The Heisenberg-Weyl And SU(2)Groups:Superposition Of Coherent States And Sub-planck Structural Analogies In Phase Space

The well-studied quantum optical Schrodinger's cat state is a superposition of two distinct states,with quantum coherence between these macroscopically distinct states being of foundational and,in the context of many quantum subjects,ranging from foun-dational to technological,e.g.,from tests of collapse models to quantum metrology.The goal of the present work is three-fold.First,for the Heisenberg-Weyl case we investigate various coherent-state superpositions of the harmonic oscillator,where position and mo-mentum operators satisfy the Heisenberg-Weyl algebra for a single degree of freedom,and act on an infinite-dimensional Hilbert space.We refer to these quantum-optical cat states as quantum dichotomous states,reflecting that the state is a superposition of two options,and we introduce the term quantum multichotomous state to refer to a super-position of multiple macroscopically distinguishable options.First,we explore various aspects of the coherent-states superpositions associated with the harmonic oscillator,related to the connection between sub-Planck structures present in their Wigner func-tion and their sensitivity to displacements(ultimately determining their metrological potential).For the harmonic oscillator,we identify phase-space structures with support smaller than the Planck action both in cat-state mixtures and superpositions,the latter known as compass states.However,as compared to coherent states,compass states are shown to have(?)-enhanced sensitivity against displacements in all phase-space di-rections(N is the average number of quanta),whereas cat states and cat mixtures show such enhanced sensitivity only for displacements along specific directions.In the second part we construct the quantum multichotomous states as a superposi-tion of Gaussian states on the position line in phase space.Using this nomenclature,a quantum tetrachotomous state(QTS)is a coherent-states superposition of four macro-scopically distinguishable states.We define,analyze,and show how to create such states,and our focus on the QTSs is due to their exhibition of much richer phenom-ena than for the quantum dichotomous states.Our characterization of the QTS involves the Wigner function,its marginal distributions,and the photon-number distribution,and we discuss the QTS's approximate realization in a multiple-coupled-well system.In third part we aim to generalize the concept of sub-Planck structures from the harmonic oscillator to the quantum states of the SU(2)types.In particular,our focus on quantum states of the(?)u(2)algebra.We find analogous results for the(?)u(2)algebra,typically associated with angular momentum.We then show that these same properties apply for analogous SU(2)states provided that(i)coherent states are restricted to the equator of the sphere that plays the role of phase space for this group,(ii)we associate the role of the Planck action to the 'size' of SU(2)coherent states in such sphere,and(iii)the role of N to the total angular momentum.