## Final Report Summary - NANOMAGNETS (Mesoscale Quantum Dissipation with Applications to Nanotechnolgy)

Project context and objectives

Quantum dissipation arising from quantum fluctuations and the quantum mechanics of macroscopic variables is important because of the ever-decreasing size (mesoscale) of the nanoparticles used in technology. The most striking example occurs in information storage by magnetic nanoparticles, where the governing factor for magnetisation reversal by macroscopic quantum tunnelling is spin size S. The S dependence, with associated large quantum effects, becomes ever more marked as one proceeds from single domain particles to molecular clusters to single molecule magnets to individual spins. Here, in the context of a general investigation of mesoscale quantum mechanics of particles and spins, we have generalised Wigner's quasi-phase space formulation of quantum mechanics without dissipation (originally used to calculate quantum corrections to classical statistical mechanics i.e. the quantum/classical borderline characteristic of the mesoscale), to systems with non-separable Hamiltonians (spins) including the effects of dissipation to the surrounding heat bath. The magnetisation relaxation of spins interacting with a thermal bath is treated via the respective evolution equations for the reduced density matrix and phase space distribution function in the high temperature and weak spin-bath coupling limits using the methods already available for classical spins. The solution of each evolution equation is written as a finite series of the polarisation operators and spherical harmonics, respectively, where the coefficients of the series (statistical averages of the polarisation operators and spherical harmonics) are found from entirely equivalent differential recurrence relations. Each system matrix possesses an identical set of eigenvalues and eigenfunctions. We have found the explicit solution of the master equation for the quasi-probability density function of spin orientations in the phase space of the polar and azimuthal angles for an arbitrary spin Hamiltonian as a finite series of spherical harmonics. The expansion coefficients (statistical averages of the spherical harmonics) may then be determined just as the classical case from a differential-recurrence relation yielding a convenient method of treating the stochastic spin dynamics for arbitrary spin number S. In the large spin number limit, the quantum differential-recurrence relation reduces to that yielded by the classical Fokker-Planck equation. We have also developed effective numerical algorithms and programmes for evaluating the spin size effects and the temperature dependence of the reversal time of the magnetisation, the switching fields and corresponding dynamic hysteresis loops of various spin systems such as molecular magnets, nanoclusters, etc. In particular, we have evaluated the time behaviour of the longitudinal component of the magnetisation and its characteristic relaxation times for a uniaxial paramagnet of arbitrary spin S in an external constant magnetic field applied along the axis of symmetry. In the large spin limit, the quantum solutions reduce to those of the Fokker-Planck equation for a classical uniaxial superparamagnet. For linear response, the results entirely agree with existing solutions. A Langevin equation for the quantum Brownian motion of a spin of arbitrary size in a uniform external DC magnetic field is also derived from the phase space master equation (in the weak coupling and narrowing limits) governing the quasiprobability distribution function of spin orientations in the configuration space of polar and azimuthal angles following methods long familiar in quantum optics. The closed system of differential-recurrence equations for the statistical moments describing magnetic relaxation of the spin is obtained as an example of possible applications of this equation. It is also shown that the signal-to-noise ratio (SNR) of the magnetic moment fluctuations in the magnetic stochastic resonance of a quantum uniaxial paramagnet of arbitrary spin value S subjected to a weak probing AC field H(t) and a DC bias magnetic field H0 displays a pronounced dependence on S. The dependence originates in the quantum dynamics of spins which differ markedly from the magnetisation dynamics of classical superparamagnets. In the large spin limit, S, the quantum solutions reduce to those for a classical uniaxial superparamagnet. Thus we have developed a comprehensive theory of the magnetisation relaxation of spin systems as a function of S spanning the entire region between individual spins and single domain particles which is necessary for the analysis of both the experimental data and the coexistence of quantum and establishing classical phenomena in spin systems (molecular magnets, nanoclusters, etc.). Hence we have achieved all the anticipated objectives set out in our project.

Address of the project website: http://perso.univ-perp.fr/kalmykov/Nanomagnets.htm

Quantum dissipation arising from quantum fluctuations and the quantum mechanics of macroscopic variables is important because of the ever-decreasing size (mesoscale) of the nanoparticles used in technology. The most striking example occurs in information storage by magnetic nanoparticles, where the governing factor for magnetisation reversal by macroscopic quantum tunnelling is spin size S. The S dependence, with associated large quantum effects, becomes ever more marked as one proceeds from single domain particles to molecular clusters to single molecule magnets to individual spins. Here, in the context of a general investigation of mesoscale quantum mechanics of particles and spins, we have generalised Wigner's quasi-phase space formulation of quantum mechanics without dissipation (originally used to calculate quantum corrections to classical statistical mechanics i.e. the quantum/classical borderline characteristic of the mesoscale), to systems with non-separable Hamiltonians (spins) including the effects of dissipation to the surrounding heat bath. The magnetisation relaxation of spins interacting with a thermal bath is treated via the respective evolution equations for the reduced density matrix and phase space distribution function in the high temperature and weak spin-bath coupling limits using the methods already available for classical spins. The solution of each evolution equation is written as a finite series of the polarisation operators and spherical harmonics, respectively, where the coefficients of the series (statistical averages of the polarisation operators and spherical harmonics) are found from entirely equivalent differential recurrence relations. Each system matrix possesses an identical set of eigenvalues and eigenfunctions. We have found the explicit solution of the master equation for the quasi-probability density function of spin orientations in the phase space of the polar and azimuthal angles for an arbitrary spin Hamiltonian as a finite series of spherical harmonics. The expansion coefficients (statistical averages of the spherical harmonics) may then be determined just as the classical case from a differential-recurrence relation yielding a convenient method of treating the stochastic spin dynamics for arbitrary spin number S. In the large spin number limit, the quantum differential-recurrence relation reduces to that yielded by the classical Fokker-Planck equation. We have also developed effective numerical algorithms and programmes for evaluating the spin size effects and the temperature dependence of the reversal time of the magnetisation, the switching fields and corresponding dynamic hysteresis loops of various spin systems such as molecular magnets, nanoclusters, etc. In particular, we have evaluated the time behaviour of the longitudinal component of the magnetisation and its characteristic relaxation times for a uniaxial paramagnet of arbitrary spin S in an external constant magnetic field applied along the axis of symmetry. In the large spin limit, the quantum solutions reduce to those of the Fokker-Planck equation for a classical uniaxial superparamagnet. For linear response, the results entirely agree with existing solutions. A Langevin equation for the quantum Brownian motion of a spin of arbitrary size in a uniform external DC magnetic field is also derived from the phase space master equation (in the weak coupling and narrowing limits) governing the quasiprobability distribution function of spin orientations in the configuration space of polar and azimuthal angles following methods long familiar in quantum optics. The closed system of differential-recurrence equations for the statistical moments describing magnetic relaxation of the spin is obtained as an example of possible applications of this equation. It is also shown that the signal-to-noise ratio (SNR) of the magnetic moment fluctuations in the magnetic stochastic resonance of a quantum uniaxial paramagnet of arbitrary spin value S subjected to a weak probing AC field H(t) and a DC bias magnetic field H0 displays a pronounced dependence on S. The dependence originates in the quantum dynamics of spins which differ markedly from the magnetisation dynamics of classical superparamagnets. In the large spin limit, S, the quantum solutions reduce to those for a classical uniaxial superparamagnet. Thus we have developed a comprehensive theory of the magnetisation relaxation of spin systems as a function of S spanning the entire region between individual spins and single domain particles which is necessary for the analysis of both the experimental data and the coexistence of quantum and establishing classical phenomena in spin systems (molecular magnets, nanoclusters, etc.). Hence we have achieved all the anticipated objectives set out in our project.

Address of the project website: http://perso.univ-perp.fr/kalmykov/Nanomagnets.htm