Energy Level Crossings Research Articles

Synchrotron radiation far infrared spectrum of the astrophysically significant Ethanol (CH3CH2OH) molecule in the gauche states in the vibrational ground state and other infrared observations

In this communication, the analyses of synchrotron radiation far infrared (FIR) spectrum corresponding to the gauche- (e1ando1) states of Ethyl Alcohol are reported. Detailed assignments have been performed for b-type transitions for K values ranging from 5 to 32 and up to a maximum J value of 50. The assignments confirmed the earlier microwave (MW) and millimeter wave (MMW) spectroscopic results. The transition wavenumbers have been used to determine the term values for the gauche-states involved for all the K and J values for which observation has been made. About 2000 spectral lines have been assigned, including some accurately calculated lines that fall in the MW and MMW regions. Although transitions between the trans-species and gauche species are not allowed in Ethanol, many resonances and level crossings of energy levels cause forbidden transitions. One such system of level crossings between K=10o1 and 12ee00 has been analyzed in detail, and the interaction coefficient determined. The state mixing seems quite strong and causes forbidden transitions (ΔK=0,2,3), including transitions for trans↔gauche, have been found. In addition, many new strong transitions have been assigned, which belong to the trans-species. To extend our work to higher wave numbers to facilitate observations made by the infrared detector on board the James Webb Space Telescope (JWST), the analysis of the torsional fundamental band transitions (around 220cm-1) for o2←e0 and o2←e1 has been taken up. Some of the assignments are discussed here. Lastly, the lower-lying vibrational bands centered around 425cm-1 (CCO– bending mode) and the band at around 800cm-1 (CH3- rocking mode) have been recorded. The CCO bending band shows an unmistakable parallel character for the trans-species. The assignment work on the CCO-bending band is in progress, and the assignments for K=0andK=1(+and-) are included in this report. Accurate term values for the CCO-bending mode could be obtained. The results allowed the determination of the leading rotational constants for the trans-species in the excited vibrational state of the CCO-bending state. The complete known assigned lines of about 16,000 lines for the parent isotopomer and about 650 lines for 13 -C1 and 13−C2 (see graphical abstract) substituted Ethanol isotopomers are gathered in Appendix I and II, respectively. These atlases should be a valuable tool for further energy level analysis and astronomical detection of Ethanol in interstellar space.

The influence of off-centre donor impurity on the tuning of the quantum ring magnetic susceptibility

The 1/N expansion method has been used to investigate the problem of an electron moving in a two-dimensional semiconductor quantum ring with an off-center impurity. The energy spectra, magnetization, and magnetic susceptibility have been calculated as a function of the magnetic field, impurity position, temperature, and quantum ring parameters. Interesting energy-level crossings attributed to QR confinement manifest themselves as low-temperature oscillations in the magnetization and magnetic susceptibility curves. On the contrary, the presence of impurity at the QR center eliminates these level crossings. This work shows that changing the impurity position can manipulate the magnetic field value correspondence to these diamagnetic-paramagnetic phase transitions.

Minimal model of excited-state symmetry breaking in symmetric dimers and covalently linked dyads.

A model of symmetry breaking (SB) charge separation in symmetric excited dyads and dimers is presented. The minimal model should include at least four basis electronic states due to a small energy gap between the locally excited and charge separated (zwitterionic) states of the chromophores. There are electronic couplings between all these states. The model includes the following interactions: (i) the Coulomb interaction between charges on the chromophores of the dyad, (ii) the interaction of the dipole moment of the asymmetric dyad with the solvent polarization, and (iii) the electronic-vibrational interaction. SB becomes possible only if the intensity of these interactions exceeds a threshold value. The threshold vanishes if there is a degeneration of the levels. Unusual resonant dependencies of the dissymmetry degree on the model parameters are revealed. Resonances arise due to the degeneration of energy levels. The ranges of the parameters in which energy level crossings occur are established. The oddity lies in the dependence of the resonance shape on the parameters of the model. A variation in the electronic couplings and the energy gap between the locally excited and ionic states, which leads to a broadening of the resonance, simultaneously leads to an increase in the resonant height. This opens up wide possibilities for controlling the charge separation degree. The predictions of the theory agree with the available experimental data. The charge separation SB is predicted to accompany by SB in the excitation distribution on the branches of dyads.

On the Concurrent Bipartite Entanglement of a Spin-1 Heisenberg Diamond Cluster Developed for Tetranuclear Nickel Complexes

The bipartite entanglement of a quantum spin-1 Heisenberg diamond cluster in the presence of the external magnetic field is quantified through the negativity, which is calculated for spin pairs from a diagonal and a side of the diamond spin cluster taking into consideration two different coupling constants. The magnetic field may cause a few crossings of energy levels of the spin-1 Heisenberg diamond cluster, which is responsible at low enough temperatures for a stepwise dependence of the negativity on the magnetic field accompanied with a drop of the negativity at respective magnetic-field-driven transitions due to emergence of mixed states. It is shown that the bipartite entanglement between spin pairs on a diagonal and a side of the diamond spin cluster is concurrent although they may eventually become both nonzero albeit not fully saturated. It is predicted that the tetranuclear nickel complex [Ni4(μ-CO3)2(aetpy)8](ClO4)4 (aetpy = 2-aminoethyl-pyridine), which represents an experimental realization of the spin-1 Heisenberg diamond cluster with two different antiferromagnetic coupling constants, exhibits a substantial bipartite entanglement between two spin-1 Ni2+ magnetic ions from a shorter diagonal of the diamond spin cluster up to temperatures approximately about 50 K and up to magnetic fields about 70 T.

Simultaneous effects of Rashba, magnetic field and impurity on the magnetization and magnetic susceptibility of a GaAs-semiconductor quantum ring

The magnetization and magnetic susceptibility of GaAs/AlGaAs quantum ring (QR) with the presence of Rashba spin-orbit interaction and acceptor Gaussian impurity have been studied. We have used the shifted 1/N expansion to calculate the energy eigenvalues as a function of the magnetic field. The obtained energies show energy level crossings; transition in the angular momentum of the ground state of the QR. Furthermore, we investigate the dependence of energy eigenvalues of QR on: confinement potential strength, Rashba parameter, and the presence of acceptor impurity. The comparisons show that the present computed results are in a good agreement with the available experimental and theoretical ones.Furthermore, we have shown the variation of the magnetization and magnetic susceptibility with the Hamiltonian variables. These magnetic quantities show an oscillating behavior associated with the phenomena of energy level crossings. The computed results indicate that as the Rashba strength increases, the positions of these oscillating peaks in the magnetic susceptibility curve shift towards a lower magnetic field value.

Aharonov-Bohm oscillations in phosphorene quantum rings: Mass anisotropy compensation by confinement potential

We consider the Aharonov-Bohm (AB) effect on a confined electron ground state in a quantum ring defined electrostatically within the phosphorene monolayer. The strong anisotropy of effective masses in phosphorene quenches ground-state oscillations for a circular ring because of interrupted persistent current circulation around the ring. An elliptic deformation of the confinement potential can compensate for the anisotropy of the effective masses and produce ground-state parity transformations with the AB periodicity. Moreover, a specific ratio of the semiaxes is determined for which the spectrum becomes identical to that of a circular quantum ring and an isotropic effective mass. We identify a generalized angular momentum operator which commutes with the continuum Hamiltonian for the chosen ratio of the semiaxes that closes the avoided crossings of energy levels for states of the same parity and spin. Ground-state oscillations for the two-electron ground state are also discussed.

Wavelength conversion of KTP crystal based Stark-chirped rapid adiabatic passage

The nonlinear wavelength conversion can generate the laser wavelengths which are not directly available, thereby broadening the laser spectrum range. However, the phase mismatch greatly limits the development and application of nonlinear optical technology. The wavelength conversion schemes in a manner analogous to population transfer in atomic rapid adiabatic passage, stimulated Raman adiabatic passage (STIRAP), and Stark chirped rapid adiabatic passage (SCRAP) provide feasible solutions for efficient and broadband wavelength conversion. The SCRAP uses the Stark shift caused by the Stark field to generate energy level crossings, therefore, the population in initial state can be efficiently converted into the target state. It does not require the two-photon resonance, and can be applied to multi-photon transition. In this paper, by approximate analogy to the adiabatic population theory, a wavelength conversion model with the KTP crystals based SCRAP is established, the influence of the coupling delay parameters, width parameters, pump intensity, temperature, and incident wavelength on the conversion process are systematically studied. The results show that the signal laser energy can be almost converted into output laser energy, while the intermediate laser energy is kept extremely low in the conversion process. The conversion process is sensitive to changes in coupling delay parameters, width parameters, and pump intensity. The farther away fromits optimal value the coupling delay parameter, the lower the conversion efficiency is. When the width parameter <inline-formula><tex-math id="M1">\begin{document}$ d_2^2 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20210887_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20210887_M1.png"/></alternatives></inline-formula> is fixed, as the width parameter <inline-formula><tex-math id="M2">\begin{document}$ d_1^2 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20210887_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20210887_M2.png"/></alternatives></inline-formula> increases, the conversion efficiency first increases to a maximum value, and then slowly decreases. At the same time, the greater the value of the width parameter <inline-formula><tex-math id="M3">\begin{document}$ d_2^2 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20210887_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11-20210887_M3.png"/></alternatives></inline-formula>, the greater the achievable maximum conversion efficiencies, and the greater the bandwidth that can achieve high-efficiency wavelength conversion. The conversion efficiency increases as the pump intensity increases. When the conversion efficiency value reaches a maximum value, the increase in pump intensity has almost no effect on the conversion efficiency. However, changes in temperature and incident wavelength have little effect on the conversion efficiency. The above research can provide a theoretical basis for the acquisition of ultraviolet to mid-infrared light sources and the manufacture of photonic devices.

Realization of an Optical Thermometer via Structural Confinement and Energy Transfer.

The influence of temperature on a variety of physiological or chemical processes has generated considerable interest, and recently noninvasive lanthanide-incorporated optical thermometers have been considered as promising candidates for monitoring its changes at different scales. Herein, a novel Bi3+-activated Sr3-xGdxGaO4+xF1-x phosphor with tunable color has been constructed by a cooperative cation-anion substitution strategy with to the replacement of [Sr2+-F-] by [Gd3+-O2-]. When x = 0, the sample Sr3GaO4F/Bi3+ possesses a peak wavelength at 438 nm, and this value will shift to 470 nm if x is equal to 1 (Sr2GdGaO5/Bi3+). In addition, photoluminescence tuning from blue to red has been realized successfully by an efficient Bi3+ → Eu3+ energy migration model in Sr2.6Gd0.4GaO4.4F0.6 samples. The specific Bi3+ → Eu3+ energy transfer has been explained by dipole-dipole interactions derived from a model of the Dexter pathway. Intriguingly, the two dopants (a blue signal from Bi3+ and a red signal from Eu3+) possess different thermal responses to increasing temperature. Accordingly, the intensity ratio values are sensitive to the temperature changes. The energy level cross relaxation causes the quenching effect of Bi3+, and the multi-phonon de-excitation mode leads to the thermal quenching of Eu3+. At room temperature (298 K), the determined maximum relative sensitivity (Sr) is 1.27% K-1. Moreover, the absolute sensitivity (Sa) is 0.067 K-1 since the temperature is elevated to 523 K. The collected results are superior to most of the reported optical thermometry materials.

Hidden symmetries, spin and charge of artificial magnetic monopoles

We discuss the non-Abelian artificial magnetic monopoles associated with n-level energy crossing in quantum systems. We found that hidden symmetries reveal themselves as observables such as spin, hypercharge, and other physical degrees of freedom. We illustrated our results on concrete examples of two and three energy-level crossings. Our findings can be helpful for modeling various phenomena in physical systems such as SU(N) topological insulators and spintronic devices based on topological insulators.

Homonuclear J-Coupling Spectroscopy at Low Magnetic Fields using Spin-Lock Induced Crossing*.

Nuclear magnetic resonance (NMR) spectroscopy usually requires high magnetic fields to create spectral resolution among different proton species. Although proton signals can also be detected at low fields the spectrum exhibits a single line if J-coupling is stronger than chemical shift dispersion. In this work, we demonstrate that the spectra can nevertheless be acquired in this strong-coupling regime using a novel pulse sequence called spin-lock induced crossing (SLIC). This techniques probes energy level crossings induced by a weak spin-locking pulse and produces a unique J-coupling spectrum for most organic molecules. Unlike other forms of low-field J-coupling spectroscopy, our technique does not require the presence of heteronuclei and can be used for most compounds in their native state. We performed SLIC spectroscopy on a number of small molecules at 276 kHz and 20.8 MHZ and show that the simulated SLIC spectra agree well with measurements.

Spectroscopic investigation and laser behaviour of Yb-doped laser ceramics based on mixed crystalline structure (ScxY1-x)2O3

We report on the spectroscopic and laser characterization of Yb-doped (ScxY1-x)2O3 ceramics with different Y3+/Sc3+ balance (x=0, 0.273, 0.508 and 0.742) and ~6 at.% Yb concentration, fabricated by solid-state vacuum sintering of laser-ablated mixed sesquioxide nanoparticles. The optical transmittances of 2-mm-thick samples with x=0, 0.273 and 0.508 were above 80% at a wavelength of 600 nm and 82% at 1100 nm which is very close to the theoretical limit. The lifetime of the 2F5/2 Yb3+ energy level, absorption and emission cross sections result to be influenced by the Sc3+ content. In particular, the upper state lifetime decreased from 781 μs to 614 μs with an increase in Sc2O3 concentration from 0 mol.% to 74.2 mol.%. Moreover, the main emission peaks of Yb3+ near 1040 nm and 1080 nm shift toward longer wavelength with an increase in Sc3+ content and progressively broaden until Y3+/Sc3+ balance reaches ~1. Concerning the laser tests, all samples showed good performance under quasi-CW laser diode end-pumping at 929 nm. For the mixed compositions, a maximum output power of 5.95 W with an optical-to-optical conversion efficiency of 58.8% was achieved in (Yb0.058Sc0.508Y0.434)2O3 ceramics. Most interestingly, we measured an end-to-end tuning range of 117.7 nm for the sample (Yb0.062Sc0.273Y0.665)2O3, which is to the best of our knowledge the broadest tuning range found in literature for an Yb doped laser ceramics. This represents the first highly efficient laser operation of compositionally disordered Yb-doped (ScxY1-x)2O3 ceramic matrices.

Self-Hybridization and Tunable Magnon-Magnon Coupling in van der Waals Synthetic Magnets

Van der Waals magnets are uniquely positioned at the intersection between two-dimensional materials, antiferromagnetic spintronics, and magnonics. The interlayer exchange interaction in these materials enables antiferromagnetic resonances to be accessed at GHz frequencies. Consequently, these layered antiferromagnets are intriguing materials out of which quantum hybrid magnonic devices can be fashioned. Here, we use both a modified macrospin model and micromagnetic simulations to demonstrate a comprehensive antiferromagnetic resonance spectra in van der Waals magnets near the ultrathin (monolayer) limit. The number of optical and acoustic magnon modes, as well as the mode frequencies, are found to be exquisitely sensitive to the number of layers. We discover a self-hybridization effect where pairs of either optical or acoustic magnons are found to interact and self-hybridize through the dynamic exchange interaction. This leads to characteristic avoided energy level crossings in the energy spectra. Through simulations, we show that by electrically controlling the damping of surface layers within heterostructures both the strength and number of avoided energy level crossings in the magnon spectra can be controlled.

Anneal-path correction in flux qubits

Quantum annealers require accurate control and optimized operation schemes to reduce noise levels, in order to eventually demonstrate a computational advantage over classical algorithms. We study a high coherence four-junction capacitively shunted flux qubit (CSFQ), using dispersive measurements to extract system parameters and model the device. Josephson junction asymmetry inherent to the device causes a deleterious nonlinear cross-talk when annealing the qubit. We implement a nonlinear annealing path to correct the asymmetry in situ, resulting in a substantial increase in the probability of the qubit being in the correct state given an applied flux bias. We also confirm the multi-level structure of our CSFQ circuit model by annealing it through small spectral gaps and observing quantum signatures of energy level crossings. Our results demonstrate an anneal-path correction scheme designed and implemented to improve control accuracy for high-coherence and high-control quantum annealers, which leads to an enhancement of success probability in annealing protocols.

Signatures of avoided energy-level crossings in entanglement indicators obtained from quantum tomograms

Extensive theoretical and experimental investigations on multipartite systems close to an avoided energy-level crossing reveal interesting features such as the extremisation of entanglement. Conventionally, the estimation of entanglement directly from experimental observation involves either one of two approaches: uncertainty-relation-based estimation that captures the linear correlation between relevant observables, or rigorous but error-prone quantum state reconstruction on tomograms obtained from homodyne measurements. We investigate the behaviour, close to avoided crossings, of entanglement indicators that can be calculated directly from a numerically-generated tomogram. The systems we study are two generic bipartite continuous-variable systems: a Bose–Einstein condensate trapped in a double-well potential, and a multi-level atom interacting with a radiation field. We also consider a multipartite hybrid quantum system of superconducting qubits interacting with microwave photons. We carry out a quantitative comparison of the indicators with a standard measure of entanglement, the subsystem von Neumann entropy (SVNE). It is shown that the indicators that capture the nonlinear correlation between relevant subsystem observables are in excellent agreement with the SVNE.

Adjustable photoluminescence of Bi3+ and Eu3+ in solid solution constructed by isostructural end components through composition and excitation-driven strategy

• From orange red to blue in one phosphor via excitation-driven strategy. • The luminescence color of Bi 3+ was fine-tuned by solid solution composition. • Bi 3+ ions are used as doping ions, which can be carried out in air. • Solid solution constructed by isostructural end components. Most lanthanide doped phosphors need to be prepared in a reducing atmosphere, such as phosphors doped by Eu 2+ and Ce 3+ ions with d-f transition. In contrast, the doping ion Bi 3+ selected in this paper has more advantages. Moreover, the 6S electrons of bare bismuth are very sensitive to the surrounding coordination field environment, which makes it highly tunable. A new type of solid solution Sr 3−x Gd x AlO 4+x F 1−x designed by the isostructural end components Sr 3 AlO 4 F and GdSr 2 AlO 5 well promotes the luminescence of Bi 3+ ion. By adjusting the concentration of Bi 3+ /Eu 3+ , the phosphor can be tuned from blue to red, which is due to the energy transfer caused by dipole–dipole interaction between Bi 3+ and Eu 3+ . Surprisingly, by means of excitation-driven, the emission peak position of Bi 3+ ions can also be moved from ~425 nm to ~437 nm, and red shift occurs, so as to realize the tuning of the fluorescence from orange red to blue. We speculate that this is induced by the presence of multiple emission centers of Bi 3+ , and different absorption strength of Bi 3+ and Eu 3+ . In addition, the thermal response of Bi 3+ and Eu 3+ ions to temperature is different. They are thermally quenched by energy level cross relaxation (ELCR) and multi-phonon deexcitation (MPD), respectively. The maximum relative sensitivity of optical thermometry based on fluorescence intensity ratio (FIR) is approximately 1.17 %K −1 at 323 K, which is higher than that of previously reported Bi 3+ and Eu 3+ co-doped phosphors. At 523 K, we also obtained a high absolute sensitivity of 0.0064 K −1 . Consequently, the solid solution phosphor prepared by us has a far-reaching application prospect in the field of temperature monitoring.

Resonance free domain for a system of Schrödinger operators with energy-level crossings

We consider a [Formula: see text] system of 1D semiclassical differential operators with two Schrödinger operators in the diagonal part and small interactions of order [Formula: see text] in the off-diagonal part, where [Formula: see text] is a semiclassical parameter and [Formula: see text] is a constant larger than [Formula: see text]. We study the absence of resonance near a non-trapping energy for both Schrödinger operators in the presence of crossings of their potentials. The width of resonances is estimated from below by [Formula: see text] and the coefficient [Formula: see text] is given in terms of the directed cycles of the generalized bicharacteristics induced by two Hamiltonians.

Enhancing Intersystem Crossing to Achieve Thermally Activated Delayed Fluorescence in a Water-Soluble Fluorescein Derivative with a Flexible Propenyl Group.

It is a challenge to rationally design an organic molecule with thermally activated delayed fluorescence (TADF) due to the intrinsically spin-forbidden transition. Meanwhile, those reported TADF organic molecules have difficulty to be directly applied in the field of biological and medical imaging because they usually have no water solubility. Here, a water-soluble TADF organic molecule DCF-BXJ was developed by introducing a flexible propenyl group into the commercial traditional fluorophore DCF (2,7-dichlorofluorescein). The flexible group provides nonradiative rotational motion, which causes an efficient energy level cross between the S1 state and the T2 state of DCF-BXJ. Results of transient absorption spectra and theoretical calculations supported that nonradiative rotational motion of the flexible group can enhance intersystem crossing (ISC) and bring out TADF. This work provides a new mechanism explanation for TADF existing in organic molecules.

Probabilistic hysteresis in an isolated quantum system: The microscopic onset of irreversibility from a quantum perspective

Recently probabilistic hysteresis in isolated Hamiltonian systems of ultracold atoms has been studied in the limit of large particle numbers, where a semiclassical treatment is adequate. The origin of irreversibility in these sweep experiments, where a control parameter is slowly (adiabatically) tuned back and forth, turned out to be a passage blue back and forth across a separatrix (integrable case) or a passage in and out of a chaotic sea in phase space (chaotic case). Here we focus on the full quantum mechanical description of the integrable system and show how the semiclassical results emerge in the limit of large particle numbers. Instead of the crossing of a separatrix in phase space, where classical adiabaticity fails, the origin of irreversibility in the quantum system is a series of avoided crossings of the adiabatic energy levels: they become so close that already for modest particle numbers the change of the external parameter has to be unrealistically slow to reach the quantum adiabatic limit of perfectly reversible evolution. For a slow but finite sweep rate we find a broad regime where the quantum results agree with the semiclassical results, but only if besides the limit $N\to \infty$ an initial ensemble of states is considered, with sufficient initial energy width. For a single initial energy eigenstate we find in contrast that the backward sweep reveals strong quantum effects even for very large particle numbers.

Direct measurement of the density and energy level of compensating acceptors and their impact on the conductivity of n-type Ga2O3 films

Intrinsic and extrinsic point defects often act as electron traps in oxide-based semiconductors and significantly impact their electrical and optical properties. Here, we show how to measure the density, energy level, and trapping cross section of the compensating acceptors that act as electron traps in Ga2O3 films, and we introduce the sheet trap number or the sheet compensating acceptor number as an essential parameter to fully describe the electrical transport properties of semiconductors. Si-doped β-Ga2O3 thin films were fabricated homoepitaxially by metalorganic chemical vapor deposition and studied by thermally stimulated luminescence spectroscopy, temperature dependent Hall-effect measurements, and secondary ion mass spectroscopy to investigate the compensating acceptor defects responsible for suppressing conductivity in the films. A deep level defect of energy in the range of 0.50–0.65 eV was identified as a compensating acceptor. The correlation between the electrical properties and its concentration and characteristics was established. This work shows how to quantify the density of compensating acceptors in semiconductors and directly relate it to the electrical transport properties, which should significantly advance the development of semiconductors and devices.

Reliability analysis of the hysteretic nonlinear energy sink in shock mitigation considering uncertainties

The reliability of the hysteretic nonlinear energy sink in shock mitigation is investigated herein. The hysteretic nonlinear energy sink is a passive vibration control device which is coupled to a primary linear oscillator. Apart from its small mass and a nonlinear elastic spring of the Duffing oscillator, it also comprises a purely hysteretic and a linear elastic spring of potentially negative stiffness. The Bouc–Wen model is used to describe the force produced by both the purely hysteretic and linear elastic springs. The hysteretic nonlinear energy sink protects the primary system through the energy pumping mechanism which transfers energy from the primary system and dissipates it in the hysteretic nonlinear energy sink. Three nonlinear equations of motion describe the resulting two-degree-of-freedom system response. The parameters of the system to be considered as uncertain are the natural frequency of the primary system and the hysteretic nonlinear energy sink linear elastic spring, which follow a normal distribution. A reliability analysis is then performed to evaluate the robustness of the coupled system in the presence of uncertainty. Specifically, the reliability index is calculated based on first passage probabilities of distinct dissipation energy level crossings using the Monte Carlo method. Several examples are examined considering various levels of initial input energy, and useful conclusions are drawn concerning the influence of uncertainty in the system robustness.