Coherent dynamics of V-type systems driven by time-dependent incoherent radiation
aa r X i v : . [ phy s i c s . c h e m - ph ] A ug Coherent dynamics of V-type systems driven by time-dependentincoherent radiation
Amro Dodin, Timur V. Tscherbul, and Paul Brumer Chemical Physics Theory Group, Department of Chemistry,and Center for Quantum Information and Quantum Control,University of Toronto, Toronto, Ontario, M5S 3H6, Canada Department of Physics, University of Nevada, Reno, NV, 89557, USA (Dated: September 4, 2018)
Abstract
Light induced processes in nature occur by irradiation with slowly turned-on incoherent light. Thegeneral case of time-dependent incoherent excitation is solved here analytically for V-type systemsusing a newly developed master equation method. Clear evidence emerges for the disappearanceof radiatively induced coherence as turn-on times of the radiation exceed characteristic systemtimes. The latter is the case, in nature, for all relevant dynamical time scales for other thannearly degenerate energy levels. We estimate that, in the absence of non-radiative relaxationand decoherence, turn-on times slower than 1 ms (still short by natural standards) induce Fanocoherences between energy eigenstates that are separated by less than 0.9 cm − . . INTRODUCTION A number of femtosecond laser spectroscopy studies [1–4] on components of light harvest-ing systems, such as the Fenna-Matthews-Olson (FMO) or PC645 photosynthetic complexes,show that irradiation with fs pulses results in coherent molecular energy transfer dynamics.These observations have been interpreted as demonstrating a role for quantum coherentdynamics in biological systems. However, as has been repeatedly argued [5–17] both for-mally and computationally, the response of a molecule to coherent laser light is dramaticallydifferent from that to natural incoherent radiation, such as sunlight. For example, in theabsence of a decohering environment the pulsed laser case shows persistent molecular coher-ences, whereas the incoherent case yields a complex mixture of molecular energy eigenstates[8, 18]. These results cast doubt on the relevance of the experimentally observed molecularcoherences to natural light induced and light harvesting processes.Such studies, barring one [5] have all relied upon the sudden turn-on of the radiation,which is both unnatural and which generates initial coherence due to the abrupt turn-onof the light. That is, they focused on the fate of coherences after they were generatedby sudden turn-on of the radiation. However, natural turn-on of light (e.g. sunrise forphotosynthesis, or the blinking of an eye for vision) is very slow by comparison with moleculartime scales, motivating further studies of the time evolution of systems subject to timedependent incoherent excitation, and of the associated Fano coherences discussed below.This study is carried out here on the generic V-system, analytically exposing the dependenceof the system evolution and the associated coherences on the turn-on time. The resultsclearly show that the slower the turn-on time, the less the generated molecular coherences.In particular, with natural turn-on times, no molecular coherences will appear between otherthan near-degenerate levels. For example, with turn-on times on the order of 1 ms, whichis still very short compared to natural turn-on times, coherences will be established onlybetween levels spaced by 0.9 cm − . A one second turn-on will only induce coherences inlevels spaced by × − cm − .To consider such coherent effects rigorously, we examine the most general picture ofweak-field incoherent light-matter interactions. This is given by the Bloch-Redfield (BR)master equations, in which the populations and coherences of a reduced density matrix aretreated on an equal footing [19]. The Pauli rate law equations underlying, for example, the2instein theory for excitation by incoherent light [20], can be obtained from the BR theoryby neglecting the non-secular terms that couple populations and coherences. However,these non-secular terms are responsible for Fano interference between different incoherentexcitation pathways [21–23] The existence of Fano coherences in incoherently driven systemshas sparked considerable interest in the context of naturally occurring LHC’s and artificialphotovoltaics [21, 22, 24] where it has been proposed as a mechanism for enhancing theefficiency of quantum heat engines [25, 26].The Fano coherences differ significantly in origin from the coherences induced by coher-ent light. As shown below, they can be understood most easily in terms of a number statepicture where absorption of light with frequency ω = ω i leads a system in the ground state | g i to make a transition to the excited statesx | e i i and gain phase according to the complexphase of the corresponding transition dipole moment [18]. For simplicity assume that thetransition dipole moments are real and positive. The Fano coherences arise due to simulta-neous excitation from the ground state to both excited states, producing a coherent in-phasesuperposition of the excited states. In contrast, the coherences arising from excitation of thesystem with coherent sources are a consequence of the phase relations between the transi-tion frequencies. In the number state (photon) picture, coherent light is given by a coherentsuperposition of number states which contains the phase information as coherences betweenthe number states at the corresponding frequencies. The radiation field coherences can thenbe “transferred” to the system through the dipole interaction.Our previous work has explored the role of the Fano coherences in the dynamical evolu-tion of the V-system in the weak pumping limit [8, 18]. We have derived the Bloch-Redfieldequations for a general class of multilevel systems [6] and identified the parameter depen-dence of the dynamical evolution of the V-system [18]. Here, these studies are significantlyextended by considering the regime of non-stationary time-dependent incoherent radiation.Section II develops a model for a time-dependent field which is then used to generalize theBloch-Redfield equations to the case of time-dependent fields in Equation (7). Equation (23)uses the BR equations to consider a weakly pumped V-system and presents the general an-alytical solution, which is examined more closely for the limiting cases of a closely spaced ∆ ≪ γ system in Equation (29c) and a system with wide level spacing ∆ ≫ γ in Equa-tion (36b), where ∆ is the excited state splitting and γ is the rate of spontaneous emission.Finally, Equation (44) summarizes our results.3ote that this paper deals with the molecular coherences generated by the incident light.Those associated with, for example, donor excitation in a donor-acceptor system, brieflydiscussed in [5], will be discussed in detail elsewhere [27, 28]. Further, these studies donot include, but do motivate including, a second bath that would model, e.g., a bosonicenvironment. In that case, where the system is coupled to two baths, the long time steadystate is a "transport problem", with flow of energy from the radiation field to the secondbath. Studies of this kind, which would extend work such as that in Ref. [29], are in progress. II. MODELLING THE TIME-DEPENDENT FIELDA. Properties of Field Dynamics
In this section, we first consider a model for a time varying radiation field correspondingto the slow turn on of incoherent light (e.g., a thermal field attenuated by a variable filter),that is an isotropic unpolarized radiation field with constant frequency distribution buttime varying intensity. Furthermore, let the radiation field be diagonal in the number staterepresentation at all times.The isotropy and unpolarized property is straightforward to implement through the fol-lowing relationship, enforced at all times t and for all non-negative integers m : h ˆ n m j λ i ( t ) = h ˆ n m k µ i ( t ) ; if | j | = | k | (1)where ˆ n j λ is the number operator for the field mode with wave vector j and linear polar-ization λ = 1 , and h ˆ A i ( t ) is the expectation value of the field operator ˆ A . That is, Eq. (1)states that all statistical moments, m , of the field mode depend only on the magnitude of thewave vector and not on its direction or polarization. Therefore, the statistical distributionof all field modes with the same wave vector magnitude is identical, realizing an isotropicand unpolarized field.The requirement for a time-independent frequency distribution is equivalent to assumingthat the temperature of a blackbody source does not change. That is, h ˆ n j λ i ( t ) h ˆ n k µ i ( t ) = Constant for all t > (2)This leads to a source with time-independent bandwidth, ∆ ω , since the bandwidth is a4roperty of the frequency distribution. Consequently, this leads to a time-independentcoherence time τ c for all t > since τ c ∆ ω ∼ [30, 31].The time-dependence of the field is characterized by the varying intensity of the incidentlight. Consider the intensity operator of a multimode field: ˆ I = ˆ E ( − ) · ˆ E (+) = X k λ ǫ k λ ξ k ˆ a k λ e iν k t ! · X j µ ǫ j µ ξ j ˆ a † j µ e − iν j t ! (3)For a given field mode with wavevector k and linear polarization λ = 1 , , ˆ a k λ and ˆ a † k λ are theannihilation and creation operators, ǫ k ,λ is the corresponding polarization vector, ν k is thefrequency of the field mode and ξ k = ( ~ ν k / ǫ V ph ) / is the electric field per photon, where V ph is the photon volume. Here, ˆ E ( ± ) are the positive and negative frequency componentsof the electric field. For an unpolarized isotropic field, Eq. (3) reduces to ˆ I = X k | ξ k | (cid:18) ˆ n k + 12 (cid:19) (4)where ˆ n k = P λ P j : | j | = | k | ˆ n j λ is the total occupation number of field modes with wave vectormagnitude k . The only part of Eq. (4) that depends on the field properties is the totaloccupation number operator. Therefore, to obtain a time-dependent intensity expectationvalue, the radiation field density matrix must change over time such that n k λ ( t ) = h ˆ n k λ i ( t ) = Tr R { ˆ n k λ ˆ ρ R ( t ) } = n k λ f ( t ) (5)where Tr R is the trace over the radiation field. The turn on function must be identical forall modes due to the restrictions from Eqs. (1) and (2). For computational simplicity weconsider a slow turn on envelope of the form f ( t ) = 1 − e − αt (6)where α is a constant characterizing the turn on rate, with a corresponding turn on timescale τ r = 1 /α . Appendix A outlines the generalization of the results for arbitrary turn onfunctions through their expansion in a Laplace-like basis.Finally, Eq. (5) gives a source that is quasi-canonical in that no coherences exist betweennumber states of the bath. Intuitively, this corresponds to a bath that is similar to theincoherent fields studied previously [18] (e.g. Blackbody radiation field) at each instant intime. 5 . Realizing the Time-Dependent Field Since our interest is in the system that is irradiated by the incoherent light, the latter actsas an time-dependent external bath that is coupled to the system. Treating a time-dependentbath within the standard framework of open quantum systems poses challenges that are notpresent in the stationary field case. That is, in standard density matrix theory the System-Bath composite is assumed to be closed, allowing for the development of the density matrixtheory in terms of the Unitary Hamiltonian evolution of the total state vector in the jointHilbert Space of the system and bath [32]. However, the time evolution of the radiation fielddescribed in Sec. IIA does not arise through the typical Hamiltonian evolution of a system + bath.Instead, we need to consider an additional environment E coupled to the radiation bath,but not to the system, in the Born-Markov approximation, and let the radiative bath be cou-pled to the system in the Born-Markov regime. This set up is sketched in Fig. 1. Intuitively,the environment corresponds to the physical system that produces the dynamics of the fieldon the system. This hierarchical approach has the benefit of allowing the use of the standardapproach to open quantum systems [32, 33], since the system-bath-environment compositeis closed and hence evolves according to unitary Hamiltonian dynamics. To obtain wellposed problems for the dynamics of the bath and the system, the system-bath-environmentis assumed to be initially in a separable state ˆ ρ = ˆ ρ S ⊗ ˆ ρ R ⊗ ˆ ρ E .The Born approximation to the system-bath coupling implies that the influence of thesystem on the bath dynamics is negligible. Therefore, we can treat the environment-bathcomposite using the standard approach to computing the reduced dynamics of the bath.We can select the environment and interaction potential such that the dynamics of the bathcorrespond to that discussed in Sec. IIA. In the resultant picture, the properties of the bathoperators can be treated by exact analogy to the system operators in a typical system-bathcase. For example, the two time correlation function for two Schrödinger Operators, ˆ A and ˆ B , on the bath Hilbert Space, H R is given by: h ˆ A ( s ) ˆ B ( t ) i = Tr R n ˆ A ˆ U ( s, t ) ˆ B ˆ U ( t,
0) ˆ ρ R (0) o (7)where ˆ ρ R (0) is the initial state of the bath, assumed to be the vacuum state in our case, and ˆ U ( s, t ) is the bath propagation operator from time s to time t . After obtaining the bath6 ig. 1. A schematic representation of the coupling between the molecular system, S, radiation bath,R, and environment, E. The system and environment are not directly coupled. The system-bathinteraction, V SR , and bath-environment interaction, V RE satisfy the Born-Markov approximation. dynamics, the resulting time-varying density matrix can be used to determine the dynamicsof the system while neglecting any further role of the environment, E. III. DERIVATION OF THE BLOCH-REDFIELD EQUATIONS FOR TIME-DEPENDENTBATHS
Traditional master equations assume a time independent bath. Therefore, to treat turn-on effects, we derive a generalization to the time dependent bath. Consider a multilevelsystem interacting with a quantized incoherent radiation field, as described in Sec. II, underthe dipole and rotating wave approximations. Such a system is characterized by the totalHamiltonian ˆ H T = ˆ H S + ˆ H R + ˆ H E + ˆ V SR + ˆ V RE (8)where ˆ H S = P i E i | i ih i | is the system Hamiltonian, ˆ H R = P k λ ~ ν k ˆ a k λ ˆ a † k λ is the radiationbath Hamiltonian and ˆ H E is the environment Hamiltonian. The operator ˆ V RE is the bath-environment interaction potential. The bosonic creation and annihilation operators of thefield mode with wavevector k , frequency ν k and polarization λ = 1 , are given by ˆ a † k λ and7 a k λ respectively. ˆ V SR is the system-bath interaction potential given by [30] ˆ V SR = − ˆ µ · X k ,λ (cid:18) ~ ν k ǫ V (cid:19) / ǫ k λ (ˆ a k λ − ˆ a † k λ ) (9)where V is the quantization volume, ˆ µ is the dipole moment operator of the system and ǫ k λ is the polarization vector of the field mode with wavevector k and polarization λ .The bath-environment interaction potential and environment Hamiltonian are not spec-ified, but are chosen to produce the bath dynamics discussed in Sec. II. Under the Born-Markov approximation, the system does not contribute to the bath evolution, induced byits interaction with the environment. In other words, the effects of ˆ V ISR and ˆ H S on thedynamics of the radiation field + environment can be neglected. This produces a densityoperator, ˆ ρ RE , on the radiation field + environment Hilbert space such that the radiationfield density matrix ˆ ρ R ( t ) = Tr E ˆ ρ RE follows the conditions described in Sec. II, where Tr E is the trace over the environment E .Transforming Eq. (9) into the interaction picture gives ˆ V ISR ( t ) = ~ X i ≤ j X k λ g ( i,j ) k λ (ˆ a k λ | j ih i | e i ( ω ij − ν k ) t + H.c. ) (10)where g ( i,j ) k λ = (cid:16) ~ ν k ǫ V (cid:17) µ ij · ǫ k λ ~ are the light-matter coupling constants and µ ij = h i | ˆ µ | j i arethe transition dipole matrix elements, assumed real.The equations of motion of the system-bath composite ˆ ρ = ˆ ρ S ⊗ ˆ ρ R in the interactionpicture are given by the Liouville Von-Neuman equation [32, 33] ˙ˆ ρ ( t ) = − i [ ˆ V ISR ( t ) , ˆ ρ (0)] − Z t dt ′ [ ˆ V ISR ( t ) , [ ˆ V ISR ( t ′ ) , ˆ ρ ( t ′ )]] (11)If the system and bath are initially in a separable state ˆ ρ (0) = ˆ ρ S (0) ⊗ ˆ ρ R (0) , the Bornapproximation states that, for weak system-bath coupling (which is valid for the naturallight excitation of LHC’s) they remain in a separable state at all times. Furthermore, thesmall system produces no back reaction on the large bath.Applying the Born approximation to Eq. (11) and tracing over the bath coordinates givesthe equations of motion for the reduced system density matrix ˆ ρ S : ˙ˆ ρ S ( t ) = − i Tr R [ ˆ V ISR ( t ) , ˆ ρ S (0) ⊗ ˆ ρ R (0)] − Z t dt ′ Tr R [ ˆ V ISR ( t ) , [ ˆ V ISR ( t ′ ) , ˆ ρ S ( t ′ ) ⊗ ˆ ρ R ( t ′ )]] (12)8here Tr R denotes a trace over the radiation field. While this equation is similar to thestandard master equation for stationary baths [32, 33], the radiation field density matrixnow carries an explicit time-dependence due to the interaction with the environment.Given Eq. (10), the double commutator in Eq. (12) gives products of ˆ V ISR at two differenttimes, with a typical term of the form: Z t dt ′ X i ≤ j X l ≤ m X k λ X k ′ λ ′ g ( i,j ) k λ g ( l,m ) k ′ λ ′ h ˆ a k λ ˆ a † k ′ λ ′ i ( t ′ )ˆ σ i,j ˆ σ m,l ˆ ρ S ( t ′ ) e − i ( ω ij − ν k ) t ′ + i ( ω lm − ν k ′ ) t . (13)Here ˆ σ i,j = | i ih j | is the “quantum jump operator” from state | j i to state | i i .Using the commutator algebra of bosonic creation and annihilation operators, and theidentity ˆ a k λ ˆ a † k ′ λ ′ = (1 + ˆ n k λ ) δ k , k ′ δ λ,λ ′ , the trace over the bath in Eq. (13) is obtained as: h ˆ a k λ ˆ a † k ′ λ ′ i ( t ) = T r R n ˆ a k λ ˆ a † k ′ λ ′ ˆ ρ R ( t ) o = δ k , k ′ δ λ,λ ′ (1 + n k λ ( t )) (14)Here we have used the normalization of the bath density matrix Tr R { ˆ ρ R ( t ) } = 1 , Eq. (5),and the linearity of the trace. Substituting Eq. (14) into Eq. (13) gives Z t dt ′ X i ≤ j X l ≤ m X k λ g ( i,j ) k λ g ( l,m ) k λ (1 + n k λ ( t ′ ))ˆ σ i,j ˆ σ m,l ˆ ρ S ( t ′ ) e − i ( ω ij − ν k ) t ′ + i ( ω lm − ν k ) t (15)Taking the continuum limit of the k summation X k → V (2 π ) Z π dφ Z π dθ sin( θ ) Z ∞ dk k (16)and noting that for an isotropic and unpolarized field n k λ depends only on k and not on theangular or polarization coordinates, Eq. (15) can be rearranged to yield π ǫ Z t dt ′ X i ≤ j X l ≤ m ˆ σ i,j ˆ σ m,l ˆ ρ S ( t ′ ) e − i ( ω ij t ′ − ω lm t ) × "X λ Z π dφ Z π dθ sin( θ )( µ ij · ǫ k λ )( µ lm · ǫ k λ ) ∞ dkk ν k (1 + n k λ ( t ′ )) e iν k ( t − t ′ ) (cid:21) (17)The summation over the angular and polarization coordinates in Eq. (17) can be evaluatedto give µ ij · µ lm . Changing variables from k to ν k = ck , Eq. (17) can be written in thefollowing form: X i ≤ j X l ≤ m µ ij · µ lm π ǫ c e i ( ω ij − ω lm ) t Z ∞ dν k ν k Z t dt ′ e i ( ω ij − ν k )( t − t ′ ) (1 + n k λ ( t ′ ))ˆ σ i,j ˆ σ l,m ˆ ρ S ( t ′ ) (18)9he exponential factor in the integrand oscillates rapidly at ν k = ω ij , so provided that n k λ varies slowly near ω ik at all times, we can make the Wigner-Weisskopff approximation bysetting [30] Z ∞ dν k ν k e − iν k ( t − t ′ ) → ω ij Z ∞ dν k e − iν k ( t − t ′ ) (19)giving X i ≤ j X l ≤ m µ ij · µ lm ω ij π ǫ c e i ( ω ij − ω lm ) t Z t dt ′ (1 + n k λ ( t ′ ))ˆ σ i,j ˆ σ m,l ˆ ρ S ( t ′ ) e iω ij t ′ Z ∞ dν k e − iν k ( t − t ′ ) (20)The ν k integral in Eq. (20) can now be evaluated as πδ ( t − t ′ ) + iP (1 / ( t − t ′ )) where P denotes the Cauchy Principal Part. Neglecting the small Lamb shift due to the imaginarypart of Eq. (19), and doing the time integral gives X i ≤ j X l ≤ m µ ij µ lm p ij,lm ω ij π ǫ c (1 + n k ij λ ( t ))ˆ σ ij ˆ σ m,l ˆ ρ S ( t ) e i ( ω lm − ω ij ) t (21)where alignment parameters, p ij,lm , for the transition dipole moments have been defined: p ij,lm = µ ij · µ lm µ ij µ lm (22)Transforming Eq. (21) back into the Schrödinger picture eliminates the oscillating phasefactor and yields the same master equations as previously reported for stationary fields [6],but with time dependent occupation numbers. That is, following the same approach as inRef. [6] we arrive at the same master equations [Eq. (17) in Ref. [6]] with the followingsubstitution. r ij = γ ij n ω ij → r ij ( t ) = γ ij n ω ij ( t ) (23)where r ij ( t ) is the pumping rate from level g i in the ground state manifold to state e j in theexcited state manifold. IV. V-SYSTEM MASTER EQUATIONS
Consider now a V-system with one ground state and two excited states (as shown inFig. 2) coupled to a time dependent incoherent radiation field. Since there is only oneground state in the V-system, we suppress the first index ( i ) which specifies the groundstate in the pumping rate r ij ( t ) and spontaneous emission rate γ ij . That is r ij ( t ) → r j ( t ) gives the pumping rate from the ground state g to the excited state e j and γ ij → γ j gives the10 γ r Δ e g e r γ Fig. 2. Schematic representation of a V-type System. ∆ is the excited state splitting, γ i is theradiative line-width, and r i is the incoherent pumping rate of excited state | e i i . spontaneous decay rate from excited state e j to the ground state g . The master equationfor such a system is given by ˙ ρ e j e j = − ( r j ( t ) + γ j ) ρ e j e j + r j ( t ) ρ gg − p ( p r ( t ) r ( t ) + √ γ γ ) ρ Re e (24a) ˙ ρ e e = −
12 ( r ( t ) + r ( t ) + γ + γ ) ρ e e − iρ e e ∆+ p p r ( t ) r ( t )(2 ρ gg − ρ e e − ρ e e ) − p √ γ γ ( ρ e e + ρ e e ) (24b)where ρ Re e is the real part of the off-diagonal (coherence) density matrix element betweenlevels e and e . In Eq. (24b), spontaneous emission processes are governed by the radiativedecay widths of the excited states, γ i = ω ge i | µ ge i | / (3 πǫ c ) , ∆ = ω e e gives the excited statesplitting and p = µ ge · µ ge / ( | µ ge || µ ge | ) measures the alignment of the | g i ↔ | e i i transitiondipole moments, µ ge i . Absorption and stimulated emission processes are parametrized bytime-dependent incoherent pumping rates of the | g i ↔ | e i i transitions, r i ( t ) = γ i ¯ n ( t ) . Herewe neglect environment-induced dephasing and relaxation processes, assuming that the ratesof excited state relaxation and dephasing are small compared to those of the radiativeprocesses (absorption, decay and stimulated emission) [18]. These effects can, however beincluded by generalizing this approach [8].As in the case of stationary pumping rates [18], applying the conservation of populationconstraint, ρ gg = 1 − ρ e e − ρ e e , and transforming into the Liouville space representation11ith state vector x = [ ρ e e , ρ e e , ρ Re e , ρ Ie e ] T yields Eq. (24b) in a vector form: ddt x = A ( t ) x + d ( t ) (25a) A ( t ) = − (2 r ( t ) + γ ) − r ( t ) − p √ γ γ (1 + ¯ n ( t )) 0 − r ( t ) − (2 r ( t ) + γ ) − p √ γ γ (1 + ¯ n ( t )) 0 − p √ γ γ (1 + 3¯ n ( t )) − p √ γ γ (1 + 3¯ n ( t )) − ¯ γ (1 + ¯ n ( t )) ∆0 0 − ∆ − ¯ γ (1 + ¯ n ( t )) (25b) d ( t ) = r ( t ) r ( t ) p p r ( t ) r ( t )0 (25c)where ¯ γ = ( γ + γ ) is the arithmetic mean of the spontaneous decay rates of the excitedstate manifold, and we have used the fact that r i ( t ) /γ i = ¯ n ( t ) .Rewriting Eq. (25c) in the form d ( t ) = [ γ , γ , p √ γ γ , T ¯ n ( t ) shows that the time-varying field pumps the system to the same statistical mixture, ρ d , in the excited manifoldas does the stationary field [18], where ρ d ∝ (1 − p )( γ | e ih e | + γ | e ih e | ) + p | φ + ih φ + | . (26)Here | φ + i = (1 / √ γ )( √ γ | e i + √ γ | e i ) is an in-phase coherent superposition of excitedenergy eigenstates. However, in contrast to the stationary field case, the rate of excitationinto this statistical mixture varies with time. Hence, although these equations look similarto the stationary case, the time-varying excitation rate in this case produces very differentdynamics than one sees in the stationary rate case.Provided the weak-pumping limit ( ¯ n ( t ) ≪ ) is satisfied at all times, the coefficient matrix A ( t ) Eq. (25b) can be perturbatively expanded in ¯ n ( t ) . This yields a time-independentcoefficient matrix A (0) to zeroth order in ¯ n ( t ) , allowing the time-dependent contributions to A ( t ) to be treated through a perturbative expansion: A ( t ) = A (0) + ¯ n ( t ) A (1) (27a)12 (0) = − γ − p √ γ γ − γ − p √ γ γ − p √ γ γ − p √ γ γ − ¯ γ ∆0 0 − ∆ − ¯ γ (27b)As a result, this yields a linear system of constant coefficient ordinary differential equa-tions with a time dependent driving term. Applying the initial conditions appropriate toexcitation from a molecule in the ground state, ρ gg (0) = 1 or x = , the dynamics of theV-system is given by the variation of parameters solution [34] x = Z t dse A (0) ( t − s ) d ( s ) → X i =1 Z t ds ( v i · d ( s )) e λ i ( t − s ) v i (28)where λ i is the i th eigenvalue of A (0) with corresponding eigenvector v i . Eq. 28 relates theeigenvalues, { λ i } , of A (0) to the timescales of the system’s evolution τ i = Re( λ i ) − and tothe frequencies of its oscillations ω i = Im( λ i ) . This solution is similar to that obtained in[18] with the crucial modification that d ( s ) is explicitly time dependent.Since A (0) is time independent, its eigenvalues, eigenvectors and normal modes are iden-tical to those calculated in the stationary field case [18], with λ i = − ¯ γ ± ∆ p p ζ − s ± p η (29a) ζ = ¯ γ ∆ p (29b) η = ∆ | γ − γ || ¯ γ − ∆ p | (29c)where ∆ p = p ∆ + (1 − p ) γ γ .The overdamped ( ∆ p / ¯ γ ≪ ) and underdamped ( ∆ p / ¯ γ ≫ ) regimes of a V-systemexcited by a field with a finite turn on time τ r of the form Eq. (5) are discussed below. Since ∆ p ≥ ∆ where the latter is the excited state level splitting, the overdamped region would berelevant to, e.g., large molecules whereas the underdamped region would correspond, e.g.,to small molecules. 13 . OVERDAMPED REGIME ∆ p ¯ γ ≪ In the overdamped regime, where ζ = ¯ γ ∆ p ≫ of Eq. (29a), the eigenvalues take thesimplified form [18] λ = − γ (30a) λ = − ∆ p γ (30b) λ , = − ¯ γ , (30c)while the normal modes are given by [18] v ∝ [ r , r , p √ r r , (31a) v ∝ [ r , r , − p √ r r , (31b) v ∝ [0 , , , (31c) v ∝ [1 , − , − γ − γ p √ γ γ , . (31d)Substituting Eqs. (30c) and (31d) for the eigenvalues and normal modes, and Eq. (5) forthe occupation number, into the variation of parameters solution Eq. (28) and doing theexponential integrals yields the V-system dynamics ρ e i e i ( t ) = 12¯ γ (cid:26) r j ∆ p γ − α (cid:20) ∆ p γ (cid:0) − e − αt (cid:1) − α (cid:18) − e − ∆2 p γ t (cid:19)(cid:21) + r i γ − α (cid:2) γ (cid:0) − e − αt (cid:1) − α (cid:0) − e − γt (cid:1)(cid:3) (cid:27) (32a) ρ e e ( t ) = p √ r r γ (cid:26) ∆ p γ − α (cid:20) ∆ p γ (cid:0) − e − αt (cid:1) − α (cid:18) − e − ∆2 p γ t (cid:19)(cid:21) − γ − α (cid:2) γ (cid:0) − e − αt (cid:1) − α (cid:0) − e − γt (cid:1)(cid:3) (cid:27) (32b)where i, j = 1 , and i = j and r i = lim t →∞ r i ( t ) = γ i ¯ n is the steady state incoherentpumping rate. Here α , we recall from Eq. (6), defines the turn on time τ r = 1 /α .Equations (32a) and (32b) show that the steady state behavior is given by lim t →∞ x ( t ) =[¯ n, ¯ n, , , is independent of α = 0 , and is identical to the steady state obtained for stationaryfields [18]. However, the non-equilibrium behavior of the system and the maximal coherencecan be markedly different for time-varying fields, as shown in Fig. 3.14 t ρ e , e ( t ) ρ e i , e i ( t ) t ρ e , e ( t ) ρ e i , e i ( t ) −3 t ρ e , e ( t ) ρ e i , e i ( t ) A. τ r ≪ τ γ ≪ τ s B. τ γ ≪ τ r ≪ τ s C. τ γ ≪ τ s ≪ τ r Fig. 3. Evolution of populations and coherences of an overdamped V-system ( ∆ p ¯ γ ≪ ) evaluatedwith aligned transition dipole moments ( p = 1 ). Here γ = 1 . γ and ∆ = 0 . . Three differentturn on regimes are shown. Panels A show the ultrafast turn on of the field with τ r = × − τ γ while Panels B and C show the intermediate ( τ r = 100 τ γ = 5 × − τ s ) and slow ( τ r = 20 τ s ) turnon regimes, respectively. Note the difference in y-axis scales for the different coherence plots. To illustrate this consider a system where the radiation field turn on time τ r = 1 /α differssignificantly from τ γ / / γ and τ s = 2¯ γ/ ∆ p . Since the overdamped regime imposes theinequality τ s ≫ τ γ / , this corresponds to a separation of timescales { τ γ / , τ s , τ r } . Thereare three possible time-orderings: the sudden turn on ( τ r ≪ τ γ / ≪ τ s ), the slow turn on( τ γ / ≪ τ s ≪ τ r ) and the intermediate ( τ γ / ≪ τ r ≪ τ s ) regimes.For the sudden turn on of the field, Eq. (32b) simplifies to ρ e i e i ( t ) = 12¯ γ (cid:20) r i (cid:0) − e − γt (cid:1) + r j (cid:18) − e − ∆2 p γ t (cid:19)(cid:21) (33a) ρ e e ( t ) = p √ r r γ (cid:18) e − ∆2 p γ t − e − γt (cid:19) (33b)under a binomial expansion to lowest contributing order in τ r /τ s ≪ . Equation (33b) isindependent of α (and hence τ r ) and is identical to the expression derived for stationaryfields, the τ r → limit [18]. The large quasistationary coherences characteristic of thisregime can clearly be seen in subplot A of Fig. 3. That is, if the field is turned on faster15han the fastest characteristic time scale of the system τ γ / / γ then the dynamics of thesystem are well approximated by the stationary field solution, Eq. (33b). In particular, thecoherences approach the same maximal value of p √ r r γ in the interval τ γ / < t < τ s = γ ∆ p asin the stationary field approximation. In this limit, the field reaches its steady state muchfaster than the system evolves, and so the very short-lived transient behavior of the field isnot reflected in the evolution of the system. Intuitively, in this regime the V-system doesnot evolve under the transient field. Instead, it evolves only under the steady state field.By contrast, if the field is turned on very slowly, τ γ / ≪ τ s ≪ τ r , the stationary fieldsolution is a very poor approximation for the system dynamics. Taking a binomial expansionto lowest contributing order in τ s /τ r ≪ , Eq. (32b) can be rewritten as ρ e i e i ( t ) = ¯ n (cid:0) − e − αt (cid:1) (34a) ρ e e ( t ) = pα √ r r ∆ p (cid:18) e − ∆2 p γ t − e − αt (cid:19) (34b)The dependence on the incoherent pumping rates, r i , in Eq. (34b) is contained in the meanthermal occupation of the field ¯ n = r i /γ i . To appreciate this result, note that when thefield is turned on adiabatically, the dynamics of the system closely resemble the incoherentexcitation produced by Pauli rate law dynamics. The rate law predicts populations evolvingto an equilibrium value of ¯ n as ρ e i e i = ¯ n (1 − e − γ i t ) . This is similar to Eq. (34a), where thepopulation of the excited states equilibrates to the same value ¯ n at the rate α . Equation (34a)may also be rewritten as ρ e i e i ( t ) = ¯ n ( t ) by substituting Eq. (5), indicating that the systemis in equilibrium at all times under the slowly-varying field. Furthermore, Eq. (34b) showsa suppression of the coherences by a factor of τ s τ r ≪ in comparison to excitation by afield with a very fast turn on time. Alternatively, writing this in terms of the characteristictimescales of the system max {| ρ Slowe e ( t ) |} = τ s τ r max {| ρ F aste e |} (35)where τ s τ r = αλ ≪ . The difference in coherence amplitude between the fast and slow turnon of the radiation field can be seen in comparing subplots A and C of Fig. 3.The suppression of the coherence amplitude under adiabatic turn-on of the field canbe understood by considering the evolution of individual components of ρ d Eq. (26)and the interactions between them. The coherences originate from the in-phase | φ + i = / √ γ )( √ γ | e i + √ γ | e i ) superposition prepared by the incident field. This super-position collapses to an equally populated incoherent mixture of excited states ρ eq =¯ n ( | e ih e | + | e ih e | ) over a time-scale τ s [8, 18]. Furthermore, the population of excitedstates enhances the decay of in-phase superpositions through the increased rate of decayprocesses. This disproportionately affects the in-phase superpositions since they exhibitconstructive interference in the decay processes which is reflected in the terms proportionalto ρ e i e i in the coherence master equations Eq. (24b). Since the | φ + i superpositions gener-ated by the incoherent field decay at a time-scale τ s leaving behind an incoherent mixtureof excited states, the | φ + i states prepared at later times will decay faster than those pre-pared at earlier times due to the increased population of the excited states. As a result,appreciable amounts of | φ + i never accumulate in the system, leading to a heavy suppressionof the coherences. This also accounts for the decay of the coherences with time-scale τ r [Eq. (34b)]. The population of excited states on this time-scale lead to an increase in thedecay rate of the coherent superpositions on a time-scale τ r . This ultimately leads to thedecay of the | φ + i components on the radiation field turn on time.Hence„ when the system is excited by a field that is turned on very slowly compared tothe system’s longest time scale (here τ s = 2¯ γ/ ∆ p ), it will evolve in constant equilibrium withthe field, producing the incoherent instantaneous steady state x ( t ) = [¯ n ( t ) , ¯ n ( t ) , , at alltimes.For completeness, consider a field in the intermediate turn on regime, τ γ / ≪ τ r ≪ τ s .Proceeding through a binomial expansion, as in the earlier cases, Eq. (32b) reduces to ρ e i e i ( t ) = 12¯ γ (cid:20) r i (cid:0) − e − αt (cid:1) + r j (cid:18) − e − ∆2 p γ t (cid:19)(cid:21) (36a) ρ e e ( t ) = p √ r r γ (cid:18) e − ∆2 p γ t − e − αt (cid:19) (36b)This implies that, in the intermediate regime the system displays the same maximal co-herence as in the fast turn on regime. However, the time scale over which it approachesits quasistationary state becomes τ r = 1 /α rather than τ γ / / γ . When the turn ontime is slower than the decay time ( τ s ) of | φ + i , the radiation field reaches a steady statebefore the | φ + i excitations decay appreciably. In contrast to the adiabatic turn on case,the survival of the early coherences in the intermediate regime allows for the maintenanceof coherences from excitations at later times. This leads to the same maximal coherence17n the intermediate regime as in the sudden turn-on case. However, note that althoughmost of the coherences are generated in the timescale τ r they decay at the same time asthose generated for the sudden turn-on, as τ s rather than ( τ r + τ s ) as may be expected apriori. This indicates that excitations to | φ + i generated at later times have a shorter decaytime than those generated at earlier times. This occurs through the same mechanism asthe decay of coherences on a time scale of τ r in the adiabatic turn-on regime. When the | φ + i excitations generated at earlier times decay to incoherent mixtures of excited states attime τ s the increase in excited state population leads to an increase in the decay rate of thecoherences. This leads to a “cascade” in which the rate of decay of the coherences increasesas more in-phase superpositions decay to incoherent mixtures of the excited eigenstates. VI. UNDERDAMPED REGIME ¯ γ ∆ p ≪ A V-system in the underdamped regime is characterized by a very small damping coeffi-cient, ζ = ¯ γ ∆ p ≪ . Taking the corresponding limit of Eq. (29a) gives the eigenvalues of anunderdamped V-system [18]: λ = − γ (37a) λ = − γ (37b) λ , = − ¯ γ ± i ∆ p (37c)Substituting Eq. (37c) into Eq. (25b), one finds the corresponding normal modes [18] v ∝ [1 , , , p √ γ γ p ] (38a) v ∝ [0 , , , p √ γ γ p ] (38b) v ∝ [0 , , , (38c) v ∝ [0 , , , − (38d)18he general solution, obtained using Eq. (28), is ρ e i e i ( t ) = ¯ nα − γ i (cid:2) α (1 − e − γ i t ) − γ i (1 − e − αt ) (cid:3) (39a) ρ Re e = p √ r r (cid:20) e − ¯ γt (¯ γ (1 − cos(∆ p t )) + ∆ p sin(∆ p t )∆ p + ¯ γ − e − ¯ γt [( α − ¯ γ ) cos(∆ p t ) + ¯ γ + ∆ p sin(∆ p t )] − αe − αt ( α − ¯ γ ) + ∆ p (cid:21) (39b) ρ Ie e ( t ) = − p √ r r (cid:20) e − ¯ γt [(∆ p (1 − cos(∆ p t )) − ¯ γ sin(∆ p t )]∆ p + ¯ γ − e − ¯ γt [∆ p (1 − cos(∆ p t )) + ( α − ¯ γ ) sin(∆ p t )]∆ p + ( α − ¯ γ ) + 12( α − γ ) (cid:2) α (1 − e − γ t ) − γ (1 − e − αt ) (cid:3) − α − γ ) (cid:2) α (1 − e − γ t ) − γ (1 − e − αt ) (cid:3) (cid:21) (39c)where ρ Re e and ρ Ie e are the real and imaginary parts of the coherence term, respectively,and where r i = lim t →∞ r i ( t ) = γ i ¯ n (as in the overdamped regime).Equation (39c) is cumbersome and does not provide much insight into the dynamics ofthe system. However, we note that the steady state of the system can easily be determinedto be the incoherent mixture lim t →∞ x ( t ) = [¯ n, ¯ n, , . This agrees with the results fromboth the overdamped regime and the stationary field case [18]. In order to obtain moreinsight into the dynamics of the V-system we consider several cases for the turn on time.First, consider a turn on time, τ r = α − , that is faster than all three of the systemtimescales, τ γ i = 1 /γ i and the period of coherence oscillations τ ∆ = 1 / ∆ p , i.e., the fast turnon regime characterized by, τ r ≪ τ ∆ ≪ τ γ i . A binomial expansion of Eq. (39c) yields thedynamics induced by a bath with a fast turn on as: ρ e i ,e i ( t ) = ¯ n (1 − e − γ i t ) (40a) ρ Re ,e ( t ) = p √ r r ∆ p e − ¯ γt sin(∆ p t ) (40b) ρ Ie ,e ( t ) = p √ r r ∆ p (cid:18) e − ¯ γt (cos(∆ p t ) − − e − γ t − e − γ t (cid:19) (40c)As expected, this is identical to the solution derived for stationary fields [18]. That is,if the field is turned on much faster than the characteristic timescales of the system, the19 −4 −2 −6−4−2024x 10 −3 t ρ e , e ( t ) −4 −2 ρ e i , e i ( t ) −4 −2 −1−0.500.51x 10 −3 t ρ e , e ( t ) −4 −2 ρ e i , e i ( t ) −4 −2 −10123x 10 −6 t ρ e , e ( t ) −4 −2 ρ e i , e i ( t ) A. τ r ≪ τ ∆ ≪ τ γ i B. τ ∆ ≪ τ r ≪ τ γ i C. τ ∆ ≪ τ γ i ≪ τ r Fig. 4. Evolution of populations and coherences of an underdamped V-system ( ∆ p ¯ γ ≫ ) evaluatedwith aligned transition dipole moments ( p = 1 ). Here γ = 1 . γ = γ and ∆ = 24 . . Threedifferent turn on regimes are shown here. Panels A show the ultrafast turn on of the field with τ r = 0 . τ ∆ while Panels B and C show the intermediate ( α = 24 τ ∆ ) and slow ( α = 100 τ γ ) turnon regimes respectively. Note the difference in y-axis scales for the coherence plots. Solid red linesindicate the real part of the coherence ρ Re e with the imaginary part ρ Ie e indicated by the dashedblue line. stationary field solution closely approximates the evolution of the system since the fieldreaches its stationary state faster than the system can evolve under the transient field.In contrast, consider a field that turns on much slower than the period of coherenceoscillations, τ r ≫ τ ∆ . In the τ ∆ ≪ τ r limit Eq. (39b) for the real part of the coherence termtakes on the much simpler form: ρ Re e ( t ) = p √ r r ∆ p α ∆ p ( e − ¯ γt cos(∆ p t ) − e − αt ) (41)which does not depend on the value of τ r relative to τ γ i . Equations (39a) and (39c) for thepopulations and the imaginary part of the coherence term depend on the magnitude of τ r relative to each of the τ γ i ’s. Equation (40a) remains an accurate solution for ρ e i e i ( t ) providedthat τ γ i ≫ τ r . In the adiabatic ( τ γ i ≪ τ r ) limit, the populations can be expressed as ρ e i e i ( t ) = ¯ n (1 − e − αt ) = ¯ n ( t ) (42)20quation (40c) remains a good approximation for the imaginary coherences provided that τ r ≪ τ γ i . More generally, ρ Ie e ( t ) depends on the magnitude of τ r relative to both τ γ i ’s.Without loss of generality, let γ > γ . This gives the dynamics of the imaginary coherencesin the following cases: ρ e e ( t ) = − p √ r r ∆ p (cid:16) α ∆ p e − ¯ γt sin(∆ p t ) − e − γ t − e − αt (cid:17) if γ ≫ α ≫ γ − p √ r r ∆ p h α ∆ p e − ¯ γt sin(∆ p t ) + α γ ( e − γ t − e − αt ) − α γ ( e − γ t − e − αt ) i if γ i ≫ α (43)Significantly, in the adiabatic limit, when the turn on time of the field far exceeds thecharacteristic timescales of the system, the coherences in Eqs. (41) and (43) are heavilysuppressed by the factor of τ ∆ τ r ≪ relative to the fast turn on case Eqs. (40b) and (40c).Therefore, the V-system is in equilibrium with the field at all times in the adiabatic limit,producing an incoherent mixture of excited states at all times.The rich dynamics of the imaginary coherences, ρ Ie e , are hidden in Fig. 4 due to theassumption of equal decay rates. Figure 5 displays the interplay between the oscillatoryand quasistationary contributions more clearly by selecting highly asymmetric decay widths γ ≫ γ .Furthermore, Eqs. (34b), (41) and (43) all display the same inverse scaling of coherenceamplitude with turn on time of the field. Hence, if a radiation field is turned on slowly, themagnitude of the coherences scales inversely with the turn on time as max {| ρ e e | ( t ) } ∝ α = 1 τ r (44)This suggests that the strong coherences observed thus far [8, 18] for ∆ = 0 arise due tothe instantaneous turn on of the radiation field and will not be retained when consideringa field with a turn on time that is slower than the radiative lifetimes of the excited states( τ r ≫ τ γ i = 1 /γ i ).In summary, in both the overdamped and underdamped regions, the Fano coherencespreviously computed in the study of V-systems with suddenly turned-on radiation [6, 8, 18]disappear if the incoherent radiation field is turned on adiabatically. Similarly, these resultsclearly indicate that coherences observed in experiments utilizing fast laser pulses (e.g. [2, 3])will not appear in nature where turn on times are essentially infinite on molecular time scales. −2 −16−14−12−10−8−6−4−20x 10 −4 t ρ I e , e ( t ) −2 −16−14−12−10−8−6−4−20x 10 −4 t ρ I e , e ( t ) −2 −16−14−12−10−8−6−4−20x 10 −6 t ρ I e , e ( t ) A. τ r ≪ τ γ ≪ τ γ C. τ γ ≪ τ γ ≪ τ r B. τ γ ≪ τ r ≪ τ γ Fig. 5. Evolution of the imaginary coherences of an underdamped V-system ( ∆ p ¯ γ ≫ ) evaluatedwith aligned transition dipole moments ( p = 1 ). Here γ = 2 . , γ = 10 − and ∆ = 24 . . Threedifferent turn on regimes are shown here. Panels A shows the ultrafast turn on of the field with τ r = 2 × − τ γ while Panels B and C show the intermediate ( τ r = 20 τ γ = 0 . τ γ ) and slow( τ r = 100 τ γ ) turn on regimes respectively. Note the difference in y-axis scales for the coherenceplots. Some explicit cases are discussed in Sec. VII below.
VII. SAMPLE LIGHT HARVESTING CASES
The above results encompass a vast range of possible systems. It is advantageous, there-fore, to focus on some simple cases to emphasize the importance of these results to moleculesof interest in, e.g., light harvesting scenarios. We address two sample questions below, beinggenerous in our requirements for coherences. Note that we assume below that the systemis isolated from an external (e.g. protein) environment, so as to focus solely on relaxationeffects due to the incoherent light. This artificial arrangement is only designed to highlightsome of the timescales associated with the above analysis.22i) Light-induced coherences have been observed experimentally in FMO, PC645 andother light harvesting complexes [1–3]. In these cases ∆ ∼ cm − and γ , due to sponta-neous emission, is on the order of ns − , which places the system in the overdamped region.For the sake of simplicity, this discussion neglects non-radiative relaxation and decay of theexcited states due to the interaction with the phonons, which play an important role inrealistic models of light-harvesting complexes [27].Given our results above we can ask, for example: what turn-on time scales would berequired to produce coherences that are even a modest of the population? Using Eq. (41)shows that | ρ e e | /ρ e i e i = γ/ (∆ p τ r ) . Hence, the turn-on time must be faster than ∼ − ns, clearly far faster than natural turn-on times. Hence, these coherences will not occur innatural light-harvesting systems.(ii) Alternatively, we might ask what coherences (that are a modest of the population)can be generated by a turn-on time of ms, still a relatively fast turn-on time on naturaltime scales.Here, using the same approach, we have | ρ e e | /ρ e i e i = γ/ (∆ p τ r ) . Requiring this ratio tobe a modest shows that states that will display coherences are separated by less than . cm − . Analogously, if we utilize a more realistic turn-on time of 1s, only levels separatedby × − cm − will display coherences. Once again, the results highlight the significanceof the slow turn-on to assessing the (lack of) involvement of coherent phenomena in naturalcases. VIII. CONCLUSION
We have presented a generalization of the Bloch-Redfield master equations to the case oftime-varying radiation fields. They are shown to be of a similar structure to the previouslystudied master equations for stationary fields [6, 8, 18], but with time-dependent incoherentpumping rates r i ( t ) . We explicitly determined the form of these master equations for the classof three-level V-systems and solved them analytically in the weak pumping limit relevant tothe natural incident light (e.g. solar radiation).Following the approach taken in the study of V-systems interacting with stationary fields[8, 18] two limiting cases were considered in detail. The underdamped regime ( ∆ p ≫ ¯ γ )characterized by oscillatory coherences and the overdamped regime ( ¯ γ ≫ ∆ p ) characterized23y quasistationary coherences. In both regimes an inverse relationship between the maximalmagnitude of the coherences and the turn on time max {| ρ e e |} ∝ /τ r in the adiabatic limitof very slow field turn-on was established. This corresponds to a V-system in equilibriumwith the radiation field at all times. In other words, the system is always approximately inthe equilibrium mixture ρ eq = [¯ n ( t ) , ¯ n ( t ) , , T .By contrast, for the very fast turn on of the radiation field, both regimes show dynamicsthat are identical to the sudden turn-on of the radiation field studied in the stationary fieldcase ( τ r → ). This limit occurs when the turn on time is much faster than any of thesystem timescales, so that the system does not evolve under the transient field. Instead itevolves under the steady state field that is reached very quickly.For intermediate turn on times, the dynamics of the system can vary from those observedin the stationary field case but they, in general, reach the same maximal coherence as in thesudden turn on case. One unexpected phenomenon observed was the synchronized decay ofthe coherences where all coherent superpositions decayed at the same time. This differs fromthe naive expectation that coherent superpositions produced at later times would decay laterthan those produced at earlier times. This synchronized decay of coherent superpositionsoccurs due to the suppression of excited state coherences by excited state populations. Whenthe coherences produced at early times decay, they lead to an increased population of theexcited state manifold. This subsequently leads to an increase in the decay rate of thecoherent superpositions, which leads to a further increase in the excited state population.Ultimately, this process leads to the run-away increase of the decay rate of the coherencesat the decay time of the first superpositions prepared by the incident field and hence thesynchronized decay of coherences.These results reveal nontrivial effects of the turn on rate of the incoherent field on thedynamics of the system. Most significantly they suggest that the significant coherencesobserved in the study of the V-system do not survive the slow turn-on of the radiation field.Moreover, in the isolated molecule case, they will not survive for a field with a turn ontime slower than the radiative lifetime of the excited states τ γ i = 1 /γ i for a V-system inthe underdamped limit or slower than the long time scale τ ∆ = 2¯ γ/ ∆ p in the overdampedlimit. This greatly restricts the class of systems that would display significant coherencesfor radiation fields with physical turn-on times.The implication of these results for pulsed laser experiments [2, 3, 35] that display co-24erences in biological molecules is profound. Specifically, they imply that illumination bynatural sunlight, where turn-on times are indeed enormously longer than all other relevantdynamical time scales, can not generate Fano coherences between other than essentiallydegenerate states. Acknowledgements
This work was supported by the US AFOSR through Contract No. FA9550-13-1-0005,and by NSERC.
Appendix A: Generality of Results
To prove the generality of the results for the exponential turn on function Eq. (5) pre-sented in the main text, consider the set, S , of all continuous driving functions, ¯ n ( t ) , suchthat the function is initially zero and evolves to a steady state value, ¯ n , in the long timelimit. That is S = n ¯ n ( t ) : [0 , ∞ ) → R | ¯ n ( t ) ∈ C ; ¯ n (0) = 0; lim t →∞ ¯ n ( t ) = ¯ n o (A1)An element of S in Eq. (A1) can, in general, be written as ¯ n ( t ) = ¯ n − g ( t ) (A2)where g ( t ) ∈ C ([0 , ∞ ) , R ) and C ([0 , ∞ ) , R ) is the set of continuous functions from theinterval [0 , ∞ ) on the real line to R which vanish at infinity.We proceed now to prove that any function g ( t ) ∈ C ([0 , ∞ ) , R ) can be written as a seriesof decaying exponentials on the positive real half-line. This can be done using the Stone-Weierstrass theorem on locally compact spaces [36]. A set of functions, A , on X is said tovanish nowhere if, for any x ∈ X , there exists a function, f ∈ A such that f ( x ) = 0 . It issaid to separate points if ∀ x = y ∈ X there exists a function g ∈ A such that g ( x ) = g ( y ) .Further, C ( X, R ) defines an algebra over R under pointwise addition and multiplication offunctions. Theorem 1. (Stone-Weierstrass)
Suppose X is a locally compact Hausdorff space and A is a subalgebra of C ( X, R ) . Then A is dense in C ( X, R ) if and only if it separates pointsand vanishes nowhere. X = [0 , ∞ ) . This is a closed subset of the locallycompact Hausdorff space R and so is itself a locally compact Hausdorff space. Define A asfollows A = span { h a ( t ) = e − at | a ∈ R + ; t ∈ [0 , ∞ ) } (A3)where R + is the set of positive real numbers. Clearly A defines a vector space over the realnumbers under pointwise addition and scalar multiplication of functions. Furthermore, alldecaying exponentials vanish at infinity so A is contained in C ([0 . ∞ ) , R ) . The product oflinear combinations of decaying exponentials produces another such linear combination ofexponentials, guaranteeing closure of A under pointwise multiplication. Therefore A definesa subalgebra of C ([0 , ∞ ) , R ) . It is trivial to show that A vanishes nowhere and separatespoints on [0 , ∞ ) .Hence, according to this theorem, A is dense in C ([0 , ∞ ) , R ) . By the definition of adense space, any function g ( t ) ∈ C ([0 , ∞ ) , R ) is either in A or is a limit point of A [37].In other words, any function, g ( t ) , vanishing at infinity on the positive real half-line can beexpressed in the following form: g ( t ) = Z ∞ daf ( a ) e − at = − (cid:18)Z ∞ daf ( a )(1 − e − at ) (cid:19) + Z ∞ daf ( a ) (A4)Substituting Eq. (A4) into Eq. (A2), and applying the initial condition ¯ n (0) = 0 yieldsthe constraint ¯ n = R ∞ daf ( a ) . This allows Eq. (A2) to be rewritten as ¯ n ( t ) = Z ∞ daf ( a )(1 − e − at ) (A5)Equation (A5) expresses a general class of driving function as a series of terms each of theform considered in the main text Eq. (5). The integral transform in Eq. (A4) is very similarto a Laplace Transform with the transformed coordinate, a , restricted to the real line ratherthan the complex plane [34]. This yields an intuitive expansion of the time-dependentoccupation number Eq. (A5) in a basis where each basis function ( f α ( t ) = 1 − e − αt ) isassociated with a characteristic turn on time τ α = 1 /α .Using Eq. (A5), the driving vector d ( t ) in Eq. (25c) for an arbitrary driving function is26iven by d ( t ) = γ γ p p γ γ ( t )0 Z ∞ daf ( a )(1 − e − at ) = d Z ∞ daf ( a )(1 − e − at ) (A6)Substituting Eq. (A6) into the general variation of parameters solution Eq. (28) yields thesolution for an arbitrary turn on function in terms of the solutions derived in the text. x ( t ) = Z ∞ daf ( a ) Z t dse A (0) ( t − s ) d (1 − e − at ) = Z ∞ daf ( a ) x a ( t ) (A7)where x a ( t ) is the solution for a turn on function ¯ n a ( t ) = (1 − e − at ) .Equation (A7) applied to the coherences indicates that any coherences observed are aresult of the components with a fast turn on time. [1] V. Tiwari, W. K. Peters, and D. M. Jonas, Proc. Natl. Acad. Sci. USA , 1203 (2013).[2] E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer, and G. D. Scholes, Nature(London) , 644 (2010).[3] G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship,and G. R. Fleming, Nature (London) , 782 (2007).[4] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature (London) , 594 (1999).[5] T. Grinev and P. Brumer, J. Chem. Phys. , 244313 (2015).[6] T. V. Tscherbul and P. Brumer, J. Chem. Phys. , 104107 (2015).[7] A. Chenu, A. M. Brańczyk, G. D. Scholes, and J. Sipe, Phys. Rev. Lett. , 213601 (2015).[8] T. V. Tscherbul and P. Brumer, Phys. Rev. Lett. , 113601 (2014).[9] Z. S. Sadeq and P. Brumer, J. Chem. Phys. , 074104 (2014).[10] J. Olšina, A. G. Dijkstra, C. Wang, and J. Cao, arXiv:1408.5385 [physics, physics:quant-ph](2014), arXiv: 1408.5385.[11] T. V. Tscherbul and P. Brumer, J. Phys. Chem. A , 3100 (2014).[12] P. Brumer and M. Shapiro, Proc. Natl. Acad. Sci. USA , 19575 (2012).[13] L. A. Pachón and P. Brumer, Phys. Chem. Chem. Phys. , 10094 (2012).[14] T. Mančal and L. Valkunas, New J. Phys. , 065044 (2010).
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