Your first spin Hamiltonian¶

Create one electronic spin $S=1/2$ in a magnetic field along $z$.

$$ H=-g\mu_B B_zS_z. $$

Fields are in T and energies in meV. Spin operators are dimensionless. For positive $g$ and $B_z$, $|+1/2\rangle$ is the ground state.

In [1]:

import plotly.io as pio

from spinfinity import SpinSys, hamil, plotting

pio.renderers.default = "plotly_mimetype+notebook"
# Use "fixed" for compact plots, or "no" to suppress display.
plotting.set_display_mode("responsive")

spin = SpinSys([0.5])
spin.B = [0.0, 0.0, 0.5]  # T
spin.BTip = [0.0, 0.0, 0.0]  # T
spin.gfactorvector = [2.0]

import plotly.io as pio from spinfinity import SpinSys, hamil, plotting pio.renderers.default = "plotly_mimetype+notebook" # Use "fixed" for compact plots, or "no" to suppress display. plotting.set_display_mode("responsive") spin = SpinSys([0.5]) spin.B = [0.0, 0.0, 0.5] # T spin.BTip = [0.0, 0.0, 0.0] # T spin.gfactorvector = [2.0]

Solve the Hamiltonian¶

calc_EigStates stores results on spin; it does not return them. Use rebuildH=True to construct the Hamiltonian from the current parameters.

E_sort contains unshifted energies, E_All is shifted to the ground state, and E_All_inGHz expresses those energy differences as ordinary frequencies.

In [2]:

hamil.calc_EigStates(spin, rebuildH=True)

print("Hamiltonian (meV):")
print(spin.H)
print("Unshifted energies (meV):", spin.E_sort)
print("Relative energies (meV):", spin.E_All)

frequency = spin.E_All_inGHz[1] - spin.E_All_inGHz[0]
print(f"Transition frequency: {frequency:.3f} GHz")

for state in spin.states:
    print(state)

hamil.calc_EigStates(spin, rebuildH=True) print("Hamiltonian (meV):") print(spin.H) print("Unshifted energies (meV):", spin.E_sort) print("Relative energies (meV):", spin.E_All) frequency = spin.E_All_inGHz[1] - spin.E_All_inGHz[0] print(f"Transition frequency: {frequency:.3f} GHz") for state in spin.states: print(state)

Hamiltonian (meV):
[[-0.02894191+0.j  0.        +0.j]
 [ 0.        +0.j  0.02894191+0.j]]
Unshifted energies (meV): [-0.02894191  0.02894191]
Relative energies (meV): [0.         0.05788382]
Transition frequency: 13.996 GHz
|0> (E = 0.0) = +1.000+0.000j|0.5>
|1> (E = 0.0579) = +1.000+0.000j|-0.5>

Inspect the eigenstates¶

The local basis is ordered from $+S$ to $-S$. Columns of eigVectors contain eigenvectors. Squared coefficients give product-basis probabilities, not thermal occupations.

The package plotting function displays automatically. We temporarily suppress that display, adjust the returned figure to the website column, and display it once.

In [3]:

hamil.plot_EigenMatrix(spin);

hamil.plot_EigenMatrix(spin);

Change a parameter and rebuild¶

Changing spin.B alone does not update cached results. Rebuild and diagonalize again. Doubling this field doubles the spin-half transition frequency.

In [4]:

spin.B = [0.0, 0.0, 1.0]
hamil.calc_EigStates(spin, rebuildH=True)

frequency = spin.E_All_inGHz[1] - spin.E_All_inGHz[0]
print(f"Transition frequency at 1 T: {frequency:.3f} GHz")

spin.B = [0.0, 0.0, 1.0] hamil.calc_EigStates(spin, rebuildH=True) frequency = spin.E_All_inGHz[1] - spin.E_All_inGHz[0] print(f"Transition frequency at 1 T: {frequency:.3f} GHz")

Transition frequency at 1 T: 27.992 GHz

Zeeman diagram¶

The response of the spin system to the external magnetic field is best visible via a Zeeman diagramm

In [5]:

hamil.plot_ZeemanDiagramm(spin);

hamil.plot_ZeemanDiagramm(spin);