凝聚态和原子物理中的多体现象

1、8.514: Many-body phenomena in condensed matter and atomic physics Problem Set #10 Due: 12/02/038.514: 凝聚态和原子物理中的多体现象 问题集 #10 提交日期：12/02/03Quantum tunneling and escape量子隧穿和逃逸Reading: S. Coleman, Aspects of Symmetry读物：S. Coleman, 对称方面的问题（这本书我没找到中文翻译版，直接翻译似乎不太好）1.Decay of a metastable state. 亚稳态的衰变Cons

2、ider a quantum particle tunneling through a barrier, from a metastable state into continuum. The WKB approach describes motion under barrier in terms of a classical trajectory in inverted potential -U (x). After replacing U (x) -U (x), one has to find a classical trajectory that rolls down a hill (o

3、riginally, the local minimum where particle is contained), bounces from the opposite side, and returns exactly to the initial point. The action S* on such bounce path determines the escape rate . 考虑穿过势垒的量子粒子隧穿，从亚稳态变到连续态。WKB方式描述了反势-U (x)中势垒下运动的经典轨道。将U (x) -U (x)，我们发现经典轨道下滚，在另一端反弹并回到原来的初始位置。反弹路径上的作用S*

4、 决定了逃逸率a) Justify this procedure using a path integral representation. Show that the bounce path action S* is identical to the WKB action . 使用路径积分表象判断这过程。证明反弹路径作用S* 与WKB作用相等。b) Find the bounce path describing quantum escape from a cubic parabola 找出描述三次抛物形量子逃逸的反弹路径 with particle initially confined at

5、 the local minimum其中，粒子最初被限制在区域最小处Figure 1: Classical bounce path for tunneling out of a metastable state.图1：亚稳态外隧穿的经典反弹路径2. Escape at finite temperature. Periodic bounce trajectories.有限温度下的逃逸。周期反弹轨道In the setting of Problem 1(Fig.1) at a finite temperature, escape may take place在问题1（图1）背景和有限温度下，逃逸可

6、能会发生 (i) via tunneling directly from the ground state, 直接从基态中隧穿，(ii) by Arrhenius thermal activation over the barrier, and, sometimes. 有时通过阿列纽斯热激发跨过势垒。(iii) in two stages, via a combination of tunneling and thermal activation. 在两个阶段中，结合隧穿和热激发。In the latter case, the particle is thermally excited to

7、a higher energy level En in the metastable well, and then tunnels through the barrier at an elevated energy. The benefit of the barrier at a higher energy being thinner and more transparent can be compensated by low thermal excitation probability.在第二种情况中，粒子在亚稳井中被势激发到更高的能级En，然后以提高的能量隧穿势垒。在高能量下势垒变得更薄和

8、更透明这一特点可以由低热激发几率来补偿。a) At finite temperature, the total tunneling probability can be written as 在有限温度下，总隧穿几率可以记为with = 1/T . Assuming that the levels En are closely spaced, so that the sum over n can be replaced by an integral, and ignoring the energy dependence of the prefactor An, show that the ma

9、ximum in the sum corresponds to the energy E for which其中 = 1/T 。假设能级En 是接近等间距的，因此对n求和可以用积分代替，并忽略能量对前因子An,的依赖，证明和的最大值对应能量E有Note that the quantity has a meaning of the classical travel time. Thus at a finite temperature the bounce paths that give the leading contribution to the escape rate are periodi

10、c functions of the imaginary time, .注意量有经典传播时间的意义，因此在有限温度下，对逃逸率作主要贡献的反弹路径是虚时间的周期函数b) Discuss the temperature dependence of the escape rate for the cubic parabola (1). Start at high temperature (small ) and show that in this case the saddle point trajectory is constant, x(t) = xmax where xmax is the

11、barrier top coordinate. Find the critical c at which the constant solution becomes unstable and turns into an oscillatory solution. Make a schematic plot of in W vs. , identify the Arrhenius and the tunneling regimes. 讨论三次抛物线（1）的逃逸率与温度的关系。由高温（小）开始，证明这情况下鞍点轨迹是常数x(t) = xmax ，其中，xmax 是势垒顶点坐标。找出常数解变得不稳定

12、并转变为振荡解的临界c。描出W- 示意图，识别阿列纽斯区域和隧穿区。3. Decay of supercurrent in a Josephson junction. Josephson结中的超流衰减Consider a Josephson junction shunted by an Ohmic resistor and biased by an external current I. The imaginary time action for this problem is考虑一个用欧姆电阻和外部偏流分路的约瑟夫结。本问题中的虚时间作用是with .其中，.a) Write down th

13、e saddle-point equation S / _= 0. By comparing with the classical equations of motion, , find the relation between m and and the junction capacitance and resistance. 写出鞍点方程S / _= 0。与经典运动方程，, ，作比较，找出m 和 的关系以及结电容值和电阻值。b) Classically, the supercurrent can flow as long as the effective potential U()-I h

14、as a local minimum. (For a stationary solution of Jc sin = I, since = 0, there is no voltage, and hence no dissipation.) This requires the external current to be in the range -Jc < I < Jc. In such a case, the decay of supercurrent is controlled by quantum and thermal fluctuations. The decay is

15、 described by the system (4) tunneling out a local minimum of U()-I. To determine the decay rate, one has to find a classical bounce trajectory in imaginary time, as it was done in problem 1 in the absence of dissipation, = 0. How are these results modified for tunneling in a dissipative system? 经典地

16、，只要有效势U()-I 存在区域最小值，超流可以流动（对Jc sin = I的定态解，由于 = 0，没有电压，因此没有耗散）这要求外部电流在-Jc < I < Jc范围内。在这情况下，超流的衰减由量子和热涨落控制。衰减由系统（4）隧穿描述。为了确定衰减率，我们必须找出虚时间下没有耗散（ = 0）的经典反弹轨迹。这结果对耗散系统下的隧穿是如何被修正的？Caldeira and Leggett pointed out that the bounce solution can be constructed in an analytic form for the highly overda

17、mped regime, , where is the time of tunneling (3) In this case, by a dimensional estimate, one shows that the kinetic energy term m2 / 2 is small compared to the dissipative term and thus can be dropped. Further simplification can be achieved near the classical stability threshold, where the effecti

18、ve potential can be replaced by a cubic parabola (1) with a .Show that in the overdamped regime, the saddle point equation S / = 0 can be satisfied by a Lorentz function ansatzCaldeira和Legget指出反弹解可以在过阻尼区（，其中是隧穿（3）的时间）的解析形式来建立。在这情况下，用尺度估计，可以证明动能项m2 / 2是小于耗散项因此可以被除去。在经典的稳定性阈值附近可以获得进一步简化，那里有效势可以用三次抛物线（1）代替。证明在过阻尼区，鞍点方程S / = 0可以由Lorentz 函数方法来满足Find A and and evaluate the action. Compare with the result found above for = 0.找出A和并估计作用。与上述 = 0.所得的结果作比较。c) Consider the decay of an overdamped Josephson junction