Generating photon-added states without adding a photon

J.D. Franson; S.U. Shringarpure

REVIEW 1 major objections 6 minor 31 references

Generating photon-added states without adding a photon

T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A post-selected optical parametric amplifier yields a continuous range of nonclassical states, including displaced-number and photon-added states, without adding a photon.

desk verdict A clean analytic result for a tunable post-selected OPA source; the physics is right, the experimental idealization is the main gap. read the letter →

arxiv 1908.08028 v2 pith:7TURFBJM submitted 2019-08-21 quant-ph

S.U. Shringarpure , J.D. Franson This is my paper · ORCID

classification quant-ph

keywords opticalparametricamplifierpost-selectionphoton-addedcoherentstatedisplacednumbernonclassicallightQ-functioncontinuous-variablequantuminformationgaintuning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to show that a single optical parametric amplifier, used with conditional measurement, can act as a tunable source of nonclassical light. A coherent state entering the signal port and a single photon entering the idler port, followed by post-selection on one idler photon at the output, yields a signal state that, by adjusting the amplifier gain, can be a coherent state, a displaced single-photon state, a photon-added coherent state, or any state in a continuous family between them. The counterintuitive result is that the amplifier adds no photons in either mode during the post-selected events, so the photon-added state emerges purely from the measurement-induced redistribution of probability amplitudes. This matters because it replaces fixed-transmittance beam splitter state-engineering methods with a single continuously tunable resource.

What carries the argument

The load-bearing object is the exact factored form of the two-mode squeezing operator (Eq. 1), where the gain $g=\cosh(\kappa t)$ and $G=\sqrt{g^2-1}$ enter separately. Acting on a single idler photon and post-selecting on a single idler photon leaves only two surviving Taylor terms, producing the central identity of Eq. (9)/(21): the output is a two-state superposition of $|\alpha/g\rangle$ and $D(\alpha/g)|1\rangle$. The displacement-operator identity $[a,D(\alpha)]=\alpha^*D(\alpha)$ then converts the superposition into the displaced-number form at $g_0$. The post-selection projection is what does the work: it forbids pair creation and annihilation, and yet the amplitude of each number state is reshaped by the gain, producing the tunable family.

What would settle it

Measure the Wigner function of the post-selected signal output at gain $g_0 = \sqrt{1+1/|\alpha|^2}$; the claim predicts a displaced single-photon Wigner function with a negative region, and if the reconstructed state shows no negativity the central identity is wrong.

Watch

Extended reading notes

Core claim

The central claim is that the post-selected signal output is exactly the normalized state $$|\psi\rangle = \frac{1}{N}\left[\left(1-\frac{G}{$g^{2}$}\right)|\$\alpha$/g\rangle - \frac{G}{$g^{2}$}\hat{n}|\$\alpha$/g\rangle\right]$$ or equivalently the two-term superposition of Eq. (21), which lies entirely in the subspace spanned by the attenuated coherent state $|\alpha/g\rangle$ and the displaced single-photon state $D(\alpha/g)|1\rangle$. Choosing the gain $g_0=\sqrt{1+1/|\alpha|^2}$ removes the coherent-state component and leaves exactly $D(\alpha/g_0)|1\rangle$, a displaced number state. In the limit $g\to\infty$ the coherent component is suppressed and the state approaches the photon-added coherent state proportional to $a^\dagger|\alpha/g\rangle$. The same formula gives a coherent state at $g=1$ and, at the gain $g_1$ of Eq. (24), a state orthogonal to a photon-added coherent state; the zeros of the Q-function track this orthogonality.

Load-bearing premise

The argument assumes the optical parametric amplifier is perfectly lossless, single-mode, phase-matched, and undepleted in its pump, so that its evolution is exactly the idealized two-mode squeezing operation used in the calculation.

Editorial extensions

If this is right

At the gain $g_0=\sqrt{1+1/|\alpha|^2}$, the output is exactly a displaced single-photon state $D(\alpha/g_0)|1\rangle$, orthogonal to the coherent state $|\alpha/g_0\rangle$, so the two states form a ready-made qubit pair.
In the limit of large gain, the same device produces a photon-added coherent state proportional to $a^\dagger|\alpha/g\rangle$, even though no photon is added.
At the gain $g_1$ given in Eq. (24), the output is orthogonal to a photon-added coherent state, providing another orthogonal pair for continuous-variable qubits.
Because the output always lies in the span of $|\alpha/g\rangle$ and $D(\alpha/g)|1\rangle$, simply varying the pump intensity sweeps through the whole family.
The success probability falls exponentially for large $|\alpha|$, so the method is practical only for moderate coherent-state amplitudes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A plausible extension is to replace the amplifier with any two-mode unitary whose post-selected subspace is small; the same mechanism would then produce families of 'virtual' state transformations without physical photon addition or subtraction.
In the displaced-number regime, a practical experiment could first locate the Q-function zero at $\alpha/g_0$ predicted by Eq. (28); finding that zero is a compact test of the two-state structure.
The continuous tunability suggests a calibration use: the same device could serve as a gain-controlled source for quantum information experiments, with the gain set by pump intensity rather than by swapping optical elements.
Chaining two post-selected amplifiers, which the authors flag for a separate paper, would plausibly generate entangled macroscopic states whose entanglement properties inherit from the two-dimensional subspace derived here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a solid theory paper, and the main caveat is the gap between the idealized OPA and any real implementation, not the algebra. I'd send it out for review.

What's actually new is the specific input combination — a coherent state in the signal mode, a single photon in the idler mode, and post-selection on one idler output photon. That gives a signal state confined to the span of an attenuated coherent state and a displaced single-photon state, Eq. (21). Varying the gain moves the output continuously through a coherent state, a displaced number state, and, in the large-gain limit, a state proportional to a-dagger |alpha/g>. The special gains g0 and g1 are explicit. This is a real, if modest, extension of the known OPA/beam-splitter equivalence and conditional state preparation lines. The derivation is self-contained and the algebra checks out; the special cases and the plots are consistent.

Soft spots, in proportion. First, the load-bearing assumption is that the OPA evolution is exactly the lossless, single-mode, phase-matched, undepleted-pump unitary of Eq. (1). Section VI analyzes detector dark counts and detection-path loss, but not loss inside the crystal, pump depletion, or multimode effects. Those will produce mixed states outside the clean two-state superposition, and Fig. 7 does not bound them. This is an applicability gap, not a mathematical error.

Second, two supporting calculations are asserted rather than shown: Eq. (24) for g1 is introduced with "It can be shown," and the fidelity contributions from the (0,2) and (1,2) outcomes are approximated by a lower bound. You cannot fully check those without doing the algebra yourself, but nothing points to them being wrong.

The exponential smallness of the success probability for large alpha is acknowledged and is a real practical limitation. The citation pattern is fine; the only self-citation is to an independent published noiseless-attenuation result, and there is no parameter fitting to data.

This paper is for people doing conditional state preparation in continuous-variable quantum information. The target states were already preparable by other means, so the value here is the single tunable source and the explicit analytic form. It deserves a serious referee; I would ask the referee to require the omitted derivations and a clearer statement of the OPA assumptions under which the Sec. VI error model holds.

Referee Report

1 major / 6 minor

Summary. The paper proposes a conditional state-preparation scheme based on an optical parametric amplifier (OPA). A coherent state |α> is injected into the signal mode, a single photon into the idler mode, and the signal output is retained only when a single photon is detected in the output idler mode. Using the factored two-mode squeezing operator (Eq. (1)), the authors derive the unnormalized post-selected signal state (Eqs. (7)-(9)) as proportional to (1 - G n-hat)|α/g>, a superposition of an attenuated coherent state and a photon-added term. They show that this state lies in the two-dimensional subspace spanned by |α/g> and D(α/g)|1> (Eq. (21)). At gain g=1 it reduces to the input coherent state; at g0 = 1/sqrt(1 - 1/|α|^2) it becomes a displaced single-photon state (Eq. (17)); and in the large-gain limit it approaches a photon-added coherent state. The paper also gives the gain g1 at which the output is orthogonal to a photon-added state, plots Q-functions for representative gains, and presents a fidelity analysis for dark counts and loss in the detection paths.

Significance. The central calculation is explicit and self-contained, and the special cases can be checked directly from Eq. (21). If the result holds, the scheme offers a single-knob (pump intensity) method to tune continuously between nonclassical states relevant to continuous-variable quantum information. The paper gives reproducible formulas for the state, the success probability, and the special gains, and the fidelity analysis in Sec. VI is a useful first step. The main caveat is that the experimental analysis assumes an ideal OPA unitary; the practical claims are therefore stronger than what the error model supports.

major comments (1)

[Sec. VI, Eqs. (29)-(30) and Fig. 7] The error model tracks only dark counts and losses in the detection record and leaves the OPA evolution as the exact lossless, single-mode, undepleted unitary of Eq. (1). Internal loss in the nonlinear crystal, pump depletion, and spatial/temporal multimode structure would modify the post-selected state itself rather than merely the detection record, so the reported fidelities are not end-to-end fidelities for a realistic amplifier. Since the abstract and title make a practical generation claim, this is a load-bearing gap; please either extend the model to include those imperfections or explicitly scope the claim to the ideal-OPA-plus-detection-errors setting.

minor comments (6)

[Sec. IV, Eq. (24)] The value of g1 is introduced with 'It can be shown' and no derivation is provided. I reproduced the result by solving the orthogonality condition, but the paper should include the one-line derivation so the reader can verify the claim without re-deriving the algebra.
[Sec. II, Eq. (10)] The success probability is also stated with 'which can be shown'; please include the norm calculation, since the typeset equation is difficult to verify as printed.
[Sec. V, Eqs. (27)-(28)] The derivations of the number-basis zero and the Q-function zero are omitted; a short derivation for each would improve reproducibility.
[Introduction, first paragraph] The sentence 'the post-selection process ensures that no photons were emitted or absorbed in either mode' is misleading, since the OPA unitary can create and annihilate pairs and the post-selection changes the signal-state amplitudes; the later discussion clarifies this, but the early sentence should be softened.
[Eq. (12)] The displaced-number-state gain g0 is real only for |α|^2 > 1; this condition should be stated explicitly.
[Fig. 3 caption] The caption 'with 10 ( 100).nα = =' is garbled; please clarify the value of α and the mean photon number used in the plot.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central result is a direct operator calculation from the standard OPA unitary, and the special states follow from derived gain conditions rather than fitted inputs or self-citation.

full rationale

The paper's central derivation starts from the factored two-mode squeezing unitary in Eq. (1), which is an externally established standard result cited to Schumaker-Caves and Caves et al. The post-selected signal state in Eqs. (7)-(9) is obtained by straightforward operator algebra: expanding the exponentials, acting on the input coherent state and single idler photon, and projecting onto a single output idler photon. No parameter is fitted to the target output state, and no empirical data are used. The special cases are derived, not assumed: the displaced number state at g0 follows by inserting Eq. (12) into Eq. (11) and using the standard commutator [a-dagger, D(alpha)]; the photon-added limit at large gain follows from Eq. (9) because the (G/g^2) n-hat term dominates as g grows. The orthogonality claims and Q-function zeros are algebraic consequences of the same expression. The only self-citations are Ref. [22], used in Sec. VI for the noiseless-attenuation error term (0,0), and Ref. [31], mentioned only as a physical analogy in the conclusions; neither is load-bearing for the main claim. The experimental analysis does assume an ideal lossless, single-mode, undepleted-pump OPA, and its fidelity model omits internal loss and pump depletion, but that is an applicability limitation, not circular reasoning. The derivation is self-contained and does not reduce to its inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters are used; the gain and coherent amplitude are controllable experimental inputs. The axioms are the standard unitary model of the OPA and the ideal single-photon input and post-selection. No new entities are introduced.

assumptions (4)

domain assumption The OPA evolution is the exact two-mode squeezing unitary given in Eq. (1).
Invoked in Section II; any loss, multimode structure, or phase-matching effects would alter the post-selected state.
domain assumption The input idler is a pure single-photon Fock state and the post-selection is a projection onto a single output idler photon.
Eqs. (2)-(3); imperfect heralding would change the output from a pure state to a mixture.
standard math Standard normal-ordering identity for the two-mode squeeze operator and the displacement-operator commutation relation [a-dagger, D(alpha)] = alpha* D(alpha) from Refs. [7,26,27].
Used in Eqs. (4)-(6) and (13)-(17) to simplify the post-selected state.
domain assumption In the experimental analysis, the SPDC source produces only one pair at a time, false counts from n>=3 are negligible, and simultaneous dark count and loss are negligible.
Section VI; these approximations are stated and define the validity of the fidelity estimate.

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Pith. "Pith review of Generating photon-added states without adding a photon." pith.science (2026). https://pith.science/paper/7TURFBJM

@misc{pith2026190808028,
  author       = {Pith},
  title        = {Pith review of: Generating photon-added states without adding a photon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7TURFBJM}},
  note         = {Machine review of arXiv:1908.08028}
}

read the original abstract

We show that a continuous range of nonclassical states of light can be generated using conditional measurements on the idler mode of an optical parametric amplifier. The output state is prepared by introducing a coherent state in the signal mode of the amplifier with a single photon in the idler mode, followed by a conditional measurement of a single photon in the output idler mode. By varying the gain of the amplifier, this approach can produce a coherent state, a photon-added state, a displaced number state, or a continuous range of other nonclassical states with intermediate properties. We note that this approach can generate a photon-added state even though the post-selected amplifier does not add any photons to the signal or idler modes. The ability to generate a continuous range of nonclassical states may have practical applications in quantum information processing.

Figures

Figures reproduced from arXiv: 1908.08028 by the authors.

**Figure 2.** The displaced number state 0 |ψ g 〉 also has the interesting property that it has the same average photon number as the initial coherent state in the input to the optical parametric amplifier. This can be shown by rearranging Eq. (12) into the form 22 0 | || αα / |1 g = − . (18) The average photon number for a displaced photon number state is given by [7] 2 ˆ ( ')| ˆ | |. ' Dn n n α α 〉 〈〉 = + (19) Combining Eqs. (1… view at source ↗

**Figure 6.** The presence of the single photon is heralded by post-selecting on a detection event in the other output path of the SPDC. We will assume that the spontaneous parametric down conversion process generates only a single pair of entangled photons at a time, which is a good approximation when the intensity of the laser used as a pump for the nonlinear crystal is sufficiently low. With this assumption, the only error in … view at source ↗

**Figure 7.** shows the lower bound on the fidelity as a function of d for several values of l. We have also plotted the actual fidelity including the contributions from the states (0, 2) and (1, 2). It can be seen that limited detection efficiency and dark counts can both have an effect on the fidelity. The detectors used in pulsed down-conversion experiments have dark counts corresponding to d as low as 6 10 , − so that dark co… view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

G. S. Agarwal and K. Tara, Physical Review A (Atomic, Molecular, and Optical Physics) 43, 492 (1991)

work page 1991

[2] [2]

Zavatta, S

A. Zavatta, S. Viciani, and M. Bellini, Science 306, 660 (2004)

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work page 2007

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Barbieri, N

M. Barbieri, N. Spagnolo, M. G. Genoni, F. Ferreyrol, R. Blandino, M. G. A. Paris, P. Grangier, and R. Tualle-Brouri, Physical Review A (Atomic, Molecular, and Optical Physics) 82, 063833 (2010)

work page 2010

[5] [5]

S. N. Filippov, V. I. Man'ko, A. S. Coelho, A. Zavatta, and M. Bellini, Physica Scripta T 2013, 014025 (2013)

work page 2013

[6] [6]

Boiteux and A

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