Bargmann invariants and off-diagonal geometric phases for multilevel quantum systems:
A unitary-group approach
N. Mukunda*
Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560012, India
Arvind†
Department of Physics, Guru Nanak Dev University, Amritsar 143005, India
S. Chaturvedi‡
Department of Physics, University of Hyderabad, Hyderabad 500046, India
R. Simon§
Institute of Mathematical Sciences, CIT Campus, Chennai 600113, India
We investigate the geometric phases and the Bargmann invariants associated with multilevel quantum
systems. In particular, we show that a full set of ‘‘gauge-invariant’’ objects for an n-level system consists of n
geometric phases and 21 (n⫺1)(n⫺2) algebraically independent four-vertex Bargmann invariants. In the process of establishing this result, we develop a canonical form for U(n) matrices that is useful in its own right.
We show that the recently discovered ‘‘off-diagonal’’ geometric phases 关N. Manini and F. Pistolesi, Phys. Rev.
Lett. 8, 3067 共2000兲兴 can be completely analyzed in terms of the basic building blocks developed in this work.
This result liberates the off-diagonal phases from the assumption of adiabaticity used in arriving at them.
I. INTRODUCTION
The notion of geometric phase, though defined originally
in the context of adiabatic, unitary, and cyclic evolution 关1兴,
has now come to be recognized as a direct consequence of
the geometry of the complex Hilbert space and that of the
associated ray space 关2兴. The quantum kinematic 关2兴 picture,
which has thus emerged, provides a much wider setting for
the notion of the geometric phase by rendering superfluous
the various assumptions that attended its original discovery
关1兴 and subsequent development 关3–5兴. In particular, the requirement of cyclic evolution is no longer necessary and it
becomes possible to ascribe a geometric phase to any open
curve in the Hilbert space that has nonorthogonal unit vectors as its end points. Further, following the quantum kinematic approach 关6兴, one is led in a natural way to the intimate
relationship that exists between the geometric phase and the
n-vertex Bargmann invariants 关7兴 and that between the geometric phase and Hamilton’s theory of turns 关8兴. As an application of this approach, the Gouy phase 共the phase jump
experienced by a focused light beam as it crosses the focus兲,
discovered over a hundred years ago, has been shown to be a
four-vertex Bargmann invariant 关9兴.
In the present work, we develop the quantum kinematic
approach for the special case of unitary evolution of an
n-level system. It turns out that, in the present context, it
becomes necessary to introduce Bargmann invariants con-
*Email address: nmukunda@cts.iisc.ernet.in
†
Email address: arvind@physics.iisc.ernet.in
Email address: scsp@uohyd.ernet.in
§
Email address: simon@imsc.ernet.in
‡
structed out of two sets of orthonormal basis vectors. We
investigate, in detail, their properties and identify and construct a full set of gauge-invariant building blocks for the
n-level system. We also develop a canonical representation
of U(n) matrices and bring out its relation to the Bargmann
invariants. This representation has recently been shown to be
extremely useful in parametrizing the Cabibbo-KobayashiMaskawa matrix that arises in the context of C P violation in
particle physics 关10兴.
As noted above, the geometric phase becomes undefined
for those open curves in the Hilbert space that have orthogonal vectors at their ends. A recent work by Manini and Pistolesi 关11兴 addresses itself precisely to such exceptional
cases. Employing the original Berry setting of adiabatic evolution for an n-level quantum system, they introduce the concept of off-diagonal geometric phases that can be meaningfully defined in such cases, and showed that these offdiagonal geometric phases play an essential role in the
interpretation of the findings of a recent experiment 关12兴.
We study the off-diagonal phases of Manini and Pistolesi
within the framework of the quantum kinematic approach
and show that, in actual fact, this approach, if suitably augmented, is robust enough to accommodate the off-diagonal
phases as well. A brief outline of this work is as follows. In
Sec. II, we quickly recapitulate the basic ingredients of the
quantum kinematic approach to geometric phases in a general setting. In Sec. III, we specialize the discussion to
n-level systems and construct the gauge-invariant objects for
this case. These turn out to be n geometric phases and a
collection of four-vertex Bargmann invariants. Next we consider the problem of counting, leading to identification of a
set of independent Bargmann invariants. This necessitates a
detailed analysis of the structure of the U(n) matrix group.
In Sec. IV, we develop a canonical representation for U(n)
matrices that is then used in Sec. V for isolating the independent four-vertex Bargmann invariants. In Sec. IV, we apply
the machinery developed in the previous sections to the offdiagonal geometric phases and show how they can be expressed in terms of the ordinary geometric phases and the
four-vertex Bargmann invariants, thus liberating these phases
from the assumptions of adiabaticity. Section VII contains
our conclusions.
II. GEOMETRIC PHASES AND BARGMANN INVARIANTS
We review very briefly the background and ingredients
that go into the definition of the quantum geometric phase
from the kinematic viewpoint and then the Bargmann invariants and their properties in the generic case. Let H be the
Hilbert space of states of some quantum system and let B be
the set of unit vectors in H,
B⫽ 兵 ␺ 苸H兩 储 ␺ 储 2 ⫽ 共 ␺ , ␺ 兲 ⫽1 其 傺H.
共1兲
The corresponding ray space 共consisting of equivalence
classes of unit vectors that differ from one another by
phases兲 and projection map are written as R and ␲ , respectively,
␲ :#B→R⫽ 共 space of unit rays兲 .
共2兲
To arrive at the concept of geometric phase, we begin with
parametrized smooth 共for our purposes continuous and once
piecewise differentiable兲 curves C of unit vectors, which may
be pictured as strings lying in B,
C⫽ 兵 ␺ 共 s 兲 苸B兩 s 1 ⭐s⭐s 2 其 傺B.
共3兲
A gauge transformation is a smooth change of phase in a
parameter-dependent manner at each point of such a curve C
to lead to another C ⬘ ,
C ⬘ ⫽ 兵 ␺ ⬘ 共 s 兲 ⫽e i ␣ (s) ␺ 共 s 兲 兩 ␺ 共 s 兲 苸C,s 1 ⭐s⭐s 2 其 傺B.
共4兲
Then C ⬘ and C share a common parametrized image curve C
in ray space,
␲ 关 C ⬘ 兴 ⫽ ␲ 关 C兴 ⫽C傺R.
共5兲
In general, we permit C and even C to be open curves.
The total, dynamical, and geometric phases are then defined as follows as functionals of appropriate arguments:
␸ tot关 C兴 ⫽arg关 ␺ 共 s 1 兲 , ␺ 共 s 2 兲兴 ,
␸ dyn关 C兴 ⫽Im
冕 冉␺
s2
s1
ds
共 s 兲,
⌬n 共 ␹ 1 , ␹ 2 , . . . , ␹ n 兲 ⬅ 共 ␹ 1 , ␹ 2 兲共 ␹ 2 , ␹ 3 兲 ••• 共 ␹ n⫺1 , ␹ n 兲
⫻共 ␹ n , ␹ 1 兲 .
共7兲
Here we assume in the generic case that no two successive
vectors in the sequence are mutually orthogonal. It is clear
that this expression is invariant under cyclic permutations of
the ␹ ’s, and also under independent phase changes of the
individual vectors. Therefore, it is actually a quantity defined
at the ray-space level. It turns out that the phase of
⌬n ( ␹ 1 , ␹ 2 , . . . , ␹ n ) is the geometric phase for suitably constructed closed ray-space curves obtained by joining each ␹ j
to the next ␹ j⫹1 共and finally ␹ n to ␹ 1 ) by any so-called
‘‘nullphase curve’’ 关6兴. A null-phase curve is a continuous
ray-space curve such that for any finite connected portion of
it the geometric phase vanishes. That is, ‘‘being in phase’’ in
the Pancharatnam sense 关13兴 becomes an equivalence relation on such curves. Examples of null-phase curves are rayspace geodesics 共with respect to the well-known Fubinistudy metric 关14,15兴兲, but the former are a much larger set
than the latter 关6兴. It should be emphasized that whereas the
Bargmann invariant 共7兲 is defined once its ‘‘vertices,’’
namely, the projections ␲ ( ␹ 1 ), ␲ ( ␹ 2 ), . . . , ␲ ( ␹ n ) in R, are
given, to interpret its phase as a geometric phase requires
that we join each ␹ j to the next ␹ j⫹1 in some definite manner, namely, by some null-phase curve, resulting in an
‘‘n-sided’’ closed figure in R.
It can now be seen that as far as phases are concerned, an
n-vertex Bargmann invariant for n⭓4 can be reduced to a
product of ⌬3 factors in the generic case 关2兴, so we can
regard the three-vertex Bargmann invariants as the primitive
ones. For example, we have
⌬4 共 ␹ 1 , ␹ 2 , ␹ 3 , ␹ 4 兲
⫽⌬3 共 ␹ 1 , ␹ 2 , ␹ 3 兲 ⌬3 共 ␹ 1 , ␹ 3 , ␹ 4 兲 / 兩 共 ␹ 1 , ␹ 3 兲 兩 2 , 共8兲
and more generally
⌬n 共 ␹ 1 , ␹ 2 , . . . , ␹ n 兲
⫽⌬3 共 ␹ 1 , ␹ 2 , ␹ 3 兲 ⌬n⫺1 共 ␹ 1 , ␹ 3 , ␺ 4 , . . . , ␹ n 兲 / 兩 共 ␹ 1 , ␹ 3 兲 兩 2 .
共9兲
冊
d␺共 s 兲
,
ds
␸ g 关 C 兴 ⫽ ␸ tot关 C兴 ⫺ ␸ dyn关 C兴 .
tional of the ray-space curve C. All three phases are, however, individually reparametrization invariant.
Now we turn to the Bargmann invariants and their connection to geometric phases. Given any sequence of n vectors ␹ 1 , ␹ 2 , . . . , ␹ n in B, the corresponding n-vertex Bargmann invariant is
共6兲
While the first two phases are functionals of C and do change
under a gauge transformation, the geometric phase ␸ g 关 C 兴 is
gauge invariant, which explains why it is written as a func-
Thus, the geometric phases of ray-space ‘‘triangles,’’ each
of whose sides is a null-phase curve, are primitive or irreducible phases, and all others can be built up from them
additively. The purpose in mentioning this is that in the particular situation we shall be dealing with later the primitive
Bargmann invariants will turn out to be ⌬4 ’s rather than
⌬3 ’s, so that situation will not be generic in the present
sense.
III. GAUGE-INVARIANT PHASES
FOR n-LEVEL SYSTEMS
We now turn to a study of phases associated with n-level
quantum systems. Thus we have an n-dimensional complex
Hilbert space Hn describing the pure states of the system.
The unit sphere in Hn and the corresponding space of unit
rays will be denoted by Bn and Rn , respectively.
If we imagine that a time-dependent Hamiltonian (n⫻n
Hermitian matrix兲 is given, then at each time its complete
orthonormal set of eigenvectors defines an orthonormal basis
for Hn . Assuming there are no degeneracies or level crossings, the eigenvalues can be arranged in, say, increasing order and at each time this basis for Hn is defined up to the
freedom of phase changes in each basis vector. As time
progresses this basis experiences a continuous unitary rotation.
In keeping with the approach of the previous section,
however, we will adopt a kinematic approach here as well
and not assume any particular Hamiltonian to be given.
Thus, we imagine that for each value of a parameter s in the
range s 1 ⭐s⭐s 2 , we have an orthonormal basis ␺ j (s), j
⫽1,2, . . . ,n for Hn ; and as s evolves this basis experiences
a continuous unitary evolution. Thus we have
„␺ j 共 s 兲 , ␺ k 共 s 兲 …⫽ ␦ jk ,
兺
␺ j 共 s 兲 ␺ j 共 s 兲 † ⫽I,
⫽ 共 ␺ j 1 , ␾ k 1 兲共 ␾ k 1 , ␺ j 2 兲
⫻ 共 ␺ j 2 , ␾ k 2 兲 ••• 共 ␺ j l , ␾ k l 兲共 ␾ k l , ␺ j 1 兲 .
共14兲
We may now regard the generic case as obtaining when every inner product ( ␺ j , ␾ k ) is nonzero. In this situation we
find that the primitive Bargmann invariants are ⌬4 ’s rather
than ⌬3 ’s, for instance,
⌬6 共 ␺ j 1 , ␾ k 1 , ␺ j 2 , ␾ k 2 , ␺ j 3 , ␾ k 3 兲
⫻⌬4 共 ␺ j 1 , ␾ k 2 , ␺ j 3 , ␾ k 3 兲 / 兩 共 ␺ j 1 , ␾ k 2 兲 兩 2 , 共15兲
s 1 ⭐s⭐s 2 .
共10兲
For ease in writing, we shall denote these vectors at the end
points s 1 and s 2 as follows:
␺ j共 s 1 兲⫽ ␺ j ,
⌬2 l 共 ␺ j 1 , ␾ k 1 , ␺ j 2 , ␾ k 2 , . . . , ␺ j l , ␾ k l 兲
⫽⌬4 共 ␺ j 1 , ␾ k 1 , ␺ j 2 , ␾ k 2 兲
j,k⫽1,2, . . . ,n,
n
j⫽1
vectors ␺ 1 , . . . , ␺ n and the n final ones ␾ 1 , . . . , ␾ n . Here
we encounter an interesting difference compared to the generic situation discussed in the previous section. Since any
two distinct ␺ j ’s 共and similarly any two distinct ␾ j ’s兲 are
orthogonal, in any Bargmann invariant ⌬n (•••) an argument
␺ j must be followed by an argument ␾ k , which must be
followed by some ␺ l , and so on. Similarly, if the first argument is some ␺ , the last one must be some ␾ . Thus in the
present context only even-order Bargmann invariants ⌬2 l
survive, a general one being
␺ j共 s 2 兲⫽ ␾ j .
共11兲
Our aim is to obtain gauge-invariant expressions and phases
in this context. We expect to be able to construct both geometric phases ␸ g 关 C 兴 for various C and Bargmann invariants.
For each value of the index j, as s varies from s 1 to s 2 , the
vector ␺ j (s) traces out a particular continuous parametrized
curve C j in Bn ,
C j ⫽ 兵 ␺ j 共 s 兲 苸Bn 兩 s 1 ⭐s⭐s 2 其 傺Bn ,
j⫽1,2, . . . ,n.
共12兲
and similarly for higher-order ⌬2 l ’s.
From this discussion, it emerges that in the present context the basic gauge-invariant expressions are the n geometric phases ␸ g 关 C j 兴 and the various four-vertex Bargmann invariants ⌬4 ( ␺ j 1 , ␾ k 1 , ␺ j 2 , ␾ k 2 ). 共Here we are using quantities
referring to the entire parameter range s 1 ⭐s⭐s 2 and to its
end points and not to any subranges.兲 An important problem
that now remains is to select out of all possible ⌬4 ’s a maximal set of independent ones as far as phases are concerned.
For this, as a first step, we turn to an interesting analysis of
the structure of the unitary matrix groups U(n).
IV. A CANONICAL REPRESENTATION FOR U„n…
MATRICES, COUNTING OF INVARIANT PHASES
␸ g 关 C j 兴 ⫽ ␸ tot关 C j 兴 ⫺ ␸ dyn关 C j 兴 ,
Referring to Eqs. 共10兲 and 共11兲, we have a parameterdependent n⫻n unitary matrix describing the transition from
the initial orthonormal basis 兵 ␺ j 其 for Hn at s⫽s 1 to the
moving basis 兵 ␺ j (s) 其 at a general s,
␸ tot关 C j 兴 ⫽arg共 ␺ j , ␾ j 兲 ,
A 共 s 兲 ⫽a jk 共 s 兲 苸U 共 n 兲 ,
This curve runs from ␺ j to ␾ j . Its image is ␲ 关 C j 兴
⫽C j 傺Rn and we have n distinct geometric phases,
␸ dyn关 C j 兴 ⫽Im
冕 冉␺
s2
s1
ds
j共 s 兲,
冊
d ␺ j共 s 兲
,
ds
j⫽1,2, . . . ,n.
共13兲
Each of these geometric phases is unchanged under arbitrary
alterations in the phase of each ␺ j (s) at each parameter
value s.
We next turn to the construction of Bargmann invariants,
the vertices of which are taken from the n initial orthonormal
a jk 共 s 兲 ⫽„␺ j , ␺ k 共 s 兲 …,
s 1 ⭐s⭐s 2 ,
a jk 共 s 1 兲 ⫽ ␦ jk .
共16兲
At s⫽s 2 we write A(s 2 )⫽A,
A⫽ 共 a jk 兲 ,
a jk ⫽ 共 ␺ j , ␾ k 兲 .
共17兲
The four-vertex Bargmann invariants of the type appearing
in Eq. 共15兲 are expressions involving products of matrix elements of A and their complex conjugates,
⌬4 共 ␺ j , ␾ k , ␺ l , ␾ m 兲 ⫽ 共 ␺ j , ␾ k 兲共 ␾ k , ␺ l 兲共 ␺ l , ␾ m 兲共 ␾ m , ␺ j 兲
⫽a jk a *
l ka l ma *
jm .
共18兲
Our problem is to determine how many algebraically independent ⌬4 ’s are there in the generic case insofar as their
phases are concerned and to find a convenient enumeration
of them. This turns out to be a somewhat intricate problem.
After some preparation in this section, the solution will be
developed in the next one.
In working with n⫻n unitary matrices it is convenient to
keep in mind the standard basis in Hn . Then U(n) is the
group of unitary transformations acting on all n dimensions.
For m⫽1,2, . . . ,n⫺1, we will denote by U(m) the unitary
group acting on the first m dimensions in Hn , leaving dimensions m⫹1,m⫹2, . . . ,n unaffected. Then we have the
inclusion relations 共the canonical subgroup chain兲
U共 1 兲 傺U共 2 兲 傺U共 3 兲 •••傺U共 n⫺1 兲 傺U共 n 兲 .
共19兲
General matrices of U(n),U(n⫺1), . . . will be written as
A n ,A n⫺1 , . . . . In a matrix A m 苸U(m), the last (n⫺m)
rows and columns are trivial, with ones along the diagonal
and zeros elsewhere. 共When no confusion is likely to arise,
A m will also denote an unbordered m⫻m unitary matrix.兲
We will now show by a recursive argument that 共almost
all兲 elements A n 苸U(n) can be expressed uniquely as n-fold
products,
A n ⫽A n 共 ␨ 兲 A n⫺1 共 ␩ 兲 A n⫺2 共 ␰ 兲 •••A 3 共 ␤ 兲 A 2 共 ␣ 兲 A 1 共 ␹ 兲 ,
共20兲
where A n ( ␨ ) is a special U(n) element determined by an
n-component complex unit vector ␨គ 苸Bn ; A n⫺1 ( ␩គ ) is a special U(n⫺1) element determined by an (n⫺1)-component
complex unit vector ␩គ 苸Bn⫺1 ; and so on down to A 2 ( ␣គ ) that
is a special U(2) element determined by a two-component
complex unit vector ␣គ 苸B2 and A 1 ( ␹ ) is a phase factor belonging to U(1). We are led to expect such a representation
for A n by the following argument. Any vector ␨គ 苸Bn can be
carried by a suitable U(n) element into the nth vector of the
standard basis, (0,0, . . . ,0,1) T and the stability group of this
vector is the subgroup U(n⫺1)傺U(n) acting on the first
(n⫺1) dimensions in Hn . Thus U(n) acts transitively on Bn
and this is just the coset space U(n)/U(n⫺1). Each coset is
thus uniquely labeled by some ␨គ 苸Bn . We, therefore, expect
that a general A n 苸U(n) is expressible as the product
A n ( ␨គ )A n⫺1 where ␨គ is the last column in A n and A n ( ␨គ ) is a
suitably chosen coset representative. Repeating this argument (n⫺1) times we are led to expect the representation
共20兲. The counting of parameters is also just right. Remembering that ␣គ , ␤គ , . . . , ␰គ , ␩គ , ␨គ are complex unit vectors of dimensions 2,3, . . . ,n⫺2,n⫺1,n and adding the U(1) phase
␹ , the number of real independent parameters adds up to n 2 ,
the dimension of U(n).
We now present the argument leading to Eq. 共20兲, yielding in the process the determination of A n ( ␨គ ). Let a generic
matrix A n ⫽(a jk )苸U(n) be given and let its last (nth兲 column be ␨គ ,
a jn ⫽ ␨ j ,
j⫽1,2, . . . ,n.
共21兲
Multiplying A n by an A n⫺1 on the right leaves this column
unchanged. We choose A n⫺1 so as to bring the nth row of A n
to a particularly simple form 共for ease in writing we keep
using A n and a jk for the U(n) element obtained at each
successive stage of the argument兲,
a nk ⫽0, k⫽1,2, . . . ,n⫺2,
a n,n⫺1 ⫽real positive⫽ 共 1⫺ 兩 ␨ n 兩 2 兲 1/2.
共22兲
The A n⫺1 used here is arbitrary up to an A n⫺2 factor on its
right. Having simplified the nth row of A n in this way, we
can determine all the other elements in the (n⫺1)th column
by imposing orthogonality of rows 1,2, . . . ,n⫺1 to row n,
a n,n⫺1 a j,n⫺1 ⫽⫺ ␨ n* ␨ j ,
j⫽1,2, . . . ,n⫺1.
共23兲
At this point the last two columns and the last row of A n are
known in terms of ␨គ .
We next use the freedom in choice of A n⫺1 mentioned
above and multiply A n on the right by a suitable A n⫺2
共unique up to an A n⫺3 on its right兲 to bring the (n⫺1)th row
of A n to a particularly simple form,
a n⫺1,k ⫽0,
k⫽1,2, . . . ,n⫺3,
a n⫺1,n⫺2 ⫽real positive.
共24兲
Normalizing this row gives a n⫺1,n⫺2 ,
a n,n⫺1 a n⫺1,n⫺2 ⫽ 共 1⫺ 兩 ␨ n 兩 2 ⫺ 兩 ␨ n⫺1 兩 2 兲 1/2.
共25兲
Next we determine all the remaining elements in the (n
⫺2)th column of A n by imposing orthogonality of rows
1,2, . . . ,n⫺2 to row (n⫺1),
2
* ␨j ,
a n⫺1,n⫺2 a j,n⫺2 ⫽⫺ ␨ n⫺1
a n,n⫺1
j⫽1,2, . . . ,n⫺2.
共26兲
At this point the last three columns and last two rows of A n
are known in terms of ␨គ .
This argument can be repeated all the way until we obtain
a matrix A n ( ␨គ )苸U(n), all of whose elements are determined
by the nth column ␨គ , namely, it serves as a coset representative in the coset space U(n)/U(n⫺1). 共In particular, the
last U(1) element A 1 ( ␹ ) is used to make a 21 real positive兲.
After some algebra, we obtain the result that the matrix
A n ( ␨គ )⫽a jk ( ␨គ ) is uniquely determined by the conditions
a jk 共 ␨គ 兲 ⫽0,
a j, j⫺1 共 ␨គ 兲 ⫽real positive,
a jn 共 ␨គ 兲 ⫽ ␨ n ,
A n ⫽A n 共 ␨គ , ␩គ , ␰គ , . . . , ␤គ , ␣គ , ␹ 兲 ⫽A n 共 ␨គ 兲 A n⫺1 ,
j⭓k⫹2,
A n⫺1 ⫽A n⫺1 共 ␩គ , ␰គ , . . . , ␤គ , ␣គ , ␹ 兲 .
j⫽2,3, . . . ,n,
共27兲
j⫽1,2, . . . ,n.
Thus A n ( ␨គ ) has vanishing matrix elements in the lower left
hand triangular portion up to two steps below the main diagonal, nonzero matrix elements appear only one step below
the main diagonal, and beyond. The explicit expressions for
the nonzero matrix elements are
a j, j⫺1 共 ␨គ 兲 ⫽ ␳ j⫺1 / ␳ j ,
* ␨ j / ␳ k ␳ k⫹1 ,
a j,k 共 ␨គ 兲 ⫽⫺ ␨ k⫹1
a jn 共 ␨គ 兲 ⫽ ␨ n ,
j⫽1,2, . . . ,n,
␳ j ⫽ 共 兩 ␨ 1 兩 2 ⫹ 兩 ␨ 2 兩 2 ⫹•••⫹ 兩 ␨ j 兩 2 兲 1/2
共28兲
Since the quantities ␳ j obey
␳ 1 ⫽ 兩 ␨ 1 兩 ⭐ ␳ 2 ⭐ ␳ 3 ⭐•••⭐ ␳ n⫺1 ⭐ ␳ n ⫽1,
共29兲
it is evident that this determination of A n ( ␨គ ) goes through
with no problems as long as ␨ 1 is nonzero, i.e., ␳ 1 ⬎0.
It may be helpful to give the expressions for A 2 ( ␣គ )
苸U(2),A 3 ( ␤គ )苸U(3) determined in this way, so as to see
the general pattern,
A 2 共 ␣គ 兲 ⫽
冉
⫺ ␣ 2* ␣ 1
␣1
兩 ␣ 1兩
冉
兩 ␣ 1兩
A 3 共 ␤គ 兲 ⫽
␣2
冊
,
兩 ␣ 1 兩 2 ⫹ 兩 ␣ 2 兩 2 ⫽1, 共30a兲
⫺ ␤ 2* ␤ 1
⫺ ␤ 3* ␤ 1
␳ 1␳ 2
␳2
␳1 /␳2
0
⫺ ␤ 3* ␤ 2
␳2
␳2
␤1
␤2
␤3
冊
,
兩 ␤ 1 兩 2 ⫹ 兩 ␤ 2 兩 2 ⫹ 兩 ␤ 3 兩 2 ⫽1,
␳ 1 ⫽ 兩 ␤ 1 兩 , ␳ 2 ⫽ 共 1⫺ 兩 ␤ 3 兩 2 兲 1/2.
A n →A n⬘ ⫽D n 共 ␪ 1 , ␪ 2 , . . . , ␪ n 兲 A n D n 共 ␪ ⬘1 , ␪ ⬘2 , . . . , ␪ ⬘n 兲 ,
共32兲
We would like to know: how many independent invariants
can we construct out of A n under these transformations, how
many of them are phases, and how can they be captured
through four-vertex Bargmann invariants? In the case of the
matrix A of Eq. 共17兲 the transformation 共32兲 amounts to
changing the phase of each ␺ j and each ␾ k independently,
j⭐k⭐n⫺1,
⫽ 共 1⫺ 兩 ␨ j⫹1 兩 2 ⫺ 兩 ␨ j⫹2 兩 2 ⫺•••⫺ 兩 ␨ n 兩 2 兲 1/2.
We are now interested in the following operation: suppose
we premultiply and post multiply A n by two independent
diagonal elements of U(n) 共a ‘‘gauge transformation’’ of
A n ),
D n 共 ␪ 1 , ␪ 2 , . . . , ␪ n 兲 ⫽diag共 e i ␪ 1 ,e i ␪ 2 , . . . ,e i ␪ n 兲 .
j⫽2,3, . . . ,n,
共31兲
共30b兲
We notice in passing that these are not elements of SU(2)
and SU(3), respectively.
Going back to the proof of Eq. 共20兲, we see that it can be
recursively established, and ␹ , ␣គ , ␤គ , . . . , ␰គ , ␩គ , ␨គ supply us
with exactly n 2 real independent parameters for A n .
1
independent quantities
Of these, the
2 n(n⫺1)
兩 ␣ 1 兩 , 兩 ␤ 1 兩 , 兩 ␤ 2 兩 , . . . , 兩 ␨ 1 兩 , 兩 ␨ 2 兩 , . . . , 兩 ␨ n⫺1 兩 are of the modulus
type and then there are 12 n(n⫹1) independent phases. We
can display a general element A n 苸U(n) 关in particular, A of
Eq. 共17兲兴 in the self-evident forms
␺ ⬘j ⫽e ⫺i ␪ j ␺ j ,
␾ k⬘ ⫽e i ␪ k⬘ ␾ k ,
a ⬘jk ⫽e i( ␪ j ⫹ ␪ k⬘ ) a jk ,
共33兲
and we seek an independent set of invariant expressions of
the form 共18兲.
First we count the expected numbers of invariants of each
kind. The real dimension of U(n) is n 2 . The number of independent ␪ ’s and ␪ ⬘ ’s in 关Eq. 共32兲兴 is (2n⫺1) because an
overall constant phase can be attributed to either the left or
the right diagonal factor. Therefore, the number of real invariants is (n⫺1) 2 . In the description 共31兲 of a general A n
苸U(n), it is clear that under 关Eq. 共32兲兴 every component of
each of ␣គ , ␤គ , . . . , ␨គ just undergoes a phase change, so
the quantities 兩 ␣ 1 兩 , 兩 ␤ 1 兩 , 兩 ␤ 2 兩 , . . . , 兩 ␨ 1 兩 , 兩 ␨ 2 兩 , . . . , 兩 ␨ n⫺1 兩 are
1
2 n(n⫺1) real independent modulus-type invariants. Thus
there must be a balance of 21 (n⫺1)(n⫺2) real independent
phase-type invariants. This is in agreement with known results 关16兴.
We shall describe in the next section a recursive procedure by which we can pick out 12 (n⫺1)(n⫺2) algebraically
independent four-vector Bargmann invariants whose phases
are the expected phase invariants associated with a general
U(n) matrix.
V. DETERMINATION OF INDEPENDENT
BARGMANN INVARIANTS
We first describe how, in a recursive manner, we can isolate the expected 21 (n⫺1)(n⫺2) independent gaugeinvariant phases for a generic A n 苸U(n) using the parametrization 共20兲, and then turn to the choice of an equal number of
independent primitive Bargmann invariants ⌬4 .
We begin with Eqs. 共20兲 and 共31兲,
A n ⫽A n 共 ␨គ 兲 A n⫺1 ,
The point is that, according to the statement accompanying
Eq. 共27兲, after removal of this diagonal factor what remains
is necessarily A n ( ␨គ ⬘ )⫽a jk ( ␨គ ⬘ ). Combining the above steps
we get
A n⫺1 ⫽A n⫺1 共 ␩គ 兲 A n⫺2 共 ␰គ 兲 •••A 2 共 ␣គ 兲 A 1 共 ␹ 兲
⫽A n⫺1 共 ␩គ , ␰គ , . . . , ␣គ , ␹ 兲 ,
共34兲
apply diagonal matrices on the left and on the right as in Eq.
共32兲, and trace the changes that occur in ␨គ and in A n⫺1 ,
A ⬘n ⫽D n 共 ␪ 1 , ␪ 2 , . . . , ␪ n 兲 A n D n 共 ␪ 1⬘ , ␪ ⬘2 , . . . , ␪ ⬘n 兲
⬘ .
⫽A n 共 ␨ ⬘ 兲 A n⫺1
共35兲
⬘ . Since the D n factors are
Our aim is to compute ␨គ ⬘ and A n⫺1
quite elementary, this can be carried through as follows:
A n⬘ ⫽D n 共 ␪ 1 , ␪ 2 , . . . , ␪ n 兲 A n 共 ␨គ 兲 A n⫺1 D n 共 ␪ 1⬘ , ␪ ⬘2 , . . . , ␪ ⬘n 兲
⫽D n 共 ␪ 1 , ␪ 2 , . . . , ␪ n 兲 A n 共 ␨គ 兲 A n⫺1 D n 共 0,0, . . . ,0,␪ n⬘ 兲
⬘ ,0兲
⫻D n 共 ␪ ⬘1 , ␪ 2⬘ , . . . , ␪ n⫺1
⫽D n 共 ␪ 1 , ␪ 2 , . . . , ␪ n 兲 A n 共 ␨គ 兲 D n 共 0, . . . ,0,␪ n⬘ 兲 A n⫺1
⬘ 兲.
⫻D n⫺1 共 ␪ 1⬘ , ␪ ⬘2 , . . . , ␪ n⫺1
共36兲
The product of the first three factors simplifies as
D n 共 ␪ 1 , ␪ 2 , . . . , ␪ n 兲 A n 共 ␨គ 兲 D n 共 0, . . . ,0,␪ n⬘ 兲
⫽D n 共 ␪ 1 , ␪ 2 , . . . , ␪ n 兲 „a jk 共 ␨គ 兲 …D n 共 0,0, . . . , ␪ ⬘n 兲
⫽b jk 共 ␨គ 兲 ,
b jn 共 ␨គ 兲 ⫽ ␨ ⬘j ⫽e i( ␪ j ⫹ ␪ n⬘ ) ␨ j ,
b jk 共 ␨គ 兲 ⫽e i ␪ j a jk 共 ␨គ 兲 ,
k⫽1,2, . . . ,n⫺1.
共37兲
Here the matrix elements a jk ( ␨គ ) are given in Eq. 共28兲 and for
simplicity the ␪ and ␪ ⬘ dependences of b jk are left implicit.
In particular, as in Eq. 共27兲,
b jk 共 ␨គ 兲 ⫽0,
k⫽1,2, . . . , j⫺2;
j⫽3,4, . . . ,n, 共38兲
while
b j, j⫺1 共 ␨គ 兲 ⫽e i ␪ j a j, j⫺1 共 ␨គ 兲 ⫽e i ␪ j ␳ j⫺1 / ␳ j ,
j⫽2,3, . . . ,n.
共39兲
Thus the matrix b jk ( ␨គ ) would have been a jk ( ␨គ ⬘ ) except for
the fact that the elements b j, j⫺1 ( ␨គ ) just below the main diagonal are not real positive but carry phases. But this can be
easily taken care of by extracting a suitably chosen diagonal
matrix on the right,
b jk 共 ␨គ 兲 ⫽„a jk 共 ␨គ ⬘ 兲 …D n 共 ␪ 2 , ␪ 3 , . . . , ␪ n ,0兲 .
共40兲
⬘
A n⬘ ⫽A n 共 ␨គ ⬘ 兲 A n⫺1
⬘ 兲
⫽„b jk 共 ␨គ 兲 …A n⫺1 D n⫺1 共 ␪ ⬘1 , ␪ 2⬘ , . . . , ␪ n⫺1
⫽A n 共 ␨គ ⬘ 兲 D n 共 ␪ 2 , ␪ 3 , . . . , ␪ n ,0兲
⬘ 兲,
⫻A n⫺1 D n⫺1 共 ␪ ⬘1 , ␪ ⬘2 , . . . , ␪ n⫺1
共41兲
so the changes induced in ␨គ and in A n⫺1 by the gauge transformation 共35兲 are
␨ ⬘j ⫽e i( ␪ j ⫹ ␪ ⬘n ) ␨ j ,
共42a兲
j⫽1,2, . . . ,n,
⬘ ⫽D n⫺1 共 ␪ 2 , ␪ 3 , . . . , ␪ n 兲 A n⫺1 D n⫺1 共 ␪ ⬘1 , ␪ 2⬘ , . . . , ␪ n⫺1
⬘ 兲.
A n⫺1
共42b兲
We see from the structure of this result that we can tackle our
problem recursively. The gauge transformation 共35兲 at the
U(n) level translates into the change ␨គ → ␨គ ⬘ given by Eq.
⬘ at the U(n
共42a兲 and a gauge transformation A n⫺1 →A n⫺1
⫺1) level given by Eq. 共42b兲. Therefore, all gauge-invariant
expressions that exist at the A n⫺1 or U(n⫺1) level survive
when we move from U(n⫺1) to U(n) and in addition as the
vector ␨គ 苸Bn becomes available, new invariant phases involving ␨គ can be constructed. The number of the latter can be
immediately computed: it is the difference between 21 (n
⫺1)(n⫺2) and 12 (n⫺2)(n⫺3), namely, the difference between the numbers of gauge-invariant phases at the U(n) and
the U(n⫺1) levels, and this is (n⫺2). Therefore, the number of new independent phase invariants involving A n ( ␨គ ),
i.e., ␨គ , in an essential way must be (n⫺2). These can now be
isolated or explicitly constructed as follows.
From Eq. 共42兲 we notice that ␪ 1 and ␪ n⬘ appears only in
the transformation law for ␨គ , not for A n⫺1 . Therefore, we
first form the (n⫺1) independent combinations ␨ *j ␨ j⫹1 to
eliminate ␪ n⬘ completely:
␨ *j ␨ j⫹1 →exp关 ⫺i 共 ␪ j ⫺ ␪ j⫹1 兲兴 ␨ *j ␨ j⫹1 ,
j⫽1,2, . . . ,n⫺1.
共43兲
*
Here ␪ 1 occurs only in the transformation law for ␨ 1 ␨ 2 , being absent as we have just mentioned in the law for A n⫺1 .
⬘ , involved
Next, we notice that the phases ␪ 1⬘ , ␪ 2⬘ , . . . , ␪ n⫺1
⬘ , are completely absent in the transformation law
in A n⫺1
共43兲 of ␨ *j ␨ j⫹1 . Let us, therefore, look at the (n⫺1)th column, say, of A n⫺1 , which, as is evident from Eq. 共34兲, is just
the (n⫺1) component complex unit vector ␩គ 苸Bn⫺1 :
A n⫺1 ⫽
冉
...
...
␩1
0
...
...
␩2
0
...
...
0
...
␩ n⫺1 0
0
1
冊
.
共44兲
The ‘‘earlier’’ columns of A n⫺1 are more complicated, as is
clear from the structure of A n⫺1 in Eq. 共34兲. From Eq. 共42b兲
we can read off the transformation law for the ␩ ’s under the
gauge transformation 共35兲:
⬘ )␩ ,
␩ ⬘j ⫽e i( ␪ j⫹1 ⫹ ␪ n⫺1
j
j⫽1,2, . . . ,n⫺1.
共45兲
involving some two adjacent rows and some two adjacent
columns. This is to be understood modulo real positive definite factors coming from the squared moduli of some of the
matrix elements of A n . The two ‘‘recursion formulas’’ that
help us achieve this simplification are
⌬j l
km ⫽⌬ j l km⫺1 ⌬ j l m⫺1m / 兩 a j,m⫺1 a l ,m⫺1 兩
2
⫺1km ⌬l ⫺1 l km / 兩 a l ⫺1,k a l ⫺1,m 兩
.
⬘ we form the (n⫺2) combinations ␩ j ␩ *j⫹1
To eliminate ␪ n⫺1
that transform thus,
␩ j ␩ *j⫹1 →e ⫺i( ␪ j⫹2 ⫺ ␪ j⫹1 ) ␩ j ␩ *j⫹1 ,
j⫽1,2, . . . ,n⫺2.
共46兲
Comparing Eqs. 共43兲 and 共46兲 we immediately obtain the
expected (n⫺2) independent 共phase-type兲 invariants involving ␨គ 苸Bn in an essential manner, namely, they can be taken
to be the complex quantities
␩ j ␩ *j⫹1 ␨ *j⫹1 ␨ j⫹2 ,
j⫽1,2, . . . ,n⫺2.
共47兲
By recursion the complete set of 21 (n⫺1)(n⫺2) independent phase-type invariants that can be formed from a generic
matrix A n 苸 U(n) can be written down in terms of the canonical parametrization 共20兲 for A n , and the list reads
␣ j ␣ *j⫹1 ␤ *j⫹1 ␤ j⫹2 ,
j⫽1,
␤ j ␤ *j⫹1 ␥ *j⫹1 ␥ j⫹2 ,
j⫽1,2,
]
␰ j ␰ *j⫹1 ␩ *j⫹1 ␩ j⫹2 ,
j⫽1,2, . . . ,n⫺3,
␩ j ␩ *j⫹1 ␨ *j⫹1 ␨ j⫹2 ,
j⫽1,2, . . . ,n⫺2.
共48兲
While here we have an explicit solution to our problem,
the difficulty is that these invariants are not directly expressed in terms of the matrix elements of A n ⫽(a jk )
苸U(n). It is true that in our parametrization ␨គ is the last,
nth, column of A n ; but the previous, (n⫺1)t column involves both ␩គ and ␨គ ; the (n⫺2)th column involves ␰គ , ␩គ , and
␨គ , and so on. The task that remains is to see how to translate
the expressions 共48兲, as far as their phases are concerned,
into an algebraically equivalent set of 21 (n⫺1)(n⫺2) expressions formed as simply as possible out of the matrix
elements of A n . We turn to this now, bringing in the fourvertex Bargmann invariants of A n .
As indicated in Eq. 共18兲, a general four-vertex Bargmann
invariant requires the choice of some two rows, say j and l
with j⬍ l , and some two columns, say k and m with k⬍m,
and use of the four matrix elements at their intersections,
⌬j l
km ⬅a jk a *
l ka l ma *
jm .
共49兲
But as far as phases go, we can see in a step by step manner
that a general ⌬ j l km reduces to a product of factors of the
simpler form,
⌬ jk ⬅⌬ j, j⫹1,k,k⫹1 ,
共50兲
⫽⌬ j l
2
,
共51兲
It, therefore, suffices to work with the (n⫺1) 2 expressions
⌬ jk ⫽a jk a *j⫹1,k a j⫹1,k⫹1 a *j,k⫹1 ,
j,k⫽1,2, . . . ,n⫺1,
共52兲
and their phases. Our goal now is to 共at least in principle and
in the generic situation兲 express 共the phases of兲 the 21 (n
⫺1)(n⫺2) complex invariants 共48兲 in terms of 共the phases
of兲 the (n⫺1) 2 complex invariants 共52兲. 共In this process any
number of real positive factors may intervene兲. Here we already have an indication that the (n⫺1) 2 expressions 共52兲
共more exactly their phases兲 cannot all be independent, the
number of independent ones being only 21 (n⫺1)(n⫺2). It
will turn out, as we indicate below, that these may be taken
to be the ⌬ jk for j⬍k⭐n⫺1. Again the proof is recursive in
nature.
Consider the (n⫺1) invariants 共47兲 that get added to all
previous ones when we make the transition U(n⫺1)
→U(n) and bring in the vector ␨គ 苸Bn . Instead of being
expressed in terms of ␨គ and ␩គ 苸Bn⫺1 , we now show that
they can be equally well expressed in terms of ␨គ and the
penultimate, i.e., (n⫺1)th column of the complete U(n) matrix A n . Let us denote this column vector by wគ 苸Bn ; it is
orthogonal to ␨គ . As noted earlier, it is easily determined in
terms of ␨គ and ␩គ or, more conveniently for our purpose, ␩គ is
expressible in terms of wគ and ␨គ . Starting with
冉
•••
•••
A n ⫽ •••
•••
•••
•••
冊
w1
␨1
w2
␨ 2 ⫽A n 共 ␨គ 兲 A n⫺1 共 ␩គ 兲 A n⫺2 ,
␨n
共53兲
wn
and transposing A n ( ␨គ ) we get
A n⫺1 共 ␩គ 兲 A n⫺2 ⫽A n 共 ␨គ 兲 † A n .
共54兲
Since the factor A n⫺2 , does not affect the last two columns
on both sides, we can use the matrix elements 共27兲 and 共28兲
of A n ( ␨គ ) to obtain
j⫹1
␩ j⫽
兺
k⫽1,2, . . .
⫺
␨ j⫹1
␳ j ␳ j⫹1
a k j 共 ␨គ 兲 * w k ⫽
␳j
␳ j⫹1
w j⫹1
j
兺
k⫽1
␨ *j w j ,
j⫽1,2, . . . ,n⫺1.
共55兲
The gauge transformation laws of ␨គ and ␩គ are given in Eqs.
共42a兲 and 共45兲, while that of wគ is seen from Eq. 共35兲 to be
⬘ )w ,
w j →e i( ␪ j ⫹ ␪ n⫺1
j
j⫽1,2, . . . ,n.
共56兲
Naturally the relations 共55兲 are consistent with these transformation laws. The combinations of ␩គ and ␨គ needed in 共47兲
are ␩ j ␨ *j⫹1 for j⫽1,2, . . . ,n⫺1. We see from Eq. 共55兲
that
they
are
real
linear
combinations
of
* ,w n ␨ n* ,
w 1 ␨ 1* ,w 2 ␨ 2* , . . . ,w n⫺1 ␨ n⫺1
␩ j ␨ *j⫹1 ⫽⫺
兩 ␨ j⫹1 兩 2
␳ j ␳ j⫹1
j
␳
兺 ␨ *j w j ⫹ ␳ j⫹1j
k⫽1
w j⫹1 ␨ *j⫹1 . 共57兲
Using both the orthogonality of wគ and ␨គ , and the reduction
process 共51兲 for ⌬4 ’s formed out of the last two columns of
A n , it is now clear that the set of complex invariants 共47兲 can
be replaced by the following set of (n⫺2)⌬4 ’s:
⌬ j,n⫺1 ⫽w j• ␨ *j w *j⫹1 ␨ j⫹1 ,
j⫽1,2, . . . ,n⫺2.
共58兲
The known algebraic independence of the set 共47兲 implies a
similar independence of these ⌬4 ’s.
To tackle the next set of (n⫺3) invariants
␰ j ␩ *j⫹1 ␰ *j⫹1 ␩ j⫹2 for j⫽1,2, . . . ,n⫺3 in the list 共48兲, we
must bring in the (n⫺2)th column of the matrix A n . Denote
this by vគ 苸Bn so that
A n⫽
冉
•
•
•
•
•
•
•
•
•
v1
w1
␨1
•
v2
w2
␨2
•
]
]
]
vn
wn
•
␨n
冊
.
共59兲
共60兲
from where we get expressions for ␰ j in terms of ␨, ␩, and vគ.
This is naturally more complicated than Eq. 共55兲 at the previous stage. In place of the real positive factors ␳ j defined in
terms of ␨គ in Eq. 共28兲, we now have similarly defined factors
␴ j in terms of ␩គ occurring in the elements of A n⫺1 ( ␩គ ). The
result of comparing the (n⫺2)th columns of both sides of
Eq. 共60兲 is
j⫹1
␴ j ␳ j⫹1
␴j
␰ j⫽
␨
v j⫹2 ⫺
␴ j⫹1 ␳ j⫹2
␴ j⫹1 ␳ j⫹1 ␳ j⫹2 j⫹2 l
兺⫽1 ␨ *l v l
⫺
␩ j⫹1
␴ j ␴ j⫹1
⫹
␩ j⫹1
␴ j ␴ j⫹1
j
␳
兺 k ␩ k* v k⫹1
k⫽1 ␳ k⫹1
j
兺
k⫽1
␩ *k ␨ k⫹1
␳ k ␳ k⫹1
k
兺
l ⫽1
␨*
l vl ,
⌬ j,n⫺2 ⫽ v j w *j v *j⫹1 w j⫹1 ,
j⫽1,2, . . . , 共 n⫺3 兲 . 共62兲
In this manner, one sees recursively that the 21 (n⫺1)(n
⫺2) independent gauge-invariant phases in a general matrix
A n 苸U(n) are the Bargmann invariants ⌬ jk for j⬍k⭐n⫺1.
In any case such a choice is permitted. However, the actual
algebraic expression of a general ⌬ jk in terms of this special
subset may be rather involved, so one may freely use all ⌬ jk
in constructing interesting gauge-invariant expressions with
various properties.
The upshot of these considerations is that the naturally
available gauge-invariant phases for the continuous unitary
evolution of an n-level quantum system, barring degeneracies and level crossings, are n geometric phases ␸ g 关 C j 兴 as
defined in Eq. 共13兲, and the (n⫺1) 2 primitive four-vertex
Bargmann invariants ⌬ jk of Eq. 共52兲; of the latter, only the
1
2 (n⫺1)(n⫺2)⌬ jk ’s for j⬍k⭐n⫺1 are independent. Any
composite expression formed out of these ingredients is, of
course, also invariant.
VI. OFF-DIAGONAL GEOMETRIC PHASES
Analogous to Eq. 共54兲 we now have
A n⫺2 共 ␰គ 兲 A n⫺3 ⫽A n⫺1 共 ␩គ 兲 † A n 共 ␨គ 兲 † A n ,
v ’s, and w’s, v ’s and ␨ ’s, and w’s and ␨ ’s. Using the reduction rules 共51兲 the v ⫺ ␨ combinations can be eliminated in
favor of the other two types. It is now clear that apart from
the ⌬ j,n⫺1 in Eq. 共58兲 that appeared at the previous stage, the
new quantities that come in now are ⌬ j,n⫺2 . But we know in
advance that at this stage only (n⫺3) new independent invariants are available. As all the rows of ⌬n are on equal
footing, we conclude that the new ⌬4 ’s to be added now to
the previous ⌬ j,n⫺1 may be taken to be
j⫽1,2, . . . ,n⫺2.
共61兲
Here next we can use Eq. 共55兲 to go from ␩គ to wគ . Then we
form the expressions ␰ j ␩ *j⫹1 and step by step work our way
up to the invariants ␰ j ␩ *j⫹1 ␰ *j⫹1 ␩ j⫹2 . We can then see that
apart from various real factors we encounter ⌬4 ’s involving
It is evident from the definitions 共6兲 that while the dynamical phase ␸ dyn关 C兴 is always numerically well defined
once the parametrized curve C is given, the total phase
␸ tot关 C兴 is only defined modulo 2 ␲ and, moreover, is undefined if the vectors ␺ (s 1 ) and ␺ (s 2 ) at the end points of C
are orthogonal. These properties naturally carry over to the
geometric phase ␸ g 关 C 兴 : only defined modulo 2 ␲ , undefined
when ␸ tot关 C兴 is undefined. The Bargmann invariants 共7兲 too
share these problems of definition as far as their phases are
concerned, which explains the limitation to generic situations.
Recently a very interesting attempt to define so-called offdiagonal geometric phases has been made to cover just these
exceptional or problematic situations 关11兴. Specifically, the
idea is to set up gauge-invariant phases associated with the
unitary evolution of an n-level quantum system, which remain well defined even when one of the eigenvectors of the
Hamiltonian at a final time t 2 , say the kth one, happens to
coincide with the jth eigenvector at the initial time t 1 , with
j⫽k. In this situation, as ␺ j (t 1 ) and ␺ k (t 2 ) are the same up
to a phase, both the geometric phases ␸ g 关 C j 兴 and ␸ g 关 C k 兴
become undefined since the inner products 关 ␺ j (t 1 ), ␺ j (t 2 ) 兴
and 关 ␺ k (t 1 ), ␺ k (t 2 ) 兴 vanish. We shall now briefly recall the
basic quantities introduced in this new approach and then
show that the usual geometric phases and Bargmann invariants as defined earlier can completely handle the new situation. It is just that they must be put together in such combi-
nations so that the potentially undefined factors in each
precisely cancel one another in exceptional situations.
The notation for the evolution of an n-level quantum system is as given in Sec. III. The quantities defined in the
off-diagonal geometric phases method, when expressed in
our notations, are
I j ⫽exp兵 ⫺i ␸ dyn关 C j 兴 其 ,
j⫽1,2, . . . ,n,
␴ jk ⫽exp兵 i arg共 ␺ j , ␾ k 兲 ⫺i ␸ dyn关 Ck 兴 其 ,
␥ jk ⫽ ␴ jk ␴ k j ,
␥ j ⫽exp兵 i ␸ g 关 C j 兴 其 ,
j⫽k,
j⫽k,
j⫽1,2, . . . ,n.
共63a兲
共63b兲
共63c兲
共63d兲
Of these, I j and ␴ jk are not gauge invariant, but ␥ jk and ␥ j
are gauge invariant. In case for some j⫽k we have
兩 ( ␺ j , ␾ k ) 兩 ⫽1, it is clear that both ␸ g 关 C j 兴 and ␸ g 关 C k 兴 become undefined, but the ‘‘off-diagonal’’ quantity ␥ jk remains
well defined.
The two-state or two-index quantity ␥ jk has been generalized to a multi-index quantity of order l as follows:
there are compensating factors from ⌬4 ( ␺ j , ␾ k , ␺ k , ␾ j ) that
precisely cancel these parts of the individual geometric
phases, so that ␥ jk remains unambiguous. The mechanism is
similar in the case of the higher-order expressions ␥ j 1 j 2 ••• j l .
It has been shown that ␥ j 1 j 2 ••• j l for l ⭓4 can be reduced
to the expressions with l ⫽2 and l ⫽3, so these are the
primitive ones. Among these, we can limit ourselves to
choices obeying j 1 ⬍ j 2 when l ⫽2 and j 1 ⫽1⬍ j 2 ⬍ j 3 when
l ⫽3, in counting independent quantities. However, the upshot of our analysis is that we can always work with just the
geometric phases ␸ g 关 C j 兴 and the independent ⌬4 ’s listed in
the previous section 共but for convenience employ all the ⌬ jk
if necessary兲. All gauge-invariant quantities can be built up
out of them, so that conceptually the off-diagonal geometric
phases are constructed out of previously known familiar
building blocks.
VII. CONCLUDING REMARKS
In the case of ␥ jk , for example, we see that when
兩 ( ␺ j , ␾ k ) 兩 ⫽1 and ␸ g 关 C j 兴 , ␸ g 关 C k 兴 become undefined because the total phases ␸ tot关 C j 兴 and ␸ tot关 Ck 兴 are undefined,
We have carried out a complete analysis of the gaugeinvariant objects for n-level quantum systems. This entails
introduction of Bargmann invariants defined over two sets of
orthonormal basis vectors, demonstration that the primitive
Bargmann invariants are four-vertex Bargmann invariants,
and finally the identification of an algebraically independent
set of four-vertex Bargmann invariants that turn out to be
(n⫺1)(n⫺2)/2 in number. In the process of achieving this
task we developed a canonical form for U(n) matrices in
terms of a sequence of complex unit vectors of dimensions
n,n⫺1, . . . ,1 which may be useful in other contexts as well.
Indeed, this form has already found application in parametrizing the CKM matrices that arise in the context of C P
violation in particle physics. The gauge-invariant building
blocks constructed here are shown to provide a complete
quantum kinematic picture of the recently discovered offdiagonal phases. The usefulness of the off-diagonal phases is
thus extended far beyond the restrictive framework of adiabatic evolution. This reinforces the view that the Bargmann
invariants and the traditional geometric phases, and suitably
constructed combinations of them, suffice in answering all
interesting questions in this domain.
关1兴 M.V. Berry, Proc. R. Soc. London, Ser. A 392, 45 共1984兲.
关2兴 N. Mukunda and R. Simon, Ann. Phys. 共N.Y.兲 228, 205 共1993兲;
ibid. 228, 269 共1993兲.
关3兴 Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593
共1987兲.
关4兴 J. Samuel and R. Bhandari, Phys. Rev. Lett. 60, 2339 共1988兲.
关5兴 Many of the early papers on geometric phase have been reprinted in Geometric Phases in Physics, edited by A. Shapere
and F. Wilczek 共World Scientific, Singapore, 1989兲; and in
Fundamentals of Quantum Optics, SPIE Milestone Series, edited by G. S. Agarwal 共SPIE Press, Bellington, 1995兲.
关6兴 E.M. Rabei, Arvind, N. Mukunda, and R. Simon, Phys. Rev. A
60, 3397 共1999兲.
关7兴 V. Bargmann, J. Math. Phys. 5, 862 共1964兲.
关8兴 W.R. Hamilton, Lectures on Quaternions 共Dublin, 1853兲; L.C.
Biedenharn and J.D. Louck, Angular Momentum in Quantum
Physics, Encyclopedia of Mathematics and its applications,
Vol. 8 共Addison-Wesley, Reading, MA, 1981兲. Hamilton’s
theory of turns has been generalized to the simplest noncompact semisimple group SU(1,1)⬃SL(2,R)⬃Sp(2,R) in R. Simon, N. Mukunda, and E.C.G. Sudarshan, Phys. Rev. Lett. 62,
1331 共1989兲; J. Math. Phys. 30, 1000 共1989兲. and to the Lorentz group in R. Simon, S. Chaturvedi, V. Srinivasan, and N.
Mukunda, Hamilton’s Turns for the Lorentz Group 共The Institute of Mathematical Sciences, Chennai, in press兲; the relationship between Hamilton’s turns and geometric phase for twolevel systems is described in R. Simon and N. Mukunda, J.
Phys. A 25, 6135 共1992兲.
␥ j 1 j 2 ••• j l ⫽ ␴ j 1 j 2 ␴ j 2 j 3 ••• ␴ j l ⫺1 j l ␴ j l j 1 ,
共64兲
and this again is gauge invariant.
We can now see that all these newly introduced gaugeinvariant off-diagonal quantities ␥ jk , ␥ j 1 j 2 ••• j l are actually
expressible completely in terms of the geometric phases and
Bargmann invariants for the n-level system, in carefully chosen combinations,
␥ jk ⫽exp兵 i arg ⌬4 共 ␺ j , ␾ k , ␺ k , ␾ j 兲 ⫹i ␸ g 关 C j 兴 ⫹i ␸ g 关 C k 兴 其 ,
␥ j 1 j 2 . . . j l ⫽exp兵 i arg ⌬2 l 共 ␾ j 1 , ␺ j 1 , ␾ j 2 , ␺ j 2 , . . . , ␾ j l , ␺ j l 兲
⫹i ␸ g 关 C j 1 兴 ⫹i ␸ g 关 C j 2 兴 ⫹•••⫹i ␸ g 关 C j l 兴 其 .
共65兲
关9兴 R. Simon and N. Mukunda, Phys. Rev. Lett. 70, 880 共1993兲.
关10兴 S. Chaturvedi and N. Mukunda, Int. J. Mod. Phys. A 16, 1481
共2001兲.
关11兴 F. Pistolesi and N. Manini, Phys. Rev. Lett. 85, 1585 共2000兲;
N. Manini and F. Pistolesi, ibid. 85, 3067 共2000兲.
关12兴 D.E. Manolpoulos and M.S. Child, Phys. Rev. Lett. 82, 2223
共1999兲.
关13兴 S. Panchratnam, Proc.-Indian Acad. Sci., Sect. A 44, 247
共1956兲; S. Ramaseshan and R. Nityananda, Curr. Sci. 55, 1225
共1986兲.
关14兴 D. Page, Phys. Rev. A 36, 3479 共1987兲.
关15兴 S. Kobayashi and K. Nomizu, Foundations of Differential Geometry 共Interscience Publishers, New York, 1969兲, Vol. II,
Chap. 9.
关16兴 H. Harari and M. Leurer, Phys. Lett. B 181, 123 共1986兲; J.F.
Nieves and P.B. Pal, Phys. Rev. D 36, 315 共1987兲.