Synthetic magnetic field effects on neutral bosonic condensates in quasi three-dimensional anisotropic layered structures

Abstract

We discuss a system of dilute Bose gas confined in a layered structure of
stacked square lattices (slab geometry). A derived phase diagram reveals a
non-monotonic dependence of the ratio of tunneling to on-site repulsion on the
artificial magnetic field applied to the system. The effect is reduced when
more layers are added, which mimics a two- to quasi-three-dimensional geometry
crossover. Furthermore, we establish a correspondence between anisotropic
infinite (quasi three-dimensional) and isotropic finite (slab geometry) systems
that share exactly the same critical values, which can be an important clue for
choosing experimental setups that are less demanding, but still leading to the
identical results. Finally, we show that the properties of the ideal Bose gas
in a three-dimensional optical lattice can be closely mimicked by finite (slab)
systems, when the number of two-dimensional layers is larger than ten for
isotropic interactions or even less, when the layers are weakly coupled.

Full text

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Synthetic magnetic field effects on neutral bosonic condensates
in quasi three-dimensional anisotropic layered structures
T. A. Zaleski
Institute of Low Temperatures and Structure Research,
Polish Academy of Sciences, POB 1410, 50-950 Wrocław 2, Poland
T. P. Polak
Adam Mickiewicz University of Poznań, Faculty of Physics, Umultowska 85, 61-614 Poznań, Poland
We discuss a system of dilute Bose gas confined in a layered structure of stacked square lattices
(slab geometry). A derived phase diagram reveals a non-monotonic dependence of the ratio of
tunneling to on-site repulsion on the artificial magnetic field applied to the system. The effect is
reduced when more layers are added, which mimics a two- to quasi-three-dimensional geometry
crossover. Furthermore, we establish a correspondence between anisotropic infinite (quasi three-
dimensional) and isotropic finite (slab geometry) systems that share exactly the same critical values,
which can be an important clue for choosing experimental setups that are less demanding, but still
leading to the identical results. Finally, we show that the properties of the ideal Bose gas in a
three-dimensional optical lattice can be closely mimicked by finite (slab) systems, when the number
of two-dimensional layers is larger than ten for isotropic interactions or even less, when the layers
are weakly coupled.
PACS numbers: 05.30.Jp, 03.75.Lm, 03.75.Nt
I.INTRODUCTION
Systems of dilute bosonic gases confined in optical lat-
tices are ideal toolboxes for testing theoretical models
and their solutions [1]. However, investigation of their
properties under external magnetic field seemed to be
precluded, since the particles used in the experiments are
uncharged atoms and thus, are not directly affected by
the magnetic field. Fortunatelly, development of various
experimental techniques allow to investigate ultracold
systems that are described by exactly the same Hamil-
tonians as the ones interacting with external magnetic
field. Furthermore, they lead to appearance of vortices in
Bose-Einstein condensates (BEC), which is a hallmark of
a superfluid in a magnetic field. One of those techniques
results from equivalence between the Lorentz force and
the Coriolis force. As a result, rotation of the condensate
(usually rotation of the masks with set of holes located
in the laser beams producing quasi-two-dimensional ro-
tating optical lattice) acts as a synthetic magnetic field
(SMF) [2–4]. However, this approach puts limit on max-
imum rotational velocity, thus large SMF (e.g. required
for quantum Hall physics) cannot be reached. To over-
come those difficulties, imprinting of the quantum me-
chanical phase is used, which is based on superimpos-
ing of an external potential on a BEC (e.g. by applying
rotating magnetic field) [5, 6]. Another approaches are
stirring a BEC with laser beam [7], or using additional
Raman lasers to coherently transfer atoms from one inter-
nal state to another. This induces a non-vanishing phase
of particles moving along a closed path, which simulates
magnetic flux through the lattice [8, 9]. Additionally,
rectification of the magnetic field in the optical lattice by
using a superlattice allows to ensure that each plaque-
tte acquires the same phase, thus simulating the uniform
magnetic field for any value of phase between 0 and π
[10].
A ground state of neutral atoms in an optical trap-
ping potential can be either superfluid (SF) or a Mott-
insulator (MI). The zero-temperature coupling (t/U) vs.
chemical potential (µ/U) phase diagram contains char-
acteristic lobes marking a quantum phase transition be-
tween SF and MI states; t is the matrix element for tun-
neling between adjacent lattice sites and U is the on-site
energy cost for multiple occupancy. For average number
of particles per site equal to one, the transition occurs
at the ratio (t/U)crit, which is strongly dependent on
the geometry of the system. Recently, many very pre-
cise calculations and measurements have been performed
to obtain correct value of the (t/U)crit. For the three-
dimensional (3D) optical lattices, most methods converge
to (t/U)3D
are more demanding because of growing influence of the
quantum fluctuations. In order to precisely describe ex-
perimental results, it is necessary to develop a theory
that contains fully tunable lattice degrees of freedom, in-
cludes effects of the synthetic magnetic field and is non-
perturbative, which allow to capture essential physics of
strongly correlated system in U/t ? 1 regime. To this
end, we constructed a theoretical field approach, in which
the dimensionality along c axis can be tuned by adding
an arbitrary number of single layers along with vari-
able tunneling between them (thus including anisotropy
of tunnelling between and within planes). In principle,
we can include sixty layers as in the experiments of the
Spielman’s group [13], or just a few as in the Krüger
setup [14], where the magnetic confinement along the z-
direction was used to localize atoms in the effectively
N = 4kBT/mω2
crit= 0.03 [11, 12], but lower-dimensional cases
zl2∼ 2 ÷ 4 central lattice planes (T is
arXiv:1102.3283v1 [cond-mat.quant-gas] 16 Feb 2011

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the temperature, ωz is the frequency at the bottom of
the lattice wells and l is the lattice period that can be
adjusted to any value higher than half of the atomic res-
onance wavelength). The Burger group [15] developed
method in which by increasing the nodal planes of the
optical lattice superposed to a 3D potential it is possible
to follow the transition from three-dimensional system
to an two-dimensional array. Therefore the system be-
comes quasi-3D rather than purely planar. That can be
used in order to look in the interference between planes
and in consequence to access to spatial coherence of such
structure.
The outline of the paper is as follows: in Sec. II we
introduce the model Hamiltonian and the effects of arti-
ficial magnetic field and briefly describe our method. In
Sec. IV, we present our results starting with the zero-
temperature phase diagrams and its dependence on the
system geometry. Furthermore, we show the influence
of the synthetic magnetic field on properties of the Bose-
Hubbard system. Also, we find a correspondence between
systems sharing the same critical properties but differing
with the geometry of the opticall lattice and interactions,
which can be a helpful clue on choosing between various
experimental setups. Finally, we conclude in Sec. V.
II. MODEL
In optical lattices, two main energy scales are set by
the hopping amplitude t (the kinetic energy of bosons
tunneling between the lattice sites), and the on-site re-
pulsive interaction U (resulting from repulsion of multiple
boson occupying the same lattice site). For t ? U, the
superfluid order is well established in zero-temperature
limit. However, for sufficiently large repulsive energy U,
the quantum phase fluctuations lead to suppression of the
long-range phase coherence resulting in SF to MI transi-
tion. The critical ratio of (t/U)cfor which this transition
occurs depends strongly on the number of bosons intro-
duced to the optical lattice (which in theoretical models
is often controled by a chemical potential µ). The syn-
thetic magnetic field B (resulting either from rotation of
the system, phase imprinting, or external electric field)
introduces the Peierls phase factor e
B = ∇ × A(r), and Φ0 = hc/e is the flux quantum,
with A(r) being the vector potential (which can be re-
alized experimentally, see Ref. [9]), and h, c and e –
Planck constant, speed of light and charge of electron,
respectively. Thus, the system can be described by the
following quantum Bose-Hubbard Hamiltonian [16, 17]
2πi
Φ0
´ri
rjA·dl, where
H =
U
2
?
?r,r??
?
r
nr(nr− 1)
−
trr?e
2πi
Φ0
´ri
rjA·dla†
rar? − µ
?
r
nr,
(1)
where a†
hilation operators that obey canonical commutation re-
lations [ar,a†
ber operator on the site r. Here, ?r,r?? denotes sum-
mation over the nearest-neighbor sites.
trr? is the hopping matrix element with the dispersion
tk = 2t?
hopping between layers and t?within the planes. Since,
we are interested in investigating the influence of the
lattice geometry on the system properties, we consider
a stack of an arbitrary number (L) of two-dimensional
planes coupled with t⊥. As a result, the values of kxand
kyare continuous (kx,y= −π,...,π), while kzis descrete
(kz=2π
axis anisotropy, which is a ratio of inter-plane to in-plane
hopping η = t⊥/t?.
A boson hopping around a lattice cell of the area of
A will gain an additional phase 2πf resulting from the
synthetic magnetic field, where f = ABe/2π?.
result, the properties of the system will be periodic with
a period corresponding to 1/f. As a result, the periodic
potential leads to splitting of Landau levels into integer
number q of sub-bands. Of special interest are the values
of the SMF which correspond to rational numbers of f ≡
p/q = 1/2,1/3,1/4,... (p is an integer), since for those
values the energy spectra and the density of states can be
obtained exactly, although it is analytically feasible only
for small values of q. In the present paper, we present
new results for f = 1/8 and f = 3/8, which up to now
have been analitically unaccessible (see, Ref. [18]). Since,
all properties of the Hamiltonian Eq. (1) are invariant
under f → −f and also under f → f + 1, it is sufficient
to consider f in the range 0 < f < 1/2. In solid state
physics obtaining f = 1/2 in experimental setup would
require magnetic field of the order of 105T. In optical
lattices however, due to larger lattice spacings (and larger
A consequently), much smaller values of B of the order
of 10−3T are required. This makes investigation of the
above-mentioned rational values of f reasonable.
To proceed, we rely on the quantum rotors approach.
The method is extensively described in Refs. [12, 17, 19,
20], so here we only summarize its main points. We use
the functional integral representation of the model with
bosonic operators becoming complex fields ar(τ) (where
τ is imaginary Matsubara’s time). The most important
element of our method is a local gauge transformation to
the new bosonic variables:
rand ar? are for the bosonic creation and anni-
r?] = δrr?, nr = a†
rar is the boson num-
Furthermore,
?
coskx+ cosky+t⊥
t?coskz
?
where t⊥ is the
Ll, where l = 0,...,L − 1). Also, we allow for c-
As a
ai(τ) = bi(τ)exp[iφi(τ)].
(2)
This allows to cast the strongly correlated bosonic prob-
lem into a system of weakly interacting bosons, sub-
merged into the bath of strongly fluctuating gauge po-
tentials on the high energy scale set by U. It also allows
to formulate the problem in the phase representation,
which is the best suited to describe MI to SF transition,
since it is governed by phase fluctuations. As a result,
the superfluid order parameter can be written as:

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3
ΨB≡ ?ai(τ)? = b0ψB,
(3)
where non-zero value of ψB=?exp[iφi(τ)]? results from
phase ordering and b0 is the amplitude of the bosonic
field:
?
The coefficient 4+2?1 −1
zero-temperature limit we arrive at the equation for the
phase order parameter:
?
with
?µ
where frac(x) = x−[x] is the fractional part of the num-
ber and [x] is the floor function which gives the greatest
integer less then or equal to x and the phase stiffness
Jk= b2
Since the number of layers L in the system is finite,
the summation in Eq. 5 runs over discrete values of kz
and continuous values of kx and ky. However, because
density of states of a single layer under SMF (ρf ) is
known, we explicitly derive the density of stated of the
whole stack of L coupled planes (for calculation details,
see Ref. [21]):
?
As a result, the critical line equation (ψB= 0) including
the effects of SMF and c-axis anisotropy reads:
b2
0= 4 + 2
?
1 −1
L
?t⊥
?t⊥
t?
?t?
U+µ
U+1
2.
(4)
Lt?is an effective number of
nearest neighbors averaged over all lattice sites. In the
1 − ψ2
B=
1
2N
k
1
?
Jk=0−Jk
U
+ υ2?µ
U
?
(5)
υ
U
?
= frac
?µ
U
?
−1
2,
(6)
0tk.
ρL
f(η,ξ) =1
L
kz
ρf(ξ − η coskz).
(7)
1 =1
2
ˆ+∞
−∞
dξ
ρL
f(η,ξ)
?
2(ξ0− ξ)b2
0
t?
U+ υ2?µ
U
?,
(8)
with ξ0being the half-width of the band dispersion for
selected value of f = p/q.
III.RESULTS
A. Phase diagram
The Eq.
temperature phase diagram of the investigated Bose-
Hubbard model from Eq. (1) as a dependence of critical
interaction on the chemical potential, SMF, number of
layers and c-axis anisotropy:
?t?
(8) allows us to calculate the zero-
U
?
c
= xc= xL
f
?µ
U,η
?
.
(9)
0 0.51 1.52 2.53
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
x1/2
µ/U
MI
SF
L=1
L=2
L=4
L
Figure 1: (Color online) The zero-temperature phase diagram
of a stack of square lattice planes (number of particles per
lattice site is nB = 1 inside the first and nB = 2 and 3
in second and third lobe, respectively) with magnetic field
f = 1/2 for various number L of layers and η = 1. Within
the MI phase the phase order parameter ΨB = 0.
The diagram is plotted in Fig. 1 for different number of
layers L, synthetic magnetic field f = 1/2, in isotropic
case (η = 1). In the weak coupling limit (t?? U), the
kinetic energy dominates and the ground state is a de-
localized superfluid, described by nonzero value of the
superfluid order parameter ΨB?= 0. On the other hand,
in the strong coupling regime (t?? U) the phase fluc-
tuation become significant and the long-range order is
destroyed leading to a series of MI lobes with fixed in-
teger filling nB = 1,2,...[12, 16]. A single-layer system
(L = 1) has a simple square (two-dimensional) geom-
metry, which results in the phase diagram with charac-
teristic narrow-edged lobes. As the number of layers is
being increased, the tops of the lobes become smooth
and their maxima deviate towards lower values of the
chemical potential µ (this effect is also clearly presented
in Ref. [22]). As a result, the phase diagram becomes
similar to the one of a cubic (three-dimensional) system.
It is important to notice, that also in the presence of
the synthetic magnetic field, the phase diagrams of the
finite L system becomes indistinguishable from the in-
finite (cubic) one for L as small as 10.
for ten layers, the value of x10
system x∞
x∞
tor approach (QRA) was verified several times [17, 19]
by comparison with the very precise numerical methods:
quantum Monte-Carlo [11] and diagrammatic perturba-
tion theory [22]. The QRA phase diagrams were also
compared to an analytical works: mean-field theory [23]
and Padé analysis [24]. From the above, it is expected
that QRA accurately captures the low energy physics of
the Bose-Hubbard model in the more demanding case
where the system is placed under an artificial magnetic
field.
For example,
1/3is 102.2% of the infinite
1/3is close to 100.1% of
1/3and for sixty – x60
1/3. The phase diagram obtained from the quantum ro-

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4
0
0
0.25
0.5
0.75
1
L
x /x
f
1/8
1/6
1/4
1/3
3/8
1/2
f
100
60
20
16
10
6
4
2
1
number of layers 1
0
L
0
0.25
0.5
0.75
1
0
1/8
1/6
1/4
1/3
3/8
1/2
f
x /x
f
0
L
L
η=1
η=0.1
Figure 2: (Color online) Ratio of critical coupling xL
the tip of the first lobe [see, Eq. (9)] of the system without
(f = 0) and with synthetic magnetic field f as a function
of number layers L for isotropic (top) η = 1 and anisotropic
(bottom) η = 0.1 case, respectively.
0/xL
fof
B. Effect of the synthetic magnetic field and the
system geometry
The transition, experimentally seen in the time-of-
flight images (the presence of sharp peaks has been con-
sidered as an unequivocal signature of superfluidity in the
Bose system), occurs rather rapidly with increasing lat-
tice depth. Because the experimental parameter V0/ER
(V0is the maximum value of the lattice depth), depends
logarithmically on U/t, the small changes of the dimen-
sionless depth of the optical lattice can cover a wide range
of the phase diagram. The phase coherent Bose gas can
be also driven into the Mott insulating phase by apply-
ing the synthetic magnetic field. The effect of the SMF is
presented in Fig. 2. The long range order is suppressed
by the phase changes imposed on the bosonic wave func-
tion and this suppression has a non-monotonic character
strongly depending on the topology of the system. The
Mott insulating phase becomes more stable, which is as
expected since the magnetic field should localize parti-
cles. In the single-layer system, the effect of the SMF
is the most pronunced and this decreases with growing
number of layers L. By adding more layers the global
coherence of the system is restored, because growing di-
mensionality entails the suppression of quantum fluctua-
tions effects. Here, the convergence of properties of the
finite system to those of the infinite (cubic) one is much
slower, although also non-trivially dependent on f (for
some values like f = 1/8,1/6 and 3/8 seems to be much
Figure 3: The evolution of the Hostadter butterfly with in-
creasing number of layers L in the isotropic case of tunneling
ratio t⊥= t?. Adding more layers simply results in additional
fractal patterns and very complex structures of L butterflies
emerging.
more pronunced than for f = 1/4,1/3 and 1/2). When
the system is anisotropic (see, the bottom plot in Fig.
2), the convergence is much faster, but still depending
on the specific value of f.
In the non-interacting (U = 0) Bose-Hubbard model
under the synthetic magnetic field, the energy spectrum
is known as the Hofstadter butterfly [25] (see, the top
plot for L = 1 in Fig. 3). The band width is strongly
dependent on f and exhibits a self-similar gap structure
for rational values of f. It has been observed, that in
the mean-field approach, the critical hopping x1
follows the bandwidth of the Hofstadter’s butterfly [23].
However, in our results the non-monotonicity of xfis not
exactly following the Hofstadter butterfly bandwidth, es-
pecially for lower values of f. Also, we calculate the band
for multiple-layer system and present it in Fig. 3. It is
clear, that increasing the number of layers (for L > 1)
does not influence the width of the band, while the xL
in Fig. 2 evidently changes while converging to the three-
dimensional (cubic) case. We would like to note that,
both densities of states for square system in the mag-
froughly
0/xL
f

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5
η=0.126
η=0.532
η=0.755
η=1
15 10 50100500 1000
0.55
0.60
0.65
0.70
0.75
x /x
0
L
f=1/2
f
15 10 50 100500 1000
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
L
f=1/4
x /x
0
f
η=1
η=0.708
η=0.584
η=0.408
η=0.195
η=0.0289
L
L
L
L
Figure 4: (Color online) The evolution of normalized critical
ratio xL
and f = 1/4 with different values of the tunneling ratio η =
t⊥/t?. Dashed lines show correspondence between systems
with finite and infinite L.
0/xL
fwith number of layers for magnetic field f = 1/2
netic field ρf and DOS of L-layer stack ρL
tained exactly in analytical form with any approxima-
tions. The Hofstadter’s band structure has also been in-
vestigated in context of quantum graphs and the integer
quantum Hall effect[26], where similar energy spectra for
two-dimensional to three-dimensional system crossover
have been observed (non-symmetricity of the spectra re-
sults from their representation as functions of wave vec-
tors instead of energy).
f(η,ξ) are ob-
C. Correspondence between finite L-layer and
infinite systems
Although, conventional optical lattices typically pos-
sess a uniform tunneling matrix elements, by changing
lasers intensity along z axis one can control the tunnel-
ing between adjacent layers. Sometimes the tunneling
rate between adjacent sites is just negligible on the time
scale of experiments [14]. On the other hand, using a
mask in the novel holographic methods [27] one can also
reproduce almost arbitrary potential. Such fully control-
lable optical environment allows us to restrict the spatial
(along z axis) degree of freedom of the particles and in-
troduce c-axis anisotropy to the system. In consequence,
the tunneling between planes can be different from that
x /x
0
f
L
L
η=0.146
η=0.146
f=1/4 f=1/4
η=0.518
η=0.856
η=0.929
η=1
151050 100500 1000
0.040
0.042
0.044
0.046
0.048
0.050
0.052
0.054
15 1050 100 500 1000
0.06
0.08
0.10
0.12
0.14
0.16
0.18
η=0.522
η=0.728
η=1
x /x
0
f
L
L
f=0
L
L
Figure 5: (Color online) The evolution of normalized critical
parameter xL
and for f = 1/4 with different values of the tunneling ra-
tio η = t⊥/t?. Dashed lines show correspondence between
systems with finite and infinite L.
f with number of layers without magnetic field
in the single layer t⊥?= t?or even completely suppressed.
However, since some experimental setups can be more
convenient to use than others, once can try to establish a
correspondence between number of layers and anisotropy,
i.e. find values of L and η that result in the systems that
share the same properties in time-of-flight experiments,
thus are interchangable.The results are presented in
Figs. 4 and 5 for two setups: when the effect of the field
is compared to the system in zero field (xL
4), and when the bare critical value of xL
(see, Fig. 5). In both cases, we calculate xL
as a function of finite number of layers L for isotropic
(η = 1) system. Then, we determine the anisotropy ratio
η for the infinite cubic system (L → ∞), which results in
the same xL
it is clear that 4-layer (quasi 2D) isotropic system corre-
sponds to 3D cubic system with anisotropy ratio equal to
η = 0.755 (f = 1/2, top plot). If the field value f = 1/4,
a 3D system of similar anisotropy (η = 0.708) shares the
properties with 10-layer isotropic one. Fig. 5 shows, that
the c-axis anisotropy is less important when the magnetic
field is turned off (values of η are closer to isotropic η = 1
case).
0/xL
fis monitored
0/xL
f, see Fig.
for xL
f
0/xL
for xL
fvalues. For example, from Fig. 4