arXiv subjects

Marilena Ligabò

Publications and source records attributed to Marilena Ligabò.

At least 19 recordsLinked to original sources

arXiv2026

Limits of Stochastic Semigroups and Block-Triangular Majorisation

We investigate limits of semigroups of stochastic matrices defined by their invariant distribution. Given probability vectors $γ(β)$ depending on a parameter $β$, we introduce a notion of convergence as $β\to\infty$ for the corresponding semigroups of $γ(β)$-preserving stochastic matrices and investigate the structure of the resulting limit. In general, the limiting semigroup differs from the semigroup preserving the limiting distribution, showing that these two operations do not commute. We develop a general framework for such limiting semigroups and study in detail the case in which the invariant distributions are Gibbs vectors at the inverse temperature $β$. We show that the limiting semigroup consists of block-upper-triangular stochastic matrices subject to additional substochasticity constraints. We characterise and enumerate their extremal elements and determine the preorder on probability vectors induced by the action of the semigroup. The resulting notion of Block-Triangular majorisation interpolates between ordinary majorisation and upper triangular (aka unordered) majorisation. We show that it is completely characterised by a finite family of monotones and analyse the corresponding behaviour of Rényi $α$-entropies as $β\to\infty$.

math-ph↗

arXiv2026

Quantum Systems with jump-discontinuous mass. I

We consider a free quantum particle in one dimension whose mass profile exhibits jump discontinuities. The corresponding Hamiltonian is a self-adjoint realisation of the kinetic-energy operator, with the specific realisation determined by the boundary conditions at the points of mass discontinuity. For a family of scale-free boundary conditions, we analyse the associated spectral problem. We find that the eigenfunctions exhibit a highly sensitive and erratic dependence on the energy. Notably, the system supports infinitely many distinct semiclassical limits, each labeled by a point on a spectral curve embedded in the two-torus. These results demonstrate a rich interplay between discontinuous coefficients, boundary data, and spectral asymptotics.

math-ph↗

arXiv2026

Wandering range of robust quantum symmetries

This paper introduces the concept of the wandering range of a robust symmetry $S$ of a Hamiltonian $H$. This quantity measures how the perturbed time evolution $\mathrm{e}^{\mathrm{i}t(H+\varepsilon V)} S \mathrm{e}^{-\mathrm{i} t(H+\varepsilon V)}$ deviates from its unperturbed counterpart $\mathrm{e}^{\mathrm{i} tH} S\mathrm{e}^{-\mathrm{i} tH} = S$. Although the wandering range does not necessarily scale linearly with the perturbation strength $\varepsilon$, we identify conditions under which this linear behavior is recovered and we obtain explicit nonperturbative bounds.

quant-ph↗

arXiv2025

Truncated quantum observables and their semiclassical limit

For quantum observables $H$ truncated on the range of orthogonal projections $Π_N$ of rank $N$, we study the corresponding Weyl symbol in the phase space in the semiclassical limit of vanishing Planck constant $\hbar\to0$ and large quantum number $N\to\infty$, with $\hbar N$ fixed. Under certain assumptions, we prove the $L^2$- convergence of the Weyl symbols to a symbol truncated (hence, in general discontinuous) on the classically allowed region in phase space. As an illustration of the general theorems we analyse truncated observables for the harmonic oscillator and for a free particle in a one-dimensional box. In the latter case, we also compute the microscopic pointwise limit of the symbols near the boundary of the classically allowed region.

math-ph↗

arXiv2024

Robustness of quantum symmetries against perturbations

We investigate quantum symmetries in terms of their large-time stability with respect to perturbations of the Hamiltonian. We find a complete algebraic characterization of the set of symmetries robust against a single perturbation and we use such result to characterize their stability with respect to arbitrary sets of perturbations.

quant-ph↗

arXiv2024

Classical echoes of quantum boundary conditions

We consider a non-relativistic particle in a one-dimensional box with all possible quantum boundary conditions that make the kinetic-energy operator selfadjoint. We determine the Wigner functions of the corresponding eigenfunctions and analyze in detail their classical limit in the high-energy regime. We show that the quantum boundary conditions split into two classes: all local and regular boundary conditions collapse to the same classical boundary condition, while singular non-local boundary conditions slightly persist in the classical limit.

quant-ph↗

arXiv2024

Random matrices associated to Young diagrams

We consider the singular values of certain Young diagram shaped random matrices. For block-shaped random matrices, the empirical distribution of the squares of the singular eigenvalues converges almost surely to a distribution whose moments are a generalisation of the Catalan numbers. The limiting distribution is the density of a product of rescaled independent Beta random variables and its Stieltjes-Cauchy transform has a hypergeometric representation. In special cases we recover the Marchenko-Pastur and Dykema-Haagerup measures of square and triangular random matrices, respectively. We find a further factorisation of the moments in terms of two complex-valued random variables that generalises the factorisation of the Marcenko-Pastur law as product of independent uniform and arcsine random variables.

math.PR↗

arXiv2024

Taming the Rotating Wave Approximation

The interaction between light and matter is one of the oldest research areas of quantum mechanics, and a field that just keeps on delivering new insights and applications. With the arrival of cavity and circuit quantum electrodynamics we can now achieve strong light-matter couplings which form the basis of most implementations of quantum technology. But quantum information processing also has high demands requiring total error rates of fractions of percentage in order to be scalable (fault-tolerant) to useful applications. Since errors can also arise from modelling, this has brought into center stage one of the key approximations of quantum theory, the Rotating Wave Approximation (RWA) of the quantum Rabi model, leading to the Jaynes-Cummings Hamiltonian. While the RWA is often very good and incredibly useful to understand light-matter interactions, there is also growing experimental evidence of regimes where it is a bad approximation. Here, we ask and answer a harder question: for which experimental parameters is the RWA, although perhaps qualitatively adequate, already not good enough to match the demands of scalable quantum technology? For example, when is the error at least, and when at most, 1%? To answer this, we develop rigorous non-perturbative bounds taming the RWA. We find that these bounds not only depend, as expected, on the ratio of the coupling strength and the oscillator frequency, but also on the average number of photons in the initial state. This confirms recent experiments on photon-dressed Bloch-Siegert shifts. We argue that with experiments reporting controllable cavity states with hundreds of photons and with quantum error correcting codes exploring more and more of Fock space, this state-dependency of the RWA is increasingly relevant for the field of quantum computation, and our results pave the way towards a better understanding of those experiments.

quant-ph↗

arXiv2023

The semiclassical limit of a quantum Zeno dynamics

Motivated by a quantum Zeno dynamics in a cavity quantum electrodynamics setting, we study the asymptotics of a family of symbols corresponding to a truncated momentum operator, in the semiclassical limit of vanishing Planck constant $\hbar\to0$ and large quantum number $N\to\infty$, with $\hbar N$ kept fixed. In a suitable topology, the limit is the discontinuous symbol $pχ_D(x,p)$ where $χ_D$ is the characteristic function of the classically permitted region $D$ in phase space. A refined analysis shows that the symbol is asymptotically close to the function $pχ_D^{(N)}(x,p)$, where $χ_D^{(N)}$ is a smooth version of $χ_D$ related to the integrated Airy function. We also discuss the limit from a dynamical point of view.

math-ph↗

arXiv2022

Stability of the gapless pure point spectrum of self-adjoint operators

We consider a self-adjoint operator $T$ on a separable Hilbert space, with pure-point and simple spectrum with accumulations at finite points. Explicit conditions are stated on the eigenvalues of $T$ and on the bounded perturbation $V$ ensuring the global stability of the spectral nature of $T+V$.

math-ph↗

arXiv2020

Hidden non-Markovianity in open quantum systems

We show that non-Markovian open quantum systems can exhibit exact Markovian dynamics up to an arbitrarily long time; the non-Markovianity of such systems is thus perfectly "hidden", i.e. not experimentally detectable by looking at the reduced dynamics alone. This shows that non-Markovianity is physically undecidable and extremely counterintuitive, since its features can change at any time, without precursors. Some interesting examples are discussed.

quant-ph↗

arXiv2019

Spectral properties of the singular Friedrichs-Lee Hamiltonian

We show that the Friedrichs-Lee model, which describes the one-excitation sector of a two-level atom interacting with a structured boson field, can be generalized to singular atom-field couplings. We provide a characterisation of its spectrum and resonances and discuss the inverse spectral problem.

math-ph↗

arXiv2019

Third-order phase transition: random matrices and screened Coulomb gas with hard walls

Consider the free energy of a $d$-dimensional gas in canonical equilibrium under pairwise repulsive interaction and global confinement, in presence of a volume constraint. When the volume of the gas is forced away from its typical value, the system undergoes a phase transition of the third order separating two phases (pulled and pushed). We prove this result i) for the eigenvalues of one-cut, off-critical random matrices (log-gas in dimension $d=1$) with hard walls; ii) in arbitrary dimension $d\geq1$ for a gas with Yukawa interaction (aka screened Coulomb gas) in a generic confining potential. The latter class includes systems with Coulomb (long range) and delta (zero range) repulsion as limiting cases. In both cases, we obtain an exact formula for the free energy of the constrained gas which explicitly exhibits a jump in the third derivative, and we identify the 'electrostatic pressure' as the order parameter of the transition. Part of these results were announced in [F. D. Cunden, P. Facchi, M. Ligabò and P. Vivo, J. Phys. A: Math. Theor. 51, 35LT01 (2018)].

math-ph↗

arXiv2018

Universality of the weak pushed-to-pulled transition in systems with repulsive interactions

We consider a $d$-dimensional gas in canonical equilibrium under pairwise screened Coulomb repulsion and external confinement, and subject to a volume constraint. We show that its excess free energy displays a generic third-order singularity separating the pushed and pulled phases, irrespective of range of the pairwise interaction, dimension and details of the confining potential. The explicit expression of the excess free energy is universal and interpolates between the Coulomb (long-range) and the delta (zero-range) interaction. The order parameter of this transition - the electrostatic pressure generated by the surface excess charge - is determined by invoking a fundamental energy conservation argument.

cond-mat.stat-mech↗

arXiv2017

Quantum cavities with alternating boundary conditions

We consider the quantum dynamics of a free nonrelativistic particle moving in a cavity and we analyze the effect of a rapid switching between two different boundary conditions. We show that this procedure induces, in the limit of infinitely frequent switchings, a new effective dynamics in the cavity related to a novel boundary condition. We obtain a dynamical composition law for boundary conditions which gives the emerging boundary condition in terms of the two initial ones.

math-ph↗

arXiv2017

On the derivation of the GKLS equation for weakly coupled systems

We consider the reduced dynamics of a small quantum system in interaction with a reservoir when the initial state is factorized. We present a rigorous derivation of a GKLS master equation in the weak-coupling limit for a generic bath, which is not assumed to have a bosonic or fermionic nature, and whose reference state is not necessarily thermal. The crucial assumption is a reservoir state endowed with a mixing property: the n-point connected correlation function of the interaction must be asymptotically bounded by the product of two-point functions (clustering property).

math-ph↗

arXiv2017

Quantum Fluctuation Relations

We present here a set of lecture notes on exact fluctuation relations. We prove the Jarzynski equality and the Crooks fluctuation theorem, two paradigmatic examples of classical fluctuation relations. Finally we consider their quantum versions, and analyze analogies and differences with the classical case.

quant-ph↗

arXiv2017

Universality of the third-order phase transition in the constrained Coulomb gas

The free energy at zero temperature of Coulomb gas systems in generic dimension is considered as a function of a volume constraint. The transition between the 'pulled' and the 'pushed' phases is characterised as a third-order phase transition, in all dimensions and for a rather large class of isotropic potentials. This suggests that the critical behaviour of the free energy at the 'pulled-to-pushed' transition may be universal, i.e., to some extent independent of the dimension and the details of the pairwise interaction.

math-ph↗