A witness for coherent electronic oscillations in ultrafast spectroscopy

We report a conceptually straightforward witness that isolates coherent electronic oscillations from their vibronic counterparts in nonlinear optical spectra of molecular aggregates: Coherent oscillations as a function of waiting time in broadband pump/broadband probe spectra correspond to coherent electronic oscillations. Oscillations in individual peaks of 2D electronic spectra do not necessarily yield this conclusion. Our witness is simpler to implement than quantum process tomography and potentially resolves a long-standing controversy on the character of oscillations in ultrafast spectra of photosynthetic light harvesting systems.

Recently, there has been considerable interest in longlived quantum superpositions of electronic states in photosynthetic molecular aggregates and their potential role in efficient energy transport in biological conditions [1,2].Evidence for such electronic coherences stems from time oscillations in peaks of two-dimensional electronic spectra (2D-ES) which persist for over 600 fs [3][4][5].However, coherences between vibronic levels involving a single electronic state exhibit similar signatures in 2D-ES [4,7,8] and have been shown to nontrivially affect energy transfer [9][10][11].Although there are additional hints that support the interpretation of the oscillations as due to electronic states (beating frequencies and comparison with all-atom simulations [12]), unambiguous tools to experimentally unravel the nature of these oscillations are required.A big step has been the observation that, under weak coupling to vibrations and negligible coherence transfer processes, electronic coherences imply oscillations in off-diagonal peaks of rephasing 2D-ES and in diagonal peaks of their non-rephasing counterparts [13], whereas general vibronic coherences show up as oscillations in any region of either spectra [14].However, the rephasing 2D-ES of the paradigmatic Fenna-Matthews-Olson (FMO) complex exhibits oscillations in both diagonal and off-diagonal peaks, indicating that systems of interest may lie in the regime of strong coupling to vibrations [15] or exhibit vibronic coherences only [8].Techniques of wavepacket reconstruction [7] or quantum process tomography (QPT) [17,18] should clearly provide an answer at a cost of several experiments.Our purpose here is to provide a practical witness for coherent electronic oscillations, which is applicable across different regimes of weak and strong coupling to vibrations.
We illustrate the witness by considering the simplest molecular exciton model, the coupled dimer [17].Its Hamiltonian is given by H 0 (R) = T N +H el (R), where T N is the nuclear kinetic energy, and H el (R) is the electronic Hamiltonian which depends on the nuclei R, H el (R) = mn V mn (R)|mn mn|+J(R)(|10 01|+|01 10|).|mn denotes the electronic state with m, n excitations in the first, second molecules, respectively (m, n ∈ {0, 1}), V mn (R) is the corresponding diabatic potential energy surface, and J(R) is the coupling between site excita-tions.Any pure state |Ψ may be expressed in terms of vibronic states, that is, product states of the electronic (system) and nuclear (bath) degrees of freedom, |Ψ = i a i |e i |N i , for coefficients a i , and {|e i , |N i } electronic and nuclear bases.A reduced electronic description of |Ψ is obtained by performing a trace over the bath, ρ el = Tr nuc (|Ψ Ψ|).We consider light-matter perturbation in the dipole approximation, H pert (s) = −µ • ǫ(r, s), where µ = e=01,10 µ eg |e g| + µ f e |f e| +h.c. is the dipole operator, and ǫ(r, s) = p=P,P ′ [ǫ p (s − t p )e p + c.c.] denotes the pump (P) and probe (P') pulses, with ǫ p (s the Gaussian time-profile.Here, λ, ω p , t p , σ, and e p , are the strength, carrier frequency, center time, width, and polarization of the p-th pulse, respectively.We shall discuss PP' spectra S P P ′ (T ) as a function of T = t P ′ − t P (waiting time) [1], which can be recovered from a 2D-ES by integration along both frequency axes (Supplementary Material [21] sec.I, SI-I).The main result of this article is: In the Condon approximation and the broadband limit (σ → 0), oscillations of S P P ′ (T ) as a function of T correspond to coherent electronic oscillations; in this limit, S P P ′ (T ) may be expressed solely in terms of reduced electronic states ρ el , so oscillations cannot be due exclusively to nuclear dynamics.
The PP' signal may be written as the sum of S SE (T ), S ESA (T ), and S GSB (T ), with separate contributions from stimulated emission (SE), excited state absorption (ESA), and ground state bleach (GSB) [3].If the initial vibrational state is known, each of these terms may be expressed as a suitable wavefunction overlap (SI-I [21]).For example, let the initial wavefunction (before any pulse) be |Ψ 0 (0) = |g |ν Treating the laser pulses pertubatively, the first order wavefunction due to P is ( = 1) |Ψ P (s) = i ´∞ −∞ ds ′ e −iH0(s−s ′ ) {µ • ǫ P (s ′ − t P )}|Ψ 0 (s ′ ) , and the second order wavefunction due to both P and P' is It can be shown that S SE (T ) = Ψ P P ′ (s)|g g|Ψ P P ′ (s) (SI-I [21] and [6,23]).
Preliminary example.-Wewill develop some intuition through an illustration, in which we focus on S SE (T ).
Consider the case where the surfaces of the singly-excited diabatic states have the same shape, and |10 in that they are delocalized due to J(0).Note that both |α and |β are coupled in the same way to the vibrational bath, and hence they form a decoherence-free subspace [25].The first order wavefunction "right before" the probe pulse may be expanded as are eigenstates of the molecular Hamiltonian H 0 (R), the excitons are the adiabatic electronic states, there is no dissipation in the electronic system, and the values |c i,m (T )| 2 are constants as a function of T , depending only on the details of P [28].The wavefunction "right after" the probe at time T is, in the Condon approximation, given by, n , where ǫp (ω) = λe −(ω−ωp) 2 σ 2 /2 is the Fourier transform of pulse p at frequency ω.This expression can be interpreted as a wavepacket in the ground state created when the probe couples the vibrational levels of the singly-excited states to the vibrational levels of the ground state via the electric dipole moment, where the amplitudes in the various vibrational levels depends on the probe's electric field at the given transition energy and the Condon overlap.Computing the norm of the resulting wavepacket, which corresponds to sums of interferences between vibrational states of the same and different excitonic states, respectively, projecting onto the same vibrational state in the ground state.Note that S SE (T ) can be written as a linear combination of elements of the full vibronic density matrix ρ(T ) correspond to vibronic coherences and oscillate at the difference frequency between the |i |m and the |j |m ′ states.When we consider the broadband (bb) limit of Eq. ( 1) , where ǫ(ω) = λ for all the ω values of interest, Crucially, Eq. ( 2) is a linear combination of elements of ρ el (T ) as opposed to the full vibronic space.In fact, the terms for i = j correspond to electronic populations and, due to the absence of electronic decoherence in this example, stay constant with respect to T .The term α|ρ el (T )|β = i c i,α (T )c * i,β (T ) corresponds to an electronic coherence between |α and |β , and shows oscillations at the single frequency ω αβ as a function of T .Hence, coherent oscillations in S bb SE (T ) are a witness for coherent electronic dynamics.Remarkably, in the additional limit where one of the excitons is dark (e.g., µ βg = 0), we have a monomer instead of a dimer, and S bb SE (T ) is a constant even in the case of large Condon displacements, where there is large vibrational motion between pump and probe.This observation for the monomer has been previously reported by Yan and Mukamel [26].
The results above can be interpreted as follows.In the Condon approximation, the probe couples only to the electronic dipole, so in the broadband limit it acts uniformly across every transition energy, and hence across every nuclear configuration within a particular electronic state.In general, S SE (T ) is a sum of multiple interferences among portions of wavepackets at different electronic and nuclear configurations.In S bb SE (T ), the probe opens only two interference pathways (just as in the double-slit experiment), via emission from the |α or the |β state, insensitive to vibrational dynamics, providing a witness for coherent electronic oscillations.
General case.-Theexample above readily generalizes to include effects of initial thermalized states of the bath, ESA and GSB contributions, and non-adiabatic effects.In the limit of broadband P (SI-II and III, [21]) and P ′ , each of the contributions to S bb P P ′ (T ) are (SI-II, [21]), , where the process matrix χ(T ) is given by, χ ijqp (T ) = Tr nuc { i|e −iH0T (|q p| ⊗ ρ B (0)) e iH0T |j } [17,18], and it is easy to see that S bb P P ′ (T ) is invariant under change of electronic basis within the singly-exicted states.Here, n | is the initial thermal vibrational ensemble in the ground electronic state.χ(T ) describes the evolution of the electronic system, assuming that the vibrational system begins in ρ B (0).If the initial state of the bath can be prepared at ρ B (0) regardless of the electronic state, as in the impulsive limit, an integrated equation of motion can be written as ρ ij (T ) = ijqp χ ijqp (T )ρ qp (0).As in the preliminary example, S bb P P ′ (T ) is a linear combination of entries of reduced states ρ el (T ), so oscillations in it are a manifestation of electronic oscillations, justifying the witness.
Given an electronic basis, any element χ ijqp (T ) can in principle exhibit oscillations.For a large variety of systems, it is however, possible to associate the largest amplitude oscillations of χ(T ) to electronic coherences in some basis.In the preliminary example, the lack of dissipation implies that χ ijqp (T ) = δ iq δ jp e −iωqpT , so the only possible oscillatory contribution to S P P ′ (T ) corresponds to χ αβαβ (T ) = χ * βαβα (T ) (excitonic coherence).In the non-adiabatic case where V 01 (R) = V 10 (R) + c, each electronic state couples differently to the vibrational modes.However, in the limit of weak system-bath coupling, the vibronic states |e |ν (e) j are still the correct eigenstates of H 0 (R) up to zeroth order in the coupling, so any oscillations in the signal will still be dominated by excitonic coherences.Finally, for intermediate and strong system-bath coupling together with a fast bath decorrelation timescale, a polaron transformation defines an electronic basis {|g , |α , | β , |f } that diagonalizes a zeroth-order electronic Hamiltonian weakly coupled to a renormalized bath ( [12] and SI-IV [21]).In this case, the highest amplitude oscillations in its S bb P P ′ (T ) would correspond to electronic coherences χ α β α β (T ) = χ * β α β α(T ).For more general aggregates, if this were an issue of interest, a partial QPT could be designed to determine the value of specific terms of χ(T ) [17,18].Numerical examples.-Wehave performed simulations for a monomer, a dimer which exhibits coherent electronic oscillations, and an incoherent dimer, where each singly-excited site is coupled to a single vibrational mode.These three examples illustrate the value of the witness (Fig. 1), as all three have oscillatory 2D-ES (Fig. 2), but the monomer and incoherent dimer do not have coherent electronic oscillations.The witness correctly shows that only the coherent dimer has a positive witness.The simulations include inhomogeneous broadening (ensembles of 500 molecules with Gaussian site disorder of standard deviation 40 cm −1 and, for the dimers, site energy correlation 0.8), thermal averaging of initial vibrational states according to a Boltzmann distribution at 273 K, isotropic averaging, and explicit inclusion of pulses with the dynamics.Roughly, there are two energy scales to consider, an average coupling J and a reorganization energy λ, in which case the impulsive limit is set by 1 σ ≫ max(J, λ).For these simulations, the pulses are within the FWHM=10-20 fs range, and cover the entire absorption spectra, respectively (SI-V, [21]).Fig. 1 shows S bb P P ′ (T ) zzzz , the witness averaged at the collinear pulse setting zzzz, for about 900 fs (top).We can associate the witness oscillations to oscillations of elements in χ(T ).We show a few representative elements of this matrix (bottom).Fig. 2 presents snapshots of the rephasing 2D-ES, S(ω τ , T, ω t ) zzzz , for a sampling of waiting times T between 71.6 and 270.6 fs (left), indicating that vibronic coherences manifest as diagonal and cross-peak oscillations [29].Notice that due to strong coupling to vibrations, the coherent dimer also exhibits oscillations in the diagonal peaks, implying the inapplicability of previous measures for this case [13,14].As another illustration, the integrated signal under the cross-peaks encircled in black is in the right plots.Note that the largest amplitude oscillations are in the monomer, which cannot have coherent electronic oscillations, showing that oscillations in peaks in the 2D-ES do not directly translate into coherent electronic dynamics, and hence are not the correct witness.
The witness is positive if, once the dc background is subtracted from S bb P P ′ (T ), there are oscillations with amplitude proportional to µ 4 , where µ is some estimate of an electronic transition dipole moment.If spurious oscillations due to finite pulse-duration are suspected, a more quantitative confirmation is the following: (a) Collect traces S bb P P (T ) at several pulse widths σ, all roughly in the broadband domain.(b) Fourier transform the data: 2π ´∞ 0 dT e iωT S bb P P (T ).(c) Locate non-zero frequencies of Sbb P P (ω T ) corresponding to oscillations between discrete states (ignore the dc component).For each of these frequencies ω T , plot Sbb P P (ω T ) as a function of σ, and linearly extrapolate to σ → 0. If the obtained intercepts are at zero within noise levels, the witness is negative .SI-II [21] displays an analytical expression for the O(σ) correction of S bb P P (T ), providing a theoretical basis for this procedure.
Although the theory has been detailed here for a dimer, the witness is applicable to larger aggregates.In the case of FMO, due to spectral congestion, it might be fruitful to focus on pairs of exciton states at a time, for instance, the first and the third exciton states, either via direct PP' measurements that cover these transitions exclusively, or alternatively, integrating windows of broadband 2D-ES corresponding to these two states only, assuming that relaxation processes do not occur outside of this spectral window.
We dedicate this letter to the late Bob Silbey who at different stages introduced J.Y.Z. to theoretical chemistry and mentored A.A.G in the field.We thank DARPA Award No. N66001-10-4060 and EFRC-DOE Award No. de-sc0001088.where |ζ corresponds to the singly excited state manifold.
• The tensor product basis {|m, ν (g) n }, where {|m } denotes electronic states in an arbitrary electronic basis (for instance, the excitonic one), and {|ν , and |f, ν Using both bases, the process matrix χ(T ) affords a compact representation, Our goal is to write S P P ′ (T ) for arbitrary bandwidth in a similar style, so that in the broadband limit, we can identify it as a linear combinations of elements of χ(T ), hence proving Eqs.(S22)-(S24) in the article.We start by rewriting Eqs.(S2)-(S7) in the vibronic bases: S n ′ e iω gn ′ ,gn T * SE and ESA processes only contribute to corrections of O(σ 2 ) via the Gaussian spectral profile of the pulses.

REQUIREMENT OF BROADBAND PUMP P
Although the conclusions of the premilinary example in the article hold even in the case of narrowband P', we also require broad bandwidth for P for two reasons: 1. Non-stationary GSB contributions.Eqs.(S5) and (S7) show that in the limit of broadband P, this pulse promotes a wavepacket to the excited states and immediately back down to |g , yielding a wavefunction |Ψ P P (t) that is proportional to the original |Ψ 0 (t P ) before any pulse (also see Eq. S24).In this limit, as emphasized in the previous section, S GSB (T ) is a constant background as a function of T , giving the opportunity to identify S P P ′ (T ) as a probe for singly-excited state dynamics.Under a narrowband P, this no longer holds, as shown by Eq. (S21), which depends on T in general.In this case, |Ψ P P (t) will be a non-stationary wavepacket in the ground electronic surface, which will manifest as time-evolving overlaps both in Ψ P P (s)|g g|Ψ P ′ P ′ (s) and in Ψ 0 (s)|g g|Ψ P P P ′ P ′ (s) (see Eq. (S10)).

Figure 1 :
Figure 1: (Top) Broadband PP' spectra as a function of waiting time T as a witness for coherent electronic oscillations.The small oscillations in (a) and (c) are due to finite pulse durations.(Bottom) The witness is a linear combination of elements of the process matrix χ(T ).Traces of a few representative elements of χ(T ) are displayed.
(g)n } refers to vibrational eigenstates of the ground vibrational Hamiltonian, H vib,00 (R) = T N + V 00 (R).Note that we can always write states in the vibronic eigenbasis in terms of the second one: |g, ν (g) n stays the same, |ζ = mn m, ν (g) n |ζ |m, ν (g) n