Quantum entanglement and statistics of photons on a beam splitter in the form of coupled waveguides

Abstract

It is well known that a beam splitter (BS) can be used as a source of photon quantum entanglement. This is due to the fact that the statistics of photons changes at the output ports of the BS. Usually, quantum entanglement and photon statistics take into account the constancy of the reflection coefficient R or the transmission coefficient T of the BS, where $$R + T = 1$$ R + T = 1 . It has recently been shown that if BS is used in the form of coupled waveguides, the coefficients R and T will depend on the photon frequencies. In this paper, it is shown that the quantum entanglement and statistics of photons at the output ports of a BS can change significantly if a BS is used in the form of coupled waveguides, where the coefficients R and T are frequency-dependent.

Quantum Entanglement of Monochromatic and Non-Monochromatic Photons on a Waveguide Beam Splitter

“…In the case of constants R and T , quantum entanglement is a periodic function contingent on , and for large a rapidly oscillating dependence, which is a negative factor for use in quantum technologies. It should be added that in [ 21 ] the simplest case of input states was analyzed in detail and it was shown that quantum entanglement and photon statistics, at the BS output ports, can be very different for non-monochromatic photons compared with monochromatic photons. In this paper a similar conclusion is drawn, but in the general case, not limited to states.…”

“…Let us choose von Neumann entropy as a measure of quantum entanglement. It is well known [ 20 , 21 , 24 , 25 , 26 ] that in this case the entropy will be .…”

“…It is well known that the wave function accounting for non-monochromaticity of photons can be represented as [ 20 , 21 , 28 , 29 ] where is determined similarly to monochromatic photons, i.e., is the state of the photons at the BS output ports; and are the initial number of photons at 1 and 2 input ports, respectively, the joint spectral amplitude (JSA) of the two-mode wave function ( ), is determined from Equation ( 2 ). In this case, the probability of detecting photons in k and states at the first and second ports of the BS, respectively, will be …”

“…In this case, as previously well known, the wave function of the photon is factorizable, i.e., , where . To calculate quantum entanglement, it is natural to use von Neumann entropy [ 1 , 20 , 21 , 24 , 25 , 26 ] Next, let us choose ( ) in the most commonly used form, this is a Gaussian distribution where is the mean frequency (expectation) and is the dispersion. Below, we will use the condition that applies to most photon sources, .…”

“…This is due to the fact that in these theories coefficients R and T are always constant for the waveguide BS. Recently, [ 20 , 21 ] the theory of a frequency-dependent BS in the form of coupled waveguides was presented. In these papers, it was shown that if the BS is represented as a coupled waveguide, the coefficients R and T depend on the frequencies of the photons fed into both ports of the BS.…”