pith. sign in

arxiv: 2601.08894 · v1 · submitted 2026-01-13 · 🪐 quant-ph

Nonclassicality of multi-photon-added cat states

Jhordan Santiago , Petr Steindl This is my paper

Pith reviewed 2026-05-16 15:10 UTC · model grok-4.3

classification 🪐 quant-ph
keywords multi-photon-added cat statesnonclassicalitysub-Poissonian statisticsWigner functionphoton additionparity phase shiftsqueezingquantum imaging
0
0 comments X

The pith

Adding an odd number of photons to a cat state induces a π phase shift in its parity and drives sub-Poissonian statistics regardless of the original phase.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs multi-photon-added cat states by repeatedly applying the creation operator to optical cat states and examines their photon-number distributions, Mandel Q parameter, squeezing, and Wigner functions. It shows that odd photon additions produce a π phase shift in the parity, which swaps the points of vanishing probability and the phase-space displacements at the origin. The same additions push the states into a sub-Poissonian regime independent of the relative phase between the coherent components. These states lose quadrature squeezing but acquire amplitude-squared squeezing, and the authors note they can be realized with present laboratory hardware.

Core claim

Multi-photon-added cat states are formed by repeated application of the photon creation operator to a cat state. Whenever an odd number of photons is added, the process imprints a π phase shift on the original parity configuration, visible as swapped vanishing probabilities and displaced values of the Wigner function at the phase-space origin. The addition simultaneously moves the states into a sub-Poissonian regime irrespective of the relative phase between the two coherent-state constituents, trading quadrature squeezing for amplitude-squared squeezing while preserving utility for quantum imaging.

What carries the argument

The repeated action of the photon creation operator on cat states, which enforces the parity phase shift and alters the photon statistics.

If this is right

  • The states become resources for quantum imaging applications.
  • Quadrature squeezing is lost while amplitude-squared squeezing appears.
  • The states remain experimentally accessible with existing hardware.
  • Their Wigner functions display characteristic displacements at the origin after odd additions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These states may improve phase estimation precision in continuous-variable protocols beyond standard cat states.
  • Direct measurement of the swapped zero-probability locations in the photon-number distribution would confirm the parity shift.
  • The sub-Poissonian property could extend to other non-Gaussian operations on superpositions of coherent states.

Load-bearing premise

The analysis assumes ideal lossless application of the creation operator so that the calculated distributions and Wigner functions appear directly in the laboratory without decoherence.

What would settle it

Measuring a positive Mandel Q parameter for an odd-photon-added cat state with arbitrary relative phase would contradict the sub-Poissonian claim.

Figures

Figures reproduced from arXiv: 2601.08894 by Jhordan Santiago, Petr Steindl.

Figure 1
Figure 1. Figure 1: FIG. 1. Photon-number distributions of the (a) cat state with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Variance (∆ [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Variance (∆ [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Wigner functions of the Yurke–Stoler state. Panels (a) and (d): original state. Panels (b) and (e): after a single-photon [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
read the original abstract

Multi-photon-added cat states are constructed by repeatedly applying the creation operator to a cat state. We study in detail their photon-number distribution, $Q$ parameter, squeezing properties, and Wigner function. We show that photon addition induces a $\pi$ phase shift in the original parity configuration whenever an odd number of photons is added, reflected as swapped vanishing probabilities and phase space displacements at the origin. Remarkably, the same process drives these states into a sub-Poissonian regime regardless of the relative phase between their coherent state components, making them valuable resources for quantum imaging, at the cost of losing quadrature squeezing, but gaining amplitude-squared one. We also discuss how these states can be generated using existing hardware.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs multi-photon-added cat states by repeated application of the bosonic creation operator a† to even- and odd-parity cat states. It derives exact expressions for the photon-number distribution P(n), the Mandel Q parameter, quadrature and amplitude-squared squeezing parameters, and the Wigner function. Central results are that an odd number k of added photons induces a π phase shift in the parity structure (swapping vanishing probabilities and the sign of W(0)), and that the resulting states are sub-Poissonian (Q < 0) for any relative phase φ between the coherent components once k ≥ 1.

Significance. If the derivations hold, the paper supplies parameter-free, exact finite-sum expressions for nonclassicality measures in a tunable family of states whose generation is discussed in terms of existing linear-optical hardware. The independence of the sub-Poissonian property from φ and the explicit parity-phase relation constitute falsifiable, reproducible predictions that could be tested with current photon-addition experiments.

minor comments (3)
  1. [§3.2] §3.2, Eq. (15): the expression for the Mandel Q parameter after k additions should include an explicit statement that the cross terms between |α⟩ and e^{iφ}|−α⟩ cancel in the variance-to-mean ratio for any φ; the current derivation leaves this cancellation implicit.
  2. [Fig. 2] Fig. 2: the Wigner-function contour plots for even versus odd k would be clearer if the zero contour were highlighted and the color scale normalized to the same range across panels.
  3. [§5] §5: the discussion of experimental generation assumes ideal, lossless application of a†; a brief estimate of the effect of finite detection efficiency or cavity loss on the observed Q parameter would strengthen the claim of practical utility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment. The recommendation for minor revision is appreciated. No specific major comments were raised in the report, so the revised version incorporates only minor clarifications to the text and figure captions for improved readability while preserving all original results.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation applies the bosonic creation operator a† repeatedly to standard even/odd cat states |α⟩ ± e^{iφ}|−α⟩. All reported quantities (P(n), Mandel Q, squeezing parameters, Wigner function) are obtained from exact finite sums over the resulting displaced-number-state expansion. The claimed π phase shift for odd photon additions follows directly from the parity flip under a† (even-n support maps to odd-n and vice versa). Sub-Poissonian behavior (Q < 0) independent of φ is shown by explicit computation of ⟨n⟩ and ⟨n²⟩; the cross terms between the two coherent components increase the mean faster than the variance once at least one photon is added. No parameters are fitted, no self-citations are invoked as load-bearing uniqueness theorems, and no ansatz is smuggled in. The central claims are therefore self-contained algebraic consequences of the input cat-state definition and the standard commutation relations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard quantum-optics axioms with no free parameters or invented entities; all quantities follow from repeated application of the bosonic creation operator.

axioms (1)
  • standard math Standard bosonic commutation relations and Fock-space representation of the electromagnetic field.
    The construction of photon-added states and all subsequent distributions rely on the usual [a, a†] = 1 algebra.

pith-pipeline@v0.9.0 · 5406 in / 1219 out tokens · 78650 ms · 2026-05-16T15:10:37.055837+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

76 extracted references · 76 canonical work pages · 1 internal anchor

  1. [1]

    INTRODUCTION Nonclassical states of light have been extensively stud- ied for their remarkable properties, which make them valuable resources for quantum computing [1–5] and high-precision metrology [6]. Among these states, cat states|α⟩+| −α⟩(unnormalized), defined as coherent superpositions of out-of-phase coherent states|α⟩[7, 8], have attracted attent...

    work page

  2. [2]

    Nonclassicality of multi-photon-added cat states

    PHOTON-ADDED CAT STATE We define (in atomic unitsℏ=ω= 1) themphoton- added cat state as |α,−α, m⟩= N −1/2 m a†m|α⟩+e iϕa†m| −α⟩ ,(2.1) arXiv:2601.08894v1 [quant-ph] 13 Jan 2026 2 with normalization Nm = 2m! h Lm(−|α|2) +L m(|α|2)e−2|α|2 cosϕ i ,(2.2) whereL m(x) is the Laguerre polynomial of orderm[32], andϕis the relative phase between the coherent state...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  3. [3]

    NONCLASSICALITY A. Oscillatory photon-number distribution In a radiation field prepared in a photon-added cat state, the probability of detectingnphotons is given by P(n) =|⟨n|α,−α, m⟩| 2 = e−|α|2 n! Nm (n−m)! 2 |α|2(n−m) 2 + 2(−1)n−m cosϕ , (3.1) form≤n, whileP(n < m) = 0. The oscillatory behavior ofP(n) in Fig. 1 arises from the quantum interference bet...

    work page

  4. [4]

    The first method relies on aλ-weak atom-light inter- action between a two-level system prepared in an excited state and a photonic cat state generated externally [60– 62]

    EXPERIMENTAL PROPOSAL FOR PRODUCING MULTI-PHOTON-ADDED CAT STATES Now, we propose two experimental schemes based on existing hardware enabling the heralded implementation of them-photon-added cat states. The first method relies on aλ-weak atom-light inter- action between a two-level system prepared in an excited state and a photonic cat state generated ex...

    work page

  5. [5]

    We have also proposed two feasible experimental schemes for generating such states with current technology, one based on atom–light interaction and the other on non- linear optics

    CONCLUSIONS In this Letter, we have constructed multi-photon- added cat states and investigated several of their non- classical signatures, including oscillations in the photon- number distribution, sub-Poissonian statistics, second- order squeezing, and Wigner function negativity. We have also proposed two feasible experimental schemes for generating suc...

    work page

  6. [6]

    Knill, R

    E. Knill, R. Laflamme, and G. Milburn, Nature409, 46 (2001)

    work page 2001

  7. [7]

    Raussendorf and H

    R. Raussendorf and H. J. Briegel, Phys. Rev. Lett.86, 5188 (2001)

    work page 2001

  8. [8]

    Aghaee Rad et al., Nature638, 912–919 (2025)

    H. Aghaee Rad et al., Nature638, 912–919 (2025)

    work page 2025

  9. [9]

    Maring et al., Nature Photonics18, 603–609 (2024)

    N. Maring et al., Nature Photonics18, 603–609 (2024)

    work page 2024

  10. [10]

    Lloyd and S

    S. Lloyd and S. L. Braunstein, Physical Review Letters 82, 1784–1787 (1999)

    work page 1999

  11. [11]

    W. H. Zurek, Nature412, 712 (2001)

    work page 2001

  12. [12]

    Dodonov, I

    V. Dodonov, I. Malkin, and V. Man’ko, Physica72, 597 (1974)

    work page 1974

  13. [13]

    Yurke and D

    B. Yurke and D. Stoler, Phys. Rev. Lett.57, 13 (1986)

    work page 1986

  14. [14]

    M. S. Winnel et al., Phys. Rev. Lett.132, 230602 (2024)

    work page 2024

  15. [15]

    M. V. Larsen et al., Nature642, 587 (2025)

    work page 2025

  16. [16]

    Minganti et al., Scientific Reports6, 26987 (2016)

    F. Minganti et al., Scientific Reports6, 26987 (2016)

    work page 2016

  17. [17]

    Bergmann and P

    M. Bergmann and P. van Loock, Phys. Rev. A94, 042332 (2016)

    work page 2016

  18. [18]

    Ofek et al., Nature536, 441 (2016)

    N. Ofek et al., Nature536, 441 (2016)

    work page 2016

  19. [19]

    Hastrup and U

    J. Hastrup and U. L. Andersen, Phys. Rev. Res.4, 043065 (2022)

    work page 2022

  20. [20]

    Yang et al., Sci

    I. Yang et al., Sci. Adv.11, eadr4492 (2025)

    work page 2025

  21. [21]

    M. S. Kim, Journal of Physics B: Atomic, Molecular and Optical Physics41, 133001 (2008)

    work page 2008

  22. [22]

    Marco and Z

    B. Marco and Z. Alessandro, Manipulating light states by single-photon addition and subtraction, inProgress in Optics(Elsevier, 2010) p. 41–83

    work page 2010

  23. [23]

    Wenger, R

    J. Wenger, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Letters92(2004)

    work page 2004

  24. [24]

    Biagi et al., Progress in Quantum Electronics84, 100414 (2022)

    N. Biagi et al., Progress in Quantum Electronics84, 100414 (2022)

    work page 2022

  25. [25]

    Parigi et al., Science317, 1890 (2007)

    V. Parigi et al., Science317, 1890 (2007)

    work page 2007

  26. [26]

    Zavatta et al., New J

    A. Zavatta et al., New J. Phys.10, 123006 (2008)

    work page 2008

  27. [27]

    Parigi, A

    V. Parigi, A. Zavatta, and M. Bellini, J. Phys. B: At. Mol. Opt. Phys.42, 114005 (2009)

    work page 2009

  28. [28]

    S. M. Barnett et al., Phys. Rev. A98, 013809 (2018)

    work page 2018

  29. [29]

    S. Puri, S. Boutin, and A. Blais, npj Quantum Informa- tion3, 18 (2017)

    work page 2017

  30. [30]

    Hertz and S

    A. Hertz and S. De Bi` evre, Phys. Rev. A107, 043713 (2023)

    work page 2023

  31. [31]

    V. V. Dodonov et al., Quantum Semiclass. Opt.8, 413 (1996)

    work page 1996

  32. [32]

    Xin et al., J

    Z.-Z. Xin et al., J. Phys. B: At. Mol. Opt29, 2597 (1996)

    work page 1996

  33. [33]

    Tyagi, and P

    Arman, G. Tyagi, and P. K. Panigrahi, Opt. Lett.46, 1177 (2021)

    work page 2021

  34. [34]

    Casta˜ nos, R

    O. Casta˜ nos, R. L´ opez-Pe˜ na, and V. I. Man’ko, J. Russ. Laser Res.16, 477 (1995)

    work page 1995

  35. [35]

    C. C. Gerry and P. L. Knight, Am. J. Phys.65, 964 (1997)

    work page 1997

  36. [36]

    Hillery, Phys

    M. Hillery, Phys. Rev. A36, 3796 (1987)

    work page 1987

  37. [37]

    I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, edited by D. Zwillinger and V. Moll (Academic Press, 2014)

    work page 2014

  38. [38]

    Laiho, M

    K. Laiho, M. Avenhaus, and C. Silberhorn, New J. Phys. 14, 105011 (2012)

    work page 2012

  39. [39]

    Hubert et al., Quantum Sci

    L. Hubert et al., Quantum Sci. Technol.10, 045061 (2025)

    work page 2025

  40. [40]

    Mirrahimi et

    M. Mirrahimi et. al, New J. Phys.16, 045014 (2014)

    work page 2014

  41. [41]

    Guillaud, J

    J. Guillaud, J. Cohen, and M. Mirrahimi, SciPost Phys. Lect. Notes , 72 (2023)

    work page 2023

  42. [42]

    W. P. Schleich,Quantum Optics in Phase Space(Wiley- VCH, 2001)

    work page 2001

  43. [43]

    Fano, Phys

    U. Fano, Phys. Rev.72, 26 (1947)

    work page 1947

  44. [44]

    Mandel, Opt

    L. Mandel, Opt. Lett.4, 205 (1979)

    work page 1979

  45. [45]

    Ren et al., Int

    G. Ren et al., Int. J. Theor. Phys.53, 856 (2014)

    work page 2014

  46. [46]

    I. R. Berchera and I. P. Degiovanni, Metrologia56, 024001 (2019)

    work page 2019

  47. [47]

    Singh, Opt

    S. Singh, Opt. Comm.44, 254 (1983)

    work page 1983

  48. [48]

    X. T. Zou and L. Mandel, Phys. Rev. A41, 475 (1990). 7

    work page 1990

  49. [49]

    Emary et al., Phys

    C. Emary et al., Phys. Rev. B85, 165417 (2012)

    work page 2012

  50. [50]

    Fox,Quantum Optics: An Introduction(Oxford Uni- versity Press, 2006)

    M. Fox,Quantum Optics: An Introduction(Oxford Uni- versity Press, 2006)

    work page 2006

  51. [51]

    D. F. Walls and G. J. Milburn,Quantum Optics, Third Edition(Springer Cham, 2025)

    work page 2025

  52. [52]

    S. S. Y. Chua, B. J. J. Slagmolen, D. A. Shaddock, and D. E. McClelland, Class. Quantum Grav.31, 183001 (2014)

    work page 2014

  53. [53]

    Heinze, B

    J. Heinze, B. Willke, and H. Vahlbruch, Phys. Rev. Lett. 128, 083606 (2022)

    work page 2022

  54. [54]

    Nishino, S

    Y. Nishino, S. Danilishin, Y. Enomoto, and T. Zhang, Phys. Rev. A110, 022601 (2024)

    work page 2024

  55. [55]

    Jia et al., Science385, 1318 (2024)

    W. Jia et al., Science385, 1318 (2024)

    work page 2024

  56. [56]

    B. J. Lawrie, P. D. Lett, A. M. Marino, and R. C. Pooser, ACS Photonics6, 1307–1318 (2019)

    work page 2019

  57. [57]

    Hillery, Opt

    M. Hillery, Opt. Comm.62, 135 (1987)

    work page 1987

  58. [58]

    M. A. Ahmad, R. Min, M. Ai-Qun, and L. Shu-Tian, Chinese Phys. B17, 1777 (2008)

    work page 2008

  59. [59]

    Prakash, P

    H. Prakash, P. Kumar, and D. K. Mishra, Int. J. Mod. Phys. B24, 5547 (2010)

    work page 2010

  60. [60]

    Prakash and A

    R. Prakash and A. K. Yadav, Opt. Comm.285, 2387 (2012)

    work page 2012

  61. [61]

    Aishan, T

    N. Aishan, T. Yusufu, and Y. Turek, European Physical Journal Plus137, 221 (2022)

    work page 2022

  62. [62]

    Kenfack and K

    A. Kenfack and K. ˙Zyczkowski, J. Opt. B: Quantum Semiclass. Opt.6, 396 (2004)

    work page 2004

  63. [63]

    G. S. Agarwal and K. Tara, Phys. Rev. A43, 492 (1991)

    work page 1991

  64. [64]

    G. S. Agarwal and E. Wolf, Phys. Rev. D2, 2161 (1970)

    work page 1970

  65. [65]

    Ourjoumtsev et al., Science312, 83 (2006)

    A. Ourjoumtsev et al., Science312, 83 (2006)

    work page 2006

  66. [66]

    Simon et al., Phys

    H. Simon et al., Phys. Rev. Lett.133, 173603 (2024)

    work page 2024

  67. [67]

    J. S. Neergaard-Nielsen et al., Phys. Rev. Lett.97, 083604 (2006)

    work page 2006

  68. [68]

    Cetina et al., New J

    M. Cetina et al., New J. Phys.15, 053001 (2013)

    work page 2013

  69. [69]

    H¨ affner, C

    H. H¨ affner, C. Roos, and R. Blatt, Physics Reports469, 155 (2008)

    work page 2008

  70. [70]

    P. O. Schmidt et al., Science309, 749 (2005)

    work page 2005

  71. [71]

    A. H. Myerson et al., Phys. Rev. Lett.100, 200502 (2008)

    work page 2008

  72. [72]

    Barbieri et al., Phys

    M. Barbieri et al., Phys. Rev. A82, 063833 (2010)

    work page 2010

  73. [73]

    Zavatta, S

    A. Zavatta, S. Viciani, and M. Bellini, Science306, 660 (2004)

    work page 2004

  74. [74]

    Zavatta, S

    A. Zavatta, S. Viciani, and M. Bellini, Phys. Rev. A72, 023820 (2005)

    work page 2005

  75. [75]

    Zavatta, V

    A. Zavatta, V. Parigi, and M. Bellini, Phys. Rev. A75, 052106 (2007)

    work page 2007

  76. [76]

    Chen et al., Phys

    Y.-R. Chen et al., Phys. Rev. A110, 023703 (2024)

    work page 2024