Quartic Double Well Research Articles

Tunneling Probability of Quantum Wavepacket in Time-Dependent Potential Well

Quantum tunneling of particles plays an important role in many chemical reactions. Studying quantum tunneling in time-dependent potential wells is tricky since most of the available solutions of time-dependent potential wells are coupled to specific properties of the Hamiltonian. Here, we investigate the tunneling probability of a quantum wavepacket in time-dependent potential well by using the split operator method. This numerical method can give us an overview of the tunneling probability of a quantum wavepacket for any temporal change in the shape of the potential well. We study a time-dependent potential well model evolving from symmetric to asymmetric quartic double well since quartic potential wells resemble certain practically available potential well models.Quanta 2024; 13: 11–19.

Manifestation of classical instability in the quantum density of states of a double well potential

Within Bohr's correspondence principle, quantum theory should reproduce the classical world when ħ→0. In practice, obtaining the complete spectrum is challenging since the discrete energies come close to each other before passing to continuum. However it is possible to identify signatures that the spectrum is approaching the continuum limit. In this work, for ħ→0, we focus our attention on the spectrum of the 1D quartic double well and find signatures of the classical instability in the quantum spectrum near the critical energy, and explore how the tunneling phenomenon remains only at energies close to this critical energy.

Toppling Pencils-Macroscopic Randomness from Microscopic Fluctuations.

We construct a microscopic model to study discrete randomness in bistable systems coupled to an environment comprising many degrees of freedom. A quartic double well is bilinearly coupled to a finite number N of harmonic oscillators. Solving the time-reversal invariant Hamiltonian equations of motion numerically, we show that for , the system exhibits a transition with increasing coupling strength from integrable to chaotic motion, following the Kolmogorov-Arnol’d-Moser (KAM) scenario. Raising N to values of the order of 10 and higher, the dynamics crosses over to a quasi-relaxation, approaching either one of the stable equilibria at the two minima of the potential. We corroborate the irreversibility of this relaxation on other characteristic timescales of the system by recording the time dependences of autocorrelation, partial entropy, and the frequency of jumps between the wells as functions of N and other parameters. Preparing the central system in the unstable equilibrium at the top of the barrier and the bath in a random initial state drawn from a Gaussian distribution, symmetric under spatial reflection, we demonstrate that the decision whether to relax into the left or the right well is determined reproducibly by residual asymmetries in the initial positions and momenta of the bath oscillators. This result reconciles the randomness and spontaneous symmetry breaking of the asymptotic state with the conservation of entropy under canonical transformations and the manifest symmetry of potential and initial condition of the bistable system.

Quartic double-well system modulation for under-damped stochastic resonance tuning

In this paper a novel methodology for stochastic resonance (SR) tuning, based on the classical quartic double-well (QDW) system modulation is proposed. The QDW modulation is accomplished by finding the optimal stable focus separation, well depth and damping parameters of the Duffing system, enhancing a weak signal buried in additive white Gaussian (AWG) noise. Hence, direct parameter tuning SR through geometric QDW system modulation is developed unlike classical SR, where linear and non-linear stiffness parameters, indirectly, modulates the QDW system. To evaluate the harmonic content of the SR response, signal-to-noise and distortion ratio (SINAD) was used thus, optimal SR response was chosen under maximum SINAD criteria. Signal-to-noise ratio (SNR) was also used to compare our methodology with state-of-the-art weak signal detection (WSD) methods of autocorrelation and discrete wavelet transform (DWT). After applying the proposed methodology on three noisy signals with different input SNR, results showed an important signal enhancement of 50dB in SNR gain, for all cases, far above the results obtained with other methods compared. Moreover, signal enhancement with the lowest harmonic distortion was achieved with SR, compared with autocorrelation and DWT, demonstrating the feasibility of SR for WSD in highly noisy environments.

Quantum control in a double well with symmetric or asymmetric driving

We investigate the coherent control of a single particle held in a quartic double well with a symmetric or asymmetric driving and exhibit time evolutions of relative probability of the particle in one of the double-well wells. It is shown analytically and numerically that the coherent destruction of tunneling could occur only for the symmetric intense driving, and application of the asymmetric sawtooth driving leads to an increase in the tunneling rate. The results agree with the recent experimental data reported by Kierig et al. [Phys. Rev. Lett. 100, 190405 (2008)]. We also exhibit the effect of multiple photon resonances on the tunneling through a dc field. Finally, we demonstrate a connection between the classically chaotic barrier crossing and quantum tunneling numerically.

Semiquantal analysis of adiabatic hydrogen transfer rate

The reaction rate of adiabatic proton/hydrogen/hydride (H) transfers in condensed phase is examined by combining the semiquantal time-dependent Hartree theory and the multidimensional transition state theory, which takes into account the zero-point effect and the dynamical modulation of the wavepacket width in the adiabatic transfer regime. By applying the theory to a model potential consisting of a quartic double well coupled linearly and quadratically (symmetrically) to external degrees of freedom, a set of compact analytical formulas was derived for the adiabatic H transfer rate. The analysis suggests that the kinetic isotope effect on the H transfer rate may exhibit a maximum as a function of the coupling strength to the external degrees of freedom measured by the reorganization energy.

Dynamical singularities for complex initial conditions and the motion at a real separatrix

This work investigates singularities occurring at finite real times in the classical dynamics of one-dimensional double-well systems with complex initial conditions. The objective is to understand the relationship between these singularities and the behavior of the systems for real initial conditions. An analytical treatment establishes that the dynamics of a quartic double well system possesses a doubly infinite sequence of singularities. These are associated with initial conditions that converge to those for the real separatrix as the singularity time becomes infinite. This confluence of singularities is shown to lead to the unstable behavior that characterizes the real motion at the separatrix. Numerical calculations confirm the existence of a large number of singularities converging to the separatrix for this and two additional double-well systems. The approach of singularities to the real axis is of particular interest since such behavior has been related to the formation of chaos in nonintegrable systems. The properties of the singular trajectories which cause this convergence to the separatrix are identified. The hyperbolic fixed point corresponding to the potential energy maximum, responsible for the characteristic motion at a separatrix, also plays a critical role in the formation of the complex singularities by delaying trajectories and then deflecting them into asymptotic regions of space from where they are directly repelled to infinity in a finite time.

Characterization of Threshold-Induced Phenomena in Deterministic Driven Devices

In the field of measurement nonlinear behavior is typical of a large set of sensors (mechanical, electrical, pneumatic, and so on). In several devices, SQUID and piezoelectric sensors being very well-known examples, the nonlinearity is strictly correlated to the hysteresis phenomenon, which can be described by introducing the quartic double well (QDW) potential in the system model. The analysis addresses the influence of potential characteristics on the performance of hysteretic measuring devices. A Brownian system with a QDW potential is used as a general prototype of hysteretic phenomena. This paper applies to the general class of hysteretic devices forced with low-amplitude signals and gives suitable outlines for the design of several measuring devices.

Properties of diffusive systems near a saddle point: application to a quartic double well

This paper aims at the analysis of diffusive properties of unidimensional mechanical systems in the environment of maxima and minima of the potential. It begins with a study of the properties of the singular solutions of the Hamilton–Jacobi–Yasue equation in the above-mentioned environment, in both strong or very small frictional forces. For the quartic symmetrical double-well potential, approximate solutions are found for local validity and the diffusion operator is then calculated in the limits of deep wells and small temperature, the regime being supposed to be aperiodic, with high or moderate values of frictional coefficient. This equation is proved to be nonunique. This operator is then reduced to second order by imposing suitable boundary conditions. Thus an appropriate eigenvalue equation is obtained to describe stationary states in the environment of extremal points of the potential energy function. The main interest of this work relies upon the fact that transition times between wells mainly depend upon fluctuations near the saddle point.

Many-bodysymbolic dynamics of a classical oscillator chain

We study a certain type of the celebrated Fermi-Pasta-Ulam particlechain, namely the inverted FPU model, where the interparticle potentialhas the form of a quartic double well. Numerical evidence is given insupport of a simple symbolic description of dynamics (in the regime ofsufficiently high potential barrier between the wells) in terms of an(approximate) Markov process. The corresponding transition matrix isformally identical to a ferromagnetic Heisenberg quantum spin-1/2chain with long-range coupling, whose diagonalization yields accurateestimates for a class of time correlation functions of the model.

Decoherence in a classically chaotic quantum system: entropy production and quantum-classical correspondence.

We study the decoherence process for an open quantum system that is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators). We carefully analyze the time dependence of the rate of entropy production showing that it has two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength); for longer times (but before equilibration) it is fixed by dynamical properties of the system (and is related to the Lyapunov exponent). The nature of the transition time between both regimes is investigated and the issue of quantum to classical correspondence is addressed. Finally, the impact of the interaction with the environment on coherent tunneling is analyzed.

Hysteresis shaping in stochastic driven systems

A large set of sensors (SQUID, piezoelectric, etc.) show a nonlinear calibration curve due to the hysteresis phenomenon. Moreover, sensors usually work in the presence of noise that very often compromises the sensitivity of the device in the low-level range. A useful way to investigate the behavior of such a complex system is to model the hysteresis with the well-known quartic double well (QDW) potential subjected to noise-driven fluctuations along with a periodic signal. The authors have previously discussed the analysis of such a system in the case when a deterministic signal is forced in. In this paper, an analysis of the QDW potential subjected to both dissipation and stochastic fluctuation along with an external deterministic forcing signal is performed to investigate the possibility of optimizing the system sensitivity, by a suitable shaping of the hysteresis, as the forcing term characteristics change. This study can help to improve the feature of a general class of devices forced with a low-amplitude signal, located in a noisy environment. Moreover, suitable issues useful during the design phase of the devices can be obtained.

Decoherence and the rate of entropy production in chaotic quantum systems

We show that for an open quantum system which is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators) the rate of entropy production has, as a function of time, two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength). For longer times (but before equilibration) there is a regime where the entropy production rate is fixed by the Lyapunov exponent. The nature of the transition time between both regimes is investigated.

Tunnel splittings and chaotic transport in periodically driven bistable systems

We study the crossover from coherent transport by tunneling to diffusive transport through classicaly chaotic phase-space regions, and the corresponding changes in the spectrum. The harmonically driven quartic double well is used as a working model and treated numerically in the framework of the Floquet formalism. We find a transition from exponentially small tunnel splittings to level separations of the order of the mean level distance, when the corresponding classical dynamics becomes dominated by chaotic diffusion.

Tunneling and the onset of chaos in a driven bistable system.

We study the interplay between coherent transport by tunneling and diffusive transport through classically chaotic phase-space regions, as it is reflected in the Floquet spectrum of the periodically driven quartic double well. The tunneling splittings in the semiclassical regime are determined with high numerical accuracy, and the association of the corresponding doublet states to either chaotic or regular regions of the classical phase space is quantified in terms of the overlap of the Husimi distribution with the chaotic layer along the separatrix. We find a strong correlation between both quantities. They show an increase by orders of magnitude as the chaotic diffusion between the wells starts to dominate the classical dynamics. We discuss semiclassical explanations for this correlation.

Quantum dynamics via mobile basis sets: The Dirac variational principle

We propose a variation on the wave-packet interpretation of quantum mechanics introduced by Heller, here using the Dirac variational principle to optimize the time evolution of all the parameters that enter our basis set. We solve the general one-dimensional problem, and test our method on the Morse oscillator and the symmetric quartic double well.

Energy spectrum of the potential V=ax2+x4

Suitable sequences of quasi-exactly solvable Hamiltonians are shown to provide stringent upper bounds to the energy eigenvalues of the bound state potential V=ax2+x4. Procedures to convert these bounds into even further improved energy estimates are developed. For the quartic anharmonic oscillator (a)0) case a simple argument is provided to indicate that the conventional small-parameter energy expansion does not converge as a Taylor series. An accurate closed-form parameterization of the entire quartic (a=0) spectrum is noted. The energy difference between the lowest-lying levels of a quartic double well (a(0) is satisfactorily recovered and for deep wells a useful expression is deduced for it empirically.

Quantum-classical crossover of the transition rate in the damped double well

The author studies the barrier-crossing rate Gamma for a particle in a symmetric double well coupled to a sluggish bath; the aim is to justify the quantum simulation procedure used in related work on the diffusion of hydrogen in metals. He uses path-integral methods to discuss the transition between the low-temperature tunnelling and the high-temperature barrier-jumping regimes, and suggests a unifying formula valid for both, which relates Gamma to the probability of finding the centroid of the quantum path at the barrier top. Numerical results for Gamma are presented for the case of the quartic double well.