arXiv subjects

A. Matveev

Publications and source records attributed to A. Matveev.

13 recordsLinked to original sources

arXiv2026

Contraction theory: Hausdorff--Riemann Measures as Set-Based Lyapunov Functions

We offer a measure-theoretic extension of the concept and theory of $k$-contraction, including their generalization on fractional dimensions $d$. The respective contraction property is defined through the exponential decay of the $d$-dimensional volume of compact sets transported by a nonlinear flow. For autonomous systems on positively invariant compact sets, we derive comprehensive, i.e., necessary and sufficient, conditions for $d$-contractivity in two complementary forms. The first is expressed in terms of the finite-time Lyapunov characteristic exponents and is akin in spirit to the first Lyapunov method. The second one is consonant with the second Lyapunov method and relies on existence of a Riemannian metric ensuring exponential decay of the metric-induced $d$-dimensional Hausdorff measure. To acquire monotone measure-theoretic-based Lyapunov functions, we introduce a family of \emph{Hausdorff-Riemann measures}, which are elliptic, metric-dependent $d$-measures that strictly decrease along the trajectories and thus may serve as Lyapunov functions. These measures enable an anytime characterization of the rate of contraction and provide constructive tools for stability analysis and feedback design. To illustrate the applicability of the approach, we derive tractable criteria for orbital stability of periodic solutions of autonomous ODE's and employ several prototypical particular examples, including a rigid body with dissipation and constant torque, the R\"ossler system, and the Langford system.

math.DS↗

arXiv2025

Towards practical non-Markovianity measures: Normalization and regularization techniques

Measures characterizing the non-Markovianity degree of the quantum dynamics have several drawbacks when applied to real devices. They depend on the chosen measurement time interval and are highly sensitive to experimental noise and errors. We propose several techniques to enhance the practical applicability of the measures and verify our findings experimentally on a superconducting transmon and a trapped-ion qubit. The time dependence can be disregarded by introduction of the measure per oscillation, while the sensitivity to noise is reduced by applying a regularization procedure. The results for both types of qubits are compared with theoretical predictions for a simple model of non-Markovianity based on qubit-qubit interaction.

quant-ph↗

arXiv2023

Long-distance chronometric leveling with a portable optical clock

We have measured the geopotential difference between two locations separated by $457~\mathrm{km}$ by comparison of two optical lattice clocks via an interferometric fiber link, utilizing the gravitational redshift of the clock transition frequency. The $^{87}$Sr clocks have been compared side-by-side before and after one of the clocks was moved to the remote location. The chronometrically measured geopotential difference of $3918.1(2.6)\,\mathrm{m^2 \, s^{-2}}$ agrees with an independent geodetic determination of $3915.88(0.30)\,\mathrm{m^2 \, s^{-2}}$. The uncertainty of the chronometric geopotential difference is equivalent to an uncertainty of $27~\mathrm{cm}$ in height.

physics.atom-ph↗

arXiv2021

Measurement of the $2$S$_{1/2}-8$D$_{5/2}$ transition in hydrogen

We present a measurement of the hydrogen $2$S$_{1/2}-8$D$_{5/2}$ transition performed with a cryogenic atomic beam. The measured resonance frequency is $\nu=770649561570.9(2.0)$ kHz, which corresponds to a relative uncertainty of $2.6\times10^{-12}$. Combining our result with the most recent measurement of the $1$S$-2$S transition, we find a proton radius of $r_p=0.8584(51)$~fm and a Rydberg constant of $R_\infty=10973731.568332(52)$ m$^{-1}$. This result has a combined 3.1~$\sigma$ disagreement with the CODATA 2018 recommended value. Possible implications of the discrepancy are discussed.

physics.atom-ph↗

arXiv2020

Remote state estimation problem: towards the data-rate limit along the avenue of the second Lyapunov method

In the context of control and estimation under information constraints, restoration entropy measures the minimal required data rate above which the state of a system can be estimated so that the estimation quality does not degrade over time and, conversely, can be improved. The remote observer here is assumed to receive its data through a communication channel of finite bit-rate capacity. In this paper, we provide a new characterization of the restoration entropy which does not require to compute any temporal limit, i.e., an asymptotic quantity. Our new formula is based on the idea of finding a specific Riemannian metric on the state space which makes the metric-dependent upper estimate of the restoration entropy as tight as one wishes.

math.OC↗

arXiv2019

Pressure Shifts in High-Precision Hydrogen Spectroscopy: I. Long-Range Atom-Atom and Atom-Molecule Interactions

We study the theoretical foundations for the pressure shifts in high-precision atomic beam spectrosopy of hydrogen, with a particular emphasis on transitions involving higher excited P states. In particular, the long-range interaction of an excited hydrogen atom in a 4P state with a ground-state and metastable hydrogen atom is studied, with a full resolution of the hyperfine structure. It is found that the full inclusion of the 4P_1/2 and 4P_3/2 manifolds becomes necessary in order to obtain reliable theoretical predictions, because the 1S ground state hyperfine frequency is commensurate with the 4P fine-structure splitting. An even more complex problem is encountered in the case of the 4P-2S interaction, where the inclusion of quasi-degenerate 4S-2P_1/2 state becomes necessary in view of the dipole couplings induced by the van der Waals Hamiltonian. Matrices of dimension up to 40 have to be treated despite all efforts to reduce the problem to irreducible submanifolds within the quasi-degenerate basis. We focus on the phenomenologically important second-order van der Waals shifts, proportional to 1/R^6 where R is the interatomic distance, and obtain results with full resolution of the hyperfine structure. The magnitude of van der Waals coefficients for hydrogen atom-atom collisions involving excited P states is drastically enhanced due to energetic quasi-degeneracy; we find no such enhancement for atom-molecule collisions involving atomic nP states, even if the complex molecular spectrum involving ro-vibrational levels requires a deeper analysis.

physics.atom-ph↗

arXiv2019

Pressure Shifts in High-Precision Hydrogen Spectroscopy: II. Impact Approximation and Monte-Carlo Simulations

We investigate collisional shifts of spectral lines involving excited hydrogenic states, where van der Waals coefficients have recently been shown to have large numerical values when expressed in atomic units. Particular emphasis is laid on the recent hydrogen 2S-4P experiment (and an ongoing 2S-6P experiment) in Garching, but numerical input data are provided for other transitions (e.g., involving S states), as well. We show that the frequency shifts can be described, to sufficient accuracy, in the impact approximation. The pressure related effects were separated into two parts, (i) related to collisions of atoms inside of the beam, and (ii) related to collisions of the atoms in the atomic beam with the residual background gas. The latter contains both atomic as well as molecular hydrogen. The dominant effect of intra-beam collisions is evaluated by a Monte-Carlo simulation, taking the geometry of the experimental apparatus into account. While, in the Garching experiment, the collisional shift is on the order of 10 Hz, and thus negligible, it can decisively depend on the experimental conditions. We present input data which can be used in order to describe the effect for other transitions of current and planned experimental interest.

physics.atom-ph↗

arXiv2017

Long-range interactions of hydrogen atoms in excited states. I. 2S-1S interactions and Dirac-delta perturbations

The theory of the long-range interaction of metastable excited atomic states with ground-state atoms is analyzed. We show that the long-range interaction is essentially modified when quasi-degenerate states are available for virtual transitions. A discrepancy in the literature regarding the van der Waals coefficient C_6(2S;1S) describing the interaction of metastable atomic hydrogen (2S state) with a ground-state hydrogen atom is resolved. In the the van der Waals range a_0 << R << a_0/alpha, where a_0= hbar/(alpha m c) is the Bohr radius and alpha is the fine structure constant, one finds the symmetry-dependent result E_{2S;1S}(R) ~ (-176.75 +/- 27.98) E_h (a_0/R)^6 (E_h denotes the Hartree energy). In the Casimir-Polder range a_0/alpha << R << hbar c/L, where L = E(2S_{1/2})-E(2P_{1/2}) is the Lamb shift energy, one finds E_{2S;1S}(R) ~ (-121.50 +/- 46.61) E_h (a_0/R)^6. In the the Lamb shift range R >> hbar c/L, we find an oscillatory tail with a negligible interaction energy below 10^(-36) Hz. Dirac-delta perturbations to the interaction are also evaluated and results are given for all asymptotic distance ranges; these effects describe the hyperfine modification of the interaction, or, expressed differently, the shift of the hydrogen 2S hyperfine frequency due to interactions with neighboring 1S atoms. The 2S hyperfine frequency has recently been measured very accurately in atomic beam experiments.

physics.atom-ph↗

arXiv2017

Long-range interactions of hydrogen atoms in excited states. II. Hyperfine-resolved 2S-2S system

The interaction of two excited hydrogen atoms in metastable states constitutes a theoretically interesting problem because of the quasi-degenerate 2P_{1/2} levels which are removed from the 2S states only by the Lamb shift. The total Hamiltonian of the system is composed of the van der Waals Hamiltonian, the Lamb shift and the hyperfine effects. The van der Waals shift becomes commensurate with the 2S-2P_{3/2} fine-structure splitting only for close approach (R < 100 a_0, where a_0 is the Bohr radius) and one may thus restrict the discussion to the levels with n=2 and J=1/2 to good approximation. Because each S or P state splits into an F=1 triplet and an F=0 hyperfine singlet (eight states for each atom), the Hamiltonian matrix {\em a priori} is of dimension 64. A careful analysis of symmetries the problem allows one to reduce the dimensionality of the most involved irreducible submatrix to 12. We determine the Hamiltonian matrices and the leading-order van der Waals shifts for states which are degenerate under the action of the unperturbed Hamiltonian (Lamb shift plus hyperfine structure). The leading first- and second-order van der Waals shifts lead to interaction energies proportional to 1/R^3 and 1/R^6 and are evaluated within the hyperfine manifolds. When both atoms are metastable 2S states, we find an interaction energy of order E_h chi (a_0/R)^6, where E_h and L are the Hartree and Lamb shift energies, respectively, and chi = E_h/L ~ 6.22 \times 10^6 is their ratio.

physics.atom-ph↗

arXiv2011

Low phase noise diode laser oscillator for 1S-2S spectroscopy in atomic hydrogen

We report on a low-noise diode laser oscillator at 972 nm actively stabilized to an ultra-stable vibrationally- and thermally compensated reference cavity. To increase the fraction of laser power in the carrier we designed a 20 cm long external cavity diode laser with an intra-cavity electro-optical modulator. The fractional power in the carrier reaches 99.9% which corresponds to a rms phase noise of $\phi^2_\textrm{rms}=1\,\textrm{mrad}^2$ in 10\,MHz bandwidth. Using this oscillator we recorded 1S-2S spectra in atomic hydrogen and have not observed any significant loss of the excitation efficiency due to phase noise multiplication in the three consecutive 2-photon processes.

quant-ph↗

arXiv2009

Testing the Stability of the Fine Structure Constant in the Laboratory

In this review we discuss the progress of the past decade in testing for a possible temporal variation of the fine structure constant $\alpha$. Advances in atomic sample preparation, laser spectroscopy and optical frequency measurements led to rapid reduction of measurement uncertainties. Eventually laboratory tests became the most sensitive tool to detect a possible variation of $\alpha$ at the present epoch. We explain the methods and technologies that helped make this possible.

quant-ph↗

arXiv2008

New Measurement of the 2S Hyperfine Interval in Atomic Hydrogen

An optical measurement of the 2S hyperfine interval in atomic hydrogen using two-photon spectroscopy of the 1S-2S transition gives a value of 177 556 834.3(6.7) Hz. The uncertainty is 2.4 times smaller than achieved by our group in 2003 and more than 4 times smaller than for any independent radio-frequency measurement. The specific combination of the 2S and 1S hyperfine intervals predicted by QED theory $D_{21}=8 f_{\rm HFS}({2S}) - f_{\rm HFS}({1S})=48 953(3)$ Hz is in good agreement with the value of 48 923(54) Hz obtained from this experiment.

physics.atom-ph↗

arXiv2008

Sub-Hz line width diode lasers by stabilization to vibrationally and thermally compensated ULE Fabry-Perot cavities

We achieved a 0.5 Hz optical beat note line width with ~ 0.1 Hz/s frequency drift at 972 nm between two external cavity diode lasers independently stabilized to two vertically mounted Fabry-Perot (FP) reference cavities. Vertical FP reference cavities are suspended in mid-plane such that the influence of vertical vibrations to the mirror separation is significantly suppressed. This makes the setup virtually immune for vertical vibrations that are more difficult to isolate than the horizontal vibrations. To compensate for thermal drifts the FP spacers are made from Ultra-Low-Expansion (ULE) glass which possesses a zero linear expansion coefficient. A new design using Peltier elements in vacuum allows operation at an optimal temperature where the quadratic temperature expansion of the ULE could be eliminated as well. The measured linear drift of such ULE FP cavity of 63 mHz/s was due to material aging and the residual frequency fluctuations were less than 40 Hz during 16 hours of measurement. Some part of the temperature-caused drift is attributed to the thermal expansion of the mirror coatings. High-frequency thermal fluctuations that cause vibrations of the mirror surfaces limit the stability of a well designed reference cavity. By comparing two similar laser systems we obtain an Allan instability of 2*10-15 between 0.1 and 10 s averaging time, which is close to the theoretical thermal noise limit.

physics.optics↗