Quantum decoherence, Zeno process, and time symmetry breaking

Time-Reversal Symmetry and Arrow of Time in Quantum Mechanics of Open Systems

It is one of the most important and long-standing issues of physics to derive the irreversibility out of a time-reversal symmetric equation of motion. The present paper considers the breaking of the time-reversal symmetry in open quantum systems and the emergence of an arrow of time. We claim that the time-reversal symmetric Schrödinger equation can have eigenstates that break the time-reversal symmetry if the system is open in the sense that it has at least a countably infinite number of states. Such eigenstates, namely the resonant and anti-resonant states, have complex eigenvalues. We show that, although these states are often called "unphysical", they observe the probability conservation in a particular way. We also comment that the seemingly Hermitian Hamiltonian is non-Hermitian in the functional space of the resonant and anti-resonant states, and hence there is no contradiction in the fact that it has complex eigenvalues. We finally show how the existence of the states that break the time-reversal symmetry affects the quantum dynamics. The dynamics that starts from a time-reversal symmetric initial state is dominated by the resonant states for t > 0 ; this explains the phenomenon of the arrow of time, in which the decay excels the growth. The time-reversal symmetry holds in that the dynamic ending at a time-reversal symmetric final state is dominated by the anti-resonant states for t < 0 .

Irreversibility and the breaking of resonance-antiresonance symmetry

We consider open quantum systems modeled as discrete lattices. Using a simple model of a single-site coupled to two leads as an example, we show that the time evolution of these systems can be analyzed in terms of an explicitly time-reversal symmetric resolution of unity. This resolution of unity includes both resonant states, which decay in the future, and anti-resonant states, which decay in the past. We show that a time-reversal invariant state contains both resonant and anti-resonant components with equal weights. However, this symmetry is automatically broken as the system evolves in time, with the resonant component becoming much larger than the anti-resonant component for t > 0 (and vice versa for t < 0). We argue that irreversibility is a manifestation of this symmetry breaking. We also compare our present approach with the subdynamics approach developed by Prof. Prigogine and collaborators. Finally, we suggest an extension of our present approach from the level of wave functions to the level of density matrices.

Study of DNA conducting properties: reversible and irreversible evolution

A hole transport through DNA base sequences was modeled. The fact that DNA consists of two polynucleotide strands was taken into consideration. Specific DNA base locations are determined in the model. The model predicts the behavior of a hole acting on the DNA chain, taking into account reversible and irreversible dynamics. It was shown that the transfer mechanisms depend on the sequence type and can be either of hopping nature or of superexchange one. Distance dependence of the hole transport relative rate on the number of hopping steps was investigated. The results obtained were compared with the experimental data. The investigation demonstrates the utilization of the formalism in practical problems for description of the charge migration through the different molecular sequences.

Reduced dynamics of two oscillators collectively coupled to a thermal bath

We study the reduced dynamics of a pair of nondegenerate oscillators coupled collectively to a thermal bath. The model is related to the trilinear boson model where the idler mode is promoted to a field. Due to nonlinear coupling, the Markovian master equation for the pair of oscillators admits non-Gaussian equilibrium states, where the modes distribute according to the Bose-Einstein statistics. These states are metastable before the nonlinear coupling is taken over by linear coupling between the individual oscillators and the field. The Gibbs state for the individual modes lies in the subspace with infinite occupation quantum number. We present the time evolution of a few states to illustrate the behaviors of the system.

Nonequilibrium transport on a quantum molecular chain in terms of the complex Liouvillian spectrum

The transport process in a molecular chain in a nonequilibrium stationary state is theoretically investigated. The molecule is interacting at both ends with thermal baths of different temperatures, while no dissipation mechanism is contained inside the molecular chain. We have first obtained the nonequilibrium stationary state outside the Hilbert space in terms of the complex spectral representation of Liouvillian. The nonequilibrium stationary state is obtained as an eigenstate of the Liouvillian, which is constructed through the collision invariant of the kinetic equation. The eigenstate of the Liouvillian contains information on the spatial correlation between the molecular chain and the thermal baths. While energy flow in the nonequilibrium state which is due to the first-order correlation can be described by the Landauer formula, the particle current due to the second-order correlation cannot be described by the Landauer formula. The present method provides a simple way to evaluate the energy transport in a molecular chain in a nonequilibrium situation.

Exact Markovian kinetic equation for a quantum Brownian oscillator

We derive an exact Markovian kinetic equation for an oscillator linearly coupled to a heat bath, describing quantum Brownian motion. Our work is based on the subdynamics formulation developed by Prigogine and collaborators. The space of distribution functions is decomposed into independent subspaces that remain invariant under Liouville dynamics. For integrable systems in Poincaré's sense the invariant subspaces follow the dynamics of uncoupled, renormalized particles. In contrast, for nonintegrable systems, the invariant subspaces follow a dynamics with broken time symmetry, involving generalized functions. This result indicates that irreversibility and stochasticity are exact properties of dynamics in generalized function spaces. We comment on the relation between our Markovian kinetic equation and the Hu-Paz-Zhang equation.

Strongly coupled matter field and nonanalytic decay rate of dipole molecules in a waveguide

The decay rate gamma of an excited dipole molecule inside a waveguide is evaluated for the strongly coupled matter-field case near a cutoff frequency omegac without using perturbation analysis. Because of the singularity in the density of photon states at the cutoff frequency, we find that gamma depends nonanalytically on the coupling constant g as g4/3, which leads to a vast increase in the decay rate.

Damped harmonic oscillator: pure states of the bath and exact master equations

Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes-grand canonical ensemble, number states, and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and momentum show a Brownian-like behavior. We derive an exact master equation for the characteristic function of the oscillator valid for arbitrary factorized initial conditions. In the case of the equilibrium bath this equation reduces to an equation of the Hu-Paz-Zhang type, while for the coherent states bath it leads to an exact stochastic master equation with a multiplicative noise.

Star-unitary transformations: from dynamics to irreversibility and stochastic behavior

We consider a simple model of a classical harmonic oscillator coupled to a field. In standard approaches, Langevin-type equations for bare particles are derived from Hamiltonian dynamics. These equations contain memory terms and are time-reversal invariant. In contrast, the phenomenological Langevin equations have no memory terms (they are Markovian equations) and give a time-evolution split in two branches (semigroups), each of which breaks time symmetry. A standard approach to bridge dynamics with phenomenology is to consider the Markovian approximation of the former. In this paper, we present a formulation in terms of dressed particles, which gives exact Markovian equations. We formulate dressed particles for Poincaré nonintegrable systems, through an invertible transformation operator Lambda introduced by Prigogine and co-workers. Lambda is obtained by an extension of the canonical (unitary) transformation operator U that eliminates interactions for integrable systems. Our extension is based on the removal of divergences due to Poincaré resonances, which breaks time symmetry. The unitarity of U is extended to "star unitarity" for Lambda. We show that Lambda-transformed variables have the same time evolution as stochastic variables obeying Langevin equations, and that Lambda-transformed distribution functions satisfy exact Fokker-Planck equations. The effects of Gaussian white noise are obtained by the nondistributive property of Lambda with respect to products of dynamical variables.