Nonlinear Schrödinger equation and two-level atoms

Czachor, Marek

doi:10.1103/physreva.53.1310

Cited by 20 publications

(36 citation statements)

References 32 publications

(52 reference statements)

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“…It is argued that the nonlinear quantum evolution introduced by consciousness is logically consistent and may exist (Czachor, 1995;Shan, 2004a). In addition, we may use the definite non-linearity element in the complete evolution equation of matter to define the consciousness property of matter.…”

Section: A Quantum Effect Of Consciousnessmentioning

confidence: 99%

Quantum Consciousness and Panpsychism: A Solution to the Hard Problem

Gao

2007

Neuroquantology

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We analyze the results and implications of the combination of quantum and consciousness. The quantum effect of consciousness is first explored. We show that the consciousness of the observer can help to distinguish the definite states and their quantum superposition, while the usual physical measuring device without consciousness can not do. The result indicates that the causal efficacies of consciousness do exist when considering the basic quantum process. Based on this conclusion, we demonstrate that consciousness is not reducible or emergent, but a new fundamental property of matter. This provides a quantum basis for panpsychism. Furthermore, we argue that the conscious process is one kind of quantum computation process based on the analysis of consciousness time and combination problem. It is shown that a unified theory of matter and consciousness should include two parts: one is the complete quantum evolution of matter state, which includes the definite nonlinear evolution element introduced by consciousness, and the other is the psychophysical principle or corresponding principle between conscious content and matter state. Lastly, some experimental suggestions are presented to confirm the theoretical analysis of the paper.

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Section: A Quantum Effect Of Consciousnessmentioning

confidence: 99%

Quantum Consciousness and Panpsychism: A Solution to the Hard Problem

Gao

2007

Neuroquantology

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“…An important drawback of such Hamiltonian generalizations is that we have to represent observables by nonlinear operators which leads to interpretational difficulties. To give an example, it is not clear which definition of a nonlinear eigenvalue is physically meaningful, or how to represent higher moments of experimentally measured random variables if nonlinear operators are involved (Czachor, 1996a). Let us therefore first consider what happens if H = H a ρ a = TrĤρ and S is an arbitrary (differentiable) function of the Casimirs C k .…”

Section: -Bracketmentioning

confidence: 99%

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Czachor¹

1999

International Journal of Theoretical Physics

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“…The physical problem turns out to be of the same type as the one with the definition of dynamics of composite systems given by Weinberg in [30] . The Weinberg definition not only led to the nowadays famous "faster-than-light telegraph" but also predicted an apparently paradoxical disagreement between the Bloch equation and Janes-Cummings approaches to two-level systems [31]. A corrected description [31] showed that the paradox is a result of a wrong formalism.…”

Section: Complete Positivity and Nonlinearitymentioning

confidence: 99%

“…The Weinberg definition not only led to the nowadays famous "faster-than-light telegraph" but also predicted an apparently paradoxical disagreement between the Bloch equation and Janes-Cummings approaches to two-level systems [31]. A corrected description [31] showed that the paradox is a result of a wrong formalism. It proved also that a precise way of describing noninteracting systems leads to a meaningful dynamics when the systems are coupled.…”

Section: Complete Positivity and Nonlinearitymentioning

confidence: 99%

Complete positivity of nonlinear evolution: A case study

Czachor

Kuna

1998

Phys. Rev. A

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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.PACS numbers: 02.30. Sa, 03.65.Bz, Db, 11.10.Lm I. COMPLETE POSITIVITY AND NONLINEARITYLinear maps that are positive but not completely positive (CP) [1][2][3][4][5][6][7][8] have been shown to play an essential role in characterization of degree of entanglement of correlated quantum systems [9][10][11], an important issue for quantum computation and cryptography. A possibility of an experimental verification of CP of quantum evolutions was discussed in the context of the neutral kaon decay problem in [12]. Although the notion of CP may seem somewhat abstract and technical, it has a simple physical interpretation for linear maps. We begin with a system, labelled "1", whose dynamics is given by some positive map φ t 1 (a) = a(t), φ t 1 : A → A where A is a set of bounded operators acting in a Hilbert space H 1 . To avoid technicalities we assume H 1 is finite dimensional. In linear quantum mechanics a reversible dynamics is given by φ t 1 (a) = U t aU −1 t where U t is unitary. We require positivity of φ t 1 since if a is a density matrix we want the same to be true for a(t). Now consider a density matrix ρ 1+2 (0) of some bigger system "1+2" consisting of the original one plus a system whose dimension is m and which evolves trivially (its U t = 1). The initial density matrix of "1+2" is of the formIn the context of CP maps it is more standard to use the isomorphic form [13]It follows, the argument continues, that since the dynamics on A is given by φ t 1 each of the entries evolves by a kl → φ t 1 (a kl ) and the whole density matrix is mapped intowhich in linear quantum mechanics reduces toIf ρ 1+2 (t) = φ t 1+2 ρ 1+2 (0) is to be a density matrix it should not lead to negative probabilities. Moreover one should be able to do the construction for any m. If this is the case the map φ t 1 is said to be CP. The dynamics one typically thinks of in quantum mechanics is linear and therefore the notion of CP was initially defined only for linear maps [14]. However there are many situations in quantum physics where the dynamics is nonlinear. A nonlinear evolution of observables in Heisenberg picture is typical of quantum optics and field theory. Nonlinearly evolving states appear in mean field theories (Hartree-type equations [15]), soliton theory (nonlinear Schrödinger equations), and various attempts of nonlinear generalizations of quantum mechanics. Although the latter theories do not yet correspond to any concrete physical situation they have led to some formal developments especially due to the famous "Einstein-PodolskyRosen malignancy" discussed by Gisin and others [16][17][18][19] (see Appendix C).The argu...

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Nonlinear Schrödinger equation and two-level atoms

Cited by 20 publications

References 32 publications

Quantum Consciousness and Panpsychism: A Solution to the Hard Problem

Quantum Consciousness and Panpsychism: A Solution to the Hard Problem

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Complete positivity of nonlinear evolution: A case study

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