====================
DROPS Representation
====================

Beyond Vectors: The DROPS Representation of Operators
=====================================================

The states of uncoupled spins ½ can be completely described using
three-dimensional Bloch vectors (commonly called “magnetization
vectors” in NMR). However, to describe the general and more
interesting case of coupled spins, the Bloch vector picture is
insufficient and more advanced concepts based on operators (such as
product operators, density operators, Hamiltonian operators etc...)
are used.  While these operators and their time evolution can easily
be calculated and manipulated by computers, they are difficult to
visualize.  In the DROPS representation [GARON2015]_, operators are
displayed as a set of droplets, where each droplet corresponds to some
operators involving a defined set of spins (individual spins, pairs of
spins, etc...)

.. drops:: I1x + I1xI2y + I2x
   :nspin: 2
   :aspect: 250
   :width: 100%
   :view: p170
   :align: center

This chapter presents the general properties of the DROPS
representation.  Familiarity with the shapes of certain characteristic
droplets requires a little practice, but it can be fun and highly
rewarding.  DROPS provides a very intuitive visualization of spin
dynamics which is helpful in understanding the concepts of magnetic
resonance spectroscopy that form the basis of modern pulse sequences
design, such as coherence order, phase cycles, polarization transfer
etc...  In the following, we summarize and illustrate the most
important properties of the DROPS display.  For the mathematical
background of the DROPS representation, see :ref:`Mathematical
Background of the DROPS Visualization`.

Droplets representing Linear Operators
======================================

The droplets in :numref:`table_drops_props_I1xyz` are closely related to
the Bloch vector picture.

.. _table_drops_props_I1xyz:

.. list-table:: Bloch-vector-similar droplets: :pton:`I1x, I1y, I1z`

   * - :drop:`/3 I1x`
     - :drop:`/3 I1y`
     - :drop:`/3 I1z`


For a linear Cartesian operator (i.e. operators such as :pton:`I1x` or
:pton:`I1z` which involve only a single spin) the corresponding
droplet consists of two spheres: a positive sphere (red) and a
negative sphere (green). As illustrated below, the same is true for
sums (:numref:`fig_drops_props_I1xz`) and differences
(:numref:`fig_drops_props_I1xy`) of such operators having real
prefactors (associated with the same spin, e.g. :math:`I_1`).

Such combinations of linear operators can also be represented as a
single vector (the Bloch vector or magnetization vector in the case of
the density operator).

.. list-table:: 

   * - .. drops:: 0.3*I1x + 0.9*I1z
          :nspin: 1
          :width: 30%
          :view: C
          :name: fig_drops_props_I1xz
     - .. drops:: 0.717*I1x - 0.717*I1y
          :nspin: 1
          :width: 30%
          :view: A
          :name: fig_drops_props_I1xy

An important property of the DROPS representation is that the Bloch
vector is inscribed in the red sphere and the base of the vector
touches the green sphere. Hence, the orientation of the droplet
(defined as the direction of the vector connecting the centers of the
green and red spheres) is always identical to the orientation of the
corresponding Bloch vector.

For a single spin, the :ref:`DROPS representation` is closely related
to the well-known vector picture. Therefore, a droplet representing
the linear operators associated with a given spin is shown in
:ref:`transparent mode <Magnetization Vectors>` and the corresponding
vector is shown simultaneously, `inside` the droplet.

|html_clear|
|latex_clear|


Droplets of Cartesian Product Operators Are Real-Valued
=======================================================

Droplets representing standard :term:`Cartesian product operators
<Cartesian Product Operator>` are real and thus can have only the
colors red (for positive values) and/or green (for negative
values). The color coding of the droplets is explained in :ref:`Color
Code for Droplets`.

:numref:`drops_props_cartreal` shows some individual Cartesian product
operators, involving 0 spins (term proportional to the identity
operator E, :numref:`drops_props_id`), a single spin (linear operators
:numref:`drops_props_ix`, :numref:`drops_props_iz`) and two spins
(bilinear operators :numref:`drops_props_ixix`,
:numref:`drops_props_ixiy`, :numref:`drops_props_ixiz`).


.. list-table:: 
   :name: drops_props_cartreal

   * - .. drops:: Id
          :prefs: Show Droplet Labels=false
          :width: 50%
          :name: drops_props_id
     - .. drops:: 2I1xI2x
          :width: 50%
          :view: p170
          :name: drops_props_ixix
   * - .. drops:: I1x
          :width: 50%
          :name: drops_props_ix
     - .. drops:: 2I1xI2y
          :width: 50%
          :view: p170
          :name: drops_props_ixiy
   * - .. drops:: I1z
          :width: 50%
          :name: drops_props_iz
     - .. drops:: 2I1xI2z
          :width: 50%
          :view: p170
          :name: drops_props_ixiz


.. |dE2| drops:: 0.5*E2
   :nspin: 1
   :width: 49%
   :caption: 


Droplets of Hermitian Operators Are Red/Green
=============================================

Droplets representing :term:`Hermitian operators <Hermitian Operator>`
(corresponding to real combinations of Cartesian product operators)
can be recognized easily because they only have the colors red and
green.  Examples of some droplets representing Hermitian operators are
shown in :numref:`drops_props_hermops`.

.. list-table:: Hermetian Operators
   :name: drops_props_hermops

   * - .. drops:: 2I1yI2x - 2I1xI2y
          :width: 50%
          :view: p170
     - .. drops:: 2I1xI2x + 2I1yI2y
          :width: 50%
          :view: p170
   * - .. drops:: 2I1xI2x - 2I1yI2y
          :width: 50%
          :view: p170
     - .. drops:: 2I1xI2y + 2I1yI2x
          :width: 50%
          :view: p170
   * - .. drops:: 2I1xI2x + 2I1yI2y + 2I1zI2z
          :width: 50%
          :view: p170
     - .. drops:: I1xI2x + I1yI2y - 2I1zI2z
          :width: 50%
          :view: Cp170

Droplets of Skew-Hermitian Operators Are Yellow/Blue
====================================================

Multiplying Hermitian operator by the imaginary unit :math:`i` results
in so-called skew-Hermitian operators. The corresponding droplet
functions are purely imaginary. Their droplets can only have the
colors yellow (if the imaginary part of the droplet function is
positive) or blue (if the imaginary part of the droplet function is
negative). Examples of some droplets representing skew-Hermitian
operators are shown in :numref:`drops_props_skew`.


.. list-table:: Skew-Hermetian Operators
   :name: drops_props_skew

   * - .. drops:: i*I1x
          :width: 50%
     - .. drops:: i*I1y
          :width: 50%
   * - .. drops:: i*2I1xI2z
          :width: 50%
          :view: p170
     - .. drops:: i*2I1yI2z
          :width: 50%
          :view: p170


Droplets of Operators Consisting of Hermitian and Skew-Hermitian Components
===========================================================================

:numref:`drops_props_skewcomp` presents important examples of
operators which are neither purely Hermitian nor purely
skew-Hermitian, i.e. they can be decomposed into Hermitian and
skew-Hermitian parts.

.. list-table:: Skew-Hermetian Operators
   :name: drops_props_skewcomp
   :class: nocol no-cap-num

   * - .. drops:: I1p
          :width: 30%
          :prefs: Show Axes=false
     - :math:`=`
     - .. drops:: I1x
          :width: 30%
          :prefs: Show Axes=false
     - :math:`+`
     - .. drops:: i*I1y
          :width: 30%
          :prefs: Show Axes=false
   * - .. drops:: 2I1pI2z
          :width: 30%
          :view: p170
          :prefs: Show Axes=false
     - :math:`=`
     - .. drops:: 2I1xI2z
          :width: 30%
          :view: p170
          :prefs: Show Axes=false
     - :math:`+`
     - .. drops:: i*2I1yI2z
          :width: 30%
          :view: p170
          :prefs: Show Axes=false
   * - .. drops:: 2I1pI2p
          :width: 30%
          :view: p170
          :prefs: Show Axes=false
     - :math:`=`
     - .. drops:: 2I1xI2x-2I1yI2y
          :width: 30%
          :view: p170
          :prefs: Show Axes=false
     - :math:`+`
     - .. drops:: i*(2I1xI2y + 2I1yI2x)
          :width: 30%
          :view: p170
          :prefs: Show Axes=false


Non-Selective Rotations of Operators Simply Rotate the Droplets
===============================================================

Rotating an operator by a given angle about a given axis also rotates
the corresponding droplets by the same angle about the same axis. This
is one of the most important features of the :ref:`DROPS
representation`. For example, rotating the operator :pton:`2I1xI2x -
2I1yI2y` by 45° about the z axis results in :pton:`2I1yI2x + 2I1xI2y`
and an additional rotation by 90° about the x axis results in
:pton:`2I1zI2x + 2I1xI2z`.

.. note:: it is far more intuitive to recognize rotations based on the
   initial and final DROPS representations rather than based on the
   initial and final product operators.

.. list-table:: Skew-Hermetian Operators
   :name: drops_props_simp_rot
   :class: nocol no-cap-num

   * - .. drops:: 2I1xI2x-2I1yI2y
          :width: 25%
          :view: p170
     - :math:`\Longrightarrow` (45°z)
     - .. drops:: 2I1yI2x+2I1xI2y
          :width: 25%
          :view: p170
     - :math:`\Longrightarrow` (90°x)
     - .. drops:: 2I1zI2x+2I1xI2z
          :width: 25%
          :view: p170


Characteristic Droplets of Cartesian Product Operators Involving Two Spins
==========================================================================

:numref:`drops_props_tso` shows all of the droplets corresponding to
two-spin operators.

.. list-table:: Two-Spin Operators
   :name: drops_props_tso
   :class: no-cap-num

   * - .. drops:: 2I1xI2x
          :width: 40%
          :view: p170
     - .. drops:: 2I1xI2y
          :width: 40%
          :view: p170
     - .. drops:: 2I1xI2z
          :width: 40%
          :view: p170
   * - .. drops:: 2I1yI2x
          :width: 40%
          :view: p170
     - .. drops:: 2I1yI2y
          :width: 40%
          :view: p170
     - .. drops:: 2I1yI2z
          :width: 40%
          :view: p170
   * - .. drops:: 2I1zI2x
          :width: 40%
          :view: p170
     - .. drops:: 2I1zI2y
          :width: 40%
          :view: p170
     - .. drops:: 2I1zI2z
          :width: 40%
          :view: p170


Droplets of Cartesian Product Operators of the Form :math:`2I_{1a}I_{2a}`
=========================================================================

As shown along the diagonal in :numref:`drops_props_tso`, the bilinear
:term:`product operators <Product Operator>` :pton:`2I1xI2x`,
:pton:`2I1yI2y`, and :pton:`2I1zI2z`, of the general form
:math:`2I_{1a}I_{2a}` with :math:`a ∈ {x, y, z}`, easily be recognized
by their elongated shapes along the :math:`a` axis . They consist of
two red (positive) lobes (oriented in the direction of :math:`a` and
:math:`-a`, respectively), and a small green (negative) toroidal shape
encircling the :math:`a` axis.

.. list-table:: Two-Spin Operators of the Form :math:`2I_{1a}I_{2a}`
   :name: drops_props_tsoaa
   :class: no-cap-num

   * - .. drops:: 2I1xI2x
          :width: 35%
          :view: p170
     - .. drops:: 2I1yI2y
          :width: 35%
          :view: p170
     - .. drops:: 2I1zI2z
          :width: 35%
          :view: p170


Droplets of Cartesian Product Operators of the Form :math:`2I_{1a}I_{2b}`
=========================================================================

The operators :math:`2I_{1a}I_{2b}` with :math:`a≠b` and :math:`a, b ∈
{x, y, z}` are represented by a droplet with two bean-shaped lobes of
opposite signs.  There is a red (positive) lobe and a green (negative)
lobe.  As all operators of this form (:math:`2I_{1a}I_{2b}`) can be
obtained by non-selective 90° rotations of e.g. :pton:`2I1xI2y` (see
:ref:`Challenge 8 <Challenge 8: Rotations of operators and
droplets>`).


Composition
-----------

Here we focus on the operator :pton:`2I1xI2y` to understand the origin
of the characteristic shape of these droplets.  :pton:`2I1xI2y` can be
expressed as a linear combination of the double-quantum operator
:pton:`DQy = I1xI2y+I1yI2x` (symmetric with respect to an exchange of
the two spins) and the zero-quantum operator :pton:`ZQy =
-I1xI2y+I1yI2x` (anti-symmetric with respect to an exchange of the two
spins):

:pton:`2I1xI2y = DQy - ZQy = DQy + (-ZQy) = (I1xI2y + I1yI2x) + (I1xI2y-I1yI2x)`

Visually:

.. list-table:: Two-Spin Operators of the Form :math:`2I_{1a}I_{2a}`
   :name: drops_props_tsoab
   :class: nocol no-cap-num

   * - .. drops:: 2I1xI2y
          :view: p170
     - :math:`=`
     - :math:`DQy`
       
       .. drops:: I1xI2y + I1yI2x
          :view: p170
     - :math:`+`
     - :math:`-ZQy`
       
       .. drops:: I1xI2y - I1yI2x
          :view: p170
                  

Anti-Phase Cartesian Product Operators
--------------------------------------

Similar to :pton:`2I1xI2y`, the antiphase operator :pton:`2I1xI2z` can
be expressed as a sum of a symmetric and an anti-symmetric term:

.. list-table:: Anti-Phase Operator :pton:`2I1xI2z`
   :name: drops_props_tsoab2
   :class: nocol no-cap-num

   * - .. drops:: 2I1xI2z
          :width: 30%
          :view: p170
     - :math:`=`
     - .. drops:: I1xI2z + I1zI2x
          :width: 30%
          :view: p170
     - :math:`+`
     - .. drops:: I1xI2z - I1zI2x
          :width: 30%
          :view: p170

Identifying Kissing Bean Droplets
---------------------------------

As illustrated in :numref:`image_drops_props_beans`, an operator of
the form :pton:`2I_{1a}I_{2b}`, where :math:`a≠b` and :math:`a, b ∈
{x, y, z}`, can always be written as a sum of the form
:math:`2I_{1a}I_{2b} = (I_{1a}I_{2b} + I_{1b}I_{2a}) + (I_{1a}I_{2b} -
I_{1b}I_{2a})`. The droplet corresponding to the symmetric term in the
first bracket is “X-shaped” and consists of two red (positive) lobes
oriented along the axis of the vector sum a+b (dashed red line) and
two green (negative) lobes oriented along the axis of the vector
difference :math:`a-b` (dashed green line). The droplet corresponding
to the anti-symmetric term in the second bracket consists of a red
(positive) and a green (negative) sphere, where the red sphere is
displaced relative to the green sphere in the direction given by the
cross product a×b (yellow arrow). In the DROPS representation of
:pton:`2I_{1a}I_{2b}`, the red (and green) lobes of
(:math:`I_{1a}I_{2b} + I_{1b}I_{2a}`) and (:math:`I_{1a}I_{2b} -
I_{1b}I_{2a}`) merge, forming the characteristic droplet consisting of
a red and green bean-shaped lobe.


.. figure:: _images/page68_drops_of_form_2I1aI2b.png
   :name: image_drops_props_beans

   Identifying kissing-bean droplets.


Right-hand Rule
---------------

The shape and color of the droplet representing a Cartesian product
operators of the form :math:`2I_{1a}I_{2b}`, where :math:`a≠b` and
:math:`a, b ∈ {x, y, z}`, can be constructed and analyzed using the
following right-hand rule:

.. image:: _images/page69_right_hand_rule.png
   :align: right

a) Point the thumb of your right hand in the direction of the unit
   vector :math:`a` and the index finger in the direction of
   :math:`b`.

b) The bisector of the angle formed by the thumb and index finger
   defines the axis of the vector sum :math:`a+b` (dashed red
   line). This defines the long axis of the bean-shaped red lobe of
   the droplet. (Orthogonal to the red axis, the long axis of the
   green bean-shaped lobe is oriented along the axis defined by
   :math:`a-b`.)

c) Relative to the center of the droplet, the red lobe is displaced in
   the direction given by the cross product :math:`a×b` (yellow
   arrow), which is given by the orientation of the middle finger of
   your right hand. (The green lobe is displaced in the opposite
   direction.)


|html_clear|
|latex_clear|


.. _drops_props_kb_summary:

Head-tilt Rule
--------------

Based on the rules summarized on the previous page, a given droplet of
an antiphase operator can be translated back into the form of a
product operator (see :ref:`challenges <Challenge 9: Recognizing
droplets of antiphase operators (A)>`).  As illustrated below, it is
even simpler to recognize which of the involved two single spin
operators is a :math:`z` operator, i.e. to determine if the coherence
is in antiphase with respect to the first or the second spin.  Imagine
the red and green lobes to be a pair of “kissing beans”.  If the
“heads” of the kissing beans are tilted to the left (relative to the z
axis), the component of the first spin operator is :math:`z` and the
component of the second spin is in the transverse plane (i.e. the
antiphase operator has the form :math:`±2I_{1z}I_{2a}` with :math:`a
\in \{x,y\}` ).  Conversely, if the “heads” are tilted to the right,
the second single spin operator is a :math:`z` operator and the
Cartesian component of the first spin is in the transverse plane
(:math:`±2I_{1z}I_{2a}`).  In the general case of operators involving
spins :math:`I_m` and :math:`I_n` with :math:`m<n`, a tilt to the left
indicates that the :math:`I_m` operator is a :math:`z` operator, and
vice versa.

.. table::
   :class: no-cap-num

   +----------------------------+------------------------+
   |Left Tilt                   |Right Tilt              |
   |                            |                        |
   |:drop:`/2^p170 2I1zI2x`     |:drop:`/2^p170 -2I1xI2z`|
   +----------------------------+------------------------+

Here, the “head-tilt rule” is shown for all antiphase operators
involving spins **I1** and **I2**.

Left tilt: Antiphase coherence of spin **I2** with respect to spin
**I1**

.. table::
   :class: no-cap-num

   +-----------------+---------------+------------------------+----------------------+
   |:drop:`/4^p170   |:drop:`/4^p170 |:drop:`/4^p170 -2I1zI2x`|:drop:`/4^p170        |
   |2I1zI2x`         |2I1zI2y`       |                        |-2I1zI2y`             |
   +-----------------+---------------+------------------------+----------------------+

Right tilt: Antiphase coherence of spin **I1** with respect to spin
**I2**

.. table::
   :class: no-cap-num

   +---------------+---------------+----------------+----------------+
   |:drop:`/4^p170 |:drop:`/4^p170 |:drop:`/4^p170  |:drop:`/4^p170  |
   |2I1xI2z`       |2I1yI2z`       |-2I1xI2z`       |-2I1yI2z`       |
   +---------------+---------------+----------------+----------------+

.. note:: (By the way, an interesting study of human head-turning
          asymmetry appeared in Nature 421, 711, 2003,
          https://doi.org/10.1038/421711a)


Droplets of Cartesian Product Operators Involving Three Spins
=============================================================

Tri-linear Cartesian product operators are represented by up to four
droplets (:math:`τ_1`, :math:`τ_2`, :math:`τ_3`, :math:`τ_4`) in the
:ref:`LISA basis`.  Here, the :ref:`DROPS representation` of
characteristic Cartesian operators are displayed for the following
cases: (a) three identical components, (b-d) two identical components
and (e) all components different.


.. list-table:: Three-Spin Operators
   :name: drops_props_threeso
   :class: nocol no-cap-num

   * - .. drops:: 4I1xI2xI3x
          :width: 40%
          :view: Dq
          :caption: :pton:`(a) \tau_1 \\ 4I1xI2xI3x`
     - .. drops:: 4I1xI2xI3y
          :width: 40%
          :view: Dq
          :caption: :pton:`(b) \tau_{1,2} \\ 4I1xI2xI3y`
     - .. drops:: 4I1xI2yI3x
          :width: 40%
          :view: Dq
          :caption: :pton:`(c) \tau_{1,2,3} \\ 4I1xI2yI3x`
   * - .. drops:: 4I1yI2xI3x
          :width: 40%
          :view: Dq
          :caption: :pton:`(d) \tau_{1,2,3} \\ 4I1yI2xI3x`
     - .. drops:: 4I1xI2yI3z
          :width: 40%
          :view: Dq
          :caption: :pton:`(e) \tau_{1,2,3,4} \\ 4I1xI2yI3z`
     - 

Rotating Droplets Involving Three Spins
---------------------------------------

Based on the characteristic droplets for trilinear Cartesian Product
operators shown on the previous page, the droplets of the remaining
trilinear Cartesian product operator can be obtained by rotations around
the x, y, or z axis (up to a possible change of sign, which simply
corresponds to a change of colors).

For example, the operator :pton:`4I1zI2yI3z` (where the first and last
components are identical) can be obtained from the given operator (c)
:pton:`4I1xI2yI3x` by a non-selective 90° rotation around the y
axis.  Hence, the DROPS represen-tation of :pton:`4I1zI2yI3z` can be
found by rotating each of the droplets of :pton:`4I1xI2yI3x` by 90°
around the y axis.

.. list-table:: Rotation of Three-Spin Operators
   :name: drops_props_threesorot
   :class: nocol no-cap-num

   * - .. drops:: 4I1xI2yI3x
          :width: 50%
          :view: Dq
     - :math:`\Longrightarrow \\ \circlearrowright` (90°y)
     - .. drops:: 4I1zI2yI3z
          :width: 50%
          :view: Dq


Droplet Symmetry and Coherence Order Rules
==========================================

The droplet of an operator with well defined coherence order :math:`p`
can be easily recognized based on the following rules:

Rule 1 - Shape Invariance of Coherence Order
--------------------------------------------

.. _drops_props_rule_1:

Disregarding the color, the shape of the droplet with a well-defined
coherence order :math:`p` is invariant under rotations around the z
axis, i.e. the shape is not changed by a z rotation.

.. _drops_props_rule_2:

Rule 2 - Color Implies Coherence Order
--------------------------------------

The coherence order :math:`p` of a droplet can be inferred from the
color of a droplet, accoring to the following rules, `2.1` - `2.6`.

.. _drops_props_rule_21:

Rule 2.1 - :math:`p=0` Droplets Are Invariant Around :math:`z`
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

A droplet of coherence order :math:`p=0` does not change its color if
it is rotated by an arbitrary angle α around the z axis, as
illustrated by the following examples in
:numref:`drops_props_table_21`.

.. list-table:: Coherence Order :math:`p=0` z-Rotations
   :name: drops_props_table_21
   :header-rows: 1
   :stub-columns: 1
   :widths: 1 5 1 5
   :class: nocol no-cap-num

   * - :math:`p`
     - Droplet
     - z-Rot
     - Droplet
   * - 0
     - .. drops:: I1xI2x + I1yI2y
          :width: 50%
          :view: Cp200
     - :math:`\xrightarrow{\alpha_z}`
     - .. drops:: I1xI2x + I1yI2y
          :width: 50%
          :view: Cp200
          :caption: :pton:`I1xI2x + I1yI2y`
   * - 0
     - .. drops:: I1pI2m
          :width: 50%
          :view: Cp200
     - :math:`\xrightarrow{\alpha_z}`
     - .. drops:: I1pI2m
          :width: 50%
          :view: Cp200
          :caption: :pton:`I1pI2m`



.. _drops_props_rule_22:

Rule 2.2 - Sign of Coherence Order
++++++++++++++++++++++++++++++++++

A droplet of defined coherence order :math:`p ≠ 0` is rainbow-colored
(where :math:`p` is a non-zero integer with either positive or
negative sign).  For positive coherence order (:math:`p>0`), the
colors evolve from red to yellow to green to blue when moving
counter-clockwise around the droplet.  For negative coherence order,
the colors progress in the opposite direction, as illustrated by the
following examples.

.. table::
   :class: no-cap-num

   +---------------------------+-----------+---------------------------+-----------+
   |Operator                   |:math:`p`  |Operator                   |:math:`p`  |
   +---------------------------+-----------+---------------------------+-----------+
   |:drop:`/3 I1p`             |:math:`+1` |:drop:`/3 I1m`             |:math:`-1` |
   +---------------------------+-----------+---------------------------+-----------+
   |:drop:`/3^p170 2I1pI2p`    |:math:`+2` |:drop:`/3^p170 2I1mI2m`    |:math:`-2` |
   +---------------------------+-----------+---------------------------+-----------+
   |:drop:`/3^p180 2I1pI2pI3p` |:math:`+3` |:drop:`/3^p180 2I1mI2mI3m` |:math:`-3` |
   +---------------------------+-----------+---------------------------+-----------+

.. _drops_props_rule_23:

Rule 2.3 - Order of Coherence Order
+++++++++++++++++++++++++++++++++++

For a non-zero, well-defined coherence order :math:`p`, the absolute
value :math:`|p|` of the coherence order of a droplet is simply given
by the number of rainbows encountered when moving once around the z
axis.  Remember that according to :ref:`(2.2) <drops_props_rule_22>`,
the sign of :math:`p` is given by the direction that the rainbow
progresses around the z axis, from red to yellow to green to blue.

.. table::
   :class: no-cap-num

   +----------------------------+-----------+-----------------------------+-----------+
   |Operator                    |:math:`p`  |Operator                     |:math:`p`  |
   +----------------------------+-----------+-----------------------------+-----------+
   |:drop:`/3 I1p`              |:math:`+1` |:drop:`/3 I1m`               |:math:`-1` |
   +----------------------------+-----------+-----------------------------+-----------+
   |:drop:`/3^p170 2I1pI2pI3m`  |:math:`+1` |:drop:`/3^p170 2I1mI2z`      |:math:`-1` |
   +----------------------------+-----------+-----------------------------+-----------+
   |:drop:`/3^p170 2I1mI2mI3z`  |:math:`-2` |:drop:`/3^p170 2I1mI2mI3m`   |:math:`-3` |
   +----------------------------+-----------+-----------------------------+-----------+

.. _drops_props_rule_24:

Rule 2.4 - Coherence Order Invariance under Z-Rotation
++++++++++++++++++++++++++++++++++++++++++++++++++++++

A droplet of coherence order :math:`p` does not change its appearance
if it is rotated by integer multiples of :math:`2\pi / p` around the z
axis. In the example shown below, :math:`p=+1` and hence rotations by
integer multiples of :math:`2π/(+1)= 2π =360°` leave the droplet
invariant.

.. table::
   :class: no-cap-num

   +-----------------------+-----------------------------------+------------------------------------+
   |                       |                                   |:drop:`exp(-i*pi/2)*I1p`            |
   |                       |:math:`\{(π/2)\}_z = 90°z`         |                                    |
   |                       |                                   |                                    |
   |                       |                                   |                                    |
   |                       |:math:`\Longrightarrow`            |                                    |
   |                       |                                   |                                    |
   |                       +-----------------------------------+------------------------------------+
   |:drop:`I1p`            |:math:`\{(π)\}_z = 180°z`          |:drop:`exp(-i*pi)*I1p`              |
   |                       |                                   |                                    |
   |                       |:math:`\Longrightarrow`            |                                    |
   |                       +-----------------------------------+------------------------------------+
   |                       |:math:`\{(3π/2)\}_z = 270°z`       |:drop:`exp(-i*3*pi/2)*I1p`          |
   |                       |                                   |                                    |
   |                       |:math:`\Longrightarrow`            |                                    |
   |                       |                                   |                                    |
   |                       +-----------------------------------+------------------------------------+
   |                       |:math:`\{(2π)\}_z = 360°z`         |:drop:`exp(-i*2*pi)*I1p`            |
   |                       |                                   |                                    |
   |                       |:math:`\Longrightarrow`            |                                    |
   |                       |                                   |                                    |
   +-----------------------+-----------------------------------+------------------------------------+


.. _drops_props_rule_242:

Rule 2.4 pt. 2
++++++++++++++

As stated in :ref:`(2.4) <drops_props_rule_24>`, a droplet of
coherence order :math:`p≠0` does not change its appearance if it is
rotated by integer multiples of :math:`2π/p` around the z axis. (Note
that for :math:`p=0`, the droplet can be rotated by any angle around
the z axis without changing its shape or color, as stated in
properties :ref:`(1) <drops_props_rule_1>` and :ref:`(2.1)
<drops_props_rule_21>`.) This is illustrated by the examples shown
below for well-defined coherence orders :math:`p` of :math:`0, +1,
+2,` and :math:`+3`.

.. table::
   :class: no-cap-num

   +-------------+---------------------------+-------------------------------+
   |:math:`p=0`  |:drop:`/2^p200 I1pI2m`     |Invariant under any rotation   |
   |             |                           |                               |
   |             |                           |                               |
   +-------------+---------------------------+-------------------------------+
   |:math:`p=+1` |:drop:`/2[Magnetization    |Invariant under :math:`\pm     |
   |             |Vectors=false] I1p`        |360°,\pm 720°,...`             |
   |             |                           |                               |
   |             |                           |                               |
   |             |                           |                               |
   +-------------+---------------------------+-------------------------------+
   |:math:`p=+2` |:drop:`/2^p170 2I1pI2p`    |Invariant under :math:`\pm     |
   |             |                           |180°,\pm 360°,...`             |
   |             |                           |                               |
   |             |                           |                               |
   +-------------+---------------------------+-------------------------------+
   |:math:`p=+3` |:drop:`/2^p170 2I1pI2pI3p` |Invariant under :math:`\pm     |
   |             |                           |120°\pm 240°,...`              |
   |             |                           |                               |
   |             |                           |                               |
   +-------------+---------------------------+-------------------------------+


.. _drops_props_rule_25:

Rule 2.5 - Hermetian Mixtures of Coherence Orders Under Z-Rotaion
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Hermitian operators comprised of a mixture of coherence orders
:math:`±p` (such as :math:`±2`) with :math:`|p|≠0`, do not change
their appearance if they or their corresponding operators are rotated
by integer multiples of :math:`2π/p = 360°/p` around the z axis. This
is illustrated by the examples shown below for coherence orders
:math:`p` of :math:`±1, ±2,` and :math:`±3`.

.. table::
   :class: no-cap-num

   +----------------+------------------------------------+----------------------------------------+
   |:math:`p=\pm 1` |:drop:`/2 (I1p+I1m)/2`              |Invariant under :math:`\pm 360°,\pm     |
   |                |                                    |720°,...`                               |
   +----------------+------------------------------------+----------------------------------------+
   |:math:`p=\pm 2` |:drop:`/2^p170 I1pI2p+I1mI2m`       |Invariant under :math:`\pm 180°,\pm     |
   |                |                                    |360°,...`                               |
   +----------------+------------------------------------+----------------------------------------+
   |:math:`p=\pm 3` |:drop:`/2^p170 I1pI2pI3p+I1mI2mI3m` |Invariant under :math:`\pm 120°,\pm     |
   |                |                                    |240°,...`                               |
   +----------------+------------------------------------+----------------------------------------+

.. _drops_props_rule_26:

Rule 2.6 - Sign Inversion of Hermetian Mixtures of Coherence Orders Under Z-Rotaion
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Hermitian operators corresponding to a mixture of coherence
orders :math:`±p` (such as :math:`±2`) with :math:`|p|≠0`, change sign
if they are rotated by :math:`360°/(2p)` around the z axis. Hence,
after the rotation the shape of the droplet is unchanged but green
lobes are now where corresponding red lobes were before the
rotation. This is illustrated by the examples shown below for
coherence orders :math:`p` of :math:`±1, ±2,` and :math:`±3`.

.. list-table:: 
   :class: no-cap-num

   * - :math:`p=\pm 1`
     - .. drops:: (I1p + I1m)/2
          :width: 30%

     - | 180° rotation
       | :math:`\rightarrow`
     - .. drops:: -(I1p + I1m)/2
          :width: 30%

   * - :math:`p=\pm 2`
     - .. drops:: I1pI2p + I1mI2m
          :width: 30%
          :view: p170

     - | 90° rotation
       | :math:`\rightarrow`
     - .. drops:: -(I1pI2p + I1mI2m)
          :width: 30%
          :view: p170

   * - :math:`p=\pm 3`
     - .. drops:: I1pI2pI3p + I1mI2mI3m
          :width: 30%
          :view: p170

     - | 60° rotation
       | :math:`\rightarrow`
     - .. drops:: -(I1pI2pI3p + I1mI2mI3m)
          :width: 30%
          :view: p170