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Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento

A graphical method to evaluate vacuum expectation values of products of creation and annihilation operators

P. P. Delsanto

Published Aug 1, 1983• 1 citation
Publisher
DOI
10.1007/bf02751780

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In the context of many-body quantum mechanics, one often encounters the need to evaluate expectation values of operators or their matrix elements. For instance, given a Hamiltonian https://www.w3.org/1998/Math/MathML" display="inline"> H ^ , we typically want to evaluate its expectation value, which is the energy of the system, https://www.w3.org/1998/Math/MathML" display="inline"> E = ⟨ H ^ ⟩ . In second quantization, the matrix element of an operator can always be expressed as a vacuum expectation value. This generally yields an expression which is hard to evaluate. Fortunately, a powerful tool to simplify these calculations exists: Wick's theorem. This theorem provides a systematic way to break down expressions involving products of creation and annihilation operators into sums of simpler terms.
The graphical technique of evaluation in the second quantization representation of matrix elements of operators associated with the interactions in atoms is developed. The diagrams corresponding to operators and many-electron states given in terms of irreducible tensorial product of creation and annihilation operators are considered. The general rules for the construction of the diagrams for operators and matrix elements are presented. The procedures of reduction of the diagrams to obtain the algebraic expression are discussed.
Motivated by the readout scheme in interferometric gravitational-wave detectors, we consider the device which measures the expectation value of the photon annihilation and creation operators for output optical field from the main interferometer. As the result, the eight-port homodyne detection is rediscovered as such a device. We evaluate the noise spectral density in this measurement. We also briefly discuss on the application of our results to the readout scheme of gravitational-wave detectors. We call this measurement scheme to measure these expectation values as "double balanced homodyne detection."