DOCSLIB.ORG

A. Operator Relations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A. Operator Relations A. Operator Relations A.1 Theorem 1 Let A and B be two non-commuting operators, then [1] α2 exp(αA)B exp(−αA)=B + α [A, B]+ [A, [A, B]] + .... (A.1) 2! Proof. Let f1(α)=exp(αA)B exp(−αA) , (A.2) then, one can expand f1 in Taylor series about the origin. We first evaluate the derivatives − − f1(α)=exp(αA)(AB BA)exp( αA) , so f1(0) = [A, B] . (A.3) Similarly − − f1 (α)=exp(αA)(A [A, B] [A, B] A)exp( αA) , so that f1 (0) = [A, [A, B]] . (A.4) Now, we write the Taylor’s expansion α2 f (α)=f (0) + αf (0) + f (0) + ... (A.5) 1 1 1 2! 1 or α2 exp(αA)B exp(−αA)=B + α [A, B]+ [A, [A, B]] + .. (A.6) 2! A particular case is when [A, B]=c,wherec is a c-number, then exp(αA)B exp(−αA)=B + αc , (A.7) in which case exp(αA) acts as a displacement operator. 364 A. Operator Relations A.2 Theorem 2: The Baker–Campbell–Haussdorf Relation Let A and B be two non-commuting operators such that [A, [A, B]] = [B,[A, B]] = 0 , (A.8) then α2 exp [α(A + B)] = exp αA exp αB exp − [A, B] (A.9) 2 α2 =expαB exp αA exp [A, B] . 2 Proof. Define f2(α) ≡ exp αA exp αB . (A.10) Then df (α) 2 =[A +exp(αA)B exp −(αA)]f (α) (A.11) dα 2 =(A + B + α [A, B])f2(α) , where in the last step, we used (A.6). Also, from the definition of f2(α), we can write df (α) 2 =exp(αA)A exp αB +exp(αA)exp(αB)B (A.12) dα =expαA exp αB [exp(−αB)A exp αB + B] = f2(α)(A + B + α [A, B]) . By comparing (A.11) and (A.12), we can see that f2(α)commuteswith (A+B +α [A, B]), thus one can integrate as a c-number differential equation, getting α2 α2 f (α)=exp (A + B)α + [A, B] =expα(A+B)exp [A, B] , (A.13) 2 2 2 thus obtaining the desired result. Another application of the Theorem 1 is taking A = aa†, (A.14) B = a or a† . As [n, a]=−a (A.15) A.3 Theorem 3: Similarity Transformation 365 and the higher order commutators also give a with alternating signs, thus α2 exp(αn)a exp(−αn)=a − αa + a + ... =exp(−α)a. (A.16) 2 Similarly exp(αn)a† exp(−αn)=exp(α)a† . (A.17) A.3 Theorem 3: Similarity Transformation exp(αA)f(B)exp−(αA)=f(exp(αA)(B)exp−(αA)) . (A.18) Proof. We start with the following identity [exp(αA)(B)exp−(αA)]n =exp(αA)B exp(−αA)exp(αA)B exp(−αA)... =exp(αA)Bn exp(−αA) . Then, for any function f(B) that can be expanded in power series, the Theorem 3 follows. As an interesting application, let us calculate exp −αa† + α∗a f(a, a†)exp αa† − α∗a = f[exp −αa† + α∗a a exp αa† − α∗a , exp −αa† + α∗a a† exp αa† − α∗a ] = f(a + α, a† + α∗) . Also exp −αa† f(a, a†)exp αa† = f(a + α, a†) , (A.19) exp (α∗a) f(a, a†)exp(−α∗a)=f(a, a† + α∗) , (A.20) exp (αn) f(a, a†)exp(−αn)=f[a exp(−α),a† exp(α)] (A.21) Other properties: One can easily show that da†l a, a†l = la†l−1 = , (A.22) da† dal a†,al = −lal−1 = − . da A more general version of the above relations is for a function f(a, a) which may be expanded in power series of a and a† ∂f(a, a†) a, f(a, a†) = , (A.23) ∂a† ∂f(a, a†) a†,f(a, a†) = − . (A.24) ∂a 366 A. Operator Relations Reference 1. Louisell, W.H.: Quantum Statistical Properties of Radiation. John Wiley, New York (1973) B. The Method of Characteristics We have a first-order partial differential equation: [1] Pp+ Qq = R, (B.1) where P = P (x, y, z),Q= Q(x, y, z),R= R(x, y, z), and ∂z ∂z p ≡ ,q ≡ , (B.2) ∂x ∂y and we wish to find a solution of (B.1), of the form z = f(x, y) . (B.3) The general solution of (B.1) is F (u, v)=0, (B.4) where F is an arbitrary function, and u(x, y, z)=c1 , (B.5) v(x, y, z)=c2 , is a solution of the equations dx dy dz = = . (B.6) P Q R Proof. If (B.5) are solutions of (B.6), then the equations ∂u ∂u ∂u dx + dy + dz =0, (B.7) ∂x ∂y ∂z and dx dy dz = = , P Q R must be compatible; thus, we must have 368 B. The Method of Characteristics Pux + Quy + Ruz =0, (B.8) and similarly for v Pvx + Qvy + Rvz =0. (B.9) On the other hand, if x and y are independent variables and z = z(x,y), then from (B.5), we get ∂z u + u =0, (B.10) x z ∂x ∂z u + u =0, y z ∂y and substituting (B.10) into (B.8) we get ∂z ∂z ∂u −P − Q + R =0, ∂x ∂y ∂z and (B.1) is satisfied. The second part of the proof is to show that the general solution of (B.1) is F (u, v)=0. (B.11) From (B.11), one writes ∂F ∂F ∂u ∂u ∂z ∂F ∂v ∂v ∂z = + + + =0, (B.12) ∂x ∂u ∂x ∂z ∂x ∂v ∂x ∂z ∂x ∂F ∂F ∂u ∂u ∂z ∂F ∂v ∂v ∂z = + + + =0. (B.13) ∂y ∂u ∂y ∂z ∂y ∂v ∂y ∂z ∂y We finally notice that (B.13) is satisfied considering (B.10). Example. Find the general solution of the equation ∂z ∂z x2 + y2 =(x + y)z. (B.14) ∂x ∂y In this case P = x2 , (B.15) Q = y2 , R =(x + y)z, and we have to find the solution of dx dy dz = = . (B.16) x2 y2 (x + y)z Reference 369 Integrating, first dx dy = , x2 y2 we get −1 −1 x + y = c1 . (B.17) On the other hand, x2 − dx − dy ( 2 1)dy dy dz = y = = , x2 − y2 x2 − y2 y2 (x + y)z from where we get x − y = c = v. (B.18) z 2 Combining (B.17) and (B.18), we get xy = c = u, (B.19) z 1 so the general solution can be put as xy x − y F , =0, (B.20) z z or if we write (B.20) in the equivalent form u = g(v) , (B.21) then the solution is xy x − y = g . (B.22) z z Reference 1. Sneddon, I.: Elements of Partial Differential Equations. (Mc-Graw Hill, New York (1957) C. Proof In this Appendix, we show the equation ⎡ ⎤ ⎣ − − − 2⎦ 2 − − δ(t tj)δ(t tk) S R ρaa = pRδ(t t ) . (C.1) j= k For regular pumping, one can put tj = t0 + jτ, where τ is the constant time interval between the atoms and t0 some arbitrary time origin [1]. In this case, there are no pumping fluctuations, and therefore, there are no correlations between the products of delta functions, that is δ(t − tj)δ(t − tk)S = δ(t − tj)S δ(t − tk)S (C.2) j,k j k = R2 . Now, we split the l.h.s. of the above equation in two parts 2 δ(t − tj)δ(t − tk)S + δ(t − tj)δ(t − tk)S = R , (C.3) j=k j=k 2 δ(t − tj)δ(t − tk)S + Rδ(t − t )=R , j= k thus proving the relation ⎡ ⎤ ⎣ − − − 2⎦ 2 − − δ(t tj)δ(t tk) S R ρaa = pRδ(t t ) j= k for p =1. In the Poissonian case, tj is totally uncorrelated from tk(j = k),so 2 δ(t − tj)δ(t − tk)S = δ(t − tj)S δ(t − tk)S = R , (C.4) j= k j k which proves (C.1) for p =0. Notice that in the above result, we are missing an atom in the second summation, so the above result is approximate, the approximation being very good when R>>1. (The error is of the order of R compared to R2.) A more general proof is found in the reference [2]. 372 C. Proof References 1. An excellent discussion on this point, as well and on noise supression in quantum optical systems is found in: Davidovich, L.: Rev. Mod. Phys., 68, 127 (1996) 2. Benkert, C., Scully, M.O., Bergou, J., Davidovich, L., Hillery, M., Orszag, M.: Phys. Rev. A, 41, 2756 (1990) D. Stochastic Processes in a Nutshell D.1 Introduction Classical Mechanics gives a deterministic view of the dynamical variables of a system. This of course is true, when one is not in a chaotic regime. On the other hand, in many cases, the system under study is only de- scribed by the time evolution of probability distributions. To show these ideas with an example, we take a look at the random walk in one dimension, by now, a classical problem [3]. A person moves in a line, taking random steps forward or backward, with equal probability, at fixed time intervals τ. Calling the position xn = na, then the probability that it occupies the site xn at time t is P (xn | t) and obeys the equation 1 1 P (x | t + τ)= P (x − | t)+ P (x | t) .
Load more

Recommended publications
  • Squeezed and Entangled States of a Single Spin
    SQUEEZED AND ENTANGLED STATES OF A SINGLE SPIN Barı¸s Oztop,¨ Alexander A. Klyachko and Alexander S. Shumovsky Faculty of Science, Bilkent University Bilkent, Ankara, 06800 Turkey e-mail: [email protected] (Received 23 December 2007; accepted 1 March 2007) Abstract We show correspondence between the notions of spin squeez- ing and spin entanglement. We propose a new measure of spin squeezing. We consider a number of physical examples. Concepts of Physics, Vol. IV, No. 3 (2007) 441 DOI: 10.2478/v10005-007-0020-0 Barı¸s Oztop,¨ Alexander A. Klyachko and Alexander S. Shumovsky It is well known that the concept of squeezed states [1] was orig- inated from the famous work by N.N. Bogoliubov [2] on the super- fluidity of liquid He4 and canonical transformations. Initially, it was developed for the Bose-fields. Later on, it has been extended on spin systems as well. The two main objectives of the present paper are on the one hand to show that the single spin s 1 can be prepared in a squeezed state and on the other hand to demonstrate≥ one-to-one correspondence between the notions of spin squeezing and entanglement. The results are illustrated by physical examples. Spin-coherent states | Historically, the notion of spin-coherent states had been introduced [3] before the notion of spin-squeezed states. In a sense, it just reflected the idea of Glauber [4] about creation of Bose-field coherent states from the vacuum by means of the displacement operator + α field = D(α) vac ;D(α) = exp(αa α∗a); (1) j i j i − where α C is an arbitrary complex parameter and a+; a are the Boson creation2 and annihilation operators.
    [Show full text]
  • Wigner Function for SU(1,1)
    Wigner function for SU(1,1) U. Seyfarth1, A. B. Klimov2, H. de Guise3, G. Leuchs1,4, and L. L. Sánchez-Soto1,5 1Max-Planck-Institut für die Physik des Lichts, Staudtstraße 2, 91058 Erlangen, Germany 2Departamento de Física, Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico 3Department of Physics, Lakehead University, Thunder Bay, Ontario P7B 5E1, Canada 4Institute for Applied Physics,Russian Academy of Sciences, 630950 Nizhny Novgorod, Russia 5Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain In spite of their potential usefulness, Wigner functions for systems with SU(1,1) symmetry have not been explored thus far. We address this problem from a physically-motivated perspective, with an eye towards applications in modern metrology. Starting from two independent modes, and after getting rid of the irrelevant degrees of freedom, we derive in a consistent way a Wigner distribution for SU(1,1). This distribution appears as the expectation value of the displaced parity operator, which suggests a direct way to experimentally sample it. We show how this formalism works in some relevant examples. Dedication: While this manuscript was under review, we learnt with great sadness of the untimely passing of our colleague and friend Jonathan Dowling. Through his outstanding scientific work, his kind attitude, and his inimitable humor, he leaves behind a rich legacy for all of us. Our work on SU(1,1) came as a result of long conversations during his frequent visits to Erlangen. We dedicate this paper to his memory. 1 Introduction Phase-space methods represent a self-standing alternative to the conventional Hilbert- space formalism of quantum theory.
    [Show full text]
  • Arxiv:2010.08081V3 [Quant-Ph] 9 May 2021 Dipole fields [4]
    Time-dependent quantum harmonic oscillator: a continuous route from adiabatic to sudden changes D. Mart´ınez-Tibaduiza∗ Instituto de F´ısica, Universidade Federal Fluminense, Avenida Litor^anea, 24210-346 Niteroi, RJ, Brazil L. Pires Institut de Science et d'Ing´enierieSupramol´eculaires, CNRS, Universit´ede Strasbourg, UMR 7006, F-67000 Strasbourg, France C. Farina Instituto de F´ısica, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, RJ, Brazil In this work, we provide an answer to the question: how sudden or adiabatic is a change in the frequency of a quantum harmonic oscillator (HO)? To do this, we investigate the behavior of a HO, initially in its fundamental state, by making a frequency transition that we can control how fast it occurs. The resulting state of the system is shown to be a vacuum squeezed state in two bases related by Bogoliubov transformations. We characterize the time evolution of the squeezing parameter in both bases and discuss its relation with adiabaticity by changing the transition rate from sudden to adiabatic. Finally, we obtain an analytical approximate expression that relates squeezing to the transition rate as well as the initial and final frequencies. Our results shed some light on subtleties and common inaccuracies in the literature related to the interpretation of the adiabatic theorem for this system. I. INTRODUCTION models to the dynamical Casimir effect [32{39] and spin states [40{42], relevant in optical clocks [43]. The main The harmonic oscillator (HO) is undoubtedly one of property of these states is to reduce the value of one of the most important systems in physics since it can be the quadrature variances (the variance of the orthogonal used to model a great variety of physical situations both quadrature is increased accordingly) in relation to coher- in classical and quantum contexts.
    [Show full text]
  • `Nonclassical' States in Quantum Optics: a `Squeezed' Review of the First 75 Years
    Home Search Collections Journals About Contact us My IOPscience `Nonclassical' states in quantum optics: a `squeezed' review of the first 75 years This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2002 J. Opt. B: Quantum Semiclass. Opt. 4 R1 (http://iopscience.iop.org/1464-4266/4/1/201) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.206.92.227 The article was downloaded on 27/08/2013 at 15:04 Please note that terms and conditions apply. INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1–R33 PII: S1464-4266(02)31042-5 REVIEW ARTICLE ‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years V V Dodonov1 Departamento de F´ısica, Universidade Federal de Sao˜ Carlos, Via Washington Luiz km 235, 13565-905 Sao˜ Carlos, SP, Brazil E-mail: [email protected] Received 21 November 2001 Published 8 January 2002 Online at stacks.iop.org/JOptB/4/R1 Abstract Seventy five years ago, three remarkable papers by Schrodinger,¨ Kennard and Darwin were published. They were devoted to the evolution of Gaussian wave packets for an oscillator, a free particle and a particle moving in uniform constant electric and magnetic fields. From the contemporary point of view, these packets can be considered as prototypes of the coherent and squeezed states, which are, in a sense, the cornerstones of modern quantum optics. Moreover, these states are frequently used in many other areas, from solid state physics to cosmology.
    [Show full text]
  • Redalyc.Harmonic Oscillator Position Eigenstates Via Application of An
    Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México Soto-Eguibar, Francisco; Moya-Cessa, Héctor Manuel Harmonic oscillator position eigenstates via application of an operator on the vacuum Revista Mexicana de Física, vol. 59, núm. 2, julio-diciembre, 2013, pp. 122-127 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57048159005 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative EDUCATION Revista Mexicana de F´ısica E 59 (2013) 122–127 JULY–DECEMBER 2013 Harmonic oscillator position eigenstates via application of an operator on the vacuum Francisco Soto-Eguibar and Hector´ Manuel Moya-Cessa Instituto Nacional de Astrof´ısica, Optica´ y Electronica,´ Luis Enrique Erro 1, Santa Mar´ıa Tonantzintla, San Andres´ Cholula, Puebla, 72840 Mexico.´ Received 21 August 2013; accepted 17 October 2013 Harmonic oscillator squeezed states are states of minimum uncertainty, but unlike coherent states, in which the uncertainty in position and momentum are equal, squeezed states have the uncertainty reduced, either in position or in momentum, while still minimizing the uncertainty principle. It seems that this property of squeezed states would allow to obtain the position eigenstates as a limiting case, by doing null the uncertainty in position and infinite in momentum. However, there are two equivalent ways to define squeezed states, that lead to different expressions for the limiting states.
    [Show full text]
  • Fp3 - 4:30 Quantum Mechanical System Symmetry*
    Proceedings of 23rd Conference on Decision and Control Las Vegas, NV, December 1984 FP3 - 4:30 QUANTUM MECHANICAL SYSTEM SYMMETRY* + . ++ +++ T. J. Tarn, M. Razewinkel and C. K. Ong + Department of Systems Science and Mathematics, Box 1040, Washington University, St. Louis, Missouri 63130, USA. ++ Stichting Mathematisch Centrum, Kruislaan 413, 1098 S J Amsterdam, The Netherlands. +++M/A-COM Development Corporation, M/A-COM Research Center, 1350 Piccard Drive, Suite 310, Rockville. Maryland 20850, USA. Abstract Cll/i(X, t) ifi at :: H(x,t) , ( 1) The connection of quantum nondemolition observables with the symmetry operators of the Schrodinger where t/J is the wave function of the system and x an equation, is shown. The connection facilitates the appropriate set of dynamical coordinates. construction of quantum nondemolition observables and thus of quantum nondemolition filters for a given Definition 1. The symmetry algebra of the Hamiltonian system. An interpretation of this connection is H of a quantum system is generated by those operators given, and it has been found that the Hamiltonian that commute with H and possess together with H a description under which minimal wave pockets remain common dense invariant domain V. minimal is a special case of our investigation. With the above definition, if [H,X1J:: 0 and [H,X2J = 1. Introduction O, then [H,[X1 ,x 2 JJ:: 0 also on D. Clearly all constants of the motion belong to the symmetry algebra In developing the theory of quantum nondemolition of the Hamiltonian. observables, it has been assumed that the output observable is given. The question of whether or not the given output observable is a quantum nondemolition Denote the Schrodinger operator by filter has been answered in [1,2].
    [Show full text]
  • Magnon-Squeezing As a Niche of Quantum Magnonics
    Magnon-squeezing as a niche of quantum magnonics Akashdeep Kamra,1, 2, a) Wolfgang Belzig,3 and Arne Brataas1 1)Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 2)Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, USA 3)Department of Physics, University of Konstanz, D-78457 Konstanz, Germany The spin excitations of ordered magnets - magnons - mediate transport in magnetic insulators. Their bosonic nature makes them qualitatively distinct from electrons. These features include quantum properties tradition- ally realized with photons. In this perspective, we present an intuitive discussion of one such phenomenon. Equilibrium magnon-squeezing manifests unique advantageous with magnons as compared to photons, in- cluding properties such as entanglement. Building upon the recent progress in the fields of spintronics and quantum optics, we outline challenges and opportunities in this emerging field of quantum magnonics. The spin excitations of ordered magnets, broadly called We find it convenient to introduce the equilibrium \magnons", carry spin information1{8 and offer a viable magnon-squeezing physics first and later place it in the path towards low-dissipation, unconventional comput- context of the more mature and widely known nonequi- ing paradigms. Their bosonic nature enables realizing librium squeezing phenomenon28,29,31,32. Magnons in a and exploiting phenomena not admitted by electrons9{15. ferromagnet admit single-mode squeezing mediated by The field of \magnonics" has made rapid progress to- the relatively weak spin-nonconserving interactions16,17, wards fundamental physics as well as potential applica- thereby providing an apt start of the discussion.
    [Show full text]
  • Operators and States
    Operators and States University Press Scholarship Online Oxford Scholarship Online Methods in Theoretical Quantum Optics Stephen Barnett and Paul Radmore Print publication date: 2002 Print ISBN-13: 9780198563617 Published to Oxford Scholarship Online: January 2010 DOI: 10.1093/acprof:oso/9780198563617.001.0001 Operators and States STEPHEN M. BARNETT PAUL M. RADMORE DOI:10.1093/acprof:oso/9780198563617.003.0003 Abstract and Keywords This chapter is concerned with providing the necessary rules governing the manipulation of the operators and the properties of the states to enable us top model and describe some physical systems. Atom and field operators for spin, angular momentum, the harmonic oscillator or single mode field and for continuous fields are introduced. Techniques are derived for treating functions of operators and ordering theorems for manipulating these. Important states of the electromagnetic field and their properties are presented. These include the number states, thermal states, coherent states and squeezed states. The coherent and squeezed states are generated by the actions of the Glauber displacement operator and the squeezing operator. Angular momentum coherent states are described and applied to the coherent evolution of a two-state atom and to the action of a beam- splitter. Page 1 of 63 PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter
    [Show full text]
  • Squeeze Operators in Classical Scenarios
    Squeeze operators in classical scenarios Jorge A. Anaya-Contreras1, Arturo Z´u˜niga-Segundo1, Francisco Soto-Eguibar2, V´ıctor Arriz´on2, H´ector M. Moya-Cessa2 1Departamento de F´ısica, Escuela Superior de F´ısica y Matem´aticas, IPN Edificio 9 Unidad Profesional ‘Adolfo L´opez Mateos’, 07738 M´exico D.F., Mexico 2Instituto Nacional de Astrof´ısica Optica´ y Electr´onica Calle Luis Enrique Erro No. 1, Sta. Ma. Tonantzintla, Pue. CP 72840, Mexico Abstract We analyse the paraxial field propagation in the realm of classical optics, showing that it can be written as the action of the fractional Fourier transform, followed by the squeeze operator applied to the initial field. Secondly, we show that a wavelet transform may be viewed as the application of a displacement and squeeze operator onto the mother wavelet function. PACS numbers: arXiv:1901.05491v1 [physics.optics] 16 Jan 2019 1 I. INTRODUCTION In the late seventies, squeezed states were introduced [1, 2]. On the one hand, Yuen [3] defined them squeezing the vacuum and then displacing the resulting state. On the other hand, Caves [4] defined them by displacing the vacuum and then squeezing the produced coherent state. Squeezed states have been shown to produce ringing revivals (a fingerprint that a squeezed state is used) in the interaction between light and matter [5]. Applications of quantum techniques in classical optics have been the subject of many studies during the last years [6, 7]. Along the same line, one of the goals of this article is to show, that in a mathematical sense, the squeeze operator could have been introduced in the description of free light propagation, i.e.
    [Show full text]
  • Number-Coherent States 55
    Open Research Online The Open University’s repository of research publications and other research outputs Quantum optical states and Bose-Einstein condensation : a dynamical group approach Thesis How to cite: Feng, Yinqi (2001). Quantum optical states and Bose-Einstein condensation : a dynamical group approach. PhD thesis The Open University. For guidance on citations see FAQs. c 2001 The Author https://creativecommons.org/licenses/by-nc-nd/4.0/ Version: Version of Record Link(s) to article on publisher’s website: http://dx.doi.org/doi:10.21954/ou.ro.0000d4a8 Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyright owners. For more information on Open Research Online’s data policy on reuse of materials please consult the policies page. oro.open.ac.uk Quantum Optical States and Bose-Einstein Condensation: A Dynamical Group Approach Yinqi Feng A thesis submitted for the degree of Doctor of Philosophy in the Faculty of Mathematics and Computing of The Open University May,2001 Contents Abstract vii Acknowledgements ix Introduction 1 I Quantum Optical States and Dynamical Groups 5 1 Displaced and Squeezed Number States 6 1.1 Conventional Coherent and Squeezed States ...... 6 1.1.1 Coherent States . ...... 8 1.1.2 Squeezed States . ........ 9 1.1.3 Group-theoretical Description . ..... 12 1.2 Photon Number States ....... ..... 19 1.2.1 Displaced Number States .. ..... 19 1.2.2 Squeezed Number States . 22 1.2.3 Displaced Squeezed Phase Number States (DSPN states) 24 1.3 Optimal Signal-to-Quantum Noise Ratio ............. 26 i 2 Kerr States and Squeezed Kerr States(q-boson Analogue) 32 2.1 Kerr States .
    [Show full text]
  • Cosmological Complexity in K-Essence
    Cosmological Complexity in K-essence a,b c d Ai-chen Li ,∗ Xin-Fei Li ,† Ding-fang Zeng ,‡ and Lei-Hua Liu a Institut de Ci`encies del Cosmos, Universitat de Barcelona, Mart´ıi Franqu`es 1, 08028 Barcelona, Spain b Departament de F´ısica Qu`antica i Astrof´ısica, Facultat de F´ısica, Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Spain c School of Science, Guangxi University of Science and Technology, 545026 Liuzhou, China d Theoretical Physics Division, College of Applied Sciences, Beijing University of Technology, China and e Department of Physics, College of Physics, Mechanical and Electrical Engineering, Jishou University, 416000 Jishou, China We calculate the cosmological complexity under the framework of scalar curvature perturbations for a K-essence model with constant potential. In particular, the squeezed quantum states are de- fined by acting a two-mode squeezed operator which is characterized by squeezing parameters rk and φk on vacuum state. The evolution of these squeezing parameters are governed by the Schrodinger¨ equation, in which the Hamiltonian operator is derived from the cosmological perturbative action. With aid of the solutions of rk and φk, one can calculate the quantum circuit complexity between unsqueezed vacuum state and squeezed quantum states via the wave-function approach. One ad- vantage of K-essence is that it allows us to explore the effects of varied sound speeds on evolution of cosmological complexity. Besides, this model also provides a way for us to distinguish the different cosmological phases by extracting some basic informations, like the scrambling time and Lyapunov exponent etc, from the evolution of cosmological complexity.
    [Show full text]
  • Arxiv:Quant-Ph/0002070V1 24 Feb 2000
    Generalised Coherent States and the Diagonal Representation for Operators N. Mukunda∗ † Dipartimento di Scienze Fisiche, Universita di Napoli “Federico II” Mostra d’Oltremare, Pad. 19–80125, Napoli, Italy and Dipartimento di Fisica dell Universita di Bologna Viale C.Berti Pichat, 8 I–40127, Bologna, Italy Arvind‡ Department of Physics, Guru Nanak Dev University, Amritsar 143005, India S. Chaturvedi§ School of Physics, University of Hyderabad, Hyderabad 500 046, India R.Simon∗∗ The Institute of Mathematical Sciences, C. I. T. Campus, Chennai 600 113, India (October 29, 2018) We consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with an unitary irreducible representation of a (compact) Lie group. We show that necessary and sufficient conditions for the possibility of such a representation can be obtained by combining Clebsch-Gordan theory and the reciprocity theorems associated with induced unitary group representation. Applications to several examples involving SU(2), SU(3), and the Heisenberg-Weyl group are presented, showing that there are simple examples of generalized coherent states which do not meet these conditions. Our results are relevant for phase-space description of quantum mechanics and quantum state reconstruction problems. I. INTRODUCTION There is a long history of attempts to express the basic structure of quantum mechanics, both kinematics and dynamics, in the c-number phase space language of classical mechanics. The first major step in this direction was taken by Wigner [1] very early in the development of quantum mechanics, during a study of quantum corrections to classical statistical mechanics.
    [Show full text]