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nature physics
https://doi.org/10.1038/s41567-023-02321-y
Artic�e
Emergence of highly coherent two-level
systems in a noisy and dense quantum
network
In the format provided by the
authors and unedited
Supplementary Information to
Emergence of highly coherent two-level systems in a noisy and
dense quantum network
1 Outline
In this supplement to Ref. [1], we provide detailed derivations of the theoretical model used to
describe and analyse the spin-echo decay. In Sec. 2, we first derive the ring-exchange dephasing.
In Sec. 3 we then briefly derive the algebraic decay of Rabi oscillations in pairs, as observed
in Fig. 4b of the main text. The central part of this supplemental text is contained in Sec. 4,
where we give more detailed derivations of the coherence-limiting mechanisms observed in the
experiments. Its main results are summarized in the Methods section of the main text. Here
we first discuss echo decay due to dipolar excitation hopping between Tb
3+
ions (Sec. 4.1),
pure dephasing due to magnetic dipolar and ring-exchange interactions between the Tb
3+
ions
(Sec. 4.2, as well as dephasing due to the host’s nuclear spins (Sec. 4.4). Having derived these
dephasing mechanisms, we explain why pairs of spins in densely doped materials, rather than
single ions in a dilute sample, optimize the abundance of qubits of a certain targeted coherence
(Sec. 4.4). Finally, we fit all our experimental data to the derived analytical/numerical model
and present the resulting plots and parameters in Sec. 5.
2 Ring-exchange interaction
In this section we derive the ring-exchange interaction between a pair of Tb
3+
ions and a third
ion using second-order perturbation theory. As a consequence of this interaction, the transition
energy for an excitation of the pair depends on the state of the neighboring ion. We refer to this
self-energy-like correction as a ‘ring-exchange’, as it arises from a virtual exchange of excitations
between the three ions.
For simplicity, we focus on clock-state ions and neglect internal dipolar fields, setting
h
i
= 0
.
The Hamiltonian of three clock-state ions in the secular approximation reads
H
=
1
2
3
X
i
=1
i
τ
x
i
+
J
pair
τ
+
1
τ
2
+
J
13
τ
+
1
τ
3
+
J
23
τ
+
2
τ
3
+
h.c.

,
(1)
with the Pauli matrices
τ
x
and
τ
±
(
τ
z
y
)
/
2
acting in the basis of the single-ion eigen-
states. The ions
1
and
2
that form the pair are assumed to be resonantly coupled, i.e.,
|
J
pair
|≫
1
2
|
1
2
|
, and more strongly coupled to each other than to the third ion,
|
J
13
| ≈
|
J
23
|≪|
J
pair
|
. We treat the couplings
J
13
,J
23
in second-order perturbation theory. The energies
for the pair transitions
|
01 + 10
⟩→|
00
,
|
11
depend on the state
s
[0
,
1]
of the third ion, and
the pair transition energies
|
01 + 10
⟩→|
ττ
(with
τ
∈{
0
,
1
}
) are found to differ by
1
V
ring
1
4
|
ε
01+10
,
1
ε
ττ,
1
|−|
ε
01+10
,
0
ε
ττ,
0
|

= (1
2
τ
)
J
13
J
23
2[(∆
pair
(1
2
τ
)
J
pair
)
3
]
.
(2)
Note that the denominator contains the detuning between the frequency of the single ion,
3
,
and the energy difference
|
ε
01+10
ε
1
τ,
1
τ
|
between
|
01 + 10
and the pair state
|
1
τ,
1
τ
that is
not
involved in the probed transition. For dipolar interactions
J
r
3
, the ring-exchange
decays as
V
ring
(
r
)
r
6
with the distance
r
between pair and single ion.
The above derivation (2) is only slightly modified in the case of a virtual exchange of an excitation
between a clock-state pair and a third ion in a different hyperfine state. One simply has to
1
The factor 4 is owed to the fact that a pair and a single ion, described as effective spins
σ
1
,
2
and Hamiltonian
H
int
=
V σ
z
1
σ
z
2
, come with energy differences
(
ε
1
,
1
ε
1
,
1
)
(
ε
1
,
1
ε
1
,
1
) = 4
V
1
replace the transition frequency of the third ion,
3
, and take into account the off-diagonal
matrix elements
m
off
(
I
z
) = ∆
/
p
2
+
h
2
(
I
z
)
of the corresponding hyperfine state in the dipolar
interaction,
J
13
,J
23
. The dephasing resulting from ring-exchange is discussed in Sec. 4.2.
We also note that the derivation of the ring-exchange can be seen as the leading term of a virial
expansion, which is an expansion at low density. There are corrections involving few (
>
3
)-body
interactions which eventually kick in at high enough density
x
=
O
(1)
, where typical distances
are of the order of the lattice constant. At low density, a pair is instead correctly described as
interacting separately with every one of its (typically single ion) neighbors.
3
nnn
pair spin-echo: Algebraic decay
Here we derive the algebraic decay of the Rabi oscillations of
nnn
pairs as observed in Fig. 4b of
the main text. The pulse scheme involves a first pulse of length
t
p
with frequency
ω
p
J
nnn
and, after a waiting time
τ
, a second
π
-pulse (approximated as instantaneous). The echo is
detected after another waiting time
τ
. We assume the inhomogeneous broadening of the pair
excitations
ω
=
ω
p
+
δω
to be a Gaussian
ρ
pair
(
ω
p
+
δω
) = exp

δω
2
2
W
2
pair

/
q
2
πW
2
pair
centered
around
ω
p
with standard deviation
W
pair
. Within the rotating-wave approximation, the echo
then takes the form
I
(
t
p
) =
Z
d
δω
2
+
δω
2
sin

p
2
+
δω
2
t
p

ρ
pair
(
ω
p
+
δω
)
,
(3)
where
is the Rabi frequency of the pair. Here we neglect decoherence, which would only
manifest itself on longer time scales than were probed in the experiment of Fig. 4b.
In the limit
W
2
pair
t
p
, only
δω
with
δω
2
t
p
/
1
contribute significantly, and a stationary
phase approximation leads to the long time asymptotics
I
(
t
p
)
1
2
s
W
2
pair
t
p
sin(Ω
t
p
+
π/
4)
1
t
p
,
(4)
with an algebraic decay of the Rabi oscillations.
4 Theoretical modelling of decoherence of Tb
3+
excitations
In this section we discuss the various decoherence mechanisms for Tb
3+
excitations with dipolar
interactions. We analyze the spin echo probed at frequency
ω
p
after a CPMG sequence involving
N
short
π
-pulses applied at times
t
= (2
n
1)
τ
(
1
n
N
). To describe the echo decay,
we take into account several decoherence mechanisms: (i) decay of excitations due to dipolar
hopping, (ii) magnetic noise from other Tb
3+
ions, (iii) ring-exchange with Tb
3+
ions, and
(iv) magnetic noise from fluorine nuclear spins. The phonon-induced relaxation rate is on the
order of milliseconds and can thus be neglected in the analysis of our experiments (Supp.Fig. 1).
Below, we sketch the derivation of each contribution in the high temperature limit (
T
), and
summarize the results that are then used to numerically fit the data. A more detailed analysis
will be published elsewhere. The numerical parameters of the different dephasing sources are
given in Sec. 5.
4.1 Lifetime of dilute spins in the presence of dipolar hopping
The crystal field splittings
i
= ∆ +
δ
i
of the ions are subject to random shifts
δ
i
. We
model their distribution
ρ
(
ω
)
as Gaussian with standard deviation
W
. The excitation energy
E
I
z
i
=
±
q
2
i
+
h
2
i
(
I
z
i
)
with the effective longitudinal field
h
i
(
I
z
i
) =
h
(
I
z
i
) +
δh
i
on a given
2
Figure 1:
Phonon relaxation times.
Comparison of
T
1
relaxation times of
nnn
pairs at
x
= 0
.
1%
(blue) to those of typical ions at
x
= 0
.
01%
(red). A simple exponential fit yields
T
1
times of
1
.
0
ms and 3.7 ms, respectively. Their ratio agrees well with the theory of decoherence
by acoustic phonons,
1
/T
1
M
2
ω
3
, where
M
is a phonon matrix element and
ω
is the transition
frequency [2], which predicts a ratio of
2
×
(35
.
53
/
27
.
89)
3
= 4
.
1
. The phonon relaxation times
are orders of magnitude longer than the relevant coherence times in the echo experiments of the
main text and can thus be neglected.
site depends on the local field internal field
δh
i
and on the nuclear spin state
I
z
i
via the hy-
perfine interaction
A
,
i.e.
h
(
I
z
i
) =
g
μ
B
2
B
z
+
A
2
I
z
i
. The algebraic tail of dipolar interactions
J
(
⃗r
) =
J
0
/r
3
(1
3 cos
2
θ
)
, with
J
0
=
μ
0
(
μ
B
g
/
2)
2
/
(4
π
)
being the dipolar interaction constant,
assures that a given Tb
3+
excitation can always find resonant sites with energy mismatch smaller
than the hopping amplitude
J
ij
=
J
(
⃗r
ij
)
m
off
(
I
z
i
)
m
off
(
I
z
i
)
,
|
E
I
z
i
E
I
z
j
|≤
2
c
res
|
J
ij
|
,
(5)
where the numerical factor
c
res
=
O
(1)
defines a precise resonance condition and the hopping
matrix element for hyperfine state
I
z
is
m
off
(
I
z
) = ∆
/
p
2
+
h
2
(
I
z
)
. Due to the smallness of
the dipolar interaction, such excitation hopping can only be resonant between Tb
3+
ions in the
same nuclear-spin state. Their effective concentration is
x
eff
=
x/
(2
I
+ 1) =
x/
4
, corresponding
to a spatial density
n
= 4
x
eff
/
(
a
2
c
)
.
We focus below on the lifetime of clock-state excitations with
m
off
(
I
z
) = 1
, neglecting weak
internal fields. The derivation also carries over to magnetized hyperfine states, modulo modified
hopping matrix elements and diagonal matrix elements
m
(
I
z
) =
|
h
(
I
z
)
|
/
p
2
+
h
2
(
I
z
)
, which
modify the disorder strength according to
W
I
z
q
W
2
+
δh
2
i
m
2
(
I
z
)
. In the derivations,
we consider excitation frequencies in different ranges to differentiate between ‘single ions’ and
pairs. The detuning of the probed frequency
ω
p
from the mean CF energy
is denoted by
ω
p
ω
p
.
4.1.1 Decay of excitations of typical single clock-state ions (
|
ω
p
|
W
)
We first consider the effect of resonant hopping on the lifetime of typical clock-state ions with
excitation energy in the middle of the spectrum of inhomogeneously broadened CF energies,
|
ω
p
|
W
. Such typical ions have no atypically close neighbors, since the dipolar interaction
with them would imply a significant shift in
|
ω
p
|
. We thus refer to these spins as ‘single ions’
(to distinguish them from ion pairs or larger ion clusters).
The typical decay time is found by counting the number of resonant sites
N
(
J
)
having in-
teractions exceeding
J
and satisfying the resonance condition of Eq. (5). The lifetime
τ
s
of
typical single ions is of the order of the inverse interaction strength with their nearest resonant
neighbor. The associated energy scale
J
res
is found from
N
(
J
res
) =
c
N
with
c
N
=
O
(2)
. The
3
frequency-dependent typical spin-flip rate is obtained as
1
τ
s
(
ω
p
)
= 2
c
τ
|
J
res
(
ω
p
)
|
=
2
ec
τ
c
res
J
typ
α
(
ω
p
)
exp

c
N
c
res
1
α
(
ω
p
)

.
(6)
The numerical factor
c
τ
relates resonant coupling and decay rate,
c
τ
2
|
J
res
(
ω
p
)
|
τ
s
(
ω
p
) =
O
(1)
.
2
The frequency-dependent disorder parameter
α
(
ω
p
)
is defined as
α
(
ω
p
) = 4
J
typ
ρ
(
ω
p
)
,
(7)
with
J
typ
the dipolar hopping between a typical site and its most strongly coupled neighbor,
J
typ
8
π
9
3
nJ
0
.
(8)
Equation (6) holds for relatively strong disorder,
α
(
ω
p
)
c
N
/c
res
. In the method section of the
main text, we abbreviate
c
1
=
c
res
/c
N
and
c
2
=
c
τ
/c
res
.
The decoherence due to excitation hopping dominates the Hahn-echo decay of clock-ions in the
range
|
ω
p
|
W
at short times
t
τ
s
(
ω
p
)
where it results in a simple exponential decay
I
dec
,
single
(
t
τ
s
(
ω
p
)) =
e
t/τ
s
(
ω
p
)
.
(9)
These times are, however, too short to be measured in our experimental set-up.
4.1.2 Decay of rare single-ion clock-states (
|
ω
p
|
W
) with atypically distant
neighbors
Echo contributions from typical clock ions decay exponentially for
t
τ
s
(
ω
p
)
. In that regime,
the echo signal is instead dominated by rare, long-lived spins, whose nearest resonant site is
atypically far away. With high probability, the nearest resonant site is a typical site, with closer
resonant neighbors. Thus, the hopping to that site is generally much slower than the subsequent
hopping away from that site. The isolated site can thus be treated as being coupled to an
energy-continuum of lifetime-broadened typical sites, and the decay rate can be calculated with
Fermi’s golden rule, similarly as in Ref. [3]. Assuming the same lifetime
τ
s
τ
s
(
ω
= ∆)
[Eq. (6)]
for all neighboring sites, each contributes to the total decay rate with a partial rate
γ
i
γ
(
r
i
i
,
i
)
1
2
2
π

J
0
r
3
i
(1
3 cos
2
θ
i
)

2
A
(
ω
p
; ∆
i
)
,
(10)
where
A
(
ω
p
;
ω
) =
1
π
1
/
(2
τ
s
)
1
/
(2
τ
s
)
2
+(
ω
ω
p
)
2
is the Lorentzian broadened density of states at energy
ω
p
. Accounting for the random spatial distribution of Tb
3+
ions, one obtains a distribution
p
(
γ
=
P
i
γ
i
) =
e
1
/
(4
γT
1
(
ω
p
))
/
p
4
πγ
3
T
1
(
ω
p
)
of decay rates, from which one can compute the
sample averaged echo signal (for
t
τ
s
)
I
dec
,
single
(
t
τ
s
(
ω
p
)) =
Z
0
dγ p
(
γ
)
e
γt
= exp
"
s
t
T
1
(
ω
p
)
#
,
(11)
where the frequency-dependent lifetime
T
1
(
ω
p
)
,
1
T
1
(
ω
p
)
= 4
π
2
J
2
typ

Z
dω ρ
(
ω
)
A
1
/
2
(
ω
p
;
ω
)

2
,
(12)
2
A self-consistent estimate based on Fermi’s golden rule yields
τ
1
s
(
ω
p
) = 2
|
J
res
(
ω
p
)
|
. Shortcomings of this
approximation are captured by the deviation of the coefficient
c
τ
from 1.
4
scales roughly as
1
T
1
(
ω
p
)
π
2
1
τ
s

ρ
(
ω
p
)
ρ
(∆)

2
for large disorder, as long as
ρ
(
ω
p
)
(∆)
is not
exponentially small. For a general disorder distribution
ρ
(
ω
)
, it has to be evaluated numerically.
For our fits we model the crossover from short times (9) to long times (11) by a simple interpo-
lating function,
I
dec
,
single
(
t
) = exp

t
τ
s
(
ω
p
)


1 +

t
τ
s
(
ω
p
)

T
1
(
ω
p
)
t

1
/
2

.
(13)
4.1.3 Decay of excitations of pairs of clock-states (
|
ω
p
|
W
)
At detunings
|
ω
p
|
W
, the signal is dominated by compact pairs of (clock-state) Tb
3+
whose dipolar interaction shifts the excitation energies,
cf.
Fig. 2a of the main text. For those,
resonant hopping to equivalent pairs is negligible, since their concentration and thus their dipolar
coupling scales as
x
2
1
. Excitations on such pairs will instead predominantly decay to non-
resonant single ions, as allowed by the lifetime-broadening of the latter. The decay follows from
Fermi’s golden rule (10) as in the preceding subsection, with only two small differences: (i) The
hopping matrix element is enhanced by a factor of
2
, leading to an additional factor of
2
in
the expression for the hopping rate. (ii) A pair realizes an effective three-level system with the
symmetric states
|
00
,
|
01 + 10
, and
|
11
, creating an additional hopping channel. Taking both
factors into account, we find the spin-echo decay of pairs due to the finite lifetime as
I
dec
,
pair
(
t
) =
Z
0
dγ p
(
γ
)
e
3
γt
= exp
"
s
t
T
1
,
pair
(
ω
p
)
#
,
(14)
with the same
p
(
γ
)
as before. The life time
T
1
,
pair
(
ω
p
)
is three times smaller than that of (the
very rare) single ions with the same excitation frequency,
cf.
Eq. (12),
1
T
1
,
pair
(
ω
p
)
= 12
π
2
J
2
typ

q
A
(
ω
p
;
ω
)

2
12
π
2
J
2
typ
A
(
ω
p
; ∆)
6
π

J
typ
ω
p

2
1
τ
s
.
(15)
Note that, in contrast to the single ions (13), there is no crossover from a short- to a long-
time regime for the decay of pairs. Indeed, the short-time regime becomes irrelevant at large
detunings for which
τ
s
(
ω
p
)
T
1
,
pair
(
ω
p
)
.
4.2 Pure dephasing from dilute spins with dipolar interactions
Apart from inducing finite lifetimes, the dipolar interactions also lead to pure dephasing (
T
2
).
Below we discuss the decoherence of a two-level systems (TLS) for the case that it is dominated
by interactions with dilute neighboring (Tb
3+
) fluctuators. We discuss couplings that decay as
general power laws of the distance
r
,
V
(
r
)
r
γ
. This covers both magnetic noise between
magnetic moments of the TLS and fluctuators (
γ
= 3
) as well as dephasing of barely magnetized
clock-state pairs via ring-exchange (
γ
= 6
). Note that even for clock states the former does not
vanish completely, as small residual moments are induced by internal magnetic fields from the
fluorine ions.
We consider a spin or spin pair as the TLS of interest and model all Tb
3+
ions as classical
random fluctuators which flip stochastically between
s
j
(
t
) =
±
1
with rate
κ
. Since the de-
phasing factors from independent sources simply multiply, it suffices to analyze the fluctuators
belonging to a single hyperfine state
I
z
. We further assume that the corresponding ions all have
the same (
I
z
-dependent) fluctuation rate
κ
and flip independently from each other. To simplify
the analysis, we treat a simple power law interaction,
V
(
⃗r
) =
V
(
γ
)
0
g
γ
(
θ
)
/r
γ
, which depends on
the distance
r
and the angle
θ
between
⃗r
and the crystallographic
c
-direction via a dimension-
less function
g
γ
(
θ
)
. For the ring-exchange interaction, we define
g
γ
=6
(
θ
) = (1
3 cos
2
θ
)
2
and
5
V
(
γ
=6)
0
=
J
2
0
/
(2∆
ω
)
, with
ω
= ∆
ω
p
E
I
z
the mismatch between the neighbor’s excita-
tion energy
E
I
z
and
ω
p
, the pair’s transition energy that is not driven. For the residual
magnetic interactions, we use
V
(
γ
=3)
0
=
J
0
m
p
m
(
I
z
)
, where
m
p
is a typical fluorine-induced mo-
ment and
m
(
I
z
)
is the moment of the neighboring fluctuator, and we define
g
γ
=3
(
θ
) = 1
3 cos
2
θ
.
In the continuum limit, similarly as was done in Ref. [4] for the Hahn-echo sequence (
N
= 1
)
with dipolar interactions (
γ
= 3
), one can integrate over the distribution of fluctuator positions,
leading to the analytic expression of the echo-decay function
I
(
t
) = exp
h
[
̄
V
(
γ
)]
3
G
γ
(
t
)
i
,
(16)
where
̄
V
(
γ
)
2
V
(
γ
)
0

cos

3
π
2
γ

Γ

3
γ

4
π
γ
n

Z
1
1
d
(cos(
θ
))
2
|
g
γ
(
θ
)
|
3

γ/
3
,
(17)
is a typical nearest neighbor interaction,
Γ
being the Euler Gamma function. The factor
G
γ
(
t
)
is the following average over spin-flip histories (denoted by
.
s
)
G
γ
(
t
)
*
Z
t
0
s
(
t
)
f
(
t
)
dt
3
+
s
,
(18)
whereby the function
f
(
t
) =
P
N
i
=0
(
1)
i
[
θ
(
t
[2
i
1]
τ
)
θ
(
t
[2
i
+ 1]
τ
)]
describes the alter-
nating spin orientation of the TLS (
f
(
t
) =
±
1
) as imposed by the CPMG sequence of
π
-pulses.
Below we give approximations of
G
γ
(
t
)
for short times (
κt
1
) and long times (
κt
N
).
4.2.1 Short-time regime
κt
1
First we consider the short-time regime
κt
1
. The echo is diminished only if any spin flips
occur at all (since for constant
s
(
t
)
one finds
R
t
0
s
(
t
)
f
(
t
)
dt
= 0
). From events with one spin
flip, one easily finds that
G
γ
(
t
)
κt
(
t/N
)
3
. Following Ref. [4], the spin-echo intensity for
short times
κt
1
is obtained fully quantitatively as
I
(
t
1
) = exp
"

t
T
s

1+3
#
,
(19)
with the short-time timescale
1
T
s
=
1
N
3
/
(3+
γ
)

γ
3 +
γ
κ
̄
V
3
(
γ
)

γ/
(3+
γ
)
.
(20)
The stretched exponential reflects the temporal growth of the range over which the interactions
have a detrimental effect, when a neighboring spin flips.
For magnetic Tb
3+
ions with dipolar interactions (
γ
= 3
), or for ring-exchange of pairs (
γ
= 6
),
we find, respectively, the short-time echo decay
I
magn
(
t
1
) = exp
"

t
T
magn
,s

2
#
,
1
T
magn
,s
=
κ
2
̄
V
(
γ
=3)
N
!
1
/
2
2
.
25
κV
(
γ
=3)
0
n
N
!
1
/
2
,
I
ring
(
t
1
) = exp
"

t
T
ring
,s

3
/
2
#
,
1
T
ring
,s
=
2
κ
3
r
̄
V
(
γ
=6)
N
!
2
/
3
2
.
44
κ
2
V
(
γ
=6)
0
n
2
N
!
1
/
3
.
(21)
6
4.2.2 Long-time regime
κt
N
In the long-time regime
κt
N
, each spin flips many times between two consecutive pulses in
the CPMG sequence. Since the
π
-pulses are only effective in cancelling noise contributions at
frequencies lower than
1
=
N/t
, the only remaining effect of the CPMG sequence lies in the
cancellation of static fields. As far as noise from fluctuators is concerned, its contribution to
echo signal suppression is essentially independent of
N
in this regime.
Instead of directly evaluating
G
γ
(
t
)
in the long-time regime, as was
e.g.
done in Ref. [4] for
γ
= 3
, we proceed in a different way. At times
κt
N
, merely TLS’s that couple only to ’weak
fluctuators’ (defined by
V
j
κ
) contribute to the echo, since TLS’s coupled to strong fluctuators
(
V
j
κ
) will already have decohered. A single weak fluctuator
i
leads to an exponential spin-
echo decay
exp(
γ
i
t
)
with the motionally narrowed decoherence rate
γ
i
= 2
V
2
i
[5]. Similarly
as in the derivation of Eq. (11), we can then derive a distribution
p
(
γ
=
P
i
γ
i
)
of the total
dephasing rate. Following the same steps, we find at long times
κt
N
I
(
t
N/κ
) = exp
"

t
T
l

3
/
(2
γ
)
#
,
(22)
with the timescale
1
T
l
=
π
Γ

γ
3
2
γ

cos

3
π
2
γ

2
γ/
3
2
̄
V
2
(
γ
)
κ
.
(23)
For magnetic Tb
3+
ions with dipolar interactions (
γ
= 3
) and ring-exchange of pairs (
γ
= 6
),
we find, respectively, the long-time asymptotics
I
magn
(
t
N/κ
) = exp
"

t
T
magn
,
l

1
/
2
#
,
1
T
magn
,
l
=
2
π
̄
V
(
γ
=3)

2
κ
65
.
3

V
(
γ
=3)
0
n

2
κ
,
I
ring
(
t
N/κ
) = exp
"

t
T
ring
,
l

1
/
4
#
,
1
T
ring
,
l
0
.
46
̄
V
(
γ
=6)

2
κ
488

V
(
γ
=6)
0
n
2

2
κ
.
(24)
4.2.3 Crossover from short to long times
Above we have derived the asymptotics of spin echos under the CPMG sequence for short (21)
and long times (24). To conveniently fit the experimental data at all times, we interpolate
between them with a form that comes as a close as possible to an exact evaluation of the
dephasing function (16). For ring-exchange noise of pairs (
γ
= 6
) we use
I
ring
(
t
) = exp

t
T
ring
,
s

3
/
2

1 +

t
T
ring
,
s

3
β/
2

T
ring
,
l
t

β/
4

1
.
(25)
where the parameter
β
tunes the sharpness of the crossover and is fit to the numerically evaluated
analytic expression of the echo decay. Since the crossover function is an approximation to the
analytical result, the fitted tuning parameter
β
depends weakly on the evaluated time-window.
We used
0
t
20
and found
β
[1
.
2
,
1
.
1
,
1
.
1
,
1
.
0
,
0
.
93]
for
N
= [1
,
2
,
3
,
4
,
5]
. The systematic
decrease of
β
with
N
correctly reflects that the crossover from short to long times becomes longer
with increasing
N
.
For magnetic noise (
γ
= 3
), Hu and Hartmann [4] had derived an analytic expression for the
Hahn-Echo (
N
= 1
), which is, however, unsuitable for numerical evaluation at long times
κt
1
.
7
To cover long times and general
N
, we resort also here to using a phenomenological crossover
function
I
magn
(
t
) = exp

t
T
magn
,
s

2

1 +

t
T
magn
,
s

2
β

T
magn
,
l
t

β/
2

1
,
(26)
with
β
[0
.
93
,
0
.
74
,
0
.
63
,
0
.
58
,
0
.
54]
obtained for
N
= [1
,
2
,
3
,
4
,
5]
from fitting to the exact
analytical function in the window
0
t
20
.
4.3 Dephasing due to the host’s nuclear spins (fluorine noise)
TLS’s with a sizeable magnetic moment create an inhomogeneous dipolar field around them-
selves. This suppresses resonant flip-flops between nuclear spins and induces a so-called ‘frozen
core’ of nuclear spins. [6][7, 8]. The main dephasing is then contributed by the many nuclear
spins at the boundary of this frozen core. However, clock states have only very small residual
magnetic moments (mostly induced by fluorine fields), and thus the frozen core barely exists.
The dipolar field of clock states only slows the dynamics of the nearest fluorine spins. As a result
the dephasing is still dominated by the most strongly coupled (i.e., the
nn
and
nnn
) fluorine,
which, however, fluctuate at a lower rate than the fluorine further away.
As a (
nn
or
nnn
) fluorine spin flips, the longitudinal fields on a nearby Tb
3+
changes – mediated
by the effective interaction
J
(
⃗r
i
) =
μ
0
μ
F
(
μ
B
g/
2)
/
(4
πr
3
i
)(1
3 cos
2
θ
i
)
m
p
(
μ
F
being the fluorine
moment). This rapidly dephases the TLS, since the coupling is much larger than the fluorine’s
fluctuation rate,
J
(
⃗r
i
)
κ
i
. A single strongly coupled fluctuator
i
leads to the Hahn-echo
decay (for
N
= 1
π
-pulses) [5]
I
i
(
t
) =
e
κ
i
t
2
λ
h
(
λ
+ 1)
e
κ
i
λt
+ (
λ
1)
e
κ
i
λt
i
e
κ
i
t

1 +
κ
i
2
J
(
⃗r
i
)
sin

2
J
(
⃗r
i
)
t


,
(27)
with
λ
q
1
[2
J
(
⃗r
i
)
i
]
2
. To describe the effect of all
nn
and
nnn
fluorine neighbors, we
multiply their echo shape, approximating their fluctuations as uncorrelated.
In Eq. (27), the clock states are assumed to have a small, but constant moment of a typical
magnitude
m
p
. However, in reality the moment is induced by the fluorine neighborhood and
thus differs from ion to ion. Accordingly, the decay function should be appropriately averaged
over the distribution of couplings.
3
The couplings are expected to have a standard deviation of
the order of the typical
J
, and thus the oscillatory terms average out on a time scale
1
/J
.
We capture this effect qualitatively by assuming a typical moment and associated couplings
J
(
⃗r
i
)
, but we retain the oscillatory term in
I
i
(
t
)
only up to
t
t
c
=
π/
(2
J
(
⃗r
i
))
and drop it for
longer times, setting
I
F,nnn
(
t
) =
Y
i
∈{
nn,nnn
}
̃
I
i
(
t
)
,
with
̃
I
i
(
t
) =
I
i
(
t
)
,
for
t <
π
2
J
(
⃗r
i
)
,
e
κ
i
t
,
for
t
π
2
J
(
⃗r
i
)
.
(28)
To generalize the decay function to a CPMG sequence with
N >
1
pulses, we observe that
the echo at short times is
I
F,nnn
(
t
)
P
i
κ
i
t
(
J
τ
)
2
P
i
κ
i
(
J
/N
)
2
t
3
, where
N
essentially just
renormalizes the couplings,
J
J
/N
(Ext.Dat.Fig. 3). This also holds for the function (27)
up to times
tJ
N
, while at longer times dephasing is unaffected by
N
and independent of
J
.
Thus both limits of the echo suppression due to fluorine under a CPMG sequence are still well
described by Eq. (28), provided all couplings are divided by
N
. This observation suggests that
3
If the moments were static in time, the echo would receive larger contributions from very small moments.
However, the moments themselves evolve as neighboring spins flip, so that atypically small moments do not
survive for long.
8
we extend this recipe to intermediate times, too, in order to obtain a reasonable approximation
for that time regime. The resulting function is used in the numerics to describe the CPMG
data of the
nnn
pair, since its echo is dominated by fluorine noise. At short times, we find that
the coherence time scales as
T
char
N
2
/
3
, the same scaling as observed in the CPMG data
(Fig. 4c). At long times, the decay function asymptotes to a simple exponential with timescale
T
F,nnn
= 1
/
(16
κ
F
)
(since there are 16 strongly coupled neighboring fluorine ions).
Since typical magnetic moments of a
nnn
pair are smaller than those of looser pairs (due to
corrections of order
J
pair
/
which are sizable for the strongly coupled
nnn
pair), we cannot use
the same fitting parameters to describe the coupling of looser pairs to fluorine. For looser pairs,
fluorine only plays a role in the long-time regime of the decay, short times being dominated
by ring-exchange. Thus, a fit cannot capture the short-time regime of Eq. (28) very well. We
therefore only aim at fitting the long-time decay. We make the phenomenological ansatz of a
stretched exponential which allows us to capture part of the crossover from the short-time regime
as well,
I
F
= exp
h
(
t/T
F
)
β
F
i
.
(29)
Here we take
T
F
and
β
F
as free fit parameters independent of the probe frequency
ω
p
, antici-
pating an exponent
β
F
close to 1.
4.4 Optimizing the abundance of coherent qubits
In a host containing nuclear spins, the coherence of qubits is ultimately bounded from above by
the magnetic noise of those spins. In a nuclear-spin free host, however, dephasing of clock state
ions will be limited only by excitation hopping and/or ring-exchange fluctuations.
The relative disorder strength seen by single ions remains moderate with increasing dilution if we
assume that disorder is dominated by strain from the dopants and thus scales as
x
, in the same
way as the dipolar couplings among dopants. Therefore single ions flip with a rate proportional
to the typical dipolar interactions,
i.e.
κ
s
x
(Eq. (6)).
In contrast, pairs dephase predominantly due to ring-exchange with their closest clock state
neighbors. That interaction scales as
V
ring
J
2
typ
x
2
(Eq. (2)). For strong dilution the
corresponding dephasing is in the motionally narrowed regime (since
V
ring
κ
s
), which leads
to very slow typical dephasing rates scaling as
1
pair
V
2
ring
s
x
3
(Eq. (24)).
These considerations imply that a sample where almost all ions have a coherence time
T
2
requires a dilution
x
1
/T
2
, whereas a sample with higher concentration
x
1
/T
1
/
3
2
hosts a
larger density
(
x
)
2
1
/T
2
/
3
2
(
1
/T
2
x
)
of pairs with equally long coherence time. We
thus reach the conclusion that to maximize the density of coherent qubits in a randomly doped
magnet, it is best to focus on pairs in high density samples, rather than to increase the dilution
such that all typical ions reach the desired degree of coherence.
5 Numerical fit of data to the theory
We now combine the different dephasing mechanisms discussed above to obtain a full model
that we use to simultaneously fit all the experimental echo data for the various frequencies and
different echo protocols. The fitting function and the detailed fitting procedure are outlined in
the method section of the main text. The fit parameters that we need to determine are: The
numerical coefficients
c
1
,
c
2
appearing in the fluctuation rate of spins, Eq. (6), the spread of CF
splittings
W
of single ions, the effective fluorine dephasing parameters of loose pairs (
β
F
,
T
F
),
the coupling parameters of the
nnn
Tb pair to nearby fluorine spins (
J
(
⃗r
nn
) and the fluctuation
rate
κ
F
of the latter). Knowing the parameters
c
1
,
c
2
and
W
allows us to calculate the Tb
3+
fluctuation rates
κ
I
z
for all hyperfine states at any concentration (assuming that disorder in
9
I
z
/
̄
V
loose pair
nnn
pair
magnetic (
γ
= 3
)
ring-exchange (
γ
= 6
)
magnetic (
γ
= 3
)
ring-exchange (
γ
= 6
)
3
/
2
1
.
2
ms
0
.
4
μ
s
1
.
9
ms
8
.
4
μ
s
1
/
2
12
.
7
μ
s
0
.
2
μ
s
21
μ
s
9
.
6
μ
s
+1
/
2
6
.
9
μ
s
2
.
2
μ
s
11
μ
s
13
.
6
μ
s
+3
/
2
5
.
1
μ
s
6
.
4
μ
s
8
.
4
μ
s
21
.
3
μ
s
Table 1: Typical inverse interaction strengths
̄
V
(Eq. 17) at Tb
3+
concentration
x
= 0
.
1%
for
a weak clock-state pair (detuning frequency
ω
= 2
π
×−
0
.
5
GHz, corresponding to 2-3 lattice
spacings between the Tb
3+
) and for the
nnn
clock-state pair (detuning
ω
= 2
π
×
7
.
61
GHz)
with neighboring Tb
3+
in various hyperfine states labelled by
I
z
.
γ
= 3
,
6
refers to the exponent
governing the spatial decay of the interactions. For the loose pair, the by far most strongly
coupled Tb
3+
ions are single clock state ions (
I
z
=
3
/
2
) and those with similar nuclear spin,
I
z
=
1
/
2
. Ring-exchange is seen to be the dominant interaction, except for the most strongly
magnetized HF species (
I
z
= 3
/
2
), for which both interactions are weak, however. For the
nnn
pair, all interactions are weak. Ring-exchange dominates for the hyperfine states
I
z
=
3
/
2
,
I
z
=
1
/
2
, while magnetic interactions dominate for neighbors having
I
z
= 1
/
2
,
I
z
= 3
/
2
.
The latter interactions leave no visible trace in the experimental echo decay though due to the
exponentially slow flip rates of these states (Table 2).
the CF splittings is dominated by elastic strain due to the random doping, and thus
W
x
).
Below we summarize the extracted timescales and show the numerical fits.
5.1 Coupling parameters of clock-state pairs with neighboring Tb
3+
ions
We consider clock state pairs having nuclear spin
I
z
=
3
/
2
, with the clock field of
B
z
= 38
mT
being applied. The interaction strengths at typical nearest neighbor distance between such a
pair and a single neighbor Tb
3+
in one of four possible hyperfine states,
cf.
Eq. (17), are listed
in Table. 1 for a pair with dipole-induced detuning
ω
=
2
π
×
0
.
5
GHz and for the
nnn
pair (
ω
= 2
π
×
7
.
61
GHz), respectively. Magnetic interactions are estimated with a typical
effective moment on the clock-state pair, which we take as the HWHM value of its (Gaussian)
distribution,
0
.
277
mT. The latter is induced by the distribution of fluorine magnetic fields,
with the fluorine spins being aligned parallel or anti-parallel to the external field (in contrast to
the magnetic hyperfine species, where nearby fluorine spins align along the stronger dipolar field
of the nearby Tb
3+
ion). Note that the typical (total) moment of the
nnn
pairs (
m
p
= 0
.
00098
)
is smaller by a factor of about
1
.
6
than that of looser pairs with smaller detuning (
m
p
= 0
.
0016
).
4
This difference is due to corrections of order
J
pair
/
which are sizable for the strongly coupled
nnn
pair. As seen from Table 1, at concentration
x
= 0
.
1%
the ring-exchange of loose pairs is
stronger than the residual magnetic interactions (except for the weak interactions with single
I
z
= +3
/
2
ions), and thus they dominate the dephasing in the relevant experimental time
window.
5.2 Spin flip rates of the various Tb
3+
hyperfine states
We used the optimal fit values
c
1
= 0
.
41
,
c
2
= 1
.
67
,
W
(
x
= 0
.
1%) = 21
MHz to extract
the fluctuation rates of the different HF states at concentration
x
= 0
.
01%
and
x
= 0
.
1%
, as
summarized in Table 2 (see also the following subsections). Note that only the moderate flipping
times of the least magnetized hyperfine species (
I
z
=
1
/
2
) are physically relevant. The more
4
We define the moment as half the difference between the moments of the upper and the lower state of the
considered transition. The typical moment is determined as the HWHM of its Gaussian distribution.
10