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Physical review. B, Condensed matter

Two-level system with nonlinear coupling

Pedro de Vries, Ad Lagendijk, Hans A. De Raedt

Published Apr 15, 1987• 8 citations
Publisher

Abstract

We present a numerical study of a two-level system coupled quadratically to a harmonic oscillator. The possibility of an instability as a function of the coupling strength is studied. The model might be relevant for the behavior of quadratically coupled polarons in solids. In the adiabatic limit the thermodynamical properties of the system can be calculated exactly over the whole coupling range. In addition, we discuss computations based on a discrete path-integral representation for the partition function. In the adiabatic limit there is a transition as a function of the coupling strength from the weak-coupling regime to a state where the oscillator mode is soft.

DOI
10.1103/physrevb.35.5425

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We present a numerical study of a two-level system coupled quadratically to a harmonic oscillator. The possibility of an instability as a function of the coupling strength is studied. The model might be relevant for the behavior of quadratically coupled polarons in solids. In the adiabatic limit the thermodynamical properties of the system can be calculated exactly over the whole coupling range. In addition, we discuss computations based on a discrete path-integral representation for the partition function. In the adiabatic limit there is a transition as a function of the coupling strength from the weak-coupling regime to a state where the oscillator mode is soft.
We present a numerically exact study of a polaron with quadratic coupling to the oscillator displacement, or X2{X}^{2}X2 polaron, using two alternative methodological developments. Our results cover both antiadiabatic and adiabatic regimes and the entire range of electron-phonon coupling g2{g}_{2}g2​, from the system's stability threshold at attractive ${g}_{2}=\ensuremath{-}1$ to arbitrary strong repulsion at ${g}_{2}\ensuremath{\gg}1$. The key properties of X2{X}^{2}X2 polarons prove dramatically different from their linear counterparts. They (i) are insensitive even to large quadratic coupling except in the antiadiabatic limit near the threshold of instability at attraction, (ii) depend only on the adiabatic ratio but are insensitive to the electron dispersion and dimension of space, and (iii) feature weak lattice deformations even at the instability point. Our results are of direct relevance to the properties of electrons at low densities in polar materials, including recent proposals for their superconducting states.
The thermodynamics of a two-level system coupled to a boson field is investigated by means of a functional-integral method. The model can be used to describe the motion of a small polaron between two sites on a molecule, or a spin-1/2 magnetic moment coupled to a phonon bath. The model is also relevant to the problem of dissipation in macroscopic quantum phenomena. We use a functional-integral method to investigate a two-level electronic system coupled linearly to a harmonic oscillator. The density matrix is cast as a functional integral over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/kB{\mathit{k}}_{\mathit{B}}kB​T. The integration variables are c numbers for the oscillator and Grassmann numbers for the electrons. The oscillator variables are integrated out, leaving an effective density matrix for the electrons containing retarded interactions. The partition function is then calculated up to second order of an expansion about the exact fermionic classical paths. Then, the free energy, specific heat, and average hopping energy are obtained. We find that the most significant reduction in the hopping energy is manifested at intermediate temperatures.
The validity of the semiclassical approximation is studied for a system comprising one quasiparticle coupled to a boson degree of freedom. Using a two-site Holstein model as an example, it is shown that the semiclassical approximation becomes exact in a nontrivial adiabatic limit. Furthermore, in the model's polaron regime, there exists a hierarchy of time scales that rationalizes the quantum dynamics of the Holstein model. For the single-mode case considered, the discrete nonlinear Schr\"odinger equation is found to be valid only in a highly limited antiadiabatic regime.
Abstract This work aims to provide an alternative approach to modeling a two-state system (qubit) coupled to a nonlinear oscillator. Within a single algebraic scheme based upon the f -deformed oscillator description, hard and soft nonlinearities are proposed to be simulated by making use of fitting algebraic models extracted from the trigonometric and modified Pöschl–Teller potentials, respectively. In the regime where the strength of the coupling is considered to be moderate, this approach allows for an analytic, albeit approximate, diagonalization process of the proposed Hamiltonian through using the Van Vleck perturbation theory and embracing the two types of nonlinear features. In the ultrastrong-coupling regime, the effect of such nonlinearities upon the squeezing and phase space properties of the ground state of the composite system is also explored by numerical means.
Recent experiments on quantum behavior in microfabricated solid-state systems suggest tantalizing connections to quantum optics. Several of these experiments address the prototypical problem of cavity quantum electrodynamics: a two-level system coupled to a quantum harmonic oscillator. Such devices may allow the exploration of parameter regimes outside the near-resonance and weak-coupling assumptions of the ubiquitous rotating-wave approximation (RWA), necessitating other theoretical approaches. One such approach is an adiabatic approximation in the limit that the oscillator frequency is much larger than the characteristic frequency of the two-level system. A derivation of the approximation is presented, together with a discussion of its applicability in a system consisting of a Cooper-pair box coupled to a nanomechanical resonator. Within this approximation the time evolution of the two-level-system occupation probability is calculated using both thermal- and coherent-state initial conditions for the oscillator, focusing particularly on collapse and revival phenomena. For thermal-state initial conditions parameter regimes are found in which collapse and revival regions may be clearly distinguished, unlike the erratic evolution of the thermal-state RWA model. Coherent-state initial conditions lead to complex behavior, which exhibits sensitive dependence on the coupling strength and the initial amplitude of the oscillator state. One feature of the regime considered here is that closed-form evaluation of the time evolution may be carried out in the weak-coupling limit, which provides insight into the differences between the thermal- and coherent-state models. Finally, potential experimental observations in solid-state systems, particularly the Cooper-pair box---nanomechanical resonator system, are discussed and found to be promising.
We use a functional-integral technique to evaluate all the terms of the semiclassical expansion series for the density matrix of a mean-field two-level system coupled to an oscillator with q2{\mathit{q}}^{2}q2 interaction. Our calculation reproduces the exact result found by direct diagonalization. We also show that our theory yields a lower free energy than that obtained by using the adiabatic approximation.