2D-DQCS uses a sequence of three coherent optical pulses to correlate certain electronic coherences in a three-level system consisting of ground state (

*g*), a manifold of single excited states (

*e*), and a manifold of double excited states (

*f*), as shown in . The time ordering of a sequence of three femtosecond laser pulses applied to this system is shown in . Pulse 1 interacts with the system to create a coherence between

*g* and

*e*. A coherence is a quantum superposition represented by an off-diagonal element of the density matrix, in this case

*ρ*_{eg} = |

*e**g*|. After a time delay,

*t*_{1}, pulse 2 interacts with the system to create an electronic coherence between

*g* and

*f*, that is, the double-quantum coherence. For simplicity,

*t*_{1} is set to be zero such that the first two pulses create the double-quantum coherence via two-field interactions. After a double coherence time delay,

*t*_{2}, the system interacts with pulse 3, which induces either a single-quantum coherence between

*g* and

*e*′ or a coherence between

*f* and

*e*. During a time period,

*t*_{3}, termed “single coherence time”, the third-order polarization,

*P*^{(3)}, is radiated in the direction of

**k**_{s} =

**k**_{1} +

**k**_{2} −

**k**_{3}, where

**k**_{1},

**k**_{2}, and

**k**_{3} are wave vectors of pulse 1, 2, and 3, respectively. The induced third-order polarization,

*P*^{(3)}(

*t*_{2},

*t*_{3}), is measured by scanning the time delays

*t*_{2} and

*t*_{3}. Then, 2D Fourier transformation of

*P*^{(3)} (

*t*_{2},

*t*_{3}) with respect to

*t*_{2} and

*t*_{3} delivers 2D spectra as a function of two frequency axes, ω

_{2} and ω

_{3}, which are conjugates to

*t*_{2} and

*t*_{3}, respectively. It should be noted that, instead of scanning the

*t*_{3} time delay, its conjugate frequency, ω

_{3}, is directly obtained using a combination of spectrograph and CCD array detector. Since one frequency axis is already obtained by a dispersed detection scheme, only the Fourier transformation along the

*t*_{2} delay is needed to produce a (ω

_{2}, ω

_{3}) 2D spectrum:

The two quantum pathways contributing to the induced third-order signal can be represented by the Feynman diagrams, displayed in . Each diagram shows the sequence of field-matter interactions and the evolution of the electron density matrix of the system during the time intervals between each interaction. The 2D-DQC signal,

*S*(

*ω*_{3},

*ω*_{2}), is given by

^{10}
The two terms in the bracket are the contributions of the pathways **R**_{1} and **R**_{2}. Both share the common term (ω_{2} − ω_{f g})^{−1}, and thus the energies of double-quantum resonances, ω_{fg}, are projected along the ω_{2} axis. In contrast, along the ω_{3} axis, **R**_{1} and **R**_{2} contribute differently to the signal, which results in their interference. The first pathway gives the energies of single-quantum resonances between ground and single excited states, i.e., ω_{3}= ω_{e'g}. The second reveals resonances between double and single excited states, ω_{3} = ω_{f e}. Thus, 2D-DQCS correlates the energies of double-quantum resonances, ω_{fg}, to those of single-quantum resonances, ω_{e’g}, and ω_{fe}, in a 2D spectrum.

To orient the reader an example of a 2D-DQCS map over a broad energy range for the molecule dibenz[

*a,h*]anthracene calculated according to

eq 3 is shown in . The electronic states and the dipole strength for transitions among them were calculated using the SAC-C/6-31G method. According to this method (note that the results are highly method-dependent because the quantum-chemical approach is approximate), there are 41 electronic states lying below 9 eV. Of these, 8 can be optically-excited from the lowest energy singlet excited state, and they are listed on the right side of the figure. Red peaks are contributed by

**R**_{1} and blue troughs (negative peaks) come from

**R**_{2}. The blue and red peaks are displaced symmetrically along the ω

_{3}-axis about the diagonal ω

_{fe} = ω

_{eg}. These features map out the energies of the |

*f* states relative to |

*e*. Certain of the |

*f* states contain significant contributions (~30%) from determinants that represent double excitation from the HOMO to LUMO. It is these states that reveal information about electron correlation, as discussed below.

It has been predicted that the two pathways **R**_{1} and **R**_{2} interfere in a unique manner to make 2D-DQCS powerful for directly probing the signatures of electron correlation effects.^{10}^{, }^{24} For independent electrons, where the correlations between electrons are totally absent, the double excited state, *f*, is given by a direct product of the single excited states, *e* and *e*′. In this case, ω_{e'g} = ω_{f e}, and, owing to exact cancellation of the two signal contributions in the bracket, the 2D-DQC signal vanishes. An example of where this model can be useful is a semiconductor, where *e* is the exciton and *f* is the biexciton. However, when quantum confinement effects are significant, say in quantum dots, the exciton acquires a notable fine structure (owing mainly to exchange interactions),^{25}^{–}^{27} and that alone ensures ω_{e’g} ≠ ω_{fe} even in a mean-field approach.^{28}

Molecules are even more challenging to understand. First, the exchange interaction (singlet-triplet splitting) is significant^{29} and this raises the energy of the first singlet excited state ω_{eg} relative to ω_{fg}/2 by a few hundred meV. Therefore, at zeroth-order, the data recorded by 2D-DQCS is captured by mean-field theoretical approaches. Second, there is substantial differential electron correlation between the three primary states, *g, e*, and *f*, probed in the experiment. Usually *f* is lowered even more relative to *e*. In some systems, like polyenes, *f* can even lie below *e*.^{22}^{, }^{30}^{, }^{31} Third, it is essential to go beyond a two-electron, four-orbital description of the system because there are prominent single-excitation configurations with energies ~2ω_{eg} and appropriate symmetries. Such configurations are already known to be very important for determining third-order nonlinear susceptibilities.^{32}^{, }^{33}

An important difference between our experimental results and the theoretical results previously reported

^{10}^{, }^{23} (cf. ) is that femtosecond laser pulses limited spectral bandwidth (130 – 150 meV FWHM), and therefore we only measure specific correlations between the first electronic excited state and the

*f* state(s) at close to double this excitation energy, as permitted by the bandwidth. Finite pulse duration effects can be included according to the modified form of

eq 3 that includes the frequency-domain pulse envelopes.

^{34} Model quantum-chemical calculations () indicate that outside this energy bandwidth there can be features of significantly greater intensity.

We have modeled a simple three-level system, , with the *f* state up-shifted relative to *e*, consistent with our experimental results. In the simulations, the generally weaker negative peak relative to the observed strong positive peak has been empirically accounted for using a scaling factor of 0.75. We have assumed Gaussian homogeneous line shapes. Transition energy gap fluctuations were incorporated by introducing the fluctuations, δω_{ij}, via energy offsets sampled from a Gaussian distribution using a Monte-Carlo procedure, e.g., ω′_{eg} = ω_{eg} + δω_{eg}. At each iteration, two random offsets are selected: δ_{1} and δ_{2}. The transition energies are then given by δω_{eg} = δ_{1}, δω_{fe} = *a*δ_{1} + (1 − |*a*|)δ_{2}, and δω_{fg} = δω_{eg} + δω_{fe}, where *a* marks the extent to which the fluctuations δω_{eg} and δω_{fe} are correlated. If *a* = 0, then the fluctuations are uncorrelated, which would be the case for two independent particle excitations. More generally, *a* can vary between *a* = 1 (entirely correlated) and *a* = −1 (anti-correlated). Simulations displayed in show that the strongly correlated case (*a* ~ 1), which yields the marked signal elongation parallel to the diagonal, is most consistent with the experimental results reported below.