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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Conditional probabilities behind a beam splitter for an amplified single photon

A. Lukš, Vlasta Peřinová

Published Jul 14, 1999• 0 citations
Publisher

Abstract

We calculate a joint distribution of the photon number and the momentum-like quadrature measured in a simple experimental scheme. We assume that any quantum state is preamplifier to compensate for the attenuation on a beam splitter. Particularly, we investigate a single input photon. We compare the measured joint distribution with quasidistributions related to partially symmetric orderings of the photon number and momentum-like quadrature operators.

DOI
10.1117/12.353099

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We calculate a joint distribution of the photon number and the momentum-like quadrature measured in a simple experimental scheme. We assume that any quantum state is preamplifier to compensate for the attenuation on a beam splitter. Particularly, we investigate a single input photon. We compare the measured joint distribution with quasidistributions related to partially symmetric orderings of the photon number and momentum-like quadrature operators.
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Quasiprobability distributions of the photon number and the momentum-like quadrature appropriate to partial symmetrizations of products of the photon-number and quadrature operators allowing these operators to alternate at most (2r-2) times are good approximations to the Wigner function for these simultaneously considered noncommuting quantities related to the fully symmetric ordering. It is shown that the r-parametrized quasiprobability distributions must take on values at multiples of 1/r. A scheme for simultaneous measurement of the photon number and the momentum-like quadrature is proposed. The r-parametrized quasidistributions can be compared with a joint distribution of the photon number and the momentum-like quadrature measured simultaneously.
State preparation via conditional output measurement on a beam splitter is studied, assuming the signal mode is mixed with a mode prepared in a Fock state and photon numbers are measured in one of the output channels. It is shown that the mode in the other output channel is prepared in either a photon-subtracted or a photon-added Jacobi polynomial state, depending upon the difference between the number of photons in the input Fock state and the number of photons in the output Fock state onto which it is projected. The properties of the conditional output states are studied for coherent and squeezed input states, and the probabilities of generating the states are calculated. Relations to other states, such as near-photon-number states and squeezed-state-excitations, are given and proposals are made for generating them by combining the scheme with others. Finally, effects of realistic photocounting and Fock-state preparation are discussed.
We study multiphoton quantum-interference effects at a beam splitter and its connection to the prospect of attaining interferometric phase-shift measurements with noise levels below the standard quantum limit. Specifically, we consider the mixing of the most classical states of light coherent states with the most nonclassical states of light number states at a 50 : 50 beam splitter. Multiphoton quantum-interference effects from mixing photon-number states of small photon numbers with coherent states of arbitrary amplitudes are dramatic even at the level of a single photon. For input vacuum and coherent states, the joint photon-number distribution after the beam splitter is unimodal, a product of Poisson distributions for each of the output modes but with the input of a single photon, the original distribution is symmetrically bifurcated into a bimodal distribution. With a two-photon-number state mixed with a coherent state, a trimodal distribution is obtained, etc. The obtained distributions are shown to be structured so as to be conducive for approaching Heisenberg-limited sensitivities in photon-number parity-based interferometry. We show that mixing a coherent state with even a single photon results in a significant reduction in noise over that of the shot-noise limit. Finally, based on the results of mixing coherent light with single photons, we consider the mixing coherent light with the squeezed vacuum and the squeezed one-photon states and find the latter yields higher sensitivity in phase-shift measurements for the same squeeze parameter owing to the absence of the vacuum state.
A quasicontinuous measurement incorporating lossless beam splitters and photodetectors is proposed. Here the measurement by photodetectors is assumed to obey the projection postulate. The photon statistics in the output state of the signal mode of a beam splitter is investigated. It is shown that under certain conditions the measurement yields results equivalent to those obtained by the continuous measurement of photon number described by the quantum Markov process. Furthermore, measurements using a nondegenerate parametric amplifier and four-wave mixer, instead of beam splitter, are considered. The relations to the continuous measurement of photon number with a quantum counter and the continuous quantum nondemolition measurement of photon number are discussed.
A scheme for generating Schr\"odinger-cat-like states of a single-mode optical field by means of conditional measurement is proposed. Feeding a squeezed vacuum into a beam splitter and counting the photons in one of the output channels, the conditional states in the other output channel exhibit a number of properties that are very similar to those of superpositions of two coherent states with opposite phases. We present analytical and numerical results for the photon-number and quadrature-component distributions of the conditional states and their Wigner and Husimi functions. Further, we discuss the effect of realistic photocounting on the states.