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Cited by 32Quantum optics and atomic interactionsQuantum Information and CryptographyQuantum Mechanics and Applications

Possibility of coherent phenomena such as Bloch oscillations with single photons viaWstates

Open access
Rai, Amit, Agarwal, G. S.·2009·Physical Review A
DOI 10.1103/physreva.79.053849
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Abstract

We examine the behavior of single photons at multiport devices and inquire if coherent effects are possible. In particular we study how single photons need to be manipulated in order to study coherent phenomena. We show that single photons need to be produced in $W$ states which lead to vanishing mean amplitude but nonzero correlations between the inputs at different ports. Such correlations restore coherent effects with single photons. As a specific example we demonstrate Bloch oscillations with single photons and thus provide strict analog of Bloch oscillation of electrons.

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Written from the paper’s full text.

TL;DR

The paper shows theoretically that single photons prepared in a W-state can exhibit coherent interference effects despite having zero mean field amplitude, because the W-state retains nonzero two-point correlations. As an application, it demonstrates that a single-photon W-state input to a coupled-waveguide array can produce Bloch oscillations analogous to electronic Bloch oscillations.

Key findings

  • Interference at the mean-intensity level requires nonzero correlation \(\langle a_r^\dagger a_s\rangle \neq 0\); coherent states satisfy this via factorization, while Fock states do not. A two-mode entangled state \((|1,0\rangle+|0,1\rangle)/\sqrt{2}\) gives \(\langle a_1^\dagger a_2\rangle = 1/2\).
  • A single-photon W-state \(|\psi\rangle=\sum_p c_p |1_p,\{0\}\rangle\) has \(\langle a_p\rangle=0\) but \(\langle a_p^\dagger a_q\rangle=c_p^* c_q\). At a linear multiport with \(b_q=\sum_p G_{p,q}a_p\), the output intensity becomes \(|\sum_p G_{p,q} c_p|^2\), mimicking a coherent input with amplitudes \(c_p\).
  • In 26 coupled waveguides with \(\alpha=\delta/J=0.5\), a W-state input produces Bloch oscillations (Fig. 7) closely matching a coherent Gaussian input with \(\sigma=3.6\), \(\bar p=13\) (Fig. 6). The approximate result is \(I_p \approx |c_{\bar p}|^2\), with \(\bar p \equiv p-(4/\alpha)\sin^2(\alpha t/2)\).
  • The Fourier distribution \(|\tilde c(k)|^2\) has a half-height width of 0.31, small compared with the \(k\) range \([-\pi,\pi]\), supporting the approximation.

Methods

  • Theoretical quantum-optics derivation for linear multiport devices, establishing the correlation condition for coherent effects and proposing W-state generation via a multiport beam splitter from a heralded single photon.
  • Bloch oscillations modeled by a coupled single-mode waveguide array with Hamiltonian \(H=\hbar\sum_p \delta_p a_p^\dagger a_p+\hbar\sum_p J(a_p^\dagger a_{p+1}+a_{p+1}^\dagger a_p)\); Heisenberg equations are solved for \(G_{p,q}(t)\) using Bessel functions for large arrays, and output intensities are computed for single-waveguide, Gaussian coherent, and W-state inputs.
  • Numerical examples use 26 waveguides, \(\alpha=0.5\), Gaussian parameters \(\sigma=3.6\) and \(\bar p=13\), plus a Fourier-space approximation.

Limitations

  • Stated: the Bessel-function solution assumes a large array of waveguides; small arrays require another method (Rai et al., Phys. Rev. A 78, 042304). The complex phase factors of \(c_p\) from propagation between the beam splitter and mirror are set to unity.
  • Stated/evident: this is a theoretical proposal; no experimental realization is reported. Feasibility is argued by citing existing heralded single-photon sources, waveguide structures, and mirror assemblies.
  • Evident: the treatment assumes ideal lossless linear optics and perfect W-state preparation and phase control; these are not analyzed quantitatively.

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

Integrity signals

Registry

Crossref · 10.1103/physreva.79.053849

Cites

39 of 39 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 2 cited sources checked · 0 supported · 0 flagged

  • Findingown data
    § II. POSSIBILITY OF COHERENT

    If the input fields are in a W state, then 〈ar†as〉 ≠ 0 for all r,s holds.

    “One may be able to find several states satisfying (12) we have found that if the input fields are in a W state, then (12) holds.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § III. COHERENT BLOCH OSCILLATIONS

    An approximate formula for the W-state output intensity is Ip ≈ |c_{p̄}|^2 with p̄ ≡ (p - 4/α sin^2(αt/2)).

    “∼= |cp̄|2, p̄ ≡ ( p− 4 α sin2( αt 2 ) ) . (28)”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § III. COHERENT BLOCH OSCILLATIONS

    With a W-state input, coherent Bloch oscillations are recovered even though incoherent single photons are used.

    “In this case we recover the coherent Bloch oscillations even though we use incoherent single photons.”
    • Also stated in: § IV. CONCLUSIONS · hedged
      “Our results show that the Bloch oscillations are possible with single photon W -state.”
    Quote in the paperSubstantial claim
  • Interpretationreasoning
    § Abstract

    The correlations in a W-state restore coherent effects with single photons.

    “Such correlations restore coherent effects with single photons.”
    • Also stated in: § I. INTRODUCTION · plain
      “The quantum correlations in the W -state are responsible for restoring the coherent effects.”
    Quote in the paperSubstantial claim
  • Interpretationreasoning
    § Abstract

    Bloch oscillations with single photons are possible provided the single photons are prepared in a W-state.

    “As a specific example we demonstrate Bloch oscillations with single photons and thus provide strict analog of Bloch oscillation of electrons.”
    • Also stated in: § IV. CONCLUSIONS · plain
      “Our results show that the Bloch oscillations are possible with single photon W -state.”
    Quote in the paperSubstantial claim
  • May combine two claims. 5
  • Classified as probably not a substantial claim. 3

  • May combine two claims.

    Findingnone given
    § Abstract

    A W-state has vanishing mean amplitude but nonzero correlations between inputs at different ports.

    “We show that single photons need to be produced in W-states which lead to vanishing mean amplitude but nonzero correlations between the inputs at different ports.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingreasoning
    § III. COHERENT BLOCH OSCILLATIONS

    The structure shown in Fig. 7 is in conformity with the approximate formula (28).

    “The structure shown in the Fig. 7 is in conformity with the approximate formula (28).”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationreasoning
    § III. COHERENT BLOCH OSCILLATIONS

    The similarity between the coherent Bloch oscillation (Fig. 6) and the W-state Bloch oscillation (Fig. 7) is striking and is because of the similarity of the results (9) and (15).

    “The similarity between Figs. 6 and 7 is striking. This is because of the similarity of the results (9) and (15).”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Premisenone given
    § I. INTRODUCTION

    In the electron case we have a quantum particle whereas in the case of a coherent beam of light we have a classical source.

    “In the latter case we have a quantum particle whereas in the case of coherent beam of light we have a classical source.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § II. POSSIBILITY OF COHERENT

    If the inputs are incoherent, then all coherent effects are lost and the output mean intensity is ∑p G*_{p,q}G_{p,q} 〈ap†ap〉.

    “On the other hand if the inputs are incoherent, then we loose all coherent effects 〈b†qbq〉 ≡ ∑ p G∗ p,qGp,q 〈ap†ap〉 . (10)”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisenone given
    § II. POSSIBILITY OF COHERENT

    If the two beams are in coherent states, then 〈a1†a2〉 = 〈a1†〉〈a2〉 and interference occurs.

    “If the two beams are in coherent states, then 〈a1†a2〉 = 〈a1†〉〈a2〉 , (4) and thus interference obviously occurs.”
    Quote in the paperSubstantial claim
6 of 6

  • The authors state this as an assumption.

    The solution for Gp,q(t) in terms of Bessel functions assumes a large array of waveguides.

    “This solution assumes a large array of waveguides.”
1 of 1

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • In photonics, a W-state can describe a single photon coherently shared among N optical modes.

    “In photonics this can describe the situation where a single photon is coherently shared among N optical modes [82,83]:”

    From Integrated photonic quantum walks (2020) · cites this paper as [82]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Output-intensity formula reverses the transfer-matrix indices

    Minor

    Equation (7) defines Gp,q with its first index denoting an input and its second an output. Its output intensity therefore involves G∗r,q Gs,q, whereas Eq. (11) uses G∗q,r Gq,s.

  • Fourier-space evolution uses αt instead of the defined ατ

    Minor

    Equation (21) defines τ=Jt and uses ατ as the evolution argument. Equation (25) instead uses αt in the corresponding Fourier-space shift and phase; these differ by a factor of J under the paper's definitions.

  • Propagator has the opposite sign from the equation of motion

    Minor

    For p=q+1, expanding Eq. (21) near t=0 gives Ġp,q(0)=+iJ. Equations (19) and (20) instead require Ġp,q(0)=−iJ, so the stated propagator does not solve the paper's equation of motion.

3 of 3

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. G. I. Taylor, 1909, Proc. Cambridge Philos. Soc., 15, 114

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Related

Authors
Rai, A., & Agarwal, G. S.
Affiliations
Oklahoma State University (US)
ORCID
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2009-05-27
Volume
79
Issue
5
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.79.053849
OpenAlex
W2036213775
Cited by
32
References
39
Citations by year
2025: 2 · 2023: 2 · 2022: 1 · 2020: 3 · 2019: 1 · 2016: 1 · 2015: 1 · 2014: 3 · 2013: 4 · 2012: 4
FWCI
2.60
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum optics and atomic interactions
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
Keywords
Bloch oscillations, single photons, W state, multiport devices, photon correlations, mean amplitude

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