We consider a Bose-Einstein condensate of ultracold atoms loaded into a
square optical lattice and subject to a static force. For vanishing atom-atom
interactions the atoms perform periodic Bloch oscillations for arbitrary
direction of the force. We study the stability of these oscillations for
non-vanishing interactions, which is shown to depend on an alignment of the
force vector with respect to the lattice crystallographic axes. If the force is
aligned along any of the axes, the mean field approach can be used to identify
the stability conditions. On the contrary, for a misaligned force one has to
employ the microscopic approach, which predicts periodic modulation of Bloch
oscillations in the limit of a large forcing.Comment: 4 pages, 3 figure

A. R. Kolovsky

Anderson

Brazhnyi

Glück

Greiner

Kolovsky

M.-C. Chung

Morsch

Raizen

Snoek

Witthaut

Zheng

English

arXiv

We consider a Bose-Einstein condensate of ultracold atoms loaded
into a square optical lattice and subject to a static force. For
vanishing atom-atom interactions the atoms perform periodic Bloch
oscillations for arbitrary direction of the force. We study 
stability of these oscillations for non-vanishing interactions,
which is shown to depend on an alignment of the force vector with
respect to the lattice crystallographic axes. If the force is
aligned along any of the axes, the mean field approach can be used
to identify the stability conditions. On the contrary, for a
misaligned force one has to employ the microscopic approach, which
predicts periodic modulation of Bloch oscillations in the limit of a
large forcing

EDP Sciences OAI-PMH repository (1.2.0)

Effect of the lattice alignment on Bloch oscillations  of a Bose-Einstein condensate in a square optical lattice

Springer - Publisher Connector

Effect of the lattice alignment on Bloch oscillations of a Bose-Einstein condensate in a square optical lattice

Chung, M. C.

Kolovsky, A. R.

Max Planck Society eDoc Server

Crossref

Eur. Phys. J. D 47, 421–425 (2008)
DOI: 10.1140/epjd/e2008-00060-0 THE EUROPEAN
PHYSICAL JOURNAL D
Eﬀect of the lattice alignment on Bloch oscillations
of a Bose-Einstein condensate in a square optical lattice
M.-C. Chung1,a and A.R. Kolovsky2,3
1 Max-Planck-Institut fu¨r Physik komplexer Systeme, 01187 Dresden, Germany
2 Kirensky Institute of Physics, 660036 Krasnoyarsk, Russia
3 Siberian Federal University, 660041 Krasnoyarsk, Russia
Received 5 November 2007 / Received in ﬁnal form 26 February 2008
Published online 4 April 2008 – c© EDP Sciences, Societa` Italiana di Fisica, Springer-Verlag 2008
Abstract. We consider a Bose-Einstein condensate of ultracold atoms loaded into a square optical lattice
and subject to a static force. For vanishing atom-atom interactions the atoms perform periodic Bloch
oscillations for arbitrary direction of the force. We study stability of these oscillations for non-vanishing
interactions, which is shown to depend on an alignment of the force vector with respect to the lattice
crystallographic axes. If the force is aligned along any of the axes, the mean ﬁeld approach can be used to
identify the stability conditions. On the contrary, for a misaligned force one has to employ the microscopic
approach, which predicts periodic modulation of Bloch oscillations in the limit of a large forcing.
PACS. 03.75.Lm Bose-Einstein condensates in periodic potentials – 03.75.Kk Dynamic properties of con-
densates
1 Introduction
Due to a virtually perfect experimental control over pa-
rameters, ultracold neutral atoms in optical lattices have
been intensively used for studying a variety of phenomena
of the condensed matter physics, including the transport
phenomena like Landau-Zener tunnelling [1,2], Wannier-
Stark ladder [3], and Bloch oscillations [3–6]. A number
of exciting experimental results were reported during the
last decade, which also stimulated the progress in the
theory [7–10]. As concerns applications, one should dis-
tinguish the latter problem of Bloch oscillations (BO) as
particularly important, because BO provide a tool for a
precision measurement of the gravitational ﬁeld and inter-
atomic interaction constant [5,6].
Until quite recently almost all theoretical, numerical
and experimental studies of BO concerned 1D or quasi 1D
lattices (see Refs. [8–10] for the contemporary reviews).
Nowadays one observes a growing interest in BO in mul-
tidimensional lattices [10–15]. In the single-particle ap-
proach this problem was considered in references [11,12].
It was shown that an increase of the lattice dimensional-
ity introduces new eﬀects not present in the 1D lattice.
Some predictions of these works were later on conﬁrmed
in the experiment with the array of optical guides [14],
where one uses a formal analogy between the Maxwell
and Schro¨dinger equations. The experiments with a Bose-
Einstein condensate (BEC) of interacting atoms addresses
the further questions [15], in particular, the question about
the stability of multidimensional BO. Indeed, it is known
a e-mail: mccchung@pks.mpg.de
that a BEC in optical lattices can be dynamically unsta-
ble, which quantum-mechanically means decoherence of
the BEC [16]. In the present work we study the condi-
tions under which the dynamical instability is suppressed
and, hence, multidimensional BO are stable. Unlike 1D
lattices, these conditions are shown to involve an align-
ment of the static force vector with respect to the crys-
tallographic axes of the lattice. We also argue in the work
that by changing the angle between the primary lattice
vectors and the force vector one may observe a transition
from the mean-ﬁeld to the microscopic Bloch dynamics.
2 The Hamiltonian
To simplify the equations we shall consider the two-
dimensional case throughout the paper – generalization
of the results in three dimensions is straightforward. The
Bose-Hubbard Hamiltonian of atoms in the tilted 2D lat-
tice reads,
̂H = − Jx
2
∑
m,l
(
aˆ†m+1,laˆm,l + h.c.
)
− Jy
2
∑
m,l
(
aˆ†m,l+1aˆm,l + h.c.
)
+
W
2
∑
m,l
nˆm,l(nˆm,l − 1)
+ d
∑
m,l
(Fxm + Fyl)nˆm,l, (1)
422 The European Physical Journal D
where Jx,y are the hopping matrix elements in x and
y-directions, W microscopic atom-atom interaction con-
stant, d lattice period, and Fx,y the projections of the
static force vector on the lattice axes. The Hilbert space
of (1) is spanned by the Fock states |n〉 ≡ |nm,l〉, where
∑
m,l nm,l = N – the total number of atoms. (Since in
the coordinate representation the Fock states are given by
the symmetrized product of the localized Wannier func-
tions, we shall refer to this basis as the Wannier basis.)
The translational invariance of the system, broken by the
static term, can be actually recovered by using the gauge
transformation [7]. Then the Hamiltonian (1) takes the
form
̂H(t) = − Jx
2
∑
m,l
(
e−iωxtaˆ†m+1,laˆm,l + h.c.
)
− Jy
2
∑
m,l
(
e−iωytaˆ†m,l+1aˆm,l + h.c.
)
+
W
2
∑
m,l
nˆm,l(nˆm,l − 1), (2)
where ωx,y = dFx,y/ are the Bloch frequencies associ-
ated with x and y component of the static force. We
also note that in stead of the Wannier basis one can
use the quasimomentum Fock basis |q〉 ≡ |qp,k〉 for the
Hamiltonian (2), which we shall refer to as the Bloch basis.
(Needless to say that in the coordinate representation the
quasimomentum Fock states are given by the symmetrized
product of the extended Bloch functions.) Formally this
corresponds to the canonical transformation
bˆp,k =
1
L
∑
m,l
exp
[
−i2π
L
(mp + kl)
]
aˆm,l,
which implicitly assumes the periodic boundary condi-
tions.
Through the paper we shall illustrate the analytical
results by simulating BO in a ﬁnite 2D lattice and it is
an appropriate place here to comment on the choice of
the system parameters. This choice is determined by two
criteria: we want our ﬁnite system to reproduce the main
features of an inﬁnite system but, to minimize computa-
tional eﬀorts, to be as small as possible. It is an advan-
tage of the periodic boundary conditions that already the
3 × 3 lattice can satisfy the former condition. For exam-
ple, if equation (8), which we shall discuss later on in Sec-
tion 3.2 and which refers to an inﬁnite lattice, is applied
to the lattice with L = 3 sites, it will give an error of less
than 5 percents. As concerns the ﬁlling factor n¯ = N/L2,
we keep it close to unity, which corresponds to its typical
value in the contemporary laboratory experiments. Note
that for the considered in the work ratio W/Jx,y < 1 the
exact value of the ﬁlling factor does not matter. It be-
comes important only if one considers larger interactions,
where the system may show diﬀerent types of the super-
ﬂuid to Mott-insulator quantum phase transitions. (For
example, for integer N/L and not equal hopping matrix
elements the system can be super-ﬂuid in one direction
but insulating in the orthogonal direction.) Since our sub-
ject is BO, we intentionally stay far from insulating phase
by choosing W smaller than either of the hopping matrix
elements.
A remark about the magnitude of the static force is in
turn. Formally, i.e., within the frame of the Bose-Hubbard
model, the static force can be arbitrary large. However,
when referring to the original system of cold atoms in
an optical lattice, the force magnitude is restricted from
above by requiring negligible Landau-Zener transitions.
The latter condition is satisﬁed by assuming ∆ > dF ,
where ∆ is the energy gap separating the ground Bloch
band from the rest of the spectrum.
To conclude this section we mention that although a
small system with L = 3 or L = 4 sites along one axis,
used in the numerical simulations below, is capable to re-
produce the features of inﬁnite 2D lattice, we cannot com-
pletely avoid the presence of ﬁnite-size eﬀects. In what
follows we shall explicitly indicate which of the numerical
results have general validity and which are artifacts due
to the system ﬁnite size.
3 Bloch dynamics
3.1 Misaligned force
We begin with the case of a strong misaligned force
dFx, dFy  Jx,y > W [17]. In order to illuminate sit-
uation, we simulate BO by numerically solving the time-
dependent Schro¨dinger equation with the Hamiltonian (2)
for the speciﬁed initial conditions. As those we consider
the superﬂuid state with qp,k = Nδp,0δk,0, which approx-
imates the ground state of the system for F = 0 and
W < Jx,y. (Substitution of this state by the exact ground
state practically does not aﬀect the ﬁnal result.) Figure 1
shows the numerical results for the 3 × 3 lattice with
7 atoms inside. The lower panel in Figure 1 depicts the
kinetic energy of the atoms, the upper and middle panels
show the order parameters ex(t) and ey(t) deﬁned as [18]
ex(t) = − 1
N
Re
⎡
⎣〈Ψ(t)|
∑
m,l
aˆ†m+1,laˆm,l|Ψ(t)〉
⎤
⎦ . (3)
(By replacing the operators in the bracket in equation (3)
with
∑
aˆ†m,l+1aˆm,l one obtains a similar expression for
the order parameter ey(t).) It is seen in the ﬁgure that BO
persist in time but are modulated with some characteristic
period. We have checked that this period is not aﬀected by
increasing the lattice size and, hence, the displayed result
has a general validity.
To prove that BO in the misaligned lattice are sta-
ble in the limit of strong forcing and to identify the
modulation period we proceed as follows. First we intro-
duce the new wave function |˜Ψ(t)〉 through the relation
|Ψ(t)〉 = ̂U0(t)|˜Ψ (t)〉, where ̂U0(t) is the evolution oper-
ator for vanishing atom-atom interactions. The function
M.-C. Chung and A.R. Kolovsky: Eﬀect of the lattice alignment on Bloch oscillations of a Bose-Einstein condensate 423
−1
0
e
x
−1
0
e
y
 
0 10 20 30 40 50 60
−2
2
t/TJ
E/
N
Fig. 1. Bloch oscillations of condensed atoms in the 2D lat-
tice: the order parameter ex(t) (top), ey(t) (middle), and the
mean kinetic energy (bottom). The system parameters are
N = 7, L = 3 (periodic boundary conditions), Jx = Jy = J ,
W = 0.2J , Fd = 20J , and F/F = (
√
2/5,
√
3/5). The time is
measured in units of the tunnelling period TJ = 2π/J .
|˜Ψ(t)〉 obviously obeys the equation,
i
∂|˜Ψ(t)〉
∂t
=
W
2
̂U †0 (t)
⎛
⎝
∑
m,l
nˆm,l(nˆm,l − 1)
⎞
⎠ ̂U0(t)|˜Ψ (t)〉.
(4)
On the other hand, the explicit form of the evolution op-
erator is given by ̂U0(t) = ̂T † ̂D(t) ̂T , where the unitary op-
erator ̂T represents the transformation from the Wannier
basis |n〉 to the Bloch basis |q〉 and the matrix of the
operator ̂D(t) is diagonal in the Bloch basis,
〈q|D(t)|q〉 = exp
[
i
Jx
dFx
N
∑
i=1
sin
(
2πpi
L
− ωxt
)
+i
Jy
dFy
N
∑
i=1
sin
(
2πki
L
− ωyt
)
]
. (5)
Note that the operator (5) tends to the identity opera-
tor for Fx, Fy → ∞. Substituting ̂U0(t) in equation (4)
by identity matrix and noting that the interaction energy
operator is diagonal in the Wannier basis with integer en-
tries, 〈n|∑m,l nˆm,l(nˆm,l−1)|n〉 =
∑
m,l n
2
m,l−N , we con-
clude that the time evolution of the wave function |˜Ψ(t)〉 is
periodic with the period TW = 2π/W [19]. Coming back
to the original wave function this result means the peri-
odic modulation of BO with the frequency ωW = W/. It
is worth stressing that the above proof assumes both Fx
and Fy to be large and, hence, the case of aligned lattices
is excluded.
3.2 Aligned force
Next we consider the situation where the force is aligned
along one of the crystallographic axes (to be certain, the
y-axis in what follows). Within the single-particle ap-
proach the static force Fy would localize the atoms in
the y-direction. Thus one may expect that if Fy is large
enough the atoms form separate BECs in the planes per-
pendicular to the force vector, weakly coupled together as
a one-dimensional BEC chain. Introducing new operators
Aˆl = 1√L
∑
m aˆm,l and Aˆ
†
l , the eﬀective Hamiltonian reads
̂Heﬀ = − Jx
∑
l
Aˆ†l Aˆl −
Jy
2
∑
l
(
e−iωytAˆ†l+1Aˆl + h.c.
)
+
Weﬀ
2
∑
l
Nˆl(Nˆl − 1), (6)
where Weﬀ = W/L. Thus we have reduced the 2D problem
to a 1D problem with the renormalized interaction con-
stant. (If one considers 3D lattices, the renormalization is
Weﬀ = W/L2.) Moreover, since the mean number of atoms
˜N in any site of the eﬀective 1D system is given by n¯L,
the occupation numbers will be macroscopically large in
the thermodynamic limit N,L →∞, n¯ = N/L2 = const.,
which justiﬁes the mean ﬁeld approach.
The mean-ﬁeld Hamiltonian of the system (6) reads
(up to the irrelevant constant terms proportional to N)
Heﬀ = −Jy2
∑
l
(
e−iωytA∗l+1Al + h.c.
)
+
g
2
∑
l
|Al|4, (7)
where Al and A∗l are pairs of the canonically conju-
gated variables and the macroscopic interaction constant
g = Weﬀ ˜N = Wn¯. Within the mean-ﬁeld approach the
border between stable and unstable (decaying) BO is know
exactly [16,20]. Namely, for J/Fd  0.5 the critical value
of nonlinearity is a linear function of the static force mag-
nitude, while for J/Fd  0.5 it additionally depends on
the value of the hopping matrix elements:
gcr ≈
{
0.33Fd, Fd  2J
0.1(Fd)2/J, Fd  2J.
(8)
Obviously, for a ﬁxed nonlinearity g the condition (8) can
also be formulated as a condition on the critical magnitude
Fcr of the static force.
The microscopic analysis of BO in the aligned lattice
conﬁrms our working hypothesis. Choosing the parame-
ters in such a way that the 1D mean-ﬁeld BO are stable,
we simulate BO of N = 7 atoms in the 2D lattice with
L = 3. The dashed line in the upper panel of Figure 2
shows the dynamics of the order parameter ex(t). It is
seen that ex(t) ≈ −1, – therefore we indeed have in-plane
BECs. We also note that the decay and revival of the or-
der parameter ey(t) in the lower panel is an artifact due
to the ﬁnite size of our lattice. We have checked numeri-
cally that increasing the lattice size modiﬁes the revivals
424 The European Physical Journal D
0 10 20 30 40 50 60
−1
0
e
x
0 10 20 30 40 50 60
−1
0
t/TJ
e
y
Fig. 2. (Color online) Dynamics of the order parameters for
F/F = (0, 1), dashed lines, and F/F = (0.001,
√
1− 0.001),
solid lines. The other parameters are the same as in Figure 1.
according to the following equation,
ey(t) = − exp
(
−2 ˜N
[
1− cos
(
Weﬀt

)])
, (9)
which can be obtained analytically on the basis of the
eﬀective Hamiltonian (6) [10]. Because Weﬀ = W/L and
˜N = n¯L, one has ey(t) = −1 in the thermodynamic limit.
The above analysis of BO in the aligned lattice relies
on the reduction of a two-dimensional system to an eﬀec-
tive one-dimensional mean-ﬁeld problem. It should be es-
pecially stressed that this reduction is possible only if the
dynamics of the reduced system is stable. If we choose the
parameters in the unstable regime the situation becomes
totally diﬀerent. Figure 3 shows the numerical results for
F = 0.2J/d < Fcr, where one-dimensional BO suﬀer
from dynamical instability. Unlike in the stable regime,
BO along y direction now excite the transverse degree of
freedom and we observe decay of the both order parame-
ters towards zero. Thus no reduction to one dimension is
possible.
3.3 Slightly misaligned force
Finally we brieﬂy analyze an experimentally important
situation of a small mismatch between the lattice axis and
the static ﬁeld vector, i.e., Fx  Fy. The solid lines in Fig-
ure 2 show the order parameters for the same dF = 20J
but Fx = 0.001F . Compared to the case Fx = 0 (dashed
lines in Fig. 2), we observe the destruction of BEC after
time t∗ ≈ 12 TJ . This critical time can be understood
in terms of the mean-ﬁeld approach as well. Indeed, it is
known that a stationary BEC is unstable for the quasimo-
mentum κ outside the ﬁrst quarter of the Brillouin zone.
Since the static force causes the linear growth of the quasi-
momentum, κx,y(t) = κx,y + Fx,yt/, the system always
−1
0
e
x
−1
0
e
y
 
0 10 20 30 40 50 60
−2
2
t/TJ
E/
N
Fig. 3. Bloch dynamics for dF = 0.2J and F/F = (0, 1). The
other parameters are the same as in Figure 1.
enter the instability region of the Brillouin zone. However,
if the static force is strong enough, the system passes the
instability region so quickly that it ‘has no time’ to decay.
(In fact this is a physical argument behind Eq. (8).) In
the considered example the strong static force ensures the
fast driving along y direction but simultaneously it slowly
brings the system to κx = π/2d along x direction. As soon
as this border of instability is reached (t∗/TJ = J/4dFx),
we observe an irreversible decay of the order parameters.
4 Conclusion
In summary, we have studied BO of a BEC of atoms in a
square lattice for both aligned and misaligned static forces.
It is shown that in the case of aligned force the system
may be reduced to a one-dimensional chain of mini BECs,
which we treat by using the mean-ﬁeld approach. Then
the stability diagram of this eﬀective 1D system deﬁnes
the critical magnitude of the static force above which BO
are stable, with no excitations of the transverse degrees of
freedom. On the contrary, in the unstable regime, F < Fcr,
BO induced by the static force excite the transverse modes
and, as a consequence, one observes BEC destruction and
decay of BO. Our studies also illuminate importance of
the alignment. The strong (F > Fcr) but slightly mis-
aligned force is shown to slowly intrigue the transverse
modes, which destabilize BO after some well-deﬁned tran-
sient time. However, if misalignment is large, BO appear
to be stable again. This case corresponds to the quantum
(not mean-ﬁeld) regime of BO, where they are modulated
with the frequency deﬁned by the microscopic interaction
constant.
Fruitful discussions with A. Buchleitner and ﬁnancial support
by Deutsche Forshungsgemeinschaft under SPP1116 are grate-
fully acknowledged.
M.-C. Chung and A.R. Kolovsky: Eﬀect of the lattice alignment on Bloch oscillations of a Bose-Einstein condensate 425
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Chung, M.

Kolovsky, A.

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Effect of the lattice alignment on Bloch oscillations of a Bose-Einstein
  condensate in a square optical lattice

Effect of the lattice alignment on Bloch oscillations of a Bose-Einstein condensate in a square optical lattice

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