Plain Text Operator Notation

The Plain Text Operator Notation (PTON) defines concise plain-text expressions for Product Operators used to describe spin experiments and states.

PTON is necessary because on most keyboards, mathematical typesetting and characters are not available, in particular the sub- and superscripts and Greek characters. PTON is defined for the most common bases used in NMR, including the Cartesian Basis, Spherical Tensor Basis, Single Element Basis, and the less common Semi-Cartesian Basis.

PTON is chosen to be as close as possible to the notation used in the literature to describe NMR experiments. Indeed, some other NMR software packages implement some of the same textual operator notation described here.

PTON is formatted here with a fixed-width font in bold for clarity, for example, this text indicates a PTON operator: 2I1xI2y for the product operator \(2I_{1x}I_{2y}\). \(2 I_{1x}\)

Cartesian Product Operators

The Cartesian product operators can be written in PTON as, for example, I1x, I1y, 2I1xI2y for \(I_{1x}\), \(I_{1y}\), \(2I_{1x}I_{2y}\), and so on. The prefix factor is a scale that ensures that the norm of the operators remains uniform. That is to say, in a three-spin system, \(|I_{1y}|\) is the same as \(|2I_{1y}I_{2x}|\) and \(|4I_{1y}I_{2x}I_{3z}|\).

Non-cartesian Product Operators

Operators of other product bases, such as the Spherical Tensor Basis, Single Element Basis and the Semi-Cartesian Basis can also be represented easily using PTON. This is done by using the characters pmab in place of the symbols \(+,-,\alpha,\beta\). Thus, the PTON I1p means \(I_{1}^{+}\), I1m is \(I_{1}^{-}\), I1a is \(I_{1}^{\alpha}\), and I1b is \(I_{1}^{\beta}\); I1xI2b is \(I_{1x}I_{2}^{\beta}\), and so on…

Quantum Information Notation

There are also some facilities in the PTON interpreter to support the notation for spin down \(\lvert\downarrow\rangle\) and spin up \(\lvert\uparrow\rangle\) pure states, also written as \(\lvert 0\rangle\) and \(\lvert 1\rangle\), respectively. These are written in pton using the vertical bar and the greater-than and less-than symbols: \(|01><11|\).

PTON for the LISA basis

The \(PTON\) has a special (and perhaps slightly awkward) notation to use operators from the LISA Basis.

The identity elements of the LISA basis are simply denoted by E followed by the length of the corresponding matrix, ie, for a single spin 1/2, E2, for a two-spin 1/2, E4, and so one. In contexts where the system size is known, Id also signifies the identity operator, \(Id\), for the system size.

The next LISA basis operators are grouped first by their linearity, or related spin, the first spin’s elements are {1}_11m, {1}_10, {1}_11p, the second spin’s _{2}_11m, _{2}_10, _{2}_11p, and so on. Since the names of these elements must be unique within PTON, a small trick is applied to distinguish the matrices associated with 1-spin systems from 2-spin and 3-spin systems: adding an underscore for every additional spin in the system, so for the first spin in a two-spin system, the associated matrices become _{1}_11m, _{1}_10, _{1}_11p, and the first spin in a three-spin system become __{1}_11m, __{1}_10, __{1}_11p. Notice the only difference in the names is the number of underscores which preceed the label. This pattern is regular for linear and bi-linear operators.

The bi-linear terms are labeled similarly, the matrices associated with both spins one and two in a two-spin system are _{1,2}_00, _{1,2}_11m,, _{1,2}_10, _{1,2}_11p, _{1,2}_22m, and so on. The same rule about prepending underscores _ to distinguish matrices of different sizes applies.

For three spin systems, the trilinear operators are named \(τ\) with a subscript. In PTON, this tau can be written as τ or \tau, giving names to their operators in PTON \tau1_11m, … through \tau4.