Set Of Coherent States Research Articles

The overlap of generalized coherent states of any rank one simple Lie algebra and its classical limit

We provide a formula for computing the overlap between two generalized coherent states of any rank one simple Lie algebra. Then, we apply our formula to spin coherent states (i.e. su(2) algebra), pseudo-spin coherent states (i.e. su(1,1) algebra), and the sl(2,R) subalgebras of Virasoro. In all these examples, we show the emergence of a semi-classical behavior from the set of coherent states and verify that it always happens when some parameter, depending on the algebra and its representation, becomes large.

Experimental quantum tomography of a homodyne detector

We suggest and demonstrate a tomographic method to characterise homodyne detectors at the quantum level. The positive operator measure associated with the detector is expanded in a quadrature basis and probed with a set of coherent states. The coefficients of the expansion are then retrieved using a least squares algorithm. Our model is general enough to describe different implementations of the homodyne setup, and it has proven capable of effectively describing the detector response to different tomographic sets. We validate the reconstructed operator measure on nonclassical states and exploit results to estimate the overall quantum efficiency of the detector.

Experimental implementation of a quantum optical state comparison amplifier.

We present an experimental demonstration of a practical nondeterministic quantum optical amplification scheme that employs two mature technologies, state comparison and photon subtraction, to achieve amplification of known sets of coherent states with high fidelity. The amplifier uses coherent states as a resource rather than single photons, which allows for a relatively simple light source, such as a diode laser, providing an increased rate of amplification. The amplifier is not restricted to low amplitude states. With respect to the two key parameters, fidelity and the amplified state production rate, we demonstrate significant improvements over previous experimental implementations, without the requirement of complex photonic components. Such a system may form the basis of trusted quantum repeaters in nonentanglement-based quantum communications systems with known phase alphabets, such as quantum key distribution or quantum digital signatures.

Interaction between the Kravchuck and Meixner oscillators coherent states and the coherent radiation field

To describe a series of phenomena that occur in micro- and nano-systems, we have used the discrete variable Kravchuk and Meixner oscillators for which some sets of coherent states and some squeezed states were established. For a system consisting of molecular oscillators interacting with the coherent radiation field, an oscillatory variation of the refractive index was established.

Coherent States Quantization for Generalized Bargmann Spaces with Formulae for Their Attached Berezin Transforms in Terms of the Laplacian on ℂ N

While dealing with a class of generalized Bargmann spaces, we rederive their reproducing kernels from the knowledge of an orthonormal basis by using an addition formula for Laguerre polynomials involving the disk polynomials. We construct for each of these spaces a set of coherent states to apply a coherent states quantization method. This provides us with another way to recover the Berezin transforms attached to these spaces. Finally, two new formulae representing these transforms as functions of the Euclidean Laplacian are established and a possible physics direction for the application of such formulae is discussed.

U(N) coherent states for loop quantum gravity

We investigate the geometry of the space of N-valent SU(2) intertwiners. We propose a new set of holomorphic operators acting on this space and a new set of coherent states which are covariant under U(N) transformations. These states are labeled by elements of the Grassmannian GrN, 2, they possess a direct geometrical interpretation in terms of framed polyhedra and are shown to be related to the well-known coherent intertwiners.

Entanglement and coherent states

We outline the generalization of our earlier result (Phys. Lett. A 372 3792–5) on the connection between coherent states and pure state entanglement given there for the symmetric part of . We show that the set of coherent states in the symmetric subspace of is identical to the set of unentangled states for any K. We point out that the deeper reason for this theorem lies in the fact that coherent states result from a set of local unitary transforms of the so-called lowest weight state that is obviously unentangled.

Complete Characterization of Quantum-Optical Processes

The technologies of quantum information and quantum control are rapidly improving, but full exploitation of their capabilities requires complete characterization and assessment of processes that occur within quantum devices. We present a method for characterizing, with arbitrarily high accuracy, any quantum optical process. Our protocol recovers complete knowledge of the process by studying, via homodyne tomography, its effect on a set of coherent states, that is, classical fields produced by common laser sources. We demonstrate the capability of our protocol by evaluating and experimentally verifying the effect of a test process on squeezed vacuum.

Vanishing of electron pair recession at central impact

Identity of electrons leads to description of their states by symmetrical or antisymmetrical combinations of free coherent states. Due to the coordinate uncertainty potential energy of the Coulomb repulsing, is limited from above and so when energy of electrons is large enough, electrons go through each other without noticing one another. We show existence of set of coherent states for which wave packages recession vanish—electrons remain close regardless of Coulomb repulsion.

Improving information/disturbance and estimation/distortion trade-offs with non-universal protocols

We analyse in detail a conditional measurement scheme based on linear optical components, feed-forward loop and homodyne detection. The scheme may be used to achieve two different tasks. On the one hand it allows the extraction of information with minimum disturbance about a set of coherent states. On the other hand, it represents a nondemolitive measurement scheme for the annihilation operator, i.e. an indirect measurement of the Q-function. We investigate the information/disturbance trade-off for state inference and introduce the estimation/distortion trade-off to assess estimation of the Q-function. For coherent states chosen from a Gaussian set, we evaluate both information/disturbance and estimation/distortion trade-offs and found that non-universal protocols may be optimized in order to achieve better performances than universal ones. For Fock number states we prove that universal protocols do not exist and evaluate the estimation/distortion trade-off for a thermal distribution.

Method of Asymptotic Dynamics in Light-Front Field Theory

We discuss the recently proposed modified approach to asymptotic dynamics in the context of light-front field theories and apply it to light-front quantum electrodynamics (LFQED) and light-front quantum chromodynamics (LFQCD). We show that in the case of LFQED, the improved method leads to the same set of coherent states as the one obtained earlier by us using the standard Kulish and Faddeev (KF) method. However, in the case of LFQCD, the improved method leads to a set of states different from the ones obtained by using the KF method.

The completeness of sequences of generalized coherent states

Analytic representations based on generalized coherent states are studied. The growth of the analytic functions is intimately connected to the completeness of the sequences of these generalized coherent states. The least density that such sequences must have in order to be overcomplete is calculated. The results generalize known results on the completeness of von Neumann lattices for the standard coherent states to other sets of coherent states.

Classical and Quantum Coherent States

p-Mechanics is a consistent physical theory which describes both quantum and classical mechanics simultaneously (V. V. Kisil, p-Mechanics as a physical theory. An Introduction, E-print:arXiv:quant-ph/0212101, 2002; International Journal of Theoretical Physics41(1), 63–77, 2002). We continue the development of p-mechanics by introducing the concept of states. The set of coherent states we introduce allows us to evaluate classical observables at any point of phase space and simultaneously to evaluate quantum probability amplitudes. The example of the forced harmonic oscillator is used to demonstrate these concepts.

Decomposition of Hilbert space in sets of coherent states

Within the generalized definition of coherent states as group orbits we study the orbit spaces and the orbit manifolds in the projective spaces constructed from linear representations. Invariant functions are suggested for arbitrary groups. The group SU(2) is studied in particular and the orbit spaces of its j=1/2 and j=1 representations completely determined. The orbits of SU(2) in CP^N can be either 2 or 3 dimensional, the first of them being either isomorphic to S^2 or to RP^2 and the latter being isomorphic to quotient spaces of RP^3. We end with a look from the same perspective to the quantum mechanical space of states in particle mechanics.

Uncertainty for spin systems

A modified definition of quantum mechanical uncertainty Δ for spin systems, which is invariant under the action of SU(2), is suggested. Its range is shown to be ℏ2j⩽Δ⩽ℏ2j(j+1) within any irreducible representation j of SU(2) and its mean value in Hilbert space computed using the Fubini–Study metric is determined to be mean(Δ)=ℏ2j(j+1/2). The most used sets of coherent states in spin systems coincide with the set of minimum Δ uncertainty states.

Generalized coherent states forSU(n) systems

Generalized coherent states are developed for SU(n) systems for arbitrary $n$. This is done by first iteratively determining explicit representations for the SU(n) coherent states, and then determining parametric representations useful for applications. For SU(n), the set of coherent states is isomorphic to a coset space $SU(n)/SU(n-1)$, and thus shows the geometrical structure of the coset space. These results provide a convenient $(2n - 1)$--dimensional space for the description of arbitrary SU(n) systems. We further obtain the metric and measure on the coset space, and show some properties of the SU(n) coherent states.

Hilbert-Schmidt distance and non-classicality of states in quantum optics

The Hilbert—Schmidt distance between two arbitrary normalizable states is discussed as a measure of the similarity of the states. Unitary transformations of both states with the same unitary operator (e.g. the displacement of both states in the phase plane by the same displacement vector and squeezing of both states) do not change this distance. The nearest distance of a given state to the whole set of coherent states is proposed as a quantitative measure of non-classicality of the state which is identical when considering the coherent states as the most classical ones among pure states and the deviations from them as non-classicality. The connection to other definitions of the non-classicality of states is discussed. The notion of distance can also be used for the definition of a neighbourhood of considered states. Inequalities for the distance of states to Fock states are derived. For given neighbourhoods, they restrict common characteristics of the state as the dispersion of the number operator and the squared deviation of the mean values of the number operator for the considered state and the Fock state. Possible modifications in the definition of non-classicality for mixed states with dependence on the impurity parameter and by including the displaced thermal states as the most classical reference states are discussed.

Generalized Phase-Space Representatives of Spin- J Operators in Terms of Bloch Coherent States

The c-number functions corresponding to the matrix elements and to the diagonal representative of the elements of the spin- J representation of the rotation group in a Bloch coherent state basis are obtained in closed form. The matrix element and the diagonal representative of the spin- J representation are simply the power 2 J and 2( J + 1)/3 of those of the spin- 1 2 case, respectively. We obtain the representatives of any spin operator by differentiation of the properly ordered exponential operators. Our methods simplify the calculations, avoiding the introduction of Schwinger bosons and the consequent projection to the physical subspace. In particular, we obtain several product theorems for the representatives of two or more operators. Use of the rotated spherical basis vectors allows us to write the representatives of the cumulants of spin operators in a compact form. In particular, the lowest three take a factorized form. Introducing the holomorphic representation, we obtain the diagonal representative from the diagonal matrix element as an integral relation without recurring to expansions in spherical harmonics. Introducing a generalized Fourier transformation in spin- J space in a way similar to the treatment by Cahill and Glauber of the boson coherent states, new analogies between the two sets of coherent states are established. Using its inversion properties we obtain in closed form relations between the c-number functions which represent an operator in spin- J space. Also,we study the generalization to the spin case of the s-ordering of Cahill and Glauber interpolating between the diagonal matrix element and the diagonal representative.

Incompleteness of Sparse Coherent States

A phase space shift of a square integrable function φ ∈ L 2( R n ) is defined by ρ( p, q)φ( x) ≔ e 2π iqx φ( x + p). Let Λ ⊂ R 2 n be a discrete subset and S Λ,φ = {ρ( p, q)φ : ( p, q) ∈ Λ}. The issue of when an arbitrary function ƒ ∈ L 2( R n ) can be expanded in terms of a particular set of coherent states, S Λ,φ arises in various subjects such as quantum mechanics and signal processing. Special cases lead to the intuition that the possibility of such a general expansion is highly dependent on the density of the set Λ. In this paper, we give rigorous results about this relationship. Our approach provides an elementary proof of a result of Rieffel as well as an extension of work of Landau. Our final result states that if S Λ,φ is a Riesz basis for L 2, then the asymptotic density of Λ is exactly one.

Quantization Without Quantization

Appropriate to a single classical degree of freedom, completely general, two-parameter "coherent states" are defined without the use of any group or any c-number to q-number transformation and in such a way that the classical phase space inherits a generically symmetry-free geometry. A coherent-state path integral provides a quantization prescription for a classical system that gives a new dimension to the generality of such a procedure. In addition, sets of coherent states may be designed so that each set is invariant under time evolution by a Hamiltonian chosen from an infinite set of distinct Hamiltonian operators. The extension of these ideas to multiple degree-of-freedom systems is indicated.