Work statistics in non-Hermitian evolutions with Hermitian endpoints

Non-Hermitian systems with specific forms of Hamiltonians can exhibit novel phenomena. However, it is difficult to study their quantum thermodynamical properties. In particular, the calculation of work statistics can be challenging in non-Hermitian systems due to the change of state norm. To tackle this problem, we modify the two-point measurement method in Hermitian systems. The modified method can be applied to non-Hermitian systems which are Hermitian before and after the evolution. In Hermitian systems, our method is equivalent to the two-point measurement method. When the system is non-Hermitian, our results represent a projection of the statistics in a larger Hermitian system. As an example, we calculate the work statistics in a non-Hermitian Su-Schrieffer-Heeger model. Our results reveal several differences between the work statistics in non-Hermitian systems and the one in Hermitian systems.

Figure 1

(a) Schematic illustration of the statistics in non-Hermitian systems. The processes with decreasing norm, conserved norm, and increasing norm are shown with a red dashed line, a black solid line, and a blue dotted line, respectively. (b) Illustration of the Hermitian Su-Schrieffer-Heeger model and two types of non-Hermitian terms. The terms in the red dashed squares and the terms in the blue dotted squares represent the loss-gain terms and nonreciprocal terms, respectively.

Figure 2

(a) The average work $W_{\mathrm{ave}}$ and the system-energy change probability $\mathrm{\Delta }{E}_s$ for $\gamma =1.9g_1$ (before the EP). (b) The average work $W_{\mathrm{ave}}$ and the system-energy change $\mathrm{\Delta }{E}_s$ for $\gamma =2.1g_1$ (after the EP). (c) The comparison between the average works $W_{\mathrm{ave}}$ of a slowly changing Hamiltonian and a fast changing Hamiltonian for $\gamma =1.9g_1$ (before the EP). (d) The comparison between the average works $W_{\mathrm{ave}}$ of a slowly changing Hamiltonian and a fast changing Hamiltonian for $\gamma =2.1g_1$ (after the EP). The inverse of the temperature $\beta$ and the energy are expressed in units of $1/g_1$ and $g_1$, respectively.

Figure 3

(a) High-temperature case $\beta =0.1/g_1$ with $\gamma =1.9$ (before the EP). (b) Low-temperature case $\beta =1000/g_1$ with $\gamma =1.9$ (before the EP). (c) High-temperature case $\beta =0.1/g_1$ with $\gamma =2.1$ (after the EP). (d) Low-temperature case $\beta =1000/g_1$ with $\gamma =2.1$ (after the EP). (e) High-temperature case $\beta =0.1/g_1$ with $\gamma =2.1$ (after the EP) and one additional round. (f) Low-temperature case $\beta =1000/g_1$ with $\gamma =2.1$ (after the EP) and one additional round. The inverse of the temperature $\beta$ and the energy are expressed in units of $1/g_1$ and $g_1$, respectively. The change of the Hamiltonian is slow.

Figure 4

The effects of the additional non-Hermitian Hamiltonian. (a) The real parts of the eigenvalues for systems with $\delta =0$ and $\delta =0.2\gamma$, respectively. (b) The imaginary parts of the eigenvalues for systems with $\delta =0$ and $\delta =0.2\gamma$, respectively. (c) The average work $W_{\mathrm{ave}}$ versus $\beta$ with $\gamma =2.1g_1$ (after the EP) and different values of $\delta$. (d) The average work $W_{\mathrm{ave}}$ versus $\delta$ with $\gamma =2.1g_1$ (after the EP) and different values of $\beta$. (e) The fluctuation of the work $\langle \mathrm{\Delta }{W}^2\rangle$ versus $\beta$ with $\delta =0,0.1\gamma$, and $0.2\gamma$. (f) The work probabilities with $\beta =1000/g_1$ and different values of $\delta$. The work and $\gamma$ are expressed in units of $g_1$, and $\delta$ is shown as the ratio to $\gamma$. The inverse of the temperature $\beta$ is expressed in units of $1/g_1$. The control function is set to be $f\left(t\right)=1$ in (a) and (b). The strength $\gamma$ of $H_{\mathrm{nh}}^1$ is $2.1g_1$ (after the EP) in (c)–(f).