PHYSICAL REVIEW A 97, 013407 (2018)

Synthetic clock transitions via continuous dynamical decoupling
D. Trypogeorgos,1,* A. Valdés-Curiel,1 N. Lundblad,2 and I. B. Spielman1
1

Joint Quantum Institute, University of Maryland and National Institute of Standards and Technology, College Park, Maryland 20742, USA
2
Department of Physics and Astronomy, Bates College, Lewiston, Maine 04240, USA(Received 27 June 2017; published 16 January 2018)
Decoherence of quantum systems due to uncontrolled fluctuations of the environment presents fundamental
obstacles in quantum science. Clock transitions which are insensitive to such fluctuations are used to improvecoherence, however, they are not present in all systems or for arbitrary system parameters. Here we create a trioof synthetic clock transitions using continuous dynamical decoupling in a spin-1 Bose-Einstein condensate inwhich we observe a reduction of sensitivity to magnetic-field noise of up to four orders of magnitude; this work
complements the parallel work by Anderson et al. [R. P. Anderson et al., following paper, Phys. Rev. A 97, 013408(2018)]. In addition, using a concatenated scheme, we demonstrate suppression of sensitivity to fluctuations in
our control fields. These field-insensitive states represent an ideal foundation for the next generation of cold-atomexperiments focused on fragile many-body phases relevant to quantum magnetism, artificial gauge fields, andtopological matter.
DOI: 10.1103/PhysRevA.97.013407
I. INTRODUCTION

II. SYSTEM

The loss of coherence due to uncontrolled coupling toa fluctuating environment is a limiting performance factor
for quantum technologies [1–4]. In select cases, first-orderinsensitive transitions, clock transitions, can mitigate the deleterious effect of the dominant noise sources, yet in most casessuch transitions are absent [5]. Remarkably, under almost all
circumstances, clock transitions can be synthesized using dynamical decoupling protocols. These protocols involve driving
the system with an external oscillatory field, resulting in a
dynamically protected dressed system. A number of dynamical
decoupling protocols, pulsed or continuous, have been shown
to isolate quantum systems from low-frequency environmental noise [6–14]. Continuous dynamical decoupling (CDD)
relies on the application of time-periodic continuous control
fields, rather than a series of quantum-logic pulses. Unlike
conventional dynamical decoupling, CDD does not require any
encoding overhead or quantum feedback measurements.
Thus far, CDD has inoculated multilevel systems in nitrogenvacancy centers in diamond, nuclear magnetic resonanceexperiments, and trapped atomic ions [15–22], from spatiotemporal magnetic-field fluctuations. We demonstrate CDD in
atomic Bose-Einstein condensates (BECs) producing a protected three-level system of dressed states, whose Hamiltonian
is fully controllable. The CDD-protected states are sensitive
to fluctuations of the amplitude of the control field, and we
demonstrate that a second coupling field protects against thosein a concatenated manner [6,10,16].

We implemented CDD using a strong radio-frequency (rf)magnetic field with strength  that linked the three mF statescomprising the F = 1 electronic ground-state manifold of87
Rb. The rf field was linearly polarized along ex and had
angular frequency ω close to the Larmor frequency ω0 =gF μB B0 from a magnetic field B0 ez ; gF is the Lande gfactor and μB is the Bohr magneton. We coupled the dressedstates using a weaker probe field with strength p , polarizedalong ey with angular frequency ω + ωp [Fig. 1(a)]. Using therotating-wave approximation (RWA) for the frame rotating at
ω (valid when ω0  ,p ,ωp ), the system is described by Ĥ = F̂z + h̄ F̂z2 h̄2 − Î + F̂x*

dtrypo@umd.edu

Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.
2469-9926/2018/97(1)/013407(7)013407-1+ p [sin(ωp t)F̂x + cos(ωp t)F̂y ],(1)

with detuning  = ω − ω0 , quadratic Zeeman shift , spin-1
angular momentum operators F̂x,y,z , and identity operator
Î. For p = 0 the resulting eigenstates, denoted by |x,|y, and |z, are linear combinations of |mF = 0,±1. The
corresponding
eigenvalues for  = 0 are ωx = 0 and ωz,y =√−( ± 42 +  2 )/2. The energy differences h̄ωxy , h̄ωzy , andh̄ωzx are only quadratically sensitive to  for   ,1 so
detuning fluctuations δ are suppressed to first order, making
these a trio of synthetic clock states. At an optimal , ωzx depends quartically on  [23,24]. For    the |x,y,z states
adiabatically connect to the corresponding |mF = 1,0,−1
states [Fig. 1(b)]. As  → 0+ and for  = 0, the |x,y,z
states continuously approach the |X,Y,Z states familiar fromquantum chemistry. In contrast, as  → ∞ they become
eigenstates of the F̂x operator: |y,x,z → |mx = +1,0,−1.

The energies are quadratic in  for    and linear for   with a slope of 7 MHz/mT.
1

Published by the American Physical Society

D. TRYPOGEORGOS et al.

PHYSICAL REVIEW A 97, 013407 (2018)(a) 40

| 1

0.4

RWA
E/h (MHz)

20

|y

0.2

|0
|x

0.0

0

0.2

20

|1
|z

0.4
40

0

(b)

1

2 3 4
B (mT)

5400

0/2 (kHz)

400

zy

400

zx
236

(kHz)

390

p /2

transition frequency (kHz)

410

234
180

xy

160
60

40

20

0
20
/2 (kHz)

40

60

FIG. 1. (a) Shown on the left is the dependence of the 5 2 S1/2 ,F = 1 ground state of 87 Rb on the magnetic field, where thequadratic dependence of the |mF = 0 state’s Zeeman shift has
been exaggerated so it is visible on the same scale. In the middle
are energies of the |x,y,z eigenstates for /2π = 200 kHz (black
curves) and  = 0 (gray curves). On the right is the TOF image of|z at  = 0, showing the constituent mF states. (b) On the left arespectroscopic data showing transitions between the |x,y,z states for
/2π = 194.5(1) kHz. The vertical scale of the middle panel (zxtransition) has only 10% the range of the other panels. The dashedlines correspond to the Hamiltonian of Eq. (1), while the solid lines
include the dependence of the quadratic shift on . Shown on theright are the representative spectra.

Unlike for the mF basis, an oscillatory magnetic field can drive
transitions between all pairs of the |x,y,z states with nonzerotransition matrix elements (see Appendix A for more details).
Our BECs had N ≈ 5 × 104 atoms and were held in acrossed dipole trap with trapping frequencies (fx ,fy ,fz ) =[42(3),34(2),133(3)] Hz.2 The B0 ≈ 3.27 mT bias field lifted
the ground-state degeneracy, giving an ω0 /2π = 22.9 MHz
Larmor frequency, with a quadratic shift /2π = 76.4 kHz. In
our laboratory the ambient magnetic-field fluctuations were
dominated by contributions from line noise giving an rms
uncertainty δ/2π = gF μB δB/ h = 0.67(3) kHz.
We used adiabatic rapid passage (ARP) to transfer atoms
initially prepared in any of the |mF = 0,−1,1 states into thecorresponding |x,y,z states. Beginning far from resonance[(t = 0)/2π ≈ −450 kHz] with all coupling fields off, we
ramped on the rf dressing field in a two-step process. We first2

All uncertainties herein represent the uncorrelated combination of
statistical and systematic uncertainties.

ramped from  = 0 to approximately half its final value in
10 ms. By increasing the magnetic field B0 , we then ramped to zero in 12 ms using a nonlinear ramp adiabatic with respect
to the relevant energy gaps. After allowing B0 to stabilize for
30 ms, we ramped the rf dressing field to its final value  in
10 ms, yielding the dynamically decoupled |y,x,z states.
We measured the population in the |x,y,z states, adiabatically deloaded them back into the mF basis by ramping B0
so that  approached its initial detuned value in 2 ms, andthen ramped off the dressing rf field in 1 ms. We obtained thespin-resolved momentum distribution using standard time-offlight (TOF) imaging techniques, with a Stern-Gerlach field
to spatially separate the spin components during the TOF.
The right panel of Fig. 1(a) shows such a TOF image fordecomposition of |z into the mF states in a typical TOF image.
We confirmed our control and measurement techniques
spectroscopically measuring the energy differences between
the |x,y,z states with our prove field. Figure 1(b) shows thedependence of the ωxy /2π , ωyz /2π , and ωzx /2π on detuningfor /2π = 194.5(1) kHz derived from spectra such as in
the side panel with coupling strength p /2π ≈ 1 kHz and/2π ≈ 9 kHz.
The dashed curves based on Eq. (1) clearly depart fromour measurements for the zx transition. This departure resultsfrom neglecting the weak dependence of the quadratic shift on bias field B0 . In near-perfect agreement with experiment,
the solid curves from the full Breit-Rabi expression accountfor this dependence.
III. ROBUSTNESSThe zx transition is remarkably robust against magneticfield variations, as commonly result from temporal and spatial
magnetic-field noise in laboratory environments [Fig. 2(a)].
We focus on the zx transition, which can be made virtuallyindependent of magnetic-field variations due to the similar
curvature of ωz () and ωx () [see the middle panel ofFig. 1(a)]. We quantified the sensitivity of this transition to field
variations with three methods corresponding to the different
markers in Fig. 2(b). In each case we measured the energy
shift from resonance as a function of detuning and then used
a fourth-order polynomial fit to extract the rms residuals δωzx
due to the known detuning noise.3 (1) Triangles denote datausing full spectroscopical measurements similar to Fig. 2(a).
(2) Squares denote data in which a detuned π pulse of theprobe field transferred atoms from |z to |x, a side-of-peaktechnique giving a signal first-order sensitive to changes in ωzx .
(3) Circles denote data using an adiabatic technique described
below. The results are not consistent with the theory simple
from Eq. (1) (dashed line) and instead require the Breit-Rabi
expression (solid line) to obtain full agreement.4
Even at our smallest coupling /2π = 69(1) kHz thetypical magnetic-field noise was attenuated by two ordersof magnitude, rendering it essentially undetectable. Ideally,3
Our procedure also quantifies the small fluctuations that survive
for spectra that are flat beyond second order, as in Eq. (1).
4
The fluctuations can be even smaller for a given  if we allow for = 0 (see Appendix A).

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PHYSICAL REVIEW A 97, 013407 (2018)(a)/2 (kHz)(a)275/2 = 231.8 kHz

252/2 = 212.2 kHz

amplitude

(kHz)

255

zx /2

272

/2 = 194.5 kHz

237
234
100
(b)

50

0
/2 (kHz)

50

100

100

(b)
relative matrix elements

E/h (kHz)

10 2

1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
0.0

10 4
10 6
reduced H
Breit-Rabi

10

8

0

1

2
/

3

4

69

87

110

138

167

194

223

0.5

1.0
time (ms)

1.5

2.0

1.0
0.8

zx

0.6

xy

0.4
zy

0.2
0.0

FIG. 2. (a) Transition frequency ωzx /2π for three values of /2π .
The dashed curves correspond to Eq. (1), while the solid curves
use the Breit-Rabi expression. (b) The change in energy from ourexperimental detuning fluctuations as measured in the mF basisis δ/2π = 0.67 kHz (red dashed line). Triangles correspond to|x,y,z spectroscopy data, squares to side-of-peak π -pulse data, andcircles to double-dressed data. The black dashed (solid) curve wascalculated using Eq. (1) (the Breit-Rabi expression). The shading of
the data points corresponds to the Rabi frequencies in Fig. 3.

0

0

100

200

300

/2 (kHz)

FIG. 3. (a) Rabi oscillations. Phase coherence is maintained
throughout the oscillations in the dressed basis, while it is quickly
lost in the mF basis. The marker size reflects the typical uncertainties
on the dressed basis oscillations. (b) Transition matrix elements forzx (blue) and zy (orange) transitions decrease monotonically withincreasing  for  = 0, while they increase for xy.

these time scales, the loss of coherence was predominantly due
to bias magnetic-field temporal noise.5
the radius of curvature of ωzx () changes sign at about
/2π = 220 kHz, leaving only a 4 contribution; however, inpractice the small dependence of  on B prevents this perfectcancellation.
We explored the strength of the probe-driven transitions
between these states by observing coherent Rabi oscillations[Fig. 3(a)] where our BEC was prepared in |z and the probe
field had strength p /2π ≈ 1 kHz. The top panel shows
Rabi oscillations between |mF = 0 and |mF = −1 states forreference and the remaining panels show oscillations between|z and |x. The observed Rabi frequency between dressedstates decreased with increasing , indicating a dependence ofthe zx transition matrix elements on . These matrix elements,
as well as those for the zy transition, decrease with increasing
 for  = 0, as shown in Fig. 3(b) and Appendix A 2.
The coherence of the Rabi oscillations for longer times waslimited by gradients in  that lead to phase separation of the
dressed states and therefore loss of contrast after a few tens of
milliseconds, but had no measurable effect on the coherenceof the oscillations. In comparison, the coherence of the Rabi
oscillation between the mF states deteriorates after 500 μs. For

IV. CONCATENATED CDD

The driving field  coupled together the |mF  states, giving
us synthetic clock states |x,y,z that were nearly insensitive
to magnetic-field fluctuations. However, the spectrum of thesestates is first-order sensitive to fluctuations δ of the driving
field. Reference [10] showed that an additional field coupling
together these |x,y,z states can produce doubly dressed statesthat are insensitive to both δ and δ: a process called
concatenated CDD. In our experiment, the probe field provided
the concatenating coupling field. Because p  , we focus
on a near-resonant two-level system formed by a single pairof dressed states, here |z and |x, which we consider as
pseudospins |↑ and |↓. These are described by the effective

5

We canceled gradient magnetic fields so that no phase separationof the bare states was observed for more than 10 s.

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D. TRYPOGEORGOS et al.
(a)

PHYSICAL REVIEW A 97, 013407 (2018)

1.0

3

0.8

0.6
/2 = 138.2 kHz

100

50

0
/2 (kHz)counts

(b)

(c)

30
20
10
0
0.0

0.2

50

0.4
0.6
population ratio

100

0.8

1.0

1.0
1
|x

Pzx
|z

0.6

0

5

0.4

0
/2 (kHz)

5

10

12

d

3 /d

(1/kHz)

0.8

0.2
0.0

0

2

4

6
p /2

8
(kHz)

and |↑ and quantified the sensitivity of this transition to largechanges in the detuning in terms of the fractional population
imbalance σ̂3  = P↓ () − P↑ (), shown in Fig. 4(a) for
/2π = 138.2(1) kHz.6 This signal is first-order sensitive toω↓,↑ and provided our third measurement of sensitivity to
detuning in Fig. 2(b) denoted by circles.
We compared the fidelity of preparing a superposition of
the |↓ and |↑ states to adiabatically preparing a similar
superposition of the the |mF = 0 and |mF = −1 states, bothwith a probe field strength of approximately 1 kHz. Figure 4(b)shows the rms deviation of the population imbalance measured
over a few hundred repetitions of the experiment. The rms
deviation for the dressed basis is 0.024(1) and is an order ofmagnitude smaller than for the mF basis 0.29(1), where it ispractically impossible to prepare a balanced superposition forthe parameters used here.7
Figure 4(c) shows the response of the ↓,↑ transition to smallchanges δ for different values of p . We prepared an equalsuperposition of |↓ and |↑ following the same procedure
as before for /2π = 138.2(1) kHz. We then measured how
the population imbalance changes for small variations of ,
the effective detuning in the twice-rotated frame, for different
probe amplitudes p . We defined a sensitivity parameterdσ̂3 /d, obtained from the linear regime of the population
imbalance measurements [see the inset in Fig. 4(c)]. The
robustness of the doubly dressed states against δ fluctuations
increased with p , thus verifying the concatenating effect of
CDD in the |x,y,z basis.
V. CONCLUSION

14

FIG. 4. (a) Fractional population imbalance of the ↓↑ transitionfor /2π = 138.2(1) kHz over detuning . The dashed curve iscalculated using Eq. (1) and the solid one using the full Breit-Rabi
expression. (b) Fidelity of preparing a balanced superposition of |↓
and |↑ (dark blue) states compared to |mF = 0 and |mF = −1
states (light blue). (c) Robustness of the ↓,↑ transition against
fluctuations δ for different probe field coupling strengths. The points
represent the slope of the fitted curves to the fractional population
imbalance (inset).

two-level Hamiltonianh̄σ̂3 + h̄ cos(ωp t)σ̂1 ,
Ĥp =
(2)
2
with energy gap  ≈ ω↓,↑ (shifted by off-resonant coupling to
the zy and xy transitions) and coupling strength  ∝ p , as set
by the matrix elements displayed in Fig. 3(b). Here σ̂1,2,3 are thethree Pauli operators. A RWA of this Hamiltonian leads to theenergy spectrum E↑,↓ ≈ ± /2 + ( )2 /2 , having again
assumed the coupling  exceeds any fluctuations in  . Thus,
the concatenated CDD field protects from the fluctuations δ
of the first dressing field in the same way that CDD provided
protection from detuning noise δ.
We produced doubly dressed states by using the probe fieldnear resonance with the ↓,↑ transition and an ARP sequence.
We started in |↓ at  = 0 and ramped on the probe field pa few milliseconds before ramping  to its final value. We
chose ARP parameters to create an equal superposition of |↓We realized a three-level system that is dynamically decoupled from low-frequency noise, measured now-allowedtransitions between all three states, and demonstrated control
techniques for creating arbitrary Hamiltonians. These techniques add no heating or loss mechanisms, yet within theprotected subspace retain the full complement of cold-atomcoherent control tools such as optical lattices and Raman
laser coupling and permit new first-order transitions that areabsent in the unprotected subspace. These transitions enable
experiments requiring a fully connected geometry as for engineering exotic states, e.g., in cold-atom topological insulators,
and two-dimensional Rashba spin-orbit coupling in ultracold
atomic systems [25,26].
The synthetic clock states form a decoherence-free subspace that can be used in quantum information tasks whereconventional clock states might be absent or incompatible
with other technical requirements [27]. Moreover, their energy
differences are proportional to the amplitude of the dressing
field, and hence tunable, so they can be brought to resonancewith a separate quantum system. The effective quantization
axis can be arbitrarily rotated so that the two systems can bestrongly coupled, pointing to applications in hybrid quantum

6
We chose the maximum value of  such that the population of |y
was negligible after deloading.
7
In Fig. 4(b), the noise in the mF basis in not Gaussian distributed
as is typical of line noise in these experiments.

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FIG. 5. Decomposition of the |x,y,z states on the mF basis for /2π = 145(1) kHz. The |mF = −1,0,1 states correspond to blue, orange,and gray, respectively.

systems [28,29]. Introducing a second coupling field shields
the system from fluctuations of the first, a process which
can be concatenated as needed. More broadly, synthetic clockstates should prove generally useful in any situation wherefluctuations of the coupling field can be made smaller thanthose of the environment.

ACKNOWLEDGMENTS

This work was partially supported by the AROs atomtronics
MURI, the AFOSRs Quantum Matter MURI, NIST, and the
NSF through the PFC at the JQI. We are grateful for various
discussions with P. Solano and thank R. P. Anderson for avery thoughtful and detailed eleventh hour reading of ourmanuscript.

linear combinations of the mF basis states:|x = |−1 + |1,˜+|y = |−1 − √|0 + |1,
(A1)
2
˜
−
|z = |−1 − √|0 + |1.
2We measured the above decomposition of the |x,y,z states to
mF states using a projective measurement by abruptly turning
off the dressing field  (see Fig. 5). For  = 0 and small
coupling / → 0 with regard to the quadratic shift the |y and|x become symmetric and antisymmetric superpositions of the|mF = −1,1 states, while |z is predominantly composed of|0,|x = |1 − |−1,|y = |1 + |−1 +APPENDIX A: THE |x, y,z DRESSED BASIS

The reduced Hamiltonian of Eq. (1) can be diagonalized
analytically. The eigenvaluesfor  = 0 are ωx = 0 and ωz,y =√˜ where ˜ = 42 +  2 is a generalized Rabi fre− ± ,
quency. The corresponding (non-normalized) eigenvectors are


|0,


(A2)


(|1 + |−1) − |0.
On the other hand, when  → ∞ they are independent of
the driving field amplitude and continuously approach the|z =FIG. 6. Transition matrix elements over  and . There is an asymmetry between coupling on the blue and red sides of the resonance that
corresponds to the counter- and corotating terms F̂− and F̂+ .
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FIG. 7. Shown on the left is the optimum response (solid line) of the zx transition to detuning fluctuations allowing for finite  comparedto  = 0 (dashed line). Shown on the right are the values of  that correspond to the minimum derivative of ωzx .

eigenstates of the F̂x operator|x = |1 − |−1,√|y = |1 + 2|0 + |−1,√|y = |1 − 2|0 + |−1.

(A3)

The states adiabatically map to the |mF  states for    asshown in Fig. 5. For /2π > 200 kHz the |x,y,z states are
not yet fully deloaded to a single mF state since the populationin other mF states is not negligible. The state |z maps to |1(|−1) for positive (negative) detuning, |y maps in the exact
opposite way to |z, and |x always maps to |0. For large
enough  the zx and xy transitions become degenerate andthe system resembles an F = 1 ground state at low magnetic
field.
1. Dependence on detuning

For nonzero values of the detuning , the eigenvalues are
the root of the characteristic cubic polynomial H (λ) = 2  +(2 + 2 )λ − λ2 − λ3 . The eigenvalues are even functions
with respect to  as can be seen by the leading-order expansion
for  → 0,
ωx = − 2 2 + O(4 ),˜( + )1˜ −ωy = (− + )
2 + O(4 ),22
˜2− − 4 +  ˜( − )1˜ +ωz = (− − )
2 + O(4 ). (A4)22˜
2
 + 4 +  
hence their resemblance to clock states. We focused on the zx
transition since the curvature of ωx and ωz has the same sign for˜ [Eq. (A4)]. Since the quadratic term changes curvature
<
it can be made arbitrarily small. However, this cancellation
does not take place when we consider the dependence of  on
 from the Breit-Rabi expression.
2. Transition matrix elements

The |X,Y,Z states transform under the application of
the spin-1 operators as j kl F̂j |k = i h̄|l, so that a resonant
probe field can induce transitions between at least one pair ofstates, irrespectively of its polarization. The transition matrix

elements between the |x,y,z states show a dependence on
both  and  (see Fig. 6). For    the matrix elements
correspond to those of the |mF  basis and x|F̂+ |y = 0 as
expected by angular momentum selection rules. When  and are comparable in magnitude all transition matrix elements
are nonzero and the states can be coupled cyclically. As   
the |z and |y states decouple and the system resembles anundressed basis following similar selection rules.
3. Optimal response to noise

The sensitivity of the zx transition to detuning fluctuations
can be optimized further by working at  = 0 as shown in
Fig. 7. This behavior can only be captured by including thedependence of the quadratic shift on  as given by the BreitRabi expression.
For small values of  the optimum value of  corresponds
to one of the concave features of the zx transition energy thatarise due to the asymmetry introduced by the quadratic shift.
As  gets larger, these features merge into a single one and the
optimum value is  ≈ 0. The deviation from  = 0 is due to an
overall tilt of the transition energy coming from the dependenceof the quadratic shift on . At the optimum point / ≈ 3 thesensitivity of the synthetic clock transition is 1.9 × 10−7 kHz;
cf. the 87 Rb clock transition, which scales as 57.5 kHz/mT2
and gives 5.8 × 10−7 kHz.
APPENDIX B: LOCATING FIELD RESONANCE

We used an iterative procedure to measure and adjust thevalue of the detuning  to account for the weak responseof the |x,y,z states to detuning variations. As most of ourexperiments were done at  = 0, we first obtained an estimateof  from the imbalance of |1 and |−1 populations fromthe decomposition of |z which should be zero for  = 0 (seeFig. 5). We then located the transition frequencies for at least
two transitions (usually zx and zy as shown in Fig. 8) using
an ARP protocol as described in the main text and varying thefrequency of the probe field. These frequencies correspond toa unique pair of  and || values which can then be used toadjust the bias magnetic field B0 so that  = 0. However, there
is an ambiguity as to the sign of  since the eigenstates areeven functions of .

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FIG. 8. Characteristic spectroscopy curves for the zx (left) and zy (middle) transitions. Two symmetric ARPs (right) define the resonant
value of the magnetic field. The width of the peak gets smaller as  gets closer to the resonant value. Here /2π = 3 kHz.

We selected a direction randomly and subsequently verified
if  = 0 using another set of spectroscopic measurements.
We fixed the value of the probe field to be a few kilohertz
above the transition frequency corresponding to  = 0 andused the same ARP sequence to transfer atoms from |z to |x.
This procedure gave two resonant values for  where atomtransfer takes place and the value where  = 0 corresponded
to their mean (see Fig. 8). Finally, we remeasured the zx and zy
transition frequencies to validate that  = 0. For higher valuesof R , using the zx transition becomes impractical due to its
insensitivity to detuning and we followed the same procedurebut using the xy transition instead.

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