Fluctuations in the detection of the HOM effect

www.nature.com/scientificreports OPEN Fluctuations in the detection of the HOM effect Dmitry N. Makarov Hong-Ou-Mandel (HOM) effect is known to be one of the main phenomena in quantum optics. It is believed that the effect occurs when two identical single-photon waves enter a 1:1 beam splitter, one in each input port. When the photons are identical, they will extinguish each other. In this work, it is shown that these fundamental provisions of the HOM interference may not always be fulfilled. One of the main elements of the HOM interferometer is the beam splitter, which has its own coefficients of reflection R = 1/2 and transmission T = 1/2. Here we consider the general mechanism of the interaction of two photons in a beam splitter, which shows that in the HOM theory of the effect it is necessary to know (including when planning the experiment) not only R = 1/2 and T = 1/2, but also their root-mean-square fluctuations R2 , T 2, which arise due to the dependence of R = R(ω1 , ω2 ) and T = T(ω1 , ω2 ) on the frequencies where ω1 , ω2 are the frequencies of the first and second photons, respectively. Under certain conditions, specifically when the dependence of the fluctuations R2 and T 2 can be neglected and R = T = 1/2 is chosen, the developed theory coincides with previously known results. The HOM effect was first experimentally demonstrated by Hong et al in 1 9871. HOM interference shows up in many instances, both in fundamental studies of quantum mechanics and in practical implementations of quantum technologies2–8. For example, one of the main practical applications of the HOM effect is to check the degree of indistinguishability of two incoming photons. When the HOM dip reaches all the way down to zero coincident counts, the incoming photons are perfectly indistinguishable, whereas if there is no dip, the photons are distinguishable. A HOM interferometer scheme was presented in1, one of the main elements of which was a beam splitter (BS). To observe quantum interference, a BS is chosen close to 1:1 (having coefficients of reflection R and transmission T close to 1/2). A theoretical explanation of the HOM effect based on constant coefficients R and T and boson statistics of photons is quite simple9,10. In this interpretation, we are not interested in what happens to the incident photons in the BS. For this, they consider BS lossless (hereinafter simply BS) as ideal, i.e. with constant coefficients R and T and BS is the source of the other two photons obeying bosonic statistics. In this case, the annihilation operators before entering 1 and 2 photons in BS represent â1 and â2, respectively, and after exiting BS is b̂1 and b̂2. The transformation from one pair of operators to another is generally described by the BS matrix (denoted as UBS ) in the form (see, e.g.11,12)      iφ √ √  T eiφ2 √R b̂1 â e 1 √ . = UBS 1 , UBS = (1) â2 −e−iφ2 R e−iφ1 T b̂2 It is easy to see that for R = T = 1/2, the photons at the output (described by the operator b̂2 b̂1) only come out in pairs from 1 or 2 ports. This analysis is fundamental to understanding the HOM effect and is not subject to any additional research. The basic scheme HOM interferometer for arbitrary photons (including quantum entangled photons) is shown in Fig. 1. In reality, the pair of photons arriving at the BS do not have a set frequency, but have a certain frequency distribution. Nonetheless, in the theoretical description (e.g.1,13–17) of the experimentally observed value P (P is the joint probability of detecting photons after exiting the BS on the output ports), the frequency distribution does not affect BS matrix UBS , because R and T are constant values. Currently, the wellknown HOM effect theories are based on calculating the value of P within the constant values of R = T = 1/2. In the work presented the coefficients R and T are variables, which significantly affects the theory of the HOM effect. The problem of interaction of two photons in BS is solved analytically, allowing the determination of the photon statistics after exiting the BS. Within the general form UBS is a BS matrix similar to Eq. (1), where R and T are some functions that depend on the frequencies of incident photons, the interaction time of two photons in BS, and on the BS material. This leads to the value of the coincidence counting probability P being calculated to take into account the dependence on the frequencies of R and T. It is shown that even in the case of identical incident photons and their average values R̄ = T̄ = 1/2 (averaging over the frequencies of incident photons), a Northern (Arctic) Federal University, Arkhangelsk 163002, Russia. email: Scientific Reports | (2020) 10:20124 | https://doi.org/10.1038/s41598-020-77189-6 1 Vol.:(0123456789) www.nature.com/scientificreports/ Figure 1.  Schematic representation of the HOM interferometer, where D1 , D2 are the first and second detectors, respectively; τ is the time delay between 1 and 2 photons and δτ is the time delay caused by the spatial displacement of the BS from the equilibrium position. zero value of P may not be observed, despite being predicted by the HOM interference theory taking into account the constants R and T. Indeed, in the case of constant coefficients, as well as without a time delay between two photons, i.e. δτ = 0 or τ = 0 and identical photons, because it is well known that P ∝ (R − T)2, for R = T = 1/2 we get P = 01,13. In our case, P ∝ (R − T)2 , which means P ∝ R2 − (R)2 or P ∝ T 2 − (T)2 (when R̄ = T̄ = 1/2) i.e. there is a fluctuation in the reflection and transmission coefficients, had was not earlier taken into account in theoretical and experimental studies. It is shown (arbitrary falling photons, including not only Fock state, but also taking into account the time delay δτ and τ ) that under certain conditions the coefficients R and T can be considered constant, and the results obtained pass into well-known approaches. The theory developed here is especially important when planning experiments in the HOM interferometer and analyzing them; because the fluctuations of R and T can be very large, the results of such experiments may not be correctly interpreted. Photons in BS It is well known that in quantum optics two modes of the electromagnetic field (two input ports) are usually considered, since even if 1 port remains unused, it should be considered as an input for vacuum fluctuations9,10. Thus, we will proceed from the fact that we have two ports at the input. Two input and output ports can be in the form of freely spreading photons or in the form of waveguides along which photons propagate. If the waveguides are connected, then we get BS in the form of a coupled waveguide. It should be added that such systems are usually studied using various simplified models, for example, Bose–Hubbar m odel18, Jaynes–Cummings model (JCM)19, Dicke m odel20 and others. We will try to approach this problem based on a complete record of t (...truncated)