Preparing entangled states by Lyapunov control

Abstract

By Lyapunov control, we present a protocol to prepare entangled states such as Bell states in the context of cavity QED system. The advantage of our method is of threefold. Firstly, we can only control the phase of classical fields to complete the preparation process. Secondly, the evolution time is sharply shortened when compared to adiabatic control. Thirdly, the final state is steady after removing control fields. The influence of decoherence caused by the atomic spontaneous emission and the cavity decay is discussed. The numerical results show that the control scheme is immune to decoherence, especially for the atomic spontaneous emission from $|2\rangle$ to $|1\rangle$. This can be understood as the state staying in an invariant subspace. Finally, we generalize this method in preparation of W state.

Appendix

In this physical model, we now demonstrate that the Bell state can also be achieved by fractional stimulated Raman adiabatic passage (f-STIRAP) in which the Stokes pulses and the pump pulses vanish simultaneously eventually. The system Hamiltonian reads

$$\begin{aligned} H_{0}=\sum _{j=1}^{2}\Big (\Omega _{j}|1\rangle _{jj}\langle 2|+g|2\rangle _{jj}\langle 0|a+ H.c.\Big ). \end{aligned}$$

(33)

In the single excitation subspace, the system eigenstate with null eigenvalue can be described by

$$\begin{aligned} |\lambda _{1}\rangle =\mathcal {N}_1(\Omega _{2}|10,0\rangle -\frac{\Omega _1\Omega _2}{g}|00,1\rangle +\Omega _{1}|01,0\rangle ). \end{aligned}$$

(34)

From this expression, it is straightforward to find that the eigenstate $|\lambda _{1}\rangle$ becomes $|10,0\rangle$ or $|01,0\rangle$ in the absence of the classical fields, and the eigenstate $|\lambda _{1}\rangle$ becomes the Bell state $|\lambda _{1}\rangle =\frac{1}{\sqrt{2}}(|10\rangle +|01\rangle )\otimes |0\rangle$ if both the classical fields $\Omega _1(t)$ and $\Omega _2(t)$ vanish adiabatically and satisfy the condition $\lim \limits _{t\rightarrow +\infty }\frac{\Omega _1(t)}{\Omega _2(t)}=1$. This is the main idea of f-STIRAP which is based on the adiabatical evolution of dark state.

Fig. 6

a The time evolution of the fidelity of the Bell state. b The time evolution of the classical fields $\Omega _j(t), j=1,2$. Other parameters are set as $\alpha =\frac{\pi }{4}$, $T=35$, and $\tau =0.6T$

Assume that the initial state is $|10,0\rangle$ again. At first, the second atom is driven by Stokes pulses, and the first atom is driven by pump pulses after a certain time delay. In order to satisfy the condition $\lim \limits _{t\rightarrow +\infty }\frac{\Omega _1(t)}{\Omega _2(t)}=1$, one can design the following waveform for the two classical fields (we can also refer to the control fields like in the main text),

$$\begin{aligned}&\Omega _{1}(t)=\Omega _{0}e^{-\frac{(t-\tau )^2}{T^2}}\sin \alpha , \nonumber \\&\Omega _{2}(t)=\Omega _{0}e^{-\frac{(t-\tau )^2}{T^2}}\cos \alpha +e^{-\frac{(t+\tau )^2}{T^2}}. \end{aligned}$$

(35)

Figure 6a displays the time evolution of the fidelity. Although the order of magnitude in the control fields almost equals each other in the f-STIRAP and Lyapunov control, the dynamics behaviors are quite different. We can find that the evolution time is much longer than that in Lyapunov control. This is a universal drawback for the f-STIRAP because it must be adiabatical during the whole evolution, while Lyapunov control is not restricted to this adiabatical condition. Additionally, the waveform of classical field $\Omega _2(t)$ is superposition of two types of pulses and different from the classical field $\Omega _1(t)$, which is required to control the temporal sequence of classical fields with high precision. Nevertheless, we can adopt a phase modulator with square pulses and only make $\pi$ phase difference for two classical fields in Lyapunov control.