Schr{\"o}dinger cat state formation in small bosonic Josephson junctions at finite temperatures and dissipation

A P Alodjants; D V Ansimov; D V Tsarev; S A Podoshvedov

REVIEW 2 major objections 4 minor 61 references

Schr{\"o}dinger cat state formation in small bosonic Josephson junctions at finite temperatures and dissipation

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Two-mode Bose condensates with attractive interactions form Schrödinger cat states only below a critical temperature set by the junction's Rabi frequency and interaction strength.

desk verdict The zero-temperature analysis is solid but the finite-temperature crossover claim is built on a misapplied metastable-decay formalism: in a symmetric double well the temperature that destroys cat coherence is set by the exponentially small tunneling splitting, not by the barrier-top frequency. read the letter →

arxiv 2507.16032 v1 pith:STQJXR3U submitted 2025-07-21 quant-ph

D V Tsarev , D V Ansimov , S A Podoshvedov , A P Alodjants This is my paper · ORCID

classification quant-ph

keywords SchrödingercatstatesbosonicJosephsonjunctiontwo-modeBose-EinsteincondensatequantumphasetransitioncriticaltemperaturetunnelingthermalactivationN00N

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

This paper claims that a small two-mode Bose-Einstein condensate with attractive interactions, realized as a bosonic Josephson junction, forms Schrödinger cat states through a zero-temperature quantum phase transition at $\lambda = 1$, where $\lambda = uN/\hbar\omega_R$ is the ratio of nonlinear interaction strength to Josephson coupling. At finite temperature the same system has a critical temperature $T_c = \hbar\omega_R\sqrt{\lambda-1}/(2\pi k_B)$, with a damping-dependent correction, below which quantum tunneling between the wells dominates over thermal activation and the cat superposition can form. The result matters because it turns the question of when macroscopic superpositions are experimentally accessible into a concrete temperature budget, and the paper estimates that for a lithium condensate this scale is about $1.6$ nK, well below the condensation temperature.

What carries the argument

The machinery is an effective-potential mapping plus the imaginary-time bounce technique. The two-mode condensate is replaced by a fictitious macroscopic particle with effective mass $m_{\rm eff}=s\hbar\sqrt{1-x^2}/\omega_R$ moving in the potential $V(x)=-(s\hbar\omega_R/2)(\lambda x^2+2\sqrt{1-x^2})$, where $x=(n_b-n_a)/N$ is the normalized particle-number imbalance. For $\lambda>1$ this potential is a double well whose minima and barrier height are analytic functions of $\lambda$. The finite-temperature analysis inverts the potential to $U(x)=-V(x)$, introduces a 'thermon' quasiparticle that oscillates near the saddle point with frequency $\omega=\omega_R\sqrt{\lambda-1}$, and identifies the crossover temperature from the standard bounce relation $T_c=\hbar\omega/(2\pi k_B)$; Ohmic dissipation is folded in through the factor $\alpha$. That conversion of a many-body superposition problem into single-particle metastable decay is what carries the quantitative claims.

What would settle it

One could measure the escape or relaxation rate of a small attractive two-mode condensate as a function of temperature and check for a crossover from Arrhenius behavior $\Gamma\simeq(\omega_0/2\pi)e^{-V_0/k_BT}$ at $T>T_c$ to temperature-independent quantum tunneling at $T<T_c$, with the crossover temperature scaling as $\sqrt{\lambda-1}$ and vanishing at $\lambda=1$; a direct probe would be to record the Wigner function of the ground state and observe whether the two peaks and interference fringes survive below $T_c$ and wash out above it.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the ground state of a two-mode BJJ with attractive interactions is governed by the parameter $\lambda$, with a quantum phase transition at $\lambda=1$: for $\lambda<1$ the state is a coherent Gaussian distribution over particle-number imbalance, while for $\lambda>1$ it becomes a superposition of two wave packets centered at $x_{\pm}=\pm\sqrt{1-\lambda^{-2}}$, and in the limit $\lambda\to\infty$ it approaches the N00N state $(|N,0\rangle+|0,N\rangle)/\sqrt{2}$. The finite-temperature extension shows that quantum tunneling, which is responsible for the cat state, dominates only for $T\le T_c=\hbar\omega/(2\pi k_B)$ with $\omega=\omega_R\sqrt{\lambda-1}$, and weak Ohmic dissipation renormalizes this to $T_c=\hbar\omega\alpha/(2\pi k_B)$ with $\alpha=\sqrt{1+(\gamma/2\omega)^2}-\gamma/2\omega$. The paper also derives the minimum bounce exponent $B_c=s\pi(\lambda-1)^{3/2}/(\alpha\lambda)$ for observing the cat state, and concludes that the required temperatures are low but experimentally reachable.

Load-bearing premise

The argument assumes that the single-particle imaginary-time bounce theory of metastable decay in an inverted potential correctly says when the many-body two-mode condensate actually forms or keeps its Schrödinger cat superposition, rather than merely describing escape from a metastable well.

Editorial extensions

If this is right

For a given junction, cooling below $T_c=\hbar\omega_R\sqrt{\lambda-1}/(2\pi k_B)$ should suffice for cat-state formation, while above $T_c$ thermal activation destroys the superposition.
The fidelity of the N00N-like state grows with $\lambda$, and the ideal N00N state is reached only as $\lambda\to\infty$, so experiments should target moderate $\lambda$ with a fidelity threshold rather than exact N00N states.
Because $T_c$ vanishes at $\lambda=1$, the quantum phase transition itself is purely quantum; approaching it from the cat side requires ever-lower temperatures.
Weak Ohmic dissipation lowers the crossover temperature by the factor $\alpha$, so stronger damping pushes cat-state formation to lower temperatures.
For the $^7$Li parameters used, the temperature scale is $T_0=\hbar\omega_R/(2\pi k_B)=1.6$ nK, so $T_c=T_0\sqrt{\lambda-1}$ is far below the $\sim100$ nK condensation temperature and within reach of current cooling.

Reading between the lines

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

A natural extension the paper does not develop is to turn the static crossover into a dynamical prediction: below $T_c$, the same bounce formalism could estimate the lifetime of the cat superposition under Ohmic noise, giving a direct decoherence time for practical preparation protocols.
The Gaussian and inverted-parabola approximations break down near $\lambda=1$ and for $\lambda\to\infty$; a numerical solution of the two-mode Schrödinger equation with a thermal bath would test whether $T_c$ and $B_c$ survive beyond the analytic approximations.
Because the transition at $T_c$ is described as second-order, one could look for critical slowing down or enhanced fluctuations in the particle-number imbalance near the crossover, similar to equilibrium phase transitions.

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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

You should know the paper's central finite-T claim is very likely wrong, despite the clean standard derivation. The authors compute the crossover temperature between thermal activation and quantum tunneling using the bounce formalism for a metastable well (Eqs. 29–37), but for λ > 1 their effective potential is a symmetric double well with degenerate minima. The low-energy physics is a coherent superposition (cat state) split by an exponentially small Δ ≃ ℏω0 exp(−S0/ℏ). At thermal equilibrium, the off-diagonal coherence is (1/2)tanh(Δ/2kBT), which is already negligible for kBT ≫ Δ. For their example (N=100, λ≈2, ωR=2π×208 Hz), Δ/kB is of order 10⁻³ nK or less, while their Tc is 1.6 nK·√(λ−1). So the paper never establishes that observing a coherent SC state requires T ≤ Tc; it just shows an escape-rate crossover that is irrelevant to the ground-state superposition. This is a load-bearing flaw, not a minor caveat.

What is actually useful: the zero-temperature effective-potential mapping (Sec. 3), the Gaussian approximation for the two localized wave packets, the Wigner function (Sec. 4) showing interference fringes, and the numerical estimates for 7Li. These parts are standard but coherent. The derivation of the λ=1 quantum phase transition is not new and is properly credited (Refs. [21,50]). The paper also contains a dimensionally wrong Eq. (19); it should be σ = (ℏ²/(2m_eff c₂))^(1/4) or similar, as printed the units do not work. That is a typo, but symptomatic of a hasty write-up.

The bounce calculation itself is routine and the self-citations to Ref. [42] are legitimate background, not circular. The problem is the physical interpretation: they conflate a decay rate from a local minimum with the coherence of an equilibrium cat state. This could be repaired if they either (a) add a slight tilt to make one well metastable and frame the result as a decay-rate crossover, or (b) explicitly compute the thermal density matrix and the temperature at which the off-diagonal purity drops. As written, the finite-temperature section cannot support their main claim.

For a serious referee: send it out? Yes, because the zero-temperature parts may be publishable and the flaw is instructive, but the paper needs major revision. The reader's conditional verdict is too generous; I would lean toward rejection unless the authors reframe the finite-T result. Who would benefit? People working on two-mode BECs and cat-state preparation, but primarily as a cautionary example of how instanton methods get misapplied to symmetric double wells.

Referee Report

2 major / 4 minor

Summary. This paper studies a two-mode bosonic Josephson junction (BJJ) with attractive interactions. It maps the Fock-space dynamics to a continuous Schrödinger equation for a fictitious particle in an effective potential, identifies a zero-temperature structural change at lambda = 1, uses Wigner functions to characterize cat and N00N-like states, and then invokes the imaginary-time bounce formalism to compute a crossover temperature Tc = hbar*omega_R*sqrt(lambda-1)/(2*pi*k_B) below which quantum tunneling is claimed to dominate and Schrödinger cat states can be observed in the presence of weak dissipation. The paper closes with numerical estimates for a 7Li condensate, finding Tc = 1.6 nK, well below the BEC temperature.

Significance. If the central finite-temperature claim were established, the paper would provide a simple quantitative temperature scale for preparing cat states in small BJJs, which would be useful for experiment planning. The paper has explicit analytic strengths: the effective-potential mapping, closed-form expressions for the barrier height and oscillation frequencies, and a Wigner-function representation of the cat-state interference are all given without fitted free parameters, and the lambda = 1 structural change is taken from prior literature rather than introduced ad hoc. However, the finite-temperature interpretation does not follow from the metastable-decay machinery used, and the quoted temperature is the wrong physical scale for coherence of the symmetric double-well ground state. As it stands, the paper's advertised main conclusion is not supported, although the zero-temperature analysis may have value as a separate contribution.

major comments (2)

[Sec. 5, Eqs. (29)-(37)] The manuscript identifies Tc = hbar*omega/(2*pi*k_B) as the threshold below which Schrödinger cat states can be observed, but the bounce formalism (Refs. [53-57]) computes the crossover between thermal activation and quantum tunneling for metastable decay, not the temperature at which the equilibrium state of a symmetric double well is a coherent cat. For the effective potential (13) with lambda > 1, the two minima are degenerate, so the low-energy eigenstates are symmetric and antisymmetric superpositions split by an exponentially small energy Delta ~ hbar*omega_0*exp(-S0/hbar). The equilibrium off-diagonal coherence between the two localized configurations is proportional to tanh(Delta/(2*k_B*T)), so the coherence is already negligible once k_B*T >> Delta. The weak-metastability condition in Eq. (32), V0/(hbar*omega_0) >> 1, makes Delta exponentially suppressed, so Delta/k_B can be orders of magnitude smaller than Tc. For the Section 5 parameters (N=100, lambda approximately 2, omega_R=2*pi*208 Hz), the two-level estimate gives Delta/k_B far below the quoted Tc = 1.6 nK. The statement in Eq. (37) that SC state observation 'may be obtained for barriers possessing B >= Bc' therefore rests on an identification between an incoherent escape-rate crossover and the existence of a coherent superposition that the paper never derives. A direct calculation of the thermal-state fidelity or coherence in the two-mode model is needed before this claim can be accepted.
[Sec. 5, numerical example after Eq. (37)] The numerical parameters are internally inconsistent. With uN/k_B = 10 nK and omega_R = 2*pi*208 Hz, the parameter lambda = uN/(hbar*omega_R) = (uN/k_B)/(hbar*omega_R/k_B) = 10/1.6 is about 6.25 at N = N0 = 100, not 'nearby the critical value lambda = 1' as stated in the text. To realize lambda approximately 1 with the same omega_R, one would need uN/k_B approximately 1.6 nK, which changes the quoted critical temperature and the associated experimental estimates. The numerical example should be reworked so that the stated values of N0, u, and omega_R are mutually consistent.

minor comments (4)

[Eq. (19)] The printed formula sigma = 2*m_eff/hbar^2*c_2^{-1/4} is dimensionally inconsistent; the harmonic-oscillator ground-state width should be sigma = (hbar^2/(2*m_eff*c_2))^{1/4}. The subsequent explicit expressions in Eqs. (20)-(21) are consistent with the corrected form, so this appears to be a typographical error that should be fixed.
[Sec. 2 and Sec. 6] The transition at lambda = 1 is repeatedly called a quantum phase transition, but for a finite-N system with N = 100 the ground-state energy is analytic in lambda; this is a finite-size crossover rather than a true phase transition. The terminology should be qualified.
[Fig. 3 caption and Abstract] There are several typos: Fig. 3 caption contains 'other other parameters', and the Abstract contains 'in the the zero temperature limit'. These should be corrected.
[Eq. (12) and surrounding text] The statement that the sqrt(1-x^2) factor in the effective mass 'plays no essential role' is not substantiated; at the minima for lambda > 1 this factor changes the effective mass by a factor lambda, which is explicitly used in Eq. (22). Please clarify the intended approximation.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: Tc is obtained from the standard instanton crossover formula with the barrier frequency computed from U(x)=-V(x); the T≤Tc-to-SC identification is an unsupported physical assumption, not an input-output equivalence.

full rationale

The derivation chain is self-contained rather than circular. Starting from the two-mode Hamiltonian (1), the continuum limit Eq. (10) and the effective potential V(x) in Eq. (13) are algebraic; the λ=1 double-well onset follows from V''(0), and the Gaussian widths, ω0, and Wigner function are computed from the model parameters. The central finite-temperature result Tc=ℏω/(2πkB) (Eq. 29) is not fitted and is not defined to be the SC condition: it is the standard crossover temperature of the bounce/instanton formalism attributed to Refs. [53–57], and ω=ω_R√(λ-1) is obtained from U''_xx(0)/m of the inverted potential U(x)=-V(x) (Eqs. 26–30). The dissipation-modified Tc (Eq. 33) comes from the same external formalism. Numerics insert reported physical parameters (a_sc, ω⊥, ω_R) and do not reverse-engineer Tc. Self-citations to Refs. [21] and [42] provide background on the spectrum and the effective-potential/inverted-potential method; the load-bearing instanton formulas are external. The genuine weakness is physical, not circular: Eq. (37) asserts that 'the SC state observation, which is based on the essentially quantum tunneling processes, may be obtained for the barriers possessing B ≥ Bc,' but the paper never derives the relation between the metastable-decay crossover and the exponentially small tunneling splitting that controls equilibrium cat coherence in a symmetric double well. That is a missing justification for the central claim, not a reduction of the prediction to its inputs. Score 1 reflects minor, non-load-bearing self-citation only.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The central derivation introduces no free parameters fitted to data; the assumptions are the two-mode and continuous-variable approximations, the application of single-particle instanton theory to the many-body model, and the Ohmic weak-dissipation model.

assumptions (5)

domain assumption Two-mode approximation is valid for the considered particle numbers (up to N ≈ 1000).
Invoked in Sec. 2 and the conclusion; the paper notes the two-mode approach breaks down for N ≳ 1000.
domain assumption The discrete three-term recurrence (3) can be accurately replaced by the continuous Schrödinger equation (10) with position-dependent mass.
Used in Sec. 3 to define the effective potential; the approximation is checked only for a few λ values in Fig. 2.
ad hoc to paper The imaginary-time bounce formalism for a single quantum particle in a double-well potential applies to the many-body BJJ system and its effective potential.
Assumed in Sec. 5 without derivation from the original two-mode Hamiltonian; the crossover temperature (29) is taken from this formalism.
domain assumption Dissipation is Ohmic and weak (γ ≪ ω), entering only through the factor α in Eq. (33).
Stated in Sec. 5; no microscopic derivation of the environment coupling is given.
domain assumption Gaussian wave-packet approximation (15) and harmonic expansion (17) are valid away from λ = 1.
Used to derive σ, C², and ω0 in Secs. 3-4; the paper acknowledges the failure near λ = 1 and for λ → ∞.

invented entities (2)

Fictitious macroscopic (FM) quantum particle
purpose: Represents the two-mode condensate population imbalance as a single particle moving in effective potential V(x).
Mathematical mapping introduced in Sec. 3 (Eqs. 11-13); not a physical entity.
Thermon quasiparticle
purpose: Describes quantum tunneling in imaginary time as harmonic oscillations in the inverted potential U(x) = -V(x).
Standard instanton-theory construct used in Sec. 5 (Eq. 26 and Fig. 2c inset); not a physical quasiparticle.

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Cite this review

Pith. "Pith review of Schr{\"o}dinger cat state formation in small bosonic Josephson junctions at finite temperatures and dissipation." pith.science (2026). https://pith.science/paper/STQJXR3U

@misc{pith2026250716032,
  author       = {Pith},
  title        = {Pith review of: Schr\"odinger cat state formation in small bosonic Josephson junctions at finite temperatures and dissipation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/STQJXR3U}},
  note         = {Machine review of arXiv:2507.16032}
}

abstract

In this work, we consider the feasibility of Schr{\"o}dinger cat (SC) and $N00N$ states formation by a convenient bosonic Josephson junction (BJJ) system in two-mode approximation. Starting with purely quantum description of two-mode Bose-Einstein condensate we investigate the effective potential approach that provides an accurate analytical description for the system with a large number of particles. We show that in the zero temperature limit SC states result from a quantum phase transition that occurs when the nonlinear strength becomes comparable with the Josephson coupling parameter. The Wigner function approach demonstrates the growth of the SC state halves separation and formation of $N00N$-like states (a Fock state superposition) with the particle number increase. We examine the possibility to attain the SC state at finite temperatures and a weak dissipation leading to appearing of some critical temperature; it defines the second-order phase transition from classical activation process to the SC state formation through the quantum tunneling phenomenon. Numerical estimations demonstrate that the critical temperature is sufficiently below the temperature of atomic condensation. The results obtained may be useful for experimental observation of SC states with small condensate Josephson junctions.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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work page 2017

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Schr¨ odinger E 1935 Naturwissenschaften, Bd.23 844

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Dodonov V V, Malkin I A and Man’ko V I 1974 Physica 72 597

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