Fast trajectory tracking of the steady state of open quantum systems

We present a control strategy to speed up the steady-state tracking for open quantum systems governed by a time-dependent Markovian master equation. With this fast trajectory tracking strategy, the quantum state strictly follows the instantaneous steady state of open quantum systems and evolves into the target state without any fidelity loss. A compact expression for the counterdiabatic term is derived in terms of a superoperator formalism in Liouville space. To be specific, we utilize a universal method to transform the counterdiabatic superoperator into the ordinary operator form. It can be seen from the operator form that in addition to the coherent control field, an incoherent control is required for the fast trajectory tracking. The control strategy is successfully applied in two examples: One is a two-level quantum system with a time-dependent Hamiltonian coupling to a heat bath, the other is a two-level system following an isothermal process.

Figure 1

(a) The final fidelity vs the change rate $\omega$ with $\omega t_f=\pi /2$; (b) the evolution of the fidelity with the change rate $\omega =\gamma$ for the adiabatic tracking control (green dash line) and the fast tracking control (blue solid line) at the zero-temperature limit ($T=0$). The control field strength is set to be ${\mathrm{\Omega }}_0=10\gamma$. The above parameters are in units of $\gamma$.

Figure 2

(a) The final fidelity vs the change rate $\omega$ at the zero-temperature limit ($T=0.1{\gamma }_0$), and (b) the fidelity vs the temperature of the bath $T$ with the change rate $\omega =10{\gamma }_0$ for the adiabatic engineering program (green dashed line) and the superadiabatic engineering program (blue solid line). The total evolution time satisfies ${\omega }_i+\omega t_f=\pi /2$ with ${\omega }_i=0.1$. The above parameters are in units of $\gamma$.

Figure 3

(a) The final fidelity $F$ vs the dimensionless amplification parameter $A$ and bath temperature $T$; (b) the final infidelity vs the dimensionless amplification parameter $A$ and bath temperature $T$ in logarithmic scale; (c) the evolution of the fidelity with $A=5$ and $T={10}^{-0.25}\gamma$. The other parameters are the same as those in Fig. 2.

Figure 4

(a) The final fidelity vs the change rate ${\omega }_0/\gamma$ and (b) the evolution of the fidelity with the change rate ${\omega }_0=10\gamma$ for the adiabatic tracking control (green dashed line) and the fast tracking control (blue solid line). The squeezed amplitude is set to be $r=2$. The above variables are in units such that $\gamma =1$.

Figure 5

The control period ${\gamma }_0{t}_f$ vs the reservoir temperature $T$ with the final infidelity ${10}^{-8}$ for the dissipative engineering (green dashed line) and the modulated tracking control (blue solid line). The change rate is $\omega =\pi /2t_f$, and the dimensionless amplification parameter is $A=4$. The decoherence strength and energy splitting used in the dissipative engineering scheme are set to be ${\gamma }_d=A{\gamma }_0$ and ${\omega }_d=(0.1+\pi /2){\gamma }_0$. The above parameters are in units of $\gamma$.

Figure 6

The final infidelity vs the control period ${\gamma }_0{t}_f$ and the dimensionless amplification parameter $A$ for the modulated tracking control. The change rate is $\omega =\pi /2t_f$, and the reservoir temperature is set to be $T={\gamma }_0$. The above parameters are in units of $\gamma$.