Coherent states for isospectral Hamiltonians

The standard quantum limit (SQL) as an error probability can be broken by means of a unitary transformation such as a Kerr medium installed before the conventional homodyne scheme. Such an improvement in the error probability cannot be realized by the conventional unitary transformer like squeezers for fixed coherent state inputs. The physical origin of this effect is nonlinear quantum interference for the components of the output state evolved from the input coherent state. We clarify the conditions for overcoming the SQL by unitary transformation and show several examples of the interaction Hamiltonian which generate unitary processes to break the SQL.

A time-dependent unitary (canonical) transformation is found which maps the Hamiltonian for a harmonic oscillator with time-dependent real mass and real frequency to that of a generalized harmonic oscillator with time-dependent real mass and imaginary frequency. The latter may be reduced to an ordinary harmonic oscillator by means of another unitary (canonical) transformation. A simple analysis of the resulting system leads to the identification of a previously unknown class of exactly solvable time-dependent oscillators. Furthermore, it is shown how one can apply these results to establish a canonical equivalence between some real and imaginary frequency oscillators. In particular it is shown that a harmonic oscillator whose frequency is constant and whose mass grows linearly in time is canonically equivalent with an oscillator whose frequency changes from being real to imaginary and vice versa repeatedly.

After a unitary transformation, exact wavefunctions of the time-dependent two-dimensional harmonic oscillator in a time-dependent magnetic field are written in the form of the product of two one-dimensional time-dependent harmonic oscillators in the (x, py) representation. Applying the Heisenberg correspondence principle (HCP) to the system, the exact classical solution is obtained from the quantum matrix elements. Meanwhile, the coherent state is discussed.

The relationship between quantum degeneracies and the emergence of classical periodic orbits in the Fock-Darwin (FD) system is systematically explored by using the ladder operators in the Cartesian coordinates. The quantum-classical connection is analytically developed with the time-dependent coherent state that is theoretically verified to correspond to the Gaussian wave packet state with unitary transformation. The time-dependent coherent state is further used to derive the stationary coherent state that can be expressed as a superposition of degenerate eigenstates. More importantly, the stationary coherent state can alternatively be expressed as an integral of the Gaussian wave packet over a periodic orbit. With the integral representation, the quantum vortex structures of the stationary coherent states can be precisely characterized up to extremely high order. The vortex arrays are generally observed when the phase structures are near the crossing points and cusps of the classical trajectories. Finally, the quantum Fourier transform is employed to verify that the circularly symmetric eigenstate physically corresponds to an ensemble of classical hypotrochoid orbits. This work can provide a pedagogical insight into the formation of Laguerre-Gaussian eigenmodes and hypotrochoid geometric modes in degenerate spherical cavities.

We present a possible construction of coherent states on the unit circle as configuration space. Our approach is based on Borel quantizations on S1 including the Aharonov–Bohm-type quantum description. Coherent states are constructed by Perelomov’s method as group-related coherent states generated by Weyl operators on the quantum phase space . Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction, we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was . The coherent states constructed in this work are shown to satisfy the resolution of unity. To compare them with canonical coherent states, some of their further properties are also studied demonstrating similarities as well as substantial differences.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Light. Volume 2. Laser Light Dynamics. By H. Haken. (Amsterdam, New York, Oxford: North-Holland, 1985.) [Pp. xxiii + 336 + subject index.] Price Dfl. 125·00/US $46.25 hard cover. Electronic Structure and Electronic Transitions in Layered Materials. Edited by V. Grasso. (Dordrecht, Boston, Lancaster: D. Reidel, 1986.) [Pp. ix + 491 + author and subject index.] Price Dfl.225.00 hard cover. The Physics and Fabrication of Microstructures and Microdevices. Edited by M. J. Kelly and C. Weisbuch. (Berlin, Heidelberg, New York: Springer-Verlag, 1986.) [Pp. xi + 460.1 Price DM 99·00 hard cover. Unitary Transformations in Solid State Physics. By M. Wagner. (Amsterdam, Oxford, New York: North-Holland, 1986.) [Pp. xvi + 341 + author and subject index.] Price Dfl.225.00 hard cover. Differential Manifolds and Theoretical Physics. By W. D. Curtis and F. R. Miller. (Orlando, San Diego and New York: Academic Press, 1985.) [Pp. xiii + 388 + subject index.] Price £69·00 hard cover. Coherent States. Applications in Physics and Mathematical Physics. By J. R. Klauder and B. Skagerstam. (Singapore: World Scientific, 1985.) [Pp. xvii + 911.] Price £6670 soft cover. (69 reprints from various journals + introduction.) International Tables for Crystallography. Edited by T. Hahn. (Dordrecht, Boston, Lancaster: D. Reidel, 1985.) [Pp. viii +119.1 Price £9·00 soft cover. (Brief Teaching Edition of Volume A—Space-Group Symmetry.) Multiple Time Scales. Edited by J. U. Brackbill and B. I. Cohen. (Orlando, San Diego, New York: Academic Press, 1985.) [Pp. xiii + 436 + subject index.] Price ·65.00 hard cover. (Thirteen reviews of applications in physics.) Silicon-on-Insulator: Its Technology and Applications. Edited by S. Furukawa. (Dordrecht, Boston, London: D. Reidel; and Tokyo: KTK Scientific Publishers, 1985.) [Pp. vii + 295 + author index.] Price £49·95 hard cover. (20 papers presented at the seminar on ‘Solid Phase Epitaxy and Interface Kinetics’ held in Oiso, Japan, June, 1983.) Layered Structures and Interface Kinetics: Their Technology and Applications. Edited by S. Furukawa. (Dordrecht, Boston, London: D. Reidel; and Tokyo: KTK Scientific Publishers, 1985.) [Pp. viii + 367 + author index.] Price £60·95 hard cover. (22 papers presented at the seminar on ‘Solid Phase Epitaxy and Interface Kinetics’ held in Oiso, Japan, June, 1983.) Superlattices and Microstructures. Volume 1. Number 1. Edited by J. D. Dow. (Orlando, London, New York: Academic Press, 1985.) [Pp. 96.] soft cover. (New interdisciplinary journal devoted to the physics, chemistry, materials science and electrical engineering of submicron structures not only in semiconductors but also in materials with metallic, insulating and superconducting properties.) Solid Clues. Quantum Physics, Molecular Biology, and the Future of Science. By G. Feinberg. (London: Heinemann, 1985.) [Pp. i + 278 + subject index.] Price £12·95 hard cover. Laser Spectroscopy of Solids. Second Printing. Edited by W. M. Yen and P. M. Selzer. (Berlin, Heidelberg, New York: Springer-Verlag, 1986.) [Pp. xiv + 311 + subject index.] Price DM79 soft cover (See Phil. Mag. B, 44, 185 for review.)

The quantum harmonic oscillator can be considered as a composite system of indistinguishable Bose-Einstein symmetric two-level-systems (quanta). In analogy to the classical Poisson limit theorem, we show that a coherent state is the limit of a sequence of homogeneous product states (coherent spin states) and discuss statistical properties of the quanta in classical and nonclassical states.

The technique regarding the integration within a normally ordered product of operators, which refers to the creation and annihilation operators of the harmonic oscillator coherent states, has proved to be very fruitful for different operator identities and applications in quantum optics. In this paper we propose a generalization of this technique by introducing a new operatorial approach—the diagonal ordering operation technique (DOOT)—regarding the calculations connected with the normally ordered product of generalized creation and annihilation operators that generate the generalized hypergeometric coherent states. We have pointed out a number of properties of these coherent states, including the case of mixed (thermal) states. At the same time, by particularizing the obtained results to the one-dimensional harmonic and pseudoharmonic oscillators, we get the well-known results achieved through other methods in the corresponding coherent states representation.

We prove that the vast majority of symmetric states of qubits can be decomposed in a unique way into a superposition of spin 1/2 coherent states. For the case of two qubits, the proposed decomposition reproduces the Schmidt decomposition and therefore, in the case of a higher number of qubits, can be considered as its generalization. We analyze the geometrical aspects of the proposed representation and its invariant properties under the action of local unitary and local invertible transformations. As an application, we identify the most general classes of entanglement and representative states for any number of qubits in a symmetric state.

The work of Carruthers & Nieto on the harmonic oscillator coherent states is combined with Schwinger’s construction of angular momentum to produce the angular momentum coherent states. It is shown that these states become the vector representatives of angular momentum in the classical limit, and so are particularly useful for discussing the transition from quantum to classical angular momentum. The uncertainty relations for angle and angular momentum are described and are compatible with the classical limit. Under rotations the coherent states transform in a manner that in the classical limit is equivalent to the transformation of vectors, and in the same limit the root mean square variation of the expectation values of the components of angular momentum become negligible in comparison with the expectation values themselves. The coupling of two angular momenta in the classical limit is investigated: it is shown that although the product of two coherent states is not itself a coherent state, it does represent a packet similar to a true coherent state, and centred on the direction of the classical resultant of the two component vectors. The properties and implications of hyperbolic angular momentum space are discussed.

In this Chapter we study the Weyl–Heisenberg group in the Schrodinger representation in arbitrary dimension n. One shows that it operates in the Hilbert space of quantum states (and on quantum operators) as a phase-space translation. Then applying it to a Schwartz class state of arbitrary profile we get a set of generalized coherent states. When we apply the Weyl–Heisenberg translation operator to the ground state of the n-dimensional Harmonic Oscillator, one gets the standard coherent states introduced by Schrodinger (Naturwissenshaften 14:664–666, 1926) in the early days of quantum mechanics (1926). Later the coherent states have been extensively studied by Glauber (Phys. Rev. 131:2766–2788, 1963; Phys. Rev. 130:2529–2539, 1963) for the purpose of quantum optics and it seems that their name comes back to this work. The standard coherent states have been generalized by Perelomov (Generalised Coherent States and Their Applications, 1986) to more general Lie groups than the Weyl–Heisenberg group.

Berezin-Klauder-Toeplitz (“anti-Wick”) or “coherent state” quantization of the complex plane, viewed as the phase space of a particle moving on the line, is derived from the resolution of the unity provided by the standard (or gaussian) coherent states. The construction of these states and their attractive properties are essentially based on the energy spectrum of the harmonic oscillator, that is on natural numbers. We follow in this work the same path by considering sequences of non-negative numbers and their associated “non-linear” coherent states. We illustrate our approach with the 2-d motion of a charged particle in a uniform magnetic field. By solving the involved Stieltjes moment problem we construct a family of coherent states for this model. We then proceed with the corresponding coherent state quantization and we show that this procedure takes into account the circle topology of the classical motion.

We construct ladder operators, C̃ and C̃†, for a multistep rational extension of the harmonic oscillator on the half plane, x ≥ 0. These ladder operators connect all states of the spectrum in only infinite-dimensional representations of their polynomial Heisenberg algebra. For comparison, we also construct two different classes of ladder operator acting on this system that form finite-dimensional as well as infinite-dimensional representations of their respective polynomial Heisenberg algebras. For the rational extension, we construct the position wavefunctions in terms of exceptional orthogonal polynomials. For a particular choice of parameters and for the three lowest weights μ = −5, −3, and 5, we construct the coherent states, eigenvectors of C̃ with generally complex eigenvalues, z, as superposition of subsets of the energy eigenvectors. Then, we calculate the properties of these coherent states, looking for classical or nonclassical behavior. We calculate the energy expectations as functions of |z|. We plot position probability densities for the coherent states and for the even and odd cat states formed from these coherent states. We plot the Wigner functions for a particular choice of z. For these coherent states on one arm of a beamsplitter, we calculate the two excitation number distributions and the linear entropies of the output states. We plot the standard deviations in x and find squeezing in the regime considered in one of the cases. By plotting the Mandel Q parameters for the coherent states as functions of |z|, we find that the number statistics is sub-Poissonian in all cases.

We present details of the method for constructing the coherent states of the quantum harmonic oscillator model that exhibits the semiconfinement effect due to a specific change of its mass by position. Its energy spectrum completely overlaps with the Hermite oscillator energy spectrum, whereas the wavefunctions of the stationary states are expressed via the generalized Laguerre polynomials. Two different methods have been applied to compute the coherent states. They are generalized and Barut-Girardello coherent states methods. In both cases, the exact expressions have been obtained. We also analyzed some limit cases, under which the constructed coherent states completely recover the coherent states of the Hermite oscillator.

In this paper, we introduce two classes of superposed states generated by a superposition of two single-mode para-Bose coherent states (CS) with arbitrary relative phase factors. The first class is superposition of two opposite para-Bose CS and second class consists of two para-Bose CS, [Formula: see text] out of phase with each other. These states are reduced to the well-known superposed single-mode CS of the simple harmonic oscillator when the deformation parameter tends to one. We study the nonclassical properties of the introduced states by evaluating their para-Bose Fock state distribution, second-order squeezing, Klyshko’s and Mandel’s parameters. We investigate the essential role of coherent and deformation parameters on their nonclassicality. We show that some nonclassical properties of these states are different from those of the even and odd coherent states. Finally, we present a simple scheme for the physical generation of the introduced states.