Observation of Geometric Phases for Three-Level Systems using NMR Interferometry
Hongwei Chen
1
, Mingguang Hu
2
, Jingling Chen
2
, and Jiangfeng Du
1∗
1
Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
2
Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, China
(Dated: January 5, 2009)
Geometric phase independent of energy and time rely only on the geometry of state space. It has
b een argued to have potential fault tolerance and plays an important role in quantum information
and quantum computation. We present the first experiment for producing and measuring an abelian
geometric phase shift in a three-level system by using NMR interferometry. In contrast to existing
exp eriments, based on the geometry of S
2
in evolving space, our experiment concerns the geometric
phase associated with SU (3)/U(2) geometry. It also provides a realistic candidate for designing
more complex geometric quantum circuits.
PACS numbers: 03.65.Vf, 76.60.-k
Introduction. —When a quantum mechanical system
evolves cyclically in time so that it returns to its initial
physical state, its wave function can acquire a geometric
phase factor in addition to the familiar dynamic phase
[1]. If the cyclic change of the system is adiabatic, this
additional factor is known as Berry’s phase [2]. Other-
wise, it is related to Aharonov-Anandan phase [3] that
has been pointed out to be a continuous version of earlier
Pancharatnam phase [4]. Geometric phases (GP) inde-
pendent of energy and time rely only on the geometry of
state space. It is therefore argued to be resilient to cer-
tain types of errors and suggest the possibility of an in-
trinsically fault-tolerant way of performing quantum gate
operations [5, 6]. This potentiality makes it important
to observe and apply GP in various quantum systems.
The GP observation for a qubit has been made in vari-
ous systems [7, 8, 9, 10, 11, 12]. The principle of them
is geometrically the same. That is, in a two-level Hilbert
space with geometry of a unit sphere S
2
, the cyclic GP
adiabatic or non-adiabatic is just one half the solid angle
subtended by closed evolution paths on the sphere.
When one generalizes to a three-level quantum sys-
tem, the four-dimensional SU (3)/U(2) geometry, equiv-
alent to a slice on sphere S
7
, should be considered as
the counterpart of S
2
in two-level systems [13, 14]. This
geometry change can be seen from the number of inde-
pendent parameters needed in fixing a state. The uni-
tary evolution for a three-level state in general belongs
to a SU(3) group operation and it is theoretically and
experimentally more complicate than that of SU(2) in
the two-level case. Besides, the cyclic GP in three-level
ray space is no longer as simply related to solid angles as
that in two-level case but referred to the Bargmann in-
variants [15, 16]. All these increased flexibilities in three-
level systems, however, may substantially provide more
choices for designing geometric quantum circuits. See the
related papers [17, 18] for merits of the GP-based geo-
metric quantum gates. Simultaneously it also indicates
an increased complexity on the experimental scheme for
observing three-level GP as pointed in [19].
In this Letter, we report an experimental observation
of the non-adiabatic cyclic GP for three-level systems by
using NMR interferometry. A two spin-1/2 interacting
system having four levels is exploited in the experiment
to contribute three target levels and a remainder level.
Unitary evolutions, belonging to SU(3), for performing
cyclic paths in the three-dimensional ray space are im-
plemented by using quantum controlled logic gates [20].
Aimed at obtaining a measurable GP, we utilize the re-
mainder level to be as a reference state that is kept un-
changed during all evolutions. The dynamical phase fac-
tor can be vanished by evolving along geodesics in the
projective Hilbert space. As a result, an observable geo-
metric phase factor appears as a local phase between the
three-level target state and the reference state.
Three-level GP. —In a three-dimensional Hilbert
space H
3
, an arbitrary three-component state can be ex-
pressed as
|ψi = e
iη
¡
e
iχ
1
cos α
1
, e
iχ
2
sin α
1
cos α
2
, sin α
1
sin α
2
¢
T
,(1)
where the real parameters have the range α
1,2
∈ [0, π/2]
and χ
1,2
∈ [0, 2π). Actually we can geometrically view
(α
1
, α
2
) and (ξ
1
, ξ
2
) as points on an octant of S
2
and a
torus (see Fig. 3), respectively. Consider a state evolving
from |ψ(s
1
)i to |ψ(s
2
)i and curve parameters s
1,2
consist
of the five parameters defined in Eq. (1). Corresponding
to this evolution, in H
3
there is a continuous piecewise
smooth parametrized curve, C =
©
ψ(s)
¯
¯
s
1
≤ s ≤ s
2
ª
,
and its image in the ray space R is likewise contin-
uous and piecewise smooth denoted by C =
©
ρ(s) =
|ψ(s)ihψ(s)|
¯
¯
s
1
≤ s ≤ s
2
ª
. Then the GP β associated
with the curve C equals to the difference between a total
phase ϕ
tot
and a dynamical phase γ
d
[14], that is,
β[C] = ϕ
tot
[C] − γ
d
[C],
ϕ
tot
[C] = arghψ(s
1
)|ψ(s
2
)i, (2)
γ
d
[C] = −
Z
s
2
s
1
dshψ(s)|i
∂
∂s
|ψ(s)i,
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