弯曲时空基本量子系统的动力学演化、几何相和量子纠缠
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摘要
量子信息科学的发展,推动了人们对各种可应用于量子信息处理的物理体系的广泛研究。然而,由于量子系统与环境之间存在着不可避免的相互作用,任何真实的量子系统都不可能是孤立系统。不同的环境可能会导致量子系统的动力学演化行为完全不同。因此,通过研究开放量子系统的动力学演化行为,我们可以获得与其耦合的环境的相关信息。在本文中,我们利用开放量子系统理论研究了一个基本量子系统的动力学演化,并讨论了其几何相和量子纠缠行为,得到了以下主要结论：
1.我们研究了Schwarzschild黑洞外部固定位置处与Unruh真空中涨落电磁场耦合的两能级原子的动力学演化问题。通过求解这个两能级原子的主方程,我们分析了其跃迁率和最终所达到的稳态。结果表明,在经历了足够长时间的演化之后,原子会热化到一个与初态无关,而与其跃迁频率有关的热混合态,这一行为与一个稳态非热平衡环境中电介质附近原子的动力学演化行为非常类似。因此,我们的结果一方面说明了Unruh真空非热平衡的本性,另一方面也表明,原则上可以在实验室中利用人工超材料构建稳态非热平衡电磁环境来模拟黑洞Hawking辐射的某些特征。
2.以开放量子系统的动力学演化为基础,我们进一步研究了一个开放两能级原子的几何相。我们分别考虑了Minkowski真空中的匀加速原子、理想无穷大全反射平面附近的惯性原子,以及Schwarzschild黑洞外部的静止原子这三种情形。对于Minkowski真空中匀加速两能级原子,我们发现其几何相存在由加速度带来的修正项,因此原则上可以利用干涉实验通过测量惯性原子和加速原子的几何相位差来验证Unruh效应。对于理想无穷大全反射平面附近的情形,原子的几何相与其到边界的距离有关,这是由于边界的存在使得真空中涨落电磁场的模式发生了改变。根据这一特点,我们提出了利用原子干涉实验测量到边界距离不同的惯性原子之间的几何相位差来直接探测被反射边界所改变的电磁场真空涨落的方案。对于Schwarzschild黑洞外部的情形,我们分别讨论了与Boulware, Unruh和Hartle-Hawking三种真空标量场耦合的静止原子的几何相。我们发现,与平直时空的结果相比,Schwarzschild黑洞外部原子的几何相存在由时空曲率反散射和Hawking辐射带来的修正项,原则上为我们观测弯曲时空量子效应提供了新的思路。
3.在单原子动力学演化研究的基础上,我们进一步研究了弯曲时空中两原子系统的动力学演化,并探讨了它们之间量子纠缠产生问题。首先,我们讨论了与Schwarzschild时空中真空标量场耦合的静止两原子系统平衡态的量子纠缠,从一个新的角度阐释了黑洞的Hawking辐射。之后,我们研究了与de Sitter不变真空中无质量共形耦合标量场相互作用的静止两原子系统初始时刻的量子纠缠产生条件以及平衡态的量子纠缠。我们的结果表明,尽管de Sitter时空中单个原子的辐射行为与平直时空热库中原子的行为相同,但是两原子之间量子纠缠的产生条件不同,因此原则上可以依此区分这两种不同的环境。
With the rapid development of quantum information science, physical systems which can be applied in quantum information processing have been extensively studied. However, any real quantum system has to be an open system because of its inevitable coupling to the external environment. Different types of environment may lead to distinct dynamical behaviors of quantum systems. Therefore, one may draw information on the nature of the environment from the dynamics of the open quantum system. In this dissertation, we study the dynamical behaviors, the geometric phase, and the quantum entanglement of an elementary quantum system. The main conclusions come as follows:
1. We investigate the dynamics of a radially polarizable two-level atom in multipolar coupling to fluctuating electromagnetic fields in the Unruh vacuum which is placed at a fixed radial distance outside a Schwarzschild black hole. With the help of the master equation, we analyze the transition rates between atomic energy levels and the steady state the atom is driven to. It is shown that, regardless of its initial state, the atom always thermalizes towards a steady thermal state at an effective temperature which depends on the transition frequency of the atom. This counterintuitive behavior is, however, in close analogy to what happens for a two-level atom in a stationary environment out of thermal equilibrium near a dielectric body of certain geometry and dielectric permittivity. On one hand, our result is a reflection of the fact that the Unruh vacuum exhibits a nonequilibrium nature. On the other hand, it suggests, in principle, a possibility to verify the peculiar features of the Hawking radiation by observing the dynamical behaviors for a two-level atom in tabletop experiments using engineered metamaterials with desired dielectric properties and superconducting circuits for an experimental implementation of two-level atoms.
2. Based on the dynamics of open quantum systems, we study the geometric phase of an open two-level atom. We consider a uniformly accelerated one in the Minkowski vacuum, an inertial one near an infinite ideal reflecting boundary, and a static one outside a Schwarzschild black hole. For a uniformly accelerated atom, it is found that there is a correction to the geometric phase due to the acceleration, and this phase variation can in principle be observed via atomic interferometry between the accelerated atom and an inertial one, thus providing an evidence of the Unruh effect. For an inertial atom whose trajectory is parallel to a reflecting boundary, we find that the geometric phase is position dependent as a result of the presence of the boundary which modifies the quantum fluctuations, hence a possible way is proposed to detect vacuum fluctuations in experiments involving geometric phase. For a static atom outside a Schwarzschild black hole, we consider the atom coupled to a bath of fluctuating scalar fields in the Boulware, Unruh and Hartle-Hawking vacua respectively. We find the geometric phase is corrected by the presence of the backscattering of spacetime curvature and the Hawking radiation. So, a measurement of the change of the geometric phase as opposed to that in a flat spacetime can in principle demonstrate the existence of the Hawking radiation.
3. Further, we have investigated the dynamics of open two-atom systems and the entanglement generation. First, we discuss the asymptotic entanglement of two mutually independent two-level atoms placed at a fixed radial distance outside a Schwarzschild black hole, and demonstrate the Hawking effect from a new perspective. Then, we study the entanglement generation between two mu-tually independent static two-level atoms in de Sitter spacetime. It is shown that, although a single atom behaves as if there were a thermal bath, the conditions for entanglement generation are different for the two situations. So, in principle, one can tell whether he is in a thermal bath or in de Sitter space by checking the entanglement creation between two initially separable static atoms in certain circumstances.
1.我们研究了Schwarzschild黑洞外部固定位置处与Unruh真空中涨落电磁场耦合的两能级原子的动力学演化问题。通过求解这个两能级原子的主方程,我们分析了其跃迁率和最终所达到的稳态。结果表明,在经历了足够长时间的演化之后,原子会热化到一个与初态无关,而与其跃迁频率有关的热混合态,这一行为与一个稳态非热平衡环境中电介质附近原子的动力学演化行为非常类似。因此,我们的结果一方面说明了Unruh真空非热平衡的本性,另一方面也表明,原则上可以在实验室中利用人工超材料构建稳态非热平衡电磁环境来模拟黑洞Hawking辐射的某些特征。
2.以开放量子系统的动力学演化为基础,我们进一步研究了一个开放两能级原子的几何相。我们分别考虑了Minkowski真空中的匀加速原子、理想无穷大全反射平面附近的惯性原子,以及Schwarzschild黑洞外部的静止原子这三种情形。对于Minkowski真空中匀加速两能级原子,我们发现其几何相存在由加速度带来的修正项,因此原则上可以利用干涉实验通过测量惯性原子和加速原子的几何相位差来验证Unruh效应。对于理想无穷大全反射平面附近的情形,原子的几何相与其到边界的距离有关,这是由于边界的存在使得真空中涨落电磁场的模式发生了改变。根据这一特点,我们提出了利用原子干涉实验测量到边界距离不同的惯性原子之间的几何相位差来直接探测被反射边界所改变的电磁场真空涨落的方案。对于Schwarzschild黑洞外部的情形,我们分别讨论了与Boulware, Unruh和Hartle-Hawking三种真空标量场耦合的静止原子的几何相。我们发现,与平直时空的结果相比,Schwarzschild黑洞外部原子的几何相存在由时空曲率反散射和Hawking辐射带来的修正项,原则上为我们观测弯曲时空量子效应提供了新的思路。
3.在单原子动力学演化研究的基础上,我们进一步研究了弯曲时空中两原子系统的动力学演化,并探讨了它们之间量子纠缠产生问题。首先,我们讨论了与Schwarzschild时空中真空标量场耦合的静止两原子系统平衡态的量子纠缠,从一个新的角度阐释了黑洞的Hawking辐射。之后,我们研究了与de Sitter不变真空中无质量共形耦合标量场相互作用的静止两原子系统初始时刻的量子纠缠产生条件以及平衡态的量子纠缠。我们的结果表明,尽管de Sitter时空中单个原子的辐射行为与平直时空热库中原子的行为相同,但是两原子之间量子纠缠的产生条件不同,因此原则上可以依此区分这两种不同的环境。
With the rapid development of quantum information science, physical systems which can be applied in quantum information processing have been extensively studied. However, any real quantum system has to be an open system because of its inevitable coupling to the external environment. Different types of environment may lead to distinct dynamical behaviors of quantum systems. Therefore, one may draw information on the nature of the environment from the dynamics of the open quantum system. In this dissertation, we study the dynamical behaviors, the geometric phase, and the quantum entanglement of an elementary quantum system. The main conclusions come as follows:
1. We investigate the dynamics of a radially polarizable two-level atom in multipolar coupling to fluctuating electromagnetic fields in the Unruh vacuum which is placed at a fixed radial distance outside a Schwarzschild black hole. With the help of the master equation, we analyze the transition rates between atomic energy levels and the steady state the atom is driven to. It is shown that, regardless of its initial state, the atom always thermalizes towards a steady thermal state at an effective temperature which depends on the transition frequency of the atom. This counterintuitive behavior is, however, in close analogy to what happens for a two-level atom in a stationary environment out of thermal equilibrium near a dielectric body of certain geometry and dielectric permittivity. On one hand, our result is a reflection of the fact that the Unruh vacuum exhibits a nonequilibrium nature. On the other hand, it suggests, in principle, a possibility to verify the peculiar features of the Hawking radiation by observing the dynamical behaviors for a two-level atom in tabletop experiments using engineered metamaterials with desired dielectric properties and superconducting circuits for an experimental implementation of two-level atoms.
2. Based on the dynamics of open quantum systems, we study the geometric phase of an open two-level atom. We consider a uniformly accelerated one in the Minkowski vacuum, an inertial one near an infinite ideal reflecting boundary, and a static one outside a Schwarzschild black hole. For a uniformly accelerated atom, it is found that there is a correction to the geometric phase due to the acceleration, and this phase variation can in principle be observed via atomic interferometry between the accelerated atom and an inertial one, thus providing an evidence of the Unruh effect. For an inertial atom whose trajectory is parallel to a reflecting boundary, we find that the geometric phase is position dependent as a result of the presence of the boundary which modifies the quantum fluctuations, hence a possible way is proposed to detect vacuum fluctuations in experiments involving geometric phase. For a static atom outside a Schwarzschild black hole, we consider the atom coupled to a bath of fluctuating scalar fields in the Boulware, Unruh and Hartle-Hawking vacua respectively. We find the geometric phase is corrected by the presence of the backscattering of spacetime curvature and the Hawking radiation. So, a measurement of the change of the geometric phase as opposed to that in a flat spacetime can in principle demonstrate the existence of the Hawking radiation.
3. Further, we have investigated the dynamics of open two-atom systems and the entanglement generation. First, we discuss the asymptotic entanglement of two mutually independent two-level atoms placed at a fixed radial distance outside a Schwarzschild black hole, and demonstrate the Hawking effect from a new perspective. Then, we study the entanglement generation between two mu-tually independent static two-level atoms in de Sitter spacetime. It is shown that, although a single atom behaves as if there were a thermal bath, the conditions for entanglement generation are different for the two situations. So, in principle, one can tell whether he is in a thermal bath or in de Sitter space by checking the entanglement creation between two initially separable static atoms in certain circumstances.
引文
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