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Showing papers on "Coherent states published in 1980"


Journal ArticleDOI
TL;DR: In this paper, the authors considered a new type of quantum nondemolition measurement called back-action-evading measurement, where the real part of the harmonic oscillator's complex amplitude is measured by a single transducer.
Abstract: The monitoring of a quantum-mechanical harmonic oscillator on which a classical force acts is important in a variety of high-precision experiments, such as the attempt to detect gravitational radiation. This paper reviews the standard techniques for monitoring the oscillator, and introduces a new technique which, in principle, can determine the details of the force with arbitrary accuracy, despite the quantum properties of the oscillator. The standard method for monitoring the oscillator is the "amplitude-and-phase" method (position or momentum transducer with output fed through a narrow-band amplifier). The accuracy obtainable by this method is limited by the uncertainty principle ("standard quantum limit"). To do better requires a measurement of the type which Braginsky has called "quantum nondemolition." A well known quantum nondemolition technique is "quantum counting," which can detect an arbitrarily weak classical force, but which cannot provide good accuracy in determining its precise time dependence. This paper considers extensively a new type of quantum nondemolition measurement—a "back-action-evading" measurement of the real part X_1 (or the imaginary part X_2) of the oscillator's complex amplitude. In principle X_1 can be measured "arbitrarily quickly and arbitrarily accurately," and a sequence of such measurements can lead to an arbitrarily accurate monitoring of the classical force. The authors describe explicit Gedanken experiments which demonstrate that X_1 can be measured arbitrarily quickly and arbitrarily accurately. In these experiments the measuring apparatus must be coupled to both the position (position transducer) and the momentum (momentum transducer) of the oscillator, and both couplings must be modulated sinusoidally. For a given measurement time the strength of the coupling determines the accuracy of the measurement; for arbitrarily strong coupling the measurement can be arbitrarily accurate. The "momentum transducer" is constructed by combining a "velocity transducer" with a "negative capacitor" or "negative spring." The modulated couplings are provided by an external, classical generator, which can be realized as a harmonic oscillator excited in an arbitrarily energetic, coherent state. One can avoid the use of two transducers by making "stroboscopic measurements" of X_1, in which one measures position (or momentum) at half-cycle intervals. Alternatively, one can make "continuous single-transducer" measurements of X_1 by modulating appropriately the output of a single transducer (position or momentum), and then filtering the output to pick out the information about X_1 and reject information about X_2. Continuous single-transducer measurements are useful in the case of weak coupling. In this case long measurement times are required to achieve good accuracy, and continuous single-transducer measurements are almost as good as perfectly coupled two-transducer measurements. Finally, the authors develop a theory of quantum nondemolition measurement for arbitrary systems. This paper (Paper I) concentrates on issues of principle; a sequel (Paper II) will consider issues of practice.

969 citations


Journal ArticleDOI
TL;DR: In this article, a class of normal ordering representations of quantum operators is introduced, that generalises the Glauber-Sudarshan P-representation by using nondiagonal coherent state projection operators.
Abstract: A class of normal ordering representations of quantum operators is introduced, that generalises the Glauber-Sudarshan P-representation by using nondiagonal coherent state projection operators. These are shown to have practical application to the solution of quantum mechanical master equations. Different representations have different domains of integration, on a complex extension of the usual canonical phase-space. The 'complex P-representation' is the case in which analytic P-functions are defined and normalised on contours in the complex plane. In this case, exact steady-state solutions can often be obtained, even when this is not possible using the Glauber-Sudarshan P-representation. The 'positive P-representation' is the case in which the domain is the whole complex phase-space. In this case the P-function may always be chosen positive, and any Fokker-Planck equation arising can be chosen to have a positive-semidefinite diffusion array. Thus the 'positive P-representation' is a genuine probability distribution. The new representations are especially useful in cases of nonclassical statistics.

491 citations


Journal ArticleDOI
TL;DR: It was shown that homodyne detection achieves the same signal-to-noise ratio as the quantum field quadrature measurement, thus providing a receiver which realizes the linear modulation TCS performance gain found in Part I.
Abstract: In Part I of this three-part study it was shown that the use of two-photon coherent state (TCS) radiation may yield siginificant performance gains in free-space optical communicatinn if the receiver makes a quantum measurement of a single field quadrature In Part II it was shown that homodyne detection achieves the same signal-to-noise ratio as the quantum field quadrature measurement, thus providing a receiver which realizes the linear modulation TCS performance gain found in Part I Furthermore, it was shown in Part il that ff homodyne detection does exactly correspond to the field quadrature measurement, then a large binary communication performance gain is afforded by homodyne detection of antipodal TCS signals The full equivalence of honmdyne detection and single-quadrature field measurement, as well as that of heterodyne detection and two-quadrature field measurement, is established Furthermore, a heterodyne configuration which uses a TCS image-band oscillator in addition to the usual coherent state local oscillator is studied This coafiguration termed TCS heterodyne detection is shown to realize all the quantum measurements described by arbitrary TCS The foregoing results are obtained by means of a representation theorem which shows that photoemissive detection realizes the photon flux density measurement

471 citations


Journal ArticleDOI
TL;DR: In this article, a generalized Heisenberg-type uncertainty relation is obtained for two arbitrary operators both in the case of pure and of mixed states, and as a rule equality is found to hold for pure quantum state only.

265 citations


Journal ArticleDOI
Barry Simon1
TL;DR: In this paper, the authors extend Lieb's limit theorem to general compact Lie groups and prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has "diagonal form") and discuss the classical limit for various continuum systems.
Abstract: We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approachS2 classical spins asL→∞] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight vector in an irreducible representation and we prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has “diagonal form”).

179 citations


Journal ArticleDOI
TL;DR: In this article, a path integral expression for the transition amplitude which connects a pair of SU(2) coherent states is derived for the simplest semisimple Lie group SU (2) and its classical consequences are investigated.
Abstract: Path integral in the representation of coherent state for the simplest semisimple Lie group SU(2) and its classical consequences are investigated. Using the completeness relation of the coherent state, we derive a path integral expression for the transition amplitude which connects a pair of SU(2) coherent states. In the classical limit we arrive at a canonical equation of motion in a ’’curved phase space’’ (two‐dimensional sphere) which reproduces the ordinary Euler’s equation of a rigid body when applied to a rotator.

108 citations


Journal ArticleDOI
TL;DR: In this paper, a spin-like, two-valued quantum number is used to enlarge the physical Hilbert space by enlarging the phase operator of an oscillator, which can be used to define a phase representation on which trigonometric functions of the phase are numbers and the number of quanta is a differential operator.

83 citations


Journal ArticleDOI
TL;DR: In this article, the minimum-uncertainty coherent-state formalism is extended to higher-dimensional systems for spherically symmetric three-dimensional potentials, where coherent states are products of an angular wave function times a radial wave function.
Abstract: The minimum-uncertainty coherent-states formalism is extended to higher-dimensional systems Specifically, for spherically symmetric three-dimensional potentials the formalism looks for coherent states which are products of an angular wave function times a radial wave function After reviewing the many studies on angular coherent states, I concentrate on the physically distinguishing radial coherent states The radial formalism is explained in detail and contrasted with the effective one-dimensional formalism The natural classical variables in the radial formalism are those which vary sinusoidally as g(E,L)theta(t), where theta(t) is the real azimuthal angular variable and g(E,L) is the number of oscillations between apsidal distances per classical orbit When changed to natural quantum operators, these operators can be given as the Hermitian sums and differences of the ''l'' raising and lowering operators The formalism is applied to the three-dimensional harmonic-oscillator and Coulomb problems

57 citations


Journal ArticleDOI
TL;DR: In this paper, the correspondence between a function f on phase space and the matrix elements is studied in some detail, and the Fourier coefficients of f with respect to an explicit orthonormal system are obtained.
Abstract: We study in some detail the correspondence between a function f on phase space and the matrix elements. (Qf)(a,b) of its quantized Qf between the coherent states ‖ a〉 and ‖ b〉. It is an integral transform: Qf(a,b) = F{a,b ‖ v}f(v) dv, which resembles in many ways the integral transform of Bargmann. We obtain the matrix elements of Qf between harmonic oscillator states as the Fourier coefficients of f with respect to an explicit orthonormal system.

53 citations


Journal ArticleDOI
TL;DR: In this paper, the authors show the time evolution of minimum-uncertainty coherent-state (MUCS) wave packets in the solvable potentials and compare the results with those which can be obtained from other types of coherent states.
Abstract: We show the time evolution of minimum-uncertainty coherent-state (MUCS) wave packets in the solvable potentials we have considered. (Numerical techniques are discussed in the appendices.) The time evolution is compared to the motion that a classical particle would have in the same potential. We make a number of observations on the conditions which can cause the states to lose their coherence more (or less) rapidly with time, and compare the MUCS results with those which can be obtained from other types of coherent states. The most physically interesting comparison is with the "continuous representation" coherent states.

53 citations


Journal ArticleDOI
TL;DR: In this paper, the cooperative resonance fluorescence steady state is discussed within the context of an operator master equation which conserves total pseudospin, and its significance in relation to a background of factorised dynamics is discussed.
Abstract: The cooperative resonance fluorescence steady state is discussed within the context of an operator master equation which conserves total pseudospin. Emphasis throughout is on quantum fluctuations and their significance in relation to a background of factorised dynamics. Atom-atom correlations are shown to play a fundamental role for systems driven beyond the linear regime. Use of the atomic coherent state representation yields a Fokker-Planck description closely allied to the dynamics for a classical angular momentum oscillator. For intense incident fields the quantum-mechanical steady state is understood in terms of diffusion both around and between classical trajectories on the Bloch sphere. In the limit of infinite systems simple closed-form expressions for steady-state features are derived. Coherent and incoherent fluorescent intensities are obtained together with the second-order correlation function for fluorescent light. Specific features are illustrated by numerical results for systems of from two to fifty atoms.

Book ChapterDOI
01 Jan 1980
TL;DR: The coherent states of the harmonic oscillator and hence the radiation field (which is considered as an assembly of oscillators) can be defined in many different, but essentially equivalent ways as mentioned in this paper.
Abstract: The coherent states of the harmonic oscillator and hence the radiation field (which is considered as an assembly of oscillators) can be defined in many different, but essentially equivalent ways.1

Journal ArticleDOI
TL;DR: In this paper, a path integral representation for the transition amplitude which joins arbitrary initial and final states is derived, and the time-dependent Hartree-Fock is naturally obtained as a classical limit.

Journal ArticleDOI
TL;DR: In this article, a quantum theory interpretation of the non-linear classical Yang-Mills field equations is given, which relates them to the matrix element of the field operators between the vacuum state and a coherent state of spin-one quanta of definite helicity.

Journal ArticleDOI
01 Jul 1980-Pramana
TL;DR: In this article, it was shown that the expectation value of the quantum Hamiltonian in any coherent state equals the energy of the classical field at which the state is peaked, and that this property can be used to characterize the usual Fock representation.
Abstract: In the usual Fock quantisation of fields in Minkowski space-time, one has the result that the expectation value of the quantum Hamiltonian in any coherent state equals the energy of the classical field at which the state is peaked. It is shown that this property can be used tocharacterise the usual Fock representation. It is also pointed out that the entire analysis goes through for a substantially more general class of systems including, in particular, Bose fields in arbitrary stationary space-times.

Journal ArticleDOI
TL;DR: In this paper, the probability of finding a harmonic oscillator in the ground state is calculated in the conventional way and in a gauge-invariant way, and the results compared.

Journal ArticleDOI
TL;DR: In this article, the time-dependent Hartee-Fock (TDHF) equations were derived for nuclear systems with internal dynamical group U(r) and the coordinates which appear in the TDHF equations are the coordinates that parameterize the U( r) coherent states.

Journal ArticleDOI
TL;DR: In this paper, a model of the free electron laser is proposed which is based on the classical current of the electron in the wiggler field interacting with a quantized radiation field, but the distribution of the electrons after the interaction is essentially quantum mechanical.

Journal ArticleDOI
TL;DR: The time-dependent Hartree-Fock solutions of the two-level Lipkin-Meshkov-Glick model are studied in this article by transforming the time-dependant Hartree Fock equations into Hamilton's canonical form and analyzing the qualitative structure of the HartreeFock energy surface in the phase space.
Abstract: The time-dependent Hartree-Fock solutions of the two-level Lipkin-Meshkov-Glick model are studied by transforming the time-dependent Hartree-Fock equations into Hamilton's canonical form and analyzing the qualitative structure of the Hartree-Fock energy surface in the phase space. It is shown that as the interaction strength increases these time-dependent Hartree-Fock solutions undergo a qualitative change associated with the ground state phase transition previously studied in terms of coherent states. For two-body interactions stronger than the critical value, two types of time-dependent Hartree-Fock solutions (the ''librations'' and ''rotations'' in Hamilton's mechanics) exist simultaneously, while for weaker interactions only the rotations persist. It is also shown that the coherent states with the maximum total pseudospin value are determinants, so that time-dependent Hartree-Fock analysis is equivalent to the coherent state method.

Book ChapterDOI
01 Jan 1980
TL;DR: In this paper, the authors presented an approximate Schrodinger evolution for Gaussian states with the Trotter Product Formula and the second order Taylor expansion of the potential about the center of the wave packet.
Abstract: For certain Gaussian states we present a simple approximate evolution which is asymptotic to the Schrodinger evolution as → O.In 3 or more dimensions our error estimates are uniform in time if the potential is suitably chosen. Consequently, our methods apply to scattering theory. The approximate evolution is obtained by using the Trotter Product Formula and the second order Taylor expansion of the potential about the center of the wave packet.

Journal ArticleDOI
TL;DR: In this article, it was shown that the usual quantum condition between the physical variables, namely the displacement and the momentum, has to be modified by the damping, and that it is possible to change to new variables which are canonical.
Abstract: Assuming a damping force proportional to the velocity, it is shown that the usual quantum condition between the physical variables, namely the displacement and the momentum, has to be modified by the damping. However, it is possible to change to new variables which are canonical. Then we can apply the usual quantum mechanics and use the usual physical interpretation through the statistical average values. The case of a one-dimensional damped harmonic oscillator under the action of an arbitrary external force has been worked out as an example. No difficulty arises and the results are reasonable.


Journal ArticleDOI
TL;DR: In this paper, the electron-field coherent quasi-classical states of a free electron laser were defined and the photon number and electron momentum were given by a Poisson distribution centered on the classical trajectories.


Journal ArticleDOI
TL;DR: In this paper, a harmonic quantum oscillator and the motion of charged particles in a magnetic field have been considered, the dependence on time of the magnetic field and the oscillator frequency being such that they permit us to keep an exact record of nonadiabaticity.
Abstract: A harmonic quantum oscillator and the motion of charged particles in a magnetic field have been considered, the dependence on time of the magnetic field and the oscillator frequency being such that they permit us to keep an exact record of non-adiabaticity. For these systems, exact solutions, motion integrals, coherent states and Green functions have been constructed and proper amplitudes and transition probabilities calculated. One of the possible practical applications of the results obtained has been pointed out, namely their use for the interpretation of experiments with electronic vibrational transitions in molecules.

Journal ArticleDOI
TL;DR: Using the coherent states method combined with the Schwinger coupled boson representation of the spin operators, it was shown that solitons can propagate in the continuum limit of the quantum ferromagnetic Heisenberg chain this article.
Abstract: Using the coherent states method combined with the Schwinger coupled boson representation of the spin operators (1965) it is shown that solitons can propagate in the continuum limit of the quantum ferromagnetic Heisenberg chain. These solitons are solutions of a nonlinear Schrodinger equation similar to that discussed by Lakshmanan (1977) for the classical Heisenberg chain. Several characteristics of the soliton, like its momentum, energy, and the number of bosonic excitations involved in the soliton formation, are calculated.

Journal ArticleDOI
TL;DR: In this article, it was shown that minimum-uncertainty coherent states for nonharmonic potentials provide a better approximation to the classical motion than do gaussians, and that Gaussians are not, as a matter of principle, allowable wave functions for the ISO system.



Journal ArticleDOI
TL;DR: In this article, a coherent state variables for the electromagnetic field and appropriate spin variables for two-level atoms are used for obtaining the dynamical evolution of the field and the atoms separately.
Abstract: A method utilising the coherent state variables for the electromagnetic field and appropriate spin variables for two-level atomsis developed for obtaining the dynamical evolution of the field and the atoms separately. The case of a single radiation mode interacting with two two-level atoms is studied. The time evolution of the photon number, starting from n photons and both atoms at their higher levels, is given. Furthermore, when the initial state consists of n photons and one of the atoms in its excited state with the other in its ground state, the temporal development of the photon number, as well as the motion of each atom, are obtained. In the latter case it becomes apparent that energy exchange takes place among the atomic systems, while the field acts as a transfer agent.