# SU(2) hyper-clocks:
quantum engineering of spinor interferences for time and frequency metrology

T. Zanon-Willette  Sorbonne Université, Observatoire de Paris, Université PSL,
CNRS, LERMA, F-75005, Paris, France<EMAIL_ADDRESS>MajuLab, International Research Laboratory IRL 3654, Université Côte d’Azur,
Sorbonne Université, National University of Singapore, Nanyang Technological
University, Singapore,  Centre for Quantum Technologies, National University
of Singapore, 117543 Singapore, Singapore  D. Wilkowski  MajuLab,
International Research Laboratory IRL 3654, Université Côte d’Azur, Sorbonne
Université, National University of Singapore, Nanyang Technological
University, Singapore,  Centre for Quantum Technologies, National University
of Singapore, 117543 Singapore, Singapore  School of Physical and Mathematical
Sciences, Nanyang Technological University, 637371 Singapore, Singapore  R.
Lefevre  Department of Physics, Royal Holloway, University of London, Royal
Holloway Egham Hill, Egham TW20 0EX, United Kingdom  A.V. Taichenachev
Novosibirsk State University, ul. Pirogova 2, 630090 Novosibirsk, Russia
Institute of Laser Physics, Siberian Branch, Russian Academy of Sciences,
prosp. Akad. Lavrent’eva 15B, 630090 Novosibirsk, Russia  V.I. Yudin
Novosibirsk State University, ul. Pirogova 2, 630090 Novosibirsk, Russia
Institute of Laser Physics, Siberian Branch, Russian Academy of Sciences,
prosp. Akad. Lavrent’eva 15B, 630090 Novosibirsk, Russia  Novosibirsk State
Technical University, prosp. Karla Marksa 20, 630073 Novosibirsk, Russia

###### Abstract

In 1949, Ramsey’s method of separated oscillating fields was elaborated
boosting over many decades metrological performances of atomic clocks and
becoming the standard technique for very high precision spectroscopic
measurements. A generalization of this interferometric method is presented
replacing the two single coherent excitations by arbitrary composite laser
pulses. The rotation of the state vector of a two-level system under the
effect of a single pulse is described using the Pauli matrices basis of the
SU(2) group. It is then generalized to multiple excitation pulses by a
recursive Euler-Rodrigues-Gibbs algorithm describing a composition of
rotations with different rotation axes. A general analytical formula for the
phase-shift associated with the clock’s interferometric signal is derived. As
illustrations, hyper-clocks based on three-pulse and five-pulse interrogation
protocols are studied and shown to exhibit nonlinear cubic and quintic
sensitivities to residual probe-induced light-shifts. The presented formalism
is well suited to optimize composite phase-shifts produced by tailored quantum
algorithms in order to design a new generation of optical frequency standards
and robust engineering control of atomic interferences in AMO physics with
cold matter and anti-matter.

††preprint: APS/123-QED

##  I Introduction

The method of separated oscillating fields was introduced by Ramsey in 1949 to
improve frequency resolution of spectroscopic measurements and collect
information about the internal structure of atoms and molecules [1, 2, 3].
Today, understanding how to improve the robustness of spectroscopy with
coherent radiation by reducing or eliminating laser probe-induced systematics
still remains a central goal in the broad and important field of robust atomic
sensors from stringent tests of fundamental physics to quantum metrology with
optical clocks and matter-wave interferometry [4].

Ramsey derived in 1950 the first original quantum mechanical description of a
spin 1/2 interferometric resonance with two separated coherent pulses by using
a Schrödinger wave-function description [2] later extending the analysis to
phase jump, pulse shapes and amplitudes [5, 6]. The Ramsey’s method became the
standard technique in atomic physics based on microwave and laser spectroscopy
and in quantum metrology with atomic beams [7] and cold atomic fountains [8]
to measure transition frequencies between particle states with very high-
precision [9]. After 70 years, Ramsey interferometry is still a powerful tool
to investigate matter-light interaction with a few particles such as in modern
cavity QED experiments on Schrödinger’s cats with Rydberg’s atoms [10, 11,
12], in quantum information with trapped ions [13] or with superconducting
qubits [14].

Figure 1: (color online) Generalized Ramsey’s method of separated oscillating
fields with angle-axis representation. Single pulses (or sets of composite
pulses), around a single Ramsey free evolution phase as $\delta\textup{T}$,
are introduced by an effective pulse area $\widetilde{\theta}_{p}$ and
$\widetilde{\theta}_{q}$ where $p\equiv 1^{\prime},2^{\prime},3^{\prime}...$,
$q\equiv 1,2,3...$ with arbitrary rotation axis orientation $\widehat{n}_{p}$
and $\widehat{n}_{q}$ around the $\widehat{m}$ axis. Single or composite laser
pulse parameters are including laser phase $\varphi_{p},\varphi_{q}$, field
excitation $\Omega_{p},\Omega_{q}$, pulse duration $\tau_{p},\tau_{q}$ and a
frequency detuning
$\delta_{p}=\delta\mp\Delta_{p},\delta_{q}=\delta\mp\Delta_{q}$ including the
uncompensated part of the light-shift $\Delta_{p},\Delta_{q}$ during pulses
[15]. Any residual light-shift is inducing a weak distortion of the angle-axis
orientation relatively to the intermediate free evolution zone.

Nevertheless, the architecture of the two-pulse Ramsey interferometer as shown
in Fig. 1 has remained untouched until 2010 when an improvement for clock
spectroscopy was proposed [15, 16] and experimentally applied to a single
trapped ion [18]. A Ramsey sequence of two coherent laser pulses is used with
a pre-compensation of the estimated light-shift including a third intermediate
pulse which is inserted to act like a spin-echo compensation of field
amplitude error [15, 16, 19, 17]. After the experimental success of the hyper-
Ramsey protocol to drastically reduce, by four orders of magnitude, the
residual light-shift on the single-ion 171Yb+ octupole clock transition [20],
more robust generalized hyper-Ramsey three-pulse protocols have been
discovered against laser pulses-induced frequency-shifts [21] including
decoherence effect [22, 23, 24]. Naturally, the question arises if there is a
way to extend interrogation protocols to any sets of arbitrary composite
pulses around a single free evolution time in a symmetrical fashion. The first
positive answer including composite laser pulses was provided through a
Cayley-Klein parametrization of rotation spinors [25]. An extended model based
on multiple hyper-Ramsey-Bordé building-blocks with two-level operators and
quantization of motion has been recently developed where arbitrary composite
optical pulses are used not only to shield quantum clock interferences against
residual light-shift but also to protect atomic matter-waves against laser
probe-induced frequency-shifts at ultra-cold temperature [4]. A complementary
approach to [4], extending [25], would be strongly helpful in designing new
sequences of laser pulses to compensate for their systematic errors in pulse
detuning, relative phase and/or pulse area through various robust quantum
control techniques [26, 27, 28, 29, 30].

The purpose of this work is to present an alternative formalism to refs [16,
25, 4] allowing a convenient derivation of generalized hyper-Ramsey clock
interferences and atomic phase-shifts with arbitrary composite pulses and axis
orientation. Exponentials of Pauli matrices [31] are used to decompose a
complex transition amplitude following an initial suggestion by I.I. Rabi,
N.F. Ramsey and J. Schwinger [32]. Multiple interactions between the two-level
system (or qubit) and laser pulses will be treated as a composition of spinor
rotations on a Bloch-sphere. Further, a recursive algorithm based on the
Euler-Rodrigues-Gibbs geometrical transformation for dual axis rotation
composition [33, 34, 35, 36, 37] is extended to qubit rotation allowing a
systematic exploration and optimization of more elaborated interrogation
protocols with multiple laser pulses.

The recursive algorithm has been inspired by composite pulses developed
originally in Nuclear Magnetic Resonance (NMR) [38, 39, 40] where composition
of two rotations with quaternion computation rules [41, 42] have already been
applied to facilitate geometrical analysis and role of symmetry in the design
of composite pulse action on nuclear spins ensemble [43, 44]. More recently,
the Schrödinger equation has been re-explored within a quaternionic
representation of Pauli spinor of an electron [45] and a quaternionic
derivation of the Ramsey transition probability has been presented [46]
providing an alternative way to compute composite rotations on the Bloch-
sphere.

The paper is organized as follow: in section II, we introduce a vectorial
representation of spinor matrix components associated to complex transition
amplitudes. Then, a compact expression of the composite phase-shift associated
to quantum interferences with multi-pulses is given in section III. From a
quantum engineering perspective, the best tailoring approach of atomic
interferences is tracked to produce, by pulse engineering methods such as
quantum control [47, 48], an optimization of some targeted performances, i.e
frequency-shift and signal amplitude of optical clocks, make more robust to
important variations of relevant experimental parameters. Finally, exact
expressions of atomic phase-shifts are derived. Here, we are mainly focusing
on specific laser pulse protocols on three-pulse and five-pulse schemes
related to the design of hyper-Ramsey composite phase-shifts for ultra-robust
optical clocks [16, 49, 25]. Such hyper-clocks produce various highly
nonlinear, flexible and robust compensation of the residual light-shift with a
different sensitivity to laser probe intensity fluctuation [50]. SU(2) hyper-
clocks are a class of optical qubit-clocks based on composite laser pulse
protocols aim at reducing laser probe-induced frequency shifts by several
order of magnitude improving the accuracy of optical clocks.

##  II Vectorial representation of generalized hyper-Ramsey spinor components

###  II.1 Pauli-spin decomposition

The model is based on a SU(2) Pauli-spin decomposition of generalized hyper-
Ramsey resonances and phase-shifts. Exact expressions are derived for spinor
components of a unitary interaction matrix describing coherent interaction
between a qubit and laser excitation pulses. The time-dependent atomic wave-
function $\Psi(t)=C_{g}(t)|g\rangle+C_{e}(t)|e\rangle$ is interacting with two
pulses labeled by $p$ and $q$ and separated by a single free evolution time T
as reported in Fig. 1, inducing a qubit rotation composition as [32]:

$\begin{split}\Psi(t)&=e^{i\widetilde{\theta}_{q}(\widehat{n}_{q}\cdot\overrightarrow{\sigma})}e^{i\theta_{m}(\widehat{m}\cdot\overrightarrow{\sigma})}e^{i\widetilde{\theta}_{p}(\widehat{n}_{p}\cdot\overrightarrow{\sigma})}\cdot\Psi(0)\\\
&=_{p}^{q}\textup{C}\cdot\Psi(0)\end{split}$ (1)

The Pauli vector is defined by
$\overrightarrow{\sigma}=\sigma_{x}\widehat{x}+\sigma_{y}\widehat{y}+\sigma_{z}\widehat{z}$.
Rotation axis definitions corresponding to Eq. 1 are introduced by
$\widehat{n}_{p}=\overrightarrow{n}_{p}/\|\overrightarrow{n}_{p}\|$,
$\widehat{m}=\overrightarrow{m}/\|\overrightarrow{m}\|$ and
$\widehat{n}_{q}=\overrightarrow{n}_{q}/\|\overrightarrow{n}_{q}\|$. Rotation
angles and angular velocities are defined by
$\widetilde{\theta}_{p}=\|\overrightarrow{n}_{p}\|\tau/2$,
$\widetilde{\theta}_{m}=\|\overrightarrow{m}\|\textup{T}/2$ and
$\widetilde{\theta}_{q}=\|\overrightarrow{n}_{q}\|\tau/2$, with cartesian unit
vector coordinates $\overrightarrow{n}_{p}=(n_{p_{x}},n_{p_{y}},n_{p_{z}})$,
$\overrightarrow{m}=(m_{x},m_{y},m_{z})$ and
$\overrightarrow{n}_{q}=(n_{q_{x}},n_{q_{y}},n_{q_{z}})$ [39, 40]. The
$2\times 2$ matrix components ${}_{p}^{q}C_{u,u^{\prime}}$ are written as:

$\begin{split}{}_{p}^{q}\textup{C}=\left(\begin{array}[]{cc}{}_{p}^{q}C_{gg}&{}_{p}^{q}C_{ge}\\\
{}_{p}^{q}C_{eg}&{}_{p}^{q}C_{ee}\\\ \end{array}\right)\end{split}$ (2)

where $u,u^{\prime}=g,e$. Relations between the components of the unitary
interaction matrix correspond to the SU(2) group, namely
${}_{p}^{q}C_{gg}=\\{_{p}^{q}C_{ee}\\}^{*}$,
${}_{p}^{q}C_{ge}=-\\{_{p}^{q}C_{eg}\\}^{*}$,
$|_{p}^{q}C_{gg}|^{2}+|_{p}^{q}C_{ge}|^{2}=1$.

Any general unitary operator corresponding to a rotation of a qubit around a
rotation axis $\widehat{n}_{l}$ with a rotation angle
$\widetilde{\vartheta}_{l}$ ($l=p,q$) is evaluated by the exponential Pauli-
spin decomposition [31, 51]:

$e^{i\widetilde{\vartheta}_{l}(\widehat{n}_{l}\cdot\overrightarrow{\sigma})}=\sigma_{0}\cos\widetilde{\vartheta}_{l}+i(\widehat{n}_{l}\cdot\overrightarrow{\sigma})\sin\widetilde{\vartheta}_{l}$
(3)

with the identity Pauli matrix $\sigma_{0}$ and satisfying the vectorial
identity relation [51]:

$(\widehat{n_{p}}\cdot\overrightarrow{\sigma})\cdot(\widehat{n}_{q}\cdot\overrightarrow{\sigma})=(\widehat{n}_{p}\cdot\widehat{n}_{q})\sigma_{0}+i(\widehat{n}_{p}\times\widehat{n}_{q})\cdot\overrightarrow{\sigma}$
(4)

The computational procedure calculates the transition probability for a
reorientation of the qubit state $u$ into a state $u^{\prime}$. It is simply
given by ${}_{p}^{q}P_{uu^{\prime}}=|_{p}^{q}C_{uu^{\prime}}|^{2}$ where each
spinor component of the unitary matrix is expressed as [19, 21, 24, 25]:

$\begin{split}{}_{p}^{q}C_{uu^{\prime}}=_{p}^{q}\widetilde{C}^{+}_{uu^{\prime}}e^{i_{p}^{q}\widetilde{\Phi}_{uu^{\prime}}^{+}}e^{i\widetilde{\theta}_{m}}+_{p}^{q}\widetilde{C}^{-}_{uu^{\prime}}e^{i_{p}^{q}\widetilde{\Phi}_{uu^{\prime}}^{-}}e^{-i\widetilde{\theta}_{m}}\end{split}$
(5)

The phase-shift difference between components
${}_{p}^{q}\widetilde{C}^{\pm}_{uu^{\prime}}$ is introduced as:

$\begin{split}{}_{p}^{q}\widetilde{\Phi}_{uu^{\prime}}&=_{p}^{q}\widetilde{\Phi}_{uu^{\prime}}^{+}-_{p}^{q}\widetilde{\Phi}_{uu^{\prime}}^{-}\end{split}$
(6)

Envelopes ${}_{p}^{q}\widetilde{C}_{uu^{\prime}}^{\pm}$ are themselves
expressed with a complex modulus as [52, 53]:

$\begin{split}{}_{p}^{q}\widetilde{C}_{uu^{\prime}}^{\pm}=\frac{1}{2}\left(\cos\widetilde{\theta}_{p}\cos\widetilde{\theta}_{q}\right)~{}_{p}^{q}C^{\pm}\sqrt{1+\tan^{2}\left({}_{p}^{q}\widetilde{\Phi}_{uu^{\prime}}^{\pm}\right)}\end{split}$
(7)

and

$\begin{split}{}_{p}^{q}C^{\pm}=&\sigma_{0}\pm\widehat{m}\cdot\overrightarrow{\sigma}\pm\left[{}_{p}^{q}\widehat{N}_{-}\times\widehat{m}\right]\cdot\overrightarrow{\sigma}-~{}_{p}^{q}\widehat{N}^{\widehat{m}}_{\bullet}\end{split}$
(8)

Atomic phase-shifts $\widetilde{\Phi}_{uu^{\prime}}(\pm)$ associated to
${}_{p}^{q}\widetilde{C}^{\pm}_{uu^{\prime}}$ are also evaluated with Pauli-
spin matrices and are expressed with a complex argument [52, 53]:

$\tan_{p}^{q}\widetilde{\Phi}_{uu^{\prime}}^{\pm}\equiv\frac{{}_{p}^{q}\widehat{N}_{+}\cdot\left[\overrightarrow{\sigma}\pm\widehat{m}\sigma_{0}\right]+~{}_{p}^{q}\widehat{N}_{\times}\cdot\left[\overrightarrow{\sigma}\mp\widehat{m}\sigma_{0}\right]}{\sigma_{0}\pm\widehat{m}\cdot\overrightarrow{\sigma}\pm\left[{}_{p}^{q}\widehat{N}_{-}\times\widehat{m}\right]\cdot\overrightarrow{\sigma}-~{}_{p}^{q}\widehat{N}^{\widehat{m}}_{\bullet}}$
(9)

where:

$\begin{split}{}_{p}^{q}\widehat{N}_{+}&\equiv\widehat{n}_{p}\tan\widetilde{\theta}_{p}+\widehat{n}_{q}\tan\widetilde{\theta}_{q}\\\
{}_{p}^{q}\widehat{N}_{-}&\equiv\widehat{n}_{p}\tan\widetilde{\theta}_{p}-\widehat{n}_{q}\tan\widetilde{\theta}_{q}\\\
{}_{p}^{q}\widehat{N}_{\times}&\equiv\widehat{n}_{p}\tan\widetilde{\theta}_{p}\times\widehat{n}_{q}\tan\widetilde{\theta}_{q}\\\
{}_{p}^{q}\widehat{N}^{\widehat{m}}_{\bullet}&\equiv\left(\widehat{n}_{p}\cdot\widehat{n}_{q}\right)_{\widehat{m},\overrightarrow{\sigma}}\tan\widetilde{\theta}_{p}\tan\widetilde{\theta}_{q}\end{split}$
(10)

with a reduced variable:

$\begin{split}\left(\widehat{n}_{p}\cdot\widehat{n}_{q}\right)_{\widehat{m},\overrightarrow{\sigma}}=&\left(\sigma_{0}\mp\widehat{m}\cdot\overrightarrow{\sigma}\right)\left(\widehat{n}_{p}\cdot\widehat{n}_{q}\right)\\\
&\pm\left[\left(\widehat{m}\cdot\widehat{n}_{p}\right)\widehat{n}_{q}+\left(\widehat{m}\cdot\widehat{n}_{q}\right)\widehat{n}_{p}\right]\cdot\overrightarrow{\sigma}\end{split}$
(11)

Frequency shifts of atomic interferences produced by the laser probe
excitation scheme are described by Eq. 9 where the influence of the light-
shift is to change simultaneously the rotation axis orientation and the
effective Rabi frequency as shown in Fig 1. This equation contains a dot-
product (scalar) term as $\widehat{N}_{\bullet}$ and a cross-product
(vectorial) term as $\widehat{N}_{\times}$ that are effectively related to a
composition rule of two unit quaternions [39, 41, 42] and to the Euler-
Rodrigues-Gibbs (ERG) formula for 3D rotation composition [33, 34, 35, 37].
Pauli-spin matrices $\sigma_{x,y,z}$ as well as the identity matrice
$\sigma_{0}$ are used as Hilbert-space pointers to individually address each
${}_{p}^{q}\widetilde{\Phi}_{uu^{\prime}}^{\pm}$ component associated to
diagonal and off-diagonal elements of the spinor matrix [54]. All
${}_{p}^{q}C_{uu^{\prime}}$ components of a rotated qubit by Ramsey
spectroscopy with composite pulses can be analytically derived using the
Pauli-spin model presented above.

Figure 2: (color online) Hyper-clock protocol classification following the
Pascal’s triangle for binomial coefficients representation. Appropriate laser
phase-jumps of $\pi$ are inserted as subscripts within specific laser pulses.
Pulse areas are here indicated in integer units of a $90^{\circ}$ reference
pulse following the time length of each single pulse. The qubit free-rotation
denoted $\delta\textup{T}$ is placed either between the first two pulses or
between the last two pulses of HR3π and HR5π protocols.

###  II.2 Euler-Rodrigues-Gibbs (ERG) transformation rules and recursive
algorithm

Turning to a generalized hyper-Ramsey resonance with an arbitrary number of
composite pulses, left and right single Pauli-spin qubits from Fig. 1 should
be now replaced by composite qubits as following:

$\begin{split}e^{i\widetilde{\theta}_{p}(\widehat{n}_{p}\cdot\overrightarrow{\sigma})}&\equiv\overrightarrow{\prod}_{l=\textup{1'}}^{\textup{p}}e^{i\widetilde{\theta}_{l}(\widehat{n}_{l}\cdot\overrightarrow{\sigma})}\\\
e^{i\widetilde{\theta}_{q}(\widehat{n}_{q}\cdot\overrightarrow{\sigma})}&\equiv\overleftarrow{\prod}_{l=\textup{1}}^{\textup{q}}e^{i\widetilde{\theta}_{l}(\widehat{n}_{l}\cdot\overrightarrow{\sigma})}\end{split}$
(12)

where each arrow indicates the direction to develop the multi-pulse product
with growing indices.

In order to track analytically the resulting phase-shift associated to
composite interferences, a recursive algorithm is presented based on iteration
of the Euler-Rodrigues-Gibbs (ERG) transformation applied to composite pulses
from left and right sides of the two-pulse interferometer. The ERG
transformation rules, acting on unit vector coordinates, for a given set of
$l\in\\{p,q\\}$ ($-$ for p and $+$ for q) composite pulses are given by:

$\displaystyle\left\\{\begin{split}\cos\widetilde{\theta}_{l}&\mapsto\cos\widetilde{\theta}_{l}\cos\widetilde{\theta}_{l+1}\left[1-_{l}^{l+1}\widehat{N}^{0}_{\bullet}\right]\\\
\widehat{n}_{l}\tan\widetilde{\theta}_{l}&\mapsto\frac{{}_{l}^{l+1}\widehat{N}_{+}\pm~{}_{l}^{l+1}\widehat{N}_{\times}}{1-_{l}^{l+1}\widehat{N}^{0}_{\bullet}}\end{split}\right.$
(13)

with
${}_{l}^{l+1}\widehat{N}^{0}_{\bullet}\equiv\widehat{n}_{l}\cdot\widehat{n}_{l+1}\tan\widetilde{\theta}_{l}\tan\widetilde{\theta}_{l+1}$.

These rules applied on ${}_{p}^{q}C_{uu^{\prime}}$ components, used as a
quantum-processing algorithm, are iterated $p-1$ and $q-1$ times when running
with an ensemble of $\\{p,q\\}$ pulses (see appendix for an example). A
different recursive algorithm has been developed in [4] related to a Möbius
transformation in conformal mapping [55], for instance see the reference note
[56]. A complete geometrical representation of the qubit dynamics is achieved
through Feynman-Vernon-Hellwarth coordinates to visualize composite rotations
on a Bloch-sphere [57]. A straightforward extension of generalized hyper-
Ramsey resonances and phase-shifts to a higher quantum J spin made of
composite qubits with equally energy spaced levels (hyper qudit-clock) is
provided by application of the Majorana formula [58, 59, 60] or by using a
polynomial matrix expansion of spin rotation [61].

Figure 3: (color online). Two dispersive error signals, calculated from Eq.
14, are plotted versus the clock frequency detuning. (a) HR3π protocol as
$90^{\prime\circ}_{\pm\pi/2}\dashv\delta\textup{T}\vdash
180_{\pi}^{\circ}90^{\circ}$, (b) HR5π protocol as
$90^{\prime\circ}_{\pm\pi/2}\dashv\delta\textup{T}\vdash
360_{\pi}^{\circ}540^{\circ}360_{\pi}^{\circ}90^{\circ}$. The reference Rabi
frequency for all pulses is $\Omega=\pi/2\tau$ ($\Omega\tau\equiv 90^{\circ}$
area in degrees) where the pulse duration reference is $\tau=3/16$ s, the free
evolution time is T$=2$ s.

##  III COMPOSITE PHASE-SHIFT OPTIMIZATION

###  III.1 Analytical formula

The Pauli-spin model is now tested in cases when composite pulses are used in
Ramsey interferometry. As a demonstration, a few composite phase-shifts are
derived following our recursive algorithm. The quantization axis is oriented
along the z axis as $\widehat{m}=(0,0,1)$ for laser pulsed qubit spectroscopy.
Normalized unitary rotation axis parameters can be introduced as
$\widehat{n}_{l_{x}}\equiv\frac{\Omega_{l}}{\omega_{l}}\cos\varphi_{l}$,
$\widehat{n}_{l_{y}}\equiv\frac{\Omega_{l}}{\omega_{l}}\sin\varphi_{l}$ and
$\widehat{n}_{l_{z}}\equiv\frac{\delta_{l}}{\omega_{l}}$ ($l\in\\{p,q\\}$)
respectively related to complex Rabi field frequency in the $x,y$ plane and
frequency detuning along the $z$ axis [62]. We therefore define the effective
Rabi field as $\omega_{l}=\sqrt{\delta_{l}^{2}+\Omega_{l}^{2}}$.

In selected interrogation schemes reported in Fig. 2, dispersive error signals
of spinor interferences are produced by subtracting two recorded transition
probabilities ${}_{p}^{q}\textup{P}_{gg}$ with additional laser phase-steps
$\pm\varphi_{l}$ opposite in sign and applied on required pulses to produce
dispersive curves or interferences [5, 21]:

$\begin{split}\Delta\textup{E}=_{p}^{q}\textup{P}_{gg}(+\varphi_{l})-_{p}^{q}\textup{P}_{gg}(-\varphi_{l}),\end{split}$
(14)

While addressing the ${}_{p}^{q}C_{gg}$ matrix element with Pauli matrices,
the atomic phase-shift expression, for a generalized hyper-Ramsey interference
with $\\{p,q\\}$ composite pulses, can always be decomposed into two
contributions:

$\begin{split}{}_{p}^{q}\widetilde{\Phi}_{gg}^{+}=&\arctan\left[\frac{\left(\right)^{p}_{z}+\left(\right)^{q}_{z}}{1-\left(\right)^{p}_{z}\left(\right)^{q}_{z}}\right]\\\
{}_{p}^{q}\widetilde{\Phi}_{gg}^{-}=&\arctan\left[\frac{\left(\right)^{p}_{y}\left(\right)^{q}_{x}-\left(\right)^{p}_{x}\left(\right)^{q}_{y}}{\left(\right)^{p}_{x}\left(\right)^{q}_{x}+\left(\right)^{p}_{y}\left(\right)^{q}_{y}}\right]\end{split}$
(15)

Few elements $\left(\right)^{p,q}_{x,y,z}$ will be given later. Note that
${}_{p}^{q}\widetilde{\Phi}_{gg}^{\pm}$ can be recast into a single canonical
expression as [52]:

$\begin{split}{}_{p}^{q}\widetilde{\Phi}_{gg}^{+}\mp_{p}^{q}\widetilde{\Phi}_{gg}^{-}=\arctan\left[\frac{\tan_{p}^{q}\widetilde{\Phi}_{gg}^{+}\mp\tan_{p}^{q}\widetilde{\Phi}_{gg}^{-}}{1\pm\tan_{p}^{q}\widetilde{\Phi}_{gg}^{+}\tan_{p}^{q}\widetilde{\Phi}_{gg}^{-}}\right]\end{split}$
(16)

Various interrogation protocols are now investigated. Two-pulse, three-pulse
and five-pulse protocols are shown in the diagram of Fig. 2; they can be
identified by the rotation angle of each pulse, expressed in terms of an
integer multiple of $90^{\circ}$.

Using this approach, pulses are classified as Ramsey $[1:1]$ (R, blue), Hyper-
Ramsey $[1:2:1]$ (HR3π, green) and high-order hyper-Ramsey $[1:4:6:4:1]$
(HR5π, red) protocols following the Pascal’s triangle for binomial
coefficients representation. They can be symmetrically read from left to right
or from right to left in the diagram of Fig. 2.

Figure 4: (color online). 2D diagrams of the hyper-Ramsey clock frequency-
shift (a) ${}_{1^{\prime}}^{2}\widetilde{\Phi}_{gg}/(2\pi\textup{T})$ and (b)
${}_{1^{\prime}}^{4}\widetilde{\Phi}_{gg}/(2\pi\textup{T})$ versus
uncompensated part of a residual light-shift $\Delta/2\pi$ along the
horizontal axis and pulse area $\Omega\tau$ along the vertical axis (see also
[21]). The only fixed parameter is the free evolution time as $\textup{T}=2$
s. Amplitude of the clock frequency-shift is indicated by a color graded scale
from -2 mHz to 2 mHz on the right side. White regions correspond to values out
of range. The reference pulse is $\Omega\tau=\alpha\times\pi/2$ where $\alpha$
is the parameter tuned along the vertical axis multiplying all single pulse
areas within a composite pulse protocol. Phase-shifts are evaluated modulo
$\pm k\pi,~{}k\in\mathbb{N}$ (see also [25]). Arrows are pointing pulse area
values for a maximum central fringe amplitude (black) or a high-order
sensitivity to the accumulated phase-shift (red).

This classification is used to identify pulse protocols which make the optical
clock sensitive only at high-order levels to the probe laser frequency shifts
where the degree of sensitivity scales with the number of pulses. This is also
remnant to the Taylor expansion of some leading coefficients of pulse
parameters from the theoretical analysis of refs [16, 23]. The free-rotation
of the qubit denoted $\delta\textup{T}$ has to be positioned between the first
two pulses (or the last two pulses) of each configuration. If not, other
protocols are generated with a different interferometric lineshape and
sensitivity to the residual light-shift (see for example two additional
examples of spinor interferences with four pulses shown in Fig. 6 from the
appendix).

The two-pulse Ramsey (R) protocol ($p=1^{\prime},q=1$) was proposed in 1949
[1]. The hyper-Ramsey interrogation scheme ($p=2^{\prime},q=1$ or
$p=1^{\prime},q=2$) originally presented in 2010, denoted as HR3π protocol, is
based on a sequence of three laser pulses [16]. It relies on replacing the
first or the second Ramsey pulse by a combination of two pulses (a composite
pulse) including an additional laser phase-step of $\pi$. The sequence of five
laser pulses ($p=4^{\prime},q=1$ or $p=1^{\prime},q=4$) is a new high-order
HR5π protocol including this time a set of more elaborated composite pulses as
$360_{\pi}^{\circ}540^{\circ}360_{\pi}^{\circ}$ replacing the intermediate
$180_{\pi}^{\circ}$ pulse. For both cases HR3π and HR5π, the interference
signal calculated using Eq. 14 is shown in Fig. 3(a) and (b) versus the clock
detuning. The three-pulse protocol generating hyper-Ramsey interferences (Fig.
3(a)) has been successfully applied on the single-ion 171Yb+ octupole clock
demonstrating a relative accuracy of $3\times 10^{-18}$ [20]. The composite
phase-shift related to these configurations is denoted
${}_{p}^{q}\widetilde{\Phi}_{gg}$. To derive the analytical expression of the
corresponding clock frequency-shift
${}_{p}^{q}\widetilde{\Phi}/(2\pi\textup{T})$, the ERG transformation rules
are iterated up to three times generating required
$\left(\right)^{p,q}_{x,y,z}$ elements with
$l=1^{\prime},2^{\prime},3^{\prime};l=1,2,3\in\\{4^{\prime},4\\}$.

###  III.2 2D map optimization of composite pulse protocols

As applications, we consider relevant phase-shifts
${}_{1^{\prime}}^{1}\widetilde{\Phi}_{gg}$,
${}_{1^{\prime}}^{2}\widetilde{\Phi}_{gg}$ and
${}_{1^{\prime}}^{4}\widetilde{\Phi}_{gg}$. Corresponding 2D diagrams
reconstructing the clock frequency-shifts
${}_{1^{\prime}}^{2}\widetilde{\Phi}_{gg}/(2\pi\textup{T})$ and
${}_{1^{\prime}}^{4}\widetilde{\Phi}_{gg}/(2\pi\textup{T})$ versus the
residual light-shift and pulse area are shown in Fig. 4(a) and (b). We choose
the amplitude of the frequency-shift to be indicated by a color graded scale
between -2 mHz and 2 mHz values as in [24].

A careful investigation of these diagrams allows us to extract some key-
parameters optimizing the robustness of the clock frequency-shift associated
to HR3π and HR5π protocols. We are able to explore wide regions of several
multiple values of the laser pulse area where the single pulse area reference
is introduced as $\Omega\tau=\alpha\times\pi/2$ ($\equiv 90^{\circ}$). The
error signal amplitude is always maximized for odd values and vanishing for
even values of this $\alpha$ parameter. By increasing the pulse area tuning
the parameter $\alpha$, the related light-shift correction is increasing
quadratically with the Rabi frequency but can still be fully compensated by
adjusting the laser frequency-step [15].

Figure 5: (color online). Central error signal interference frequency-shift
${}_{p}^{q}\widetilde{\Phi}_{gg}/(2\pi\textup{T})$ versus residual
uncompensated part of the light-shift (a) for Ramsey (R) protocol
(${}_{1^{\prime}}^{1}\widetilde{\Phi}_{gg}$ with blue dash-dotted line), HR3π
protocol (${}_{1^{\prime}}^{2}\widetilde{\Phi}_{gg}$ with a continuous green
line) and HR5π protocol (${}_{1^{\prime}}^{4}\widetilde{\Phi}_{gg}$ with a red
short-dotted line). (b) Zoom of clock frequency-shifts emphasizing the linear
dependence $\propto\Delta/\Omega$, the cubic dependence
$\propto(\Delta/\Omega)^{3}$ and the quintic dependence
$\propto(\Delta/\Omega)^{5}$ versus residual uncompensated part of the light-
shift $\Delta/2\pi$. Same laser parameters as in Fig. 3 with a fixed pulse
area parameter $\alpha=1$ for all curves.

The clock-frequency-shift compensation can be made more robust by slightly
shifting the pulse area parameter from $\alpha=1,3$ (black arrow) to
$\alpha=4/3,8/3$ (red arrows) for the HR3π protocol (black and red arrows are
pointing different pulse area values reported in Fig. 4(a)). These particular
pulse areas are strongly upgrading the compensation of the clock frequency-
shift up to two or three orders of magnitude but are associated to a relative
reduction in the central fringe amplitude by $\sim 30\%$. As an example,
injecting a residual light-shift of $\Delta/2\pi\sim 200$ mHz with a free
evolution time of T$=2$ s using the HR3π protocol, the composite clock
frequency-shift ${}_{1^{\prime}}^{2}\widetilde{\Phi}_{gg}/(2\pi\textup{T})$
drops from $1.3$ mHz (for $\alpha=1$) to -$0.043$ mHz (for $\alpha=4/3$) and
even -$2.5$ $\mu$Hz (for $\alpha=8/3$) if adjusting carefully the light-shift
pre-compensation. Injecting the same initial residual light-shift with the
HR5π protocol gives a clock frequency-shift
${}_{1^{\prime}}^{4}\widetilde{\Phi}_{gg}/(2\pi\textup{T})$ about $0.11$ mHz
(for $\alpha=1$) collapsing to $1.5$ $\mu$Hz (for $\alpha=3$) with no
reduction in the signal amplitude.

We note that these specific values of the pulse area are steering the clock
frequency-shift to locking-points offering a better compensation of the
systematics synchronized with the laser probe intensity fluctuation when a
pulse area modification of a few percents as
$\Delta\theta/\theta\equiv\Delta\alpha/\alpha<\pm 5\%$ is tolerated
(identified by violet and pink color graded regions in Fig. 4(a) and (b) where
an abrupt flip is observed between positive and negative clock frequency-shift
values of similar amplitude). As a difference to the HR3π scheme, the HR5π
protocol gets exceptional pulse area values where the fringe interference
contrast is maximum and the clock frequency-shift exhibits a higher order
dependence to the residual light-shift (black and red arrows are pointing to
the same pulse area values denoted as exceptional points reported in Fig.
4(b)). Our new HR5π protocol is simultaneously optimizing the error signal
amplitude and the robustness of quantum interferences against probe-induced
uncompensated residual light-shifts.

Clock frequency-shifts of quantum interferences versus the residual light-
shift are finally simulated and reported in Fig. 5(a) and (b) with a reference
pulse fixed to $\Omega\tau=\pi/2$ ($\equiv 90^{\circ}$) following the
horizontal axis of Fig. 4 taking $\alpha=1$). While the Ramsey clock
frequency-shift is a linear function of the residual light-shift affecting the
quantum states, the cubic sensitivity from a three-pulse scheme turns to
collapse to a high-order quintic sensitivity to residual light-shifts under a
five-pulse protocol as reported in Fig. 5(b).

The required elements needed to calculate
${}_{1^{\prime}}^{1}\widetilde{\Phi}_{gg}$ are given by:

$\begin{split}\begin{array}[]{c}\left(\right)^{1^{\prime}}_{x}=\widehat{n}_{1^{\prime}_{x}}\tan\widetilde{\theta}_{1^{\prime}},\hskip
14.22636pt\left(\right)^{1}_{x}=\widehat{n}_{1_{x}}\tan\widetilde{\theta}_{1}\\\
\left(\right)^{1^{\prime}}_{y}=\widehat{n}_{1^{\prime}_{y}}\tan\widetilde{\theta}_{1^{\prime}},\hskip
14.22636pt\left(\right)^{1}_{y}=\widehat{n}_{1_{y}}\tan\widetilde{\theta}_{1}\\\
\left(\right)^{1^{\prime}}_{z}=\widehat{n}_{1^{\prime}_{z}}\tan\widetilde{\theta}_{1^{\prime}},\hskip
14.22636pt\left(\right)^{1}_{z}=\widehat{n}_{1_{z}}\tan\widetilde{\theta}_{1}\end{array}\end{split}$
(17)

By applying Eq. 16, the original Ramsey clock frequency-shift reduces to:

$\begin{split}{}_{1^{\prime}}^{1}\widetilde{\Phi}_{gg}&=\varphi_{1}-\varphi_{1^{\prime}}+\frac{\delta_{1^{\prime}}}{\omega_{1^{\prime}}}\tan\widetilde{\theta}_{1^{\prime}}+\frac{\delta_{1}}{\omega_{1}}\tan\widetilde{\theta}_{1}\\\
&=\varphi_{1}-\varphi_{1^{\prime}}+\phi_{1^{\prime}}+\phi_{1}\end{split}$ (18)

in accordance with [19, 25, 49].

The required elements needed to calculate
${}_{1^{\prime}}^{2}\widetilde{\Phi}_{gg}$ or
${}_{2^{\prime}}^{1}\widetilde{\Phi}_{gg}$ are:

$\begin{split}\begin{array}[]{c}\left(\right)^{2^{\prime}}_{x}=\frac{\widehat{n}_{1^{\prime}_{x}}\tan\widetilde{\theta}_{1^{\prime}}+\widehat{n}_{2^{\prime}_{x}}\tan\widetilde{\theta}_{2^{\prime}}+(\widehat{n}_{1^{\prime}_{z}}\widehat{n}_{2^{\prime}_{y}}-\widehat{n}_{1^{\prime}_{y}}\widehat{n}_{2^{\prime}_{z}})\tan\widetilde{\theta}_{1^{\prime}}\tan\widetilde{\theta}_{2^{\prime}}}{1-\left(n_{1^{\prime}_{x}}n_{2^{\prime}_{x}}+n_{1^{\prime}_{y}}n_{2^{\prime}_{y}}+n_{1^{\prime}_{z}}n_{2^{\prime}_{z}}\right)\tan\widetilde{\theta}_{1^{\prime}}\tan\widetilde{\theta}_{2^{\prime}}}\\\
\left(\right)^{2^{\prime}}_{y}=\frac{\widehat{n}_{1^{\prime}_{y}}\tan\widetilde{\theta}_{1^{\prime}}+\widehat{n}_{2^{\prime}_{y}}\tan\widetilde{\theta}_{2^{\prime}}+(\widehat{n}_{1^{\prime}_{x}}\widehat{n}_{2^{\prime}_{z}}-\widehat{n}_{1^{\prime}_{z}}\widehat{n}_{2^{\prime}_{x}})\tan\widetilde{\theta}_{1^{\prime}}\tan\widetilde{\theta}_{2^{\prime}}}{1-\left(n_{1^{\prime}_{x}}n_{2^{\prime}_{x}}+n_{1^{\prime}_{y}}n_{2^{\prime}_{y}}+n_{1^{\prime}_{z}}n_{2^{\prime}_{z}}\right)\tan\widetilde{\theta}_{1^{\prime}}\tan\widetilde{\theta}_{2^{\prime}}}\\\
\left(\right)^{2^{\prime}}_{z}=\frac{\widehat{n}_{1^{\prime}_{z}}\tan\widetilde{\theta}_{1^{\prime}}+\widehat{n}_{2^{\prime}_{z}}\tan\widetilde{\theta}_{2^{\prime}}+(\widehat{n}_{1^{\prime}_{y}}\widehat{n}_{2^{\prime}_{x}}-\widehat{n}_{1^{\prime}_{x}}\widehat{n}_{2^{\prime}_{y}})\tan\widetilde{\theta}_{1^{\prime}}\tan\widetilde{\theta}_{2^{\prime}}}{1-\left(n_{1^{\prime}_{x}}n_{2^{\prime}_{x}}+n_{1^{\prime}_{y}}n_{2^{\prime}_{y}}+n_{1^{\prime}_{z}}n_{2^{\prime}_{z}}\right)\tan\widetilde{\theta}_{1^{\prime}}\tan\widetilde{\theta}_{2^{\prime}}}\end{array}\end{split}$
(19)

and

$\begin{split}\begin{array}[]{c}\left(\right)^{2}_{x}=\frac{\widehat{n}_{1_{x}}\tan\widetilde{\theta}_{1}+\widehat{n}_{2_{x}}\tan\widetilde{\theta}_{2}-(\widehat{n}_{1_{z}}\widehat{n}_{2_{y}}-\widehat{n}_{1_{y}}\widehat{n}_{2_{z}})\tan\widetilde{\theta}_{1}\tan\widetilde{\theta}_{2}}{1-\left(n_{1_{x}}n_{2_{x}}+n_{1_{y}}n_{2_{y}}+n_{1_{z}}n_{2_{z}}\right)\tan\widetilde{\theta}_{1}\tan\widetilde{\theta}_{2}}\\\
\left(\right)^{2}_{y}=\frac{\widehat{n}_{1_{y}}\tan\widetilde{\theta}_{1}+\widehat{n}_{2_{y}}\tan\widetilde{\theta}_{2}-(\widehat{n}_{1_{x}}\widehat{n}_{2_{z}}-\widehat{n}_{1_{z}}\widehat{n}_{2_{x}})\tan\widetilde{\theta}_{1}\tan\widetilde{\theta}_{2}}{1-\left(n_{1_{x}}n_{2_{x}}+n_{1_{y}}n_{2_{y}}+n_{1_{z}}n_{2_{z}}\right)\tan\widetilde{\theta}_{1}\tan\widetilde{\theta}_{2}}\\\
\left(\right)^{2}_{z}=\frac{\widehat{n}_{1_{z}}\tan\widetilde{\theta}_{1}+\widehat{n}_{2_{z}}\tan\widetilde{\theta}_{2}-(\widehat{n}_{1_{y}}\widehat{n}_{2_{x}}-\widehat{n}_{1_{x}}\widehat{n}_{2_{y}})\tan\widetilde{\theta}_{1}\tan\widetilde{\theta}_{2}}{1-\left(n_{1_{x}}n_{2_{x}}+n_{1_{y}}n_{2_{y}}+n_{1_{z}}n_{2_{z}}\right)\tan\widetilde{\theta}_{1}\tan\widetilde{\theta}_{2}}\end{array}\end{split}$
(20)

By fixing $\theta_{2^{\prime}}\equiv 0$ into Eq. 19 while inserting
$\widehat{n}_{1_{y}}=\widehat{n}_{2_{y}}\equiv 0$ in Eq. 20, the hyper-Ramsey
clock frequency-shift becomes identical to [19].

Required elements needed to calculate
${}_{1^{\prime}}^{4}\widetilde{\Phi}_{gg}$ or
${}_{4^{\prime}}^{1}\widetilde{\Phi}_{gg}$ are rapidly increasing in size and
are not given here. They can be derived applying two-times the ERG
transformation rules on Eq. 19 and Eq. 20 (see the appendix section for
analytics). This five-pulse protocol has been also derived with the other
recursive algorithm [56] following [4] confirming the accuracy of the plots
reported in Fig. 5(a) and (b).

##  IV Conclusion

A SU(2) formulation of hyper-Ramsey interferences with composite phase-shifts
has been presented. Hyper-clock interrogation protocols and their
interferometric dependence to light-shift have been classified by analogy with
a Pascal’s triangle representation of doublet, triplet and quintet splitting
patterns from spin-spin interaction in proton NMR multiplet spectroscopy [63].
Such a representation may ease the search for new and more efficient
interrogation protocols of ultra-narrow optical clock transitions while
offering a framework to derive analytically the phase-shifts associated to
arbitrary laser pulses, showing the remarkable intrinsic robustness introduced
in quantum metrology by hyper qubit-clocks. In the present work, a five-pulse
protocol is discovered to be a high-order version of the hyper-Ramsey three-
pulse scheme demonstrating a quintic sensitivity to residual probe-induced
light-shifts with a maximum signal amplitude.

The Pauli-spin model, complementary to analytical tools introduced in [4]
describing hyper-Ramsey-Bordé matter-wave interferometry, uses another
recursive algorithm connected to rotation composition rules of unit-
quaternions (or versors) algebra in a four dimensional space [41]. Natural
extension to SU(3) composite phase-shifts via three-level state interferences
(hyper qutrit-clock) may be also explored [64] using a compact representation
of Gell-Mann spin matrices [65, 66, 67]. Composite phase-shifts would
certainly be an advantage to qudit multiple rotations exposed to detrimental
ac Stark-shifts for robust quantum computation [68]. The next generation of
quantum clocks will irrevocably bring a relative level of accuracy below 10-18
through very long coherence times [69, 70, 71], probably supported by
robustness against noise with programmable quantum circuit technologies [72,
73], quantum non demolition measurements [74, 75] and state entanglement [76].
At this future level of accuracy, hyper-clocks with an optimal control of
composite phase-shifts should reduce the influence of laser-probe-intensity
fluctuations [50] while offering an additional toolbox for a fine tuning
control of the optical clock frequency in trapped multi-ion clocks [77] and in
optical lattice clocks [78].

This work in parallel with [4] should serve as quantum engineering methods to
explore cooperative composite pulse protocols [24, 79] dedicated to robust
control algorithms upgrading performances of optical frequency standards [80],
quantum computation with qubits and qudits immune to light-shift [68, 81],
robust quantum sensing [82] and pushing further high-precision laser
spectroscopy with cold molecules [83] and cold anti-matter [84].

##  V Acknowledgment

T.Z.W. is deeply grateful to Dr J.-P. Karr, Dr E. de Clercq, Pr M. Cahay, the
Wilkowski lab teams with Sr(I) and Sr(II) projects for discussion, comments
and criticism. V.I.Yudin was supported by the Russian Foundation for Basic
Research (Grant Nos. 20-02-00505 and 19-32-90181) and Foundation for the
Advancement of Theoretical Physics and Mathematics ”BASIS”. A.V. Taichenachev
acknowledges financial support from Russian Science Foundation through the
grant 20-12-00081. T.Z.W. acknowledges Sorbonne Université and MajuLab for
supporting a twelve months visiting research associate professorship at center
for quantum technologies (CQT) in Singapore.

## Appendix

### A: building-block for $\tan_{p}^{q}\widetilde{\Phi}_{uu^{\prime}}^{\pm}$

In this section, the decomposition of Eq. 9 from the main text is explicitly
provided with cartesian axis coordinates $\widehat{n}_{p_{x,y,z}}$,
$\widehat{n}_{q_{x,y,z}}$ and $\widehat{m}_{x,y,z}$. The interferometric
composite phase-shift numerator and denominator can be explicitly developed
using the Pauli matrices. For the diagonal phase-shift, we obtain the
numerator components to build
$\left\\{\tan_{p}^{q}\widetilde{\Phi}_{gg}^{\pm}\right\\}_{N}$:

$\begin{split}\widehat{n}_{p}\cdot\left(\overrightarrow{\sigma}\pm\widehat{m}\sigma_{0}\right)=&\widehat{n}_{p_{z}}\pm(\widehat{m}_{x}\widehat{n}_{p_{x}}+\widehat{m}_{y}\widehat{n}_{p_{y}}+\widehat{m}_{z}\widehat{n}_{p_{z}})\\\
\widehat{n}_{q}\cdot\left(\overrightarrow{\sigma}\pm\widehat{m}\sigma_{0}\right)=&\widehat{n}_{q_{z}}\pm(\widehat{m}_{x}\widehat{n}_{q_{x}}+\widehat{m}_{y}\widehat{n}_{q_{y}}+\widehat{m}_{z}\widehat{n}_{q_{z}})\\\
\left(\widehat{n}_{p}\times\widehat{n}_{q}\right)\cdot\left(\overrightarrow{\sigma}\mp\widehat{m}\sigma_{0}\right)=&\widehat{n}_{p_{x}}\widehat{n}_{q_{y}}-\widehat{n}_{q_{x}}\widehat{n}_{p_{y}}\\\
&\mp\widehat{m}_{x}(\widehat{n}_{q_{z}}\widehat{n}_{p_{y}}-\widehat{n}_{q_{y}}\widehat{n}_{p_{z}})\\\
&\mp\widehat{m}_{y}(\widehat{n}_{q_{x}}\widehat{n}_{p_{z}}-\widehat{n}_{q_{z}}\widehat{n}_{p_{x}})\\\
&\mp\widehat{m}_{z}(\widehat{n}_{p_{x}}\widehat{n}_{q_{y}}-\widehat{n}_{q_{x}}\widehat{n}_{p_{y}})\end{split}$
(21)

and the denominator components to build
$\left\\{\tan_{p}^{q}\widetilde{\Phi}_{gg}^{\pm}\right\\}_{D}$:

$\begin{split}\sigma_{0}\pm\widehat{m}\overrightarrow{\sigma}=&1\pm\widehat{m}_{z}\\\
\left[\widehat{n}_{p}\times\widehat{m}\right]\cdot\overrightarrow{\sigma}=&\widehat{m}_{y}\widehat{n}_{p_{x}}-\widehat{m}_{x}\widehat{n}_{p_{y}}\\\
\left[\widehat{n}_{q}\times\widehat{m}\right]\cdot\overrightarrow{\sigma}=&\widehat{m}_{y}\widehat{n}_{q_{x}}-\widehat{m}_{x}\widehat{n}_{q_{y}}\\\
\left(\widehat{n}_{p}\cdot\widehat{n}_{q}\right)_{\widehat{m},\overrightarrow{\sigma}}=&\widehat{n}_{p_{x}}\widehat{n}_{q_{x}}+\widehat{n}_{p_{y}}\widehat{n}_{q_{y}}+\widehat{n}_{p_{z}}\widehat{n}_{q_{z}}\\\
&\mp\widehat{m}_{z}\left(\widehat{n}_{p_{x}}\widehat{n}_{q_{x}}+\widehat{n}_{p_{y}}\widehat{n}_{q_{y}}+\widehat{n}_{p_{z}}\widehat{n}_{q_{z}}\right)\\\
&\pm\widehat{n}_{q_{z}}\left(\widehat{m}_{x}\widehat{n}_{p_{x}}+\widehat{m}_{y}\widehat{n}_{p_{y}}+\widehat{m}_{z}\widehat{n}_{p_{z}}\right)\\\
&\pm\widehat{n}_{p_{z}}\left(\widehat{m}_{x}\widehat{n}_{q_{x}}+\widehat{m}_{y}\widehat{n}_{q_{y}}+\widehat{m}_{z}\widehat{n}_{q_{z}}\right)\end{split}$
(22)

For the off-diagonal complex phase-shift, we obtain the numerator components
for $\left\\{\tan_{p}^{q}\widetilde{\Phi}_{eg}^{\pm}\right\\}_{N}$:

$\begin{split}\widehat{n}_{p}\cdot\left(\overrightarrow{\sigma}\pm\widehat{m}\sigma_{0}\right)=&\widehat{n}_{p_{x}}+i~{}\widehat{n}_{p_{y}}\\\
\widehat{n}_{q}\cdot\left(\overrightarrow{\sigma}\pm\widehat{m}\sigma_{0}\right)=&\widehat{n}_{q_{x}}+i~{}\widehat{n}_{q_{y}}\\\
\left(\widehat{n}_{p}\times\widehat{n}_{q}\right)\cdot\left(\overrightarrow{\sigma}\mp\widehat{m}\sigma_{0}\right)=&\left(\widehat{n}_{q_{y}}\widehat{n}_{p_{z}}-\widehat{n}_{q_{z}}\widehat{n}_{p_{y}}\right)\\\
&+i~{}\left(\widehat{n}_{q_{z}}\widehat{n}_{p_{x}}-\widehat{n}_{q_{x}}\widehat{n}_{p_{z}}\right)\end{split}$
(23)

and the denominator components for
$\left\\{\tan_{p}^{q}\widetilde{\Phi}_{eg}^{\pm}\right\\}_{D}$:

$\begin{split}\sigma_{0}\pm\widehat{m}\overrightarrow{\sigma}=&\pm(\widehat{m}_{x}+i~{}\widehat{m}_{y})\\\
\left[\widehat{n}_{p}\times\widehat{m}\right]\cdot\overrightarrow{\sigma}=&(\widehat{m}_{z}\widehat{n}_{p_{y}}-\widehat{m}_{y}\widehat{n}_{p_{z}})+i~{}\left(\widehat{m}_{x}\widehat{n}_{p_{z}}-\widehat{m}_{z}\widehat{n}_{p_{x}}\right)\\\
\left[\widehat{n}_{q}\times\widehat{m}\right]\cdot\overrightarrow{\sigma}=&(\widehat{m}_{z}\widehat{n}_{q_{y}}-\widehat{m}_{y}\widehat{n}_{q_{z}})+i~{}\left(\widehat{m}_{x}\widehat{n}_{q_{z}}-\widehat{m}_{z}\widehat{n}_{q_{x}}\right)\\\
\left(\widehat{n}_{p}\cdot\widehat{n}_{q}\right)_{\widehat{m},\overrightarrow{\sigma}}=&\mp\left(\widehat{m}_{x}+i~{}\widehat{m}_{y}\right)\left(\widehat{n}_{p_{x}}\widehat{n}_{q_{x}}+\widehat{n}_{p_{y}}\widehat{n}_{q_{y}}+\widehat{n}_{p_{z}}\widehat{n}_{q_{z}}\right)\\\
&\pm\left(\widehat{n}_{q_{x}}+i~{}\widehat{n}_{q_{y}}\right)\left(\widehat{m}_{x}\widehat{n}_{p_{x}}+\widehat{m}_{y}\widehat{n}_{p_{y}}+\widehat{m}_{z}\widehat{n}_{p_{z}}\right)\\\
&\pm\left(\widehat{n}_{p_{x}}+i~{}\widehat{n}_{p_{y}}\right)\left(\widehat{m}_{x}\widehat{n}_{q_{x}}+\widehat{m}_{y}\widehat{n}_{q_{y}}+\widehat{m}_{z}\widehat{n}_{q_{z}}\right)\end{split}$
(24)

where $N,D$ stands for the numerator and the denominator of the quantity
$\tan_{p}^{q}\widetilde{\Phi}_{uu^{\prime}}^{\pm}$ and all elements have to be
associated to $\tan\widetilde{\theta}_{p}$ and $\tan\widetilde{\theta}_{q}$.
Now, we proceed by fixing the orientation axis $\widehat{m}=(0,0,1)$ as in the
main text. We explicitly derive the diagonal phase-shift expressions
${}_{p}^{q}\widetilde{\Phi}_{gg}^{\pm}$ with the help of Eq. 21 and Eq. 22:

$\begin{split}\tan_{p}^{q}\widetilde{\Phi}_{gg}^{+}&=\frac{\widehat{n}_{p_{z}}\tan\widetilde{\theta}_{p}+\widehat{n}_{q_{z}}\tan\widetilde{\theta}_{q}}{1-\widehat{n}_{p_{z}}\widehat{n}_{q_{z}}\tan\widetilde{\theta}_{p}\tan\widetilde{\theta}_{q}}\\\
\tan_{p}^{q}\widetilde{\Phi}_{gg}^{-}&=\frac{\widehat{n}_{p_{y}}\widehat{n}_{q_{x}}-\widehat{n}_{p_{x}}\widehat{n}_{q_{y}}}{\widehat{n}_{p_{x}}\widehat{n}_{q_{x}}+\widehat{n}_{p_{y}}\widehat{n}_{q_{y}}}\end{split}$
(25)

Using normalized parameters from the main text
$\widehat{n}_{l_{x}}\equiv\frac{\Omega_{l}}{\omega_{l}}\cos\varphi_{l}$,
$\widehat{n}_{l_{y}}\equiv\frac{\Omega_{l}}{\omega_{l}}\sin\varphi_{l}$ and
$\widehat{n}_{l_{z}}\equiv\frac{\delta_{l}}{\omega_{l}}$ with ($l=p,q$), we
obtain, with Eq. 16, the overall Ramsey phase-shift
${}_{p}^{q}\widetilde{\Phi}_{gg}$:

$\begin{split}{}_{p}^{q}\widetilde{\Phi}_{gg}&=\varphi_{q}-\varphi_{p}+\phi_{p}+\phi_{q}\\\
&=\varphi_{1}-\varphi_{1^{\prime}}+\phi_{1^{\prime}}+\phi_{1}\end{split}$ (26)

where we use
$\phi_{l}=\frac{\delta_{l}}{\omega_{l}}\tan\widetilde{\theta}_{l}$. Indeed, we
have recovered the Ramsey phase-shift by fixing $p=1^{\prime}$ and $q=1$ as
two single pulses.

### B: $\left(\right)^{4^{\prime}}_{x,y,z}$ and $\left(\right)^{4}_{x,y,z}$
elements for ${}_{1^{\prime}}^{4}\widetilde{\Phi}_{gg}$ and
${}_{4^{\prime}}^{1}\widetilde{\Phi}_{gg}$

The ERG transformation through Eq.(15) is applied twice on numerator and
denominator elements from Eq. 19 and Eq. 20 with $p=4^{\prime},q=4$ pulses.

The transformation gives for the set of $p=4^{\prime}$ composite pulses:

$\displaystyle\left\\{\begin{split}\widehat{n}_{2^{\prime}}\tan\widetilde{\theta}_{2^{\prime}}&\mapsto\frac{{}_{2^{\prime}}^{3^{\prime}}\widehat{N}_{+}-~{}_{2^{\prime}}^{3^{\prime}}\widehat{N}_{\times}}{1-_{2^{\prime}}^{3^{\prime}}\widehat{N}^{0}_{\bullet}}\\\
\widehat{n}_{3^{\prime}}\tan\widetilde{\theta}_{3^{\prime}}&\mapsto\frac{{}_{3^{\prime}}^{4^{\prime}}\widehat{N}_{+}-~{}_{3^{\prime}}^{4^{\prime}}\widehat{N}_{\times}}{1-_{3^{\prime}}^{4^{\prime}}\widehat{N}^{0}_{\bullet}}\\\
\end{split}\right.$ (27)

The transformation gives for the set of $q=4$ composite pulses:

$\displaystyle\left\\{\begin{split}\widehat{n}_{2}\tan\widetilde{\theta}_{2}\mapsto\frac{{}_{2}^{3}\widehat{N}_{+}+~{}_{2}^{3}\widehat{N}_{\times}}{1-_{2}^{3}\widehat{N}^{0}_{\bullet}}\\\
\widehat{n}_{3}\tan\widetilde{\theta}_{3}\mapsto\frac{{}_{3}^{4}\widehat{N}_{+}+~{}_{3}^{4}\widehat{N}_{\times}}{1-_{3}^{4}\widehat{N}^{0}_{\bullet}}\\\
\end{split}\right.$ (28)

New expressions for components are thus:

$\begin{split}\widehat{n}_{2^{\prime}_{x}}\tan\widetilde{\theta}_{2^{\prime}}&\mapsto\frac{\widehat{n}_{2^{\prime}_{x}}\tan\widetilde{\theta}_{2^{\prime}}+\widehat{n}_{3^{\prime}_{x}}\tan\widetilde{\theta}_{3^{\prime}}+\left(\widehat{n}_{2^{\prime}_{z}}\cdot\widehat{n}_{3^{\prime}_{y}}\tan\widetilde{\theta}_{3^{\prime}}-\widehat{n}_{2^{\prime}_{y}}\cdot\widehat{n}_{3^{\prime}_{z}}\tan\widetilde{\theta}_{3^{\prime}}\right)\tan\widetilde{\theta}_{2^{\prime}}}{1-\left(\widehat{n}_{2^{\prime}_{x}}\cdot\widehat{n}_{3^{\prime}_{x}}\tan\widetilde{\theta}_{3^{\prime}}+\widehat{n}_{2^{\prime}_{y}}\cdot\widehat{n}_{3^{\prime}_{y}}\tan\widetilde{\theta}_{3^{\prime}}+\widehat{n}_{2^{\prime}_{z}}\cdot\widehat{n}_{3^{\prime}_{z}}\tan\widetilde{\theta}_{3^{\prime}}\right)\tan\widetilde{\theta}_{2^{\prime}}}\\\
\widehat{n}_{2^{\prime}_{y}}\tan\widetilde{\theta}_{2^{\prime}}&\mapsto\frac{\widehat{n}_{2^{\prime}_{y}}\tan\widetilde{\theta}_{2^{\prime}}+\widehat{n}_{3^{\prime}_{y}}\tan\widetilde{\theta}_{3^{\prime}}+\left(\widehat{n}_{2^{\prime}_{x}}\cdot\widehat{n}_{3^{\prime}_{z}}\tan\widetilde{\theta}_{3^{\prime}}-\widehat{n}_{2^{\prime}_{z}}\cdot\widehat{n}_{3^{\prime}_{x}}\tan\widetilde{\theta}_{3^{\prime}}\right)\tan\widetilde{\theta}_{2^{\prime}}}{1-\left(\widehat{n}_{2^{\prime}_{x}}\cdot\widehat{n}_{3^{\prime}_{x}}\tan\widetilde{\theta}_{3^{\prime}}+\widehat{n}_{2^{\prime}_{y}}\cdot\widehat{n}_{3^{\prime}_{y}}\tan\widetilde{\theta}_{3^{\prime}}+\widehat{n}_{2^{\prime}_{z}}\cdot\widehat{n}_{3^{\prime}_{z}}\tan\widetilde{\theta}_{3^{\prime}}\right)\tan\widetilde{\theta}_{2^{\prime}}}\\\
\widehat{n}_{2^{\prime}_{z}}\tan\widetilde{\theta}_{2^{\prime}}&\mapsto\frac{\widehat{n}_{2^{\prime}_{z}}\tan\widetilde{\theta}_{2^{\prime}}+\widehat{n}_{3^{\prime}_{z}}\tan\widetilde{\theta}_{3^{\prime}}+\left(\widehat{n}_{2^{\prime}_{y}}\cdot\widehat{n}_{3^{\prime}_{x}}\tan\widetilde{\theta}_{3^{\prime}}-\widehat{n}_{2^{\prime}_{x}}\cdot\widehat{n}_{3^{\prime}_{y}}\tan\widetilde{\theta}_{3^{\prime}}\right)\tan\widetilde{\theta}_{2^{\prime}}}{1-\left(\widehat{n}^{\prime}_{2_{x}}\cdot\widehat{n}_{3^{\prime}_{x}}\tan\widetilde{\theta}_{3^{\prime}}+\widehat{n}_{2^{\prime}_{y}}\cdot\widehat{n}_{3^{\prime}_{y}}\tan\widetilde{\theta}_{3^{\prime}}+\widehat{n}_{2^{\prime}_{z}}\cdot\widehat{n}_{3^{\prime}_{z}}\tan\widetilde{\theta}_{3^{\prime}}\right)\tan\widetilde{\theta}_{2^{\prime}}}\end{split}$
(29)

where $\widehat{n}_{3^{\prime}_{x,y,z}}\tan\widetilde{\theta}_{3^{\prime}}$
axial components are replaced by:

$\begin{split}\widehat{n}_{3^{\prime}_{x}}\tan\widetilde{\theta}_{3^{\prime}}&\mapsto\frac{\widehat{n}_{3^{\prime}_{x}}\tan\widetilde{\theta}_{3^{\prime}}+\widehat{n}_{4^{\prime}_{x}}\tan\widetilde{\theta}_{4^{\prime}}+\left(\widehat{n}_{3^{\prime}_{z}}\cdot\widehat{n}_{4^{\prime}_{y}}\tan\widetilde{\theta}_{4^{\prime}}-\widehat{n}_{3^{\prime}_{y}}\cdot\widehat{n}_{4^{\prime}_{z}}\tan\widetilde{\theta}_{4^{\prime}}\right)\tan\widetilde{\theta}_{3^{\prime}}}{1-\left(\widehat{n}_{3^{\prime}_{x}}\cdot\widehat{n}_{4^{\prime}_{x}}\tan\widetilde{\theta}_{4^{\prime}}+\widehat{n}_{3^{\prime}_{y}}\cdot\widehat{n}_{4^{\prime}_{y}}\tan\widetilde{\theta}_{4^{\prime}}+\widehat{n}_{3^{\prime}_{z}}\cdot\widehat{n}_{4^{\prime}_{z}}\tan\widetilde{\theta}_{4^{\prime}}\right)\tan\widetilde{\theta}_{3^{\prime}}}\\\
\widehat{n}_{3^{\prime}_{y}}\tan\widetilde{\theta}_{3^{\prime}}&\mapsto\frac{\widehat{n}_{3^{\prime}_{y}}\tan\widetilde{\theta}_{3^{\prime}}+\widehat{n}_{4^{\prime}_{y}}\tan\widetilde{\theta}_{4^{\prime}}+\left(\widehat{n}_{3^{\prime}_{x}}\cdot\widehat{n}_{4^{\prime}_{z}}\tan\widetilde{\theta}_{4^{\prime}}-\widehat{n}_{3^{\prime}_{z}}\cdot\widehat{n}_{4^{\prime}_{x}}\tan\widetilde{\theta}_{4^{\prime}}\right)\tan\widetilde{\theta}_{3^{\prime}}}{1-\left(\widehat{n}_{3^{\prime}_{x}}\cdot\widehat{n}_{4^{\prime}_{x}}\tan\widetilde{\theta}_{4^{\prime}}+\widehat{n}_{3^{\prime}_{y}}\cdot\widehat{n}_{4^{\prime}_{y}}\tan\widetilde{\theta}_{4^{\prime}}+\widehat{n}_{3^{\prime}_{z}}\cdot\widehat{n}_{4^{\prime}_{z}}\tan\widetilde{\theta}_{4^{\prime}}\right)\tan\widetilde{\theta}_{3^{\prime}}}\\\
\widehat{n}_{3^{\prime}_{z}}\tan\widetilde{\theta}_{3^{\prime}}&\mapsto\frac{\widehat{n}_{3^{\prime}_{z}}\tan\widetilde{\theta}_{3^{\prime}}+\widehat{n}_{4^{\prime}_{z}}\tan\widetilde{\theta}_{4^{\prime}}+\left(\widehat{n}_{3^{\prime}_{y}}\cdot\widehat{n}_{4^{\prime}_{x}}\tan\widetilde{\theta}_{4^{\prime}}-\widehat{n}_{3^{\prime}_{x}}\cdot\widehat{n}_{4^{\prime}_{y}}\tan\widetilde{\theta}_{4^{\prime}}\right)\tan\widetilde{\theta}_{3^{\prime}}}{1-\left(\widehat{n}_{3^{\prime}_{x}}\cdot\widehat{n}_{4^{\prime}_{x}}\tan\widetilde{\theta}_{4^{\prime}}+\widehat{n}_{3^{\prime}_{y}}\cdot\widehat{n}_{4^{\prime}_{y}}\tan\widetilde{\theta}_{4^{\prime}}+\widehat{n}_{3^{\prime}_{z}}\cdot\widehat{n}_{4^{\prime}_{z}}\tan\widetilde{\theta}_{4^{\prime}}\right)\tan\widetilde{\theta}_{3^{\prime}}}\end{split}$
(30)

and

$\begin{split}\widehat{n}_{2_{x}}\tan\widetilde{\theta}_{2}&\mapsto\frac{\widehat{n}_{2_{x}}\tan\widetilde{\theta}_{2}+\widehat{n}_{3_{x}}\tan\widetilde{\theta}_{3}-\left(\widehat{n}_{2_{z}}\cdot\widehat{n}_{3_{y}}\tan\widetilde{\theta}_{3}-\widehat{n}_{2_{y}}\cdot\widehat{n}_{3_{z}}\tan\widetilde{\theta}_{3}\right)\tan\widetilde{\theta}_{2}}{1-\left(\widehat{n}_{2_{x}}\cdot\widehat{n}_{3_{x}}\tan\widetilde{\theta}_{3}+\widehat{n}_{2_{y}}\cdot\widehat{n}_{3_{y}}\tan\widetilde{\theta}_{3}+\widehat{n}_{2_{z}}\cdot\widehat{n}_{3_{z}}\tan\widetilde{\theta}_{3}\right)\tan\widetilde{\theta}_{2}}\\\
\widehat{n}_{2_{y}}\tan\widetilde{\theta}_{2}&\mapsto\frac{\widehat{n}_{2_{y}}\tan\widetilde{\theta}_{2}+\widehat{n}_{3_{y}}\tan\widetilde{\theta}_{3}-\left(\widehat{n}_{2_{x}}\cdot\widehat{n}_{3_{z}}\tan\widetilde{\theta}_{3}-\widehat{n}_{2_{z}}\cdot\widehat{n}_{3_{x}}\tan\widetilde{\theta}_{3}\right)\tan\widetilde{\theta}_{2}}{1-\left(\widehat{n}_{2_{x}}\cdot\widehat{n}_{3_{x}}\tan\widetilde{\theta}_{3}+\widehat{n}_{2_{y}}\cdot\widehat{n}_{3_{y}}\tan\widetilde{\theta}_{3}+\widehat{n}_{2_{z}}\cdot\widehat{n}_{3_{z}}\tan\widetilde{\theta}_{3}\right)\tan\widetilde{\theta}_{2}}\\\
\widehat{n}_{2_{z}}\tan\widetilde{\theta}_{2}&\mapsto\frac{\widehat{n}_{2_{z}}\tan\widetilde{\theta}_{2}+\widehat{n}_{3_{z}}\tan\widetilde{\theta}_{3}-\left(\widehat{n}_{2_{y}}\cdot\widehat{n}_{3_{x}}\tan\widetilde{\theta}_{3}-\widehat{n}_{2_{x}}\cdot\widehat{n}_{3_{y}}\tan\widetilde{\theta}_{3}\right)\tan\widetilde{\theta}_{2}}{1-\left(\widehat{n}_{2_{x}}\cdot\widehat{n}_{3_{x}}\tan\widetilde{\theta}_{3}+\widehat{n}_{2_{y}}\cdot\widehat{n}_{3_{y}}\tan\widetilde{\theta}_{3}+\widehat{n}_{2_{z}}\cdot\widehat{n}_{3_{z}}\tan\widetilde{\theta}_{3}\right)\tan\widetilde{\theta}_{2}}\end{split}$
(31)

where $\widehat{n}_{3_{x,y,z}}\tan\widetilde{\theta}_{3}$ axial components are
replaced by:

$\begin{split}\widehat{n}_{3_{x}}\tan\widetilde{\theta}_{3}&\mapsto\frac{\widehat{n}_{3_{x}}\tan\widetilde{\theta}_{3}+\widehat{n}_{4_{x}}\tan\widetilde{\theta}_{4}-\left(\widehat{n}_{3_{z}}\cdot\widehat{n}_{4_{y}}\tan\widetilde{\theta}_{4}-\widehat{n}_{3_{y}}\cdot\widehat{n}_{4_{z}}\tan\widetilde{\theta}_{4}\right)\tan\widetilde{\theta}_{3}}{1-\left(\widehat{n}_{3_{x}}\cdot\widehat{n}_{4_{x}}\tan\widetilde{\theta}_{4}+\widehat{n}_{3_{y}}\cdot\widehat{n}_{4_{y}}\tan\widetilde{\theta}_{4}+\widehat{n}_{3_{z}}\cdot\widehat{n}_{4_{z}}\tan\widetilde{\theta}_{4}\right)\tan\widetilde{\theta}_{3}}\\\
\widehat{n}_{3_{y}}\tan\widetilde{\theta}_{3}&\mapsto\frac{\widehat{n}_{3_{y}}\tan\widetilde{\theta}_{3}+\widehat{n}_{4_{y}}\tan\widetilde{\theta}_{4}-\left(\widehat{n}_{3_{x}}\cdot\widehat{n}_{4_{z}}\tan\widetilde{\theta}_{4}-\widehat{n}_{3_{z}}\cdot\widehat{n}_{4_{x}}\tan\widetilde{\theta}_{4}\right)\tan\widetilde{\theta}_{3}}{1-\left(\widehat{n}_{3_{x}}\cdot\widehat{n}_{4_{x}}\tan\widetilde{\theta}_{4}+\widehat{n}_{3_{y}}\cdot\widehat{n}_{4_{y}}\tan\widetilde{\theta}_{4}+\widehat{n}_{3_{z}}\cdot\widehat{n}_{4_{z}}\tan\widetilde{\theta}_{4}\right)\tan\widetilde{\theta}_{3}}\\\
\widehat{n}_{3_{z}}\tan\widetilde{\theta}_{3}&\mapsto\frac{\widehat{n}_{3_{z}}\tan\widetilde{\theta}_{3}+\widehat{n}_{4_{z}}\tan\widetilde{\theta}_{4}-\left(\widehat{n}_{3_{y}}\cdot\widehat{n}_{4_{x}}\tan\widetilde{\theta}_{4}-\widehat{n}_{3_{x}}\cdot\widehat{n}_{4_{y}}\tan\widetilde{\theta}_{4}\right)\tan\widetilde{\theta}_{3}}{1-\left(\widehat{n}_{3_{x}}\cdot\widehat{n}_{4_{x}}\tan\widetilde{\theta}_{4}+\widehat{n}_{3_{y}}\cdot\widehat{n}_{4_{y}}\tan\widetilde{\theta}_{4}+\widehat{n}_{3_{z}}\cdot\widehat{n}_{4_{z}}\tan\widetilde{\theta}_{4}\right)\tan\widetilde{\theta}_{3}}\end{split}$
(32)

Phase-shifts expressions ${}_{1^{\prime}}^{4}\widetilde{\Phi}_{gg}$ and
${}_{4^{\prime}}^{1}\widetilde{\Phi}_{gg}$ can thus be analytically obtained
encapsulating Eq. 29 with Eq. 30 and Eq. 31 with Eq. 32 into Eq. 19 and Eq.
20.

Figure 6: (color online). Examples of dispersive error signals based on
${}_{2^{\prime}}^{2}C_{gg}$, calculated with Eq. 33 and applying the ERG
transformation through Eq. 36. (a)
$90^{\prime\circ}_{\pm\pi/2}90^{\prime\circ}\dashv\delta\textup{T}\vdash
270^{\circ}$, (b)
$90^{\prime\circ}_{\pm\pi/2}90^{\prime\circ}\dashv\delta\textup{T}\vdash
180_{\pi}^{\circ}90^{\circ}$. The Rabi frequency for all pulses is
$\Omega=\pi/2\tau$ where the pulse duration reference is $\tau=3/16$ s, the
free evolution time is T$=2$ s.

### C: spinor interferences with composite pulses

We apply our ERG algorithm following Eq. 13 to derive transition probabilities
of composite spinor interferences based on Eq. 5. We use a quantization axis
projection with $\widehat{m}=(0,0,1)$ leading to
$\widetilde{\theta}_{m_{z}}=\delta\textup{T}/2$. We first derive the Ramsey
formula with two pulses based on the coefficient ${}_{1^{\prime}}^{1}C_{gg}$
as following:

$\begin{split}{}_{1^{\prime}}^{1}C_{gg}&=_{1^{\prime}}^{1}\widetilde{C}^{+}_{gg}e^{i_{1^{\prime}}^{1}\widetilde{\Phi}_{gg}^{+}}e^{i\widetilde{\theta}_{m_{z}}}+_{1^{\prime}}^{1}\widetilde{C}^{-}_{gg}e^{i_{1^{\prime}}^{1}\widetilde{\Phi}_{gg}^{-}}e^{-i\widetilde{\theta}_{m_{z}}}\\\
{}_{1^{\prime}}^{1}\widetilde{C}_{gg}^{\pm}&=\frac{1}{2}\left(\cos\widetilde{\theta}_{1^{\prime}}\cos\widetilde{\theta}_{1}\right)~{}_{1^{\prime}}^{1}C^{\pm}\sqrt{1+\tan^{2}\left({}_{1^{\prime}}^{1}\widetilde{\Phi}_{gg}^{\pm}\right)}\end{split}$
(33)

where phase-shifts are given by:

$\begin{split}{}_{1^{\prime}}^{1}\widetilde{\Phi}_{gg}^{+}=&\arctan\left[\frac{\widehat{n}_{1^{\prime}_{z}}\tan\widetilde{\theta}_{1^{\prime}}+\widehat{n}_{1_{z}}\tan\widetilde{\theta}_{1}}{1-\widehat{n}_{1^{\prime}_{z}}\widehat{n}_{1_{z}}\tan\widetilde{\theta}_{1^{\prime}}\tan\widetilde{\theta}_{1}}\right]\\\
{}_{1^{\prime}}^{1}\widetilde{\Phi}_{gg}^{-}=&\arctan\left[\frac{\left(\widehat{n}_{1^{\prime}_{y}}\widehat{n}_{1_{x}}-\widehat{n}_{1^{\prime}_{x}}\widehat{n}_{1_{y}}\right)\tan\widetilde{\theta}_{1^{\prime}}\tan\widetilde{\theta}_{1}}{\left(\widehat{n}_{1^{\prime}_{x}}\widehat{n}_{1_{x}}+\widehat{n}_{1^{\prime}_{y}}\widehat{n}_{1_{y}}\right)\tan\widetilde{\theta}_{1^{\prime}}\tan\widetilde{\theta}_{1}}\right]\end{split}$
(34)

and:

$\begin{split}{}_{1^{\prime}}^{1}C^{+}&=2-2\widehat{n}_{1^{\prime}_{z}}\tan\widetilde{\theta}_{1^{\prime}}\cdot\widehat{n}_{1_{z}}\tan\widetilde{\theta}_{1}\\\
{}_{1^{\prime}}^{1}C^{-}&=-2\widehat{n}_{1^{\prime}_{x}}\tan\widetilde{\theta}_{1^{\prime}}\cdot\widehat{n}_{1_{x}}\tan\widetilde{\theta}_{1}-2\widehat{n}_{1^{\prime}_{y}}\tan\widetilde{\theta}_{1^{\prime}}\cdot\widehat{n}_{1_{y}}\tan\widetilde{\theta}_{1}\end{split}$
(35)

We now proceed with our ERG algorithm to obtain coefficients of the
${}_{2^{\prime}}^{2}C_{gg}$ amplitude of transition using $p=2^{\prime}$
pulses on the left arm and $q=2$ pulses on the right arm of the spectroscopic
pulse scheme. We get:

$\begin{split}\cos\widetilde{\theta}_{1^{\prime}}&\mapsto\cos\widetilde{\theta}_{1^{\prime}}\cos\widetilde{\theta}_{2^{\prime}}\left(1-\widehat{n}_{1^{\prime}}\tan\widetilde{\theta}_{1^{\prime}}\cdot\widehat{n}_{2^{\prime}}\tan\widetilde{\theta}_{2^{\prime}}\right)\\\
\cos\widetilde{\theta}_{1}&\mapsto\cos\widetilde{\theta}_{1}\cos\widetilde{\theta}_{2}\left(1-\widehat{n}_{1}\tan\widetilde{\theta}_{1}\cdot\widehat{n}_{2}\tan\widetilde{\theta}_{2}\right)\\\
\widehat{n}_{1^{\prime}_{x,y,z}}\tan\widetilde{\theta}_{1^{\prime}}&\mapsto\left(\right)^{2^{\prime}}_{x,y,z}\\\
\widehat{n}_{1_{x,y,z}}\tan\widetilde{\theta}_{1}&\mapsto\left(\right)^{2}_{x,y,z}\\\
{}_{1^{\prime}}^{1}\widetilde{\Phi}_{gg}^{\pm}&\mapsto_{2^{\prime}}^{2}\widetilde{\Phi}_{gg}^{\pm}\end{split}$
(36)

where modified elements $\left(\right)^{2^{\prime}}_{x,y,z}$ and
$\left(\right)^{2}_{x,y,z}$ are given by Eq. 19 and Eq. 20. We have plotted
two examples of arbitrary composite spinor interferences based on four pulses
versus the clock detuning in Fig. 6.

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