Number-phase Uncertainty Relation Research Articles

Number-phase uncertainty relations and bipartite entanglement detection in spin ensembles

We present a method to detect bipartite entanglement based on number-phase-like uncertainty relations in split spin ensembles. First, we derive an uncertainty relation that plays the role of a number-phase uncertainty for spin systems. It is important that the relation is given with well-defined and easily measurable quantities, and that it does not need assuming infinite dimensional systems. Based on this uncertainty relation, we show how to detect bipartite entanglement in an unpolarized Dicke state of many spin-1/2 particles. The particles are split into two subensembles, then collective angular momentum measurements are carried out locally on the two parts. First, we present a bipartite Einstein-Podolsky-Rosen (EPR) steering criterion. Then, we present an entanglement condition that can detect bipartite entanglement in such systems. We demonstrate the utility of the criteria by applying them to a recent experiment given in K. Lange et al. [Science 360, 416 (2018)] realizing a Dicke state in a Bose-Einstein condensate of cold atoms, in which the two subensembles were spatially separated from each other. Our methods also work well if split spin-squeezed states are considered. We show in a comprehensive way how to handle experimental imperfections, such as the nonzero particle number variance including the partition noise, and the fact that, while ideally BECs occupy a single spatial mode, in practice the population of other spatial modes cannot be fully suppressed.

Intelligent states for a number-operator–annihilation-operator uncertainty relation

Recently a new uncertainty relation was found as an alternative to a number-phase uncertainty relation for a harmonic oscillator. In this paper we determine numerically, via the discrete-variable-representation method known from quantum chemistry, the exact states that saturate this new uncertainty relation. We analyze the physical properties of the states and compare them to the intelligent states of the Pegg-Barnett uncertainty relation. We find that for a given set of expectation values of the physical parameters, which are the particle number and the two quadratures, the two kinds of intelligent states are equivalent. The intelligent states are the eigenstates corresponding to the lowest eigenvalue of a Hermitian operator, which, when interpreted as a Hamiltonian of a physical sytem, describes a nonlinear driven harmonic oscillator, for example, a Duffing oscillator for a certain parameter range. Hence, our results can be interpreted as the determination of the ground state of such physical systems in an explicit analytic form as well. As the Pegg-Barnett intelligent states we use are expressed in terms of a coherent-state superposition facilitating experimental synthesis, we relate the states determined here to experimentally feasible ones.

Number-phase entropic squeezing and nonclassical properties of a three-level atom interacting with a two-mode field: intensity-dependent coupling, deformed Kerr medium, and detuning effects

In this paper, we follow the model presented in our previous work [J. Opt. Soc. Am. B30, 1109 (2013)], in which the interaction between a Λ-type three-level atom and a quantized two-mode radiation field in a cavity in the presence of nonlinearities was studied. After giving a brief review of the procedure for obtaining the state vector of the atom-field system, we investigated some further interesting and important physical features of the whole system state, which are of particular interest to the quantum optics field of research, i.e., the number-phase entropic uncertainty relation (based on the two-mode Pegg–Barnett formalism) and a few of the nonclassicality signs that consist of sub-Poissonian statistics, Cauchy–Schwartz inequality, and two types of squeezing phenomena. During our presentation, the effects of intensity-dependent coupling, deformed Kerr medium, and the detuning parameters on the depth and domain of each of the mentioned nonclassical criteria of the considered quantum system are described in detail. The results show that each of the mentioned nonclassicality aspects can be obtained by appropriately choosing the related parameters.

Number-phase uncertainty relations in terms of generalized entropies

Number-phase uncertainty relations are formulated in terms of unified entropies which form a family of two-parametric extensions of the Shannon entropy. For two generalized measurements, unified-entropy uncertainty relations are given in both the state-dependent and state-independent forms. A few examples are discussed as well. Using the Pegg--Barnett formalism and the Riesz theorem, we obtain a nontrivial inequality between norm-like functionals of generated probability distributions in finite dimensions. The principal point is that we take the infinite-dimensional limit right for this inequality. Hence number-phase uncertainty relations with finite phase resolutions are expressed in terms of the unified entropies. Especially important case of multiphoton coherent states is separately considered. We also give some entropic bounds in which the corresponding integrals of probability density functions are involved.

Chirality and angular momentum in optical radiation

This paper develops, in precise quantum electrodynamic terms, photonic attributes of the "optical chirality density", one of several measures long known to be conserved quantities for a vacuum electromagnetic field. The analysis lends insights into some recent interpretations of chiroptical experiments, in which this measure, and an associated chirality flux, have been treated as representing physically distinctive "superchiral" phenomena. In the fully quantized formalism the chirality density is promoted to operator status, whose exploration with reference to an arbitrary polarization basis reveals relationships to optical angular momentum and helicity operators. Analyzing multi-mode beams with complex wave-front structures, notably Laguerre-Gaussian modes, affords a deeper understanding of the interplay between optical chirality and optical angular momentum. By developing theory with due cognizance of the photonic character of light, it emerges that only the spin angular momentum of light is engaged in such observations. Furthermore, it is shown that these prominent measures of the helicity of chiral electromagnetic radiation have a common basis, in differences between the populations of optical modes associated with angular momenta of opposite sign. Using a calculation of the rate of circular dichroism as an example, with coherent states to model the electromagnetic field, it is discovered that two terms contribute to the differential effect. The primary contribution relates to the difference in left- and right- handed photon populations; the only other contribution, which displays a sinusoidal distance-dependence, corresponding to the claim of nodal enhancements, is connected with the quantum photon number-phase uncertainty relation. From the full analysis, it appears that the term "superchiral" can be considered redundant.

Photon statistics, parity measurements, and Heisenberg-limited interferometry: example of the two-mode SU(1,1) SU(1,1) coherent states

Parity measurement-based quantum optical interferometry and metrology is known to be sensitive to the joint photon number distribution, , of the two-mode quantum states inside the interferometer. In cases where the statistics display a clustering of the populations around the borders (i.e. along the axes ) with a plateau over most of the interior, Heisenberg-limited (or super-sensitive) measurements of phase shifts are enabled. This follows from the heuristic number-phase uncertainty relation because of the uncertainty in the location of the photons being ΔN ∼ . As an example of the connection between the photon statistics and Heisenberg-limited phase uncertainty, we study parity measurement-based interferometry of the two-mode Barut–Girardello coherent states for SU(1,1)SU(1,1) obtained from the coupling together of two representations of SU(1,1) each associated with single-mode boson realizations with Bargmann indices as discussed by Gerry and Benmoussa [Phys. Rev. A 2000, 62, 033812]. These states generally have distributions with clustered populations along the borders. We demonstrate that these states do, indeed, lead to super-sensitive measurements and also lead to super-resolved interference fringes.

Generalized coherent states for solvable quantum systems with degenerate discrete spectra and their nonclassical properties

In this paper, the generalized coherent state for quantum systems with degenerate spectra is introduced. Then, the nonclassicality features and number-phase entropic uncertainty relation of two particular degenerate quantum systems are studied. Finally, using the Gazeau–Klauder coherent states approach, the time evolution of some of the nonclassical properties of the coherent states corresponding to the considered physical systems are discussed.

Number-operator–annihilation-operator uncertainty as an alternative for the number-phase uncertainty relation

We consider a number operator-annihilation operator uncertainty as a well behaved alternative to the number-phase uncertainty relation, and examine its properties. We find a formulation in which the bound on the product of uncertainties depends on the expectation value of the particle number. Thus, while the bound is not a constant, it is a quantity that can easily be controlled in many systems. The uncertainty relation is approximately saturated by number-phase intelligent states. This allows us to define amplitude squeezing, connecting coherent states to Fock states, without a reference to a phase operator. We propose several setups for an experimental verification.

Number-phase entropic uncertainty relations and Wigner functions for solvable quantum systems with discrete spectra

In this Letter, the “number-phase entropic uncertainty relation” and the “number-phase Wigner function” of generalized coherent states associated to a few solvable quantum systems with non-degenerate spectra are studied. We also investigate time evolution of “number-phase entropic uncertainty” and “Wigner function” of the considered physical systems with the help of temporally stable Gazeau–Klauder coherent states.

Cooper-Pair Number-Phase Quantization for Inductance Coupling Circuit Including Josephson Junctions

Based on the entangled state representation and a bosonic phase operator formalism, we tackle with Cooper-pair number-phase quantization for the inductance coupling circuit including Josephson junctions, and then investigate how Josephson current equations change due to the presence of the coupling inductance and obtain bosonic operator Faraday formula, as well as the corresponding number-phase uncertainty relation.

Number-Phase Uncertainty Relation

Number-phase uncertainty relation is examined on the basis of an orthogonal phase state and a quasi-probability function. The measure of number fluctuation is not the usual standard deviation but a kind of Fisher information. This is a direct generalization to a discrete system of the corresponding quantity used recently by Hall in a theory of uncertainty for a coordinate-momentum system. It is essential to separate the “classical” part of a phase operator with the aid of the orthogonal phase state.

Nonclassical statistical properties of finite-coherent states in the framework of the Jaynes–Cummings model

Adopting the framework of the nonresonant Jaynes–Cummings model, we investigate the nonclassical statistical properties of coherent states defined in a finite-dimensional Hilbert space, considering the single-mode cavity field prepared in a finite and discrete harmonic oscillator-like coherent state with a small average number of photons. Explicit expressions for the time evolution of various functions characterizing the quantum state, such as the Mandel's Q parameter, the photon-number distribution and its respective entropy, the discrete Wigner function, and the number-phase uncertainty relation, are investigated in detail. We also show that the atomic inversion possesses regular structures with collapses and revivals in the Rabi oscillations since the detuning between atom and field is large as compared to the coupling constant, i.e., ( κ/2 g) 2⪢1. The numerical and analytical results obtained in this work turn evident the quantum interference effects between the components of the finite-coherent states. Furthermore, we present a discussion about unitary depolarizers in finite-dimensional Hilbert space and their connection to quantum information theory.

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.

Entropic uncertainty relations for nonextensive quantum scattering

In this paper the angle-angular momentum entropic uncertainty relations are obtained for Tsallis-like entropies for nonextensive quantum scattering of spinless particles. The number-phase entropic uncertainty relations are also proved for nonextensive quantum scattering. Numerical results on the experimental tests of these entropic uncertainty relations, for the nonextensive ( q≠1) statistics case are obtained by calculations of Tsallis-like scattering entropies from the 48 experimental sets of the pion-nucleus phase shifts.

Phase properties of even and odd nonlinear coherent states

Using the Pegg–Barnett formalism we obtain phase probability distributions of the even and odd nonlinear coherent states. These distributions are then used to examine whether or not the even and odd nonlinear coherent states exhibit number/phase squeezing. We also examine whether these states are intelligent states with respect to the number-phase uncertainty relation.

Q-phase-coherent states and their squeezing properties

Some kinds of q-phase-coherent states of a q-harmonic osscillator in a finite-dimensional Hilbert space are constructed. Some properties of these states are discussed. Second-order squeezing properties of these states with respect to the phase quadrature operators are studied. The number-phase squeezing and number-phase uncertainty relations are also studied in detail for a two-state system. Some new number-phase minimum uncertainty states are found.

Number-phase uncertainty relations: Verification by balanced homodyne measurement

It is shown that fundamental uncertainty relations between photon number and canonical phase of a single-mode optical field can be verified by means of balanced homodyne measurement. All the relevant quantities can be sampled directly from the measured phase-dependent quadrature distribution.

Coherent States and Number-Phase Uncertainty Relations

The number-phase uncertainty relations arerevisited in view of the recent discovery of a propercovariant phase observable. The high-amplitude limits ofthe coherent-state expectations of the moment operators of the phase observable are determined and thebehavior of the number-phase uncertainty product in thatlimit is investigated.

Quantum phase and dynamical properties of light in the Jaynes-Cummings model via phase coupling

It is pointed out here that quantum phase properties, as well as dynamical properties, of light exhibit a sensitivity to the relative phase between the atomic dipole and light when the atom is prepared in a coherent superposition of its states in the two-level Jaynes-Cummings model interacting with a coherent state of light. The existence of the coherent-trapping of light is demonstrated by looking not only at the variance of the light phase about the coherent state value, and the number-phase uncertainty relation, but also at the factorization of the electric-field operator, and the photon-number distribution. We find also the factorization expression of the compound state of the atom-field system. We show once more that the phase properties of light based on the phase operator formalism of Pegg-Barnett and the coherence properties of light based on the coherence theory of Glauber are complementary in description of the optical quantum coherence.

Even and Odd Phase Coherent States Associated with the Pegg-Barnett Hermitian Phase Theory

Even and odd phase coherent states associated with the Pegg-Barnett Hermitian phase theory are introduced. Even and odd coherent-state formalism of the Pegg-Barnett phase operator is given in terms of the projection operator in the even and odd coherent-state space. The number-phase uncertainty relation is investigated for these states.