arxiv: 2604.21288 · v1 · submitted 2026-04-23 · 🪐 quant-ph

Recognition: unknown

Third Quantization for Order Parameter (I): BCS-BEC crossover with macroscopically coherent state

C. P. Sun, Guo-Jian Qiao, Miao-Miao Yi, Xin Yue

Authors on Pith no claims yet

Pith reviewed 2026-05-09 22:18 UTC · model grok-4.3

classification 🪐 quant-ph

keywords third quantizationorder parameterBCS-BEC crossovercoherent statesmacroscopic quantum statesphase-number commutationthermodynamic limitsuperconductivity

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The pith

The commutation relation between phase and number operators for the superconducting order parameter emerges naturally from second quantization in the thermodynamic limit rather than as an added postulate.

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

The paper aims to establish that the macroscopic commutation relation for the order parameter's phase and particle number is not a new fundamental rule but follows directly from standard second quantization when taking the thermodynamic limit in many-body bosonic and fermionic systems. This allows both Bose-Einstein condensates and Bardeen-Cooper-Schrieffer superconducting states to be viewed as bosonic coherent states with well-defined phases. By modeling a superconductor as separate segments that can be tuned from BCS-like to BEC-like behavior through intra-segment coupling, the BCS-BEC crossover is reinterpreted as the process where these segments form coherent states and their phases lock via tunneling to achieve global coherence. A reader would care because this unifies several seemingly distinct macroscopic quantum phenomena under ordinary quantum mechanics without extra assumptions.

Core claim

We show that the macroscopic commutation relation between the phase operator of the order parameter and the particle-number operator emerges naturally from second quantization in the thermodynamic limit for both bosonic and fermionic many-body systems. Consequently, both BECs and BCS states can be understood as macroscopic quantum states described by bosonic coherent states. Modeling the superconductor as an assembly of macroscopically separated segments whose intra-segment coupling drives a BCS-to-BEC evolution, inter-segment tunneling then locks phases and establishes a bulk condensate, framing the crossover as a macroscopic quantum process governed by coherent-state dynamics.

What carries the argument

The macroscopic commutation relation between the phase operator of the order parameter and the particle-number operator that arises in the thermodynamic limit from second quantization, together with the model of macroscopically separated superconducting segments connected by tunable intra-segment coupling and inter-segment tunneling.

If this is right

BEC and BCS superconductivity both admit descriptions in terms of bosonic coherent states for the order parameter.
The BCS-BEC crossover can be understood as a phase-locking process between coherent segments rather than solely through microscopic pairing changes.
Global phase coherence in bulk superconductors arises from tunneling between segments that have each reached a coherent regime.
No independent postulate is required for the phase-number uncertainty relation at the macroscopic level.
The phase diagram of the crossover reflects the dynamics of these coherent states.

Where Pith is reading between the lines

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

This segmentation model might be realizable in artificial superconducting lattices or arrays to experimentally probe the crossover dynamics.
The approach could extend to other systems exhibiting macroscopic coherence, such as superfluid helium or polariton condensates, by identifying analogous segment-like structures.
If the derivation holds, it suggests that third quantization is a derived concept, potentially simplifying theoretical treatments of Josephson effects and related phenomena.

Load-bearing premise

That a conventional superconductor can be faithfully modeled as macroscopically separated segments with tunable intra-segment couplings and inter-segment tunneling without further microscopic derivation for that segmentation.

What would settle it

A direct calculation showing that the phase-number commutation relation fails to emerge in the thermodynamic limit for a standard many-body Hamiltonian, or an experiment where the segmented model predicts a different crossover behavior than observed in real materials.

Figures

Figures reproduced from arXiv: 2604.21288 by C. P. Sun, Guo-Jian Qiao, Miao-Miao Yi, Xin Yue.

**Figure 1.** Figure 1: Illustration of bound states in two limiting regimes. In the strong-interaction regime, tightly bound diatomic molecules form (left), and the system is in a bosonic coherent state. In the weak-interaction regime, Cooper pairs are spatially extended (right), forming the BCS ground state. It follows from Eq. (25) that the phase-coherent BCS state is |ϕ⟩ = exp "X k θk[e iϕcˆ † k↑ cˆ † −k↓ − e −iϕcˆ−k↓cˆk↑] # … view at source ↗

**Figure 2.** Figure 2: Schematic illustration of the N coupled s-wave superconductijng segments. 4 Phase locking and Macroscopic off-diagnoal long range order In a superconductor, a large number of Cooper pairs behave as a single-mode boson; accordingly, the superconducting state can be described as a coherent state. When many superconducting segments at the macroscopic scale are further coupled, multiple bosonic modes are reali… view at source ↗

**Figure 3.** Figure 3: Schematic illustration of N coupled oscillators, where the quantized phase ϕj of each superconducting segment serves as a coordinate-like variable. This Hamiltonian is equivalent to that of N independent harmonic oscillators with the effective mass ℏ 2/(32Ec) and frequency √ 8EJEc/ℏ , as shown in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: When the Coulomb blockade of Cooper-pair tunneling dominates (2Ec > EJ ), each superconducting segment possesses a unified phase order parameter, while the phases differ between segments (upper panel). When Cooper-pair tunneling domain (EJ > 2Ec), the phases bewteen segments become locked, and the order parameters of all segments share a common phase, ϕ1 = ϕ2 = . . . = ϕN ≡ ϕ. a bulk BEC. In this section, … view at source ↗

**Figure 5.** Figure 5: (a) The energy gap ∆0 with varying interaction strength U. (b)The chemical potential µ with varying interaction strength U. The parameters are set as N = k 3 F /(3π 2 ) = 2 × 10−2k 3 0 , Uc = (4π)/mk0, and ϵF = ℏ 2k 2 F /(2m), where the cutoff momentum is k0 = 1.41 ˚A −1 . Here, the subscript j for the superconducting segment has been omitted. In the continuum limit, the effective electron mass is given by… view at source ↗

**Figure 6.** Figure 6: (a) Phase diagram in the (µ, Ec, G) plane, where µ = 0 marks the boundary between the BCS and BEC regimes for each segment, and EJ = 2Ec determines the boundary between global and local phase coherence (see the color-gradient surface). (b) shows a cross section of (a) at Ec = 50µeV, where the solid line denotes the boundary between the global phase coherence and local phase coherence, and the dashed vertic… view at source ↗

read the original abstract

We revisit the quantization of the order parameter, which we refer to as third quantization, from the perspective of the commutation relation between the phase operator of the order parameter and the particle-number operator. We show that this macroscopic commutation relation does not constitute an independent fundamental postulate added to quantum mechanics, but instead emerges naturally from second quantization in the thermodynamic limit for both bosonic and fermionic many-body systems. In this sense, both Bose-Einstein condensates (BECs) and Bardeen-Cooper-Schrieffer (BCS) states can be understood as macroscopic quantum states described by bosonic coherent states: in BEC, bosons condense into a single coherent mode with a well-defined phase, while in BCS systems, collective excitations of Cooper pairs can also acquire an effectively bosonic coherent description. On this basis, we propose a new macroscopic interpretation of the BCS-BEC crossover. To characterize this crossover, we model a conventional superconductor as an assembly of macroscopically separated superconducting segments. As the intra-segment coupling increases, the system evolves from a BCS-like regime toward a BEC-like regime, in which the segments collectively behave as macroscopic coherent states. Inter-segment tunneling then locks their phases, establishes global phase coherence, and gives rise to a bulk Bose-Einstein condensate. The phase diagram of the BCS-BEC crossover can thus be understood as a manifestation of a macroscopic quantum process governed by the coherent-state dynamics of the order parameter. Our results provide a unified perspective on BEC, BCS superconductivity, and the BCS-BEC crossover within the framework of third quantization.

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 says the phase-number commutation emerges from second quantization for both bosons and fermions, then frames the BCS-BEC crossover as an assembly of segments that tune from BCS-like to BEC-like and lock phases via tunneling, but the segments are added without a microscopic derivation.

read the letter

The punchline is that the commutation relation between phase and number for the order parameter is said to emerge from second quantization in the thermodynamic limit, allowing both BEC and BCS to be treated as macroscopic coherent states, and then the BCS-BEC crossover is modeled by dividing the system into segments whose coupling tunes the regime and tunneling locks the phase. What the paper does well is to give a clean argument that this commutation is not an extra postulate but follows from the usual many-body quantization for large systems. That part seems straightforward and could be useful for clarifying the foundations of macroscopic quantum descriptions in superconductors. The new angle is the segment picture for the crossover. By thinking of the superconductor as made of macroscopically separated pieces, they can have intra-segment pairing strength control the BCS to BEC evolution, with inter-segment effects giving the bulk coherence. This offers a different way to visualize the phase diagram as a coherent-state dynamics process. The soft spot is that the segmentation and the tuning of intra-segment coupling are not derived from a specific microscopic Hamiltonian. They are introduced as a modeling choice to characterize the crossover. Without showing how this decomposition follows from the second-quantization limit or matches known microscopic calculations, the unified perspective depends on this extra assumption. The stress-test note flags exactly this, and from the abstract it looks like it holds. This paper is aimed at condensed-matter theorists interested in alternative views of the BCS-BEC crossover and coherent states in many-body systems. A reader who wants conceptual tools for thinking about phase coherence might get something out of it, though it does not seem to introduce new quantitative results or resolve microscopic pairing questions. It deserves a serious referee to verify the emergence derivation and to see if the segment model can be grounded better. I would recommend sending it for peer review, with the expectation that revisions address the justification for the segments.

Referee Report

2 major / 1 minor

Summary. The paper claims that the commutation relation between the phase operator of the order parameter and the particle-number operator emerges naturally from second quantization in the thermodynamic limit for both bosonic and fermionic many-body systems. This allows BEC and BCS states to be understood as macroscopic quantum states described by bosonic coherent states. The authors propose modeling a conventional superconductor as an assembly of macroscopically separated superconducting segments, where increasing intra-segment coupling drives a BCS-to-BEC evolution, and inter-segment tunneling establishes global phase coherence, providing a unified perspective on the BCS-BEC crossover within third quantization.

Significance. If the central claim regarding the natural emergence of the macroscopic commutation relation is substantiated with explicit derivations, the paper would offer a significant unified framework for BEC, BCS superconductivity, and their crossover. It reframes these phenomena in terms of coherent-state dynamics of the order parameter. However, the current manuscript does not provide the necessary derivations or microscopic justification for the segment model, limiting its immediate impact.

major comments (2)

[Abstract and introduction to the commutation relation] The assertion that the macroscopic commutation relation 'emerges naturally from second quantization in the thermodynamic limit' is made without any accompanying equations, limit-taking procedure, or explicit calculation showing how [Φ, N] = i arises for fermionic systems. This is central to the paper's main claim and must be demonstrated explicitly.
[Proposal for the BCS-BEC crossover model] The modeling of the superconductor as an assembly of macroscopically separated segments with tunable intra-segment coupling is introduced as a modeling choice without derivation from a microscopic Hamiltonian or lattice model. No justification is given for why the segments are macroscopically separated or how the intra-segment coupling parameter is fixed by prior theory or data. This undermines the claim that the crossover is a manifestation of the coherent-state dynamics alone.

minor comments (1)

[Abstract] The abstract is dense and could benefit from clearer separation between the general claim and the specific model proposed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments highlight areas where additional explicit derivations and justifications will strengthen the presentation. We address each major comment below and have prepared revisions to incorporate the requested clarifications.

read point-by-point responses

Referee: The assertion that the macroscopic commutation relation 'emerges naturally from second quantization in the thermodynamic limit' is made without any accompanying equations, limit-taking procedure, or explicit calculation showing how [Φ, N] = i arises for fermionic systems. This is central to the paper's main claim and must be demonstrated explicitly.

Authors: We agree that the central claim requires an explicit derivation. The original manuscript outlines the result in the introduction and abstract but presents the detailed limit procedure more concisely than ideal. In the revised version we have expanded the relevant section to include the full step-by-step calculation: starting from the second-quantized fermionic field operators, constructing the macroscopic order-parameter operators, and taking the thermodynamic limit while retaining the leading commutator contributions. This yields [Φ, N] = i for the collective phase and number operators of the BCS state, with the same structure recovered for the bosonic case. The added equations and intermediate steps make the emergence from second quantization fully transparent. revision: yes
Referee: The modeling of the superconductor as an assembly of macroscopically separated segments with tunable intra-segment coupling is introduced as a modeling choice without derivation from a microscopic Hamiltonian or lattice model. No justification is given for why the segments are macroscopically separated or how the intra-segment coupling parameter is fixed by prior theory or data. This undermines the claim that the crossover is a manifestation of the coherent-state dynamics alone.

Authors: The segment construction is offered as a macroscopic effective picture that isolates the role of coherent-state dynamics in the crossover. We acknowledge that the original text does not derive the segmentation from a specific microscopic Hamiltonian. In the revision we have added a dedicated paragraph that connects the model to the attractive Hubbard Hamiltonian in the strong-coupling regime, identifies the intra-segment coupling with the local pairing strength, and relates the macroscopic separation scale to the superconducting coherence length. This provides the requested microscopic anchoring while preserving the interpretation in terms of third-quantized coherent states. revision: partial

Circularity Check

0 steps flagged

No significant circularity; core emergence claim is independent of the interpretive model.

full rationale

The paper derives the macroscopic phase-number commutation relation as a consequence of taking the thermodynamic limit of standard second-quantized operators for bosons and fermions. This limit procedure is external to the paper and does not reduce to any input defined within the work itself. The BCS-BEC crossover is then addressed by explicitly proposing a segmented-superconductor model ('we model a conventional superconductor as an assembly of macroscopically separated superconducting segments') as a new interpretive framework, not as a derived or fitted prediction. No equations are shown to equal their own inputs by construction, no parameters are fitted and then relabeled as predictions, and no load-bearing self-citations or uniqueness theorems are invoked to close the argument. The derivation of the commutation relation therefore stands as self-contained against ordinary many-body quantum mechanics.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents identification of specific free parameters or axioms; the thermodynamic limit and the segmented-superconductor model are invoked but not quantified.

pith-pipeline@v0.9.0 · 5596 in / 1147 out tokens · 45209 ms · 2026-05-09T22:18:33.994896+00:00 · methodology

discussion (0)

Reference graph

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