Thermodynamics of an Open $\mathcal{PT-}$Symmetric Quantum System

TL;DR

This work develops a Hermitian basis for a subclass of PT-symmetric Hamiltonians obeying {H,H†}=I and constructs a generalized density matrix ρ_G(t) in that basis to study ergotropy. It provides analytic expressions for ρ_G(t) and the passive state, and shows how ergotropy behaves under open-system dynamics when the system couples to a Jaynes-Cummings type bath, highlighting non-Markovian revivals and enhanced work extraction near Hermitian normal points. The authors verify the first, second, and third laws of quantum thermodynamics for the PT-symmetric open system, demonstrating thermodynamic consistency across regimes of non-Hermiticity. The results point to the viability of PT-symmetric systems as quantum batteries with tunable ergotropy and controlled thermodynamic performance.

Abstract

For a subclass of a general $\mathcal{PT}-$symmetric Hamiltonian obeying anti-commutation relation with its conjugate, a Hermitian basis is found that spans the bi-orthonormal energy eigenvectors. Using the modified projectors constructed from these eigenvectors, the generalized density matrix of the $\mathcal{PT}-$symmetric evolution is calculated, and subsequently, ergotropy for a closed system is obtained. The $\mathcal{PT}-$symmetric system, in an open system scenario, is studied to understand ergotropy under different regimes of non-Hermiticity of the Hamiltonian. The consistency of the three laws of thermodynamics for the $\mathcal{PT}-$symmetric system in an open system scenario is also analyzed.

Figures (4)

• Figure 1: A diagrammatic representation of the open system scheme we apply for a $\mathcal{PT}-$symmetric Hamiltonian.
• Figure 2: Variation of ergotropy $\mathcal{W}(\rho_G(t))$ in an open system. The initial states are the excited state $\rho_G^2$ in (a), the state $\rho_G^3$ in (b), defined in Eq. \ref{['Initialstates']}. The Hamiltonian parameter $s$ is kept fixed at one, and $r$ is varied to render different regimes of Hermiticity. The coupling constant is $g=0.5$, and the bath frequency, dimension, and temperature are consecutively $\omega_c=2,~d_B=15,~T=10$.
• Figure 3: Different thermodynamic quantities are plotted when the initial state is the excited state $\rho_G^2$. (a) is for the Hermitian limit at $r=0$, (b) is for the non-Hermitian Hamiltonian at $r=0.5$, and (c) is for when the Hamiltonian approaches the exceptional limit, $r=0.95$. All cases have $s=1$. The coupling constant is $g=0.5$, and the bath frequency, dimension, and temperature are consecutively, $\omega_c=2,~d_B=15,~T=10$
• Figure 4: Entropy production ($\Sigma$) for the $\mathcal{PT}-$symmetric open quantum system. The initial states [Eq. \ref{['Initialstates']}] are the excited state $\rho_G^2$ in (a), the state $\rho_G^3$ in (b), and the ground state $\rho_G^1$ in (c). The parameter $s$ is kept fixed at one, and $r$ is varied to render different regimes of non-Hermiticity. The coupling constant is $g=0.5$, and the bath frequency, dimension, and temperature are consecutively, $\omega_c=2,~d_B=15,~T=10$.