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Physica status solidi. C, Conferences and critical reviews/Physica status solidi. C, Current topics in solid state physics

Quantum dynamics of damped and driven anharmonic oscillators

Michael J. Schrapp, Elena del Valle, Jonathan James Finley, Fabrice P. Laussy

Published Apr 12, 2012• 1 citation
Publisher

Abstract

Abstract We study various techniques to characterize the quantum states of a strongly dissipative system, namely, the an‐harmonic oscillator. One cannot directly access the quantum states themselves but only transitions between them. Various methods bring various advantages and inconvenients. We compare several observables under coherent and incoherent excitation. Incoherent excitation leads to huge broadening while coherent excitation quickly dresses the system at the energies required to access the excited states. Less perturbative methods such as four wave mixing appear to be superior but are, by construction, only usable for the lowest rungs. (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssc.201100195

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Abstract We study various techniques to characterize the quantum states of a strongly dissipative system, namely, the an‐harmonic oscillator. One cannot directly access the quantum states themselves but only transitions between them. Various methods bring various advantages and inconvenients. We compare several observables under coherent and incoherent excitation. Incoherent excitation leads to huge broadening while coherent excitation quickly dresses the system at the energies required to access the excited states. Less perturbative methods such as four wave mixing appear to be superior but are, by construction, only usable for the lowest rungs. (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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The present review is devoted to the study of certain aspects of anharmonic, time-dependent and damped oscillator(s) system using different theoretical techniques. A theoretical understanding of these systems is important for application in many problems in physics, mechanics and other fields. We discuss in detail the difficulties in the theoretical analysis of the problem. In particular we discuss here the regular, well-behaved perturbative solution, the large quantum number behaviour of anharmonic oscillator(s) using the technique of coherent states, exact solution of quantum anharmonic oscillators, the electromagnetic radiation emitted by a charged particle executing damped anharmonic oscillator motion using Krylov-Bogoliubov approximation method, use of invariants to obtain solution and coherent states of time-dependent oscillator(s), the derivation of perturbative frequencies of a damped coupled anharmonic oscillators system using suitable canonical transformation in the framework of Hamilton-Jacobi formalism and the quantisation and construction of coherent states of a damped oscillator using time-dependent operators.
Abstract. The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a damped oscillator is determined by an arbitrary effective susceptibility that obeys Kramers-Kronig relations. This approach offers an alternative description of nano-mechanical oscillators and opto-mechanical systems. PACS numbers: 03.65.-w, 03.65.Yz, 03.70.+k 1.
Abstract We discuss, from a theoretical point of view, the four wave mixing spectroscopy on an ensemble of p‐doped quantum dots in a magnetic field slightly tilted from the in‐plane configuration. We describe the system evolution in the density matrix formalism. In the limit of coherent ultrafast optical driving, we obtain analytical formulas for the single system dynamics and for the response of an inhomogeneously broadened ensemble. The results are compared to the previously studied time‐resolved Kerr rotation spectroscopy on the same system. We show that the Kerr rotation and four wave mixing spectra yield complementary information on the spin dynamics (precession and damping). (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract Four‐wave mixing (FWM) optical response from a sample containing a non‐homogeneously broadened ensemble of vertically stacked pairs of quantum dots is studied theoretically. The effects of collective interaction between the dots and their electromagnetic environment are analyzed. It is shown that the shape of the FWM echo is affected by the coupling between the two dots and, to some extent, by the statistics of the ensemble. In the time‐integrated signal, oscillations on picosecond delay time scale appear which lead to a reduction of the coherent response for coupled dots. For longer delays, the evolution is dominated by radiative decay, with the decay rate increasing from Γ to 2Γ(superradiance effect) as the coupling becomes stronger. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a damped oscillator is determined by an arbitrary effective susceptibility that obeys the Kramers–Kronig relations. This approach offers an alternative description of nano-mechanical oscillators and opto-mechanical systems.