††thanks: G.A.S-L and H.M contributed equally.

Current affiliation: ]Propulsion Laboratory, California Institute of
Technology, Pasadena, California 91109, USA ††thanks: G.A.S-L and H.M
contributed equally.

Current affiliation: ]Los Alamos National Laboratory, Los Alamos, NM 87545,
USA

# Decoherence-free radiofrequency dressed subspaces

G.A. Sinuco-Leon  Department of Physics & Astronomy, University of Sussex,
Falmer, Brighton, BN1 9QH, UK  H. Mas  [  Institute of Electronic Structure
and Laser, Foundation for Research and Technology-Hellas, Heraklion 70013,
Greece  Department of Physics, University of Crete, Heraklion, Greece  S.
Pandey  [  Institute of Electronic Structure and Laser, Foundation for
Research and Technology-Hellas, Heraklion 70013, Greece  Department of
Materials Science and Technology, University of Crete, Heraklion, Greece  G.
Vasilakis  Institute of Electronic Structure and Laser, Foundation for
Research and Technology-Hellas, Heraklion 70013, Greece  B.M. Garraway
Department of Physics & Astronomy, University of Sussex, Falmer, Brighton, BN1
9QH, UK  W. von Klitzing  Institute of Electronic Structure and Laser,
Foundation for Research and Technology-Hellas, Heraklion 70013, Greece

###### Abstract

We study the spectral signatures and coherence properties of radiofrequency
dressed hyperfine Zeeman sub-levels of 87Rb. Experimentally, we engineer
combinations of static and RF magnetic fields to modify the response of the
atomic spin states to environmental magnetic field noise. We demonstrate
analytically and experimentally the existence of “magic” dressing conditions
where decoherence due to electromagnetic field noise is strongly suppressed.
Building upon this result, we propose a bi-chromatic dressing configuration
that reduces the global sensitivity of the atomic ground states to low-
frequency noise, and enables the simultaneous protection of multiple
transitions between the two ground hyperfine manifolds of atomic alkali
species. Our methods produce protected transitions between any pair of
hyperfine sub-levels at arbitrary (low) DC-magnetic fields.

RF-dressing, noise protection, dynamical decoupling, qdits

##  I Introduction

The sensitivity to environmental field fluctuations of atomic transitions in
quantum systems represents a major challenge for improving the accuracy and
reliability of applications such as atomic clocks Bloom  _et al._ (2014), low-
frequency field sensing Baumgart  _et al._ (2016), and quantum memories Zhao
_et al._ (2009) and information processors Preskill (1998); Gottesman (1999);
Sinuco-León and Garraway (2016). The sensitivity problem arises since,
typically, the transition frequency between pairs of quantum states is
affected by temporal and spatial variations of the electromagnetic
environment, which results in a rapid dephasing of the system’s wavefunction.
The ubiquity of this problem, identified in ultracold atomic ensembles
Santarelli  _et al._ (1999), superconducting devices Meier and Loss (2005),
nitrogen-vacancy centres in diamond Golter and Wang (2014) and doped silicon
Ladd  _et al._ (2005), has driven the development of various flavours of
dynamical coherent control (pulsed Viola and Knill (2003), concatenated Cai
_et al._ (2012) and continuous Bermudez  _et al._ (2012); Laucht  _et al._
(2017)) that isolate such systems from unwanted noise sources, and improve
their coherence time by several orders of magnitude Timoney  _et al._ (2011).
In atomic systems, in particular, reduction of the frequency broadening can be
achieved by applying electromagnetic fields (DC magnetic field Matthews  _et
al._ (1999); Szmuk  _et al._ (2015), microwaves Sárkány  _et al._ (2014),
radio-frequency Kazakov and Schumm (2015), optical radiation Katori  _et al._
(2003)) tuned to suppress the differential Zeeman or Stark energy shifts
between selected pairs of states (clock/qubit states), which, effectively,
protects the transition frequency against field fluctuations.

Here we demonstrate the control of the magnetic field sensitivity of the
electronic ground state of 87Rb, dressed by a radiofrequency magnetic field.
The dressing is achieved by applying an oscillating magnetic field,
$\bm{B}_{\textrm{RF}}$, with frequency, $\omega_{\textrm{RF}}$, close to the
Larmor frequency (or Zeeman splitting) and controlled by an applied static
magnetic field, $\bm{B}_{\textrm{DC}}$. We use microwave spectroscopy Haroche
_et al._ (1970); Sinuco-León  _et al._ (2019) to determine the energy
difference and line-width of transitions between dressed states Garraway and
Perrin (2016); Pandey  _et al._ (2019), and study their dependence with
respect to the dressing configuration. We observe that, in general, for any
pair of dressed states there is a dressing condition for which the atoms
decouple from fluctuations of the static field, which results in a significant
reduction of the line broadening. In our setup, using a linearly polarised RF
dressing field, we find that the broadening of dressed transitions lines
($\sim 0.1~{}$kHz) is one order of magnitude smaller than that of equivalent
bare transitions ($\sim 1.0~{}$kHz), limited mainly by amplitude noise of our
RF generator Morizot  _et al._ (2008). We also find that the optimal dressing
condition (i.e. that with reduced magnetic field sensitivity) depends on the
selected pair of dressed states as a result of the difference of the
gyromagnetic factors of the two electronic ground state hyperfine manifolds,
non-linear Zeeman shifts Sinuco-León and Garraway (2012) and Bloch-Siegert
shift effects Bloch and Siegert (1940).

While the majority of existing techniques for noise suppression focuses on
qubit and clock applications, there is growing interest in manipulating qudit
systems in atomic Smith  _et al._ (2013); Trypogeorgos  _et al._ (2018) and
solid-state Kucsko  _et al._ (2018) platforms, which requires the control of
$d$ internal states Choi  _et al._ (2017); Campbell (2014); Lanyon  _et al._
(2009). In this paper, we propose a dressing scheme that reduces the
sensitivity of all possible hyperfine transitions in the electronic ground
state of 87Rb. Our scheme exploits the possibility of addressing independently
each hyperfine manifold by tuning the frequency of each circular component
($\sigma_{\pm}$) of the dressing field Mas  _et al._ (2019).

As a figure of merit in the improvement in the stability against low-frequency
noise, and thus of the atomic resilience to decoherence O’Malley  _et al._
(2015); Brouard and Plata (2003), we evaluate the root-mean-square average of
the first derivative of the atomic transitions frequencies with respect to
static field, $\left\langle\alpha_{DC}\right\rangle$ (see Sec. IV). Using the
Rotating Wave Approximation (RWA) and neglecting the quadratic Zeeman shift,
we found a first estimate of the ratio between the $\sigma_{\pm}$ frequencies
that minimises $\left\langle\alpha_{DC}\right\rangle$. This optimal condition
should be corrected to take into account effects from non-linear Zeeman and
Bloch-Siegert shifts, which we explain qualitatively and evaluate numerically.
We found that, although it is not possible to fully cancel the influence of
magnetic field fluctuations for all transitions, our scheme defines dressed
states with a magnetic sensitivity smaller than that possible with bare and
single frequency dressed atoms. This scheme can be used to improve the
robustness of qudits encoded in the electronic ground state of alkali atoms,
for which control protocols have been recently demonstrated using microwave
pulses Smith  _et al._ (2013); Anderson  _et al._ (2015).

The structure of this paper is as follows. Section II reviews the formalism we
use to define the basis of dressed states. In section III we present
experimental results for the dependence of the transition frequencies between
dressed states with respect to the applied static magnetic field, which
reflects into a dependence of the linewidth and decoherence. Following this,
in section IV we propose a bichromatic dressing configuration that leads to an
overall reduction of the magnetic sensitivity (on average) of all possible
transitions between dressed states. The closing section (V) presents the
conclusions of our work.

##  II  RF-dressing of the ground state manifold of alkali atoms

The internal dynamics of an alkali atom in its electronic ground state
interacting with a magnetic field, $\bm{B}(t)$, are governed by the
Hamiltonian:

$H(t)=\frac{A}{\hbar^{2}}\bm{I}\cdot\bm{J}+\frac{\mu_{\textrm{B}}}{\hbar}(g_{I}\bm{I}+g_{J}\bm{J})\cdot\bm{B}(t)$
(1)

where $A$ is the hyperfine coupling, $\mu_{\textrm{B}}$ is the Bohr magneton
and the factors $g_{I}$ and $g_{J}$ are the nuclear ($\bm{I}$) and electronic
($\bm{J}$) Landé g-factors, with corresponding angular momentum operators
$\bm{I}$ and $\bm{J}$. The coupling between nuclear and electronic magnetic
momenta defines two hyperfine manifolds (given $J=1/2$) with different total
angular momentum, $F=I\pm 1/2$, which are split by an energy gap of
$\hbar\omega_{\textrm{hfs}}=A\left(I+1/2\right)$. This splitting defines a
natural basis of states to describe the atomic dynamics, labelled by the total
angular momentum ($F$) and its projection along a quantisation axis ($m$),
$\\{\left|F,m\right\rangle\\}$.

Here we study RF-dressed atoms of 87Rb prepared by the magnetic field:

$\bm{B}(t)=B_{\textrm{DC}}\,\hat{\bm{e}}_{z}+\sum_{i\in\\{x,y,z\\}}B_{\textrm{RF},i}\cos(\omega_{\textrm{RF}}t+\phi_{i})\hat{\bm{e}}_{i}$
(2)

where the quantisation axis $z$ is defined along the direction of the static
field $\bm{B}_{\textrm{DC}}$.

The dressed basis is defined as the set of solutions of the Schrödinger
equation resulting from Eqs. (1) and (2), with the form:

$\left|\Psi_{F\bar{m}}\right\rangle=e^{-i\bar{E}_{F,\bar{m}}t/\hbar}\left|F,\bar{m}\right\rangle$
(3)

where $\bar{E}_{F,\bar{m}}$ is the corresponding dressed energy Shirley
(1965). The ket on the right hand side of Eq. (3) is a time-periodic linear
combination of the bare states, which can be expressed as the Fourier series:

$\left|F,\bar{m}\right\rangle=\sum_{m}\sum_{n}U_{m,\bar{m}}^{n}e^{in\omega_{\textrm{RF}}t}\left|F,m\right\rangle$
(4)

The existence of this solution of the Schrödinger equation is guaranteed by
the Floquet-Bloch theorem Weinberg  _et al._ (2017).

In our experiments, see Sec. III, we transform the eigenstates of the static
Hamiltonian, $\\{\left|F,m\right\rangle\\}$ into the dressed states
$\\{\left|F,\bar{m}\right\rangle\\}$, by adiabatically varying the DC and RF
fields until reaching a final dressing configuration. We achieve high fidelity
transformations from the bare to the dressed basis by defining temporal
trajectories of the field configuration that avoid the degeneracy of the
dressed energies $\\{\bar{E}_{F,\bar{m}}\\}$ Weinberg  _et al._ (2017).
Furthermore, the absence of level crossings during the switch-over to the
dressed basis allows us to establish a one-to-one correspondence between the
bare and the dressed basis, such that we can use the same set of quantum
numbers for labelling the two bases.

In the limit of weak dressing and linear Zeeman shift, the Fourier
coefficients in Eq. (4) can be approximated using the Rotating Wave
Approximation (RWA) with the dressed energy Series (1978) (see appendix A):

$\bar{E}_{F,\bar{m}}=E_{F}+\frac{g_{F}}{|g_{F}|}\bar{m}\sqrt{(\hbar\omega_{0}-\hbar\omega_{\textrm{RF}}^{\textrm{sgn}(g_{F})})^{2}+2\left|\hbar\Omega_{\textrm{RF}}^{\textrm{sgn}(g_{F})}\right|^{2}}$
(5)

where $E_{F}=A(F(F+1)-I(I+1)-J(J+1))/2$,
$\omega_{0}=|\mu_{B}g_{F}B_{\textrm{DC}}/\hbar|$ the Larmor angular frequency,
$\omega_{\textrm{RF}}^{\textrm{sgn}(g_{F})}$ denotes the rotating frequency of
the $\sigma_{\textrm{sgn}(g_{F})}$-polar component of the RF field, defined
with respect to the local direction of the static magnetic field ($z-$axis),
and the Rabi coupling:

$\Omega_{\textrm{RF}}^{\textrm{sgn}(g_{F})}=\frac{\mu_{B}g_{F}}{2^{3/2}\hbar}\left(-\textrm{sgn}(g_{F})B_{\textrm{RF},x}e^{-i\phi_{x}}+iB_{\textrm{RF,y}}e^{-i\phi_{y}}\right)$
(6)

with $\textrm{sgn}(g_{F})\in\\{+1,-1\\}$.

The general definition of the dressed states, $\left|F,\bar{m}\right\rangle$
as the Fourier series in Eq. (4) allows us to calculate corrections to the RWA
and include the full dependence of the Zeeman shifts with the static magnetic
field. This formulation can be extended to the case of polychromatic dressing
with $N$ inconmensurable frequencies, using the multidimensional Fourier
decomposition of the dressed state:

$\left\langle
F,m\right|\left.F,\bar{m}\right\rangle=\sum_{\vec{n}}U_{m,\bar{m}}^{\vec{n}}e^{i\vec{n}\cdot\vec{\omega}t}$
(7)

where the $N$-dimensional vectors
$\vec{\omega}=(\omega_{1},\omega_{2},...\omega_{N})$ and $\vec{n}\in Z^{N}$
Sinuco-León (2020). We apply this formalism in Sec. IV to evaluate the dressed
energies when applying a bichromatic RF field.

###  II.1 Resonant condition in RF-dressed states

In the experiments described in Sec. III, we perform microwave spectroscopy of
the dressed atoms by applying a short and weak MW pulse to the dressed
ensemble, followed by the detection of the fraction of atoms remaining in (and
transferred to) the initial (the final) hyperfine manifold Sinuco-León  _et
al._ (2019).

To calculate the response of the dressed atom to this pulse, we should express
the MW coupling in the dressed basis, using the transformation rule:

$\bar{H}_{\textrm{MW}}=U_{F}^{\dagger}(t)H_{\textrm{MW}}U_{F}(t)$ (8)

where $H_{\textrm{MW}}$ is the atomic coupling to the microwave field
expressed in the basis of Zeeman states (see Eq. (1)), and $U_{F}(t)$ is the
unitary time-dependent transformation defined in Eq. (4).

The harmonic components of $U_{F}(t)$ combine with the oscillation of the MW
field, to produce a resonant condition for transitions between dressed states,
$\left|F,\bar{m}\right\rangle\leftrightarrow\left|F^{\prime},\bar{m}^{\prime}\right\rangle$:

$\omega_{\textrm{MW}}=n\omega_{\textrm{RF}}+\omega_{\bar{m}^{\prime},\bar{m}}$
(9)

with $n\in Z$ and the transition angular frequency:

$\omega_{\bar{m}^{\prime},\bar{m}}=\frac{\bar{E}_{F^{\prime},\bar{m}^{\prime}}-\bar{E}_{F,\bar{m}}}{\hbar}$
(10)

where the dressed energy $\bar{E}_{F,\bar{m}}$ is defined in Eq. (3).

Using the Rotating Wave Approximation and considering near-resonant RF
dressing, this resonant condition becomes (see Appendix A):

$\displaystyle\omega_{\textrm{MW}}$ $\displaystyle=$ $\displaystyle
n\omega_{\textrm{RF}}+\omega_{\textrm{hfs}}+\frac{g_{F+1}}{|g_{F+1}|}\bar{m}^{\prime}\sqrt{2}|\Omega_{\textrm{RF}}^{+}|-\frac{g_{F}}{|g_{F}|}\bar{m}\sqrt{2}|\Omega_{\textrm{RF}}^{-}|+\left(\frac{g_{F+1}}{|g_{F+1}|}\frac{\bar{m}^{\prime}}{|\Omega_{\textrm{RF}}^{+}|}-\frac{g_{F}}{|g_{F}|}\frac{\bar{m}}{|\Omega_{\textrm{RF}}^{-}|}\right)\frac{\omega_{\textrm{RF}}^{2}}{2^{3/2}}$
$\displaystyle-\left(\frac{\bar{m}^{\prime}g_{F+1}}{|\Omega_{\textrm{RF}}^{+}|}-\frac{\bar{m}g_{F}}{|\Omega_{\textrm{RF}}^{-}|}\right)\omega_{\textrm{RF}}\frac{\mu_{\textrm{B}}B_{\textrm{DC}}}{\sqrt{2}\hbar}+\frac{1}{2^{3/2}}\left[\frac{g_{F+1}}{|g_{F+1}|}\frac{\bar{m}^{\prime}g_{F+1}^{2}}{|\Omega_{\textrm{RF}}^{+}|}-\frac{g_{F}}{|g_{F}|}\frac{\bar{m}g_{F}^{2}}{|\Omega_{\textrm{RF}}^{-}|}\right]\left(\frac{\mu_{\textrm{B}}B_{\textrm{DC}}}{\hbar}\right)^{2}$

where $\omega_{\textrm{hfs}}=(I+1/2)A/\hbar$ is the hyperfine splitting.

Experimentally, we scan the MW frequency and determine the resonant condition
as a function of the static magnetic field $B_{\textrm{DC}}$. Thus, we
determine the energy difference between dressed states
$\hbar\omega_{\bar{m}^{\prime},\bar{m}}=(\bar{E}_{F^{\prime},\bar{m}^{\prime}}-\bar{E}_{F,\bar{m}})$,
observing a quadratic dependence of which we give details below.

##  III  Spectroscopy and protection of RF-dressed 87Rb

In this section, we describe the experimental procedure to achieve
decoherence-free pairs of dressed Zeeman sub-levels with an ultra-cold cloud
of 87Rb atoms. We focus on the dependence of the microwave resonances with the
dressing configuration, which is here controlled by the applied static
magnetic field. The main components of our experimental setup are a crossed
red-detuned optical trap, a pair of coils that define the quantization axis,
and three pairs of coils to produce static and RF magnetic fields. For a
detailed description we refer to Ref. Sinuco-León  _et al._ (2019).

We use trapped ultra-cold atoms so as not to be limited by the free-fall time
of the cloud. Instead, our background collisional lifetimes close to two
minutes impose no limit on detecting the target coherence times in this work,
e.g. $\tau_{coherence}\approx 100$ ms. Furthermore, using an optical dipole
trap, instead of a magnetic trap Mas  _et al._ (2019), we can generate a
state-independent trapping potential and ensure that the applied RF and DC
magnetic fields are nearly homogeneous over the atomic sample. All these
features of our setup enable a clean interrogation of the energy difference
between RF-dressed states, limited mainly by fluctuations of the dressing
field amplitude and polarization.

In our system, the quantisation axis is defined by a homogeneous DC field
driven by a pair of Helmholtz coils. However, since we do not implement any
magnetic compensation or shielding, both the Earth’s magnetic field and fields
generated by the equipment nearby, contribute to the total static field. We
determine this offset field, $\bm{B}_{\textrm{DC}}^{\textrm{offset}}$ by
measuring the resonant frequency of transitions between bare states with a
known applied DC field, and fitting the Breit-Rabi formula. These measurements
are done in between the experimental runs that register the dressed spectrum
(for details see Mas (2019)), and we find an offset field with
$B_{\textrm{DC},z}^{\textrm{offset}}=(-0.252\pm 0.014)~{}$G and a component in
the x-y plane of magnitude $B_{\textrm{DC},\perp}^{\textrm{offset}}=(0.264\pm
0.012)~{}$G. This field adds to the one produced by the Helmholtz coils to
define total magnetic field
$\bm{B}_{DC}=\bm{B}_{DC}^{H}+\bm{B}_{DC}^{\textrm{offset}}$. We denote its
magnitude by $B_{DC}$.

###  III.1  Preparation of the atomic cloud

Our initial sample is a cold cloud of 87Rb prepared in the bare electronic
ground state $|F=1,m=-1\rangle$ (in Fig. 1(a)) via the methods described in
detail in Ref. Sinuco-León  _et al._ (2019). This cloud is evaporated in a
hybrid crossed dipole trap plus a quadrupole (with the gradient
$\alpha=10~{}$G/cm) until reaching a temperature of $100$ nK with a typical
population of $n=$ $2\times 10^{5}$ atoms in the bare state $|1,-1\rangle$. A
bias field in the vertical direction $B_{\textrm{DC}}^{H}\approx 1.2$ G,
produced by a pair of Helmholtz coils, is present at this point too. The
dipole trap has been tuned at $\alpha=2~{}$G/cm so that the trapping
frequencies are $\omega_{\rho}/2\pi\approx 180$ Hz and
$\omega_{\textrm{axial}}/2\pi\approx 30$ Hz. We subsequently switch off the
current to the quadrupole coils, leaving a residual magnetic quadrupole field
with a maximal measured field-gradient of $\alpha<0.05~{}$G/cm.

In the next step, the atoms are dressed through the procedure sketched in Fig.
1(b): the current in the pair of Helmholtz coils is ramped up to increase the
applied static field from $\bm{B}_{\textrm{DC}}^{H}=0\hat{\bm{z}}~{}$G to
$\bm{B}_{\textrm{DC}}^{H}=5\hat{\bm{z}}~{}$G in $\Delta t=200~{}$ms. Then a
linearly polarised RF-field in the $x$ direction with angular frequency
$\omega_{\textrm{RF}}/2\pi\approx 2.3~{}$MHz and Rabi coupling
$\sqrt{2}|\Omega_{\textrm{RF}}^{\pm}|/2\pi\approx 350~{}$kHz is switched on,
followed by an adiabatic linear ramp (of duration $\Delta t=0.4~{}$s) from
$\bm{B}_{\textrm{DC}}^{H}=5\hat{\bm{z}}~{}$G to
$\bm{B}_{\textrm{DC}}^{H}=3\hat{\bm{z}}~{}$G. Then, the magnitude of the
Helmholtz field, $B_{\textrm{DC}}^{H}$, is increased to a final value so that
the total field $B_{\textrm{DC}}$ is within the range $[3.1,3.3]~{}$G. Finally
we probe the RF-dressed atom with a MW pulse and take quasi-simultaneous
independent absorption images of the $F=1$ and $F=2$ manifolds, shortly after
switching off the dipole trap that holds the cloud Mas (2019). We will refer
to the number of atoms measured in $F=2$ ($F=1$) as $n_{2}$ ($n_{1}$), and the
fraction of atoms in $F=2$ as $f_{2}=n_{2}/(n_{1}+n_{2})$.

Figure 1:  (a) Schematic of the energy level diagram that indicates available
microwave transitions from the initial dressed state
$|1,\bar{m}_{F}=-1\rangle$ state to the upper $F=2$ manifold dressed sub-
levels. The solid (dashed) arrow lines indicate allowed (non-allowed)
transitions for the field configuration shown in the inset. Both the MW and
the DC field point in the z direction (purple and green lines, respectively).
The RF-field is linearly polarised along the x direction. (b) Schematic of the
experimental sequence. The red dotted line tracks the DC-field, the dashed
blue line tracks the RF frequency and the purple dashed line tracks the RF-
amplitude, represented by the Rabi frequency. All curves are shown in
frequency units. The greyed-out areas represent the MW pulse (green) and
imaging (blue). The shaded yellow are represents the adiabatic dressing of the
atoms. The ramp times of
$B_{\textrm{DC}},\nu_{\textrm{RF}}=\omega_{\textrm{RF}}/(2\pi)$ and
$\Omega_{\textrm{RF}}$ are those described in Sec. III. The extent of the “MW
on” and “Imaging” greyed-out areas does not represent a real duration.

###  III.2  Protected transitions with ultra-cold RF-dressed atoms

After preparing the atomic cloud in the dressed state $|1,\bar{m}=-1\rangle$,
we fix the dressing frequency $\omega_{\textrm{RF}}^{\pm}/2\pi=2.263410$ MHz
and Rabi couplings $\sqrt{2}|\Omega_{\textrm{RF}}^{\pm}|/2\pi\approx
350~{}$kHz. Then we determine the resonant frequency of three transitions in
the vicinity of the zero field hyperfine splitting Sinuco-León  _et al._
(2019) as functions of the applied static field $B_{\textrm{DC}}^{H}$. For
each value of the total static magnetic field, $B_{\textrm{DC}}$, we extract
the resonant transition frequency after fitting Lorentzian curves to
measurements of the transferred atomic state population following a short
microwave pulse and scanning the microwave frequency. The transitions
investigated have the final states $|2,\bar{m}=1\rangle$ (at
$\omega_{\textrm{MW}}\approx\omega_{\textrm{hfs}}$), $|2,\bar{m}=2\rangle$ (at
$\omega_{\textrm{MW}}\approx\omega_{\textrm{hfs}}+\sqrt{2}|\Omega_{\textrm{RF}}^{+}|$)
and $|2,0\rangle$ (at
$\omega_{\textrm{MW}}\approx\omega_{\textrm{hfs}}-\sqrt{2}|\Omega_{\textrm{RF}}^{+}|$),
and the corresponding resonant frequencies can be estimated from Eq. (9).
Results of these measurements are shown in Fig. 2. When sampling the total
field $B_{\textrm{DC}}$ near the resonant condition of the RF field with the
Larmor frequency, we observed a linear dependence of the resonant transition
frequency to the state $|2,\bar{m}=1\rangle$, and a quadratic dependence for
the transitions to $|2,\bar{m}=2\rangle$ and $|2,\bar{m}=0\rangle$, which is
in qualitative agreement with Eqs. (5)-(9) after taking into account the
difference between Landé factors of the two hyperfine manifolds. Even though
we represented the transition to the $|2,1\rangle$ state as non-allowed in
Fig.1, this occurs only in the idealised case where all fields are aligned as
the inset of this figure indicates, see Ref. Sinuco-León  _et al._ (2019).

Figure 2:  Detuning of the resonant frequencies of the transitions between the
dressed states (a) $\left|1,-1\right\rangle\rightarrow\left|2,2\right\rangle$
(b) $\left|1,-1\right\rangle\rightarrow\left|2,-1\right\rangle$ (c)
$\left|1,-1\right\rangle\rightarrow\left|2,0\right\rangle$, as functions of
the applied static magnetic field. The symbols correspond to experimental
values obtained from spectral signals. The solid lines are fits to our model,
in Sec. II, which takes into account non-linear Zeeman shifts and beyond RWA
effects. The long-dashed (short-dashed) lines are fits to a model considering
the RWA and non-linear (linear) Zeeman shifts. The amplitude of the RF field
is the only free parameter of our models since we measure the total static
magnetic field. For the full model (long-dashed) we obtained
$B_{\textrm{RF}}^{+}=(1.07\pm 0.09)~{}$G, while for for the solid lined
$B_{\textrm{RF}}^{+}=(1.08\pm 0.09)~{}$G, with
$B_{\textrm{RF}}^{\textrm{sgn}(g_{F})}=\sqrt{2}|\Omega_{\textrm{RF}}^{\textrm{sgm}(g_{F})}/(\mu_{B}g_{F})|$
The short-dashed line is evaluated using $B_{\textrm{RF}}^{+}=1.075~{}$G and
the measured DC magnetic field.

We also observe that the transition lines from the dressed state
$|1,\bar{m}=-1\rangle$ to $|2,\bar{m}=0\rangle$ and $|2,\bar{m}=2\rangle$
narrow down to a linewidth of only $\Delta\nu\approx 110~{}$Hz (for a MW pulse
duration of $\Delta t=10~{}$ms) when the static field is adjusted to the
extrema of the quadratic line-shifts, even without using active stabilization
of $B_{\textrm{DC}}$ or magnetic shielding.

In Fig. 3 we show the line-shape of the transition
$|1,\bar{m}=-1\rangle\rightarrow|2,\bar{m}=2\rangle$ for three particular
total fields $B_{\textrm{DC}}=3.197~{}$G, $3.216~{}$G and $3.247~{}$G. This
figure shows the fraction of atoms transferred to the $F=2$ manifold as a
function of the applied microwave frequency, in the vicinity of the hyperfine
splitting frequency, $\omega_{\textrm{hfs}}$. The most striking feature of
these measurements is the narrowing of the line-shape as the static field
approaches the turning point of the quadratic line-shift in Fig. 2(a), where
the transition becomes protected against fluctuations of the DC field.

Figure 3:  Experimental demonstration of line narrowing in the dressed RF
system. The measurements (dots) of the transition
$|1,\bar{m}=-1\rangle\rightarrow|2,\bar{m}=2\rangle$ for three different total
fields $B_{\textrm{DC}}$: $3.195~{}$G (red), $3.247~{}$G (blue) and
$3.216~{}$G (black). This last value corresponds to the minimum of the
parabola in Fig. 2(a). In all cases the MW pulse duration is $\Delta t=10$ ms.
The horizontal axis is the detuning
$\Delta\omega_{0}=\omega_{\textrm{MW}}-\omega_{0}$ with respect to the minimum
of the parabola, which we have labeled $\omega_{0}$. The vertical axis shows
the fraction of measured population in $F=2$, that we defined as $f_{2}$ in
III.1. The lines are fits to Lorentzian curves for the three total magnetic
fields: $B_{\textrm{DC}}=3.195~{}$G (dashed, red), $B_{\textrm{DC}}=3.247~{}$G
(dotted, blue) and $B_{\textrm{DC}}=3.216~{}$G (solid, black). One can see
that the randomness of the transition increases as one interrogates further
away from the condition tha minimize the parabola in Fig. 2(a). This is due to
the higher sensitivity to DC magnetic field fluctuations. The fitted
Lorentzian linewidths are: $\Delta\nu=(132\pm 13)~{}$Hz, $\Delta\nu=(347\pm
54)~{}$Hz and $\Delta\nu=(427\pm 189)~{}$Hz for the black, red, and blue
curves, respectively.

Our data in Fig. 2(b) shows that the transition to $|2,\bar{m}=1\rangle$
presents a remarkably different behaviour compared to the other two. In this
case, the resonant frequency presents a linear dependence with the static
field in the vicinity of the single MW photon transition (i.e. with $n=0$ in
Eq. (9)), preventing us from finding a condition to null the DC field
sensitivity of this transition. However, as shown in Fig. 4(a), the resonant
frequency displays a turning point when the magnitude of the DC field enables
a two-photon transition, which corresponds to the resonant condition with
$n=1$ in Eq. (9) Sinuco-León  _et al._ (2019). Under this condition, we
consistently measured coherence times $>50$ ms in a Ramsey type experiment in
the crossed-dipole trap, which quickly shortens when setting the DC field away
from the turning point Merkel  _et al._ (2019).

Figure 4:  Line-shift and Ramsey fringes observed in 87Rb dressed with the
Rabi frequency $\Omega_{\textrm{RF}}/2\pi=300~{}$KHz and the RF frequency
$\omega_{\textrm{RF}}/2\pi=2.27~{}$MHz, as functions of the static magnetic
field. (a) Shows the line-shift for the
$\left|1,\bar{m}=-1\right\rangle\rightarrow\left|2,\bar{m}=1\right\rangle$
transition. We observe a second weaker peak in the range of the static field
explored (see Appendix B). Inset: shows a typical spectral signal with two
Lorentzian peaks matched to data in the main panel. (b) Shows Ramsey fringes
of the atomic population of the upper hyperfine manifold, $n_{F=2}$, when
$B_{\textrm{DC}}=2.87~{}$G (red line) and $B_{\textrm{DC}}=2.56~{}$G (blue
line). The vertical dashed and dotted lines in (a) indicate the values of the
magnetic fields used to obtain the Ramsey fringes in (b). Also in (b), the red
dots with errors bars are RMS values and standard deviations, respectively,
obtained from 5 repetitions per point. Red and blue dots without error bars
are single measurements of the total number of atoms in the $F=2$ manifold,
$n_{2}$. To highlight the contrast between the noise-sensitive (red dots) and
the protected (blue data) dressing configurations, we only show the first
$65~{}$ms of a longer experimental run (not shown). Considering Ramsey fringes
modelled with exponential decay, the protected configuration (blue dots)
displays a dephasing time of $\tau=17~{}$ms while the noise-sensitive case
shows a dephasing time of $\tau=1.65~{}$ms.

Figures 4(a)-(b) illustrate the behaviour of the transition
$\left|1,\bar{m}=-1\right\rangle\rightarrow\left|2,\bar{m}=1\right\rangle$ far
from the resonant point of the RF frequency with the Larmor frequency and with
atoms trapped in a crossed-dipole potential, as before. The panel (a) of Fig.
4 shows the dependence with the static field of the detuning of the resonant
frequency defined as
$\delta_{\textrm{MW}}=(\omega_{\textrm{MW}}-\omega_{\textrm{hfs}}-\omega_{\textrm{RF}})/2\pi$,
where $\omega_{\textrm{hfs}}$ is the hyperfine splitting frequency and
$\omega_{\textrm{RF}}/2\pi=2.27~{}$MHz. The black dots correspond to the
center of the Lorentzian fits in Fig. 4, and the black dotted line is included
as a guide to the eye. We also observe the intermittent appearance of a
second, weaker peak (see Appendix B) marked by grey squares and a dashed line.
This feature comes from transitions from the bare state
$\left|1,1\right\rangle$, which becomes populated by non-adiabatic effects
when sweeping the applied DC field Burrows  _et al._ (2017). In the same
panel, the vertical dashed line (blue) indicates the turning point of the
curve, where $\partial\delta_{\textrm{MW}}/\partial B_{\textrm{BC}}=0$, and
the vertical dotted line (red) points to a static field with a comparatively
large gradient of $\delta_{\textrm{MW}}$.

Fig. 4(b) shows two sets of data in a Ramsey-type experiment. The blue set of
data is taken at the turning point of the inverted parabola in panel (a),
corresponding to $B_{\textrm{DC}}=2.56~{}$G. The red set of data is taken with
$B_{\textrm{DC}}=2.87~{}$G, i.e. where the vertical red dotted line indicates
in (a). For these experiments, we detune the driving MW pulse from each
transition by $\delta\nu_{R}=\nu_{\textrm{MW}}-\nu_{0}$ -where $\nu_{0}$ is
the transition frequency, and $\nu_{\textrm{MW}}$ is the MW frequency- and
then drive the transition off-resonantly with two MW pulses separated by
$\tau_{\text{Ramsey}}$. We then scan $\tau_{\text{Ramsey}}$ and observe the
free temporal evolution of the population in the upper dressed state
$\left|2,1\right\rangle$. In both cases we fit the atom number to
$f_{2}=(n_{2}/2)\left[1+e^{-\frac{t^{2}\sigma^{2}}{2}}\cos{\left(2\pi
t\delta_{\textrm{R}}\right)}\right]$, where $\sigma$ is the standard deviation
of a Gaussian-shaped noise profile associated with the two-level system
defined by the states $\left|1,\bar{m}=-1\right\rangle$ and
$\left|2,\bar{m}=1\right\rangle$. The fit to the blue set of data gives
$\delta_{\textrm{R}}=216\pm 2$ Hz and $\sigma=79\pm 8$ Hz, whereas the fit to
the red set of data gives $\delta_{\textrm{R}}=1972\pm 35$ Hz and
$\sigma=883\pm 112$ Hz. The MW pulse detuning in the red (blue) set of data
was chosen to be $\delta_{\textrm{R}}=2000$ Hz ($\delta_{\textrm{R}}=200$ Hz),
which agrees with the fit estimate. The difference in dephasing rates of the
two Ramsey oscillations is a result of the distinct DC-field noise sensitivity
of the dressed transitions (quantified by the gradient
$\partial\delta_{\textrm{MW}}/\partial B_{\textrm{BC},z}$) at the chosen
static fields on Fig. 4(a). The different amplitude of the oscillations is due
to the relative shift of the driving field detuning with respect to resonant
driving, which in the case of the red curve it is as much as $2~{}$kHz.

In both cases, the fluctuations of the RF-field still shift the transitions
and contribute to the broadening of the line-shape. For instance, in the
transition
$\left|1,\bar{m}=-1\right\rangle\rightarrow\left|2,\bar{m}=1\right\rangle$, we
measured a line shift of $7~{}$Hz per kHz of Rabi frequency caused by the RF
amplitude noise. It is evident in any case that the DC-noise fluctuations
result in a much larger dispersion of the measurements as the lines are
further detuned from the extreme of the parabola-shaped line-shift. This
produces a better Lorentzian fit for the black data set in Fig. 3, and worse
fits for the blue and red data sets. Both Fig. 3 and Fig. 4(b) show that
transitions between Zeeman sub-states can be narrowed down in the RF-dressed
regime by choosing field parameters where the sensitivity to DC-fields of the
differential energy shifts become minimal, although the optimal configuration
depends on the pair of states involved.

##  IV  Simultaneous DC protection of multiple dressed transitions

The protection against DC-field fluctuations demonstrated in the previous
section (Sec. III) occurs at different DC-fields for different transitions. In
particular, we observed how the dressing configuration can be adjusted to
reduce strongly the sensitivity of the dressed transitions
$\left|F=1,\bar{m}=-1\right\rangle\rightarrow\left|F=2,\bar{m}=2\right\rangle$
(see Fig. 3) and
$\left|F=1,\bar{m}=-1\right\rangle\rightarrow\left|F=2,\bar{m}=1\right\rangle$
(see Fig. 4). This configuration is possible thanks to the nearly identical
dependence of the dressed energies involved with the applied static field.
These dependencies result in a condition where, to first order, variations of
the static field do not affect the transition angular frequency:

$\left.\frac{\partial\omega_{\bar{m}^{\prime},\bar{m}}}{\partial
B_{\textrm{DC}}}\right|_{B_{\textrm{DC}}^{0}}=0.$ (12)

at a particular value of the static field, $B_{\textrm{DC}}^{0}$. In this
section, we propose to use a combination of two RF fields to manipulate the
dressed energy of the hyperfine manifolds $F=1$ and $F=2$ independently of
each other, and investigate how this scheme can be used to reduce the DC-
magnetic field sensitivity of several transitions at the same value of the
static field.

Considering only the RWA, the upper and lower hyperfine manifolds are dressed
independently by different circular component of the RF dressing field. Also,
assuming that a static magnetic field of amplitude $B_{\textrm{DC}}^{0}$
produces a linear Zeeman shift, we find that all transitions between dressed
states become protected when the frequency of each polar component is set
according to:

$\displaystyle\hbar\omega_{\textrm{RF}}^{\textrm{sgn}(g_{F+1})}$
$\displaystyle=$ $\displaystyle\mu_{\textrm{B}}|g_{F+1}|B_{\textrm{DC}}^{0}$
(13) $\displaystyle\hbar\omega_{\textrm{RF}}^{\textrm{sgn}(g_{F})}$
$\displaystyle=$ $\displaystyle\mu_{\textrm{B}}|g_{F}|B_{\textrm{DC}}^{0}$
(14)

i.e., the resonant dressing of each hyperfine manifold ensures that Eq. (12)
is satisfied exactly for all transitions.

Furthermore, the RWA also tells us that when the polar components of the
dressing field satisfy the condition:

$\frac{B^{\textrm{sgn}(g_{F+1})}_{\textrm{RF}}}{B^{\textrm{sgn}(g_{F})}_{\textrm{RF}}}=\frac{\bar{m}}{\bar{m}^{\prime}}\frac{g_{F}}{g_{F+1}},$
(15)

with
$B_{\textrm{RF}}^{\textrm{sgn}(g_{F})}=\sqrt{2}|\Omega_{\textrm{RF}}^{\textrm{sgn}(\pm)}/(\mu_{B}g_{F})|$,
all transitions between states with the same sign of the magnetic moment are
also stable to second order:

$\left.\frac{\partial^{2}\omega_{\bar{m}^{\prime},\bar{m}}}{\partial
B_{\textrm{DC}}^{2}}\right|_{B_{\textrm{DC}}^{0}}=0$ (16)

The mechanism behind the reduction of the sensitivity of multiple transitions
is shown in Fig. 5, where we plot the detuning of the resonant frequencies of
all 15 possible transitions,
$\delta_{\bar{m}^{\prime},\bar{m}}=(\omega_{\bar{m}^{\prime},\bar{m}}-\omega_{\textrm{hfs}})/2\pi$,
as functions of the DC field. To see the effect, in Fig. 5(a), we use
$B_{\textrm{DC}}^{0}=6.0~{}$G and $B_{\textrm{RF}}^{+}=0.12~{}$G along with
Eqs. (13) - (15) for the frequencies and amplitudes of the circular components
of the RF field. In this case, the resonant frequency of all transitions,
$\delta_{\bar{m}^{\prime},\bar{m}}$, display one equilibrium point at our
chosen value $B_{\textrm{DC}}^{0}=6.0~{}$G (i.e.
$\left.\partial\delta_{\bar{m}^{\prime},\bar{m}}/\partial
B_{\textrm{DC}}\right|_{B_{\textrm{DC}}^{0}}=0,~{}\forall\\{\bar{m}^{\prime},\bar{m}\\}$),
which corresponds to ensuring first order (i.e. linear) protection against
noise in the static field. In Fig. 5(b), we plot the dressed energy calculated
taking into account non-linear Zeeman shifts and beyond RWA effect, using the
method outlined in Sec. II. In contrast to the previous case, now the
equilibrium points of $\delta_{\bar{m}^{\prime},\bar{m}}$ occur at different
values of the static field, significantly distinct from $B_{\textrm{DC}}$. We
conclude that, in general, the conditions Eqs. (13)-(15), do not protect the
states against fluctuations of the DC field at the applied field
$B_{\textrm{DC}}$ because of non-linear effects breaking the regularity of the
energy spectrum assumed by our RWA model. Note, however, that with these
conditions all resonant conditions experience a similar, but not identical,
shift towards $B_{\textrm{DC}}\approx 5.93~{}$G.

Figure 5:  Resonant frequencies of all hyperfine transitions in RF dressed
87Rb, as functions of the static magnetic field. An offset is applied to each
resonant frequency to define a simple vertical scale with a similar range for
all cases. In all panels, the dressing field configuration is given by Eqs.
(13) - (15), with $B_{\textrm{DC}}^{0}=6.0~{}$G, $\bar{m}=-\bar{m}^{\prime}=1$
and the $B_{\textrm{RF}}^{+}=0.12~{}$G. The dressed energies are calculated
using (a) the RWA and (b) the dressing scheme in Eq. (7) with two frequencies.
Solid, dashed and short-dashed lines correspond to transitions with the
initial states $|F=1,\bar{m}\rangle=$ $|1,-1\rangle$, $|1,0\rangle$,
$|1,1\rangle$, respectively. The labels indicate the final states as
$|F=2,\bar{m}^{\prime}=N-\bar{m}\rangle$. In panel (b), the non-linear Zeeman
shift and beyond RWA effects break the degeneracy of the transitions observed
in panel (a).

Figure 6:  Average sensitivity $\left\langle\alpha_{\textrm{DC}}\right\rangle$
of the dressed atomic transitions with respect to the static field (Eq. (17)),
calculated using the field configuration given by the RWA Eqs. (13) -(15).
Panel (a) shows $\left\langle\alpha_{\textrm{DC}}\right\rangle$ as a function
of the ratio $B_{\textrm{RF}}^{+}/B_{\textrm{DC}}$ for the DC fields $0.5~{}$G
(short-dashed, red), $1.70~{}$G (dashed, green), $3.08~{}$G (dotted, blue),
$4.28~{}$G (dot-dashed, cyan) and $6.0~{}$G (solid, black). The inset shows
the expanded region of weak fields. Panel (b) shows
$\left\langle\alpha_{\textrm{DC}}\right\rangle$ as a function of the static
field, $B_{\textrm{DC}}$, for RF fields of amplitude $0.5~{}$G (short-dashed,
red), $1.70~{}$G (dashed, green), $3.08~{}$G (dotted,blue), $4.28~{}$G
(dashed-dot, cyan) and $6.0~{}$G (solid, black).

We quantify the average DC sensitivity of the dressed atom using the root-
mean-square (RMS) of the first derivative of the 15 microwave transition
frequencies with respect to the static field:

$\left\langle\alpha_{\textrm{DC}}\right\rangle=\frac{1}{2\pi}\sqrt{\frac{1}{15}\sum_{\bar{m}=-1}^{1}\sum_{\bar{m}^{\prime}=-2}^{2}\left(\frac{\partial\omega_{\bar{m}^{\prime},\bar{m}}}{\partial
B_{\textrm{DC}}}\right)^{2}}$ (17)

The dressed transition frequencies in Fig. 5(b) indicate that the non-linear
Zeeman shifts and beyond RWA effects frustrate the exact first order stability
expected when applying the RWA when we obtain
$\left\langle\alpha_{\textrm{DC}}\right\rangle=0$.

Setting the dressing frequencies and amplitudes according to Eqs. (13)-(14),
we evaluated numerically $\left\langle\alpha_{\textrm{DC}}\right\rangle$ for a
typical range of experimentally relevant parameters. Our numerical results, in
Fig. 6, show that, when using the conditions indicated by the RWA and for
sufficiently strong RF fields (e.g.
$B_{\textrm{RF}}^{+}>~{}0.08B_{\textrm{DC}}$), the atomic sensitivity
$\left\langle\alpha_{\textrm{DC}}\right\rangle$ depends linearly on the
amplitude of the RF field and is independent of the applied static field. For
weaker RF fields, $\left\langle\alpha_{\textrm{DC}}\right\rangle$ depends on
the static field since both beyond-RWA and non-linear Zeeman effects become
comparable.

The residual sensitivity observed in Fig. 6 can be reduced by adjusting the
dressing frequencies to bring them back into resonance after taking into
account energy shifts induced by the driving. Such energy shifts occur
because, while each circular component of the RF field dresses one hyperfine
manifold, they also cause off-resonant perturbations of the other one Bloch
and Siegert (1940). Using a second order perturbative expansion of the RWA
dressed energy, these energy shifts translate into a correction of the
resonant condition given by (see Appendix C):

$\displaystyle\hbar\Delta\omega_{\textrm{RF}}^{\ell_{F}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{|\mu_{\textrm{B}}g_{F}B^{-\ell_{F}}_{\textrm{RF}}|^{2}}{\mu_{\textrm{B}}|g_{F}|B_{\textrm{DC}}+\hbar\omega_{\textrm{RF}}^{-\ell_{F}}}$
(18)

with $\ell_{F}=\textrm{sgn}(g_{F})$ and $F=I\pm 1/2$.

With these arguments as a guide for our calculations, we numerically optimise
the combination of frequencies that minimise
$\left\langle\alpha_{\textrm{DC}}\right\rangle$, using field amplitudes at the
ratio given by Eq. (15). The fractional frequency shift with respect to the
conditions given by the RWA is shown in Fig. 7, which is in qualitative
agreement with Eqs. (18). Note that, although the relative frequency shift is
of order $\sim 10^{-3}$, this translates into an important correction in
absolute terms, typically corresponding to a shift of a few kHz of the RF
frequency.

Figure 7:  Fractional correction of the frequencies required to reduce the
average atomic linear sensitivity with respect to the static field: frequency
of the $\sigma_{\textrm{sgn}(g_{F+1})}$ (dashed) and
$\sigma_{\textrm{sgn}(g_{F})}$ (solid) polar components of the dressing
fields. This shift is independent of the applied static field.

When using the optimised frequencies,
$\left\langle\alpha_{\textrm{DC}}\right\rangle$ is reduced by one order of
magnitude and becomes weakly dependent on the amplitude of the RF field (since
we compensate for their main contribution to the energy shifts). It also
becomes linearly dependent on the static field, as shown in Fig. 8 111When
using the RWA conditions, we have
$\frac{\partial\left\langle\alpha_{\textrm{DC}}\right\rangle}{\partial
B_{\textrm{DC}}}\approx 0$ and
$\frac{\partial\left\langle\alpha_{\textrm{DC}}\right\rangle}{\partial
B_{\textrm{RF,+}}}>0$. With the optimized RF frequencies
$\frac{\partial\left\langle\alpha_{\textrm{DC}}\right\rangle}{\partial
B_{\textrm{DC}}}>0$ and
$\frac{\partial\left\langle\alpha_{\textrm{DC}}\right\rangle}{\partial
B_{\textrm{RF,+}}}\approx 0$..

Figure 8:  Average sensitivity $\left\langle\alpha_{\textrm{DC}}\right\rangle$
of the dressed atomic transitions with respect to the static field (Eq. 17),
calculated using the field configuration optimised numerically. Panel (a)
shows $\left\langle\alpha_{\textrm{DC}}\right\rangle$ as a function of the
ratio $B_{\textrm{RF}}^{+}/B_{\textrm{DC}}$ for the DC fields $0.5$ G (short-
dashed, red), $1.70$ G (dashed, green), $3.08$ G (dotted, blue), $4.28$ G
(dot-dashed, cyan) and $6.0$ G (solid, black). Panel (b) shows
$\left\langle\alpha_{\textrm{DC}}\right\rangle$ as a function of the static
field, $B_{\textrm{DC}}$, for RF fields of amplitude $0.5$ G (short-dashed,
red), $1.70$ G (dashed, green), $3.08$ G (dotted, blue), $4.28$ G (dot-dashed,
cyan) and $6.0$ G (solid, black).

The effect of using corrected RF frequencies is presented in Fig. 9, where we
plot the $15$ dressed transition frequencies as functions of the static
magnetic field. Note that in this case the extrema of all curves
$\omega_{\bar{m}^{\prime},\bar{m}}$ return to the vicinity of
$B_{\textrm{DC}}=~{}6.0$G (compare with Figs. 5(a) and (b)).

Figure 9:  Detuning of the resonant frequencies of all hyperfine transitions
in RF dressed 87Rb, as functions of the static magnetic field. The field
configuration is numerically optimised to minimise the RMS of the linear
sensitivity of all bichromatic dressed transitions (see text). In all cases
$B_{\textrm{DC}}^{0}=6.0~{}$G and the $B_{\textrm{RF}}^{+}=0.12~{}$G. Solid,
dashed and short-dashed lines correspond to transitions with the initial
states $|F=1,\bar{m}\rangle=$ $|1,-1\rangle$, $|1,0\rangle$, $|1,1\rangle$,
respectively. The labels on the right hand side indicate the final states as
$|F=2,\bar{m}^{\prime}=N-\bar{m}\rangle$.

Finally, we illustrate quantitatively the improved stability enabled by the
optimised bichromatic RF-dressing using the fractional frequency fluctuation
due to noise in the DC field, defined by:

$\sigma_{{\bar{m}^{\prime},\bar{m}}}=\frac{1}{\omega_{\bar{m}^{\prime},\bar{m}}}\frac{\partial\omega_{\bar{m}^{\prime},\bar{m}}}{\partial
B_{\textrm{DC}}}\times\Delta B_{\textrm{DC}}$ (19)

For concreteness, in Table 1, we consider all transitions between the state
$\left|1,\bar{m}=-1\right\rangle$ to the five states of the upper hyperfine
manifold, and perform this calculation for four comparable field
configurations: (A) bare atom (B) monochromatic linearly polarised RF dressing
with frequency
$\omega_{\textrm{RF}}=\mu_{\textrm{B}}|g_{2}-g_{1}|B_{\textrm{DC}}/(2\hbar)$
and $B_{\textrm{RF},x}=0.2B_{\textrm{DC}}$, (C) bichromatic dressing with the
dressing field given by the RWA, Eqs. (13)-(15) and using
$B_{\textrm{RF}}^{+}=0.1B_{\textrm{DC}}$ and (D) optimised bichromatic driving
with $B_{\textrm{RF}}^{+}=0.1B_{\textrm{DC}}$. In all these cases, we consider
DC fluctuations of amplitude $\Delta B_{\textrm{DC}}=0.1~{}$mG.

The best stability is obtained for the transition
$\left|1,m=-1\right\rangle\rightarrow\left|2,m=1\right\rangle$ of the bare
atom (A) at the magic field $B_{\textrm{DC}}=3.22$ G. However, all dressing
configurations considered provide a global improvement over the fluctuations
of two orders of magnitude for all transitions. In particular, the optimized
bichromatic driving configuration (D) defines all transitions with the same
level of protection.

$B_{\textrm{DC}}~{}$ | FS | A | B | C | D
---|---|---|---|---|---
| $\left|2,-2\right\rangle$ | $2.7\times 10^{-7}$ | $5.0\times 10^{-9}$ | $6.8\times 10^{-9}$ | $3.3\times 10^{-11}$
| $\left|2,-1\right\rangle$ | $1.8\times 10^{-7}$ | $4.5\times 10^{-9}$ | $4.5\times 10^{-9}$ | $4.6\times 10^{-11}$
$0.5~{}$ | $\left|2,0\right\rangle$ | $9.2\times 10^{-8}$ | $4.0\times 10^{-9}$ | $2.2\times 10^{-9}$ | $6.6\times 10^{-11}$
| $\left|2,1\right\rangle$ | $3.1\times 10^{-10}$ | $3.6\times 10^{-9}$ | $2.0\times 10^{-11}$ | $6.6\times 10^{-11}$
| $\left|2,2\right\rangle$ | $9.1\times 10^{-8}$ | $3.2\times 10^{-9}$ | $2.2\times 10^{-9}$ | $2.3\times 10^{-11}$
| $\left|2,-2\right\rangle$ | $2.7\times 10^{-7}$ | $4.8\times 10^{-9}$ | $6.7\times 10^{-9}$ | $1.9\times 10^{-10}$
| $\left|2,-1\right\rangle$ | $1.8\times 10^{-7}$ | $4.2\times 10^{-9}$ | $4.2\times 10^{-9}$ | $3.5\times 10^{-10}$
$3.2~{}$ | $\left|2,0\right\rangle$ | $9.1\times 10^{-8}$ | $3.7\times 10^{-9}$ | $1.8\times 10^{-9}$ | $4.2\times 10^{-10}$
| $\left|2,1\right\rangle$ | $4.0\times 10^{-12}$ | $3.3\times 10^{-9}$ | $3.4\times 10^{-10}$ | $3.7\times 10^{-10}$
| $\left|2,2\right\rangle$ | $9.1\times 10^{-8}$ | $3.0\times 10^{-9}$ | $2.3\times 10^{-9}$ | $1.7\times 10^{-10}$
| $\left|2,-2\right\rangle$ | $2.7\times 10^{-7}$ | $4.7\times 10^{-9}$ | $6.5\times 10^{-9}$ | $4.0\times 10^{-10}$
| $\left|2,-1\right\rangle$ | $1.8\times 10^{-7}$ | $3.8\times 10^{-9}$ | $3.7\times 10^{-9}$ | $7.9\times 10^{-10}$
$7.0~{}$ | $\left|2,0\right\rangle$ | $9.1\times 10^{-8}$ | $3.2\times 10^{-9}$ | $1.3\times 10^{-9}$ | $9.2\times 10^{-10}$
| $\left|2,1\right\rangle$ | $4.2\times 10^{-10}$ | $2.9\times 10^{-9}$ | $7.8\times 10^{-10}$ | $7.9\times 10^{-10}$
| $\left|2,2\right\rangle$ | $9.1\times 10^{-8}$ | $2.8\times 10^{-9}$ | $2.5\times 10^{-9}$ | $3.8\times 10^{-10}$
Table 1:  Comparison of the fractional frequency fluctuations
($\Delta\omega_{\bar{m}^{\prime},\bar{m}}/\omega_{\bar{m}^{\prime},\bar{m}}$)
for the transitions from the state $\left|1,-1\right\rangle$ to all final
states (FS) of the $F=2$ manifold of Zeeman sub-levels (see Eq. (19)). The
transition frequencies and sensitivities are calculated for transitions
between (A) bare states, (B) monochromatic RF dressed states, and bichromatic
RF dressed states with $\omega_{\textrm{RF}}^{\pm}$ (C) given by the RWA (Eqs.
(13)-(14)) and (D) corrected to minimise
$\left\langle\alpha_{\textrm{DC}}\right\rangle$, as shown in Fig. 7. In all
cases, the field fluctuations are $\Delta B=0.1~{}$mG. In (B) the dressing
fiield is $B_{\textrm{RF}}=0.2B_{\textrm{DC}}$, while in (C) and (D) we use
$B_{\textrm{RF}}^{+}=0.1B_{\textrm{DC}}$. In (D),
$\Delta\omega_{\textrm{RF}}^{\ell}/\omega_{\textrm{RF}}^{\ell}=2.46\times
10^{-3}~{}$ and $2.50\times 10^{-3}$, for $\ell=+$ and $-$, respectively.

Noise in the amplitude of the dressing field contributes to the linewidth of
the spectral lines of transitions between dressed states. Temporal variations
of each polar component of the RF field affect the transition frequencies by
modifying the dressed energy of the manifold it dresses and induce corrections
to the perturbative energy shift of the other hyperfine manifold. We use the
RMS variation of the transition frequencies
($\omega_{\bar{m},\bar{m}^{\prime}}/(2\pi)$) with respect to each polar
component to quantify these effects in Appendix C. The average first-order
sensitivity of the transition frequencies are of order $\approx 10^{2}~{}$Hz
mG-1; much larger than the residual DC sensitivity
$\left\langle\alpha_{\textrm{DC}}\right\rangle\lesssim 10~{}$Hz mG-1 achieved
by tuning the frequencies of each polar component of the RF field. Thus, the
improved stability of the bichromatic dressing scheme becomes useful when
there is equally good stability of the RF source. The bi-chromatic dressing
configuration has been demonstrated in Mas  _et al._ (2019) for the case of
atoms trapped in a magnetic quadrupole field. The two circular components can
be produced by using two pairs of Helmholtz coils that point along the $x$ and
$y$ directions, respectively. Each pair of coils is driven with an RF-signal
that contains both frequencies, with the appropriate phases (for details, see
Mas  _et al._ (2019)).

##  V Conclusions

Using an ultra-cold atomic cloud of 87Rb in an optical dipole trap, we showed
characteristic features of transitions between RF dressed states of the ground
state hyperfine manifolds. First, considering the trio of transitions between
dressed states
$\left|1,\bar{m}=-1\right\rangle\rightarrow\\{\left|2,\bar{m}=0\right\rangle,\left|2,\bar{m}=1\right\rangle,\left|2,\bar{m}=2\right\rangle\\}$,
we observe a quadratic dependence of their transition frequencies as functions
of total magnetic field $B_{\textrm{DC}}$, with a significantly weaker
curvature when $\Delta\bar{m}=0$. We found that our measurements can be
explained quantitatively only after taking into account non-linear Zeeman
shifts and beyond-RWA effects. However, a good qualitative description of the
observed quadratic differential energy shifts can be obtained using well-known
expressions for the dressed energies valid in the regimes of linear Zeeman
shift and RWA.

We also study the coherence and linewidth of the transitions between RF-
dressed states
$\left|1,\bar{m}=-1\right\rangle\rightarrow\left|2,\bar{m}=1\right\rangle$ and
$\left|1,\bar{m}=-1\right\rangle\rightarrow\left|2,\bar{m}=2\right\rangle$, as
functions of the applied static field. We observe a significant increase in
the decay time of Ramsey-type fringes for the transition
$\left|1,\bar{m}=-1\right\rangle\rightarrow\left|2,\bar{m}=1\right\rangle$ at
a particular “magic” point. Following the same method, we observed a
significant reduction of the linewidth of the transition
$\left|1,\bar{m}=-1\right\rangle\rightarrow\left|2,\bar{m}=2\right\rangle$,
reaching fluctuations of the order $\Delta\nu/\nu\approx 10^{-8}$. These
experimental results demonstrate how monochromatic RF dressing can be tuned to
produce pairs of selected transitions protected against DC-field noise. To
further reduce the transition linewidth to the order of Hz with this scheme,
the fluctuations of the amplitude of the RF fields need to be stabilised at
the level of order $~{}10\mu$G. Recent reports using one Sinuco-León  _et al._
(2019), two Mas  _et al._ (2019), three Harte  _et al._ (2018) and four
Bentine  _et al._ (2017) RF-frequency dressing components also find this to be
a limiting factor, and indicate that active control of the RF-field amplitude
is therefore necessary Merkel  _et al._ (2019).

Furthermore, we propose a bi-chromatic RF-dressing configuration to reduce the
global sensitivity of the dressed atom to noise in the static field. We
demonstrate that by independently tuning the frequencies of the two circular
components of the RF field, it is possible to reduce the average linear DC
sensitivity to the level of Hz mG-1. This dressing scheme enables the
protection at arbitrary DC-magnetic fields of up to
$(2F+1)\times(2F^{\prime}+1)=15$ atomic microwave transitions in RF-dressed
87Rb, only limited by the noise in the RF generator. Also, this bichromatic
dressing configuration can stabilise more than just one single atomic
transition, which is useful to define stable qdits with $d>2$. Such systems
present advantages for applications in quantum metrology Giovannetti  _et al._
(2011) and enhanced fault-tolerance for quantum information Campbell (2014);
Lanyon  _et al._ (2009). In addition, since the energies of the dressed states
can be tuned precisely, these dressed states are attractive also for
applications in quantum simulations Doria  _et al._ (2011) and for hybrid
quantum systems that include devices operating in the RF/MW regime (e.g.
superconducting resonators, NV centres and trapped ions). Further research
will be directed to determine multiparametric magic configurations in the
regime of strong RF-dressing, where multi-level atomic transitions can be made
less sensitive to noise in DC and AC fields.

##  VI Acknowledgements

This work is supported by the project “HELLAS-CH” (MIS 5002735) which is
implemented under the “Action for Strengthening Research and Innovation
Infrastructures”, funded by the Operational Programme “Competitiveness,
Entrepreneurship and Innovation” (NSRF 2014-2020) and co-financed by Greece
and the European Union (European Regional Development Fund). We acknowledge
financial support from the Greek Foundation for Research and Innovation
(ELIDEK) in the framework of project, _Guided Matter-Wave Interferometry_
under grant agreement number 4823 and General Secretariat for Research and
Technology (GSRT). GV received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie Grant
Agreement No 750017. This work has been supported by the UK EPSRC grant
EP/M013294/1 and the University of Sussex. The authors would like to
acknowledge the contribution of the COST Action CA16221.

Authors contributions. G.A.S-L and H.M contributed equally. H.M. and G.A.S-L.
conceived and developed the main ideas of the paper. H.M. carried out the
experimental measurements. H.M., S.P., G.V. and W.v.K. built the experimental
setup. G.A.S.-L. and H.M. performed the data analysis. G.A.S.-L. was
responsible for the theoretical and numerical work. B.M.G. and W.v.K.
supervised the development of the work. All authors contributed to the result
discussion and paper writing.

##  Appendix A  Dressed states in the Rotating Wave Approximation

In the main text, Eq. (4) defines the unitary transformation, $U_{F}$, between
the bare and dressed basis as:

$\left\langle
F,m\right.\left|\bar{F},\bar{m}\right\rangle=\sum_{n}e^{in\omega_{\textrm{RF}}t}U_{F,m;\bar{F},\bar{m}}^{n}$
(20)

The Fourier coefficients of this expansion are determined by the Schrödinger
equation and the condition Eq. (3), which translates into:

$U_{F}^{\dagger}(t)\left[H(t)-i\hbar\partial_{t}\right]U_{F}(t)=\sum_{\bar{F},\bar{m}}\bar{E}_{\bar{F},\bar{m}}\left|\bar{F},\bar{m}\right\rangle\left\langle\bar{F},\bar{m}\right|$
(21)

This last expression gives us a straightforward physical interpretation of the
dressed basis: the dressed states are the eigenenergy states observed in a
frame of reference where the Hamiltonian is time-independent. This concept is
commonly used in quantum physics in the figure of Rotating (or Resonant) Wave
Approximation (RWA), where, after moving to a rotating frame of reference the
time-dependence of the Hamiltonian either cancels completely or can be
neglected following perturbative arguments Series (1978).

In the present case, we consider oscillating fields with a frequency
comparable to the Zeeman splitting induced by a static magnetic field, but
much smaller than the hyperfine splitting
($\hbar\omega_{\textrm{RF}}\approx\mu_{\textrm{B}}g_{F}B_{\textrm{DC}}\ll\Delta
E_{\text{Hyperfine}}$). Under these conditions, we can neglect the inter-
manifold coupling and apply the RWA within each hyperfine manifold, where the
transformation to the dressed basis can be written as a combination of
rotations in the space of angular momentum Sinuco-León  _et al._ (2019):

$U_{F}(t)=e^{i\theta_{y}F_{y}}e^{-i\frac{g_{F}}{|g_{F}|}\omega_{\textrm{RF}}tF_{z}}$
(22)

with

$\tan(\theta_{y})=\frac{\sqrt{2}|\Omega_{\textrm{RF}}^{\textrm{sgn}(g_{F})}|}{\omega_{0}-\omega_{\textrm{RF}}}$
(23)

where $\hbar\omega_{0}=|\mu_{\textrm{B}}g_{F}B_{\textrm{DC}}|$ defines the
Larmor frequency and the Rabi frequency
$|\Omega_{\textrm{RF}}^{\textrm{sgn}(g_{F})}|$ is defined in Eq. (6). The
corresponding dressed energies are:

$\bar{E}_{F,\bar{m}}=E_{F}+\textrm{sgn}(g_{F})\bar{m}\sqrt{(\hbar\omega_{0}-\hbar\omega_{\textrm{RF}})^{2}+2|\hbar\Omega_{\textrm{RF}}^{\textrm{sgn}(g_{F})}|^{2}}$
(24)

where $E_{F}=A\left(F(F+1)-I(I+1)-J(J+1)\right)/2$ is the hyperfine splitting.
This dressed energy leads to the dependence of the resonant condition with
respect to the field configuration:

$\displaystyle\omega_{\textrm{MW}}$ $\displaystyle=$ $\displaystyle
n\omega_{\textrm{RF}}+\frac{(I+1/2)A}{\hbar}$
$\displaystyle+\frac{g_{F+1}}{|g_{F+1}|}\bar{m}^{\prime}\sqrt{2}|\Omega_{\textrm{RF}}^{+}|-\frac{g_{F}}{|g_{F}|}\bar{m}\sqrt{2}|\Omega_{\textrm{RF}}^{-}|$
$\displaystyle+\left(\frac{g_{F+1}}{|g_{F+1}|}\frac{\bar{m}^{\prime}}{|\Omega_{\textrm{RF}}^{+}|}-\frac{g_{F}}{|g_{F}|}\frac{\bar{m}}{|\Omega_{\textrm{RF}}^{-}|}\right)\frac{\omega_{\textrm{RF}}^{2}}{2^{3/2}}$
$\displaystyle-\left(\frac{\bar{m}^{\prime}g_{F+1}}{|\Omega_{\textrm{RF}}^{+}|}-\frac{\bar{m}g_{F}}{|\Omega_{\textrm{RF}}^{-}|}\right)\omega_{\textrm{RF}}\frac{\mu_{\textrm{B}}B_{\textrm{DC}}}{\sqrt{2}\hbar}$
$\displaystyle+\frac{1}{2^{3/2}}\left[\frac{g_{F+1}}{|g_{F+1}|}\frac{\bar{m}^{\prime}g_{F+1}^{2}}{|\Omega_{\textrm{RF}}^{+}|}-\frac{g_{F}}{|g_{F}|}\frac{\bar{m}g_{F}^{2}}{|\Omega_{\textrm{RF}}^{-}|}\right]\left(\frac{\mu_{\textrm{B}}B_{\textrm{DC}}}{\hbar}\right)^{2}$

which is valid for inter-manifold
transitions,$\left|F,\bar{m}\right\rangle\rightarrow\left|F+1,\bar{m}^{\prime}\right\rangle$
and in the vicinity of the condition of resonant RF dressing,
$\mu_{\textrm{B}}|g_{F}B_{\textrm{DC}}|\approx\hbar\omega_{\textrm{RF}}$.

This formulation can be easily extended to situations with polychromatic
driving. The time-evolution operator should be then expressed as a
multidimensional Fourier series with as many dimensions as the number of
applied fields with incommensurately frequencies. More explicitly, the dressed
state defined in Eq. (20) becomes:

$\left\langle
F,m\right.\left|\bar{F},\bar{m}\right\rangle=\sum_{\vec{n}}e^{i\vec{n}\vec{\omega}t}U_{F,m;\bar{F},\bar{m}}^{\vec{n}}$
(26)

with $\vec{\omega}=(\omega_{1},\omega_{2},\ldots)$ as the vector formed with
all the applied frequencies and $\vec{n}$ a vector with integer components.
After plugging this Ansatz in the Schrödinger equation Eq. (21) we obtain a
standard eigen problem defining the coefficients
$U_{F,m;\bar{F},\bar{m}}^{\vec{n}}$ and generalized dressed energies
$\bar{E}_{\bar{F},\bar{m}}$.

##  Appendix B  Microwave spectroscopy of the dressed transition
$\left|1,-1\right\rangle\rightarrow\left|2,1\right\rangle$ near the two-photon
condition

We measured the line-shift of the transition
$|1,\bar{m}=-1\rangle\rightarrow|2,\bar{m}=1\rangle$ in Fig. 4(a). We
intermittently observed a second weaker peak, with some examples in Fig. 10.
Here, we interrogate the first group of transitions, i.e. those with $N=1$,
following Eq. (9), with $\omega_{\textrm{RF}}/2\pi=2.27$ MHz. A qualitative
analysis indicates that the each peak is produced by atoms in different
dressed states, with the intensity of each corresponding to their population.
This second resonance indicates that some atoms do not follow adiabatically
our dressing sequence Burrows  _et al._ (2017).

Figure 10:  Spectra for selected values of DC field in Fig. 4(a). From top to
bottom, $B_{\textrm{DC}}=2.63$ G, $2.66$ G, $2.69$ G, $2.72$ G and $2.75$ G.
The Rabi frequency is $\Omega_{\textrm{RF}}/2\pi$ kHz and the RF frequency is
$\omega_{\textrm{RF}}/2\pi=2.27$ MHz. We observe the appearance of a second,
usually weaker transition in some cases. Just like in the rest of the text,
$n_{2}$ is the atom number measured with absorption imaging in $F=2$. Each
point corresponds to one measurement. We fit Lorentzian curves for each
observed peak. We observe the intermittent appearance of a second, weaker
peak. We attribute this second peak to transitions from atoms that did not
follow adiabatically our dressing sequence.

##  Appendix C  Off-resonant corrections to the RWA and linear sensitivity of
the bichromatic driven atom to RF amplitude noise

The susceptibility of the dressed transitions with respect to variations of
the dressing parameters can be calculated correcting the RWA dressed energy by
including perturbative shifts of the Zeeman states:

$\displaystyle\bar{E}_{F,\bar{m}}$ $\displaystyle=$ $\displaystyle
E_{F}+\frac{g_{F}}{|g_{F}|}\bar{m}\times$
$\displaystyle\sqrt{(\mu_{\textrm{B}}|g_{F}|B_{\textrm{DC}}-\hbar\omega_{\textrm{F}}+\hbar\Delta_{F})^{2}+2\hbar|\Omega_{\textrm{RF}}^{\textrm{sgn}(g_{F})}|^{2}}$

with $E_{F}=A(F(F+1)-I(I+1)-J(J+1))/2$ and where $\Delta_{F}$ has
contributions from the fields which are counter-rotating in the dressed frame
of reference. The total shift can be approximated by:

$\displaystyle\hbar\Delta_{F}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{g_{F}}{|g_{F}|}\frac{|\mu_{\textrm{B}}g_{F}(B^{-\textrm{sgn}(g_{F})}_{\textrm{RF}}+\Delta
B_{\textrm{RF}}^{\textrm{sgn}(g_{F})})|^{2}}{\mu_{\textrm{B}}|g_{F}|B_{\textrm{DC}}+\hbar\omega_{0}}$
(27)

where $\omega_{0}=(\omega_{+}+\omega_{-})/2$ and $\Delta
B_{\textrm{RF}}^{\ell}$ is the counter-rotating component of the field
oscillating at frequency $\omega_{\ell}$.

Another important contribution to the broadening of the transition lines is
their instability with respect to variations in the amplitude of the dressing
fields. In this case, we can distinguish four contributions emerging from the
decomposition of the variations of each dressing frequency into $\sigma_{+}$
and $\sigma_{-}$ polarisations. To evaluate the effects of fluctuations of the
RF field, we split the noise of each polar component of the RF field into co-
rotating and counter-rotating contributions:

$B_{\textrm{RF}}^{\ell}=B_{\textrm{RF},0}^{\ell}+\delta
B_{\textrm{RF},\ell}^{\ell}+\delta B_{\textrm{RF},-\ell}^{\ell}$ (28)

where $\ell\in{+,-}$, $B_{\textrm{RF},0}^{\ell}$ is the RMS value of the
field, and $\delta B_{\textrm{RF},(-)\ell}^{\ell}$ is the component of the
fluctuation co-rotating (counter-rotating) with the $\sigma_{\ell}$
($\sigma_{-\ell}$) component of the RF field. As in Sec. IV, we quantify these
effects by defining average linear sensitivities,
$\left\langle\alpha_{\textrm{RF},m}^{\ell}\right\rangle$ for each component of
the ($m\in{\\{+,-\\}}$) of the two dressing fields $\ell\in{\\{+,-\\}}$:

$\left\langle\alpha_{\textrm{RF},\ell}^{m}\right\rangle=\sqrt{\frac{1}{15}\sum_{m=-1}^{1}\sum_{m^{\prime}=-2}^{2}\left(\frac{\partial\omega_{\bar{m},\bar{m}^{\prime}}}{\partial\delta
B_{\textrm{RF},m}^{\ell}}\right)^{2}}$ (29)

In Fig. 11 we show the atomic sensitivity associated with the co-rotating
noisy components of the dressing fields ($m=\ell$), when using the optimised
frequency configuration. The co-rotating noise modifies the amplitude of the
dressing field, leading to a sensitivity that weakly depends on the static
field.

Figure 11:  RMS susceptibility of the dressed transition frequencies of 87Rb
with respect to co-rotating variations of the (a) $\sigma_{+}$ and (b)
$\sigma_{-}$ dressing fields as functions of the dressing field amplitude. The
static field are $0.5~{}$G (dashed line), $1.5~{}$G (long-dashed line),
$3.0~{}$G (dotted line), $4.5~{}$G(dashed-dotted line) and $6.0~{}$G (solid
line). For each dressing configuration, we evaluate the optimal combination of
$\sigma_{\pm}$ frequencies that minimise
$\left\langle\alpha_{\textrm{DC}}\right\rangle$ Eq. (17) Figure 12:  RMS
linear sensitivity of the dressed transition frequencies of 87Rb with respect
to the counter-rotating polar components (a) $\sigma_{+}$ (b) $\sigma_{-}$ of
the variations of the dressing fields $\sigma_{-}$ (a) and $\sigma_{+}$ (b).
The scaled amplitudes of the dressing fields are $0.02~{}B_{\textrm{DC}}$
(dashed line), $0.03~{}B_{\textrm{DC}}$ (long-dashed line),
$0.1~{}B_{\textrm{DC}}$ (dotted line), $0.15~{}B_{\textrm{DC}}$ (dashed-dotted
line) and $0.2~{}B_{\textrm{DC}}$ (solid line). For each dressing
configuration, we evaluate the optimal combination of $\sigma_{\pm}$
frequencies that minimise $\left\langle\alpha_{\textrm{DC}}\right\rangle$.

Another effect due to noise of the dressing field comes from the counter-
rotating noisy component, which causes off-resonant energy shifts similar to
the ones described before in Eq (27), but this time oscillating at the same
frequency of the corresponding dressing component. In Fig. 12 we show the a
RMS linear susceptibility of all transitions,
$\left\langle\alpha_{\textrm{RF},-\ell}^{\ell}\right\rangle$, induced by
counter-rotating variations of the dressing fields, using the frequency
configuration that minimise the sensitivity to static fields.

The scaling and order of magnitude of
$\left\langle\alpha_{\textrm{RF},\ell}^{m}\right\rangle$ can be obtained by
calculating second order perturbative energy shifts and applying the RWA (see
Appendix B). Note that the sensitivity to variations of the dressing fields is
two orders of magnitude larger than the sensitivity to variations of the
static field with the optimised parameters (compare with Fig. 8). To reduce
the line broadening of resonant transitions to the level of a few Hz, the
fluctuations of dressing RF fields should of order $\approx 10~{}\mu$G, which
corresponds to a relative amplitude fluctuation of the order
$10^{-5}-10^{-6}$.

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of $k$ lines. Intuitively, this means that the variable $x_{p}$ “should” be
accessed by a circuit or a branching program at least $k$ times to produce
this implicant.

######  Example 3 (Many short lines).

Instead of $2$-dimensional tensors (matrices), as in the case of the function
$\mathrm{Lines}_{n,2}$ used in Lemma 8, one can consider $k$-dimensional
tensors for $k\geqslant 3$. Let $n$ be of the form $n=m^{k}$. As the
underlying set $P$ of points, take the set $P=[m]^{k}$ of $k$-tuples
$p=(p_{1},\ldots,p_{k})\in[m]^{k}$, and consider the family ${\cal L}\subseteq
2^{P}$ of all (combinatorial) lines, where the _line_ in the $i$th direction
through a point $p\in P$ is the set
$L=\big{\\{}(p_{1},\ldots,p_{i-1},\ast,p_{i+1},\ldots,p_{k})\colon\ast=1,\ldots,n\big{\\}}$
of $|L|=m$ points. Thus, we have $|{\cal L}|=km^{k-1}=(k/m)|P|$ distinct
lines, no two sharing more than one point. Since each point belongs to exactly
$k$ lines (there are $k$ possible positions for $\ast$), the family ${\cal L}$
is $k$-regular. Hence, the corresponding (to this family ${\cal L}$) blocking
lines function $\mathrm{Lines}_{n,k}(x):=f_{{\cal L}}(x)$ can be computed by a
monotone read-$k$ $(\lor,\land)$ circuit of size $\leqslant m|{\cal
L}|=k|P|=km^{k}=kn$; in particular,
$\mathsf{B}_{k}(\mathrm{Lines}_{n,k})\leqslant kn$ holds. Note that in this
family ${\cal L}$ of lines, two points ($k$-tuples) $p\neq q\in[m]^{k}$ are
collinear iff they differ in exactly one position. In particular, any set of
_pairwise_ collinear points must entirely lie in one line. In the case $k=2$
(matrices, as in Lemma 8), smallest blocking sets consist of the entries of a
permutation matrix.

######  Example 4 (Few long lines).

Let $n$ be of the form $n=\binom{m}{k}$, were $m$ is divisible by $k$. As the
underlying set $P$ of points, take the collection $P=\binom{[m]}{k}$ of all
$|P|=n$ $k$-element sets $p\subseteq[m]:=\\{1,\ldots,m\\}$. For $i\in[m]$, let
the line in the $i$th direction be the set $L_{i}=\\{p\in P\colon i\in p\\}$
of $|L_{i}|=\binom{m-1}{k-1}$ points containing $i$. Let ${\cal
L}=\\{L_{1},\ldots,L_{m}\\}$ be the family of all $m$ lines (in all $m$
directions). Since every point $p\in P$ consists of $k$ _distinct_ elements of
$[m]$, each of them belongs to exactly $k$ lines; hence, the family ${\cal L}$
is $k$-regular For the corresponding (to this family ${\cal L}$) blocking
lines function $\mathrm{Cov}_{n,k}(x):=f_{{\cal L}}(x)$, we have an upper
bound $\mathsf{B}_{k}(\mathrm{Cov}_{n,k})\leqslant m|{\cal
L}|=m\binom{m-1}{k-1}=k\binom{m}{k}=k|P|=kn$. In this family ${\cal L}$ of
lines, two points $p\neq q\in\binom{[m]}{k}$ are collinear iff $p\cap
q\neq\emptyset$, and a set $S\subseteq\binom{[m]}{k}$ of points ($k$-element
subsets of $[m]$) is a blocking set iff the union of these subsets is the
entire set $[m]$. Hence, smallest blocking sets $S\subseteq P$ consist of
$|S|=m/k$ points forming a partition of $[m]$ into $m/k$ disjoint blocks of
size $k$. In the case $k=2$, points $p\in P$ correspond to the edges
$p=\\{i,j\\}$ of the complete graph $K_{m}$ on $[m]$, and the line $L_{i}$ in
the $i$th direction is the set of all $m-1$ edges incident with vertex $i$.
Then $\mathrm{Cov}_{n,2}(x)=1$ iff the subgraph $G_{x}$ of $K_{m}$ specified
by $x$ has no isolated vertices. Smallest blocking sets in this case are
perfect matchings in $K_{m}$.

#### Acknowledgments

I am thankful to both referees for very useful comments and suggestions.

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