The Wave Functional of the Vacuum in a Resonator

A. Friedricha, D. Molla, M. Freybergera, L. Plimaka, W.P. Schleicha, b, c aInstitut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQST), Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany bTexas A&M AgriLife Research, Texas A&M University, College Station, Texas 77843-4242, USA cHagler Institute for Advanced Study and Department of Physics and Astronomy, Institute for Quantum Science and Engineering (IQSE), Texas A&M University, College Station, Texas 77843-4242, USA

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We show that despite fundamentally different situations, the wave functional of the vacuum in a resonator is identical to that of free space. The infinite product of the Gaussian ground state wave functions defining the wave functional of the vacuum translates into an exponential of a sum rather than an integral over the squares of mode amplitudes weighted by the mode volume and power of the mode wave number. We express this sum by an integral of a bilinear form of the field containing a kernel given by a function of the square root of the negative Laplacian acting on a transverse delta function. For transverse fields, it suffices to employ the familiar delta function, which allows us to obtain explicit expressions for the kernels of the vector potential, the electric field, and the magnetic induction. We show for the example of the vector potential that different mode expansions lead to different kernels. Lastly, we show that the kernels have a close relationship with the Wightman correlation functions of the fields.

DOI:10.12693/APhysPolA.143.52 topics: wave functional, vacuum, Wightman tensor, cavity quantum electrodynamics