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Physical Review A

Comparison between quantum and approximate semiclassical dynamics of an externally driven spin-harmonic oscillator system

Robert I. Cukier, Manuel Morillo

Published Jan 14, 2000• 4 citations
Publisher

Abstract

A Hamiltonian describing the interaction of a two-level system with an oscillator and an external field that induces transitions between the two levels is analyzed using the quantum mechanical (QM) solution and a semiclassical (SC) approximation. The Hamiltonian can describe coherent optical transitions and coherences in electron transfer processes. We focus on strongly driven systems where coherent oscillations of the populations of the two-level system are important and show that in this regime the SC approximation is a good representation of the QM behavior. The results are interpreted by constructing appropriate Landau-Zener-Massey parameters.

DOI
10.1103/physreva.61.024103

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A Hamiltonian describing the interaction of a two-level system with an oscillator and an external field that induces transitions between the two levels is analyzed using the quantum mechanical (QM) solution and a semiclassical (SC) approximation. The Hamiltonian can describe coherent optical transitions and coherences in electron transfer processes. We focus on strongly driven systems where coherent oscillations of the populations of the two-level system are important and show that in this regime the SC approximation is a good representation of the QM behavior. The results are interpreted by constructing appropriate Landau-Zener-Massey parameters.
Abstract We analyze the dynamics of a two-level quantum system (TLS) under the influence of a strong sinusoidal driving signal whose origin is the interaction of the two-level system with a quantum field. In this approach the driving field is replaced by a harmonic oscillator that is either strongly coupled to the TLS or populated with a large number of photons. Starting from the Rabi model, we derive expressions for the TLS’s oscillation frequencies and compare the results with those obtained from the model where the driving signal is treated classically. We show that in the limits of weak coupling and large photon number, the well-known expression for the Rabi frequency in the strong driving regime is recovered. In the opposite limit of strong coupling and small photon number, we find differences between the predictions of the semiclassical and quantum models. The results of the quantum picture can therefore be understood as Landau–Zener–Stueckelberg interferometry in the fully quantum regime.
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.
A system of N two-level atoms in a resonator is considered interacting with a resonant eigenmode field and with an external coherent field, with a frequency slightly different from the frequency of the atomic transition. A model Hamiltonian is constructed for describing the slow quantum dynamics of the system, and a set of closed c-number equations for time-dependent quantum expectation values is derived in the boson and spin coherent states. In the region of parameters for developed chaos in the semiclassical limit (when the radiation field is considered classically) we show that the semiclassical approximation is violated by quantum effects at the time scale ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Elzxh}}}$\ensuremath{\sim}lnN. We also show that quantum correlations grow exponentially during the time interval 0${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Elzxh}}}$. An additional averaging over the initial density matrix is considered. The connection of our model with the Dicke model [Phys. Rev. 93, 99 (1956)] is demonstrated, and the possiblity of measuring the quantum chaos effects considered in this paper is discussed.
We analyze the dynamics of a two-level system subject to driving by large-amplitude external fields, focusing on the resonance properties in the case of driving around the region of avoided level crossing. In particular, we consider three main questions that characterize resonance dynamics: (1) the resonance condition, (2) the frequency of the resulting oscillations on resonance, and (3) the width of the resonance. We identify the regions of validity of different approximations. In a large region of the parameter space, we use a geometric picture in order to obtain both a simple understanding of the dynamics and quantitative results. The geometric approach is obtained by dividing the evolution into discrete time steps, with each time step described by either a phase shift on the basis states or a coherent mixing process corresponding to a Landau-Zener crossing. We compare the results of the geometric picture with those of a rotating wave approximation. We also comment briefly on the prospects of employing strong driving as a useful tool to manipulate two-level systems.
Systems of interacting harmonic oscillators have recently received considerable attention as models for describing a variety of physical problems. We have investigated the validity of the rotating-wave approximation which constitutes the traditional approach to the solution of the dynamical problem by comparing it with the exact solution. A numerical comparison has been made and the limits of validity of the rotating-wave approximation has been established in terms of the strength of oscillator interaction.In particular, the time development of dynamical operators and certain transition probabilities have been compared. In the region where the rotating-wave approximation is valid, the time evolution of the quasiprobability distribution $P(\ensuremath{\alpha}, t)$ of one oscillator is given for several initial conditions. A counting scheme similar to the argument given by Feynman for the driven harmonic oscillator, is proposed for the interpretation of the time-dependent transition amplitudes between number states.
Two basic physical models, a two-level system and a harmonic oscillator, are realized on the mesoscopic \nscale as coupled qubit and resonator. The realistic system includes moreover the electronics for controlling the \ndistance between the qubit energy levels and their populations and to read out the resonator's state, as well as the \nunavoidable dissipative environment. Such rich system is interesting both for the study of fundamental quantum \nphenomena on the mesoscopic scale and as a promising system for future electronic devices. We present recent \nresults for the driven superconducting qubit–resonator system, where the resonator can be realized as an LC \ncircuit or a nanomechanical resonator. Most of the results can be described by the semiclassical theory, where a \nqubit is treated as a quantum two-level system coupled to the classical driving field and the classical resonator. \nApplication of this theory allows to describe many phenomena for the single and two coupled superconducting \nqubits, among which are the following: the equilibrium-state and weak-driving spectroscopy, Sisyphus damping \nand amplification, Landau–Zener–Stückelberg interferometry, the multiphoton transitions of both direct and ladder-type \ncharacter, and creation of the inverse population for lasing.