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Physical Review A

Quasicontinuous measurements of photon number

Masashi Ban

Published Jun 1, 1994• 42 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.49.5078

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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.
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.
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.
The preparation, or generation of coherent states, squeezed states, and photon number states is discussed. The quantum noise is evaluated for various simultaneous measurements of two quadrature components: heterodyning, the beam splitter followed by two single quadrature measurements, the parametric amplifier, the (degenerate and/or nondegenerate) four-wave mixer, the Brillouin and Raman amplifiers, and the laser amplifier. A quantum nondemolition measurement followed by a measurement of the conjugate variable is also categorized as a simultaneous measurement. It is shown that, for all of these schemes, the minimum uncertainty product of the measured variables is exactly equal to that required for a simultaneous measurement of two noncommuting variables. On the other hand, measurements of a single quadrature component are noise-free. Such measurements are degenerate heterodyning, degenerate parametric amplification, and cavity degenerate four-wave mixing and photon counting by a photomultiplier or avalanche photodiode. The Heisenberg uncertainty principle and the quantum-mechanical channel capacity of Shannon are discussed to address the question "How much information can be transmitted by a single photon?" The quantum-mechanical channel capacity provides an upper bound on the achievable information capacity and is ideally realized by photon number states and photon counting detection. Its value is $\frac{\ensuremath{\hbar}\ensuremath{\omega}}{(\mathrm{ln}2)kT}$ bit per photon. The use of coherent or squeezed states and a simultaneous measurement of two quadrature field components or the measurement of one single quadrature field component does not achieve the ultimate limit.
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.