DOCSLIB.ORG

Squeezed and Entangled States of a Single Spin

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. In the case of spin-s system specified by the three spin operators Sx;Sy;Sz that obey the commutation relations [Sj;Sk] = ijk`S`; (2) the lowest spin state s (Sz s = s s ) plays the role of the vacuum state, so thatj − thei spin-coherentj − i − statej − i has the form α = exp(αS+ α∗S ) s (3) j i − − j − i where 1 S = (Sx iSy) (4) ± p2 ± are the spin-ladder operators (compare with Eq. (1)). For extension of the notion of coherence on arbitrary Lie groups we refer to [5]. Since the Hermitian spin operators Sx;Sy;Sz form an orthogonal basis of the su(2) algebra, the spin-displacement operator (exponen- tial in Eq. (3) belongs to the SU(2) group, so that the spin-coherent 442 Concepts of Physics, Vol. IV, No. 3 (2007) Squeezed and entangled states of a single spin state (3) can be considered as the SU(2) rotation of the minimal state s . In other words, there is a direction ~eα in the three-dimensional j− i spin space (new quantization axis) such that the operator Sz;~eα (z- component of the spin vector, which is now directed along the ~eα axis, has the same eigenvalue s in the state α . As well as conventional coherent− states, thej i spin-coherent states (3) can be specified by uncertainties of spin operators measured in those states. In view of the commutation relation (2), the Heisenberg uncertainty relation in an arbitrary state has the form 1 2 V (Sj; )V (Sk; ) S` :; (5) ≥ 4jh j j ij where V (X; ) X2 X 2 ≡ h j j i − h j j i denotes the variance (uncertainty). Thus, in the coherent state (3), the uncertainty of the corresponding z-component V (Sz;~eα ; α) = 0 and α Sz;~e α = s. In turn, the variances of the two spin operators jh j α j ij orthogonal to Sz;~eα are both equal to s=2. An important characteristics of states is provided by the total variance (see Ref. [6] and references therein) X V( ) = V (Sj; ): (6) j=x;y;z By definition, this quantity is invariant under rotations of the basis of the su(2) algebra. Thus, for an arbitrary spin-coherent state V(α) = s. It can be easily seen that s V( ) s(s + 1); (7) ≤ ≤ so that the spin-coherent state corresponds to the minimal total vari- ance. It is also easily seen that any state of spin-1=2 system is a spin-coherent one. Spin-squeezed states | In the case of Bose-fields, the simplest squeezed state is defined by action of the squeezing operator (ζ) on the vacuum state [1] S 1 2 1 +2 ζ field = exp ζ∗a ζa vac (8) j i 2 − 2 j i Concepts of Physics, Vol. IV, No. 3 (2007) 443 Barı¸s Oztop,¨ Alexander A. Klyachko and Alexander S. Shumovsky (for recent review, see [7]). For the two field quadratures Q = (a+ +a) and P = i(a a+) that obey the uncertainty relation − − V (Q; )V (P; ) 1; ≥ in the state (8) we have 2 ζ 2 ζ V (Q; ζ) = e− j j;V (P; ζ) = e j j; so that uncertainty of one of the quadratures can be squeezed with si- multaneous enlargement of uncertainty of the other quadrature under the condition that V (Q; ζ)V (P; ζ) = 1. The attempts to define spin-squeezed states in direct analogy with the Bose-field squeezed states have failed (see critical review in Ref. [8]). An approach has been proposed in Ref. [8] considers the mean- spin vector S¯ with components Sj (j = x; y; z). Its direction again is chosen to be the z-axis. Ah spinj j statei is called to be squeezed whenever the uncertainty in one of the spin components orthogonal to this direction is smaller than s=2. For the spin-squeezed state, the inequality (7) takes the form Vcoh = s < V(ζ) s(s + 1) = Vmax: (9) ≤ Measure of spin squeezing | With the inequality (9) in hands, we can define a measure of squeezing of an arbitrary state of spin 1. It is possible to carry the interval of this inequality into [0; 1] such as V( ) Vcoh 0 < − 1: Vmax Vcoh ≤ − Thus, the measure ν( ) of squeezing becomes V( ) Vcoh ν( ) = − ; (10) Vmax Vcoh − which takes the value 0 for spin-coherent states (no squeezing) and 1 for the maximally squeezed state. Taking into account that X S2 = s(s + 1) h j j j i j=x;y;z 444 Concepts of Physics, Vol. IV, No. 3 (2007) Squeezed and entangled states of a single spin for any state , we can now rewrite Eq. (10) as follows S¯2 ν( ) = 1 ; (11) − s2 ¯2 P 2 where S j Sj is the squared length of the mean-spin vector. ≡ h j j i Entanglement in spin systems | According to the modern theory of quantum entanglement (see [6] and references therein), the com- pletely entangled (CE) states of a given system are defined to be the states with the maximal amount of total uncertainty calculated for the set of basic observables. In the case of a single spin, the basic observables are given by the operators Sx;Sy;Sz, so that V( CE) = max V( ) = s(s + 1); where maximum is taken over all states in the (2s + 1)-dimensional Hilbert space of spin states. This definition assumes the following condition CE Sj CE = 0 j = x; y; z ; (12) h j j i 8 that can be used as the operational definition of complete entan- glement [9] (also see discussion in Appendix A). All other entangled states can be obtained from the completely entangled states by means of SLOCC (stochastic local operations assisted by classical commu- nications) [10]. Thus, by definition, maximally squeezed spin states coincide with the completely entangled spin states. Moreover, it has been shown in Ref. [11] that the square root of the measure of squeezing (11) gives the measure of the amount of entanglement carried by a pure spin state: s ¯2 p S µ( ) ν( ) = 1 : (13) − s2 Let us now note that an organic connection between the notions of entanglement and squeezing is under discussion for some time [12, 13]. In the most of works, a kind of \two-mode" entanglement [14] was considered for the two-qubit (two spin-1=2) systems. In the papers Concepts of Physics, Vol. IV, No. 3 (2007) 445 Barı¸s Oztop,¨ Alexander A. Klyachko and Alexander S. Shumovsky [15, 17, 16], the possibility of entanglement of a single spin-1 object has been shown. It should be stressed that our present consideration shows equivalence of the spin squeezing and spin entanglement for a single object with any s = 1. At s > 1, squeezing implies entangle- ment but not vice versa. Squeezed states of spin-1 | As an illustrative physical example of some considerable interest, examine now squeezing of a single spin-1 object. Stress that hereafter by \spin" we mean an object with the dynamic symmetry given by the SU(2) group, so that it can be a real spin, isospin, angular momentum, etc. A general pure state in the three-dimensional Hilbert 3 space has the form H 1 1 X X 2 = s s ; s = 1: (14) j i j i j j s= 1 s= 1 − − In the basis s , spin operators have the following representation j i Sx = ( + 1 + 1 ) 0 + 0 ( +1 + 1 ); j i j − i h j j i h j h− j Sy = i( + 1 1 ) 0 + i 0 ( +1 1 ); (15) − j i − j − i h j j i h j − h− j Sz = + 1 +1 1 1 : j ih j − j − ih− j Since the spin-displacement operator in Eq. (3) has the form 0 iφ 2iφ 1 1 [1 + cos( α p2)] e sin( α p2) e [1 cos( α p2)] 2 p2 2 −iφ j j j j iφ − j j B e sin( α p2) cos( α p2) e sin( α p2) C B p2 p2 C @ −−2iφ j j −iφ j j j j A e [1 cos( α p2)] e sin( α p2) 1 [1 + cos( α p2)] 2 − j j − p2 j j 2 j j the coherent states for spin 1 have the following general form e2iφ eiφ α = [1 cos( α p2)] + 1 + sin( α p2) 0 j i 2 − j j j i p2 j j j i 1 + [1 + cos( α p2)] 1 : (16) 2 j j j − i Here φ = arg α.
Load more

Recommended publications
  • 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]
  • 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]
  • A Point and Local Position Operator Bernice Black Durand Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1971 A point and local position operator Bernice Black Durand Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Elementary Particles and Fields and String Theory Commons Recommended Citation Durand, Bernice Black, "A point and local position operator " (1971). Retrospective Theses and Dissertations. 4876. https://lib.dr.iastate.edu/rtd/4876 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. 71-26,850 DURAND, Bernice Black, 1942- A POINT AND LOCAL POSITION OPERATOR. Iowa State University, Ph.D., 1971 Physics, elementary particles University Microfilms, A XEROX Company, Ann Arbor, Michigan A point and local position operator by Bernice Black Durand A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: High Energy Physics Approved: Signature was redacted for privacy. In Charge of Major Work Signature was redacted for privacy. Head of Major Department Signature was redacted for privacy. Iowa State University Of Science and Technology Ames, Iowa 1971 PLEASE NOTE: Some pages have
    [Show full text]
  • Decoupling Light and Matter: Permanent Dipole Moment Induced Collapse of Rabi Oscillations
    Decoupling light and matter: permanent dipole moment induced collapse of Rabi oscillations Denis G. Baranov,1, 2, ∗ Mihail I. Petrov,3 and Alexander E. Krasnok3, 4 1Department of Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden 2Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny 141700, Russia 3ITMO University, St. Petersburg 197101, Russia 4Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas 78712, USA (Dated: June 29, 2018) Rabi oscillations is a key phenomenon among the variety of quantum optical effects that manifests itself in the periodic oscillations of a two-level system between the ground and excited states when interacting with electromagnetic field. Commonly, the rate of these oscillations scales proportionally with the magnitude of the electric field probed by the two-level system. Here, we investigate the interaction of light with a two-level quantum emitter possessing permanent dipole moments. The semi-classical approach to this problem predicts slowing down and even full suppression of Rabi oscillations due to asymmetry in diagonal components of the dipole moment operator of the two- level system. We consider behavior of the system in the fully quantized picture and establish the analytical condition of Rabi oscillations collapse. These results for the first time emphasize the behavior of two-level systems with permanent dipole moments in the few photon regime, and suggest observation of novel quantum optical effects. I. INTRODUCTION ment modifies multi-photon absorption rates. Emission spectrum features of quantum systems possessing perma- Theory of a two-level system (TLS) interacting with nent dipole moments were studied in Refs.
    [Show full text]
  • Phase-Space Formulation of Quantum Mechanics and Quantum State
    formulation of quantum mechanics and quantum state reconstruction for View metadata,Phase-space citation and similar papers at core.ac.uk brought to you by CORE ysical systems with Lie-group symmetries ph provided by CERN Document Server ? ? C. Brif and A. Mann Department of Physics, Technion { Israel Institute of Technology, Haifa 32000, Israel We present a detailed discussion of a general theory of phase-space distributions, intro duced recently by the authors [J. Phys. A 31, L9 1998]. This theory provides a uni ed phase-space for- mulation of quantum mechanics for physical systems p ossessing Lie-group symmetries. The concept of generalized coherent states and the metho d of harmonic analysis are used to construct explicitly a family of phase-space functions which are p ostulated to satisfy the Stratonovich-Weyl corresp on- dence with a generalized traciality condition. The symb ol calculus for the phase-space functions is given by means of the generalized twisted pro duct. The phase-space formalism is used to study the problem of the reconstruction of quantum states. In particular, we consider the reconstruction metho d based on measurements of displaced pro jectors, which comprises a numb er of recently pro- p osed quantum-optical schemes and is also related to the standard metho ds of signal pro cessing. A general group-theoretic description of this metho d is develop ed using the technique of harmonic expansions on the phase space. 03.65.Bz, 03.65.Fd I. INTRODUCTION P function [?,?] is asso ciated with the normal order- ing and the Husimi Q function [?] with the antinormal y ordering of a and a .
    [Show full text]
  • Correspondence Rules in SU (3)
    Lakehead University Knowledge Commons,http://knowledgecommons.lakeheadu.ca Electronic Theses and Dissertations Electronic Theses and Dissertations from 2009 2018 Correspondence rules in SU (3) Nunes Martins, Alex Clesio http://knowledgecommons.lakeheadu.ca/handle/2453/4322 Downloaded from Lakehead University, KnowledgeCommons Correspondence Rules in SU(3) by Alex Clésio Nunes Martins A thesis presented in conformity with the requirements for the degree of Masters of Physics Department of Physics Lakehead University Canada c Alex Clésio Nunes Martins 2018 Abstract In this thesis, I present a path to the correspondence rules for the generators of the su(3) symmetry and compare my results with the SU(2) correspondence rules. Using these rules, I obtain analytical expressions for the Moyal bracket between the Wigner symbol of a Hamiltonian H^ , where this Hamiltonian is written linearly or quadratically in terms of the generators, and the Wigner symbol of a general operator B^. I show that for the semiclassical limit, where the SU(3) representation label λ tends to infinity, this Moyal bracket reduces to a Poisson bracket, which is the leading term of the expansion (in terms of the semiclassical parameter ), plus correction terms. Finally, I present the analytical form of the second order correction term of the expansion of the Moyal bracket. 1 Acknowledgements Firstly, I would like to thank my wife, Michelle Goertzen Martins. Her continuous support, patience, advice, courage, love and kindness were essential for the completion of this thesis. Specially in those moments when I thought I would not be able to keep moving forward, she was my pillar of strength and reminded me that taking rests, enjoying family, reconnecting to my roots by playing and watching the beautiful game, learning new things, trying a different approach and checking for minus signs in my calculations would help me to see clearly every single step of this arduous but incredible journey.
    [Show full text]
  • Appendix: Special Mathematical Functions
    Appendix: Special Mathematical Functions For completeness this appendix briefly lists without further discussion the definiti­ tions and some important properties of the special functions occurring in the book. More detailed treatments can be found in the relevant literature. The "Handbook of Mathematical Functions" [AS70], the "Tables" by Gradshteyn and Rhyzik [GR65] and the compilation by Magnus, Oberhettinger and Soni [M066] are particularly useful. Apart from these comprehensive works it is worth mention­ ing Appendix B in "Quantum Mechanics I" by Messiah [Mes70], which describes a selection of especially frequently used functions. A.I Legendre Polynomials, Spherical Harmonics The Ith Legendre Polynom P,(x) is a polynomial of degree 1 in x, 1 d' 2 , P,(x) = 2'11 dx'(x - 1), 1 = 0, 1, ... (A.1) It has 1 zeros in the interval between -1 and +1; for even (odd) I, P,(x) is an even (odd) function of x. The associated Legendre junctions P"m(x) , Ixl ~ 1, are products of (1 - x2)m/2 with polynomials of degree 1 - m (m = 0, ... , I) , (A.2) The spherical harmonics Yi,m(fJ, c/» are products of exp (imc/» with polyno­ mials of degree m in sin fJ and of degree 1- m in cos fJ, where the fJ-dependence is given by the associated Legendre functions (A.2) as functions of x = cos fJ. For m ;::: 0, 0 ~ fJ ~ 7r we have Yi,m(fJ, c/» = (_1)m [(2~: 1) ~~: :;!! f/2 P',m(cos fJ)eim.p (A.3) = (_I)m [(2/+1) (/_m)!]1/2 . mfJ dm n( fJ) im.p 47r (1 + m)! sm d(cos fJ)m .q cos e The spherical harmonics for negative azimuthal quantum numbers are obtained via (A.4) 302 Appendix A reflection of the displacement vector x = r sin 0 cos </>, y = r sin 0 sin </>, z = r cos 0 at the origin (cf.
    [Show full text]
  • Quantum Physics of Light-Matter Interactions
    Quantum Physics of Light-Matter Interactions FAU - Summer semester 2019 Claudiu Genes Introduction This is a summary of topics covered during lectures at FAU in the summer semesters 2018 and 2019. A first goal of the class is to introduce the student to the physics of fundamental processes in light-matter interactions such as: • stimulated emission/absorption, spontaneous emission • motional effects of light onto atoms, ions and mechanical resonators (optomechanics) • electron-vibrations-light coupling in molecules The main purpose of the class is to build a toolbox of methods useful for tackling real applications such as cooling, lasing, etc. Some of the general models and methods introduced here and used throughout the notes are: • quantum master equations • the Jaynes (Tavis)-Cummings Hamiltonian • the Holstein Hamiltonian • the radiation pressure Hamiltonian • quantum Langevin equations • the polaron transformation Among others, some of the applications presented either within the main course or as exercises cover aspects of: • laser theory • Doppler cooling, ion trap cooling, cavity optomechanics with mechanical resonators • electromagnetically induced transparency • optical bistability • applications of subradiant and superradiant collective states of quantum emitters Here is a short list of relevant textbooks. • Milburn • Gardiner and Zoller • Zubairy Other references to books and articles are listed in the bibliography section at the end of the course notes. As some numerical methods are always useful here is a list of useful programs and platforms for numerics in quantum optics: • Mathematica • QuantumOptics.jl - A Julia Framework for Open Quantum Dynamics • QuTiP - Quantum Toolbox in Python The exam will consist of a 90 mins written part and 15 mins oral examination.
    [Show 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).
    [Show full text]
  • Generalized Squeezed States from Generalized Coherent States
    LA-UR-93-3731 Proceedings of the International Symposium on Coherent States: Past, Present, and Future (to be published) GENERALIZED SQUEEZED STATES FROM GENERALIZED COHERENT STATES Michael Martin Nieto1 Theoretical Division, Los Alamos National Laboratory University of California Los Alamos, New Mexico 87545, U.S.A. ABSTRACT Both the coherent states and also the squeezed states of the harmonic oscillator have long been understood from the three classical points of view: the 1) displacement operator, 2) annihilation- (or ladder-) opera- tor, and minimum-uncertainty methods. For general systems, there is the same understanding except for ladder-operator and displacement-operator squeezed states. After reviewing the known concepts, I propose a method for obtaining generalized minimum-uncertainty squeezed states, give ex- amples, and relate it to known concepts. I comment on the remaining concept, that of general displacement-operator squeezed states. arXiv:hep-th/9311039v1 6 Nov 1993 1 Introduction As we all know, Glauber, Klauder, and Sudarshan produced the modern era’s seminal works on coherent states of the harmonic oscillator [1, 2, 3, 4]. There came to be three classical definitions of these coherent states, from 1) the displacement-operator method, 2) the annihilation-operator method which I will relabel the ladder-operator method, and 3) the minimum-uncertainty method. Today coherent states are important in many fields of theoretical and experimental physics [5, 6]. Generalizations of these states appeared in two areas. One was to “two-photon states” [7], states which were rediscovered under many names. In 1979 they were 1Email: [email protected] 1 first called “squeezed states” [8].
    [Show full text]
  • Theoretical Quantum Optics and Quantum Information (SS 2017) Exercise II (Discussion: 17.05.17)
    Theoretical Quantum Optics and Quantum Information (SS 2017) Exercise II (discussion: 17.05.17) 1. Coherent states (a) Eigenstates of ^by Show that an eigenstate of ^by cannot exist. Hint: Expand a putative eigenstate of ^by in the basis of Fock states jni 1 X jβi = jni hnjβi : n=0 Then you will see that the creation operator produces a new superposition of occupations jni with different range of occupancy than jβi. (b) Action of ^by Show that @ α∗ ^by jαi = + jαi @α 2 where jαi is a coherent state, i.e. ^b jαi = α jαi. Use the Wirtinger derivative @ 1 @ @ = − i : @(x + iy) 2 @x @y Note that the chain rule of differentiation does not hold in general for matrix functions! (c) Superposition of coherent states Calculate the probability distribution of the photon number and the mean photon number for the state j ±i / (jαi ± |−αi); (1) which is a superposition of two coherent states. Start by normalizing j ±i. Hint: The resulting prefactor can be simplified by the use of hyperbolic func- tions. 2. Two coupled harmonic oscillators Consider a system that consists of two coupled harmonic oscillators with frequen- cies !1 and !2. The first oscillator corresponds to a mode of an electromagnetic field inside a cavity, which couples via radiation pressure to a mechanical oscilla- tor, as shown in the figure. The parameter χ characterizes the coupling strength between them. The dynamics of such a system is governed by the standard Hamiltonian in optomechanics 2 ^ X ^y^ ^y^ ^y ^ H = ~ !ibi bi − ~χb1b1 b2 + b2 : i=1 ^y ^ Here, bi and bi, i 2 f1; 2g denote the bosonic creation and annihilation operator of the corresponding harmonic oscillator.
    [Show full text]
  • Irreducible Tensor Operators and the Wigner-Eckart Theorem †
    Copyright c 2019 by Robert G. Littlejohn Physics 221A Fall 2019 Notes 19 Irreducible Tensor Operators and the Wigner-Eckart Theorem † 1. Introduction The Wigner-Eckart theorem concerns matrix elements of a type that is of frequent occurrence in all areas of quantum physics, especially in perturbation theory and in the theory of the emis- sion and absorption of radiation. This theorem allows one to determine very quickly the selection rules for the matrix element that follow from rotational invariance. In addition, if matrix elements must be calculated, the Wigner-Eckart theorem frequently offers a way of significantly reducing the computational effort. We will make quite a few applications of the Wigner-Eckart theorem in this course. The Wigner-Eckart theorem is based on an analysis of how operators transform under rotations. It turns out that operators of a certain type, the irreducible tensor operators, are associated with angular momentum quantum numbers and have transformation properties similar to those of kets with the same quantum numbers. An exploitation of these properties leads to the Wigner-Eckart theorem. 2. Definition of a Rotated Operator We consider a quantum mechanical system with a ket space upon which rotation operators U(R), forming a representation of the classical rotation group SO(3), are defined. The representation will be double-valued if the angular momentum of the system is a half-integer. In these notes we consider only proper rotations R; improper rotations will be taken up later. The operators U(R) map kets into new or rotated kets, ψ′ = U(R) ψ , (1) | i | i where ψ′ is the rotated ket.
    [Show full text]