Table Of ContentResonant control of cold-atom transport through two optical lattices with a constant
relative speed
M.T. Greenaway1, A.G. Balanov2, and T.M. Fromhold1
1School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
2Department of Physics, Loughborough University, Leicestershire, LE11 3TU, United Kingdom
(Dated: December 11, 2013)
Weshowtheoreticallythatthedynamicsofcoldatomsinthelowestenergybandofastationary
optical lattice can be transformed and controlled by a second, weaker, periodic potential moving
at a constant speed along the axis of the stationary lattice. The atom trajectories exhibit complex
behavior, which depends sensitively on the amplitude and speed of the propagating lattice. When
3 the speed and amplitude of the moving potential are low, the atoms are dragged through the
1 static lattice and perform drifting orbits with frequencies an order of magnitude higher than that
0
corresponding to the moving potential. Increasing either the speed or amplitude of the moving
2
lattice induces Bloch-like oscillations within the energy band of the static lattice, which exhibit
n complex resonances at critical values of the system parameters. In some cases, a very small change
a in these parameters can reverse the atom’s direction of motion. In order to understand these
J dynamics we present an analytical model, which describes the key features of the atom transport
7 and also accurately predicts the positions of the resonant features in the atom’s phase space. The
abrupt controllable transitions between dynamical regimes, and the associated set of resonances,
] provide a mechanism for transporting atoms between precise locations in a lattice: as required
s
for using cold atoms to simulate condensed matter or as a stepping stone to quantum information
a
g processing. The system also provides a direct quantum simulator of acoustic waves propagating
- through semiconductor nanostructures in sound analogs of the optical laser (SASER).
t
n
PACSnumbers: 37.10.Jk,05.45.-a,63.20.kd
a
u
q
. I. INTRODUCTION transport through dissipative optical lattices [19]. For
t
a single-atom transport, both lensing [20] and the dynam-
m ics and control [21–26] of BECs have been observed and
Cold atoms in optical lattices created by two or more
studied.
-
d counter-propagatinglaserbeamsprovideaclean,flexible,
n physical environment for studying quantum particles in Inthispaper,weproposeamethodtocontrolprecisely
o periodic potentials. Key advantages of such systems in- the dynamics of atoms moving through a stationary op-
c clude the tunability of the lattice, low dissipation, and tical lattice by introducing an additional moving peri-
[
theabilitytodirectlycontrolandmeasuretheatoms’in- odic optical potential. We demonstrate that the com-
1 ternal state. When the atoms are cold enough for the de bined effect of the two lattices induces multiple sharp
v Broglie wavelength, corresponding to the atoms’ center- resonances, which can be exploited to tailor the atomic
6 of-mass motion, to span several wells of the lattice po- motion. Also, this system has a direct counterpart in
3 tential, the associated energy eigenvalues form bands of semiconductor physics. Specifically, it models the elec-
2
allowed energies separated by band gaps. The dynamics tron dynamics induced by an acoustic wave propagating
1
of atoms in energy bands is of particular interest due to through semiconductor nanostructures [27]: a topic of
.
1 the similarities with the behavior of electrons in solids growing interest due to the recent development of co-
0
[1–6]. For example, atoms confined to a single energy herent sources of phonons (SASERs) [28–30], which are
3
bandandsubjecttoaconstantforcehavebeenshownto analogs of the optical laser.
1
: BraggreflectandperformmanyBlochoscillations[7–12]: The paper has the following structure. In Section II,
v
aphenomenonwellknown, buthardertorealise, insolid
i weintroducethesemiclassicalequationsofmotionforan
X state physics [13, 14].
atom in a stationary optical lattice driven by a propa-
r Recently, there has been increased experimental inter- gating optical potential. Analysis of the equations re-
a
est in nonlinear phenomena relating to the transport of veals the distinct dynamical regimes of this system and
atomsinopticallattices. Inparticular,ithasbeenshown the abrupt transitions between them. In Section III, we
that moving optical lattices can transport cold atoms study how the speed of the propagating potential affects
alongdistancesasfaras20cm[15]. Inaddition,unusual the average velocity of atoms in the stationary lattice.
band structure properties have been revealed by experi- SectionIVexploreshowtheatom’sinitialpositionaffects
mentswithamovingopticallattice[16],criticalvelocities transitions between the various dynamical regimes. In
in such systems have been identified [17], the stability of Section V, we compare our semiclassical results with full
superfluid currents in systems of ultracold bosons have quantum calculations. Finally, in Section VI, we draw
beenstudied[18], andthephenomenonofstochasticres- conclusions and propose experiments to study the trans-
onances has been predicted to play an important role in port phenomena that we have identified.
2
(a) 700 3
600 v
M
V) 500 2 rmax=1 rmax= 2
(peOL 340000 -1)mm s 1 rmin=1 rmin= 2
V 200 (⟩t0
100 ⟨vx UMb
0 -1
-2 -1 0 1 2 Ud
x / d M
(b) 225 -2
0 50 100 150 200
220 U (peV)
) M
V
e 215
p FIG. 2: Solid curve shows (cid:104)v (cid:105) versus the moving optical
( x t
) x 210 potentialamplitude,UM. Vertical(red)dashedlinemarksthe
p onset of atom dragging (labelled Ud ). Dot-dashed vertical
E( 205 line(blue)markstheonsetofBlochoMscillations(labelledUb ).
M
200 Arrows highlight resonant peaks in (cid:104)vx(cid:105)t, explained in the
text. For guidance, the upper dotted line shows (cid:104)v (cid:105) = v
x t M
-0.5 -0.25 0 0.25 0.5 and the lower dotted line shows (cid:104)v (cid:105) =0.
x t
p d/h
x
(c) 700
600 number k =2π/λ , λ is the wavelength, and ω =
) M M M M
V 500 k v istheangularfrequency. Suchafieldcanbegener-
e M M
(p 400 ated by counter-propagating laser beams whose frequen-
M ciesareslightlydetunedbyδω. ThevelocityoftheMOP
V 300
+ is then v =δω/2k [15, 16, 19].
OL 200 M M
V The semiclassical Hamiltonian for an atom in the low-
100
estenergybandofthestationaryopticallattice,andsub-
0 ject to the moving potential, is H(x,p ,t) = E(p ) +
-10 -5 0 5 10 x x
V (x,t). The corresponding Hamilton’s equations of
x / d M
motion are
FIG. 1: (a) solid curve shows the potential energy, V , of a
OL
23Na atom in the stationary optical lattice. Shaded regions
∂p U
show energy ranges of first and second energy bands. (b) x = Mk cos[k (x+x )−ω t] (1)
E(p ) curve calculated for the first energy band. (c) Solid ∂t 2 M M 0 M
x
curve: totalpotential,V +V ,oftheatomatt=0. Dashed ∂x ∆d (cid:18)p d(cid:19)
OL M v = = sin x . (2)
curve: totalpotentialafterhalfatemporalperiodofVM (i.e. x ∂t 2(cid:126) (cid:126)
at t=π/ω ).
M
We solve Eqs. (1) and (2) numerically to calculate the
trajectories starting from rest with x = x(t = 0) = 0
0
II. SEMICLASSICAL ATOM TRAJECTORIES and v (t = 0) = 0. To characterise the transport of an
x
atom through the SOL, we calculate the time averaged
We consider a cloud of cold, non-interacting, sodium velocity, (cid:104)v (cid:105) , over a period of 0.25 s.
x t
atoms [11, 31–34] initially in the lowest energy band of a The solid curve in Fig. 2 shows (cid:104)v (cid:105) calculated as a
x t
1D stationary optical lattice (SOL), taking the potential functionofU forλ =20dandv =2.5mms−1. The
M M M
energyprofileofeachatomtobeV (x)=V sin2(πx/d) curve reveals that for very low U , (cid:104)v (cid:105) increases expo-
OL 0 M x t
[Fig. 1(a)]. The lattice period d = 294.5 nm and its nentially from 0 with increasing U , until U reaches a
M M
depth is V = 563 peV = 5.5E , where E = 103 peV critical value, Ud ≈2.5 peV = 2.43×10−2E (red ver-
0 R R M R
is the recoil energy. The energy versus crystal momen- ticaldashedline),atwhichpoint(cid:104)v (cid:105) =v (markedby
x t M
tum dispersion relation for the lowest energy band is, to upper horizontal dotted line). As U increases beyond
M
goodapproximation,E(p )=E +∆[1−cos(p d/(cid:126))]/2 Ud , (cid:104)v (cid:105) remains pinned at v until U reaches a sec-
x 0 x M x t M M
[Fig. 1(b)], where E ≈ 200 peV and ∆ = 24.4 peV ond critical value, Ub ≈ 16 peV = 0.156E , (blue ver-
0 M R
is the band width. To this system we apply an ad- tical dot-dashed line) where the average velocity begins
ditional moving optical potential (MOP), which propa- to decrease abruptly with increasing U . Thereafter,
M
gatesalongthexaxisatvelocity,v ,creatingaposition increasing U gives rise to a series of resonant peaks
M M
andtime(t)dependentpotentialenergyfield,V (x,t)= (arrowed in Fig. 2).
M
U [1−sin(k x−ω t)]/2 [Fig. 1(c)], where the wave To understand the form of the (cid:104)v (cid:105) versus U curve,
M M M x t M
3
in Fig. 3 we show atom trajectories for three values of 1.2(a) 20 (b)
U corresponding to three distinct dynamical regimes.
M 15
Figure 3(a) shows x versus t for a trajectory within 0.8 V)
t(<heUlodw,fiwelhdicrhegiismmeawrkheednbUyMre=d1dpasehVed=li9n.7e4in×F10ig−.3E2)R. x (μm) p) (pex105
In thMis case, the atom orbit comprises a uniform drift, 0.4 E' (
0
with mean velocity ≈75 µm s−1, and superimposed pe-
riodic oscillations. The electron velocity vx > 0 when 00 5 10 -05.25 0 0.25 0.5
VM(x,t)<UM/2 [white regions in Fig. 3(a)] and vx <0 30(c) 20 (d)
when V (x,t) > U /2 [gray regions in Fig. 3(a)]. We 15
M M
explain the form of this trajectory by considering atom 20 V)10
mpsiocosatilitoiHonanimnisitlxhto(cid:48)e(ntri)eas=ntifxsr(aHtm)(cid:48)−e(xvoM(cid:48)f,tpth.e)IMn=tOEhPi(cid:48)s(,pfirna)mw+heV,icthh(etxhs(cid:48)e)e,mawtiochlmears’es- x (μm)10 E' (p) (pex 05
x x M -5
E(cid:48)(p ) = E(p )−v p is the effective dispersion rela-
tionxand the pxotentiMal,xVM(x(cid:48))=UM[1−sin(kMx(cid:48))]/2, 010 5 10 -100.25 0 0.25 0.5
which does not explicitly depend on time. The trans- (e) (f)
formed semiclassical equations of motion are B 75
0 B V)
∂∂ptx = U2MkMcos[kM(x(cid:48)+x0)] (3) x (μm) -1 F F F B B E' (p) (pex 0
∂x(cid:48) ∆d (cid:18)p d(cid:19) F -75
v(cid:48) =v −v = = sin x −v . (4)
x x M ∂t 2(cid:126) (cid:126) M -20 5 10 -3 -2 1 0 1 2 3
t (ms) pd/h
x
SinceH(cid:48)istime-independent,itisaconstantofmotion
butdoesnotequalthetotalsystemenergy,H. Forinitial FIG. 3: x(t) [(a), (c) and (e)] and E(cid:48)(px) [(b), (d) and (f)]
conditions x(t=0)=x0 and px(t=0)=p0 we obtain curves calculated for UM =1 peV = 9.74×10−3ER [(a) and
(b)], 10 peV = 9.74×10−2E [(c) and (d)], and 150 peV =
R
1.46E [(e) and (f)]. In (a), (c) and (e), V (x,t) is <U /2
R M M
H(cid:48)(x(cid:48),p )= UM [1−sin(k x )]. (5) [>UM/2],withinthewhite[gray]regions. In(b),(d)and(f),
x 2 M 0 horizontal dashed lines show ±UM/2 and filled circles mark
whenE(cid:48)(p )=±U /2and,therefore,thelimitsoftheorbits
x M
Therefore, if the initial position of the atom is x =0,
0 in p . Arrows, labels, and brackets are discussed in the text
x
the constant of motion is H(cid:48) = UM/2, which means and Appendices A and B.
that the kinetic energy in the moving frame E(cid:48)(p ) =
x
H(cid:48)(x(cid:48),p )−V (x(cid:48))=U /2−V (x(cid:48)) can only take val-
x M M M
ues between ±U /2, marked by the horizontal dashed The LD regime persists with increasing U until the
M M
lines in Fig. 3(b) for U = 1 peV. This figure reveals value of −U /2 [lower dashed line in Fig. 3(b)] falls
M M
that if p = 0, the atom can only access a very limited, below the value of the local minimum in E(cid:48)(p ) [ar-
0 x
almost linear, region of E(cid:48)(p ) near p =0 (between the rowed in Fig. 3(b)]. Using the fact that in this regime
x x
horizontal dashed lines), where v(cid:48) =∂E(cid:48)(p )/∂p ≈−ν, |p |(cid:28)π(cid:126)/d, we find that the p value corresponding to
x x x x x
with ν being a positive constant. Consequently, in the the local minimum of E(cid:48)(p ) is p ≈ 2(cid:126)2v /∆d2. Sub-
x x M
restframe,themeanvelocityoftheatom,(cid:104)v (cid:105) ≈v −ν, stituting this value into the expression for E(cid:48)(p ), we
x t M x
is smaller than v . This can be seen from Fig. 3(a), obtain the following estimate for the critical amplitude
M
where the mean (drift) velocity of the atom orbit, is less of the moving lattice, Ud , above which the atom can
M
thantheslopeofthegreyandwhitestripes,whichequals reach and traverse the local minimum in E(cid:48)(p ):
x
v . As the atom traverses successive minima and max-
M
ima [white and gray stripes in Fig. 3(a)] of the mov-
1
inglattice,thecorrespondingforcechangessign,thereby Ud = × (6)
M 1+sink x
causing the atom to oscillate between the extremal p M 0
x
values marked by the circles in Fig. 3(b) and, in real (cid:20)4(cid:126)2v2 (cid:18) (cid:18)2(cid:126)v (cid:19)(cid:19)(cid:21)
M −∆ 1−cos M .
space, oscillate around the mean drift [Fig. 3(a)]. As ∆d2 ∆d
U increases [i.e. the horizontal dashed lines in Fig.
M
3(b)movefurtherapart],theatomcanaccessmoreofthe Fortheparametersconsidered,Eq. (6)givesUd ≈2.5
M
E(cid:48)(p ) dispersion curve. As a result, ν becomes smaller peV = 2.43 × 10−2E , which compares well with the
x R
and (cid:104)v (cid:105) increases towards v , as shown by the section transition to the region where (cid:104)v (cid:105) = v in Fig. 2 (i.e.
x t M x t M
of the black curve in Fig. 2 to the left of the vertical to the right of the vertical red dashed line).
reddashedline. Atommotioninthis‘LinearDispersion’ ThesolidcurveinFig. 3(c)showsthetrajectoryofthe
(LD) regime is described in detail in Appendix A. atom when U = 10 peV = 9.74×10−2E > Ud . We
M R M
4
find regular, almost periodic, oscillations superimposed E(cid:48)(p ) ≈ −U /2 [lower dashed line in Fig. 3(f)] and
x M
on a uniform drift, indicated by the dashed line in Fig. maximum attainable value of p [right-hand filled circle
x
3(c). The slope of the line is v , indicating that the in Fig. 3(f)]. Thereafter, p starts to decrease, trigger-
M x
atom is dragged through the SOL by the MOP. This is ing another burst of Bloch oscillations, until p reaches
x
confirmedbythefactthatthetrajectoryisconfinedtoa its minimum possible value [left-hand filled circle in Fig.
single minimum in the MOP potential where V (x,t)< 3(f)]. Subsequently, p increases again and the cycle re-
M x
0 [gray region in Fig. 3(c)], which traps the atom and peats. The details of the form of this orbit are given in
transports it through the SOL. Fig. 3(d) shows that the Appendix B.
atom is able to oscillate in a parabolic region of E(cid:48)(p ) The analysis in the appendix shows that the atom’s
x
between the two p values (filled circles) where E(cid:48)(p )= average velocity will exhibit a resonant peak whenever
x x
U /2. Since the atom remains in the almost parabolic U just exceeds a local maximum, or falls below a local
M M
region of E(cid:48)(p ), x(cid:48)(t) is an almost harmonic function of minimum, inE(cid:48)(p ), thusmaximisingthetime theatom
x x
t. Therefore we can approximate the trajectory in Fig. spendsinaregionofE(cid:48)(p )wherev(cid:48) =dE(cid:48)(p )/dp >0,
x x x x
3(c)bytheexpressionx(t)≈v t+λ [1−cos(ω t)]/4, andsoalsomaximising(cid:104)v (cid:105) =(cid:104)v(cid:48)(cid:105) +v . Theminima
M M R x t x t M
where ω is the frequency for motion to and fro across and maxima in the dispersion curve occur, respectively,
R
the potential well [27]. In this ‘Wave Dragging’ (WD) at p values given by
x
regime, the atom is trapped in a single well of the MOP,
butasthewellmoves,itdragstheatomthroughtheSOL
2π(cid:126) 2(cid:126)2v
with(cid:104)vx(cid:105)t =vM (upperhorizontaldottedlineinFig. 2). prmin ≈r + M, (8)
IncreasingU intheWDregimeinitiallyhasnoqual- x min d ∆d2
M
itative effect on the trajectories, which continue to be
and
dragged and are of the form x(t) = v t+f(t). How-
M
ever, as U increases, the atom can access increasingly
M
hnaornm-poanriacb.oElivcepnaturtaslloyf,tEh(cid:48)o(upxg)h,aUndMtbheucsofm(et)slbaercgoemenesoulegshs prxmax ≈−(2rmax−1)πd(cid:126) − 2(cid:126)∆2dv2M, (9)
fortheatomtotraversethefirstlocalmaximumofE(cid:48)(p )
x
[arrowed in Fig. 3(d)]. Thereafter, the atom can reach where rmin and rmax are integers (= 1,2,3...) labelling
theedgeofthefirstBrillouinzoneanditstrajectoryinpx the minima and maxima, respectively, in E(cid:48)(px). The
changesabruptlyfromclosedtoopen. Inthisregime,the corresponding critical values of UM, obtained from set-
atomcantraverseseveralminizones,allowingittoBragg ting E(cid:48)(px)=E(px)−vMpx =±UM/2, are
reflect and perform Bloch oscillations. The first local
maximumofE(cid:48)(p )occurswhenp ≈(cid:126)π/d−2(cid:126)2v /∆d2
x x M
[27]. Substituting this value into E(cid:48)(px) we find the fol- UMrmin ≈−2E(prxmin)+2vMprxmin, (10)
lowing approximate expression for the wave amplitude,
Ub , corresponding to the transition between the WD and
M
and ‘Bloch Oscillation’ (BO) regimes, i.e. for which the
value of U /2, shown by the upper dashed line in Fig.
M Urmax ≈2E(prmax)−2v prmax. (11)
3(d) reaches the local maximum in E(cid:48)(p ) [27]: M x M x
x
The values of Umax and Umin for r and r = 1
M M max min
2 (cid:20) (cid:126)πv 2(cid:126)2v2 (cid:21) and 2 are shown in Fig. 2 by the labelled arrows and
UMb ≈ 1−sink x ∆− dM + ∆d2M . (7) correspond very well with the positions of the resonant
M 0
peaks in (cid:104)v (cid:105) .
x t
The vertical blue dot-dashed line in Fig. 2 shows the
value of Ub obtained from Eq. (7) for x = 0, which
M 0
coincides exactly with the abrupt suppression of (cid:104)v (cid:105) . III. VARIATION OF MOVING LATTICE SPEED
x t
Figure 3(e) shows the atom’s trajectory when U =
M
150 peV = 1.46E > Ub . In this regime, the atom The parameters of a moving optical potential can be
R M
undergoes fast oscillations [bracketed in Fig 3(e)], inter- varied over a wide range [15, 33]. For example, it has
rupted by jumps in the orbit (arrowed). Initially, for been shown that v can be increased to ∼ 50 mm s−1
M
t=0 and x(t=0)=p (t=0)=0, the driving force on [15] and U to ∼500 peV [33].
x M
the atom is maximal, and therefore p increases rapidly. The color map in Fig. 4 shows the variation of the
x
As a result, the atom traverses several local maxima and average velocity of the atom, (cid:104)v (cid:105) , with both v and
x t M
minima in E(cid:48)(p ) [between circles in Fig. 3(f)], Bragg U . The figure reveals a complex patchwork of distinct
x M
reflecting and performing Bloch oscillations [within left- dynamical regimes. At the boundaries between these
most bracketed region of trajectory in Fig. 3(e)] as it regimes, the average velocity changes abruptly and may
does so. As E(cid:48)(p ) decreases, V increases to keep H(cid:48) even change sign, indicating that the atom’s response to
x M
constant. Eventually, V attains its maximum value of the moving wave depends critically on both the ampli-
M
U /2correspondingtotheminimumattainablevalueof tude and velocity of that wave.
M
5
dynamicschangestraightfromtheLDtotheBOregime,
200 i.e. without traversing the WD region.
4 IntheBOregime,weexpecttheatomtohaveitshigh-
est value of (cid:104)v (cid:105) when U = Ud since here the atom
x t M M
160 spendsmosttimeintheregionoftheE(cid:48)(p )curvewhere
3 x
the gradient of E(cid:48)(p ), and thus v(cid:48), is positive. This is
x x
⟨
v confirmedbythecolormapinFig. 4,whichrevealsanar-
x
V) 120 2t⟩ row region of high (cid:104)vx(cid:105)t values (red), where (cid:104)vx(cid:105)t ∼vM,
pe (m jusAtlasbooivnecltuhdeeddoitnteFdibgl.ac4kacruervbelaaclkonagnwdhriecdhdUoMt-d=asUhMded.
(M 1 m curvesshowing,respectively,Urmin [Eq. (10)]andUrmax
U 80 s [Eq. (11)] for r and r =M1,2 and 3. These cuMrves
-1 min max
0 ) show good correspondence with the positions of (cid:104)vx(cid:105)t
peaks in the color map, which appear as blue / green
40 BO or yellow stripes for UM (cid:38)40 peV = 0.390ER.
-1
WD
LD
0 IV. EFFECT OF INITIAL POSITION
0 2 4 6 8 10
v (mm s-1) Equations(6)and(7)indicatethatthetransitionsbe-
M
tween distinct dynamical regimes depend on the initial
FIG.4: Colormap(scaleright)showingthevariationof(cid:104)vx(cid:105)t position of the atom x(t = 0) = x0. To illustrate this,
with v and U . Dotted black curve shows Ud versus v , Fig. 5showsacolormapof(cid:104)v (cid:105) versusx andU when
M M M M x t 0 M
and marks the onset of dragged atom trajectories [Eq. (6)]. v = 2.5 mm s−1 and, as before, λ = 20d. The color
M M
Black solid curve shows UMb versus vM [Eq. (7)]. Black dot- map has an intricate structure which, we now explain,
dashed curves show UMrmin versus vM when rmin = 1,2 and is associated with the transition between the various dy-
3 (bottom to top) [Eq. (10)]. Red dot-dashed curves mark
namical regimes.
values of Urmax when r = 1,2 and 3 (bottom to top)
M max ThesolidcurveinFig. 2correspondstoaverticalslice
[Eq. (11)]. Labels LD, WD and BO mark regions of ‘Linear
through the color map in Fig. 5, when x =0, i.e. along
Dispersion’, ‘Wave Dragging’ and ‘Bloch Oscillations’. 0
themiddle of the five vertical blacklines. Increasing U
M
along this line, induces a transition from the LD into
theWDregimeoncrossingtheblackdottedcurvewhere
The structure of the color map in Fig. 4 can be un-
(cid:104)v (cid:105) = v . The WD regime appears as the dark red
x t M
derstood by considering the transitions between the LD,
regioninthecolormap. WithfurtherincreaseofU ,we
M
WD and BO regimes discussed in the previous section.
enter the BO regime (on crossing the black solid curve),
The dotted black curve in Fig. 4 shows Ud versus v
M M wherethecolormapchangesabruptlyfromredtoyellow
[Eq. (6)],markingthetransitionbetweentheLDandthe
and the atom’s velocity falls dramatically. Within the
WD regimes. For v (cid:46) 3.5 mm s−1 (i.e. to the left of
M BO regime there is a series of resonances, which appear
the intersect between the solid and dotted black curves),
asyellowareasinFig. 5andcorrespondtothosearrowed
increasing U across the black dotted curve induces a
M in Fig. 2. As explained in Section II and Appendix B,
clear transition from the LD regime, where (cid:104)v (cid:105) ≈ 0
x t these resonances occur whenever U coincides with a
M
(blue in Fig. 4), to the WD regime, where (cid:104)vx(cid:105)t ≈ vM local extremum in E(cid:48)(p ).
x
(yellow / red in Fig. 4).
The variation of Ud with x [Eq. (6)], shown by the
The black solid curve in Fig. 4 shows UMb versus vM dotted curve in Fig.M5, agrees0 well with the transition
[Eq. (7)] and thus marks the transition from the WD from the LD to the WD regime seen in the color map.
regime into the BO regime. The region of the color map As x increases from 0 to λ /4 the transition to the
0 M
below the black solid curve and above the black dotted WD regime shifts slowly to lower Ud , which attains a
M
curve corresponds to the WD regime where (cid:104)vx(cid:105)t = vM. minimum value of 1.3 peV = 1.27×10−2ER when x0 ≈
At the lower edge of this region, the color map changes λ /4. For x > λ /4, Ud increases slowly until x =
M 0 M M 0
fromblue(low(cid:104)vx(cid:105)t)tored(high(cid:104)vx(cid:105)t)asUM increases. λM/2, at which point the dynamics are approximately
Asv increases, therangeofU overwhichdragging equivalent to those at x =0. Conversely, decreasing x
M M 0 0
occursdecreasesandvanishesatv =∆d/(cid:126)π =3.5mm below 0 makes Ud increase rapidly until x ≈−3λ /16
M M 0 M
s−1, when Ud =Ub (crossing of solid and dotted black whenthedottedcurveinFig. 5diverges. Asx decreases
M M 0
curves in Fig. 4). At this v value, the tilt of E(cid:48)(p ) is further, weenteraregimeinwhichthereisnotransition
M x
large enough to ensure that the value of U needed for to the WD regime. When x ≈ −5λ /16, the dotted
M 0 M
the atom to traverse the first local minimum in E(cid:48)(p ) curve re-emerges and the value of Ud decreases rapidly
x M
is the same, or larger, as that required to traverse the with decreasing x until x = −λ /2, which is exactly
0 0 M
first local maximum. Therefore, when U ≈ Ud , the equivalent to x = λ /2, meaning that the color maps
M M 0 M
6
unabletotraversethelocalmaximuminE(cid:48)(p )[arrowed
x
200 in Fig. 3(d)] and thus cannot Bloch oscillate. Figure 5
revealsthattheBlochoscillationregimereappearsinthe
2 parameter space when x ≈ 5λ /16. Thereafter, Ub
0 M M
160 (right-hand segment of the black solid curve) decreases
with increasing x until x = λ /2, which is equivalent
0 0 M
1 ⟨
v to x0 = −λM/2 (color maps are identical on left- and
x
V) 120 t⟩ right-hand edges of Fig. 5).
pe rmax= 3 r = 3 0 (m theABsOx0rdegeicmreeasoecscufrrosmat0s,lothwelytrdaencsrietaiosninfgroUmtvhaeluWesDunto-
U (M 80 rmax= 2 min -1m s-1 tciulrUveMds(cxr0o)ss=inUFMbi(gx.05)).(wAhserxe0tdheecrbelaasceksdfuotrttMehdera,ntdhesotwlido
rmin= 2 ) curvescrossagainwhenx0 ≈−3λM/8. Forx0 valuesbe-
tweenthesetwocrossingpoints,thereisnowavedragging
40 rmax= 1 BO -2 regimeand,forgivenx0,increasingUM inducesatransi-
rmin= 1 tionstraightfromtheLDtotheBOregimeuponcrossing
WD theblacksolidcurve. Asx0 approaches−λM/4then,for
0 LD -3 any given U , the atom cannot traverse the local mini-
M
-λ /2 -λ /4 0 λ /4 λ /2 mum in E(cid:48)(p ) [arrowed in Fig. 3(b)] and thus also does
M M M M x
x0 not reach the first local maximum in E(cid:48)(px) [arrowed in
Fig. 3(d)]. Therefore, the atom will only change direc-
FIG. 5: Color map (scale right) showing the variation of the tion when it reaches the peak in E(cid:48)(px) occurring when
average atom velocity with x and U . Black dotted curve p ≈−(cid:126)π/d, resulting in the dramatic change from neg-
0 M x
shows the value of UMd (x0) representing the onset of dragged ative to positive velocity [transition from blue to yellow
atomtrajectories[Eq. (6)]. Blacksolidcurvemarksthevalues regions in Fig. 5] when U ≈45 peV = 0.438E .
M R
ofUMb (x0)[Eq. (7)]. Blackdot-dashedcurvesshowUMrmin(x0) As discussed previously, the strong resonant features
when r = 1,2 and 3 [Eq. (10)]. Red dot-dashed curves
min in (cid:104)v (cid:105) versus U and x in Fig. 5 occur when the
are the values of Urmax(x ) where r = 1,2 and 3 [Eq. x t M 0
M 0 max atom accesses new peaks in E(cid:48)(p ) [see Eqs. (10) and
(11)]. Forguidance,verticalblacklinesshowthepositionsof x
(11)]. The variation of Urmin and Urmax with x shown
the center of five wells in the SOL, located at −2d, −d, 0, M M 0
by the black and red dot-dashed curves respectively in
d and 2d. Labels LD, WD and BO mark regions of ‘Linear
Dispersion’, ‘Wave Dragging’ and ‘Bloch Oscillations’. Fig. 5 for (bottom to top) rmin and rmax = 1,2 and 3
agreewellwiththeyellowregionsofthecolormapwhere
(cid:104)v (cid:105) increasesabruptly. Notethatneartheseresonances,
x t
the color map changes abruptly between yellow and blue
on the left- and right-hand edges of Fig. 5 are identical.
indicating that (cid:104)v (cid:105) changes suddenly from positive to
Equation (6) shows that as v increases from 0, x t
M negative: i.esmallchangesinx orU switchtheatom’s
Ud (x = 0) also increases and, therefore, the range 0 M
M 0 direction of travel.
of U values corresponding to the WD regime is re-
M
duced. Figure 5 and Eq. (6) show, in addition, that as
x →−λ /4,Ud (x )→∞,meaningthatforanygiven
0 M M 0 V. QUANTUM CALCULATIONS
v and U the atom will never enter the WD regime.
M M
We understand this in the effective dispersion curve pic-
In this section, we investigate the full quantum-
ture by noting, from Eq. (5), that when x = −λ /4,
0 M
H(cid:48) = U and so E(cid:48)(p ) can only take values between 0 mechanical description of the semiclassical dynami-
M x
cal regimes revealed in Sections II-IV. The quantum-
and U . Therefore, the atom cannot traverse the local
M
minimum in E(cid:48)(p ) when p ≈ 0 [arrowed in Fig. 3(b)] mechanical Hamiltonian for the system is
x x
and thus cannot enter the WD regime.
The variation of Ub with x [Eq. (7)], shown by the (cid:126)2 ∂2
M 0 Hˆ =− +V (x)+V (x,t), (12)
blacksolidcurveinFig. 5,isinexcellentagreementwith 2m ∂x2 OL M
a
thetransitionfromtheWDregime(redregioninFig. 5)
totheBOregime(aboveboththeblackdottedandsolid where ma is the mass of a single 23Na atom. We solved
curves in Fig. 5). As x increases from 0, Ub increases the corresponding time-dependent Schr¨odinger equation
0 M
until x ≈ 3λ /16 where, for the range of U shown,
0 M M
there is no transition to the BO regime above the red
∂ψ(x)
region. Equation (7) shows that as x → λ /4, Ub → i(cid:126) =Hˆ(x,t)ψ(x), (13)
0 M M ∂t
∞, implying that trajectories with x =λ /4 are never
0 M
abletoBlochoscillateforanyv orU . Physicallythis numericallyusingtheCrank-Nicolsonmethod[35]. Note
M M
isbecausewhenx =λ /4,accordingtoEq. (5),E(cid:48)(p ) that we consider the mean-field interaction to be negli-
0 M x
only takes values between −U and 0, so the atom is gible compared with the external potential. This can be
M
7
realisedinexperimentbyusinganappliedmagneticfield 30(a) 30 (b)
to tune the Feshbach resonance so that the inter-atomic
scattering length is zero. By only considering a non- 20 20
interactingatomcloudwecanpresentafullanddetailed
descriptionofthefundamentaldynamicalphenomenaex- μm) 10 10
hibited by the system. If atom interactions are included x (
we expect more complex dynamics which would deserve 0 0
further study and analysis.
-10 -10
First, we found the form of the initial wavepacket by
evolving Eq. (13) in imaginary time, using an initial 8 30
(c) (d)
Gaussian wavefunction with a full width half maximum
value, fx =2d, to produce a state similar to the ground 4 20
state of an atom cloud held in the SOL by a harmonic
trap. To study the atomic dynamics, we then integrated m) 10
μ 0
Eq. (13) numerically, using this wavepacket as the ini- x (
tial state, which, when fx = 2d, extends across ∼ 5 0
wells of the SOL. Figure 6 shows the time-evolution of -4
the wavepacket in each of the three distinct dynamical -10
regimes corresponding to the semiclassical trajectories -8
0 5 10 0 5 10
shown in Fig. 3 (vM =2.5 mm s−1, λM =20d) (see also t (ms) t (ms)
Supplemental Information [36]). The central black curve
in Fig. 6(a) corresponds to the semiclassical trajectory FIG.6: Gray-scalemaps[coloronline: white=0,blue=high]
in the LD regime (UM = 1 peV, x0 = 0) shown in Fig. spheoVwi=ng9.t7h4e×ev1o0l−ut3iEon oafndthfe a=tom2dd(eLnDsitryegfoimr:e)(;a)(bU)MU ==1
3(a) and Supplemental Information, Movie 1 [36]. Also R x M
10 peV = 9.74×10−2E and f = 2d (WD regime); (c)
shown are semiclassical trajectories calculated for (from R x
U = 150 peV = 1.46E and f = 2d (BO regime); (d)
bottom to top) x = −2d,−d,0,d and 2d, i.e. for orbits M R x
0 U = 25 peV = 0.243E , f = 8d which encompasses both
M R x
starting at the centers of the 5 wells that are spanned
the Wave Dragging and Bloch Oscillation regimes. See text
by the initial wavefunction. In this case, most of the forfurtherdetails. Solidcurvesin(a)-(c)showcorresponding
wavepacket, shown by the gray-scale map in Fig. 6(a), semiclassicaltrajectorieswith,frombottomtotop,x =−2d,
0
does not follow the semiclassical trajectories. Instead, it −d, 0, d and 2d. Movies showing the time evolution of the
spreads rapidly with only a small fraction of the wave- wavepackets are shown in the Supplemental Material [36].
function being dragged through the lattice.
From Fig. 5, at first sight we might expect that the
wavepacket lies entirely in the LD regime because, at the atom density evolution reveals that, in this regime,
U = 1 peV = 9.74×10−3E , the vertical lines mark- thewavepacketisdraggedthroughtheSOLbytheMOL.
M R
ing the centers of the potential wells spanned by the ThereasonforthisisclearfromFig. 5,whichrevealsthat
wavepacketalllieinthelightgreenregionbelowtheblack forU =10peVthefullspreadofthewavefunction(be-
M
dotted curve, i.e. fully within the LD regime. It is clear tweentheleft-andright-mostdashedverticallinesinFig.
from Fig. 6(a), however, that the wavepacket does not 5)lieswithintheredwavedraggingregime. ForthisU
M
follow a LD semiclassical trajectory. This is because for value, the moving lattice is strong enough to dominate
suchalowU valuethespreadingofthewavepacket,re- any spreading caused by the zero point energy.
M
sulting from position-momentum uncertainty, dominates Figure 6(c) and Supplemental Information, Movie 3
the dynamics obtained from the semiclassical analysis. [36], shows the wavepacket evolution for U = 150 peV
M
From the uncertainty principle, fx∆px ≈ (cid:126), where ∆px = 1.46ER, corresponding to the BO regime [e.g. the
is the spread of the momentum of the wavepacket, and orbit in Fig. 3(e)]. The five semiclassical trajectories
the corresponding zero point energy of the wavepacket shown in Fig. 6(c) broadly encompass the evolution of
is Ezp = ∆p2x/2ma. When fx = 2d, Ezp ≈ 2.5 peV = the negative-going part of the wavepacket. One might
2.43×10−2ER. Thisenergyislargeenoughtoallowpart expect that the energy band would break for this large
of the wavepacket to enter the wave dragging regime, as value of U , causing the wavepacket to no longer track
M
observedinFig. 6(a). Thiszeropointenergydetermines the semiclassical dynamics. However, we note that the
alowerlimitofUM belowwhichthesemiclassicalanalysis relatively long wavelength λM = 20d of the moving lat-
becomes invalid due to being dominated by wavepacket tice ensures that the energy drop across a single well
expansion. within the SOL is ≈ 45 peV = 0.438E , which is small
R
Figure 6(b) and Supplemental Information, Movie 2 enough to keep the band intact. Note also for high val-
[36], shows the evolution of the wavepacket when U = ues of U the approximation that the atom will remain
M M
10peV=9.74×10−2E ,correspondingtothesemiclassi- in the lowest energy miniband becomes invalid, we esti-
R
caltrajectoryintheWDregimeshowninFig. 3(c). The matethisupperlimitonU byassumingthatinterband
M
striking similarity between the semiclassical orbits and tunnelling will occur when the amplitude of the moving
8
wave is approximately equal to the energy gap between atom trajectories is predictable, and can be tailored by
the first and second minibands. The two minibands cal- small changes in the MOP parameters, the system pro-
culated for the stationary optical lattice are ≈ 250 peV videsamechanismformovingatomcloudsbetweenwell-
apart[seeFig. 1(a)]. Thereforeweexpectinterbandtun- defined points in stationary optical lattices. It may also
nelling to occur when U (cid:38) 250 peV. It appears, from provide a sensitive method to split atom clouds, with
M
our numerical simulations, that for U (cid:38) 200 peV = potential uses in atom interferometry and in mapping
M
1.95E , the wavepacket dynamics start to deviate from the initial atom cloud dynamically rather than via opti-
R
the semiclassical paths due to band breakdown. cal imaging. The dynamics described in this paper are
Thesensitivedependenceoftheatomdynamicstothe similar to those expected for electrons in acoustically-
initial position, x , shown in Fig. 5, has an interesting driven superlattices [27]. Consequently, the cold-atom
0
consequence. Namely, if the atom cloud is large enough, system that we have studied also provides a quantum
it can simultaneously span more than one dynamical simulator for acoustically-driven semiconductor devices
regime. For example, in Fig. 6(d) and Supplemental in which the electron orbits are much harder to observe
Information, Movie 4 [36], we show the wavepacket evo- directly.
lution for U = 25 peV = 0.243E , with an initial
M R
atom cloud width f = 8d and center of mass posi-
x
tion x0 = 1.5d. The figure shows that the atom cloud Appendix A: Detailed description of atom motion in
splits, with the lower part performing Bloch oscillations the Linear Dispersion regime
around x = 0 and the rest being dragged through the
SOL by the MOL. The reason is, as can be seen from In the LD regime, initially, for t = 0 and x(cid:48) = 0,
Fig. 5, that the upper part of the initial atom cloud the force applied to the atom by the moving optical lat-
with x (cid:38) 1.5d lies in the wave dragging regime whilst tice, is maximal, which makes p increase until E(cid:48)(p )=
the lower part with x (cid:46) 1.5d is in the Bloch oscillation −U /2[lowerfilledcircleinFig.x3(b)]. Atthispointx,the
M
regime. Our semiclassical analysis suggests that approx- atom’svelocityintherest frame,v =∂E(cid:48)(p )∂p +v ,
x x x M
imately half the cloud is in the wave dragging regime. attains its maximum value for the specified initial con-
This estimate agrees well with the numerical evolution dition. The potential energy V (x(cid:48)) = U (to keep H(cid:48)
M M
of the wavefunction, which reveals that after 10 ms, the constant) is also maximal, and so the force is zero in the
fractionofthecloudinthewavedraggingregimeis0.495. moving frame. However, in the moving frame x(cid:48) contin-
This result suggests that information about the shape ues to decrease, which eventually makes the force nega-
of the initial atom cloud could be obtained by measur- tive(−dV (x(cid:48))/dx(cid:48) <0),soacceleratingtheatominthe
M
ing the number of atoms, NU, that end up in the upper negative px direction.
(dragged) part of the split cloud, as a function of x .
0 The atom continues to move in the negative p direc-
x
For example, as x0 increases so that more of the initial tionuntilE(cid:48)(p )=U /2[upperfilledcircleinFig. 3(b)]
x M
wavepacket lies within the red WD regime in Fig. 5, i.e.
where the atom’s velocity in the rest frame is minimal.
so that the initial wavepacket is shifted across the black
Theatomthenreturnstoitsinitialconditionwithp =0
0
solid curve, more atoms will end-up in the upper part
andthecyclerepeatswiththeatomtraversingsuccessive
of the split cloud. More quantitatively, we expect that
regions where V < U /2 [white regions in Fig. 3(a)]
N (x ) ≈ (cid:82)∞ |Ψ(x ,t=0)|2dx where x satisfies M M
U 0 xS(UM) 0 S and VM > UM/2 [gray regions in Fig. 3(a)]. Since the
Ub (x )=U , i.e. marks the position of the solid curve region of the dispersion curve accessed by the atom is
M S M
in Fig. 5 for given U . not completely linear, the velocities at the extremities of
M
the orbit are slightly different, causing the atom to have
non-zero average velocity.
VI. CONCLUSIONS
We have shown that a periodic optical potential mov- Appendix B: Detailed description of atom motion in
ing through a stationary optical lattice can control the the Bloch Oscillation regime
atom orbits in real and momentum space, and induce
transitions between three distinct dynamical regimes. In In the BO regime, the atom can access E(cid:48)(p ) across
x
a semiclassical picture, the atom can be confined to a severalBrillouinzones,whichmakes(cid:104)v (cid:105) varyinacom-
x t
small region of the stationary optical lattice dispersion plex way with increasing U . For U = 150 peV =
M M
curve, dragged through the stationary optical lattice by 1.46E ,thelowestpossiblevalueofE(cid:48)(p )[lowerdashed
R x
themovingwave, orperformBloch-likeoscillations. The line in Fig. 3(f)] occurs just below a local minimum in
crossover between these orbital types creates a rich pat- the curve [right-hand arrow in Fig. 3(f)]. The velocity
tern of transport regimes, with multiple resonances that of the atom in the moving frame, ∂E(cid:48)(p )/∂p , is neg-
x x
cause abrupt changes in the magnitude and sign of the ative at the right-hand filled circle, indicating that the
atom’s average velocity. These resonances occur when- atom is moving in the negative x(cid:48) direction. However,
everthereisanabruptchangeinthenumberofBrillouin since V (x(cid:48)) is maximal at this point, the moving lat-
M
zones that the atom can access. Since the form of the tice exerts little force on the atom. Consequently, p de-
x
9
creasesonlyslowlyand,asaresult,theatomalsolingers the atom take a large jump backwards along the x−axis
in the region of E(cid:48)(p ) where v(cid:48) (and thus v ) is positive [arrowslabelled‘B’inFig. 3(e)]. Figure3(e)revealsthat
x x x
[within the region marked by the right-hand bracket in for U = 150 peV, the atom jumps backwards further
M
Fig. 3(f)]. Overall, as the atom moves away from the than it jumps forwards, so giving the atom an overall
right-hand filled circle in Fig. 3(f), it spends more time negative average velocity.
with positive than negative velocity causing the atom to
jump forwards along the x−axis [arrows labelled ‘F’ in IfU hasavaluethatjustexceedsalocalmaximumor
M
Fig. 3(e)]. fallsjustbelowalocalminimumintheE(cid:48)(p )curve,then
x
Conversely, the maximum attainable value of E(cid:48)(p ) thetimethattheatomspendsintheregionoftheE(cid:48)(p )
x x
[upper dashed line in Fig. 3(f)] occurs just below the curve, where dE(cid:48)(p )/dp =v(cid:48) >0, is maximised. Con-
x x x
localmaximuminE(cid:48)(p ),markedbytheleft-handarrow sequently, the magnitude of the forward jump in the or-
x
in Fig. 3(f), where V is maximal. Since the force on bit,andtherefore(cid:104)v (cid:105) =(cid:104)v(cid:48)(cid:105) +v ,ismaximised. This
M x t x t M
theatomislowinthispartoftheE(cid:48)(p )curve,theatom phenomenonproducestheseriesofpeaksobservedinthe
x
remains in the negative velocity region of E(cid:48)(p ) [left- (cid:104)v (cid:105) versus U curve [see arrowed peaks in Fig. 2], the
x x t M
hand bracket in Fig. 3(f)] for a long time. This makes positions of which are discussed in the main text.
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