Gravitational Decoherence and Stochastic Tidal Phase Diffusion of Galactic Dark Matter Solitons: Microscopic Lindblad Evolution and Pulsar Timing Observables

Wave or Fuzzy Dark Matter (ψψψDM), consisting of ultralight bosons of mass ma∼10−22 eVm_a ~ 10⁻²² eVma∼10−22 eV, is conventionally modeled as an isolated, eternally coherent macroscopic Bose-Einstein condensate (τdecoh→∞τdecoh → ∞τdecoh→∞) governed by the unitary Gross-Pitaevskii-Poisson system. Real galactic halos, however, are non-isolated environments permeated by stochastic gravitational tidal fluctuations δΦ(r,t)δΦ(r, t)δΦ(r,t) sourced by giant molecular clouds, stellar clusters, and satellite mergers. In this work, we formulate the open quantum system dynamics of a galactic ψψψDM soliton interacting with an external stochastic gravitational bath. Invoking Kohn’s theorem, we show that uniform dipole fluctuations translate the center of mass without internal heating, whereas stochastic tidal shear drives spatial phase diffusion governed by a Lindblad-form master equation. Key Findings: Environment-Dependent Coherence: We derive the closed-form spatial decoherence rate Γgrav(Δr)=(ma/ℏ)2SΦ(0)[(Δr)2/((Δr)2+λc2)]Γgrav(Δ r) = (m_a/)^2 S_Φ(0) [ (Δ r)^2 / ((Δ r)^2 + λ_c^2) ]Γgrav(Δr)=(ma/ℏ)2SΦ(0)[(Δr)2/((Δr)2+λc2)], where λc∼50–100 pcλ_c ~ 50--100 pcλc∼50–100 pc is the clump correlation length. In dwarf spheroidal galaxies, low baryonic noise preserves quantum purity (τdecoh>500 Gyrτdecoh > 500 Gyrτdecoh>500 Gyr, Ppure>99.9%Pₚᵤᵣₑ > 99.9%Ppure>99.9%). In baryonic spiral disks and nuclear clusters, tidal heating induces partial decoherence (τdecoh∼1.3–60 Gyrτdecoh ~ 1.3--60 Gyrτdecoh∼1.3–60 Gyr). Resolution of the Overdense Core Tension: Tidal phase diffusion puffs the soliton core by 5%–45%5%--45%5%–45% and suppresses central peak densities by up to 15×15×15×, naturally resolving the overdense core tension with Milky Way Gaia kinematics without requiring fine-tuned feedback. Falsifiable Pulsar Timing Signature: We prove that phase diffusion broadens the characteristic f0=2ma/h≈48.36 nHzf_0 = 2 m_a / h ≈ 48.36 nHzf0=2ma/h≈48.36 nHz scalar field oscillation into an environment-dependent Lorentzian power spectral density with a 10-order-of-magnitude radial linewidth gradient across the Galaxy (Δf/f0∼10−10Δ f / f_0 ~ 10⁻¹⁰Δf/f0∼10−10 at the Galactic center down to <10−19< 10⁻¹⁹<10−19 in the outer halo). This spectral broadening provides an unambiguous signature distinguishing ultralight dark matter from stochastic gravitational waves in upcoming data from the International Pulsar Timing Array (IPTA DR3) and the Square Kilometre Array (SKA).