Nonlinear order separation in two-dimensional electronic spectroscopy quantifies properties of higher-excited states

This paper demonstrates a technique to separate multiple nonlinear orders in two-dimensional electronic spectroscopy by varying pump pulse intensities, enabling the quantitative characterization of highly excited states, such as transition dipole moments and energy levels, in a squaraine dimer with excellent agreement between theory and experiment.

The Gist

Imagine you are trying to listen to a specific conversation in a crowded, noisy room. Usually, the loudest voices (the "first-order" signals) drown out the quieter whispers of the people standing further back. In the world of ultrafast laser spectroscopy, scientists have long struggled with this: when they shine powerful laser pulses at molecules to see how they behave, the strongest signal they get is a mix of everything happening at once. The "higher-order" whispers—information about the molecule's most excited, energetic states—are buried under the noise of the louder, lower-energy interactions.

This paper introduces a clever trick to separate the voices from the noise, allowing scientists to hear the quiet whispers clearly. Here is how they did it, using simple analogies.

The Problem: The "Volume Knob" Dilemma

Think of a molecule like a piano. When you hit a key gently (low laser intensity), you hear a single note. If you hit it harder (higher intensity), you might hear the main note plus some harmonics or overtones. In traditional experiments, scientists usually turn the volume up just enough to get a clear sound, but this creates a messy mix where the main note and the overtones are blended together. They can't tell which sound belongs to which part of the piano.

Furthermore, if they turn the volume up too high, the piano might start to distort or break (saturation), adding even more confusing noise.

The Solution: The "Intensity Cycling" Recipe

The authors developed a method called intensity cycling. Imagine you are trying to figure out the recipe for a soup, but you can only taste the final pot. Instead of guessing, you make four different batches of soup, each with a slightly different amount of salt (laser intensity).

1. Batch 1: A tiny pinch of salt.
2. Batch 2: A medium pinch.
3. Batch 3: A large pinch.
4. Batch 4: A very large pinch.

Because the "flavor" of the salt changes in a predictable mathematical way depending on how much you add, the scientists can use a mathematical recipe (a "Vandermonde matrix," which is just a fancy way of saying a specific set of equations) to work backward. By comparing the four batches, they can mathematically subtract the "salt" to isolate exactly how much of the flavor came from the first pinch, the second, and so on.

In the lab, they did this with laser pulses. They shot the laser at a squaraine dimer (a molecule made of two linked dye parts) at four specific, carefully calculated energy levels. By combining the results, they could mathematically separate the signal into distinct "layers":

• Layer 1 (The 2nd Order): The basic interaction (what we usually see).
• Layer 2 (The 4th Order): The next level of complexity.
• Layer 3 & 4 (The 6th & 8th Orders): The deepest, most complex layers.

The Discovery: Hearing the "Hidden Rooms"

Once they separated the layers, they looked at a specific molecule called a squaraine dimer. Think of this molecule as a two-story house.

• The Ground Floor: This is where the molecule usually sits. When excited, it goes to the "first floor" (a singly excited state). This is what standard spectroscopy sees.
• The Attic (The Hidden Room): This is the "doubly excited state" or "biexciton." It's a high-energy state where the molecule is vibrating wildly. Usually, this room is invisible because the signal is too weak and gets lost in the noise of the ground floor.

By isolating the higher-order layers (the 4th, 6th, and 8th orders), the scientists could finally "see" into the attic. They found:

1. The Energy of the Attic: They measured exactly how much energy it takes to get the molecule to that high-energy state.
2. The Doorway Strength: They calculated how "easy" it is for the molecule to jump from the first floor to the attic (the transition dipole moment). They found this connection is about twice as strong as the connection from the ground floor to the first floor.
3. The "Ghost" of the Attic: Even though the molecule relaxes (calms down) very quickly (in about 100 femtoseconds, which is a quadrillionth of a second), the higher-order signals revealed that a tiny "ghost" of that high-energy state was still lingering, providing clues about the molecule's internal structure.

The Verification: The "Digital Twin"

To make sure they weren't just seeing ghosts, the scientists built a digital twin of the molecule on a computer. They programmed the computer with the laws of physics and the specific shape of their laser pulses.

When they ran the simulation, the computer generated its own "layers" of signals. The result was a perfect match: the real-world data and the computer model looked identical. This confirmed that their method of separating the signals was accurate and that the information they extracted about the high-energy states was real.

The Bottom Line

This paper doesn't just show a new way to take pictures of molecules; it shows a way to unmix the picture. By systematically changing the laser's intensity and using math to separate the layers, they turned a blurry, mixed-up signal into a clear, high-definition view of a molecule's most energetic and hidden states. They proved that by listening to the "quiet whispers" (higher-order signals), we can learn about the "loudest, most energetic parts" of a molecule that were previously impossible to study in isolation.

Technical Summary

Technical Summary: Nonlinear Order Separation in Two-Dimensional Electronic Spectroscopy

Problem Statement
Two-dimensional (2D) electronic spectroscopy is a powerful tool for resolving ultrafast energy transfer and separating homogeneous and inhomogeneous broadening. However, a persistent challenge in the field is the overlap of signals from different orders of nonlinearity. While standard 2D experiments typically target third-order signals (involving two pump interactions and one probe interaction), increasing pump intensity to improve signal-to-noise ratios inevitably introduces higher-order contributions (fifth-order, seventh-order, etc.). These higher-order signals contain valuable information about exciton–exciton interactions and properties of highly excited states, but they are usually mixed with the desired lower-order signals, making quantitative extraction of specific quantum-mechanical properties difficult. Furthermore, in multi-quantum (nQ) spectroscopy, even the lowest-order contributions for a specific nQ position can be contaminated by even higher orders, complicating the interpretation of data regarding multiply excited states.

Methodology
The authors address this challenge by applying and extending a technique known as "intensity cycling." This approach involves systematically varying the intensity of the excitation (pump) pulses and using linear combinations of the resulting measurements to mathematically separate the nonlinear orders of response.

1. Experimental Setup: The study utilizes a squaraine dimer (dSQBC) dissolved in toluene. The experiment employs a pump–probe geometry where the pump pulses ($\vec{k}_1 = \vec{k}_2$) and probe pulse ($\vec{k}_3$) are non-collinear. Signals are detected in the phase-matching direction $-\vec{k}_1 + \vec{k}_2 + \vec{k}_3$.
2. Intensity Variation: The researchers define a dimensionless intensity $I = \lambda^2$, where $\lambda$ scales the pump pulse amplitudes. The total signal $S(\tau, T, t, I)$ is modeled as a power series in $I$: $S = \sum S^{(2m)} I^m$. By measuring the 2D spectra at four distinct, optimized pump intensities (1.74 nJ, 7.35 nJ, 12.9 nJ, and 15 nJ), they solve a system of linear equations (using a Vandermonde matrix inversion) to isolate the individual nonlinear order signals $S^{(2)}$, $S^{(4)}$, $S^{(6)}$, and $S^{(8)}$.
3. Optimization: To minimize errors, the optimal intensities were determined by balancing random noise (measured in signal-free regions of the 2D map) against systematic errors arising from saturation behavior (determined via intensity-dependent transient absorption measurements).
4. Simulation and Modeling: The experimental data were compared against simulations generated using the Ultrafast Spectroscopy Suite (UFSS). The theoretical model treats the dimer as two coupled monomers, each with a three-level electronic system coupled to specific vibrational modes and a Markovian bath. The simulations explicitly account for the experimental pulse shapes and chirp to ensure quantitative agreement.

Key Contributions and Results
The paper demonstrates the successful separation of nonlinear orders up to the eighth order ($S^{(8)}$) in both single-quantum (1Q) and double-quantum (2Q) positions.

• Separation of Orders: The method successfully isolated signals up to $S^{(8)}$. Self-consistency checks confirmed the quality of the separation; for instance, the $S^{(2)}$ signal was zero at the 2Q position (as expected, since the lowest order for 2Q is $S^{(4)}$), and the $S^{(4)}$ signals showed consistent scaling across different spectral positions.
• Quantitative Extraction of Properties: By comparing the isolated experimental orders with simulations, the authors extracted specific quantum-mechanical parameters:
  • From Linear and Third-Order ($S^{(2)}$) Data: They determined energy levels, relative dipole couplings of singly excited states, and the biexciton binding energy (approx. 0.05 eV).
  • From Higher-Order Data ($S^{(4)}$ and above): They determined the coupling strengths between singly excited states and doubly excited states. Specifically, by analyzing the ratio of peak intensities between $S^{(2)}$ and $S^{(4)}$ signals, they constrained the ratio of the pump-weighted average dipole strengths of the two singly excited states ($d_j/d_i$) to be approximately 2 (within a range of 1.7 to 2.2).
• Insight into Highly Excited States: The study revealed that while the system largely relaxes to the lowest-energy exciton ($|i\rangle$) by the population time of 100 fs, the higher-order spectra ($S^{(4)}$) retain signatures of the transition dipole moments connecting higher-energy singly excited states ($|j\rangle$) to doubly excited states. The analysis of Liouville pathways (via Feynman diagrams) showed that specific contributions (ESA2 and SE2) in the higher-order signals, which are otherwise canceled out by negated pathways (NESA and NSE) in a fully relaxed system, provide the necessary information to probe these couplings.
• Peak Shifts: A subtle shift in the 1Q peak position from $S^{(2)}$ to $S^{(4)}$ was observed and attributed to a small residual population of the biexciton state ($|ij\rangle$) at the measurement time, which prevents the complete cancellation of certain pathways.

Significance
The paper claims that this work establishes higher-order spectroscopy as a unique extension of multidimensional spectroscopy. By removing the artifacts caused by mixed nonlinear orders, the technique enables the quantitative analysis of 2D electronic spectra. The authors state that this approach provides access to properties of highly excited states—such as transition dipole moments and energy levels—that are encoded in successive orders of nonlinearity but are typically inaccessible in standard third-order experiments. The successful quantitative agreement between the separated experimental data and the theoretical model validates the method as a robust tool for deriving system properties in complex quantum systems.