PHYSICAL REVIEW A 88, 023611 (2013)
Double Landau-Zener interferometry in a dc-ac–driven Bloch-Zener oscillation
Y. Mizumoto
1,2,*
and Y. Kayanuma
1,2,3,†
1
Research Organization for the 21st Century, Osaka Prefecture University, Sakai 599-8531, Japan
2
CREST, Japan Science and Technology Agency, Kawaguchi, Saitama 332-001, Japan
3
Materials and Structures Laboratory, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama, Japan
(Received 21 February 2013; revised manuscript received 29 July 2013; published 16 August 2013)
The effect of an additional high frequency alternating field on the Bloch-Zener oscillation of a quantum particle
is investigated theoretically. The effective magnitude of the band gap is reduced by the ac field, and even collapsed
to zero by appropriately tuning the amplitude and the frequency of the oscillation. This results in the enhancement
of the Zener tunneling with dramatic changes of the Bloch-Zener oscillation. The temporal behavior of the wave
packet is clarified as due to the interplay of the Landau-Zener interference in the repeated level crossings with two
different time scales, namely, the period of the Bloch oscillation, and the period of the high frequency ac field.
DOI: 10.1103/PhysRevA.88.023611 PACS number(s): 03.75.Dg, 03.75.Lm, 03.65.Vf, 78.47.jh
I. INTRODUCTION
Since the early days of quantum mechanics, it has been
predicted that electrons in the periodic lattice potentials of
solids will exhibit a peculiar temporal response to a constant
and uniform driving force: Unlike the particles in free space,
they oscillate back and forth without accelerating infinitely.
This Bloch oscillation [1] has been realized in semiconductor
superlattices [2] and also in bulk crystals under an intense
terahertz field [3]. The oscillation lasts only a few cycles in
many cases. This is due to the existence of ultrafast scatterings
and decoherence in solids.
A way to observe the Bloch oscillation avoiding the
dissipations inherent in real solids is to use simulators which
mimic the Schr¨ odinger equation by the Maxwell equation in
appropriately designed optical waveguides. A number of ex-
perimental realizations of the analogs of the Bloch oscillation
by this method have been presented with various settings [4].
On the other hand, thanks to the recent developments in the
technique of quantum simulators using ultracold atoms loaded
into optical lattices, very long lived Bloch oscillations of
atoms have been realized [5]. Actually more than a thousand
times oscillations are reported in the vertical optical lattice
loaded with strontium atoms [6]. This opens up a new field of
investigation on the macroscopic scale of quantum dynamics
of the Bloch particles, thus shedding light on the behavior of
the Bloch electrons in solids.
If the next lowest Bloch band is located above the lowest one
with a small energy gap, the particle may make nonadiabatic
transitions at the band edge between the two bands due to
the Zener tunneling [7]. As a consequence, the trajectory of
the oscillatory motion of the particle will be more abundant
of structures than that of a simple Bloch oscillation [8]. The
Bloch-Zener oscillation is of interest because of the intriguing
interference effect [9]. The experimental studies of the Bloch-
Zener oscillation have been carried out for optical waveguides
[10] and for optical lattices [11]. It has become a subject of
attention also because it carries information on the magnitude
of the energy gap. In fact, the dynamics of the wave packet of
*
mizumoto@pe.osakafu-u.ac.jp
†
kayanuma.y.aa@m.titech.ac.jp
electrons around the Dirac points in graphene or topological
insulators has attracted interest both experimentally [12] and
theoretically [13]. It is pointed out that the Zener transition of
the wave packet around the Dirac point in the two-dimensional
lattice structure becomes a signature of the merging transitions
of the lowest two bands [13].
The effect of an oscillating external field on the dynamics
of one-dimensional Bloch particles has long been a subject
of interest. One of the remarkable effects is the dynamic
localization (DL) predicted theoretically for a tight-binding
model [14], and observed experimentally in the cold atoms in
optical lattice potentials [15]. It was shown that the interference
effect between the infinite transition paths plays an essential
role in the DL [16], and the DL can be understood on the
same basis as the coherent destruction of tunneling [17] for
two-level systems. In connection with this, the present authors
theoretically found a phenomenon called the dynamical band
gap collapse (BGC) [18]. The band gap in the one-dimensional
tight-binding model can be effectively reduced to zero by
applying an appropriately tuned ac field. The dynamical
BGC is also a manifestation of the interference effect of the
Landau-Zener transition paths in the momentum space [19].
In the present work, we propose a coherent control of
the Bloch-Zener oscillation with a high frequency ac field
applied in addition to the static field. The effect of the ac
field on the single band Bloch oscillation has been studied
experimentally [20] and theoretically [21]. The application of
the high frequency ac field on the Bloch-Zener system will
open up a new aspect of the quantum phase coherence.
II. MODEL AND NUMERICAL RESULTS
For the sake of definiteness, consider a quantum particle on
a linear chain described by the tight-binding Hamiltonian with
alternating site energies [9],
H = H
0
+ V,
H
0
=−
B
2
j
(|j + 1j |+|j j + 1|), (1)
V =
j
(−1)
j
|j j |,
023611-1 1050-2947/2013/88(2)/023611(6) ©2013 American Physical Society