Programmable non-Gaussian quantum light source with state and temporal-waveform tunability

Akihiro Machinaga; Daichi Okuno; Hiroko Tomoda; Hirotaka Terai; Masahiro Yabuno; Shigehito Miki; Shuntaro Takeda; Takahiro Kashiwazaki; Takeshi Umeki; Yu Nishizawa

arxiv: 2605.02536 · v1 · submitted 2026-05-04 · 🪐 quant-ph

Programmable non-Gaussian quantum light source with state and temporal-waveform tunability

Hiroko Tomoda , Yu Nishizawa , Akihiro Machinaga , Takahiro Kashiwazaki , Takeshi Umeki , Shigehito Miki , Masahiro Yabuno , Hirotaka Terai

show 2 more authors

Daichi Okuno Shuntaro Takeda

This is my paper

Pith reviewed 2026-05-08 18:38 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum light sourcenon-Gaussian statesheraldingtemporal waveformprogrammablesingle-photonSchrödinger cat

0 comments

The pith

A heralding scheme lets users independently set both the quantum state and temporal waveform of non-Gaussian light without direct losses to the output.

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

The paper sets out a programmable source that can produce different non-Gaussian quantum states and shape their time profiles at will inside one device. Control is achieved by changing only the light sent to the detector that heralds the output photon, leaving the desired light untouched and therefore free of added loss. A working prototype shows single-photon, cat, and two-photon states emerging in several non-standard waveforms while state quality stays the same. This matters because photonic quantum processors and networks often need light that matches the exact timing and state required by the next operation; fixed sources force extra hardware or compromises. If the method scales, it removes a long-standing barrier between the light one can generate and the light one actually needs for a given task.

Core claim

The central claim is that a heralding architecture can engineer both the quantum state and its temporal waveform to arbitrary user targets by acting only on the heralding arm. Because the heralded light itself is never filtered or shaped directly, optical loss in the signal path is avoided. The prototype implements this for single-photon, Schrödinger-cat, and two-photon states and produces each in multiple unconventional temporal waveforms while preserving the original state quality.

What carries the argument

The heralding scheme that indirectly sets both state and waveform by acting on the heralding channel alone.

If this is right

Single-photon, Schrödinger cat, and two-photon states can each be emitted in a chosen temporal waveform.
Optical loss in the heralded channel is avoided because only the heralding light is modified.
Both state and waveform become independently programmable inside one platform.
The source becomes a versatile tool that can be matched to the timing and state needs of specific photonic quantum tasks.

Where Pith is reading between the lines

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

This approach may reduce the number of lossy components needed downstream when quantum light is fed into larger circuits or networks.
Real-time feedback could allow the waveform or state to be adjusted on the fly during a protocol.
Similar heralding control might be tested on other non-Gaussian resources such as higher-photon states or continuous-variable states.

Load-bearing premise

That manipulation of the heralding channel alone can produce any chosen combination of quantum state and temporal waveform while leaving the heralded state's quality unchanged.

What would settle it

Demonstrating that a change in the chosen waveform for a fixed target state produces a measurable drop in the heralded state's fidelity or purity.

Figures

Figures reproduced from arXiv: 2605.02536 by Akihiro Machinaga, Daichi Okuno, Hiroko Tomoda, Hirotaka Terai, Masahiro Yabuno, Shigehito Miki, Shuntaro Takeda, Takahiro Kashiwazaki, Takeshi Umeki, Yu Nishizawa.

**Figure 1.** Figure 1: FIG. 1. Schematic image of programmable non-Gaussian view at source ↗

**Figure 2.** Figure 2: FIG. 2. Comparison between the conventional and proposed quantum light source schemes. view at source ↗

**Figure 3.** Figure 3: FIG. 3 view at source ↗

**Figure 4.** Figure 4: FIG. 4. Experimental results view at source ↗

read the original abstract

A versatile quantum light source capable of programmably generating a variety of quantum light is a key enabler for photonic quantum technologies. In particular, independent control over both the output quantum state and its temporal waveform is essential for realizing diverse functionalities and enhancing processing performance. However, conventional sources of optical non-Gaussian states, a crucial resource for photonic quantum information processing, typically emit fixed states with predetermined temporal waveforms, lacking their programmability. Here, we propose a programmable non-Gaussian quantum light source that offers independent and arbitrary tunability of both the quantum state and the temporal waveform within a single platform. As a distinctive feature, our approach employs a heralding scheme in which these two properties are indirectly engineered to user-defined targets by manipulating the light in the heralding channel, thereby avoiding optical losses associated with direct manipulation of the heralded quantum light. We develop a prototype and demonstrate the generation of single-photon, Schr\"odinger cat, and two-photon states in a variety of unconventional temporal waveforms without degradation in state quality. This platform provides a versatile tool for tailoring quantum light to specific applications, significantly expanding the capabilities of photonic quantum technologies.

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.

Desk Editor's Note private letter to a colleague

The paper shows a working prototype that tunes non-Gaussian states and waveforms independently by shaping the herald photon, which is useful if the measurements confirm no quality loss across the reported cases.

read the letter

The main takeaway is that they built and tested a source where you adjust the herald arm to set both the output state (single-photon, cat, or two-photon) and its temporal waveform at the same time, without directly touching the signal light. This sidesteps the usual loss problems that come with post-selection or filtering on the heralded mode itself. They report a prototype that produces those states in several non-standard waveforms and claim the state quality holds up. That combination of independent control in one platform is the concrete advance over fixed non-Gaussian sources. The experimental implementation looks solid on the basic level: they have the hardware (pulse shapers, modulators, detectors) and show the states are generated as intended. The citation pattern is standard for quantum optics and does not rely on circular claims. The soft spot is exactly the one the stress-test note flags. For single-photon states the mapping from herald shaping to signal waveform is straightforward, but for cat and two-photon states the joint spectral amplitude means any residual distinguishability or mode mismatch in the herald arm can degrade negativity or bunching without obvious signs in marginal data. The abstract says “without degradation,” so the paper must include the actual Wigner or g^(2) numbers across the different waveforms; if those numbers are only shown for the simplest case and the rest is stated without the same checks, the independence claim is weaker than presented. Minor issues like error bars or full reproducibility details can be fixed in revision. This is for groups doing photonic quantum information experiments who need flexible sources rather than one-off fixed states. A reader building or using non-Gaussian resources will find the methods and the prototype results worth looking at. It is worth sending to peer review because the experimental demonstration is real and the idea is practical, even if the reviewers will want tighter data on the higher-order states.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes a programmable non-Gaussian quantum light source that achieves independent tunability of the output quantum state (single-photon, Schrödinger cat, two-photon) and its temporal waveform within a single platform. The key innovation is a heralding scheme in which both properties are engineered indirectly by manipulating the heralding channel, avoiding losses from direct manipulation of the heralded light. A prototype is developed and claimed to demonstrate the generation of the listed states in a variety of unconventional temporal waveforms without degradation in state quality.

Significance. If the experimental results are robust, the platform would provide a versatile tool for tailoring non-Gaussian resources to specific applications in photonic quantum technologies, expanding capabilities beyond fixed-state sources. The avoidance of direct manipulation losses is a practical advantage for integration.

major comments (1)

The central claim that state quality is preserved 'without degradation' for Schrödinger cat and two-photon states when the heralding waveform is shaped is load-bearing but requires explicit quantitative support. The manuscript must show that metrics such as Wigner-function negativity (for cat states) or g^(2)(0) (for two-photon states) remain constant across the reported waveforms, confirming that the heralding-channel manipulation introduces no residual distinguishability or back-action on the heralded mode via the joint spectral amplitude.

minor comments (1)

The abstract refers to 'a variety of unconventional temporal waveforms' without naming them; the main text should include concrete examples (e.g., specific pulse shapes or chirp parameters) with corresponding data.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the work's significance and for the detailed, constructive comment. We address the major point below and will revise the manuscript accordingly to strengthen the presentation of our results.

read point-by-point responses

Referee: The central claim that state quality is preserved 'without degradation' for Schrödinger cat and two-photon states when the heralding waveform is shaped is load-bearing but requires explicit quantitative support. The manuscript must show that metrics such as Wigner-function negativity (for cat states) or g^(2)(0) (for two-photon states) remain constant across the reported waveforms, confirming that the heralding-channel manipulation introduces no residual distinguishability or back-action on the heralded mode via the joint spectral amplitude.

Authors: We agree that explicit quantitative metrics are required to substantiate the no-degradation claim. Our experimental data show that Wigner negativity for the cat states and g^(2)(0) for the two-photon states remain constant (within experimental uncertainty) across the demonstrated waveforms, consistent with the theoretical prediction that the heralding-channel manipulation does not introduce back-action on the heralded mode through the joint spectral amplitude. However, these metrics were not displayed in a single comparative plot in the original manuscript. In the revised version we will add a new figure (or panel) that directly plots these quantities versus waveform parameter for each state, together with a brief theoretical paragraph confirming the absence of residual distinguishability. This revision will make the supporting evidence fully explicit. revision: yes

Circularity Check

0 steps flagged

No circularity in experimental demonstration of heralded non-Gaussian states

full rationale

The paper describes an experimental prototype using a heralding scheme to control quantum state and temporal waveform via manipulation in the heralding channel. No mathematical derivation chain exists that reduces predictions or results to inputs by construction, self-citation, or fitted parameters. Claims rest on physical implementation and measurements of single-photon, cat, and two-photon states, relying on established quantum optics without self-referential logic or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides insufficient detail to identify specific free parameters, axioms, or invented entities; the work builds on standard heralded quantum state generation techniques.

pith-pipeline@v0.9.0 · 5546 in / 1252 out tokens · 130919 ms · 2026-05-08T18:38:59.799347+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith.Cost (J = ½(x+x⁻¹)−1) washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

this source can programmably generate any quantum state of the form |ψ⟩ = Ŝ(r) Σ_{n=0}^N C_n |n⟩ in an arbitrary real-valued TW ... C_0,...,C_N ∈ C are complex superposition coefficients
IndisputableMonolith.Foundation.GeneralizedDAlembert / AxiomDischargePlan (cosh/cos/const classification of d'Alembert solutions) aczel_kannappan_via_cases unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

f(t) = g(T1 − t) ∝ e^{γ(t−T1)} ... TW f(t) ∝ e^{γt} m_env(t)
IndisputableMonolith.Foundation (no-adjustable-parameter chain) reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We choose the parameters (r0, r1, T) to achieve the target squeezing level e^{-2 r_out} ... some flexibility remains in the choice of (r0, r1, T)

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

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Projective measurement 2
[47]

Transformation of time-frequency characteristics 4
[48]

Projector onto quantum state in target temporal waveform 4
[49]

Two-mode Gaussian entangled state preparation 6
[50]

Principle of state control 8

Final heralded states 7 B. Principle of state control 8
[51]

Type of heralded state 8
[52]

Experimental details 12 A

How to set experimental parameters for target state 11 II. Experimental details 12 A. Setup details 12 B. Experimental parameters 14 C. Measurement & Analysis 15 References 17 I. FORMULA TION OF QUANTUM ST A TE GENERA TION IN PROPOSED SETUP In this section, we formulate the generation of quantum states in the proposed setup. We analyze the temporal wavefo...
[53]

NX k=1 cj′′kˆak(Tj′′) ! + αj′′ h(Tj′′) #   NO j′=2 | 0⟩j′ (14) = NO j=1 j⟨ 0|

Projective measurement Here we calculate the projective measurement 1⟨ϕProj| in Channel 1 in Eq. (1). To detect a total of N photons using on/off detectors arranged in parallel, the beam in Channel 1 is split into N channels (Channels 1– N) using beam splitters (BSs), as illustrated in the green region of Fig. S1. This process is modeled by successive BS ...
[54]

Transformation of time-frequency characteristics Next, we consider the transformation ˆW1 of time-frequency characteristics, realized through modulation and filtering as shown in Fig. S1. Following Ref. [2], transmission through a filter in Channel 1 can be formulated as a quantum operation involving two channels: ˆG† 1,anc-g[g]ˆa1(t) ˆG1,anc-g[g] = Z dτ ...
[55]

Based on the results in Secs

Projector onto quantum state in target temporal waveform We then calculate the projector incorporating the transformation of the time-frequency characteristics, denoted as 1⟨ϕProj| ˆW1. Based on the results in Secs. I A 1 and I A 2, this is given as follows: 1⟨ϕProj| ˆW1 ∝ 1⟨ 0|    NO j=1 1√ N ˆa1(Tj) + αjh(Tj)    anc-m⟨ 0| anc-g⟨ 0| ˆG1,anc-g[g] ˆM...
[56]

We now focus on the entangled states |ΦEnt⟩0,1 on which the projector acts, and discuss a similar decomposition into the mode f1 and the remaining modes {fl}∞ l=2

Two-mode Gaussian entangled state preparation So far, we have shown that 1⟨ϕProj| ˆW1 can be decomposed into a component acting on the mode f1 and components acting on the remaining modes {fl}∞ l=2. We now focus on the entangled states |ΦEnt⟩0,1 on which the projector acts, and discuss a similar decomposition into the mode f1 and the remaining modes {fl}∞...
[57]

(36) and the two-mode entangled state in Eq

Final heralded states In this section, we derive the final heralded state using the results of the projector in Eq. (36) and the two-mode entangled state in Eq. (46). The final state, |ψ⟩0 defined in Eq. (1), is expressed as follows: |ψ⟩0 ∝ NX n=0 C ′ n(α1, α2, . . . , αN) f1 1⟨n| ! ˆBf1 0,1(κ0) ˆSf1 0 (r0) ˆSf1 1 (r1) |0⟩f1 0 |0⟩f1 1 | {z } State in the ...
[58]

The temporal correlation function βj(t − t′) (j = 0, 1) is sufficiently narrow compared to the target TW f1, such that βj(t − t′) ≈ rjδ(t − t′)
[59]

This condition is approximately satisfied when the resolution is much shorter than the timescales of both the target TW f1 and the filter response time constant 1/γ

The detectors have effectively infinite temporal resolution. This condition is approximately satisfied when the resolution is much shorter than the timescales of both the target TW f1 and the filter response time constant 1/γ. Under these assumptions, we can realize a TW whose timescale is sufficiently longer than both the photon-pair correlation time and...
[60]

Type of heralded state Here, we derive an expression for the final heralded state |ψ⟩0 in the target TW f1 and show that it can always be written in the form ˆS0(rout)PN n=0 Cn |n⟩0. From Eq. (47), the final state |ψ⟩0 can be expressed as |ψ⟩0 ∝ NX n=0 C ′ n 1⟨n|ΦEnt⟩0,1, (48) |ΦEnt⟩0,1 = ˆB0,1(κ0) ˆS0(r0) ˆS1(r1) |0⟩0 |0⟩1 , (49) where we omit the TW lab...
[61]

How to set experimental parameters for target state In this section, we prove that our light source can generate any target state of the form given in Eq. (75). To this end, we describe below how to choose the experimental parameters ( r0, r1, T) and {αj}N j=1 so as to generate an arbitrary target state
[62]

(75), characterized by the parameters rout ∈ R and C0, C1,

We consider the generation of a target state of the form given in Eq. (75), characterized by the parameters rout ∈ R and C0, C1, . . . , CN ∈ C (CN ̸= 0)
[63]

We choose the parameters (r0, r1, T) to achieve the target squeezing level e−2rout based on Eq. (71). Any value of rout can be realized by an appropriate choice of ( r0, r1, T). At the end of this section, we mention the remaining flexibility in (r0, r1, T) under Eq. (71)
[64]

(50) and (51), thereby determining the coefficients {C ′′ j (n)}0≤j≤n≤N and {P (n)}N n=0

Once ( r0, r1, T) are fixed, the state |ψ(n)⟩0 heralded by n-photon detection and the corresponding probabil- ity P (n) are determined from Eqs. (50) and (51), thereby determining the coefficients {C ′′ j (n)}0≤j≤n≤N and {P (n)}N n=0
[65]

(74), using the fact that {P (n)}N n=0 and {C ′′ n (n)}N n=0 are nonzero, as discussed in Sec

Equation (74) relates the following quantities: • {C ′ n}N n=0 (to be determined, tunable via the experimental parameters {αj}N j=1) • {Cn}N n=0 (specified by the target state) • {C ′′ j (n)}0≤j≤n≤N and {P (n)}N n=0 (determined in the previous step) In Eq. (74), using the fact that {P (n)}N n=0 and {C ′′ n (n)}N n=0 are nonzero, as discussed in Sec. I B 1...
[66]

(35) and (36), the coefficients{C ′ n}N n=0 are related to the displacement amplitudes{αj}N j=1 as follows: NO j=1 1√ N c′ ˆa1[f1] + αj ∝ NX n=0 C ′ n√ n! (ˆa1[f1])n

From Eqs. (35) and (36), the coefficients{C ′ n}N n=0 are related to the displacement amplitudes{αj}N j=1 as follows: NO j=1 1√ N c′ ˆa1[f1] + αj ∝ NX n=0 C ′ n√ n! (ˆa1[f1])n . (79) 12 By replacing the operator ˆa1[f1] in Eq. (79) with a complex variable, the left-hand side becomes a product of N linear terms, while the right-hand side represents an N-th...

2000
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Projector onto quantum state in target temporal waveform 4

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Experimental details 12 A

How to set experimental parameters for target state 11 II. Experimental details 12 A. Setup details 12 B. Experimental parameters 14 C. Measurement & Analysis 15 References 17 I. FORMULA TION OF QUANTUM ST A TE GENERA TION IN PROPOSED SETUP In this section, we formulate the generation of quantum states in the proposed setup. We analyze the temporal wavefo...

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NX k=1 cj′′kˆak(Tj′′) ! + αj′′ h(Tj′′) #   NO j′=2 | 0⟩j′ (14) = NO j=1 j⟨ 0|

Projective measurement Here we calculate the projective measurement 1⟨ϕProj| in Channel 1 in Eq. (1). To detect a total of N photons using on/off detectors arranged in parallel, the beam in Channel 1 is split into N channels (Channels 1– N) using beam splitters (BSs), as illustrated in the green region of Fig. S1. This process is modeled by successive BS ...

[53] [54]

Transformation of time-frequency characteristics Next, we consider the transformation ˆW1 of time-frequency characteristics, realized through modulation and filtering as shown in Fig. S1. Following Ref. [2], transmission through a filter in Channel 1 can be formulated as a quantum operation involving two channels: ˆG† 1,anc-g[g]ˆa1(t) ˆG1,anc-g[g] = Z dτ ...

[54] [55]

Based on the results in Secs

Projector onto quantum state in target temporal waveform We then calculate the projector incorporating the transformation of the time-frequency characteristics, denoted as 1⟨ϕProj| ˆW1. Based on the results in Secs. I A 1 and I A 2, this is given as follows: 1⟨ϕProj| ˆW1 ∝ 1⟨ 0|    NO j=1 1√ N ˆa1(Tj) + αjh(Tj)    anc-m⟨ 0| anc-g⟨ 0| ˆG1,anc-g[g] ˆM...

[55] [56]

We now focus on the entangled states |ΦEnt⟩0,1 on which the projector acts, and discuss a similar decomposition into the mode f1 and the remaining modes {fl}∞ l=2

Two-mode Gaussian entangled state preparation So far, we have shown that 1⟨ϕProj| ˆW1 can be decomposed into a component acting on the mode f1 and components acting on the remaining modes {fl}∞ l=2. We now focus on the entangled states |ΦEnt⟩0,1 on which the projector acts, and discuss a similar decomposition into the mode f1 and the remaining modes {fl}∞...

[56] [57]

(36) and the two-mode entangled state in Eq

Final heralded states In this section, we derive the final heralded state using the results of the projector in Eq. (36) and the two-mode entangled state in Eq. (46). The final state, |ψ⟩0 defined in Eq. (1), is expressed as follows: |ψ⟩0 ∝ NX n=0 C ′ n(α1, α2, . . . , αN) f1 1⟨n| ! ˆBf1 0,1(κ0) ˆSf1 0 (r0) ˆSf1 1 (r1) |0⟩f1 0 |0⟩f1 1 | {z } State in the ...

[57] [58]

The temporal correlation function βj(t − t′) (j = 0, 1) is sufficiently narrow compared to the target TW f1, such that βj(t − t′) ≈ rjδ(t − t′)

[58] [59]

This condition is approximately satisfied when the resolution is much shorter than the timescales of both the target TW f1 and the filter response time constant 1/γ

The detectors have effectively infinite temporal resolution. This condition is approximately satisfied when the resolution is much shorter than the timescales of both the target TW f1 and the filter response time constant 1/γ. Under these assumptions, we can realize a TW whose timescale is sufficiently longer than both the photon-pair correlation time and...

[59] [60]

Type of heralded state Here, we derive an expression for the final heralded state |ψ⟩0 in the target TW f1 and show that it can always be written in the form ˆS0(rout)PN n=0 Cn |n⟩0. From Eq. (47), the final state |ψ⟩0 can be expressed as |ψ⟩0 ∝ NX n=0 C ′ n 1⟨n|ΦEnt⟩0,1, (48) |ΦEnt⟩0,1 = ˆB0,1(κ0) ˆS0(r0) ˆS1(r1) |0⟩0 |0⟩1 , (49) where we omit the TW lab...

[60] [61]

How to set experimental parameters for target state In this section, we prove that our light source can generate any target state of the form given in Eq. (75). To this end, we describe below how to choose the experimental parameters ( r0, r1, T) and {αj}N j=1 so as to generate an arbitrary target state

[61] [62]

(75), characterized by the parameters rout ∈ R and C0, C1,

We consider the generation of a target state of the form given in Eq. (75), characterized by the parameters rout ∈ R and C0, C1, . . . , CN ∈ C (CN ̸= 0)

[62] [63]

We choose the parameters (r0, r1, T) to achieve the target squeezing level e−2rout based on Eq. (71). Any value of rout can be realized by an appropriate choice of ( r0, r1, T). At the end of this section, we mention the remaining flexibility in (r0, r1, T) under Eq. (71)

[63] [64]

(50) and (51), thereby determining the coefficients {C ′′ j (n)}0≤j≤n≤N and {P (n)}N n=0

Once ( r0, r1, T) are fixed, the state |ψ(n)⟩0 heralded by n-photon detection and the corresponding probabil- ity P (n) are determined from Eqs. (50) and (51), thereby determining the coefficients {C ′′ j (n)}0≤j≤n≤N and {P (n)}N n=0

[64] [65]

(74), using the fact that {P (n)}N n=0 and {C ′′ n (n)}N n=0 are nonzero, as discussed in Sec

Equation (74) relates the following quantities: • {C ′ n}N n=0 (to be determined, tunable via the experimental parameters {αj}N j=1) • {Cn}N n=0 (specified by the target state) • {C ′′ j (n)}0≤j≤n≤N and {P (n)}N n=0 (determined in the previous step) In Eq. (74), using the fact that {P (n)}N n=0 and {C ′′ n (n)}N n=0 are nonzero, as discussed in Sec. I B 1...

[65] [66]

(35) and (36), the coefficients{C ′ n}N n=0 are related to the displacement amplitudes{αj}N j=1 as follows: NO j=1 1√ N c′ ˆa1[f1] + αj ∝ NX n=0 C ′ n√ n! (ˆa1[f1])n

From Eqs. (35) and (36), the coefficients{C ′ n}N n=0 are related to the displacement amplitudes{αj}N j=1 as follows: NO j=1 1√ N c′ ˆa1[f1] + αj ∝ NX n=0 C ′ n√ n! (ˆa1[f1])n . (79) 12 By replacing the operator ˆa1[f1] in Eq. (79) with a complex variable, the left-hand side becomes a product of N linear terms, while the right-hand side represents an N-th...

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