The zeroing of second order correlation functions between output fields after interferences in a 50/50 beam splitter has been accepted decades-long in the quantum optics community as an indicator of the quantum nature of lights. But, a recent work [1] presented some notable discussions and experiments that classical electromagnetic fields can still exhibit the zero correlation under specific conditions. Here, we examine analytically classical and quantum electromagnetic field interferences in a 50/50 beam splitter in the context of the second order correlation function for various input conditions. Adopting the Heisenberg picture in quantum electromagnetics, we examine components of four-term interference terms in the numerator of second order correlation functions and elucidate their physical significance. As such, we reveal the fundamental difference between the classical and quantum interference as illustrated by the Hong-Ou-Mandel (HOM) effect. The quantum HOM effect is strongly associated with: (1) the commutator relation that does not have a classical analogue; (2) the property of Fock states needed to stipulate the one-photon quantum state of the system; and (3) a destructive wave interference effect. Here, (1) and (2) imply the indivisibility of a photon. On the contrary, the classical HOM effect requires the presence of two destructive wave interferences without the need to stipulate a quantum state.
The zeroing of second order correlation functions between output fields after interferences in a 50/50 beam splitter has been accepted decades-long in the quantum optics community as an indicator of the quantum nature of lights. But, a recent work [1] presented some notable discussions and experiments that classical electromagnetic fields can still exhibit the zero correlation under specific conditions. Here, we examine analytically classical and quantum electromagnetic field interferences in a 50/50 beam splitter in the context of the second order correlation function for various input conditions. Adopting the Heisenberg picture in quantum electromagnetics, we examine components of four-term interference terms in the numerator of second order correlation functions and elucidate their physical significance. As such, we reveal the fundamental difference between the classical and quantum interference as illustrated by the Hong-Ou-Mandel (HOM) effect. The quantum HOM effect is strongly associated with: (1) the commutator relation that does not have a classical analogue; (2) the property of Fock states needed to stipulate the one-photon quantum state of the system; and (3) a destructive wave interference effect. Here, (1) and (2) imply the indivisibility of a photon. On the contrary, the classical HOM effect requires the presence of two destructive wave interferences without the need to stipulate a quantum state.
Two-photon quantum interference at a beam splitter, commonly known as Hong–Ou–Mandel interference, is a fundamental demonstration of the quantum mechanical nature of electromagnetic fields and a key component of various quantum information processing protocols. The phenomenon was recently demonstrated with microwave-frequency photons by Lang et al (2013 Nature Phys. 9 345–8). This experiment employed circuit QED systems as sources of microwave photons, and was based on the measurement of second-order cross-correlation and auto-correlation functions of the microwave fields at the outputs of the beam splitter using linear detectors. Here we present the calculation of these correlation functions for the cases of inputs corresponding to: (i) trains of pulsed Gaussian or Lorentzian single microwave photons and (ii) resonant fluorescent microwave fields from continuously driven circuit QED systems. In both cases, the signature of two-photon quantum interference is a suppression of the second-order cross-correlation function for small delays. The experiment described in Lang et al (2013) was performed with trains of Lorentzian single photons, and very good agreement with experimental data is obtained. The results are relevant not only to interference experiments using circuit QED systems, but any such setup with highly controllable sources and time-resolved detection.
When quantum state amplitudes interfere, surprising non-classical features emerge which emphasis the roles of indistinguishability and discreteness in quantum mechanics. A famous example in quantum optics is the Hong Ou Mandel interference effect,a major ingredient in current quantum information processing using photonics. Traditionally the HOM features interference between amplitudes for two one-photon number states. Surprisingly, interference can be manifested when one amplitude represents that most classical of light field states, the coherent state, provided the partner state is non-classical (eg a single photon state or an odd photon number state). Imposing such nonclassical features on an otherwise classical state is the focus of this article. Recently, the HOM effect has been generalized to the multi-photon case, termed the extended HOM effect by the authors.The implication of the extended HOM effect is that if an odd parity state, comprising only odd numbers of photons, enters one input port of a 50:50 beam splitter, then regardless of the state entering the other input port, be it pure or mixed, there will no output coincident counts. In this work, we explain the extended HOM as arising from a sequence of pairwise HOM-like complete destructive interferences occurring simultaneously in the multicomponent amplitude for the output coincidence counts. We first demonstrate this diagrammatically in order to build physical intuition, before developing a general analytical proof. We then examine the case of a single photon interacting with a coherent state (and idealized laser), and consider prospects for experimental detection by including the effect of imperfect detection efficiency. This work highlights the importance of the non-classicality of light, and in particular the interference effects stemming from the discreteness of photon quanta.
Two-photon interference effects arise because photons are indistinguishable particles. In the wellknown Hong-Ou-Mandel (HOM) effect, the transmission of two photons at a beam splitter interferes destructively with the reflection of both photons, requiring both photons to "bunch up" by leaving the beam splitter on the same side. Here, we show that the interference between locally propagating photons and photons exchanged by a mode swap can be implemented by post-selecting spatially separated photon outputs of a four-path interferometer. Even though the photons detected at spatially separated locations must have travelled along paths that never met up at the same beam splitter, the Hong-Ou-Mandel effect can be observed in correlations between the output ports that originate from the association of detection events with non-local output modes defined by the two single photon inputs. Local phase shifts can be used to map out non-classical correlations between the photons detected at different output locations, clarifying the role of linear optics in generating entanglement between spatially separated photons. Our work thus establishes a fundamental relation between multiphoton interference and entanglement, opening the door to new possibilities in optical quantum technologies.
Two-photon quantum interference at a beam splitter, commonly known as Hong–Ou–Mandel interference, is a fundamental demonstration of the quantum mechanical nature of electromagnetic fields and a key component of various quantum information processing protocols. The phenomenon was recently demonstrated with microwave-frequency photons by Lang et al (2013 Nature Phys. 9 345–8). This experiment employed circuit QED systems as sources of microwave photons, and was based on the measurement of second-order cross-correlation and auto-correlation functions of the microwave fields at the outputs of the beam splitter using linear detectors. Here we present the calculation of these correlation functions for the cases of inputs corresponding to: (i) trains of pulsed Gaussian or Lorentzian single microwave photons and (ii) resonant fluorescent microwave fields from continuously driven circuit QED systems. In both cases, the signature of two-photon quantum interference is a suppression of the second-order cross-correlation function for small delays. The experiment described in Lang et al (2013) was performed with trains of Lorentzian single photons, and very good agreement with experimental data is obtained. The results are relevant not only to interference experiments using circuit QED systems, but any such setup with highly controllable sources and time-resolved detection.
Hong-Ou-Mandel (HOM) effect is known to be one of the main phenomena in quantum optics. 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. 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 $. In this work, the general mechanism of the interaction of two photons in a beam splitter is considered using an analytical solution, 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 $ ΔR ^ 2, ΔT ^ 2 $, which arise due to the dependence of $R = R(ω_1, ω_2) $ and $ T = T (ω_1, ω_2) $ 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 $ ΔR^2 $ and $ ΔT^2 $ can be neglected and $ R=T=1/2 $ is chosen, the developed theory coincides with previously known results.
We investigate the recent multiphoton extension [Alsing et al., Phys. Rev. A 105, 013712 (2022)] of the celebrated two-photon Hong-Ou-Mandel (HOM) effect [Hong, Ou, and Mandel, Phys. Rev. Lett. 59, 2044 (1987)] for the quantum interference of amplitudes that lead to complete destructive interference for coincident counts for the output of a balanced, lossless 50:50 beam splitter (BS). We examine the extended HOM (eHOM) effect by first investigating diagrammatically the multiphoton scattering of photons from the two input ports to the two output ports for Fock number basis states (FS) ${|n,m\ensuremath{\rangle}}_{12}$, and show that destructive interference occurs pairwise within the quantum amplitudes with many scattering components, generalizing the two-photon HOM effect. This analysis leads us to a general, simple analytical derivation of the eHOM effect for FS/FS inputs, which has profound implications for arbitrary bipartite photonic input states constructed from such basis states: if the input state to one input port of the BS is of odd parity, i.e., constructed only of odd numbers of photons, then regardless of the input state to the second 50:50 BS port, there will be a central nodal line (CNL) of zeros in the joint output probability distribution along the main diagonal for coincidence detection. We further examine the case of a single photon interacting with a coherent state (an idealized laser) and consider prospects for experimental detection by including additional effects such as imperfect detection efficiency and time delay between the detected output photons. This work highlights the importance of the nonclassicality of light, and in particular the interference effects stemming from the discreteness of photon quanta.