(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

An Introduction to Noncommutative Differential Geometry and its Physical Applications: Extensions of Space-Time

We show how the dependence of phase space volume $\Omega(N)$ of a classical system on its size $N$ uniquely determines its extensive entropy. We give a concise criterion when this entropy is not of Boltzmann-Gibbs type but has to assume a {\em generalized} (non-additive) form. We show that generalized entropies can only exist when the dynamically (statistically) relevant fraction of degrees of freedom in the system vanishes in the thermodynamic limit. These are systems where the bulk of the degrees of freedom is frozen and is practically statistically inactive. Systems governed by generalized entropies are therefore systems whose phase space volume effectively collapses to a lower-dimensional 'surface'. We explicitly illustrate the situation for binomial processes and argue that generalized entropies could be relevant for self organized critical systems such as sand piles, for spin systems which form meta-structures such as vortices, domains, instantons, etc., and for problems associated with anomalous diffusion.

When do generalized entropies apply? How phase space volume determines entropy

It is known that the anti-Wick (or standard coherent state) quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator (up to the addition of a constant). In this work, we show that these two issues are not necessarily coupled: there exists a family of separable Hilbert spaces, including the usual Fock-Bargmann space, and in each element in this family there exists an overcomplete set of unit-norm states resolving the unity. With the exception of the Fock-Bargmann case, they all produce non-canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant. The statistical aspects of these non-equivalent coherent state quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.

https://iopscience.iop.org/article/10.1088/1751-8113/43/30/305304/pdf

Complex and real Hermite polynomials and related quantizations

In this paper, we investigate two different entanglement measures, the negativity and concurrence, in the case of pure and mixed states of two-qubit system basing on the spin coherent states. For two-qubit pure states, the negativity is the same as the concurrence. For mixed states, using a simplified expression of concurrence in Wootters' measure of entanglement, we write the bounds of Verstraete et al.1 as a function of some new parameters and we compare the both measures for a class of mixed states.

a Comparative Study of Negativity and Concurrence Based on Spin Coherent States

We construct a simple time stable two-parametric coherent states (CS) for the hydrogen atom, which is a system that presents degenerate levels. Analyzing the position variance for these CS, we are able to identify regions in the parameter space where the system behaves closer to a classical system.

Coherent states for a degenerate system: The hydrogen atom

Generalized Heisenberg algebra and algebraic method: The example of an infinite square-well potential

We propose a deformed version of the generalized Heisenberg algebra by using techniques borrowed from the theory of pseudo-bosons. In particular, this analysis is relevant when non self-adjoint Hamiltonians are needed to describe a given physical system. We also discuss relations with nonlinear pseudo-bosons. Several examples are discussed.

Generalized Heisenberg algebra and (non linear) pseudo-bosons

We present a list of formulae useful for Weyl-Heisenberg integral quantizations, with arbitrary weight, of functions or distributions on the plane. Most of these formulae are known, others are original. The list encompasses particular cases like Weyl-Wigner quantization (constant weight) and coherent states (CS) or Berezin quantization (Gaussian weight). The formulae are given with implicit assumptions on their validity on appropriate space(s) of functions (or distributions). One of the aims of the document is to accompany a work in progress on Weyl-Heisenberg integral quantization of dynamics for the motion of a point particle on the line.

Weyl-Heisenberg integral quantization(s): a compendium

The Berezin–Klauder–Toeplitz (“anti-Wick”) quantization or “noncommutative reading” 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 the natural numbers. This work is an attempt for following the same path by considering sequences of non-negative numbers which are not “too far” from the natural numbers. In particular, we examine the consequences of such perturbations on the noncommutative reading of the complex plane in terms of its probabilistic, functional, and localization aspects.

E. M. F. Curado

Papers

Coherent states for a degenerate system: The hydrogen atom

Generalized Heisenberg algebra and algebraic method: The example of an infinite square-well potential

Generalized Heisenberg algebra and (non linear) pseudo-bosons

Weyl-Heisenberg integral quantization(s): a compendium

Noncommutative reading of the complex plane through Delone sequences