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Conventional Glauber coherent states (CS) can be defined in severalequivalent ways, e.g., by displacing the vacuum or, explicitly, by theirinfinite Poissonian expansion in Fock states. It is well known that thesedefinitions become inequivalent if applied to finite $d$-level systems(qudits). We present a comparative Wigner-function description of the qudit CSdefined (i) by the action of the truncated displacement operator on the vacuumand (ii) by the Poissonian expansion in Fock states of the Glauber CS truncatedat $(d-1)$-photon Fock state. These states can be generated from a classicallight by its optical truncation using nonlinear and linear quantum scissorsdevices, respectively. We show a surprising effect that a macroscopicallydistinguishable superposition of two qudit CS (according to both definitions)can be generated with high fidelity by displacing the vacuum in the quditHilbert space. If the qudit dimension $d$ is even (odd) then the superpositionstate contains Fock states with only odd (even) photon numbers, which can bereferred to as the odd (even) qudit CS or the female (male) Schr\"odinger catstate. This phenomenon can be interpreted as an interference of a single CSwith its reflection from the highest-energy Fock state of the Hilbert space, asclearly seen via phase-space interference of the Wigner function. We alsoanalyze nonclassical properties of the qudit CS including their photon-numberstatistics and nonclassical volume of the Wigner function, which is aquantitative parameter of nonclassicality (quantumness) of states. Finally, westudy optical tomograms, which can be directly measured in the homodynedetection of the analyzed qudit cat states and enable the completereconstructions of their Wigner functions.

physical review. a, atomic, molecular, and optical physics/physical review, a, atomic, molecular, and optical physics

Phase-space interference of states optically truncated by quantum  scissors: Generation of distinct superpositions of qudit coherent states by  displacement of vacuum