Charge transfer and coherence dynamics of tunnelling system coupled to a harmonic oscillator
aa r X i v : . [ c ond - m a t . o t h e r] A p r Charge transfer and coherence dynamics oftunnelling system coupled to a harmonic oscillator
S Paganelli and S Ciuchi
Dipartimento di Fisica, Universit`a dell’Aquila, Via Vetoio,I-67100 L’Aquila, ItalyCRS SMC, INFM-CNR, Roma ItalyE-mail: [email protected],[email protected]
Abstract.
We study the transition probability and coherence of a two-site system, interactingwith an oscillator. Both properties depend on the initial preparation. The oscillator isprepared in a thermal state and, even though it cannot be considered as an extendedbath, it produces decoherence because of the large number of states involved in thedynamics. In the case in which the oscillator is initially displaced a coherent dynamicsof change entangled with oscillator modes takes place. Coherency is however degradedas far as the oscillator mass increases producing a increasingly large recoherencetime. Calculations are carried on by exact diagonalization and compared with twosemiclassical approximations. The role of the quantum effects are highlighted in thelong-time dynamics, where semiclassical approaches give rise to a dissipative behaviour.Moreover, we find that the oscillator dynamics has to be taken into account, even in asemiclassical approximation, in order to reproduce a thermally activated enhancementof the transition probability.PACS numbers: 71.38.Ht,63.20.kd,63.20.kk,03.65.Yz
Submitted to:
J. Phys.: Condens. Matter
1. Introduction
The quantum dynamics of a charge, moving between two potential minima, is stronglyinfluenced by the variations of the surrounding geometrical configuration. The potentialthe charge is put in, is usually produced by heavy degrees of freedom (a lattice, amolecular structure or an environment) evolving as well. In many cases, only one normalmode of the heavy system is expected to be coupled with the tunnelling charge. Thisoccurs for very simple molecular or mesoscopic structure, or when the time-scales ofheavy and light systems’ dynamics are so different to allow the coupling with a singlecollective mode.As an example, we can consider the transport of a charge between localized sites,in a crystal, affected by coupling with optical phonon modes, hereafter we shall referto this picture also for the choice of the notation. Moreover, there are a lot of other harge transfer and coherence of tunnelling system coupled to an oscillator antiadiabatic regime. While, in solid state physicsdispersionless oscillators can modelize optical phonons of the systems whose frequenciesare often much lower than hopping amplitudes of the itinerant electrons, leading toadiabatic regime.To perform an extensive study of coherence and tunnelling, as system’s parametersspan all the accessible phase diagram, we choose a simple model of a single tunnellingsystem, coupled to a single oscillator. Such a model has been widely analyzed, indifferent regimes and approximations, because of its importance both as a buildingblock of cluster expansion in a lattice model[6], and as a model for chemical reactionsand charge transfer in organics. Following a block diagonalization technique, introducedby Fulton and Gouterman [7], we calculate quantum propagators. These results hasbeen used for calculating exactly the finite temperature spectral functions [8] and tocharacterize the band-width behaviour with temperature. Here, we study the quantumdynamics of the system, considering the transition probability and coherence. For thispurpose, we study the reduced density matrix, taking into account two different initialsystem preparations: i) the electron preparation, in which oscillator is taken from anthermal equilibrium distribution in absence of interaction, and ii) polaron preparation, inwhich oscillator’s initial state is taken from a thermal equilibrium distribution in absenceof tunnelling. Notice that the two considered preparations are also representative ofsystems in which the charge is promoted to a given energy level with (polaron) orwithout (electron) vibronic relaxation. Reduction is obtained by tracing out the bosonicdegrees of freedom. The particle transitions are characterized by the diagonal elements, harge transfer and coherence of tunnelling system coupled to an oscillator
2. The Model
The model we shall consider is described by the following Hamiltonian H = ω a † a − J σ x − ˜ gσ z ( a † + a ) , (1)describing a spin-1 / ω . The modelcan be associated to a large number of physical systems [10, 11] but, for sake of clearness,we shall refer to an electron, in the tight binding approximation, moving in a two-sitelattice and interacting with it by the local distortion of the lattice site [8]. In particular,it can be shown that it is equivalent to the Holstein two-site model [12, 13, 8], withoperators a and a † referring to the relative phonon coordinate and providing that harge transfer and coherence of tunnelling system coupled to an oscillator σ z = c † c − c † c and σ x = c † c + c † c . The center of mass coordinate can easily be decoupled (for a moredetailed discussion see [8, 14]). Therefore throughout this paper we shall refer the wordelectron to the tunneling system and the word phonon to the oscillator.The strength of the electron-phonon interaction is given by the constant ˜ g = g/ √ J is the electron wave function overlap or hopping and 2 J is the tight binding half-bandwidth.Beside the temperature, we can choose two parameters that characterize the modeli) the bare e-ph coupling constant λ = g / ( ω J ) given by the ratio of the polaron energy( E p = − g /ω ) to the hopping J and ii) the adiabatic ratio γ = ω /J .In terms of these parameters we can define a weak-coupling λ < λ > γ < γ > λ as coupling constant we may choose anothercombination which is more appropriate in the so called atomic ( J = 0) limit i.e. α = q λ/ (2 γ ) (see Appendix).
3. Reduced Density matrix
The study of the charge dynamics is not trivial because, in general, it is entangledwith the harmonic oscillator. The time dependent correlation functions of the two-siteHolstein model has been investigated in the past [15] and also a short time transferdynamics has been introduced in [16].In this paper, we introduce a density matrix approach for the charge dynamicsover a very large time range. Hereafter, we shall assume that charge and oscillator areinitially separated, being the former localized on the first site and the latter in a mixedthermal state. The corresponding density matrix is ρ (0) = X n e − βω n Z | φ n ih φ n | ⊗ | ih | , (2)where we used the notation | i = c † | i and β is the inverse temperature. The state | φ n i depends on the choice of the initial preparation [17], in this paper we study twodifferent situations obtained from two different limiting regimes:(i) electronic preparation (el): the electron is initially free ( g = 0) and the oscillator isat its thermal equilibrium ρ ( el ) (0) = X n e − βω n Z | n ih n | ⊗ | ih | , (3)(ii) polaronic preparation (pol): electron is initially localized ( J = 0) on a given site(say 1), while the oscillator is displaced accordingly (see Appendix) ρ ( pol ) (0) = X n e − βω n Z | ψ n ih ψ n | ⊗ | ih | , (4) harge transfer and coherence of tunnelling system coupled to an oscillator g , in the first case, and J , in the second one,and letting evolve the density matrix with the Hamiltonian (1) ρ ( t ) = e − iHt ρ (0) e iHt .The temperature enters only in the initial state trough the incoherent distribution ofthe initial oscillator states in both preparation.Tracing over the oscillator degree of freedom, we obtain the electron reduced densitymatrix ρ ( el ) ( t ) = Tr ph { ρ ( t ) } , (5)which, in terms of the oscillator’s number states, is ρ ( el ) ( t ) = X n,m e − βω n Z h m | e − iHt | n, ih n, | e iHt | m i , (6)To characterize the motion of the polaron we cannot reduce the density matrix bytracing out the phonon degrees of freedom, this is because the polaron itself containsphonons. In order to understand better the polaron dynamics, let us first apply a Lang-Firsov transformation D (see Appendix), the new fermionic particle corresponds to apolaron, so the density matrix with the initial localized polaron can be written as ρ ( pol ) ( t ) = T r ph { D † ρ ( t ) D } , (7)and reads, in terms of the oscillator’s number states, as ρ ( pol ) ( t ) = X n,m e − βω n Z h m | e − i ¯ Ht | n, ih n, | e i ¯ Ht | m i . (8) In this paper, we will study two measures: one for transition probability and the otherfor the degree of coherence. The diagonal elements of the reduced density matrix, inthe site basis, represent the population of each site. The transition probability from site1 to site 2 is given by w , ( t ) = h | ρ ( t ) | i . (9)where ρ is the reduced density matrix in any of the previously introduced preparations.The off-diagonal elements of the reduced density matrix represent the quantuminterference between localized amplitudes. However their knowledge is not sufficient todetermine whether the state is pure or not. Suppose that initial state is pure, if diagonalelements do not evolve in time, the suppression of the off-diagonal elements implies theevolution into a mixed state. In this particular case, the knowledge of off-diagonalelements also determines the purity of the system. In the more general case in whichall the elements of ρ evolve, the choice of the off-diagonal elements obviously dependson the basis. A basis independent measure for purity (called purity itself) is P ( t ) = Tr ρ ( t ) . (10)where again ρ is the reduced density matrix. It is easy to see that 1 / ≤ P ≤ P = 1 if and only if the state is pure and P = 1 / harge transfer and coherence of tunnelling system coupled to an oscillator t it is found useful to consider the time averaged transition probability andcoherence, defined as¯ Q ( t ) = 1 t Z t dt ′ Q ( t ′ ) , (11)where Q can be either w , or P .
4. Methods
In this section we present the methods which we use to get the reduced density matricesfor both initial preparations.
As shown by Fulton and Gouterman [7], a two-level system coupled to an oscillatorin such a manner that the total Hamiltonian displays a reflection symmetry, may besubjected to a unitary transformation which diagonalizes the system with respect tothe two-level subsystem [7, 18, 19, 20]. This method can be generalized to the N-sitesituation, if the symmetry of the system is governed by an Abelian group [19].In particular, an analytic method for calculating the Green functions of the two- siteHolstein model is given in [8, 21]. Here, the Hamiltonian is diagonalized in the fermionsubspace by applying a Fulton Gouterman (FG) transformation. So the initial problemis mapped into an effective anharmonic oscillator model. It is possible to introducedifferent FG transformations for the electron and the polaron. The new problem resultsto be very simplified and very suitable to be numerically implemented. Analyticalcontinued-fraction results exist for the electron case[8, 21].In this section, we briefly recall the FG transformations method. The densitymatrix elements are given explicitly in terms of effective Hamiltonians and calculatedby means of exact diagonalizationThe FG transformation we use for the electronic case is V = 1 √ − a † a − − a † a ! , (12)the new Hamiltonian ˜ H = V HV − becomes diagonal in the electron subspace˜ H = H + H − ! , (13)the diagonal elements, corresponding to the bonding and antibonding sectors of theelectron subspace, being two purely phononic Hamiltonians H ± = ω a † a ∓ J ( − a † a − ˜ g ( a † + a ) . (14)The operator ( − a † a is the reflection operator in the vibrational subspace and it satisfiesthe condition ( − a † a a ( − a † a = − a . A wide study of the eigenvalue problem was carried harge transfer and coherence of tunnelling system coupled to an oscillator H ± is approximately diagonalized by applying adisplacement, the dynamics is reconstructed by the calculated eigenvectors and energies.The evaluation of the polaron Green function can be done on the same footings,but the expression involves also the non diagonal elements of the resolvent operators,causing an exponential increasing of the numerical calculations.To avoid this problem, we first perform the LF transformation and then apply, onthe resulting Hamiltonian (.4), a different FG transformation V = 1 √ − ( − a † a ( − a † a ! . (15)The new Hamiltonian ˜ H LF = V ¯ HV − is¯ H LF = ¯ H +
00 ¯ H − ! , (16)where ¯ H ± = ω a † a + J ( − a † a e ∓ α ( a † − a ) + E p / , (17)is real and symmetric but not tridiagonal in the basis of the harmonic oscillator, thematrix elements of ¯ H ± are given in [8].In order to write down the density matrix elements, let us introduce the followingnotation: R ( ± ) m,n ( t ) = h m | e − iH ± t | n i (18)¯ R ( ± ) m,n ( t ) = h m | e − i ¯ H ± t | n i , (19) N m,n , ( t ) = h m, | e − iHt | n, i = 12 h R (+) m,n ( t ) + R ( − ) m,n ( t ) i (20) N m,n , ( t ) = h m, | e − iHt | n, i = ( − m h R (+) m,n ( t ) − R ( − ) m,n ( t ) i , (21) M m,n , ( t ) = h ψ m , | e − iHt | ψ n , i = 12 h ¯ R (+) m,n ( t ) + ( − m + n ¯ R ( − ) m,n ( t ) i (22) M m,n , ( t ) = h ψ m , | e − iHt | ψ n , i = 12 h ( − n ¯ R ( − ) m,n ( t ) − ( − m ¯ R (+) m,n ( t ) i . (23)The reduced electron density matrix elements are ρ ( el )1 , ( t ) = X n,m e − βω n Z | N m,n , ( t ) | ρ ( el )2 , ( t ) = X n,m e − βω n Z N m,n , ( t ) N ∗ m,n , ( t ) , (24)the calculation for the polaron case gives ρ ( pol )1 , ( t ) = X n,m e − βω n Z | M m,n , ( t ) | ρ ( pol )2 , ( t ) = X n,m e − βω n Z M m,n , ( t ) M ∗ m,n , ( t ) . (25) harge transfer and coherence of tunnelling system coupled to an oscillator
8A qualitative insight into the relevant timescales involved in the evolutions of ρ ( el ) and ρ ( pol ) can be gained by looking at the behaviour of the spectral functions of themodel (1) studied in our previous work [8]. In terms of the Fourier transform of function N m,n , ( t ), the electron spectral function A ( ω ) can be defined as A ( ω ) = − π Im X n e − βω n Z N n,n , ( ω ) . (26)Analogous equation holds for the polaron spectral function relating it to the function M , ( ω ).An example of A ( ω ) is reported in figure 1. We notice that three energy scales(depicted schematically in figure 1) can be associated A ( ω ) [8]. One is the separationof the low lying energy level ∆ E , the other is the phonon energy ω and finally thereis the tunnelling J . They are depicted schematically in figure 1. These energy scalesdefine three different timescales:i) τ J = 2 πJ − ,ii) τ ω = 2 πω − iii) τ Q = 2 π ∆ E − .As it is reasonable from the relation between spectral functions and reduced densitymatrix (26), these characteristic timescales are recovered in the reduced density matrixevolution. -7 -6 -5 -4 -3 -2 -1 +1 -4 -3 -2 -1 0 1 2 3 A ( ω ) ω ω -6 -5 -4 -3 -2 -1 -3.10 -5 -5 ω -E ∆ E Figure 1.
Electron spectral function for γ = 0 . λ = 2and T = 0 see ref. [8]. The case, in which a light quantum particle interacts with much more massive particles,is very common in solid state and molecular physics. We discuss the adiabatic regime ,meaning that, in a characteristic time for the the light particle dynamics, the heavydegrees of freedom can be considered approximately quiet. Here, we describe the SAapproach in its basic formulation for the dynamics. harge transfer and coherence of tunnelling system coupled to an oscillator H = p m + mω x − ¯ g √ xσ z − J σ x − ω , (27)with ¯ g = g √ mω . In the adiabatic limit ( γ ≪ x as a classical parameter.Within this approximation, we put ω = q k/m → m is the ion mass) and theHamiltonian becomes H ad = k x − ¯ g √ xσ z − J σ x . (28)The eigenvalues can be expressed trough the classical displacement xV ± ( x ) = k x ± Ω( x ) , (29)with Ω( x ) = q ¯ g x + J . The lowest branch ( − ) of (29) defines an adiabatic potentialwhich has a minimum at x = 0 as far as λ < λ >
1, it becomes double wellpotential with minima at ± x m , x m = q ¯ g k − J ¯ g , in this case the electron is mostlylocalized on a given site. The quantum fluctuations are able to restore the symmetry inanalogy to what happens for an infinite lattice [23]. It is worth noticing that, in this limit,Hamiltonian (1) is equivalent to the adiabatic version of the spin-boson Hamiltonian[24, 25].The temporal evolution is given by e − iH ad t = e − i kx t [cos Ω( x ) t + i ( ¯ gx √ x ) σ z + J Ω( x ) σ x ) sin Ω( x ) t ] , (30)so the density matrix dynamics can be explicitly calculatedThe electronic initial preparation, corresponds to the density matrix ρ (0) = | ih | s kβ π Z dx e − βk x | x ih x | , (31)tracing out the phonon we obtain the electron reduced density matrix with elements ρ ( el )2 , = s βJ λ π Z du e − βJλ u sin ( J t q u λ + 1)1 + λ u ρ ( el )1 , = − i s βJ λ π Z du e − βJλ u sin(2 J t √ u λ + 1)2 √ λ u + 1 (32)where the scaled lenght u = xk √ / ¯ g was introduced.In the same way, we can introduce the polaronic preparation ρ (0) = | ih | e − β ¯ g k s kβ π Z dxe − β ( k x − ¯ g √ x ) | x ih x | , (33)It is worth noting that, in the adiabatic limit, we cannot define the polaronic dynamics,as introduced in (7), because the operator D is not defined for ω = 0. In this limit, harge transfer and coherence of tunnelling system coupled to an oscillator ρ ( pol )2 , = s βJ λ π Z ∞−∞ du e − βJλ ( u − sin ( J t q ( uλ ) + 1)( uλ ) + 1 ρ ( pol )1 , = s βJ λ π Z ∞−∞ du e − βJλ ( u − uλ sin ( J t q ( λu ) + 1)(( λu ) + 1) − i sin(2 J t q ( λu ) + 1)2 q ( λu ) + 1 (34)It is possible to show that ρ ( pol )2 , is actually the adiabatic limit of the diagonalelement of the reduced polaronic density matrix, while this is not true for the off-diagonal elements.We want to stress that, in the SA approach, the phonon is completely static becauseits momentum p has been neglected. Here, only the initial phonon distribution plays arole, but during electron hopping, oscillator is taken to be fixed. To account for dynamics of the slow variable, a mixed quantum-classical dynamicscan be introduced. In the past, several schemes for quantum-classical dynamicshas been proposed, for example starting from the Born-Oppenheimer (SA) adiabaticapproximation for the ground state at each step and using a density functionalHamiltonian [26, 27]. Another approach, good for a short time dynamics, consistsin a mapping from the Heisenberg equations to a classical evolution by an average overthe initial condition[28, 29]. Some schemes are based on the evolution of the densitymatrix coupled to a classical bath [30, 31]. A systematic expansion over the mass ratiohas also been done, starting from partial Wigner transform of the Liouville operator, in[32, 33, 34]. The QC approximation we use is essentially that of refs. [30, 31].Let us consider Hamiltonian (27), where x and p are assumed to be classicalvariables which can be represented as the components of a vector u . Then a QC statevector can be introduced as v = u ⊗ σ = xpσ x σ y σ z . (35)The classical variables evolve with the Ehrenfest equations ˙ x = pm ˙ p = mω x − ¯ g √ h σ z i , (36) harge transfer and coherence of tunnelling system coupled to an oscillator ˙ σ x = −√ gxσ y ˙ σ y = √ gxσ x − J σ z ˙ σ z = 2 J σ y . (37)To give a unified description of the overall evolution, we define a Liouvillian operator L = L x + L p + L σ with L σ = − i −√ gx √ gx − J J (38)and L x = ˙ x ∂∂x L p = ˙ p ∂∂p . So, the time evolution is given by v ( t ) = e i L t v (0) . (39)The numerical integration can be implemented using the symmetries Trotterbreakup formula [35, 36] v ( t ) ≃ (cid:16) e i L σ ǫ e i L p ǫ e i L x ǫ e i L p ǫ e i L σ ǫ (cid:17) N v (0) (40)with ǫ = t/N . All the density matrix elements, can be expressed in terms of elementsof v ( t ).
5. Results
In figure 2 is shown the time behaviour of the purity P (Eq. (10)) as well as the transitionprobability ( (9)) obtained in the antiadiabatic regime when the phonon frequency ( ω )is much larger than electron hopping J for both (el) (3) and (pol) (4) initial preparations.Timescales defined in section are shown as vertical lines, the time scale is logarithmicto better show the much different time domains. We consider two parameters sets atseveral temperatures. One characteristic of strong coupling (left panels) and the otherof weak coupling (right panel). The same sets of parameters and temperatures is usedin figure 3 where with show the time-averaged P and w . Let us first discuss the strongcoupling regime.It is known that, in the antiadiabatic regime, the polaron is a well defined quasi-particle at strong coupling [37], in the sense that, in the polaronic spectral function,almost all the spectral weight is contained in the polaronic peak. This has also beenshown for a two-site model [8, 14, 15, 38, 39, 40]. On the contrary, in the electronspectral function, the total spectral weight is distributed between a large number offrequencies [8].From the point of view of transition probability and purity, the strong dependenceon initial preparation can be seen comparing the low temperature evolution for boththe actual (figure 2 left panel) and the time-averaged (figure 3 left panel) quantities. harge transfer and coherence of tunnelling system coupled to an oscillator τ J , and becomes a mixed ensemble, even at zero temperature. ‡ On the same timescale, transition probability approaches 1 / T increases. Such a suppression results from destructive interference betweentime evolution of the different terms appearing in (6) when excited oscillator statesare initially populated. Referring to the spectral analysis [8] and to figure 1, thisphenomenon must be ascribed to the superposition of a large number of high frequencyexcitations.Even if the transfer does not have a regular shape, one can see some high frequencyoscillations of period τ ω . These frequencies correspond to the energy separation betweentwo adjacent electronic bands [8].On the contrary, the polaron preparation (see figure 2, top left panel) evolves ina state which is completely coherent at zero temperature. The frequency associated topolaron transfer is equal to the renormalized tunnelling J ∗ , as predicted by the HLFA(see .6). So, the state is pure and delocalized. The polaron state remains coherenteven for temperatures comparable with ω , but higher frequency modulation appearsmaking the state oscillate from a pure to a mixed one. Nevertheless, it is possible to seean overall modulation of the transition probability with the same period τ Q even at thelargest temperature. This is in contrast with the HLFA at T = 0 (.5) which predicts thatthe polaron band decreases with temperature and consequently τ Q increases. Howeverthe purity decreases as temperature increases, as shown in figure 2. This is an effectof the broadening of the polaron band that is observed as temperature increases [8].Indeed, a distribution of spectral weight among several poles around the polaron bandoccurs as an effect of increasing vibronic excitations (ref. [8] figure 3 upper panel). Thisleads to a decoherence effect due to destructive interference between these oscillatingcontributions to purity (see Eqs. (18,25)). For high temperature ( T ≫ ω ), the statebecomes completely mixed and the evolution of the polaron is analogous to that of theelectron. This is evident from the highest temperatures curves shown in figure 3, leftpanels upper and bottom left. We conclude that the main source of decoherence istemperature for polaron, while the electron decoheres even at zero temperature due tothe coupling with the vibronic mode.This is also found in the weak coupling regime (electron preparation Figs. 2,3bottom right panels). Here electron coherence approaches a value which is larger in ‡ Notice that, in this case, the analysis of purity and transition probability alone in principle do notallow to determine which states the mixture is composed of (pointer states). In particular whether thestates are localized or not. However, a straightforward analysis of non diagonal elements of reduceddensity matrix shows that the states are indeed localized. harge transfer and coherence of tunnelling system coupled to an oscillator P ( po l ) W , ( po l ) t 0.5 0.6 0.7 0.8 0.9 1 P ( po l ) W , ( po l ) t 0.5 0.6 0.7 0.8 0.9 1 P ( e l ) W , ( e l ) t 0.5 0.6 0.7 0.8 0.9 1 P ( e l ) W , ( e l ) t Figure 2.
Polaron (top) and electron (bottom) populations and purity. Left panelsantiadiabatic and strong coupling regime: γ = 10 λ = 40. Right panels antiadiabaticand weak coupling regime γ = 10 λ = 10. Curves are for T /ω = 0 . T /ω = 0 . T /ω = 2 . T /ω = 10 . τ ω , τ J , τ Q . average than that at strong coupling (figure 3) and decreases as temperature increases.We see (Figs. 2,3 right panels) that the polaron purity differs qualitatively from theelectron only near zero temperature, while averaged polaron and electron transitionprobabilities is essentially the same for all showed temperatures. Indeed, a sufficientlyweak interaction is not able, at zero temperature, to excite many vibrational states,so the electron decoherence is essentially given by the small perturbation of the lowestoscillator’s states. On the other hand, the polaron is not formed (we are below thepolaron crossover) and the charge does not acquire much coherence by moving withthe oscillation cloud. Spectral analysis shows (ref. [8] figure 3 lower panel) that inweak coupling HLFA is qualitatively recovered, we have a polaron band narrowing astemperature is raised up in contrast with the strong coupling behaviour.It is worth stressing that, in both weak and strong coupling regimes, at hightemperature, the increasing number of oscillator states involved in the initial stateproduce decoherence on timescale τ J . Decoherence can be partial but nonethelessno environment is needed to explain the decoherence process. The only source ofdecoherence are the states populated by the initial thermal distribution harge transfer and coherence of tunnelling system coupled to an oscillator P _ ( po l ) W _ , ( po l ) t 0.5 0.6 0.7 0.8 0.9 1 P _ ( po l ) W _ , ( po l ) t 0.5 0.6 0.7 0.8 0.9 1 P _ ( e l ) W _ , ( e l ) t 0.5 0.6 0.7 0.8 0.9 1 P _ ( e l ) W _ , ( e l ) t Figure 3.
Polaron (top) and electron (bottom) time-averaged populations and purity.Parameters and labels are the same of figure 2.
Results form the Exact Diagonalization method is reported in figure 4, the averagedquantities are shown in 5 in the same way we did in the antiadiabatic case. Notice thatnow the shortest timescale is τ J .Let us first discuss the strong coupling case. We see that, in contrast withantiadiabatic regime, there is a marked dependency on temperature of both electron andpolaron properties. More specifically, polaron preparation no longer evolve coherentlyat low temperature.In the first time scale, τ J , the particle is localized (its transition probability isextremely low) but the state keeps on being quite pure. The polaron is trappedinside the initial site and both transition probability and coherence evolve initially withcharacteristic time τ J independently on temperature. At intermediate timescale τ ω temperature induces delocalization while the coherence decreases. In this time regime,the polaron transition probability is related to the quasi classical motion of the oscillatorand depends strongly on temperature.This can be seen in figure 6, where we plot the temperature dependence of the levelreached by the averaged transition probability on the time scale τ ω (inset). Since thereis no clear plateau in the averaged transition probability for times greater than τ ω , thechoice of the transition probability level it is rather arbitrary. We choose the value of¯ w at τ ω . We see that this transition probability level pass from a low temperature harge transfer and coherence of tunnelling system coupled to an oscillator P ( po l ) W , ( po l ) t 0.5 0.6 0.7 0.8 0.9 1 P ( po l ) W , ( po l ) t 0.5 0.6 0.7 0.8 0.9 1 P ( e l ) W , ( e l ) t 0.5 0.6 0.7 0.8 0.9 1 P ( e l ) W , ( e l ) t Figure 4.
Polaron (top) and electron (bottom) populations and purity. Left panelsAdiabatic and strong coupling regime: γ = 0 . λ = 2. Right panels adiabatic andweak coupling regime γ = 0 . λ = 0 .
5. Curves are for
T /ω = 0 . T /ω = 0 . T /ω = 2 . T /ω = 10 . τ J , τ ω , τ Q . behaviour, which is temperature independent, to a temperature dependent behaviourtrough a wide crossover.At very low temperature, after the characteristic time τ ω , coherence and transitionprobability reach a quasi stationary value that is essentially dominated by fast tunnellingof the charge between the two sites, with a given phonon displacement. Oncetemperature increases, classical activation processes of the phonon coordinate becomeseffective, producing an increase in transition probability as well as a decrease of thepurity. As we shall see in the next section, this thermal activated behaviour is to beascribed to the phonon classical hopping between two adiabatic minima and disappearsin the SA where such hopping events are absent.As far as electron is concerned, we can see that the site occupancy begins tooscillate coherently, with period τ J , with a damping increasing with temperature. Inthe same timescale, the time averaged value shows a saturation at low temperature. Ata temperature independent intermediate timescale τ ω , time averaged coherence reachesa very slowly decreasing level which decreases with increasing temperature. Quantumoscillations still exist further in time, but the remaining coherence slows down at hightemperature, so that the long timescale ( τ Q ) seems to be not relevant in this case. Theelectron’s tendency to coherently hop is suppressed by decoherence, induced by excited harge transfer and coherence of tunnelling system coupled to an oscillator P _ ( po l ) W _ , ( po l ) t 0.5 0.6 0.7 0.8 0.9 1 P _ ( po l ) W _ , ( po l ) t 0.5 0.6 0.7 0.8 0.9 1 P _ ( e l ) W _ , ( e l ) t 0.5 0.6 0.7 0.8 0.9 1 P _ ( e l ) W _ , ( e l ) t Figure 5.
Polaron (top) and electron (bottom) time-averaged populations and purity.Parameters and labels are the same of figure 4 W _ , ( po l ) t 0.1 1 0.01 0.1 1 10 ω /T Figure 6.
Time averaged transition probability for λ = 2 . γ = 0 .
1, temperatures arefrom bottom to upper curves
T /ω = 0 . , . , . , . , . , .
0. Vertical lines marksfrom left to right timescales τ J , τ ω and τ Q respectively. Inset: levels reached at time τ ω (arrows in the main panel) as a function of the inverse temperature. harge transfer and coherence of tunnelling system coupled to an oscillator P ( po l ) W , ( po l ) t 0.5 0.6 0.7 0.8 0.9 1 P ( po l ) W , ( po l ) t 0.5 0.6 0.7 0.8 0.9 1 P ( e l ) W , ( e l ) t 0.5 0.6 0.7 0.8 0.9 1 P ( e l ) W , ( e l ) t Figure 7.
Polaron (top) and electron (bottom) populations and purity in the adiabaticstrong coupling regime γ = 0 . λ = 2. Left panels low temperature T /ω = 0 .
1. Rightpanels high temperatures
T /ω = 10 .
0. Curves refers to ED (black), QC (blue), SA(green) approximations. Vertical lines marks from left to right the timescales τ J , τ ω , τ Q . In this section, we show a comparison between the results obtained in the threedifferent ways described before: exact diagonalization (ED) by means of mappingintroduced in Sec. 5.1, the quantum-classical (QC) dynamics approach described inSect. 4.3 and the static (SA) approximation (Sect. 4.2). We shall limit ourselves to anadiabatic case ( γ = 0 .
1) with electron-phonon interaction strong enough to allow thepolaron formation ( λ = 2).In figure 7, is reported the exact dynamics given by the three different techniques,while in figure 8 is shown the time average. Remember that, as far as the polaron is harge transfer and coherence of tunnelling system coupled to an oscillator P _ ( po l ) W _ , ( po l ) t 0.5 0.6 0.7 0.8 0.9 1 P _ ( po l ) W _ , ( po l ) t 0.5 0.6 0.7 0.8 0.9 1 P _ ( e l ) W _ , ( e l ) t 0.5 0.6 0.7 0.8 0.9 1 P _ ( e l ) W _ , ( e l ) t Figure 8.
Polaron (top) and electron (bottom) time-averaged populations and purity.Parameters and labels are the same of figure 7 W _ , ω /T SA (el)SA (pol)ED (pol)QC (pol) Figure 9.
Time average transition probability level at the timescale τ ω in theadiabatic regime ( λ = 2 . , γ = 0 .
1) as a function of inverse temperature for differentpreparations and different approximations. concerned, in both SA and QC approximation, the purity represents that of the electronwith a initially displaced phonon distribution.At low temperature (left panels), and within the τ J timescale, the classical phononis almost freezed, and so both SA and QC approximation are equivalent. Nevertheless,the ED behaviour, of both polaron and electron preparation, is quite different becauseof quantum fluctuations. In particular, for short timescales, one can see that the ED harge transfer and coherence of tunnelling system coupled to an oscillator T greater than ω , when SA and QCreproduce the ED transition probability within the τ J timescale, as results evident in theright panels of Figs. (7-8). The QC approach remains a good description also for highertimescales. Notice that the same occurs in the transport of extended system whereclassical incoherent transport is achieved when T is greater than 0 . ω [41]. It is worthnoticing that for very high temperature ( T ≃ J , T /ω = 10 in figs. 8,9) the polaronis not formed, its dynamics approaches that of the electron in an initially delocalizedphonon distribution. As a result, the QC purity approaches the ED’s.At high temperature, the oscillator dynamics plays a relevant role, QC is a muchbetter approximation of ED than SA. This fact can be understood by realizing that themain temperature effect is the damping of the coherent tunnelling oscillations. Oncethese oscillation are sufficiently suppressed, the phonon driven dynamics prevails. Inthe SA framework, the initial thermal distribution of the phonon coordinate makes theelectron thermalizes irreversibly in a time that is the shorter the greater the temperature.Before this adiabatic thermalization, i.e, in a tunnelling period, the SA is still a goodapproximation.Afterwards, the hopping of the oscillator coordinate into the other minimum of theadiabatic potential (equation (29)) takes place. The charge degree of freedom followswhile w , saturates on average. In extended systems this regime corresponds activatedmobility regime [42, 43, 44]. Since SA completely neglects the oscillator’s dynamics,it does not predict correctly w , , as can be seen in figure 9. The QC approximation,instead, gives a correct qualitative prediction. harge transfer and coherence of tunnelling system coupled to an oscillator
6. Conclusions
In this paper, we have studied a simplified model to treat the dynamics of a tunnellingcharge interacting with a vibrational degree of freedom. We introduced a reduced densitymatrix approach to characterize the charge dynamics. Temperature is introducedby taking an initial equilibrium distribution of the oscillator. Both the transitionprobability and the purity are studied, in order to connect the charge transfer withits coherence.Due to the simplicity of our model, we were able to span all the parameter’s spaceeven at high temperature and strong coupling and to study the role of initial preparation.Moreover, we can explore a temporal range which is very large, compared with thetypical time scales that can be obtained in models where the charge is coupled with amany degree of freedom oscillators’ bath [17, 45, 46].As in any finite system, in our model, transition probability and purity can beexpressed as a superposition of many non commensurate by oscillations. We thereforeexpect an oscillatory behaviour in our quantities of interests. However the initial thermaldistribution of the oscillator states induces decoherence on intermediate timescale, dueto the strong interaction with the oscillator. This phenomena occurs depending oninitial preparation of the system.We find that, in the antiadiabatic and strong coupling regime, the polaron exhibitsa coherent tunnelling dynamics over time scales of the inverse polaron renormalizedband. The coherent behaviour is lost out of the polaronic phase, i.e. increasing thetemperature or decreasing the coupling. Electron evolves though partially incoherentdynamics. In the adiabatic strong coupling regime, temperature enhances the incoherentpolaron charge transfer. The opposite occurs in the electron preparation.In the adiabatic regime, two common approximations has been compared withexact results, the aim is to highlight the limits of validity of these approximationsand to provide a simple testing tool, the two-site model, for generalizations to otherextended models. As expected, a dynamical semi-classical approximation gives goodestimates for both coherence and tunnelling amplitude, at high temperature T ≫ ω .Quite unexpectedly, it allows for a good approximation of the transition probabilityat low temperature, as far as time averaged quantities are concerned. However, sucha quasiclassical approximation fails approaching the anti-adiabatic regime where nonadiabatic transition are expected to contribute significantly to charge dynamics. Thissimplified model could serve to test approximate schemes to deal with this regime [45].To conclude, we have shown that a non dissipative evolution of a tunneling system,strongly coupled to a single oscillator, can give rise to decoherence phenomena whenthe initial distribution of the oscillator is thermal and when the oscillator distributionis not initially equilibrated in the presence of the charge, that is sufficiently far fromthe thermal equilibrium distribution in the presence of the charge. These decoherencephenomena are independent on the presence of a dissipative bath. Thus, in a nonequilibrium experiment in which a charge is introduced in a molecular system and harge transfer and coherence of tunnelling system coupled to an oscillator Acknowledgments
Authors wish to thank Sara Bonella, Carlo Pierleoni and Sergio Caprara for fruitfuldiscussions and suggestions.
AppendixAtomic limit
In the atomic limit, the Hamiltonian is diagonalized by the so-called Lang-Firsov (LF)transformation D = e ασ z ( a † − a ) . (.1)This transformation shifts the phonon operators by a quantity α , while the electronoperators are transformed in new fermionic operators, with energy E p , associated toa quasi-particle called polaron[47, 12]. This particle can be tough as a charge movingtogether with a dressing cloud of oscillator quanta, α represents the mean number ofphonons in the polaron cloud.The atomic Hamiltonian H = ω a † a − ˜ gσ z ( a † + a ) , (.2)after the LF transformation ¯ H = D † H D becomes:¯ H = ω a † a + E p / , (.3)the eigenvalues E n = ω n + E p / | ψ jn , j i = D | n, j i = ¯ c † j | n i , were the index n = 0 , . . . , ∞ refers to the photon number, j = 1 , c † j is the polaron creation operator ¯ c † j = Dc † j D † = c † j exp { ( − j α ( a † − a ) } .In the case of finite J , the hopping term is not diagonalized by (.1) and the newHamiltonian ¯ H = D † HD becomes¯ H = ω a † a − J ( σ x cosh(2 α ( a † − a )) ++ i σ y sinh(2 α ( a † − a ))) + E p / . (.4)Depending on the choice of the parameters, the problem will be better described bya electron or polaron excitation picture. In particular, in the weak coupling limit, boththe small polaron and the electron are good quasiparticle while, in the intermediate andstrong coupling regimes, the polaron behaviour prevails [39].The different regimes was widely studied, in literature, both for the two site problemand the extended case. The anti-adiabatic case was first studied in small J perturbation harge transfer and coherence of tunnelling system coupled to an oscillator J issubstituted by an effective hopping integral obtained by averaging the displacementexp [2 α ( a † − a )] on the thermal distribution of phonons. The resulting effective hoppingintegral is J ∗ = J exp( − α ( n B ( T ) + 1 / , (.5)where n B is the Bose occupation number. At zero temperature, the well knownexponential reduction of the bandwidth is obtained J ∗ = J exp( − α ) (.6)and, as the temperature is increased ( T /J >> γ ), the bandwidth decreases rapidly.As we shown in ([8]) and in the present paper, this is a good approximation at zerotemperature but it becomes inadequate at finite T where incoherent processes turns outto be important. References [1] Jean J M 1994
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