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Completeness of coherent state subsystems

Karlheinz Gröchenig, Antti Haimi, José Luis Romero

Published Jul 8, 2015• 1 citation
OpenAlex

Abstract

We investigate the completeness of coherent states subsystems with respect to several classes of states on rational lattices. Our main results show that the phase-space shifts of every finite linear combination of Hermite functions with respect to a rational lattice are complete in L^2(R), thus generalizing a remark of von Neumann (and proved by Bargmann, Perelomov et al.). An analogous result is proven for windows that factor into certain rational functions and the Gaussian. The results are also interesting from a conceptual point of view since they show a vast difference between the completeness and the frame property of a discrete set of coherent states.

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The growth of Bargmann functions is intimately connected to the density of the zeros of these functions and to the completeness of sequences of coherent states. Using these ideas we find the least density that a sequence of coherent states must have in order to be overcomplete within the space of Bargmann functions of an order not exceeding (and of a type not exceeding if of order ). These results generalize known results on the completeness of von Neumann lattices. The practical significance of this formalism in the context of quantum optics is also discussed.
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We provide (partial) reconstruction formulas and discrete Fourier transforms for wave functions in standard Fock-Bargmann (holomorphic) phase-number representation from a finite number NNN of phase samples {θk=2πk/N}k=0N−1\{θ_k=2πk/N\}_{k=0}^{N-1}{θk​=2πk/N}k=0N−1​ for a given mean number ppp of particles. The resulting Coherent State (CS) subsystem S={∣zk=p1/2eiθk>}{\cal S}=\{|z_k=p^{1/2}e^{iθ_k}>\}S={∣zk​=p1/2eiθk​>} is complete (a frame) for truncated Hilbert spaces (finite number of particles) and reconstruction formulas are exact. For an unbounded number of particles, S{\cal S}S is "almost complete" (a \textit{pseudo-frame}) and partial reconstruction formulas are provided along with an study of the accuracy of the approximation, which tends to be exact when $p
A general set of states on a lattice in the phase plane is considered. The discrete set of von Neumann coherent states for a harmonic oscillator is a particular case of the above set. The kq\mathrm{kq}kq representation is used in an elementary proof of completeness and orthogonality of states on a discrete phase plane lattice.