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Journal of Modern Optics

Photon statistics of conditional output states of lossless beam splitter

Masashi Ban

Published Jan 1, 1996• 50 citations
Publisher

Abstract

When an observation is carried out at one of the two output ports of a lossless beam splitter, photon statistics of a conditional output state in the unobserved output port are investigated for several input states. Observations at the output port are carried out by means of photon counting, heterodyne detection and homodyne detection. The dependence of the average value of the photon number and the Q factor of the conditional output state on the measurement outcome are investigated in detail. The statistical distance between the input and conditional output states is also considered. Furthermore, it is shown that the relationship between the input and conditional output states of the beam splitter is characterized as a conditional isometric process.

DOI
10.1080/09500349608232803

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When an observation is carried out at one of the two output ports of a lossless beam splitter, photon statistics of a conditional output state in the unobserved output port are investigated for several input states. Observations at the output port are carried out by means of photon counting, heterodyne detection and homodyne detection. The dependence of the average value of the photon number and the Q factor of the conditional output state on the measurement outcome are investigated in detail. The statistical distance between the input and conditional output states is also considered. Furthermore, it is shown that the relationship between the input and conditional output states of the beam splitter is characterized as a conditional isometric process.
Using the quantum theory of a lossless beam splitter, we determine the joint and marginal photon-number distributions at its output ports, for two arbitrary (not necessarily independent) input fields. The joint probability at the output of the beam splitter turns out to be a Fourier series in the relative phase between the output beams. Photomixing with the number states exhibits no phase sensitivity. An elementary example of phase sensitive dependence in the photon-number domain is obtained with the Bernoulli states, which are a subclass of the binomial states.1 For a joint input Bernoulli state, the probability of detecting one photon (or zero photons) at either output port varies sinusoidally with the relative phase. For independent Bernoulli states incident on the beam splitter, the results differ and depend on whether the input is in a pure or mixed state. Photomixing of other nonclassical states exhibit higher orders of phase-induced modulation of the output photon-number probabilities.
For optical homodyning, the matrix representation of a lossless beam splitter belongs to the SU(2) group of unimodular second-order unitary matrices. The connection between this group and the rotation group in three dimensions permits the field density operators at the input and output ports of the beam splitter to be related by means of well-known angular-momentum transformations. This, in turn, provides the joint output photon-number distribution, which may be written as a Fourier series in the relative phase shift imparted by the beam splitter, for a general joint state at its inputs. The series collapses to a single term if one of the input fields is diagonal in the number-state representation. If the inputs to both ports are further restricted to be pure number states, the joint, as well as the marginal photon-number distributions, turn out to be directly proportional to the square of Jacobi polynomials in the beam-splitter transmittance. These photon-number probabilities are invariant to a set of physical and time-reversal symmetries. When one of the input photon-number states is the vacuum, the beam splitter simply deletes photons from the other port in Bernoulli fashion, as if they were classical particles. The output photon number is then described by the binomial distribution. If the inputs at the two ports are different number states, neither of which is the vacuum, the photon-number distribution is expressible in terms of summed and weighted products of the results for photomixing with the vacuum. If the inputs at the two ports are identical number states, and a beam splitter of transmittance \ensuremath{\tau}=(1/2 is used, the photon-number distribution assumes a simple but interesting form. It vanishes for odd photon numbers, indicating that the photons assemble in pairs at each output port. Finally, it is shown that homodyning quantum fluctuations can be reduced by using a balanced photomixer for arbitrary input states.
Eight-port homodyne detection may be used to perform a joint measurement on a single-mode input to a beam splitter; the xxx quadrature is measured at one output port, and the ppp quadrature is measured at the other. This is possible even when the input mode contains only one photon. We show to what extent this "joint measurement of a single photon" may be reconciled with the semiclassical notion that the photon may only exit from one of the beam splitter output ports.
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 investigate the photon statistics at the outputs of a four-port beam splitter, when a strong coherent field is incident on one input port and various other fields enter at the second inner port. Some non-classical interference effects are encountered. In particular, there is a large difference between the cases when a one photon field enters at input port 2 and when a field in a coherent state with an average of one photon enters at port 2. In the former case the probability P(m) of the difference m between the detected photons at output ports 3 and 4 has a minimum at m = 0 whenever the quantum efficiency exceeds one third. As the quantum efficiency falls from one to zero, the form of the probability distribution P(m/1) changes drastically.
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.