Molecular decision trees realized by ultrafastelectronic spectroscopyBarbara Frescha, Dawit Hilufb, Elisabetta Collinic, R. D. Levineb,d,e,f,1, and F. Remaclea,b

aDepartment of Chemistry, University of Liège, B4000 Liège, Belgium; bThe Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel;cDepartment of Chemical Sciences, University of Padova, 35131 Padova, Italy; and dCrump Institute for Molecular Imaging and eDepartment of Molecular andMedical Pharmacology, David Geffen School of Medicine, and fDepartment of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095

Contributed by R. D. Levine, August 9, 2013 (sent for review June 11, 2013)

The outcome of a light–matter interaction depends on both thestate of matter and the state of light. It is thus a natural setting forimplementing bilinear classical logic. A description of the state ofa time-varying system requires measuring an (ideally complete) setof time-dependent observables. Typically, this is prohibitive, but inweak-field spectroscopy we can move toward this goal becauseonly a finite number of levels are accessible. Recent progress in non-linear spectroscopies means that nontrivial measurements can beimplemented and thereby give rise to interesting logic schemeswhere the outputs are functions of the observables. Lie algebraoffers a natural tool for generating the outcome of the bilinearlight–matter interaction.We showhow to synthesize these ideas byexplicitly discussing three-photon spectroscopy of a bichromophoricmolecule for which there are four accessible states. Switching logicwould use the on–off occupancies of these four states as outcomes.Here, we explore the use of all 16 observables that define the time-evolving state of the bichromophoric system. The bilinear laser–system interactionwith the three pulses of the setup of a 2D photonecho spectroscopy experiment can be used to generate a rich paral-lel logic that corresponds to the implementation of a moleculardecision tree. Our simulations allow relaxation by weak couplingto the environment, which adds to the complexity of the logicoperations.

finite-state machines | Shannon decomposition | parallelmultivalued logic | quaternary logic

The need to further reduce the physical dimensions of a com-puting node is well recognized (1–4). A direct way to go below a

nanometer scale is to use a molecule or an artificial atom—aquantum dot—as a switch (5–11). Molecules can also respond inmore interesting ways than switching. So for some time we havefollowed a program of seeking to implement an entire logic circuiton an atom or molecule and to concatenating such units. To do so,we used intramolecular dynamics resulting from the responseof a molecule to a perturbation—the inputs to the computation—that can be electrical (12) or optical (13) or chemical (14), etc. Inprinciple, such an approach can implement finite-state logic(15) because typically the response of a molecule depends on itsinitial state as well as on the applied perturbation.In finite-state machines, the execution of the logic relies on

transitions between states. The operation is inherently parallel(15). A simple situation is when a molecule relaxes after beingperturbed by an optical (16, 17) or an electrical pulse (18, 19). Inoptical molecular implementations, several states can be simul-taneously addressed, which leads to massively parallel linear finite-state machines (16, 17). The other mode of operation is to provideinputs at each machine cycle. If the molecule is switched betweentwo states, the ability to encode a dependence on the initial statemeans that one can implement memristor logic (20). More elab-orated memory integrated units like set–reset machines can alsobe implemented optically using a stimulated Raman adiabaticpassage excitation pulse (21, 22) or can be based on molecularredox processes (23–25). Here, we propose the realization ofa molecular tree as a logic machine that accepts several inputs and

processes them by performing bilinear operations that are de-pendent on the input and on the state of the machine.One interpretation of a decision tree is that it provides a

graphical representation of the truth table that defines a digitalfunction. As an example, we show in Fig. 1A the truth table of theexclusive-OR (XOR) operation between two Boolean variables.The XOR function is an important two-input gate that performsthe addition modulo two (denoted by the symbol ⊕) of the twoBoolean variables. Its truth table provides the value of the functionfor each of the four combinations of inputs. The truth table canalso be represented as a rooted binary tree (Fig. 1B) composed ofthree levels. Each nonterminal node (denoted by S) splits theXOR function by assigning a precise value to one of the inputvariable: starting from the root node, if x1 is true (i.e., x1 = 0), thenwe follow the tree down to the left to assign the value to the nextinput x2; otherwise, x1 is true (i.e., x1 = 1), and we proceed downto the right. The second level performs an analog splitting ac-cording to the value of x2, thus generating four leaves (terminalnodes) labeled by the result of the XOR operation on the specificpairs of inputs corresponding to each path.We here propose and illustrate molecular realizations of decision

trees based on nonlinear spectroscopy. Among these spectroscopictechniques, four-wave mixing, in particular, two-dimensional pho-ton echo (2D-PE), is a powerful tool that provides information ondissipation, dephasing, solvation, and (electronic or vibrational)interstate coupling mechanisms in various multistate materialsystems (26–30). When the system dynamics is characterized interms of the time-dependent density matrix, four-wave–mixingspectroscopy monitors populations and coherences in real time.We take advantage of the richness of the information provided

by 2D-PE (31) for implementing molecular decision trees. Anadvantage of weak-field spectroscopy is that the dynamics reflectsthe detailed structure of the chromophoric molecules that are

Significance

One possible way to reduce the physical dimensions of a com-puting node is to instruct a molecule to evaluate a complicatedlogic function. This is even more so if several such functions areprocessed in parallel. The interaction between light andmatter isa suitable route because it is bilinear, depending on both theproperties of the laser and of themolecule; the outcomedependson the initial state of the molecule and there can be more thanone distinct path leading to the readout signal. Two-dimensionalphoton spectroscopy is shown to have four paths originating fromeach interaction, thereby enabling, as shown in SI Text, quaternarylogic. In the main text, we discuss the simpler case of binary logic.

Author contributions: B.F., R.D.L., and F.R. designed research; B.F., D.H., R.D.L., and F.R.performed research; B.F., D.H., E.C., R.D.L., and F.R. analyzed data; and B.F., E.C., R.D.L.,and F.R. wrote the paper.

The authors declare no conflict of interest.

See Commentary on page 17167.1To whom correspondence should be addressed. E-mail: rafi@fh.huji.ac.il.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1314978110/-/DCSupplemental.

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used because the laser field as used is not strong enough to distortthe molecule beyond recognition. Another advantage is that wecan trace the different paths that connect the initial state of themolecule to the readout. This enables us to relate each path toa path down a decision tree (DT) and, becausemore than one pathconnects the initial state of the molecule to the readouts, to im-plement computations in parallel. In the language of opticalspectroscopy, we can say that each such computational path cor-responds uniquely to a Feynman diagram as discussed for 2D-PEspectroscopy, for example in refs. 32 and 33 (Fig. S1). Ideally, wewould like to resolve the separate paths in a given experiment. Thisideal is not possible with current implementation of 2D electronicspectroscopy, which limits the range of functions that can becomputed. The number of paths that can be simultaneously re-solved can be increased using polarization (34) or 3D Fouriertransform spectroscopy (35).For the purpose of logic, we find it useful to describe the time

evolution in the so-called Heisenberg picture. In this way ofthinking, an elementary act of perturbation of the system is rep-resented as a bilinear operation. It is bilinear because the interac-tion consists of taking the commutator of the perturbation withthe density matrix of the system. So the operation is linear in thestate of the system and linear in the perturbation. The applicationof the commutation relation at each photon–molecule interactionprocess in a 2D-PE experiment generates a commutator tree asshown in Fig. 2.For a weak optical perturbation, it is realistic to allow the system

to have only a finite number of states that can be accessed by thephotons. In third-order nonlinear spectroscopy of a bichromo-phoric molecule, there will be four states. This suggests that we usea Lie algebraic approach based on the group SU (4), which arenorm-conserving, hence the S, unitary transformations among fourstates. One can use such a group to implement exact dynamics(36–38). To mimic a weak-field experiment such as 2D-PE, weonly need to implement perturbation theory to third order, thirdcorresponding to three photon processes. The structure of the Liealgebra and the nature of the inputs is then imprinted on thestructure of the dynamics and thus on the logic. In particular,we show how a decision tree is generated by the evolution of anexcitonic-dimer system probed by 2D-PE experiment.The first section explains the Lie algebraic background of the

present work with an emphasis on the analogy between the bi-linear operations of the logic and the bilinear operation (i.e., thecommutator) of the algebra that is implied by the quantum me-chanical Heisenberg equation of motion. In the second section, werecast the evolution of the density matrix during the PE sequenceof laser pulses and the corresponding detected signal in a form thatmakes the connection with the implemented logic more trans-parent. In the third section, we give specific examples of theimplementation of decision trees in the 2D-PE spectra calculated

for a bichromophoric dimer system. Concluding remarks andperspectives are given in the last section.

The Lie Algebraic Description of Photon Echo SpectroscopyThe 2D-PE spectroscopy of coupled chromophores has been pre-viously modeled at different levels of sophistication (32, 33, 39, 40).We consider here the model of an excitonically coupled hetero-dimer and describe its coupling to environment within the Redfieldapproach. Our technical purpose is to calculate the evolution of thedensity matrix during the pulse sequence defining the PE experi-ment and the resulting signal. This is summarized in the next sec-tion by emphasizing the underlying Lie algebraic structure, whilefull details are given in SI Text, Section 1 and Section 2.The first step is to introduce the Hamiltonian describing the two

interacting chromophores and their coupling with the externalfield showing that they can be written as a linear combination ofelements of the SU (4) Lie algebra.We note that it thereby follows(36–38) that the time evolution of any observable relevant to thesystem can be so described in closed form. In particular, theevolution of the density matrix of the bichromophore can berepresented as a linear combination of the elements of the al-gebra. This structure of the density matrix is preserved also whenwe include relaxation described by a Redfield equation in theBloch limit.The dimer system is composed by two interacting chromophoric

units that we describe in the excitonic representation, whose levelstructure is shown in Fig. 2B. The electronic Hamiltonian in thesite basis is standard (Eqs. S10–S13 in SI Text). It includes twostates for each chromophore whose energies are corrected for thesolvent contribution and the coulombic interaction that couplesthe excited state of each chromophore and leads to exciton basis.In the exciton basis, the four states shown in Fig. 2B give rise

to 16 operators, Eij = jiih jj, where i; j= 0; 1; 2; 3, that form a setthat is closed under commutation:

�Eij;Ekl

�= δjkEil − δliEjk: [1]

This closure is central for our purpose because we will draw ananalogy between the commutation of two operators and thebilinear operations of the logic. In the excitonic basis, the set of16 operators Eij is complete, meaning that any operator of in-terest can be written as a linear combination of these elemen-tary operators. The excitonic Hamiltonian, Hel, is clearly of the

Fig. 1. Mapping of the truth table of the XOR function into a decision tree.(A) Truth table of the XOR binary function. (B) The corresponding decisiontree; see Fig. 2 below for the correspondence to a commutator tree.

Fig. 2. Mapping a 2D-PE probing of a bichromophoric system onto a de-cision tree. (A) Time ordering of the pulse sequence. (B) Level structure ofthe electronic states of the heterodimer in the excitonic basis. (C) Commu-tator tree resulting from application of the dipole operator to the densitymatrix at third order. (D) The Feynman diagram corresponding to the pathmarked in red in the tree of C.

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required form. The full Hamiltonian is HðtÞ= Hel − μ · ξðtÞ withξðtÞ= P3

n= 1ðEnðtÞexpð−iϕn − iωnt+ iknrÞ+ c:c:Þ being the exter-nal electric field. Each pulse is characterized by its frequencyωn, wavevector kn, and an envelope, EnðtÞ, centered at timet= τ1; τ2; τ3, respectively. The field–matter interaction is de-scribed by the action of the transition dipole operator, μ, whichin term of the basis operators reads as follows:

μ=Xi;j

μijEij; [2]

where the μi j’s are the transition dipoles. The key technical pointis that throughout the time evolution the density matrix, ρ, of thesystem is also a linear combination of the basis operators:

ρ=Xi; j

ρi jðtÞEij; [3]

with time-dependent expansion coefficients. The proof is a directconsequence of the closure property (Eq. 1). The expansionabove for the density matrix is clearly valid at the initial timewhen ρ= j0ih0j. To propagate the density matrix forward in time,we need to commute it with the Hamiltonian, for example:

�μ; ρ

�=

Xijkl

μijρkl�δjkEil − δliEjk

�; [4]

where ρkl and μij are the coefficients of the operators Ekl and Eijin the density matrix and the dipole operator, Eqs. 3 and 2. Thecomplete equation of motion is as follows:

∂∂tρðtÞ= −

iZ

hH; ρðtÞ

i−1ZRρðtÞ; [5]

where the first term on the right is the quantum mechanicalevolution of the chromophore, whereas the Redfield tensor, R,describes the dissipative dynamics induced by the coupling withthe environment. The tetradic relaxation matrix, Rij;kl, is the trans-fer rate from ρkl to ρij and its explicit form is given in SI Text,Section 2. In the so-called Bloch limit, the only nonzero elementsof the Redfield tensor are population decay from state i, Rii;ii,population transfer from i to j (i ≠ j), Rii; jj, and coherence dephas-ing Rij;ij. It is clear by inspection that including the Redfield re-laxation does not violate the closure meaning that the rate ofchange of the density matrix, ∂ρ=∂t, remains a linear combinationof the elementary operators, Eij.

Evolution of the Density Matrix and the 2D-PE SignalDuring the excitation sequence (Fig. 2A), the system interacts withthree ultrashort laser pulses. A polarization is created, whichgenerates the signal that can be probed in several (phase-matching)directions. Because the interactions are weak, we use the standardperturbational approach based on the expansion of the densitymatrix in orders of the electric field, leading to the following ex-pression for the total third-order polarization (33):

Pð3ÞðtÞ=Trnμρð3ÞðtÞ

o;

μ being the transition dipole operator (Eq. 2). The final spec-trum is equivalent to a double Fourier transform of the time-dependent signal:

SkSðω1;T;ω2Þ=ZZ

dtdτ expð−iω1τÞ expðiω2tÞiPð3ÞkSðτ;T; tÞ; [6]

where the pulse delay times τ andT are defined in Fig. 2A. With theconvention used in Eq. 6, the rephasing signal emitted in the di-rection kS = − k1 + k2 + k3 appears at positive frequencies ðω1;ω2Þ.

To calculate the time evolution of the third-order densitymatrix, which defines the signal through Eq. 6, we apply the ro-tating wave approximation (RWA) and assume the impulsive limitfor short laser pulses. In this case, the time ordering of theinteractions is well defined and the (positive) propagation timesτ;T; t start at the midpoint for the corresponding δ -pulse. The threeinteractions with the fields result in three commutation operationsaccording to Eq. 4, whereas the time evolution operator in theabsence of the external field is defined by the evolution Eq. 5 asUðtÞ= expð−iHt=Z− Rt=ZÞ. In the Heisenberg picture for the di-pole operator (i.e., μðtÞ= UðtÞ†μUðtÞ), the third-order densitymatrix can be written as a function of the propagation times asfollows:

ρð3Þðτ;T; tÞ= ði=ZÞ3Uðτ+T + tÞ3hμðτ+TÞ�μðτÞ�μð0Þ; ρð0Þ��iU†ðτ+T + tÞ: [7]

Switching to a Liouville space notation (33), Eq. 7 takes a morecompact form as follows:

ρð3Þðt;T; τÞ= GðtÞνGðTÞνGðτÞνρð0Þ=Xij

cjiðt;T; τÞEij; [8]

with ν= ½μ; � being the operator generating the action of the tran-sition dipole. The time propagation between the laser pulses isgenerated by the operator, GðtÞ, G= expð−ðit=ZÞ½H;  �− ðt=ZÞRÞ.Eq. 8 gives an intuitive understanding of the dynamics of thedensity matrix during the PE experiment: starting from the initialstate ρð0Þ the system experiences instant interactions with theexternal field and subsequent free evolution in the time intervalsbetween them. The “history” of each element of the density matrixduring the experiment is imprinted in the time-dependent expan-sion coefficients cjiðt;T; τÞ of the rhs of Eq. 8. It is understood thatEqs. 7 and 8 are valid in the impulsive limit of the pulse–matterinteraction; beyond such approximation a threefold convolutionwith the pulse profiles has to be performed (SI Text, Section 1).The structure of the transition dipole matrix reflects the allowed

transitions showed in the scheme of the energetic levels given inFig. 2. Because at the initial time, t =0, before any interaction, thesystem is in its stationary ground state, i.e., ρð0Þ=E00, the firstinteraction excites the one exciton transitions (i.e., E0j, j = 1, 2)(Fig. 2C). According to Eq. 5, the time evolution during τ is adamped oscillation with the exciton frequencies ±ωα;β = ðe1;2 −e0Þ=Z and a decoherence rate determined by the correspondingelement of the Redfield matrix. The second interaction transformsthese coherences in population states (Ejj, j = 0, 1, 2), or electroniccoherences (E12, E21). The transitions between the ground and thedouble exciton states (E03, E30) do not survive the RWA and thusdo not contribute to the observed spectrum. The evolution duringT includes the decay of the electronic coherences and the pop-ulation transfer between the two excitonic states promoted by thecoupling with the bath (FRET). Finally, the third interactionbrings back the system into coherence states (E0j, Ej0, E3j, Ej3, j =1, 2), which evolve during the detection time, t, generating theemitted signal. Because each pulse excites all of the allowedtransitions in its spectral bandwidth, each specific sequence ofthe states of the system driven by a specific sequence of inter-actions with the three fields constitutes a particular path. In Fig. 2,for example, we highlighted one of the possible paths by selecting aspecific sequence of transitions among the elements Eij of thedensity matrix. The possible paths deriving from the evolutionof the density matrix during the PE sequence are more commonlyrepresented as double-sided Feynman diagrams (33). The Feyn-man diagram corresponding to the highlighted sequence of in-teractions is shown in Fig. 2D. In the language of pump-probe, itcorresponds to the ground-state bleaching contribution to the di-agonal peak (ωβ, ωβ) of the rephasing spectrum. For a dimer

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system, all of the paths contributing to the PE spectrum have beenspecified and analyzed previously (see, e.g., refs. 39 and 41) andtheir corresponding Feynman diagrams are reported in Fig. S1.In the time evolution of the density matrix, following the com-

mutation relation (4) and the structure of the dipole matrix in theexciton basis, each interaction with the field connects one elementEij with four other elements Ekl, defining the commutator tree.The complete commutator tree, only sketched in Fig. 2C, is shownin Fig. S2. The number of connections induced by each interactionis determined by the structure of the dipole operator for thebichromophoric model.

Logic Decision TreesThe simultaneous occurrence of the different paths contributing tothe PE signal is appealing because it represents a realization ofparallel logic operation at the molecular level. To build a bridgebetween the physical evolution of density matrix during the 2D-PEexperiment and the classical logic of a decision tree, we first in-troduce some basic concepts that generalize the example discussedin Fig. 1. A switching (Boolean) function of n variables is a mapð0; 1Þn → ð0; 1Þ. It is completely characterized by its truth table,which lists the value of the function for each of the 2n possiblecombination of inputs. Any switching function can be decomposedaccording to Shannon decomposition rule (42) as follows:

F = xi f0 ⊕ xif1; [9]

where f0 =Fðxi = 0Þ; f1 =Fðxi = 1Þ are known as the negative andthe positive Shannon cofactor, respectively. By applying the Shan-non expansion recursively to a Boolean logic function, we canrepresent the function by a binary decision tree (BDT). Fig. 1above shows the Shannon tree corresponding to the two variablefunction XOR. If we label each edge with a literal of a variable asin Fig. 1, the product of the literals from the root node to a ter-minal node along a specific path represents a minterm. For ex-ample, the leftmost path of the diagram in Fig. 1 represents theproduct x1x2. The tree is thus equivalent to the minterm expansionof the function F; for the XOR function of Fig. 1, we have thefollowing:

F = f00x1x2 ⊕ f01x1x2 ⊕ f10x1x2 ⊕ f11x1x2 = x1x2 ⊕ x1x2; [10]

where the second equality gives the sum-of-product (SOP) formof the function F in the XOR representation. Notice that thefunction and its minterm expansion are unchanged if the “exclu-sive or” is replaced by the OR operation. This is because twodifferent minterms are never both true for the same input assign-ment. Generalized Shannon expansion and SOP forms for multi-valued logic function have also been developed (43–46). For thesake of simplicity, we will first give an example of connectionbetween the physical measurement in 2D-PE and the decompo-sition of a logic function within the simpler framework of the two-valued Boolean logic. However, the complexity of the physical

signal reflected in the commutator tree that represents the evo-lution of the full density matrix (Fig. S2) invites an analysis interms of multivalued logic. We then discuss an example of multi-valued logic by implementing a decision diagram for an integerfunction. A different way of exploiting the possibility of encodingmultivalued variable is given in SI Text, Section 4, and Fig. S3 withthe realization of a quaternary decision tree.From the analysis of the dynamical evolution of the density

matrix during the interaction with the three laser pulses presentedin the previous section (Fig. 2), each interaction with the fieldtriggers a branching of the path followed by the system accordingto the structure of the dipole operator. Let us now map thecomponents of the 2D-PE signal derived in the previous sectioninto a BDT logic data structure. To this end, we consider a subsetof all of the possible paths from the ground state of the dimer,ρð0Þ, E00 = 1, to the final coherence state, which generates thesignal inω2. Different subsets can be extracted. Here, we choose asexample the subset of eight different paths shown in Fig. 3A in theform of a decision tree. These paths contribute to the rephasingsignal emitted by the dimer system in the phase-matching directionks = − k1 + k2 + k3. In Fig. 3A, each path starting from the rootnode and ending in a leaf node of the tree corresponds to aFeynman diagram. The notation of the nodes follows from thecommutator tree as shown in Fig. 2. The notation used for eachpath is given in Fig. S1.The first level of the tree, corresponding to the assignment of x1,

can be translated to the proposition “the interacting photon hasenergy Zωβ.” In the path leading from E00 to E02, x1 is realized,whereas in the path leading from E00 to E01, not(x1), x1 is realized,because the transition corresponds to the absorption of Zωα. Thenthe system evolves during τ until the second interaction occurs,which assigns in a similar way the value of x2. The interaction withthe second pulse drives the system in an excited population state(E11 or E22) or in a coherent superposition of exciton states (E12and E21). During the second evolution period, corresponding tothe evolution during the population time, T, dephasing and pop-ulation transfer occur until the system interacts with the third laserpulse. The assignment of x3 corresponds to the proposition “theinteracting photon is adsorbed by the system.” The branches forwhich x3 is realized correspond to the contribution of the excited-state absorption Feynman diagrams (Rb and Rg), whereas thebranches in which x3 is realized correspond to the stimulatedemission contributions (Ra and Rh). The Feynman diagramscorresponding to the ground state bleach (Rc andRd) are not usedin the mapping of the logic function.Each Feynman path contributing to the signal plays the role of

a minterm in the logical plane. The next step is to “read” thecoefficients of the minterm decomposition from the measuredspectrum. In the Shannon decomposition (Eq. 10), these coef-ficients are simply the Boolean value of the function for the cor-responding input string.The spectroscopic observables are not directly the Feynman

paths but rather the intensities of the four peaks, C12, C21, D1,

Fig. 3. (A) The paths that corresponds to the de-cision tree of a function of three Boolean variables,FT(x1, x2, x3), where T is the population time. Theassignment of the Boolean variable at each layer isrealized through the interaction of the laser pulsewith the dimer and corresponds to the action of thetransition dipole superoperator ντi for i = 1, 2, 3. Thecoefficients of the minterm decomposition are readin the intensities of the spectral peaks (Right Inset).The peaks to which each path contributes are listedin the leaf nodes. (B) Interpretation of the decisiontree as a three-cycle nonlinear finite-state machine.

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and D2, in the 2D spectrum shown schematically in the Inset ofFig. 3A. Because we use classical Boolean logic, where the var-iables values are 0 or 1, we use for assigning the output the powerspectrum in the phase-matching direction ks = − k1 + k2 + k3. Allof the paths are included in the computed spectrum so that itcorresponds to an experimental measurement. The four in-tensities computed for a given value of the relaxation time T aremapped to a Boolean value that is equal to 1 when the intensity ishigher than a threshold value and 0 otherwise. The spectral po-sition to which each path contributes is reported in Fig. 3A. Allof the paths contributing to the same spectral position are as-signed the same Boolean value, depending if the peak is high orlow for the relaxation time, T, considered. This logic assignment isdictated by the fact that, in currently performed 2D-PE experi-ments, one cannot resolve each Feynman path individually. Itrestricts the class of function of three Boolean variables that canbe implemented to functions for which two set of three mintermshave the same value: Fðx1; x2; x3Þ, Fðx1; x2; x3Þ, and Fðx1; x2; x3Þ,which contribute to the spectral position C12 and Fðx1; x2; x3Þ,Fðx1; x2; x3Þ, and Fðx1; x2; x3Þ, which correspond to the spectralposition C21. See SI Text, Section 3, for more details. Advances inexperimental schemes toward the unraveling of the differentpathways, as e.g., the use of polarization control (34) or 3Dspectra (35), could lead in the future to the capability of repre-senting all of the 256 functions of three Boolean variables.The Shannon decomposition (Eq. 9) implemented at each cycle,

is bilinear in the variable and the cofactor of the function. Physi-cally, the decomposition is implemented by the commutator,v= ½μ;� (Eq. 4), which is bilinear in the states of the machine, Eij,and the dipole operator. The implementation of the decision treethat we propose above can therefore be interpreted as a nonlinearthree-cycle finite-state machine, as schematically shown in Fig. 3B.The machine receives two inputs, xi and its negation xi, at eachcycle i = 1, 2, 3. In the first step, the logic unit performs thecofactoring of the function FTðx1; x2; x3Þ with respect to the firstinput, x1, generating the two new functions, f0ðx2; x3Þ=Fðx1 =0; x2; x3Þ and f1ðx2; x3Þ=Fðx1 = 1; x2; x3Þ. This is a bilinear logicoperation between the variable and the functions f0 and f1 of thetwo remaining variables: FTðx1; x2; x3Þ= x1f0ðx2; x3Þ⊕ x1f1ðx2; x3Þ.The functions f0 and f1 are stored in the memory register M1 andwill be cofactored with respect to x2 in the second cycle. In thesecond cycle, f0 and f1 are decomposed to obtain the four func-tions, f00, f01, f10, f11 (state M2). In the final cycle, the decom-position with respect to x3 generates the vector of the functionvalues, O in Fig. 3B, which are outputted in the form of the cor-responding spectral position to be read.Which function is computed can be designed by adjusting the

value of the relaxation time, T. We report in Fig. 4 the 2D-PEspectra calculated for different values of the population time T:T = 23 fs (Fig. 4A), T = 46 fs (Fig. 4B), and T = 138 fs (Fig. 4C).

The periodic intensity variation of the two cross-peaks (“quantumbeats”) that is clearly visible in the spectra of Fig. 4 A and B is amanifestation of the electronic coherence terms, which are presentin the pathsRh andRg. For these paths, the system is in an exciton-coherence state (E12 or E21) during the population time T, and thepath contributions oscillate with a period determined by thedifference of the one exciton frequencies, i.e., T12 = 2π=ðωα −ωβÞ.The chosen T values for the spectra of Fig. 4 A and B correspondto T12=2 and T12, respectively. Parameters which give a large ex-citon splitting (a short modulation period) have been taken fromref. 40 to emphasize the periodic behavior. In the spectrum cor-responding to the longest population time (Fig. 4C), the intensityof the D1 diagonal peak decreases due to the population transferto the lower energy exciton. Together with the three spectra, wereport the truth tables of the corresponding 3-bit functions de-termined according to the binary tree structure of Fig. 3. The in-tensities of the four peaks of a single spectrum determine thevalues of the function for all of the possible combinations ofinputs. Because these values are also the coefficients of theShannon decomposition of the function, it is easy to rewrite thefunctions in their canonical (sum of product or sum of minterm)form, as it is shown in Fig. 4 under each spectrum.Because the peaks in the spectrum are characterized by well-

defined relative intensities, they are well suited to be mapped intointeger values and not only to the binary outputs (0, 1). An in-teger number represents a string of binary values through its bi-nary encoding (e.g., the binary representation of the integer 5 is101 because 5 = 1 × 22 + 0 × 21 + 1 × 20). In many cases, rep-resentations of Boolean functions at the integer level (denoted inthe computer representation as the word level) are convenientbecause more compact. For example, m output switching functions

Fig. 4. Two-dimensional maps computed for different values of the population time, T. (A) T = 23 fs, (B) T = 46 fs, and (C) T = 138 fs. Each T value correspondsto the realization of the decision tree of a function of three Boolean variables, F23fs, F48fs, and F138fs. Their truth tables are shown on the Right for eachspectrum, and their minterm forms are reported under the spectrum. Parameters used for simulating the spectrum are given in SI Text.

Fig. 5. Spectral implementation of a word decision tree for a three input–two output function. (A) Computed 2D map for T = 115 fs; the outputs areread as four valued integers (0, 1, 2, 3). (B) The truth table for the integerrepresentation, FZ = F02

0 + F121, and for the word output (F0, F1).

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ðF0;F1; :::;Fm−1Þ each of n-variables, can be represented by theinteger valued function, FZðxÞ, whose values at x= ðx1; :::xnÞ areFZðxÞ=

Pm−1i= 0 Fiðx1; :::; xnÞ2i. The example that we discuss here is

that of word-level decision diagrams. They admit integer valuesin the terminal nodes and can thus be used to represent multiple-output functions. By using the same BDT as Fig. 3 but assigningto the spectrum a four-valued reading, we can represent a threeinput–two output function. An example is reported in Fig. 5,where we show the spectrum computed for a population time of115 fs in Fig. 5A. By assigning the relative intensity of the fourpeaks a value among (0, 1, 2, 3), we can read the integer valuedfunction reported in the truth table of Fig. 5B, where the lastcolumn is the binary translation that corresponds to the twooutput functions ðF0;F1Þ.Concluding RemarksOur proposal can be scaled up in two different ways: by increasingthe number of the interactions or more simply by increasing thenumber of accessible transitions. The first method would increasethe number of levels of the DT (the so-called depth of the DT),whereas the second will increase the number of outgoing edges foreach node (the so-called width of the DT). Although fifth-orderoptical spectroscopies are still technologically challenging, the

studies of large aggregates with more chromophoric units has beenperformed by 2D-PE and seems to us the most straightforwardgeneralization of the idea illustrated here for a dimer system. Thenumber of different transitions triggered by each interaction withthe external field defines the set of logical values that each variablecan assume (i.e., the radix of the logic), so that the binary imple-mentation presented here can be generalized to multivalued logic.As an example in SI Text, Section 4, we sketch the realization ofa quaternary decision tree by considering the whole set of Feyn-man diagrams contributing to the signal emitted by the dimersystem in all of the four phase-matching directions. Notice that,within this interpretation of the rephasing and not-rephasingspectra, we have access to 26 = 64 evaluations of a switchingfunction simultaneously performed by the evolution of the systemduring the 2D-PE experiment.

ACKNOWLEDGMENTS. This work is supported by the European CommissionFuture and Emerging Technologies Proactive Project MULTI (317707). F.R. isa Director of Research with Fonds National de la Recherche Scientifique(Belgium). B.F. thanks the University of Liège for a postdoctoral fellowship.The work of D.H. and R.D.L. is also supported by The James Franck Programfor Laser–Matter Interaction. E.C. appreciates the support of European Re-search Council Starting Grant “Quantum-Coherent Drive of Energy TransferAlong Helical Structures by Polarized Light.”

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