Fluctuations of work in nearly adiabatically driven open quantum systems

We extend the quantum jump method to nearly adiabatically driven open quantum systems in a way that allows for an accurate account of the external driving in the system-environment interaction. Using this framework, we construct the corresponding trajectory-dependent work performed on the system and derive the integral fluctuation theorem and the Jarzynski equality for nearly adiabatic driving. We show that such identities hold as long as the stochastic dynamics and work variable are consistently defined. We numerically study the emerging work statistics for a two-level quantum system and find that the conventional diabatic approximation is unable to capture some prominent features arising from driving, such as the continuity of the probability density of work. Our results reveal the necessity of using accurate expressions for the drive-dressed heat exchange in future experiments probing jump time distributions.

Figure 1

Schematic illustration of the effective energy level diagram of an externally driven two-level quantum system coupled to an environment. (a) A two-level system $S$ with level separation $\hslash {\omega }_0$ between $|e\rangle$ and $|g\rangle$ undergoing environment-induced transitions at rates ${\Gamma }_{(0,i)}$ experiences an external drive $D$. (b) After transformation ${\widehat{D}}_1$, the environment induces transitions at rates ${\Gamma }_{(1,i)}$ between states $|e^{\left(1\right)}\rangle$ and $|g^{\left(1\right)}\rangle$ of the driven system $S+D$ separated by $\hslash {\omega }_{01}^{\left(1\right)}$. (c) Further transformation ${\widehat{D}}_2$ yields transitions at rates ${\Gamma }_{(2,i)}$ between states $|e^{\left(2\right)}\rangle$ and $|g^{\left(2\right)}\rangle$ separated by $\hslash {\omega }_{01}^{\left(2\right)}$.

Figure 2

Adiabatic (solid line) and first superadiabatic (dashed line) parameters during the driving cycle. We assume that ${\lambda }_0=\hslash {\omega }_0/2$ and ${\omega }_d=3{\omega }_0/10$. The length of the sinusoidal driving cycle is defined by $T_{\mathrm{drive}}=2\pi /{\omega }_d$.

Figure 3

Population of the excited diabatic state, ${\rho }_{ee}$, using dynamics calculated with $n$ basis rotations in the adiabatic renormalization scheme. The length of a single sinusoidal cycle is $T_{\mathrm{drive}}=2\pi /{\omega }_d$. The parameters are selected such that ${\omega }_d=3{\omega }_0/10$, ${\lambda }_0=\hslash {\omega }_0/2$, $g=\hslash {\omega }_0/\left(5\sqrt{2}\right)$, $T=\hslash {\omega }_0/\left(2k_B\right)$, $T_0=2T$, and $\mu =1/\left(\hslash {\omega }_0^2\right)$. The system is initialized to the Gibbs state corresponding to the above bath temperature. The driving protocol is discretized using ${10}^7$ equidistant time steps and the number of trajectories is ${10}^7$.

Figure 4

Probability density of work $d_n\left(W^{\left(n\right)}\right)$ using different numbers of basis rotations, $n$, in the adiabatic renormalization scheme. The solid line (blue) corresponds to $n=0$, the dashed line (red) to $n=1$, and the dash-dotted line (green) to $n=2$. The density function is calculated with a bin size ${10}^{-2}/\left(\hslash {\omega }_0\right)$ and the parameters are selected to be the same as in Fig. 3.

Figure 5

Probability density of work $d_2^{\prime }\left(W^{\left(n\right)}\right)$ using an $n\mathrm{th}\text{−}\mathrm{order}$ work variable while fixing the dynamics to $n=2$. The solid line (blue) corresponds to $n=0$, the dashed line (red) to $n=1$, and the dash-dotted line (green) to $n=2$. The density function is calculated with a bin size ${10}^{-2}/\left(\hslash {\omega }_0\right)$ and the parameters are selected to be the same as in Fig. 3.