Super Bloch oscillations with modulated interaction

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by  Elena Díaz

Super Bloch oscillations with modulated interaction
Elena D´ıaz,1 Alberto Garc´ıa Mena,1 Kunihiko Asakura,1,2 and Christopher Gaul1,3
1GISC, Departamento de F´ısica de Materiales, Universidad Complutense, E-28040 Madrid, Spain
2Yonago National College of Technology, 683-8502, Yonago, Japan
3CEI Campus Moncloa, UCM-UPM, Madrid, Spain
(Dated: January 22, 2013)
WestudysuperBlochoscillationsofultracoldatomsinashakenlatticepotential,subjectedtoa
harmonically modulated mean-field interaction. Usually, any interaction leads to the decay of the
wavepacketanditssuperBlochoscillation. Here,weusethephasesofinteractionandshakingwith
3 respect to the free Bloch oscillation as control parameters. We find two types of long-living cases:
1 (i) suppression of the immediate broadening of the wave packet, and (ii) dynamical stability of all
0 degrees of freedom. The latter relies on the rather robust symmetry argument of cyclic time [Gaul
2 et al., Phys. Rev. A 84, 053627 (2011)].

n
PACSnumbers: 03.75.Lm;52.35.Mw;37.10.Jk
a
J

1
Experiments with ultracold atoms in optical lattices spaceregionswith,say,positivemassthaninregionswith
2
are an ideal testing ground for many problems of con- negativemass,whichresultsinadriftofthewavepacket

] densed-matter physics [1, 2]. In addition to good mea- in the direction of the force. As the relative phase be-
s
surement access, these systems offer flexible manipula- tween BO and shaking changes, the time-averaged mass
a
g tion of the system parameters, which opens the way to becomes negative and the drift gets reversed, which re-
- newperspectivesandeffects. Oneofthemostprominent sults in an SBO at the beating frequency.
t
n examples is the observation of Bloch oscillations (BOs) SBOshavebeenmostlystudiedinthelinear,noninter-
a ofcoldatomicgases[3]andofBose-Einsteincondensates actingcase[9,10]. Butultracoldbosonsopenmoreinter-
u
(BECs) [4] in optical lattices, subjected to an external esting possibilities. Feshbach resonances can be used to
q
. force F. The semiclassical explanation for BOs is the arbitrarilychangethes-wavescatteringlength[11],even
at following: thequasimomentum(cid:126)kincreaseslinearlywith time-dependently [12–14]. Generically, the interaction
m time, but because of the dispersion relation of a tight- leads to dephasing and decay of the wave packet. How-
- binding model with hopping amplitude J and lattice pe- ever, the interplay of modulated interactions and BOs
d
riod a, E(k) = 2J[1 cos(ka)], the group velocity is a has already been investigated, and an infinite family of
n −
sinusoidal function of time. The particle does not follow
o
c the potential gradient but stays localized and performs
[ an oscillatory motion. 60 (a)
2 By optical means or by magnetic levitation, the lat- 40 |Ψn|2 0.025
v tice potential can be shaken with frequency ω. This 20 0.02
0.015
5 results in a renormalization of the hopping amplitude, n 0 0.01
0 -20 0.005
which can even be suppressed, the so-called dynamic 0
4 -40
localization [5, 6]. Semiclassically, a harmonic shaking
1 -60
. F(t) = ∆F sin(ωt) causes the quasimomentum (cid:126)k =
2 (cid:82) dt(cid:48)F(t(cid:48)) to oscillate rapidly and to explore k-space re- 1.5 (b) g(t) cospa 1+∆σ
1 1 ¯h
2 gions with renormalized or even negative effective mass 0.5
0
1 meff(k) ∝ 1/cos(ka). Time-averaging cos(ka) leads to -0.5
: the renormalization of the hopping J /J = J (∆F/ω), -1
v eff 0
-1.5
i where J0 is the Bessel function of the first kind. Thus, 0 5 10 15 20
X
dependingonthestrengthandthefrequencyoftheshak-
t/T
B
r ing, the effective hopping J can be suppressed or even
a eff
negative,freezingorinvertingthecenter-of-massmotion. FIG. 1. (Color online) Stable interacting SBO. The initial
wave function has Gaussian shape with width σ = 20a and
Themodulationoftheforcearoundafinitemeanvalue 0
quasimomentum p(0) = (cid:126)π/2a. Force (2) with F = 0.2J/a,
leadstoasuperpositionoftheBOwithaslowoscillation 0
∆F =0.6F ,l/ν =4/5,andφ =0;interaction(3)withg =
0 F 0
of large real-space amplitude, similar to the one shown 1,φ=0. (a)Real-spacedensity|Ψ (t)|2 fromtheintegration
n
in Fig. 1. This phenomenon is known as quasi BO [7] or ofEq.(1). (b)Timeevolutionofsomemagnitudesofinterest:
superBO(SBO)[8]. Itcanbeexplainedwithasemiclas- theinteractiong(t),theinversemassterm∼cos(pa)andthe
sicalreasoning, too: thequasimomentumperformssmall variation of the real-space width ∆σ(t) = σ(t)−σ(0). The
oscillations around its linearly increasing mean value. gray circles mark the points of symmetry used in the cyclic-
time approach.
Then, during one Bloch cycle, it spends more time in k-
2
(harmonic)modulationsthatleadtoaperiodictimeevo- 0.4 20
lution of the wave packet has been found [15–17]. Here, 0) 0.2 ×
N( 0
weconsiderSBOsinthepresenceofamodulateds-wave T)/IP-0.2 -1 -0.5 0 0.5 1
scatteringlength. Thisproblemismorecomplexbecause N(-0.4
already the linear problem contains two frequencies and ∆IP--00..68 (a)
twophasesfromtheBOandtheshaking,respectively. In
0.4
theremainderofthisBriefReport,wetackletheproblem 20
0) 0.2 ×
with a full integration of the discrete Gross-Pitaevskii N( 0
P -1 -0.5 0 0.5 1
equation and a cyclic-time argument similar to that of T)/I-0.2
Ref. [17], for one particular phase of the shaking. After- N(-0.4
wardswestudyfrequenciesandphasesnotcoveredbythe ∆IP-0.6 (b)
-0.8
cyclic-time argument by means of collective coordinates
φ/π
ofaGaussianwavepacketandlinear stability analysis of
the infinite wave packet.
FIG. 2. (Color online) Relative change of the momentum
Model. Our starting point is the mean-field tight- IPN as a function of the phase φ for (a) φ = 0 and
F
binding equation of motion as in Refs. [15–17], but now (b)φ =−π/2. Discs(withlinesasguidetotheeye): results
F
the tilt F may be time dependent: fromintegrationofEq.(1)afterintegrationtimesT =TSBO,
15T ,and30T (fromtoptobottom). Solidline: Result
SBO SBO
i(cid:126)Ψ˙ = J(Ψ +Ψ )+F(t)anΨ +g(t)Ψ 2Ψ . from the collective coordinates theory (11). Regardless of a
n n+1 n−1 n n n
− | | (1) possibleinitialincrease,theIPNalwaysgoesdowninthelong
run, except for those points where ∆IPN=0.
ThewavefunctionΨ isnormalizedtoone,implyingthat
n
the time-dependent interaction parameter g(t) contains
the total particle number. We choose the force as
for φ = 0 the phase φ = 0 from Fig. 1 is one of the
F
two stable points, where the IPN stays constant. Simi-
F(t)=F +∆F cos(ωt+φ ). (2)
0 F
larly, in the case φ = π/2 of Fig. 2(b), two “stable”
F
−
The mean value F defines the Bloch frequency ω = points arise as well. However, their stability behavior is
0 B
F a/(cid:126). This and the frequency of the modulation ω different and will be explained in what follows.
0
are the two frequencies of interest. For concreteness, we Cyclic-time formalism for SBOs. Weexpressthedis-
choose a fixed frequency ratio ω/ωB =l/ν =4/5 for the crete wave function Ψn of (1) in terms of a continuous
rest of this Brief Report. Thus, the super Bloch period wave function A(z,t):
T is five Bloch periods. We now search for suitable
SBO Ψ (t)=A(na x(t),t)eip(t)na/(cid:126)+iφ(t) . (5)
modulations of g(t) that allow a periodic time evolution n
−
of (1). A constant interaction parameter leads sooner
With the semiclassical equations p˙ = F(t), x˙ = v =
g
or later to a decay of the SBO [8], but harmonic modu- −
2Jasin(pa/(cid:126))/(cid:126), and (cid:126)φ˙ = 2Jcos(pa/(cid:126)), the envelope
lations around zero may counteract this effect [15]. We
function obeys the nonlinear Schr¨odinger equation
choose the simplest commensurate modulation
i(cid:126)A˙ = Jcos(pa/(cid:126))A(cid:48)(cid:48)+g(t)A2A , (6)
g(t)=g0sin(ωBt/ν+φ). (3) − | |
where higher derivatives of A have been neglected. The
Throughout the paper and we vary only the phase φ.
factor in front of the second derivative takes the role of
Figure1(a)showsthetimeevolutionofthewavefunction
the mass term Jcos(pa/(cid:126)) = (cid:126)2/2m(t). Taking into ac-
for φ = φ = 0, a particular choice of parameters for
F
countthecommensurabilityofthefrequenciesω andω ,
which SBOs turn out to be stable. B
we express the time dependence in terms of the slower
In general the SBO decays with time. This decay is
time τ =ω t/ν, which increases by 2π during one super
most conveniently observed via the broadening of the B
Bloch period T = νT . After integrating the force
wave-function momentum distribution, observed via the SBO B
(2), we find the inverse mass term as
inverse participation number (IPN) of Fourier modes
(cid:88) cos(pa/(cid:126))=sin(cid:8)ντ + a∆F [sin(lτ +φ ) sinφ ](cid:9),
IPN= |Ψk|4. (4) (cid:126)ω F − F (7)
k
where we have chosen the initial condition such that
Adecreaseofthisnumbermeansmomentumbroadening ap(τ = 0) = π(cid:126)/2. This can always be achieved by
and decay of the wave packet. Figure 2 shows the varia- choosing the origin of time.
tion∆IPN(T)=IPN(T) IPN(0)afteracertaintimeT We know from previous works [15–17] that BOs can
−
for φ = 0, π/2 as a function of the interaction phase exist in the presence of suitably tuned AC interactions.
F
−
φ. For most of the phases, the IPN goes down, meaning Now, we construct a similar cyclic-time formalism for
that the wave packet broadens. Figure 2(a) shows that SBOs. The basic idea is to separate all terms of Eq.
3
30 1
(6) into the form η˙f(η), where η is a harmonic function
of τ and f an arbitrary function. Then η˙ is eliminated 25 0.1 |Ψk|2
and Eq. (6) is solved as a function of η only. As η is a 0.01
periodicfunctionoft, thetimeevolutionofAisperiodic 20
0.001
too, and we have indeed found a case of stable SBOs in
t
the presence of a non-zero interaction g(t). 15 0.0001
TSBO
The inverse mass term (7) can only be brought into 1e-05
10
the desired form η˙f(η) if φ =0 [or any phase shift that
F 1e-06
allows for a common node of sinντ and sinlτ]. Then, 5
1e-07
cos(pa/(cid:126))=sin(τ) (cosτ), (8) (a) (b) (c)
F 0 1e-08
-0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5
which is the most general 2π-periodic function that re- (k p)a (k p)a (k p)a
− − −
spects the odd symmetry with respect to τ =0 mod π.
FIG. 3. (Color online) Momentum-space portraits in the ref-
This determines η = cosτ. Thus, the left hand side of
(6) is A˙ = η˙∂ A. Finally, we may choose any modu- erenceframeofthecenterofmomentumpforshaking(2)and
η interaction (3), with σ = 20a, F = 0.2J/a, ∆F = 0.6F ,
0 0 0
lated interaction of the form g = η˙g˜(η). In particular,
g = 1. (a) Unstable sine shaking φ = −π/2 with φ = 0;
0 F
g(t) = g0sin(F0t/ν), i.e., Eq. (3) with φ = 0 fulfills the (b) sine shaking with phase φ = −0.26π adjusted according
cyclic-time condition. The symmetries involved in this to Eq. (11); (c) stable case of Fig. 1, φ =0, φ=0.
F
argument can be observed in the lower panel of Fig. 1:
g(t) and cos(cid:0)p(t)a/(cid:126)(cid:1) share the nodes of η˙, marked with
gray circles, and are odd with respect to these points. regard, we can compute analytical results in the limit
All physical quantities, e.g. the width variation ∆σ, are of a wide wave packet w(t) σ2 1. Then, Eqs. (3)
≈ 0 (cid:29)
functions of η only and are thus even with respect to the and (10c) give b(t)≈−aνg0/[8√πσ03ωB]cos(ωBt/ν+φ).
mentioned points. With (10d), and assuming that l ν =1, this yields
| − |
So far, we have explained the stability observed in
Fig. 1 and the stable points of Fig. 2(a). It turns out (cid:90) dtw˙ (l ν) √πa3Jg0 J (cid:0)δFν(cid:1)sin(φ φ ),
that the phase φF = −π/2 of shaking considered in TSBO w ≈ − σ03((cid:126)ωB/ν)2 1 l − 0
Fig. 2(b) cannot be written in the form required by the (11)
above cyclic-time argument, because the inverse mass with φ0 =(l ν)[φF +a∆F sin(φF)/(cid:126)ω]. Depending on
term cosap/(cid:126) does not exhibit points with odd symme- thephaseφo−ftheinteraction, thewavepacketcontracts
try. Nevertheless, there are two particular phase shifts, or spreads after a full super Bloch cycle TSBO. This is
φ 0.26π and φ 0.74π, where the broadening of the also reflected in the IPN shown in Fig. 2. The dynamics
≈− ≈
wave packet is suppressed. One can understand this us- can be strictly periodic only for φ = φ0 mod π. With
ing a collective-coordinates theory. The idea is to reduce the values l=4, ν =5, φF = π/2, and a∆F =0.75(cid:126)ω,
−
thecomplexityoftheproblembyparametrizingthewave this yields φ0 =0.74π, in agreement with Fig. 2(b).
function by only two coordinates, its position x and its In Fig. 3, the momentum-space portrait of SBOs is
width √w [15, 18]: shownfordifferentphasesoftheshakingandoftheinter-
(cid:18) (cid:19) action. Panel (a) shows an unstable case of sine shaking
1 na x
Ψ (t)= − eipna/(cid:126)+ib(na−x)2/(cid:126). (9) with immediate broadening of the momentum distribu-
n
√4w A √w
tion. Inpanel(b)thephaseoftheinteractionisadjusted
(u) is a continuous normalized Gaussian wave function as required by the collective-coordinates criterion of Eq.
A
withvarianceone. Theequationsofmotionforthecollec- (11) and it shows the behavior presented by the two ap-
tivecoordinates,xandw,andtheirconjugatemomenta, parently “stable” points in panel (b) of Fig. 2. Here,
p and b, are theimmediatebroadeningisindeedsuppressed;butona
longer time scale ( 20T = 100T ), side peaks grow
SBO B
p˙ = F(t), (10a) ≈
farfromthecentralwavepacket. Thisisanindicatorfor
−
(cid:126) x˙ (cid:20) 1/4+4b2w2/(cid:126)2(cid:21) adynamicalinstabilityanddoesnothappeninthestable
=2asin(pa/(cid:126)) 1 , (10b)
J a − 2w/a2 case with cosine shaking (see Fig. 1), whose k-space por-
a2 1 16w2b2/(cid:126)2 a3g(t) trait is shown in Fig. 3(c) for comparison. In this case,
b˙ = − cos(pa/(cid:126))+ , (10c) there is neither immediate broadening nor an instability.
J 4w2/a4 8√πJw3/2
Theextremelysmallresidualbroadening,isduetoeffects
(cid:126)w˙ wb
=8 cos(pa/(cid:126)) . (10d) beyond the approximation made in Eq. (6).
Ja2 (cid:126) Linear stability analysis. Collective coordinates rely
A necessary condition for stability within the collective- on a smooth Gaussian profile of the wave function and
coordinatesapproximationisthatw(t)returnstoitsini- cannotdescribefeaturesonshortlengthscales. Weclose
tial value after a full super Bloch period T . In this this gap with a linear stability analysis of the infinitely
SBO
4
π 1.01
to the destruction of such oscillations in general. There-
fore, our proposal to tune the atomic interaction time
0.5π 1.005
dependently is important to reduce this effect and to al-
φ 0 1 low for the detection of such oscillatory dynamics of a
condensate. The results of this work can be considered
0.5π 0.995
− (a) toimprovethequalityofSBOsinordertobeusedtoen-
π 0.99 gineermatter-wavetransportovermacroscopicdistances
−
π 1.01
in lattice potentials in a more reliable way, which should
0.5π 1.005 also be relevant for atom interferometry [20].
ThisworkwassupportedbyMinisteriodeEconom´ıay
φ 0 1
Competitividad (project MAT2010-17180). Research of
0.5π 0.995 C.G. was supported by a PICATA postdoctoral fellow-
−
(b) ship from the Moncloa Campus of International Excel-
π 0.99
− lence (UCM-UPM).
0 0.5 1 1.5 2 2.5 3
qa
FIG.4. (Coloronline)Stabilityprediction∆ (black: stable,
q
white: unstable)forfluctuationmodeq=k−pasafunction
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| |
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