Showing papers on "Coherent states published in 1968"

Phase and Angle Variables in Quantum Mechanics

Abstract: The quantum-mechanical description of phase and angle variables is reviewed, with emphasis on the proper mathematical description of these coordinates. The relations among the operators and state vectors under consideration are clarified in the context of the Heisenberg uncertainty relations. The familiar case of the azimuthal angle variable $\ensuremath{\phi}$ and its "conjugate" angular momentum ${L}_{z}$ is discussed. Various pitfalls associated with the periodicity problem are avoided by employing periodic variables ($sin\ensuremath{\phi}$ and $cos\ensuremath{\phi}$ to describe the phase variable. Well-defined uncertainty relations are derived and discussed. A detailed analysis of the three-dimensional harmonic oscillator excited in coherent states is given. A detailed analysis of the simple harmonic oscillator is given. The usual assumption that a (Hermitian) phase operator $\ensuremath{\varphi}$ (conjugate to the number operator $N$) exists is shown to be erroneous. However, cosine and sine operators $C$ and $S$ exist and are the appr\'opriate phase variables. A Poisson bracket argument using action-angle (rather $J$, $cos\ensuremath{\varphi}$, $sin\ensuremath{\varphi}$) variables is used to deduce $C$ and $S$. The spectra and eigenfunctions of these operators are investigated, along with the important "phase-difference" periodic variables. The properties of the oscillator variables in the various types of states are analyzed with special attention to the uncertainty relations and the transition to the classical limit. The utility of coherent states as a basis for the description of the evolution of the density matrix is emphasized. In this basis it is easy to identify the classical Liouville equation in action-angle variables along with quantum-mechanical "corrections." Mention is made of possible physical applications to superfluid systems.

Minimum Uncertainty Product, Number‐Phase Uncertainty Product, and Coherent States

Abstract: The number‐phase uncertainty products proposed by Carruthers and Nieto are studied to determine whether they are minimized by coherent states. It is found that coherent states do not minimize these products. States that do minimize some of the uncertainty products are constructed. Variational techniques for the study of arbitrary uncertainty products are developed.

Coherent Soft‐Photon States and Infrared Divergences. I. Classical Currents

Abstract: As a first step toward a treatment of soft‐photon processes which is free of infrared divergences and avoids the necessity of introducing a fictitious photon mass, the specification of asymptotic photon states belonging to non‐Fock representations is discussed. As in the work of Chung, a basis consisting of generalized coherent states is used, but in contrast to his work, these states are rigorously defined in terms of von Neumann's infinite tensor product. It is shown that the states must be given an additional label which serves to distinguish various ``weakly equivalent'' vectors, and which corresponds formally to an infinite phase factor. A nonseparable Hilbert space HIR is defined (as a subspace of the infinite tensor‐product space) which may be regarded as the space of all possible asymptotic photon states. The interaction of the electromagnetic field with a prescribed classical current distribution is discussed, and it is shown that a unitary S operator, all of whose matrix elements are finite, may...

Coherent Soft-Photon States and Infrared Divergences. III. Asymptotic States and Reduction Formulas

Abstract: In paper II of this series, the mass-shell singularities of the Green's functions of quantum electrodynamics were investigated. In this paper, the relationship between these singularities and the asymptotic states of the theory is studied by considering the nature of the intermediate states that can contribute to the corresponding discontinuity functions. The basic principle underlying this work is that the asymptotic states of the theory should not be specified a priori but should be determined from the structure of the Green's functions themselves. The pure soft-photon asymptotic states, which can be created from the vacuum by operators constructed from the soft-photon part of the electromagnetic field, are studied first. These states are defined by appropriate weak limits and are shown to span a space with the same structure as in the noninteracting case. Next, states containing a single particle (massive particle or hard photon), together with soft photons, are investigated. These states can appear as intermediate states in the two-point function. They are again defined by weak limits, and are shown to be stable in the absence of external currents. It is demonstrated that the near-mass-shell components of the field operator, acting on the vacuum or on a soft-photon coherent state, yield a state containing one particle and a soft-photon coherent state. Finally, the analysis is extended to two-particle and multiparticle states. The only essentially new feature here is the appearance of factors related to the "Coulomb phases." General reduction formulas are obtained that permit matrix elements between arbitrary asymptotic states to be extracted from the Green's functions. In effect, these matrix elements may be identified with the coefficients not of poles but of branch-point singularities.

Coherent Soft-Photon States and Infrared Divergences. IV. The Scattering Operator

Abstract: In Paper III of this series, the asymptotic states of quantum electrodynamics were defined in terms of the mass-shell singularity structure of the Green's functions. In this paper the reduction formulas obtained are used to derive simple expressions for the matrix elements of the scattering operator. This operator is defined on the space of asymptotic states, which is the direct product of the Fock space of the particles (massive particles and hard photons) with the nonseparable Hilbert space, defined in Paper I, which is spanned by the soft-photon coherent states. It is shown that the scattering operator so defined is gauge-invariant, Lorentz-invariant, unitary, crossing-symmetric, and independent of the choice of the small parameter that defines the separation between hard and soft photons. For a given initial state, the only nonvanishing scattering matrix elements are those to final states in a specific equivalence class, and conditions for states to be equivalent in this sense are obtained. The relationship between these matrix elements and physically measurable cross sections is discussed. In this way, results obtained by conventional methods are reproduced, but in addition questions inaccessible to such methods, such as the effect of an infinite number of soft photons in the initial state, may be investigated.

Properties of the generalized coherent state

Abstract: Properties of the generalized coherent states of the electromagnetic field, introduced by Titulaer and Glauber, are investigated. It is shown that, in contradistinction to photon-counting measurements, the average value of the electromagnetic field and the electron scattering amplitudes depend on the detailed form of these states.

Coherent states in the theory of superfluidity.

Abstract: It is shown that the coherent-state representation of a many-boson wave function may be identified with the order-parameter function conventionally used to describe a superfluid. The statistical mechanics of the many-boson system is reformulated in terms of the coherent states, and a theory of the Ginzburg-Landau form is recovered in an obvious approximation. The formalism is particularly useful for describing metastable states of finite superflow and the fluctuations which may cause spontaneous decay of such states.

Normal and antinormal correlations for the superposition of thermal and coherent fields

Abstract: The sixth-order normal correlation function for the superposition of thermal and coherent fields is calculated in coherent state formalism with the help of a method which enables us to understand why the semiclassical and quantum descriptions of light are equivalent. An experiment for measuring this sixth-order correlation function, which provides the possibility of examining the coherent state aspects of a laser, is proposed. Antinormal correlation functions for the superposition of thermal and coherent fields are also calculated and general relations between normal and antinormal correlation functions, which generalize the relations obtained in recent papers, are derived. A graphic method for computing the moments of the integrated intensity of arbitrary order for the superposition of thermal and coherent fields is introduced.

Convergence of the Sudarshan Expansion for the Diagonal Coherent-State Weight Functional

Abstract: The mathematical properties of the original expansion derived by Sudarshan for the diagonal coherent‐state weight functional are discussed. It is shown that, for stationary fields, the expansion is a generalized function in the space Z′(R2). The validity of this method of defining the weight functional in the case of arbitrary density operators and its relationship to other approaches to the problem of the diagonal representation is briefly considered.

Nonperturbative theory of inhomogeneous effects in gas lasers

Abstract: The self-consistent field approximation (SCFA)[1]-[4] is used to calculate the steady-state mode energies and frequency shifts for a gas laser. The SCFA is a statistical approximation which was shown[3] to be accurate for small values of a parameter ( \gamma^{2}N ), which is always small for gas lasers. The SCFA is simpler and more convenient for studying the complicated dynamical effects that occur in gas lasers than the more recent[5]-[8] quantum theoretical models. It does not, of course, give any new description of the laser photon statistics, since the approximation treats the field as a pure coherent state[9] and is, therefore, equivalent to the semiclassical theory. The main results of this paper are to show how dynamical effects can be calculated nonperturbatively for arbitrary values of the field strength, and to analyze the different roles played by different kinds of inhomogeneous effects in gas lasers. These inhomogeneous effects are due to the dependence of the atomic inversion density on the atomic positions and velocities. Some numerical results are given which show that the nonperturbative theory can explain some effects not previously explained by perturbation theories. [1], [2].

Some connections between the classical theory of fluctuations and quantum field theory

Abstract: A number of authors have noticed that the ground state of the harmonic oscillator can be represented as the motion of a classical charged particle under the influence of a fluctuating electromagnetic field, provided the effect of radiative reaction is taken into account. Their work is reviewed and some new results are obtained which extend the treatment to excited states.

Singular response of an ideal bose gas

Abstract: The concept of singular response is defined and used for the investigation of the Bose-Einstein condensation of an ideal gas. It is shown that, apart from the well-known symmetry that is broken by the transition by the state withk=0, also states withk ≠ 0 participate in the condensation. The new condensed state is shown to possess off-diagonal longrange order and to be a product of Glauber coherent states withk=0 andk ≠ 0.