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probabilistic content of the coherent states and describe their links with three important quantum distributions, namely, the P , Q distribution and the Wigner distribution Appendix A is devoted to probabilities and will also help the reader grasp these essential aspects Finally, we indicate the way in which coherent states naturally occur in the Feynman path integral formulation of quantum mechanics In more mathematical language, we tentatively explain in intelligible terms the coherent state properties such as (CS1) (CS4) and others characterizing on a mathematical level the standard coherent states
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4 Coherent States in Quantum Information 4 gives an account of a recent experimental evidence of a feedback-mediated quantum measurement aimed at discriminating between optical coherent states under photodetection The description of the experiment and of its theoretical motivations is aimed at counterbalancing the abstract character of the mathematical formalism presented in the previous two chapters 5 Coherent States: a General Construction In 5 we go back to the formalism by presenting a general method of construction of coherent states, starting from some observations on the structure of coherent states as superpositions of number states Given a set X, equipped with a measure and the resulting Hilbert space L2 (X , ) of square-integrable functions on X, we explain how the choice of an orthonormal system of functions in L2 (X , ), precisely { j (x) | j index set J }, X j (x) j (x) (dx) = j j , carry2 ing a probabilistic content, j J | j (x)| = 1, determines the family of coherent
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states |x = j j (x)| j The relation to the underlying existence of a reproducing kernel space will be clari ed This coherent state construction is the main guideline ruling the content of the subsequent chapters concerning each family of coherent states examined (in a generalized sense) As an elementary illustration of the method, we present the coherent states for the quantum motion of a particle on the circle
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6 Spin Coherent States 6 is devoted to the second most known family of coherent states, namely, the so-called spin or Bloch or atomic coherent states The way of obtaining them follows the previous construction Once they have been made explicit, we describe their main properties: that is, we depict and comment on the sequence of properties like we did in the third chapter, the link with SU (2) representations, their classical aspects, and so on 7 Selected Pieces of Applications of Standard and Bloch Coherent States In 7 we proceed to a (small, but instructive) panorama of applications of the standard coherent states and spin coherent states in some problems encountered in physics, quantum physics, statistical physics, and so on The selected pa-
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1 Introduction
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pers that are presented as examples, despite their ancient publication, were chosen by virtue of their high pedagogical and illustrative content
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Application to the Driven Oscillator This is a simple and very pedagogical model for which the Weyl Heisenberg displacement operator de ning standard coherent states is identi ed with the S matrix connecting ingoing and outgoing states of a driven oscillator Application in Statistical Physics: Superradiance This is another nice example of application of the coherent state formalism The object pertains to atomic physics: two-level atoms in resonant interaction with a radiation eld (Dicke model and superradiance) Application to Quantum Magnetism We explain how the spin coherent states can be used to solve exactly or approximately the Schr dinger equation for some systems, such as a spin interacting with a variable magnetic eld Classical and Thermodynamical Limits Coherent states are useful in thermodynamics For instance, we establish a representation of the partition function for systems of quantum spins in terms of coherent states After introducing the so-called Berezin Lieb inequalities, we show how that coherent state representation makes crossed studies of classical and thermodynamical limits easier 8 SU(1, 1), SL(2, R), and Sp(2, R) Coherent States 8 is devoted to the third most known family of coherent states, namely, the SU (1, 1) Perelomov and Barut Girardello coherent states Again, the way of obtaining them follows the construction presented in 5 We then describe the main properties of these coherent states: probabilistic interpretation, link with SU (1, 1) representations, classical aspects, and so on We also show the relationship between wavelet analysis and the coherent states that emerge from the unitary irreducible representations of the af ne group of the real line viewed as a subgroup of SL(2, R) ~ SU (1, 1) 9 SU(1, 1) Coherent States and the In nite Square Well In 9 we describe a direct illustration of the SU (1, 1) Barut Girardello coherent states, namely, the example of a particle trapped in an in nite square well and also in P schl Teller potentials of the trigonometric type 10 SU(1, 1) Coherent States and Squeezed States in Quantum Optics 10 is an introduction to the squeezed coherent states by insisting on their relations with the unitary irreducible representations of the symplectic groups S p(2, R) SU (1, 1) and their importance in quantum optics (reduction of the uncertainty on one of the two noncommuting observables present in the measurements of the electromagnetic eld)
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