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
+