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arXiv:1509.02057 · Physical Review A

Quantum speed limit for a relativistic electron in a uniform magnetic field

David Velasco Villamizar, Eduardo I. Duzzioni

Published Oct 9, 2015• 29 citations
arXivHTMLPDF

Abstract

We analyze the influence of relativistic effects on the minimum evolution time between two orthogonal states of a quantum system. Defining the initial state as a homogeneous superposition between two Hamiltonian eigenstates of an electron in a uniform magnetic field, we obtain a relation between the minimum evolution time and the displacement of the mean radial position of the electron wave packet. The quantum speed limit time is calculated for an electron dynamics described by Dirac and Schr\"odinger-Pauli equations considering different parameters, such as the strength of magnetic field and the linear momentum of the electron in the axial direction. We highlight that when the electron undergoes a region with extremely strong magnetic field the relativistic and nonrelativistic dynamics differ substantially, so that the description given by the Schr\"odinger-Pauli equation enables the electron to travel faster than $c$, which is prohibited by Einstein's theory of relativity. This approach allows a connection between the abstract Hilbert space and the space-time coordinates, besides the identification of the most appropriate quantum dynamics used to describe the electron motion.

DOI
10.1103/physreva.92.042106

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We analyze the influence of relativistic effects on the minimum evolution time between two orthogonal states of a quantum system. Defining the initial state as a homogeneous superposition between two Hamiltonian eigenstates of an electron in a uniform magnetic field, we obtain a relation between the minimum evolution time and the displacement of the mean radial position of the electron wave packet. The quantum speed limit time is calculated for an electron dynamics described by Dirac and Schr"odinger-Pauli equations considering different parameters, such as the strength of magnetic field and the linear momentum of the electron in the axial direction. We highlight that when the electron undergoes a region with extremely strong magnetic field the relativistic and nonrelativistic dynamics differ substantially, so that the description given by the Schr"odinger-Pauli equation enables the electron to travel faster than ccc, which is prohibited by Einstein's theory of relativity. This approach allows a connection between the abstract Hilbert space and the space-time coordinates, besides the identification of the most appropriate quantum dynamics used to describe the electron motion.
Abstract In this article we present a detailed description, using ladder operators, of an electron in a uniform magnetic field evolving under the Schrödinger equation. We go on to describe the same physical system in terms of relativistic quantum mechanics using the Dirac equation and to compare the two models in detail. The main differences between these two quantum mechanical approaches are discussed and we observe specifically how the relativistic phenomena modify the description of this particular quantum system by isolating effects which only exist in the relativistic model.
We consider a quantum particle as a wave packet in the coordinate space. When the conjugate wave packet in the momentum space is considered, we find that the group velocities of these two wave packets, which describe the particle dynamics, are in agreement with the Hamilton equations only if in the time dependent phases one considers the Lagrangian instead of the Hamiltonian which leads to the conventional Schrodinger equation. We define a relativistic quantum principle asserting that a quantum particle has a finite frequency spectrum, with a cutoff propagation velocity c as a universal constant not depending on the coordinate system, and that any time dependent phase variation is the same in any system of coordinates. From the time dependent phase invariance, the relativistic kinematics is obtained. We consider two types of possible interactions: 1) An interaction with an external field, by a modification of the time dependent phase differential with the terms proportional to the differentials of the space-time coordinates multiplied with the components of this field four-potential, and 2) an interaction by a deformation of the space-time coordinates, due to a gravitational field. From the invariance of the time dependent phase with field components, we obtain a mechanical force of the form of Lorentz’s force, and three Maxwell equations: The Gauss-Maxwell equations for the electric and magnetic fluxes, and the Faraday-Maxwell equation for the electromagnetic induction. When the fourth equation, Ampere-Maxwell, is considered, the interaction field takes the form of the electromagnetic field. For a low propagation velocity of the particle waves, we get a packet of waves with the time dependent phases proportional to the relativistic Hamiltonian, as in Dirac’s famous theory of spin, and a slowly-varying amplitude with a phase proportional to the momentum and this velocity. In the framework of our theory, the spin is obtained as an all quantum effect, without any additional assumption to the quantum theory. When a space-time deformation is considered in the time dependent phase of a quantum particle, from the group velocity we get the particle dynamics according to the general theory of relativity. In this way, the relativistic dynamics, the electromagnetic field, and the spin of a quantum particle are obtained only from the invariance of the time dependent phases of the particle wave functions.
In this article we present a detailed description, using ladder operators, of an electron in a uniform magnetic field evolving under the Schrödinger equation.We go on to describe the same physical system in terms of relativistic quantum mechanics using the Dirac equation and to compare the two models in detail.The main differences between these two quantum mechanical approaches are discussed and we observe specifically how the relativistic phenomena modify the description of this particular quantum system by isolating effects which only exist in the relativistic model.
Quantum speed limit (QSL) defines the theoretical upper bound on how fast a quantum system can evolve between states. It imposes a fundamental constraint on the rate of quantum information processing. For a relativistic spin-up electron in a uniform magnetic field, QSL increased with the magnetic field strength till around 101510^{15}1015 Gauss, before saturating at a saturated QSL (SQSL) of 0.2407c, where c is the speed of light. We show that by using variable magnetic fields, it is possible to surpass this limit, achieving SQSL upto 0.4-0.6c. To attain this quantum phenomenon, we solve the evolution equation of relativistic electron in spatially varying magnetic fields and find that the energies of various electron states become non-degenerate as opposed to the constant magnetic field case. This redistribution of energy is the key ingredient to accomplish higher QSL and, thus, a high information processing speed. We further explore how QSL can serve as a bridge between relativistic and non-relativistic quantum dynamics, providing insights via the Bremermann-Bekenstein bound, a quantity which constrains the maximal rate of information production. We also propose a practical experimental setup to realize these advancements. These results hold immense potential for propelling fields of quantum computation, thermodynamics and metrology.
Quantum theory sets a limit on the minimum time required to transform from an initial state to a target state. It is known as quantum speed limit time. quantum speed limit time can be used to determine the rate of quantum evolution for closed and open quantum systems. Given that in the real world we are dealing with open quantum systems, the study of quantum speed limit time for such systems has particular importance. In this work we consider the topological qubit realized by two Majorana modes. We consider the case in which the topological qubit is influenced by fermionic and bosonic environment. Fermionic and bosonic environments are assumed to have Ohmic-like spectral density. The quantum speed limit time is investigated for various environment with different Ohmic parameter. It is observed that for super-Ohmic environment with increasing Ohmic parameter the quantum speed limit time gradually reaches to a constant value and thus the speed of evolution reaches to a uniform value. The effects of external magnetic field on the evolution rate are also studied. It is observed that with increasing magnitude of magnetic field, the quantum speed limit time decreases
Abstract We probe the quantum speed limit (QSL) of an electron when it is trapped in a non-uniform magnetic field. We show that the QSL increases to a large value, but within the regime of causality, by choosing a proper variation in magnetic fields. We also probe the dependence of QSL on spin of electron and find that it is higher for spin-down electron in the relativistic regime. This can be useful in achieving faster speed of transmission of quantum information. Further, we use the Bremermann–Bekenstein bound to find a critical magnetic field that bridges the gap between non-relativistic and relativistic treatments and relates to the stability of matter. An analytical framework is developed. We also provide a plausible experimental design to supplement our theory.