Quantum heat engines: Limit cycles and exceptional points

We show that the inability of a quantum Otto cycle to reach a limit cycle is connected with the propagator of the cycle being noncompact. For a working fluid consisting of quantum harmonic oscillators, the transition point in parameter space where this instability occurs is associated with a non-Hermitian degeneracy (exceptional point) of the eigenvalues of the propagator. In particular, a third-order exceptional point is observed at the transition from the region where the eigenvalues are complex numbers to the region where all the eigenvalues are real. Within this region we find another exceptional point, this time of second order, at which the trajectory becomes divergent. The onset of the divergent behavior corresponds to the modulus of one of the eigenvalues becoming larger than one. The physical origin of this phenomenon is that the hot and cold heat baths are unable to dissipate the frictional internal heat generated in the adiabatic strokes of the cycle. This behavior is contrasted with that of quantum spins as working fluid which have a compact Hamiltonian and thus no exceptional points. All arguments are rigorously proved in terms of the systems' associated Lie algebras.

Figure 1

Comparison between a normal cycle, in the left panel, and a divergent cycle for which the steady state will never be reached, in the right panel. The dashed curves represent the frequency dependence of the thermal equilibrium value of $\langle \widehat{N}\rangle$ for the temperatures of the hot and cold heat reservoirs. The thin black rectangle inscribed between these curves is the long-time limit trajectory. The times allocated for the adiabatic processes are ${\tau }_{HC}={\tau }_{CH}=0.1$. For the left panel ${\tau }_H={\tau }_C=2$, while for the right panel ${\tau }_C=0.4$ and ${\tau }_H=0.29$. The values of the other parameters are listed in Sec. 2d.

Figure 2

Left panel: power landscape for ${\tau }_{HC}={\tau }_{CH}=0.1$. The white regions correspond to choices of parameters for which the system is not able to converge to a limit cycle, as for the trajectory shown in Fig. 1. Right panel: eigenvalues of the $3\times 3$ block $\tilde{\mathit{U}}$ of the time-evolution matrix for one cycle. The dark and bright shades of the same hue indicate the real and imaginary parts of the same eigenvalues, respectively. Dashed multicolored lines indicate that the curves of the corresponding colors overlap. When one of the eigenvalues, in this case $u_{+}$, has modulus greater than 1, the limit cycle can never be reached. This figure corresponds to the segment highlighted by the horizontal red line shown in the left panel, ${\tau }_C=0.4$.

Figure 3

Left panel: eigenvalues of $\tilde{\mathbf{\Omega }}$. Dashed segments indicate coincidence of the relevant colors. As can be noticed, in the middle region delimited by the thick vertical lines the eigenvalues are purely real. Outside of this region the real part of all the eigenvalues is equal to $-\mathrm{\Gamma }({\tau }_H+{\tau }_C)$. At the transition between these two regions all the eigenvalues are exactly equal to $-\mathrm{\Gamma }({\tau }_H+{\tau }_C)$. Right panel: the blue curve shows the absolute value of the determinant $det\left(\mathit{T}\right)$ of the matrix $\mathit{T}$ having as columns the eigenvectors of the time-evolution matrix $\mathit{U}\left(\tau \right)$. When the determinant is zero we are at an exceptional point, i.e., non-Hermitian degeneracy. For both panels ${\tau }_{HC}={\tau }_{CH}=0.1$ and ${\tau }_C=0.4$, as in Fig. 2.

Figure 4

Left panel: power landscape for two coupled spins in presence of an oscillating magnetic field. This system can never exhibit divergent behavior and indeed the white regions visible in Fig. 2 are not present here. The adiabat times are: ${\tau }_{HC}={\tau }_{CH}=0.64$. Right panel: eigenvalues of the $3\times 3$ block $\tilde{\mathit{U}}$ of the time-evolution matrix for one cycle. For the spin system the moduli of the eigenvalues are always equal to $e^{-\mathrm{\Gamma }({\tau }_H+{\tau }_C)}<1$, ensuring the existence of a limit cycle. This panel corresponds to the segment highlighted by the horizontal red line shown in the left panel, i.e., ${\tau }_C=1.6$.