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International Journal of Modern Physics B

Homodyne Photocurrent, Symmetries in Photon Mixing and Number State Synthesis

Matteo G. A. Paris

Published Jun 30, 1997• 7 citations
Publisher

Abstract

Lossless beam splitter has been studied by the exact diagonalization of the homodyne photocurrent. The probability of totally symmetric and totally antisymmetric output from a balanced beam splitter fed by arbitrary number states has been evaluated. Efficiency of photon mixing to synthesize highly excited number states has been also discussed.

DOI
10.1142/s0217979297000988

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International Journal of Quantum Information•Dec 1, 2017•13 citations
Lossless beam splitter has been studied by the exact diagonalization of the homodyne photocurrent. The probability of totally symmetric and totally antisymmetric output from a balanced beam splitter fed by arbitrary number states has been evaluated. Efficiency of photon mixing to synthesize highly excited number states has been also discussed.
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.
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.
In this work, we present a comprehensive analytical study of a lossless quantum beam splitter (BS) with transmission coefficient T and reflection coefficient R. Using the unitary operator representation and binomial expansion, we derive the exact probability distribution P_k for detecting k photons in one output port when the input ports are prepared in Fock states |N>_0 |M>_1. The balanced configuration R = T = 1/2 and N = M is examined in detail: we prove that destructive quantum interference suppresses all odd photon numbers at the outputs, leaving only even photon counts. The special case N = M = 1 reproduces the famous Hong-Ou-Mandel (HOM) dip with P_1 = 0. Beyond photon statistics, we demonstrate that the beam splitter functions as a deterministic, tunable source of photon-number entanglement. By tracing over one output mode, we construct the reduced density matrix rho_2 and compute the von Neumann entropy S = - sum_k P_k log2 P_k, a direct measure of entanglement between the two output ports. For the balanced case with N = M = 1, we obtain S = 1 ebit, i.e., maximal Bell-state entanglement. For larger equal photon numbers, the entropy increases monotonically. Our results transform the passive beam splitter into a deterministic source of photon-number entanglement, with direct applications in optical quantum computing, interferometry, and quantum information processing. Numerical three-dimensional plots of the probability distribution are provided, and all derivations are supported by rigorous algebra and a full appendix.
Preparation of entangled states of photons are useful for quantum computing and communication. In this paper, we present a simplistic protocol of entanglement generation using beam splitters with suitable reflectivity. The photons in an initial state with fully classical probability distribution pass through an optical network, made up of sequential beam splitters and are prepared in maximally entangled states. We also present the detailed theoretical analysis of entangled state generation, for an arbitrary number of photons, fed through the input ports of the beam splitters with equal probability.
Homodyne detection is the most effective detection scheme employed in quantum optics to characterize quantum states. It is based on mixing at a beam splitter the signal to be measured with a coherent state, called the “local oscillator,” and on evaluating the difference of the photocurrents of two photodiodes measuring the outputs of the beam splitter. If the local oscillator is much more intense than the field to be measured, the homodyne signal is proportional to the signal-field quadratures. If the local oscillator is less intense, the photodiodes can be replaced with photon-number-resolving detectors, which have a smaller dynamics but can measure the light statistics. The resulting new homodyne-like detector acquires a hybrid nature, being it capable of yielding information on both the particle-like (statistics) and wave-like (phase) properties of light signals. The scheme has been tested in the measurement of the quadratures of coherent states, bracket states and phase-averaged coherent states at different intensities of the local oscillator.