(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

Functional equations in several variables

Preface 1. Planck's radiation law and the Einstein coefficients 2. Quantum mechanics of the atom-radiation interaction 3. Classical theory of optical fluctuations and coherence 4. Quantization of the radiation field 5. Single-mode quantum optics 6. Multimode and continuous-mode quantum optics 7. Optical generation, attenuation and amplification 8. Resonance fluorescence and light scattering 9. Nonlinear quantum optics Index

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

In two recent articles we have examined a generalization of the binomial distribution associated with a sequence of positive numbers, involving asymmetric expressions of probabilities that break the symmetry {\it win-loss}. We present in this article another generalization (always associated with a sequence of positive numbers) that preserves the symmetry {\it win-loss}. This approach is also based on generating functions and presents constraints of non-negativeness, similar to those encountered in our previous articles.

Symmetric generalized binomial distributions

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.

Noncommutative reading of the complex plane through Delone sequences

In a recent article generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal probability distributions was a key point to allow to give them a statistical interpretation in terms of probabilities. In this article we present an approach based on generating functions that solves the previous difficulties. Our main theorem makes explicit the conditions under which those formal probability distributions are always non-negative. Therefore, the constraints of non-negativeness are automatically fulfilled giving a complete characterization in terms of generating functions. A large number of analytical examples becomes available.

Generating functions for generalized binomial distributions

Part of the difficulties in implementing communication in quantum information stems from the fragility of Schrodinger's cat-like superpositions. A recent experiment in quantum optics by Cook et al (2007 Nature 446 774) has proved the feasibility of a feedback-mediated quantum measurement for discriminating between optical coherent states under photodetection. Minimizing the error in receiver measurement over all possible POVMs leads to the so-called quantum error probability or 'Helstrom bound', and CMG measurements validate the theoretical prediction by Helstrom, Dolinar and Geremia concerning this bound. In this work, we present some preliminary theoretical and numerical explorations concerning the properties of the Helstrom bound in binary (or multibinary) communication involving non-Poissonian or nonlinear coherent states.

Nonlinear coherent states for optimizing quantum information

We examine a generalization of the binomial distribution associated with a strictly increasing sequence of numbers and we prove its Poisson-like limit. Such generalizations might be found in quantum optics with imperfect detection. We discuss under which conditions this distribution can have a probabilistic interpretation.

Ligia M. C. S. Rodrigues

Papers

Symmetric generalized binomial distributions

Noncommutative reading of the complex plane through Delone sequences

Generating functions for generalized binomial distributions

Nonlinear coherent states for optimizing quantum information

On a Generalization of the Binomial Distribution and Its Poisson-like Limit