Ordinary Coherent States Research Articles

Coherent distributions of the symmetry algebra of spinor regularization generator and hydrogen atom

A new approach is developed for solving spectral problems for operators with continuous spectrum, which consists in the integral transform of the problem by using coherent Schwartz distributions. The constructed family of coherent distributions is a complete analogue of the family of ordinary coherent states. More precisely, it satisfies all Gazeau–Klauder axioms satisfied by the usual coherent states. But in contrast to the coherent states belonging to the point spectrum of the annihilation operator (or operators), the coherent distributions belong to the continuous spectrum of some Hermitian operators. Therefore, the coherent distributions work better than the coherent states as the kernel of the integral representation of generalized eigenfunctions of operators with continuous spectrum. In this work, this approach is demonstrated with an example of solving a basic problem of quantum mechanics, i.e., the problem of the continuous part of the spectrum of the Hamiltonian of the hydrogen atom.

System of six-level atom interacting with a quantized field in the existence of time-varying coupling

In the present manuscript, we extend the Jaynes-Cummings model in the framework of six-level atom (LA) system in the presence of a quantized field describes by an ordinary coherent state or a superposition state. We investigate the dynamics of the population, coherence and quantum entanglement in terms of the system parameters. We analyze the time variation of the quantifiers in the presence and absence of time-varying coupling of the six-LA–field system and make comparison among the quantifiers. Furthermore, we show how the quantifiers can be controlled according to the values of the system parameters. The obtained results illustrate that the proposed system provides interesting values of the quantum coherence and entanglement which can be considered as good candidates for executing quantum optics and information schemes.

Coherent Schwartz distributions of the Heisenberg algebra and inverted oscillator

An unusual family of generalized coherent states of the Heisenberg algebra is constructed. It consists of two families of functionals on the Schwartz space. Each of them has the key properties of ordinary coherent states: it intertwines the representations of a given algebra, has the completeness property, and minimizes the product of variances in the Heisenberg relation. However, unlike the usual coherent states that are eigenstates for the annihilation operator, the constructed distributions belong to the continuous spectrum of Hermitian generators of the Heisenberg algebra. The inner product of functionals from different families has the main properties of the overlap function of ordinary coherent states: it is continuous in parameters, satisfies the reproducing identity, and has the corresponding geometric meaning. The application of unusual coherent states of the continuous spectrum is demonstrated by an example of solving the spectral problem for an inverted oscillator.

Bi-coherent states as generalized eigenstates of the position and the momentum operators

In this paper, we show that the position and the derivative operators, {{hat{q}}} and {{hat{D}}}, can be treated as ladder operators connecting various vectors of two biorthonormal families, {{{mathcal {F}}}}_varphi and {{{mathcal {F}}}}_psi . In particular, the vectors in {{{mathcal {F}}}}_varphi are essentially monomials in x, x^k, while those in {{{mathcal {F}}}}_psi are weak derivatives of the Dirac delta distribution, delta ^{(m)}(x), times some normalization factor. We also show how bi-coherent states can be constructed for these {{hat{q}}} and {{hat{D}}}, both as convergent series of elements of {{{mathcal {F}}}}_varphi and {{{mathcal {F}}}}_psi , or using two different displacement-like operators acting on the two vacua of the framework. Our approach generalizes well- known results for ordinary coherent states.

Quantum collapse-revival effect in a supersymmetric Jaynes–Cummings model and its possible application in supersymmetric qubits

In the photon-atom interaction, one of the most interesting phenomena can be collapse-revival effect in population inversion of atomic levels. Since a single-photon-transition Jaynes–Cummings model can be generalized to a multiphoton-transition Jaynes–Cummings model, of which the Hamiltonian can exhibit a supersymmetric Lie algebra structure, in this paper, quantum collapse and revival in atomic population inversion, which is a collaborative effect of a variety of quantum Rabi oscillations with various quantum Rabi frequencies, is studied for the supersymmetric multiphoton-transition Jaynes–Cummings model. In the literature, it has been pointed out that quantum collapse and revival might find an application in qubit design. We expect that a supersymmetric two-level system might serve as such a kind of qubits. Compared with ordinary two-level qubits, more information could be stored in the basis vectors of supersymmetric qubits, and this could increase the degrees of freedom of tensor product (multi-component extension) of basis vectors of this multiphoton-atom interacting system. For this reason, some characteristics of the collapse-revival effect driven by ordinary coherent states and Yurke–Stoler coherent states will be addressed in this paper. The influence of various characteristic parameters of this supersymmetric Jaynes–Cummings model and the quantized optical fields on the quantum collapse and revival in the population inversion will be indicated in some numerical illustrative examples.

Quantum statistical properties of photon-added spin coherent states

The photon-added spin coherent state as a new kind of coherent state has been defined by iterated actions of the proper raising operator on the ordinary spin coherent state. In this paper, the quantum statistical properties of photon-added spin coherent states such as photon number distribution, second-order correlation function and Wigner function are studied. It is found that the Wigner function shows the negativity in some regions and the second-order correlation function is less than unity. Therefore, the photon-added spin coherent state is a nonclassical state.

New mathematics for the nonadditive Tsallis’ scenario

In this paper, we investigate quantum uncertainties in a Tsallis’ nonadditive scenario. To such an end we appeal to [Formula: see text]-exponentials (qEs), that are the cornerstone of Tsallis’ theory. In this respect, it is found that some new mathematics is needed and we are led to construct a set of novel special states that are the qE equivalents of the ordinary coherent states (CS) of the harmonic oscillator (HO). We then characterize these new Tsallis’ special states by obtaining the associated (i) probability distributions (PDs) for a state of momentum [Formula: see text], (ii) mean values for some functions of space an momenta and (iii) concomitant quantum uncertainties. The latter are then compared to the usual ones.

Deformed quons and bi-coherent states

We discuss how a q-mutation relation can be deformed replacing a pair of conjugate operators with two other and unrelated operators, as it is done in the construction of pseudo-fermions, pseudo-bosons and truncated pseudo-bosons. This deformation involves interesting mathematical problems and suggests possible applications to pseudo-hermitian quantum mechanics. We construct bi-coherent states associated to D -pseudo-quons, and we show that they share many of their properties with ordinary coherent states. In particular, we find conditions for these states to exist, to be eigenstates of suitable annihilation operators and to give rise to a resolution of the identity. Two examples are discussed in details, one connected to an unbounded similarity map, and the other to a bounded map.

State-projective scheme for generating pair coherent states in traveling-wave optical fields

The pair coherent states of a two-mode quantized electromagnetic field introduced by Agarwal [Phys. Rev. Lett. 57, 827 (1986)] have yet to be generated in the laboratory. The states can mathematically be obtained from a product of ordinary coherent states via projection onto a subspace wherein identical photon number states of each mode are paired. We propose a scheme by which this projection can be engineered. The scheme requires relatively weak cross-Kerr nonlinearities, the ability to perform a displacement operation on a beam mode, and photon detection ability able to distinguish between zero and any other number of photons. These requirements can be fulfilled with currently available technology or technology that is on the horizon.

Coherent states in noncommutative quantum mechanics

Gazeau–Klauder coherent states in noncommutative quantum mechanics are considered. We find that these states share similar properties to those of ordinary canonical coherent states in the sense that they saturate the related position uncertainty relation, obey a Poisson distribution, and possess a flat geometry. Using the natural isometry between the quantum Hilbert space of Hilbert-Schmidt operators and the tensor product of the classical configuration space and its dual, we reveal the inherent vector feature of these states.

Beam splitting and entanglement: Generalized coherent states, group contraction, and the classical limit

Recently, Markham and Vedral [Phys. Rev. A 67, 042113 (2003)] investigated the effect of beam splitting on the spin, or SU(2), coherent states for a single mode field. The spin coherent state is a binomial coherent state related to the Holstein-Primakoff realization of the su(2) Lie algebra given in terms of a set of single mode bose annihilation and creation operators. Upon beam splitting, the ordinary (or Glauber) coherent states merely split into products of ordinary coherent states with reduced amplitudes without becoming entangled, as one would expect for a classical-like field. The above authors expected the spin coherent states to go over to the ordinary coherent states in the limit of high spin, $j\ensuremath{\rightarrow}\ensuremath{\infty}$, and thus to become product states after beam splitting. But this expectation was not confirmed through numerical calculation of the entropy which, instead of going to zero, leveled off with increasing spin. In this paper we find similar behavior for SU(1,1) coherent states of the Perelomov type for large Bargman index $k$, but also find that the Barut-Girardello SU(1,1) coherent states appear to rapidly become product states after beam splitting for increasing $k$. We explain these results by showing that, in reality, neither the spin coherent states nor the Perelomov SU(1,1) coherent states go over to ordinary coherent states in the limits of large $j$ or $k$, and that the Barut-Girardello coherent states merely go over to the vacuum in the large $k$ limit. Finally, we examine the correct limiting procedure for obtaining separable states (i.e., products of coherent states) upon beam splitting by performing contractions of the su(2) and su(1,1) Lie algebras and of their associated coherent states.

The Fuzzy Sphere: From The Uncertainty Relation To The Stereographic Projection

On the fuzzy sphere, no state saturates simultaneously all the Heisenberg uncertainties. We propose a weaker uncertainty for which this holds. The family of states so obtained is physically motivated because it encodes information about positions in this fuzzy context. In particular, these states realize in a natural way a deformation of the stereographic projection. Surprisingly, in the large $j$ limit, they reproduce some properties of the ordinary coherent states on the non commutative plane.

PARASUPERSYMMETRIC COHERENT STATES FOR LANDAU LEVELS WITH DYNAMICAL SYMMETRY GROUP H4

The eigenstates and their degeneracy for parasupersymmetric Hamiltonian of arbitrary order p, corresponding to the motion of a charged particle with spin [Formula: see text] on the flat surface in the presence of a constant magnetic field along z-axis, are calculated. The eigenstates are expressed in terms of Landau levels quantum states with dynamical symmetry group H4. Furthermore, parasupersymmetric coherent states with multiplicity degeneracy are derived for an ad hoc lowering operator of the eigenstates in terms of ordinary coherent states of Landau Hamiltonian.

Nonlinear Effects in Quantum Dynamics of Atom Laser: Mean-Field Approach**The project partially supported by National Natural Science Foundation of China

Quantum dynamics and statistics of an atom laser with nonlinear binary interactions are investigated in the framework of mean-field approximation. The linearized effective Hamiltonian of the system is accurately solvable. It is shown that, although the input radio frequency field is in an ordinary Glauber coherent state, the output matter wave will periodically exhibit quadrature squeezing effects purely originated from the nonlinear atom-atom collisions.

Gazeau-Klauder coherent states in one-mode systems with periodic potential

Gazeau-Klauder coherent states for a one-mode system with sinusoidal potential, are introduced. Their quantum statistical properties and their uncertainties are studied. The effect of dissipation on these states is estimated. The evolution of the ordinary (Glauber) coherent states in this system, is also studied. It is shown that these states evolve into superpositions of many macroscopically distinguishable states (`multi-Schrodinger cats').

New coherent string states and minimal uncertainty in WZWN models

We study the properties of exact (all level k) quantum coherent states in the context of string theory on a group manifold (WZWN models). Coherent states of WZWN models may help to solve the unitarity problem: having positive norm, they consistently describe the very massive string states (otherwise excluded by the spin-level condition). These states can be constructed by (at least) two alternative procedures: (i) as the exponential of the creation operator on the ground state, and (ii) as eigenstates of the annihilation operator. In the k→∞ limit, all the known properties of ordinary coherent states are recovered. States (i) and (ii) (which are equivalent in the context of ordinary quantum mechanics and string theory in flat spacetime) are not equivalent in the context of WZWN models. The set (i) was constructed by the authors in the previous article. In this paper we provide the construction of states (ii), we compare the two sets and discuss their properties. We analyze the uncertainty relation, and show that states (ii) satisfy automatically the minimal uncertainty condition for any k; they are thus quasiclassical, in some sense more classical than states (i) which only satisfy it in the k→∞ limit. Modification to the Heisenberg relation is given by 2 H/k , where H is connected to the string energy.

COHERENT STATES AND THE QUANTIZATION OF (1+1)-DIMENSIONAL YANG–MILLS THEORY

This paper discusses the canonical quantization of (1+1)-dimensional Yang–Mills theory on a spacetime cylinder from the point of view of coherent states, or equivalently, the Segal–Bargmann transform. Before gauge symmetry is imposed, the coherent states are simply ordinary coherent states labeled by points in an infinite-dimensional linear phase space. Gauge symmetry is imposed by projecting the original coherent states onto the gauge-invariant subspace, using a suitable regularization procedure. We obtain in this way a new family of "reduced" coherent states labeled by points in the reduced phase space, which in this case is simply the cotangent bundle of the structure group K. The main result explained here, obtained originally in a joint work of the author with B. Driver, is this: The reduced coherent states are precisely those associated to the generalized Segal–Bargmann transform for K, as introduced by the author from a different point of view. This result agrees with that of K. Wren, who uses a different method of implementing the gauge symmetry. The coherent states also provide a rigorous way of making sense out of the quantum Hamiltonian for the unreduced system. Various related issues are discussed, including the complex structure on the reduced phase space and the question of whether quantization commutes with reduction.

Entropic uncertainty relations and the number-phase Wigner function for nonlinear and multiphoton coherent states

The number-phase entropic uncertainty relation for the multiphoton coherent state (MCS) and nonlinear coherent state (NCS) are studied and compared with an ordinary coherent state (CS). We show that the MCS has higher (lower) number (phase) entropy while the NCS has lower (higher) number (phase) entropy in comparision to the CS. We also discuss the number-phase Wigner function for these states which gives a graphical representation of the complimentary nature of the number and phase properties for these states. With the help of this Wigner representation a photon number cat formation in the NCS is discussed.

Quantum coherent string states inAdS3and theSL(2,R)Wess-Zumino-Witten-Novikov model

In this paper we make the connection between semiclassical string quantization and exact conformal field theory quantization of strings in 2+1 anti--de Sitter spacetime. More precisely, considering the Weiss-Zumino-Witten-Novikov model corresponding to $\mathrm{SL}(2,R)$ and its covering group, we construct quantum coherent string states, which generalize the ordinary coherent states of quantum mechanics, and show that in the classical limit they correspond to oscillating circular strings. After quantization, the spectrum is found to consist of two parts: A continuous spectrum of low mass states (partly tachyonic) satisfying the standard spin-level condition necessary for unitarity $|j|<k/2,$ and a discrete spectrum of high mass states with asymptotic behavior ${m}^{2}{\ensuremath{\alpha}}^{\ensuremath{'}}\ensuremath{\propto}{N}^{2}$ $(N$ positive integer). The quantization condition for the high mass states arises from the condition of finite positive norm of the coherent string states, and the result agrees with our previous results obtained using semiclassical quantization. In the $\stackrel{\ensuremath{\rightarrow}}{k}\ensuremath{\infty}$ limit, all the usual properties of coherent or quasiclassical states are recovered. It should be stressed that we consider the circular strings only for simplicity and clarity, and that our construction can easily be used for other string configurations too. We also compare our results with those obtained in the recent paper by Maldacena and Ooguri, hep-th/0001053.

Low-temperature quasi-classical approximation for quantum macroscopic phenomena

Quantum systems with two distinct time scales for “fast” (quantum) and “slow” (quasi-classical) degrees of freedom with the respective frequencies Ω and ω≪Ω are considered. A general path-integral representation for a partition function involves ordinary coherent states for Bose and Fermi degrees of freedom as well as generalized coherent states for a nontrivial algebra of variables. Path-integral averaging over “fast” variables results in a nonlocal (in imaginary time) effective action for a “slow” (quasi-classical) variable, which becomes local in the low-temperature approximation. This low-temperature adiabatic path-integral approach, expressed by the inequality βΩ≫1≫ω/Ω, yields an appropriate basis for a subsequent quasi-classical description. Theories of three complex condensed matter systems are developed with the “slow” and “fast” subsystems represented by localized and band electrons in the lattice Anderson model, by condensate and noncondensate bosons in the Bogoliubov model (with broken translational symmetry), and by soft-mode phonons and electrons induced in large molecules by an electron-phonon optical transition. The corresponding quantum macroscopic phenomena are the Kondo-type reorganization of the spectrum of correlated electrons, Bose condensation in nonuniform media, and chaotization of the phonon spectrum in molecules.