For the d-mode damped quantum harmonic oscillator at finite temperature, we ask whether the Lindblad-resolvent ratio derived for the 1D case (Dai 2026) generalises to a dimension-dependent limit, as a naive "phase-space dimension" intuition might suggest. We establish that the limit is exactly -2, independent of d, for the collective polynomial perturbation. The argument rests on a collective-coordinate reduction theorem: under the symmetric d-mode thermal Lindblad (same γ, n̄, β across modes), the ratio depends only on the single bosonic mode, which is itself in a 1D thermal state at the same β under the original Lindblad. The d-mode problem reduces exactly to the 1D problem with rescaling. The dimensionless ratio is therefore strictly invariant under d. We further show: (i) For the asymmetric sum-of-powers observable, the finite-p ratio has a closed-form d-correction, but the correction decays geometrically (factor ~1/4 per +2 in p), so the limit is still -2. (ii) For anisotropic temperatures, the symmetry is broken at every p, but our numerical evidence (d=2, p up to 12) is consistent with, but far from establishing, the same -2 limit. We discuss why the "phase-space dim = 2d" intuition fails, and give this as a mathematically explicit example of dimensional reduction associated with the collective-observable sector.
For the d-mode damped quantum harmonic oscillator at finite temperature, we ask whether the Lindblad-resolvent ratio derived for the 1D case (Dai 2026) generalises to a dimension-dependent limit, as a naive "phase-space dimension" intuition might suggest. We establish that the limit is exactly -2, independent of d, for the collective polynomial perturbation. The argument rests on a collective-coordinate reduction theorem: under the symmetric d-mode thermal Lindblad (same γ, n̄, β across modes), the ratio depends only on the single bosonic mode, which is itself in a 1D thermal state at the same β under the original Lindblad. The d-mode problem reduces exactly to the 1D problem with rescaling. The dimensionless ratio is therefore strictly invariant under d. We further show: (i) For the asymmetric sum-of-powers observable, the finite-p ratio has a closed-form d-correction, but the correction decays geometrically (factor ~1/4 per +2 in p), so the limit is still -2. (ii) For anisotropic temperatures, the symmetry is broken at every p, but our numerical evidence (d=2, p up to 12) is consistent with, but far from establishing, the same -2 limit. We discuss why the "phase-space dim = 2d" intuition fails, and give this as a mathematically explicit example of dimensional reduction associated with the collective-observable sector.
For the d-mode damped quantum harmonic oscillator at finite temperature, we ask whether the Lindblad-resolvent ratio Δ_q*/g_F derived for the 1D case (Dai 2026, DC paper / paper #52; with p → ∞ limit −2) generalises to a dimension-dependent limit −2d, as a naive "phase-space dimension" intuition might suggest. We establish that the limit is exactly −2, independent of d, for the collective polynomial perturbation A_p = (Σ_i (a_i + a_i†))^p. The argument rests on a collective-coordinate reduction theorem: under the symmetric d-mode thermal Lindblad (same n̄, γ, ω across modes), A_p depends only on the single bosonic mode b = (1/√d) Σ_i a_i, which is itself in a 1D thermal state at the same β under the original Lindblad. The d-mode problem reduces exactly to the 1D problem with rescaling A_p ↦ d^{p/2}(b+b†)^p. The dimensionless ratio Δ_q*/g_F is therefore strictly invariant under d. We further show: (i) For the asymmetric sum-of-powers observable B_p = Σ_i (a_i + a_i†)^p, the finite-p ratio has a closed-form d-correction DC(B_p) = DC_{1D}(p) − (d−1)⟨x^p⟩²_{1D}/g_{F,1D}(p), but the correction decays geometrically, so the limit is still −2. (ii) For anisotropic temperatures β_i ≠ β_j, the symmetry is broken at every p, but numerical evidence (d=2, p up to 12) is consistent with the limit again being −2, suggesting the −2 ceiling reflects a single-mode property of the thermal Lindblad eigenstructure, not phase-space geometry. We discuss why the "phase-space dim = 2d" intuition fails, and propose this as a precise mathematical instantiation of emergence: the d-mode system, in the collective-observable sector, behaves identically to a 1-mode system at the same β.
For the damped quantum harmonic oscillator at finite temperature, we study the ratio Δ_q⋆/g_F between a Lindblad-resolvent-induced quadratic form W_q⋆(A) = ⟨A · K_β⋆(A)⟩{ρ_β} (with K_β⋆ = (−ℒ⋆)^{−1} − I, ℒ⋆ the Heisenberg-picture Lindblad generator that admits the thermal state ρ_β as a stationary state) and the exponential-family Fisher variance g_F(A) = Var{ρ_β}(A), evaluated on polynomial perturbations A_p = (a + a†)^p. We establish five exact results. Theorem A (hypergeometric decomposition): the projector weight R_l onto the l-th Hermite eigenmode of −ℒ⋆ admits a closed hypergeometric form. Theorem B (closed-form factorial moments): all factorial moments admit closed product formulas. Theorem 1 (DC ratio): the resolvent-to-Fisher ratio equals −2 for every polynomial order p. Theorem 2 (off-diagonal closed form): gives the off-diagonal deviation matrix. Corollary 3: establishes negative semidefiniteness of the full deviation matrix. Proofs use only Hermite orthogonality, OU spectral decomposition, and the Bény–Osborne Wasserstein gradient flow formulation.
Collective-Coordinate Dynamics and Radiative Decoherence in a Driven-Dissipative Polariton CondensateGeometric constraints on centre-of-mass path coherence and a numerical test of self-confinement This paper asks whether a room-temperature driven-dissipative exciton-polariton condensate can supply a localized collective coordinate able to occupy a resolvable two-path superposition. For the platform considered and for ordinary radiative coupling, the answer is negative for two independent reasons, and the paper states precisely where the formal construction holds and where its physical realization fails. Radiative which-path decoherence. Treating the condensate as an open system with a mode-resolved loss channel, the coherence between two localized branch states decays at ΓCOM = Nγ[1 − Re χ(d)], where χ(d) is the overlap of the photon states emitted by the two branches. The controlling variable is environmental distinguishability rather than mechanical recoil: a photon that displaces the centre of mass by a negligible amount can still record, at full strength, which branch the emitter occupied. Where ordinary out-coupling preserves transverse-mode information, two requirements act on the same length scale and conflict — branch resolvability requires a separation of several core widths, indistinguishability requires a separation well below one — so that, for a sech profile and the working convention d ≳ 3ξ, the dimensionless ratio ΓCOM/(Nγ) is bounded below by about 0.7. The coefficient is geometric; the absolute rate retains its dependence on N and γ. Numerical test of self-confinement. The operating point inferred from the measured group velocity on the CsPbBr3 microwire platform lies in the positive-effective-mass regime, where repulsive polariton interactions do not support a bright soliton. A free-propagation integration anchored to that regime shows a packet seeded at the analytic width broadening from 2.2 µm to 25.8 µm over 6 ps, so the proposed self-confined realization is not present there. A negative-mass run is retained only as a control of the propagation solver. Corrected collective wavelength. The de Broglie wavelength of the collective coordinate follows from the aggregate mass M* = N mLP, not from the single-polariton value: λCOM = h/(M*vg) = λLP/N ≈ 0.11 nm. The corresponding Fraunhofer scale far exceeds the demonstrated 60 µm propagation length, so no far-field centre-of-mass double-slit geometry is proposed for this platform. Supporting these results, the collective-coordinate construction is retained as a conditional formal sector for a rigid bright soliton: translational symmetry protects free motion exactly, the Kohn conditions protect harmonic centre-of-mass motion, and a hydrodynamic Madelung cross-check reproduces the same statement independently. In the aperture region the adiabaticity parameter is of order unity on the platform considered, so no perturbative mass-renormalization coefficient is assigned there. The paper distinguishes first-order field coherence, which a driven condensate can maintain and which is experimentally measured, from coherence between two centre-of-mass configurations, which the radiative channel destroys. The argument is independent of any interpretation of quantum mechanics. Engineering of the output reservoir so that the two branches emit into nearly indistinguishable environmental modes is identified as the route by which radiative which-path decoherence could be suppressed. Changes in version 2.0 Version 2.0 revises the central claim of version 1.0 and should be read in place of it. The title and the subject of the paper have both changed. Retracted. The suppressed decoherence rate of version 1.0 and the 1/N estimate on which it rested. That estimate measures the back-action of a quantum jump on the mean position; it does not measure the which-path information deposited in the environment, which scales with the number of constituents rather than against it. The coherence time reported in version 1.0, and the transit-time margin derived from it, do not hold. Withdrawn. The claim that the structure traverses a double slit in a coherent superposition of paths; the claim that a transition between interpretations of quantum mechanics emerges as a necessary structure of the dynamics; the reduced fringe visibility previously advanced as the primary experimental signature, which is a property of the order parameter rather than of a path superposition; and the self-confinement property, shown numerically not to hold at the operating point of the platform. The acronym under which the construction was introduced is not used further. Removed from scope. The admissibility layer and the interpretive material associated with it. This version is self-contained within condensed-matter physics and quantum decoherence, and its results depend on no unpublished work. Corrected. Parameter provenance. Version 1.0 combined quantities measured on four distinct platforms and conflated the cavity photon lifetime, the polariton lifetime and the first-order coherence time. All platform inputs are now drawn from one system and tabulated with operational definition, origin and status as measured, derived or chosen. The de Broglie wavelength of the collective coordinate is corrected. The description of the soliton solution set as a symplectic manifold is weakened to a finite-dimensional collective-coordinate manifold, and the claim to a first explicit formulation of the mapping as an identity is withdrawn; the Ehrenfest theorem is stated as a consistency check rather than as the derivation. Retained. The strong-coupling analysis, the conditional collective-coordinate reduction, the exact symmetry sectors with their hydrodynamic cross-check, and the parametric relation between mass and nonlinearity, the last now confirmed against a measurement on the platform.
The time evolution of the density matrix of the damped harmonic oscillator is studied within the Lindblad theory for open quantum systems. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation of the damped harmonic oscillator. Illustrative examples for specific initial conditions of the density matrix are provided. It is also shown that various master equations for the damped quantum oscillator, for damped collective modes in deep inelastic collisions of heavy ions and in different models of quantum optics are particular cases of the Lindblad equation and that only some of these equations satisfy the quantum mechanical constraints on the diffusion coefficients.
In the framework of the Lindblad theory for open quantum systems the damping of the harmonic oscillator is studied. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger and Heisenberg representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that the majority of these equations are not satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are also obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions. A comparative study is made for the Glauber P representation, the antinormal ordering Q representation and the Wigner W representation. It is proven that the variances for the damped harmonic oscillator found with these representations are the same. By solving the Fokker-Planck equations in the steady state, it is shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided.
We establish an exact, non-heuristic operator-theoretic framework for spectral gap amplification, uniting finite-dimensional discrete Laplacian networks with infinite-dimensional continuous Schrödinger operators. By utilizing an exact Feshbach-Schur reduction and a reduced Birman-Schwinger formulation, we isolate the spectral shift equation governing structural coherence. We prove that the structural admissibility condition (γΔ > β²) is sufficient and sharp within the monotone resolvent bound for strict spectral gap amplification, yielding a closed-form optimal lower bound. We establish necessity under commuting constraints and derive the exact geometric admissibility condition for simple sectors. Advancing beyond finite matrices, we extend these exact bounds to self-adjoint operators with compact resolvents. We employ regularized Fredholm determinants (det2) to track exact analytic zero-crossings in Hilbert space, removing the trace-class requirement and securing an exact topological boundary for the invariant subspace independently of ambient dimension. Finally, we formulate the Grand Unified Ω-Σ Operator Theorem. We prove that this invariant admissibility law universally governs phase closure across both discrete networks and continuous quantum manifolds. The geometric boundary γΔ > β² dictates that an injected constraint field (the Ω-layer) will only stabilize a system if its spectral alignment with the target invariant subspace (γ) combined with the intrinsic spectral isolation of that space (Δ) strictly dominates the squared variance of orthogonal leakage (β²). This establishes that dynamic coherence is not merely persistence, but an exactly tunable operator geometry spanning from discrete nxn matrices to infinite-dimensional Hilbert space.