Download Phase-controlled localization and directed

Phase-controlled localization and
directed transport in an optical bipartite
lattice
Kuo Hai,1,∗ Yunrong Luo,1 Gengbiao Lu,2 and Wenhua Hai1,3
1 Department of physics and Key Laboratory of Low-dimensional Quantum Structures and
Quantum Control of Ministry of Education, Hunan Normal University, Changsha 410081,
China
2 Department of Physics and Electronic Science, Changsha University of Science and
Technology, Changsha 410004, China
3 [email protected]
∗ [email protected]
Abstract: We investigate coherent control of a single atom interacting
with an optical bipartite lattice via a combined high-frequency modulation.
Our analytical results show that the quantum tunneling and dynamical
localization can depend on phase difference between the modulation
components, which leads to a different route for the coherent destruction
of tunneling and a convenient phase-control method for stabilizing the
system to implement the directed transport of atom. The similar directed
transport and the phase-controlled quantum transition are revealed for
the corresponding many-particle system. The results can be referable for
experimentally manipulating quantum transport and transition of cold atoms
in the tilted and shaken optical bipartite lattice or of analogical optical
two-mode quantum beam splitter, and also can be extended to other optical
and solid-state systems.
© 2014 Optical Society of America
OCIS codes: (020.1670) Coherent optical effects; (020.1475) Bose-Einstein condensates;
(020.0020) Atomic and molecular physics; (270.0270) Quantum optics.
References and links
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1.
Introduction
Quantum control of tunneling processes of particles plays a major role in different areas of
physics, optics and chemistry [1–4]. As early as 1986, Dunlap and Kenkre studied theoretically the quantum motion of a charged particle on a discrete lattice driven by an ac field [5],
and found the surprising result that particle transport can be completely suppressed when ratio of the strength and the frequency of the ac field takes some special values. This effect of
extreme localization was later found to be associated with the coherent destruction of tunneling (CDT) [2, 6, 7] at a collapse point of the Floquet quasienergy spectrum [8], and has also
been observed in different systems [9–12]. Generally, the dynamical localization (DL) refers
to the phenomenon wherein a particle initially localized in a lattice can transport within a finite distance and periodically return to its original state. There has been growing interest in the
quantum control of electrons in semiconductor superlattices or arrays of coupled quantum dots
from both theoretical and experimental sides [2, 13]. Most of the DL and CDT of the electronic
systems are generic and also can occur in atomic [8, 9, 14, 15] and optical [10, 16–20] systems.
Recently, different routes to CDT were found by considering, respectively, the priori prescribed number of bosons of a many-boson system [21, 22], the distinguishable intersite separations of a bipartite lattice [23, 24], the variable driving symmetry of a two-frequency driven
particle in a double-well [11, 25], and the different combined modulations to the different systems [26, 27]. The CDT mechanism has been applied to various physical fields such as the
qubit control [28, 29], the quantum tunneling switch [30], and the directed transport in a bipartite lattice via the selective CDT to the two different barriers [23, 24]. It is worth noting that the
CDT and DL mechanisms can also be applied to coherently control instability of the periodically driven double-well system [31], optical lattice system [32, 33] and fiber system [34]. In
the sense of Lyapunov, by the instability we mean that the small initial deviation from a given
solution grows without upper limit, which could lead to destruction of the solution behavior.
It was found that instability of the bipartite lattice systems depends on different signs of the
effective tunneling rates of two nearest-neighbor barriers [32, 33]. Therefore, one can stabilize
the systems by tuning the effective tunneling rates. For a single particle held in a simple lattice
with a single intersite separation, the nearest-neighbor barriers are the same such that the instability cannot be shown. For a bipartite lattice system, such an instability may be induced and
suppressed alternately by adjusting the driving parameters, resulting in the directed transport
of particles. Here our aim is finding a new route of CDT and supplying a simple stabiliza#201376 - $15.00 USD Received 14 Nov 2013; revised 22 Dec 2013; accepted 11 Feb 2014; published 18 Feb 2014
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24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.004277 | OPTICS EXPRESS 4279
tion method to realize the directed transport and quantum transition of cold atoms in a driven
bipartite lattice.
The coherent control of an ac driven particle in a single-band lattice with a single intersite
separation has been investigated widely in the nearest-neighbor tight binding (NNTB) approximation [5, 26]. More recently, a bipartite optical lattice or double-well train with two different intersite separations has been realized experimentally by superimposing two laser beams
with two different wavelengths [35, 36]. Such a system was applied to induce the ratchetlike effect [23, 24], to transport quantum information [37] and to realize two-qubit quantum
gates [38]. The periodic modulation is usually applied to the potential tilt (bias) between the
lattice sites [5, 39, 40] or the tunnel coupling [41–44]. The combined modulations between
both have also been adopted to produce the exact solutions for a phase controllable lattice system [26] or for an analytically solvable two-level system [27, 45]. The adjustments of driving
parameters can be performed in a nonadiabatic [23, 37] or adiabatic manner [46, 47]. On the
other hand, the directed transports have been investigated for a single classical particle in a
spatially periodic potential [48] and for a mean-field treated Bose-Einstein condensate loaded
in an optical superlattice [49, 50].
In this work, we firstly consider a single atom held in an optical bipartite lattice with two
different separations a and b and driven by a combined modulation of two resonant external
fields with a phase difference between the bias and coupling. Such a system can also be regarded
as an atomic analog of the optical two-mode quantum beam splitter [35]. In the high-frequency
regime and NNTB approximation, we derive an analytical general solution for the probability
amplitude of the particle in any localized state in which the quantum tunneling and stability
can depend on the phase difference between the two modulation components. A new route
of CDT and a simple method for stabilizing the system to perform the directed transport are
found by adjusting the phase difference nonadiabatically. Such a phase-adjustment may be
more convenient in experiments compared to the usual amplitude- and interaction-modulations.
Finally, we suggest a scheme for extending the results to the phase-controlled directed transport
and quantum transition between the superfluid and Mott insulator for the corresponding manyparticle system. The results can be readily amenable with existing experimental setups [39,
40, 42–44] on the periodically tilted and shaken optical lattices [26] and could be applied to
simulating the different optical systems [16, 17] and solid-state systems [13].
2.
General solution in the high-frequency regime
We consider a driven and tilted bipartite lattice in the form of a train of double wells formed by
the tilted laser standing wave
V (x,t) = V1 (t) cos(kL x − θ ) +V2 (t) cos(2kL x − 2θ ) + ε (t)x,
which consists of the long lattice of wave-vector kL , the short lattice of wave vector 2kL and the
linear potential. Here θ denotes the laser phase [43], the potential tilt between the lattice sites
takes the form [39,40,51] ε (t) = −ε0 cos(ω t) with amplitude ε0 and frequency ω , and the timeperiodic lattice depths reads [37, 43, 44] Vi (t) = Vi0 + δ Vi cos(mω t − φ ) for m = 0, 1, 2, ..., and
with constants Vi0 and δ Vi . The nonzero m means the frequency resonance between the modulation components. Such a lattice can be realized experimentally by a periodically shaken optical
lattice [37, 43, 44], and by moving the position of a retroreflecting mirror which is mounted on
a piezoelectric actuator [39] or by imposing a phase modulation to one of the standing wave
component fields [51]. A single particle of mass M is initially placed near the lattice center,
as shown in Fig. 1 [35], where we have adopted the spatial coordinate normalized by kL−1 and
the phase θ = 4.6, and selected the suitable driving parameters and initial time t0 = π /(2ω ) to
make ε (t0 ) = 0 and V1 (t0 ) = 1, V2 (t0 ) = 2. The different separations a and b can be adjusted
#201376 - $15.00 USD Received 14 Nov 2013; revised 22 Dec 2013; accepted 11 Feb 2014; published 18 Feb 2014
(C) 2014 OSA
24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.004277 | OPTICS EXPRESS 4280
b 0
x
a
Fig. 1. A single particle is initially placed in the driven bipartite lattice centered at coordinate 0 with two different separations a and b, where the curve denotes the initial potential
V (x, t0 ) = cos(x − 4.6) + 2 cos(2x − 9.2). Hereafter all the quantities plotted in the figures
are dimensionless.
by changing the laser wave vector kL and amplitudes Vi (t) [37, 43]. Quantum dynamics of such
a system is governed by the Hamiltonian [23, 26, 33]
H(t) =
∑ Ji j (t)(b†i b j + H.C.) + ε (t) ∑ xn b†n bn .
(i, j)
(1)
n
Here (i, j) means the nearest-neighbor site pairs. Signs b†j and b j are, respectively, the particle
creation and annihilation operators in the site j. The spatial location of the nth lattice site
reads [5]
n(a + b)/2
for even n,
2
xn = x|w(x − xn )| dx =
(n + 1)a/2 + (n − 1)b/2
for odd n,
where w(x − xi ) is the Wannier function. Expressing the lattice depths in terms of the recoil
energy Er = (h̄kL )2 /(2M), the tunnel coupling is calculated by the formula [5, 43]
Ji j (t) =
d2
w∗ (x − xi ) − 2 +V1 (t) cos(kL x − θ ) +V2 (t) cos(2kL x − 2θ ) w(x − x j )dx.
dx
Generally, the Ji j (t) for an even i has only a small time-independent difference from that for
an odd i and it will henceforth be renormalized, so we can take [23, 43] Ji j (t) = J(t) = J0 +
δ J cos(mω t − φ ), where constant J0 is from the terms of kinetic energy and of Vi0 , the shaking
intensity δ J is proportional to the driving amplitudes [43] δ Vi . To simplify, we have set h̄ = 1
and normalized energy and time by Er and ω0−1 = 10(h̄)/Er , which are determined by the laser
wave vector and atomic mass. The parameters J0 , δ J and (ε0 xn ) are in units of ω0 = Er /10 with
xn being normalized by the wave length λs = π /kL of the short lattice. Thus all the parameters
are dimensionless throughout this paper. The experimentally achievable parameter regions may
be selected as [42–44] λs ∼ 800nm, J0 ∼ ω0 , δ J < J0 , ε0 λs ∼ ω ∈ [0, 100](ω0 ), and a, b ∼ λs .
Letting |n be the localized state at the site n, we expand the quantum state |ψ (t) as the
linear superposition |ψ (t) = ∑n cn (t)|n. Combining this with Eq. (1), from the time-dependent
Schrödinger equation i ∂∂t |ψ (t) = H(t)|ψ (t) we derive the coupled equations of the probability
amplitudes [5, 33, 47]
iċn (t) = J(t)(cn+1 + cn−1 ) − ε0 cos(ω t)xn cn ,
(2)
where the dot denotes the derivative with respect to time. To solve Eq. (2), we make the function
transformation cn (t) = An (t) exp(iε0 ω −1 xn sin ω t) which leads Eq. (2) to the form
iȦn (t) = J(t)(An+1 ein sin ω t + An−1 e−in−1 sin ω t ).
(3)
#201376 - $15.00 USD Received 14 Nov 2013; revised 22 Dec 2013; accepted 11 Feb 2014; published 18 Feb 2014
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24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.004277 | OPTICS EXPRESS 4281
In this equation, we have defined n = εω0 (xn+1 − xn ) such that there are the relations n =
ε0
ε0
ε0
ε0
ω a, n−1 = ω b for even n, and n = ω b, n−1 = ω a for odd n.
We focus our attention on the situation of high-frequency regime with ω 1. The selective CDT has been illustrated analytically and numerically under this limit for an amplitude
modulation [23]. Here we shall give a general analytical solution of the system, which reveals
the phase-controlled CDT and directed transport for the considered combined modulation, then
extend the results to a many-particle system. Note that in Eq. (3), An (t) may be treated as a set
of slowly varying functions of time, and the coupling
F(t, m, φ , n ) = J(t, m, φ )ein sin ω t
1
= J0 + δ J[ei(mω t−φ ) + e−i(mω t−φ ) ] ∑ Jl (n )eil ω t
2
l
is a rapidly oscillating function which can be replaced by its time-average
F(m, φ , n )
=
=
1
J0 J0 (n ) + δ J[eiφ + (−1)m e−iφ ]Jm (n )
2
⎧
for even m,
⎨ δ J cos φ Jm (n )
J0 J0 (n ) +
⎩
iδ J sin φ Jm (n )
for odd m
(4)
with Jm (n ) = (−1)m J−m (n ) = (−1)m Jm (−n ) being the mth Bessel function of the
first kind [33]. Similarly, the time-average of F(t, φ , −n−1 ) = J(t)e−in−1 sin ω t in Eq. (3) reads
F(m, φ , −n−1 ), which is evaluated from Eq. (4) by using −n−1 instead of n . For an even
(odd) n, F(m, φ , n ) and F(m, φ , −n−1 ) are associated with the effective tunneling rates of
the lattice separations a (b) and b (a), respectively. Clearly, they may be real or complex,
corresponding to the even or odd m. Given Eq. (4), Eq. (3) is transformed to
iȦn (t) = F(m, φ , n )An+1 + F(m, φ , −n−1 )An−1 .
(5)
Comparing this equation with Eq. (9) of Ref. [33], we find that the former can become the latter
by using the new effective tunneling rates instead of the old. Thus we can construct the exact
general solution of Eq. (5) by applying the same discrete Fourier transformation [5, 33]
A(k,t) = ∑ An (t)e−ink = Ae (k,t) + Ao (k,t)
n
to transform Eq. (5) into the equations
∗
iȦe (k,t) = f (k)Ao (k,t), iȦo (k,t) = f (k)Ae (k,t)
with Ae and Ao being the sums of even terms and odd terms respectively in the Fourier series.
From the two first order equations of Ae and Ao we derive the second order equation Ä(k,t) =
−| f (k)|2 A(k,t) with the well-known general solution
A(k,t) = α (k)ei| f (k)|t + β (k)e−i| f (k)|t .
Inserting this solution into the inverse Fourier transformation, we immediately obtain the general solution of Eq. (5) as
An (t) =
1
2π
π
−π
[α (k)ei| f (k)|t + β (k)e−i| f (k)|t ]eink dk.
(6)
#201376 - $15.00 USD Received 14 Nov 2013; revised 22 Dec 2013; accepted 11 Feb 2014; published 18 Feb 2014
(C) 2014 OSA
24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.004277 | OPTICS EXPRESS 4282
Here f (k) takes the form
f (k) = J+ cos k + iJ− sin k,
J± = F(m, φ , n ) ± F(m, φ , −n−1 ),
(7)
∗
where | f (k)| and f (k) are the corresponding modulus and complex conjugate, α (k) and β (k)
are adjusted by the initial conditions. It should be noticed that the parameters J± in Eq. (7) may
be complex, while the same parameters in Ref. [33] are real. Such a difference may result in
some different properties of the solutions. Without loss of generality, let the initially occupied
state be |ψ (t0 = 0) = |N with a fixed integer N, namely the initial conditions read AN (0) =
1, An=N (0) = 0. Combining the conditions with the discrete Fourier transformation and general
solution of A(k,t) produces
A(k, 0)
=
α (k) + β (k) = e−iNk ,
iȦ(k, 0) = −| f (k)|[α (k) − β (k)] = i ∑ Ȧn (0)e−in k
n
= F(m, φ , N )e
−i(N+1)k
+ F(m, φ , −N−1 )e−i(N−1)k .
The final equation is derived from the initial conditions and Eq. (5). Solving the two equations
of α (k) and β (k) yields
1 | f (k)|e−iNk − F(m, φ , N )e−i(N+1)k
α (k) =
2| f (k)|
− F(m, φ , −N−1 )e−i(N−1)k ,
1 | f (k)|e−iNk + F(m, φ , N )e−i(N+1)k
β (k) =
2| f (k)|
(8)
+ F(m, φ , −N−1 )e−i(N−1)k .
Given the general solution (6) with Eqs. (7) and (8), we can investigate the general transport
characterization for the different initial conditions |ψ (t0 ) and the different effective tunneling rates F(m, φ , n ) and F(m, φ , −n−1 ). In the general cases, the particle may be in the
expanded states or localized states, depending on the system parameters.
3.
Unusual transport phenomena
We are interested in the unusual transport phenomena such as the DL, CDT, instability and
directed transport. It will be found that such unusual phenomena can be controlled under the
initial condition |ψ (t0 ) and for some special parameter sets with different phases. The routes
for implementing the phase-controlled transport are very different from that of the previously
considered case δ J = 0 with a constant tunneling rate J0 [23, 33].
Phase-controlled CDT. The CDT conditions mean the zero effective tunneling rates
F(m, φ0 , n ) = F(m, φ0 , −n−1 ) = 0 in Eq. (5), and f (k) = 0 in Eq. (7). Substituting the latter
into Eq. (6), the probability amplitudes becomes some constants An (t) = An (t0 ) determined by
the initial conditions, which means the occurrence of CDT. Applying the CDT conditions to
Eq. (4), we get
J0 J0 (n ) + δ J cos φ0 Jm (n ) = J0 J0 (n−1 ) + δ J cos φ0 Jm (n−1 ) = 0
for an even m, and
J0 J0 (n ) + iδ J sin φ0 Jm (n ) = J0 J0 (n−1 ) − iδ J sin φ0 Jm (n−1 ) = 0
#201376 - $15.00 USD Received 14 Nov 2013; revised 22 Dec 2013; accepted 11 Feb 2014; published 18 Feb 2014
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24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.004277 | OPTICS EXPRESS 4283
Fm,Φ,n' 0.5
Fn 0.3
0.1
0
0.1
Fn1 0
1
Φ
2 Φ0
3
Fig. 2. The effective tunneling rates as functions of phase for the parameters J0 = 1, δ J =
0.8, m = 2, n = 2.01717, n−1 = 5.37977. The solid and dashed curves describe
F(m, φ , n ) and F(m, φ , −n−1 ) respectively, which have the same zero point φ0 ≈ 2.4.
for an odd m. Therefore, we can arrive at or deviate from the CDT conditions by fixing the
parameters m, J0 , δ J, n , n−1 and adjusting the phase to arrive at or deviate from the
phase φ0 . In the general case, J0 (n ) = J0 (n−1 ), for an even m the above CDT conditions
imply −δ J cos φ /J0 = J0 (n )/Jm (n ) = J0 (n−1 )/Jm (n−1 ). For example, applying
the parameters J0 = 1, δ J = 0.8, m = 2 to the CDT conditions produces the required values
n ≈ 2.01717, n−1 ≈ 5.37977. Adopting these parameters, we plot the effective tunneling
rates as functions of phase, as in Fig. 2. It is shown that the effective tunneling rates are tunable by varying the phase, and the CDT conditions F(m, φ0 , n ) = F(m, φ0 , −n−1 ) = 0 are
established at the phase φ0 ≈ 2.4.
As a simplest example, we can fix the lattice separations a, b and tune the ratio ε0 /ω to
obey J0 (n ) = J0 (n−1 ) = 0 with n = εω0 a ≈ 2.4048, n−1 = εω0 b ≈ 5.5201, then Eq. (4)
becomes
⎧
for even m,
⎨ δ J cos φ Jm (n )
(9)
F(m, φ , n ) =
⎩
iδ J sin φ Jm (n )
for odd m,
Thus we can achieve the CDT by varying value of the phase to φ0 = π /2 for an even m or to
φ0 = 0 for an odd m in a nonadiabatic manner [23, 37].
Phase-controlled DL and selective CDT. The DL conditions mean one of the two effective
tunneling rates vanishing, which leads to selective CDT to the two different barriers [23, 24].
When F(m, φ , −n−1 ) = 0 and |ψ (0) = |N are set, from Eqs. (7) and (8) we obtain the
constant modulus | f (k)| = |F(m, φ , n )| and the periodic functions
α (k) =
F(m, φ , N ) −i(N+1)k
1 −iNk
e
e
−
,
2
2|F(m, φ , n )|
β (k) =
1 −iNk
F(m, φ , N ) −i(N+1)k
e
e
+
2
2|F(m, φ , n )|
of k. Inserting these into Eq. (6) produces the probability amplitudes
An=N,N+1 (t) = 0, AN (t) = cos(ω1t),
AN+1 (t) = −i
F(m, φ , N )
sin(ω1t).
|F(m, φ , N )|
(10)
They describe Rabi oscillation of the particle between the localized states |N and |N + 1 with
oscillating frequency ω1 = |F(m, φ , N )|. Here the selective CDTs between the states |N − 1
#201376 - $15.00 USD Received 14 Nov 2013; revised 22 Dec 2013; accepted 11 Feb 2014; published 18 Feb 2014
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24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.004277 | OPTICS EXPRESS 4284
and |N, and between the states |N + 1 and |N + 2 occur. Similarly, taking F(m, φ , n ) = 0
leads to the probability amplitudes
An=N,N−1 (t) = 0, AN (t) = cos(ω2t),
AN−1 (t) = −i
F(m, φ , −N−1 )
sin(ω2t),
|F(m, φ , −N−1 )|
(11)
which describe Rabi oscillation of the particle between the localized states |N and |N − 1
with oscillating frequency ω2 = |F(m, φ , −N−1 )|. Here the DL conditions and the different
oscillating frequencies are modulated by the phase for a set of other parameters.
Phase-controlled instability. Now we prove that the solutions of Eq. (5) are unstable under
the condition
F(m, φc , N ) = −F(m, φc , −n−1 )
(12)
for the phase φ = φc . In fact, when Eq. (12) is satisfied, Eq. (5) can be written as
1
dAn (τ )/d τ = [An−1 (τ ) − An+1 (τ )]
2
with variable τ = 2iF(m, φ , N )t being proportional to time, whose general solution is wellknown as
An (τ ) = Bn Jn (τ ) + Dn Nn (τ ),
where Jn (τ ) and Nn (τ ) are the Bessel and Neuman functions respectively, and Bn , Dn the
expansion coefficients determined by means of the initial conditions. For the complex variable τ
with nonzero
√ imaginary part, this general solution has the asymptotic property limt→∞ An (τ ) ∼
limt→∞ eτ / 2πτ = ∞ of the Bessel function, which implies that the initially small deviation
δ An (τ ) from the given solution An (τ ) can grow exponentially fast, meaning the Lyapunov
instability of solution An (τ ). In fact, by using δ An (τ ) + An (τ ) instead of An (τ ) in the linear Eq.
(5), we can find that the deviated solution possesses the same form as that of the given solution,
δ An (τ ) = δ Bn Jn (τ ) + δ Dn Nn (τ )
with constants δ Bn and δ Dn determined by the initial deviation, and has the same asymptotic
property, namely limt→∞ δ An (τ ) tends to infinity exponentially fast. Obviously such an instability can be controlled by tuning the phase to arrive at or deviate from φc in the condition (12),
as shown in Fig. 3(a).
Phase-controlled directed transport. For a set of given parameters J0 , δ J, n and n−1 , we
define two different phases φ1 and φ2 to obey
F(m, φ1 , −n−1 )
J0 J0 (−n−1 ) + δ J cos φ1 Jm (−n−1 ) = 0,
F(m, φ2 , n )
= J0 J0 (n ) + δ J cos φ2 Jm (n ) = 0
=
(13)
for an even m, which result in the selective CDT between the localized states |n and |n − 1 or
between the localized states |n and |n + 1, respectively. By nonadiabatically tuning the phase
to alternately change between φ1 and φ2 just after the two different time intervals T1 = π /ω1 and
T2 = π /ω2 , the original tunneling rate becomes the continuous and piecewise analytic function
⎧
⎨ J(t, m, φ1 ) for t ∈ [n T, n T + T1 ],
(14)
J(t, m, φ ) =
⎩
J(t, m, φ2 ) for t ∈ [n T + T1 , (n + 1)T ]
#201376 - $15.00 USD Received 14 Nov 2013; revised 22 Dec 2013; accepted 11 Feb 2014; published 18 Feb 2014
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24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.004277 | OPTICS EXPRESS 4285
2
0.1
0.05
0
Fn a
Fn1 Φ1 2.05 Φc
2.35
Φ
Φ2
Jt,Φ
Fm,Φ,n' 0.15
b
JΦ1 1.5
1
0.5
JΦ2 0
25
25.1
T1
t
25.3 25.4
Fig. 3. The effective tunneling rates versus phase (a) and the time evolutions of the original
tunneling rates (b) for the parameters J0 = 1, δ J = 0.8, m = 2, ω = 30, n = 2, n−1 =
2.2. In (a), the solid curve describes F(m, φ , n ) with zero point φ2 ≈ 2.49 and the dashed
curve labels −F(m, φ , −n−1 ) with zero point φ1 ≈ 1.93 . The phase value φc ≈ 2.17 corresponds to the cross point of the two curves, where the instability condition (12) holds. In
(b), the solid and dashed curves are associated with the original tunneling rates J(t, φ1 ) and
J(t, φ2 ), respectively. At the time t = T1 = π /ω1 = 25.2001, the J(t, φ1 ) is nonadiabatically
changed to J(t, φ2 ).
with T = T1 + T2 and n = 0, 1, 2, .... The instability condition (12) will be reached in each
process changing phase from φi to φ j for i, j = 1, 2. As an example, this is indicated by φc in
Fig. 3(a) for the parameter set J0 = 1, δ J = 0.8, m = 2, ω = 30, n = 2, n−1 = 2.2, where
Eqs. (12) and (13) give φc ≈ 2.17, φ1 ≈ 1.93 and φ2 ≈ 2.49, and the Rabi frequencies and halfperiods of Eqs. (10) and (11) read ω1 = F(m, φ1 , n ) ≈ 0.124666, ω2 = |F(m, φ2 , −n−1 )| ≈
0.140933 and T1 ≈ 25.2001, T2 ≈ 22.2914, respectively. The corresponding adjustment to the
time-dependent tunneling rate J(t, φ ) = 1 + 0.8 cos(60t + φ ) is exhibited in Fig. 3(b) for the
time interval including t = T1 , which only transforms the phase of J(t, φ ) from φ1 to φ2 and does
not change its magnitude. Clearly, at the time t = T1 + T2 , the tunneling rate will change from
J(t, φ2 ) to J(t, φ1 ). By repeatedly using such operations, the initially stable Rabi oscillation
is broken under the conditions (12), then the instability is suppressed by the conditions (13)
with phase φ1 or φ2 such that the particle is forced to transit repeatedly between the two stable
oscillation states |ψn (t) = An |n + An+1 |n + 1 and |ψn (t) = An |n + An +1 |n + 1 with the
amplitudes and frequencies of Eqs. (10) and (11) for (n, n ) = (N, N + 1); (N + 1, N + 2); ... that
lead to the directed motion toward the right [23,33]. Because the phase-transformation does not
change the magnitude of J(t, φ ) at the operation moment, it supplies a more convenient method
to manipulate the directed transport, compared to the previous amplitude-modulation schemes
to the coupling [37] or to the tilt [23].
4.
Extension to a many-particle system
It is worth noting that the method realizing the directed transport can be extended to controlling
the transport of a many-particle system in an optical bipartite lattice, where a Bose-Hubbard
interaction energy Hb = 12 U0 ∑n n̂(n̂ − 1) for n̂ = b†n bn should be added into Eq. (1). It is well
known that for an undriven simple lattice system with J = J0 , ε = ε0 , a = b and the interaction strength U0 > 0, the characteristic parameter is the ratio [39, 40] r = U0 /J0 . For U0 J0
the ground state of the system describes a superfluid, whereas it has the properties of a Mott
insulator for U J0 . Quantum transition between the superfluid and Mott insulator can occur
at the critical value r = rc . It has been argued that in the presence of a high-frequency driving the system behaves similar to the undriven system, but with the tunneling rate J0 of the
#201376 - $15.00 USD Received 14 Nov 2013; revised 22 Dec 2013; accepted 11 Feb 2014; published 18 Feb 2014
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24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.004277 | OPTICS EXPRESS 4286
latter being replaced by the effective tunneling rate Je f f = J0 J0 (ε0 /ω ) for the simple lattice
system. Thus one can control the quantum transition and transport by adjusting the driving parameters [39, 40]. Another interesting scheme for controlling the quantum transition has also
been suggested in which the periodically modulated interactions were applied [52]. In the Mott
insulating state, average atomic current along any direction approximately vanishes.
However, in the case of many particles held in the considered optical bipartite lattice, the
effective tunneling rates of the lattice separations a and b can be different, F(m, φ , n ) =
F(m, φ , −n−1 ). After transformation into the interaction picture by the unitary operator
Û = exp − i ε (t)dt ∑ xn b†n bn ,
n
from Eq. (1) and the considered Bose-Hubbard energy we arrive at the transformed interaction
Hamiltonian
Hint = Û † HI Û =
1
∑ F(m, φ , i j )(b†i b j + H.C.) + 2 U0 ∑ n̂(n̂ − 1)
(i, j)
(15)
n
in the high-frequency limit. Here we have set
HI =
1
∑ J(t, m, φ )(b†i b j + H.C.) + 2 U0 ∑ n̂(n̂ − 1),
(i, j)
n
and i j = n and −n−1 alternately. In such a system, we have two parameter ratios r1 =
U0 /F(m, φ , n ) and r2 = U0 /F(m, φ , −n−1 ), whose values are associated with the following
cases:
Case 1. The system becomes a Mott insulator with r1 rc and r2 rc ;
Case 2. There is a superfluid state with r1 rc and r2 rc ;
Case 3. There exist two different states for r1 rc , r2 < rc or r1 < rc , r2 rc , respectively.
We call the two states the local superfluid states, which may be gone through in the process of
transformations between case 1 and case 2.
When the ratios ri are changed between the case 1 and case 2, the system undergoes a quantum transition between the superfluid and Mott insulator. Because the ratios depends on the
effective tunneling rates and the latter can be tuned by varying the driving phase, we can control the quantum transition by the phase-modulation. On the other hand, to realize the directed
transport of the many-particle system, the possible experiment can begin by loading a BoseEinstein condensate into the long lattice of wave-vector kL , then one can increase the lattice
depth to make the atomic sample in the Mott insulating state with a single atom per well [42,43].
Further one can divide every well into a double-well by ramping up the short lattice of wavevector 2kL and tilting the double-well train, that achieve the load of single atoms into the “left”
sides of tilted double-wells [43]. In such a state with a single atom per double-well such that
the interatomic interaction could be negligible and the phase-modulation method for the single atom system becomes valid for the many-body system. Therefore, we can tune the driving
phase according to Eqs. (13) and (14) such that the parameter ratios change alternately between
r1 < rc , r2 rc and r1 rc , r2 < rc , leading to the nonzero average atomic current along a single direction. Such a phase-controlled directed transport of many particles will form a stronger
particle current compared to the single particle case. Particularly, the dynamics of the superfluid and Mott insulator is very different from that of the single-atom case. The scheme of the
phase-controlled quantum transition and directed transport through the local superfluid states
also differs from the previous amplitude-modulation [39, 40] and interaction-modulation [52]
#201376 - $15.00 USD Received 14 Nov 2013; revised 22 Dec 2013; accepted 11 Feb 2014; published 18 Feb 2014
(C) 2014 OSA
24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.004277 | OPTICS EXPRESS 4287
proposals on controlling quantum transitions and transports in simple optical lattices. In the latter case, the local superfluid states and the directed motion cannot exist for a symmetric driving.
It is interesting for us to compare our results with those of a repulsive Bose-Einstein condensate loaded in the spatially asymmetric optical superlattice without tilt potential and treated in
mean-field approximation [50]. Differing from the quantum states governed by the spatially
discrete and linear Eq. (2), the governing Schrödinger equation of the latter system are spatially
continuous and nonlinear. Therefore, quantum transition and directed transport of the former
system can be described by the ratios of the interaction strength and the effective tunneling rates
associated with the time-periodic lattice tilt, while the directed atomic current of the latter system is expressed by the time-averaged momentum expectation and this transport phenomenon
can occur only if the interaction strength exceeds a critical value for a symmetric driving [50].
Thus one can steer the directed current by adjusting the s-wave scattering length to vary the
interaction strength.
5.
Conclusions and discussion
We have investigated the coherent control of a single atom held in the optical bipartite lattice with two different separations a and b and driven by a combined modulation of two
resonant external fields with a phase difference between the bias and coupling. In the highfrequency regime and NNTB approximation, we derive an analytical general solution of the
time-dependent Schrödinger equation, which quantitatively describes the dependence of the
tunneling dynamics on the phase difference between the modulation components. It is demonstrated that a new route of CDT (or DL) can be formed by tuning the phase to make two (or one)
of the effective tunneling rates of the lattice separations a and b vanishing. When the two effective tunneling rates are adjusted to go through the values of the same magnitude and opposite
signs, the system loses its stability. The phase-controlled selective CDT enables the system to be
stabilized and the directed tunneling of the particle to be coherently manipulated. In the process
of control, the appropriate operation times are fixed by the two tunneling half-periods. The theoretical results have also been extended to the phase-controlled directed transport and quantum
transition between the superfluid and Mott insulator for the corresponding many-particle system. The analytic results based on the high-frequency approximation should have an advantage
over a direct numerical integration of the Schröedinger equation for transparently predicting or
explaining the experimental results.
The obtained results can be tested by employing the current accessible experimental setups [39, 40, 42, 43], and may be applicable for investigating atomic transport, quantum transition and quantum information processing [37, 38]. In experiments, the periodic modulations
of the tunneling rate [37, 43, 44] and the potential tilt [12, 39, 51] have been realized through
different methods. In order to experimentally implement the directed transport of the bipartite
lattice systems, the effective tunneling rate must be changed alternately between two different
values. The previous schemes [23,37] suggested that one can achieve the modulation by rapidly
changing the amplitudes of driving fields. In such a process, the deviation from the probability
amplitude of the initial Rabi-oscillation state crossing the separation a linearly grows in time,
then the system is stabilized to the final oscillation state crossing the separation b. This mechanism is similar to the resonance transition in standard quantum mechanics [33]. In contrast,
when the phase-modulation is adopted, the deviation from the the initial state can grow exponentially fast, as indicated below Eq. (12), that may result in faster transition to the final state.
The difference between both implies that the phase-modulation offers new dynamics of tunneling. In addition, in Fig. 3 we show that the phase-transformation does not change the magnitude
of the original tunneling rate at the operation moment. Therefore, the phase-modulation scheme
of the periodic shaking may be more convenient and experimentally feasible compared to the
#201376 - $15.00 USD Received 14 Nov 2013; revised 22 Dec 2013; accepted 11 Feb 2014; published 18 Feb 2014
(C) 2014 OSA
24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.004277 | OPTICS EXPRESS 4288
previous amplitude modulations. The directed tunneling is related to the ratchetlike effect of
quantum particles in the periodically tilted and shaken optical bipartite lattices and could be
well suited to simulating the different optical systems [16,17] and solid-state systems [13]. Particularly, the optical analog of such a system may be realized by an optical two-mode quantum
beam splitter [35].
Acknowledgments
This work was supported by the NNSF of China under Grant Nos. 11204027, 11175064 and
11205021, the Construct Program of the National Key Discipline of China, the Hunan Provincial NSF (11JJ7001) and the Scientific Research Fund of Hunan Provincial Education Department (12B082).
#201376 - $15.00 USD Received 14 Nov 2013; revised 22 Dec 2013; accepted 11 Feb 2014; published 18 Feb 2014
(C) 2014 OSA
24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.004277 | OPTICS EXPRESS 4289