**Marie Curie Actions **

**Project No: **230785
NANOMAGNETS - Mesoscale Quantum Dissipation with Applications to Nanotechnology**
2009-2012**

**
Project Co-ordinator:**

Yuri P. Kalmykov, LAMPS, Université de Perpignan Via Domitia, 52, Avenue Paul Alduy, 66860, Perpignan Cedex, France

Tel. +(33)-468662062; FAX
+(33)-468662234;

e-mail:
kalmykov@univ-perp.fr

WEB: http://lamps.univ-perp.fr/kalmykov/

**
Partners: **

**
**

**
William T.
Coffey**,
Department of Electronic and Electrical Engineering, Trinity College, Dublin 2,
Ireland

**
Serguey V. Titov**,
Institute
of Radio Engineering and Electronics of the Russian Academy of Sciences,
Vvedenskii Square 1, Fryazino, Moscow Region, 141190, Russian Federation

**
**

**Summary description of the project 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 magnetization 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 (separable and
additive Hamiltonians) and spins it is proposed to generalize 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 results, obtained by
(a) matrix continued fraction methods of solution of the appropriate master
equations (b) computer simulation and (c) quantum Kramers escape rate theory
will be compared with suitable experimental observations of the reversal time
of the magnetization and the dynamic susceptibilities of nanomagnets (molecular
magnets, nanoclusters, and nanoparticles).

**Description of the work performed since the beginning of the project:**

We have
derived an evolution equation for the reduced density matrix for a spin system
with a non-axially symmetric Hamiltonian in contact with a thermal bath using
the projection operator technique. Hence we have obtained the master equation
for the time evolution of the quasiprobability density function of spin
orientations in the phase space representation of the polar and azimuthal
angles for spin systems subject to a magnetic field. We have also elaborated a
theory of magnetic relaxation of spin systems for both axially and non-axially
symmetric magnetic anisotropy potentials. Moreover, we have derived recurrence
equations for the statistical moments from the evolution equation for the
density matrix and the corresponding master equation. We have also developed
effective numerical algorithms and programs for evaluating the spin size effects
and the temperature dependence of the reversal time of the magnetization, the
switching fields and corresponding dynamic hysteresis loops of various spin
systems such as molecular magnets, nanoclusters, etc.

**Description of the main results achieved so far: **

The
nonlinear relaxation of quantum 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 (analogous
to the Fourier series representation of the classical distribution function) of
the polarization operators and spherical harmonics, respectively, where the
coefficients of the series (statistical averages of the polarization operators
and spherical harmonics) are found from entirely equivalent differential
recurrence relations. Each system matrix has an identical set of eigenvalues
and eigenfunctions. For illustration, the time behavior of the longitudinal
component of the magnetization and its characteristic relaxation times are
evaluated for a *uniaxial *paramagnet
of arbitrary spin *S* and generalized
to the non-axially symmetric problem of a paramagnet of arbitrary spin *S* subjected to a dc magnetic field of
arbitrary direction and orientation. We
have evaluated effects of strong magnetic fields on the magnetization
relaxation of spin systems with non-axially symmetric magnetic anisotropy
potentials (biaxial, cubic, etc.). We have developed a comprehensive theory of
the magnetization 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 classical phenomena in spin systems (molecular magnets,
nanoclusters, etc.). In particular, we have developed numerical algorithms for
the estimation of the observables (the reversal time, dynamic susceptibility,
etc.) of spin systems such as molecular magnets, nanoclusters, etc. We have
estimated spin size effects and the temperature dependence of the switching
fields and corresponding hysteresis loops of molecular magnets. We have also
estimated spin size effects on the reversal time of the magnetization of various
spin systems such as molecular magnets, nanoclusters, etc. Thus we have
achieved all the anticipated objectives set out in our project.

We have
submitted for publication 4 papers and have published 6 papers in primary physical journals. We have
also given 8 talks and presented 6 posters at 11 conferences and workshops.

**Publications:**

*We have published and
have submitted for publication 10 papers in primary physical journals:*

1. Yu. P. Kalmykov, W. T. Coffey, and S. V. Titov, *"Phase space Langevin equation for spin relaxation in a dc
magnetic field", EPL *2009, v. 88, No. 1, p. 17002_(6 pages). http://dx.doi.org/10.1209/0295-5075/88/17002

2. Yu.
P. Kalmykov, S. V. Titov, and W. T. Coffey, "*Nonlinear longitudinal relaxation of a quantum superparamagnet with arbitrary spin value S: Phase space and
density matrix formulations*", *Phys. Rev. B *2010, v. 81, No. 9, p. 094432_(14 pages). http://link.aps.org/doi/10.1103/PhysRevB.81.094432

3. Yu.
P. Kalmykov, S. V. Titov, and W. T. Coffey, *"**Spin size effects in stochastic resonance in
quantum uniaxial superparamagnets**"*, *Phys. Rev. B *2010, v. 81, No. 17, p.
172411. http://link.aps.org/PhysRevB.81.172411

4. Yu.
P. Kalmykov, S. V. Titov, and W. T. Coffey, *"Quantum
effects in stochastic resonance in uniaxial superparamagnets subjected to a dc
bias magnetic field",* *J. Phys.
Cond. Matter *2010, v. 22, No. 37, p. 376001_(7 pages). http://stacks.iop.org/0953-8984/22/376001

5. Yu.
P. Kalmykov, B. P. Mulligan, S. V. Titov, and W. T. Coffey, "*Master equation in phase space for a spin in an
arbitrarily directed uniform external field* ", *J. Stat.* *Phys. *2010, v. 141, p. 589-606. http://www.springerlink.com/content/a76058036p637187/

6. Yu. P. Kalmykov, W. T. Coffey, and S. V. Titov, "*Statistical moment
equations for stochastic spin dynamics in phase space: A uniaxial paramagnet subjected to a dc
bias field of arbitrary orientation"*, *Phys.** **Rev. B*** **2012, v. 86, No. 10, p.
104435_(12 pages). http://link.aps.org/doi/10.1103/PhysRevB.86.104435

7. W.
T. Coffey, Yu. P. Kalmykov, S. V. Titov,
and William J. Dowling, *"Longest relaxation time of relaxation processes for
classical and quantum Brownian motion in a potential: escape rate theory
approach"*, *Advances
in Chemical Physics*, 2013, Vol. 152, pp. 111-309, Series Ed. S. A.
Rice, Wiley, New York.

8. Yu. P. Kalmykov, W. T. Coffey, and S. V. Titov, "*Statistical moment
equations for stochastic spin dynamics in phase space: A uniaxial paramagnet subjected to a dc
bias field of arbitrary orientation"*, *Phys. Rev. B*** **2012, v. 86, No. 10, p.
104435_(12 pages). http://link.aps.org/doi/10.1103/PhysRevB.86.104435

9. Yu.
P. Kalmykov, W. T. Coffey, and S. V. Titov, "*Dynamic magnetic
hysteresis of nanomagnets with arbitrary spin value S**"*, *Phys. Rev. B. *2012, in preparation.

10. Yu.
P. Kalmykov, W. T. Coffey, and S. V. Titov, "*Nonlinear
longitudinal relaxation of a uniaxial nanomagnet with arbitrary spin value S in
a dc magnetic field of arbitrary orientation**"*, *Phys. Rev. B. *2012, in preparation.

*We have given 8 talks
and presented 6 posters at 11 conferences and workshops.*

1. W. T. Coffey, Yu. P. Kalmykov, B. P.
Mulligan, and S. V. Titov *(oral)*, *"Phase-space description of
spin dynamics", *Annual German Physical Soc. Spring Meeting, Dresden,
22-27 March 2009, DY 13.4 Tue 15:30 ZEU 255, p. 25. http://www.dpg-verhandlungen.de/2009/dresden/dy13.pdf

2. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov *(poster)*,
*"Spin dynamics in phase-space",* 5th
International Workshop on Nanomagnetism & Superconductivity, Spain,
Comaruga 2009, http://www.ub.edu/gmag/comaruga/ .

3. W. T. Coffey, Yu. P. Kalmykov, B. Mulligan, and S.
V. Titov *(poster)*, "*Spin dynamics: quantum master equations in
phase space for axially symmetric potentials*", Tunneling and
Scattering in Complex Systems - From Single to Many Particle Physics, Seminar
and Workshop, Dresden, Germany, September 07 - 25, 2009.

4. B. P.J. Mulligan, W. T. Coffey, Yu. P. Kalmykov,
and S. V. Titov* (oral)*, *"Phase space master equations for the
Lipkin-Meshkov Hamiltonian ", *Annual German Physical Soc. Spring
Meeting, Regensburg, 21-26 March 2010, DY 4.6 Mon 15:15 H47, http://www.dpg-verhandlungen.de/2010/regensburg/dy4.pdf

5. B. P.J.
Mulligan, W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov* (poster)*, *"*Spin
dynamics: quantum master equation in phase space fora spin in a uniform
external field* ", *Annual German Physical Soc. Spring Meeting,
Regensburg, 21-26 March 2010, DY 6.9, Mon 16:00 Poster B2, http://www.dpg-verhandlungen.de/2010/regensburg/dy6.pdf

6. W. T.
Coffey, Yu. P. Kalmykov, B. P. J. Mulligan, and S. V. Titov* (poster)*,
*"Spin dynamics of a uniaxial paramagnet: Phase space formulation"*,
Theoretical, Computational, and Experimental Challenges to Exploring Coherent
Quantum Dynamics in Complex Many-Body Systems, May 9-12 2010, Dublin, Ireland. http://www.cecam.org/workshop-3-460.html

7. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov *(oral)*,
*“Nonlinear magnetic relaxation of quantum superparamagnets: Phase-space
method”*, 7th Int. Conf. on Fine Particle
Magnetism (ICFPM-2010), Uppsala, Sweden, June 21-24, 2010, p. 124. http://www-conference.slu.se/icfpm2010/

8. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov *(oral)*,
*"Nonlinear magnetic relaxation of quantum superparamagnets: Phase-space
description",* 6th International Workshop on
Nanomagnetism & Superconductivity, Spain, Comaruga 2010, p. 35 http://www.ub.edu/gmag/comaruga/ .

9. Yu. P. Kalmykov, B. P.J. Mulligan, S. V. Titov,* *and
W. T. Coffey, *(poster)*, *"*Spin dynamics in phase space*",
*Annual German Physical Soc. Spring Meeting, Dresden, 13-17 March 2011, MA
19.84, Tue 10:45 Poster P2, http://www.dpg-verhandlungen.de/2011/dresden/ma19.pdf

10. Yu. P. Kalmykov, B. P. J. Mulligan, S. V. Titov,*
*and W. T. Coffey, *(oral)*, *"Master equation in phase space
for a spin in an arbitrarily directed uniform external field", *Annual
German Physical Soc. Spring Meeting, Dresden, 13-17 March 2011, MA 45.5, Wed
18:30 http://www.dpg-verhandlungen.de/2011/dresden/ma45.pdf

11. W. T. Coffey, Y. P. Kalmykov , S. V. Titov L.
Cleary, and W. J. Dowling *(oral)*, *"Quantum master equation in
phase space applied to the Brownian motion in a tilted periodic
potential", *Annual German Physical Soc. Spring Meeting, Dresden, 13-17
March 2011, DY 33.3, Thu 14:30 http://www.dpg-verhandlungen.de/2011/dresden/dy33.pdf

12. B. P. J. Mulligan, Yu. P. Kalmykov, S. V. Titov,*
*and W. T. Coffey *(poster)*, *"Spin dynamics in phase
space", *Noise in Non-Equilibrium Systems: From Physics to Biology,
International Workshop – April 11 - 14, 2011, On the occasion of Prof. Peter
Hänggi's 60th birthday, http://www.mpipks-dresden.mpg.de/~nines11/

13. W. T. Coffey, L. Cleary, W. Dowling, Yu. P.
Kalmykov, and S. V. Titov *(oral)*, *"Phase space master equations
for quantum Brownian motion in a periodic potential: comparison of various
kinetic models", *Madrid Workshop on Open Quantum Systems, 3-5 October
2011, Madrid, p. 36. http://fama.iff.csic.es/con/MWOQS-2011/workshop-files/Book_of_Abstracts_MWOQS2011.pdf

14. W. T. Coffey, Yu. P. Kalmykov, and S.
V. Titov *(oral)*, "*Statistical moment equations for
stochastic spin dynamics in phase space: A uniaxial paramagnet subjected to a
dc bias field of arbitrary orientation"*, Annual German Physical Soc.
Spring Meeting: Magnetism Division, Regensburg, 11-16 March 2013, MA 43.10, Thu
17:15 http://www.dpg-verhandlungen.de/2013/regensburg/ma43.pdf