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

Abstract: Wave or Fuzzy Dark Matter (ψ\psiψDM), consisting of ultralight bosons of mass ma∼10−22 eVm_a \sim 10^{-22}\text{ eV}ma∼10−22 eV, is conventionally modeled as an isolated, eternally coherent macroscopic Bose-Einstein condensate (τdecoh→∞\tau_{\text{decoh}} \to \inftyτdecoh→∞) 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)\delta\Phi(\mathbf{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 ψ\psiψDM 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)]\Gamma_{\text{grav}}(\Delta r) = (m_a/\hbar)^2 S_\Phi(0) [ (\Delta r)^2 / ((\Delta r)^2 + \lambda_c^2) ]Γgrav(Δr)=(ma/ℏ)2SΦ(0)[(Δr)2/((Δr)2+λc2)], where λc∼50–100 pc\lambda_c \sim 50\text{--}100\text{ pc}λc∼50–100 pc is the clump correlation length. In dwarf spheroidal galaxies, low baryonic noise preserves quantum purity (τdecoh>500 Gyr\tau_{\text{decoh}} > 500\text{ Gyr}τdecoh>500 Gyr, Ppure>99.9%P_{\text{pure}} > 99.9\%Ppure>99.9%). In baryonic spiral disks and nuclear clusters, tidal heating induces partial decoherence (τdecoh∼1.3–60 Gyr\tau_{\text{decoh}} \sim 1.3\text{--}60\text{ Gyr}τdecoh∼1.3–60 Gyr). Resolution of the Overdense Core Tension: Tidal phase diffusion puffs the soliton core by 5%–45%5\%\text{--}45\%5%–45% and suppresses central peak densities by up to 15×15\times15×, 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 \approx 48.36\text{ nHz}f0=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∼10−10\Delta f / f_0 \sim 10^{-10}Δf/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).