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The DOI matches its Crossref record; 36 of the 36 references it cites were found in a registry or an index; none of the 11 sources its claims cite is checked for support yet.

Quantum-clustered two-photon walks

Open access
Simon, David S., Osawa, Shuto, Sergienko, Alexander V.·2020·Physical Review A
DOI 10.1103/physreva.101.032118
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Abstract

We demonstrate a previously unknown two-photon effect in a discrete-time quantum walk. Two identical bosons with no mutual interactions nonetheless can remain clustered together as they walk on a lattice of directionally reversible optical four-ports acting as Grover coins; both photons move in the same direction at each step due to a two-photon quantum interference phenomenon reminiscent of the Hong-Ou-Mandel effect. The clustered two-photon amplitude splits into two localized parts, one oscillating near the initial point and the other moving ballistically without spatial spread, in solitonlike fashion. But the two photons are always clustered in the same part of the superposition, leading to potential applications for transport of entanglement and opportunities for novel two-photon interferometry experiments.

Overview

AI snapshot

Written from the paper’s full text.

TL;DR

Two indistinguishable non-interacting photons in a discrete-time quantum walk on a chain of directionally-unbiased Grover four-ports remain spatially clustered, with the two-photon amplitude splitting into one oscillating and one ballistic localized component. The effect is a persistent quantum-walk analog of Hong-Ou-Mandel interference, and polarization entanglement is preserved for certain input states.

Key findings

  • For two indistinguishable photons entering ports 1 and 2 of the same four-port, first-step transition probabilities are \(P(LL\to LL)=1/2\), \(P(LL\to RR)=1/2\), and \(P(LL\to LR)=0\), so the photons always step in the same direction.
  • After the first step, the clustered two-photon amplitude becomes an equal superposition of two localized states: one moves ballistically without spatial spread, while the other oscillates between lattice sites \(m=0\) and \(m=1\).
  • The two photons are always found in the same part of the superposition; measurement finds either two photons or none at a given horizontal location, and the photons never separate into different components.
  • For polarization-entangled input states \(|A_{12}^{+}\rangle\) and \(|B_{12}^{+}\rangle\), clustering persists and entanglement remains undiminished; perfect state transport occurs probabilistically with 50% probability for \(|\psi_t\rangle\).

Methods

  • Theoretical/analytic study of a discrete-time quantum walk on a dual-rail or ladder chain, with a four-dimensional Grover coin at each vertex represented by the unitary matrix \(U\).
  • Initial state: two indistinguishable right-moving photons injected simultaneously into ports 1 and 2 of the \(m=0\) multiport, far from chain ends; evolution analyzed via tensor products \(U\otimes U\) and tabulated transition amplitudes.
  • Additional analyses cover polarization-entangled input states and coherent-state inputs, including balanced and unbalanced coherent states.

Limitations

  • The paper states that it assumes an idealized system, neglecting losses and other imperfections; experimental implementations on integrated optical chips will be limited by losses, decoherence, and chip imperfections.
  • The paper notes that quantum walks of single photons and entangled photon pairs have typically been implemented for lengths on the order of five to ten time steps, though longer walks have been proposed.
  • The text makes evident that this specific two-photon clustered walk is not experimentally demonstrated here; clustering occurs only for particular input states (\(|A_{12}^{+}\rangle\) and \(|B_{12}^{+}\rangle\)), while other \(A\), \(B\), and \(C\) states decluster.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physreva.101.032118

Cites

36 of 36 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Verification

Claims the paper makes

0 of 11 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    The two photons are always clustered in the same part of the superposition.

    “But the two photons are always clustered in the same part of the superposition”
    • Also stated in: § II. SETUP AND MAIN RESULT. · plain
      “measurement of the two particles will always find them in the same part of the superposition; one will never be found in the oscillating portion and the other in the ballistic portion.”
    • Also stated in: § B. Subsequent steps · plain
      “the two photons are always found clustered in the same localized part of the distribution and never separate from each other.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The system is idealized, with losses and other imperfections neglected.
  • Findingreasoning

    Regardless of which port a photon enters, exit amplitudes at all outputs are real and equal in magnitude.

    “Regardless of which port a photon enters, exit amplitudes at all outputs are real and equal in magnitude.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    Perfect state transport (PST) occurs probabilistically with 50% probability for |ψ_t, delivering two entangled photons to the same horizontal location at the same time.

    “Perfect state transport (PST) occurs probabilistically (50% probability) for |ψ t , delivering two entangled photons to the same horizontal location at the same time.”
    Quote in the paperSubstantial claim
  • Findingnone given

    Which part of the superposition the two photons are found to be clustered in is completely random.

    “Which part of the superposition the two photons are found to be clustered in is completely random, just as the output port in which the two photons are clustered in the HOM effect is purely random.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The system is idealized, with losses and other imperfections neglected.
  • Findingreasoning

    Assuming no loss and ideal measuring devices, measurement at any horizontal location always finds either two photons or none.

    “Assuming no loss and ideal measuring devices, measurement at any horizontal location always finds either two photons or none.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The system is idealized, with losses and other imperfections neglected.
  • It may state two separate claims as one. 23
  • Probably not a substantial claim. 7

  • It may state two separate claims as one.

    Findingreasoning

    These effects depend only on indistinguishable photons being inserted into the same Grover coin vertex simultaneously; entanglement is not required.

    “These effects depend only on indistinguishable photons being inserted into the same Grover coin vertex simultaneously; entanglement is not required.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The system is idealized, with losses and other imperfections neglected.

  • It may state two separate claims as one.

    Findingnone given

    If the particles are entangled, the pair moves as a single unit, with undiminished entanglement.

    “If the particles are entangled, the pair moves as a single unit, with undiminished entanglement.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The resulting state is a superposition of two spatially-localized two-photon clusters, one confined near the origin, the other moving monotonically away.

    “The resulting state is a superposition of two spatially-localized twophoton clusters, one confined near the origin, the other moving monotonically away.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The state is totally reflecting at each step, and the amplitude acts as if it is confined in a virtual cavity, oscillating between lattice sites m = 0 and m = 1.

    “The state is totally reflecting at each step, and the amplitude acts as if it is confined in a virtual cavity, oscillating between lattice sites m = 0 and m = 1.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The system is idealized, with losses and other imperfections neglected.

  • It may state two separate claims as one.

    Findingreasoning

    For unbalanced coherent state input, after one step there is both rightward transmission and leftward reflection, and the amplitudes on the two right-moving lines are equal.

    “There is now both rightward transmission and leftward reflection; however notice that the amplitudes on the two right-moving lines are of equal amplitude again.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: Coherent state amplitudes are inherently fluctuating objects.

  • It may state two separate claims as one.

    Findingreasoning

    The HOM quantum interference effect occurs just once, whereas clustering of the walk persists indefinitely, over an arbitrarily long sequence of steps.

    “However, the HOM quantum interference effect occurs just once, whereas clustering of the walk persists indefinitely, over an arbitrarily long sequence of steps.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The system is idealized, with losses and other imperfections neglected.

  • It may state two separate claims as one.

    Findingreasoning

    For A12- input, clustering does not occur, and the two photons always go in opposite directions at the first step.

    “Not only does clustering not occur, but the two photons always go in opposite directions at this first step.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Findingreasoning

    For the product state A11+, cross terms mix left-moving (1 and 2) exit states with right-moving (3 and 4) states, so no clustering occurs.

    “Since cross terms occur that mix left-moving (1 and 2) exit states with right-moving (3 and 4) states, no clustering occurs.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The B12+ state splits into a difference of reflecting and transmitting states, again leading to photon clustering.

    “In a similar manner, B 12 + splits into a difference of reflecting and transmitting states: |B 12 + → |ψ t -|ψ r ,(B7) again leading to photon clustering.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The clustered two-photon amplitude splits into two localized parts, one oscillating near the initial point, and the other moving ballistically without spatial spread, in soliton-like fashion.

    “The clustered two-photon amplitude splits into two localized parts, one oscillating near the initial point, and the other moving ballistically without spatial spread, in soliton-like fashion.”
    • Also stated in: § II. SETUP AND MAIN RESULT. · plain
      “It breaks after the first step into a sustainable superposition of two distinct localized states (Fig. 5).”
    • Also stated in: § II. SETUP AND MAIN RESULT. · plain
      “The two halves of the clustered amplitude do not spread as they propagate, exhibiting soliton-like behavior.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The system is idealized, with losses and other imperfections neglected.

  • It may state two separate claims as one.

    Findingreasoning

    For the two-photon entangled input state considered, the photons cluster: both exit left (ports 1 and 2) or both exit right (ports 3 and 4).

    “It is readily seen that once again the photons cluster: both exit left (ports 1 and 2) or both exit right (3 and 4).”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    The polarization-entanglement remains undiminished as the walk proceeds, and clustered two-photon states remain polarization-entangled between upper and lower edges as they propagate.

    “the polarization-entanglement remains undiminished as the walk proceeds.”
    • Also stated in: § IV. POLARIZATION-ENTANGLED INPUT · plain
      “clustered two-photon states remain polarization-entangled between upper and lower edges as they propagate.”
    Quote in the paperSubstantial claim

  • It may state two separate claims as one.

    Findingreasoning

    If the photons are distinguishable, each exhibits an independent quantum walk and can later be detected in widely separated spatial regions; lack of destructive interference leads to nonzero off-diagonal amplitudes indicating they may become spatially separated.

    “if the photons are distinguishable, then each exhibits an independent quantum walk and can later be detected in widely separated spatial regions.”
    • Also stated in: § B. Subsequent steps · hedged
      “In contrast, if the two photons are distinguishable (Fig. 7), the lack of destructive interference leads to the appearance of nonzero off-diagonal amplitudes, indicating that the photons may become spatially separated.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The system is idealized, with losses and other imperfections neglected.

  • It may state two separate claims as one.

    Findingreasoning

    Taking the sum of these, the amplitudes |11, |22, and |12 cancel out at each step, so that |ψt simply reproduces itself, but shifted one step to the right.

    “Taking the sum of these as in Eq. 13, one finds that the amplitudes |11 , |22 , and |12 cancel out at each step, so that |ψ t simply reproduces itself, but shifted one step to the right:”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The system is idealized, with losses and other imperfections neglected.

  • It may state two separate claims as one.

    Findingreasoning

    Inserting phase shifts of π/2 to an upper line and -π/2 to the corresponding lower line converts the reflecting and transmitting states of Eqs. 13 and 14 into each other, allowing the experimenter to controllably switch back and forth between ballistic and oscillating behavior.

    “inserting phase shifts of π 2 to an upper line and -π 2 to the corresponding lower line converts the reflecting and transmitting states of Eqs. 13 and 14 into each other, allowing the experimenter to controllably switch back and forth between ballistic and oscillating behavior.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

    Interpretationreasoning

    Quantum walks can probe large regions faster than classical walks, leading to quantum speedups of walk-based algorithms.

    “Consequently, quantum walks can probe large regions faster than classical walks, leading to quantum speedups of walk-based algorithms.”
    Quote in the paperNot a substantial claim

  • It may state two separate claims as one.

    Interpretationreasoning

    Even very small relative phase shifts will destroy the clustering effect, leading to a dip in the two-photon interference when the two halves of the superposition are brought back together, allowing sensitive detection of external fields in the relevant region.

    “even very small relative phase shifts will destroy the clustering effect, leading to a dip in the two-photon interference (similar to the Hong-Ou-Mandel dip) when the two halves of the superposition are brought back together, allowing sensitive detection of external fields in the relevant region.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: External magnetic fields can cause different polarization-dependent phase shifts for vertically and horizontally polarized photons.

  • It may state two separate claims as one.

    Interpretationreasoning

    By adding phase shifts in the lines between the multiports, the flow of the photon pairs can be controlled; they can be stopped at a desired location (oscillating between two adjacent multiports) or their direction of motion can be reversed.

    “by adding phase shifts in the lines between the multiports the flow of the photon pairs can be controlled; they can be stopped at a desired location (oscillating between two adjacent multiports) or their direction of motion can be reversed.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.

    Interpretationnone given

    If the photons are entangled then they remain together with entanglement undiminished as they move.

    “However, if the photons are entangled then they remain together with entanglement undiminished as they move”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The system is idealized, with losses and other imperfections neglected.

  • It may state two separate claims as one.

    Premisecitation

    The symmetric part of a single photon entering the system will always continue rightward at each step, while the antisymmetric portion reflects at each step, leading to oscillations.

    “The symmetric part will always continue rightward at each step, while the antisymmetric portion reflects at each step, leading to oscillations.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [35]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim.

    Interpretationcitation

    Entangled pairs can be delivered in a controllable manner to distant locations for standard applications like entanglement swapping, quantum repeaters, or to control the flow of entanglement for two-photon interference effects in quantum networks.

    “Entangled pairs can be delivered in a controllable manner to distant locations for standard applications like entanglement swapping, quantum repeaters, or to control the flow of entanglement for two-photon interference effects in quantum networks [39].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [39]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Interpretationreasoning

    The Grover four-port selects out only the portion of the reflected state that is antisymmetric under interchange of upper and lower lines, but the input contained only a symmetric part.

    “Equivalently, the Grover four-port selects out only the portion of the reflected state that is antisymmetric under interchange of upper and lower lines, but the input contained only a symmetric part.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation

    In a Hadamard walk, the walker's amplitude produces two peaks that move ballistically in opposite directions, so the distribution's width grows linearly in time, σ ∼ t.

    “In a Hadamard walk, the walker's amplitude produces two peaks that move ballistically in opposite directions (Fig. 2(a)), so the distribution's width grows linearly in time, σ ∼ t.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [5]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [6]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim.

    Premisecitation

    Perfect state transport has been demonstrated for single photon states in various systems.

    “Whereas perfect state transport has been demonstrated for single photon states in various systems [36][37][38]”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [36]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [37]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [38]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Premisereasoning

    The HOM effect: two identical photons incident on different inputs of a 50/50 beam splitter always leave at the same port; no coincidences between output ports.

    “no coincidences are seen between 3 and 4; the two photons always leave at the same port.”
    • Also stated in: § I. INTRODUCTION · plain
      “Coincidences between the two output ports are absent because of cancelations between the two indistinguishable processes on the upper line of Fig. 1(b).”
    • Also stated in: § I. INTRODUCTION · plain
      “As a result, the two photons always remain clustered together in the same output spatial mode.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [1]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Interpretationreasoning

    This provides a means of probing the region that the ballistic portion has traveled through, allowing new two-photon sensing methods.

    “This provides a means of probing the region that the ballistic portion has traveled through, allowing new two-photon sensing methods.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

    Interpretationnone given

    A similar conclusion holds for output measured at some distance to the left of the initial point.

    “A similar conclusion will hold for output measured at some distance to the left of the initial point.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation

    Quantum walks are formally equivalent to a universal quantum computer.

    “they have been shown to be formally equivalent to a universal quantum computer [2][3][4]”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [2]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [3]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [4]

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

28 of 28

  • The authors state this as an assumption.

    The system is idealized, with losses and other imperfections neglected.

    “In what follows, we assume an idealized system, neglecting losses and other imperfections.”
    “Assuming no loss and ideal measuring devices, measurement at any horizontal location always finds either two photons or none.”

  • The authors state this as an assumption.

    The two photons are in product states in the considerations above.

    “In all of the considerations above, the two photons were assumed to be in a product states.”

  • The paper relies on this without support.

    Coherent state amplitudes are inherently fluctuating objects.

    “Coherent state amplitudes are inherently fluctuating objects, so arranging equal amplitudes in the two lines over multiple steps may seem unlikely.”

  • The paper relies on this without support.

    External magnetic fields can cause different polarization-dependent phase shifts for vertically and horizontally polarized photons.

    “they may gain different polarization-dependent phase shifts due to external magnetic fields”
4 of 4

Citations

Cites

Flagged first, then by citations

Research works printed but not linked: 4

  1. A. Ambainis, 2001, Proceedings of the Thirty-third Annual ACM Symposium on Theory of Computing (STOC '01)
  2. S.-Y. Shiau, 2005, Quantum Info. Comput., 5, 492
  3. C. Moore, 2002, Proceedings, 6th International Workshop on Randomization and Approximation Techniques in Computer Science, (RANDOM 02)
  4. R. Boyd, 2008, Nonlinear Optics

Cited by

Sorted by citations

Related

Metadata

Authors
Simon, D. S., Osawa, S., & Sergienko, A. V.
Affiliations
Boston University (US)
ORCID
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2020-03-30
Volume
101
Issue
3
ISSN
  • 0556-2791 (ISSN-L)
  • 2469-9926
  • 2469-9934
DOI
10.1103/physreva.101.032118
OpenAlex
W3011691438
Cited by
20
References
36
Citations by year
2026: 2 · 2025: 4 · 2024: 8 · 2023: 1 · 2022: 1 · 2021: 3 · 2020: 1
FWCI
0.51
Type
journal article
Language
en
Open access
Licence
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • National Science Foundation (ECCS-1640968)
  • Air Force Office of Scientific Research (FA9550-18-1-0056)
  • Northrop Grumman
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
Keywords
discrete-time quantum walks, two-photon interference, Hong-Ou-Mandel interference, identical bosons, two-photon interferometry

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