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Phase Change during a Cyclic Quantum Evolution
循环量子过程中的相位变化
VOLUME 58.NUMBER 16 PHYSICAL REVIEW LETTERS 20 APRIL 1987 (Received 29 December 1986)PACS numbers: 03.65-w
Y. Aharonov and J. Anandan，Department of Physics and Astronomy, University of South Carolina卡罗来纳州, Columbia哥伦比亚, South Carolina 29208
A new geometric phase factor is defined for any cyclic evolution of a quantum system.
一个新的几何相因子被定义为任意一个量子系统的循环演化.
This is independent of the phase factor relating the initial- and final-state vectors and the Hamiltonian, for a given projection of the evolution on the projective space of rays of the Hilbert space.
它是关于投射Hilbert空间的初末态和Hamilton量的独立的相因子。
Some applications, including the Aharonov-Bohm effect, are considered for the special case of adiabatic evolution, this phase factor is a gauge-invariant generalization of the one found by Berry.
一些应用，包括AB效应，被认为是绝热演化，即由Berry发现的、作为一个规范不变量的推广的相因子的特殊情况。
A type of evolution of a physical system which is often of interest in physics is one in which the state of the system returns to its original state after an evolution.
物理学中感兴趣的往往是经过演化后回到初始状态的那类系统。
We shall call this a cyclic evolution.
我们称它为循环演化。
An example is periodic motion, such as the precession of a particle with intrinsic spin and magnetic moment in a constant magnetic field.
一个例子是周期运动，如一个带有自旋和磁矩的粒子在恒定磁场中的进动。
Another example is the adiabatic evolution of a quantum system whose Hamiltonian H returns to its original value and the state evolves as an eigenstate of the Hamiltonian and returns to its original state.
另一个例子是Hamilton量回到其初始值，且Hamilton量的本征态回到初始状态的量子系统的绝热演化
A third example is the splitting and recombination of a beam so that the system may be regarded as going backwards in time along one beam and returning along the other beam to its original state at the same time.
第三个例子是光束的分离和重组。这一系统可被视为沿着一束光去，同时沿着另一束光返回到初始状态？
Now, in quantum mechanics, the initial- and final-state vectors of a cyclic evolution are related by a phase factor [tex]e^{i\phi } [/tex], which can have observable consequences.
现在，在量子力学中，循环演化的初始末态矢量就被可观察的相因子[tex]e^{i\phi } [/tex]联系起来
An example, which belongs to the second category mentioned above, is the rotation of a fermion wave function by 2π rad by adiabatic rotation of a magnetic field through 2π rad so that φ=±π.
一个例子，它属于上述第二类，是由磁场旋转2π弧度的绝热转动引起费米子波函数的2π转动，使得φ=±π
Recently, Berry has shown that when H, which is a function of a set of parameters [tex]R^i[/tex] , undergoes adiabatic evolution along a closed curve Γ in the parameter space, then a state that remains an eigenstate of H(R) corresponding to a simple eigenvalue [tex]E_n (R)[/tex] develops a geometrical phase [tex]\gamma _n[/tex] which depends only on Γ.
最近Berry表明，当一个参数[tex]R^i[/tex] 的集合的函数H沿参数空间中的封闭曲线Γ演化时，则原来的一个平凡的的本征值[tex]E_n (R)[/tex]对应的本征态H(R)将会产生一个只依赖于Γ的几何相因子[tex]\gamma _n[/tex]
Simon has given an interpretation of this phase as due to holonomy in a line bundle over the parameter space.
Simon解释说，这一相因子的存在是由于参数空间上线丛的和乐（群）。
Anandan and Stodolsky have shown how the Berry phases for the various eigenspaces can be obtained from the holonomy in a vector bundle.
Anandan和Stodolsky给出了如何从一个向量丛上的和乐获得各种本征空间的Berry相位的方法。
For the adiabatic motion of spin, this is determined by a rotation angle α, due to the parallel transport of a Cartesign frame with one axis along the spin direction, which contains the above-mentioned rotation by 2π radians as a special case.
对于绝热自旋运动，这是由笛卡尔坐标系和一个沿自旋方向的轴的平行移动引起的一个旋转角α决定的，其中包含上述2π弧度的转动的特殊情况。
The result of a recent experiment to observe Berry's phase for light can also be understood as a rotation of the plane of polarization by this angle α.
最近有一项观察光的Berry相位的实验，也可以理解为由这个角α引起的偏振面的一个旋转
In this Letter, we consider the phase change for all cyclic evolutions which contain the three examples above as special cases.
在这篇论文中，我们考虑了包含上述三个例子作为特殊情况的所有循环演化的相位变化
We show the existence of a phase associated with cyclic evolution, which is universal in the sense that it is the same for the infinite number of possible motions along the curves in the Hilbert space [tex]{\cal H}[/tex] which project to a given closed curve [tex]{\hat C}[/tex] in the projective Hilbert space [tex]\wp[/tex] of rays of [tex]{\cal H}[/tex] and the possible Hamiltonians H(t) which propagate the state along these curves.
我们证明了与循环演化相关的相位的存在性。对于沿着Hilbert空间[tex]{\cal H}[/tex]的曲线，即投射Hilbert空间[tex]\wp[/tex]中的一条给定的封闭曲线[tex]{\hat C}[/tex]的无数可能的运动与沿着这些曲线传播状态的可能的Hamilton量H（t）来说，它们在普遍的意义上是相同的。
This phase tends to the Berry phase in the adiabatic limit if H(t)=H[R(t)] is chosen accordingly. For an electrically charged system, we formulate this phase gauge invariantly and show that the Aharonov-Bohm (AB) phase due to the electromagnetic field may be regarded as a special case.
如果选择H(t)≡H[R(t)]，则这一相位在绝热极限下趋于Berry相位。对于带电系统，我们发现这一相位具有规范不变性，并且可以由此导出AB相位，因为电磁场U(1)可作为（一般规范场）的一种特殊情况
This generalizes the gauge-noninvariant result of Berry that the AB phase due to a static magnetic field is a special case of his phase.
这一发现推广了Berry相位规范不变性的结果，使得AB相由于静磁场而成为新相位的特殊情况
This also removes the mystery of why the AB phase, even in this special case, should emerge from Berry's expression even though the former is independent of this adiabatic approximation.
这也消除了为什么AB相，即使在这种特殊情况下，应该走出Berry的表达式的神秘，尽管前者在这个绝热近似下是独立的
Suppose that the normalized state [tex]\left| {\psi (t)} \right\rangle \in {\cal H}[/tex] evolves according to the Schrödinger equation
[tex]H(t)\left| {\psi (t)} \right\rangle = i\hbar (d/dt)\left| {\psi (t)} \right\rangle[/tex] (1)
Such that [tex]\left| {\psi (\tau )} \right\rangle = e^{i\phi } \left| {\psi (0)} \right\rangle[/tex] ,φ real
假设标准态 根据Schrödinger方程（1）演化，以至于[tex]\left| {\psi (\tau )} \right\rangle = e^{i\phi } \left| {\psi (0)} \right\rangle[/tex]，其中φ为实数。
Let [tex]\Pi :{\cal H} \to \wp[/tex] be the projection map defined by
[tex]\Pi (\left| \psi \right\rangle ) = \{ \left| {\psi '} \right\rangle :\left| {\psi '} \right\rangle = c\left| \psi \right\rangle \}[/tex]
c is a complex number.
令[tex]\Pi :{\cal H} \to \wp[/tex]为由
[tex]\Pi (\left| \psi \right\rangle ) = \{ \left| {\psi '} \right\rangle :\left| {\psi '} \right\rangle = c\left| \psi \right\rangle \}[/tex]
定义的投影图，其中c是一个复数。
Then [tex]\left| {\psi (t)} \right\rangle[/tex] defines a curve [tex]C:[0,\tau ] \to {\cal H}[/tex] with [tex]\hat C \equiv \Pi (C)[/tex] being a closed curve in [tex]\wp [/tex] .
那么[tex]\left| {\psi (t)} \right\rangle[/tex]定义了一条曲线[tex]C:[0,\tau ] \to {\cal H}[/tex]使得[tex]\hat C \equiv \Pi (C)[/tex]成为[tex]\wp [/tex]中一条封闭曲线。
Conversely given any such curve C, we can define a Hamiltonian function H (t) so that (1) is satisfied for the corresponding normalized [tex]\left| {\psi (t)} \right\rangle[/tex].
反之，给定任何此类曲线C，我们可以定义一个Hamilton函数H（t），使（1）满足相应的标准的[tex]\left| {\psi (t)} \right\rangle[/tex]
Now define
[tex]\left| {\tilde \psi (t)} \right\rangle = e^{ - if(t)} \left| {\psi (t)} \right\rangle[/tex]
such that f(τ)-f(0)=φ. Then
[tex]\left| {\tilde \psi (\tau )} \right\rangle = \left| {\tilde \psi (0)} \right\rangle[/tex]
and from (1),
[tex]- \frac{{df}}{{dt}} = \frac{1}{\hbar }\left\langle {\psi (t)} \right|H\left| {\psi (t)} \right\rangle - \left\langle {\tilde \psi (t)} \right|i\frac{d}{{dt}}\left| {\tilde \psi (t)} \right\rangle[/tex](2)
现在定义
[tex]\left| {\tilde \psi (t)} \right\rangle = e^{ - if(t)} \left| {\psi (t)} \right\rangle[/tex]
使得f(τ)-f(0)=φ.接着，由
[tex]\left| {\tilde \psi (\tau )} \right\rangle = \left| {\tilde \psi (0)} \right\rangle[/tex]
和（1），可得
[tex]- \frac{{df}}{{dt}} = \frac{1}{\hbar }\left\langle {\psi (t)} \right|H\left| {\psi (t)} \right\rangle - \left\langle {\tilde \psi (t)} \right|i\frac{d}{{dt}}\left| {\tilde \psi (t)} \right\rangle[/tex] （2）
Hence, if we remove the dynamical part from the phase φ by defining
[tex]\beta \equiv \phi + \hbar ^{ - 1} \int_0^\tau {\left\langle {\psi (t)} \right|H\left| {\psi (t)} \right\rangle dt}[/tex] (3)
It follows from (2) that
[tex]\beta = \int_0^\tau {\left\langle {\tilde \psi (t)} \right|i\frac{d}{{dt}}\left| {\tilde \psi (t)} \right\rangle dt}[/tex] (4)
因此，如果我们通过定义
[tex]\beta \equiv \phi + \hbar ^{ - 1} \int_0^\tau {\left\langle {\psi (t)} \right|H\left| {\psi (t)} \right\rangle dt}[/tex]
把动力学部分从总相位φ中分离出来，那么根据（2）式可得
[tex]\beta = \int_0^\tau {\left\langle {\tilde \psi (t)} \right|i\frac{d}{{dt}}\left| {\tilde \psi (t)} \right\rangle dt}[/tex] （4）
Now, clearly, the same [tex]\left| {\tilde \psi (t)} \right\rangle[/tex] can be chosen for every curve C for which [tex]\Pi (C) = \hat C[/tex] , by appropriate choice of f(t).
现在，很显然，通过适当选择f(t)，相同的[tex]\left| {\tilde \psi (t)} \right\rangle[/tex]可以选为任意满足[tex]\Pi (C) = \hat C[/tex]的曲线C
Hence β, defined by (3), is independent of φ and H for a given curve [tex]{\hat C}[/tex] .
因此，对于任意给定的曲线[tex]{\hat C}[/tex]，由（3）式定义的β是独立于φ和H的。
Indeed, for a given [tex]{\hat C}[/tex] , H(t) can be chosen so that the second term in (3) is zero, which may be regarded as an alternative definition of β.
事实上，对于一条给定的曲线[tex]{\hat C}[/tex] ，可以选择使（3）式第二项为零的H(t)，这可作为β的另一种定义。
Also, from (4), β is independent of the parameter t of [tex]{\hat C}[/tex] , and is uniquely defined up to 2πn (n =integer).
另外，从（4）可知，β是独立于[tex]{\hat C}[/tex]的参数t的，是唯一定义多达2πn组（n =整数）。
Hence [tex]e^{i\beta }[/tex] is a geometric property of the unparametrized image of [tex]{\hat C}[/tex]in [tex]\wp[/tex] only.
因此[tex]e^{i\beta }[/tex]只是[tex]\wp[/tex]中[tex]{\hat C}[/tex]的一个非参数化图像的几何属性
Consider now a slowly varying H(t),with [tex]H(t)\left| {n(t)} \right\rangle = E_n (t)\left| {n(t)} \right\rangle[/tex] for a complete set [tex]\{ \left| {n(t)} \right\rangle \}[/tex] .
现在考虑一个缓慢变化的H(t)，完备集[tex]\{ \left| {n(t)} \right\rangle \}[/tex] 满足[tex]H(t)\left| {n(t)} \right\rangle = E_n (t)\left| {n(t)} \right\rangle[/tex]
If we write
[tex]\left| {\psi (t)} \right\rangle = \sum\limits_n {a_n (t)\exp ( - \frac{i}{\hbar }\int {E_n dt} )} \left| {n(t)} \right\rangle[/tex]
And use (1) and the time derivative of the eigenvector equation, we have
[tex]\dot a_m = - a_m \left\langle {m}
\mathrel{\left | {\vphantom {m {\dot m}}}
\right. \kern-\nulldelimiterspace}
{{\dot m}} \right\rangle - \sum\limits_{n \ne m} {a_n \frac{{\left\langle m \right|\dot H\left| n \right\rangle }}{{E_n - E_m }}\exp [\frac{i}{\hbar }\int {(E_m - E_n )dt} ]}[/tex]
Where the dot denotes time derivative.
如果我们写出
[tex]\left| {\psi (t)} \right\rangle = \sum\limits_n {a_n (t)\exp ( - \frac{i}{\hbar }\int {E_n dt} )} \left| {n(t)} \right\rangle[/tex]
并利用（1）式和本征矢方程的时间导数，我们就得到
[tex]\dot a_m = - a_m \left\langle {m}
\mathrel{\left | {\vphantom {m {\dot m}}}
\right. \kern-\nulldelimiterspace}
{{\dot m}} \right\rangle - \sum\limits_{n \ne m} {a_n \frac{{\left\langle m \right|\dot H\left| n \right\rangle }}{{E_n - E_m }}\exp [\frac{i}{\hbar }\int {(E_m - E_n )dt} ]}[/tex] (5)
其中矢量上面的“点”表示时间导数。
Suppose that
[tex]\sum\limits_{n \ne m} {\left| {\frac{{\hbar \left\langle m \right|\dot H\left| n \right\rangle }}{{(E_n - E_m )^2 }}} \right|} {\rm < < 1}[/tex] (6)
Then if [tex]a_n (0) = \delta _{mn}[/tex] ，the last term in (5) is negligible and the system would therefore continue as an eigenstate of H(t), to a good approximation.
假设
[tex]\sum\limits_{n \ne m} {\left| {\frac{{\hbar \left\langle m \right|\dot H\left| n \right\rangle }}{{(E_n - E_m )^2 }}} \right|} {\rm < < 1}[/tex]
那么，如果[tex]a_n (0) = \delta _{mn}[/tex] ，（5）式最后一项是微不足道的，因此该系统仍然是H（t）的本征态，这是一个良好的近似
In this adiabatic approximation, (5) yields
[tex]a_m (t) \approx \exp ( - \int {\left\langle {m}
\mathrel{\left | {\vphantom {m {\dot m}}}
\right. \kern-\nulldelimiterspace}
{{\dot m}} \right\rangle dt} )a_m (0)[/tex]
在这种绝热近似，由（5）可得
[tex]a_m (t) \approx \exp ( - \int {\left\langle {m}
\mathrel{\left | {\vphantom {m {\dot m}}}
\right. \kern-\nulldelimiterspace}
{{\dot m}} \right\rangle dt} )a_m (0)[/tex]
For a cyclic adiabatic evolution, the phase
[tex]i\int_0^\tau {\left\langle {m}
\mathrel{\left | {\vphantom {m {\dot m}}}
\right. \kern-\nulldelimiterspace}
{{\dot m}} \right\rangle dt}[/tex]
is independent of the chosen [tex]\left| {m(t)} \right\rangle[/tex] and Berry regarded this as a geometrical property of the parameter space of which H is a function.
对于一个循环绝热演化，相位
[tex]i\int_0^\tau {\left\langle {m}
\mathrel{\left | {\vphantom {m {\dot m}}}
\right. \kern-\nulldelimiterspace}
{{\dot m}} \right\rangle dt}[/tex]
是独立于[tex]\left| {m(t)} \right\rangle[/tex] 的选择的。Berry认为这是一个函数H的参数空间的几何性质。
But this phase is the same as (4) on our choosing [tex]\left| {\tilde \psi (t)} \right\rangle = \left| {m(t)} \right\rangle[/tex] in the present approximation.
但是，在目前的近似下，这一相位和我们的选择[tex]\left| {\tilde \psi (t)} \right\rangle = \left| {m(t)} \right\rangle[/tex]相同。
But β，defined by (3), does not depend on any approximation; so (4) is exactly valid.
但由（3）式定义的β不依赖于任何近似，所以（4）是非常有效的..
Moreover, [tex]\left| {\psi (t)} \right\rangle[/tex] need not be an eigenstate of H(t), unlike in the limiting case studied by Berry.
进一步，[tex]\left| {\psi (t)} \right\rangle[/tex]不必是H（t）的本征态，这和Berry研究的极限情况不同。
Also, the two examples below will show respectively that it is neither necessary nor sufficient to go around a (nontrivial) closed curve in parameter space in order to have a cyclic evolution, with our associated geometric phase β.
此外，下面的两个例子将分别显示，为了得到一个循环演化，及与我们相关的几何相位β，我们既无必要，也不足以左右参数空间的一条（平凡的）封闭曲线。
For these reasons, we regard β as a geometric phase associated with a closed curve in the projective Hilbert space and not the parameter space, even in the special case considered by Berry.
基于这些原因，我们把β看作与投射Hilbert空间（而不是参数空间）的一条封闭曲线相关的几何相位，甚至在Berry考虑的特殊情况也如此。
But given a cyclic evolution, an H(t) which generated this evolution can be found so that the adiabatic approximation is valid.
但给定一个循环演化，那么可以发现由这个演化产生的H（t）能够使绝热近似是有效的。
Then β can be computed with the use of the expression given by Berry in terms of the eigenstates of this Hamiltonian.
之后，β就可以由Berry给出的Hamilton量本征态的表达式来计算
We now consider two examples in which the phase βemerges naturally and is observable, in principle, even though the adiabatic approximation is not valid.
现在，我们来考虑两个这样的例子，在这里相位β是自然出现并且可观察的，原则上，它甚至适用于绝热近似无效的情况。
Suppose that a spin -1/2 particle with a magnetic moment is in a homogeneous magnetic field B along the z axis.
假设将一个自旋-1 /2，具有磁矩粒子放入沿z轴的均匀磁场B中。
Then the Hamiltonian in the rest frame is [tex]H_1 = - \mu B\sigma _z[/tex], where
\[
\sigma _z =
\left(
\begin{array}{*{20}c}
1 & 0 \\
0 & { - 1} \\
\end{array}
\right)
\]
在静止坐标系中，Hamilton量为 [tex]H_1 = - \mu B\sigma _z[/tex] 。其中
\[
\sigma _z =
\left(
\begin{array}{*{20}c}
1 & 0 \\
0 & { - 1} \\
\end{array}
\right)
\]
Also,
\[
\left| {\psi (t)} \right\rangle =
\left(
\begin{array}{*{20}c}
{\cos \frac{\theta }{2}} \\
{\sin \frac{\theta }{2}}
\end{array}
\right)
\]
So that
\[
\left| {\psi (t)} \right\rangle = \exp (\mu Bt\sigma _z \frac{i}
{\hbar })
\]
\[
=
\left(
\begin{array}{*{20}c}
\exp (\mu Bt\frac{i}{\hbar })\cos \frac{\theta }{2} \\
\exp (\mu Bt\frac{i}{\hbar })\sin \frac{\theta }{2}
\end{array}
\right)
\]
which corresponds to the spin direction being always at an angle θ to the z axis.
另外，
\[
\left| {\psi (t)} \right\rangle =
\left(
\begin{array}{*{20}c}
{\cos \frac{\theta }{2}} \\ {\sin \frac{\theta }{2}}
\end{array}
\right)
\]
使得
\[
\left| {\psi (t)} \right\rangle = \exp (\mu Bt\sigma _z \frac{i}
{\hbar })
\]
\[
=
\left(
\begin{array}{*{20}c}
\exp (\mu Bt\frac{i}{\hbar })\cos \frac{\theta }{2} \\ \exp (\mu Bt\frac{i}{\hbar })\sin \frac{\theta }{2}
\end{array}
\right)
\]
自旋方向到z轴夹角总是θ
This evolution is periodic with period [tex]\tau = \pi \hbar /\mu B[/tex].
这一演化是周期性的，周期为[tex]\tau = \pi \hbar /\mu B[/tex]
Then from (3), for each cycle, β=π(1-cosθ), up to the ambiguity of adding 2πn.
接着，由（3）可得，对于每一个循环，β=π(1-cosθ)，每次增加2πn
Hence, β is1/2 of the solid angle subtended by a curve traced on a sphere, by the direction of the spin state, at the center.
因此，β是对向球上一条曲线的立体角的1 /2，由自旋态的方向，在中心
This is like the Berry phase except that in the latter case (1)the solid angle is subtended by a curve traced by the magnetic field B'(t) which is large [i.e., [tex]\mu B'/\hbar > > \omega[/tex], the frequency of the orbit of B'(t)]so that the adiabatic approximation is valid, and (2) [tex]\left| {\psi (t)} \right\rangle[/tex] is assumed to be an eigenstate of this Hamiltonian.
这就好比除了Berry相位在后一种情况（1）对向磁场B'（t）追踪一条曲线的立体角是巨大的[即，[tex]\mu B'/\hbar > > \omega[/tex] ，其中ω为B’(t)的轨道频率]，这样绝热近似是有效的，并且（2）中[tex]\left| {\psi (t)} \right\rangle[/tex] 被假设为一本Hamilton量的本征态
Indeed, we may substitute such a Hamiltonian for the above [tex]H_1[/tex] , or add it to [tex]H_1[/tex] with [tex]\omega = 2\mu B/\hbar[/tex] , without changing β，in this approximation.
事实上，我们可以用上述[tex]H_1[/tex] 替代这样一个Hamilton量，或在不改变β这样的近似下用[tex]\omega = 2\mu B/\hbar[/tex]把它增加到[tex]H_1[/tex]
The spin state will also move through the same closed curve in the projective Hilbert space as above if the magnetic field [tex]B = (B_0 \cos \omega t,B_0 \sin \omega t,B_3 )[/tex] with
[tex]\cot \theta = (B_3 - \hbar \omega /2\mu )/B_0[/tex]
where [tex]B_0 \ne 0[/tex] .
如果磁场[tex]B = (B_0 \cos \omega t,B_0 \sin \omega t,B_3 )[/tex]满足
[tex]\cot \theta = (B_3 - \hbar \omega /2\mu )/B_0[/tex]
其中[tex]B_0 \ne 0[/tex]
那么自旋态还将上述射影希尔伯特空间中相同的封闭曲线移动
And β is the same for all such Hamiltonians.
对所有这样的Hamilton来说，β是相同的
This illustrates the statement earlier that β is the same for all curves C in H with the same [tex]\hat C \equiv \Pi (C)[/tex]
这说明了早些时候的声明中的所有β为H曲线具有相同的[tex]\hat C \equiv \Pi (C)[/tex]
Also,β may be interpreted as arising from the holonomy transformation, around the closed curve on the above sphere traced by the direction of the spin state, due to the curvature on this sphere, which is a rotation.
此外，β可被有趣地解释为转平行输运所产生的和乐，上述球的周围被自旋态的方向上的封闭曲线环绕，由于球的曲率，它是一个旋转，
By varying appropriately a magnetic field applied to the two arms of a neutron interferometer with polarized neutrons, it is possible to make the dynamical part of β[the last term in (3)] the same for the two beams.
通过应用极化中子中子干涉仪来适当改变磁场，它有可能使两束光的动力部分β[（3）最后一项]相同
Then the phase difference between the two beams is just the geometrical phase, which is observable in principle, from the interference pattern, even when the magnetic field is varied nonadiabatically.
然后，两束光之间的相位差别仅仅是几何相位，这在原则上是可以从干涉图案中观察到的，甚至当磁场非绝热变化时也成立
In particular, a phase difference of ±πrad would correspond to a 2π-rad rotation of the fermion wave function, which is thus observable.
特别是，±π弧度的相位差将对应费米子波函数的2π弧度的转动，因此它是可观察的
As our second example, suppose that the magnetic field is [tex]B(t) = B_0 + B_1 (t)[/tex] , where [tex]B_0[/tex]is constant and [tex]B_1 (t)[/tex] rotates slowly in a plane containing [tex]B_0[/tex] with [tex]\left| {B_1 (t)} \right| = \left| {B_0 } \right|[/tex] .
作为我们的第二个例子，假设磁场为[tex]B(t) = B_0 + B_1 (t)[/tex] ，其中[tex]B_0[/tex] 是常数，[tex]B_1 (t)[/tex]在包含[tex]B_0[/tex] （满足[tex]\left| {B_1 (t)} \right| = \left| {B_0 } \right|[/tex] ）的平面内缓慢转动
Suppose that at time t the angle between [tex]B_1[/tex] and [tex]B_0[/tex] is π-θ(t) and the spin state [tex]\left| {\psi (t)} \right\rangle[/tex] is in an approximate eigenstate of H(t)=μB·σ , where σ are the Pauli spin matrices.
假设t时刻[tex]B_1[/tex]与[tex]B_0[/tex] 间夹角为π-θ(t)，在绝热近似下自旋态[tex]\left| {\psi (t)} \right\rangle[/tex]是H(t)=μB·σ 的本征态，其中σ是Pauli自旋矩阵
For 0≤θ<<1 , the adiabatic condition (6) gives [tex]0 \le - \hbar \dot \theta /\mu B_0 \theta < < 1[/tex] ，assuming [tex]\dot \theta \le 0[/tex] .
Hence [tex]\theta > > \theta _0 \exp ( - \mu B_0 t/\hbar ) > 0[/tex]. So θ can never become zero.
令0≤θ<<1，绝热条件（6）给出[tex]0 \le - \hbar \dot \theta /\mu B_0 \theta < < 1[/tex] ，假定[tex]\dot \theta \le 0[/tex] 因此[tex]\theta > > \theta _0 \exp ( - \mu B_0 t/\hbar ) > 0[/tex] 所以θ永远不会变成零。
That is, if B(T)=0 for some T then the adiabatic approximation, as defined above, cannot be satisfied, regardless of how slowly [tex]B_1 (t)[/tex] rotates.
这说明，如果某些T使得B(T)=0，那么不管[tex]B_1 (t)[/tex] 多么缓慢的旋转，上面定义的绝热近似都不能满足
However, because of conservation of angular momentum, [tex]\left| {\psi (t)} \right\rangle[/tex] remains an eigenstate of H(t) even at t=T.
然而，由于角动量守恒，即使t=T，[tex]\left| {\psi (t)} \right\rangle[/tex]仍然是H(t)的本征态
But if θ changes monotonically then a level crossing occurs at the point of degeneracy (B=0) so that the energy eigenvalue corresponding to [tex]\left| {\psi (t)} \right\rangle[/tex] changes sign at t=T.
但是，如果θ单调变化，使得水平交叉发生在简并点（B=0），这样，相应于[tex]\left| {\psi (t)} \right\rangle[/tex] 的能量本征值在t = T处变化显著
For each rotation of [tex]B_1[/tex] by 2πrad，[tex]\left| \psi \right\rangle[/tex] rotates by πrad, so that the system returns to its original state after two rotations of B(t).
对于[tex]B_1[/tex]的每一个2π弧度旋转，[tex]\left| \psi \right\rangle[/tex] 的π弧度旋转，从而使系统经过B(t)的两次旋转返回到其原始状态
For this cyclic evolution, our β=π which can be seen from the fact that a spin-1/2 particle acquires a phase π during a rotation, or that the curve [tex]\hat C[/tex] on the projective Hilbert space, which is a sphere, is a great circle, subtending a solid angle 2π at the center.
对于这个循环演化，我们的β为π，这可以从以下事实看出：自旋为-1/2粒子在旋转后获得相位差π，或者说，投射希尔伯特空间上的曲线[tex]\hat C[/tex] ，在球面上是一个大圆，对位在中心的立体角2π
This example is similar to Berry's phase in that [tex]\left| {\psi (t)} \right\rangle[/tex] is always an eigenstate of H(t), even though Berry's prescription cannot be applied here because of the crossing of the point of degeneracy at which the adiabatic approximation breaks down.
这个例子类似于Berry相位，[tex]\left| {\psi (t)} \right\rangle[/tex]总是H(t)的本征态，即使Berry的方法在这里不能应用（因为对简并点处的绝热近似破坏了）
Consider now a system with electric charge q for which
[tex]H = H_k (p - (q/c)\hat A(t),R_i ) + q\hat A_0 (t)[/tex] in (1).
现在考虑一个带电荷q的系统，满足由（1）式定义的
[tex]H = H_k (p - (q/c)\hat A(t),R_i ) + q\hat A_0 (t)[/tex]
Here,
[tex]\left\langle x \right|\hat A_\mu (t)\left| {\psi (t')} \right\rangle = A_\mu (x,t)\psi (x,t')[/tex]
where [tex]A_\mu (x,t)[/tex] is the usual electromagnetic four-potential, and [tex]R_i[/tex] are some parameters.
这里，
[tex]\left\langle x \right|\hat A_\mu (t)\left| {\psi (t')} \right\rangle = A_\mu (x,t)\psi (x,t')[/tex]
其中[tex]A_\mu (x,t)[/tex] 是通常的四维电磁势，[tex]R_i[/tex]是一些参数
Under a gauge transformation,
[tex]\left| {\psi (t)} \right\rangle \to \exp [i(q/c)\hat \Lambda (t)]\left| {\psi (t)} \right\rangle[/tex],
[tex]\hat A_0 (t) \to \hat A_0 (t) - c^{ - 1} \partial \hat \Lambda (t)/\partial t[/tex]
And
[tex]H_k (t) \to \exp [i(q/c)\hat \Lambda (t)]H_k (t)\exp [ - i(q/c)\hat \Lambda (t)][/tex]
在规范变换 下，
[tex]\left| {\psi (t)} \right\rangle \to \exp [i(q/c)\hat \Lambda (t)]\left| {\psi (t)} \right\rangle[/tex],
[tex]\hat A_0 (t) \to \hat A_0 (t) - c^{ - 1} \partial \hat \Lambda (t)/\partial t[/tex]
并且
[tex]H_k (t) \to \exp [i(q/c)\hat \Lambda (t)]H_k (t)\exp [ - i(q/c)\hat \Lambda (t)][/tex]
As before, define
[tex]\left| {\tilde \psi (t)} \right\rangle = e^{ - if(t)} \left| {\psi (t)} \right\rangle[/tex]
如前所述，定义
[tex]\left| {\tilde \psi (t)} \right\rangle = e^{ - if(t)} \left| {\psi (t)} \right\rangle[/tex]，
If we require that [tex]\left| {\tilde \psi } \right\rangle[/tex] undergo the same gauge transformation as [tex]\left| {\psi (t)} \right\rangle[/tex]，f(t) is gauge invariant. Then, from (1)，
[tex]\frac{{df}}{{dt}} = \left\langle {\tilde \psi (t)} \right|\frac{d}{{dt}} - \frac{q}{\hbar }\hat A_0 (t)\left| {\tilde \psi (t)} \right\rangle - \frac{1}{\hbar }\left\langle {\psi (t)} \right|H_k (t)\left| {\psi (t)} \right\rangle[/tex] （7）
如果我们要求[tex]\left| {\tilde \psi } \right\rangle[/tex]像[tex]\left| {\psi (t)} \right\rangle[/tex] 一样经历相同的规范变换，f（t）是规范不变量，那么由（1）得
[tex]\frac{{df}}{{dt}} = \left\langle {\tilde \psi (t)} \right|\frac{d}{{dt}} - \frac{q}{\hbar }\hat A_0 (t)\left| {\tilde \psi (t)} \right\rangle - \frac{1}{\hbar }\left\langle {\psi (t)} \right|H_k (t)\left| {\psi (t)} \right\rangle[/tex] (7)
We consider now a cyclic evolution so that
[tex]\left| {\psi (\tau )} \right\rangle = e^{i\phi } \exp ( - \frac{{iq}}{\hbar }\int_0^\tau {\hat A_0 dt} )\left| {\psi (0)} \right\rangle[/tex]
Choose f(t) so that φ=f(τ)-f(0). Then
[tex]\left| {\tilde \psi (\tau )} \right\rangle = \exp ( - \frac{{iq}}{\hbar }\int_0^\tau {\hat A_0 dt} )\left| {\tilde \psi (0)} \right\rangle[/tex]
现在，我们考虑一个循环演化，使得
[tex]\left| {\psi (\tau )} \right\rangle = e^{i\phi } \exp ( - \frac{{iq}}{\hbar }\int_0^\tau {\hat A_0 dt} )\left| {\psi (0)} \right\rangle[/tex]
选择f(t)使φ=f(τ)-f(0).于是，
[tex]\left| {\tilde \psi (\tau )} \right\rangle = \exp ( - \frac{{iq}}{\hbar }\int_0^\tau {\hat A_0 dt} )\left| {\tilde \psi (0)} \right\rangle[/tex]
So we now define the gauge –invariant generalization of (3) as
[tex]\beta \equiv \phi + \frac{1}{\hbar }\int_0^\tau {\left\langle {\psi (t)} \right|H_k (t)\left| {\psi (t)} \right\rangle dt}[/tex] (8)
因此，我们现在定义（3）式的规范不变量的推广为
[tex]\beta \equiv \phi + \frac{1}{\hbar }\int_0^\tau {\left\langle {\psi (t)} \right|H_k (t)\left| {\psi (t)} \right\rangle dt}[/tex] (8)
Which on use of (7) gives
[tex]\beta = \int_0^\tau {\left\langle {\tilde \psi (t)} \right|\frac{d}{{dt}} - \frac{q}{\hbar }\hat A_0 (t)\left| {\tilde \psi (t)} \right\rangle dt}[/tex] (9)
由（7）给出
[tex]\beta = \int_0^\tau {\left\langle {\tilde \psi (t)} \right|\frac{d}{{dt}} - \frac{q}{\hbar }\hat A_0 (t)\left| {\tilde \psi (t)} \right\rangle dt}[/tex] （9）
Here, [tex]\left| {\tilde \psi (\tau )} \right\rangle[/tex] is obtained by parallel transport of [tex]\left| {\tilde \psi (0)} \right\rangle[/tex], with respect to the electromagnetic connection, along the congruence of lines parallel to the time axis.
在这里，[tex]\left| {\tilde \psi (\tau )} \right\rangle[/tex]由[tex]\left| {\tilde \psi (0)} \right\rangle[/tex] 的平行输运得到，对于电磁联络来说，沿线平行于时间轴的线汇。
We could have chosen, instead, any other congruence of paths from t=0 to t=τ in our definition of φ and therefore [tex]\left| {\tilde \psi (\tau )} \right\rangle[/tex].
相反，在我们φ及由此得到的[tex]\left| {\tilde \psi (\tau )} \right\rangle[/tex] 的定义中，我们可以选择任何从t=0到t =τ的其他路径。
This would correspondingly change β，which therefore depends on the chosen congruence.
这将相应地改变依赖于所选一致性的β，
But, again, βis independent of φ and H(t) for all the motions in [tex]{\cal H}[/tex] that project to the same closed curve [tex]{\hat C}[/tex] in [tex]\wp[/tex] , for a given chosen congruence.
但是，再说一次，对于一个给定的满足选择一致性的[tex]\wp[/tex] 中投射到相同封闭曲线[tex]{\hat C}[/tex] 的[tex]{\cal H}[/tex]中的所有运动，β是独立于φ和H（T）的。
Both β and φ, which satisfies
[tex]e^{ - i\phi } = \left\langle {\psi (\tau )} \right|\exp ( - \frac{{iq}}{c}\int_0^\tau {\hat A_0 dt} )\left| {\psi (0)} \right\rangle[/tex] ,
are gauge invariant.
β和φ都满足
[tex]e^{ - i\phi } = \left\langle {\psi (\tau )} \right|\exp ( - \frac{{iq}}{c}\int_0^\tau {\hat A_0 dt} )\left| {\psi (0)} \right\rangle[/tex]
是规范不变量。
In the adiabatic limit, [tex]\left| {\tilde \psi (t)} \right\rangle[/tex] can be chosen to be an eigenstate of [tex]H_k (t)[/tex] and (9) is then a gauge-invariant generalization of the Berry phase.
在绝热极限下， [tex]\left| {\tilde \psi (t)} \right\rangle[/tex]可以选成[tex]H_k (t)[/tex] 的一个本征态，而（9）式是Berry相位规范不变量的推广
We illustrate this by means of the AB effect. Berry has obtained the AB phase from the gauge-noninvariant expression (4) with [tex]\left| {\tilde \psi (t)} \right\rangle[/tex] an eigenstate of H(t), for a stationary magnetic field, in a special gauge.
我们通过AB效应来说明。对于稳恒磁场的特殊规范，Berry已经从表达式（4）的规范不变性和H(t)的一个本征态[tex]\left| {\tilde \psi (t)} \right\rangle[/tex] 中得到AB相
But a gauge can be chosen so that the AB phase is included in the dynamical phase instead of the geometrical phase (4).
但是，可以选择一个规范，使得AB相位包含动力学相位而不含几何相位
Also, in general, there is no cyclic evolution in an AB experiment.
此外，一般情况下，在AB实验中没有循环演化
But our β defined by Eq. (8) or (9) is gauge invariant and includes the AB phase in the special case to be described now.
但是，我们由方程 （8）或（9）定义的β是规范不变量，包括现在描述的特殊情况下的AB相。
Suppose that a charged-particle beam is split into two beams at t =0 which, after traveling in field-free regions, are recombined so that they have the same state at t=τ.
假设一个带电粒子束在t= 0时分成两束，其中，在自由场区域运行后，使他们在t=τ再次重合时拥有相同的状态
It is assumed here that the splitting and the subsequent evolution of the two beams occur under the action of two separate Hamiltonians.
据推测，分裂和随后的两束的演化在两个独立的的Hamilton量下进行
This is possible if we restrict ourselves to the Hilbert space of a subset of the degrees of freedom of a given system, as in the example considered by Aharonov and Vardi.
如Aharonov 和Vardi所考虑的例子，如果我们把自己限制在一个给定系统自由度的Hilbert空间的一个子集中，这是有可能的
This belongs to the third example of a cyclic evolution mentioned at the beginning of this Letter.
这是在这篇论文开头提到的循环演化的第三个例子
The wave function of each beam at τ,assuming that it has a fairly well defined momentum, is
[tex]\psi _i (x,\tau ) = \exp ( - \frac{i}{\hbar }\int_0^\tau {E_i dt} )\exp ( - \frac{{iq}}{c}\int_{\gamma _i } {A_\mu dx^\mu } )\exp (\frac{i}{\hbar }\int_{\gamma _i } {p \cdot dx} )\psi (x,0)[/tex]
where [tex]\gamma _i[/tex] is a space-time curve through the beam and p represents the approximate kinetic of the momentum beam.
假设每一束在τ时刻的波函数都有一个相当好的确定的动量，为
[tex]\psi _i (x,\tau ) = \exp ( - \frac{i}{\hbar }\int_0^\tau {E_i dt} )\exp ( - \frac{{iq}}{c}\int_{\gamma _i } {A_\mu dx^\mu } )\exp (\frac{i}{\hbar }\int_{\gamma _i } {p \cdot dx} )\psi (x,0)[/tex]
其中[tex]\gamma _i[/tex]是通过粒子束的时空曲率，p代表粒子束的的近似运动学动量
Hence on using (8), we have
[tex]\beta = - \frac{q}{c}\oint_\gamma {A_\mu dx^\mu } + \frac{1}{\hbar }\oint_\gamma {p \cdot dx}[/tex] (10)
where γ is the closed curve formed from [tex]\gamma _1[/tex] and [tex]\gamma _2[/tex].
因此利用（8），我们有
[tex]\beta = - \frac{q}{c}\oint_\gamma {A_\mu dx^\mu } + \frac{1}{\hbar }\oint_\gamma {p \cdot dx}[/tex] （10）
其中γ是由[tex]\gamma _1[/tex]和[tex]\gamma _2[/tex]形成的封闭曲线.
But this is only an approximate treatment and a more careful investigation of this problem is needed.
但是，这只是一个近似处理，有必要对这个问题进行更仔细的研究
In conclusion, we note that [tex]{\cal H}^ * = {\cal H} - \{ 0\}[/tex] is a principal fiber bundle over [tex]\wp[/tex] with structure group C*( the group of nonzero complex numbers), and the disjoint union of the rays in [tex]{\cal H}[/tex] is the natural line bundle over [tex]\wp[/tex] whose fiber above any [tex]p \in \wp[/tex] is p itself.
总之，我们注意到[tex]{\cal H}^ * = {\cal H} - \{ 0\}[/tex] 是[tex]\wp[/tex]上带有结构群C*(这个群的非零复数)的主纤维丛，[tex]{\cal H}[/tex]中不相交的射线是[tex]\wp[/tex] 上纤维为p本身（其中 ）的自然的线丛。
Then, clearly, β，given by (4), arises from the holonomy due to a connection in either bundle such that [tex]\left| {\psi (t)} \right\rangle[/tex]is parallel transported if
[tex]\left\langle {\psi (t)} \right|{d \mathord{\left/
{\vphantom {d {dt}}} \right.
\kern-\nulldelimiterspace} {dt}}\left| {\psi (t)} \right\rangle = 0[/tex]（11）
于是，很显然，如果
[tex]\left\langle {\psi (t)} \right|{d \mathord{\left/
{\vphantom {d {dt}}} \right.
\kern-\nulldelimiterspace} {dt}}\left| {\psi (t)} \right\rangle = 0[/tex]
那么由（4）式定义的β源自其他丛上联络的和乐，它使得[tex]\left| {\psi (t)} \right\rangle[/tex]是平行输运
i.e., the horizontal spaces are perpendicular to the fibers with respect to the Hilbert space inner product.
也就是说，水平空间垂直于纤维相对于希尔伯特空间的内积
Condition (11) was used by Simon to define a connection on a line bundle over parameter space, which is different from the above bundles.
条件（11）被Simon用来定义一个参数空间中的线丛（不同于上面的丛）上的联络
The real part of (11)says that [tex]\left\langle {{\psi (t)}}
\mathrel{\left | {\vphantom {{\psi (t)} {\psi (t)}}}
\right. \kern-\nulldelimiterspace}
{{\psi (t)}} \right\rangle[/tex] is constant during parallel transport.
（11）式的实部表明，[tex]\left\langle {{\psi (t)}}
\mathrel{\left | {\vphantom {{\psi (t)} {\psi (t)}}}
\right. \kern-\nulldelimiterspace}
{{\psi (t)}} \right\rangle[/tex]是平行输运中的常数
Since this is true also during any time evolution determined by (1)，we may restrict consideration to the subbundle
[tex]{\cal F} = \{ \left| \psi \right\rangle \in {\cal H}:\left\langle {\psi }
\mathrel{\left | {\vphantom {\psi \psi }}
\right. \kern-\nulldelimiterspace}
{\psi } \right\rangle = 1\}[/tex]
of [tex]{\cal H}^ *[/tex]. This [tex]{\cal F}[/tex] is the Hopf bundle over [tex]\wp[/tex] .
由于这在由（1）决定的任何时间演化都成立，我们可以考虑[tex]{\cal H}^ *[/tex]的限制子丛
[tex]{\cal F} = \{ \left| \psi \right\rangle \in {\cal H}:\left\langle {\psi }
\mathrel{\left | {\vphantom {\psi \psi }}
\right. \kern-\nulldelimiterspace}
{\psi } \right\rangle = 1\}[/tex]
这个[tex]{\cal F}[/tex]就是[tex]\wp[/tex] 上的Hopf丛
Then the imaginary part of (11) defines the horizontal spaces in [tex]{\cal F}[/tex] which determine a connection.
然后，（11）式的虚部定义了确定一个联络的[tex]{\cal F}[/tex] 中的水平空间
This is the usual connection in [tex]{\cal F}[/tex] and [tex]e^{i\beta }[/tex] is the holonomy transformation associated with it.
这就是[tex]{\cal F}[/tex] 中通常的联络，而[tex]e^{i\beta }[/tex] 是关于这个联络的和乐变换（群）
If [tex]{\cal H}[/tex] has finite dimension N then [tex]\wp[/tex] has dimension N-1. For N=2, [tex]\wp[/tex] is the complex projective space [tex]P_1 (C)[/tex] which is a sphere with the Fubini-study metric on [tex]\wp[/tex] being the usual metric on the sphere.
如果[tex]{\cal H}[/tex]是有限的N维，则[tex]\wp[/tex]是N-1维的。对于N=2，[tex]\wp[/tex]是复射影空间[tex]P_1 (C)[/tex] ，即Fubini研究的球面上的通常度量
Opposite points on this sphere represent rays containing orthogonal states.
在这个球面上相对的点代表包含正交态的射线
Our geometric phase can then be obtained from the holonomy angle α associated with parallel transport around a closed curve on this sphere like in Ref.4.
于是，我们的几何相位就可以从围绕球面（就像在参考文献4中的）上的一条封闭曲线的平行移动关联的和乐角α中得到
It is a pleasure to thank Don Page for suggesting the relevance of the Hopf bundle and the Fubini-Study metric to this work.
最后，要感谢Don在这项研究中关于Hopf丛和Fubini度量的启发
1 Y. Aharonov and L. Susskind, Phys. Rev. 158, 1237 (1967).
2 M. V. Berry, Proc. Roy. Soc. London, Ser. A 392, 45 (1984).
3 B. Simon, Phys. Rev. Lett. 51, 2167 (1983).
4 J. Anandan and L. Stodolsky, Phys. Rev. D 35 2597 (1987).
5 R. Y. Chino and Y.-S. Wu, Yhys. Rev. Lett. 57, 933
(1986); A. Tomita and R. Y. Chino, Phys. Rev. Lett. 57, 937(1986).
6 Y. Aharonov and D. Bohm, Phys. Rev. 11S, 485 (1959).
7 See, for example, L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968), pp. 289一291.
8 An experiment of this type has been done to measure Berry's phase (ω=0) using nuclear magnetic resonance by D. Suter, G. Chingas, R. A. Harris, and A. Pines, to be published. One of us (J.A.) wishes to thank A. Pines for a discussion during which it was realized that the same type of experiment can be used to measure the geometric phaseβintroduced in the present Letter far nonadiabatic cyclic evolutions as well.
关于用核磁共振测量Berry相因子（ω=0）的实验D. Suter, G. Chingas, R. A. Harris和A. Pines已经做过，即将发表。我们中的一个想感谢A. Pines的讨论。这期间人们意识到同类实验也可以用来测量最近论文中涉及的非绝热循环演化的几何相因子β
9 In this proof, in Ref. 2, the eigenfunctions, in the absence of the electromagnetic field, are in effect assumed to be real, in order that Eq. (34) is valid. Since the coefficients of the stationary Schrodinger equation are then real, it is always possible to find real solutions. Then, for any eigenfunction belonging to a given eigenvalue to be necessarily a real function multiplied by exp(iλ) (λ=const), it is necessary and sufficient that the eigenvalue is simple. But in our treatment of the AB effect, it is not necessary to make this assumption.
在这个证明中，在参考文献2中，在电磁场的情况下，为使方程（34）是有效的，本征函数实际上被认为是实的。因为定态薛定谔方程的系数是实的，总可以找到真正的解决办法。对于任何一个属于某一给定本征值值的本征函数，必须是一个实函数乘以exp(iλ)（λ为常数），它是本征值为单值的充分必要条件。但是，在我们对AB效应的处理中，没有必要作出这个假设
10 Y. Aharonov and M. Vardi, Phys. Rev. D 20, 3213 (1979).
11 See, S. Kobayashi and K. Nomizu, Foundations of
Differential Geometry (Interscience, New York, 1969), Vol. 2.
2011年4月25日 11:20
可惜，很多公式转换过来时出错了……
2016年12月21日 08:30
以前的博文怎么看不见了？
2018年10月31日 11:11
惊人的文章感谢分享。
2020年6月11日 11:36
Pines for a discussion during which it was realized that the same type of experiment can be used to measure the geometric phaseβintroduced in the present Letter far nonadiabatic cyclic evolutions as well.
2021年3月02日 15:51
Pines for a discussion during which it was realized that the same type of experiment can be used to measure the geometric phaseβintroduced in the present Letter far nonadiabatic cyclic evolutions as well.
2022年3月22日 17:18
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