evolution of ˆρ s t I In Eq. 43 is slow since it is governe by the weak interaction between the system an the bath. On the other han, t epenences of the other terms in the integran the kernel are governe by Ĥ0 an thus they are fast at the scale of ˆρ s t I. The analysis shows that the kernel in Eq. 43 is localize in the region t t 1/ω max, where ω max is the maximal frequency of the bath excitations. Thus in the integral over t one can make the short-memory approximation ˆρ s t I ˆρ s t I after which the time integral can be calculate explicitly. Returning to the original reuce ensity operator c.f. Eq. 35, an computing the erivative with the help of Eq. 33, one obtains ˆρ s t = Û0tˆρ s t I Û 0 t, 44 tˆρ st = i [Ĥs, ˆρ s t] + Û0t tˆρ st = i [Ĥs, ˆρ s t] 1 t 0 tˆρ st I ]] t Tr b [ˆV, [ˆV t t I, ˆρ s tˆρ eq b Û 0 t, 45 46

9 or, with τ t t an ropping the subscript s, ˆρ s t ˆρt, tˆρt = i [Ĥs, ˆρt] + ˆRˆρt, 47 where the first term with commutator is the conservative term an ˆRˆρt 1 [ ]] t τ 1 Tr b [ˆV, ˆV τ I, ˆρtexp Ĥb 48 Z b k B T escribes the relaxation of the small system. Here, for time-inepenent problems, 0 ˆV τ I = Û 0 τˆv Û0τ = e iĥ0τ ˆV e iĥ0τ. 49 If the Hamiltonian of the small system Ĥs epens on time, this time epenence is typically slow in comparizon to the frequency of the bath excitations ω max, so that uring the short times τ 1/ω max the change of Ĥs is negligibly small. Thus one can simply use Eq. 49 with Ĥs = Ĥst. 8 B. From the DOE to DME Now one can go over from the ensity operator ˆρ to the ensity matrix ρ mn using Eq. 0 an the notations H s,mn ψ m Ĥs ψ n, V m,n ψ m φ ˆV ψ n φ, 50 where φ are eigenfunctions of Ĥb. Then the partition function of the bath becomes Z b = φ e Ĥb/k BT φ = e E /kbt. 51 The conservative term of the resulting DME has three ifferent forms in three ifferent cases. If the small-system states ψ m are time inepenent an form a so-calle natural basis unrelate to Ĥs, the DME has the form t ρ mn = i H s,ml ρ ln ρ ml H s,ln + ψ m ˆRˆρ ψ n. 5 l The natural basis is inconvenient for the evaluation of the relaxation term since the relaxation of the small system takes place not between the states ψ m but between the eigenstates of Ĥs: Ĥ s χ α = ε α χ α, e iĥsτ/ χ α = e iεατ/ χ α. 53 The basis of χ α will be calle iagonal basis since the Hamiltonian matrix χ α Ĥs χ β = ε α δ is iagonal. In the iagonal basis the DME has the form t ρ = iω ρ + χ α ˆRˆρ χ β, 54 where ω are transition frequencies between the energy levels of the small system, ω ε α ε β, 55 an the relaxation term can be conveniently evaluate see next section. If Ĥ s epens on time, one can use the aiabatic basis of the states χ α t efine as In this basis ρ acquires aitional non-aiabatic terms: i.e., Ĥ s t χ α t = ε α t χ α t. 56 t ρ = t χ α ˆρ χ β = χ α ˆρ χ β + χ α ˆρ χ β i χ α [Ĥs, ˆρ] χ β + χ α ˆRˆρ χ β 57 t ρ = γ χα χγ ργβ + ρ αγ χ γ χβ iω ρ + χ α ˆRˆρ χ β. 58 As argue at the en of the preceing section, calculation of the relaxation term χ α ˆRˆρ χ β is not complicate by the time epenence of Ĥs. This calculation will be one in the next section where Eq. 54 will be use for brevity. The terms ue to the time epenence of Ĥs in Eq. 58 can be ae if neee.

10 9 C. DME in the iagonal basis Using Eqs. 54 an 48 an inserting summation over intermeiate states yiels t t ρ = iω ρ 1 τ 1 0 Z b α β { χ α φ ˆV χ α φ χ α φ e iĥ0τ/ ˆV e iĥ0τ/ χ β φ χ β φ ˆρe Ĥb/k BT χ β φ χ α φ ˆV χ α φ χ α φ ˆρe Ĥb/k BT χ β φ χ β φ e iĥ0τ/ ˆV e iĥ0τ/ χ β φ χ α φ e iĥ0τ/ ˆV e iĥ0τ/ χ α φ χ α φ ˆρe Ĥb/k BT χ β φ χ β φ ˆV χ β φ } + χ α φ ˆρe Ĥb/k BT χ α φ χ α φ e iĥ0τ/ ˆV e iĥ0τ/ χ β φ χ β φ ˆV χ β φ. 59 Since the states are eigenfunctions of Ĥs an Ĥb, this simplifies to t t ρ = iω ρ 1 τ 1 0 Z b α β { e i ε α E +ε β +E τ/ e E /kbt V α,α V α,β ρ β βδ e i ε β E +ε β+e τ/ e E /kbt V α,α ρ α β δ V β,β e i εα E +ε α +E τ/ e E /kbt V α,α ρ α β δ V β,β + e i ε α E +ε β +E τ/ e E /kbt ρ αα δ V α,β V β,β } 60 an further to t t ρ = iω ρ 1 τ 1 0 Z b α β { e i ε α E +ε β +E τ/ e E /kbt V α,α V α,β ρ β β e i ε β E +ε β+e τ/ e E /kbt V α,α ρ α β V β,β e i εα E +ε α +E τ/ e E /kbt V α,α ρ α β V β,β + e i ε α E +ε β +E τ/ e E /kbt ρ αα V α,β V β,β }. 61 Since the time kernel is sharply localize, the integration over τ can be extene to the interval 0,. Further, the relaxation of the small system that we are mainly intereste in is ue to the real part of the bath coupling term in Eq. 61. Its imaginary part is in most cases only a small correction to the first conservative term in this equation. Thus taking into account the time symmetry of F ij τ one can replace t 0 τe i ε α E +ε β +E τ/ πδ ε α E + ε β + E etc. After renaming inices in Eq. 61 α λ, β α in the first term an β λ, α β in the forth term one obtains the DME in the form 6 t ρ = iω ρ + R,α β ρ α β, 63 α β where R,α β = π Z b { γ e E /kbt δ ε α ε γ + E E V α,γ V γ,α δ β β γ e E /kbt δ ε β ε γ + E E δαα V β,γ V γ,β + e E /kbt [ δ ε β ε β + E E + δ εα ε α + E E ] V α,α V β,β }. 64

11 10 Eq. 63 can be written in a more compact form where t ρ = Φ,α β ρ α β, 65 α β Φ,α β iω δ αα δ ββ + R,α β. 66 In the case of time-epenent Hamiltonian one has to a the non-aiabatic terms from Eq. 58 to the above DME. The asymptotic solution of Eq. 63 is ρ eq = 1 exp ε α δ 67 Z s k B T that escribes thermal equilibrium. This will be shown in Sec. II E. D. Secular approximation an Fermi golen rule One can transform Eq. 63 back into the interaction representation using Eqs. 35 an 0. In the iagonal basis, the relation between ρ t an ρ t I has the simple form ρ t I = χ α e iĥst/ ˆρte iĥst/ χ β = e iω t ρ t. 68 Computing the time erivative of ρ t I an using Eq. 63, one arrives at the equation t ρ t I = α β e iω ω α β t R,α β ρ α β t I, 69 where the conservative term isappeare an the relaxation term has an explicit time epenence. While the change of the ensity matrix ue to the relaxation is slow, the oscillation in the terms with ω ω α β are generally fast. These fast oscillating terms average out an make a negligible contribution into the ynamics of the small system. In general, all transition frequencies are nonegenerate, so that one can rop all terms with α α an β β, if α β. In the equations for the iagonal terms ρ αα t I one can keep only iagonal terms with α = β. This is the secular approximation that tremenously simplifies the DME. In the secular approximation one has t ρ t I = δ R αα,α α ρ α α t I + 1 δ R, ρ t I α = δ R αα,α α ρ α α t I + R, ρ t I. 70 α α Simplifying Eq. 64 an using the Hermiticity V α,α = Vα,α 71 one obtains R αα,α α α α = π Z b e E /kbt δ ε α ε α + E E V α,α Γ αα 7 an R, = π Z b γ { γ e E /kbt δ ε α ε γ + E E V α,γ e E /kbt δ ε β ε γ + E E V β,γ + e E /kbt δ E E V α,α V β,β }. 73

12 11 Rearranging the terms one obtains R, = Γ, 74 where Γ = Γ + 1 Γ α α + Γ β β. 75 α α β β Here Γ α α is efine by Eq. 7 an Γ = π Z b e E /kbt δ E E V α,α V β,β. 76 Using these results in Eq. 70 an changing to ρ, one obtains the secular DME in the iagonal basis in the form t ρ = iω + Γ ρ + δ α α Γ αα ρ α α. 77 In Eq. 7, Γ αα is the rate of quantum transitions α α in the small system, accompanie by an appropriate transition in the bath so that the total energy is conserve. To the contrary, Γ α α in Eq. 74 is the rate of quantum transitions α α in the small system. One can relate both rates as Γ α α = π e E /kbt δ ε α ε α + E E V Z b = e E +εα ε α /kbt δ ε α ε α + E E V Z b 78 or Γ α α = e εα ε α /kbt Γ αα. 79 This is the so-calle etaile-balance relation that ensures that asymptotically the small system reaches the thermal equilibrium escribe by Eq. 67. If ε α < ε α, then at low temperatures the rate of transitions α α with increasing energy is exponentially small. Note that the transition rates Γ α α an Γ αα correspon to the Fermi golen rule. The quantity Γ of Eq. 76 is the ephasing rate that turns to zero for the iagonal elements of the ensity matrix n α ρ αα, 80 the populations of states α. The ephasing rate is not relate to any transitions of the small system. Its origin is moulating its transition frequencies ω by fluctuations of the bath. One can see that in the ecouple equations for noniagonal terms of the ensity matrix α β in Eq. 77 there are only outgoing terms, so that the noniagonal terms ten to zero asymptotically. The iagonal terms of the DME satisfy the system of rate equations The asymptotic solution satisfies t n α = Γ αα n α Γ α αn α. 81 α α n α = Γα α = e ε α /kbt 8 n α Γ αα e εα/kbt that correspons to the thermal equilibrium. The equation for noniagonal elements can be written in the form with Γ given by Eq. 75. t ρ = iω + Γ ρ 83

13 1 E. Analysis of the non-secular DME We have seen above that within the secular approximation noniagonal DM elements are ecouple from iagonal ones an oscillate with ecay inepenently of each other. In the full DME of Eq. 63, noniagonal DM elements are couple to the iagonal elements. Nevertheless, they approach zero at equilibrium, in spite of iagonal elements being nonzero. This points at an interesting feature of the full DME that has to be worke out in more etail. Separating iagonal an noniagonal elements in the relaxation term of Eq. 63, one obtains t ρ = iω ρ + α β R,α β 1 δα β ρα β + α R,α α ρ α α = iω ρ + α β R,α β 1 δα β ρα β + π Z b { γ e E /kbt δ ε β ε γ + E E V α,γ V γ,β ρ ββ γ e E /kbt δ ε α ε γ + E E V α,γ V γ,β ρ αα + γ + γ e E /kbt δ ε β ε γ + E E V α,γ V γ,β ρ γγ e E /kbt δ ε α ε γ + E E V α,γ V γ,β ρ γγ }. 84 With the use of the energy conservation this can be rewritten as t ρ = iω ρ + α β R,α β 1 δα β ρα β + π e E /kbt V α,γ V γ,β Z b γ [ δ ε β ε γ + E E ρ γγ e ε β ε γ/k BT ρ ββ ] + δ ε α ε γ + E E ρ γγ e εα εγ/kbt ρ αα. 85 One can see that as the iagonal DM elements approach their equilibrium values, they cease to rive noniagonal elements because of the etaile balance relation. Thus the equilibrium solution of the full DME is Eq. 67. F. Semi-secular approximation While the secular approximation neglects the interaction between iagonal an slow noniagonal DM elements, the full non-secular formalism involves a big N N matrix that has to be iagonalize. In important particular cases such as thermal activation over a barrier or tunneling, the eigenvalues of the DM span a broa range from very fast to very slow, the latter being of a primary importance in relaxation. Because of this, one has to o numerical calculations with increase precision that makes them very slow. This ifficulty can be overcome with the help of the semi-secular approximation that consiers couple equations for iagonal an slow noniagonal DM elements plus ecouple equations for the fast DM elements. The easiest way to implement it is to inclue the iagonal an subiagonal terms ρ αα an ρ α,α±1 into the slow group, because in most situations there are only two levels that come close to each other, making ρ α,α+1 or ρ α,α 1 slow. Implementation of the semi-secular DME in the case of time-inepenent Ĥs will be one in Sec. III A3. G. Transformation to the natural basis One can transform the DME, Eq. 77 to the natural basis using Eqs. 0 an 19 in the form ρ mn = ψ m ˆρ ψ n = ψ m χ α ρ χ β ψ n. 86

14 13 The inverse transformation has the form ρ = χ α ˆρ χ β = mn χ α ψ m ρ mn ψ n χ β. 87 The general DME in the iagonal basis, Eq. 63, can be transforme to the natural basis with the help of Eq. 86 as follows This yiels where H s,mm where t ρ mn = ψ m χ α t ρ χ β ψ n = i ψ m χ α ε α ε β ρ χ β ψ n + ψ m χ α R, ρ α β χ β ψ n α β = i ψ m χ α ε α χ α ψ m ρ m n ψ n χ β χ β ψ n m n + i ψ m χ α χ α ψ m ρ m n ψ n χ βε β χ β ψ n m n + ψ m χ α R,α β χ α ψ m ρ m n ψ n χ β χ β ψ n. 88 m n α β t ρ mn = i ψm ψ m Ĥ s an H s,mm ρ m n + i ρ mn H s,n n + R mn,m n ρ m n, 89 m n m n R mn,m n T mn,m n ;,α β R,α β, 90,α β T mn,m n ;,α β ψ m χ α χ β ψ n χ α ψ m ψ n χ β. 91 Aitionally, the matrix elements in the Fermi-golen-rule transition rate, Eq. 7, that are efine with respect to the iagonal basis, can be expresse through those with respect to the natural basis. Similarly to Eq. 87 one obtains V α,α = χ α ψ m V m,m ψ m χ α. 9 mm

15 14 III. TIME-DEPENDENT PROBLEMS In this section we consier the DME an its solution in three important cases: i free evolution for time-inepenent Ĥ s ; ii fast resonance perturbation an iii perioic or nonperioic slow perturbation. In the secon case it is sufficient to keep the perturbation in the conservative part of the DME only, while in the thir case moification of the relaxation terms is require, as well. In all these cases the solution can be obtaine by matrix algebra. To the contrast, problems with a large temporal change of Ĥ s cannot be solve by matrix algebra. The secular, non-secular, an semi-secular versions of the DME yiel the same results except for the case of anomalously close energy levels e.g., tunnel split levels. In the latter case the semi-secular DME is preferre, while in general the fastest an easiest secular DME is the best choice. A. Free evolution 1. Non-secular DME The solution of time-inepenent DME, Eq. 63, that can be rewritten as t ρ = Φ,α β ρ α β, Φ,α β = iω δ αα δ ββ + R,α β, 93 α β is a linear combination of time exponentials with exponents being eigenvalues of the matrix Φ builing the DME. To bring the DME into a stanar form, it is convenient to introuce the compoun inex a efine by a = α + Nβ 1, 94 where N is the size of the ensity matrix, i.e., α, β = 1,,...,N. Then a = 1,,..., N. Inversion of Eq. 94 yiels a 1 a 1 α = 1 + NFrac, β = 1 + Int. 95 N N With the inex a, the ensity matrix ρ becomes a vector with the components ρ a while Φ,α β with the elements Φ aa : becomes a matrix t ρ = Φ ρ, t ρ a = a Φ aa ρ a. 96 The eigenvalue problem for the DME can be written as Φ R µ = Λ µ R µ, Λ µ = Γ µ + iω µ, 97 where R µ is the right eigenvector corresponing to the eigenvalue Λ µ an µ = 1,,...,S + 1. Since Φ is a non-hermitean matrix, right eigenvectors iffer from left eigenvectors that satisfy L µ Φ = Λ µ L µ. Left an right eigenvectors satisfy orthonormality an completeness relations L µa R νa = δ µν, L µa R µa = δ aa. 98 a In general, L µ an R µ are not Hermitean conjugate, see, e.g., Eq All real parts of the eigenvalues are positive, Γ µ > 0. There are N purely real eigenvalues, one of which is zero an correspons to thermal equilibrium. We assign the zero eigenvalue the inex µ = 1. Complex eigenvalues occur in complex conjugate pairs. The solution of the DME with the initial conitions can be written in the form µ ρt = µ R µ e Λµt L µ ρ0. 99 The fully vectorize form of this equation is ρt = E Wt E 1 ρ0, 100

16 where E is right-eigenvector matrix compose of all eigenvectors R µ staning vertically, E 1 is the left-eigenvector matrix, compose of all left eigenvectors lying horizontally, an Wt is the iagonal matrix with the elements e Λµt. In fact, The asymptotic value of ρt is escribe by the zero eigenvalue Λ 1 = 0, E Wt E 1 = expφt. 101 ρ = R 1 L 1 ρ0. 10 Here ρ = ρ eq shoul be satisfie, where ρ eq follows from Eq. 67, an this result shoul be inepenent of ρ0. Thus one conclues that L 1a = δ αaβa, R 1a = 1 exp ε αa δ αaβa, 103 Z s k B T where αa an βa are given by Eq. 95. This means that L 1 is relate to the normalization of the DM while R 1 contains the information about the equilibrium state. One obtains 15 L 1 ρ0 = a L 1a ρ a 0 = δ ρ 0 = α ρ αα 0 = an ρ = R 1 = ρ eq, as it shoul be. Note that R 1 an L 1 satisfy the orthonormality conition in Eq. 98, L 1a R 1a = 1 exp ε αa δ αaβa = 1 exp ε α δ = Z s k B T Z s k B T a a The time epenence of any physical quantity A is given by Eq. 4 that can be rewritten in the form At = A βα ρ t = a A a ρ a t, 106 where A a A βaαa, A βα β Â α. 107 Writing Eq. 106 in the vector form as At = A ρt one obtains At = µ A R µ e Λµt L µ ρ0 108 or, in the fully vectorize form, At = A E Wt E 1 ρ Since the time epenence of observables in the course of evolution of the ensity matrix is escribe by more than one exponential, one nees an appropriate efinition of the relaxation rate or relaxation time. A convenient way is to use the integral relaxation time efine as the area uner the relaxation curve τ int 0 t [At A ]. 110 A0 A One can check that in the case of a single exponential, At = A + [A0 A ] e Γt, the result is τ int = 1/Γ. From Eq. 108 one obtains At A = µ 1A R µ e Λµt L µ ρ0 111 an thus τ int = N µ= A R µλ 1 µ L µ ρ0 N µ= A R. 11 µl µ ρ0 This formula cannot be fully vectorize since summation skips the static eigenvalue µ = 1.

17 16. Secular DME Within the secular approximation, one has to consier the ynamics of iagonal an noniagonal components of the ensity matrix separately. The former is escribe by Eqs where the vector ρ is replace by the vector of the iagonal components n = {n α } = {ρ αα } an the N N matrix Φ is replace by the N N matrix Φ sec having matrix elements = 1 δ αα Γ αα δ αα Φ sec αα γ Γ γα, 113 as follows from Eqs. 77 or 81. All eigenvalues of Φ sec are positive reals, except for one zero eigenvalue, Λ 1 = 0. Eq. 103 becomes simply L 1α = 1, R 1α = 1 exp ε α. 114 Z s k B T If the initial conition is a iagonal matrix, the non-iagonal elements o not arise ynamically an hence they can be roppe. Then the time epenence of any quantity A is escribe by At = α A αα n α t = A E Wt E 1 n0, 115 c.f. Eqs For the integral relaxation time in the case of a purely iagonal evolution one obtains τ int = N µ= A R µ Λ 1 µ L µ n0 N µ= A R, 116 µl µ n0 c.f. Eq. 11. If the initial state is a non-iagonal ensity matrix, one has to a the corresponing trivial terms following from Eq. 83, At = A E Wt E 1 n0 + α β A βα ρ 0e iω + Γ t. 117 Then Eq. 116 is generalize to N µ= τ int = A R µλµ 1 L µ n0 + α β A βαρ 0iω + Γ 1 N µ= A R µl µ n0 + α β A. 118 βαρ 0 Obviously this expression is real. 3. Semi-secular DME Within the semisecular approximation introuce in Sec. II F, the slow group being forme by iagonal an subiagonal DM elements, α β 1, the equations of motion for the latter have the form t ρ = Φ;α,α 1ρ α,α 1 + Φ ;α α ρ α α + Φ ;α,α +1ρ α,α +1 α 119 that is a subset of Eq. 93. In labeling matrix elements, one can introuce the compoun inex a = α 1 + β. 10 Here α = 1,,...,N an β = α 1, α, α + 1, so that a takes the values a = 1,...,3N. In terms of a one has a a α = 1 + Int, β = α 1 + 3Frac,

18 17 an Eq. 119 can be rewritten as t ρslow a = a Φaa ρ slow a, 1 where Φ aa = Φ αa,βa;αa,βa. The solution of Eq. 1 is similar to that of Eq. 96. On the other han, there are uncouple DM elemens with α β, like in the secular approximation. Instea of Eq. 117 one has an instea of Eq. 118 one has τ int = At = A E Wt E 1 ρ slow 0 + 3N α β >1 A βα ρ 0e iω + Γ t 13 µ= A R µ Λ 1 µ Lµ ρ slow 0 + α β >1 A βαρ 0iω + Γ 1 3N µ= A R µ L µ ρ slow 0 + α β >1 A. 14 βαρ 0 Let Ĥs contain a perioic perturbation B. Resonant perturbation ˆV t = ˆV 0 e iωt + ˆV 0 eiωt 15 with the frequency ω > 0 close to one of transition frequences ω ηη > 0. For the resonance to occur, the latter shoul excee the relaxation rates, so that one can use the secular DME, Eq. 77. Since ˆV 0 is small, one can isregar it in the relaxation terms an use the iagonal basis corresponing to the time-inepenent part of Ĥ s. Combining Eqs. 5 an 77, one obtains t ρ = i V t αα ρ α β ρ V t β β α β iω + Γ ρ + δ Γ αγ ρ γγ. 16 Here one shoul rop all noniagonal DM elements but ρ ηη an ρ η η since the former are not excite by the resonance perturbation an vanish with time. As a result, one obtains a couple system of equations for the iagonal elements ρ αα an the elements ρ ηη an ρ η η = ρ ηη of the form t ρ ηη t ρ ηη = i t ρ η η γ α = i e iωt V 0,ηη ρ η η e iωt ρ ηη V 0,ηη iω ηη + Γ ηη e iωt V 0,ηη ρ η η eiωt ρ ηη = i e iωt ˆV 0 η η ρ ηη ˆV 0 + Γηγ ρ γγ Γ γη ρ ηη η η γ η ρ ηη e iωt ρ η η V 0,ηη + γ η Γη γρ γγ Γ γη ρ η η, 17 plus Eq. 81 for all iagonal elements with α η, η. Also terms such as e ˆV iωt 0 ρ ηη η η e iω+ωηη t in the secon equation have been roppe. Neglecting such terms that oscillate out is the so-calle rotating-wave approximation that is similar to the secular approximation. It is convenient to introuce the slow noniagonal elements ρ ηη an ρ η η via ρ ηη = ρ ηη e iωt, ρ η η = ρ η ηe iωt. 18

19 With the use of ˆV 0 = V η 0,ηη the DME acquires the form η t ρ ηη = i V 0,ηη ρη η ρ [ ηη + i ω ω ηη Γ ] ηη ρ ηη t ρ ηη = Im ρ ηη V0,ηη + Γηγ ρ γγ Γ γη ρ ηη t ρ η η γ η = Im ρ ηη V 0,ηη + γ η Γη γρ γγ Γ γη ρ η η t ρ αα = Γαγ ρ γγ Γ γα ρ αα, α η, η. 19 γ α Here Γ ηη is given by Eq. 75. In the strong-ephasing case Γ ηη Γ γη, ˆV 0,ηη, the noniagonal element ρ ηη quickly reaches its quasistationary value that can be obtaine from the first equation of Eq. 19 by setting ρ ηη /t = 0. This results in ρη η ρ ηη ρ ηη = 1 V 0,ηη ω ω ηη + i Γ ηη Substituting this into the equations for the iagonal DM elements, one obtains ˆV 0,ηη t ρ ηη = ρη η ρ Γ ηη ηη Γηγ ρ γγ Γ γη ρ ηη t ρ η η = ˆV 0,ηη ω ω ηη + Γ ηη + γ η ρη η ρ Γ ηη ηη ω ω ηη + Γ + Γη γρ γγ Γ γη ρ η η ηη γ η t ρ αα = Γαγ ρ γγ Γ γα ρ αα, α η, η. 131 γ α Note that the absorbe power of the fiel acting on the system is given by ˆV 0,ηη P = ω ηη ρη η ρ Γ ηη ηη ω ω ηη + Γ, 13 ηη since every transition η η absorbs the quant of energy equal to ω ηη. Eq. 131 can be simplifie by introucing the experimentally measure power P an thus eliminating the matrix element ˆV 0,ηη an the ephasing rate Γ ηη. Still the equations are rather complicate so that one cannot fin a simple general solution even for the stationary state. Let us consier at first the moel with only two levels, η an η. Here with ρ ηη = 1 ρ η η an the inuce-transition rate Eq. 131 reuces to a single equation Λ ˆV 0,ηη Γ ηη ω ω ηη + Γ ηη 133 t ρ η η = Λ ρ η η ρ ηη + Γη ηρ ηη Γ ηη ρ η η = Λ ρ η η 1 + Γ η η Γ η η + Γ ηη ρ η η = Λ + Γ η η Λ + Γ η η + Γ ηη ρ η η 134 that escribes relaxation with the rate Λ + Γ η η + Γ ηη towars the stationary state escribe by ρ η η = Λ + Γ η η, Γ ηη = Γ η η exp ω ηη Λ + Γ η η + Γ ηη k B T 135

20 19 for the groun-state population. The limiting cases of this formula are { ] ρ η η = ρ eq η η [1 = + exp ω 1 ηη k BT, Λ = 0 1/, Λ, 136 as it shoul be. For the population ifference from Eq. 135 one obtains ρ η η ρ ηη = ρ η η 1 = Γ η η Γ ηη Λ + Γ η η + Γ ηη = ρ eq η η ρeq ηη 1 + Λ/Γ η η + Γ ηη. 137 One can see that the population ifference is reuce by the resonance perturbation. The absorbe power is then given by ω ηη ρ P = ω ηη ρη η ρ ηη eq η Λ = η ρeq ηη Λ 1 + Λ/Γ η η + Γ ηη. 138 It increases with Λ an reaches an asymptotic maximal value. To relate the populations irectly to the measure absorbe power P, it is most convenient to rewrite Eq. 131 for the two-level system in the form This equation has a stationary solution t ρ η η = P + Γ η ηρ ω ηη Γ ηη ρ η η ηη = P + Γ η η Γ η η + Γ ηη ρ ω η η. 139 ηη ρ η η = Γ η η P/ ω ηη Γ η η + Γ ηη = ρ eq η η P/ ω ηη. 140 Γ η η + Γ ηη Note that the maximal absorbe power correspons to the saturation, ρ η η = ρ ηη, i.e., ρ η η = 1/, wherefrom follows P max = ω ηη Γ η η Γ ηη = ω [ ηη Γ η η 1 exp ω ] ηη k B T that also can be obtaine from Eq Consier now a physical quantity  that has the expectation value given by Eq. 4. The contributions from the noniagonal DM elements oscillate in time an thus average out. The result is ue to the iagonal elements only, With the help of Eq. 140 it can be rewritten as 141 A = A ηη ρ ηη + A η η ρ η η = A ηη + A η η A ηηρ η η 14 A = A eq + A, 143 where A eq = A ηη + A η η A ηηρ eq η η A = η η + A ηη exp 1 + exp ω ηη k BT ω ηη k BT 144 is the equilibrium value an the eviation from the equilibrium is given by A = A η η A ηη Γ η η + Γ ηη P ω ηη Thus the total relaxation rate between the states η an η can be expresse by the formula. 145 Γ = Γ η η + Γ ηη = A ηη A η η A P ω ηη 146

21 through the measure P an A. If the system has more than two levels but the levels η an η are the lowest two levels an the temperature is low, so that the populations of all other levels are small, one can still consier an effective two-level moel in which relaxation between η an η can be assiste by the upper levels. Instea of the irect relaxation η η that can have a small rate the system can be thermally excite from η to some high level α an then fall own to η. This is the Orbach mechanism that has the characteristic Arrhenius temperature epenence of the rate. Using the effective two-level moel is justifie by the fact that the upper levels o not contribute to physical quantities A at low temperatures. 0 C. Linear response In this section we consier a small harmonic perturbation of Eq. 15 acting on the small system, similarly to the preceing section. However, the frequency ω of the perturbation oes not nee to be close to the resonance with any transition ω. If ω ω for all α, β, the response of the small system is ue to the relaxation an it epens on the relation between ω an the relaxation rate Γ. To make an account of this effect, one has to inclue ˆV t into the relaxation terms of the ensity-matrix equation. Temporal change of ˆV t changes the instantaneous equilibrium to which the system relaxes that gives rise to a issipative ynamics. On the other han, one has to inclue ˆV t into the conservative term of the DME as well, where it can work as a resonance perturbation. The secular approximation will be use below for simplicity. Using the aiabatic basis of Eq. 56 an combining Eq. 58 with Eqs. 81 an 83, one can write the secular DME in the form t ρ αα = γ χα χγ ργα + ρ αγ χ γ χ α + α α Γ α α e ε α εα/kbt ρ α α ρ αα 147 for iagonal terms an t ρ = γ χα χγ ργβ + ρ αγ χ γ χβ iω + Γ ρ 148 for noniagonal terms. If time erivatives of the aiabatic states are small, the ensity matrix in the aiabatic basis ρ is close to its instantaneous quasiequilibrium form given by Eq. 67. The time epenence of the aiabatic states rives the DM out of the instantaneous equilibrim, ρ lagging behin time-epenent ρ eq. On the other han, since ˆV t is a small perturbation, ρ eq eviates from the full equilibrium ρ0eq in the absense of ˆV t at linear orer in ˆV t. Thus the eviation from the full equilibrium, δρ ρ ρ 0eq 149 is also linear in ˆV t. Since the ifference e ε α εα/kbt ρ α α ρ αα an the noniagonal elements ρ are small, one oes not have to expan ω an the relaxation rates. The only term to expan is e ε α εα/kbt, where one can use, at linear orer in ˆV t, ε α t = ε 0 α + δε 0 α t, δε α t = χ 0 α ˆV t χ 0 α = V t 0 αα. 150 Thus e ε α εα/kbt = e ε 0 α ε 0 α /kbt 1 + V t0 α α V t0 αα. 151 k B T The riving term expans as follows: χα χγ ργβ + ρ αγ χ γ χβ γ = γ χ χγ 0eq α ρ γβ + ρ 0eq αγ χ γ χβ = χ α χ β ρ 0eq ββ + ρ 0eq αα χ α χ β = χα χ β ρ 0eq αα ρ0eq ββ, 15

22 where χ χβ α = δ was use. One can see that in the iagonal equations this term isappears. Thus for the iagonal equations the linearize DME has the form t δρ αα = Γ α α e ε0 α ε0 α /kbt δρ α α δρ αα α α 1 + α α Γ α αe ε0 α ε0 α /kbt ρ 0eq V t 0 α α V t0 αα α α k B T 153 or t ρ αα = α Γ αα ρ α α Γ α αρ αα + f αα, 154 where f αα e ε0 α /kbt Z s α Γ α α V t 0 α α V t0 αα. 155 k B T Linearization of noniagonal equations yiels t ρ = iω + Γ ρ + f, 156 where f χ α χβ ρ 0eq αα ρ 0eq ββ. 157 One can see that within the secular approximation the riving term is conservative in the noniagonal equations an issipative in the iagonal equations. The ensity matrix ρ in the equations above is still efine with respect to the aiabatic basis χ α, although the relaxation term is alreay expane aroun the unperturbe states χ 0 α. Let us now make transformation to ρ 0 efine with respect to the unperturbe iagonal basis χ 0 α. The relations between the two ensity matrices have the form ρ 0 = χ 0 χ α χ α ρ α β χ 0 β β, 158 α β c.f. Eq. 86, an ρ = χ 0 χ α α ρ 0 α β χ0 χβ β. 159 α β Differentiating the first equation over time one obtains, at linear orer in ˆV t, t ρ0 = χ 0 χ α χ α ρ α β χ 0 β β + χ 0 χ α χ α ρ α β χ 0 β β + χ 0 χ α χ α ρ α β χ 0 β β α β α β α β = χ 0 α χ α ρ0eq α β + ρ 0eq χ χ 0 β β + χ 0 χ α χ α ρ α β χ 0 β β α β α β = χ 0 α χβ ρ 0eq αα ρ 0eq ββ + χ 0 χ α χ α ρ α β χ 0 β β. 160 α β For α = β the first term in this equation isappears an for α β it cancels a similar term in Eq Thus Eq. 156 with α β transforms as t ρ0 = χ 0 α χ α iω α β + Γ χ α β ρ α β χ 0 β β α β = α χ α iω α β + Γ α β χ α χ 0 α ρ 0 α β χ0 β χ β χβ χ 0 β. α β α β χ 0

23 In this expression the projectors such as χ 0 α χ α are linear in ˆV t, if the inices o not coincie. Thus there are two types of contributions: i All inices pairwise coincie an thus ρ 0 α β ρ0 ˆV t or ii α = β so that ρ 0 α β ρ0eq α α an one of the pair of inices in projectors o not coincie. This yiels i.e., t ρ0 = iω 0 0 α β + Γ α β χ 0 α χ α α β ρ 0 iω α β + Γ α β ρ 0eq α α χ0 α χβ χβ χ 0 β χ 0 α χ α iω α β + Γ α β χ α χ 0 β ρ 0eq χ β β χ 0 β β α β = iω 0 0 α β + Γ χ 0 α χ 0 α t ρ0 α β ρ 0 χβ Γββ ρ 0eq ββ χβ Γββ ρ 0eq ββ = iω 0 0 α β + Γ χ 0 Γ αα ρ 0eq αα χ α χ 0 β Γ αα ρ 0eq αα χ α χ 0 β α β ρ 0 iω + Γ ρ 0eq α iω + Γ χ 0 χ α β α χβ Γββ ρ 0eq χ 0 ββ χ α β Γαα ρ 0eq αα iω + Γ χ 0 α χβ ρ 0eq χ 0 αα + χ α β αα χ 0 The small projections χ 0 α χβ can be calculate with the help of the perturbative expansion that yiels χ α = χ 0 α χ 0 α χ = β χ 0 χ = α β + χ χ α ˆV t χ α α α χ 0 α ˆV t χ 0 ε 0 β ε0 α χ 0 α Thus the DME for noniagonal components has the form where f 0 [ i + t ρ0 = iω 0 Γ 0 ρ 0eq ω 0 αα ρ0eq ββ β χβ ρ 0eq ββ, 161 ρ 0eq ββ. 16 ε 0 α ε 0 α, 163 = V t0 ω 0 ˆV t χ 0 β = V t ε 0 α ε 0 β ω Γ ρ 0 + f0, 165 Γ αα ρ 0eq αα Γ ββ ρ 0eq ββ ω 0 ] V t 0 As the secular approximation implies Γ ω, relaxation terms in this equation can be neglecte, f 0 i ρ 0eq αα ρ 0eq ββ. 166 V t 0 /. 167 The conservative part of this riving term coul actually be transforme exactly using the relation between the two

24 3 bases, Eqs. 5 an 58. Transformation of the iagonal Eq. 154 is as follows: t ρ0 αα = χ 0 χ α χ α ρ α α χ 0 α α α = [ ] χ 0 α χ α Γ α α ρ α α Γ α α ρ α α + f α α χ α α α = α Γ αα ρ α α Γ α αρ αα + f αα = α Γ αα α β χ α χ 0 α ρ 0 α β χ0 β χ α Γ α α α β χ α χ 0 α χ 0 α ρ 0 α β χ0 β χ α + f αα 0 = Γ αα ρ 0 α α Γ α αρ 0 αα + f αα α Since there are no corrections to Γ αα linear in V t, this finally yiels t ρ0 αα = Γ 0 αα ρ0 α α Γ0 α α ρ0 αα + f αα α where f αα 0 e ε 0 α /kbt Z s α α Γ 0 α α V t 0 α α V t0 αα. 170 k B T One can see that this transformation was trivial. In the sequel we will rop the superscript 0. Let us now solve Eqs Similarly to Sec. III A, one can write Eq. 169 in the vectorize form t δn = Φsec δn + f iag, 171 where δn α δρ αα an the elements of Φ sec are given by Eq Since f contains positively- an negativelyrotating terms, the stationary solution of this equation has the form with ± iωδn ± = Φ sec δn ± + f iag,± 17 f ± αα = e εα/kbt Z s α α V ± Γ α α α α V ± αα. 173 k B T Here V + 0,αγ V 0,αγ an V 0,αγ ˆV 0 = V0,γα. The solution of this equation can be expane over the set of right αγ eigenvectors efine by an equation similar to Eq. 97, δn ± = µ C ± µ R µ. 174 Inserting this into Eq. 17, multiplying from left by the left eigenvector L ν an using orthogonality in Eq. 98, one obtains an Now the final result for the populations is ± iωc ± ν = Λ ν C ± ν + L ν f iag,± 175 C ± ν = L ν f iag,± Λ ν ± iω. 176 δn ± = µ L µ f iag,± Λ µ ± iω R µ. 177

25 4 At equilibrium, ω = 0, this expression shoul reuce to the static result δn α = ± L µ f iag,± R µα = e εα/kbt Λ µ Z s µ δεα k B T + δz s Z s 178 that can be proven to satisfy Eq. 169 in the static case. Using Eq. 173 an ± V αα ± ientity µ R µα Λ µ δε δε Γ α α α α k B T α α = δε α, one obtains the L µα = δε α k B T δz s Z s 179 that shoul be satisfie by the matrix solution an can be use for checking. Noniagonal components of the DME satisfy the equations ± iωδρ ± = iω + Γ δρ ± + f±, 180 where f ± 1 i + Γ ω V ± ρ eq αα ρeq ββ. 181 These equations have the solution = f ± Γ + iω ± iω. 18 δρ ± For a physical quantity  the linear response has the form At = e iωt A + ω + e iωt A ω, 183 where A ± ω = N A R µ L µ f iag,± + A βα f ± Λ µ= µ ± iω Γ α β + iω ± iω 184 c.f. Eq Here A α = A αα. The zero eigenvalue, µ = 1 an Λ 1 = 0, oes not make a contribution to this formula since L 1 given by Eq. 114 is orthogonal to f iag,± efine by Eq Physically relevant are real an imaginary parts of the linear response A ω = Re A + ω + Re A ω A ω = Im A + ω + Im A ω. 185