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Entropy

Wigner Functions of Time-Dependent Cat-like Even/Odd Superpositions of Nonlinear Coherent States

Miguel Citeli de Freitas, V. V. Dodonov

Published Mar 3, 2026• 1 citation
Publisher

Abstract

We calculate and plot 2D slices of the Wigner functions of several families of highly excited even and odd superpositions of nonlinear coherent states, looking for conditions under which such superpositions can be interpreted as models of the "Schrödinger cat" states. The decisive factor seems to be the form of the number distribution function over the Fock basis: it must have well localized peaks. Otherwise, no "cat-like" structures are observed.

DOI
10.3390/e28030287

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We calculate and plot 2D slices of the Wigner functions of several families of highly excited even and odd superpositions of nonlinear coherent states, looking for conditions under which such superpositions can be interpreted as models of the "Schrödinger cat" states. The decisive factor seems to be the form of the number distribution function over the Fock basis: it must have well localized peaks. Otherwise, no "cat-like" structures are observed.
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We investigate the time evolution of Morse coherent states in the potential of the NO molecule. We present animated wave functions and Wigner functions of the system exhibiting spontaneous formation of Schrödinger-cat states at certain stages of the time evolution. These nonclassical states are coherent superpositions of two localized states corresponding to two di.erent positions of the center of mass. We analyze the degree of nonclassicality as the function of the expectation value of the position in the initial state. Our numerical calculations are based on a novel, essentially algebraic treatment of the Morse potential.
Higher-generation Schrödinger cat states of the quantized electromagnetic field can be produced in a high-Q cavity, starting from a coherent state, through the passage of prepared Rydberg atoms interacting dispersively across it. These states are natural generalizations of the even and odd coherent states, the N th-generations corresponding to specific superpositions of 2 N states on a circle in phase space with well defined parity, and present very peculiar properties. Their photon statistics interchange between super- and sub-Poissonian behaviours and the nature of the photon bunching oscillates as the field intensity in the cavity is varied. For higher-generation even states, the minimum value of the Mandel factor almost reaches −1.0 and the state represents the Fock state |2 N ). Squeezing properties and the Wigner function of these higher-generation Schrödinger cat states are also considered.
Canonical coherent states of a quantum harmonic oscillator have been generalized by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Superpositions of these states are considered in the present scenario as a generalization of the optical Schrödinger cat states. The Fock space is assumed to be canonical or finite-dimensional. The photon number distribution of these generalized Schrödinger cat states departs from the Poisson statistics in various ways for high photon numbers. For small nonvanishing values of the label, the photon number distribution is sub-Poissonian (nonclassical) or super-Poissonian, according to the interference properties. In fact, the sub- or super-Poissonian statistics is determined by the interplay between the relative phase and a critical value of the phase. The photon number distribution is uniquely sub-Poissonian for large values of the label.
Wigner functions in phase space are reconstructed for the excited coherent states，which are generated by applying the inverse of k-Boson operators repeatedly to the coherent states. The non-classicality of these states is discussed by calculating their Wigner functions in Fock-state space. Numerical results show that these excited coherent states always reveal non-classical characteristics，no matter whether the excited number k is even or odd，and also the corresponding non-classicality is more obvious with the number k increasing.
We demonstrate that superpositions of coherent states and displaced Fock states, also referred to as generalized Schr\"odinger cat states, can be generated by application of a deformed Glauber displacement operator on number states, coherent states, and displaced Fock states. Based on such a deformed displacement operator we introduce a deformed version of the so-called Glauber photonic lattices. These novel lattices are endowed with alternating positive and negative coupling coefficients and give rise to classical analogs of Schr\"odinger cat states. Finally, we show that the analytic propagator of these new Glauber-Fock arrays explicitly contains the Wigner operator opening the possibility to emulate Wigner functions of the quantum harmonic oscillator in the classical domain.