arXiv subjects
Publications and source records attributed to Marek Czachor.
We show that extensive thermostatistics based on Renyi entropy and Kolmogorov-Nagumo averages can be expressed in terms of Tsallis non- extensive thermostatistics. We use this correspondence to generalize thermostatistics to a large class of Kolmogorov-Nagumo means and suitably adapted definitions of entropy.
The paper is an extension of quant-ph/9912102. The new framework is tested on the 2-photon spontaneous emission and blackbody radiation. The new effects are rather subtle. The probability of the 2-photon emission is shown to consist of a product of several terms: One which is identical to this from standard quantum optics and the remaining ones formally resembling detector inefficiencies (here arising from nontrivial vacuum structure). The blackbody distribution is indistinguishable from the Planck law for T T_{critical} the maximum of the distribution gets lowered and shifts towards higher frequencies. T_{critical} is a parameter that, in principle, should be observable. No normal ordering of operators is needed and vacuum energy is nonzero but finite. Vacuum is represented by a subspace spanned by ground states of the oscillators and is not equivalent to the cyclic vector of the GNS construction. The non-CCR algebra is discussed in more detail.
A new scheme of field quantization is proposed. Instead of associating with different frequencies different oscillators we begin with a single oscillator that can exist in a superposition of different frequencies. The idea is applied to the electromagnetic radiation field. Using the standard Dirac-type mode-quantization of the electromagnetic field we obtain several standard properties such as coherent states or spontaneous and stimulated emission. As opposed to the standard approach the vacuum energy is finite and does not have to be removed by any ad hoc procedure.
Starting with a time-independent Hamiltonian $h$ and an appropriately chosen solution of the von Neumann equation $i\dotρ(t)=[ h,ρ(t)]$ we construct its binary-Darboux partner $h_1(t)$ and an exact scattering solution of $i\dotρ_1(t)=[h_1(t),ρ_1(t)]$ where $h_1(t)$ is time-dependent and not isospectral to $h$. The method is analogous to supersymmetric quantum mechanics but is based on a different version of a Darboux transformation. We illustrate the technique by the example where $h$ corresponds to a 1-D harmonic oscillator. The resulting $h_1(t)$ represents a scattering of a soliton-like pulse on a three-level system.
We review some recent developments in the theory of nonlinear von Neumann equations. We distinguish between the von Neumann equation (which can be nonlinear) and the Liouville equation (which should be linear). Explicit examples illustrate the technique of binary Darboux integration of nonlinear density matrix equations and special attention is payed to the problem of how to find physically nontrivial `self-scattering' solutions.
Generalized Euler-Arnold-von Neumann density matrix equations can be solved by a binary Darboux transformation given here in a new form: $ρ[1]=e^{P\ln(μ/ν)}ρe^{-P\ln(μ/ν)}$ where $P=P^2$ is explicitly constructed in terms of conjugated Lax pairs, and $μ$, $ν$ are complex. As a result spectra of $ρ$ and $ρ[1]$ are identical. Transformations allowing to shift and rescale spectrum of a solution are introduced, and a class of stationary seed solutions is discussed.
Cryptographic scheme proposed by Bennett, Brassard, and Mermin [Phys. Rev. Lett. {\bf 68}, 557 (1992)] is reformulated in a version involving two polarizing Mach-Zehnder interferometers. Such a form, although physically equivalent to the original one, makes its security explicit, suggestive and easy to explain to non-experts.
Combination of the Liouville equation with the q-averaged energy $U_q = _q$ leads to a microscopic framework for nonextensive q-thermodynamics. The resulting von Neumann equation is nonlinear: $i\dotρ=[H,ρ^q]$. In spite of its nonlinearity the dynamics is consistent with linear quantum mechanics of pure states. The free energy $F_q=U_q-TS_q$ is a stability function for the dynamics. This implies that q-equilibrium states are dynamically stable. The (microscopic) evolution of $ρ$ is reversible for any q, but for $q\neq 1$ the corresponding macroscopic dynamics is irreversible.
A new form of a binary Darboux transformation is used to generate analytical solutions of a nonlinear Liouville-von Neumann equation. General theory is illustrated by explicit examples.
Simple Hartree-type equations lead to dynamics of a subsystem that is not completely positive in the sense accepted in mathematical literature. In the linear case this would imply that negative probabilities have to appear for some system that contains the subsystem in question. In the nonlinear case this does not happen because the mathematical definition is physically unfitting as shown on a concrete example.
A general method for extending a non-dissipative nonlinear Schrödinger and Liouville-von Neumann 1-particle dynamics to an arbitrary number of particles is described. It is shown at a general level that the dynamics so obtained is completely separable, which is the strongest condition one can impose on dynamics of composite systems. It requires that for all initial states (entangled or not) a subsystem not only cannot be influenced by any action undertaken by an observer in a separated system (strong separability), but additionally that the self-consistency condition $Tr_2\circ ϕ^t_{1+2}=ϕ^t_{1}\circ Tr_2$ is fulfilled. It is shown that a correct extension to $N$ particles involves integro-differential equations which, in spite of their nonlocal appearance, make the theory fully local. As a consequence a much larger class of nonlinearities satisfying the complete separability condition is allowed than has been assumed so far. In particular all nonlinearities of the form $F(|ψ(x)|)$ are acceptable. This shows that the locality condition does not single out logarithmic or 1-homeogeneous nonlinearities.
The nonlinear algorithms proposed recently by Abrams and Lloyd [Report No. quant-ph/9801041] are fast but make an explicit use of an arbitrarily fast unphysical transfer of information within a quantum computer. It is shown that there exists a simplification of the second Abrams-Lloyd algorithm which eliminates the unphysical effect but keeps the algorithm fast.
Recenty Abrams and Lloyd have proposed a fast algorithm that is based on a nonlinear evolution of a state of a quantum computer. They have explicitly used the fact that nonlinear evolutions in Hilbert spaces do not conserve scalar products of states, and applied a description of separated systems taken from Weinberg's nonlinear quantum mechanics. On the other hand it is known that violation of orthogonality combined with the Weinberg-type description generates unphysical, arbitrarily fast influences between noninteracting systems. It was not therefore clear whether the algorithm is fast because arbitrarily fast unphysical effects are involved. In these notes I show that this is not the case. I analyze both algorithms proposed by Abrams and Lloyd on concrete, simple models of nonlinear evolution. The description I choose is known to be free of the unphysical influences (therefore it is not the Weinberg one). I show, in particular, that the correct local formalism allows even to simplify the algorithm.
We discuss a simple example demonstrating that spontaneous emission from "space-time-superposed" atomic center-of-mass wave packets is nontrivially and time-dependent modified with respect to the standard dipole-pattern typical of "space-superposed" wave packets. Our approach provides an approximate description of a nonsimultaneous interaction of electromagnetic field with different parts of a wave packet.
The spectrum of a density matrix $ρ(t)$ is conserved by a Lie-Nambu dynamics if $ρ(t)$ is a self-adjoint and Hilbert-Schmidt solution of a nonlinear triple-bracket equation. This generalizes to arbitrary separable (positive- and indefinite-metric) Hilbert spaces the previous result which was valid for finite-dimensional Hilbert spaces.
Nonlinear Doebner-Goldin [Phys. Rev. A 54, 3764 (1996)] gauge transformations (NGT) defined in terms of a wave function $ψ(x)$ do not form a group. To get a group property one has to consider transformations that act differently on different branches of the complex argument function and the knowledge of the value of $ψ(x)$ is not sufficient for a well defined NGT. NGT that are well defined in terms of $ψ(x)$ form a semigroup parametrized by a real number $γ$ and a nonzero $λ$ which is either an integer or $-1\leq λ\leq 1$. An extension of NGT to projectors and general density matrices leads to NGT with complex $γ$. Both linearity of evolution and Hermiticity of density matrices are gauge dependent properties.
Linear quantum mechanics can be regarded as a particular example of a nonlinear Nambu-type theory. Some elements of this approach are presented.
A classical system violating the Bell inequality is discussed. The system is local, deterministic, observers have free will, and detectors are ideal so that no data are lost. The trick is based on two elements. First, a state of one observer is locally influenced by a "particle". Second, random variables used in the experiment are complementary. A relation of this effect to nonlocality is discussed.