少数自由度导致的退相干和量子—经典过渡
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摘要
本文研究了少数自由度导致的退相干和量子-经典过渡。研究结果表明：与一个粒子的相互作用能够破坏系统的量子相干性,使得系统的量子行为过渡到经典极限。
一.本文研究了一个质量非常小的粒子导致的退相干。其中,环境和系统都是kicked rotor模型。kicked rotor经典混沌运动的特征是经典扩散,即动量平方的系综平均值随时间线性增长。由于量子相干性抑制波函数在动量空间的扩散,量子动量平方的平均值随时间的增长趋近于饱和值。这个现象就是动力学局域化。本论文研究的情况是：第二个转子的质量(m2)远远小于第一个的质量(m1)。它们之间的相互作用使得第二个转子的经典运动处于强混沌状态。在这种情况下,这个质量非常小的转子更相当于噪声。本文的研究结果表明：这种量子噪声有效得促进了系统的退相干。随着耦合强度的增加,系统的量子行为由局域化状态逐渐过渡到经典扩散。在这个转变过程中,两个转子之间的纠缠增强。本论文对两种不同的耦合势的数值计算都证明了相同的结果。
二.本文研究了对经典运动几乎没有影响的相互作用导致的退相干。本文以两个耦合的kicked rotor模型为例,研究了耦合强度(ε)和第二个转子的质量(m2)与有效普朗克常数(h)成比例变化的情况。在h减小的过程中,耦合强度逐渐减弱,第二个转子也逐渐消失。在半经典临域(h→0),第一个转子的经典运动几乎不受外界影响。本文的研究表明：随着h的减小,即使第二个转子逐渐消失,它与第一个转子的纠缠确逐渐增强。这种很强的纠缠能够抑制系统波函数的相干性,导致量子-经典过渡。
三.本文研究了无限深势阱内少数自由度导致的退相干。模型是：无限深势阱内以排斥势相耦合的两个kicked particle。排斥势是物理系统中普遍存在的相互作用。随着第二个粒子质量(m2)的减小,它对第一个粒子经典运动的影响也减小。当它们的质量相差几个数量级(m2《m1)时,第一个粒子经典运动几乎不受影响。当m2非常小时,第一个粒子的排斥力使得这个粒子的经典运动处于强混沌状态。第二个粒子的混沌运动有效得促进系统的退相干。本文研究了第一个粒子的量子-经典过渡在参数h和m2空间的相图。结果表明：系统的量子-经典过渡出现三种不同的区间：局域化,过渡区间和经典扩散。本文的数值结果表明：环境自由度的混沌运动使得系统的量子态与局域化状态之间的距离以指数函数形式衰减。
We investigate the quantum to classical transition in chaotic systems. The typical character of classical chaos is the exponential deviation of nearby trajectories. Such exponential dependence on initial conditions is not exhibited by quantum dynamics. Due to unitary time evolution, the inner product of two quantum states remains the same as the initial value. This demonstrates that the distance between two initial states is unchanged during quantum evolutions. Such stability of quantum behaviors is very different from classically chaotic dynamics. Decoherence theory gives an reasonable explanation of quantum to classical transition. The unavoidable interaction between system and environment makes entanglement between them. Such entanglement suppresses the quantum coherence of the system, and consequently leads to the quantum to classical transition.
We investigate the decoherence caused by an environment consisting of only one degree of freedom. We are interested in the case that the mass of the environment is much smaller than that of the system. Our results show that the influence from a small weightless particle makes the decoherence of the system, even when the classical motion is almost unaffected.
We firstly investigate the quantum to classical transition in a system of two coupled kicked rotors. In this system, the mass of the second kicked rotor is smaller than that of the first one by several orders. For large coupling strength, the entanglement between the two rotors rapidly becomes strong with time evolution. This process is accompanied by the disappearance of quantum coherence of the first rotor. The decoherence results in the emergence of classical diffusion from quantum dynamics.
We further investigate the case that both the mass m2 of the second rotor and the coupling strengthεchange proportionally with the effective Planck's constant h. With the decrease of h, both the second rotor and the coupling become small, so that the classical diffusion of the first rotor is almost unaffected. In the limit h→0, this small weightless part is strongly entangled with the first rotor. This makes decoherence and the quantum to classical transition of the first rotor.
The last system which we investigated consists of two kicked particles in an infinite square wall. They are coupled by repulsive potential. In classical dynamics, as the mass of particle 2 m2 decreases its effect on particle 1 decreases. When the mass of particle 2 is smaller than that of particle 1 by several orders m2<
一.本文研究了一个质量非常小的粒子导致的退相干。其中,环境和系统都是kicked rotor模型。kicked rotor经典混沌运动的特征是经典扩散,即动量平方的系综平均值随时间线性增长。由于量子相干性抑制波函数在动量空间的扩散,量子动量平方的平均值随时间的增长趋近于饱和值。这个现象就是动力学局域化。本论文研究的情况是：第二个转子的质量(m2)远远小于第一个的质量(m1)。它们之间的相互作用使得第二个转子的经典运动处于强混沌状态。在这种情况下,这个质量非常小的转子更相当于噪声。本文的研究结果表明：这种量子噪声有效得促进了系统的退相干。随着耦合强度的增加,系统的量子行为由局域化状态逐渐过渡到经典扩散。在这个转变过程中,两个转子之间的纠缠增强。本论文对两种不同的耦合势的数值计算都证明了相同的结果。
二.本文研究了对经典运动几乎没有影响的相互作用导致的退相干。本文以两个耦合的kicked rotor模型为例,研究了耦合强度(ε)和第二个转子的质量(m2)与有效普朗克常数(h)成比例变化的情况。在h减小的过程中,耦合强度逐渐减弱,第二个转子也逐渐消失。在半经典临域(h→0),第一个转子的经典运动几乎不受外界影响。本文的研究表明：随着h的减小,即使第二个转子逐渐消失,它与第一个转子的纠缠确逐渐增强。这种很强的纠缠能够抑制系统波函数的相干性,导致量子-经典过渡。
三.本文研究了无限深势阱内少数自由度导致的退相干。模型是：无限深势阱内以排斥势相耦合的两个kicked particle。排斥势是物理系统中普遍存在的相互作用。随着第二个粒子质量(m2)的减小,它对第一个粒子经典运动的影响也减小。当它们的质量相差几个数量级(m2《m1)时,第一个粒子经典运动几乎不受影响。当m2非常小时,第一个粒子的排斥力使得这个粒子的经典运动处于强混沌状态。第二个粒子的混沌运动有效得促进系统的退相干。本文研究了第一个粒子的量子-经典过渡在参数h和m2空间的相图。结果表明：系统的量子-经典过渡出现三种不同的区间：局域化,过渡区间和经典扩散。本文的数值结果表明：环境自由度的混沌运动使得系统的量子态与局域化状态之间的距离以指数函数形式衰减。
We investigate the quantum to classical transition in chaotic systems. The typical character of classical chaos is the exponential deviation of nearby trajectories. Such exponential dependence on initial conditions is not exhibited by quantum dynamics. Due to unitary time evolution, the inner product of two quantum states remains the same as the initial value. This demonstrates that the distance between two initial states is unchanged during quantum evolutions. Such stability of quantum behaviors is very different from classically chaotic dynamics. Decoherence theory gives an reasonable explanation of quantum to classical transition. The unavoidable interaction between system and environment makes entanglement between them. Such entanglement suppresses the quantum coherence of the system, and consequently leads to the quantum to classical transition.
We investigate the decoherence caused by an environment consisting of only one degree of freedom. We are interested in the case that the mass of the environment is much smaller than that of the system. Our results show that the influence from a small weightless particle makes the decoherence of the system, even when the classical motion is almost unaffected.
We firstly investigate the quantum to classical transition in a system of two coupled kicked rotors. In this system, the mass of the second kicked rotor is smaller than that of the first one by several orders. For large coupling strength, the entanglement between the two rotors rapidly becomes strong with time evolution. This process is accompanied by the disappearance of quantum coherence of the first rotor. The decoherence results in the emergence of classical diffusion from quantum dynamics.
We further investigate the case that both the mass m2 of the second rotor and the coupling strengthεchange proportionally with the effective Planck's constant h. With the decrease of h, both the second rotor and the coupling become small, so that the classical diffusion of the first rotor is almost unaffected. In the limit h→0, this small weightless part is strongly entangled with the first rotor. This makes decoherence and the quantum to classical transition of the first rotor.
The last system which we investigated consists of two kicked particles in an infinite square wall. They are coupled by repulsive potential. In classical dynamics, as the mass of particle 2 m2 decreases its effect on particle 1 decreases. When the mass of particle 2 is smaller than that of particle 1 by several orders m2<
引文
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