The time evolution of a prepared vibrational state (system) coupled to a reservoir with a dense spectrum of its vibrational states has been investigated under the assumption that the reservoir has an equidistant spectrum, and the system-reservoir coupling matrix elements are independent of the reservoir states. The analytical solution manifests three regimes of the evolution for the system: (i) weakly damped oscillations; (ii) a multicomponent Loschmidt echo in recurrence cycles; and (iii) overlapping recurrence cycles. We find the characteristic critical values of the system-reservoir coupling constant for the transitions between these regimes. Stochastic dynamics occurs in regime (iii) due to an unavoidably, in any real system, coarse graining of the time or energy measurements or the initial condition uncertainty. At any finite accuracy, one can always find the cycle number k c when the dynamics of the system for k>k c cannot be determined uniquely from the spectrum, and, in this sense, the long-time system evolution becomes chaotic. Even though a specific toy model is investigated here, when properly interpreted, it yields quite a reasonable description for a variety of physically relevant phenomena, such as the complex vibrational dynamics of nanoparticles with characteristic interlevel spacing on the order of 10 cm−1 observed using subpicosecond spectroscopy methods.
The time evolution of a prepared vibrational state (system) coupled to a reservoir with a dense spectrum of its vibrational states has been investigated under the assumption that the reservoir has an equidistant spectrum, and the system-reservoir coupling matrix elements are independent of the reservoir states. The analytical solution manifests three regimes of the evolution for the system: (i) weakly damped oscillations; (ii) a multicomponent Loschmidt echo in recurrence cycles; and (iii) overlapping recurrence cycles. We find the characteristic critical values of the system-reservoir coupling constant for the transitions between these regimes. Stochastic dynamics occurs in regime (iii) due to an unavoidably, in any real system, coarse graining of the time or energy measurements or the initial condition uncertainty. At any finite accuracy, one can always find the cycle number k c when the dynamics of the system for k>k c cannot be determined uniquely from the spectrum, and, in this sense, the long-time system evolution becomes chaotic. Even though a specific toy model is investigated here, when properly interpreted, it yields quite a reasonable description for a variety of physically relevant phenomena, such as the complex vibrational dynamics of nanoparticles with characteristic interlevel spacing on the order of 10 cm−1 observed using subpicosecond spectroscopy methods.
A dynamic problem for a system coupled to a reservoir possessing a dense discrete spectrum of states has been analytically solved under two simplifying assumptions proposed by Zwanzig [15], according to which the reservoir spectrum is equidistant and all the system-reservoir coupling matrix elements are identical (i.e., independent of reservoir states). It is demonstrated that a multicomponent Loschmidt echo arises in each recurrence cycles, the number of components being equal to the cycle number. At a certain critical cycle number, the components of neighboring cycles exhibit mixing. As a result, the dynamics of the system transforms from a regular to stochastic-like dynamics, in which an arbitrarily small coarse graining (inherent in any real system) of the results of measurements or uncertainty in the initial conditions leads to (i) the loss of one-to-one correspondence between the discrete spectrum of eigenvalues and the state of the system and (ii) the loss of invariance with respect to the time reversal. Interrelation between the mixing of cycles with the entanglement of trajectories, on the one hand, and the overlap of resonances in classical systems with mixing, on the other hand, is discussed. The properties of the proposed model are consistent with a variety of the kinetic regimes of vibrational relaxation (from exponential decay to irregular, weakly damping oscillations) observed in various objects. Common features are average distances between neighboring levels on the order of 1–10 cm −1 and recurrence cycles on a time scale of 10 –13 –10 −11 s, which is studied using femtosecond spectroscopy techniques.
Abstract Small (but still containing many, about 102–104, atoms) quantum systems (traditionally termed nano-systems) are dramatically different from their macroscopic or genuine microscopic (atomic) cousins. Microscopic molecular systems (with a few atoms) obey a regular quantum dynamics (described by time dependent Schrödinger equation), whereas in macroscopic systems with continuous energy spectra, one can expect, also regular, although typically relaxation, dynamic behavior. The topic of our paper is in between these limits. Nano-scale systems are characterized by small (but finite) mean interlevel spacing. In such a case with recurrence periods in picosecond range, Loschmidt echo and double resonance phenomena come into the game. System behavior becomes nontrivial and manifests a sort of transitions between regular and chaotic dynamics. We show that such dynamic transitions occur when the Loschmidt echo time of life exceeds the typical recurrence cycle period. We illustrate this behavior in the frame work of a few versions of the exactly solvable quantum problem, proposed long ago by Zwanzig. It is based on the study of time evolution of the initially prepared vibrational state coupled to a reservoir with dense spectrum of its vibrational states. In the simplest version of the Zwanzig model, the reservoir has an equidistant spectrum, and the system—reservoir coupling matrix elements are independent of the reservoir states. We generalize the model to include into consideration the coupling of the initially prepared single state to system phonon excitations. The coupling results to temperature dependent broadening and decay of the echo components. Another generalization is to replace a single level by two states coupled to the Zwanzig reservoir. We anticipate that the basic ideas inspiring our work can be applied to a large variety of interesting for the applications nanosystems (e.g., dissipative free propagation of excitations along molecular chains, or as a model for exchange reactions).
The quantum dynamics of a simple model of a nanoparticle has been investigated. This model suggests that the initially prepared state (system) interacts with the other states (reservoir) forming a dense discrete spectrum. In contrast to our previous papers concerned with this problem [1, 2], in which only the dynamics of the system has been studied, the present paper is devoted to the description of the evolution of reservoir states. In the initial recurrence cycles, the reverse transitions from the reservoir to the system generate a double resonance (an echo at frequencies of the reservoir states transitions). Since different states of the reservoir are depleted at different instants of time, the Loschmidt echo in the system is inhomogeneously broadened, whereas the double resonances remain narrower and more intense. Apart from the main resonances, there arise satellites due to the redistribution of the populations between the reservoir states during the cycle. In mixing cycles regime, the regular evolution of the reservoir states (like the system state) transforms into a stochastic-like evolution. It is noted that the predicted double resonances can be experimentally detected and used in analyzing vibrational relaxation of large molecules and nanoparticles.
A quantum problem of the evolution of a system coupled to a reservoir for which the interlevel spacing monotonically increases or decreases with increasing energy was solved. It was demonstrated that, despite the absence of an unambiguous definition of the recurrence cycle period, there is a wide range of parameters within which the basic feature of the evolution of system with an equidistant spectrum persist: the existence of the Loschmidt echo and mixing of the cycles that initiates the transition from a regular to a stochastic dynamic behavior. The results predict the existence of nonergodic nanosystems in which, as they evolve, energy periodically concentrates on certain vibrational degrees of freedom.
Small (but still containing many, about 10 2 -10 4 , atoms) quantum systems (traditionally termed nano-systems) are dramatically different from their macroscopic or genuine microscopic (atomic) cousins.Microscopic molecular systems (with a few atoms) obey a regular quantum dynamics (described by time dependent Schrodinger equation), whereas in macroscopic systems with continuous energy spectra, one can expect, also regular, although typically relaxation, dynamic behavior.The topic of our paper is in-between these limits.Nano-scale systems are characterized by small (but finite) mean inter-level spacing.In such a case with recurrence periods in pico-second range, Loschmidt echo and double resonance phenomena come into the game.System behavior becomes non-trivial and manifests a sort of transitions between regular and chaotic dynamics.We show that such dynamic transitions occur when the Loschmidt echo time of life exceeds the typical recurrence cycle period.We illustrate this behavior in the frame work of a few versions of the exactly solvable quantum problem, proposed long ago by Zwanzig.It is based on the study of time evolution of the initially prepared vibrational state coupled to a reservoir with dense spectrum of its vibrational states.In the simplest version of the Zwanzig model, the reservoir has an equidistant spectrum, and the system -reservoir coupling matrix elements are independent of the reservoir states.We generalize the model to include into consideration the coupling of the initially prepared single state to system phonon excitations.The coupling results to temperature dependent broadening and decay of the echo components.Another generalization is to replace a single level by two states coupled to the Zwanzig reservoir.We anticipate that the basic ideas inspiring our work can be applied to a large variety of interesting for the applications nano-systems (e.g., dissipative free propagation of excitations along molecular chains, or as a model for exchange reactions).
In the framework of a simple spin-boson Hamiltonian we study an interplay between dynamic and spectral roots to stochastic-like behavior. The Hamiltonian describes an initial vibrational state coupled to discrete dense spectrum reservoir. The reservoir states are formed by three sequences with rationally independent periodicities 1; 1 ± δ typical for vibrational states in many nanosize systems (e.g., large molecules containing CH2 fragment chains, or carbon nanotubes). We show that quantum evolution of the system is determined by a dimensionless parameter δΓ, where Γ is characteristic number of the reservoir states relevant for the initial vibrational level dynamics. When δΓ > 1 spectral chaos destroys recurrence cycles and the system state evolution is stochastic-like. In the opposite limit δΓ < 1 dynamics is regular up to the critical recurrence cycle k c and for larger k > k c dynamic mixing leads to quasi-stochastic time evolution. Our semi-quantitative analytic results are confirmed by numerical solution of the equation of motion. We anticipate that both kinds of stochastic-like behavior (namely, due to spectral mixing and recurrence cycle dynamic mixing) can be observed by femtosecond spectroscopy methods in nanosystems in the spectral window 1011–1013 s−1