A Pseudo-Hermitian Hybrid Model at Finite Temperature: The Role of the Exceptional Points

TL;DR

The paper addresses thermodynamics of a non-Hermitian, pseudo-Hermitian hybrid system formed by an ensemble of nitrogen-vacancy centers coupled to a superconducting flux qubit at finite temperature. It constructs an exact grand partition function by decomposing the NV and SFQ spaces into irreducible representations, analyzes zeros of the partition function in the broken-PT-symmetric phase, and links them to exceptional points via Yang–Lee theory, revealing first-order transitions. Through Maxwell construction and spinodal analysis, it shows metastable heterogeneous phases below a critical temperature Tc(α,g) and demonstrates EP-driven thermodynamic cycles (Carnot and Stirling) with efficiencies comparable to or exceeding classical limits, especially near EPs; a rescaling scheme is developed to extend results to larger Hilbert spaces. The results provide a framework for the thermodynamics of pseudo-Hermitian systems and may guide experiments in NV–SFQ hybrids, with potential connections to fluctuation relations such as Jarzynski’s equality.

Abstract

We study a hybrid system formed by an ensemble of colour nitrogen-vacancy centres in diamond interacting with a superconducting flux-qubit at finite temperature. The presence of impurities in the system is modelled through pseudo-hermitian Hamiltonian, by introducing an asymmetry parameter in the interaction between the superconducting flux qubit and the ensemble of colour nitrogen-vacancy centres in diamond. We construct the exact grand partition function of the system, and from it we derive the thermodynamic quantities, e.g. entropy, internal energy, and Helmholtz free energy. In the broken symmetry phase, we observe the existence of zeros in the partition function. These zeros are related to the existence of complex-pair-conjugate eigenvalues with real parts lying among the low levels of energy. In line with the Yang-Lee framework, these zeros in the complex plane signal phase transitions, and the proposed hybrid model exhibits transitions of first-order. To account for metastable regions in parameter space, we perform a Maxwell construction and a spinodal-decomposition analysis. We determine the critical temperature at which the first zero of the partition-function appears, as a function of the asymmetry parameter and the coupling constant of the interaction between the ensemble of colour nitrogen-vacancy centres in diamond and the superconducting flux-qubit. We also design a Carnot cycle that traverses Exceptional Points in the broken symmetry phase for temperatures above the critical value, achieving the same efficiency as the classical Carnot cycle. Furthermore, we implement a Stirling cycle whose efficiency surpasses its classical counterpart, particularly when operating near Exceptional Points. Finally, we outline how the model can be scaled to larger Hilbert-space dimensions.

Figures (12)

• Figure 1: Position of first EPs above and below $\alpha = 1$ as a function of $g$. The system consists $N_{NVs}=8$ NVs and $N_p=2$ pairs to model the SFQ. For the model parameter presented in the text, $D=2.878$ [GHz], $E=0.26$ [GHz], $G=1.73$ [GHz], $\epsilon_2=-\epsilon_1=1$ [GHz].
• Figure 2: The figure shows the spectrum and the critical temperature as a function of $\alpha$ of a system of $N_S=8$ NVs and $N_p=2$ pairs to model the SFQ. For the model parameter presented in the text, $D=2.878$ [GHz], $E=0.26$ [GHz], $G=1.73$ [GHz], $\epsilon_2=-\epsilon_1=1$ [GHz]. The upper panels display the real contribution to the eigenenergies of the system as a function of the asymmetry parameter $\alpha$. The middle panels show the imaginary part of the eigenenergies as a function of the parameter $\alpha$. The lower panels depict the critical temperature as a function of $\alpha$, that is, the temperature of the first zero of the partition function. Red lines are drawn to show the position of the limits of the critical domain in $\alpha$. Dotted lines correspond to the position of the EPs. The left column corresponds to a coupling between the NVs and the SFQ of $g=1$ [GHz], while for the middle column $g=G$, and for the right column $g=2G$, respectively.
• Figure 3: The figure shows the behaviour of the free energy, $F$, the entropy, $S$, and the internal energy, $U$, as a function of the temperature in units of $D$, $T_r=T/D$, in the absence of interaction between the NVs and the SFQ. The system is that of the previous Figures, $N_S=8$ NVs and $Np=2$ pairs to model the SFQ. For the model parameter presented in the text, $D=2.87$ [GHz], $E=0.26$ [GHz], $G=1.73$ [GHz], $\epsilon=1$ [GHz]. The left Panels correspond to values of the coupling constant $g=1.0$ [GHz], while the right Panels correspond to values of $g=1.73$ [GHz].
• Figure 4: The figure shows the essentials in the construction of the spinodal decomposition. We plot the free energy as a function of $\alpha$ for three different values of $T_r$. We draw with green dots the line that connects the minima of $F$ at different temperatures, while those connecting the inflexion points are depicted in red. The absolute value of the slopes of the black lines that connect the points A and B, and B and C, respectively, give the equilibrium pressure of the heterogeneous system.
• Figure 5: The figure shows the behaviour of the isotherms of the Helmholtz free energy, $F$, in the range of temperatures from $T_r=0.04$ to $0.07$, to show the evolution of minima and of inflexion points of $F$.