A quantum dynamical problem has been analytically solved for a two-level system where localized states L 0 and R 0 are strongly coupled with reservoirs of local oscillations {L n } and {R n }. It is additionally assumed that the spectra of reservoirs are equidistant and the coupling constants are the same. It has been shown that the evolution of states L 0 and R 0 in recurrence cycles depends on three independent factors, which characterize exchange with the two-level system, exchange of L 0 with {L n } (R 0 with {R n }) and the phonon-induced decay of {L n } and {R n }. In addition to coherent oscillations with the frequency of the two-level system, Δ, and dissipative tunneling with a rate Δ2/πC 2 (where C is the matrix element of the coupling of L 0 and R 0 with L n and R n ), a new regime appears where L-R transitions are induced by the partial recovery of the populations of L 0 and R 0 in each recurrence cycle due to synchronous transitions from reservoirs. These transitions induce repeating changes in the populations of the states of the two-level system (Loschmidt echo). The number and width of the echo components increase with the cycle number. Evolution becomes irregular because of the mixing of the contributions from pulses of the neighboring cycles, when the cycle number k exceeds the critical value k c = π2 C 2. Unlike the populations, their cycle-average values remain regular at k ≫ k c. When Δ ≪ πC 2, the cycle-average populations oscillate with a frequency of ΔΩ/πC 2 irrespective of mixing. The frequency of oscillations of the populations of the states {L n } and {R n } is approximately nΩ(Δ/2πC 2)2, where Ω is the spacing between the neighboring levels of the reservoir and nΩ is the difference between the energies of the states L 0 and L n . The appearance of the mentioned low-frequency oscillations is due to the formation of collective states of the two-level system that are “dressed” by the reservoir. The predicted oscillations can be detected by femtosecond spectroscopy methods.
A quantum dynamical problem has been analytically solved for a two-level system where localized states L 0 and R 0 are strongly coupled with reservoirs of local oscillations {L n } and {R n }. It is additionally assumed that the spectra of reservoirs are equidistant and the coupling constants are the same. It has been shown that the evolution of states L 0 and R 0 in recurrence cycles depends on three independent factors, which characterize exchange with the two-level system, exchange of L 0 with {L n } (R 0 with {R n }) and the phonon-induced decay of {L n } and {R n }. In addition to coherent oscillations with the frequency of the two-level system, Δ, and dissipative tunneling with a rate Δ2/πC 2 (where C is the matrix element of the coupling of L 0 and R 0 with L n and R n ), a new regime appears where L-R transitions are induced by the partial recovery of the populations of L 0 and R 0 in each recurrence cycle due to synchronous transitions from reservoirs. These transitions induce repeating changes in the populations of the states of the two-level system (Loschmidt echo). The number and width of the echo components increase with the cycle number. Evolution becomes irregular because of the mixing of the contributions from pulses of the neighboring cycles, when the cycle number k exceeds the critical value k c = π2 C 2. Unlike the populations, their cycle-average values remain regular at k ≫ k c. When Δ ≪ πC 2, the cycle-average populations oscillate with a frequency of ΔΩ/πC 2 irrespective of mixing. The frequency of oscillations of the populations of the states {L n } and {R n } is approximately nΩ(Δ/2πC 2)2, where Ω is the spacing between the neighboring levels of the reservoir and nΩ is the difference between the energies of the states L 0 and L n . The appearance of the mentioned low-frequency oscillations is due to the formation of collective states of the two-level system that are “dressed” by the reservoir. The predicted oscillations can be detected by femtosecond spectroscopy methods.
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 dynamical problem is solved for a system coupled to an equidistant-spectrum bath with the energy difference Ω between the neighboring levels n and n + 1 and the coupling matrix elements C 2 = C 2(1 + Δ−2 n 2)−1 constraining the energy interval comprising the bath states interacting with the system. The evolution in the strong-coupling limit is determined by two parameters, Γ = πC 2/Ω ≫ 1 and α = Γ/Δ. If α ≠ 0, then the decrease in the population in the initial cycle with a period of 2π/Ω is not exponential and the effective rate constant increases with time. The results qualitatively explain the appearance of nonexponential relaxation regimes for a dense-spectrum nanosystem and predict the possibility of the multiple recovery of the initial-state population.
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).
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 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.
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.