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Interference of identical particles: collective phases, "circle dancing" and graph theory

V. S. Shchesnovich, M. E. O. Bezerra

Published Jul 12, 2017• 1 citation
OpenAlex

Abstract

To describe how distinguishability affects multiparticle interference of identical bosons (or fermions) is an important fundamental problem. We propose a solution based the weighted graph theory. We show, for instance, that quantum interference of $N$ independent identical particles with no pair of them being in orthogonal internal states is governed by $(N-1)(N-2)/2$ independent triad phases, i.e., no genuine $N$-particle interference, beyond the three-particle interference recently observed, exists with $N\ge 4$ partially distinguished particles. Surprisingly, deterministic distinguishability, preventing quantum interference with two or three particles, can break this rule: setting each particle to be deterministically distinguishable from all others except two, one can switch off the $R$-particle interference for $3\le R\le N-1$, whereas leaving switched on a novel (circle-dance) interference of $N\ge 4$ particles due to the $N$th order quantum correlations. Our results can be useful in quantum computation, quantum information, and quantum metrology with single photons.

Related papers

Collective phases of identical particles interfering on linear multiports

V. S. Shchesnovich, M. E. O. Bezerra

We introduce collective geometric phases of bosons and fermions interfering on a linear unitary multiport, where each phase depends on the internal states of identical particles (i.e., not affected by the multiport) and corresponds to a cycle of the symmetric group. We show that quantum interference of $N$ particles in generic pure internal states, i.e., with no pair being orthogonal, is governed by $(N\ensuremath{-}1)(N\ensuremath{-}2)/2$ independent triad phases (each involving only three particles). The deterministic distinguishability, preventing quantum interference with two or three particles, allows for the genuine $(N\ensuremath{\ge}4)$-particle phase (interference) on a multiport: setting each particle to be deterministically distinguishable from all others except two by their internal states allows for a novel (circle-dance) interference of $N\ensuremath{\ge}4$ particles governed by a collective $N$-particle phase, while simultaneously preventing the $R$-particle interference for $3\ensuremath{\le}R\ensuremath{\le}N\ensuremath{-}1$. The genuine $N$-particle interference manifests the $N\mathrm{th}$-order quantum correlations between identical particles at a multiport output; it does not appear in the marginal probability for a subset of the particles, e.g., it cannot be detected if at least one of the particles is lost. This means that the collective phases are not detectable by the usual ``quantumness'' criteria based on the second-order quantum correlations. The results can be useful for quantum computation, quantum information, and other quantum technologies with single photons.

arXiv:1707.03893•Physical Review A•Sep 6, 2018•28 citations

Distinguishability and Many-Particle Interference

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Quantum interference of two independent particles in pure quantum states is fully described by the particles' distinguishability: the closer the particles are to being identical, the higher the degree of quantum interference. When more than two particles are involved, the situation becomes more complex and interference capability extends beyond pairwise distinguishability, taking on a surprisingly rich character. Here, we study many-particle interference using three photons. We show that the distinguishability between pairs of photons is not sufficient to fully describe the photons' behavior in a scattering process, but that a collective phase, the triad phase, plays a role. We are able to explore the full parameter space of three-photon interference by generating heralded single photons and interfering them in a fiber tritter. Using multiple degrees of freedom-temporal delays and polarization-we isolate three-photon interference from two-photon interference. Our experiment disproves the view that pairwise two-photon distinguishability uniquely determines the degree of nonclassical many-particle interference.

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Quantum interference of two independent photons in pure quantum states is fully described by the particles' distinguishability: the closer the particles are to being identical, the higher the degree of quantum interference. When more than two particles are involved, the situation becomes more complex and interference capability extends beyond pairwise distinguishability. Here, we study many-particle interference using three photons. We show that the three distinct distinguishabilities between pairs of photons are not sufficient to fully describe the photons' behaviour in a scattering process, but that additionally a fourth parameter, a collective phase, governs the scattering process. We use heralded single photons and a fibre-optic tritter to explore this full parameter space for the first time and thereby demonstrate genuine three-photon interference. Further, our work challenges the view that using time delays between identical photons alone suffices to fully study multi-photon interference.

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Complementarity between bosonic and fermionic many-body interferences with partially distinguishable particles

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It is well known that bosons and fermions exhibit opposite behaviors when experiencing interference, in the sense that bosons have a tendency to bunch whereas fermions have a tendency to antibunch. Recently, this complementarity was mathematically characterized in [arXiv:2312.17709] by means of an identity relating the transition probabilities of both types of particles in a linear interferometer. Here, we show that such a complementarity still holds even when particles become partially distinguishable, for example, when they have slightly different polarizations or time delays. Namely, we establish a relation that combines bosonic and fermionic multiparticle interferences in an arbitrary linear interferometer, in the presence of partial distinguishability. Incidentally, this also provides a new mathematical identity relating the permanent and determinant of tensors of order 3. Importantly, this complementarity has direct operational consequences in quantum metrology. Indeed, we show that the correlation matrices for bosonic and fermionic particle number distributions at the output of the interferometer obey a simple sum rule: their sum equals twice the correlation matrix for classical particles. This, in turn, constraints the achievable quantum Fisher information in phase-estimation protocols, highlighting a trade-off whereby greater indistinguishability enhances bosonic sensitivity whereas reduced indistinguishability can benefit fermionic schemes.

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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.

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Interference of identical particles from entanglement to boson-sampling

Malte C. Tichy

Progress in the reliable preparation, coherent propagation and efficient detection of many-body states has recently brought collective quantum phenomena of many identical particles into the spotlight. This tutorial introduces the physics of many-boson and many-fermion interference required for the description of current experiments and for the understanding of novel approaches to quantum computing. The field is motivated via the two-particle case, for which the uncorrelated, classical dynamics of distinguishable particles is compared to the quantum behaviour of identical bosons and fermions. Bunching of bosons is opposed to anti-bunching of fermions, while both species constitute equivalent sources of bipartite two-level entanglement. The realms of indistinguishable and distinguishable particles are connected by a monotonic transition, on a scale defined by the coherence length of the interfering particles. As we move to larger systems, any attempt to understand many particles via the two-particle paradigm fails: In contrast to two-particle bunching and anti-bunching, the very same signatures can be exhibited by bosons and fermions, and coherent effects dominate over statistical behaviour. The simulation of many-boson interference, termed Boson-Sampling, entails a qualitatively superior computational complexity when compared to fermions. The hierarchy between bosons and fermions also characterises multipartite entanglement generation, for which bosons again clearly outmatch fermions. Finally, the quantum-to-classical transition between many indistinguishable and many distinguishable particles features non-monotonic structures. While the same physical principles govern small and large systems, the deployment of the intrinsic complexity of collective many-body interference makes more particles behave differently.

arXiv:1312.4266•Journal of Physics B Atomic Molecular and Optical Physics•May 7, 2014•96 citations
To describe how distinguishability affects multiparticle interference of identical bosons (or fermions) is an important fundamental problem. We propose a solution based the weighted graph theory. We show, for instance, that quantum interference of NNN independent identical particles with no pair of them being in orthogonal internal states is governed by (N−1)(N−2)/2(N-1)(N-2)/2(N−1)(N−2)/2 independent triad phases, i.e., no genuine NNN-particle interference, beyond the three-particle interference recently observed, exists with N≥4N\ge 4N≥4 partially distinguished particles. Surprisingly, deterministic distinguishability, preventing quantum interference with two or three particles, can break this rule: setting each particle to be deterministically distinguishable from all others except two, one can switch off the RRR-particle interference for 3≤R≤N−13\le R\le N-13≤R≤N−1, whereas leaving switched on a novel (circle-dance) interference of N≥4N\ge 4N≥4 particles due to the NNNth order quantum correlations. Our results can be useful in quantum computation, quantum information, and quantum metrology with single photons.
We introduce collective geometric phases of bosons and fermions interfering on a linear unitary multiport, where each phase depends on the internal states of identical particles (i.e., not affected by the multiport) and corresponds to a cycle of the symmetric group. We show that quantum interference of NNN particles in generic pure internal states, i.e., with no pair being orthogonal, is governed by $(N\ensuremath{-}1)(N\ensuremath{-}2)/2$ independent triad phases (each involving only three particles). The deterministic distinguishability, preventing quantum interference with two or three particles, allows for the genuine $(N\ensuremath{\ge}4)$-particle phase (interference) on a multiport: setting each particle to be deterministically distinguishable from all others except two by their internal states allows for a novel (circle-dance) interference of $N\ensuremath{\ge}4$ particles governed by a collective NNN-particle phase, while simultaneously preventing the RRR-particle interference for $3\ensuremath{\le}R\ensuremath{\le}N\ensuremath{-}1$. The genuine NNN-particle interference manifests the NthN\mathrm{th}Nth-order quantum correlations between identical particles at a multiport output; it does not appear in the marginal probability for a subset of the particles, e.g., it cannot be detected if at least one of the particles is lost. This means that the collective phases are not detectable by the usual ``quantumness'' criteria based on the second-order quantum correlations. The results can be useful for quantum computation, quantum information, and other quantum technologies with single photons.
Quantum interference of two independent particles in pure quantum states is fully described by the particles' distinguishability: the closer the particles are to being identical, the higher the degree of quantum interference. When more than two particles are involved, the situation becomes more complex and interference capability extends beyond pairwise distinguishability, taking on a surprisingly rich character. Here, we study many-particle interference using three photons. We show that the distinguishability between pairs of photons is not sufficient to fully describe the photons' behavior in a scattering process, but that a collective phase, the triad phase, plays a role. We are able to explore the full parameter space of three-photon interference by generating heralded single photons and interfering them in a fiber tritter. Using multiple degrees of freedom-temporal delays and polarization-we isolate three-photon interference from two-photon interference. Our experiment disproves the view that pairwise two-photon distinguishability uniquely determines the degree of nonclassical many-particle interference.
Quantum interference of two independent photons in pure quantum states is fully described by the particles' distinguishability: the closer the particles are to being identical, the higher the degree of quantum interference. When more than two particles are involved, the situation becomes more complex and interference capability extends beyond pairwise distinguishability. Here, we study many-particle interference using three photons. We show that the three distinct distinguishabilities between pairs of photons are not sufficient to fully describe the photons' behaviour in a scattering process, but that additionally a fourth parameter, a collective phase, governs the scattering process. We use heralded single photons and a fibre-optic tritter to explore this full parameter space for the first time and thereby demonstrate genuine three-photon interference. Further, our work challenges the view that using time delays between identical photons alone suffices to fully study multi-photon interference.
It is well known that bosons and fermions exhibit opposite behaviors when experiencing interference, in the sense that bosons have a tendency to bunch whereas fermions have a tendency to antibunch. Recently, this complementarity was mathematically characterized in [arXiv:2312.17709] by means of an identity relating the transition probabilities of both types of particles in a linear interferometer. Here, we show that such a complementarity still holds even when particles become partially distinguishable, for example, when they have slightly different polarizations or time delays. Namely, we establish a relation that combines bosonic and fermionic multiparticle interferences in an arbitrary linear interferometer, in the presence of partial distinguishability. Incidentally, this also provides a new mathematical identity relating the permanent and determinant of tensors of order 3. Importantly, this complementarity has direct operational consequences in quantum metrology. Indeed, we show that the correlation matrices for bosonic and fermionic particle number distributions at the output of the interferometer obey a simple sum rule: their sum equals twice the correlation matrix for classical particles. This, in turn, constraints the achievable quantum Fisher information in phase-estimation protocols, highlighting a trade-off whereby greater indistinguishability enhances bosonic sensitivity whereas reduced indistinguishability can benefit fermionic schemes.
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.
Progress in the reliable preparation, coherent propagation and efficient detection of many-body states has recently brought collective quantum phenomena of many identical particles into the spotlight. This tutorial introduces the physics of many-boson and many-fermion interference required for the description of current experiments and for the understanding of novel approaches to quantum computing. The field is motivated via the two-particle case, for which the uncorrelated, classical dynamics of distinguishable particles is compared to the quantum behaviour of identical bosons and fermions. Bunching of bosons is opposed to anti-bunching of fermions, while both species constitute equivalent sources of bipartite two-level entanglement. The realms of indistinguishable and distinguishable particles are connected by a monotonic transition, on a scale defined by the coherence length of the interfering particles. As we move to larger systems, any attempt to understand many particles via the two-particle paradigm fails: In contrast to two-particle bunching and anti-bunching, the very same signatures can be exhibited by bosons and fermions, and coherent effects dominate over statistical behaviour. The simulation of many-boson interference, termed Boson-Sampling, entails a qualitatively superior computational complexity when compared to fermions. The hierarchy between bosons and fermions also characterises multipartite entanglement generation, for which bosons again clearly outmatch fermions. Finally, the quantum-to-classical transition between many indistinguishable and many distinguishable particles features non-monotonic structures. While the same physical principles govern small and large systems, the deployment of the intrinsic complexity of collective many-body interference makes more particles behave differently.