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Optics Communications

Relation between input and output states for a beam splitter

Z. Y. Ou, C.K. Hong, L. Mandeļ

Published Jul 1, 1987• 158 citations
Publisher

Abstract

The state of the optical field at the output of a beam splitter is expressed directly in terms of the state at the input via the diagonal coherent state representation. The conclusion is illustrated for a two-photon Fock state, and it is shown that an output having some features of the two-photon singlet state can be produced.
DOI
10.1016/0030-4018(87)90271-9

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By making use of the diagonal coherent state representation of an optical field, together with the operator relations between dynamic variables at the input and output of a beam splitter, we have derived a general relation between density operators of input and output states. As an illustration of the general procedure we consider two orthogonally polarized photons at the beam splitter input and derive the output state. We show that in the special case in which the reflectivities and transmissivities of the beam splitter are equal, this state exhibits certain features of the well-known singlet state for orthogonally polarized photons.
The second-order coherence of optical fields at output ports of a beam splitter is investigated. The properties of the second-order coherence are not only strongly dependent on the input states but also quite involved with the nature of the beam splitter. When parameters of both the input states and the beam splitter are chosen appropriately, the fields at the output ports will exhibit quantum behaviour. Between the two output modes, there stands mode–mode correlation which can even be anticorrelation. Some nonclassical properties, such as the photon antibunching effect of the input fields, may be destroyed by the beam splitter.
Truncated output states can be generated by the conditional measurement if the coherent state and arbitrary Fock state are regarded as the input states of beam splitters.The properties of this kind of truncated optical states are studied in this paper.The results have shown that the output states in one out port have stronger nonclassical properties of squeeze and sub-Poissonian distribution when zero photon is measured in the other out port,and the nonclassical properties can become stronger with the increase of input number state.When m(m＞0) photon is measured in one of the out ports,each output state after conditional measurement has the same property in optical field but also exhibits apparent nonclassical effects such as squeezing and sub-Poissonian distribution.It should be noted particularly that the number state corresponding to the measurement in output state after conditional measurement would be truncated if the single number state is input.
We report preparation and characterization of coherent superposition states t[0>+alpha]1> of the electromagnetic field by conditional measurements on a beam splitter. This state is generated in one of the beam splitter output channels if a coherent state [alpha> and a single-photon Fock state [1> are present in the two input ports and a single photon is registered in the other beam splitter output. The single photon thus plays a role of a "catalyst:" it is explicitly present in both the input and the output channels of the interaction yet facilitates generation of a nonclassical state of light.
A beam splitter is one of the most important devices in an optics laboratory because of its handiness and versatility; equivalent devices are found in various quantum systems to couple two subsystems or to interfere them. While it is normal that two independent input fields are superposed at the beam splitter to give correlated outputs, identical Gaussian states interfere there to produce totally independent output fields. We prove that the Gaussian states with same variances are the only states which bring about factorizable output fields.
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.