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arXiv:1505.02698 · Journal of the Optical Society of America B

Signatures of entanglement in an optical tomogram

M. Rohith, C. Sudheesh

Published Jan 5, 2016• 24 citations
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Abstract

We theoretically study the optical tomograms of maximally entangled states generated at the output modes of a beam splitter. We consider even and odd coherent states in one of the input modes and vacuum state in the other input mode of the beam splitter. 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. Two distinct types of optical tomograms are observed in any one of the output modes of the beam splitter based on the quadrature measurement in the other output mode if the output modes are entangled. The different features shown by the optical tomograms are verified by investigating the photon-number statistics of the corresponding state. We analyze the effect of decoherence on the optical tomograms of the entangled states.

DOI
10.1364/josab.33.000126

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We theoretically study the optical tomograms of maximally entangled states generated at the output modes of a beam splitter. We consider even and odd coherent states in one of the input modes and vacuum state in the other input mode of the beam splitter. 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. Two distinct types of optical tomograms are observed in any one of the output modes of the beam splitter based on the quadrature measurement in the other output mode if the output modes are entangled. The different features shown by the optical tomograms are verified by investigating the photon-number statistics of the corresponding state. We analyze the effect of decoherence on the optical tomograms of the entangled states.
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.
Abstract Several nonclassical effects displayed by wave packets governed by nonlinear Hamiltonians can be identified and assessed directly from tomograms without attempting to reconstruct the Wigner function or the density matrix explicitly. We have demonstrated this for both single-mode and bipartite systems. We have shown that a wide spectrum of effects such as the revival phenomena, quadrature squeezing, and Hong–Mandel and Hillery type higher-order squeezing in a generic single-mode system and the double-well Bose–Einstein condensate (BEC) can be obtained from appropriate tomograms in a straightforward manner. We have examined the manner in which decoherence affects the nature of the state of a generic single-mode system at specific instants during temporal evolution. We have investigated entropic squeezing of the subsystem state of a bipartite system as it evolves in time, solely from tomograms. The procedures that we have demonstrated can be readily adapted to multimode systems. Further, for the double-well BEC we have identified an indicator of entanglement between subsystems that can be obtained directly from the tomogram. This mirrors the qualitative behavior of the subsystem von Neumann entropy and the subsystem linear entropy.
We have examined both single and entangled two-mode multiphoton coherent states and shown how the ‘Janus-faced’ properties between two partner states are mirrored in appropriate tomograms. Entropic squeezing, quadrature squeezing and higher-order squeezing properties for a wide range of nonclassical states are estimated directly from tomograms. We have demonstrated how squeezing properties of two-mode entangled states produced at the output port of a quantum beamsplitter are sensitive to the relative phase between the reflected and transmitted fields. This feature allows for the possibility of tuning the relative phase to enhance squeezing properties of the state. Finally, we have examined the manner in which decoherence affects squeezing and the changes in the optical tomogram of the state due to interaction with the environment.
Photon subtraction is useful to produce nonclassical states of light addressed to applications in photonic quantum technologies. After a very accelerated development, this technique makes possible obtaining either single photons or optical cats on demand. However, it lacks theoretical formulation enabling precise predictions for the produced fields. Based on the representation generated by the two-mode SU(2) coherent states, we introduce a model of entangled light beams leading to the subtraction of photons in one of the modes, conditioned to the detection of any photon in the other mode. We show that photon subtraction does not produce nonclassical fields from classical fields. It is also derived a compact expression for the output field from which the calculation of conditional probabilities is straightforward for any input state. Examples include the analysis of squeezed-vacuum and odd-squeezed states. We also show that injecting optical cats into a beam splitter gives rise to entangled states in the Bell representation.
Multiphoton interference results in a wide variety of non-classical photon number statistics, including characteristic signatures of entanglement between two or more sets of optical modes. Here, we consider the photon number statistics observed after applying discrete Fourier transformations (DFTs) to bipartite entangled states generated using single photon sources and beam splitters. It is shown that the output photon number states of DFTs are eigenstates of a translational mode shifting operator in the input. The complex eigenvalues of the mode shift can be identified by a phase number K obtained from the output photon distribution. For each output distribution, the possible input states are limited to mode shift eigenstates with the same K-value. Using this mode shift rule, we can identify the quantum coherence between different photon number distributions in the input with experimentally observable K-values in the output of the DFT. In the case of multi-photon entanglement obtained by post-selection and beam splitting single photons, this coherence is non-local, resulting in correlated pairs of K-values that always sum up to zero. We can therefore observe both the correlations between the input photon number distributions and a complementary correlation between the output photon numbers of two DFTs to obtain a reliable characterization of the entanglement between the two multi-mode multi-photon systems. Importantly, the K-value allows a classification of large sets of possible photon number distributions, resulting in a significant simplification of the experimental evaluation of the multi-photon output statistics and opening up the road towards more efficient applications of non-classical multi-photon states
The behavior of general nonclassical two-mode Gaussian states at a beam splitter is investigated. Single-mode nonclassicality as well as two-mode entanglement of both input and output states are analyzed suggesting their suitable quantifiers. These quantifiers are derived from local and global invariants of linear unitary two-mode transformations such that the sum of input (or output) local nonclassicality measures and entanglement measure gives a global invariant. This invariant quantifies the global nonclassicality resource. Mutual transformations of local nonclassicalities and entanglement induced by the beam splitter are analyzed considering incident noisy twin beams, single-mode noisy squeezed vacuum states, and states encompassing both squeezed states and twin beams. A rich tapestry of interesting nonclassical output states is predicted.