This record contains the manuscript “Frozen Density-Selected Unitaries: Exact Action on Marginals, Operator Regularity, and Controlled Singular-Kernel Extensions”. The paper studies a one-step class of density-selected phase transformations in bosonic quantum dynamics. The transformation is frozen in the sense that the density profile used to generate the phase is fixed externally during the imprint, rather than being updated through a self-consistent feedback loop. The manuscript develops three complementary results. First, it proves the exact action of the frozen product-imprint map on reduced marginals of bosonic many-body states. Second, it establishes regularity properties of the associated operator map in the bounded-kernel regime, including Lipschitz continuity and differentiability with respect to the prescribed input density. Third, it gives a controlled extension to singular kernels, where operator-norm convergence is generally too strong and strong-operator convergence becomes the natural topology. The paper is intended as an operator-theoretic foundation for density-selected phase-imprint models and as a bridge toward later mean-field and Hartree-type analyses. It separates the frozen external-density layer from genuinely self-consistent feedback mechanisms and clarifies which conclusions belong to each regime.
This record contains the manuscript “Propagation of Chaos after a Frozen Density-Selected Phase Imprint”. Abstract. We study a bosonic mean-field system after a single frozen density-selected phase imprint, with phase profile determined by the pre-imprint density. The main result isolates the exact frozen-density interface at the imprint time and shows that the one-particle defect is transported through the jump without additional loss. On each fixed observation window, this reduces the post-imprint analysis to the standard Hartree propagation-of-chaos framework and yields the usual O(N^{-1}) trace-norm convergence in the regular bounded-integrable kernel regime. The manuscript also records two controlled extensions. First, finite-duration imprints are compared with the same frozen benchmark through area matching, with the dynamic-reference mismatch quantified separately. Second, for prescribed finite kick schedules independent of N, the same bookkeeping yields a conservative fixed-schedule extension on fixed windows. In addition, a hypothesis-dependent weakly singular branch is included under explicit extra spacetime assumptions; this branch is conditional and is stated separately from the unconditional regular result. The paper is intended as a rigorous bridge result: it identifies the exact frozen-density jump law, the induced defect transport, and the subsequent standard Hartree comparison. It does not claim a refreshed-generator theorem, a continuum-drive limit, or a uniform-in-the-number-of-kicks result. Files in this record include the main manuscript and supporting source materials.
We study finite-dimensional open quantum systems in contact with macroscopic equilibrium systems at their boundaries such that the system density matrix $\ensuremath{\rho}$ evolves via a Lindbladian, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}=\ensuremath{-}i[H,\ensuremath{\rho}]+\mathcal{D}\ensuremath{\rho}$. Here
H is the Hamiltonian of the system and
D is the dissipator. We consider the case where the system consists of two parts, the ``boundary''
A and the ``bulk''
B, and
D acts only on
A, so $\mathcal{D}={\mathcal{D}}_{A}\ensuremath{\bigotimes}{\mathcal{I}}_{B}$, where
IB is the identity superoperator on part
B. Let
DA be ergodic, so ${\mathcal{D}}_{A}{\ensuremath{\pi}}_{A}=0$ only for one unique density matrix ${\ensuremath{\pi}}_{A}$. We show that any stationary density matrix $\overline{\ensuremath{\rho}}$ on the full system which commutes with
H must be of the product form $\overline{\ensuremath{\rho}}={\ensuremath{\pi}}_{A}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{B}$ for some ${\ensuremath{\rho}}_{B}$. This rules out finding any
DA that has the Gibbs measure ${\ensuremath{\rho}}_{\ensuremath{\beta}}={e}^{\ensuremath{-}\ensuremath{\beta}H}/Z(\ensuremath{\beta})$ as a stationary state with $\ensuremath{\beta}\ensuremath{\ne}0$, unless there is no interaction between parts
A and
B. We give criteria for the uniqueness of the stationary state $\overline{\ensuremath{\rho}}$ for systems with interactions between
A and
B. Related results for nonergodic cases are also discussed.
This preprint develops the operator level formulation of nonlocal morphing, also described as density imprinting, in quantum dynamics. The paper studies the mathematical structure of the corresponding unitary transformation and clarifies the assumptions under which the construction is well defined. The manuscript analyzes the basic operator properties of the density selected phase imprint, its stability under changes of the input density and kernel, and its relation to mean field and Hartree type descriptions. It also discusses the distinction between a single frozen imprint, repeated prescribed imprints, and possible dynamical extensions. The work is intended as a foundational operator theoretic layer for a broader research program on density selected phase transformations, mean field propagation, and physically motivated imprinting protocols. The presentation emphasizes mathematical transparency, reproducibility, and a clear separation between rigorously established operator results and later many body or experimental extensions. Keywords: nonlocal morphing; density imprinting; unitary operators; operator theory; quantum dynamics; mean field models; Hartree theory; phase transformations; reproducible research.
We formulate a regular-regime theorem for continuum density-selected phase drives refreshed along a Hartree reference path. In contrast to a prescribed continuum drive, the density path is generated by the reference Hartree solution itself, namely ρt = |φt|2 . The microscopic comparison is nevertheless made against the corresponding time-dependent Hartree-refreshed reference field, not against a fully microscopic feedback law depending on the evolving reduced density of the same N-body state. In this regular finite-window setting we isolate three mechanisms: construction of the Hartree-refreshed reference branch, convergence of finite refreshed phase schedules to an abelian multiplication product integral, and a Pickl-type propagation-of-condensation estimate for the associated many-body reference dynamics.