IMPERIAL COLLEGE PRESS eBooks

Dissipation and quantization

Published Feb 1, 2011• 0 citations

Abstract

We show that the dissipation term in the Hamiltonian for a couple of classical damped-amplified oscillators manifests itself as a geometric phase and is actually responsible for the appearance of the zero point energy in the quantum spectrum of the 1D linear harmonic oscillator.We also discuss the thermodynamical features of the system.Our work has been inspired by 't Hooft proposal according to which information loss in certain classical systems may lead to "an apparent quantization of the orbits which resembles the quantum structure seen in the real world".

DOI: 10.1142/9781848162815_0009

Related papers

Dissipation, coherence and entanglement

Massimo Blasone A, Giuseppe Vitiello

The study of the damped harmonic oscillator shows that dissipation could be seen at the origin of the zero point energy, which is the signature of quantum behavior. This is in accord with ’t Hooft proposal that loss of information in a completely deterministic dynamics would play a rôle in the quantum mechanical nature of our world. We show the equivalence, within quite general conditions, between the pair of a damped oscillator and its time-reversed image and electrodynamics. The ground state of the damped-amplified oscillator pair appears to be a finite temperature coherent two-mode squeezed state with fractal self-similarity properties and the modes are maximally entangled. Temperature is strictly related to the zero point energy.

International Journal of Geometric Methods in Modern PhysicsFeb 25, 20200 citations

Comment on "Quantization of the damped harmonic oscillator" [Serhan et al, J. Math. Phys. 59, 082105 (2018)]

Zafar Ahmed, Sachin Kumar, A. Baishya

A recent paper [J. Math. Phys. {\bf 59}, 082105 (2018)] constructs a Hamiltonian for the (dissipative) damped harmonic oscillator. We point out that non-Hermiticity of this Hamiltonian has been ignored to find real discrete eigenvalues which are actually non-real. We emphasize that non-Hermiticity in Hamiltonian is crucial and it is a quantal signature of dissipation.

arXiv:1902.04895Feb 13, 20190 citations

Dissipation in Quantum Mechanics. The Harmonic Oscillator

I. R. Senitzky

The need for a quantum-mechanical formalism for systems with dissipation which is applicable to the radiation field of a cavity is discussed. Two methods that have been used in this connection are described. The first, which starts with the classical Newtonian equation of motion for a damped oscillator and applies the conventional formal quantization techniques, leads to an exact solution; but subsequent discussion shows that this method is invalid, the results being unacceptable from a quantum-mechanical viewpoint. The second method, which considers the interaction of two systems, the lossless oscillator and the loss mechanism, is adopted in the present article. No special model is used for the loss mechanism, but this mechanism is assumed to have a large number of densely-spaced energy states.The approximations with respect to the loss mechanism that underlie the concept of dissipation are discussed. These approximations are then applied to the analysis, and a differential equation for a coordinate operator of the harmonic oscillator is obtained which has the formal appearance of the Newtonian equation of motion for a driven damped harmonic oscillator, the driving term being an operator referring to the loss mechanism. The presence of the driving term is responsible for the difference between the present theory and that of the first method mentioned above. A solution of the differential equation for the coordinate operator is given explicitly. An examination of the physical significance of the solution shows that the driving term is responsible not only for the thermal fluctuations which are due to the loss mechanism, but also for the proper commutation relationship of the conjugate coordinates of the oscillator and for its zero-point fluctuations.A generalization of the solution to provide for a classical driving force and coupled atomic systems is given. The results are then restated in a form that refers to the loss mechanism only through the two parameters by which it is usually described---the dissipation constant and the temperature.

Physical ReviewJul 15, 1960466 citations

Quantization of the linearly damped harmonic oscillator

H. Dekker

A previously developed novel theory for the formal canonical quantization of classically dissipating systems will be the starting point for a detailed discussion of the quantum statistical aspects of the simple linearly damped harmonic oscillator. The formalism essentially involves complex classical canonical coordinates and momenta, and thus quite naturally leads to the possibility of creation and annihilation operators. Furthermore, the occurrence of quantal noise operators appears to be of principal importance for the conservation of the fundamental commutator in the course of time, as will be expressed in a simple fluctuation-dissipation relation. Making a canonical transformation back to the real, Cartesian Hermitian position and momentum an "effective" Hermitian Hamiltonian will be derived, with which a transformation is made from the Heisenberg frame to the Schr\"odinger frame where the density operator equation will be computed. This will make it obvious that no proper Schr\"odinger equation exists for the dissipative subsystem on its own, thus reflecting an incomplete knowledge. The master equation will then be translated into its Wigner representation. The intimate connection between the diffusion coefficients in the resulting Fokker-Planck equation and the uncertainty relation will be demonstrated in a clear fashion.

Physical Review ANov 1, 1977167 citations

Quantum damped oscillator I: dissipation and resonances

Chruscinski, D, Jurkowski, J

Quantization of a damped harmonic oscillator leads to so called Bateman's dual system. The corresponding Bateman's Hamiltonian, being a self-adjoint operator, displays the discrete family of complex eigenvalues. We show that they correspond to the poles of energy eigenvectors and the corresponding resolvent operator when continued to the complex energy plane. Therefore, the corresponding generalized eigenvectors may be interpreted as resonant states which are responsible for the irreversible quantum dynamics of a damped harmonic oscillator.

Jan 1, 200527 citations

Interpreting Liouvillian Eigenmodes: Quantized Orbits in Dissipative Systems

Ashlin V Thomas, Felix Fritzsch, Masudul Haque +1 more

In closed quantum dynamics, energy eigenstates represent physical excitations which, for systems with a classical limit, localize on quantized orbits in phase space. Such a picture is lacking in open systems: Liouvillians have distinct right and left eigenstates that decay with time and are difficult to interpret. We recover a physical correspondence by introducing a gauge-invariant quasiprobability measure combining right and left eigenstates that interprets a Liouvillian mode as a set of coherences, allowing one to visualize the modes as quasiprobability distributions in a ``doubled'' phase space. Using this measure we show that, remarkably, a damped oscillator retains its quantized ``orbits'' for a large class of linear and nonlinear loss, with measurable signatures. A thermal bath broadens these orbits, similar to energy levels, leading to a characteristic multistage relaxation. This provides an experimentally accessible open-system analog of the eigenstate-orbit correspondence underlying semiclassical quantization. For quadratic systems, our results yield dissipative analogs of invariant tori - a central concept in Hamiltonian systems.

quant-pharXiv:2606.20271Jun 19, 20260 citations