Approach to the quantum evolution for underdamped, critically damped, and overdamped driven harmonic oscillators using unitary transformation

Exact solution of the Schrödinger equation is derived for underdamped, critically damped, and overdamped harmonic oscillators with a driving force. A unitary operator transforming Hamiltonian into a simple form is introduced. The transformed Hamiltonian, represented in terms of a modified frequency ω, is identical with the Hamiltonian of the standard harmonic oscillator for the underdamped oscillator, with the Hamiltonian of a free particle for the critically damped oscillator, and with the Hamiltonian of a system with a harmonic parabolic potential for the overdamped oscillator. The eigenvalues of underdamped oscillator are discrete while those of the critically damped and the overdamped oscillators are continuous.     

Structure of resonances and formation of stationary points in symmetrical chains of bilinear oscillators

Dynamics of strongly nonlinear systems can in many cases be modelled by bilinear oscillators, which are the oscillators whose springs have different stiffnesses in compression and tension. This underpins the analysis of a wide range of phenomena, from oscillations of fragmented structures, connections and mooring lines to deformation of geological media. Single bilinear oscillators were studied previously and the presence of multiple resonances both super- and sub-harmonic was found. Less attention was paid to systems of multiple bilinear oscillators that describe many natural and engineering processes such as for example the behaviour of fragmented solids. Here we fill this gap concentrating on the simplest case – 1D symmetrical chains of bilinear oscillators. We show that the presence and structure of resonances in a symmetric chain of bilinear oscillators with fixed ends depends upon the number of oscillating masses. Two elementary chains act as the basic ones: a single mass bilinear chain (a mass connected to the fixed points by two bilinear springs) that behaves as a linear oscillator with a single resonance and a two mass chain that is a coupled bilinear oscillator (two masses connected by three bilinear springs). The latter has multiple resonances. We demonstrate that longer chains either do not have resonances or get decomposed, in the resonance, into either the single mass or two mass elementary chains with stationary masses in between. The resonance frequencies are inherited from the basic chains of decomposition. We show that if the number of masses is odd the chain can be decomposed into the single mass bilinear chains separated by stationary masses. It then inherits the resonances of the single mass bilinear chain. The chains with the number of masses minus 2 divisible by 3 can be decomposed into the two mass bilinear chains separated by stationary masses and inherit the resonances of the two mass chains. The chains whose lengths satisfy both criteria (such as chains with 5, 11, 17 … masses) allow both types of resonances.     

Resonance tongues in a system of globally coupled FitzHugh-Nagumo oscillators with time-periodic coupling strength

We investigate the dynamics of a population of globally coupled FitzHugh-Nagumo oscillators with a time-periodic coupling strength. While for synchronizing global coupling, the in-phase state is always stable, the oscillators split into several cluster states for desynchronizing global coupling, most commonly in two, irrespective of the coupling strength. This confines the ability of the system to form n:m locked states considerably. The prevalence of two and four cluster states leads to large 2:1 and 4:1 subharmonic resonance regions, while at low coupling strength for a harmonic 1:1 or a superharmonic 1:m time-periodic coupling coefficient, any resonances are absent and the system exhibits nonresonant phase drifting cluster states. Furthermore, in the unforced, globally coupled system the frequency of the oscillators in a cluster state is in general lower than that of the uncoupled oscillator and strongly depends on the coupling strength. Periodic variation of the coupling strength at twice the natural frequency causes each oscillator to keep oscillating with its autonomous oscillation period.     

外力驱动下耦合弹簧振子的法诺共振

法诺共振是物理体系中普遍存在的一种非对称共振现象,它最早起源于量子物理,其微观图像是原子谱线中窄的分离态与宽的连续态之间的相干干涉.本文利用经典力学体系中两个弹簧振子的耦合,使其中一个弹簧振子受到周期性外力的驱动,成功类比了量子力学中的法诺共振现象.通过分析每个弹簧振子的动力学方程,严格求解它们的振动公式,从而得到每个弹簧振子的振幅和位相与外部驱动力频率之间的关系.结果表明,耦合体系中受外力驱动的那个弹簧振子既可以发生非对称的法诺共振,又可以发生对称的洛仑兹共振,而另一个弹簧振子只能发生洛仑兹共振.本文的推导与分析能够使读者更好地理解法诺共振现象及其激发条件.     

Accurate transport cross sections for the Lennard-Jones potential

It is well-known that in open quantum systems resonances can coalesce at an exceptional point, where both the energies and the wave functions coincide. In contrast to the usual behaviour of the scattering amplitude at one resonance, the coalescence of two resonances invokes a pole of second order in the Green’s function, in addition to the usual first order pole. We show that the interference due to the two pole terms of different order gives rise to patterns in the scattering cross section which closely resemble Fano-Feshbach resonances. We demonstrate this by extending previous work on the analogy of Fano-Feshbach resonances to classical resonances in a system of two driven coupled damped harmonic oscillators.     

Experimental circuit realization of zero-dispersion nonlinear resonance via the generic input-output analysis

When a nonlinear oscillator is driven by an external force which is calculated on the basis of the zero-dispersion nonlinear resonance (ZDNR) theory, it can be transparent, i.e., the output of the response system is the same as the magnitude of driving force without any dispersion or distortion. In fact, this is a unique phenomenon, called zero-dispersion nonlinear resonance (ZDNR). Herein, we used the generic input-output analysis to realize the ZDNR circuit experiment of two unidirectional coupling self-excited oscillators. The experimental observations also show good agreement with numerical simulations.     

Irreversible quantum graphs

Irreversibility is introduced to quantum graphs by coupling the graphs to a bath of harmonic oscillators. The interaction which is linear in the harmonic oscillator amplitudes is localized at the vertices. It is shown that for sufficiently strong coupling, the spectrum of the system admits a new continuum mode which exists even if the graph is compact, and a single harmonic oscillator is coupled to it. This mechanism is shown to imply that the quantum dynamics is irreversible. Moreover, it demonstrates the surprising result that irreversibility can be introduced by a ‘bath’ which consists of a single harmonic oscillator.     

Features of Phase Bunching of Weakly Excited Oscillators under the Action of an Alternating Force

Using the Duffing equation, we consider resonance bunching of oscillators with small initial amplitudes under the action of an alternating force. The relative phases of the oscillators and of the force are calculated analytically. It is shown that all the initially unphased oscillators acquire close phases and, consequently, close amplitudes over a short time comparable with the period of fast oscillations. The amplitudes can be calculated using simple equations of an oscillator that is in phase with the external force. A physical explanation of the bunching is given.     

Enhancing energy harvesting by coupling monostable oscillators

The performance of a ring of linearly coupled, monostable nonlinear oscillators is optimized towards its goal of acting as energy harvester – through piezoelectric transduction – of mesoscopic fluctuations, which are modeled as Ornstein-Uhlenbeck noises. For a single oscillator, the maximum output voltage and overall efficiency are attained for a soft piecewise-linear potential (providing a weak attractive constant force) but they are still fairly large for a harmonic potential. When several harmonic springs are linearly and bidirectionally coupled to form a ring, it is found that counter-phase coupling can largely improve the performance while in-phase coupling worsens it. Moreover, it turns out that few (two or three) coupled units perform better than more.     

Universal and deterministic manipulation of the quantum state of harmonic oscillators: a route to unitary gates for Fock state qubits

We present a simple quantum circuit that allows for the universal and deterministic manipulation of the quantum state of confined harmonic oscillators. The scheme is based on the selective interactions of the referred oscillator with an auxiliary three-level system and a classical external driving source, and enables any unitary operations on Fock states, two by two. One circuit is equivalent to a single qubit unitary logical gate on Fock states qubits. Sequences of similar protocols allow for complete, deterministic, and state-independent manipulation of the harmonic oscillator quantum state.     

Systems of oscillators with statistical energy exchange in collisions

Classical transition probabilities are derived for the exchange of energy between sets of weakly coupled harmonic oscillators, subject to the assumption that energy is ‘statistically redistributed’ in all collisions. In the case of single oscillators the result is particularly simple and an explicit solution of the relaxation equation for such a system is derived. A ‘mean first passage time’ for the dissociation of truncated harmonic oscillators is also obtained using Widom's theory and the limiting case of equilibrium reaction is discussed.

The same approach is sketched briefly for the case of quantized systems in which statistical redistribution of energy is allowed.

Within the general scope of the model, these transition probabilities represent the most efficient conceivable coupling between system and heat bath.     

Linear and nonlinear resonance features of an erbium-doped fibre ring laser under cavity-loss modulation

The continuous-wave output of a single-mode erbium-doped fibre ring laser when subjected to cavity-loss modulation is found to exhibit linear as well as nonlinear resonances. At sufficiently low driving amplitude, the system resembles a linear damped oscillator. At higher amplitudes, the dynamical study of these resonances shows that the behaviour of the system exhibits features of a nonlinear damped oscillator under harmonic modulation. These nonlinear dynamical features, including harmonic and subharmonic resonances, have been studied experimentally and analysed with the help of a simple time-domain and frequency-domain information obtained from the output of the laser. All the studies are restricted to the modulation frequency lying in a regime near the relaxation oscillation frequency.     

非平衡功与自由能差的关系

以一个受迫谐振子为对象证明了Jarzynski等式,即两平衡态的自由能差可以用它们之间非平衡功的指数平均来表示,这里两个态的转换是在有限时间完成的.当外部驱动力是一个周期信号时,发现外界对系统所做的广义功存在一个共振现象.     

Stochastic resonance of a damped oscillator with frequency fluctuation driven by a periodic external force

<正>Considering a damped linear oscillator model subjected to a white noise with an inherent angular frequency and a periodic external driving force,we derive the analytic expression of the first moment of output response,and study the stochastic resonance phenomenon in a system.The results show that the output response of this system behaves as a simple harmonic vibration,of which the frequency is the same as the external driving frequency,and the variations of amplitude with the driving frequency and the inherent frequency present a bona fide stochastic resonance.     

Spectroscopy of a driven solid-state qubit coupled to a structured environment

We study the asymptotic dynamics of a driven spin-boson system where the environment is formed by a broadened localized mode. Upon exploiting an exact mapping, an equivalent formulation of the problem in terms of a quantum two-state system (qubit) coupled to a harmonic oscillator which is itself Ohmically damped, is found. We calculate the asymptotic population difference of the two states in two complementary parameter regimes. For weak damping and low temperature, a perturbative Floquet-Born-Markovian master equation for the qubit-oscillator system can be solved. We find multi-photon resonances corresponding to transitions in the coupled quantum system and calculate their line-shape analytically. In the complementary parameter regime of strong damping and/or high temperatures, non-perturbative real-time path integral techniques yield analytic results for the resonance line shape. In both regimes, we find very good agreement with exact results obtained from a numerical real-time path-integral approach. Finally, we show for the case of strong detuning between qubit and oscillator that the width of the n-photon resonance scales with the nth Bessel function of the driving strength in the weak-damping regime.     

Unraveling the primary mechanisms leading to synchronization response in dissimilar oscillators

We study how the phenomenon of response to synchronization arises in sets of pulse-coupled dissimilar oscillators. One of the sets is constituted by oscillators that can easily synchronize. Conversely, the oscillators of the other set do not synchronize. When the elements of the first set are not synchronized, they induce oscillation death in the constituents of the second set. By contrast, when synchronization is achieved in oscillators of the first set, those of the second set recover their oscillatory behavior and thus, responding to synchronization. Additionally, we found another interesting phenomenon in this type of systems, namely, a new control of simultaneous firings in a population of similar oscillators attained by means of the action of a dissimilar oscillator.     

Theoretical Study of Millimeter Wave Harmonic Oscillator Stabilized with a Transmission Cavity

An equivalent circuit model of millimeter wave second harmonic oscillator stabilized with a transmission cavity has been proposed for constructing analytical formulations between performance parameters of the oscillator and parameters of the circuit. The model consists of an equivalent circuit of fundamental wave and that of second harmonic wave. Each of the circuits comprises circuit models of main cavity, transmission waveguide, and transmission cavity. Absorbing material placed between the transmission waveguide and the transmission cavity can suppress additional resonances originated from transmission cavity. The behavior of the second harmonic oscillator can be effectively described by the circuit model. Furthermore, based on this model, mechanical tuning characteristics have been studied at first, and then analytical formulas for quality factor and efficiency depending on circuit parameters have been derived. The circuit parameters can be conveniently extracted by electromagnetic field simulation. Hence the formulas exhibit both compact form and enough accuracy. Thereafter, general rules of performance parameters varying with circuit parameters have been deduced for the harmonic oscillators. Then some design considerations have been derived according to the corresponding analysis. The equivalent circuit model is useful for designing and adjusting millimeter wave second harmonic stabilizing oscillator with a transmission cavity.     

Heisenberg-limited Measurements and Sub-Planck Structures in Phase Space

It is shown how sub-Planck phase-space structures can be used to achieve Heisenberg-limited sensitivity in weak force measurements. Nonclassical states of harmonic oscillators, such as superpositions of coherent states, are shown to be useful for the measurement of weak forces that cause translations or rotations in phase space, which is done by entangling the quantum oscillator with a two-level system. This method is closely related to the Loschmidt echo techniques employed in nuclear magnetic resonance experiments. Implementations of this strategy in cavity QED and ion traps are described.     

周期外力对频率涨落的过阻尼谐振子所作的功和能量随机共振

研究了周期外力对频率涨落的过阻尼谐振子系统作功的特点. 结果揭示了瞬时功率随时间的周期变化出现不对称性. 研究结果还揭示周期外力一个周期对系统所做的功随乘法噪声强度的变化出现非单调行为, 系统是否出现能量随机共振与抑制并存, 由乘法噪声与加法噪声之间互关联的符号决定. 关键词： 过阻尼谐振子 频率涨落 周期外力作功 能量随机共振与抑制     

Nonlinear resonance phenomena of a doped fibre laser under cavity-loss modulation: Experimental demonstrations

Our experiments with an erbium-doped fibre ring laser (CW, single transverse mode and multiaxial mode) with an intracavity LiNbO₃ electro-optic modulator (EOM ) display the characteristic features of a nonlinear oscillator (e.g., harmonic and period-2 sub-harmonic resonances ) when the EOM driver voltage is modulated periodically. Harmonic resonance leads to period-1 bistability and hysteresis. Inside the period-2 sub-harmonic resonance region, the laser exhibits Feigenbaum sequence and generalized bistability.