Geometric phase of the gyromotion for charged particles
in a time-dependent magnetic field
Jian Liu
1
and Hong Qin
2
1
State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University,
Beijing 100871, China
2
Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA and Department of
Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
(Received 9 March 2011; accepted 9 June 2011; published online 28 July 2011)
We study the dynamics of the gyrophase of a charged particle in a magnetic field which is uniform
in space but changes slowly with time. As the magnetic field evolves slowly with time, the
changing of the gyrophase is composed of two parts. The first part is the dynamical phase, which is
the time integral of the instantaneous gyrofrequency. The second part, called geometric gyrophase,
is more interesting, and it is an example of the geometric phase which has found many important
applications in different branches of physics. If the magnetic field returns to the initial value after a
loop in the parameter space, then the geometric gyrophase equals the solid angle spanned by the
loop in the parameter space. This classical geometric gyrophase is compared with the geometric
phase (the Berry phase) of the spin wave function of an electron placed in the same adiabatically
changing magnetic field. Even though gyromotion is not the classical counterpart of the quantum
spin, the similarities between the geometric phases of the two cases nevertheless reveal the similar
geometric nature of the different physics laws governing these two physics phenomena.
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American Institute of Physics. [doi:10.1063/1.3609830]
I. INTRODUCTION
A moving classical charged particle is exerted the Lor-
entz force in a magnetic field, and it follows the so-called
gyromotion with a helical-like orbit. In a strongly magne-
tized plasma, the fast gyromotion of particles makes the
study of magnetized plasmas cumbersome because of the
mixture of different temporal and spatial scales. Magnetohy-
drodynamics and traditional gyrokinetics theories choose to
remove the fast gyromotion, by averaging out the gyrophase
variable, to simplify the problem for both analytical and nu-
merical purposes. While yielding many important results,
these ingenious methods ignore some physics carried by
gyrophase, which is sometimes pivotal. The gyrophase sur-
vives in modern gyrokinetics, which rigorously decouples,
instead of eliminates, the gyrophase from other slow compo-
nents of the particle dynamics.
1–3
The gyrophase contributes
its due part in many interesting phenomena such as the polar-
ization density, the shear Alfve´n waves, and the radio-
frequency wave heating. In the gyrokinetic theory, we begin
to pay attention to the physics of the gyrophase, especially
its responses on high frequency electro-magnetic field.
4–6
On
the other hand, the gyrophase in slowly changing magnetic
fields contains no little physical meanings. We will reveal an
interesting physics of the gyrophase in an adiabatically
changing magnetic field in this paper.
We study the dynamics of the gyrophase of a charged
particle in a magnetic field which is uniform in space but
changes slowly with time. If the magnetic field does not
change with time, the particle will rotate with a constant
angular velocity X ¼ qB=m in the plane perpendicular to the
magnetic field, where q is the electric charge carried by the
particle, m is the mass of the particle, and B is the magnitude
of the magnetic field. The gyrophase of the particle at time t
can be easily written as h(t) ¼ qBt=m, given h ¼ 0 when
t ¼ 0. As the magnetic field evolves slowly with time, the
changing of the gyrophase turns out to be composed of
two parts. The first part is the dynamical phase
h
d
ðtÞ¼
Ð
t
0
X
d
ðt
0
Þdt
0
, which is simply the time integral of the
instantaneous angular velocity X
d
(t) ¼ qB(t)=m resulting
from the Lorentz force. Of course, this dynamical phase is of
no surprise. What is interesting is that in addition to the dy-
namics phase, the gyrophase contains another part h
g
, which
is called the geometric phase. The name comes from its ele-
gant geometric meaning and its geometric origin, the non-
commutativity of the rotation operations.
7
If the evolution of
the magnetic field B(t) forms a closed loop C in the parame-
ter space composed of (B
x
, B
y
, B
z
), the magnitude of the geo-
metric phase equals to the solid angle a spanned by the loop
C (see Fig. 1). Its value depends only on the closed path C
instead of other physical ingredients such as the particle’s
mass, velocity, the changing rate of the field, etc. This geo-
metric phase associated with the gyrophase is an example of
the general geometric phase in physics.
As early as in 1958, Lyman Spitzer anticipated the exis-
tence of geometric phase through studying rotation trans-
forms and invented the Figure-8 stellarator.
8
In 1984, Berry
studied the quantum adiabatic system.
9
According to the adi-
abatic theorem first introduced by Born and Fock,
10
if the
initial state of a system is an eigenstate of its initial Hamilto-
nian, it should stay at its corresponding eigenstate of the
1070-664X/2011/18(7)/072505/7/$30.00
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2011 American Institute of Physics18, 072505-1
PHYSICS OF PLASMAS 18, 072505 (2011)