DOCSLIB.ORG

Theory and Design of Quantum Devices in Circuit QED

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Theory and Design of Quantum Devices in Circuit QED C HRISTIAN THEORY AND DESIGN OF QUANTUM K. A NDERSEN DEVICES IN CIRCUIT QED T The information technology of the future will contain a vast number of quantum ele- HEORY AND DESIGN OF QUANTUM DEVICES IN CIRCUIT ments with the promise of faster computers and more secure communication. The com- puter chips embedded into almost all technology today are made of silicon transistors and similarly will the quantum technology of tomorrow rely on mass produced quan- tum circuits. Circuit QED is one framework for building quantum circuits in which the information is encoded in superconducting quantum bits and operations are performed with the help of modern microwave technology. The aim of this thesis is to propose and describe devices that can be integrated into these circuits and, thus, enable new quantum technologies to emerge. A cornerstone in these devices is the so-called Josephson junc- tion, which works as a non-linear electrical component. It is shown in this thesis, that by combining Josephson junctions and microwave resonators, novel quantum effects, such as dynamics in the ultrastrong coupling regime and quantum-classical dynamics, can be explored. Moreover, this thesis describes a proposal for a microwave detector using current-biased Josephson junctions and a recent experiment which confirms the abilities of the device. Josephson junctions also play a key role in a proposed efficient approach to classical control of quantum devices working within the cryogenic environment of the quantum system. Finally, the thesis also considers the construction of desirable con- trollable couplings in circuit QED. Combined, these proposed designs offer a novel per- spective on the quantum information technology of the future using superconducting QED quantum circuits. D ISSERTATION AU AARHUS CHRISTIAN KRAGLUND ANDERSEN UNIVERSITY PHDDISSERTATION DEPARTMENT OF PHYSICS AND ASTRONOMY Theory and design of quantum devices in circuit QED Christian Kraglund Andersen P h D t h e s i s J u l y 2 0 1 6 Supervisor: Klaus Mølmer Department of Physics and Astronomy Aarhus University English summary The information technology of the future will contain a vast number of quantum elements with the promise of faster computers and more secure communication. The computer chips embedded into almost all technology today are made of silicon transistors and similarly will the quantum tech- nology of tomorrow rely on mass produced quantum circuits. Circuit QED is one framework for building quantum circuits in which the information is encoded in superconducting quantum bits and operations are performed with the help of modern microwave technology. The aim of this thesis is to propose and describe devices that can be integrated into these circuits and, thus, enable new quantum technologies to emerge. A cornerstone in these devices is the so-called Josephson junction, which works as a non-linear electrical component. It is shown in this thesis, that by combin- ing Josephson junctions and microwave resonators, novel quantum effects, such as dynamics in the ultrastrong coupling regime and quantum-classical dynamics, can be explored. Moreover, this thesis describes a proposal for a microwave detector using current-biased Josephson junctions and a recent experiment which confirms the abilities of the device. Josephson junctions also play a key role in a proposed efficient approach to classical control of quantum devices working within the cryogenic environment of the quantum system. Finally, the thesis also considers the construction of desirable controllable couplings in circuit QED. Combined, these proposed designs offer a novel perspective on the quantum information technology of the future using superconducting quantum circuits. i Dansk resumé (Danish summary) Fremtidens informationsteknologi kommer til at indeholder mange kvan- teelementer, hvilket vil medføre hurtigere computere og mere sikker kom- munikation. De computerchips der er inde i stort set al teknologi idag er bygget af silliciumtransistore, og på samme vis vil morgendagens kvan- teteknologi bygge på masseproducerede kvantekredsløb. Circuit QED er navnet på en en metode til at bygge disse kvantekredsløb, hvori kvantein- formation er gemt i superledende kvante-bits og hvor kvanteoperationer bliver udført ved hjælp af moderne mikrobølgeteknologi. Formålet med denne afhandling er at foreslå og beskrive enheder, der kan blive inte- greret i disse kredsløb og som dermed muliggør ny kvanteteknologi. En grundsten i disse enheder er den såkaldte Josephson junction, der virker som et ikke-liniært elektronisk element. Ved at kombinerer Josephson junctions med mikrobølgeresonatorer vil denne afhandling vise at det er muligt at udforske nye kvanteeffekter for eksempel ultrastærk kobling og kvante-klassisk dynamik. Derudover foreslåes det at en Josephson junction påtvunget en elektrisk strøm kan blive brugt som en mikrobølgedetektor og et nyligt udførst eksperiment viser at denne ide virker. Josephson junctions spiller også en afgørende rolle i en ny fremgangsmåde for klassisk kontrol af kvanteenheder, der bliver forslået i denne afhandling og som virker med samme teknologi som kvantesystemerne selv er implementeret med. Tilsidst kigger denne afhandling på hvordan forskellige ønskede koblinger i circuit QED kan kontrolleres. Tilsammen giver alle de forslåede design et nyt perspektiv på fremtidens kvanteinformationsteknologi implementeret med superledende kvantekredsløb. ii Preface In this thesis you find a presentation of research completed during my PhD studies performed at the Department of Physics and Astronomy at Aarhus University. The studies was funded by the Faculty of Science and Technology and further supported by the Danish Ministry of Higher Education and Science. The research was carried out between August 2011 and July 2016 under the supervision of Klaus Mølmer. Additionally, 5 months was spent during 2015 in the group of Alexandre Blais at University of Sherbrooke. The experimental data used for analysis in Secs. 3.2 and 4.3 was obtain by the groups of Michel Devoret at Yale and Evgeni Il’ichev at IPHT Jena respectively. On the next pages you will find a list of publications produced during my studies. The articles [1] and [4] are less related to the main topic of this thesis and, as such, will not be covered to any extend. The same goes for [15] which is an outreach article written in Danish. The papers [11–13] fits well within the scope of the thesis, but due to constrains of length these publications will not be included even though some of the general techniques used in these publications are presented in Chapter 2. The rest of the publications will be covered during the thesis, however not all details of all the papers will be presented and I encourage the reader to consult the papers themselves for additional details. As of writing, the papers [10–12, 14] are not published yet. I would like to thank Klaus Mølmer for his supervision during my studies. He has been a great supervisor; always ready with an impressive intuition and ready to challenge my ideas, but most importantly he has supported me greatly in developing and working out my own ideas. I am also very grateful to Alexandre Blais for the hospitality and our collaboration during my stay in Sherbrooke; his insight and expertise as well as his keen eye for details have taught me a lot. A thanks also goes to the all people I have worked with and discussed physics with over the years. There are too many people with whom I have enjoyed good iii iv discussions about physics, but in particular I would like to mention Jerome Bourassa, Samuel Boutin, Benjamin J. Chapman, Arne Grimsmo, The Jenses, Archana Kamal, Joseph Kerckhoff, Konrad W. Lehnert, Gregor Oelsner, Shruti Puri, Malte Tichy and Andrew C.J. Wade. Finally, a very big thanks goes to Eliska Greplova for supporting me in writing this thesis, for providing the drawing for the cover and for countless of valuable input and feedback. List of Publication [1] Christian Kraglund Andersen and Klaus Mølmer. Squeezing of collective excitations in spin ensembles. Phys. Rev. A, 86, 043831, (2012) [2] Christian Kraglund Andersen and Klaus Mølmer. Effective description of tunneling in a time-dependent potential with applications to voltage switching in Josephson junctions. Phys. Rev. A, 87, 052119, (2013) [3] Christian Kraglund Andersen, Gregor Oelsner, Evgeni Il’ichev, and Klaus Mølmer. Quantized resonator field coupled to a current-biased josephson junction in circuit QED. Phys. Rev. A, 89, 033853, (2014) [4] Malte C. Tichy and Christian Kraglund Andersen. Comment on ”Contextuality in bosonic bunching”. Phys. Rev. Lett., 113,138901, (2014) [5] Christian Kraglund Andersen and Klaus Mølmer. Circuit QED flip-flop memory with all-microwave switching. Phys. Rev. Applied, 3,024002, (2015) [6] Christian Kraglund Andersen and Klaus Mølmer. Multifrequency modes in superconducting resonators: Bridging frequency gaps in off-resonant couplings. Phys.Rev. A, 91,023828, (2015) [7] Christian Kraglund Andersen, Joseph Kerckhoff, Konrad W. Lehnert, Benjamin J. Chapman, and Klaus Mølmer. Closing a quantum feedback loop inside a cryostat: Autonomous state preparation and long-time memory of a superconducting qubit. Phys. Rev. A, 93,012346, (2016). v [8] G Oelsner, CK Andersen, M Rehák, M Schmelz, S Anders, M Grajcar, U Hübner, K Mølmer, and E Il’ichev. Detection of weak microwave fields with an underdamped Josephson junction. arXiv:1605.05935, (2016) [9] Christian Kraglund Andersen and Alexandre Blais. Ultrastrong coupling dynamics
Load more

Recommended publications
  • Solving the Quantum Scattering Problem for Systems of Two and Three Charged Particles
    Solving the quantum scattering problem for systems of two and three charged particles Solving the quantum scattering problem for systems of two and three charged particles Mikhail Volkov c Mikhail Volkov, Stockholm 2011 ISBN 978-91-7447-213-4 Printed in Sweden by Universitetsservice US-AB, Stockholm 2011 Distributor: Department of Physics, Stockholm University In memory of Professor Valentin Ostrovsky Abstract A rigorous formalism for solving the Coulomb scattering problem is presented in this thesis. The approach is based on splitting the interaction potential into a finite-range part and a long-range tail part. In this representation the scattering problem can be reformulated to one which is suitable for applying exterior complex scaling. The scaled problem has zero boundary conditions at infinity and can be implemented numerically for finding scattering amplitudes. The systems under consideration may consist of two or three charged particles. The technique presented in this thesis is first developed for the case of a two body single channel Coulomb scattering problem. The method is mathe- matically validated for the partial wave formulation of the scattering problem. Integral and local representations for the partial wave scattering amplitudes have been derived. The partial wave results are summed up to obtain the scat- tering amplitude for the three dimensional scattering problem. The approach is generalized to allow the two body multichannel scattering problem to be solved. The theoretical results are illustrated with numerical calculations for a number of models. Finally, the potential splitting technique is further developed and validated for the three body Coulomb scattering problem. It is shown that only a part of the total interaction potential should be split to obtain the inhomogeneous equation required such that the method of exterior complex scaling can be applied.
    [Show full text]
  • Zero-Point Energy of Ultracold Atoms
    Zero-point energy of ultracold atoms Luca Salasnich1,2 and Flavio Toigo1 1Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM, Universit`adi Padova, via Marzolo 8, 35131 Padova, Italy 2CNR-INO, via Nello Carrara, 1 - 50019 Sesto Fiorentino, Italy Abstract We analyze the divergent zero-point energy of a dilute and ultracold gas of atoms in D spatial dimensions. For bosonic atoms we explicitly show how to regularize this divergent contribution, which appears in the Gaussian fluctuations of the functional integration, by using three different regular- ization approaches: dimensional regularization, momentum-cutoff regular- ization and convergence-factor regularization. In the case of the ideal Bose gas the divergent zero-point fluctuations are completely removed, while in the case of the interacting Bose gas these zero-point fluctuations give rise to a finite correction to the equation of state. The final convergent equa- tion of state is independent of the regularization procedure but depends on the dimensionality of the system and the two-dimensional case is highly nontrivial. We also discuss very recent theoretical results on the divergent zero-point energy of the D-dimensional superfluid Fermi gas in the BCS- BEC crossover. In this case the zero-point energy is due to both fermionic single-particle excitations and bosonic collective excitations, and its regu- larization gives remarkable analytical results in the BEC regime of compos- ite bosons. We compare the beyond-mean-field equations of state of both bosons and fermions with relevant experimental data on dilute and ultra- cold atoms quantitatively confirming the contribution of zero-point-energy quantum fluctuations to the thermodynamics of ultracold atoms at very low temperatures.
    [Show full text]
  • Discrete Variable Representations and Sudden Models in Quantum
    Volume 89. number 6 CHLMICAL PHYSICS LEITERS 9 July 1981 DISCRETEVARlABLE REPRESENTATIONS AND SUDDEN MODELS IN QUANTUM SCATTERING THEORY * J.V. LILL, G.A. PARKER * and J.C. LIGHT l7re JarrresFranc/t insrihrre and The Depwrmcnr ofC7temtsrry. Tire llm~crsrry of Chrcago. Otrcago. l7lrrrors60637. USA Received 26 September 1981;m fin11 form 29 May 1982 An c\act fOrmhSm In which rhe scarrcnng problem may be descnbcd by smsor coupled cqumons hbclcd CIIIW bb bans iuncltons or quadrature pomts ISpresented USCof each frame and the srnrplyculuatcd unitary wmsformatlon which connects them resulis III an cfliclcnt procedure ror pcrrormrnpqu~nrum scxrcrrn~ ca~cubr~ons TWO ~ppro~mac~~~ arc compxcd wrh ihe IOS. 1. Introduction “ergenvalue-like” expressions, rcspcctnely. In each case the potential is represented by the potcntud Quantum-mechamcal scattering calculations are function Itself evahrated at a set of pomts. most often performed in the close-coupled representa- Whjle these models have been shown to be cffcc- tion (CCR) in which the internal degrees of freedom tive III many problems,there are numerousambiguities are expanded in an appropnate set of basis functions in their apphcation,especially wtth regardto the resulting in a set of coupled diiierentral equations UI choice of constants. Further, some models possess the scattering distance R [ 1,2] _The method is exact formal difficulties such as loss of time reversal sym- IO w&in a truncation error and convergence is ob- metry, non-physical coupling, and non-conservation tained by increasing the size of the basis and hence the of energy and momentum [ 1S-191. In fact, it has number of coupled equations (NJ While considerable never been demonstrated that sudden models fit into progress has been madein the developmentof efficient any exactframework for solutionof the scattering algonthmsfor the solunonof theseequations [3-71, problem.
    [Show full text]
  • The S-Matrix Formulation of Quantum Statistical Mechanics, with Application to Cold Quantum Gas
    THE S-MATRIX FORMULATION OF QUANTUM STATISTICAL MECHANICS, WITH APPLICATION TO COLD QUANTUM GAS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Pye Ton How August 2011 c 2011 Pye Ton How ALL RIGHTS RESERVED THE S-MATRIX FORMULATION OF QUANTUM STATISTICAL MECHANICS, WITH APPLICATION TO COLD QUANTUM GAS Pye Ton How, Ph.D. Cornell University 2011 A novel formalism of quantum statistical mechanics, based on the zero-temperature S-matrix of the quantum system, is presented in this thesis. In our new formalism, the lowest order approximation (“two-body approximation”) corresponds to the ex- act resummation of all binary collision terms, and can be expressed as an integral equation reminiscent of the thermodynamic Bethe Ansatz (TBA). Two applica- tions of this formalism are explored: the critical point of a weakly-interacting Bose gas in two dimensions, and the scaling behavior of quantum gases at the unitary limit in two and three spatial dimensions. We found that a weakly-interacting 2D Bose gas undergoes a superfluid transition at T 2πn/[m log(2π/mg)], where n c ≈ is the number density, m the mass of a particle, and g the coupling. In the unitary limit where the coupling g diverges, the two-body kernel of our integral equation has simple forms in both two and three spatial dimensions, and we were able to solve the integral equation numerically. Various scaling functions in the unitary limit are defined (as functions of µ/T ) and computed from the numerical solutions.
    [Show full text]
  • (2021) Transmon in a Semi-Infinite High-Impedance Transmission Line
    PHYSICAL REVIEW RESEARCH 3, 023003 (2021) Transmon in a semi-infinite high-impedance transmission line: Appearance of cavity modes and Rabi oscillations E. Wiegand ,1,* B. Rousseaux ,2 and G. Johansson 1 1Applied Quantum Physics Laboratory, Department of Microtechnology and Nanoscience-MC2, Chalmers University of Technology, 412 96 Göteborg, Sweden 2Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, 75005 Paris, France (Received 8 December 2020; accepted 11 March 2021; published 1 April 2021) In this paper, we investigate the dynamics of a single superconducting artificial atom capacitively coupled to a transmission line with a characteristic impedance comparable to or larger than the quantum resistance. In this regime, microwaves are reflected from the atom also at frequencies far from the atom’s transition frequency. Adding a single mirror in the transmission line then creates cavity modes between the atom and the mirror. Investigating the spontaneous emission from the atom, we then find Rabi oscillations, where the energy oscillates between the atom and one of the cavity modes. DOI: 10.1103/PhysRevResearch.3.023003 I. INTRODUCTION [43]. Furthermore, high-impedance resonators make it pos- sible for light-matter interaction to reach strong-coupling In the past two decades, circuit quantum electrodynamics regimes due to strong coupling to vacuum fluctuations [44]. (circuit QED) has become a field of growing interest for quan- In this article, we investigate the spontaneous emission tum information processing and also to realize new regimes of a transmon [45] capacitively coupled to a 1D TL that is in quantum optics [1–11].
    [Show full text]
  • 3 Scattering Theory
    3 Scattering theory In order to find the cross sections for reactions in terms of the interactions between the reacting nuclei, we have to solve the Schr¨odinger equation for the wave function of quantum mechanics. Scattering theory tells us how to find these wave functions for the positive (scattering) energies that are needed. We start with the simplest case of finite spherical real potentials between two interacting nuclei in section 3.1, and use a partial wave anal- ysis to derive expressions for the elastic scattering cross sections. We then progressively generalise the analysis to allow for long-ranged Coulomb po- tentials, and also complex-valued optical potentials. Section 3.2 presents the quantum mechanical methods to handle multiple kinds of reaction outcomes, each outcome being described by its own set of partial-wave channels, and section 3.3 then describes how multi-channel methods may be reformulated using integral expressions instead of sets of coupled differential equations. We end the chapter by showing in section 3.4 how the Pauli Principle re- quires us to describe sets identical particles, and by showing in section 3.5 how Maxwell’s equations for electromagnetic field may, in the one-photon approximation, be combined with the Schr¨odinger equation for the nucle- ons. Then we can describe photo-nuclear reactions such as photo-capture and disintegration in a uniform framework. 3.1 Elastic scattering from spherical potentials When the potential between two interacting nuclei does not depend on their relative orientiation, we say that this potential is spherical. In that case, the only reaction that can occur is elastic scattering, which we now proceed to calculate using the method of expansion in partial waves.
    [Show full text]
  • Path Integral in Quantum Field Theory Alexander Belyaev (Course Based on Lectures by Steven King) Contents
    Path Integral in Quantum Field Theory Alexander Belyaev (course based on Lectures by Steven King) Contents 1 Preliminaries 5 1.1 Review of Classical Mechanics of Finite System . 5 1.2 Review of Non-Relativistic Quantum Mechanics . 7 1.3 Relativistic Quantum Mechanics . 14 1.3.1 Relativistic Conventions and Notation . 14 1.3.2 TheKlein-GordonEquation . 15 1.4 ProblemsSet1 ........................... 18 2 The Klein-Gordon Field 19 2.1 Introduction............................. 19 2.2 ClassicalScalarFieldTheory . 20 2.3 QuantumScalarFieldTheory . 28 2.4 ProblemsSet2 ........................... 35 3 Interacting Klein-Gordon Fields 37 3.1 Introduction............................. 37 3.2 PerturbationandScatteringTheory. 37 3.3 TheInteractionHamiltonian. 43 3.4 Example: K π+π− ....................... 45 S → 3.5 Wick’s Theorem, Feynman Propagator, Feynman Diagrams . .. 47 3.6 TheLSZReductionFormula. 52 3.7 ProblemsSet3 ........................... 58 4 Transition Rates and Cross-Sections 61 4.1 TransitionRates .......................... 61 4.2 TheNumberofFinalStates . 63 4.3 Lorentz Invariant Phase Space (LIPS) . 63 4.4 CrossSections............................ 64 4.5 Two-bodyScattering . 65 4.6 DecayRates............................. 66 4.7 OpticalTheorem .......................... 66 4.8 ProblemsSet4 ........................... 68 1 2 CONTENTS 5 Path Integrals in Quantum Mechanics 69 5.1 Introduction............................. 69 5.2 The Point to Point Transition Amplitude . 70 5.3 ImaginaryTime........................... 74 5.4 Transition Amplitudes With an External Driving Force . ... 77 5.5 Expectation Values of Heisenberg Position Operators . .... 81 5.6 Appendix .............................. 83 5.6.1 GaussianIntegration . 83 5.6.2 Functionals ......................... 85 5.7 ProblemsSet5 ........................... 87 6 Path Integral Quantisation of the Klein-Gordon Field 89 6.1 Introduction............................. 89 6.2 TheFeynmanPropagator(again) . 91 6.3 Green’s Functions in Free Field Theory .
    [Show full text]
  • Path-Integral Formulation of Scattering Theory
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Paul Finkler Papers Research Papers in Physics and Astronomy 10-15-1975 Path-integral formulation of scattering theory Paul Finkler University of Nebraska-Lincoln, [email protected] C. Edward Jones University of Nebraska-Lincoln M. Misheloff University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/physicsfinkler Part of the Physics Commons Finkler, Paul; Jones, C. Edward; and Misheloff, M., "Path-integral formulation of scattering theory" (1975). Paul Finkler Papers. 11. https://digitalcommons.unl.edu/physicsfinkler/11 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Paul Finkler Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. PHYSIC AL REVIE% D VOLUME 12, NUMBER 8 15 OCTOBER 1975 Path-integral formulation of scattering theory W. B. Campbell~~ Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, California 91125 P. Finkler, C. E. Jones, ~ and M. N. Misheloff Behlen Laboratory of Physics, University of Nebraska, Lincoln, Nebraska 68508 (Received 14 July 1975} A new formulation of nonrelativistic scattering theory is developed which expresses the S matrix as a path integral. This formulation appears to have at least two advantages: (1) A closed formula is obtained for the 8 matrix in terms of the potential, not involving a series expansion; (2) the energy-conserving 8 function can be explicitly extracted using a technique analogous to that of Faddeev and Popov, thereby yielding a closed path- integral expression for the T matrix.
    [Show full text]
  • Relativistic Dynamics for Spin-Zero and Spin-Half Particles
    Pramana- J. Phys., Vol. 27, No. 6, December 1986, pp. 731-745. © Printed in India. Relativistic dynamics for spin-zero and spin-half particles J THAKUR Department of Physics, Patna University, Patna 800005, India MS received 17 May 1986; revised 28 August 1986 Abstract. The classical and quantum mechanics of a system o f directly interacting relativistic particles is discussed. We first discuss the spin-zero case, where we basically follow Rohrlich in introducing a set of covariant centre of mass (CM) and relative variables. The relation of these to the classic formulation of Bakamjian and Thomas is also discussed. We also discuss the important case of relativistic potentials which may depend on total four-momentum squared. We then consider the quantum mechanical case of spin-half particles. The negative energy difficulty is solved by introducing a number of first class constraints which fix the spinor structure of physical solutions and ensure the existence of proper CM and relative variables. We derive the form of interactions consistent with Lorentz invariance, space inversion, time reversal and charge conjugation and with the above mentioned first class constraints and find that it is analogous to that for the non-relativistic case. Finally the relationship of the present work with some previous work is briefly discussed. Keywords. Direct interactions; spin zero; spin half; covariant centre of mass; first class constraints. PACS Nos 03-20;, 03-65; 11-30 1. Introduction There has been considerable progress recently in formulating a Hamiltonian theory of directly interacting relativistic particles. By direct interaction we mean an interaction between particles which does not require, for its description, the aid of an intervening field.
    [Show full text]
  • Scattering Theory, Born Approximation
    LectureLecture 55 ScatteringScattering theory,theory, BornBorn ApproximationApproximation SS2011: ‚Introduction to Nuclear and Particle Physics, Part 2‘ SS2011: ‚Introduction to Nuclear and Particle Physics, Part 2‘ 1 ScatteringScattering amplitudeamplitude We are going to show here that we can obtain the differential cross section in the CM frame from an asymptotic form of the solution of the Schrödinger equation: (1.1) Let us first focus on the determination of the scattering amplitude f (θ, φ), it can be obtained from the solutions of (1.1), which in turn can be rewritten as 2 2μE (1.2) where k = 2 h The general solution of the equation (1.2) consists of a sum of two components: 1) a general solution to the homogeneous equation: (1.3) In (1.3) is the incident plane wave 2) and a particular solution of (1.2) with the interaction potential 2 GeneralGeneral solutionsolution ofof SchrSchröödingerdinger eq.eq. inin termsterms ofof GreenGreen‘‘ss functionfunction The general solution of (1.2) can be expressed in terms of Green’s function. (1.4) is the Green‘s function corresponding to the operator on the left side of eq.(1.3) The Green‘s function is obtained by solving the point source equation: (1.5) (1.6) (1.7) 3 GreenGreen‘‘ss functionfunction A substitution of (1.6) and (1.7) into (1.5) leads to (1.8) The expression for can be obtained by inserting (1.8) into (1.6) (1.9) (1.10) To integrate over angle in (1.10) we need to make the variable change x=cosθ (1.11) 4 MethodMethod ofof residuesresidues Thus, (1.9) becomes (1.12) (1.13) The integral in (1.13) can be evaluated by the method of residues by closing the contour in the upper half of the q-plane: The integral is equal to 2π i times the residue of the integrand at the poles.
    [Show full text]
  • Scattering Theory with Path Integrals R
    JOURNAL OF MATHEMATICAL PHYSICS 55, 032106 (2014) Scattering theory with path integrals R. Rosenfelder Particle Theory Group, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland (Received 27 June 2013; accepted 15 February 2014; published online 13 March 2014) Starting from well-known expressions for the T-matrix and its derivative in standard nonrelativistic potential scattering, I rederive recent path-integral formulations due to Efimov and Barbashov et al. Some new relations follow immediately. C 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4867605] I. INTRODUCTION Traditionally, the path-integral method in quantum physics has been applied mostly to bound- state problems. Following the pioneering work of Ref. 1, there has been renewed interest in the last few years to use it also for scattering problems2–4 in nonrelativistic physics. This offers the chance of finding new approximation methods6, 7 or trying to evaluate the involved path integrals by stochastic methods although the oscillating nature of the real-time path integral presents a great challenge (see, e.g., Ref. 8). It should be also kept in mind that the path-integral approach, where one integrates functionally over the degrees of freedom weighted by the exponential of the classical action, is much more general than a Schrodinger¨ description thus allowing an immediate generalization to many-body or field-theoretical problems.9–11 In the following, however, only single- channel nonrelativistic scattering in a local potential V (x) is considered with the aim to find practical path-integral representations for the scattering amplitude free of infinite (time-)limits. Extensions to nonlocal potentials have been considered in Ref.
    [Show full text]
  • Applications of Quantum Mechanics University of Cambridge Part II Mathematical Tripos
    Preprint typeset in JHEP style - HYPER VERSION Lent Term, 2017 Applications of Quantum Mechanics University of Cambridge Part II Mathematical Tripos David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/aqm.html [email protected] { 1 { Recommended Books and Resources There are many good books on quantum mechanics. Here's a selection that I like: • Griffiths, Introduction to Quantum Mechanics An excellent way to ease yourself into quantum mechanics, with uniformly clear expla- nations. For this course, it covers both approximation methods and scattering. • Shankar, Principles of Quantum Mechanics • James Binney and David Skinner, The Physics of Quantum Mechanics • Weinberg, Lectures on Quantum Mechanics These are all good books, giving plenty of detail and covering more advanced topics. Shankar is expansive, Binney and Skinner clear and concise. Weinberg likes his own notation more than you will like his notation, but it's worth persevering. This course also contains topics that cannot be found in traditional quantum text- books. This is especially true for the condensed matter aspects of the course, covered in Sections 3, 4 and 5. Some good books include • Ashcroft and Mermin, Solid State Physics • Kittel, Introduction to Solid State Physics • Steve Simon, Solid State Physics Basics Ashcroft & Mermin and Kittel are the two standard introductions to condensed matter physics, both of which go substantially beyond the material covered in this course. I have a slight preference for the verbosity of Ashcroft and Mermin. The book by Steve Simon covers only the basics, but does so very well.
    [Show full text]