Investigating the dissipation of heat and quantum information from DNA-scaffolded chromophore networks

Coherent, quantum-mechanical processes are important for quantum information applications, such as communications, sensing, and computation.¹ DiVincenzo formalized material requirements for quantum-information: (a) scalability, (b) the ability to initialize the system to a particular quantum state, (c) long decoherence times compared to gate-operation times, (d) the existence of universal quantum gates, and (e) measurement capabilities.² Various types of systems are under consideration for these applications, including optical cavities,³ trapped ions,⁴ spins,⁵ diamond-lattice nitrogen vacancies,⁶ and superconductors.⁷ However, while each of these systems has numerous advantages for these applications, they also have disadvantages in at least one of DiVincenzo’s conditions. These trade-offs mean that there is no universal best choice of quantum-information material; rather, it depends on the fit for the particular application. Here, another class of systems is considered, which are coupled aggregates of organic chromophore sites. Although traditionally considered a weaker competitor than these others because of their fast decoherence rates, these systems, nonetheless, have advantages. They are in the intermediate regime between bulk semiconductor and isolated molecular systems, which allows their nanoscale characteristics and corresponding dynamics to be tuned. By carefully designing and constructing molecular aggregates, particular electronic couplings and site energies can be controlled in order to create a reactivity of choice, in principle. In contrast, the delocalized excitations of bulk semiconductors make them vulnerable to defects anywhere in the material, which can cause localization, while the isolation of individual molecular sites makes them difficult to scale. Furthermore, the fast dynamics of electronic processes (compared to, for example, spin dynamics) are advantageous because more operations could be performed per second in a system with universally faster dynamics.⁸ 

However, in practice, the electronic states within organic molecular networks are often considered less competitive for many of these processes because they tend to dissipate quantum information into their environment rapidly due to their strong system–environment coupling and rapid thermal fluctuations.⁹ This problem is not unique to organic systems, and it also exists in systems such as semiconductor quantum dots.¹⁰ In a comparison of decoherence rates, electronic systems in organic aggregates are orders of magnitude faster than those of the previously listed systems, and they are unlikely to catch up. However, they can also tolerate faster decoherence, in principle: DiVincenzo’s rule c only specifies that the decoherence rates must be long compared to the gate-operation times, which would be similarly fast for these electronic systems.⁸ As a result, the need exists to slow the dissipation of quantum information in scaffolded organic molecular networks.

In particular, it would be convenient if these processes could be slowed using controllable nanoscale parameters, such as the constituent sites’ positions, orientations, and energy levels. One of the key features that tunes the electronic dynamics of molecular networks is their aggregate structure based on their impact on the Coulombic coupling between the molecular sites.¹¹ Using DNA nanotechnology to produce scaffolds for covalently attached chromophore networks, the aggregate structures can be designed and synthesized algorithmically,^12,13 and these structures can implement complex, programmatic material functions.^14,15 Biological systems exhibit many examples, such as pigment–protein complexes that perform solar water-oxidation and energy-harvesting.^16,17 In these systems, their electronic and structural parameters have been optimized over billions of years, in some cases,¹⁸ to perform these functions. Furthermore, even subtle structural changes in these systems can result in large functional changes.^19,20 After initial measurements of coherent excitonic motion in photosynthetic proteins,^21–23 these functions were pursued in artificial molecular systems for applications such as coherent energy transport and quantum-information technology.²⁴ Although coherent quantum-mechanical phenomena are not known to contribute to biological fitness,²⁵ artificial molecular systems can still harness these phenomena to perform useful functions. However, the interactions between the electronic (or vibronic) system and its environment, which contribute effects such as decoherence, spectral line broadening, and the dissipation of heat and quantum information, must be optimized to make these quantum-mechanical applications possible.²⁶ 

The goal of this work is to recognize whether simple changes to the molecular network’s nanoscale structure can tune these system–environment dynamics significantly. Because the network’s aggregate structure can be tuned using approaches such as DNA nanotechnology techniques, if these structural changes can result in significant benefits to figures of merit, such as quantum information dissipation, then it would bolster the case for using these materials for quantum-information applications. In analogy to natural pigment–protein systems, DNA nanotechnology provides structural control over modular chromophore networks.²⁷ Using these methods, large libraries of these systems can be produced rationally and efficiently,^28–30 including both structural variations and the modular tuning of each site’s energy levels.^31,32 When the Coulombic coupling between the vibronic states is strong compared to the system–environment coupling, these systems gain the potential for quantum-mechanical functions, such as coherent energy transport.^33–35 As a result, DNA nanotechnology has been proposed for the implementation of devices such as quantum logic gates^8,14,15 as well as customizable molecular optoelectronics.³⁶ 

Approaches have been proposed to slow decoherence in perturbative environments, such as using the Kuramoto effect to synchronize the electronic and/or vibrational oscillators,^37–40 reservoir engineering to produce quantum-information backflow from non-Markovian environmental baths into the system,^41,42 and delayed quantum-feedback to stabilize quantum coherences dynamically.^43–46 As an early step, heat dissipation in DNA-scaffolded chromophore networks must be better understood. Therefore, another goal of this study is to observe energy transport from the environment into the system and to understand how to tune this process.

The dissipation of photoexcitation energy from the system to its environmental baths (or vice versa) is described by heat currents, whose dynamics can include both coherent and incoherent motion.⁴⁷ These heat currents between the system and its environment are related to changes in the von Neumann entropy, which, in turn, describes the amount of quantum information in the system.⁴⁸ Entropy is known to increase as a function of the population differences between states, and it can also be bolstered by their quantum coherences.⁴⁹ Additionally, heat currents have been studied previously in organic chromophore networks connected by organic linking chains, for applications such as heat rectification.^50,51 When the heat transport is non-resonant, it occurs by either energy dissipation through vibrational states or collisions between vibrational excitations.⁵² The energy-transport mechanisms in molecular electronics are also subject to interference effects arising from coherent motion.^53,54 When the scaffolded chromophores are individually excited, their interface with a connected organic chain represents a thermal conductance boundary. Energy transport through this boundary occurs using the anharmonic couplings of the atomic vibrational modes.⁵⁵ These findings indicate that the particular scaffold and chromophore network characteristics can impact the extent of heat and quantum-information dissipation from the molecular network to its environment.

In this study, DNA-scaffolded molecular networks are investigated theoretically to understand the vibronic dynamics, as well as the heat currents and quantum-information dissipation. In particular, a three-chromophore system is investigated (Fig. 1). Attached to a double-helix DNA scaffold, sites 1–3 contain Cyanine 3 (Cy3), Cyanine 3.5 (Cy3.5), and Cyanine 5 (Cy5), respectively. Cy3.5 is on the first strand by itself, Cy5 is located in-register (zero base-pair separation) on the complementary strand, and Cy3 is located two base-pairs apart from Cy5 on its same strand. DNA-scaffolded systems containing each of these monomers, as well as the combined Cy3Cy3.5Cy5 trimer network, are investigated. To study these systems, first, their nanoscale structural and energetic characteristics are understood using genetic algorithms and spectroscopic calculations in comparison to experimental measurements. These methods have been used previously on homogeneous DNA-scaffolded chromophore pairs by computing their linear absorption and circular dichroism spectra in comparison to experimental measurements.^56,57 The genetic algorithm approach was chosen to optimize the nanoscale parameters because it scales well in high-dimensional search spaces and resists local optima.⁵⁸ Subsequently, these characteristics are used as inputs to calculate the dynamics between the system and environmental bath using the hierarchical equations of motion (HEOM).^59,60 Finally, these results are related to the quantum-information dissipation and the entanglement among the vibronic excited states.

Dye-labeled DNA strands were obtained from Integrated DNA Technologies (Coralville, IA, USA) and dissolved in deionized water without further purification. The DNA structures were composed of a double-stranded DNA structure prepared in 2.5 X PBS (phosphate buffered saline: 0.345 M NaCl, 0.00675 M KCl; 7.4 pH) with an annealing program, in which the solution was heated to 95 °C for 5 min and then lowered 1 °C per minute until the temperature reached 4 °C. The two DNA strands used for the chromophore networks are listed in Table I. Here, the dyes are covalently attached to DNA through double phosphoramidite attachment chemistry with three-carbon spacers, fully occupying the position of a nucleotide in each case. Similar sequences were utilized for the monomer samples.

Samples were prepared at 5 µM, and their linear absorption and circular dichroism spectra were measured using a JASCO J-1500 spectrophotometer, performed at 20 °C, 1 nm step intervals, at 100 nm/min, 4 s digital integration time, three accumulations, in a 10 mm path length quartz cuvette.

Genetic algorithms use a stochastic optimization approach to funnel the search toward an optimal solution. However, a recurring problem with this method is that it tends to cluster the guesses toward local optima over many iterations, reducing the efficiency of searching over the configuration space. Delaunay clearing procedures are used to make the search resistant to trapping in local optima.⁶² In a standard clearing procedure, most of the guesses that are nearly identical to each other are deleted, which causes the algorithm not to cluster as strongly into these local minima over subsequent iterations. However, because guesses are being deleted, it lowers the efficiency of the search. In the Delaunay clearing method, Delaunay triangulation is used to identify empty regions of the configuration space, and the cleared guesses are randomly reassigned into these empty regions instead of being deleted. As a result, the guesses are retained, while still resisting local optima. The following shortcuts were included to reduce the computational expense of this method: (a) only the 100 highest-scoring members (not including near-duplicates) were included in the triangulation; (b) only four randomly selected variables were triangulated at a time; and (c) triangulation was only performed every third iteration of the genetic algorithm. Near duplicates are determined by applying the distance formula for all of the varying parameters combined, with a cutoff of 1. This is a heuristic cutoff; therefore, the units were allowed to be mixed.

The cumulant expansion model described by Mukamel²⁶ was used to calculate spectra in the genetic algorithm. This approach was described previously, but it will be summarized here.⁵⁶ It was used because it is relatively computationally efficient, which is necessary because the genetic algorithm computes many iterations of these calculations. In this method, the spectral time-domain line shapes $ϱt$ are determined by Eq. (14), and they are related by the Fourier transform to the optical spectra in the frequency domain. According to this model, the vibronic populations evolve according to the states’ angular frequencies ω. However, stochastic changes in the state energies due to environmental perturbations transiently perturb these frequencies, which impacts the population and coherence trajectories.

The structure of the three-site system was deduced by measuring linear absorption and circular dichroism spectra (Fig. 2) and then optimizing the computed spectra in comparison to the measured spectra through optimization of the nanoscale parameters. Note that the bars shown in Fig. 2 correspond to vibronic eigenstates, while the vibronic states (not their eigenstates) will be discussed in the remainder of this report. A genetic algorithm was used to perform this optimization (see the section titled Methods). The Cy3.5 and Cy5 sites are positioned zero base-pairs apart (i.e., in-register on the complementary DNA duplex strands), while Cy3 is positioned two base pairs away on the same strand as Cy5. The structure obtained by the genetic algorithm procedure is consistent with this configuration because Cy3.5 and Cy5 are highly intercalated, while Cy3 is spatially offset. The nanoscale parameters corresponding to these spectra are shown in Table II, and the corresponding energy levels of the vibronic states are shown in Fig. 3.

The vibronic states correspond to the excitation of one of the three possible electronic states. Each electronic state contains three vibrational sub-levels, for a total of 27 possible permutations of the vibrational configurations within the three electronic states. For example, the configuration for vibronic state 1 is (0, 0, 0). For vibronic states 1–27, Cy3 is electronically excited and the vibrational configuration ranges systematically from (0, 0, 0) to (2, 2, 2). States 28–54 or 55–81 involve the electronic excitation of Cy3.5 or Cy5, respectively, with the same progression of the vibrational configurations in each case. As a result, 81 vibronic states were included in the system Hamiltonian overall (see the section titled Methods). For example, in the Cy3.5 section of Fig. 3, the lowest-energy vibronic state corresponds to a configuration of (0, 0, 0). The next three states are clustered at similar energies. In each of these vibronic states, Cy3.5 is electronically excited, and the system contains one quantum of vibrational energy. This energy quantum can be stored either in the excited vibrational state of Cy3.5 yielding the (0, 1, 0) configuration or in the electronic ground-states of Cy5 or Cy3 to yield (0, 0, 1) or (1, 0, 0), respectively. Because there are only three permutations in one-vibrational-quantum configurations, this cluster only contains three states. The following cluster likewise contains two vibrational quanta, yielding six possible configurations: (0, 2, 0), (0, 1, 1), (1, 1, 0), (0, 0, 2), (1, 0, 1), and (2, 0, 0). For the subsequent clusters of states, this pattern continues with additional vibrational quanta, up to the maximum of six quanta that can be contained by this system. The reason that the energy levels in these clusters are so close together is that the vibrational energies for each of the three sites are similar to each other, as will be discussed subsequently in the section titled Results.

The vibronic population dynamics were calculated using HEOM (see the section titled Methods). The initial state for these calculations assumes that state 28 was fully occupied, which corresponds to the blue solid line in Fig. 4(b). Figure 4(b) contains the population dynamics obtained from HEOM, using the optimized nanoscale parameters for the three-site system obtained from the genetic algorithm (Table II). This system will subsequently be referred to as “unchanged,” in comparison to several similar systems whose aggregation characteristics have been varied. In addition to Fig. 4, all 81 of the normalized vibronic population and heat current dynamics for each of the systems are displayed in Figs. S1 and S2, respectively, in the supplementary material. At room temperature, the oscillations that occurred within approximately the first 200 fs indicated coherent energy transport between the vibronic states. These coherent motions decay at similar time scales to those measured in photosynthetic proteins.⁸³ 

The lines in Fig. 4 tend to cluster into groups. The states with the (0, 0, 0) vibrational configurations, with populations represented by the blue solid lines in each panel of Fig. 4, tend to be the lowest-energy state represented in each panel with the highest population after 1 ps. This result is attributed to downhill energy-transport. Although energy gradually transports to the minimum-energy eigenstates, the vibronic states are not eigenstates, and therefore, their equilibrium energy distribution will depend on their overlap with the lowest-energy eigenstates. Furthermore, 1 ps is not necessarily sufficient to reach these equilibrium configurations. Because state 28 is initially fully populated, its population quickly decays from 1 to ∼0.05 after 1 ps [Fig. 4(b)]. This electronic energy transfers partially into Cy5, as observed by the coincident rise of the blue solid peak corresponding to state 55 from 0 to ∼0.2, on a similar time scale [Fig. 4(b)]. In contrast, state 1, corresponding to the electronic excitation of Cy3, rises to ∼0.1 via coherent dynamics within femtoseconds of excitation, but then does not experience net incoherent energy transport on subsequent time scales of up to 1 ps. This result indicates that, when Cy3.5 is initially excited, Cy5 accumulates energy mainly from incoherent transport (e.g., Förster Resonance Energy Transfer, or FRET), while Cy3 does mainly from coherent transport. These results indicate that, by designing systems with variable base-pair separation, the transport mechanism will be tuned between coherent and incoherent motion.

The higher vibronic states undergo transport through both coherent and incoherent mechanisms, as evidenced by the combination of both damped sinusoidal and exponential contributions to their line shapes (Fig. 4). When Cy3 becomes electronically excited, the lower vibronic states undergo mainly coherent energy transport within tens of femtoseconds after excitation [e.g., the solid lines in Figs. 2(a)–2(c)]. In contrast, the higher vibronic states also exhibit incoherent dynamics, as indicated by their exponential line shape components. For example, comparing the sudden, sinusoidal rise of population density in vibronic state 1 at early time delays [Fig. 4(c), blue solid line] with the gradual, exponential rise over hundreds of femtoseconds in vibronic state 55 [Fig. 4(c), blue solid line], these differences in the dynamics suggest that the higher-lying vibrational populations assist in energy transport through an incoherent transport mechanism, while the lower-lying vibrational populations can participate more in coherent transport.

Next, the branching ratios for the energy transport to each site are determined. Summing all of the vibronic state populations for which each site is electronically excited (e.g., summing the populations for states 1–27 when Cy3 is excited, etc.) indicates that 18% of the electronic excited-state population remains in Cy3.5, while 58% goes to Cy5 and 24% goes to Cy3 after 1 ps. While the lowest-lying vibronic states for Cy3.5 and Cy5 are still transporting energy after 1 ps [Figs. 2(a) and 2(b)], in the higher vibronic states, this transport has already occurred within ∼400 fs for the vibronic states where Cy5 is electronically excited [Fig. 4(e)] and ∼200 fs when either Cy3.5 or Cy3 are electronically excited [Figs. 2(d) or 2(f), respectively]. These time scales are approximately uniform for the many higher-lying vibronic states, when comparing the vibronic populations corresponding to the same electronic excitation. Therefore, even after only 200 fs, the respective excitation distribution for Cy3, Cy3.5, and Cy5 is 25%, 19%, and 56%, respectively. These numbers are very similar to those observed after 1 ps time delay. Therefore, although the lowest vibronic states in each sub-panel of Fig. 4 are still transporting energy after 1 ps, the vast majority of energy transport from one site to another occurs within 200 fs, assisted by the higher vibronic states.

For example, Fig. 5 displays the vibronic heat currents for the chromophore network, corresponding to the nanoscale parameters listed in Table II. Like the vibronic population curves in Fig. 4, these heat current curves contain sinusoidal oscillations at early time delays, plus longer-lived exponential contributions. These sinusoidal oscillations are due to coherent interactions between the system and environment, while the exponential contributions are due to incoherent energy-transfer processes. For the vibronic states where Cy3.5 is electronically excited [Figs. 5(b) and 5(e)], the heat currents are almost entirely positive-valued. While state 28’s heat current is initially 1.6 × 10⁻³ cm⁻¹/fs since this was the only initially populated state, within femtoseconds, it decreases to less than 0.2 × 10⁻³ cm⁻¹/fs. Meanwhile, the higher-energy vibronic states never have heat currents larger than ∼0.01 × 10⁻³ cm⁻¹/fs. In contrast, for the vibronic states where Cy5 is electronically excited [Figs. 5(c) and 5(f)], the lowest energy vibronic state 55 becomes negative after ∼300 fs, eventually becoming the most strongly negative at approximately −0.07 × 10⁻³ cm⁻¹/fs after 1 ps. The heat current for state 57 also becomes negative, though only to approximately −0.01 × 10⁻³ cm⁻¹/fs. These negative heat currents tend to appear as populations are accumulated in the lowest vibronic states for particular site excitations, as shown by the blue solid curves in Figs. 5(a) and 5(c). Otherwise, the heat currents remain positive for most of the other states. Likewise, for the vibronic states where Cy3 is electronically excited [Figs. 5(c) and 5(f)], state 1 has a positive heat current that persists for ∼250 fs, before turning negative. Unlike when Cy5 was electronically excited, many of the higher vibronic states when Cy3 was excited also have negative heat currents for most of the 1 ps time window, though with a smaller magnitude. Therefore, the general picture of the heat currents in this system is that Cy3.5 is dissipating heat into the environment, while Cy3 and Cy5 are absorbing it into their lowest electronic excited states, and Cy3 is additionally absorbing it in the higher vibronic states. Furthermore, in comparison to Fig. 4, these results indicate that the vibronic population dynamics and heat-current dynamics do not coincide one-to-one with each other and that the heat current typically decays much faster than the electronic populations.

In order to gain insights into how the vibronic population and heat-current dynamics depend on the structural parameters for this representative system, the deviations in these dynamics are analyzed while varying the relative distances or orientations. In particular, the HEOM calculations were performed on alternative systems with the Cy3.5-Cy3 distance either halved (0.5r_Cy3.5,Cy3) or doubled (2r_Cy3.5,Cy3) or the Cy3.5-Cy5 relative θ (θ_Cy3.5,Cy5 + $π2$⁠) or ϕ (ϕ_Cy3.5,Cy5 + $π2$⁠) angle increased by $π2$⁠, compared to the unchanged system parameters obtained by the genetic algorithm. The Cy3.5’s position and orientation were constant in each of these variations. Cy3.5 and Cy3 were chosen for the displacement comparisons because Cy3.5 and Cy5 were already positioned so closely to each other that it was physically implausible to move them significantly closer. Meanwhile, Cy3.5 and Cy5 were chosen for the orientation adjustments because their proximity made the effects on their coupling more significant.

For each of these variations, the calculated population and heat current dynamics are shown for vibronic states 55–81 (Fig. 6). These vibronic states correspond to the electronic excitation of the site with the smallest optical gap, Cy5. It therefore includes the entire system’s minimum-energy state (state 55, blue solid line), where the most population tends to accumulate over time. In these calculations, Cy3.5’s lowest-energy vibronic state (state 28) was initially populated. In the population dynamics for state 55, 0.5r_Cy3.5,Cy3 is populated more quickly than 2r_Cy3.5,Cy3. The initial dynamics of the blue solid line in 0.5r_Cy3.5,Cy3 feature a sudden rise to 0.15, which is not present in 2r_Cy3.5,Cy3. Nonetheless, after 1 ps, this line in both samples reaches a similar population of ∼0.35–0.4. Therefore, the distance between these sites affects how much time it takes for the excitation energy to transport down to the minimum-energy states; however, it does not greatly affect their populations at a 1 ps time delay.

In contrast, the heat-current dynamics are much different in these two cases. The sudden positive peak for state 55 in 0.5r_Cy3.5,Cy3 indicates that this state initially dissipates a relatively large quantity of energy into its environmental bath. Subsequently, this peak decreases, indicating a decrease in energy dissipation from the system to the bath. At later times, the negative heat current indicates that the site receives energy from its environment, in net. In contrast, for 2r_Cy3.5,Cy3, the heat current gradually increases for 400 fs, before peaking and decreasing gradually at later times. These results indicate that the heat current of the lowest-energy vibronic state is very sensitive to the distance between sites, producing qualitatively different behavior even, while the population dynamics exhibit similar results after 1 ps, as discussed in the previous paragraph. In contrast, the higher-energy vibronic states show qualitatively similar dynamics in the two systems, with a sudden rise at early times, followed by a decay at larger times. The vibronic heat currents in both samples that correspond to the lower-energy states eventually show slightly negative values, while those of higher-energy states remain positive. While the qualitative behaviors of these populations are similar for these two systems, 2r_Cy3.5,Cy3 has about half the peak heat current values as 0.5r_Cy3.5,Cy3. Therefore, for these higher-energy states, the heat currents are increased as the molecular distance is decreased. Whereas this result would have been expected for population decay dynamics, due to the decreased Coulombic coupling between the sites, it was less clear how these distances should affect the heat currents from the system to the bath reservoirs. Nonetheless, the same trends in the heat currents between the system and baths are also evident.

These dynamics were also investigated as a function of the relative Cy3.5-Cy5 orientation [Figs. 6(e)–6(h)]. In the population dynamics [Figs. 6(e) and 6(g)], the line for state 55 rises for the first picosecond to become the most populated state. However, it rises to ∼0.35 when θ is increased by $π2$⁠, compared to ∼0.2 when ϕ is increased by $π2$ instead. Aside from this difference in height, the features are qualitatively very similar in both instances. The higher-lying vibronic states have similar populations after 1 ps in both cases; however, their rising dynamics change with the peak maxima occurring near 400 fs in Fig. 6(e), while they all continue to rise for at least 1 ps in Fig. 6(g). Despite these differences in the electronic dynamics, the heat-transfer dynamics have similar patterns. However, they differ in height, with the θ-shifted system resulting in approximately a 50%–75% enhancement of the peak heights, compared to the ϕ-shifted case.

Considering the population and heat current dynamics together (Figs. 4 and 5), the lowest-energy vibronic state (state 55) accumulates both the population from the other vibronic states and also heat currents from the environmental bath. As the vibronic population transfers into state 55, corresponding to (0, 0, 0), its heat current also becomes negative after the first 300 fs, when the environment begins transferring energy to this state’s population. Meanwhile, many of the other states show different behaviors, with some becoming slightly negative while others remaining positive (Fig. 5). As shown in Fig. 6, these behaviors are also influenced by the position and orientation of the chromophore sites within the scaffolded network, though the relationship between these properties and the heat current is not simple.

In contrast, for IC, the τ₂ contribution is not apparent in the 0.5r_Cy3.5,Cy3 system, but it is 27 or 219 fs for the unchanged or 2r_Cy3.5,Cy3 systems, respectively. As the distance between Cy3.5 and Cy3 doubles from the unchanged system to 2r_Cy3.5,Cy3, the IC’s τ₂ rate decreases by approximately a factor of 8, implying an inverse cubic dependence of this time constant on distance, which is consistent with the coupling frequency’s dependence on distance in a Förster resonance energy-transfer mechanism. Extrapolating this relationship, 0.5r_Cy3.5,Cy3 should have a time constant of about 4 fs, which would be difficult to distinguish in the fitting procedure from the 8 fs signal that was already found for the IC’s τ₁ value. For the 0.5r_Cy3.5,Cy3 sample, this similar value of the fitted τ₁ component and the hypothetical τ₂ component is likely to explain why the 0.5r_Cy3.5,Cy3 fit lacks this τ₂ component in the IC fit. Meanwhile, the orientation of the Cy3.5 and Cy5 sites also affects the rates. For the S data, rotation along θ by $π2$ raises the τ₁ rate from 3.19 to 8.31 fs, while rotation along ϕ by $π2$ raises it to 13.3 fs and lowers the pre-exponential weight A₁ to approximately half. Likewise, these rotations raise the τ₂ values from 49.4 fs in the unchanged system to 85.1 or 134 fs, respectively. However, in this case, it also changes the pre-exponential weighting A₂ from −5.6 × 10⁻⁵ to −6.6 × 10⁻⁵ or $−13×10−5cm−1Kfs$ for θ_Cy3.5,Cy5 + $π2$ or ϕ_Cy3.5,Cy5 + $π2$⁠, respectively.

The values for τ₃ exceed the 1 ps calculation window and cannot be discussed further; however, their pre-exponential weights A₃ all decline in comparison with the unchanged sample. Because these are the only decaying contributions within the fit, they are responsible for the cross-overs that occur after several hundred femtoseconds in Fig. 8(a), where the ϕ_Cy3.5,Cy5 + $π2$ and θ_Cy3.5,Cy5 + $π2$ curves eventually surpass the 2r_Cy3.5,Cy3 curve, despite slower starts and smaller maximum heights. Meanwhile, for the IC data, the lifetime is increased to 83 or 116 fs when the orientation of Cy5 is rotated by $π2$ along the θ or ϕ angles, respectively. Once again, the pre-exponential weights A₂ only change a small amount (from −0.16 to $−0.13cm−1Kfs$⁠) from the rotation of along θ, but by a larger amount (to $−0.27cm−1Kfs$⁠) from the rotation along ϕ. In the IC data, the pre-exponential weights A₃ are relatively insensitive to the change in structure after 1 ps. Therefore, by tuning these nanoscale parameters, the rate at which quantum information and quantum entanglement are formed and dissipated across the molecular network can be tuned by the particular molecular aggregation characteristics. Where it was possible, physical attributions have been discussed above. However, in many cases, the trends are less clear, and the impact on the dynamics appears to be more complicated than simple physical rules of thumb can explain. In these cases, finding the particular physical rationale for the structural dependencies in the dynamics will require more investigation in future work.

To obtain more specific physical insights, the change in entropy ΔS due to heat transfer between the system and environment is considered next [Eq. (30)]. This metric correlates directly to the heat current by the second law of thermodynamics, and it therefore isolates the dissipation mechanism of quantum-information from these particular physical processes. This ΔS value describes the amount of quantum information dissipating from the system to the environment if positive or vice versa if negative.⁴⁷ The results in Fig. 9(a) are obtained by summing the heat-current contributions from all of the vibronic states for each system independently. Visually, these system configurations have differing dynamics, with the unchanged system’s component decaying much faster than the others for example. However, fitting these lines to 2–3 exponential components reveals that they share very similar time constants and differ primarily by the pre-exponential weights of these contributions (Table III).

For the first time constant τ₁, the unchanged and 0.5r_Cy3.5,Cy3 systems have the smallest time constants of 5–6 fs, while that of the 2r_Cy3.5,Cy3 system was slower at 9 fs. In contrast, both the θ_Cy3.5,Cy5 + $π2$ and ϕ_Cy3.5,Cy5 + $π2$ systems lack this time constant contribution. Whereas the pre-exponential weight A₁ was ∼4 × 10⁻⁶ $cm−1Kfs$ in the unchanged sample, changing the distance in either 0.5r_Cy3.5,Cy3 or 2r_Cy3.5,Cy3 resulted in a decrease of A₁ to ∼1.5 × 10⁻⁶ $cm−1Kfs$⁠. This A₁ term explains why the unchanged sample’s ΔS value declines the fastest. Likewise, the second rate constant τ₂ is very similar among the sample configurations, at 79–87 fs. However, once again, the A₂ values differ significantly, with the unchanged sample exhibiting the lowest value at 1 × 10⁻⁶ $cm−1Kfs$⁠. While the 0.5r_Cy3.5,Cy3 and 2r_Cy3.5,Cy3 samples have nearly double that value, the θ_Cy3.5,Cy5 + $π2$ or ϕ_Cy3.5,Cy5 + $π2$ samples have triple or quadruple that value, respectively. Overall, a higher weight of A₁ corresponds to a lower weight of A₂. The process corresponding to τ₂ appears to be approximately coincident with the coherent system–bath dynamics shown by the oscillations in the residuals of Fig. 9(a), and therefore, it may be related to coherent transport effects. However, because the exact shape of this residual is highly dependent on the particular fit, it is difficult to fit a time constant to the residual signal envelope, so precise comparisons of time constants are not made here. For τ₃, the values range from 240 to 600 fs. The unchanged sample has the largest rate constant of 596 fs, 2r_Cy3.5,Cy3 has an intermediate constant of 476 fs, and 0.5r_Cy3.5,Cy3, θ_Cy3.5,Cy5 + $π2$⁠, and ϕ_Cy3.5,Cy5 + $π2$ have the smallest constants of 297, 242, and 273 fs, respectively. Except for the unchanged system, the A₃ weights are similar to one another within a range of 1.5 × 10⁻⁶ to 2.3 × 10⁻⁶ $cm−1Kfs$⁠. In contrast, the unchanged system has a much smaller value of 0.3 × 10⁻⁶ $cm−1Kfs$⁠, which is attributed to its comparatively large A₁ that caused more signal depletion earlier.

The quantum information dissipated from these ΔS contributions over the first picosecond is obtained by integrating the area under the ΔS curves. These integrated results are shown in Fig. 8(b). The unchanged system dissipates the least quantum information by a factor of ∼1/3, compared to the other systems. Whereas its integrated area is 2.52 × 10⁻⁴ $cm−1K$⁠, those of the 0.5r_Cy3.5,Cy3, 2r_Cy3.5,Cy3, θ_Cy3.5,Cy5 + $π2$⁠, and ϕ_Cy3.5,Cy5 + $π2$ systems are 7.63 × 10⁻⁴, 8.84 × 10⁻⁴, 7.94 × 10⁻⁴, and 7.26 × 10⁻⁴ $cm−1K$⁠, respectively [Fig. 8(b)]. The much smaller area for the unchanged system explains why its S and IC values are the highest in Fig. 7 because less energy dissipation due to weaker system–environment heat currents allows more quantum information and stronger entanglement to be retained in the system. Likewise, the system with the lowest retention of S and IC is 2r_Cy3.5,Cy3, which also has the largest area under the ΔS curve. These ΔS area trends also correlate to the slightly higher ϕ_Cy3.5,Cy5 + $π2$ curve compared to θ_Cy3.5,Cy5 + $π2$ after 1 ps in the S and IC plots within Fig. 7. However, 0.5r_Cy3.5,Cy3 deviates from this correspondence, because it has the second-highest S and IC scores after 1 ps despite only having the median ΔS area score. This deviation occurs in both panels of Fig. 7 because of an especially large initial rise of the line corresponding to 0.5r_Cy3.5,Cy3, compared to those of the other systems (excluding the unchanged system). While it is difficult to assign a physical mechanism to this particular deviation, the general trend is nonetheless significant, with four out of the five systems following a simple correlation between the system’s entropy or entanglement and the ΔS area, after 1 ps.

To explain why the unchanged sample has the fastest ΔS value decline in Fig. 8(a), corresponding to its largest A₁ value in Table III, consider the input parameters for the HEOM calculation. Only the non-zero vibronic couplings between distinct states, contained in the off-diagonal elements within the Hamiltonian, differ among the systems considered in these HEOM calculations. Because each system’s Hamiltonian contains a very large number (4374) of these non-zero couplings, rather than considering them individually, their distributions are considered using histograms (Fig. 9). The order of these distribution widths corresponds to the order of the A₁ magnitudes for ΔS in Table III. The unchanged sample has the largest distribution of vibronic coupling strengths, with magnitudes near 750 cm⁻¹, corresponding to the largest A₁ value for the ΔS dynamics shown in Table III. The 0.5r_Cy3.5,Cy3 and 2r_Cy3.5,Cy3 distributions are the next widest, corresponding to the next-fastest decay at early time delays in Fig. 8(a). Finally, the θ_Cy3.5,Cy5 + $π2$ and ϕ_Cy3.5,Cy5 + $π2$ distributions are the most narrow, corresponding to no discernable A₁ contribution for the ΔS dynamics in Table III.

Next, the backflow of heat and quantum information is considered as another means to prolong the quantum information in these systems. While the system as a whole is dissipating quantum information over time, for at least some of the calculated time window, some of the individual vibronic states are receiving this information from the environment instead. This backflow suggests that quantum reservoir engineering could enhance the robustness of coherences in perturbative environments.⁸⁹ The heat current dynamics in Fig. 5 reveal that, after dissipating quantum information (as indicated by a positive heat current) for ∼300 fs, the heat current becomes negative and therefore vibronic state 55 experiences a backflow of quantum information from its environment. There are numerous other states whose heat currents become negative within Fig. 5 as well. This backflow has been attributed to non-Markovian bath contributions, whose memory effects can (a) revive quantum information within the open quantum system and (b) improve quantum entanglement even within perturbative environments.⁴² Meanwhile, Fig. 6 shows that the intensity of the negative signal and the time delay before it changes from positive to negative signals (i.e., when backflow turns on) can be tuned by the particular aggregate structure, as well. This result suggests the promise of tuning the backflow processes using structural DNA nanotechnology. For example, state 28’s heat current becomes negative in the unchanged sample after about 300 fs (Fig. 5). For 0.5r_Cy3.5,Cy3, the heat current turns negative, and therefore, quantum-information backflow begins at ∼700 fs (Fig. 6). In contrast, by doubling the distance or changing the relative Cy3.5-Cy5 angles compared to the unchanged system, a rising contribution is added to the heat current dynamics of state 55, prolonging its positive valuation, and thus, backflow never occurs within the first picosecond. Once again, the unchanged sample has the biggest advantage in terms of backflow, coinciding with its largest IC score and least quantum information dissipation (smallest ΔS area) in Fig. 8, followed by 0.5r_Cy3.5,Cy3 in both respects. These results are consistent with the work of Mirkin et al.,⁴² which found that the presence of quantum information backflow increases the I-concurrence values. This relation may help explain why the I-concurrence values are the highest in the unchanged sample, followed by the 0.5r_Cy3.5,Cy3 sample. This observation indicates an opportunity, in future work, to understand how to optimize these backflow processes in order to bolster the longevity of quantum information and entanglement in open quantum systems interacting with perturbative environments.

The vibronic population and heat–current dynamics were investigated theoretically for a DNA-scaffolded chromophore network involving three chromophore sites and 81 vibronic states. This investigation found that modifications to the organic molecular network’s structure can significantly impact the heat currents, as well as the dissipation dynamics of quantum information and entanglement. These dynamics were calculated after modification of the positions and orientations of the chromophore sites in order to understand whether these structural input parameters could significantly affect the heat currents and dissipation of quantum information and the dynamics of quantum-mechanical entanglement. For the systems studied here, at room-temperature, the population transported from its initial (0, 0, 0) excited-state population (state 28) to the other vibronic states, and predominantly to the lowest-lying (0, 0, 0) excited state (state 55), over the first picosecond (Fig. 4).

The entanglement and quantum-information dissipation dynamics were also obtained from these vibronic dynamics. The trends, correlations, and differences between these figures of merit were investigated. The heat current in the lowest-energy excited state (state 55) was found to be highly dependent on the nanoscale parameters, from heat-current values that reached maxima almost immediately after excitation for the unchanged system to ∼400 fs for the 2r_Cy3.5,Cy3 sample (Fig. 6). Backflow of heat from the environment to the system was observed in some states, indicated by a negative heat current. This backflow can increase the system’s quantum-coherence lifetimes⁴² and therefore represents a direction for optimizing coherences in highly perturbative systems.

Meanwhile, the entropy and entanglement dynamics also depend on the nanoscale parameters. From one initially excited state, the energy transferred across the network initially within several femtoseconds. Subsequently, it transferred further with a slower, more highly variable time constant that was consistent with the r⁻³ distance dependence of the coupling in Förster resonance energy transfer. This entanglement, once formed, persisted for the picosecond calculation time window. The entropy also exhibited qualitatively similar transient characteristics (Fig. 7). As shown in Fig. 8, the entropy dissipation rates could be drastically impacted by variation of the structural configuration. The unchanged sample had a much smaller entropy dissipation (ΔS area) than the other structural configurations, which did not show a simple linear dependence with respect to distance or angular offset, suggesting that more subtle relations governed these processes. Based on the contents of Table III, the main difference between the more strongly and weakly dissipating variations was not their rate constants, but rather the distribution of their pre-exponential weights. For instance, although all of the systems instantaneously dissipated approximately the same amount of quantum information at time-zero [as indicated by their same value at t = 0 in Fig. 8(a)], the unchanged system most rapidly ceased dissipating its quantum information as time elapsed. It did so not because its rate constants were substantially faster than the other systems, but because it had a more substantial weighting in the A₁ pre-exponential parameter, which corresponded to the fastest rate constant (τ₁).

Based on Table III, halving or doubling the distance between Cy3.5 and Cy3 decreased A₁ to about 1/3 of its unchanged value, indicating a non-trivial (but nonetheless important) relationship between the inter-site distances and quantum information dissipation in this system. Increasing either the ϕ_Cy3.5,Cy5 or θ_Cy3.5,Cy5 parameters by $π2$ was sufficient to eliminate A₁ altogether, most drastically increasing the duration and ΔS area of the quantum-information dissipation. Meanwhile, a decrease in A₁ coincided with an increase in A₂, which drastically affected the curve decay in Fig. 7 because τ₂ is ∼10× slower than τ₁.

Overall, these results indicate that (a) minimizing the integrated heat current promotes longer-lived quantum information and entanglement in the system, (b) variations of the molecular network structure can significantly tune the heat-current decay rates, and (c) particularly strong mitigation of quantum information loss occurs when either the maximum coupling between vibronic states or the distribution width of such couplings is increased. Combined, these points validate the concept of tuning the heat and entropy dissipation by structural configuration, for instance, using scaffolds created by DNA nanotechnology, therefore opening the way for future optimization studies of these highly configurable systems.

See the supplementary material for this report, which displays the normalized vibronic population and heat current dynamics for the systems under discussion.

B.S.R. was supported by the Jerome and Isabella Karle Distinguished Scholar Fellowship at the U.S. Naval Research Laboratory. This work was supported by the U.S. Naval Research Laboratory Institute for Nanoscience and the Office of Naval Research.

The authors have no conflicts to disclose.

Brian S. Rolczynski: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Sebastián A. Díaz: Investigation (supporting); Resources (equal); Writing – original draft (supporting); Writing – review & editing (supporting). Ellen R. Goldman: Resources (equal); Writing – review & editing (supporting). Igor L. Medintz: Funding acquisition (equal); Resources (equal); Supervision (equal); Writing – review & editing (supporting). Joseph S. Melinger: Funding acquisition (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.