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Quantum Probability Cancellation Due to a Single-Photon State

Z. Y. Ou

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

When an N-photon state enters a lossless symmetric beamsplitter from one input port, the photon distribution for the two output ports has the form of Bernouli Binormial, with highest probability at equal partition (N/2 at one outport and N/2 at the other). However, injection of a single photon state at the other input port can dramatically change the photon distribution at the outputs, resulting in zero probability at equal partition. Such a strong deviation from classical particle theory stems from quantum probability amplitude cancellation. The effect persists even if the N-photon state is replaced by an arbitrary state of light. A special case is the coherent state which corresponds to homodyne detection of a single photon state and can lead to the measurement of the wave function of a single photon state.

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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.

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quant-pharXiv:2605.02151•May 5, 2026•0 citations
When an N-photon state enters a lossless symmetric beamsplitter from one input port, the photon distribution for the two output ports has the form of Bernouli Binormial, with highest probability at equal partition (N/2 at one outport and N/2 at the other). However, injection of a single photon state at the other input port can dramatically change the photon distribution at the outputs, resulting in zero probability at equal partition. Such a strong deviation from classical particle theory stems from quantum probability amplitude cancellation. The effect persists even if the N-photon state is replaced by an arbitrary state of light. A special case is the coherent state which corresponds to homodyne detection of a single photon state and can lead to the measurement of the wave function of a single photon state.
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.
A lossless beamsplitter has certain (complex-valued) probability amplitudes for sending an incoming photon into one of two possible directions. We use elementary laws of classical and quantum optics to obtain general relations among the magnitudes and phases of these probability amplitudes. Proceeding to examine a pair of (nearly) single-mode wavepackets in the number-states n1 and n2 that simultaneously arrive at the splitter's input ports, we find the distribution of photon-number states at the output ports using an argument inspired by Feynman's scattering analysis of indistinguishable Bose particles. The result thus obtained coincides with that of the standard quantum-optical treatment of beamsplitters via annihilation and creation operators â and â†. A simple application of the Feynman method provides a form of justification for the Bose enhancement implicit in the well-known formulas ân=nn−1 and â†n=n+1n+1.
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.
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.
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.