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arXiv:1507.03724 · Physical Review A

Visualizing revivals and fractional revivals in a Kerr medium using an optical tomogram

M. Rohith, C. Sudheesh

Published Nov 11, 2015• 37 citations
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Abstract

We study theoretically optical tomograms of the time-evolved states generated by the evolution of different kinds of initial wave packets in a Kerr medium. An exact analytical expression for the optical tomogram of the quantum state at any instant during the evolution of a generic initial wave packet is derived in terms of Hermite polynomials. The time evolution of the optical tomogram is discussed for three kinds of initial states: a coherent state, an $m$-photon-added coherent state, and even and odd coherent states. We show the manifestation of revival and fractional revivals in the optical tomograms of the time-evolved states. We find that the optical tomogram of the time-evolved state at the instants of fractional revivals shows structures with sinusoidal strands. The number of sinusoidal strands in the optical tomogram of the time-evolved state at $l$-subpacket fractional revivals is $l$ times the number of sinusoidal strands present in the optical tomogram of the initial state. We also investigate the effect of decoherence on the optical tomograms of the states at the instants of fractional revivals for the initial states considered above. We consider amplitude decay and phase damping models of decoherence and show the direct manifestations of decoherence in the optical tomogram.

DOI
10.1103/physreva.92.053828

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We study theoretically optical tomograms of the time-evolved states generated by the evolution of different kinds of initial wave packets in a Kerr medium. An exact analytical expression for the optical tomogram of the quantum state at any instant during the evolution of a generic initial wave packet is derived in terms of Hermite polynomials. The time evolution of the optical tomogram is discussed for three kinds of initial states: a coherent state, an mmm-photon-added coherent state, and even and odd coherent states. We show the manifestation of revival and fractional revivals in the optical tomograms of the time-evolved states. We find that the optical tomogram of the time-evolved state at the instants of fractional revivals shows structures with sinusoidal strands. The number of sinusoidal strands in the optical tomogram of the time-evolved state at lll-subpacket fractional revivals is lll times the number of sinusoidal strands present in the optical tomogram of the initial state. We also investigate the effect of decoherence on the optical tomograms of the states at the instants of fractional revivals for the initial states considered above. We consider amplitude decay and phase damping models of decoherence and show the direct manifestations of decoherence in the optical tomogram.
By analyzing the optical tomogram of a linear superposition of coherent states, we show that distinctive signatures of the macroscopic superposition states are displayed directly in the optical tomograms of the states. We also study the effect of decoherence on the optical tomograms of the macroscopic superposition states. Since the wave packet fractional revivals are associated with the generation of macroscopic superposition states, these signatures help in visualizing the revivals and fractional revivals occurring in a nonlinear medium directly in the optical tomogram of the time-evolved state. We found that the optical tomogram of the time-evolved state at the instants of fractional revivals show structures with sinusoidal strands. Using a class of initial superposed wave packets evolving in the Kerr-like medium, we further show that the condition for the occurrence of fractional revival phenomenon depends on the number of wave packets composing the initial superposition state. In the case of a two-mode electromagnetic field, we investigate the entanglement of the state directly using the optical tomogram. We have shown that the signatures of entanglement can be observed directly in the single-mode optical tomogram of the state without reconstructing the density matrix of the system. We also analyze the effect of decoherence on the optical tomograms of the entangled states. Further, we examine the optical tomograms of the entangled states generated using a beam splitter with a Kerr medium placed into one of its input modes. We have found the signatures of entanglement in the optical tomogram for the entangled states generated at the instants of two- and three-subpacket fractional revival times.
We show in the framework of a tractable model that revivals and fractional revivals of wave packets afford clear signatures of the extent of departure from coherence and from Poisson statistics of an initial state of the radiation field propagating in a nonlinear medium. We establish this by considering a generic Hamiltonian which models wave packet propagation in a Kerr-like medium, and comparing the non-classical effects displayed by an initial photon-added coherent state with that of an inital ideal coherent state.
We study theoretically the dynamics of entangled states created in a beam splitter with a nonlinear Kerr medium placed into one input arm. Entanglement dynamics of initial classical and nonclassical states are studied and compared. Signatures of revival and fractional revival phenomena exhibited during the time evolution of states in the Kerr medium are captured in the entangled states produced by the beam splitter. Dynamics of entanglement shows local minima at the instants of fractional revivals. These minima correspond to the generation of two-component Schrödinger cat states or multi-component Schrödinger cat-like states if the initial state considered is a coherent state. Maximum entanglement is obtained at the instants of collapses of wave packets in the medium. Our analysis shows increase in entanglement with increase in the degree of nonclassicality of the initial states considered. We show that the states generated at the output of the beam splitter using initial nonclassical states are more robust against decoherence due to photon absorption by an environment than those formed by an initial classical state.
The reliable quantification of nonclassicality in quantum states under realistic decoherence remains a central challenge in advancing quantum technologies. Conventional quantifiers such as Wigner negativity, Mandel's QQQ-parameter, nonclassical depth, etc., are often experimentally intractable, non-unique, or insensitive to key quantum signatures. We demonstrate that tomogram-based measures, the homodyne nonclassical area and sum tomographic entropy, offer a robust, experimentally accessible alternative for quantifying nonclassicality dynamics, as they can be directly obtained from optical tomograms via balanced homodyne detection, avoiding density matrix reconstruction and ensuring feasibility. We study coherent, photon-added coherent, and even coherent states evolving in Kerr and cubic nonlinear systems, with environmental effects modelled using the Lindblad master equation under amplitude and phase damping. The homodyne nonclassical area, which quantifies the excess quadrature variance beyond that of a coherent state, tracks both the onset and decay of nonclassicality, clearly identifying fractional revivals, wave packet splitting, and macroscopic superpositions. We find that amplitude damping drives a rapid monotonic decay toward the vacuum, while phase damping allows partial revival features to survive longer. Complementing this, the sum tomographic entropy derived from conjugate-quadrature tomograms captures higher-order fractional revivals and phase-space interference through persistent entropy minima under weak damping. Our results establish homodyne-based quantifiers as powerful, real-time, and experimentally viable tools for tracking nonclassical dynamics in nonlinear optical media, offering a compelling alternative to conventional, experimentally challenging measures.
Wave packet revivals and fractional revivals are striking quantum interference phenomena that can occur under suitable conditions in a system with a nonlinear spectrum. In the framework of a specific model (the propagation of an initially coherent wave packet in a Kerr-like medium), it is shown that distinctive signatures of these revivals and fractional revivals are displayed by the time evolution of the expectation values of physical observables and their powers, i.e., by experimentally measurable quantities. Moreover, different fractional revivals can be selectively identified by examining appropriate higher moments.
We study the dynamics of superposed wave packets in a specific nonlinear Hamiltonian which models the wave packet propagation in Kerr-like media and the dynamics of Bose–Einstein condensates. We show the dependence of fractional revival times on the initial superposed wave packets using analysis based on the expectation values, Rényi entropy and Wigner function. We also show how the selective identification of fractional revivals using moments of appropriate observables depends on the number of wave packets present in the initial state.