# 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 \(2I_{1x}I_{2y}\).

## 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`

.