Non-Markovian dynamics of the electronic subsystem in a laser-driven molecule: Characterization and connections with electronic-vibrational entanglement and electronic coherence
aa r X i v : . [ qu a n t - ph ] N ov Non-Markovian dynamics of the electronic subsystem in a laser-driven molecule:Characterization and connections with electronic-vibrational entanglement andelectronic coherence
Mihaela Vatasescu ∗ Institute of Space Sciences, INFLPR, MG-23, 77125 Bucharest-Magurele, Romania
Non-Markovian quantum evolution of the electronic subsystem in a laser-driven molecule is charac-terized through the appearance of negative decoherence rates in the canonical form of the electronicmaster equation. For a driven molecular system described in a bipartite Hilbert space H = H el N H vib of dimension 2 × N v , we derive the canonical form of the electronic master equation, deducing thecanonical measures of non-Markovianity and the Bloch volume of accessible states. We find thatone of the decoherence rates is always negative, accounting for the inherent non-Markovian charac-ter of the electronic evolution in the vibrational environment. Enhanced non-Markovian behavior,characterized by two negative decoherence rates, appears if there is a coupling between the elec-tronic states g , e , such that the evolution of the electronic populations obeys d ( P g P e ) /dt > molecule, we analyze non-Markovian dynamics under laser pulses ofvarious strengths, finding that the weaker pulse stimulates the bigger amount of non-Markovianity.Our results show that increase of the electronic-vibrational entanglement over a time interval iscorrelated to the growth of the total amount of non-Markovianity calculated over the same intervalusing canonical measures, and connected with the increase of the Bloch volume. After the pulse,non-Markovian behavior is correlated to electronic coherence, such that vibrational motion in theelectronic potentials which diminishes the nuclear overlap, implicitly increasing the linear entropyof entanglement, brings a memory character to dynamics. I. INTRODUCTION
Memory effects in the dynamics of open quantumsystems [1] have been extensively studied over thepast decade, through new concepts proposed to tacklequantum non-Markovianity and examination of non-Markovian behavior in various scenarios involving openquantum systems [2–5]. The classical definition of aMarkovian process, implying a memoryless time evolu-tion in a classic stochastic process, cannot be simply ex-tended to the quantum regime, where the correspondingquantum probabilities have to be associated with mea-surement schemes. Definition of quantum Markovianityconstitutes a recent research area [3, 4], and is still adebated subject [6–8].We have to note the multiplicity of approaches toquantum non-Markovianity [3–5, 8]: as deviation fromsemigroup dynamics [9], based on the backflow of infor-mation from the environment to the open system [10],as deviation from completely positive divisibility [11],based on the quantum Fisher information flow [12], us-ing entanglement-based measures [11, 13] or quantummutual-information-based measures [14], related to thedynamical behavior of the volume of accessible states[15], and based on quantifiers of the negative rates inthe canonical form of the time-local master equation[16]. Recent proposals use the spectral properties of dy-namical maps [17], and the process tensor framework ∗ mihaela [email protected] [6, 18] to characterize non-Markovian behavior. Thesealternative approaches imply different non-Markovianityconcepts and propose various measures or witnesses ofquantum non-Markovianity. Comparative studies [19–21] show them as offering different perspectives on thecomplex manifestation of quantum memory effects.Non-Markovian quantum dynamics typically occurswhen open quantum systems are coupled to structuredor finite reservoirs, due to strong system-environment in-teractions, large initial system-environment correlations,or low temperature environments. In contrast to Marko-vian (memoryless) evolution of an open quantum sys-tem weakly coupled to a noisy environment, character-ized by decoherence and dissipation, non-Markovian dy-namics of an open system can lead to revivals of its char-acteristic quantum properties, such as quantum coher-ence and entanglement [4, 22, 23]. Recent developmentsin experimental techniques allowing control and modi-fication of the dynamical properties of various environ-ments through quantum reservoir engineering [24] bringforward non-Markovian open quantum systems interact-ing with controllable environments [25–27]. These ex-perimental advances are motivating investigations on therole of non-Markovianity as a resource for quantum in-formation processing [28, 29] or quantum metrology [30].Understanding memory effects in various quantum sce-narios, such as non-Markovianity studies in driven openquantum systems [31], contributes to the recent attemptsto design non-Markovian systems which could be usefulas resources in quantum technologies [22, 23, 25].Molecular physics has a long tradition in treatingsystem-bath interactions, including non-Markovian influ-ences of the environment [32]. Non-Markovian effectsoperate in various molecular processes, such as electrontransfer in complex molecular systems [33], environment-assisted quantum transport [34] in molecular junctions[35], or excitonic energy transfer in photosynthetic com-plexes [36]. Possible applications of non-Markovianity in-clude, for example, the use of certain molecular systemsas quantum probes to reveal characteristic features oftheir environments [4, 5], or utilization of memory effectsin the design of functional artificial biomaterials [37].Current efforts trying to exploit non-Markovianity asa resource for quantum control [38] rely on the under-standing of memory effects as related to a backflow ofinformation from the environment to the system, capableof restoring system coherence. In this sense, recent in-vestigations of strategies for quantum control of memoryeffects in molecular open-quantum systems seek to pro-tect the central system from dissipation and decoherenceby increasing non-Markovian bath response [39]. Non-Markovianity enhancement leading to longer decoherencetimes of the central system could be exploited to increasethe robustness of molecular alignment-orientation [39] orto preserve coherence of molecular qubits.Electronic coherences play an essential role in chem-ical and biological processes, and their function is cur-rently being investigated in new domains like attochem-istry or quantum biology. Recent works on electron dy-namics in molecules explore the mechanisms influenc-ing electronic decoherence and the role played by nu-clear motion in this process, especially in the presence ofstrong nonadiabatic couplings [40]. On the other hand,understanding quantum coherence contributions to elec-tronic energy transport in molecular aggregates and bio-logical systems is a major goal in quantum biology [41].Energy transport is examined using open quantum sys-tem approaches to treat electronic-vibrational dynamicsin large molecules, in which an open ”system” contain-ing relevant molecular electronic states is coupled to abath of harmonic vibrational modes [42]. Studies of non-Markovianity in photosynthetic complexes have shown asignificant non-Markovian information flow between elec-tronic and phononic degrees of freedom, which could playan important role in energy transfer, as well as corre-lations between non-Markovian behavior and long-livedquantum coherence [43].Approaches to quantum non-Markovianity using quan-tum information concepts have been recently developedin the theory of open quantum systems, bringing newframeworks for molecular processes with memory. Non-Markovianity is recognized as a highly context-dependentconcept, whose understanding should not be based solelyon the evolution of the system density operator; infact, system-environment correlations are of direct rele-vance to grasp non-Markovianity more broadly [8]. Thisis also our approach here: We will characterize non-Markovianity of the electronic subsystem in a diatomicmolecule using canonical measures, and subsequently we proceed to understand the dynamic meaning of non-Markovian behavior by relating it to quantum correla-tions in the molecular system, namely entanglement withthe vibrational environment [44] and electronic coher-ence.We consider a diatomic molecule described in a bi-partite Hilbert space H = H el N H vib of the electronicand vibrational degrees of freedom, driven by a laserpulse which couples the electronic states inducing trans-fer of population and influencing the vibrational dy-namics. We shall analyze the electronic subsystem asa driven open quantum system in the vibrational en-vironment. Non-Markovianity of the electronic dynam-ics will be characterized using the approach introducedby Hall et al. in Ref. [16], which employs the canon-ical form of the time-local master equation describingthe open system dynamics to define non-Markovianityquantifiers based on the occurrence of negative decoher-ence rates. We derive the canonical measures of non-Markovianity for a 2-dimensional electronic subsystem ofa laser-driven molecule, and connect non-Markovian be-havior with temporal behaviors of electronic-vibrationalentanglement (quantified using linear entropy and vonNeumann entropy) and electronic coherence (measuredwith l norm and Wigner-Yanase skew information).The canonical measures [16] provide a complete de-scription of non-Markovianity in terms of canonical de-coherence rates. Additionally, we shall also refer to theBloch volume of accessible states as a non-Markovianitywitness [15, 16]. Unlike the canonical measures, theBloch volume is only a possible witness, and does notalways detect non-Markovian behavior [3, 16]. Never-theless, examination of non-Markovianity using differentmeasures, besides being interesting in itself, will help todistinguish non-Markovianity regimes in the dynamicalevolution, highlighting an ”enhanced non-Markovian be-havior” which is detected by both measures.The paper is structured as follows. Sec. II introducesthe non-Markovianity approach used in this paper, basedon the occurrence of negative decoherence rates in thetime-local master equation. The definitions of the canon-ical measures of non-Markovianity [16] and the Blochvolume characterization of non-Markovianity [15, 16] arepresented. Sec. III describes our model, allowing us tocharacterize non-Markovian dynamics of the electronicsubsystem of a laser-driven molecule. We derive thecanonical form of the master equation for a 2-dimensionalelectronic subsystem of a laser-driven molecule, and de-duce the canonical non-Markovianity measures and theBloch volume. Sec. IV contains a theoretical analysis ofthe relations between enhancement of non-Markovianityand dynamical behaviors of the electronic-vibrational en-tanglement and electronic coherence. Sec. V shows thatenhanced non-Markovian behavior in the electronic evo-lution increases the uncertainty on the electronic en-ergy. Sec. VI examines non-Markovian behavior of theelectronic subsystem and its connections with electronic-vibrational entanglement and electronic coherence, tak-ing as example the coupling of two electronic states inthe Cs molecule by laser pulses of several strengths.The time evolutions during the pulse and after pulseare simulated numerically, being analyzed using the non-Markovianity measures, the entropies of entanglementand the measures of electronic coherence. Our conclu-sions are exposed in Sec. VII. The paper includes an ap-pendix which discusses the conditions determining the in-crease of distinguishability between two electronic states. II. CANONICAL FORM FOR ALOCAL-IN-TIME MASTER EQUATION ANDNEGATIVE DECOHERENCE RATES
The concept of quantum Markovianity implicitly usedhere is related to the concept of divisibility of dynam-ical maps [3, 4, 45]. We briefly recall the notion ofdivisibility, which is central to the definition of quan-tum (non)Markovianity in models using time-local mas-ter equations. Considering a dynamical map Λ( t, ρ ( t ) = Λ( t, ρ (0) of anopen system state ρ ( t ), Λ( t,
0) is a t -parametrized fam-ily of completely positive and trace preserving (CPTP)maps. Λ( t,
0) is defined to be divisible if it can bewritten as a composition of two trace-preserving maps,Λ( t,
0) = Λ( t, t ′ )Λ( t ′ , t ≥ t ′ ≥
0, mean-ing that the two-parameter family Λ( t, t ′ ) has to existfor all t, t ′ . The positivity (P) or complete positivity(CP) of Λ( t, t ′ ) lead to the notions of a P-divisible orCP-divisible family of dynamical maps. P divisibilityand CP divisibility of a quantum process were both usedto define the quantum dynamics of a process as beingMarkovian, and to build connections between the quan-tum and the classical concepts of Markovianity [3, 4, 46].Moreover, the notion of k -divisibility of a dynamical map(with 1 ≤ k ≤ n an integer, n the dimension of theopen system, 1-divisibility corresponding to P divisibil-ity, and n -divisibility corresponding to CP divisibility)was introduced to define a ”degree of non-Markovianity”of a quantum evolution [20], as well as the notions of”weak non-Markovianity” (for processes which are onlyP-divisible) and ”essential non-Markovianity” (for pro-cesses which are not even P-divisible).A variety of theoretical and numerical methods areused to treat the dynamics of open quantum systems andto reveal the presence of memory effects [1–5], such asNakajima-Zwanzig projection operator techniques [47],the time-convolutionless (TCL) projection operator tech-nique [48], or stochastic wave-function techniques [49–51].Quantum memory effects attached to an open systemdynamics can be studied either using a non-local mas-ter equation with a memory kernel (obtained throughthe Nakajima-Zwanzig projection operator technique),or, equivalently, using the local in time equation given bythe time-convolutionless (TCL) projection operator tech-nique. Both approaches support an investigation of non- Markovian effects [1, 52]. In the second approach, TCLprovides a local-in-time first-order differential equation˙ ρ ( t ) = L ( t ) ρ ( t ) for the reduced density ρ ( t ) characterizingthe open system, on the condition that a certain operatorinverse exists [2, 4]. For a time-local equation which doesnot involve a memory kernel and an integration over thepast history of the system, the non-Markovian characterof the dynamics appears in the explicit time-dependenceof the generator L ( t ), which keeps the memory aboutthe starting point [51, 52]. The time-local generator L ( t )obtained with TCL method is defined by a perturbationexpansion with respect to the strength of the system-environment coupling, which does not guarantee the com-plete positivity of the resulting map Λ( t,
0) describing theevolution of the open system state between 0 and t : ρ ( t )= Λ( t, ρ (0) [1, 2].If the requirements for preservation of the Hermiticityand the trace of ρ ( t ) are imposed on the generator L ( t )of the time-local master equation ˙ ρ ( t ) = L ( t ) ρ ( t ), one ob-tains a general structure of the master equation (Eq. (7)),which is a generalization of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form for a memoryless mas-ter equation [2–4, 51]. Moreover, the diagonalization pro-cedure leading to this GKSL-like structure provides aunique, and then canonical form of the master equation,which can be used to characterize non-Markovianity ofthe time evolution [16].The derivation of the canonical form for a general time-local master equation ˙ ρ ( t ) = L ( t ) ρ ( t ) comes as a straight-forward extension of the GKSL approach [53, 54]. Weshall briefly sketch the main steps, referring to Refs. [1–3, 16] for a detailed demonstration.Let us consider an open system described in a Hilbertspace of finite dimension d . A complete set of N := d basis operators { G n } N − n =0 is introduced, having theproperties G = ˆ I/ √ d ; G n = G + n ; Tr[ G m G n ] = δ mn , (1)with ˆ I being the identity operator. G n are orthonormaltraceless operators (excepting G , for which Tr[ G ] = 1).A general master equation ˙ ρ ( t ) = L ( t ) ρ ( t ) can be writtenin the following form [1, 16]: dρdt = − i ~ [ H ( t ) , ρ ( t )]+ N − X i,j =1 D ij ( t )[ G i ρ ( t ) G j − { G j G i , ρ ( t ) } ] , (2)with the operator H ( t ) being Hermitian, and D ij ( t ) beingthe time-dependent elements of the Hermitian decoher-ence matrix D . The Hermiticity property of the decoher-ence matrix leads to the existence of a unique canonicalform of the master equation, which follows using the di-agonal form of D [16]: D ij ( t ) = N − X k =1 U ik ( t ) γ k ( t ) U ∗ jk ( t ) , (3)where γ k ( t ) are the real eigenvalues of the decoher-ence matrix D , and U ik ( t ) are the elements of the uni-tary matrix formed by the eigenvectors of D , such that P N − k =1 U ik U ∗ jk = δ ij . Let us note that the trace of thedecoherence matrix D equals the sum of the decoherencerates γ k ( t ) : Tr[ D ] = X k γ k . (4)If one defines the time-dependent decoherence operators L k ( t ) ( k = 1 , .., N − L k ( t ) := N − X i =1 U ik ( t ) G i , (5)which form an orthonormal basis set of traceless opera-tors Tr[ L + j ( t ) L k ( t )] = δ jk ; Tr[ L k ( t )] = 0 , (6)Eq. (2) can be written in the canonical form [16]: dρdt = − i ~ [ H ( t ) , ρ ]+ N − X k =1 γ k ( t )[ L k ( t ) ρL + k ( t ) − { L + k ( t ) L k ( t ) , ρ } ] . (7)The canonical form (7) is similar to the Lindblad formof a memoryless master equation, but the Hamiltonian H ( t ), the decoherence operators L k ( t ), and the decoher-ence rates γ k ( t ) are time-dependent. Moreover, the deco-herence operators L k ( t ) correspond to a set of orthogonaldecoherence channels, and the time-dependent decoher-ence rates γ k ( t ) obtained as eigenvalues of the decoher-ence matrix are uniquely determined and can be negative [16].Formulation of necessary and sufficient conditions un-der which the dynamics described in Eq. (7) is completelypositive remains an open problem [2, 4]. If the rates arepositive for all times, γ k ( t ) ≥
0, the dynamics is com-pletely positive, being in Lindblad form for each fixed t [2]. However, there are cases where the rates γ k ( t ) maybecome temporarily negative without violating completepositivity [3, 4].For a master equation in the GKSL-form (7) with time-dependent coefficients, it can be shown that the corre-sponding dynamical map satisfies CP-divisibility if andonly if γ k ( t ) ≥ γ k ( t ) ≥ γ k ( t ) ≥ γ k ( t ) < et al. [16] have shown that for a finite-dimensionalsystem, the criterion for non-Markovianity based on theviolation of CP divisibility, proposed by Rivas et al. [11],is equivalent to the criterion based on the negativity ofthe decoherence rates appearing in the canonical form ofthe master equation.We employ the canonical measures [16] to detect andquantify non-Markovianity. Because of their sensitivityto individual canonical decoherence rates, they are ableto completely detect non-Markovian behavior when sev-eral decoherence channels are present. Additionally, theBloch volume of accessible states is also used as a non-Markovianity witness [3]. The two following sections ex-pose the definitions of the canonical measures and Blochvolume, respectively. A. Negative decoherence rates and canonicalmeasures of non-Markovianity
Since the appearance of negative decoherence rates inthe canonical form (7) of the master equation is a featureof non-Markovianity, Hall et al [16] define several mea-sures of non-Markovianity as functions of the negativecanonical decoherence rates γ k ( t ). These definitions areintroduced in the following and will be employed in ouranalysis.For an individual channel k with decoherence rate γ k ( t ), non-Markovianity can be described using the func-tion [16] f k ( t ) := max[0 , − γ k ( t )] = 12 [ | γ k ( t ) | − γ k ( t )] , (8)which is 0 if the decoherence rate γ k ( t ) is positive, and | γ k ( t ) | if the decoherence rate is negative.The canonical measure of non-Markovianity at time t is defined as the sum of the individual channels measures: f ( t ) = X k f k ( t ) . (9)Hall et al. [16] have shown that their canonical measure f ( t ) coincides, up to a multiplicative factor 2 /d depend-ing on the dimension d of the system, with the trace-normmeasure of non-Markovianity g ( t ) proposed by Rivas etal. [11]: g ( t ) = 2 d − f ( t ).One can also define a total amount of non-Markovianity in a channel k over the time interval [t,t’][16] as the integral F k ( t, t ′ ) = Z t ′ t f k ( s ) ds, (10)and a total amount of non-Markovianity over the timeinterval [t,t’] by F ( t, t ′ ) = X k F k ( t, t ′ ) = Z t ′ t f ( s ) ds. (11)Ref. [16] also defines a non-Markov index n ( t ) as thenumber of strictly negative decoherence rates: n ( t ) := { k : γ k ( t ) < } . (12)The orthogonality of the decoherence channels allows theinterpretation of the non-Markov index n ( t ) as the di-mension of the space of non-Markovian evolution, orthog-onal to the Markovian region [16]. B. Bloch volume characterization ofnon-Markovianity
Lorenzo et al. [15] proposed a geometrical characteri-zation of non-Markovianity based on the increase of thevolume of states dynamically accessible to the system.The proposal originates in the observation that for adynamical map corresponding to a Markovian quantumevolution the volume of physical states decreases mono-tonically in time, as there is no recovery of information,energy, or coherence by the system. On the contrary,a time evolution leading to a growth in the volume ofaccessible states reveals physical effects associated withnon-Markovianity.Ref. [16] shows that, for a d -dimensional quantum sys-tem which can be represented by a generalized Bloch vec-tor of dimension d −
1, the Bloch volume V ( t ) at time t is only sensitive to the sum of the canonical decoherencerates, P k γ k ( t ), as follows: V ( t ) = V exp " − d Z t ds X k γ k ( s ) , (13)with V being the initial volume at the time t = 0.Consequently, the Bloch volume can increase at time t , becoming a witness of non-Markovianity, if and only ifthe sum of the canonical decoherence rates is negative: P k γ k ( t ) < III. NON-MARKOVIANITY IN THEREDUCED TIME EVOLUTION OF THEELECTRONIC SUBSYSTEM OF ALASER-DRIVEN MOLECULE
We will now consider the time evolution of the elec-tronic subsystem of a molecule driven by a laser pulsewhich creates entanglement between electronic and vi-brational degrees of freedom. We treat the electronicsubsystem as an open quantum system in the vibrationalenvironment. A non-Markovian character of the elec-tronic system dynamics is expected, since the vibrationalenvironment is a dynamical one, being structured by thevibrational motion in the electronic molecular potentialscoupled by the laser pulse. Therefore, the non-Markovianeffects in the electronic evolution will be determined bythe traits of the vibrational dynamics and of the driv-ing field. This section exposes our model, allowing usto characterize non-Markovianity of the electronic evo-lution using the measures introduced in the precedentsection. We begin by describing the theoretical model ofa diatomic molecule driven by a coupling between elec-tronic states, such that several electronic states couldbe populated. The intramolecular dynamics of such amolecule is characterized by electronic-vibrational entan-glement and electronic coherence [58, 59]. Subsequently,we will deduce the canonical form of the master equa-tion for a 2-dimensional electronic subsystem, buildingthe non-Markovianity measures from the canonical deco-herence rates.We consider a diatomic molecule described in the Born-Oppenheimer (BO) approximation [60], neglecting therotational degree of freedom, such that the molecularsystem is described by states | Ψ el,vib ( t ) > of the Hilbertspace H = H el N H vib .We assume the molecule driven by the total Hamilto-nian ˆH = ˆ H mol + ˆ W ( t ) , (14)where the molecular Hamiltonian ˆ H mol = ˆ H el + ˆ T R is thesum of the electronic Hamiltonian ˆ H el and the nuclearkinetic-energy ˆ T R . ˆ W ( t ) describes a time-dependent cou-pling of the electronic states of the molecule [61]. Thedynamics of the molecular system is obtained from thevon Neumann equation i ~ d ˆ ρ el,vib ( t ) dt = [ ˆH , ˆ ρ el,vib ( t )] , (15)where ˆ ρ el,vib ( t ) = | Ψ el,vib ( t ) >< Ψ el,vib ( t ) | is a pure stateof the bipartite system (el N vib).A detailed description of the molecular model can befound in previous papers [58, 59], where we have analyzedentanglement and coherence of pure states | Ψ el,vib ( t ) > created by laser pulses. The molecular state | Ψ el,vib ( t ) > has the form | Ψ el,vib ( t ) > = N el X α =1 | α > O | ψ α ( t ) >, (16)the summation being over the populated electronic chan-nels α = 1 , N el . We recall that the molecular wavefunction Ψ el,vib ( ~r i , R, t ) depends on the electronic co-ordinates { ~r i } (expressed in the molecule-fixed coordi-nate system), the internuclear distance R , and the time t . The electronic states | α > = φ elα ( ~r i ; R ) (dependingparametrically on R) are orthonormal eigenstates of theelectronic Hamiltonian ˆ H el satisfying the clamped nucleielectronic Schr¨odinger equation ˆ H el | α > = V α ( R ) | α > ,which gives the adiabatic potential-energy surfaces V α ( R )as eigenvalues of ˆ H el [60]. | ψ α ( t ) > designates the vibra-tional wave packet ψ α ( R, t ) corresponding to the elec-tronic state | α > . A. The electronic subsystem as an open quantumsystem entangled with the vibrational environment
We will follow the electronic subsystem dynamicsin relation to dynamical behaviors of the electronic-vibrational entanglement and electronic coherence. Thereduced time evolution of the electronic subsystem is de-rived from the unitary dynamics (Eq. (15)) of the molec-ular system described by the molecular density operatorˆ ρ el,vib = | Ψ el,vib ( t ) >< Ψ el,vib ( t ) | , obtained with Eq. (16)as ˆ ρ el,vib ( t ) = N el X α,β | α >< β | O | ψ α ( t ) >< ψ β ( t ) | . (17)Therefore, the reduced electronic density operatorˆ ρ el =Tr vib (ˆ ρ el,vib ) is [59]ˆ ρ el ( t ) = N el X α,β | α >< β | < ψ β ( R, t ) | ψ α ( R, t ) > . (18)ˆ ρ el ( t ) describes an electronic subsystem which is entan-gled with the vibrational environment [58]. For N el pop-ulated states, the linear entropy L ( t ) = 1 − Tr el (ˆ ρ el ( t ))of the electronic-vibrational entanglement has the expres-sion [59]: L ( t ) = 2 N el X α,β,α = β [ P α ( t ) P β ( t ) − | < ψ α ( R, t ) | ψ β ( R, t ) > | ] . (19)In Eq. (19), P α ( t )= < ψ α ( R, t ) | ψ α ( R, t ) > is the popula-tion of the electronic state | α > , and the total population obeys the normalization condition P N el α =1 P α ( t ) = 1. Theother term appearing in Eq. (19) involves the off-diagonalelements < α | ˆ ρ el ( t ) | β > = < ψ β ( R, t ) | ψ α ( R, t ) > , whichare giving the coherence of the reduced electronic stateˆ ρ el ( t ). Using the l norm definition of coherence [62], oneobtains as measure of the electronic coherence: C l (ˆ ρ el ) = N el X α,β,α = β | < ψ α ( R, t ) | ψ β ( R, t ) > | . (20)In the following we suppose an electronic subsystemof dimension dim( H el ) = 2, and we derive the canonicalform of the master equation which describes its evolution. B. The master equation for a two-dimensionaldriven electronic subsystem
We consider a diatomic molecule in which two elec-tronic states | g >, | e > are coupled by a laser pulse, suchthat a pure molecular state | Ψ el,vib ( t ) > is created: | Ψ el,vib ( t ) > = | g > O | ψ g ( R, t ) > + | e > O | ψ e ( R, t ) > . (21)The quantum dynamics of the molecular system drivenby the Hamiltonian (14) is given by the time-dependentSchr¨odinger equation: i ~ ∂∂t | Ψ el,vib ( t ) > = [ ˆ H mol + ˆ W ( t )] | Ψ el,vib ( t ) > . (22)Projecting Eq. (22) on the electronic states | g >, | e > ,and taking into account the BO approximation (i.e. <α | ˆ H mol | α > = ˆ T R + V α ( R ) and < α | ˆ H mol | β > = 0), aswell as the off-diagonal nature of the coupling (i.e. <α | ˆ W ( t ) | α > = 0), where | α >, | β > generically designatethe electronic adiabatic states, one obtains i ~ ∂∂t (cid:18) ψ g ( R, t ) ψ e ( R, t ) (cid:19) = (23) (cid:18) ˆ T R + V g ( R ) W ( R, t ) W ∗ ( R, t ) ˆ T R + V e ( R ) (cid:19) (cid:18) ψ g ( R, t ) ψ e ( R, t ) (cid:19) . Eq. (23) describes the vibrational dynamics of the wavepackets ψ g,e ( R, t ) moving in the electronic potentials V g ( R ) and V e ( R ), which are coupled by W ( R, t )=
We shall derive here the canonical form of the mas-ter equation for the 2-dimensional electronic subsystemˆ ρ el ( t ). The master equation (25) wil be used to deduceboth (2) and (7) forms, in order to obtain the decoher-ence matrix D and the decoherence rates γ k ( t ).As dim( H el )= 2, the orthornormal basis { G i } i =0 canbe chosen as { ˆ I/ √ , σ i / √ } , with { σ i } i =1 , , being thePauli operators: σ = | e >< g | + | g >< e | , σ = − i | e >
00 0 1 , (41)with γ , being the decoherence rates given in Eq. (39)and D , D , D being the elements of the decoher-ence matrix shown in Eqs. (33 - 36). n and n are realnormalization factors (with n + n = 1) given by theexpressions: n = γ − D γ − γ ; n = D − γ γ − γ . (42)The time dependent decoherence operators { L i ( t ) } i =1 , , , corresponding to orthogonal decoher-ence channels are obtained using Eq. (5) as L ( t ) = n √ σ + γ − D D σ ) , (43) L ( t ) = n √ σ + γ − D D σ ) , (44) L ( t ) = 1 √ σ . (45)Finally, we obtain the canonical form for the masterequation of the reduced electronic density operator ˆ ρ el ( t )(24): d ˆ ρ el dt = − i ~ [ H ( t ) , ˆ ρ el ( t )]+ X i =1 γ i ( t )[ L i ( t )ˆ ρ el L + i ( t ) − { L + i ( t ) L i ( t ) , ˆ ρ el } ] , (46)with the operator H ( t ) having the matrix (31), the de-coherence rates { γ i ( t ) } i =1 , , given in Eqs. (39, 40), andthe decoherence operators { L i ( t ) } i =1 , , determined byEqs. (43-45).The sum of the canonical decoherence rates is the traceof the decoherence matrix given by Eqs. (33 - 37): X i γ i ( t ) = Tr[ D ( t )]= 12 dP g dt ( P g − P e ) (cid:20) P g P e + 1 | < ψ g | ψ e > | (cid:21) , (47)becoming zero at instants t for which dP g /dt = 0 or P g ( t ) = P e ( t ).The Bloch volume of the accessible states, obtainedwith Eq. (13), is V ( t ) = V ( t ) exp " − Z tt ds X i γ i ( s ) . (48) As already discussed, if the sum P i γ i ( t ) of the canoni-cal decoherence rates is negative, the Bloch volume of theaccessible states increases, witnessing non-Markovianity.Therefore, a first indication on the non-Markovian behav-ior is given by Eq. (47) which shows that a growth of theBloch volume, V ( t ) > V ( t ), appears if dP g dt ( P g − P e ) < P g ( t ) + P e ( t ) = 1 implies dP g dt ( P g − P e ) = − ddt ( P g P e ) . (49)Therefore, the condition to have P i γ i ( t ) <
0, leading toa growth of the Bloch volume, can also be expressed as ddt ( P g P e ) > D. Decoherence rates and canonical measures ofnon-Markovianity for the electronic system
Let us analyze the signs of the decoherence rates γ i ( t )given by Eqs. (39,40). Since P g P e ≥ | < ψ g | ψ e > | ,and with Eq. (49), it appears that the sign of γ ( t ) de-pends on the time evolution of the electronic populations P g ( t ) , P e ( t ) as follows:sgn[ γ ( t )] = sgn (cid:20) dP g dt ( P g − P e ) (cid:21) = − sgn (cid:20) ddt ( P g P e ) (cid:21) . (50)On the other hand, Eq. (39) can be written as γ , ( t ) = 12 P g P e (cid:12)(cid:12)(cid:12)(cid:12) dP g dt (cid:12)(cid:12)(cid:12)(cid:12) × (cid:26) sgn (cid:20) dP g dt ( P g − P e ) (cid:21) | P g − P e | ± p r ( t ) (cid:27) , (51)with r ( t ) = 4 P g P e ( dP g /dt ) | d < ψ g | ψ e > /dt | | < ψ g | ψ e > | . (52)Taking into account that 0 ≤ | P g − P e | ≤
1, it becomesobvious that γ ( t ) is always positive, and γ ( t ) is alwaysnegative: γ ( t ) > γ ( t ) < . (53)Consequently, we will distinguish four cases:(i) If dP g dt ( P g − P e ) >
0, or equivalently, ddt ( P g P e ) < one negative decoherence rate , γ ( t ) <
0, andthe non-Markov index defined by Eq. (12) is n ( t ) = 1.Eq. (47) shows that the sum of the decoherence ratesis positive, P i γ i ( t ) >
0, leading to a diminution of theBloch volume.The non-Markovianity measure obtained withEqs. (8,9) is f ( t ) = f ( t )= | γ ( t ) | . Using Eq. (51) wefind f ( t ) = 12 P g P e (cid:12)(cid:12)(cid:12)(cid:12) dP g dt (cid:12)(cid:12)(cid:12)(cid:12) hp r ( t ) − | P g − P e | i . (54)(ii) If dP g dt ( P g − P e ) <
0, or equivalently, ddt ( P g P e ) > two negative decoherence rates , γ ( t ) < γ ( t ) <
0. The dimension of the space of non-Markovianevolution, given by the non-Markov index [16], becomes n ( t ) = 2. The non-Markovianity measure is obtainedfrom the negative decoherence rates using Eqs. (8) and(9), as f ( t ) = f ( t ) + f ( t )= | γ ( t ) | + | γ ( t ) | . UsingEqs. (51,40) we find f ( t ) = 12 P g P e (cid:12)(cid:12)(cid:12)(cid:12) dP g dt (cid:12)(cid:12)(cid:12)(cid:12) h | P g − P e | + p r ( t ) i + 1 P g P e d ( P g P e ) dt L ( t )[ C l (ˆ ρ el )] . (55)In Eq. (55), L ( t ) and C l (ˆ ρ el ) are the linear entropyof the electronic-vibrational entanglement and the elec-tronic coherence, respectively, whose expressions can bederived from Eqs. (19,20) for N el = 2.Moreover, the sum of the decoherence rates is nega-tive , P i γ i ( t ) <
0, which means that the Bloch volume of the dynamically accessible states increases (Eq. 13),witnessing non-Markovianity. We distinguish this caseas indicating enhancement of non-Markovianity .(iii) If P g ( t ) = P e ( t ), the decoherence rates are γ ( t ) =0, and γ ( t ) = − γ ( t ). The sum of the decoherence ratesbecomes zero, P i γ i ( t ) = 0. Using Eq. (51), the non-Markovianity measure f ( t ) = | γ ( t ) | becomes f ( t ) = 12 P g P e (cid:12)(cid:12)(cid:12)(cid:12) dP g dt (cid:12)(cid:12)(cid:12)(cid:12) p r ( t ) . (56)(iv) If dP g dt = 0. This condition corresponds to extremain the evolution of the electronic populations during thepulse, or to constant populations after pulse. The deco-herence rates become γ ( t ) = 0, and γ ( t ) = − γ ( t ), with P i γ i ( t ) = 0. Eq. (39) gives γ , ( t ) = ± | < ψ g | ψ e > | (cid:12)(cid:12)(cid:12)(cid:12) d < ψ g | ψ e >dt (cid:12)(cid:12)(cid:12)(cid:12) , (57)and f ( t ) = | γ ( t ) | .Let us consider the case of a molecule with constantpopulations in the electronic states g, e (it can be amolecule after the action of a laser pulse): dP g dt = 0for all t . Therefore, the Bloch volume of the dynami-cally accessible states remains constant, V ( t )= V . For W ( R, t ) = 0, Eqs. (57) and (23) give an alternative formof the decoherence rates as γ , ( t ) = ± ~ | < ψ g | V e ( R ) − V g ( R ) | ψ e > || < ψ g | ψ e > | . (58)Writing the complex overlap of the vibrational packetsas < ψ g | ψ e > = | < ψ g | ψ e > | exp( iα ( t )), with α ( t ) a realfunction, the non-Markovianity measure f ( t ) = | γ ( t ) | ob-tained using Eq. (57) becomes f ( t ) = s(cid:18) | < ψ g | ψ e > | d | < ψ g | ψ e > | dt (cid:19) + (cid:18) dαdt (cid:19) . (59) Eq. (59) is useful for understanding the relation between f ( t ) and the electronic coherence | < ψ g | ψ e > | . It ap-pears that if at an instant t m one has ( d | <ψ g | ψ e > | dt ) t m = 0(an extremum in the time evolution of the coherence),but | < ψ g | ψ e > | t m = 0, one obtains a minimum of thefunction f ( t ), which becomes f ( t m ) = | dαdt | t m . On thecontrary, at an instant t M for which | < ψ g | ψ e > | t M → d | <ψ g | ψ e > | dt ) t M = 0),the function f ( t ) has a maximum, becoming f ( t M )= q (cid:0) dαdt (cid:1) t M . Eq. (59) shows that in a molecule withconstant electronic populations, the non-Markovianitymeasure f ( t ) can be seen as a measure of the tempo-ral behavior of the electronic coherence, having minimawhen the electronic coherence has maxima, and attain-ing maximum values whenever the overlap of the vibra-tional packets tends to zero, | < ψ g | ψ e > | →
0. Atthe same time, as we have shown previously [59], if theelectronic populations are constant, the time variationsof the coherence | < ψ g | ψ e > | completely determine thetemporal evolution of the linear entropy of entanglement L ( t ) (see Eq. (61)), which becomes maximum when co-herence attains a minimum. Therefore, the maxima ofthe non-Markovianity measure f ( t ) correspond to max-ima of the electronic-vibrational entanglement measuredby the linear entropy.These results make explicit the fundamental non-Markovian character of the electronic subsystem evolu-tion. Indeed, we have shown that one of the decoherencerates is always negative: γ ( t ) <
0. Besides this inherentnon-Markovianity, the character of the electronic evolu-tion becomes strongly non-Markovian under the condi-tion ( P g − P e ) dP g /dt < d ( P g P e ) /dt >
0, which sup-poses an exchange of population between the electronicchannels. In the following, d ( P g P e ) /dt will be called thenon-Markovianity factor.The condition ( P g − P e ) dP g /dt < dP g /dt )= − sgn[ P g ( t ) − P e ( t )]. Therefore, it appears that thenon-Markovian character of the dynamics is strengthenedwhen the transfer of population between the two elec-tronic channels is such as the larger population decreases(i.e., the smaller electronic population increases). Thiscondition, describing an evolution oriented to the equal-ization of the electronic populations, is in fact a condi-tion indicating the increase of the electronic-vibrationalentanglement, which becomes maximum when the elec-tronic populations are equal [58]. This observation willbe developed in the following sections.0 IV. CONNECTING NON-MARKOVIANITY OFTHE ELECTRONIC EVOLUTION WITHELECTRONIC-VIBRATIONALENTANGLEMENT AND ELECTRONICCOHERENCE
We will now analyze enhancement of non-Markovianity, determined by the condition d ( P g P e ) /dt >
0, in relation to the evolutions of theelectronic-vibrational entanglement and the electroniccoherence. The key observation is that the quantity P g ( t ) P e ( t ) is connected to measures of entanglement andcoherence in the molecular system.The electronic-vibrational entanglement in the bipar-tite molecular state | Ψ el,vib ( t ) > given by Eq. (21) canbe analyzed using the von Neumann entropy S vN (ˆ ρ el ( t ))or the linear entropy L ( t ) of the reduced density op-erator ˆ ρ el . In previous works [58, 59] we have investi-gated the results given by these two entanglement mea-sures. Both of them depend on the temporal behaviorof the electronic populations, but only L ( t ) depends onthe electronic coherence. The von Neumann entropy ofthe electronic-vibrational entanglement has the followingexpression [58]: S vN (ˆ ρ el ( t )) = − P g ( t ) log P g ( t ) − P e ( t ) log P e ( t ) . (60)For N el = 2, the linear entropy L ( t ) = 1 − Tr(ˆ ρ el ( t ))obtained with Eq. (19) becomes L ( t ) = 2 P g ( t ) P e ( t ) − | < ψ g ( R, t ) | ψ e ( R, t ) > | , (61)and, with Eq. (20), the l norm measure of the electroniccoherence is C l (ˆ ρ el ) = 2 | < ψ g ( R, t ) | ψ e ( R, t ) > | . (62)Therefore, Eq. (61) can be read as a relation between thephenomena of electronic-vibrational entanglement, non-Markovianity of the electronic evolution, and electroniccoherence. Indeed, Eqs. (61) and (62) lead to ddt [ P g ( t ) P e ( t )] = 12 dLdt + 12 C l (ˆ ρ el ) dC l (ˆ ρ el ) dt . (63)In the following, Eq. (63) will be used to explorethe relations between enhancement of non-Markovianity( d ( P g P e ) /dt > dL/dt > dC l (ˆ ρ el ) /dt > L ( t ) and C l ( t ) can be given. Using Eq. (47), the sum ofthe decoherence rates becomes X i γ i ( t ) = − d [ln( P g P e )] dt L ( t ) + [ C l (ˆ ρ el )] [ C l (ˆ ρ el )] , (64)and, with Eq. (40), γ ( t ) can be written γ ( t ) = − d [ln( P g P e )] dt L ( t )[ C l (ˆ ρ el )] . (65) Besides the l norm measure of the electronic coher-ence, C l (ˆ ρ el ), we shall use the Wigner-Yanase skew in-formation I S (ˆ ρ el , ˆ H el )= − Tr el [ √ ˆ ρ el , ˆ H el ] for the elec-tronic state ˆ ρ el , with respect to the electronic Hamilto-nian ˆ H el , to additionally characterize electronic subsys-tem coherence [59]. The skew information I S is a mea-sure of coherence as asymmetry relative to a group oftranslations [64–66], quantifying the coherence of a statewith respect to a certain Hamiltonian eigenbasis. Thisnotion of coherence was termed unspeakable [64], to showits structural relation to the eigenvalues of the observablewhich defines the basis relative to which coherence is de-fined [67]. It is a notion of coherence closely related to thecontext of quantum speed limits [65, 66]. In particular, I S (ˆ ρ el , ˆ H el ) characterizes the coherence of the reducedelectronic state ˆ ρ el relative to the eigenbasis {| g >, | e > } of the electronic Hamiltonian ˆ H el , whose eigenvalues arethe electronic potentials V g ( R ) , V e ( R ). The skew infor-mation I S (ˆ ρ el , ˆ H el ) has the following expression [59]: I S (ˆ ρ el , ˆ H el ) = [ V g ( R ) − V e ( R )] | < ψ g ( R, t ) | ψ e ( R, t ) > | p L ( t ) . (66) I S (ˆ ρ el , ˆ H el )= I S ( R, t ) appears as a product between afunction of the internuclear distance R (depending on theelectronic potentials difference at given R ) and a functionof time t , a factorization which reflects the BO approx-imation. It can be said that I S ( R, t ) is a measure ofthe unspeakable electronic coherence which characterizesthe reduced electronic state ˆ ρ el at a given internucleardistance R . Let us observe that the time behavior of I S is determined by the time evolutions of the electroniccoherence C l ( t ) and the linear entropy of entanglement L ( t ). Our aim is to investigate non-Markovian behaviorin relation to various quantum correlations in the molec-ular system, and we find it useful to also examine thismeasure of correlations, which combines coherence andentanglement.Eqs. (66) and (62) determine the relation betweenthe time variations of the electronic coherences I S ( R, t ), C l ( t ), and of the linear entropy of entanglement L ( t ):1 I S ∂ I S ∂t = 2 C l dC l dt − √ L (1 + √ L ) dLdt . (67)We shall analyze the condition d ( P g P e ) /dt > l norm C l , and ”unspeakable” [64],quantified by the skew information I S ). Eqs. (67) and(61) give: d ( P g P e ) dt = √ L + L + 2 P g P e √ L (1 + √ L ) (cid:18) dLdt (cid:19) + C l I S (cid:18) ∂ I S ∂t (cid:19) , (68)1 TABLE I. Connections between the time behavior of the electronic-vibrational entanglement ( dL/dt ), the enhancement ofnon-Markovianity in the evolution of the electronic subsystem ( d ( P g P e ) /dt > l norm C l ( t ) and skew information I S ( t ), respectively. dLdt d ( P g P e ) dt dC l dt ∂ I S ∂t (1) > < ⇒ < < > > ⇒ >
0, if >
0, if d ( P g P e ) dt > dLdt √ L (1+ √ L ) dLdt < √ L + L +2 P g P e d ( P g P e ) dt < C l dC l dt <
0, if √ L (1+ √ L ) dLdt > √ L + L +2 P g P e d ( P g P e ) dt > C l dC l dt <
0, if < d ( P g P e ) dt < dLdt (3) < > ⇒ > > < < ⇒ <
0, if >
0, if − dLdt < − d ( P g P e ) dt − √ L (1+ √ L ) dLdt > − √ L + L +2 P g P e d ( P g P e ) dt > − C l dC l dt <
0, if − √ L (1+ √ L ) dLdt < − √ L + L +2 P g P e d ( P g P e ) dt < − C l dC l dt >
0, if > − dLdt > − d ( P g P e ) dt d ( P g P e ) dt = √ L + L + 2 P g P e C l (cid:18) dC l dt (cid:19) − √ L (1 + √ L ) I S (cid:18) ∂ I S ∂t (cid:19) . (69)Table I systematizes the relations between enhance-ment of non-Markovianity ( d ( P g P e ) /dt >
0) and the dy-namics of the quantum correlations measured using L ( t ), C l ( t ), and skew information I S (ˆ ρ el , ˆ H el ). This analy-sis is performed using Eqs. (63,67,68,69). Observing thatnon-Markovian behavior accompanies the phenomenon ofelectronic-vibrational entanglement, we have considereddefinite signs for dL/dt and d ( P g P e ) /dt , in order to de-duce the compatible behaviors of electronic coherences.Table I shows the following relations among phenomena:(1) Entanglement growth ( dL/dt >
0) accompanied bydiminution of non-Markovianity ( d ( P g P e ) /dt <
0) has tobe associated with a decrease of both electronic coher-ences ( C l and I S ).(2) When both entanglement and non-Markovianityincrease ( dL/dt > d ( P g P e ) /dt > C l may either increase (if d ( P g P e ) dt > dLdt ), ordecrease (if the opposite relation is true). If dC l /dt > L ( t ), P g ( t ) P e ( t ), and C l ( t ), as it is shown in the fourth col-umn of the Table I. On the contrary, if dC l /dt <
0, theskew information can only decrease, ∂ I S /∂t < dL/dt <
0) isaccompanied by enhanced non-Markovian behavior( d ( P g P e ) /dt >
0) only if the electronic coherences ( C l and I S ) increase.(4) When both entanglement and non-Markovianitydecrease ( dL/dt < d ( P g P e ) /dt < C l may either increase or decrease. As in thecase (2), we will have several possibilities, shown in theTable.We observe a notable difference between the cases (2)and (4), with dL/dt , d ( P g P e ) /dt having the same sign,and cases (1) and (3), with them having opposite signs.The numerical results presented in Sec. VI will show thatcases (2) and (4) represent the rule, and cases (1) and (3)are the exception, because enhanced non-Markovian be-havior is deeply connected with increase of entanglement,as already explained in Sec. III D.It is interesting to compare the time behaviors of thetwo electronic coherences: Even if the skew informationhas the tendency to follow the C l time behavior, its sen-sitivity to entanglement brings cases in which the increaseof the electronic coherence C l is accompanied by the de-2crease of I S , or the opposite. The conditions of possi-bility leading to these situations appear in the cases (2)and (4), specified in Table I.The aim of this analysis is to gain insight into themeaning of non-Markovianity in relation to entangle-ment and coherence. An interesting question would beif the model used here to characterize non-Markovianityallows us to relate non-Markovian behavior to a back-flow of information from environment to the system.More specifically, the question is if any of the conditions d ( P g P e ) /dt > dL/dt >
0, or dC l /dt > et al. [10] identify as an essential fea-ture of non-Markovian behavior the existence of a re-versed flow of information from the environment to theopen system, a ”backflow” which is manifested in thegrowth of distinguishability between quantum states ofthe open system. In the Appendix we show that thetrace distance between ˆ ρ el ( t ) and a state ˆ ρ el ( t ) with co-herence C l ( t ) = 0 is increased when d ( P g P e ) /dt > dC l /dt >
0. In general (see the appendix), the con-dition d ( P g P e ) /dt > D (ˆ ρ el ( t ) , ˆ ρ el ( t )), contributing with a positive term at therate of change dD (ˆ ρ el ( t ) , ˆ ρ el ( t )) /dt given by Eq. (A.3).Regarding the condition dL/dt >
0, Sec. III D explainedthat the condition ( P g − P e ) dP g /dt < d ( P g P e ) /dt > dL/dt > dS vN /dt >
0) will appear clearly in the numerical results presentedin Sec. VI.This theoretical analysis, grounded on the analytic for-mulas relating the non-Markovianity factor d ( P g P e ) /dt with the time behaviors of entanglement and coherence,will be completed in Sec. VI with an examination ofnumerical results for the canonical measures of non-Markovianity obtained from simulations of the moleculardynamics in a laser-driven molecule. V. NON-MARKOVIANITY AND QUANTUMUNCERTAINTY ON THE ELECTRONICENERGY
If ˆ ρ el,vib ( t ) is a pure state, the uncertainty on the elec-tronic energy (i.e. the mean square deviation from theaverage value) is given by [59](∆ ˆ H el ) = I S (ˆ ρ el,vib , ˆ H el O ˆ I v )= [ V g ( R ) − V e ( R )] P g ( t ) P e ( t ) , (70)where I S (ˆ ρ el,vib , ˆ H el N ˆ I v ) is the Wigner-Yanase skewinformation for the molecular state ˆ ρ el,vib with respectto the electronic Hamiltonian ˆ H el . Consequently, en-hancement of non-Markovianity in the electronic evolu- V ( R ) ( un it s o f c m - )
10 20 30 40 50 60 70R (units of a ) -2000
100 200 t(ps) a Σ u+ (6s,6s)1 g (6s,6p ) h ω L e(t) V g (R)V e (R) FIG. 1. (Color online) a Σ + u (6 s, s ) and 1 g (6 s, p / ) elec-tronic potentials of Cs , coupled at a internuclear distanceof about R c ≈ a by a pulse with frequency ω L / π andenvelope e ( t ) shown in the inset. The energy origin is takento be the dissociation limit E s +6 s = 0 of the a Σ + u (6 s, s )potential. tion increases uncertainty on the electronic energy (andinversely, growing uncertainty on the electronic energyreflects a non-Markovian behavior in the electronic evo-lution): d ( P g P e ) dt > ⇐⇒ ∂ (∆ ˆ H el ) ∂t > . (71)The Wigner-Yanase skew information I S (ˆ ρ el , ˆ H el ) is alsorecognized as a measure of the quantum uncertainty ofˆ H el in the state ˆ ρ el [68]. Let us observe that Eq. (69)connects the time behavior of the uncertainty on the elec-tronic energy in the pure molecular state ˆ ρ el,vib ( t ) withbehavior of the quantum uncertainty I S (ˆ ρ el , ˆ H el ) in thereduced state ˆ ρ el . VI. NON-MARKOVIAN DYNAMICS OF THEELECTRONIC SUBSYSTEM IN ALASER-DRIVEN MOLECULE: ANALYSIS FROMSIMULATIONS OF MOLECULAR DYNAMICS
This section will present results obtained from the sim-ulation of the intramolecular dynamics for a diatomicmolecule which is under the action of a laser pulse cou-pling two electronic states. Non-Markovian behavior ofthe electronic subsystem is characterized using the canon-ical measures of non-Markovianity f ( t ) and F ( t , t )= R t t f ( t ) dt , calculated using the equations established inSec. III D. We will also examine the time behavior of theBloch volume V ( t ) of the accessible states, obtained usingEqs. (47,48), as well as the dynamics of the electronic-vibrational entanglement and the electronic coherence in3 L ( t ) , S v N ( t ) I S ( t ) | < Ψ g | Ψ e > |
50 100 150 200 250 300 350 400t(ps) f( t ) P g ( t ) , P e ( t ) -0.0200.02 d ( P g P e ) / d t P g (t)P e (t) S vN (t) L(t) (a)(b)(c)(d)(e)(f) T Rv e v g FIG. 2. (Color online) Results characterizing the vibrationaldynamics in the electronic potentials g = a Σ + u and e = 1 g of Cs coupled by a pulse with envelope e ( t ) (Fig. 1), fora coupling strength W L = 3 .
29 cm − . Time evolutionsduring the pulse ( t <
250 ps) and after pulse ( t > P g ( t ) and P e ( t ) (two specific Rabi periods T Rv e ,v g , of47 . . d ( P g P e ) /dt (non-Markovianityis enhanced if d ( P g P e ) /dt > L ( t ) and von Neumann entropy S vN ( t )of the electronic-vibrational entanglement. (d) Time evolu-tion of the skew information I S ( t ) = I S ( R, t ) / [∆ V ( R )] . (e)Time evolution of the electronic coherence C l ( t ) / | <ψ g ( t ) | ψ e ( t ) > | . (f) Non-Markovianity measure f ( t ). Thefilled surface shows the integral R f ( t ) dt . the molecule. Non-Markovian behavior during time evo-lution will be connected with the dynamics of quantumcorrelations.As a model system, we consider the Cs moleculein which the electronic states g = a Σ + u (6 s, s ) and e = 1 g (6 s, p / ) are coupled by a laser pulse. In previ-ous works [58, 69, 70] , we have analyzed the vibrationaldynamics in these electronic potentials for various con-ditions of coupling, and we shall refer to these works fordetails of the molecular model, including definitions ofthe characteristic times of dynamics, such as vibrationaland Rabi periods.Let us suppose the electronic states g = a Σ + u (6 s, s )and e = 1 g (6 s, p / ) coupled by an electric field withtemporal amplitude E ( t ) = E e ( t ) cos ω L t . The field am-plitude E = p I/cǫ depends on the laser intensity I , e ( t ) is the temporal envelope of the pulse, and ω L / π isthe frequency of the field, such as the photon energy ~ ω L couples the electronic potentials V g ( R ) and V e ( R ) at a in-ternuclear distance of about R c ≈ a , as it is shown inFig. 1. Using the rotating wave approximation with the | ψ g ( R , t ) | , | ψ e ( R , t ) | t(ps) t(ps) FIG. 3. (Color online) Time evolution (80 - 190 ps) of thevibrational wave packets | Ψ g ( R, t ) | (full line) and | Ψ e ( R, t ) | (dotted line) in g = a Σ + u (6 s, s ) and e = 1 g (6 s, p / )electronic potentials coupled by a pulse with envelope e ( t )(Fig. 1), for a coupling strength W L = 3 .
29 cm − . frequency ω L / π , and a transformation of the radial wavefunctions with appropriate phase factors, one obtains thetypical Eq. (23) for the vibrational wave packets ψ g ( R, t )and ψ e ( R, t ) whose dynamics takes place in the diabaticelectronic potentials crossing in R c [69]. The coupling be-tween the electronic channels is W ( t ) = W L e ( t ), with thestrength W L = − E D ~e L ge , where D ~e L ge is the transitiondipole moment between the ground g and the excited e electronic states, for a polarization ~e L of the electric field[69]. Here the R -dependence of the transition dipole mo-ment is neglected, and several coupling strengths W L areconsidered, for the same pulse envelope e ( t ) (representedin Fig. 1).The intramolecular dynamics is obtained usingEq. (23), which is solved numerically by propagating intime an initial wave function (here the initial state is thevibrational eigenstate with v e = 142 of the 1 g (6 s, p / )potential) on a spatial grid with length L R . The MappedSine Grid (MSG) method [71, 72] is used to representthe radial dependence of the wave packets, and the timepropagation uses the Chebychev expansion of the evolu-tion operator [73, 74]. The electronic populations P g ( t ), P e ( t ) are calculated from the vibrational wave packetsas P g,e ( t ) = R L R | Ψ g,e ( R ′ , t ) | dR ′ , and the electroniccoherence (62) is obtained from the overlap of the vi-brational wave packets calculated on the spatial grid: < ψ g ( t ) | ψ e ( t ) > = R L R Ψ ∗ g ( R ′ , t )Ψ e ( R ′ , t ) dR ′ . These re-sults are used to calculate the canonical decoherence ratesand measures of non-Markovianity, as well as the en-tropies of the electronic-vibrational entanglement and theskew information.We begin by analyzing dynamics for a coupling4strength W L = 3 .
29 cm − (corresponding to a pulse in-tensity I ≈ . for a linear polarization vec-tor ~e L [75]), for which the results are given in Figs. 2,3and 4. Fig. 2 shows the time evolutions of several sig-nificant quantities: electronic populations P g ( t ), P e ( t ),”non-Markovianity factor” d ( P g P e ) /dt , entropies L ( t )and S vN ( t ) of the electronic-vibrational entanglement,electronic coherence C l ( t ) and skew information I S ( t ),as well as the non-Markovianity measure f ( t ). The ver-tical dotted lines in the figure help us to observe thecorrelations between the temporal variations of all theseproperties. Figs. 3 and 4 show the time evolution of thevibrational wave packets | Ψ g ( R, t ) | and | Ψ e ( R, t ) | , duringthe pulse and after pulse.The pulse, which operates from 50 to 250 ps (see theenvelope e ( t ) in Fig. 1), couples the two electronic statesactivating a vibrational dynamics which involves severalvibrational levels of each surface, with vibrational peri-ods of about 11 ps in the 1 g electronic potential (thevibrational levels v e = 140 up to 143 are implied), andbetween 33 and more than 100 ps in the a Σ + u poten-tial (corresponding mainly to the vibrational levels from v g = 43 up to 49). The pulse produces a rich vibra-tional dynamics, implying transfer of population betweenthe electronic states, inversion of population, and beatswith various Rabi periods T Rv e ,v g [70] between the popu-lated vibrational levels of the excited and ground states.These phenomena are visible in Fig. 2(a), where typi-cal Rabi periods can be identified, such as T Rv e ,v g = 47 . v e = 142 of 1 g and v g = 47 of a Σ + u ) and T Rv e ,v g = 16 . v e = 142, v g = 45). The timeevolution of the wave packets in Figs. 3 and 4 allows usto observe the relation between the population transferbetween electronic channels and the vibrational motionin the potential wells. Let us briefly decipher the dy-namics from these results. The pulse begins by transfer-ring electronic population from e = 1 g state ( P e (0) = 1)to g = a Σ + u state, the populations becoming equals atabout 80 ps. This process, taking place from 50 to 80ps, increases entanglement (Fig. 2(c)), and is associatedwith a strong non-Markovian behavior (Fig. 2(f)). After80 ps, P g ( t ) > P e ( t ), and the population transfer from e to g continues with the diminution of the entanglementand the non-Markovianity measure f ( t ). The inversion ofpopulation is almost completed at 100 ps, and the trans-fer is inverted, producing a non-Markovianity maximumbetween 100 and 110 ps (Fig. 2(f)), followed by stabi-lization of populations with small Rabi beatings between110 and 130 ps. The vibrational motion inside the a Σ + u potential empties the transfer zone located around thecrossing point R c ≈ a (see Fig. 3(f), t=140 ps), there-fore between 130 and 140 ps the population is transferredfrom 1 g to a Σ + u , diminishing the entanglement and thefunction f ( t ). Between 160 and 190 ps, the pulse againtransfers again population from the g = a Σ + u state to the e = 1 g state, increasing the entanglement and the non-Markovianity function f ( t ) (this process is temporarilystopped around 170 ps by the vibration of the g = a Σ + u | ψ g ( R , t ) | , | ψ e ( R , t ) | t(ps) t(ps) FIG. 4. (Color online) Continuation of Fig. 3: time evolution(200 - 370 ps) of the vibrational wave packets | Ψ g ( R, t ) | (fullline) and | Ψ e ( R, t ) | (dotted line) for a coupling W L = 3 . − . (a-e) Time evolution during the pulse. (f-j) Time evo-lution after pulse. packet, as shown in Fig. 3(h)). Finally, before the endof the pulse, the massive transfer of population from the g = a Σ + u state to the e = 1 g state, between 200 and 220ps, increases the entanglement and has a notable non-Markovian character (see Figs. 2(a,c,f) and 4(a-c)).Let us observe more closely the influence exerted bythis dynamics of transfer and vibration on the non-Markovian character of the electronic evolution. Letus analyze the evolution during the pulse ( t < dL ( t ) /dt > dS vN ( t ) /dt > d ( P g P e ) /dt >
0, and whenever entan-glement decreases ( dL ( t ) /dt < dS vN ( t ) /dt < d ( P g P e ) /dt < t , t ] when the condition of en-hanced non-Markovian behavior d ( P g P e ) /dt > F ( t , t )= R t t f ( t ) dt becomes significantly bigger(for example, the intervals 100-110 ps, 120-130 ps, 145-155 ps, 160-190 ps, or 203-220 ps). On the contrary,if the entanglement decreases during the time interval[ t , t ], F ( t , t ) is drastically diminished, approaching 0(between 130-145 ps, for example).After pulse ( t >
250 ps), the electronic populationsbecome constant, and d ( P g P e ) /dt = 0. Vibrational mo-tion in the electronic potentials leads to oscillations of theelectronic coherence, and implicitly of the linear entropy5 L ( t ) , S v N ( t ) I S ( t ) | < Ψ g | Ψ e > |
50 100 150 200 250 300 350 400t(ps) f( t ) P g ( t ) , P e ( t ) -0.0200.02 d ( P g P e ) / d t P g (t)P e (t) S vN (t) L(t) (a)(b)(c)(d)(e)(f)
FIG. 5. (Color online) Results for a coupling strength 4 W L =13 .
16 cm − between the electronic states g = a Σ + u and e = 1 g of Cs coupled by a pulse with the same envelope e ( t ) shown in Fig. 1. Evolutions during the pulse and af-ter pulse. (a) Time evolutions of the populations P g ( t ) and P e ( t ). (b) Time evolution of the ”non-Markovianity fac-tor” d ( P g P e ) /dt . (c) Time evolutions of the linear entropy L ( t ) and von Neumann entropy S vN ( t ) of the electronic-vibrational entanglement. (d) Time evolution of the skewinformation I S ( t ) = I S ( R, t ) / [∆ V ( R )] . (e) Time evolutionof the electronic coherence C l ( t ) / | < ψ g ( t ) | ψ e ( t ) > | . (f)Non-Markovianity measure f ( t ). The filled surface shows theintegral R f ( t ) dt . L ( t ). The non-Markovianity measure is deduced fromEq. (57) as f ( t ) = | <ψ g | ψ e > | (cid:12)(cid:12)(cid:12) d<ψ g | ψ e >dt (cid:12)(cid:12)(cid:12) , taking the form(59) as function of the electronic coherence | < ψ g | ψ e > | .The results shown in Figs. 2(e,f) confirm the analysismade in Sec. III D for a molecule with constant electronicpopulations: indeed, the non-Markovianity measure f ( t )has minima when the electronic coherence | < ψ g | ψ e > | has maxima (for example, at t=250 ps, 280 ps, 385 ps),and attains maximum values when | < ψ g | ψ e > | → ψ g ( R, t ) vibration inside the a Σ + u poten-tial. This vibrational motion (during which the vibra-tional wave packets explore the electronic potentials) di-minishes coherence, increasing the electronic-vibrationalentanglement and bringing a memory character to dy-namics.Let us observe the evolution of the two ”electronic co-herences”, C l ( t ) and the skew information I S ( t ), shownin Figs. 2(e,d), respectively. During the pulse, they man-
50 100 150 200 250 L ( t ) , S v N ( t )
200 250
50 100 150 200 250 -0.0200.02 d ( P g P e ) / d t
50 100 150 200 250 t(ps) f( t )
50 100 × V ( t ) / V ( t )
100 150 200 t =76 ps 3 1 (a)t =62 ps t =100 ps t =195 ps S vN (t) (b)(c) (d) (e)(f)1 3 FIG. 6. (Color online) Results during the pulse for a cou-pling 4 W L = 13 .
16 cm − .(a) Time evolutions of the lin-ear entropy L ( t ) and von Neumann entropy S vN ( t ) of theelectronic-vibrational entanglement. (b) Time evolution ofthe ”non-Markovianity factor” d ( P g P e ) /dt (non-Markovianityis enhanced for d ( P g P e ) /dt > t , V ( t ) / V ( t ). Three time periods (withappropriated initial times t ) are considered: (c) beginning ofthe pulse [50 − − − f ( t ). The filled surface shows the in-tegral R f ( t ) dt . ifest similar behaviors, so we do not observe the excep-tions signaled in the Table I for the cases (2) and (4).After pulse, their temporal behaviors are also similar, but I S ( t ) → C l ( t ) has smallvalues (for example, 260-270 ps, or 360-370 ps). At thesame time, these intervals are also the periods when thenon-Markovianity measure F ( t , t )= R t t f ( t ) dt attainsthe bigger values after pulse (see Figs. 2(d,e,f)).We will now analyze the results obtained for a muchbigger coupling strength, 4 W L = 13 .
16 cm − , which areshown in Figs. 5 (evolution during and after pulse) and6 (detailed evolution during the pulse). The transferof population between the electronic channels becomesmore intense and fast, and then the ”non-Markovianityfactor” d ( P g P e ) /dt varies more rapidly (Figs. 5(a,b)).As in the case discussed previously, the increase ofthe electronic-vibrational entanglement ( dL ( t ) /dt > dS vN ( t ) /dt >
0) is completely correlated with the posi-tivity of the ”non-Markovianity factor” ( d ( P g P e ) /dt > d ( P g P e ) /dt <
0. Thedotted vertical lines in Figs. 5(b,c) and 6(a,b) clearlyshow these correlations. Nevertheless, in this case ex-ceptions from this rule can be observed: indeed, as it6 L ( t ) e ( t ) | < Ψ g | Ψ e > |
50 100 150 200 250 300 350 400 450t(ps)03 f( t ) L L W L L W L L (a)(b)(c)(d) FIG. 7. (Color online) Results for the coupling strengths W L = 3 .
29 cm − (thin line), 2 W L (dashed line), and 4 W L (thick line) between the electronic states g = a Σ + u and e = 1 g of Cs (Fig. 1). The dashed vertical line at t = 250 ps in-dicates the end of the pulse. (a) Pulse envelope e ( t ). (b)Time evolution of the linear entropy L ( t ) of the electronic-vibrational entanglement. (c) Time evolution of the elec-tronic coherence C l ( t ) / | < ψ g ( t ) | ψ e ( t ) > | . (d) Non-Markovianity measure f ( t ). is shown in Figs. 6(a,b), one can distinguish small peri-ods of time corresponding to the cases (1) and (3) an-alyzed in the Table I. Figs. 6(a,b,f) also show that, aspreviously, when entanglement increases and the condi-tion d ( P g P e ) /dt > R f ( t ) dt issignificantly increased.Figs. 6(c-e) show time evolutions of the Bloch vol-ume reported at an initial time t , V ( t ) / V ( t ), corre-sponding to three periods belonging to the time interval[50 , t : (c) beginning of the pulse [50 − t = 62 ps, t = 76 ps); (d) the period of constantstrength [100 − t = 100 ps); (e) end of the pulse,[195 − t = 195 ps). From the theoretical anal-ysis exposed in Sec. III D, it is expected that the Blochvolume will increase, witnessing non-Markovianity, onlyif d ( P g P e ) /dt >
0. This is exactly what we observe inFigs. 6(a-f): increase of the Bloch volume is correlated toincrease of entanglement, the condition of enhanced non-Markovian behavior d ( P g P e ) /dt >
0, and the increase ofthe integral R f ( t ) dt .Non-Markovianity evolution after pulse is shown inFig. 5(f). The function f ( t ) evolves in the manner previ-ously analyzed, with pronounced maxima correspondingto the electronic coherence | < ψ g | ψ e > | minima.The results obtained for three strengths of the cou-pling ( W L = 3 .
29 cm − , 2 W L , and 4 W L ) and the samepulse envelope are compared in Fig. 7, which exposes the linear entropy L ( t ) of the electronic-vibrational en-tanglement, the electronic coherence | < ψ g | ψ e > | , andthe non-Markovianity measure f ( t ). The total amountof non-Markovianity F ( t i , t f )= R t f t i f ( t ) dt over the timeinterval [ t i , t f ] was calculated for several time intervals,corresponding to the beginning of the pulse ([50 , , , , F (50,250 ps), decreases with the increase of thecoupling W L , but, after pulse, the values F (250,495 ps)calculated for the three strengths of the coupling attainsimilar values.Therefore, we find that during the pulse action, it isthe weaker pulse which stimulates the bigger amount ofnon-Markovianity. This behavior is related to the Rabiperiods of the population exchange between electronicchannels, with a weak coupling enabling a more power-ful presence of the vibrational environment. Indeed, astrong coupling induces a stronger electronic coherence(see Fig. 7 (c)), favoring the transfer of population be-tween channels (localized around the crossing point ofthe electronic potentials) over the vibrational motion inthe molecular potentials. A fast transfer of populationcorresponding to a strong coupling (i.e. small Rabi pe-riod) has the effect of ”locking” the population in thetransfer zone, inhibiting vibration. By contrast, a slowertransfer of population, produced by a weak pulse, giveswave packets more time to explore the electronic poten-tials, increasing gradually the entanglement and enhanc-ing non-Markovian behavior. VII. CONCLUSIONS
We have examined non-Markovian behavior in the re-duced time evolution of the electronic subsystem of alaser-driven molecule, as an open quantum system en-tangled with the vibrational environment.Non-Markovianity was characterized using the canon-ical measures defined in Ref. [16] as functions of the neg-ative decoherence rates appearing in the correspondingcanonical master equation. The canonical measures pro-vide a complete description of non-Markovian behavior,being sensitive to individual decoherence rates when sev-eral decoherence channels are present. The Bloch vol-ume of accessible states was also considered as a non-Markovianity witness, even if it does not always detectnon-Markovian behavior, being only sensitive to the sumof the decoherence rates [16]. The use of different non-Markovianity measures helped to highlight the enhancednon-Markovian behavior, detected by both measures andgenerally accompanied by the increase of the electronic-vibrational entanglement.For a laser-driven molecule described in a bipartiteHilbert space H = H el N H vib with dimension 2 × N v , wehave derived the canonical form of the electronic mas-7 TABLE II. The total amount of non-Markovianity over the time interval [ t i , t f ], F ( t i , t f )= R t f t i f ( t ) dt , calculated for varioustime intervals (during the pulse with the envelope e ( t ) shown in Fig. 7(a), and after pulse), and for the strengths W L = 3 . − , 2 W L , and 4 W L of the coupling. F (50,100 ps) F (100,195 ps) F (195,250 ps) F (50,250 ps) F (250,495 ps) W L W L W L ter equation, deducing the canonical decoherence ratesas functions of the electronic populations P g ( t ) , P e ( t )and of the electronic coherence (Eqs. (39), (40)). Sub-sequently, the canonical measures of non-Markovianityand the Bloch volume of dynamically accessible stateswere obtained. We found that one of the decoherencerates is always negative, accounting for the inherent non-Markovian character of the electronic evolution. More-over, a second decoherence rate becomes negative if thecondition d ( P g P e ) /dt > f ( t ) can be seen as a measure of the temporal be-havior of the electronic coherence (which determines theevolution of L ( t ), the linear entropy of entanglement),having minima when the electronic coherence has max-ima ( L ( t ) minima), and attaining maximum values when-ever the overlap of the vibrational packets tends to zero( L ( t ) maxima). This signifies that vibrational motionwhich explore the electronic potentials diminishing nu-clear overlap (i.e. increasing the linear entropy of entan-glement) brings a memory character to dynamics.The condition d ( P g P e ) /dt > d ( P g P e ) /dt , the time behaviorof linear entropy of entanglement ( dL/dt ), and behaviorsof speakable and unspeakable [64] electronic coherences,measured by l norm C l ( t ) and skew information I S ( t ),respectively. We have also discussed the possibility ofrelating the conditions d ( P g P e ) /dt > dL/dt >
0, or dC l /dt > d ( P g P e ) /dt > D (ˆ ρ el ( t ) , ˆ ρ el ( t )), and is closely related to the conditionof increase of entanglement, dL ( t ) /dt > g = a Σ + u (6 s, s ) and e = 1 g (6 s, p / ) ofthe Cs molecule, coupled by a laser pulse. The mo-tion of the vibrational wave packets in the electronicmolecular potentials coupled by the laser pulse was sim-ulated numerically, for several strengths of the pulse.The non-Markovian behavior, characterized using thecanonical measures and the Bloch volume, was analyzedin relation to dynamics of the electronic-vibrational en-tanglement and electronic coherence in the molecule.We found that increase of electronic-vibrational entan-glement ( dL ( t ) /dt > dS vN ( t ) /dt >
0) is corre-lated with the positivity of the non-Markovianity factor( d ( P g P e ) /dt > t , t ], given by the integral F ( t , t )= R t t f ( t ) dt , where f ( t ) is the canonical measure of non-Markovianity, defined from the appearance of negativedecoherence rates in the canonical master equation.We have shown that the total amount of non-Markovianity corresponding to the pulse action decreaseswith the increase of the coupling. Nevertheless, the val-ues F ( t , t ) corresponding to evolutions after pulses aresimilar, probably because analogous domains of vibra-tional levels are populated, and therefore a similar vi-brational dynamics is activated. The fact that duringthe pulse action, it is the weaker pulse which stimulatesthe bigger amount of non-Markovianity, has to be re-lated to the Rabi periods characterizing the exchange ofpopulation between electronic channels, and influencingvibration in the electronic potentials. A weak pulse givesmore time to vibrational wave packets to explore the elec-tronic potentials, leading to entanglement increase andenhancement of non-Markovianity.In conclusion, in a molecule (here with two populatedelectronic states), the evolution of the electronic subsys-tem has an inherent non-Markovian character due to thedynamics of the vibrational environment, even if thereis no exchange of population between electronic chan-nels, but only vibrational motion in the electronic po-tentials. Enhanced non-Markovian behavior of the elec-tronic dynamics arises if there is a coupling between8electronic channels such that the evolution of electronicpopulations obeys d ( P g P e ) /dt >
0, and it appears asa dynamical property associated with the increase ofthe electronic-vibrational entanglement . Several non-Markovianity regimes, determined by the sign of thenon-Markovianity factor d ( P g P e ) /dt , were analyzed inSec. III D and Sec. IV.A key motivation shaping the present work was to ex-amine non-Markovian behavior of the electronic evolu-tion in relation to the dynamics of the quantum corre-lations in the molecular system. In this sense, obser-vation of the correlation phenomena accompanying en-hancement of non-Markovianity reveals appropriate waysto understand non-Markovian behavior. Therefore, ifthe non-Markovian character of the electronic dynam-ics cannot be separated from the presence of the elec-tronic coherence, the most significant relation is betweennon-Markovianity and entanglement dynamics: We haveshown that non-Markovianity of the electronic evolutionis essentially a dynamical property generated during theincrease of electronic-vibrational entanglement. ACKNOWLEDGMENTS
This work was supported by the LAPLAS 4 andLAPLAS 5 programs of the Romanian National Author-ity for Scientific Research.
Appendix: Distinguishability between two electronicstates, ˆ ρ el ( t ) and ˆ ρ el ( t ) Distinguishability between two electronic states ˆ ρ el ( t )and ˆ ρ el ( t ) can be analyzed using as measure the tracedistance D (ˆ ρ el ( t ) , ˆ ρ el ( t )) between the two states, definedas [2, 10] D (ˆ ρ el ( t ) , ˆ ρ el ( t )) = 12 Tr el | ˆ ρ el ( t ) − ˆ ρ el ( t )) | . (A.1)Taking into account the matrix of the electronic densitygiven by Eq. (24), one obtains [2] D (ˆ ρ el ( t ) , ˆ ρ el ( t )) = q [ P g ( t ) − P g ( t )] + | C ( t ) − C ( t ) | . (A.2)In Eq. (A.2), P g ( t ) − P g ( t ) is the difference of thepopulations between t and t , and C ( t ) − C ( t ) is thedifference between the complex nondiagonal elements C ( t )= < ψ g ( t ) | ψ e ( t ) > = | C ( t ) | exp[ iα ( t )] of the electronicdensity matrix (24) at t and t . The l norm measure ofthe electronic coherence is C l (ˆ ρ el )= 2 | C ( t ) | .We look for the conditions determining an increaseof the trace distance, i.e. a positive rate of change dD (ˆ ρ el ( t ) , ˆ ρ el ( t )) /dt >
0. From Eq. (A.2) one obtainsthe following equation giving the rate of change of thetrace distance, dD (ˆ ρ el ( t ) , ˆ ρ el ( t )) /dt : D (ˆ ρ el ( t ) , ˆ ρ el ( t )) dD (ˆ ρ el ( t ) , ˆ ρ el ( t )) dt = [ P g ( t ) − P g ( t )] dP g ( t ) dt + | C ( t ) | d | C ( t ) | dt −| C ( t ) | d | C ( t ) | dt cos[ α ( t ) − α ( t )] −| C ( t ) || C ( t ) | sin[ α ( t ) − α ( t )] dα ( t ) dt . (A.3)As it could be expected, Eq. (A.3) shows that dD (ˆ ρ el ( t ) , ˆ ρ el ( t )) /dt is an oscillating function, which be-comes positive or negative depending on the evolution atthe instant t and on the initial state at t . Nevertheless,some interesting observations can be made.Let us consider the right hand side of Eq. (A.3). Thefirst term becomes positive, [ P g ( t ) − P g ( t )] dP g ( t ) /dt > dP g /dt )= sgn[ P g ( t ) − P g ( t )], i.e. on those in-tervals [ t , t ] of the time evolution on which a smallerpopulation at t is increased at t ( P g ( t ) < P g ( t ), dP g ( t ) /dt >
0) or a larger population at t is dimin-ished at t ( P g ( t ) > P g ( t ), dP g ( t ) /dt < P g − P e ) dP g /dt < t , t ] when the condi-tion ( P g − P e ) dP g /dt < d ( P g P e ) /dt >
0) is fulfilled,also [ P g ( t ) − P g ( t )] dP g ( t ) /dt > C l dC l /dt ) /
4, and it becomes positive if theelectronic coherence increases, dC l /dt > C ( t ) and C ( t ), andcan be characterized as ”easily oscillating” terms, whosesigns are rapidly changing.Let us suppose that the electronic state ˆ ρ el ( t ) is astate with electronic coherence | C ( t ) | = 0. There-fore, the last two terms become 0, and Eq. (A.3) showsthat the trace distance between ˆ ρ el ( t ) and another stateˆ ρ el ( t ) will increase ( dD (ˆ ρ el ( t ) , ˆ ρ el ( t )) /dt >
0) in a in-terval [ t , t ] in which the conditions d ( P g P e ) /dt > dC l /dt > ρ el ( t ) and a state ˆ ρ el ( t ) with co-herence C l ( t ) = 0 is increased when d ( P g P e ) /dt > dC l /dt >
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