In this study, we analyze the temperature dependence of Exceptional Points (EPs) in a pseudo-Hermitian hybrid model composed of a superconducting flux qubit (SFQ) interacting with an ensemble of nitrogen-vacancy (NV) color centers in diamond [1–2]. This model is relevant not only due to its experimental realizability [1], but also as a representative example of a many-body Hamiltonian exhibiting a nontrivial interplay of distinct physical effects.

The interaction between the NV ensemble and the SFQ is modeled by introducing an asymmetry parameter [2] that modifies the balance between the creation and annihilation processes of NVs coupled to the SFQ. This asymmetry accounts for the presence of impurities within the NV ensemble, such as P1 centers [3].

As reported in [4], temperature effects on the zero-field splitting states of the NV ensemble are negligible below T=200 KT = 200~\mathrm{K}T=200 K. Therefore, we assume that temperature has an insignificant effect on the NV ensemble. This is not the case for the SFQ, as superconductivity is well known to be strongly temperature-dependent.

We investigate how the location of the EPs and the extent of each symmetry phase evolve with temperature by explicitly incorporating the temperature dependence of the superconducting gap in the SFQ . To characterize the system’s evolution and the impact of thermal effects, we analyze several observables, including the survival probability, expectation values of spin components, spin squeezing, and the SU(2) Wigner function. The initial state is prepared as a spin coherent state for the NV ensemble, with the SFQ in its ground state.

Our results show that increasing temperature tends to expand the region of parameter space corresponding to the exact-symmetry phase [5].

[1] X. Zhu et al., Nature 478, 211 (2011).

[2] R. Ramírez, M. Reboiro, and D. Tielas, The European Physical Journal D 74, 193 (2020).

[3] V. Stepanov and S. Takahashi, Phys. Rev. B 94, 024421 (2016).

[4] M. C. Cambria et al., Phys. Rev. B 108, L180102 (2023)

[5] to be published.