arXiv subjects

M. Shapiro

Publications and source records attributed to M. Shapiro.

At least 19 recordsLinked to original sources

arXiv2025

Generalized chord diagrams and weight systems

Weight systems are functions on chord diagrams satisfying Vassiliev's $4$-term relations. They originate in the theory of finite type knot invariants. Recent developments in understanding weight systems arising from Lie algebras are based on extending these weight systems from chord diagrams (which can be interpreted as involutions without fixed points, considered modulo cyclic shifts) to arbitrary permutations (also modulo cyclic shifts). We suggest relations for functions on permutations, which generalize Vassiliev's relations. We show that the $gl$- and $so$- weight systems satisfy these relations. We also analyze certain properties of these weight systems and study realted Hopf algebras of permutations.

math.CO↗

arXiv2024

Geometric leaf of symplectic groupoid

We consider the symplectic groupoid of pairs $(B, A)$ with $A$ real unipotent upper-triangular matrix and $B\in GL_n$ being such that $\tilde A=BAB^T$ is also a unipotent upper-triangular matrix. Fock and Chekhov defined a Poisson map of Teichm\"uller space ${\mathcal T_{g,s}$ of genus $g$ surfaces with $s$ holes into the space of unipotent upper-triangular $n\times n$ matrices whose image forms the \emph{geometric locus}. The elements of geometric locus satisfy \emph{rank condition}. We describe the Hamiltonian reduction of the Poisson cluster variety of symplectic groupoid by the rank condition for $n=5$ and $6$. In both cases, we analyze the induced cluster structures on the results of Hamiltonian reduction and recover celebrated cluster structure on ${\mathcal T}_{2,1}$ for $n=5$ and ${\mathcal T}_{2,2}$ for $n=6$.

math.QA↗

arXiv2020

Noncommutative Networks on a Cylinder

In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder $\Sigma$, which gives rise to the quasi Poisson bracket of G.Massuyeau and V.Turaev on the group algebra $\mathbf k\pi_1(\Sigma,p)$ of the fundamental group of a surface based at $p\in\partial\Sigma$. This bracket also induces a noncommutative Goldman Poisson bracket on the cyclic space $\mathcal C_\natural$, which is a $\mathbf k$-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative $r$-matrix formalism. This gives a more conceptual proof of the result of N. Ovenhouse that traces of powers of Lax matrix form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on $\mathcal C_\natural$.

math.QA↗

arXiv2020

Darboux coordinates for symplectic groupoid and cluster algebras

Using Fock--Goncharov higher Teichm\"uller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the $\mathcal A_n$ groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric $R$-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for $\mathcal A_3$ and $\mathcal A_4$ in terms of geodesic functions for Riemann surfaces with holes. We represent braid-group transformations for $\mathcal A_n$ via sequences of cluster mutations in the special $\mathbb A_n$-quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit.

math.QA↗

arXiv2016

Localization Lifetime of a Many-Body System with Periodic Constructed Disorder

We show that, in a many-body system, all particles can be strongly confined to the initially occupied sites for a time that scales as a high power of the ratio of the bandwidth of site energies to the hopping amplitude. Such time-domain formulation is complementary to the formulation of the many-body localization of all stationary states with a large localization length. The long localization lifetime is achieved by constructing a periodic sequence of site energies with a large period in a one-dimensional chain. The scaling of the localization lifetime is independent of the number of particles for a broad range of the coupling strength. The analytical results are confirmed by numerical calculations.

cond-mat.mes-hall↗

arXiv2015

Femtosecond lasing from a fluorescent protein in a one dimensional random cavity

We present evidence of ultrafast random lasing from the fluorescent protein DsRed2 embedded in a random one-dimensional cavity. Lasing is achieved when a purified protein solution, placed inside a layered random medium, is optically excited with a femtosecond pump pulse in the direction perpendicular to the plane of random layers. We demonstrate that pumping with ultrashort pulses resulted in a lasing threshold two orders of magnitude lower than that found for nanosecond excitation.

physics.optics↗

arXiv2011

Stern-Gerlach deflection of field-free aligned paramagnetic molecules

The effects of laser-induced prealignment on the deflection of paramagnetic molecules by inhomogeneous static magnetic field are studied. Depending on the relevant Hund's coupling case of the molecule, two different effects were identified: either suppression of the deflection by laser pulses (Hund's coupling case (a) molecules, such as ClO), or a dramatic reconstruction of the broad distribution of the scattering angles into several narrow peaks (for Hund's coupling case (b) molecules, such as O2 or NH). These findings are important for various applications using molecular guiding, focusing and trapping with the help of magnetic fields.

physics.chem-ph↗

arXiv2009

Complete transfer of populations from a single state to a pre-selected superposition of states using Piecewise Adiabatic Passage: Experiment

We demonstrate a method of adiabatic population transfer from a single quantum state into a coherent superposition of states. The transfer is executed with femtosecond pulses, spectrally shaped in simple and intuitive manner, which does not require iterative feedback-controlled loops. In contrast to non-adiabatic methods of excitation, our approach is not sensitive to the exact value of laser intensity. We show that the population transfer is complete, and analyze the possibility of controlling the relative phases and amplitudes of the excited eigenstates. We discuss the limitations of the proposed control methods due to the dynamic level shifts and suggest ways of reducing their influence.

quant-ph↗

arXiv2009

Quantum Measurements of Coupled Two-Level Systems

We propose an approach to measuring nonresonant coupled systems, which gives a parametrically smaller error than the conventional fast projective measurements. The approach takes into account that, due to the coupling, excitations are not entirely localized on individual systems. It combines high spectral selectivity of the detector with temporal resolution and uses the ideas of the quantum diffusion theory. The results bear on quantum computing with perpetually coupled qubits.

quant-ph↗

arXiv2007

Population transfer between two quantum states by piecewise chirping of femtosecond pulses: Theory and experiment

We propose and demonstrate the method of population transfer by piecewise adiabatic passage between two quantum states. Coherent excitation of a two-level system with a train of ultrashort laser pulses is shown to reproduce the effect of an adiabatic passage, conventionally achieved with a single frequency-chirped pulse. By properly adjusting the amplitudes and phases of the pulses in the excitation pulse train, we achieve complete and robust population transfer to the target state. The effect is demonstrated experimentally by observing piecewise excitation of Rubidium atoms from 5s_1/2 to 5p_1/2 electronic state. We show that similarly to the conventional adiabatic passage, the piecewise process is insensitive to the total excitation energy as long as the adiabaticity conditions are satisfied. The piecewise nature of the process suggests that robust and selective population transfer could be implemented in a variety of complex quantum systems beyond the two-level approximation.

physics.atom-ph↗

arXiv2007

Overlapping resonances in the control of intramolecular vibrational redistribution

Coherent control of bound state processes via the interfering overlapping resonances scenario [Christopher et al., J. Chem. Phys. 123, 064313 (2006)] is developed to control intramolecular vibrational redistribution (IVR). The approach is applied to the flow of population between bonds in a model of chaotic OCS vibrational dynamics, showing the ability to significantly alter the extent and rate of IVR by varying quantum interference contributions.

quant-ph↗

arXiv2006

On total reality of meromorphic functions

We show that if a meromorphic function of degree at most four on a real algebraic curve of an arbitrary genus has only real critical points then it is conjugate to a real meromorphic function after a suitable projective automorphism of the image.

math.AG↗

arXiv2006

First steps towards total reality of meromorphic functions

It was earlier conjectured by the second and the third authors that any rational curve $g:{\mathbb C}P^1\to {\mathbb C}P^n$ such that the inverse images of all its flattening points lie on the real line ${\mathbb R}P^1\subset {\mathbb C}P^1$ is real algebraic up to a linear fractional transformation of the image ${\mathbb C}P^n$. (By a flattening point $p$ on $g$ we mean a point at which the Frenet $n$-frame $(g',g'',...,g^{(n)})$ is degenerate.) Below we extend this conjecture to the case of meromorphic functions on real algebraic curves of higher genera and settle it for meromorphic functions of degrees $2,3$ and several other cases.

math.AG↗

arXiv2005

A few riddles behind Rolle's theorem

We call a smooth function of one variable a degree n pseudopolynomial if its n-th derivative has no (real) zeros. An n pseudopolynomial is called hyperbolic if it has exactly n simple zeros. In this short note we describe the necessary and sufficient conditions on the arrangements of 6 points consisting of 3 zeros of pseudopolynomials of degree 3, two zeros of their 1st derivatives and 1 zero of their second derivatives. Besides the standard Rolle's inequalities the restrictions include additional quadratic inequalities of geometric origin. This text is an easy reading accessible for undergraduates. We formulated also two questions not solved yet.

math.CA↗

arXiv2005

Many-particle confinement by constructed disorder and quantum computing

Many-particle confinement (localization) is studied for a 1D system of spinless fermions with nearest-neighbor hopping and interaction, or equivalently, for an anisotropic Heisenberg spin-1/2 chain. This system is frequently used to model quantum computers with perpetually coupled qubits. We construct a bounded sequence of site energies that leads to strong single-particle confinement of all states on individual sites. We show that this sequence also leads to a confinement of all many-particle states in an infinite system for a time that scales as a high power of the reciprocal hopping integral. The confinement is achieved for strong interaction between the particles while keeping the overall bandwidth of site energies comparatively small. The results show viability of quantum computing with time-independent qubit coupling.

cond-mat.mes-hall↗

arXiv2005

Scaling in activated escape of underdamped systems

Noise-induced escape from a metastable state of a dynamical system is studied close to a saddle-node bifurcation point, but in the region where the system remains underdamped. The activation energy of escape scales as a power of the distance to the bifurcation point. We find two types of scaling and the corresponding critical exponents.

cond-mat.mes-hall↗

arXiv2004

Many-particle localization by constructed disorder: enabling quantum computing with perpetually coupled qubits

We demonstrate that, in a many-particle system, particles can be strongly confined to their sites. The localization is obtained by constructing a sequence of on-site energies that efficiently suppresses resonant hopping. The time during which a many-particle state remains strongly localized in an infinite chain can exceed the reciprocal hopping frequency by $\agt 10^6$ already for a narrow energy bandwidth. The results show viability of quantum computers with time-independent qubit coupling.

cond-mat.mes-hall↗

arXiv2004

Strong many-particle localization and quantum computing with perpetually coupled qubits

We demonstrate the onset of strong on-site localization in a one-dimensional many-particle system. The localization is obtained by constructing, in an explicit form, a bounded sequence of on-site energies that eliminates resonant hopping between both nearest and remote sites. This sequence leads to quasi-exponential decay of the single-particle transition amplitude. It also leads to strong localization of stationary many-particle states in a finite-length chain. For an {\it infinite} chain, we instead study the time during which {\it all} many-particle states remain strongly localized. We show that, for any number of particles, this time exceeds the reciprocal frequency of nearest-neighbor hopping by a factor $\sim 10^5$ already for a moderate bandwidth of on-site energies. The proposed energy sequence is robust with respect to small errors. The formulation applies to fermions as well as perpetually coupled qubits. The results show viability of quantum computing with time-independent qubit coupling.

quant-ph↗