Strongly interacting quantum systems are at the heart of many physical processes, ranging from photosynthesis to quantum information. We demonstrate that many characteristic features of strongly coupled systems can be derived classically for a simple coupled harmonic oscillator. This model system is used to derive frequency splittings and to analyze adiabatic and diabatic transitions between the coupled states. A classical analog of the Landau–Zener formula is derived. The classical analysis is intuitive and well suited for introducing students to the basic concepts of strongly interacting systems.
Strongly interacting quantum systems are at the heart of many physical processes, ranging from photosynthesis to quantum information. We demonstrate that many characteristic features of strongly coupled systems can be derived classically for a simple coupled harmonic oscillator. This model system is used to derive frequency splittings and to analyze adiabatic and diabatic transitions between the coupled states. A classical analog of the Landau–Zener formula is derived. The classical analysis is intuitive and well suited for introducing students to the basic concepts of strongly interacting systems.
Abstract Coupled systems subject to dissipation exhibit two different regimes known as weak coupling and strong coupling. Two damped coupled harmonic oscillators (CHOs) constitute a model system where the key features of weak and strong coupling can be identified. Several of these features are common to classical and quantum systems, as a number of quantum-classical correspondences have shown. However, the condition defining the boundary between weak and strong coupling is distinct in classical and quantum formalisms. Here we describe the origin of two widely used definitions of strong coupling. Using a classical CHO model, we show that energy exchange cycles and avoided resonance crossings signal the onset of strong coupling according to one criterion. From the classical CHO model we derive a non-Hermitian Hamiltonian describing open quantum systems. Based on the analytic properties of the Hamiltonian, we identify the boundary between weak and strong coupling with a different feature: a non-Hermitian degeneracy known as the exceptional point. For certain parameter ranges the classical and quantum criterion for strong coupling coincide; for other ranges they do not. Examples of systems in strong coupling according to one or another criterion, but not both, are illustrated. The framework here presented is suitable for introducing graduate or advanced undegraduate students to the basic properties of strongly coupled systems, as well as to the similarities and subtle differences between classical and quantum descriptions of coupled dissipative systems.
The time-dependent behavior of a two-level system interacting with a quantum oscillator system is analyzed in the case of a coupling larger than both the energy separation between the two levels and the energy of the quantum oscillator ($\ensuremath{\Omega}<\ensuremath{\omega}<\ensuremath{\lambda}$, where $\ensuremath{\Omega}$ is the frequency of the transition between the two levels, $\ensuremath{\omega}$ is the frequency of the oscillator, and $\ensuremath{\lambda}$ is the coupling between the two-level system and the oscillator). Our calculations show that the amplitude of the expectation value of the oscillator coordinate decreases as the two-level system undergoes the transition from one level to the other, while the transfer probability between the levels is staircaselike. This behavior is explained by the interplay between the adiabatic and nonadiabatic regimes encountered during the dynamics with the system acting as a quantum counterpart of the Landau-Zener model. The transition between the two levels occurs as long as the expectation value of the oscillator coordinate is driven close to zero. On the contrary, if the initial conditions are set such that the expectation values of the oscillator coordinate are far from zero, the system will remain locked on one level.
They have analytically tested several quantum transitions in molecular dynamics. In addition, they have introduced applications of theoretical formalism in physics to economics. First, they have completed the semiclassical theory of nonadiabatic transitions in the exponential potential model. They treated a two-state quantum system, which commonly appears as a subsystem in many-state problems. They were concerned with one-dimensional Schroedinger equation, although the type or semiclassical approach which they used can be applied to higher-dimension potential surfaces by means of Fourier transform. In particular, they solved the case of repulsive exponentially growing potentials, that is the forward-back scattering. They re-expressed the nonadiabatic transition matrix, which describes the transition between the two states for one-direction passage, in terms of contour integrals in the complex coordinate plane. The integrands are adiabatic momenta derived from the eigenvalues of potential matrix. Such type of concept has the advantage that it can be applied to other models than just the exponential potential model. Based on this, they have obtained an analytical tool for describing one-transition point class of potential curve systems. Examples are Rosen-Zener-Demkov model and Landau-Zener-Stueckelberg model. The formalism, which they contributed to, is useful in describing physical processes such as the charge transfer, and it is suitable for shaping external fields or modulating their frequencies in order to achieve a desired state or a product with the theoretical 100% probability. Second, they have derived the first exact analytical solution for a diabatically avoided crossing model. The physical system is described with one-dimensional coupled potential well and barrier which do not intersect in the whole coordinate range. On the basis of special mathematical functions they derived the complete quantum mechanical solution for the wave functions and the transmission coefficient. They also analyzed this model semiclassically and compared it with the exact results. This comparison yields interesting implications on the validity and accuracy of semiclassical approximation. In particular, they solved the case of dramatically avoided crossing with exponentially steep potential curves, and obtained the probability of one-way passage through the interaction region. The overall behavior corresponds to the probability of tunneling through the single potential barrier. In addition, there are complete reflection resonances at discrete energies, resulting from the trapping of a wave in the potential well. The formalism of diabatically avoided crossing model is useful in laser control of molecular reactions. This is because the potential energy curves in realistic molecular systems can be shifted by the photon energy; the curves are coupled by the field intensity via the dipole moment. That is why the new topology of potential energy curves, which was so far broadly omitted, becomes practically interesting due to the development in lasers techniques. Third, they treated several two-state quantum systems common in molecular physics. The semiclassical conditions for the complete transmission and the complete reflection in one-dimensional scattering were given analytically. They discovered the possibility of these phenomena in energy ranges in which the complete reflection or the complete transmission are not commonly expected to occur. In particular, they have made use of the broadly shared assumption that the transitions between the states are localized phenomena. This enables them to separate the whole coordinate region into distinct subsets and to analyze the physics in each subset separately. Applying this approach, they obtained a fully algebraic problem, which can be treated e.g. by diagrammatic techniques. Let them also mention that the complete transmission or complete reflection conditions simplify the wave function, and hence the mathematical solution. The formalism which they used while deriving the semiclassical conditions for the resonant phenomena mentioned above can be further generalized. This will probably yield a whole spectrum of tools in the laser control of molecular processes. Fourth, they developed a powerful technique for solving integral equations with singular kernel. One of the physical formulations of scattering is the time-independent approach based on the Lippmann-Schwinger equation for the Green operator. They have developed a detailed technique how to treat the singularity in this equation. In particular, any scattering problem can be divided into two parts. The first one is the geometry of the system and the static interaction of its components. The second one is more dynamical: the singular equation for the Green operator including the variable energy of the scattering. They fix the most costly calculations in the static part. Then they solve the singularity analytically and thus obtain the cross section as a function of energy. The so far common approach is quite opposite: to fix the singularity first and then to recalculate heavily all the static part for any new value of energy. Obviously, their approach saves a considerable amount of computational time; therefore it is a good standpoint for treating large systems. The formalism above was proved to be extremely efficient taking the example of dissociative recombination H2+ + e- → H + H* because this is a diatomic system, quantum chemical data are easily available in the literature and they could demonstrate the superiority of their approach to the other approaches. Fifth, they suggested applications of basic physical concepts to economics. They introduced aggregate quantities in financial markets and interpreted these quantities as potential energy surfaces. They also formalized interactions between cyclic processes in terms of state to state transitions. Summarizing, they treated the exponential potential model of nonadiabatic transitions, introduced a new exactly solvable model of diabatically avoided crossing, discussed the semiclassical conditions for complete transmission and complete reflection from the viewpoint of laser control, contributed to scattering calculations, and introduced some interdisciplinary applications.
In this chapter we introduce the basic concept of a quantum harmonic oscillator, and how such oscillators couple to their environment. Our purpose is not to exhaustively cover all aspects of such systems, but rather to provide the foundation material required to treat the quantum mechanics of mechanical resonators and optical cavities – and their interactions – in later chapters. The quantum fluctuation-dissipation theorem is derived relating the power spectral density of environmental forcing of an oscillator to the rate of energy decay from the oscillator into the environment. This is in direct analogy to the familiar fluctuation-dissipation theorem in classical physics. However, in contrast to the classical case, the noncommutation of position and momentum introduces an asymmetry between the positive and negative frequency components of the power spectrum. This provides a natural definition for the effective temperature of the oscillator. The quantum Langevin approach is introduced as a method to model the open dynamics of quantum systems, both with and without recourse to the rotating wave approximation.
A comprehensive investigation of the transition dynamics of a level-crossing system with quantum dissipation is carried out for the Landau-Zener model coupled with a system of phonons. Analytical study by the formal perturbation expansion series with respect to the off-diagonal matrix element shows that the transition dynamics is characterized by the competition between the energy fluctuation and the energy dissipation. Closed expressions of the transition rate are derived for various limiting situations in a unified way. In the case of strong decoherence, a formula of the transition rate is obtained, which covers the high-temperature limit and the low-temperature, strong-coupling limit. Numerical calculation by utilizing the damping hyperoperator technique is performed, which clarifies the overall features of the time evolution of the system.
This textbook provides a concise yet comprehensive introduction to the principles, concepts, and methods of quantum mechanics. It covers the basic building blocks of quantum mechanics theory and applications, illuminated throughout by physical insights and examples of quantum mechanics, such as the one-dimensional eigen-problem, the harmonic oscillator, the Aharonov-Bohm effect, Landau levels, the hydrogen atom, the Landau-Zener transition and the Berry phase. This self-contained textbook is suitable for junior and senior undergraduate students, in addition to advanced students who have studied general physics (including classical mechanics, electromagnetics, and atomic physics), calculus, and linear algebra. Key features: Presents an accessible and concise treatment of quantum mechanics Contains a wealth of case studies and examples to illustrate concepts Based off the author's established course and lecture notes