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Optics Express

Formation of Schrödinger-cat states in the Morse potential: Wigner function picture

Péter Földi, Attila Czirják, Balázs Molnár, M. G. Benedict

Published Apr 22, 2002• 21 citations
Publisher

Abstract

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.

DOI
10.1364/oe.10.000376

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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.
The Wigner distribution function for the time-dependent quadratic Hamiltonian system in the coherent Schrodinger cat state is investigated. The type of state we consider is a superposition of two coherent states, which are by an angle of π out of phase with each other. The exact Wigner distribution function of the system is evaluated under a particular choice of phase, δc, q. Our development is employed for two special cases, namely, the Caldirola–Kanai oscillator and the frequency stable damped harmonic oscillator. On the basis of the diverse values of the Wigner distribution function that were plotted, we analyze the nonclassical behavior of the systems.
The transient evolution of nonclassical radiation fields interacting with an atom in a cavity is correlated to the atomic dynamics. Connection between the atomic phases of collapse and revival, and the Wigner function pattern is explored. Initial classical coherent field can evolve into nonclassical field. Schrodinger cat state field is generate during the atomic collapse phase. For initial Schrodinger cat field state, the state dissolves with time but revives during the atomic phase that corresponds with the initial phase. More intricate but orderly characteristics are found between the collapse-revival dynamics and the patterns of the Wigner function for initial thermal field.
Abstract Superposition of two coherent states, the Schrodinger’s cat state, can exhibit different nonclassical properties having foundational applications in quantum information processing. We consider the ‘superposition of Schrodinger’s cat state with the vacuum state (SCVS)’ of the optical field. We discuss different witness of nonclassicality properties such as lower- and higher-order squeezing (viz., squeezing, Hong & Mandel’s fourth-order squeezing, amplitude-squared squeezing) and sub-Poissonian photon statistics. Further, we discuss the negativity of the Wigner function of SCVS indicating the nonclassicality of the state under investigation. We find that the vacuum state contribution in SCVS exhibits different nonclassicalities under some conditions stronger where the nonclassicalities exhibited by the state without vacuum state contribution are weaker.
The decay of coherence when a quantum system interacts with a much larger environment is usually described by a master equation for the system reduced density matrix and emphasizes the evolution of an entire ensemble. We consider two methods that have been developed recently to simulate the evolution of single realizations. Quantum-state diffusion involves both diffusion, where the individual quantum trajectory fluctuates through a Wiener process deriving from the environment, and localization to a coherent state, an eigenstate of the relevant Lindblad operator describing the coupling of the system to the environment. We demonstrate the localization process for different initial states and utilize the Wigner function to depict this localization in phase space. We concentrate on quantum states that can be expressed as a superposition of appropriate coherent states. For an initial superposition of two coherent states (a Schr\"odinger ``cat''), one of the two components will dominate the evolution. For initial Fock states, which can be described as a continuous superposition of coherent states on a ring, localization takes place when one coherent state is selected from that ring where each component has nearly the same energy as the original Fock state. We also consider the localization from a nonclassical squeezed ground state, which can be expressed as a superposition of coherent states along a line in phase space. The second simulation method considered is the state vector Monte Carlo, or ``quantum jump,'' approach, which relates to the direct counting of decay quanta. In the case of an initial Schr\"odinger ``cat,'' we find that when no quantum is detected the ``cat'' shrinks, but when a quantum is detected, the Schr\"odinger ``cat'' ``jumps'' from one type of ``cat'' to another with different internal phase. For an initial squeezed state we show how quantum jumps lead to individual realizations which are superpositions of two squeezed states.