Operator.m

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

Syntax

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

Parameters

This function supports three types of calls:

1. If operators is a string and spins is a string

                    operators='Lz'; spins='13C';

the function returns the sum of the corresponding single-spin operators (Hilbert space) or superoperators (Liouville space) on all spins of that type. Valid labels for states in this type of call are 'E' (identity), 'Lz', 'Lx', 'Ly', 'L+', 'L-', 'Tl,m' (irreducible spherical tensor, l and m are integers), 'CTx', 'CTy', 'CTz', 'CT+', 'CT-' (central transition operators in the Zeeman basis). Valid labels for spins are standard isotope names, as well as 'electrons', 'nuclei', and 'all'.

2. If operators is a string and spins is a vector

                    operators='Lz'; spins=[1 2 4];

the function returns the sum of all single-spin operators (Hilbert space) or superoperators (Liouville space) for all spins with the specified numbers. Valid labels for operators are the same as in Item 1 above.

3. If operators is a cell array of strings and spins is a cell array of numbers

                    operators={'Lz','L+'}; spins={1,2};

then a product operator (Hilbert space) or its superoperator (Liouville space) is produced. In the case above, Spinach will generate LzS+ in Hilbert space or its specified superoperator in Liouville space. Valid labels for operators are the same as in Item 1 above.

Bosonic modes - cavities, phonon modes, and transmons - have their own operator labels: 'E' (identity), 'C' (creation), 'A' (annihilation), 'N' (population number), products such as 'CCAA', and 'BL#' for the projector onto the #-th Fock level, counted from 1, so that 'BL1' is the vacuum. These may be mixed with spin labels in product operator calls, for example operators={'L+','A'}; spins={1,2} builds one of the two flip-flop terms of a Jaynes-Cummings coupling.

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 ignored, and the operator itself is always returned.

The format parameter refers to the format of the output:

            'csc' - returns a Matlab sparse matrix

            'xyz' - returns a [rows, cols, vals] array

Outputs

   A   - a CSC sparse (default) or a [rows, cols, vals] repre-
         sentation of a spin operator or superoperator.

Examples

1. L+ on spin 3

    Lp=operator(spin_system,{'L+'},{3});

2. A sum of Lx on all 15N spins in the system

    Lx=operator(spin_system,'Lx','15N');

3. AxBx between spin 2 and spin 5

    AxBx=operator(spin_system,{'Lx','Lx'},{2,5});

Operators will be generated in Hilbert space and superoperators in Liouville space.

Notes

WARNING: do not try to obtain product commutation superoperators by multiplying up single-spin commutation superoperators. 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 Section 3 above.

Operator caching is supported, add 'op_cache' to sys.enable array to enable; make sure your scratch storage is fast.

See also

unit_state.m, unit_oper.m, mprealloc.m, singlet.m, equilibrium.m, state.m, human2opspec.m, bos2ist.m, boson_mono.m, boson_ortho.m, centrans.m, ct2ist.m, enlev2bm.m, enlev2ist.m, hamiltonian.m, kinetics.m, lindbladian.m, oper2bm.m, oper2ist.m, orientation.m, propagator.m, relaxation.m, sin_tran.m, weyl.m, coherent.m, device.m, Kernel_functions

Version 2.8, authors: Ilya Kuprov, Luke Edwards, Dmitry Savostyanov