Coherent State

7 Key excerpts on "Coherent State"

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  Coherent States: Past, Present And Future - Proceedings Of The International Symposium

  Past, Present, and Future

  • John R Klauder, Michael Robert Strayer, Da-hsuan Feng(Authors)
  • 1994(Publication Date)
  • World Scientific

    (Publisher)

  We will not try to make a general enough definition of the concept of Coherent States (for such an attempt see e.g. the introduction of Ref.[5]). We rather intend to illustrate in some rather elementary and explicit examples the use of the concept of Coherent States and, when appropriate, stress the physics involved. One basic issue, which one may raise in any elementary presentation of quantum mechanics, is the transition to the domain of classical physics, a topic which has been 'Email address: [email protected]. Research supported by the Swedish National Research Council under contract no. 8244-311 and the Knut and Alice Wallenberg Foundation. Coherent StateS: PAST, PRESENT, AND PUTURE (pp. 469-506) © 1994 World Scientific Publishing Company 470 discussed at great detail during this wonderful symposium. Quantum mechanics we believe is, after all, the fundamental framework for the description of all known natural physical phenomena. Still we are, however, very often puzzled about the role of con-cepts from the domain of classical physics within this framework. The interpretation of the theoretical framework of quantum mechanics is, of course, directly connected to the classical picture of physical phenomena. We often talk about quantization of the classical observables in particular so with regard to classical dynamical systems in the Hamiltonian formulation as has so beautifully been discussed by Dirac (see e.g. Ref.[7]) and others. I believe that the concept of Coherent States is very useful in trying to orient the inquiring mind in this jungle of conceptually difficult issues when connecting classical pictures of physical phenomena with the fundamental notion of quantum-mechanical probability-amplitudes and probabilities. As is well-known, Coherent States appears in a very natural way when consider-ing the classical limit of quantum electrodynamics (QED).

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  The Dirac Equation and its Solutions

  • Vladislav G. Bagrov, Dmitry Gitman(Authors)
  • 2014(Publication Date)
  • De Gruyter

    (Publisher)

  10 Coherent states

  10.1 Introduction

  Coherent States (CS ) play an important role in modern quantum theory due to their fundamental theoretical importance and wide range of applications, for example in semiclassical descriptions of quantum systems, in quantization theory, in condensed matter physics, in radiation theory, in quantum computations and so on, see, for example Refs. [164, 216, 252]. The first example of CS was given by Schrödinger for the harmonic oscillator [289]. Especially useful CS are in the theory of the electromagnetic field. Such states were introduced and systematically studied in the works by Schwinger [293], Rashevsky [272], and Glauber [184]. Perelomov and Gilmore [169, 263] pointed out the group-theoretical aspect of CS. This enabled them to generalize the construction of CS. Using Perelomov’s method, CS for SU(N) and SU(N, 1) groups were constructed in Refs. [178, 179]. Their application to semiclassical descriptions of the quantum rotator was considered in [180]. As they are usually labeled by phase space variables, CS allow one to construct classical symbols of operators, thus they obtain a special status of a quantizer à la Berezin–Klauder [87, 89, 90, 164, 217, 218]. Developing the Glauber and Malkin–Man’ko initial approach [110, 239, 240, 242], Dodonov and Man’ko had constructed CS for arbitrary nonrelativistic systems with quadratic Hamiltonians [125]. Some nontrivial generalizations of the Glauber approach are developed by Klauder and Gazeau (see [165]). For constructing CS of relativistic particles described by the K–G or Dirac equation, new methods were required. For specific cases, the problem was solved in Refs. [34, 35, 63, 122, 123], where two methods were proposed. Namely, in Refs. [34, 35, 63] the light-cone formulation was used for this purpose, whereas Ref. [122, 123] dealt with the proper-time method.

  It should be noted that a universally accepted definition of CS for an arbitrary physical system is still lacking. Nevertheless, constructing CS one always tries to maintain basic properties of already known CS for simple system. In particular, CS have to form a complete system, they have to minimize uncertainty relations for some physical quantities (e.g. coordinates and momenta) at a fixed time instant and mean values of some physical quantities, calculated with respect to time-dependent CS and have to move along the corresponding classical trajectories. It is also desirable for time-dependent CS to maintain their form under the time evolution, such that this evolution affects only their parameters. Besides, CS have to be labeled by parameters that have a direct classical analog, say, by points in a phase space.

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  Coherent States: Applications In Physics And Mathematical Physics

  Applications in Physics and Mathematical Physics

  • John R Klauder, Bo-sture Skagerstam(Authors)
  • 1985(Publication Date)
  • World Scientific

    (Publisher)

  3 Coherent States combined with the variation-al method have been used to treat an isovector meson field interacting with a static source. 4 It has also been shown that Coherent States can be used to describe equilibrium states of boson fields. 5 Apart from their applications, Coherent States themselves have been intensively studied since the pioneering work of Bargmann and Segal. 6 In 1971, Bargmann, Butera, Ghiraradeilo, Klauder, andPerelomov proved that a certain sub set of coher-ent states form a complete, butnotovercomplete, set. 7 This will be important for our considera-tions. In the present paper we shall study the semi-classical treatment of a charged Schrbdinger parti-cle interacting with a c-number radiation field. We shall derive the corresponding Schrodinger equation from the fully quantized theory, in which the radiation field is also quantized, by using co-herent states to describe the radiation field. To avoid trouble due to the above-mentioned over-completeness of Coherent States we shall use the complete subset of Coherent States of Bargmann et al. and Perelomov, which we shall call VNLCS (see Sec. III). In order to introduce such a sub set of 22 Coherent States (VNLCS) in a consistent manner into the quantized theory, we shall use the time-dependent Schrodinger equation written in projec-tion form, 8 which is a generalization of the Hill-Wheeler method. 9 This method has been used to obtain a microscopic nuclear theory for low-ener-gy phenomena from a unified point of view. 8 As a charged Schrodinger particle we shall con-sider a nonrelativistic atomic electron interacting with a strong radiation field. In Sec. II we shall formulate the problem quantitatively by defining the semiclassical treatment of the system. In Sec. Ill we shall introduce Coherent States, es-pecially VNLCS, as the basis states for the radia-tion field. In Sec. IV we shall rewrite the Schro-dinger equation for the system in the projection form.

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  Introductory Quantum Optics

  • Christopher Gerry, Peter Knight(Authors)
  • 2004(Publication Date)
  • Cambridge University Press

    (Publisher)

  In fact, all states of light must have some quantum features as the quantum theory of light is more fundamental than the classical theory. However, the quantum features of light are generally difficult to observe. We shall deal with some of these features in Chapter 7 . 48 Coherent States Fig. 3.2. Phase distributions for Coherent States with θ = 0 for (a) ¯ n = 2 and (b) ¯ n = 10. E ( t ) t Fig. 3.3. Coherent State expectation value of the electric field as a function of time for a fixed position showing the quantum fluctuations. The fluctuations of the field are the same at all times such that the field is as close to a classical field as is possible for any quantum state. 3.2 Displaced vacuum states We have discussed two ways in which the Coherent States may be defined: as right eigenstates of the annihilation operator and as states that minimize the uncer-tainty relation for the two orthogonal field quadratures with equal uncertainties (identical to those of a vacuum state) in each quadrature. There is, in fact, a third 3.2 Displaced vacuum states 49 definition that leads to equivalent states. This involves the displacement of the vacuum. As we will show, this is closely related to a mechanism for generating the Coherent States from classical currents. The displacement operator ˆ D ( α ) is defined as [ 2 ] ˆ D ( α ) = exp( α ˆ a † − α ∗ ˆ a ) (3.30) and the Coherent States are given as | α = ˆ D ( α ) | 0 . (3.31) To see this, consider the identity (the disentangling theorem) e ˆ A + ˆ B = e ˆ A e ˆ B e − 1 2 [ ˆ A , ˆ B ] = e ˆ B e ˆ A e 1 2 [ ˆ A , ˆ B ] (3.32) valid if [ ˆ A , ˆ B ] = 0 but where also [ ˆ A , [ ˆ A , ˆ B ]] = [ ˆ B , [ ˆ A , ˆ B ]] = 0 . (3.33) With ˆ A = α ˆ a † and ˆ B = − α ∗ ˆ a , [ ˆ A , ˆ B ] = | α | 2 and Eq. ( 3.33 ) holds. Thus ˆ D ( α ) = e α ˆ a † − α ∗ ˆ a = e − 1 2 | α | 2 e α ˆ a † e − α ∗ ˆ a .

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  Coherent States in Quantum Physics

  • Jean-Pierre Gazeau(Author)
  • 2009(Publication Date)
  • Wiley-VCH

    (Publisher)

  In both approaches, however, the probability distribution shows a Gaussian behavior for large values of J, and this explains the similarity of the results. It would appear that allowing for generalized phase and amplitude behavior in the definition of Coherent States has led us closer to the idealized goal of a set of Coherent States adapted to a chosen system and having a large number of properties in common with the associated classical system, despite being fully quantum in their characteristics. 165 10 Squeezed States and Their SU(1, 1) Content 10.1 Introduction This chapter is devoted to another occurrence of SU (1, 1) in the construction of various popular quantum states. The so-called squeezed states, which are to be de- scribed here, pertain again to quantum optics and have been raising interest in quantum optics and other fields for the last three decades. Squeezed states, a name given by Hollenhorst [139], were initially introduced in quantum optics [140] for dealing with processes in which emission or absorption of two photons is involved. The formalism underlying the process involves the square of raising and lowering operators, a 2 and a † 2 , for each mode of the quantized electromagnetic field. In this regard, squeezed states might be viewed as a sort of “two-photon” Coherent States. Their mathematical properties had actually been investigated before [141–143]. The experimental evidence for such states was provided by Slusher et al. [144]. Inter- esting applications to quantum nondemolition in view of detecting gravitational waves were envisaged as early as 1981 [145, 146]. In addition, it is interesting to learn from Nieto [147] that these states were already known in 1927, 1 year after the Schrödinger states. For more details on theoretical and experimental aspects of squeezed states, see [25] and the recent report by Dell’Annoa et al.

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  An Introduction to Condensed Matter Physics for the Nanosciences

  • Arthur McGurn(Author)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 6 on lasers and in the discussion of problems related to the interactions of light with matter.

  5.1 Quantized Electromagnetic Waves

  The quantization of electromagnetic waves is intimately related to the quantization of the mechanical harmonic oscillator [3 ,7 ,8 and 9 ]. This was discussed by both Planck and Einstein early in the development of quantum mechanics. Planck originally noted the nature of quantum phenomena in his quantization scheme, which postulated that the energy changes within the electromagnetic modes of a system occur in integer multiples of

  ℏ ω

  . Einstein latter extended this idea to treat the energy contained in the vibrational modes of materials. Both of these results are ultimately statements of the linearity in the nature of the excitations of electromagnetic and acoustic systems, and similarly are applied in a variety of magnetic, ferroelectric, and general media supporting Bose–Einstein excitations. Consequently, as a starting point a review is given of the properties of a one-dimensional mechanical harmonic oscillator. This is then extended to the problem of the quantization of the electromagnetic fields.

  In the simplest formulation of classical mechanics, the one-dimensional harmonic oscillator is described by a linear differential equation given by [3 ,8 ,9 ]

  x ¨

  +

  ω 2

  x = 0

  (5.1)

  where ω is the angular frequency of the oscillator. The general harmonic solution is then written as a linear combination of a set of exponential harmonic solutions, i.e.,

  x

  ( t )

  =

  x 0

  e

  − i ω t

  +

  x 0 *

  e

  i ω t

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  Basic Optics

  Principles and Concepts

  • Avijit Lahiri(Author)
  • 2016(Publication Date)
  • Elsevier

    (Publisher)

  Chapter 8

  Quantum Optics

  Abstract

  Quantum optics deals with processes and phenomena where the quantum states of the electromagnetic field are of central relevance. This chapter begins with a brief summary of the way states and observables are described in the classical theory, explaining the concepts of pure and mixed states and those relating to states of composite systems. The equations describing the time evolutions of pure and mixed states in the phase spaces of systems are stated. The corresponding features of a quantum system are outlined, indicating the contrast between classical and quantum systems. The concept of entanglement as a nonclassical correlation in a composite quantum system is briefly explained.

  The quantum mechanics of the harmonic oscillator, which is of fundamental relevance in quantum optics, is outlined, highlighting the concepts of number states, Coherent States, and squeezed states, comparing the number distributions (mean and variance of the number operator) in these three types of states. The concept of classical and nonclassical states of the quantum harmonic oscillator is explained with reference to the P-representation.

  The mode decomposition of the electromagnetic field in free space (under periodic boundary conditions) and in a cavity is explained, first in the classical context and then in the quantum mechanical description, where the concept of photons is of central relevance. The modes appear as independent harmonic oscillators, each with its own creation and annihilation operators and with its own photon number distribution. Single-mode number states, Coherent States, and squeezed states of the electromagnetic field are described, along with the statistics of field fluctuations in these states. Multimode states are briefly described, and the wave packet number states (including photon-pair states), Coherent States, chaotic states, and squeezed states in the continuous-mode description are also explained. The distinction between classical and nonclassical states of the electromagnetic field is explained, and the optical equivalence theorem is outlined. The quantum mechanical description of field transformations by beam splitters and Mach-Zehnder interferometers is explained in brief.

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