Optimal Lyapunov-based quantum control for quantum systems

Quantum Lyapunov control was developed in order to transform a quantum system from arbitrary initial states to a target state. The idea is to find control fields that steer the Lyapunov function to zero as $t\to \infty$, meanwhile the quantum system is driven to the target state. In order to shorten the time required to reach the target state, we propose two designs to optimize Lyapunov control in this paper. The first design makes the Lyapunov function decrease as fast as possible with a constraint on the total power of control fields, and the second design has the same purpose but with a constraint on each control field. Examples of a three-level system demonstrate that the evolution time for Lyapunov control can be significantly shortened, especially when high control fidelity is required. Besides, this optimal Lyapunov-based quantum control is robust against uncertainties in the free Hamiltonian and decoherence in the system compared to conventional Lyapunov control. We apply our optimal design to cool a nanomechanical resonator where a shorter cooling time is found with respect to the cooling time by the conventional Lyapunov design.

Figure 1

(a) $P_f$ (solid line) and $|\dot{V}|$ (dashed line) as a function of time with the conventional control design $f_n\left(t\right)=-K{T}_n$. (b) Control fields (in units of $\omega$) for $W_{\max }=0.0001$, and $S_{\max }=0.007$. These fields are switched off when $P_f\ge 0.999$ at $t=191$.

Figure 2

(a) $P_f$ (solid line) and $|\dot{V}|$ (dashed line) as a function of time with the control design Eq. (12) under constraint A. (b) Control fields (in units of $\omega$) with $W_{\max }=0.0001$. These fields are switched off at $t=90$ when $P_f\ge 0.999$.

Figure 3

(a) $P_f$ (solid line) and $|\dot{V}|$ (dashed line) as a function of time with bang-bang control Eq. (15) under constraint B. (b) Control fields (in units of $\omega$) with $S=0.007$. These fields are switched off at $t=74$ when $P_f\ge 0.999$.

Figure 4

Evolution of $D=1-|\langle \phi |{\phi }_f\rangle {|}^2$ for the three control designs with logarithmic scale. (a),(b), and (c) correspond to the conventional control field design, the design under constraint A, and that under constraint B, respectively. Each picture is a result averaged over 50 random initial states. The convergence speed of (b) and (c) is faster than that of (a) as these figures show.

Figure 5

The average fidelity verses uncertainties in the free Hamiltonian, (a) for ${\lambda }_1$ and (b) for ${\lambda }_3$. The blue triangle, green star, and black circle represent the fidelity obtained by bang-bang control design, design with power constraint, and the conventional one, respectively.

Figure 6

The average fidelity as a function of the decoherence rates, here we choose ${\gamma }_2={\gamma }_3=\gamma$. The blue triangle, green star, and black circle represent bang-bang control design, design with power constraint, and the conventional one, respectively.

Figure 7

(a) Phonon number versus time for the conventional Lyapunov design (upper half) and optimal Lyapunov design (lower half). The evolution time for $\langle n_a\rangle =0.05$ is $t=37.5$ (upper) and $t=13.4$ (lower), respectively. (b) Time-dependent coupling for the conventional design (upper) and the optimal design (lower), where $g\left(t\right)$ for the two designs share the same maximal strength $|g_{\max }|/\omega =1.91$. $g\left(t\right)$ for the optimal design is shut off after $t=13.4$.