含时驱动的Dicke模型的混沌特性

Chaos propeties of the time-dependent driven Dicke model

依据Dicke量子相变首次被观测的实验装置,我们通过调节抽运激光强度来实现原子-场集体耦合强度的单频非绝热调制,即实现含时驱动的Dicke模型.当驱动的耦合强度为零时,系统回退到标准的Dicke模型.从刻画的庞加莱截面图观测到:在临近相变点附近系统由经典规则轨道向混沌变化,超辐射区对应着相空间的完全混沌;当静态耦合和驱动耦合同时存在时,系统显示出丰富的混沌运动.通过调节振荡频率,系统可在正常相区间从经典规则轨道变到混沌再变到经典规则轨道.

Now, many different approaches have been presented to study the different semi-classical models derived trom the Dicke Hamiltonian, which reflect a fact that the quantum-mechanical spin possesses no direct classical analog. The Hartree-Fock-type approximation is one of the widely used approaches, with which we derive the Heisenberg equations of motion for the system and replace the operators in these equations with the corresponding expectation values. In the present paper, we investigate the role of quantum phase transition in determining the chaotic property of the time- dependent driven Dicke model. The semi-classical Hamiltonian is derived by evaluating the expectation value of the Dicke Hamiltonian in a state, which is a product state of photonic and atomic coherent states. Based on the inverse of the relations between the position-momentum representation and the Bosonic creation-annihilation operators, the Hamiltonian is rewritten in the position-momentum representation and it undergoes a spontaneous symmetry-breaking phase transition, which is directly analogous to the quantum phase transition of the quantum system. In order to depict the Poincare sections, which are used to analyze the trajectories through the four-dimensional phase space, we give the equations of motion of system from the derivatives of the semi-classical Hamiltonian for a variety of different parameters and initial conditions. According to the Dicke quantum phase transition observed from the experimental setup, we study the effect of a monochromatic non-adiabatic modulation of the atom-field coupling in Dicke model （i.e., the driven Dicke model） on the system chaos by adjusting the pump laser intensity . The change from periodic track to chaotic figure reflects the quantum properties of the system, especially the quantum phase transition point, which is a key position for people to analyse the shift from periodic orbit to chaos. In an undriven case, the system reduces to the standard Dicke model. We discover from the Poincare sections that the system undergoes a change from the classical periodic orbit to a number of chaotic trajectories and in the superradiant phase area, the whole phase space is completely chaotic. When the static and driving coupling both exist, the system shows rich chaotic motion. The ground state properties are mainly determined by the static coupling, while the orbit of the system is adjusted by the driving coupling. If the static coupling is greater than the critical coupling, the system displays completely chaotic images in the Poincare sections, and the periodic orbits in the chaos can also be adjusted by the strong driving coupling. While the static coupling is less than the critical coupling, the system can also show the chaotic images by adjusting the driving coupling strength and oscillation frequency. Moreover, by tuning the oscillation frequency, the Poincare sections may change from the classical orbits to the chaos, and back to the classical orbits in the normal phase of the system.