pulsed_field.m

Magnetisation dynamics under a time-dependent magnetic field along the Z axis of the laboratory frame with spin-phonon relaxation, as measured in pulsed-field magnetometry of molecular magnets. The field profile is replaced by a staircase; on each stair the Hamiltonian is constant, the spin-phonon dissipator is rebuilt in the eigenbasis of that Hamiltonian, and the density matrix is propagated in that eigenbasis by a symmetric split: exact coherent phases for half a stair, the dissipative step to second order in the dissipator times the stair width, and the phases again. The dissipator times the stair width must be small; the coherent part is treated exactly for any stair width. The dissipator is applied as Hilbert space matrix products (see rlx_phonon.m), so the cost of a stair is cubic in the dimension of the Hilbert space.

The function removes the unit-field Zeeman term from the Hamiltonian received from the context, starts from the thermal equilibrium state of the field-free Hamiltonian, and on every stair evaluates the field profile at the stair midpoint, diagonalises the stair Hamiltonian, obtains the dressed coupling operator from rlx_phonon.m in that eigenbasis, and records the expectation values of the coil operators every parameters.nout stairs.

Syntax

        answer=pulsed_field(spin_system,parameters,H,R,K)

Parameters

   parameters.field_prof - function handle returning the field
                           in Tesla at a time in seconds

   parameters.hzeeman    - Zeeman operator per Tesla, rad/s/T,
                           Hilbert space, supplied by the con-
                           text when 'zeeman_op' is requested
                           in parameters.needs

   parameters.timestep   - stair width, seconds

   parameters.nsteps     - number of stairs

   parameters.coil       - Hermitian Hilbert space observable
                           operator or a cell array of them

   parameters.phonon_x   - spin-phonon coupling operator, see
                           rlx_phonon.m

   parameters.phonon_i0  - phonon spectral density prefactor,
                           see rlx_phonon.m

   parameters.phonon_alpha - phonon spectral density exponent,
                             1 or above, see rlx_phonon.m

   parameters.nout       - number of stairs between recorded
                           observable values

   H - Hamiltonian received from the context function, Hilbert
       space, containing the Zeeman term at sys.magnet=1 Tesla;
       the function removes that term and adds the field of
       each stair itself

   R - relaxation superoperator received from the context
       function; ignored, the spin-phonon dissipator is built
       here at every stair

   K - kinetics superoperator received from the context
       function; ignored

Outputs

   answer.t     - column of recording times, seconds

   answer.field - column of field values at those times, Tesla

   answer.obs   - matrix of observable expectation values, one
                  column per coil, at the recording times

Examples

See the pulsed-field magnetometry case studies in the examples/giant_spin/case_studies/ho_twelfth_rank directory: a J=8 holmium giant spin under linear, piecewise linear, spline, and sinusoidal field profiles, its powder average, a dimer with four exchange coupling tensor types, and a manganese trimer in effective bases, all with thermal equilibrium curves for comparison.

Notes

The sequence works in zeeman-hilb formalism under the crystal and powder contexts, which assemble the anisotropic part of the Hamiltonian; the liquid context drops that part, and with it the crystal field of a giant spin. The context must be called with the labframe assumption set, so that H and the Zeeman operator are built consistently; the powder context must be called with parameters.sum_up=false because the answer is a structure; additional rotating frames (parameters.rframes) and frequency offsets (parameters.offset) are not supported because the field operator is added in the laboratory frame. The temperature of the phonon bath is inter.temperature.

sys.magnet must be 1 Tesla, so that parameters.hzeeman is the Zeeman operator per Tesla; the Hamiltonian received from the context then contains the Zeeman term at 1 Tesla, which this function removes before adding the field on each stair. The initial state is the thermal equilibrium of the field-free Hamiltonian at the temperature of the phonon bath.

The Hamiltonian on each stair uses the field at the midpoint of the stair; answer.field is the profile evaluated at the recording times, which are the ends of the recorded stairs.

See also

rlx_phonon.m, crystal.m, powder.m, fieldscan_magn.m, stevens.m, stev2sph.m, equilibrium.m, Built-in_experiments

Version 2.13, authors: Ilya Kuprov