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Physical Review B

Quantum relaxation dynamics of magnetic moments in a radiative reservoir

Yuh‐Jen Cheng

Published Dec 29, 2004• 1 citation
Publisher

Abstract

A quantum Langevin equation is derived to describe the relaxation dynamics of a magnetic moment in a static magnetic field and radiative reservoir. The damping and fluctuation forces are derived from the radiative interaction between magnetic moment and surrounding reservoir. Through the use of a symmetrized interaction Hamiltonian, the damping force is identified due to the combination of radiation self-reaction and reservoir fluctuations. The radiation self-reaction is a quantum version of its classical counterpart, whereas the reservoir fluctuations are solely a quantum effect resulting from the quantization of the electromagnetic field. The relative magnitude between these two effects changes during the relaxation process. This equation shows that the relative magnitude of the quantum correction to the classical Landau-Lifshitz description is inversely proportional to the system angular momentum quantum number.

DOI
10.1103/physrevb.70.224435

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A quantum Langevin equation is derived to describe the relaxation dynamics of a magnetic moment in a static magnetic field and radiative reservoir. The damping and fluctuation forces are derived from the radiative interaction between magnetic moment and surrounding reservoir. Through the use of a symmetrized interaction Hamiltonian, the damping force is identified due to the combination of radiation self-reaction and reservoir fluctuations. The radiation self-reaction is a quantum version of its classical counterpart, whereas the reservoir fluctuations are solely a quantum effect resulting from the quantization of the electromagnetic field. The relative magnitude between these two effects changes during the relaxation process. This equation shows that the relative magnitude of the quantum correction to the classical Landau-Lifshitz description is inversely proportional to the system angular momentum quantum number.
The physical processes responsible for the evolution of a small system S weakly coupled to a large reservoir R are analysed through the coupled Heisenberg equations of the problem. The respective contributions of reservoir fluctuations (R fluctuates and polarizes S) and self-reaction (S fluctuates and polarizes R which reacts back on S) are identified with a general method developed in a previous paper dealing with radiative processes (contributions of vacuum fluctuations and radiation reaction). The Hamiltonian and non Hamiltonian parts of the evolution of S are made explicit and related to correlation functions and polarizabilities of S and R. The limit where S is a quasi-classical system is investigated, and the evolution of the energy distribution function of S is shown to be described by a Fokker-Planck equation, the drift and diffusion terms of which are respectively associated with self-reaction and reservoir fluctuations. These results are finally applied to the problem of the spontaneous emission of a large angular momentum (Dicke's model of superradiance).
An effective model for describing the relativistic quantum dynamics of a radiating electron is developed via a relativistic generalization of the Lindblad master equation. By incorporating both radiation reaction and vacuum fluctuations into the Dirac equation within an open quantum system framework, our approach captures the Zitterbewegung of the electron, ensuing noncommutativity of its effective spatial coordinates, and provides the quantum analogue of the Landau-Lifshitz (LL) classical equation of motion with radiation reaction. We develop the corresponding phase-space representation via the relativistic Wigner function and derive the semiclassical limit through a Foldy-Wouthuysen transformation. The latter elucidates the signature of quantum vacuum fluctuations in the LL equation, and shows its relationship with the corrected Sokolov equation. Our results offer a robust framework for investigating quantum radiation reaction effects in ultrastrong laser fields.
The multiplicative stochastic process treatment of the time development of the density matrix for a subsystem in contact with a heat reservoir is applied to the specific problem of the relaxation of a nuclear magnetic moment which is interacting with a fluctuating magnetic environment. A model for the fluctuating interaction Hamiltonian, appropriate for the magnetic moment case, is presented, and the Bloch equations for nuclear magnetic relaxation are constructed as a consequence. Agreement with empirical observations is noted.
Electromagnetic interactions of a single charged particle in the presence of damping are investigated theoretically in this work. Classically, by considering the explicit example of a particle in a constant magnetic field, it is shown that the minimal coupling prescription is inadequate for the infinitely large number of Hamiltonians from which the equations of motion in the presence of damping are derivable by Hamilton's equations. The use of minimal coupling procedure to switch on electromagnetic interactions in phenomenological quantum equations of damped motion is shown to lead to incorrect equations in the classical limit. It is necessary to consider the microscopic models of damping to allow for electromagnetic interactions in these systems. Two microscopic models for damping involving a linear coupling with a heat bath of oscillators are considered in detail and it is shown that correct Heisenberg equations of motion follow upon elimination of bath degrees of motion so that damping and Lorentz force do not affect each other. However, the field couples to the mechanical momentum as mv-(e/c)A rather than to the canonical momentum. The linearly coupled quantal case is analogous to the classical situation. The Schr\"odinger-Langevin equation contains an extra term Fr{\mathcal{F}}^{\mathrm{r}}Fr A\ensuremath{\cdot}dr'; however, the gauge invariance character and the Aharonov-Bohm effect are unaffected. Two nonlinear models of damping are studied by considering the exchange of bosons in the different states of the central particle. In this situation the Lorentz force is not recovered in the classical limit.
It is generally considered that there exists in quantum radiation theory an indetermination in the separation of the respective effects of vacuum fluctuations and radiation reaction. We show in this paper that such an indetermination can be removed by imposing to the corresponding rates of variation to be Hermitian (this is necessary if we want them to have a physical meaning). Such a procedure is generalized to the case of a small system S interacting with a large reservoir R and allows the separation of two types of physical processes, those where R fluctuates and polarizes S (effects of reservoir fluctuations), those where it is S which polarizes R (effects of self reaction). We apply this procedure to an atomic electron interacting with the radiation field and we then identify the contribution of vacuum fluctuations and self reaction to radiative corrections and spontaneous emis- sion of radiation. The analysis of the results obtained in this way allows us to specify the physical pictures which must be associated with the various radiative processes.
Abstract Radiation damping of an accelerated particle coupled with the field is a classical problem. Still there are difficulties. Time symmetry is broken. So the radiation damping belongs to a class of phenomena that includes transport properties. In a series of our recent articles we proposed a radically new approach based on the extension of the dynamics of integrable systems. There, we considered a simple model: a harmonic oscillator interacting with a field. For integrable systems, it is well known that there exists a unitary transformation U. However, in the radiation damping we have resonances between the action of a charged particle and the actions of the field modes. This makes the system an example of Poincaré nonintegrable systems. We extended the unitary operator to a nonunitary operator Λ. This changes the dynamical description of radiation damping. Once we know the Hamiltonian, we can write the Hamilton equations. But we have the opportunity to go to new descriptions. The invertible Λ transformation gives many new aspects, which are hidden in the initial description. For example, we showed that there are fluctuations in the emitted field. In the current article we apply the Λ transformation to describe the evolution of the emitted field from the charged particle and show that the fluctuations lead to a Langevin equation for the field modes. As a consequence, we have a stochastic wave equation that is an extension of Maxwell–Lorentz equation for classical matter–field interacting systems. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004