Continuous-Time Quantum Walk in Glued Trees: Localized State-Mediated Almost Perfect Quantum-State Transfer.

In this work, the dynamics of a quantum walker on glued trees is revisited to understand the influence of the architecture of the graph on the efficiency of the transfer between the two roots. Instead of considering regular binary trees, we focus our attention on leafier structures where each parent node could give rise to a larger number of children. Through extensive numerical simulations, we uncover a significant dependence of the transfer on the underlying graph architecture, particularly influenced by the branching rate (M) relative to the root degree (N). Our study reveals that the behavior of the walker is isomorphic to that of a particle moving on a finite-size chain. This chain exhibits defects that originate in the specific nature of both the roots and the leaves. Therefore, the energy spectrum of the chain showcases rich features, which lead to diverse regimes for the quantum-state transfer. Notably, the formation of quasi-degenerate localized states due to significant disparities between M and N triggers a localization process on the roots. Through analytical development, we demonstrate that these states play a crucial role in facilitating almost perfect quantum beats between the roots, thereby enhancing the transfer efficiency. Our findings offer valuable insights into the mechanisms governing quantum-state transfer on trees, with potential applications for the transfer of quantum information.

Optimized excitonic transport mediated by local energy defects: Survival of optimization laws in the presence of dephasing.

In an extended star with peripheral defects and a core occupied by a trap, it has been shown that exciton-mediated energy transport from the periphery to the core can be optimized [S. Yalouz et al., Phys. Rev. E 106, 064313 (2022)2470-004510.1103/PhysRevE.106.064313]. If the defects are judiciously chosen, then the exciton dynamics is isomorphic to that of an asymmetric chain and a speedup of the excitonic propagation is observed. Here we extend this previous work by considering that the exciton in both an extended star and an asymmetric chain is perturbed by the presence of a dephasing environment. Simulating the dynamics using a Lindblad master equation, two questions are addressed: How does the environment affect the energy transport on these two networks? and Do the two systems still behave equivalently in the presence of dephasing? Our results reveal that the timescale for the exciton dynamics strongly depends on the nature of the network. But quite surprisingly, the two networks behave similarly regarding the survival of their optimization law. In both cases, the energy transport can be improved using the same original optimal tuning of energy defects as long as the dephasing remains weak. However, for moderate or strong dephasing, the optimization law is lost due to quantum Zeno effect.

Quantum self-trapping on a star graph.

The attractive Bose-Hubbard model is applied for describing the two-exciton dynamics in a nonlinear quantum star graph. When the excitons are created on the core of the star, it is shown that the interplay between the complex architecture of the network and the nonlinearity favors the occurrence of a real quantum self-trapping. Quite weak in the small nonlinearity limit, this self-localization is enhanced as the nonlinearity increases. This feature originates in the restructuring of the two-exciton eigenstates whose localized nature intensifies with the nonlinearity. Nevertheless, the quantum self-trapping is never complete since it is impossible to localize the entire exciton density, even in the strong nonlinearity limit.

Two-exciton bound state quantum self-trapping in an extended star graph.

An attractive Bose-Hubbard model is applied for describing quantum self-trapping in an extended star graph. In the strong coupling limit and when two excitons are created on the core of the star, the dynamics is dominated by pair states whose properties are governed by the branch number N. When N = 2, the star reduces to a linear chain so that the energy does not self-localize. Conversely, when N ≥ 3, restructuring of the eigenstates arises and a low-energy state occurs describing a pair localized on the core of the star. Preferentially excited, this localized state gives rise to quantum self-trapping of the energy, a process that intensifies as N increases.

Extended star graph as a light-harvesting-complex prototype: Excitonic absorption speedup by peripheral energy defect tuning.

We study the quantum dynamics of a photoexcitation uniformly distributed at the periphery of an extended star network (with N_{B} branches of length L_{B}). More specifically, we address here the question of the energy absorption at the core of the network and how this process can be improved (or not) by the inclusion of peripheral defects with a tunable energy amplitude Δ. Our numerical simulations reveal the existence of optimal value of energy defect Δ^{*} which depends on the network architecture. Around this value, the absorption process presents a strong speedup (i.e., reduction of the absorption time) provided that L_{B}≤L_{B}^{*} with L_{B}^{*}≈12.5/ln(N_{B}). Analytical and numerical developments are then conducted to interpret this feature. We show that the origin of this speedup takes place in the hybridization of two upper-band excitonic eigenstates. This hybridization is important when L_{B}≤L_{B}^{*} and vanishes almost totally when L_{B}>L_{B}^{*}. These structural rules we draw here could represent a potential guide for the practical design of molecular nanonetwork dedicated to the realization of efficient photoexcitation absorption.

Continuous-time quantum walk on an extended star graph: Disorder-enhanced trapping process.

Using a tight binding model, we investigate the dynamics of an exciton on a disordered extended star graph whose central site acts as an energy trap. When compared with what happens in an ordered network, our results reveal that the disorder drastically improves the excitonic absorption that becomes complete. Moreover, we show the occurrence of an optimal disorder for which the absorption time is strongly minimized, a surprising effect that originates in a disorder-induced restructuring process of the exciton eigenstates. Finally, we also show the existence of an optimal value of the absorption rate that reduces even more the absorption time. The resulting superoptimized trapping process is interpreted as a positive interplay between both the disorder and the so-called superradiance transition.

Continuous-time quantum walk on an extended star graph: Trapping and superradiance transition.

A tight-binding model is introduced for describing the dynamics of an exciton on an extended star graph whose central node is occupied by a trap. On this graph, the exciton dynamics is governed by two kinds of eigenstates: many eigenstates are associated with degenerate real eigenvalues insensitive to the trap, whereas three decaying eigenstates characterized by complex energies contribute to the trapping process. It is shown that the excitonic population absorbed by the trap depends on the size of the graph, only. By contrast, both the size parameters and the absorption rate control the dynamics of the trapping. When these parameters are judiciously chosen, the efficiency of the transfer is optimized resulting in the minimization of the absorption time. Analysis of the eigenstates reveals that such a feature arises around the superradiance transition. Moreover, depending on the size of the network, two situations are highlighted where the transport efficiency is either superoptimized or suboptimized.

Exciton-phonon dynamics on complex networks: Comparison between a perturbative approach and exact calculations.

A method combining perturbation theory with a simplifying ansatz is used to describe the exciton-phonon dynamics in complex networks. This method, called PT^{*}, is compared to exact calculations based on the numerical diagonalization of the exciton-phonon Hamiltonian for eight small-sized networks. It is shown that the accuracy of PT^{*} depends on the nature of the network, and three different situations were identified. For most graphs, PT^{*} yields a very accurate description of the dynamics. By contrast, for the Wheel graph and the Apollonian network, PT^{*} reproduces the dynamics only when the exciton occupies a specific initial state. Finally, for the complete graph, PT^{*} breaks down. These different behaviors originate in the interplay between the degenerate nature of the excitonic energy spectrum and the strength of the exciton-phonon interaction so that a criterion is established to determine whether or not PT^{*} is relevant. When it succeeds, our study shows the undeniable advantage of PT^{*} in that it allows us to perform very fast simulations when compared to exact calculations that are restricted to small-sized networks.

Exciton-phonon system on a star graph: A perturbative approach.

Based on the operatorial formulation of the perturbation theory, the properties of an exciton coupled with optical phonons on a star graph are investigated. Within this method, the dynamics is governed by an effective Hamiltonian, which accounts for exciton-phonon entanglement. The exciton is dressed by a virtual phonon cloud whereas the phonons are clothed by virtual excitonic transitions. In spite of the coupling with the phonons, it is shown that the energy spectrum of the dressed exciton resembles that of a bare exciton. The only differences originate in a polaronic mechanism that favors an energy shift and a decay of the exciton hopping constant. By contrast, the motion of the exciton allows the phonons to propagate over the graph so that the dressed normal modes drastically differ from the localized modes associated to bare phonons. They define extended vibrations whose properties depend on the state occupied by the exciton that accompanies the phonons. It is shown that the phonon frequencies, either red shifted or blue shifted, are very sensitive to the model parameter in general, and to the size of the graph in particular.

Disorder-enhanced exciton delocalization in an extended dendrimer.

The exciton dynamics in a disordered extended dendrimer is investigated numerically. Because a homogeneous dendrimer exhibits few highly degenerate energy levels, a dynamical localization arises when the exciton is initially located on the periphery. However, it is shown that the disorder lifts the degeneracy and favors a delocalization-relocalization transition. Weak disorder enhances the delocalized nature of the exciton and improves any quantum communication, whereas strong disorder prevents the exciton from propagating in accordance with the well-known Anderson theory.

Exciton localization-delocalization transition in an extended dendrimer.

Exciton-mediated quantum state transfer between the periphery and the core of an extended dendrimer is investigated numerically. By mapping the dynamics onto that of a linear chain, it is shown that a localization-delocalization transition arises for a critical value of the generation number G(c) ≈ 5. This transition originates in the quantum interferences experienced by the excitonic wave due to the multiple scatterings that arise each time the wave tunnels from one generation to another. These results suggest that only small-size dendrimers could be used for designing an efficient quantum communication protocol.

A qubit coupled with confined phonons: the interplay between true and fake decoherence.

The decoherence of a qubit coupled with the phonons of a finite-size lattice is investigated. The confined phonons no longer behave as a reservoir. They remain sensitive to the qubit so that the origin of the decoherence is twofold. First, a qubit-phonon entanglement yields an incomplete true decoherence. Second, the qubit renormalizes the phonon frequency resulting in fake decoherence when a thermal average is performed. To account for the initial thermalization of the lattice, the qua- ntum Langevin theory is applied so that the phonons are viewed as an open system coupled with a thermal bath of harmonic oscillators. Consequently, it is shown that the finite lifetime of the phonons does not modify fake decoherence but strongly affects true decoherence. Depending on the values of the model parameters, the interplay between fake and true decoherence yields a very rich dynamics with various regimes.

The reduced dynamics of an exciton coupled to a phonon bath: a new approach combining the Lang-Firsov transformation and the perturbation theory.

To go beyond the Born approximation, a new method is introduced for describing the reduced dynamics of an exciton coupled to a phonon bath. Two unitary transformations are applied for accounting for the exciton-phonon entanglement through a dual dressing mechanism affecting both the exciton and the phonons. In doing so, one obtains an analytical expression of the exciton reduced density matrix without integrating numerically any generalized master equation. Therefore, by using a quite simple model that can be solved exactly, it has been shown that the proposed method is particularly suitable for describing the exciton dynamics over a rather broad region in the parameter space. However, although the method shows many strengths, it also exhibits weaknesses and it accidentally breaks down owing to the occurrence of specific resonances.

Vibrons in finite size molecular lattices: a route for high-fidelity quantum state transfer at room temperature.

A communication protocol is proposed in which vibron-mediated quantum state transfer takes place in a molecular lattice. We consider two distant molecular groups grafted on each side of the lattice. These groups form two quantum computers where vibrational qubits are implemented and received. The lattice defines the communication channel along which a vibron delocalizes and interacts with a phonon bath. Using quasi-degenerate perturbation theory, vibron-phonon entanglement is taken into account through the effective Hamiltonian concept. A vibron is thus dressed by a virtual phonon cloud whereas a phonon is clothed by virtual vibronic transitions. It is shown that three quasi-degenerate dressed states define the relevant paths followed by a vibron to tunnel between the computers. When the coupling between the computers and the lattice is judiciously chosen, constructive interference takes place between these paths. Phonon-induced decoherence is minimized and a high-fidelity quantum state transfer occurs over a broad temperature range.

Energy transfer in finite-size exciton-phonon systems: confinement-enhanced quantum decoherence.

Based on the operatorial formulation of the perturbation theory, the exciton-phonon problem is revisited for investigating exciton-mediated energy flow in a finite-size lattice. Within this method, the exciton-phonon entanglement is taken into account through a dual dressing mechanism so that exciton and phonons are treated on an equal footing. In a marked contrast with what happens in an infinite lattice, it is shown that the dynamics of the exciton density is governed by several time scales. The density evolves coherently in the short-time limit, whereas a relaxation mechanism occurs over intermediated time scales. Consequently, in the long-time limit, the density converges toward a nearly uniform distributed equilibrium distribution. Such a behavior results from quantum decoherence that originates in the fact that the phonons evolve differently depending on the path followed by the exciton to tunnel along the lattice. Although the relaxation rate increases with the temperature and with the coupling, it decreases with the lattice size, suggesting that the decoherence is inherent to the confinement.

Quantum decoherence in finite size exciton-phonon systems.

Based on the operatorial formulation of the perturbation theory, the properties of a confined exciton coupled with phonons in thermal equilibrium is revisited. Within this method, the dynamics is governed by an effective Hamiltonian which accounts for exciton-phonon entanglement. The exciton is dressed by a virtual phonon cloud whereas the phonons are clothed by virtual excitonic transitions. Special attention is thus paid for describing the time evolution of the excitonic coherences at finite temperature. As in an infinite lattice, temperature-enhanced quantum decoherence takes place. However, it is shown that the confinement softens the decoherence. The coherences are very sensitive to the excitonic states so that the closer to the band center the state is located, the slower the coherence decays. In particular, for odd lattice sizes, the coherence between the vacuum state and the one-exciton state exactly located at the band center survives over an extremely long time scale. A superimposition involving the vacuum and this specific one-exciton state behaves as an ideal qubit insensitive to its environment.

Vibron-phonon coupling strength in a finite size lattice of H-bonded peptide units.

An attempt is made to measure the vibron-phonon coupling strength in a finite size lattice of H-bonded peptide units. Within a finite temperature density matrix approach, we compare separately the influence of both the vibron-phonon coupling and the dipole-dipole interaction on the coherence between the ground state and a local one-vibron state. Due to the confinement, it is shown that the vibron-phonon coupling yields a series of dephasing-rephasing mechanisms that prevents the coherence to decay. Similarly, the dipole-dipole interaction gives rise to quantum recurrences for specific revival times. Nevertheless, intense recurrences are rather rare events so that the coherence behaves as a random variable whose most probable value vanishes. By comparing the degree of the coherence for each interaction, a critical coupling chi*(L) is defined to discriminate between the weak and the strong coupling limits. Its size dependence indicates that the smaller the lattice size is, the weaker the vibron-phonon coupling relative to the dipole-dipole interaction is.

Vibron phonon in a lattice of H-bonded peptide units: A criterion to discriminate between the weak and the strong coupling limit.

Based on dynamical considerations, a simple and intuitive criterion is established to measure the strength of the vibron-phonon coupling in a lattice of H-bonded peptide units. The main idea is to compare separately the influence of both the vibron-phonon coupling and the dipole-dipole interaction on a specific element of the vibron reduced density matrix. This element, which refers to the coherence between the ground state and a local excited amide-I mode, generalizes the concept of survival amplitude at finite temperature. On the one hand, when the dipole-dipole interaction is neglected, it is shown that dephasing-limited coherent dynamics is induced by the vibron-phonon coupling. On the other hand, when the vibron-phonon coupling is disregarded, decoherence occurs due to dipole-dipole interactions since the local excited state couples with neighboring local excited states. Therefore, our criterion simply states that the strongest interaction is responsible for the fastest decoherence. It yields a critical coupling chi( *) approximately 25 pN at biological temperature.

Parametric resonance-induced time-convolutionless master equation breakdown in finite size exciton-phonon systems.

A detailed analysis is performed to show that the second order time-convolutionless master equation fails to describe the exciton-phonon dynamics in a finite size lattice. To proceed, special attention is paid to characterizing the coherences of the exciton reduced density matrix. These specific elements measure the ability of the exciton to develop superimpositions involving the vacuum and the one-exciton states. It is shown that the coherences behave as wavefunctions whose dynamics is governed by a time-dependent effective Hamiltonian defined in terms of the so-called time-dependent relaxation operator. Due to the confinement, quantum recurrences provide to the relaxation operator an almost periodic nature, so the master equation reduces to a linear system of differential equations with almost periodic coefficients. We show that, in accordance with the Floquet theory, unstable solutions emerge due to parametric resonances involving specific frequencies of the relaxation operator and specific excitonic eigenfrequencies. These resonances give rise to an unphysical exponential growth of the coherences, indicating the breakdown of the second order master equation.

Phonon anharmonicity-induced decoherence slowing down in exciton-phonon systems.

Based on a generalized Fröhlich model, a time-convolutionless master equation is established for studying the dynamics of an exciton coupled with anharmonic phonons. Special attention is paid to describing the influence of the phonon anharmonicity on specific elements of the exciton reduced density matrix. These elements, called coherences, characterize the ability of the exciton to develop quantum states that are superimpositions involving the vacuum and the local one-exciton states. Whether the phonons are harmonic or not, it is shown that dephasing limited-coherent motion takes place. The coherences irreversibly decrease with time, the decay rate being the so-called dephasing rate, so that they experience a localization phenomenon and propagate over a finite length scale. However, it is shown that the phonon anharmonicity softens the influence of the phonon bath and reduces the dephasing rate. A slowdown in the decoherence process appears so that the coherences are able to explore a larger region along the lattice. Moreover, the phonon anharmonicity modifies the way the dephasing rate depends on both the adiabaticity and the temperature. In particular, the dephasing rate increases linearly with the temperature in the weak anharmonicity limit whereas it becomes almost temperature-independent in the strong anharmonicity limit. Note that the present formalism is applied to describe amide-I excitons (vibrons) in a lattice of H-bonded peptide units.