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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Two alternative expansion formulas of density operator in quantum optics

Hong-Yi Fan

Published Oct 28, 1992• 0 citations
Publisher
DOI
10.1117/12.130397

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An alternative method for evaluating the canonical density matrix (x mod exp(- beta H) mod x') for a one-dimensional harmonic oscillator is proposed. The method uses the coherent states, i.e. the eigenstates of the annihilation operator a, as the basis of the expansion of exp(- beta H)) mod x') in place of the Hermite functions which are used in the conventional treatment. It is shown that the use of the coherent states makes it much easier to evaluate the density matrix than the other methods. As a direct extension of the method, the density matrix for a three-dimensional harmonic oscillator placed in a uniform magnetic field is recalculated in the creation-annihilation operator formalism. Some other uses of the coherent states in operator problems, such as the Wigner distribution function, are also discussed. Two alternative derivations of the Gaussian integral formula are also given, which plays an important role in the present problem as well as in various other problems in physics.
Two calculation formulas of expansion each other for normal ordering and antinormal ordering of creation operator and annihilation operator are given in this paper as () n( +) m=∑pk=0m!n!(m-k)!(n-k)!k!( +) m-k () n-k  and ( +) n() m=∑pk=0(-1) km!n!(m-k)!(n-k)!k!() m-k ( +) n-k , where p=min(n,m). It is right that we prove by using mathematical inductive method. The two formulas of this paper are used in a wide class of study of quantum statistical properties for various excited optical fields.