By Jean-Pierre Gazeau

This self-contained creation discusses the evolution of the suggestion of coherent states, from the early works of Schr?dinger to the newest advances, together with sign research. An built-in and glossy method of the software of coherent states in lots of various branches of physics, it moves a stability among mathematical and actual descriptions. break up into elements, the 1st introduces readers to the main usual coherent states, their foundation, their building, and their program and relevance to numerous chosen domain names of physics. half II, commonly in accordance with contemporary unique effects, is dedicated to the query of quantization of varied units via coherent states, and exhibits the hyperlink to systems in sign research.

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Standard) coherent states and their time evolution. It offers the possibility to select the energy ~ |z|2 : the vacuum z = 0 is shown on the top, and an example of CS with intermediate z =/ 0 is shown in the middle figure. The shape of a perfect Poisson law is easily recognized for these two cases. One can see in the third figure (bottom) to what extent a small pertubation of the Poisson distribution destroys the “coherent” Gaussian shape of CS states. 33 34 3 The Standard Coherent States: the (Elementary) Mathematics One notices that this physical quantity can be viewed as the average value of the number operator in the coherent state |z 0 , I0 = z 0 |N |z 0 (lower symbol).

The D-“function” (it is actually an operator, also named displacement operator) D (z) = exp (za † – z¯ a) is essential in determining the properties and applications of the coherent states. Using the Weyl formula, one can also write D (z) = e – |z|2 2 † i e za e –¯z a = e – 2 q p e i pQ e –iqP . 52) With this formulation and from a|0 = 0 ⇒ e –¯z a |0 = |0 , one gets a new deﬁnition of the standard coherent states, precisely the one that emerges from group theory [10, 11]: |z = D (z)|0 . 53) Hence, the family of coherent states is to be viewed as the orbit of the Fock vacuum under the action of operators D (z), z ∈ C.

The respective expressions of the electric and magnetic ﬁeld operators are easily derived from the vector potential: E =– 1 ∂A , c ∂t B = ∇ ~ A. 21 22 2 The Standard Coherent States: the Basics Finally, the electromagnetic ﬁeld Hamiltonian is given by H= 1 2 E 2 + B 2 d3r = 1 2 ωk a †k a k + a k a †k . k Let us now decompose the electric ﬁeld operator into positive and negative frequencies. † E = E (+) + E (–) , E (–) = E (+) , ωk a k u k (r )e –iωk t . 51) k and the derived sequence of correlation functions G (n) .