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arXiv:1407.5535 · Physical Review A

Macroscopic quantum effects for classical light

Nikolai I. Petrov

Published Oct 13, 2014• 18 citations
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Abstract

Optical analogies of macroscopic quantum effects (Schr\"odinger cat states, squeezing, collapse, and revival) for light beams propagating in an inhomogeneous linear medium are demonstrated theoretically using exact analytical solutions of the wave equation. It is shown that the coherent superposition of macroscopically distinguishable states is generated via mode interference from an initial off-axis single wave packet. Squeezed cat states with a fidelity >99% arise periodically and disappear rapidly within limited intervals of a propagation distance. Collapse and revival of wave packets at long-term nonparaxial evolution due to mode interference is demonstrated. Oscillations of the beam trajectory occur with extremely small amplitude, of the order of $10{}^{\ensuremath{-}19}$ m, which is typical of the estimated displacement caused by cosmic gravitational waves in gravity-wave detectors.

DOI
10.1103/physreva.90.043814

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Optical analogies of macroscopic quantum effects (Schr"odinger cat states, squeezing, collapse, and revival) for light beams propagating in an inhomogeneous linear medium are demonstrated theoretically using exact analytical solutions of the wave equation. It is shown that the coherent superposition of macroscopically distinguishable states is generated via mode interference from an initial off-axis single wave packet. Squeezed cat states with a fidelity >99% arise periodically and disappear rapidly within limited intervals of a propagation distance. Collapse and revival of wave packets at long-term nonparaxial evolution due to mode interference is demonstrated. Oscillations of the beam trajectory occur with extremely small amplitude, of the order of 10\ensuremath−1910{}^{\ensuremath{-}19}10\ensuremath−19 m, which is typical of the estimated displacement caused by cosmic gravitational waves in gravity-wave detectors.
Treating the ideal coherent state as a reference state, the effects due to departure from coherence of an initial wave packet propagating through a nonlinear medium, were examined, specifically in the context of non-classical effects such as revivals, fractional revivals, squeezing and higher-order squeezing during its temporal evolution. Further, these studies were extended to examine the role of quantum entanglement in bipartite systems. The dynamics of quantum expectation values were tracked carefully in various cases and the conditions under which a wave packet spreads chaotically, were investigated.
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.
Here it is shown that, for a wide class of nonlinear Hamiltonians, an essentially classical initial state will evolve into the coherent superposition of distinguishable quantum states. The detection of the interference fringes can be performed by beating the light leaving the nonlinear optical medium with light from a local oscillator. A model for the loss in the detection apparatus is introduced. The squeezed-vacuum technique is introduced finally as a means of enhancing the interference fringes displaying the distinguishable quantum state.
We propose a scheme to generate a displaced macroscopic superposition of phase squeezed coherent states by incorporating an optical parametric oscillator (OPO) and a nonlinear Kerr medium in a Mach-Zhender interferometer configuration. We show that the phase squeezed coherent state generated by the OPO evolves into a macroscopic superposition of phase squeezed coherent states on spending a specific amount of time inside a Kerr medium. The phase space displacement is achieved by the interference of the intense reference beam with the signal in the final beam splitter of the interferometer. The noise of the reference beam contaminating the signal is prevented by making this beam splitter highly reflective. We have calculated the photon number uncertainty of the outcoming beam and have shown that it is smaller than the usual amplitude squeezed coherent state's value.
Nondegenerate parametric oscillation and four-wave mixing generate correlated twin-beams that exhibit squeezing in the intensity difference fluctuations. Using a simple theoretical model for parametric amplification we predict that the twin-beam devices generate macroscopic states that violate Bell’s inequality. In four-wave mixing, or parametric oscillation with an amplitude-squeezed pump, the twin-beams may also exhibit squeezing in the intensity of each beam. We predict that the use of these intensity-squeezed devices is advantageous in the preparation of the macroscopic quantum states. It is also possible to use nondegenerate systems to generate macroscopic superposition (Schrodinger cat) states.
Pure states are usually used to observe quantum phenomena. In this study, we show that a quantum superposition of spatially displaced mixed cat states can be generated within an optical waveguide via nonparaxial unitary evolution of the initial low coherence (low purity) light beam. It is shown that highly mixed Schrodinger cat states can be observed at a well-defined propagation distance. The importance of the long-term decoherence and recoherence of the original wave packet in observing the mixed cat states is demonstrated. The Heisenberg and Schrodinger-Robertson uncertainty relations for mixed cat states are evaluated. We have demonstrated the feasibility of our method using accurate numerical simulations for the parameters of the source and optical waveguide available in practice.