We propose a novel approach to intrinsic decoherence without adding new assumptions to standard Quantum Mechanics. We generalize the Liouville equation just by requiring the dynamical semigroup property of time evolution and dropping the unitarity requirement. With no approximations and specific statistical assumptions we find a generalized Liouville equation which depends on two characteristic time t1 and t2 and reduces to the usual equation in the limit t1 = t2 --> 0. However, for t1 and t2 arbitrarily small but finite, our equation can be written as a finite difference equation which predicts state reduction to the diagonal form in the energy representation. The rate of decoherence becomes faster at the macroscopic limit as the energy scale of the system increases. In our approach the evolution time appears, a posteriori, as a statistical variable as if time evolution would take place randomly at average intervals t2, each evolution having a time width t1. A generalized Tam Mandelstam inequality is derived. The relation with previous work by Milburn is discussed. The agreement with recent experiments on damped Rabi oscillations is described.
We propose a novel approach to intrinsic decoherence without adding new assumptions to standard Quantum Mechanics. We generalize the Liouville equation just by requiring the dynamical semigroup property of time evolution and dropping the unitarity requirement. With no approximations and specific statistical assumptions we find a generalized Liouville equation which depends on two characteristic time t1 and t2 and reduces to the usual equation in the limit t1 = t2 --> 0. However, for t1 and t2 arbitrarily small but finite, our equation can be written as a finite difference equation which predicts state reduction to the diagonal form in the energy representation. The rate of decoherence becomes faster at the macroscopic limit as the energy scale of the system increases. In our approach the evolution time appears, a posteriori, as a statistical variable as if time evolution would take place randomly at average intervals t2, each evolution having a time width t1. A generalized Tam Mandelstam inequality is derived. The relation with previous work by Milburn is discussed. The agreement with recent experiments on damped Rabi oscillations is described.
We propose a novel approach to intrinsic decoherence without adding new assumptions to standard quantum mechanics. We generalize the Liouville equation just by requiring the dynamical semigroup property of time evolution and dropping the unitarity requirement. With no approximations and statistical assumptions we find a generalized Liouville equation which depends on two characteristic time t1 and t2 and reduces to the usual equation in the limit t1 = t2 -> 0. However, for t1 and t2 arbitrarily small but finite, our equation can be written as a finite difference equation which predicts state reduction to the diagonal form in the energy representation. The rate of decoherence becomes faster at the macroscopic limit as the energy scale of the system increases. In our approach the evolution time appears, a posteriori, as a statistical variable with a Poisson-gamma function probability distribution as if time evolution would take place randomly at average intervals t2 each evolution having a time width t1. This view point is supported by the derivation of a generalized Tam Mandelstam inequality. The relation with previous work by Milburn, with laser and micromaser theory and many experimental testable examples are described.
Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.
The chrono-topology, as introduced axiomatically in a different context, is also supported by Liouville's theorem of statistical mechanics. It is shown that, if time is quantized, the distribution function (d.f.) becomes real. An elementary solution, g, of the classical Liouville equation has been found in phase-space and time, which can be used to construct any differentiable d.f, F(g), satisfying the same Liouville equation. The conditions imposed on F(g) are reality and additivity. The reality requirement, {Im F(g)=0) quantizes: (i) F(g) and makes it time-independent, (ii). The time variable, (iii) The energy. As a verification of chronotopology, the Planck constant h has been calculated on the basis of the time quantization. The d.f. F(g) becomes, after the time quantization, a real generalized Maxwell-Boltzmann d.f, F(g) = exp[g(p, g; l1,l2,..,lN)], depending on Ν quantum numbers. These facts are significant for quantum theory, because they uncover an intrinsic relationship between Liouville's theorem and quantum mechanics.
Although the statistical mechanics of periodically driven ("Floquet") systems in contact with a heat bath has some formal analogy with the traditional statistical mechanics of undriven systems, closer examination reveals radical differences. In Floquet systems all quasienergies epsilon_{j} can be placed in a finite frequency interval 0< or =epsilon_{j}<omega (with omega the driving frequency and variant Planck's over 2pi=1 ). Therefore, if we describe a Floquet system approximately by restricting its available state space to be spanned by a finite number N of basis states, then the number of near degeneracies ( |epsilon_{j}-epsilon_{k}|< or =delta for arbitrarily small fixed delta ) in this interval grows without limit as N is increased. As we noted in a previous paper, this leads to pathologies, including drastic changes in Floquet states, as N tends to infinity. In earlier work on Floquet systems in contact with a heat bath these difficulties were often put aside by fixing N while taking the coupling to the bath to be smaller than any quasienergy difference. This led to a simple explicit theory for the reduced density matrix, with some major differences from the usual time-independent statistical mechanics. We show that, for weak but finite coupling between system and heat bath, the accuracy of a calculation within the truncated space spanned by the N lowest energy eigenstates of the undriven system is limited, as N increases indefinitely, only by the usual Born-Markov approximation, which neglects bath memory effects. As we seek higher accuracy by increasing N , we inevitably encounter quasienergy differences smaller than the system-bath coupling. We therefore derive here the equations for the steady-state reduced density matrix without restriction on the size of quasienergy splittings. In general, this matrix is no longer diagonal in the Floquet states. We analyze, in particular, the behavior near a weakly avoided crossing, where near degeneracies of quasienergies appear. In spite of the Floquet state pathologies, the explicit form of our results for the density matrix gives a consistent prescription for the statistical mechanics of periodically driven systems in the limit as N approaches infinity.
A model for intrinsic decoherence in quantum mechanics is proposed, based on a simple modification of unitary Schr\"odinger evolution. On sufficiently small time scales the system evolves by a random sequence of unitary phase changes generated by the Hamiltonian. The Schr\"odinger equation is obtained to zeroth order in the expansion parameter. Higher-order corrections lead to a loss of coherence in the energy basis. The rate of coherence loss becomes very large as the energy scale of the system is increased. The expansion parameter determines an uncertainty in the time step on very short times scales. A number of testable consequences are derived including anomalous dispersion of a free particle, decay of oscillatory systems, destruction of interference-fringe visibility, and a phase shift of interference fringes.