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arXiv:1806.01236 · Physical Review A

Discriminating distinguishability

Stasja Stanisic, Peter S. Turner

Published Oct 19, 2018• 18 citations
arXivHTMLPDF

Abstract

Particle distinguishability is a significant challenge for quantum technologies, in particular for photonics, where the Hong-Ou-Mandel (HOM) effect clearly demonstrates it is detrimental to quantum interference. We take a representation theoretic approach in first quantization, separating particles' Hilbert spaces into degrees of freedom that we control and those we do not, yielding a quantum-information-inspired bipartite model where distinguishability can arise as correlation with an environment carried by the particles themselves. This makes it clear that the HOM experiment is an instance of a (mixed) state discrimination protocol, which can be generalized to interferometers that discriminate unambiguously between ideal indistinguishable states and interesting distinguishable states, leading to bounds on the success probability of an arbitrary HOM generalization for multiple particles and modes. After setting out the first quantized formalism in detail, we consider several scenarios and provide a combination of analytical and numerical results for up to nine photons in nine modes. Although the quantum Fourier transform features prominently, we see that it is suboptimal for discriminating completely distinguishable states.

DOI
10.1103/physreva.98.043839

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Particle distinguishability is a significant challenge for quantum technologies, in particular for photonics, where the Hong-Ou-Mandel (HOM) effect clearly demonstrates it is detrimental to quantum interference. We take a representation theoretic approach in first quantization, separating particles' Hilbert spaces into degrees of freedom that we control and those we do not, yielding a quantum-information-inspired bipartite model where distinguishability can arise as correlation with an environment carried by the particles themselves. This makes it clear that the HOM experiment is an instance of a (mixed) state discrimination protocol, which can be generalized to interferometers that discriminate unambiguously between ideal indistinguishable states and interesting distinguishable states, leading to bounds on the success probability of an arbitrary HOM generalization for multiple particles and modes. After setting out the first quantized formalism in detail, we consider several scenarios and provide a combination of analytical and numerical results for up to nine photons in nine modes. Although the quantum Fourier transform features prominently, we see that it is suboptimal for discriminating completely distinguishable states.
This article is a detailed introduction to Hong-Ou-Mandel (HOM) interference, in which two photons interfere on a beamsplitter in a way that depends on the photons' distinguishability. We begin by considering distinguishability in the polarization degree of freedom. We then consider spectral distinguishability, and show explicitly how to calculate the HOM dip for three interesting cases: 1) photons with arbitrary spectral distributions, 2) spectrally entangled photons, and 3) spectrally mixed photons.
Usually fermions and bosons behave very differently. Regarding distinguishability, they behave in one and the same way. Two-identical-particle quantum theory is discussed on the example of the deuteron and that of distant correlations, as well as on the example of the real interference and quantum erasure experiments in the Hong–Ou–Mandel two-photon interferometer (with and without polarizers and analyzers). Distinguishability and indistinguishability of two identical photons in the experiments are illustrated. It is demonstrated, using a simple formalism valid simultaneously for fermions and for bosons, that there exists an isomorphism giving theoretical foundation to distinguishing two identical particles. But the distinguishing has a limited range of exact applicability. In most cases of interest, only the indistinguishability approach has a first-principle (and universal) validity.
We show that it is possible for completely distinguishable particles to interfere postselectively without operating on, or indeed having any knowledge of, the distinguishing degree of freedom. In particular, we find a family of three-mode spatial interferometers that, upon inputting, say, a red photon in port 1 and a green photon in port 2, produce a state such that when vacuum is detected at one output, the two photons in the other two outputs will pass a HOM test, despite their frequency degree of freedom remaining untouched. In doing so we develop a general approach to distinguishability based on the Schmidt decomposition between particles' "System" and "Label" degrees of freedom, corresponding to what has been called unitary-unitary duality in many-body physics. This also gives insight into the relationship between the first and second quantized pictures, which is useful in bringing other quantum information concepts across from the former to the latter.
Quantum interference constitutes one of the sharpest divides between our classical intuition and the weirdness of the quantum world. Whenever there are multiple paths leading to the same outcome, it is the interplay of associated probability amplitudes that determines measurement results. In the double slit experiment, wave-like interference governs the distribution of particles on a detection screen. Extending to multiple particles allows behaviour that cannot be described by a classical wave, such as the bunching of photon pairs in Hong-Ou-Mandel interference. In both these examples, the ability to distinguish the interfering paths leads to degradation of interference and a return to behaviour described by probabilities. Understanding how this transition evolves for larger multiparticle quantum systems is of fundamental interest and also vital for developing photonic quantum technologies that rely on many-photon interference. In this thesis we investigate the role of distinguishability in the interference of three and four independent photons by using polarisation and time delays. For three photons, a new type of collective distinguishing phase appears that goes beyond a pairwise description of the similarity of quantum states. We use nonlinear photon sources and a fibre interferometer to experimentally demonstrate the ability to tune and isolate this phase in interference statistics. Extending to four photons we show that, contrary to intuition from two-photon experiments, the interference of photons prepared in distinguishable states is possible due to a four-particle phase. This is confirmed experimentally using a bulk four-mode interferometer. We then extend our study to include the effects of state impurity in the interference of three photons. New properties of this interference enable diagnosis of realistic experimental imperfections and characterisation tasks impossible using only two photons. Finally, we link discussions of state distinguishability to the group representations underpinning the exchange symmetry of multiparticle wavefunctions.
Quantum interference forces two indistinguishable input photons to depart a beamsplitter in the same (of two possible) spatial output modes; this is known as the Hong-Ou-Mandel (HOM) effect. It provides a quantitative way of measuring the distinguishability between two photons and also has been utilised to determine their relative temporal delays [1, 2]. HOM interferometry offers numerous advantages over its classical counterpart, most importantly it lacks any dependence on the phase of the photons and therefore can potentially provide reliable measurements in environments where the phase is unstable. To date there have been very few investigations on the precision that is achievable with HOM interferometry. Experiments with collinear geometries have yielded sensitivities as low as 0.1fs however these approaches are limited to measurements of birefringence and other polarisation dependent effects [3, 4], as the precision derives from the inherently stable shared path of the photon pairs. Typical non-shared path HOM interferometers exhibit resolution of roughly 1–3 fs (0.3–1 μm).
The Hong-Ou-Mandel (HOM) effect is a quintessential process in various quantum information technologies and quantum optics applications. In this work, we investigate multi-photon interference, developing a model for the simultaneous characterization of polarization and spectro-temporal mode mismatch on the coincidence probabilities including the effects of realistic imperfections of devices used in HOM experiments. We also study the coincidence probability for coherent states as a function of source intensity, as well as spectro-temporal and polarization mismatch of the incident beams. We apply our model to the case of multi-photon interference from independent sources and analyze the consequences of mode mismatch in various instances that occur in quantum networking including entanglement swapping, quantum key distribution, quantum sensing, quantum optical classification, and photonic quantum computing.