Coherent State Basis Research Articles

Fermion self-energy correction in light-front QED using coherent state basis

We discuss IR divergences in lepton mass renormalization in Light-Front Quantum Electrodynamics (LFQED) in Feynman gauge. We consider LFQED with Pauli-Villars fields and using old-fashioned time ordered perturbation theory (TOPT), we show that our earlier result regarding cancellation of true IR divergences upto $O(e^4)$ in coherent state basis holds in Feynman gauge also.

On the Coherent State Method in Constructing Representations of Quantum Superalgebras

In recent years, one of the new applications of the coherent state method was to con-struct representation of superalgebras and quantum superalgebras. Following this stream, we hada contribution to working out explicit representation of Uq[gl(2|1)]. Up to now, Uq[gl(2|1) is stillthe biggest quantum superalgebra representations in coherent state basis of which can be built. Inthis article, we will show some detailed techniques used in our previous work but useful for ourfurther investigations. The newest results on building representations in a coherent state basis ofUq[osp(2|2)], which has the same rank as Uq[gl(2|1)], are also briey exposed.

Einstein–Podolsky–Rosen-Like Correlation on a Coherent-State Basis and Inseparability of Two-Mode Gaussian States

The strange property of the Einstein–Podolsky–Rosen (EPR) correlation between two remote physical systems is a primitive object on the study of quantum entanglement. In order to understand the entanglement in canonical continuous-variable systems, a pair of the EPR-like uncertainties is an essential tool. Here, we consider a normalized pair of the EPR-like uncertainties and introduce a state-overlap to a classically correlated mixture of coherent states. The separable condition associated with this state-overlap determines the strength of the EPR-like correlation on a coherent-state basis in order that the state is entangled. We show that the coherent-state-based condition is capable of detecting the class of two-mode Gaussian entangled states. We also present an experimental measurement scheme for estimation of the state-overlap by a heterodyne measurement and a photon detection with a feedforward operation.

Infrared divergences in light-front QED and coherent state basis

We present a next to leading order calculation of electron mass renormalization in Light-Front Quantum Electrodynamics (LFQED) using old-fashioned time ordered perturbation theory (TOPT). We show that the true infrared divergences in $\delta m^2$ get canceled up to $O(e^4)$ if one uses coherent state basis instead of fock basis to calculate the transition matrix elements.

Nonadiabatic dynamics in open quantum-classical systems: Forward-backward trajectory solution

A new approximate solution to the quantum-classical Liouville equation is derived starting from the formal solution of this equation in forward-backward form. The time evolution of a mixed quantum-classical system described by this equation is obtained in a coherent state basis using the mapping representation, which expresses N quantum degrees of freedom in a 2N-dimensional phase space. The solution yields a simple dynamics in which a set of N coherent state coordinates evolves in forward and backward trajectories, while the bath coordinates evolve under the influence of the mean potential that depends on these forward and backward trajectories. It is shown that the solution satisfies the differential form of the quantum-classical Liouville equation exactly. Relations to other mixed quantum-classical and semi-classical schemes are discussed.

Energy localization in maximally entangled two- and three-qubit phase space

Motivated by the Möbius transformation for symmetric points under the generalized circle in the complex plane, the system of symmetric spin coherent states corresponding to antipodal qubit states is introduced. In terms of these states, we construct the maximally entangled complete set of two-qubit coherent states, which in the limiting cases reduces to the Bell basis. A specific property of our symmetric coherent states is that they never become unentangled for any value of ψ from the complex plane. Entanglement quantifications of our states are given by the reduced density matrix and the concurrence determinant, and it is shown that our basis is maximally entangled. Universal one- and two-qubit gates in these new coherent state basis are calculated. As an application, we find the Q symbol of the XYZ model Hamiltonian operator H as an average energy function in maximally entangled two- and three-qubit phase space. It shows regular finite-energy localized structure with specific local extremum points. The concurrence and fidelity of quantum evolution with dimerization of double periodic patterns are given.

Coherent state excitons in anisotropic quantum dots: classical and quantum correlations

We investigate the entanglement dynamics of excitons confined in two adjacent quantum dots, which we describe through their algebraic properties using in the z-direction. We use two explicit forms of coherent states: the Perelomov and Barut–Girardello states to represent the electronic component of the excitonic state in the z-direction. Our results show that in a coherent state basis, the concurrence of an excitonic-qubit pair shows subtle variations which are dependent on the algebraic parameters of the group. These variations also appear in the classical and quantum correlations, present in the qubit–qubit density matrix that is associated with the purely Förster-coupled excitonic-qubit pair. A brief discussion of a plausible experimental technique of detecting excitonic coherent states is provided.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Vortex Images, q-Calculus and Entangled Coherent States

The two circles theorem for hydrodynamic flow in annular domain bounded by two concentric circles is derived. Complex potential and velocity of the flow are represented as q-periodic functions and rewritten in terms of the Jackson q-integral. This theorem generalizes the Milne-Thomson one circle theorem and reduces to the last on in the limit q → ∞. By this theorem problem of vortex images in annular domain between coaxial cylinders is solved in terms of q-elementary functions. An infinite set of images, as symmetric points under two circles, is determined completely by poles of the q-logarithmic function, where dimensionless parameter q = r22/r21 is given by square ratio of the cylinder radii. Motivated by Möbius transformation for symmetrical points under generalized circle in complex plain, the system of symmetric spin coherent states corresponding to antipodal qubit states is introduced. By these states we construct the maximally entangled orthonormal two qubit spin coherent state basis, in the limiting case reducible to the Bell basis. Average energy of XYZ model in these states, describing finite localized structure with characteristic extremum points, appears as an energy surface in maximally entangled two qubit space. Generalizations to three and higher multiple qubits are found. We show that our entangled N qubit states are determined by set of complex Fibonacci and Lucas polynomials and corresponding Binet-Fibonacci q-calculus.

Representations of quantum superalgebra Uq[gl(2|1)] in a coherent state basis and generalization

The coherent state method has proved to be useful in quantum physics and mathematics. This method, more precisely, the vector coherent state method, has been used by some authors to construct representations of superalgebras but almost, to our knowledge, it has not yet been extended to quantum superalgebras, except $U_q[osp(1|2)]$, one of the smallest quantum superalgebras. In this article the method is applied to a bigger quantum superalgebra, namely $U_q[gl(2|1)]$, in constructing $q$--boson-fermion realizations and finite-dimensional representations which, when irreducible, are classified into typical and nontypical representations. This construction leads to a more general class of $q$--boson-fermion realizations and finite-dimensional representations of $U_q[gl(2|1)]$ and, thus, at $q=1$, of $gl(2|1)$. Both $gl(2|1)$ and $U_q[gl(2|1)]$ have found different physics applications, therefore, it is meaningful to construct their representations.

Holographic Wilsonian flows and emergent fermions in extremal charged black holes

We study holographic Wilsonian RG in a general class of asymptotically AdS backgrounds with a U(1) gauge field. We consider free charged Dirac fermions in such a background, and integrate them up to an intermediate radial distance, yielding an equivalent low energy dual field theory. The new ingredient, compared to scalars, involves a ‘generalized’ basis of coherent states which labels a particular half of the fermion components as coordinates or momenta, depending on the choice of quantization (standard or alternative). We apply this technology to explicitly compute RG flows of charged fermionic operators and their composites (double trace operators) in field theories dual to (a) pure AdS and (b) extremal charged black hole geometries. The flow diagrams and fixed points are determined explicitly. In the case of the extremal black hole, the RG flows connect two fixed points at the UV AdS boundary to two fixed points at the IR AdS2 region. The double trace flow is shown, both numerically and analytically, to develop a pole singularity in the AdS2 region at low frequency and near the Fermi momentum, which can be traced to the appearance of massless fermion modes on the low energy cut-off surface. The low energy field theory action we derive exactly agrees with the semi-holographic action proposed by Faulkner and Polchinski in [21]. In terms of field theory, the holographic version of Wilsonian RG leads to a quantum theory with random sources. In the extremal black hole background the random sources become ‘light’ in the AdS2 region near the Fermi surface and emerge as new dynamical degrees of freedom.

Elementary gates for quantum information with superposed coherent states

We propose a new way of implementing several elementary quantum gates for qubits in the coherent state basis. The operations are probabilistic and employ single photon subtractions as the driving force. Our schemes for single-qubit phase gate and two-qubit controlled phase gate are capable of achieving arbitrarily large phase shifts with currently available resources, which makes them suitable for the near-future tests of quantum information processing with superposed coherent states.

Effective classical Liouville-like evolution equation for the quantum phase-space dynamics

A model for the evolution of a quantum particle in the phase plane is derived. The quasi-distribution function is decomposed into a suitable over-complete basis of coherent states. In this approach, a coarse grain length scale is introduced by using the spread of the coherent states. The obtained evolution system is completely equivalent to the Wigner formulation of the quantum mechanics. It shows some analogy with the von Neumann approach and the particle in the cell method in classical plasma physics. Differing from the usual formulation of the quantum phase space, given in terms of infinite-order partial differential equations, in this model, the evolution equations of the second differential order are expressed by a hierarchy of coupled functions (the first term being the Husimi function). The resulting formulation reveals itself to be particularly close to the classical description of the particles motion. This formal analogy is useful to gain new physical insights and to profit from numerical methods developed for classical systems.

On Husimi Distribution for Systems with Continuous Spectrum

We propose a procedure to generalize the Husimi distribution to systems with continuous spectrum. We start examining a pioneering work, by Gazeau and Klauder, where the concept of coherent states for systems with discrete spectrum was extended to systems with continuous one. In the present article, we see the Husimi distribution as a representation of the density operator in terms of a basis of coherent states. There are other ways to obtain it, but we do not consider here. We specially discuss the problem of the continuous harmonic oscillator.

Representations of the orthosymplectic Lie superalgebra and paraboson coherent states

We introduce and obtain two-mode paraboson coherent states. In appropriate subspaces these coherent states provide a decomposition of unity where the measure, when expressed using the cat-type states, is positive definite. Bicoherent states where the mutually commuting lowering operators are diagonalized are also obtained. Matrix elements in the coherent state basis are calculated.

Qubit phase space:SU(n)coherent-statePrepresentations

We introduce a phase-space representation for qubits and spin models. The technique uses an $\mathrm{SU}(n)$ coherent-state basis and can equally be used for either static or dynamical simulations. We review previously known definitions and operator identities, and show how these can be used to define an off-diagonal, positive phase-space representation analogous to the positive-$P$ function. As an illustration of the phase-space method, we use the example of the Ising model, which has exact solutions for the finite-temperature canonical ensemble in two dimensions. We show how a canonical ensemble for an Ising model of arbitrary structure can be efficiently simulated using SU(2) or atomic coherent states. The technique utilizes a transformation from a canonical (imaginary-time) weighted simulation to an equivalent unweighted real-time simulation. The results are compared to the exactly soluble two-dimensional case. We note that Ising models in one, two, or three dimensions are potentially achievable experimentally as a lattice gas of ultracold atoms in optical lattices. The technique is not restricted to canonical ensembles or to Ising-like couplings. It is also able to be used for real-time evolution and for systems whose time evolution follows a master equation describing decoherence and coupling to external reservoirs. The case of $\mathrm{SU}(n)$ phase space is used to describe $n$-level systems. In general, the requirement that time evolution be stochastic corresponds to a restriction to Hamiltonians and master equations that are quadratic in the group generators or generalized spin operators.

Husimi distribution for the linear rigid rotator

We find the Husimi distribution for the linear rigid rotator. The Husimi function can be viewed in a variety of ways. In particular, we obtain it as the expectation value of the density operator in a suitable basis of coherent states. We employ the Schwinger's oscillator model for the construction of the angular momentum coherent states.

Bosonization, coherent states and semiclassical quantum Hall skyrmions

We bosonize (2+1)-dimensional fermionic theory using coherent states. The gauge-invariant subspace ofboson–Chern–Simons Hilbert space is mapped to fermionic Hilbert space. This subspace isthen equipped with a coherent state basis. These coherent states are labelled by a dynamicspinor field. The label manifold could be assigned a physical meaning in terms of densityand spin density. A path-integral representation of the evolution operator in terms of thesephysical variables is given. The corresponding classical theory when restricted to LLL isdescribed by spin fluctuations alone and is found to be the NLSM with Hopf term. Theformalism developed here is suitable to study quantum Hall skyrmions semiclassicallyand/or beyond the hydrodynamic limit. The effects of Landau level mixing orthe presence of slowly varying external fields can also be easily incorporated.

A charged particle in a magnetic field: a review of two formalisms of coherent states and the Husimi function

In this work, we review two formalisms of coherent states for the case of a particle in a magnetic field. We focus our revision on both pioneering (Feldman and Kahn 1970 Phys. Rev. B 1 4584) and recent (Kowalski and Rembieliński 2005 J. Phys. A: Math. Gen. 38 8247) formulations of coherent states for this problem. We introduce a general application, the Husimi function, which takes into account collective and environmental effects, and which can be derived in a variety of ways. However, we calculate this function as the expected value of the density operator in a basis of coherent states.

On Bargmann representations of the Wigner function

By using the localized character of canonical coherent states, we give a straightforward derivation of the Bargmann integral representation of the Wigner function (W). A non-integral representation is presented in terms of a quadratic form W ∝ V†FV, where F is a self-adjoint matrix whose entries are tabulated functions and V is a vector depending, in a simple recursive way, on the derivatives of the Bargmann function. Such a representation may be of use in numerical computations. We discuss a relation involving the geometry of the Wigner function and the spatial uncertainty of the coherent state basis we use to represent it.

Fisher information, delocalization and the semiclassical description of molecular rotation

We discuss phase-space delocalization for the rigid rotor within a semiclassical context by recourse to the Husimi distributions of both the linear and the 3D-anisotropic instances. The Husimi function can be viewed in a variety of ways. In particular, we obtain it as the expectation value of the density operator in a suitable basis of coherent states. Our treatment is based upon the concomitant Fisher information measures. The pertinent Wehrl entropy is also investigated in the linear case.