Work and its fluctuations in a driven quantum system

We analyze work done on a quantum system driven by a control field. The average work depends on the whole dynamics of the system, and is obtained as the integral of the average power operator. As a specific example we focus on a superconducting Cooper-pair box forming a two-level system. We obtain expressions for the average work and work distribution in a closed system, and discuss control field and environment contributions to the average work for an open system.

Figure 1

Schematic presentation of avoided crossing with the eigenstates of energies $E_g$ and $E_e$ in a CPB as a function of the normalized gate charge $q$. The phases which contribute to the interference are explicitly indicated.

Figure 2

(a) Average work $\langle W\rangle$ normalized to $E_C$ for different initial states: $\alpha =1/\sqrt{2}$ and $\gamma =0$ (blue oscillating curve), ground state $\alpha =1$ and $\gamma =0$ (top black exponentially decaying curve), and excited state $\alpha =0$ and $\gamma =0$ (lower black exponentially decaying curve). The purple curve denotes the behavior $\sqrt{P_{\text{LZ}}(1-P_{\text{LZ}})}$. (b) Work variance $\langle \delta W^2\rangle$ normalized to $E_C^2$. The blue solid lines contain the oscillatory behavior for initial state $\alpha =1/\sqrt{2}$ and $\gamma =0$. The black dashed line is obtained for the initial state, $\alpha =1$ and $\gamma =0$ (ground state). We have used $\epsilon =E_J/\left(2E_C\right)=0.05$. Inset: Circuit scheme of the Cooper-pair box (CPB) connected to a dissipative environment $R$.

Figure 3

(a) Numerically calculated (solid lines) dissipated average heat during the sweep in an open CPB as a function of the sweep time when the system is initially in the ground state. The system-environment coupling constant is chosen to be $C_g/C_{\Sigma }=0.05$ here, $E_C/k_B=1$ K, and the environment resistance is $R=1\times {10}^4$ $\Omega$. The different curves from top to bottom correspond to $\epsilon =0.05$, $0.0375$, and $0.025$. The dashed line is the analytic approximation of Eq. (10). (b) Dissipated average heat at $E_C\mathcal{T}/\hslash =150,100$, and 50 from top to bottom as a function of ${\epsilon }^2$. The other parameters and the line conventions are as in (a).