Jarzynski Equality in $\mathcal{P}\mathcal{T}$-Symmetric Quantum Mechanics

We show that the quantum Jarzynski equality generalizes to $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics with unbroken $\mathcal{P}\mathcal{T}$ symmetry. In the regime of broken $\mathcal{P}\mathcal{T}$ symmetry, the Jarzynski equality does not hold as also the $\mathcal{C}\mathcal{P}\mathcal{T}$ norm is not preserved during the dynamics. These findings are illustrated for an experimentally relevant system—two coupled optical waveguides. It turns out that for these systems the phase transition between the regimes of unbroken and broken $\mathcal{P}\mathcal{T}$ symmetry is thermodynamically inhibited as the irreversible work diverges at the critical point.

Figure 1

Irreversible work $⟨W_{\mathrm{irr}}⟩$ [Eq. (27)] for the linear protocol [Eq. (26)] and ${\alpha }_0=1/2$, which is a system starting in the regime of unbroken $\mathcal{P}\mathcal{T}$ symmetry. Parameters are $\beta =1$, $\hslash =1$, $\tau =1$, and $\kappa =1$. The inset illustrates the validity of the $\mathcal{P}\mathcal{T}$-symmetric Jarzynski equality [Eq. (15)].

Figure 2

Real part (red solid line) and imaginary part (purple dashed line) of the irreversible work $⟨W_{\mathrm{irr}}⟩$ [Eq. (27)] for the linear protocol [Eq. (26)] and ${\alpha }_0=3/2$, which is a system starting in the regime of broken $\mathcal{P}\mathcal{T}$ symmetry. Parameters are $\beta =1$, $\hslash =1$, $\tau =1$, and $\kappa =1$. The inset illustrates the violation of the Jarzynski equality [Eq. (15)] in the real part (blue solid line) and the imaginary part (green dashed line).