# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

1 Page

### hw3

Course: EEAP 248, Fall 2009
School: Stanford
Rating:

#### Document Preview

of Fundamentals Noise Processes EE248/Fall 2007 Instructor: Professor Yoshihisa Yamamoto Please address homework questions to Kristiaan De Greve (kdegreve@stanford.edu). Problem Set 3 Due date: November 5, 2008 Problem I: External Circuit Current (Rs = 0 case) (15 points) Look at the vacuum diode (Fig.1). Suppose an electron is emitted from the cathode and is in transit to the anode. Assume the source resistance...

Register Now

#### Unformatted Document Excerpt

Coursehero >> California >> Stanford >> EEAP 248

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
of Fundamentals Noise Processes EE248/Fall 2007 Instructor: Professor Yoshihisa Yamamoto Please address homework questions to Kristiaan De Greve (kdegreve@stanford.edu). Problem Set 3 Due date: November 5, 2008 Problem I: External Circuit Current (Rs = 0 case) (15 points) Look at the vacuum diode (Fig.1). Suppose an electron is emitted from the cathode and is in transit to the anode. Assume the source resistance Rs is zero. Show that the external short-circuit current due to this moving charge carrier is given by i(t) = qv(t) d where q is the fundamental unit of charge, v(t) is the electron drift velocity, and d is the distance between the two electrodes. HINT: Calculate the energy gained by the electron from the electric field associated with this moving charge, and equate it to the energy provided by the external constant applied voltage V. Problem II: Surface Charges and External Circuit Current (25 points) In a vacuum diode (Fig.1), a single electron is emitted from the cathode at t = 0. For each of the following three cases, calculate the surface charges of the cathode and anode, as well as the external circuit current as a function of time. (1) The electron drift velocity is initially zero at the cathode and is accelerated by the constant applied electric field, and RC t . This might be the case if the initial kinetic energy of the electron is small relative to the energy gained in transit from the field. (2) The electron drift velocity is assumed to be constant over the electron's transit from the cathode to the anode, as is the case if the initial kinetic energy is large relative to qV . In addition, assume RC t . (3) RC t . In this case, we assume the electron transit to be an impulsive event. 1 2 cathode charge QC(t) -q v anode charge QA(t) i(t) capacitance C d Rs V Figure 1. A vacuum diode. Problem III: Equipartition theorem (30 points) A small system A, with allowed energies 1 , 2 , ..., is placed in thermal contact with a heat reservoir B. In other words, although their total energy E0 remains constant, A and B are free to exchange energy. It can be shown that the probability distribution for finding system A with energy i is P ( i) = e- i /kB - i /kB ie Here, kB is Boltzmann's constant, and is the temperature of the reservoir B. This distribution is known as the canonical distribution, and characterizes the distribution for a system that can exchange energy, but not particles, with a reservoir. Note that the probability of occupation decreases exponentially with the energy of the eigenstate; the exponential in the numerator is known as the Boltzmann factor. (1) Show that the probability of finding the system A with energy i is given by the canonical distribution above. HINT: The key to this probelm is to note that if system A has energy i , system B must have energy E0 - i . We postulate that the probability of system A having energy i is proportional to the number of internal microstates of the reservoir with energy E0 - i , which we label as (Eo - i ). Now, as system A is much smaller than B, we assume that i E0 , implying that log[(Eo - i )] can be expanded as a power series around E0 . Note also that two systems with system energies Ei , Ej are considered to be in thermal equilibrium (and hence have the same temperature ) if and only if the derivatives of ln i (Ei ) and ln j (Ej ) with respect to their respective system energies Ei , Ej are equal. From that, an appropriate definition of the temperature can be derived, which you will need in your derivation. (2) Let us now say that the system of system can A be described by a set of continuous, classical dynamical variables x1 , x2 , ..., xn and corresponding momenta p1 , p2 , ..., pn . (As an example, A might consist of a collection of n/3 gaseous 3 molecules, and {xi , pi } may represent the position and momentum of each molecule along a prescribed coordinate axis.) We make two assertions: The energy of system A can be split into two terms, only one of which is dependent on pi : = E(pl ) + (q1 , q2 , ..., qn ; p1 , p2 , ..., pl-1 , pl+1 , ..., pn ) E(pl ) is quadratic in pl : E(pl ) = bp2 l where b is a given constant. Under these conditions, show that the statistical average of E(pl ) is given by This result is known as the equipartition theorem. It states that the mean 1 value of each independent quadratic term in the energy is 2 kB . HINT: The statistical average of quantity a is given by 1 k . 2 B a = aP (a)dq1 ...dqn dp1 ...dpn . (For this formula to be appropriate, we must assume that the density of accessible states in phase space is constant.) (3) If A is a quantum mechanical system with a discrete eigenenergy spectrum, does the equipartition theorem hold ? Discuss the conditions under which your result is valid. Problem IV: Macroscopic conductor (30 points) At finite temperature, the electron collisions with the lattice vibrations and impurities induce fluctuations in the electron's velocity and position. The electronic motion is well explained by a Brownian particle model. The net microscopic electronic thermal agitation drives a fluctuating voltage across the resistor. In this problem, you will derive the Johnson-Nyquist formula for equilibrium thermal noise using the fluctuationdissipation theorem. Consider a parallel resistor-inductor circuit. R is the resistance and L is the inductance. Given the voltage across the resistor V0 (t), the Kirchhoff voltage law tells you that L dI(t) = V0 (t), where I(t) is the current of the circuit. Suppose V0 (t) decomposes dt into a slowly-varying part V (t) and a rapidly-varying part v(t). (1) Show that the Langevin equation for this circuit is L dI(t) = -RI(t) + v(t) dt 4 I (t) + V0 (t) - R L Figur...

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Stanford - EEAP - 248
Fundamentals of Noise ProcessesEE/AP 248 , Fall 2008 Instructor: Professor Yoshihisa YamamotoPlease address homework questions to Kristiaan De Greve (kdegreve@stanford.edu).Problem Set 4Due date: December 1, 2008This short problem set is intend
Stanford - EEAP - 248
Fundamentals of Noise ProcessesEE248/Fall 2008 Instructor: Professor Yoshihisa YamamotoPlease address homework questions to Kristiaan De Greve (kdegreve@stanford.edu).Problem Set 5Due date: December 10, 2008 Reading assignment: review chapter 8
Stanford - EEAP - 248
Fundamentals of Noise ProcessesEE/AP 248 , Fall 2008 Instructor: Professor Yoshihisa YamamotoPlease address homework questions to Kristiaan De Greve (kdegreve@stanford.edu).Problem Set 6Due date: December 10, 2008In conclusion of this course, I
Stanford - EEAP - 248
Chapter 1 Mathematical Methods 1.1 Time average vs. ensemble averageNoise is only statistically characterized. We cannot discuss a single event at a certain time.time average: Averaged quantity of a single system over a time interval.(directly re
Stanford - EEAP - 248
Chapter 2 Quantum Statistics 2.1 Symmetrization postulateIdentical quantum particles are indistinguishable. We label a particle by a detection event, i.e. each &quot;click&quot; defines a particle 1 and 2.classical picture:50-50% beam splitter particle 1 f
Stanford - EEAP - 248
Chapter 3 Circuit Theory Noise in Linear Networks An inherently noisy active or passive network A noise-free network with external noise generatorsequivalent noise resistance (conductance) equivalent noise temperature3.1 Thermal noise of two-ter
Stanford - EEAP - 248
Chapter 4 Macroscopic ConductorsJ. B. Johnson (1927): Nature 119, 50 (1927) An open-circuit voltage noise spectral density Sv() is independent of the material and the measurement frequency, and is equal to 4kBTR.Rv(t) Microscopic theoryA. Ein
Stanford - EEAP - 248
Chapter 5 Mesoscopic Conductors 5.1 Mesoscopic electronsA. Effective mass equationballistic transport in a mesoscopic conductordeBroglie device phase momentum scattering wavelength size coherence length length p/h (Fermi wavelength )single part
Stanford - EEAP - 248
Chapter 6 Macroscopic pn JunctionsFET monopolar (either electron or hole) device simple macroscopic conductor (thermal noise limited) (shot noise suppressed by Pauli principle) pn junction devices Light emitting diodes (LED) Semiconductor lasers Bip
Stanford - EEAP - 248
Chapter 7 Mesoscopic pn JunctionsMesoscopic p+-N Junction with (weak forward bias)N I P+ Pelectron+ + +xn(t)xu-before emission holeelectronnegligible (xp(t)after emissionThermionic emission rate:cross-sectionA* : Richardson c
Stanford - EEAP - 248
Chapter 8 Noise of pn junction devices 8.1 Noise of a semiconductor laserelectron fluxRsp n photon fluxA. Noise equivalent circuit of a semiconductor laser vn iL in L is Rs i R C ia -Ra ioiout Roinput current fluctuationinternal electron
Stanford - EEAP - 248
Chapter 9 1/f Noise and Random Telegraph Signals1/f noise is a ubiquitous type of fluctuation. It appears in all types of carbon resistors single crystal semiconductors (Ge, Si, GaAs, .) p-n junction devices (bipolar transistors, semiconductor la
Stanford - EEAP - 248
Chapter 10 Negative Conductance Oscillators: LasersEssential components for negative conductance oscillators: frequency selective circuit LC circuit, delayed feedback loop, Fabry-Perot, ring, distributed feedback (DFB) cavities, . amplifying eleme
Stanford - AP - 388
Appendix A. Quantization of an Ensemble of Atoms A.1 Quantization of orbital angular momentumA.1.1 Commutation relationr p (momentum vector) r q (coordinate vector)0r r r orbital angular momentum l = q p ^ ^ ^ ^ l^ = q p - q px y z z y^ ^ ^
Stanford - AP - 388
Appendix C Mesoscopic Noise C.1 Mesoscopic electronsA. Effective mass equationdeBroglie momentum scattering phase wavelength device length size coherence length p/hsingle particle Schrdinger equationperiodic latticephononelectrostatic conf
Stanford - AP - 388
Mesoscopic Physics and NanostructuresAP388/EE338B Spring 2008 Instructor: Professor Yoshihisa YamamotoPlease address homework questions to Georgios Roumpos (roumpos@stanford.edu).Problem Set 1Due date: April 29th, 2008 Tuesday This problem set i
Stanford - AP - 388
Mesoscopic Physics and NanostructuresAP388/EE338B Spring 2008 Instructor: Professor Yoshihisa YamamotoPlease address homework questions to Georgios Roumpos (roumpos@stanford.edu).Problem Set 2Due date: May 29th, 2008 ThursdayThis problem set i
Stanford - AP - 388
Mesoscopic Physics and NanostructuresAP388/EE338B Spring 2008 Instructor: Professor Yoshihisa YamamotoPlease address homework questions to Georgios Roumpos (roumpos@stanford.edu).Problem Set 3Due date: Wednesday June 11th 2008, noonThis proble
Stanford - AP - 388
Stanford - AP - 388
-
Stanford - AP - 388
PRL 97, 227401 (2006)PHYSICAL REVIEW LETTERSweek ending 1 DECEMBER 2006Optical Detection and Ionization of Donors in Specific Electronic and Nuclear Spin StatesA. Yang,1 M. Steger,1 D. Karaiskaj,1 M. L. W. Thewalt,1,* M. Cardona,2 K. M. Itoh,3
Stanford - AP - 388
RESEARCH ARTICLESCoherent Manipulation of Coupled Electron Spins in Semiconductor Quantum DotsJ. R. Petta,1 A. C. Johnson,1 J. M. Taylor,1 E. A. Laird,1 A. Yacoby,2 M. D. Lukin,1 C. M. Marcus,1 M. P. Hanson,3 A. C. Gossard3We demonstrated coherent
Stanford - AP - 388
PHYSICAL REVIEW B 71, 014401 (2005)Coherence time of decoupled nuclear spins in siliconT. D. Ladd,* D. Maryenko, and Y. YamamotoQuantum Entanglement Project, SORST, JST, Edward L. Ginzton Laboratory, Stanford University, Stanford, California 9430
Stanford - AP - 388
letters to nature.Strong coupling in a single quantum dotsemiconductor microcavity system J. P. Reithmaier1, G. Sek1*, A. Loffler1, C. Hofmann1, S. Kuhn1, S. Reitzenstein1, L. V. Keldysh2, V. D. Kulakovskii3, T. L. Reinecke4 &amp; A. Forchel1 Tec
Stanford - AP - 388
Vol 439|12 January 2006|doi:10.1038/nature04446LETTERSA semiconductor source of triggered entangled photon pairsR. M. Stevenson1, R. J. Young1,2, P. Atkinson2, K. Cooper2, D. A. Ritchie2 &amp; A. J. Shields1Entangled photon pairs are an important re
Stanford - AP - 388
letters to naturefamily formation predict that the memory of spin of the original unshattered parent body is lost3, and existing models of spin angular momentum suggest that collisional evolution randomizes asteroid spin vectors regardless of their
Stanford - AP - 388
PRL 99, 016405 (2007)PHYSICAL REVIEW LETTERSweek ending 6 JULY 2007Quantum Simulator for the Hubbard Model with Long-Range Coulomb Interactions Using Surface Acoustic WavesTim Byrnes,1,2 Patrik Recher,2,3 Na Young Kim,3 Shoko Utsunomiya,1 and
Stanford - AP - 388
Chapter 1. Nuclear Magnetic Resonance (NMR) 1.0 Historical background Gorter (Leiden): Paramagnetic relaxation Prediction of spin resonance (1936) Rabi (Columbia): Magnetic resonance experiment with molecular beam World War II: Microwave componen
Stanford - AP - 388
Chapter 2. Electron Spin Resonance (ESR)The major difference between electron and nuclear magnetic resonance is that the nuclear properties are hardly affected by the surroundings whereas the electronic properties strongly dependent on the surroundi
Stanford - AP - 388
Chapter 3. Double ResonanceExcite one resonant transition of a system while simultaneously monitoring a different transition.nuclear polarization enhancing sensitivity simplifying spectra unraveling complex spectraThree broad categories:1. Pou
Stanford - AP - 388
Chapter 4. Optical Processes in Semiconductors 4.1 Two-dimensional electron and hole gases in quantum wells4.1.1 Conduction electrons rDouble heterostructure (Alferov, Hayashi, Panish, 1970)zA: narrow bandgap semiconductor B: wide bandgap semic
Stanford - AP - 388
Chapter 5. Exciton Polaritons ck photon = nexciton strong coupling of excitons and photonsupper polaritonlower polaritonnew quasi-particle (polariton)k5.1 Classical (dielectric) theory of exciton polaritons polarization: P = Edispl
Stanford - AP - 388
Chapter 6. Exciton Bose Einstein CondensationIn this chapter we discuss ideal quantum gases which are systems of non-interacting quantum particles but nevertheless are in thermal equilibrium. Density operator: grand-canonical ensemble=^ H: ^ N:
Stanford - AP - 388
Chapter 7. Two Dimensional Electron Gas System (2DEG) 7.1 Two-dimensional electron gas (2-DEG)Si MOSFET (metal-oxide-semiconductor field effect transistor) GaAs HEMT (high electron mobility transistor)S. M. Sze, Physics of Semiconductor Devices (J
Stanford - AP - 388
Chapter 8. Electron Transport as a Transmission Problem (Landauer-Bttiker Formula) 8.1 Quantum unit of conductancereflection less contact:reflectioncut-offChapter8-1There are reflections for higher-order transverse modes when electrons enter
Stanford - AP - 388
Chapter 9. Quantum Hall Effect 9.1 Magneto electric subbandsEffective mass equation:2 i + eA Es + + U ( y ) ( x, y ) = E ( x, y ) 2m ()quantization energy along z-directionvector potential Px x P -i y y-iconstant magnetic fie
Stanford - AP - 227
Exciton Binding Energy11s Exciton Wavefunction2Exciton Binding Energy in 2D System3Absorption Spectrum of Excitons and Continuum states4Modified Density of Field States in Planar MicrocavitymirrorL=QW2cavitymirror A planar m
Stanford - AP - 227
Appendix A. Quantization of an Ensemble of Atoms A.1 Quantization of orbital angular momentumA.1.1 Commutation relationp (momentum vector) q (coordinate vector)0orbital angular momentum l = q p lx = q y pz qz p y ly = qz p x q x pz
Stanford - AP - 227
Phys. Rev. 69, 37 - 38 (1946)Phys. Rev. 69, 127 - 127 (1946)1nuclear spins1H 13C 19F 27Al 29Si 31Pabundance (%) 99.985 1.11 100 100 4.7 100spin 1/2 1/2 1/2 5/2 1/2 1/2nuclear g-factor gn 2.7927 0.70238 2.628 3.6414 -0.55525 1.1316reson
Stanford - AP - 227
To appear in Nature (2009)Optimized Dynamical Decoupling in a Model Quantum MemoryMichael J. Biercuk, Hermann Uys, Aaron P. VanDevender, Nobuyasu Shiga, Wayne M. Itano &amp; John J. Bollinger1RabiRamsey23Refocusing and Decoupling for Nucle
Stanford - AP - 227
Qubit Relaxation and Dynamical DecouplingThaddeus Ladd April 8, 2009 In this chapter, we discuss single-spin or single-qubit relaxation processes (T1 , T2 , T2 ) and how they may be changed by dynamical decoupling, the application of rapid control
Stanford - AP - 227
APPPHYS227 Handout: QM review of basic atomic structureOrbital angular momentum, in general Angular momentum operator in QM is defined just the same way as in classical mechanics: r r L p , i x, y, z and p are the `vector' position and m
Stanford - AP - 227
Physical representations of qubits|i = c0 |0i + c1 |1icomputational basis states States of what? A photon an atom an exciton a group of N atoms A photon, an atom, an exciton, a group of N atoms, . What kinds of states? Orthogonal states cert
Stanford - AP - 227
APPPHYS227 Handout: Basic models of cavity QEDIn this short set of notes we will introduce the elementary models of cavity QED in a rather modern way, starting with the Jaynes-Cummings Hamiltonian, adding decoherence terms, and then deriving the Max
Stanford - AP - 227
Recap: qubit subspaces in hydrogenlike atoms At i l l Atomic levels provide basis states for optical/microwave/rf qubit encodings id b i t t f ti l/ i / f bit di A variety of experimental techniques for coherent manipulation are knowndirect opti
Stanford - AP - 227
Class discussion on 4/21 utilized notes from 4/2 as well as the following papers: &quot;Quantum jumps and conditional spin dynamics in a strongly coupled atomcavity system&quot; http:/arxiv.org/abs/0901.3738v1 &quot;Quantum Rabi Oscillation: A Direct Test of Fi
Stanford - AP - 227
Bloch Equations for a two-level atom Recall the general relation between spin-1/2 operators and the Pauli matrices, Sz z 2 2 1 0 0 -1 , Sx x 2 2 0 1 1 0 , Sy y 2 2 0 -i i 0 .Here the matrix forms of the spin operators are written in a b
Stanford - AP - 227
Class discussions on 4/30 and 5/1 utilized the following references: &quot;Investigation of the spectrum of resonance fluorescence induced by a monochromatic field&quot; http:/link.aps.org/doi/10.1103/PhysRevLett.35.1426 &quot;Spontaneous singleatom phase switc
Stanford - AP - 227
JaynesCummings ladder, vacuum Rabi splittingResonant case =0, JaynesCummings Hamiltonian couples |g,n+1i and |e,ni Jaynes Cummings Hamiltonian couples |g n+1i and |e ni statesCsH = g(a + a )h h |e, 1i |g, 2ih |e, 0i |g, 1i 2 2g h 1 (|
Stanford - AP - 227
Laser spectroscopy of trapped ions/atoms in the Lamb-Dicke regimeMechanical harmonic oscillator Recall that for a quantized simple harmonic oscillator, a a , p -i m a - a , 2m 2 where a and a are creation and annihilation operators for the lad
Stanford - AP - 227
The Hamiltonian for a spin in a magnetic field, written in the interaction picture (rotating frame) is H S B eff , B eff B 0 z 1 x. If we assume L then H 1 x . 2 The Schrdinger Equation is of course d | i H | , dt which has
Stanford - AP - 227
Class discussions on 1 May and 5 May (ion traps) utilized the following papers: A. BenKish et al., &quot;Experimental Demonstration of a Technique to Generate Arbitrary Quantum Superposition States of a Harmonically Bound Spin1/2 Particle,&quot; Phys. Rev. L
Stanford - AP - 227
APPPHYS227 Homework #1Due Thursday 9 April, at the beginning of class1. Atomic structure a. What is the Rydberg constant? How precisely has it been measured? (Note that both the Wikipedia entry and my AP227 handout are a bit out of date.) b. What
Stanford - AP - 227
APPPHYS227 Homework #4Due Thursday 30 April1. Bloch Equations Derive the `torque' part of the Bloch equations, d v B, v dt starting from the Master Equation -i H, , H -S B. 2. Cavity damping Starting from the Master Equation 2aa
Stanford - AP - 227
APPPHYS227 Homework #5Due Thursday 7 May1. Mollow Triplet In the article &quot;Investigation of the Spectrum of Resonance Fluorescence Induced by a Monochromatic Field&quot; (http:/link.aps.org/doi/10.1103/PhysRevLett.35.1426), can you explain the basic fea
Stanford - AP - 227
APPPHYS227 Homework #6Due Thursday 14 May1. Quantum logic Using methods discussed in the 5/7 class notes, show how the Hadamard gate U had 1 2 1 -1 1 1 ,can be generated by evolving the general two-level Hamiltonian H -S B for some value o
Stanford - AP - 227
APPPHYS 227Quantum Device Physics of Atomic and Semiconductor SystemsStanford University, Spring Quarter 2009Course objectives o introduce atomic, semiconductor systems for quantum information processing o introduce key experimental techniques for
Stanford - AP - 227
Quantum Device Physics of Atomic and Semiconductor SystemsAP227 Spring 2009 Instructors: Hideo Mabuchi, Yoshihisa YamamotoProblem Set 2Due date: Thursday, April 16, 2009Problem I: Hyperfine Splitting in a Semiconductor The hyperfine splitting of
Stanford - AP - 227
Quantum Device Physics of Atomic and Semiconductor SystemsAP227 Spring 2009 Instructors: Hideo Mabuchi, Yoshihisa YamamotoProblem Set 3Due date: Thursday, April 23, 2009Problem I: Phase Decoherence Coherence of a spin-1/2 particle is lost due to
Stanford - AP - 227
Quantum Device Physics of Atomic and Semiconductor SystemsAP227 Spring 2009 Instructors: Hideo Mabuchi, Yoshihisa YamamotoProblem Set 7Due date: Thursday, May 21, 2009Problem I: Phase Decoherence Explain the physical reason for the following fac
Stanford - AP - 227
Chapter 1. Nuclear Magnetic Resonance (NMR) 1.0 Historical background Gorter (Leiden): Paramagnetic relaxation Prediction of spin resonance (1936) Rabi (Columbia): Magnetic resonance experiment with molecular beam World War II: Microwave componen