Operator.m

Generates Hilbert space operators and Liouville space superoperators from their human-readable descriptions. Syntax:

    A=operator(spin_system,operators,spins,operator_type)

The function supports two types of calls:

1. If an operator is given as a single string and spins are named by passing a single string, e.g.

    sum_Lz=operator(spin_system,'Lz','13C');

then the function returns the sum of the corresponding single-spin operators (in Hilbert space) or superoperators (in Liouville space) on all spins with that name. Valid labels for operators in this type of call are

    'E', 'Lz', 'L+', 'L-', 'Tl,m'

where l and m are integers. In the latter case a spherical tensor operator or superoperator is returned. Valid labels for spins are standard isotope names as well as

    'electrons', 'nuclei', 'all'

2. If the operators are supplied as a cell array of strings and spins as a cell array of numbers, a product operator (in Hilbert space) or superoperator (in Liouville space) is produced, e.g.

    LzSp=operator(spin_system,{'Lz','L+'},{1,2});

will return LzL+ operator in Hilbert space and [LzL+,_] commutation superoperator in Liouville space, with Lz on spin 1 and L+ on spin 2. Valid labels for operators in the cell array are

    'E', 'Lz', 'L+', 'L-', 'Tl,m'

where l and m are integers. In the latter case a spherical tensor operator or superoperator is included into the product.

In Liouville space calculations, operator_type can be set to:

                 'left' - produces left side product superoperator

                'right' - produces right side product superoperator

                 'comm' - produces commutation superoperator (default)

                'acomm' - produces anticommutation superoperator

In Hilbert space calculations operator_type parameter is not permitted.

WARNING: do not try to obtain product commutation superoperators by multiplying them up! It is easy to see that

\({{\hat{\hat{O}}}^{2}}=\left[ \hat{O},\left[ \hat{O},\_ \right] \right]\ne \left[ {{{\hat{O}}}^{2}},\_ \right]\)

If you require a commutation superoperator corresponding to a multi-spin operator, use the syntax given in the Section 2 above.

Examples

1. Generating the AxBx operator in a two-spin system

Both components are Cartesian and must therefore be translated into the convention above:

\(\begin{matrix} {{{\hat{A}}}_{\text{X}}}=\frac{{{{\hat{A}}}_{+}}+{{{\hat{A}}}_{-}}}{2};\text{ }{{{\hat{B}}}_{\text{X}}}=\frac{{{{\hat{B}}}_{+}}+{{{\hat{B}}}_{-}}}{2} \\ {{{\hat{A}}}_{\text{X}}}{{{\hat{B}}}_{\text{X}}}=\frac{1}{4}\left( {{{\hat{A}}}_{\text{+}}}{{{\hat{B}}}_{+}}+{{{\hat{A}}}_{+}}{{{\hat{B}}}_{-}}+{{{\hat{A}}}_{-}}{{{\hat{B}}}_{+}}+{{{\hat{A}}}_{-}}{{{\hat{B}}}_{-}} \right) \\ \end{matrix}\)

The Spinach code would therefore be:

    AxBx=(operator(spin_system,{'L+','L+'},{1,2})+...
          operator(spin_system,{'L+','L-'},{1,2})+...
          operator(spin_system,{'L-','L+'},{1,2})+...
          operator(spin_system,{'L-','L-'},{1,2}))/4;

Revision 3142, authors: Ilya Kuprov, Luke Edwards