Quantum state measurement

Size: px
Start display at page:

Download "Quantum state measurement"

Transcription

1 Quantum state measurement Introduction The rotation properties of light fields of spin are described by the 3 3 representation of the 0 0 SO(3) group, with the generators JJ ii we found in class, for instance JJ zz = Then, 0 0 ee iiii 0 0 the rotation operator for the same zz axis is RR zz (ββ) = 0 0. It would appear that 0 0 ee iiii the light field, having spin, would require a basis set made of three states. However, the field component along the beam momentum is AA aaaaaaaaaa kk = 0 for the radiation electromagnetic fields (the most common case, in the far-field region, away from the sources) and two states are sufficient for the basis set. It is possible to describe the rotation properties of these transverse light states in a basis set made of only two kets with mm = ±, rather than three, because the mm = 0 component is not present. There are many possible two-state basis sets, but we will use the one made of the HH, VV states (horizontally- and vertically-polarized light), because it maps well to a method of calculating the polarization of light in classical optics, called the Jones calculus (Appendix). In quantum mechanics, light states αα are written in this basis set as αα = aa HH + bbee iiii VV or represented by αα aa bbee iiii, where aa, bb are real numbers. We set aa 2 + bb 2 = to have normalized states, as usual. Since optical elements, for instance QWP or HWP phase retarders, vary the polarization of light, they must be described by operators, acting on these quantum states of light. It turns out that we can simply take the matrices describing these optical elements in classical Jones calculus as the representations of operators HHHHHH, QQQQQQ in the HH, VV basis set. Specifically, for a half-wave- and quarter-wave-plate with the fast axis at an angle φφ from the horizontal, HHHHHH cos 2φφ sin 2φφ (φφ) and sin 2φφ cos2 φφ QQQQQQ (φφ) (cos φφ)2 + ii(sin φφ) 2 ( ii) cos φφ sin φφ ( ii) cos φφ sin φφ ii(cos φφ) 2 + (sin φφ) 2, respectively. Example: a QWP is followed by a HWP in the light beam. We can find the final state αα by calculating αα = HHHHHH (φφ 2 )QQQQQQ (φφ )(aa HH + bbee iiii VV ) and confirm that its projection along the horizontal axis is given in the three following cases by: Page

2 φφ = 0, φφ 2 = 0. Then, αα aa iiiiee iiii and HH αα 2 = aa 2 φφ = ππ, φφ 4 2 = 0. Then, αα + ii ii aa 2 + ii ii bbee iiii and HH αα 2 = aa( + ii) + 2 bbeeiiii ( ii) 2 = ( + 2aaaa sin φφ) 2 φφ = ππ, φφ 4 2 = ππ. Then, 8 αα aa ii ii bbee iiii and HH αα 2 = aa + bbeeiiii 2 = ( + 2aaaa cos φφ) 2 Note: φφ corresponds to the QQWWWW, φφ 2 corresponds to the HHWWWW. These measurements are useful if we want to find the initial quantum state of light αα. We require three measurements, since we need two real numbers aa, bb and one phase φφ. The three measurements above are necessary and sufficient (the condition aa 2 + bb 2 = is applied to normalize the intensity measurements). 2 Setup We will use the setup with the AA, BB, BB detectors (we do not need the AA detector) The polarizing beam splitter (PBS) in the BB beam lets the pp polarization through to BB, while the ss polarization component is reflected to BB Post the door with the warning sign. Turn on the 405 nm laser and add the BBO crystal to its post Turn on the FPGA card Turn on the detectors for AA, BB, BB As usual with these measurements, the overhead light and other sources (phones, flashlights, etc.) must be off at all times. Start the vi called QSM_rs232_2.6 (it will use the Lab View version 20) Add the 80 nm HWP in the infrared BB beam. Find the zero orientation of HWP (when the fast axis is horizontal), by finding its rotation angle that maximizes AAAA. Enter this number in the vi window. Similarly, add the 80 QWP and find its zero orientation (fast axis horizontal) when AAAA is maximum. Enter this number in the vi window. The detections in the BB, BB channels are conditioned by the detection in the AA channel and therefore we are looking at a single-photon beam. The combination of QWP, HWP, PBS and the two detectors BB, BB form an apparatus that can be applied to measuring quantum states of different polarizations for the single-photon infrared BB beam. Page 2

3 3 Experiments To vary the light beam quantum state, additional polarization elements between the crystal and the HWP, QWP you aligned must be inserted We call these new elements QQQQPP gg, HHHHPP gg for state generator-qwp, and generator- HWP Add the HHHHPP gg (no QQQQPP gg inserted). How should we rotate this to obtain the + 45 state after it, with the polarization of the single-photon beam rotated by +45 degrees (as looking along the beam direction) from the horizontal? Classical Jones calculus works fine here. Measure the aa, bb, φφ parameters of this state by doing the set of three measurements from the introduction and compare to the expected result. Replace the HHHHPP gg with the QQQQPP gg and generate the left-circularly polarized state LL. How should we orient the QWP (classical Jones calculus work well here)? Measure the aa, bb, φφ parameters of the state by doing the set of three measurements and compare to expectations. Turn off the detectors, the 405 nm laser, and return the BBO crystal to the jar. 4 Conclusion The measurements in this experiment can be described with the classical Jones calculus. Why do we do these complicated polarization measurements of quantum states and call a matrix already well-known in classical optics, a representation of an operator? These measurements will be useful in a famous experiment (Hardy s test ) that allowed an experimental verification of one astonishing consequence of quantum mechanics postulates. 5 Appendix Light and optical elements in Jones calculus of classical optics A matrix formalism is introduced in polarization calculations of classical optics, to speed up calculations. Classical polarization states of light are replaced with 2-element columns, with the two elements corresponding to the horizontal and vertical component of the light, respectively. Each optical element is also replaced with a 2 2 matrix. The advantage of this method is that Page 3

4 the action of a series of optical elements on a light state can be replaced by the matrix obtained by multiplying the matrices corresponding to individual elements (the ``Jones calculus ). For instance, a HWP with the fast axis horizontal (this would be the zero of the angle θθ) is replaced with the matrix HWP(0) = 0. As you can see what this matrix does to a light 0 state h vv is to reverse the sign of the vv component, equivalent to multiplication by eeiiii. In other words, the vertical component has been delayed (fast axis horizontal) by λλ (half-wave plate) or 2 ii2ππ λλ 2 by a phase ee λλ = ee iiii. Similarly, a QWP with fast axis horizontal has a matrix QWP(0) = 0. There is no quantum mechanics here, just mathematical convenience. 0 ii To find the matrices corresponding to optical elements rotated in an orientation with the fast axis at an arbitrary angle, we can apply the classical rotation transformation. For instance, cos φφ sin φφ cos φφ sin φφ 2φφ sin 2φφ HHHHHH(φφ) = HWP(0) = cos sin φφ cos φφ sin φφ cos φφ sin 2φφ cos2 φφ. When θθ = ππ 2 (fast axis vertical) we get HHHHHH ππ 0 = 2 0 as expected. When θθ = ππ we get HHHHHH 4 ππ = 4 0 0, or when acting on a horizontal state 0 we get 0, as expected. Similarly, cos φφ sin φφ cos φφ sin φφ QQQQQQ(φφ) = QWP(0) sin φφ cos φφ sin φφ cos φφ = (cos φφ)2 + ii(sin φφ) 2 ( ii) cos φφ sin φφ ( ii) cos φφ sin φφ ii(cos φφ) 2 + (sin φφ) 2. Light in quantum mechanics Following the qualitative arguments in Sakurai Ch., RCP and LLLLLL = ii HH ± the RCP/LCP basis set HH are complete and orthogonal. Page 4 and VV ii ii VV. Then, in. Both { HH, VV } and { LL, RR } basis sets Example: the representation of spin angular momentum in the LL, RR basis set is JJ zz 0 0. The representation of the rotation operator about the zz axis is then DD zz(rr) = ee iijj zzφφ = iiiijj zz + 2 ( ii)2 JJ 2 zz + = cos φφ ii sin φφ JJ zz ee iiii 0 (compare to the iiii 0 ee expression in the Introduction section). To find the representation of this operator in the HH, VV basis set from its representation in the LL, RR basis set, apply DD = UU + DDDD

5 2 ii ii ee iiii 0 ii φφ sin φφ 0 eeiiii = cos. The result is consistent with ii sin φφ cos φφ expectations from the ``active rotations (see Sakurai [3..3]). Exercise: verify that the representations of a phase retarder in the HH, VV basis set and the classical Jones calculus give the same answer in one specific case. Page 5

6 Name Phys-602 Quantum Mechanics Laboratory Quantum state measurement lab report Date of measurements: Page 6

Elastic light scattering

Elastic light scattering Elastic light scattering 1. Introduction Elastic light scattering in quantum mechanics Elastic scattering is described in quantum mechanics by the Kramers Heisenberg formula for the differential cross

More information

(1) Introduction: a new basis set

(1) Introduction: a new basis set () Introduction: a new basis set In scattering, we are solving the S eq. for arbitrary VV in integral form We look for solutions to unbound states: certain boundary conditions (EE > 0, plane and spherical

More information

Parametric down-conversion

Parametric down-conversion Parametric down-conversion 1 Introduction You have seen that laser light, for all its intensity and coherence, gave us the same PP(mm) counts distribution as a thermal light source with a high fluctuation

More information

(2) Orbital angular momentum

(2) Orbital angular momentum (2) Orbital angular momentum Consider SS = 0 and LL = rr pp, where pp is the canonical momentum Note: SS and LL are generators for different parts of the wave function. Note: from AA BB ii = εε iiiiii

More information

Quantum Mechanics. An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc.

Quantum Mechanics. An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc. Quantum Mechanics An essential theory to understand properties of matter and light. Chemical Electronic Magnetic Thermal Optical Etc. Fall 2018 Prof. Sergio B. Mendes 1 CHAPTER 3 Experimental Basis of

More information

CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I

CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 1 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles 5.6 Uncertainty Principle Topics 5.7

More information

Grover s algorithm. We want to find aa. Search in an unordered database. QC oracle (as usual) Usual trick

Grover s algorithm. We want to find aa. Search in an unordered database. QC oracle (as usual) Usual trick Grover s algorithm Search in an unordered database Example: phonebook, need to find a person from a phone number Actually, something else, like hard (e.g., NP-complete) problem 0, xx aa Black box ff xx

More information

Chem 263 Winter 2018 Problem Set #2 Due: February 16

Chem 263 Winter 2018 Problem Set #2 Due: February 16 Chem 263 Winter 2018 Problem Set #2 Due: February 16 1. Use size considerations to predict the crystal structures of PbF2, CoF2, and BeF2. Do your predictions agree with the actual structures of these

More information

Optical pumping and the Zeeman Effect

Optical pumping and the Zeeman Effect 1. Introduction Optical pumping and the Zeeman Effect The Hamiltonian of an atom with a single electron outside filled shells (as for rubidium) in a magnetic field is HH = HH 0 + ηηii JJ μμ JJ BB JJ μμ

More information

Atomic fluorescence. The intensity of a transition line can be described with a transition probability inversely

Atomic fluorescence. The intensity of a transition line can be described with a transition probability inversely Atomic fluorescence 1. Introduction Transitions in multi-electron atoms Energy levels of the single-electron hydrogen atom are well-described by EE nn = RR nn2, where RR = 13.6 eeee is the Rydberg constant.

More information

Angular Momentum, Electromagnetic Waves

Angular Momentum, Electromagnetic Waves Angular Momentum, Electromagnetic Waves Lecture33: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay As before, we keep in view the four Maxwell s equations for all our discussions.

More information

Mathematics Ext 2. HSC 2014 Solutions. Suite 403, 410 Elizabeth St, Surry Hills NSW 2010 keystoneeducation.com.

Mathematics Ext 2. HSC 2014 Solutions. Suite 403, 410 Elizabeth St, Surry Hills NSW 2010 keystoneeducation.com. Mathematics Ext HSC 4 Solutions Suite 43, 4 Elizabeth St, Surry Hills NSW info@keystoneeducation.com.au keystoneeducation.com.au Mathematics Extension : HSC 4 Solutions Contents Multiple Choice... 3 Question...

More information

Variations. ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra

Variations. ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra Variations ECE 6540, Lecture 02 Multivariate Random Variables & Linear Algebra Last Time Probability Density Functions Normal Distribution Expectation / Expectation of a function Independence Uncorrelated

More information

Coulomb s Law and Coulomb s Constant

Coulomb s Law and Coulomb s Constant Pre-Lab Quiz / PHYS 224 Coulomb s Law and Coulomb s Constant Your Name: Lab Section: 1. What will you investigate in this lab? 2. Consider a capacitor created when two identical conducting plates are placed

More information

CHAPTER 4 Structure of the Atom

CHAPTER 4 Structure of the Atom CHAPTER 4 Structure of the Atom Fall 2018 Prof. Sergio B. Mendes 1 Topics 4.1 The Atomic Models of Thomson and Rutherford 4.2 Rutherford Scattering 4.3 The Classic Atomic Model 4.4 The Bohr Model of the

More information

Radiation. Lecture40: Electromagnetic Theory. Professor D. K. Ghosh, Physics Department, I.I.T., Bombay

Radiation. Lecture40: Electromagnetic Theory. Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Radiation Zone Approximation We had seen that the expression for the vector potential for a localized cuent distribution is given by AA (xx, tt) = μμ 4ππ ee iiiiii dd xx eeiiii xx xx xx xx JJ (xx ) In

More information

Cold atoms in optical lattices

Cold atoms in optical lattices Cold atoms in optical lattices www.lens.unifi.it Tarruel, Nature Esslinger group Optical lattices the big picture We have a textbook model, which is basically exact, describing how a large collection of

More information

Photons in the universe. Indian Institute of Technology Ropar

Photons in the universe. Indian Institute of Technology Ropar Photons in the universe Photons in the universe Element production on the sun Spectral lines of hydrogen absorption spectrum absorption hydrogen gas Hydrogen emission spectrum Element production on the

More information

Magnetic Force and Current Balance

Magnetic Force and Current Balance Pre-Lab Quiz / PHYS 224 Magnetic Force and Current Balance Name Lab Section 1. What do you investigate in this lab? 2. Consider two parallel straight wires carrying electric current in opposite directions

More information

Haar Basis Wavelets and Morlet Wavelets

Haar Basis Wavelets and Morlet Wavelets Haar Basis Wavelets and Morlet Wavelets September 9 th, 05 Professor Davi Geiger. The Haar transform, which is one of the earliest transform functions proposed, was proposed in 90 by a Hungarian mathematician

More information

14. Matrix treatment of polarization

14. Matrix treatment of polarization 14. Matri treatment of polarization This lecture Polarized Light : linear, circular, elliptical Jones Vectors for Polarized Light Jones Matrices for Polarizers, Phase Retarders, Rotators (Linear) Polarization

More information

SECTION 7: FAULT ANALYSIS. ESE 470 Energy Distribution Systems

SECTION 7: FAULT ANALYSIS. ESE 470 Energy Distribution Systems SECTION 7: FAULT ANALYSIS ESE 470 Energy Distribution Systems 2 Introduction Power System Faults 3 Faults in three-phase power systems are short circuits Line-to-ground Line-to-line Result in the flow

More information

Chapter 4: Polarization of light

Chapter 4: Polarization of light Chapter 4: Polarization of light 1 Preliminaries and definitions B E Plane-wave approximation: E(r,t) ) and B(r,t) are uniform in the plane ^ k We will say that light polarization vector is along E(r,t)

More information

PHL424: Nuclear Shell Model. Indian Institute of Technology Ropar

PHL424: Nuclear Shell Model. Indian Institute of Technology Ropar PHL424: Nuclear Shell Model Themes and challenges in modern science Complexity out of simplicity Microscopic How the world, with all its apparent complexity and diversity can be constructed out of a few

More information

Dressing up for length gauge: Aspects of a debate in quantum optics

Dressing up for length gauge: Aspects of a debate in quantum optics Dressing up for length gauge: Aspects of a debate in quantum optics Rainer Dick Department of Physics & Engineering Physics University of Saskatchewan rainer.dick@usask.ca 1 Agenda: Attosecond spectroscopy

More information

Gradient expansion formalism for generic spin torques

Gradient expansion formalism for generic spin torques Gradient expansion formalism for generic spin torques Atsuo Shitade RIKEN Center for Emergent Matter Science Atsuo Shitade, arxiv:1708.03424. Outline 1. Spintronics a. Magnetoresistance and spin torques

More information

Lecture 22 Highlights Phys 402

Lecture 22 Highlights Phys 402 Lecture 22 Highlights Phys 402 Scattering experiments are one of the most important ways to gain an understanding of the microscopic world that is described by quantum mechanics. The idea is to take a

More information

ECEN 4606, UNDERGRADUATE OPTICS LAB

ECEN 4606, UNDERGRADUATE OPTICS LAB ECEN 4606, UNDERGRADUATE OPTICS LAB Lab 6: Polarization Original: Professor McLeod SUMMARY: In this lab you will become familiar with the basics of polarization and learn to use common optical elements

More information

Lab #13: Polarization

Lab #13: Polarization Lab #13: Polarization Introduction In this experiment we will investigate various properties associated with polarized light. We will study both its generation and application. Real world applications

More information

POLARIZATION OF LIGHT

POLARIZATION OF LIGHT POLARIZATION OF LIGHT OVERALL GOALS The Polarization of Light lab strongly emphasizes connecting mathematical formalism with measurable results. It is not your job to understand every aspect of the theory,

More information

General Strong Polarization

General Strong Polarization General Strong Polarization Madhu Sudan Harvard University Joint work with Jaroslaw Blasiok (Harvard), Venkatesan Gurswami (CMU), Preetum Nakkiran (Harvard) and Atri Rudra (Buffalo) December 4, 2017 IAS:

More information

Modeling of a non-physical fish barrier

Modeling of a non-physical fish barrier University of Massachusetts - Amherst ScholarWorks@UMass Amherst International Conference on Engineering and Ecohydrology for Fish Passage International Conference on Engineering and Ecohydrology for Fish

More information

Secondary 3H Unit = 1 = 7. Lesson 3.3 Worksheet. Simplify: Lesson 3.6 Worksheet

Secondary 3H Unit = 1 = 7. Lesson 3.3 Worksheet. Simplify: Lesson 3.6 Worksheet Secondary H Unit Lesson Worksheet Simplify: mm + 2 mm 2 4 mm+6 mm + 2 mm 2 mm 20 mm+4 5 2 9+20 2 0+25 4 +2 2 + 2 8 2 6 5. 2 yy 2 + yy 6. +2 + 5 2 2 2 0 Lesson 6 Worksheet List all asymptotes, holes and

More information

The Bose Einstein quantum statistics

The Bose Einstein quantum statistics Page 1 The Bose Einstein quantum statistics 1. Introduction Quantized lattice vibrations Thermal lattice vibrations in a solid are sorted in classical mechanics in normal modes, special oscillation patterns

More information

CHAPTER 2 Special Theory of Relativity

CHAPTER 2 Special Theory of Relativity CHAPTER 2 Special Theory of Relativity Fall 2018 Prof. Sergio B. Mendes 1 Topics 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 Inertial Frames of Reference Conceptual and Experimental

More information

Experiment 6 - Tests of Bell s Inequality

Experiment 6 - Tests of Bell s Inequality Exp.6-Bell-Page 1 Experiment 6 - Tests of Bell s Inequality References: Entangled photon apparatus for the undergraduate laboratory, and Entangled photons, nonlocality, and Bell inequalities in the undergraduate

More information

Module 7 (Lecture 25) RETAINING WALLS

Module 7 (Lecture 25) RETAINING WALLS Module 7 (Lecture 25) RETAINING WALLS Topics Check for Bearing Capacity Failure Example Factor of Safety Against Overturning Factor of Safety Against Sliding Factor of Safety Against Bearing Capacity Failure

More information

Integrating Rational functions by the Method of Partial fraction Decomposition. Antony L. Foster

Integrating Rational functions by the Method of Partial fraction Decomposition. Antony L. Foster Integrating Rational functions by the Method of Partial fraction Decomposition By Antony L. Foster At times, especially in calculus, it is necessary, it is necessary to express a fraction as the sum of

More information

Physics 371 Spring 2017 Prof. Anlage Review

Physics 371 Spring 2017 Prof. Anlage Review Physics 71 Spring 2017 Prof. Anlage Review Special Relativity Inertial vs. non-inertial reference frames Galilean relativity: Galilean transformation for relative motion along the xx xx direction: xx =

More information

Worksheets for GCSE Mathematics. Algebraic Expressions. Mr Black 's Maths Resources for Teachers GCSE 1-9. Algebra

Worksheets for GCSE Mathematics. Algebraic Expressions. Mr Black 's Maths Resources for Teachers GCSE 1-9. Algebra Worksheets for GCSE Mathematics Algebraic Expressions Mr Black 's Maths Resources for Teachers GCSE 1-9 Algebra Algebraic Expressions Worksheets Contents Differentiated Independent Learning Worksheets

More information

Magnetism of materials

Magnetism of materials Magnetism of materials 1. Introduction Magnetism and quantum mechanics In the previous experiment, you witnessed a very special case of a diamagnetic material with magnetic susceptibility χχ = 1 (usually

More information

ABSTRACT OF THE DISSERTATION

ABSTRACT OF THE DISSERTATION A Study of an N Molecule-Quantized Radiation Field-Hamiltonian BY MICHAEL THOMAS TAVIS DOCTOR OF PHILOSOPHY, GRADUATE PROGRAM IN PHYSICS UNIVERSITY OF CALIFORNIA, RIVERSIDE, DECEMBER 1968 PROFESSOR FREDERICK

More information

Quantum information processing using linear optics

Quantum information processing using linear optics Quantum information processing using linear optics Karel Lemr Joint Laboratory of Optics of Palacký University and Institute of Physics of Academy of Sciences of the Czech Republic web: http://jointlab.upol.cz/lemr

More information

17. Jones Matrices & Mueller Matrices

17. Jones Matrices & Mueller Matrices 7. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates - the rotation matrix Stokes Parameters and unpolarized light Mueller Matrices R. Clark Jones (96-24) Sir George G. Stokes (89-93)

More information

CS Lecture 8 & 9. Lagrange Multipliers & Varitional Bounds

CS Lecture 8 & 9. Lagrange Multipliers & Varitional Bounds CS 6347 Lecture 8 & 9 Lagrange Multipliers & Varitional Bounds General Optimization subject to: min ff 0() R nn ff ii 0, h ii = 0, ii = 1,, mm ii = 1,, pp 2 General Optimization subject to: min ff 0()

More information

OBE solutions in the rotating frame

OBE solutions in the rotating frame OBE solutions in the rotating frame The light interaction with the 2-level system is VV iiiiii = μμ EE, where μμ is the dipole moment μμ 11 = 0 and μμ 22 = 0 because of parity. Therefore, light does not

More information

Interaction with matter

Interaction with matter Interaction with matter accelerated motion: ss = bb 2 tt2 tt = 2 ss bb vv = vv 0 bb tt = vv 0 2 ss bb EE = 1 2 mmvv2 dddd dddd = mm vv 0 2 ss bb 1 bb eeeeeeeeeeee llllllll bbbbbbbbbbbbbb dddddddddddddddd

More information

Math 171 Spring 2017 Final Exam. Problem Worth

Math 171 Spring 2017 Final Exam. Problem Worth Math 171 Spring 2017 Final Exam Problem 1 2 3 4 5 6 7 8 9 10 11 Worth 9 6 6 5 9 8 5 8 8 8 10 12 13 14 15 16 17 18 19 20 21 22 Total 8 5 5 6 6 8 6 6 6 6 6 150 Last Name: First Name: Student ID: Section:

More information

Inferring the origin of an epidemic with a dynamic message-passing algorithm

Inferring the origin of an epidemic with a dynamic message-passing algorithm Inferring the origin of an epidemic with a dynamic message-passing algorithm HARSH GUPTA (Based on the original work done by Andrey Y. Lokhov, Marc Mézard, Hiroki Ohta, and Lenka Zdeborová) Paper Andrey

More information

Big Bang Planck Era. This theory: cosmological model of the universe that is best supported by several aspects of scientific evidence and observation

Big Bang Planck Era. This theory: cosmological model of the universe that is best supported by several aspects of scientific evidence and observation Big Bang Planck Era Source: http://www.crystalinks.com/bigbang.html Source: http://www.odec.ca/index.htm This theory: cosmological model of the universe that is best supported by several aspects of scientific

More information

7.3 The Jacobi and Gauss-Seidel Iterative Methods

7.3 The Jacobi and Gauss-Seidel Iterative Methods 7.3 The Jacobi and Gauss-Seidel Iterative Methods 1 The Jacobi Method Two assumptions made on Jacobi Method: 1.The system given by aa 11 xx 1 + aa 12 xx 2 + aa 1nn xx nn = bb 1 aa 21 xx 1 + aa 22 xx 2

More information

Rotational Motion. Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition

Rotational Motion. Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition Rotational Motion Chapter 10 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 We ll look for a way to describe the combined (rotational) motion 2 Angle Measurements θθ ss rr rrrrrrrrrrrrrr

More information

Review for Exam Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa

Review for Exam Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa 57:020 Fluids Mechanics Fall2013 1 Review for Exam3 12. 11. 2013 Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa 57:020 Fluids Mechanics Fall2013 2 Chapter

More information

General Strong Polarization

General Strong Polarization General Strong Polarization Madhu Sudan Harvard University Joint work with Jaroslaw Blasiok (Harvard), Venkatesan Gurswami (CMU), Preetum Nakkiran (Harvard) and Atri Rudra (Buffalo) May 1, 018 G.Tech:

More information

Charge carrier density in metals and semiconductors

Charge carrier density in metals and semiconductors Charge carrier density in metals and semiconductors 1. Introduction The Hall Effect Particles must overlap for the permutation symmetry to be relevant. We saw examples of this in the exchange energy in

More information

Lecture No. 1 Introduction to Method of Weighted Residuals. Solve the differential equation L (u) = p(x) in V where L is a differential operator

Lecture No. 1 Introduction to Method of Weighted Residuals. Solve the differential equation L (u) = p(x) in V where L is a differential operator Lecture No. 1 Introduction to Method of Weighted Residuals Solve the differential equation L (u) = p(x) in V where L is a differential operator with boundary conditions S(u) = g(x) on Γ where S is a differential

More information

Some Aspects of Oscillatory Visco-elastic Flow Through Porous Medium in a Rotating Porous Channel

Some Aspects of Oscillatory Visco-elastic Flow Through Porous Medium in a Rotating Porous Channel Some Aspects of Oscillatory Visco-elastic Flow Through Porous Medium in a Rotating Porous Channel RITA CHOUDHURY, HILLOL KANTI BHATTACHARJEE, Department of Mathematics, Gauhati University Guwahati-78 4

More information

Schemes to generate entangled photon pairs via spontaneous parametric down conversion

Schemes to generate entangled photon pairs via spontaneous parametric down conversion Schemes to generate entangled photon pairs via spontaneous parametric down conversion Atsushi Yabushita Department of Electrophysics National Chiao-Tung University? Outline Introduction Optical parametric

More information

Lesson 7: Linear Transformations Applied to Cubes

Lesson 7: Linear Transformations Applied to Cubes Classwork Opening Exercise Consider the following matrices: AA = 1 2 0 2, BB = 2, and CC = 2 2 4 0 0 2 2 a. Compute the following determinants. i. det(aa) ii. det(bb) iii. det(cc) b. Sketch the image of

More information

Doppler Correction after Inelastic Heavy Ion Scattering 238 U Ta system at the Coulomb barrier

Doppler Correction after Inelastic Heavy Ion Scattering 238 U Ta system at the Coulomb barrier Doppler-Corrected e - and γ-ray Spectroscopy Physical Motivation In-beam conversion electron spectroscopy complements the results obtained from γ-spectroscopy A method for determining the multipolarity

More information

Independent Component Analysis and FastICA. Copyright Changwei Xiong June last update: July 7, 2016

Independent Component Analysis and FastICA. Copyright Changwei Xiong June last update: July 7, 2016 Independent Component Analysis and FastICA Copyright Changwei Xiong 016 June 016 last update: July 7, 016 TABLE OF CONTENTS Table of Contents...1 1. Introduction.... Independence by Non-gaussianity....1.

More information

Neutron ββ-decay Angular Correlations

Neutron ββ-decay Angular Correlations Neutron ββ-decay Angular Correlations Brad Plaster University of Kentucky Figure Credit: V. Cirigliano, B. Markisch, et al. Figure Credit: https://www.physi.uniheidelberg.de/forschung/anp/perkeo/index.php

More information

QUANTUM MECHANICS AND ATOMIC STRUCTURE

QUANTUM MECHANICS AND ATOMIC STRUCTURE 5 CHAPTER QUANTUM MECHANICS AND ATOMIC STRUCTURE 5.1 The Hydrogen Atom 5.2 Shell Model for Many-Electron Atoms 5.3 Aufbau Principle and Electron Configurations 5.4 Shells and the Periodic Table: Photoelectron

More information

Lecture 3. STAT161/261 Introduction to Pattern Recognition and Machine Learning Spring 2018 Prof. Allie Fletcher

Lecture 3. STAT161/261 Introduction to Pattern Recognition and Machine Learning Spring 2018 Prof. Allie Fletcher Lecture 3 STAT161/261 Introduction to Pattern Recognition and Machine Learning Spring 2018 Prof. Allie Fletcher Previous lectures What is machine learning? Objectives of machine learning Supervised and

More information

Fermi Surfaces and their Geometries

Fermi Surfaces and their Geometries Fermi Surfaces and their Geometries Didier Ndengeyintwali Physics Department, Drexel University, Philadelphia, Pennsylvania 19104, USA (Dated: May 17, 2010) 1. Introduction The Pauli exclusion principle

More information

(1) Correspondence of the density matrix to traditional method

(1) Correspondence of the density matrix to traditional method (1) Correspondence of the density matrix to traditional method New method (with the density matrix) Traditional method (from thermal physics courses) ZZ = TTTT ρρ = EE ρρ EE = dddd xx ρρ xx ii FF = UU

More information

Last Name _Piatoles_ Given Name Americo ID Number

Last Name _Piatoles_ Given Name Americo ID Number Last Name _Piatoles_ Given Name Americo ID Number 20170908 Question n. 1 The "C-V curve" method can be used to test a MEMS in the electromechanical characterization phase. Describe how this procedure is

More information

Module 6 (Lecture 22) LATERAL EARTH PRESSURE

Module 6 (Lecture 22) LATERAL EARTH PRESSURE Module 6 (Lecture ) LATERAL EARTH PRESSURE 1.1 LATERAL EARTH PRESSURE DUE TO SURCHARGE 1. ACTIVE PRESSURE FOR WALL ROTATION ABOUT TOP- BRACED CUT 1.3 ACTIVE EARTH PRESSURE FOR TRANSLATION OF RETAINING

More information

INTRODUCTION TO QUANTUM MECHANICS

INTRODUCTION TO QUANTUM MECHANICS 4 CHAPTER INTRODUCTION TO QUANTUM MECHANICS 4.1 Preliminaries: Wave Motion and Light 4.2 Evidence for Energy Quantization in Atoms 4.3 The Bohr Model: Predicting Discrete Energy Levels in Atoms 4.4 Evidence

More information

10.4 The Cross Product

10.4 The Cross Product Math 172 Chapter 10B notes Page 1 of 9 10.4 The Cross Product The cross product, or vector product, is defined in 3 dimensions only. Let aa = aa 1, aa 2, aa 3 bb = bb 1, bb 2, bb 3 then aa bb = aa 2 bb

More information

EPR paradox, Bell inequality, etc.

EPR paradox, Bell inequality, etc. EPR paradox, Bell inequality, etc. Compatible and incompatible observables AA, BB = 0, then compatible, can measure simultaneously, can diagonalize in one basis commutator, AA, BB AAAA BBBB If we project

More information

SECTION 8: ROOT-LOCUS ANALYSIS. ESE 499 Feedback Control Systems

SECTION 8: ROOT-LOCUS ANALYSIS. ESE 499 Feedback Control Systems SECTION 8: ROOT-LOCUS ANALYSIS ESE 499 Feedback Control Systems 2 Introduction Introduction 3 Consider a general feedback system: Closed-loop transfer function is KKKK ss TT ss = 1 + KKKK ss HH ss GG ss

More information

Discovery of the Higgs Boson

Discovery of the Higgs Boson Discovery of the Higgs Boson Seminar: Key Experiments in Particle Physics Martin Vogrin Munich, 22. July 2016 Outline Theoretical part Experiments Results Open problems Motivation The SM is really two

More information

Lise Meitner, Otto Hahn. Nuclear Fission Hans-Jürgen Wollersheim

Lise Meitner, Otto Hahn. Nuclear Fission Hans-Jürgen Wollersheim Lise Meitner, Otto Hahn Nuclear Fission Hans-Jürgen Wollersheim Details of the 252 Cf decay α s: 96.9% SF: 3.1% T 1/2 = 2.647 a Q α = 6.217 MeV E α = 6.118 MeV α α α α α-decay of 252 Cf Mass data: nucleardata.nuclear.lu.se/database/masses/

More information

Chap. 5. Jones Calculus and Its Application to Birefringent Optical Systems

Chap. 5. Jones Calculus and Its Application to Birefringent Optical Systems Chap. 5. Jones Calculus and Its Application to Birefringent Optical Systems - The overall optical transmission through many optical components such as polarizers, EO modulators, filters, retardation plates.

More information

" = Y(#,$) % R(r) = 1 4& % " = Y(#,$) % R(r) = Recitation Problems: Week 4. a. 5 B, b. 6. , Ne Mg + 15 P 2+ c. 23 V,

 = Y(#,$) % R(r) = 1 4& %  = Y(#,$) % R(r) = Recitation Problems: Week 4. a. 5 B, b. 6. , Ne Mg + 15 P 2+ c. 23 V, Recitation Problems: Week 4 1. Which of the following combinations of quantum numbers are allowed for an electron in a one-electron atom: n l m l m s 2 2 1! 3 1 0 -! 5 1 2! 4-1 0! 3 2 1 0 2 0 0 -! 7 2-2!

More information

Impact of Chamber Pressure on Sputtered Particle Energy

Impact of Chamber Pressure on Sputtered Particle Energy Wilmert De Bosscher Chief Technology Officer +32 9381 6177 wilmert.debosscher@soleras.com Impact of Chamber Pressure on Sputtered Particle Energy Tampa, October 18 th, 2017 Background Why Sputtering at

More information

Classical RSA algorithm

Classical RSA algorithm Classical RSA algorithm We need to discuss some mathematics (number theory) first Modulo-NN arithmetic (modular arithmetic, clock arithmetic) 9 (mod 7) 4 3 5 (mod 7) congruent (I will also use = instead

More information

Review for Exam Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa

Review for Exam Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa Review for Exam2 11. 13. 2015 Hyunse Yoon, Ph.D. Adjunct Assistant Professor Department of Mechanical Engineering, University of Iowa Assistant Research Scientist IIHR-Hydroscience & Engineering, University

More information

10.4 Controller synthesis using discrete-time model Example: comparison of various controllers

10.4 Controller synthesis using discrete-time model Example: comparison of various controllers 10. Digital 10.1 Basic principle of digital control 10.2 Digital PID controllers 10.2.1 A 2DOF continuous-time PID controller 10.2.2 Discretisation of PID controllers 10.2.3 Implementation and tuning 10.3

More information

Testing Heisenberg s Uncertainty Principle with Polarized Single Photons

Testing Heisenberg s Uncertainty Principle with Polarized Single Photons Testing Heisenberg s Uncertainty Principle with Polarized Single Photons Sofia Svensson sofia.svensson95@gmail.com under the direction of Prof. Mohamed Bourennane Quantum Information & Quantum Optics Department

More information

Building Blocks for Quantum Computing Part V Operation of the Trapped Ion Quantum Computer

Building Blocks for Quantum Computing Part V Operation of the Trapped Ion Quantum Computer Building Blocks for Quantum Computing Part V Operation of the Trapped Ion Quantum Computer CSC801 Seminar on Quantum Computing Spring 2018 1 Goal Is To Understand The Principles And Operation of the Trapped

More information

Wave Motion. Chapter 14 of Essential University Physics, Richard Wolfson, 3 rd Edition

Wave Motion. Chapter 14 of Essential University Physics, Richard Wolfson, 3 rd Edition Wave Motion Chapter 14 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 Waves: propagation of energy, not particles 2 Longitudinal Waves: disturbance is along the direction of wave propagation

More information

A Posteriori Error Estimates For Discontinuous Galerkin Methods Using Non-polynomial Basis Functions

A Posteriori Error Estimates For Discontinuous Galerkin Methods Using Non-polynomial Basis Functions Lin Lin A Posteriori DG using Non-Polynomial Basis 1 A Posteriori Error Estimates For Discontinuous Galerkin Methods Using Non-polynomial Basis Functions Lin Lin Department of Mathematics, UC Berkeley;

More information

EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique. Jeudi 18 juin Salle B-508.

EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique. Jeudi 18 juin Salle B-508. EXAMEN GÉNÉRAL DE SYNTHÈSE ÉPREUVE ÉCRITE Programme de doctorat en génie physique Salle B-58 de 9h3 à 3h3 NOTES : No documentation allowed. A non-programmable calculator is allowed. The candidate can answer

More information

Experimental generalized contextuality with single-photon qubits: supplementary material

Experimental generalized contextuality with single-photon qubits: supplementary material Experimental generalized contextuality with single-photon qubits: supplementary material XIANG ZHAN 1, ERIC G. CAVALCANTI 2, JIAN LI 1, ZHIHAO BIAN 1, YONGSHENG ZHANG 3,4*, HOWARD M. WISEMAN 2,5,, AND

More information

A Simple and Usable Wake Vortex Encounter Severity Metric

A Simple and Usable Wake Vortex Encounter Severity Metric A Simple and Usable Wake Vortex Encounter Severity Metric Ivan De Visscher Grégoire Winckelmans WaPT-Wake Prediction Technologies a spin-off company from Université catholique de Louvain (UCL) WakeNet-Europe

More information

PHY103A: Lecture # 4

PHY103A: Lecture # 4 Semester II, 2017-18 Department of Physics, IIT Kanpur PHY103A: Lecture # 4 (Text Book: Intro to Electrodynamics by Griffiths, 3 rd Ed.) Anand Kumar Jha 10-Jan-2018 Notes The Solutions to HW # 1 have been

More information

Quantization of electrical conductance

Quantization of electrical conductance 1 Introduction Quantization of electrical conductance Te resistance of a wire in te classical Drude model of metal conduction is given by RR = ρρρρ AA, were ρρ, AA and ll are te conductivity of te material,

More information

Lecture 15: Scattering Rutherford scattering Nuclear elastic scattering Nuclear inelastic scattering Quantum description The optical model

Lecture 15: Scattering Rutherford scattering Nuclear elastic scattering Nuclear inelastic scattering Quantum description The optical model Lecture 15: Scattering Rutherford scattering Nuclear elastic scattering Nuclear inelastic scattering Quantum description The optical model Lecture 15: Ohio University PHYS7501, Fall 017, Z. Meisel (meisel@ohio.edu)

More information

Acceleration to higher energies

Acceleration to higher energies Acceleration to higher energies While terminal voltages of 20 MV provide sufficient beam energy for nuclear structure research, most applications nowadays require beam energies > 1 GeV How do we attain

More information

Photon Interactions in Matter

Photon Interactions in Matter Radiation Dosimetry Attix 7 Photon Interactions in Matter Ho Kyung Kim hokyung@pusan.ac.kr Pusan National University References F. H. Attix, Introduction to Radiological Physics and Radiation Dosimetry,

More information

Jones vector & matrices

Jones vector & matrices Jones vector & matrices Department of Physics 1 Matrix treatment of polarization Consider a light ray with an instantaneous E-vector as shown y E k, t = xe x (k, t) + ye y k, t E y E x x E x = E 0x e i

More information

Yang-Hwan Ahn Based on arxiv:

Yang-Hwan Ahn Based on arxiv: Yang-Hwan Ahn (CTPU@IBS) Based on arxiv: 1611.08359 1 Introduction Now that the Higgs boson has been discovered at 126 GeV, assuming that it is indeed exactly the one predicted by the SM, there are several

More information

Problem 4.1 (Verdeyen Problem #8.7) (a) From (7.4.7), simulated emission cross section is defined as following.

Problem 4.1 (Verdeyen Problem #8.7) (a) From (7.4.7), simulated emission cross section is defined as following. Problem 4.1 (Verdeyen Problem #8.7) (a) From (7.4.7), simulated emission cross section is defined as following. σσ(νν) = AA 21 λλ 2 8ππnn 2 gg(νν) AA 21 = 6 10 6 ssssss 1 From the figure, the emission

More information

Gravitation. Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition

Gravitation. Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition Gravitation Chapter 8 of Essential University Physics, Richard Wolfson, 3 rd Edition 1 What you are about to learn: Newton's law of universal gravitation About motion in circular and other orbits How to

More information

Polarization. Polarization. Physics Waves & Oscillations 4/3/2016. Spring 2016 Semester Matthew Jones. Two problems to be considered today:

Polarization. Polarization. Physics Waves & Oscillations 4/3/2016. Spring 2016 Semester Matthew Jones. Two problems to be considered today: 4/3/26 Physics 422 Waves & Oscillations Lecture 34 Polarization of Light Spring 26 Semester Matthew Jones Polarization (,)= cos (,)= cos + Unpolarizedlight: Random,, Linear polarization: =,± Circular polarization:

More information

TECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES

TECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES COMPUTERS AND STRUCTURES, INC., FEBRUARY 2016 TECHNICAL NOTE AUTOMATIC GENERATION OF POINT SPRING SUPPORTS BASED ON DEFINED SOIL PROFILES AND COLUMN-FOOTING PROPERTIES Introduction This technical note

More information

Jasmin Smajic1, Christian Hafner2, Jürg Leuthold2, March 23, 2015

Jasmin Smajic1, Christian Hafner2, Jürg Leuthold2, March 23, 2015 Jasmin Smajic, Christian Hafner 2, Jürg Leuthold 2, March 23, 205 Time Domain Finite Element Method (TD FEM): Continuous and Discontinuous Galerkin (DG-FEM) HSR - University of Applied Sciences of Eastern

More information

Testing The Existence of Single Photons

Testing The Existence of Single Photons Testing The Existence of Single Photons Quynh Nguyen and Asad Khan Method of Experimental Physics Project, University of Minnesota. (Dated: 12 May 2014) We demonstrated the existence of single photon by

More information