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Control of nanorobot motion in a quantum alpha field

Branko M. Novakovic, Dubravko Majetić, Josip Kasać, Danko Brezak

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

An alpha field is the field that can be described by two dimensionless field parameters α and α'.A quantum alpha field is the alpha field in which quantum effects can be occurred.In classical mechanics, a nanorobot has, at every moment, an exact position and an exact momentum, following the Newton's laws of motion.In quantum mechanics, nanorobots do not have exactly determined properties and when they are measured, the result is randomly drawn from a probability distribution.The classical approaches to control of nanorobot motion in the multipotential field have been presented in the authors previous papers.In this paper we continue to investigate the related control algorithms for a nanorobot motion in a quantum alpha field.In that sense the nanorobot dynamics has been described by the Schrödinger equation in an alpha field that predicts what the probability distributions are.Meanwhile, it cannot fundamentally predict the exact result of each measurement.Thus, the problem is: how to control of a nanorobot motion in the case when quantum effects are occurred.Some approaches are discussed in this paper.

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An alpha field is the field that can be described by two dimensionless field parameters α and α'.A quantum alpha field is the alpha field in which quantum effects can be occurred.In classical mechanics, a nanorobot has, at every moment, an exact position and an exact momentum, following the Newton's laws of motion.In quantum mechanics, nanorobots do not have exactly determined properties and when they are measured, the result is randomly drawn from a probability distribution.The classical approaches to control of nanorobot motion in the multipotential field have been presented in the authors previous papers.In this paper we continue to investigate the related control algorithms for a nanorobot motion in a quantum alpha field.In that sense the nanorobot dynamics has been described by the Schrödinger equation in an alpha field that predicts what the probability distributions are.Meanwhile, it cannot fundamentally predict the exact result of each measurement.Thus, the problem is: how to control of a nanorobot motion in the case when quantum effects are occurred.Some approaches are discussed in this paper.
We consider a toy model for the study of monitored dynamics in many-body quantum systems. We study the stochastic Schrödinger equation resulting from continuous monitoring with a rate Γ of a random Hermitian operator, drawn from the Gaussian unitary ensemble (GUE) at every time t. Due to invariance by unitary transformations, the dynamics of the eigenvalues {λα}α=1n of the density matrix decouples from that of the eigenvectors, and is exactly described by stochastic equations that we derive. We consider two regimes: in the presence of an extra dephasing term, which can be generated by imperfect quantum measurements, the density matrix has a stationary distribution, and we show that in the limit of large size n→∞ it matches with the inverse-Marchenko–Pastur distribution. In the case of perfect measurements, instead, purification eventually occurs and we focus on finite-time dynamics. In this case, remarkably, we find an exact solution for the joint probability distribution of λ’s at each time t and for each size n. Two relevant regimes emerge: at short times tΓ=O(1), the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the n→∞ limit. In that case, all moments of the density matrix become self-averaging and it is possible to exactly characterize the entanglement spectrum. In the limit of large times tΓ=O(n), one enters instead a regime in which the eigenvalues are exponentially separated log(λα/λβ)=O(Γt/n), but fluctuations ∼O(Γt/n) play an essential role. We are still able to characterize the asymptotic behaviors of the entanglement entropy in this regime.
We show that the Schrödinger equation describes the ensemble mean dynamics of solitons in a Galilean invariant field theory where we interpret solitons as particles. On a zero background, solitons move classically, following Newton's second law; however, on a nonzero amplitude chaotic background, their momentum and position fluctuate fulfilling an exact uncertainty relation, which gives rise to the emergence of quantum phenomena. The Schrödinger equation for the ensemble of solitons is obtained from this exact uncertainty relation, and the amplitude of the background fluctuations is what corresponds to the value of ℏ . We confirm our analytical results running simulations of solitons moving against a potential barrier and comparing the ensemble probabilities with the predictions of the time-dependent Schrödinger equation, providing a deterministic version of the quantum tunneling effect. We conclude with a discussion of how our theory does not present statistical independence between measurement and experiment outcomes.
This paper presents a novel approach to modeling dynamic quantum control scenarios involving nonlinear quantum systems. The core claim is the establishment of a model capable of accurately describing the behavior of such systems under dynamic control. This model is built upon the introduction of a nonlinear Schrödinger equation, coupled with principles from dynamic control theory. The methodology allows for detailed analysis and manipulation of nonlinear quantum systems, expanding the theoretical foundation for quantum control and manipulation research. We explore the challenges and potential solutions in designing control strategies for systems exhibiting complex, nonlinear interactions. The presented framework offers a foundation for further research into advanced quantum control techniques. The key mathematical components are detailed within the subsequent sections, including the nonlinear Schrödinger equation, control vector formulation, and analysis of system dynamics. The focus is on providing a rigorous mathematical framework rather than a specific experimental implementation.
Abstract It is well-known that the problem of the collision between two particles interacting according to the inverse square law is exactly soluble on the wave mechanics, and that the solution yields the same scattering laws as the classical theory. If, however, the two particles are identical, e.g., two electrons or two α-particles, this is not necessarily the case; for the wave functions used must be antisymmetrical or symmetrical in the co-ordinates of the two particles; and this may affect the scattering laws. In this paper we shall discuss the collision between two particles possessing spin, such as electrons, and also between two particles without spin, such as α-particles. Assuming an inverse square law force between the particles, and neglecting the actual spin forces, we shall deduce from the symmetry properties of the wave functions a scatter­ing law differing considerably from the classical. We shall also mention the various methods by which the effect could be observed, and give some experi­mental evidence in its favour. The application of the exclusion principle to collision problems has been discussed by the author in a previous paper. Suppose we wish to describe the motion of two particles interacting in any field of force. We obtain a solution w(r1 r2) of the wave equation, where r1 refers to the position of the first particle, and r2 to that of the second. If we did not use antisymmetrical wave functions, we should argue that the probability that the first particle should be at r1 and the second at r 2 would be |w (r1 r2)|2, and therefore the probability that one particle should be at r1 and the other at r2 would be |w (r1 r2)|2 + |w(r2 r1)|
Most quantum systems considered for control by external fields are plagued by a serious lack of complete information about the underlying Hamiltonian. Traditional feedback control techniques are generally not appropriate due to the latter problem, as well as the ultrafast nature of typical quantum dynamics phenomena and the fact that observations of the quantum system will inevitably lead to a disturbance which may often be contradictory to the desired control. In contrast, learning control techniques have a special role to play in the manipulation of quantum dynamics phenomena. The unique capabilities of quantum systems making them amenable to learning control are (a) the ability to have very large numbers of identical systems for submission to control, (b) the high duty cycle of laboratory laser controls, and (c) the ability to observe the impact of trial controls at ultrafast time scales. Various learning algorithms have been proposed to guide this control process. The paper discusses these proposals, as well as some new perspectives.