Bateman's dual oscillator embeds a damped harmonic oscillator and an independent anti-damped adjoint coordinate in a conservative doubled system. We develop a split-quaternionic formulation that remains consistent from the classical equations through ordinary complex canonical quantization. The doubled dynamics is written as a two-sided hypercomplex evolution whose conserved indefinite quadratic form is the split norm, while the sign of the generator square unifies the underdamped, overdamped, and critical regimes. The Bateman Lagrangian is recovered as a real scalar action on the split-complex configuration subalgebra. After quantization, operator-valued split quaternions belong to the complexified algebra ℂ⊗_ℝℍ_s≃ M_2(ℂ). Noncommuting mechanical components require a symmetrized quantum split norm, which exactly reproduces the symmetrically ordered canonical Hamiltonian. Null idempotents resolve the crossed canonical structure of the doubled phase space. In particular, damping yields [mẏ,mẋ]=iħ mγ and a Robertson–Schrödinger uncertainty relation. The limit γ→0 retains the doubled positive/negative geometry; the ordinary single-oscillator Heisenberg relation is recovered only after selecting the positive normal mode, or by undoubling before quantization. The scalar split action gives the exact quadratic propagator and Gaussian evolution. A formal trace over the auxiliary coordinate is finally compared with high-temperature Caldeira–Leggett dynamics, clarifying that the doubled model is a coherent zero-noise embedding rather than a microscopic reservoir theory.

A single operational protocol based on free evolution and projective measurements yields inequivalent quantum time distributions through distinct post-processing procedures. We construct an activity-based time-of-flow (TF) distribution and a presence-based quantum stroboscopic (QS) distribution, providing complementary operational notions of time. Applied to tunneling, the regional QS mean saturates, whereas the TF mean first decreases in the Hartman regime and then grows for larger barrier widths. Within this framework, we provide an operational interpretation of the Hartman effect in terms of quantum time distributions associated with flow through the exit region and occupation within the barrier, capturing the mechanism of early penetration, dominant reflection, and spectrally filtered transmission.

In this article, we propose a unified framework for cosmological expansion and inflation at the level of background dynamics, by modeling both the inflaton field and dark energy as a four-dimensional continuous medium, whose elastic deformation is described by a covariant vector field. Focusing on homogeneous and isotropic background cosmology, we show that for a bulk modulus K = 1.64 & times; 10(109) N m(-2), the dark-energy density decreases by a factor of 10(122), while the scale factor expands 10(28) times over similar to 10(-42) seconds during primordial inflation. For illustrative parameter values, our analysis suggests three potential new physical phenomena for future investigation, including longitudinal elastic modes, frequency red-shifts in early-universe light, and improved fits to supernova curves. At the end of the paper, we discuss the challenges of applying the framework to inflationary perturbations, particularly the need for a consistent theory capable of producing a nearly scale-invariant power spectrum, as well as of addressing reheating, and identifying these as key directions for future work.

We show that Bose–Einstein condensation produces a cusp in the time-of-arrival (TOA) statistics of a harmonically trapped gas released into free fall. In the semiclassical long time-of-flight regime, with ε=σ_V/√(2gH)≪1, both the mean and standard deviation of the arrival time distribution, which are governed by the longitudinal velocity variance, remain continuous, but acquire a cusp whose one-sided slope ratio is universal within the ideal-gas far-field limit, ℛ_∞=2.5556…, and equals the trapped-gas specific-heat ratio C(Tc^-)/C(Tc^+). Finite atom number rounds the cusp and weak interactions perturb it only weakly, leaving a measurable time-domain signature of condensation.

We introduce a general framework for defining context-dependent time distributions in quantum systems using projective measurements. The time-of-flow (TF) distribution, derived from population transfer rates into a measurement subspace, yields a time-energy uncertainty relation of the form L\T L\H >= h/(6 3)80, where 80 quantifies the net population transfer measured by the projector. This bound applies to arbitrary projectors under unitary dynamics and reveals that time uncertainty is inherently measurement dependent. We demonstrate the framework with two applications: a general time-of-arrival-energy uncertainty relation and a driven three-level system under detuned coherent driving. The TF framework unifies timing observables across spin, atomic, and matter-wave systems and offers an experimentally accessible route to probing quantum timing in controlled measurements.

The probability distribution of a time measurement T x at position x can be inferred from the probability distribution of a position measurement X t at time t as given by the Born rule [1, 2]. In an application to free-fall, this finding has been used to predict the existence of a mass-dependent positive relative shift with respect to the classical time-of-arrival in the long time-of-flight regime for dropped quantum particles [1]. The present paper extends these results in two important directions. We first show that for a Gaussian quantum particle of mass m dropped in a uniform gravitational field g , the uncertainties about time and position measurements are related by the relation Δ T x Δ X t ≥ ℏ 2 m g This novel form of uncertainty relation suggests that choosing the initial state so as to obtain a lower uncertainty in the measured position leads to a higher uncertainty in the measured arrival time. Secondly, we examine the case of a free-falling particle starting from a non-Gaussian initial superposed state, for which we predict the presence of gravitationally induced interferences and oscillations in the mean time-of-arrival as a function of the detector’s position that can be interpreted as the signature of a Zitterbewegung-like effect.

We propose a general and experimentally accessible framework to quantify transition timing in discrete quantum systems via the time-of-flow (TF) distribution. Defined from the rate of population change in a target state, the TF distribution can be reconstructed through repeated projective measurements at discrete times on independently prepared systems, thus avoiding Zeno inhibition. In monotonic regimes, it admits a clear interpretation as a time-of-arrival (TOA) or time-of-departure (TOD) distribution. We apply this approach to optimize time-dependent Hamiltonians, analyze shortcut-to-adiabaticity (STA) protocols, study non-adiabatic features in the dynamics of a three-level time-dependent detuning model, and derive a transition-based quantum speed limit (TF-QSL) for both closed and open quantum systems. We also establish a lower bound on temporal uncertainty and examine decoherence effects, demonstrating the versatility of the TF framework for quantum control and diagnostics. This method provides both a conceptual tool and an experimental protocol for probing and engineering quantum dynamics in discrete-state platforms.

Using standard results from statistics, we show that for Gaussian quantum systems the distribution of a time measurement at a fixed position can be directly inferred from the distribution of a position measurement at a fixed time as given by the Born rule. In an application to a quantum particle of mass m falling in a uniform gravitational field g, we use this approach to obtain an exact explicit expression for the probability density of the time of arrival (TOA). In the long time-of-flight approximation, we predict that the average positive relative shift with respect to the classical TOA in case of a zero initial mean velocity is asymptotically given by delta = q(2)/(2) when the factor q = h/ 2m sigma root 2gx << 1 (semiclassical regime) and by delta = pi q when q >> 1 (quantum regime), where sigma is the width of the initial Gaussian wave packet and x is the mean distance to the detector. We also discuss experimental conditions under which these predictions can be tested.

The uncertainty principle is one of the characteristic properties of quantum theory, where it signals the incompatibility of two types of measurements. In this paper, we argue that measurements of time-of-arrival $T_x$ at position $x$ and position $X_t$ at time $t$ are mutually exclusive for a quantum system, each providing complementary information about the state of that system. For a quantum particle of mass $m$ falling in a uniform gravitational field $g$, we show that the corresponding uncertainty relation can be expressed as $\Delta T_x \Delta X_t \geq \frac{\hbar}{2mg}$. This uncertainty relationship can be taken as evidence of the presence of a form of epistemic incompatibility in the sense that preparing the initial state of the system so as to decrease the measured position uncertainty will lead to an increase in the measured time-of-arrival uncertainty. These findings can be empirically tested in the context of ongoing or forthcoming experiments on measurements of time-of-arrival for free-falling quantum particles.

Using standard results from statistics, we show that for any continuous quantum system (Gaussian or otherwise) and any observable (A) over cap (position or otherwise), the distribution pi a(t) of time measurement at a fixed state a can be inferred from the distribution rho t(a) of a state measurement at a fixed time t via the transformation pi a(t) proportional to | partial derivative -infinity rho t (u)du|. This finding suggests that the answer to the long-lasting time-of-arrival problem is in fact secretly hidden within the Born rule and therefore does not require the introduction of a time operator or a commitment to a specific (e.g., Bohmian) ontology. The generality and versatility of the result are illustrated by applications to the time of arrival at a given location for a free particle in a superposed state and to the time required to reach a given velocity for a free-falling quantum particle. Our approach also offers a potentially promising new avenue toward the design of an experimental protocol for the yet-to-be-performed observation of the phenomenon of quantum backflow.

The complex Fourier transform of the two-point correlator of the energy spectrum of a quantum system is known as the spectral form factor (SFF). It constitutes an essential diagnostic tool for phases of matter and quantum chaos. In black hole physics, it describes the survival probability (fidelity) of a thermofield double state under unitary time evolution. However, detailed properties of the SFF of isolated quantum systems with generic spectra are smeared out by large temporal fluctuations, whose minimization requires disorder or time averages. This requirement holds for any system size, that is, the SFF is non-self averaging. Exploiting the fidelity-based interpretation of this quantity, we prove that using filters, disorder and time averages of the SFF involve unitarity breaking, i.e., open quantum dynamics described by a quantum channel that suppresses quantum noise. Specifically, averaging over Hamiltonian ensembles, time averaging, and frequency filters can be described by the class of mixed-unitary quantum channels in which information loss can be recovered. Frequency filters are associated with a time-continuous master equation generalizing energy dephasing. We also discuss the use of eigenvalue filters. They are linked to non-Hermitian Hamiltonian evolution without quantum jumps, whose long-time behavior is described by a Hamiltonian deformation. We show that frequency and energy filters make the SFF self-averaging at long times.

The Kibble-Zurek mechanism (KZM) predicts that the average number of topological defects generated upon crossing a continuous or quantum phase transition obeys a universal scaling law with the quench time. Fluctuations in the defect number near equilibrium are approximately of Gaussian form, in agreement with the central limit theorem. Using large deviations theory, we characterize the universality of fluctuations beyond the KZM and report the exact form of the rate function in the transverse-field quantum Ising model. In addition, we characterize the scaling of large deviations in an arbitrary continuous phase transition, building on recent evidence establishing the universality of the defect number distribution.

M. Beau, 2 A. del Campo, 4, 1 D. J. Frantzeskakis, T. P. Horikis, and P. G. Kevrekidis Department of Physics, University of Massachusetts, Boston, Massachusetts 02125, USA Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Road, Dublin 4, Ireland Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg Donostia International Physics Center, E-20018 San Sebastián, Spain Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece Department of Mathematics, University of Ioannina, Ioannina 45110, Greece Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA

We find the complete family of many-body quantum Hamiltonians withground-state of Jastrow form involving the pairwise product of a pairfunction in an arbitrary spatial dimension. The parent Hamiltoniangenerally includes a two-body pairwise potential as well as a three-bodypotential. We thus generalize the Calogero-Marchioro construction forthe three-dimensional case to an arbitrary spatial dimension. Theresulting family of models is further extended to include a one-bodyterm representing an external potential, which gives rise to anadditional long-range two-body interaction. Using this framework, weprovide the generalization to an arbitrary spatial dimension ofwell-known systems such as the Calogero-Sutherland and Calogero-Mosermodels. We also introduce novel models, generalizing the McGuiremany-body quantum bright soliton solution to higher dimensions andconsidering ground-states which involve e.g., polynomial, Gaussian,exponential, and hyperbolic pair functions. Finally, we show how thepair function can be reverse-engineered to construct models with a givenpotential, such as a pair-wise Yukawa potential, and to identify modelsgoverned exclusively by three-body interactions.

In this article, we introduce higher derivative Lagrangians of this form $\alpha_1 A_{\mu}(x)\dot{x}^\mu$, $\alpha_2 G_{\mu}(x)\ddot{x}^\mu$, $\alpha_3 B_{\mu}(x)\dddot{x}^\mu$, $\alpha_4 K_{\mu}(x)\ddddot{x}^\mu$, $\cdots$, that generalize the electromagnetic interaction to higher order derivatives. We show that odd order Lagrangians describe interactions analog to electromagnetism while even order Lagrangians are similar to gravitational interaction. From this analogy, we formulate the concept of the generalized induction principle assuming the coupling between the higher fields $U_{(n),\mu}(x),\ n\geq1$ and the higher currents $j^{(n)\mu}=\rho(x)d^nx^{\mu}/ds^n$, where $\rho(x)$ is the spatial density of mass ($n$ even) or of electric charge ($n$ odd). In short, this article is an invitation to reflect on a generalization of the concept of force and of inertia. We discuss the implications of these paradigms more in depth in the last section of the paper.

We introduce a model of trapped bosons with contact interactions as well as Coulomb repulsion or gravitational attraction in one spatial dimension. We find the exact ground-state energy and many-body wave function. The density profile and the pair-correlation function are sampled using Monte Carlo method and show a rich variety of regimes with crossovers between them. Strong attraction leads to a trapped McGuire quantum soliton. Weak repulsion results in an incompressible Laughlin-like fluid with flat density, well reproduced by a Gross-Pitaevskii equation with long-range interactions. Stronger repulsion induces Friedel oscillations and the eventual formation of a Wigner crystal.

We consider the nonadiabatic energy fluctuations of a many-body system in a time-dependent harmonic trap. In the presence of scale-invariance, the dynamics becomes self-similar and the nondiabatic energy fluctuations can be found in terms of the initial expectation values of the second moments of the Hamiltonian, square position, and squeezing operators. Nonadiabatic features are expressed in terms of the scaling factor governing the size of the atomic cloud, which can be extracted from time-of-flight images. We apply this exact relation to a number of examples: the single-particle harmonic oscillator, the one-dimensional Calogero-Sutherland model, describing bosons with inverse-square interactions that includes the non-interacting Bose gas and the Tonks-Girdardeau gas as limiting cases, and the unitary Fermi gas. We illustrate these results for various expansion protocols involving sudden quenches of the trap frequency, linear ramps and shortcuts to adiabaticity. Our results pave the way to the experimental study of nonadiabatic energy fluctuations in driven quantum fluids.

We introduce an exact mapping between the Dirac equation in (1 + 1)-dimensional curved spacetime (DCS) and a multiphoton quantum Rabi model (QRM). A background of a (1 + 1)-dimensional black hole requires a QRM with one- and two-photon terms that can be implemented in a trapped ion for the quantum simulation of Dirac particles in curved spacetime. We illustrate our proposal with a numerical analysis of the free fall of a Dirac particle into a (1 + 1)-dimensional black hole, and find that the Zitterbewegung effect, measurable via the oscillatory trajectory of the Dirac particle, persists in the presence of gravity. From the duality between the squeezing term in the multiphoton QRM and the metric coupling in the DCS, we show that gravity generates squeezing of the Dirac particle wave function.

We introduce general bounds for the parameter estimation error in nonlinear quantum metrology of many-body open systems in the Markovian limit. Given a k-body Hamiltonian and p-body Lindblad operators, the estimation error of a Hamiltonian parameter using a Greenberger-Horne-Zeilinger state as a probe is shown to scale as N^{-[k-(p/2)]}, surpassing the shot-noise limit for 2k>p+1. Metrology equivalence between initial product states and maximally entangled states is established for p≥1. We further show that one can estimate the system-environment coupling parameter with precision N^{-(p/2)}, while many-body decoherence enhances the precision to N^{-k} in the noise-amplitude estimation of a fluctuating k-body Hamiltonian. For the long-range Ising model, we show that the precision of this parameter beats the shot-noise limit when the range of interactions is below a threshold value.