Download Berry`s Phase

Berry’s Phase
Erik Lötstedt
[email protected]
November 16, 2004
Abstract
The purpose of this report is to introduce and discuss Berry’s phase
in quantum mechanics. The quantum adiabatic theorem is discussed and
Berry’s phase is introduced and examined in some detail. Also the geometrical meaning of Berry’s phase, such as Berry’s connection and curvature are included. Finally experimental verifications of Berry’s phase are
pointed out.
Contents
1 Introduction
2
2 The quantum adiabatic theorem
2.1 Adiabatic processes . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The adiabatic theorem . . . . . . . . . . . . . . . . . . . . . . . .
2
2
2
3 Berry’s phase
3.1 Anholonomic processes . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Derivation of Berry’s phase . . . . . . . . . . . . . . . . . . . . .
3
3
3
4 Geometrical aspects of Berry’s phase
4.1 Berry’s connection and curvature . . .
4.2 Holonomy . . . . . . . . . . . . . . . .
4.3 Example of Berry’s phase . . . . . . .
4.4 The monopole analogy . . . . . . . . .
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5 Experimental measurement
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6 Further reading
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1
1
Introduction
Berry’s phase, discovered by Berry in 1984 [3], points out that the phase of a
quantum system is in some cases a nonarbitrary quantity that can be measured.
This phase is geometrical in its character and introduces a geometrical approach
to quantum theory. The ideas behind are simple, so you may find it remarkable
that the discovery was made so many years after the invention of quantum
mechanics.
2
The quantum adiabatic theorem
In this section we follow [8].
2.1
Adiabatic processes
An adiabatic process is one where there are two time scales, one external
timescale Te , characteristic for the change rate of the environment of the system,
and one internal timescale Ti for the system itself. A process is called adiabatic
if Ti Te . A familiar classical example is a pendulum. Here Ti would be the
period of one oscillation and Te the time for the environment of the pendulum to
change. We could have, for example, a pendulum with a slowly changing length
or a pendulum in a slowly changing gravitational field. In the classical case,
we expect the system to behave as if it were
p solved with the external time held
fixed. The pendulum has period T = 2π L/g , so with apgradually changing
length L(t), the period of the pendulum would be T = 2π L(t)/g.
2.2
The adiabatic theorem
When speaking about quantum adiabatic processes, we mean that the Hamiltonian H changes gradually with time. The time period Te under which the
Hamiltonian changes must be much longer than the internal time scale Ti of
the wave function. The quantum adiabatic theorem now says that if the Hamiltonian is changed slowly from one initial form H i to a final form H f , the nth
eigenstate |ψni i of H i will be time translated via the Schrödinger equation into
the nth eigenstate |ψnf i of H f . Using formulas we have
H(ti )|ψn (ti )i = En (ti )|ψn (ti )i,
H(tf )|ψn (tf )i = En (tf )|ψn (tf )i
(1)
if the adiabatic approximation holds. Here H(ti,f ) = H i,f , ψn (tf,i ) = ψ f,i and
En (tf,i ) = Enf,i . We assume that the spectrum of the Hamiltonian H(t) is discrete and nondegenerate for ti < t < tf , not to have any problems with ordering
of the states. Although not treated here, the nondegeneracy requirement can
be relaxed, leading to natural appearance of non-Abelian gauge fields [14].
The adiabatic theorem tells us that there will be no transitions from an eigenf
state |ψni i of the initial Hamiltonian H i to a different eigenstate |ψm
i of the final
f
Hamiltonian H . To prove the adiabatic theorem is not that easy, even if it
2
sounds reasonable. One proof can be found in [9]. In [8], a simpler proof is
given, with the assumption that the Hamiltonian is of the form
H(t) = f (t)V,
(2)
where f rises gradually from f (ti ) = 0 to f (tf ) = 1. It is now possible to use
perturbation theory to show that for φ(ti ) = ψni no transitions occur:
1, if m = n,
f 2
hφ(tf )|ψm
i =
(3)
0, if m 6= n.
3
Berry’s phase
In this section we follow [3], [7] and [8].
3.1
Anholonomic processes
An anholonomic process is a process where a system does not return to its
original state when the external parameters of the system are changed so that
they eventually come back to where they started. Again, a familiar example
is the Foucault pendulum, where the change is also adiabatic. The direction
of swing does not return to its starting value after travelling once around the
Earth at a circle of latitude, see [4]. In geometry, anholonomy is expressed by
parallel transport round a circuit. This geometrical aspect will be treated in
section 4.
3.2
Derivation of Berry’s phase
We now ask what will happen with a quantum system, transported adiabatically round some circuit in parameter space. We let the Hamiltonian H depend
on a collection of N parameters R = (R1 , R2 , . . . , RN ) and let R depend adiabatically on time, that is R = R(t) changes slowly with t. Furthermore, we
assume that the Hamiltonian does not commute with itself at different times,
[H(R(t)), H(R(t0 ))] 6= 0. Now, if the system at time t = 0 is in the nth eigenstate (assumed normalized to one),
H(R(0))|n, R(0)i = En (R(0))|n, R(0)i,
(4)
where |ψ(0)i = |n, R(0)i, what about the solution at later times, |ψ(t)i? The
solution |ψ(t)i satisfies the Schrödinger equation
H(R(t))|ψ(t)i = i~
d
|ψ(t)i,
dt
(5)
and under the adiabatic assumption, the adiabatic theorem tells us that the
eigenstate |n, R(0)i continues to be an eigenstate at later times:
H(R(t))|n, R(t)i = En (R(t))|n, R(t)i.
(6)
The obvious guess for the solution to the Schrödinger equation is that the state
picks up a dynamical phase factor
Z
i t
0
0
|ψ(t)i = exp −
En (R(t ))dt |n, R(t)i.
(7)
~ 0
3
This is, however, not correct for a Hamiltonian satisfying [H(t), H(t0 )] 6= 0.
Inserting (7) into the Schrödinger equation (5), it is easily seen that the guess
(7) is not a solution. It turns out that we have to add an extra phase to the
state:
Z
i t
0
0
En (R(t ))dt exp [iγn (t)] |n, R(t)i.
(8)
|ψ(t)i = exp −
~ 0
The function γn is found by requiring that |ψ(t)i satisfies the Schrödinger equation. Careful calculations lead to (see [8], page 336-339)
Z t
Z R(t)
d
∂
hn, R(t0 )| 0 |n, R(t0 )idt0 = i
hn, R|
γn (t) = i
|n, RidRµ ,
(9)
dt
∂Rµ
0
R(0)
where summation over µ is implied. This extra phase factor γn is called Berry’s
phase, first published by Berry in 1984 [3].
In order to obtain an adiabatic cyclic evolution, the Hamitonian is now
brought back to its initial form, so that R(0) = R(T ) for some T large enough
for the adiabatic approximation to hold. The phase change of the state vector,
RT
besides the dynamical phase − ~i 0 dt0 En (R(t0 )), is
I
∂
|n, RidRµ ,
(10)
γn (C) = i hn, R|
∂R
µ
C
where C is a closed curve in parameter space. For the state vector we then have
"
#
Z
i T
|ψ(T )i = exp iγn (C) −
En (R(t0 ))dt0 |ψ(0)i.
(11)
~ 0
Note that γn only depends on the path taken and not on the elapsed time
T . Neither does it depend on the choice of phase for the eigenstates |n, Ri, a
statement that will be further clarified in section 4. In fact, Berry’s phase is a
geometrical quantity, depending on the geometry of the parameter space. Even
though the state returns to the same physical state, the state “remembers” its
evolution through Berry’s phase. If the dynamic phase contains information of
how long time the evolution took, the geometric phase provides the best answer
to the question: where did the system go?
We also want to check that γn is real, not to violate the normalization of |ψi.
That this is really the case is seen by
∗
∂
∂
∂
hn, R|n, Ri = hn, R|
|n, Ri + hn, R|
|n, Ri,
(12)
0=
∂Rµ
∂Rµ
∂Rµ
∂
|n, Ri is imaginary. This also shows that Berry’s phase vanishes
so hn, R| ∂R
µ
when the eigenfunctions |n, Ri are real.
Since Berry’s phase is just a phase factor, you would think that it is of no physical significance. However, it is possible to measure Berry’s phase, for example
by dividing a beam of particles into two, and let one of the beams pass through
an adiabatically changing potential. When the two beams are combined, Berry’s
phase can be measured by looking at constructive and destructive interference.
Another example of a measurement of Berry’s phase is found in [6], discussed
in section 5.
4
4
Geometrical aspects of Berry’s phase
In this section we follow [10] and [12]. We will see that equation (6) can be
used to describe a line bundle over the parameter space and that Berry’s phase
can be described as holonomy associated with the connection induced by the
adiabatic limit. Explained in the language of bundles, the geometric nature of
Berry’s phase becomes evident.
4.1
Berry’s connection and curvature
Let the parameter R = (R1 , . . . , RN ) be the local coordinate of the manifold M
(a space looking locally like RN ) describing the parameter space. To each point
R of M we associate the normalized eigenstate |n, Ri of the Hamiltonian H(R).
Since the state |n, Ri cannot be distinguished from u|n, Ri, where u ∈ U (1),
the group of unit length complex numbers, the physical state is an equivalence
class:
[|n, Ri] = u|n, Ri u ∈ U (1) .
(13)
Thus, at each point R we attach all states u|n, Ri, and this is the fibre of our U (1)
bundle. A fibre bundle is a space that locally looks as a direct product between
two manifolds, the base space and the fibre, but not globally. In our case,
the bundle locally looks like the direct product M × U (1), but may globally be
twisted in some way. This topological information is contained in the connection,
or gauge field, of the bundle. If the bundle is twisted, it will not be possible to
have a non-singular gauge field defined on all of M . We then locally define a
connection one form (a gauge field), called Berry’s connection, as
A = Aµ dRµ = hn, R|d|n, Ri,
(14)
∂
µ
is the exterior derivative in parameter space. This A can
where d = ∂R
µ dR
be checked to satisfy all demands on a connection. In particular, A transforms
under a gauge transformation as
A → A + g −1 dg,
(15)
where g belongs to the gauge group U (1). Note that the gauge group is Abelian.
The corresponding field strength, called Berry’s curvature, is given by
F = dA =
∂
∂
hn, R| ν |n, Ri (dRµ ⊗ dRν − dRν ⊗ dRµ ) .
µ
∂R
∂R
(16)
Using the connection and the curvature, we can write Berry’s phase (10) as
Z
Z
γn (C) = i
A = i F,
(17)
C
S
where S is the surface bounded by the loop C.
4.2
Holonomy
What a connection does is to provide a rule for parallel transport. Parallel
transport in a fibre bundle along a curve C in M is provided by the so-called
horizontal lift of the curve C. A horizontal lift is a curve in the bundle, which
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projected down onto M gives C and at each point is horizontal to the fibre. If
a vector |φi belonging to the fibre of R is parallel transported around a circuit
back to the fibre of R, then, in general, it does not return to the same point on
the fibre. The difference is called a holonomy (it should be anholonomy, really,
but in mathematics this is the term used) and we will show that this can be
identified with Berry’s phase.
First, we redefine the Hamiltonian to get rid of the dynamic phase:
Hn0 (R) = H(R) − En (R),
(18)
so that
Hn0 (R)|n, Ri = 0.
Hn0 (R)|ψn (t)i
The solution to the new Schrödinger equation
becomes
|ψn (t)i = exp [iγn (t)] |n, R(t)i.
(19)
=
d
|ψn (t)i
i~ dt
now
(20)
In order to parallel transport a vector around a curve, we let C = R(t), t ∈ [0, T ]
be a curve in M . Our rule for parallel transport of a vector |φ(t)i is given by
hφ(t)|
d
|φ(t)i = 0.
dt
(21)
If the parallel transported vector |φ(t)i is written as
|φ(t)i = u(R(t))|n, R(t)i,
(22)
where u(R(0)) = 1, then u ∈ U (1) satisfies
du(R(t))
d
u(R(t))−1 = −hn, R(t)| |n, R(t)i = −A(d/dt).
(23)
dt
dt
R
If we let u = exp[iϕ], then the solution of equation (23) is ϕ(t) = i C hn, R|d|n, Ri.
Thus, we can identify γn with ϕ and the parallel transported vector is given by
(22):
|φ(t)i = exp[iγn (t)]|n, R(t)i,
(24)
precisely what we obtained for the solution (20) to the Schrödinger equation.
4.3
Example of Berry’s phase
Here we will show an example of Berry’s phase. The adiabatically changing
parameters of the Hamiltonian will be a magnetic field. Consider a spin 12
particle in a magnetic field given by B = (B1 , B2 , B3 ). The Hamiltonian is
given by
B3
B1 − iB2
H(B) = Bi σi =
,
(25)
B1 + iB2
−B3
where σi are the Pauli sigma matrices. Here we have absorbed the constant
involving the gyromagnetic ratio and ~ into B. The Hamiltonian H has two
eigenvalues, E± = ±|B|. As in (18), we redefine the Hamiltonian to be
0
H±
(B) = H(B) − E± .
6
(26)
0
Now, consider the Hamiltonian H+
(B). One normalized eigenstate with eigenvalue zero is
1
|B| + B3
|1+, Bi = p
.
(27)
B1 + iB2
2|B|(|B| + B3 )
In order to observe Berry’s phase, it is convenient to introduce spherical coordinates for the magnetc field, B1 = |B| sin θ cos φ, B2 = |B| sin θ sin φ and
B3 = |B| cos θ with 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π. The eigenstate becomes
cos(θ/2)
|1+, Bi =
.
(28)
exp(iφ) sin(θ/2)
We now let θ and φ depend adiabatically on time, fix the length of |B| = B0
and let the magnetic field vector B sweep out a closed curve on the sphere with
radius B0 . In three dimensions, formula (10) can be reformulated with Stokes’s
theorem to read
Z
∂
∂
h1+, R|
|1+, Ridaj ,
(29)
γ+ = i εijk
∂B
∂B
k
i
S
where S is the area on the sphere bounded by the circuit, and da the area
element. Calculating the curl in spherical coordinates, we obtain
Z
1
Bj
1
γ+ = −
daj = − Ω,
(30)
2 S |B|3
2
where Ω is the solid angle of the path, as seen from the origin. (Remember daj =
|B|Bj sin θdθdφ = |B|Bj dΩ.) This is very much like the classical pendulum,
which, if carried around a circuit on a sphere, will have its swinging direction
altered with the solid angle of the circuit.
4.4
The monopole analogy
We note that in the above example, the point R = 0 is a singularity. All the
eigenvalues of the Hamiltonian become degenerate (H 0 is just the zero matrix)
and the adiabatic assumption breaks down. We will note that this singularity
behaves like a magnetic monopole.
cos(θ/2)
Looking at the state |1+, Bi =
, we observe that it is
exp(iφ) sin(θ/2)
not defined for θ = π (where B3 = −|B|). In order to obtain a well-defined
state on the whole sphere, we use the U (1) freedom and take the eigenvector
exp[−iφ] cos(θ/2)
|2+, Bi = exp[−iφ]|1+, Bi =
,
(31)
sin(θ/2)
defined everywhere on the sphere except at θ = 0. Berry’s gauge fields corresponding to the above states are computed to be
B2 dB1 − B1 dB2
i
= (1 − cos θ) dφ,
2B(B + B3 )
2
i
A2 = h2+, B|d|2+, Bi = − (1 + cos θ) dφ.
2
A1 = h1+, B|d|1+, Bi = −i
7
(32)
The gauge fields cover the whole sphere and are related by the gauge transformation
A2 = A1 + exp[iφ]d exp[−iφ] = A1 − idφ.
(33)
These gauge fields A1 and A2 describe a Wu-Yang magnetic monopole of strength
− 21 . A magnetic monopole is a magnetic point charge, or an Abelian gauge theory defined on R3 with the origin removed and is a simple example of a twisted
U (1) bundle, see [10]. If the field strength is calculated, we indeed obtain
F1 = dA1 =
~
iB
i
(B1 dB2 ∧dB3 +B2 dB3 ∧dB1 +B3 dB1 ∧dB2 ) =
, (34)
3
2|B|
2|B|3
the magnetic field of a magnetic point − 21 charge. Berry’s phase can now be
interpreted as the magnetic flux through the area enclosed by the circuit.
5
Experimental measurement
When
R t trying to measure Berry’s phase, the fact that the dynamical phase
− ~i 0 dt0 En (R(t0 )) will be much larger than the geometrical phase γn suggests
that the measurement will be difficult. However, measurements of Berry’s phase
that have been performed show that it is possible to separate out the two contributions. For an experiment realizing the example in 4.3, neutrons moving
through a rotating magnetic field, see [6]. Here a superposition of spin eigenstates is used, so that
X
|ψi i =
cn |ni,
(35)
n
which after cycling the magnetic field becomes
X
|ψf i =
exp [iθn + iγn ] cn |ni,
(36)
n
where iθn is the dynamic phase. If now an observable A which do not commute
with the Hamiltonian is measured, then Berry’s phase can be read from the
oscillating part of the expectation value of A:
X
hψf |A|ψf i =
c∗n cm exp [i(θm − θn + γm − γn )] hn|A|mi.
(37)
n, m
For an experiment with bosons, see [13]. Here a beam of photons are sent into a
coiled fibre. After travelling through the coil the direction of linear polarization
will be rotated by plus or minus the “solid angle” of the coil. This is Berry’s
phase for light. In this case, Berry’s phase can be understood on a classical
level, in fact the parallel transport law for the electric field can be derived from
the classical Maxwell equations [5].
6
Further reading
In this report we have, like Berry in his original paper [3], only considered
adiabatic transport of vectors, depending very much on the Hamiltonian of the
system. It is possible to generalize this and associate a geometric phase with
8
the physical motion of the quantum state itself instead of the motion of the
Hamiltonian. By disregarding which Hamiltonian that caused the motion, it is
possible to relax the conditions that the state should be an eigenstate of the
Hamiltonian and that the motion is adiabatic. Berry’s case then appears as a
limit. It also becomes possible to consider cyclic motions in isolated systems,
see [1] and [2]. So far we have always had a time-dependent Hamiltonian, which
corresponds to a system interacting with the environment. In fact, it turns out
to be possible [11] to associate a geometric phase with every path, closed or not,
using the natural metric associated with the Hilbert space to close the path with
a geodesic curve.
References
[1] Y. Aharonov and J. Anandan, Physical Review Letters 58, 1593 (1987).
[2] J. Anandan, Physics Letters A 129, 201 (1988).
[3] M. V. Berry, Proceedings of the Royal Society of London A 392, 45 (1984).
[4] M. V. Berry, Physics Today December Issue, 34 (1990).
[5] M. V. Berry, Nature 326, 277 (1987).
[6] T. Bitter and D. Dubbers, Physical Review Letters 59, 251 (1987).
[7] A. Bohm, B. Kendrick and M. E. Loewe, The Berry Phase in Molecular
Physics, Center for Particle Theory, University of Texas, DOE-ER40200235 (1990).
[8] D. J. Griffiths, Introduction to Quantum Mechanics, chapter 10, Prentice
Hall (1995).
[9] A. Messiah, Quantum Mechanics, volume II, chapter XVII, section 12, John
Wiley and Sons, New York (1962).
[10] M. Nakahara, Geometry, Topology and Physics, second edition, chapter 10,
Institute of Physics Publishing (2003).
[11] J. Samuel and R. Bhandari, Physical Review Letters 60, 2339 (1988).
[12] B. Simon, Physical Review Letters 51, 2167 (1983).
[13] A. Tomita and R. Y. Chiao, Physical Review Letters 57, 937 (1986).
[14] F. Wilczek and A. Zee, Physical Review Letters 52, 2111 (1984).
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