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开放量子系统是量子物理非常重要的研究领域, 是新兴量子科技发展不可或缺的量子平台. 但是完整描述开放系统的非平衡动力学过程往往具有一定的难度. 例如, 在某些情况下系统在弛豫至稳态之前往往会经历漫长的动力学过程, 即亚稳动力学过程. 完整描述这种复杂而又缓慢的动力学过程往往非常困难. 针对该问题, 本文研究了如何在刘维尔的慢变本征模式子空间中对亚稳动力学过程进行低维度的近似描述, 从而简化计算难度. 然后, 针对电磁感应透明(EIT)条件下的里德堡原子系统, 研究了其亚稳动力学过程的有效描述, 并讨论了该有效描述和真实动力学之间的误差. 本文的研究为建立开放多体系统动力学过程的有效简化描述提供了一种可行的思路和方法.Open quantum systems, which are coupled to an external bath, are a critical research field of quantum physics. The steady state, which is a state that any initial state converges after a long time, usually attracts the most interest. In contrast, there are relatively few studies on the nonequilibrium dynamical processes of quantum many-body systems. This is mainly due to the fact that quantum many-body systems generally have interactions, and the Hilbert space required for a complete description of their dynamical processes will grow exponentially with the number of particles and the computational difficulty will increase dramatically as well. Hence the complete description of their dynamical processes has been a difficult problem.With the advancing quantum technologies, there is increasing interest in the nonequilibrium dynamics of open quantum many-body systems. A common phenomenon is that of metastability, where the system initially relaxes into long-lived states, with subsequent converging to the final stationarity at much longer times. In this paper, we establish a low dimensional approximation to describe the metastability dynamics in Markovian open quantum systems, based on the spectra of the Liouvillian super-operator. The separation of time scales implies a splitting in the spectrum, and this spectral division allows us to eliminate the fast decay modes by perturbation method, and then we establish the effective description in the low-lying eigenmodes subspace. Furthermore, we study the dynamics process for the Rydberg atomic systems under electromagnetically induced transparency (EIT) conditions and find that the system can process metastable dynamics if the interactions between the atoms are considered. We compare the effective dynamics in the subspace and the actual dynamics in the full space, and the results show that the effective dynamics works well under the condition that the perturbation approximation holds. Our work provides a feasible idea and method to establish an effective and simplified description of the dynamical process of open quantum many-body systems.
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Keywords:
- Metastable dynamics /
- Relaxation /
- Electromagnetically induced transparency (EIT) /
- Liouvillian superoperator
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图 1 双原子系统的亚稳态动力学过程示意图 (a) 四能级原子的能级结构与光场激发构型图; (b) 系统的动力学过程, 其中红色曲线对应图c中的参数条件, 蓝色曲线对应图d中的参数条件; (c-d) 刘维尔超算子的能谱, 红色点表示稳态所对应的本征值($ \lambda_0=0 $), 绿色点表示衰减模式所对应的本征值($ \lambda_{i > 0} $), 其中图c中无相互作用$ V/\gamma=0 $, 图(d)中相互作用$ V/\gamma=0.6 $, 其他参数均为$ \Omega_c/\gamma=2.0 $, $ \Omega_p/\gamma=0.1 $, $ \delta/\gamma=0 $
Fig. 1. Metastable dynamics of two-atom system: (a) energy levels and transitions of four-levels atom; (b) The dynamic process of the system, where the red and blue curves correspond to the parameters in (c) and (d) respectively; (c-d) The spectrum of the Liouvillian superoperator, the red points indicate the eigenvalue $ \lambda_0= 0 $ corresponding to the steady state, and the green points are the eigenvalues ($ \lambda_{i > 0} $) corresponding to the decaying mode, note that there is no interaction in (c) and $ V/\gamma=0 $, while the interaction in (d) with parameter $ V/\gamma=0.6 $, other parameters are $ \Omega_c/\gamma=2.0 $, $ \Omega_p/\gamma=0.1 $, $ \delta/\gamma=0 $.
图 2 刘维尔能谱随相互作用 $ V/\gamma $ 的变化. 参数 $ \Omega_c/\gamma=2.0 $, $ \Omega_p/\gamma=0.1 $, $ \delta/\gamma=0 $.
Fig. 2. Spectrum of the Liouvillian as a function of $ V/\gamma $. The parameters are $ \Omega_c/\gamma=2.0 $, $ \Omega_p/\gamma=0.1 $, $ \delta/\gamma=0 $.
图 3 50种随机初始态情况下保真度$ F(t) = {{\rm{Tr}}}[\sqrt{\rho(t)\rho_ {\rm{eff}}(t)}] $随时间的变化. 横轴上的红色空心圆、红色实心圆和黑色方块所标注的时间分别对应于文中所定义的时间尺度$ \tau'', $$ \tau', \tau $. 参数(a) $ \Omega_p/\gamma=0.2 $; (b) $ \Omega_p/\gamma=0.5 $, 其他参数为$ \Omega_c/\gamma=2.0 $, $ \delta/\gamma=0 $, $ V/\gamma=0.6 $.
Fig. 3. Fidelity $ F(t) = {{\rm{Tr}}}[\sqrt{\rho(t)\rho_ {\rm{eff}}(t)}] $ in 50 random initial states. red open circles、red filled circles and black squares on the x axis correspond the time scales $ \tau'', \tau', \tau $ respectively.The parameters are (a) $ \Omega_p/\gamma=0.2 $; (b) $ \Omega_p/\gamma= $$ 0.5 $, and other parameters are $ \Omega_c/\gamma=2.0 $, $ \delta/\gamma=0 $, $ V/\gamma=0.6 $.
图 4 有效动力学的稳态与真实稳态的差异 $ ||\delta \rho_{ss}|| = $$ ||\rho_{ss}-\rho_{ss}^ {\rm{eff}}|| $随(a) $ \Omega_p $和(b) δ的变化, 其他参数同图3.
Fig. 4. Difference between the steady state of the effective dynamics and the real steady state $ ||\delta \rho_{ss}|| = ||\rho_{ss}-\rho_{ss}^ {\rm{eff}}|| $ as a function of parameters (a) $ \Omega_p $ and (b) δ, other parameters are the same as Fig. 3.
图 5 不同初始态在慢变子空间的动力学演化. 图中黑色的小点表示子空间中的任意纯态, 绿色圆点表示系统最终的稳态, 红色、蓝色、橙色和绿色曲线表示从几种不同的初始态开始的动力学过程, 参数设置为 (a) $ \Omega_p/\gamma=0.1, \Omega_c/\gamma =2, \delta/\gamma =0 $, $ V/\gamma=0.6 $; (b) $ \Omega_p/\gamma=0.6, \Omega_c/\gamma =2, \delta/\gamma =0 $, $ V/\gamma=0.6 $.
Fig. 5. Dynamical evolution of different initial states in the metastabel subspace. The black dots indicate arbitrary pure states in the subspace, the green dot indicates the steady state of the system, and the red, blue, orange and green curves indicate the dynamical processes starting from several different initial states. The parameters are (a) $ \Omega_p/\gamma=0.1, \Omega_c/\gamma =2, \delta/\gamma =0 $, $ V/\gamma=0.6 $; (b) $ \Omega_p/\gamma=0.6, \Omega_c/\gamma =2, \delta/\gamma =0 $, $ V/\gamma=0.6 $.
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