DOCSLIB.ORG

Thomas Precession

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Thomas Precession FYGB08 Project Thomas precession Author: SSN: Daniel Kronberg 850810-5999 Abstract This project is a derivation of Thomas precession, a relativistic correction to a vector traveling along a curvilinear trajectory. We consider the precession as a consequence of consecutive restricted Lorentz transformations between instantly inertial reference frames along the trajectory. These tranformations ~v×~v_ result in an observed retrograde precession of the vector, with the angular velocity ~!T = −(γ − 1) v2 . 1 Introduction This project is an attempt at a straightforward explanation of Thomas precession by discussing Thomas- Wigner rotation algebraically as a consequence of consecutive restricted Lorentz transformations. Fermi- Walker transport is approached in a fairly conceptual manner as a motivation of how results based on a set of inertial reference frames can be applied to accelerated systems. It is possible to derive the Thomas angular velocity directly from the equations given in the Fermi-Walker section, but here we have chosen to derive it from Thomas-Wigner rotation. Illustrated reference frames are shown as a tetrad of vectors, where the f0g-vector is a time-like vector, Minkowski-orthogonal to the three space-like vectors f1; 2; 3g. 2 Lorentz transformations Lorentz transformations are used to connect observations between inertial reference frames. In particular they give the correct transformation rules at relativistic velocities. A single Lorentz transformation along a given direction is called a restricted Lorentz transformation, or boost, and affects the time component and ~v the spatial component parallel to the velocity. If the basis vectors are chosen so thatx ^ = j~vj , then a boost has the form ~vx^ t^0 = γ 1 − c2 x^0 = γ(^x − ~vt) y^0 =y ^ z^0 =z ^ 1 ~v γ = ; β = ; p1 − β2 c where ~v is the boost velocity, and c is the speed of light. A transformation from the primed frame into some other frame with a collinear velocity has the same form, and is equivalent to a single boost from the first frame with a total velocity given by β + β0 β00 = : (1) 1 + ββ0 Any boost can be represented by a symmetric matrix, and the general boost form is 2 3 γ −γβx −γβy −γβz 2 6 βx βxβy βxβz 7 6−γβx 1 + (γ − 1) β2 (γ − 1) β2 (γ − 1) β2 7 L = 6 2 7 : 6 βy βx βy βy βz 7 6−γβy (γ − 1) β2 1 + (γ − 1) β2 (γ − 1) β2 7 4 2 5 βz βx βz βy βz −γβz (γ − 1) β2 (γ − 1) β2 1 + (γ − 1) β2 However, the product of two non-collinear boosts is not equivalent to any single boost, but instead to some boost and some rotation. For example, two perpendicular boosts 2 3 2 3 γ −γβ 0 0 γ0 0 −γ0β0 0 6 7 6 7 6−γβ γ 0 07 6 0 1 0 07 6 7 0 6 7 L = 6 7 ;L = 6 7 6 0 0 1 07 6−γ0β0 0 γ0 07 4 5 4 5 0 0 0 1 0 0 0 1 1 have the products 2 3 γγ0 −γβ −γγ0β0 0 6 7 6−γγ0β γ γγ0ββ0 07 0 6 7 LL = 6 7 (2) 6 −γ0β0 0 γ0 07 4 5 0 0 0 1 2 3 γγ0 −γγ0β −γ0β0 0 6 7 6 −γβ γ 0 07 0 6 7 L L = 6 7 ; (3) 6−γγ0β0 γγ0ββ0 γ0 07 4 5 0 0 0 1 which are not symmetric; notice further that the product is non-commutative. As a result, the coordinate axes of the final reference frame are not parallel with those of the first, but are instead rotated by some angle θ about an axis orthogonal to the boosts. We can hence express the product of two boosts as a product of a boost and a rotation. This rotation is called Thomas rotation, Wigner rotation, or Thomas-Wigner rotation, after Llewellyn Thomas and Eugene Wigner. We will in the next section derive the explicit expression of the rotation matrix represnting the Thomas-Wigner rotation. 3 Thomas-Wigner rotation Consider two inertial reference frames, observed from the lab frame ^l. When applying boosts first to the e^ frame and then frome ^ to thee ^0 frame, we will find that even though the coordinate axes of ^l ande ^ are parallel to each other, and the coordinate axes ofe ^ ande ^0 are also parallel to each other, the axes of ^l and e^0 are not in general parallel. 0 e^0 e^0 1 0 e^3 0 e^2 00 β β0 L0 00 Rθ L ^l0 e^0 β L ^l1 e^1 ^l3 e^3 ^l2 e^2 Figure 1: Three inertial reference frames with individual relative velocities. The Lorentz transformation from ^l toe ^0 is given by the matri L0L. As previously discussed, this can be 0 00 written as the product of a rotation and a boost, i.e. L L = RθL . In order to find the rotation angle θ between the two frames, we follow the method outlined in [1] by introducing an arbitrary rotation matrix and using it to find an expression for the boost L00. Since the coordinate axes can always be chosen parallel 2 to the boosts, it is sufficient to consider rotation in thex ^ ^ y^ plane. 2 3 1 0 0 0 6 7 60 cos(θ) sin(θ) 07 6 7 Rθ = 6 7 60 − sin(θ) cos(θ) 07 4 5 0 0 0 1 2 3 γγ0 −γγ0β −γ0β0 0 6 7 6 γ(γ0β0 sin(θ) − β cos(θ)) −γ(γ0ββ0 sin(θ) − cos(θ)) −γ0 sin(θ) 07 00 −1 0 6 7 L = Rθ L L = 6 7 (4) 6−γ(β sin(θ) + γ0β0 cos(θ)) γ(sin(θ) + γ0ββ0 cos(θ)) γ0 cos(θ) 07 4 5 0 0 0 1 By requiring that the matrix L00, as a boost, be symmetric, the elements of the rotational matrix can be determined 8 −γ0 sin(θ) = γ(sin(θ) + γ0ββ0 cos(θ)) <> −γγ0β = γ(γ0β0 sin(θ) − β cos(θ)) :>−γ0β0 = −γ(β sin(θ) + γ0β0 cos(θ)) =) γ + γ0 γγ0ββ0 cos(θ) = ; sin(θ) = − : (5) γγ0 + 1 γγ0 + 1 Substituting the resulting expressions into L00 gives us 2 3 γγ0 −γγ0β −γ0β0 0 6 0 γ2γ02β2 γγ02ββ0 7 00 6−γγ β 1 + γγ0+1 γγ0+1 07 L = 6 0 0 7 : (6) 6 0 0 γγ02ββ0 γ (γ+γ ) 7 6 −γ β 0 0 07 4 γγ +1 γγ +1 5 0 0 0 1 This method can be applied to any two boosts by splitting them up into collinear and perpendicular components and adding their velocities according to equation (1). For three or more arbitrary boosts, the expressions are more complex, but in the situations that are interesting when discussing Thomas precession, we need only deal with motion in a plane. 4 Fermi-Walker transport The results from the previous section can be applied easily when dealing only with inertial frames, however, many interesting systems are accelerated, and it is not immediately evident how Thomas-Wigner rotation would apply to such systems. In order to describe accelerated systems, we will consider Fermi-Walker transport, a form of parallel-transport used to move a vector along a curvilinear trajectory with the minimum possible amount of rotation. In this model, we treat the trajectory of the accelerated system as consisting of an infinite series of instantaneously inertial frames, each related to its neighbors by an infinitesimal boost in the direction of acceleration. This is a commonly used approximation in relativity theory [2]. 3 0 e^0 0 e^3 0 e^0 e^1 2 e^0 e^1 e^3 e^2 Figure 2: Two instantaneously inertial frames on a curved world line, representing an accelerated system. Let us describe Fermi-Walker transport in some more detail, following [3]. Consider an accelerated particle and its world-line, observed from the Thomas conventional frame; an inertial frame related to the lab frame by a single boost to some initial frame of the particle. Then consider the space-like hyperplane that is orthogonal to the time-axis. Any purely space-like vector ~q in the accelerated system must be continuously projected into the hyperplane of the next inertial frame. Each infinitesimal projection will be in the direction of the time-axis, and so the evolution of the vector ~q will be d~q = ke^ dτ 0 where τ is the proper time and k is some proportonality factor. Further, ~q · e^0 = 0 and d d~q d^e (~q · e^ ) = · e^ + ~q · 0 = k + ~q ·~b = 0 dτ 0 dτ 0 dτ ~ d^e0 where b = dτ is the world acceleration ande ^0 · e^0 = 1. Therefore the Fermi-Walker transport equation for the vector ~q is d~q = −(~q ·~b)^e dτ 0 ~ ~ By differentiating two space-like vectors, ~q = −(~q · b)^e0 and ~p = −(~p · b)^e0 , we find that the scalar product is preserved by Fermi-Walker transport. d (~q · ~p) = −(~q ·~b)(~p · e^ ) − (~q · e^ )(~p ·~b) = 0 dτ 0 0 Therefore the change in any space-like vector parallel-transported along this curved world-line is a rotation only. The length of the vector remains constant. With the basic concepts of Fermi-Walker transport in mind, we can now apply the results to the description of Thomas-Wigner rotation. 5 Thomas precession As a vector is Fermi-Walker transported over a curvilinear trajectory, it will experience countless infinitesimal Thomas-Wigner rotations. Since the rotation angle is dependent on the relative velocities of the boosts, the percieved rotation angle will depend on the reference system from which the parallel-transported frame is observed.
Load more

Recommended publications
  • UC Berkeley UC Berkeley Electronic Theses and Dissertations
    UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Applications of Magnetic Resonance to Current Detection and Microscale Flow Imaging Permalink https://escholarship.org/uc/item/2fw343zm Author Halpern-Manners, Nicholas Wm Publication Date 2011 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Applications of Magnetic Resonance to Current Detection and Microscale Flow Imaging by Nicholas Wm Halpern-Manners A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Chemistry in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Alexander Pines, Chair Professor David Wemmer Professor Steven Conolly Spring 2011 Applications of Magnetic Resonance to Current Detection and Microscale Flow Imaging Copyright 2011 by Nicholas Wm Halpern-Manners 1 Abstract Applications of Magnetic Resonance to Current Detection and Microscale Flow Imaging by Nicholas Wm Halpern-Manners Doctor of Philosophy in Chemistry University of California, Berkeley Professor Alexander Pines, Chair Magnetic resonance has evolved into a remarkably versatile technique, with major appli- cations in chemical analysis, molecular biology, and medical imaging. Despite these successes, there are a large number of areas where magnetic resonance has the potential to provide great insight but has run into significant obstacles in its application. The projects described in this thesis focus on two of these areas. First, I describe the development and implementa- tion of a robust imaging method which can directly detect the effects of oscillating electrical currents. This work is particularly relevant in the context of neuronal current detection, and bypasses many of the limitations of previously developed techniques.
    [Show full text]
  • Thomas Precession and Thomas-Wigner Rotation: Correct Solutions and Their Implications
    epl draft Header will be provided by the publisher This is a pre-print of an article published in Europhysics Letters 129 (2020) 3006 The final authenticated version is available online at: https://iopscience.iop.org/article/10.1209/0295-5075/129/30006 Thomas precession and Thomas-Wigner rotation: correct solutions and their implications 1(a) 2 3 4 ALEXANDER KHOLMETSKII , OLEG MISSEVITCH , TOLGA YARMAN , METIN ARIK 1 Department of Physics, Belarusian State University – Nezavisimosti Avenue 4, 220030, Minsk, Belarus 2 Research Institute for Nuclear Problems, Belarusian State University –Bobrujskaya str., 11, 220030, Minsk, Belarus 3 Okan University, Akfirat, Istanbul, Turkey 4 Bogazici University, Istanbul, Turkey received and accepted dates provided by the publisher other relevant dates provided by the publisher PACS 03.30.+p – Special relativity Abstract – We address to the Thomas precession for the hydrogenlike atom and point out that in the derivation of this effect in the semi-classical approach, two different successions of rotation-free Lorentz transformations between the laboratory frame K and the proper electron’s frames, Ke(t) and Ke(t+dt), separated by the time interval dt, were used by different authors. We further show that the succession of Lorentz transformations KKe(t)Ke(t+dt) leads to relativistically non-adequate results in the frame Ke(t) with respect to the rotational frequency of the electron spin, and thus an alternative succession of transformations KKe(t), KKe(t+dt) must be applied. From the physical viewpoint this means the validity of the introduced “tracking rule”, when the rotation-free Lorentz transformation, being realized between the frame of observation K and the frame K(t) co-moving with a tracking object at the time moment t, remains in force at any future time moments, too.
    [Show full text]
  • Newtonian Gravity and Special Relativity 12.1 Newtonian Gravity
    Physics 411 Lecture 12 Newtonian Gravity and Special Relativity Lecture 12 Physics 411 Classical Mechanics II Monday, September 24th, 2007 It is interesting to note that under Lorentz transformation, while electric and magnetic fields get mixed together, the force on a particle is identical in magnitude and direction in the two frames related by the transformation. Indeed, that was the motivation for looking at the manifestly relativistic structure of Maxwell's equations. The idea was that Maxwell's equations and the Lorentz force law are automatically in accord with the notion that observations made in inertial frames are physically equivalent, even though observers may disagree on the names of these forces (electric or magnetic). Today, we will look at a force (Newtonian gravity) that does not have the property that different inertial frames agree on the physics. That will lead us to an obvious correction that is, qualitatively, a prediction of (linearized) general relativity. 12.1 Newtonian Gravity We start with the experimental observation that for a particle of mass M and another of mass m, the force of gravitational attraction between them, according to Newton, is (see Figure 12.1): G M m F = − RR^ ≡ r − r 0: (12.1) r 2 From the force, we can, by analogy with electrostatics, construct the New- tonian gravitational field and its associated point potential: GM GM G = − R^ = −∇ − : (12.2) r 2 r | {z } ≡φ 1 of 7 12.2. LINES OF MASS Lecture 12 zˆ m !r M !r ! yˆ xˆ Figure 12.1: Two particles interacting via the Newtonian gravitational force.
    [Show full text]
  • Theoretical Foundations for Design of a Quantum Wigner Interferometer
    IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 55, NO. 1, FEBRUARY 2019 Theoretical Foundations for Design of a Quantum Wigner Interferometer Marco Lanzagorta and Jeffrey Uhlmann Abstract— In this paper, we discuss and analyze a new where is referred to as the Wigner angle of rotation. The quantum-based approach for gravimetry. The key feature of this value of completely encodes the net rotation of the qubit design is that it measures effects of gravitation on information due to interaction with a gravitational field. In most realistic encoded in qubits in a way that can provide resolution beyond the de Broglie limit of atom-interferometric gravimeters. We contexts it would be extremely difficult, if not impossible, show that it also offers an advantage over current state-of-the-art to estimate/predict with any practical fidelity the value of gravimeters in its ability to detect quadrupole field anomalies. a priori from extrinsic measurements of the gravitational This can potentially facilitate applications relating to search field. Although the gravitational field could be thought of and recovery, e.g., locating a submerged aircraft on the ocean negatively as “corrupting” the state the qubit, it is also floor, based on the difference between the specific quadrupole signature of the object of interest and that of other objects in possible to interpret the altered state of the qubit positively the environment. as being a measurement of that field. In the case of a qubit orbiting the earth at the radius of a typical GPS satellite, Index Terms— Quantum sensing, gravimetry, qubits, quadru- −9 pole field anomaly, atom interferometry, Wigner gravimeter.
    [Show full text]
  • Chapter 5 the Relativistic Point Particle
    Chapter 5 The Relativistic Point Particle To formulate the dynamics of a system we can write either the equations of motion, or alternatively, an action. In the case of the relativistic point par- ticle, it is rather easy to write the equations of motion. But the action is so physical and geometrical that it is worth pursuing in its own right. More importantly, while it is difficult to guess the equations of motion for the rela- tivistic string, the action is a natural generalization of the relativistic particle action that we will study in this chapter. We conclude with a discussion of the charged relativistic particle. 5.1 Action for a relativistic point particle How can we find the action S that governs the dynamics of a free relativis- tic particle? To get started we first think about units. The action is the Lagrangian integrated over time, so the units of action are just the units of the Lagrangian multiplied by the units of time. The Lagrangian has units of energy, so the units of action are L2 ML2 [S]=M T = . (5.1.1) T 2 T Recall that the action Snr for a free non-relativistic particle is given by the time integral of the kinetic energy: 1 dx S = mv2(t) dt , v2 ≡ v · v, v = . (5.1.2) nr 2 dt 105 106 CHAPTER 5. THE RELATIVISTIC POINT PARTICLE The equation of motion following by Hamilton’s principle is dv =0. (5.1.3) dt The free particle moves with constant velocity and that is the end of the story.
    [Show full text]
  • Magnetic Moment of a Spin, Its Equation of Motion, and Precession B1.1.6
    Magnetic Moment of a Spin, Its Equation of UNIT B1.1 Motion, and Precession OVERVIEW The ability to “see” protons using magnetic resonance imaging is predicated on the proton having a mass, a charge, and a nonzero spin. The spin of a particle is analogous to its intrinsic angular momentum. A simple way to explain angular momentum is that when an object rotates (e.g., an ice skater), that action generates an intrinsic angular momentum. If there were no friction in air or of the skates on the ice, the skater would spin forever. This intrinsic angular momentum is, in fact, a vector, not a scalar, and thus spin is also a vector. This intrinsic spin is always present. The direction of a spin vector is usually chosen by the right-hand rule. For example, if the ice skater is spinning from her right to left, then the spin vector is pointing up; the skater is rotating counterclockwise when viewed from the top. A key property determining the motion of a spin in a magnetic field is its magnetic moment. Once this is known, the motion of the magnetic moment and energy of the moment can be calculated. Actually, the spin of a particle with a charge and a mass leads to a magnetic moment. An intuitive way to understand the magnetic moment is to imagine a current loop lying in a plane (see Figure B1.1.1). If the loop has current I and an enclosed area A, then the magnetic moment is simply the product of the current and area (see Equation B1.1.8 in the Technical Discussion), with the direction n^ parallel to the normal direction of the plane.
    [Show full text]
  • Electron Spins in Nonmagnetic Semiconductors
    Electron spins in nonmagnetic semiconductors Yuichiro K. Kato Institute of Engineering Innovation, The University of Tokyo Physics of non-interacting spins Optical spin injection and detection Spin manipulation in nonmagnetic semiconductors Physics of non-interacting spins 2 In non-magnetic semiconductors such as GaAs and Si, spin interactions are weak; To first order approximation, they behave as non-interacting, independent spins. • Zeeman Hamiltonian • Bloch sphere • Larmor precession ∗ • , , and • Bloch equation The Zeeman Hamiltonian 3 Hamiltonian for an electron spin in a magnetic field magnetic moment of an electron spin : magnetic moment : magnetic field : Landé g-factor (=2 for free electrons) : Bohr magneton 58 eV/T , ) : electronic charge : Planck constant : free electron mass : spin operator : Pauli operator The Zeeman Hamiltonian 4 Hamiltonian for an electron spin in a magnetic field magnetic moment of an electron spin : magnetic moment : magnetic field : Landé g-factor (=2 for free electrons) : Bohr magneton 2 58 eV/T : electronic charge : Planck constant : free electron mass : spin operator 2 : Pauli operator The energy eigenstates 5 Without loss of generality, we can set the -axis to be the direction of , i.e., 0,0, 1 Spin “up” |↑ |↑ 2 1 2 2 1 Spin “down” |↓ |↓ 2 1 2 2 Spinor notation and the Pauli operators 6 Quantum states are represented by a normalized vector in a Hilbert space Spin states are represented by 2D vectors Spin operators are represented by 2x2 matrices.
    [Show full text]
  • 4 Nuclear Magnetic Resonance
    Chapter 4, page 1 4 Nuclear Magnetic Resonance Pieter Zeeman observed in 1896 the splitting of optical spectral lines in the field of an electromagnet. Since then, the splitting of energy levels proportional to an external magnetic field has been called the "Zeeman effect". The "Zeeman resonance effect" causes magnetic resonances which are classified under radio frequency spectroscopy (rf spectroscopy). In these resonances, the transitions between two branches of a single energy level split in an external magnetic field are measured in the megahertz and gigahertz range. In 1944, Jevgeni Konstantinovitch Savoiski discovered electron paramagnetic resonance. Shortly thereafter in 1945, nuclear magnetic resonance was demonstrated almost simultaneously in Boston by Edward Mills Purcell and in Stanford by Felix Bloch. Nuclear magnetic resonance was sometimes called nuclear induction or paramagnetic nuclear resonance. It is generally abbreviated to NMR. So as not to scare prospective patients in medicine, reference to the "nuclear" character of NMR is dropped and the magnetic resonance based imaging systems (scanner) found in hospitals are simply referred to as "magnetic resonance imaging" (MRI). 4.1 The Nuclear Resonance Effect Many atomic nuclei have spin, characterized by the nuclear spin quantum number I. The absolute value of the spin angular momentum is L =+h II(1). (4.01) The component in the direction of an applied field is Lz = Iz h ≡ m h. (4.02) The external field is usually defined along the z-direction. The magnetic quantum number is symbolized by Iz or m and can have 2I +1 values: Iz ≡ m = −I, −I+1, ..., I−1, I.
    [Show full text]
  • Covariant Calculation of General Relativistic Effects in an Orbiting Gyroscope Experiment
    PHYSICAL REVIEW D 67, 062003 ͑2003͒ Covariant calculation of general relativistic effects in an orbiting gyroscope experiment Clifford M. Will* McDonnell Center for the Space Sciences, Department of Physics, Washington University, St. Louis, Missouri 63130 ͑Received 17 December 2002; published 26 March 2003͒ We carry out a covariant calculation of the measurable relativistic effects in an orbiting gyroscope experi- ment. The experiment, currently known as Gravity Probe B, compares the spin directions of an array of spinning gyroscopes with the optical axis of a telescope, all housed in a spacecraft that rolls about the optical axis. The spacecraft is steered so that the telescope always points toward a known guide star. We calculate the variation in the spin directions relative to readout loops rigidly fixed in the spacecraft, and express the variations in terms of quantities that can be measured, to sufficient accuracy, using an Earth-centered coordi- nate system. The measurable effects include the aberration of starlight, the geodetic precession caused by space curvature, the frame-dragging effect caused by the rotation of the Earth and the deflection of light by the Sun. DOI: 10.1103/PhysRevD.67.062003 PACS number͑s͒: 04.80.Cc I. INTRODUCTION by the on-board telescope, which is to be trained on a star IM Pegasus ͑HR 8703͒ in our galaxy. One important feature of Gravity Probe B—the ‘‘gyroscope experiment’’—is a this star is that it is also a strong radio source, so that its NASA space experiment designed to measure the general direction and proper motion relative to the larger system of relativistic effect known as the dragging of inertial frames.
    [Show full text]
  • Theory of Angular Momentum and Spin
    Chapter 5 Theory of Angular Momentum and Spin Rotational symmetry transformations, the group SO(3) of the associated rotation matrices and the 1 corresponding transformation matrices of spin{ 2 states forming the group SU(2) occupy a very important position in physics. The reason is that these transformations and groups are closely tied to the properties of elementary particles, the building blocks of matter, but also to the properties of composite systems. Examples of the latter with particularly simple transformation properties are closed shell atoms, e.g., helium, neon, argon, the magic number nuclei like carbon, or the proton and the neutron made up of three quarks, all composite systems which appear spherical as far as their charge distribution is concerned. In this section we want to investigate how elementary and composite systems are described. To develop a systematic description of rotational properties of composite quantum systems the consideration of rotational transformations is the best starting point. As an illustration we will consider first rotational transformations acting on vectors ~r in 3-dimensional space, i.e., ~r R3, 2 we will then consider transformations of wavefunctions (~r) of single particles in R3, and finally N transformations of products of wavefunctions like j(~rj) which represent a system of N (spin- Qj=1 zero) particles in R3. We will also review below the well-known fact that spin states under rotations behave essentially identical to angular momentum states, i.e., we will find that the algebraic properties of operators governing spatial and spin rotation are identical and that the results derived for products of angular momentum states can be applied to products of spin states or a combination of angular momentum and spin states.
    [Show full text]
  • Optical Detection of Electron Spin Dynamics Driven by Fast Variations of a Magnetic Field
    www.nature.com/scientificreports OPEN Optical detection of electron spin dynamics driven by fast variations of a magnetic feld: a simple ∗ method to measure T1 , T2 , and T2 in semiconductors V. V. Belykh1*, D. R. Yakovlev2,3 & M. Bayer2,3 ∗ We develop a simple method for measuring the electron spin relaxation times T1 , T2 and T2 in semiconductors and demonstrate its exemplary application to n-type GaAs. Using an abrupt variation of the magnetic feld acting on electron spins, we detect the spin evolution by measuring the Faraday rotation of a short laser pulse. Depending on the magnetic feld orientation, this allows us to measure either the longitudinal spin relaxation time T1 or the inhomogeneous transverse spin dephasing time ∗ T2 . In order to determine the homogeneous spin coherence time T2 , we apply a pulse of an oscillating radiofrequency (rf) feld resonant with the Larmor frequency and detect the subsequent decay of the spin precession. The amplitude of the rf-driven spin precession is signifcantly enhanced upon additional optical pumping along the magnetic feld. Te electron spin dynamics in semiconductors can be addressed by a number of versatile methods involving electromagnetic radiation either in the optical range, resonant with interband transitions, or in the microwave as well as radiofrequency (rf) range, resonant with the Zeeman splitting1,2. For a long time, the optical methods were mostly represented by Hanle efect measurements giving access to the spin relaxation time at zero magnetic feld3. New techniques giving access both to the spin g factor and relaxation times at arbitrary magnetic felds have been very actively developed.
    [Show full text]
  • Thomas Rotation and Quantum Entanglement
    Proceedings of SAIP2015 Thomas Rotation and Quantum Entanglement. JM Hartman1, SH Connell1 and F Petruccione2 1. University of Johannesburg, Johannesburg, South Africa 2. University of Kwa-Zulu Natal, South Africa E-mail: [email protected] Abstract. The composition of two non-linear boosts on a particle in Minkowski space-time are not commutative. This non-commutativity has the result that the Lorentz transformation formed from the composition is not a pure boost but rather, a combination of a boost and a rotation. The rotation in this Lorentz transformation is called the Wigner rotation. When there are changes in velocity, as in an acceleration, then the Wigner rotations due to these changes add up to form Thomas precession. In curved space-time, the Thomas precession combines with a geometric effect caused by the gravitationally curved space-time to produce a geodetic effect. In this work we present how the Thomas precession affects the correlation between the spins of entangled particles and propose a way to detect forces acting on entangled particles by looking at how the Thomas precession degrades the entanglement correlation. Since the Thomas precession is a purely kinematical effect, it could potentially be used to detect any kind of force, including gravity (in the Newtonian or weak field limit). We present the results that we have so far. 1. Introduction While Bell's theorem is well known and was originally presented in John Bell's 1964 paper as a way to empirically test the EPR paradox, this was only for non-relativistic quantum mechanics. It was only fairly recently that people have started investigating possible relativistic effects on entanglement and EPR correlations beginning with Czachor [1] in 1997.
    [Show full text]