D Witthaut, F Keck, H J Korsch and S MossmannFB Physik, University of Kaiserslautern, D-67653 Kaiserslautern, GermanyE-mail: [email protected]
New Journal of Physics 6 (2004) 41Received 20 January 2004Published 8 April 2004Online at http://www.njp.org/DOI: 10.1088/1367-2630/6/1/041
Abstract. Bloch oscillations in a two-dimensional periodic potential undera (relatively weak) static force are studied for separable and non-separablepotentials. The dynamics depends sensitively on the direction of the staticfield with respect to the lattice. Almost dispersionless periodic motion of thewavepackets is observed, as well as breathing modes. The origin of the weakdispersion is analysed. In addition, the coherent Bloch–Zener oscillation for adouble-period potential inducing an additional Rowland ghost gap is discussed.
Recently, an increasing interest in the dynamics of Bloch oscillations can be observed, motivated,e.g. by ongoing studies of the dynamics of ultra-cold atoms or Bose–Einstein condensates inoptical lattices, both in experiment and theory (see e.g. [1]–[3]).
Most surprisingly, however, the dynamics of particles in space-periodic structures under theinfluence of a (possibly time-dependent) linear force, the so-called Wannier–Stark systems, hasbeen investigated almost exclusively for one-dimensional systems. Except for a few papers bythe founding fathers of this field like Wannier [4], the fascinating behaviour in higher dimensionshas only been explored very recently [5]–[11]. In [5, 8], the classical–quantum correspondencefor strong fields has been studied which is mainly governed by decay phenomena and chaoticscattering. A theoretical analysis of the two-dimensional tight-binding and single-band modelin terms of infinite-variable Bessel functions can be found in [7] and, finally, a dynamical studyof the two-dimensional tight-binding model has shown Bloch oscillations and, in a speciallyconstructed two-band model, coherent Bloch–Zener oscillations on two bands [9]. Further studieswere carried out from the solid state point of view. In fact, these studies focused on Blochoscillations in quantum-dot superlattices in two or three dimensions [10] and on the damping ofthe oscillations [11].
In the present paper, selected numerical studies of the dynamics of wavepackets in twospace dimensions for relatively weak forces will be presented and discussed, without anytight-binding or single-band approximation. It will turn out that the dynamical propertiescalculated in the single-band (resp. the tight-binding approximation) as described in [9]essentially give very similar results to the full quantum simulation presented in this paper.It is the hope of the authors to stimulate in this way experimental studies. Furthermore,we will discuss in detail the suppression of dispersion in two-dimensional Wannier–Starksystems.
Throughout this paper we will use dimensionless units where the period of the lattice is 2πin both lattice directions and the mass is equal to unity, i.e. the Hamiltonian is
H = p2x
2+p2y
2+ V(x, y) + Fxx + Fyy. (1)
Of course, it is always possible to make a coordinate transformation from a lattice withdifferent periods along the coordinate axes to a lattice with only one period. But thenone has to consider different effective masses for the coordinate directions. Here we willrestrict ourselves to the case that both periods are equal (resp. the effective mass isunity). This appears to be reasonable as far as optical lattices are concerned. Potentials ofthe form
V(x, y) = V0[cos2 kx + cos2 ky] or V(x, y) = V0[cos kx + cos ky]2 (2)
can be easily generated for a system of cold atoms in lattices generated by crossed laser beams,where k is the laser wavenumber (see e.g. [12] and references therein). By a scaling and rotationof the variables, both potentials can be brought to the form
V(x, y) = V0(cos x + cos y + ε cos x cos y) (3)
with ε = 0 or 1. The energy unit is chosen to satisfy V0 = 1. A more detailed discussion ofthe scaled coordinates can be found in [9]. Here we only point out that, if ε is given, the onlyindependent parameters of the system are the field components Fx, Fy and the value of the scaledPlanck constant h, which can be changed experimentally by varying the intensity of the laser
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Figure 1. Colour map of the potential (3) for ε = 0 (left; egg-crate); ε = −1(centre; quantum dot); ε = +1 (right; quantum antidot); maxima are colouredred, minima blue.
field. In our calculations we chose h = 3.3806, which corresponds to the Kasevich experiment[13]. However, in section 3, we used h = 2.
The potential (3) is separable for ε = 0; it shows a typical egg-crate structure with alternatingminima and maxima. For ε = −1, we have an array of minima on a flat background, in theterminology of semiconductor physics a quantum dot, and for ε = +1 an array of maxima, aquantum-antidot potential. A more general case is considered in section 3. Figure 1 shows thethree potentials. The dynamics strongly depends on the direction of the field with respect to thelattice. Here we will assume a rational ratio
Fx/Fy = q/r with coprime q, r ∈ N. (4)
Thus, with the definition of the field strength
F =√F 2x + F 2
y , (5)
the field can be written as(FxFy
)= F√
q2 + r2
(q
r
). (6)
This system has been previously investigated in a tight-binding approximation [9].For all the cases studied in the following sections, an initial minimum uncertainty Gaussian
wavepacket
ψ(x, y) = (2πσ2)−1/2exp
{−[(x− x0)2 + (y − y0)
2]
4σ2+
ip0,xx
h+
ip0,yy
h
}(7)
centred at (x0, y0) with initial momentum (p0,x, p0,y) and width σ = �x = �y is propagatedin time by means of a split-operator method [14]. We consider two cases which lead toquite distinct dynamics: broad wavepackets, which populate initially several potential wellscoherently with sharply localized momentum and, alternatively, initially strongly localizedpackets with a broad momentum distribution. We mainly concentrate on the first case, where
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
we will demonstrate the oscillatory motion of the centre of the wavepacket (section 2.1) withsurprisingly small dispersion, which is analysed in detail in section 2.3. In addition, for adouble-period potential, such a wavepacket can show coherent Bloch–Zener oscillations ondifferent energy surfaces. This is investigated in section 3. In the second case, discussed insection 2.2, two-dimensional breathing wavepackets are found whose mean position is fixedin space.
Finally, some remarks concerning the numerics are appropriate: In sections 2.2 and 3,the coordinate plane with a length of 320π is discretized with 2048 steps in each direction,�x = �y = 5π/32, which leads to a resolution in momentum space of �k = 1/160. In theremaining sections the coordinate space has length 200π and a discretization of�x = 25π/128(�k = 1/100). The Bloch period is discretized in 16 384 (section 3), 4096 (section 2.2), (resp.1024 time steps) (otherwise). The system shows only a small decay of the wavepacket for theparameter values chosen in this paper. Nevertheless, through our numerical method, periodicboundary conditions are imposed on the system so that the decaying fractions reappear. To avoidthis non-physical behaviour, we multiply after every time step the wavefunction with a functionthat is unity for the main part of coordinate space and vanishes smoothly at the boundaries.But the lattice is large enough so that the propagation of the main part of the wavepacket is notinfluenced by this finite-size effect.
2. Two-dimensional Bloch oscillations
In this section, we discuss various aspects of two-dimensional Bloch oscillations. First, thedynamics of a wavepacket with a spatially broad initial distribution is considered. For a separablepotential and a rational field direction, the mean value of such a wavepacket follows a periodictrajectory which is the superposition of two perpendicular oscillations with commensurablefrequencies, i.e. a Lissajous oscillation. This behaviour can be described using the band modelof the field-free case together with quasi-classical considerations as discussed in section 2.1. ALissajous-like oscillation is also found in the non-separable case. For initially strongly localizedwavepackets, the oscillatory motion of the mean value is replaced by an oscillation of the widthof the wavepacket. This will be discussed in section 2.2. Eventually, in section 2.3, the problemof dispersion is analysed using an approach that takes into account the full Wannier–Starkdynamics.
2.1. Lissajous oscillations
As already mentioned above, a broad initial wavepacket shows Lissajous oscillations of itsmean value. For our calculations we considered the Gaussian wavepacket (7) with σ = 7π/
√2.
Due to the discrete translational invariance of the potential (3) in the field-free case, there existsan additional quantum number, the quasi-momentum κ.
In the following it will turn out to be useful to discuss the dynamics in terms of thedispersion relation E(κ), i.e. the eigenenergies of the field-free system which depend on thequasi-momentum κ and form the well-known Bloch bands displayed in figure 2 for potential (3).The symmetries of the potential translate into the relations
E(κx, κy) = E(−κx,−κy) and E(κx, κy) = E(κx,−κy). (8)
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Figure 2. Dispersion relation for the field-free case for the separable potential(ε = 0, left) and the non-separable potential (ε = +1, right).
M
κy
κxXΓ
Figure 3. Brillouin zone for a two-dimensional lattice. The dashed red lineillustrates the ‘quasi-classical’ motion of the wavepacket for the field directionFx/Fy = 2/1.
Figure 2 shows the dispersion relation for a path along the lines connecting the highly symmetricpoints �, X and M in the first Brillouin zone (see also figure 3) where we use the notation ofsolid-state physics (see e.g. [15]).
In a quasi-classical approximation, the quasi-momentum κ follows the Bloch equation [16]
hκ(t) = −F . (9)
In the examples discussed below, the field strength is chosen as F = 0.03 and the directionisFx/Fy = q/r, i.e. the wavepacket moves in the direction of the dashed line in the Brillouin zoneas shown in figure 3 for the case q/r = 2/1 and κ(0) = 0. In the following, we will compare theseparable potential for ε = 0 with the non-separable case ε = +1.
Separable case ε = 0: In the separable case, the dynamics is simply a superposition of twoindependent motions evolving in the x- and y-direction, respectively, each of them executing aone-dimensional Bloch oscillation as studied in a preceding paper [17]. The initial populationof the higher bands rapidly decays to infinity and the remaining ground-band projection of theinitial wavepacket oscillates with the Bloch period TBx = h/Fx in the x-direction extending overa region Lx = �/Fx where � is the width of the ground band in the field-free case. The samehappens in the y-direction, however with period TBy = h/Fy and Ly = �/Fy. For rational field
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Figure 4. Wavepacket dynamics for Fx/Fy = 2/1 (separable potential ε = 0).Shown is a snapshot at time t = 0.16TB. The animation lissajous210-sep.avishows the Bloch oscillation over two Bloch periods.
directionsFx/Fy = q/r, one can define a total Bloch periodTB as the least common multiple of thetwo one-dimensional Bloch periods. For the discussed field direction q/r = 2/1, we have TB =TBy = 2TBx and Ly = 2Lx. The computation shows that the wavefunction indeed follows such aLissajous trajectory as can be seen from the animation lissajous210-sep.avi (a snapshot attime t = 0.16TB is shown in figure 4). At the beginning of the motion (at the �-point in figure 2),we observe the emission of four similar wavepackets in both directions of the coordinate axes,which are mainly given by the projections of the initial wavepacket on the higher Bloch bands(cf the discussion in [17]). After one-fourth of the Bloch period, at the turning point of themotion in x-direction, we observe the emission of a second wavepacket. At this time the quasi-momentum κ(t) has reached the boundary of the Brillouin zone at κ = (− 1
2 ,− 14). In figure 2 this
is in the middle between the points X and M (see also figures 3 and 6). At this point, a fractionof the wavepacket tunnels to the first excited band. The tunnelling probability, i.e. the fractionof the emitted wavepacket, is given by
P = exp
(− πδ2
2h2F
)(10)
([18], see also the detailed recent analysis in [19]), where δ denotes the energy gap between theground and the first excited band. Here, the splitting is δ ≈ 0.998, yielding a small tunnellingprobability of P ≈ 0.010, in good agreement with the simulation. At the turning points ofthe motion, the wavepacket mainly localizes at the minima of the potential, which is wellknown from the one-dimensional case [17]. Keeping in mind that there are two differentBloch periods, it is clear that this localization appears as a pattern of stripes. After a fullBloch period TB, the initial wavepacket is reconstructed, more precisely its ground bandprojection.
Quantum-antidot potential ε = +1: In spite of the fact that the potential is no longerseparable, one finds a very similar Bloch oscillation of Lissajous type as in the separable case
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Figure 5. Wavepacket dynamics for Fx/Fy = 2/1 (quantum-antidot potential,ε = +1). A snapshot at time t = 0.41TB is shown. In the animationlissajous210-ep1.avi, one clearly sees the splitting of the emitted wavepacketas discussed in the text.
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
t/TB
E(κ
)
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
t/TB
E(κ
)
Figure 6. Dispersion relation E(κ(t)) with Fx/Fy = 2/1 for the separablepotential (ε = 0, left) and for the quantum-antidot potential (ε = +1, right).
above. Reconsidering figure 2, we observe that the dispersion relation of the ground band isalmost identical in both cases. Differences become important only for higher bands. Thus theBloch oscillation is very similar for the separable and non-separable cases. Only the decay ofsmall fractions of the wavepacket by Bloch–Zener tunnelling shows characteristic differencesfor the two cases. At time t = TB/4 = TBx/2, i.e. at the outer turning point of the x-motion(see figure 3), we observe the emission of a wavepacket as in the separable case. Because ofthe smaller band gap of δ ≈ 0.788, this emission is much stronger than in the separable case.The emitted wavepacket immediately splits into two daughter wavepackets (see figure 5), whichis due to the band splitting between the first and second excited band as can be deduced fromfigure 6, where the dispersion relation E(κ(t)) is plotted. The width of the gap is δ ≈ 0.395at κ = (0.34,−0.34), leading to an approximately equal intensity of the daughter wavepacketsin both bands. A similar process happens again at time t = 3TB/4, but this time the daughter
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Figure 7. Trajectories for the mean position (〈x〉(t), 〈y〉(t)) of the wavepacketfor field directions Fx/Fy = 1/1 (blue curve) and Fx/Fy = 2/1 (red and magentacurves). In the second case, one of the wavepackets (magenta) has non-zero initialmomenta leading to a phase shift of π/2. Shown is the evolution over two Blochperiods.
wavepacket does not split up because the band gap between the first and the second excited bandsis sufficiently large.
Different field directions: In the separable case ε = 0, the dynamics of the wavepacket is asuperposition of one-dimensional Bloch oscillations. Thus the trajectories of the centre of thewavepacket trace out Lissajous figures. The shape of these Lissajous figures is determined bythe ratio of the one-dimensional Bloch frequencies, i.e. the field direction, and the phase shift�φ between the one-dimensional oscillations which is determined by the initial momentumcomponents of the wavepacket:
�φ = 2π(p0,x − p0,y)/h. (11)
Three examples of such Lissajous figures are displayed in figure 7. The motion of the positionexpectation value of the wavepacket (〈x〉(t), 〈y〉(t)) is shown for a field direction q/r = 1/1 (bluecurve) and a field direction q/r = 2/1 (red and magenta curves). For the trajectory coloured inmagenta, the initial momentum is chosen to be (p0,x, p0,y) = (h/4, 0) which leads to a phaseshift of �φ = π/2 and the wavepacket moves on a characteristic figure-eight-shaped curve. Asthe dynamics of the centre of the wavepacket is very similar in the non-separable case, one canalso observe such Lissajous oscillations in these systems.
2.2. Breathing modes
A wavepacket that is initially strongly localized in coordinate space (σ � π) and, consequently,has a broad momentum distribution does not show Bloch oscillations but instead a breathingbehaviour. The time dependence of the width of the wavepacket can be understood in a tight-binding model, which has been studied for one-dimensional systems [17, 20]. Such breathing
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Figure 8. Breathing modes for the separable potential ε = 0 (left, see animationbreathing-sep.avi) and the non-separable case ε = +1 (middle and right, seeanimation breathing-ep1.avi). Shown are snapshots at times t = 0.5TB (leftand middle) and t = TB (right).
modes have also been studied previously in two dimensions in terms of electron localization insemiconductor superlattices in a single-band approach [10]. Breathing modes in two dimensionsare illustrated in figure 8 for the separabl case (left) and for the non-separable case ε = +1(middle and right). The field has the strength F = 0.015 and the direction q = r = 1. Thewavepacket is strongly localized at t = 0, which corresponds to a small dot at (x, y) = (0, 0)in figure 8.
For a separable potential, the breathing mode (figure 8, left) is simply a superpositionof two one-dimensional breathing modes. The non-separable case (figure 8, middle and right)shows a similar breathing behaviour. Nevertheless, we observe some differences in the decayingfraction of the wavepacket and the reconstruction after integer multiples of the Bloch period. Inthe separable case, decay can be observed mainly in the coordinate directions. After one Blochperiod, the wavepacket is essentially reconstructed, showing only weak fractions spread out alongthe coordinate directions. In the non-separable case, there are also fractions decaying alongand perpendicular to the field direction. Furthermore, the wavepacket shows weak dispersionperpendicular to the field direction.
2.3. Dispersion
In the one-dimensional case, a wavepacket in a Wannier–Stark system shows no systematicdispersion. In spite of the fact that the momentary width �xt =
√〈(x− 〈x〉)2〉 can oscillate
strongly, it remains bounded. This can be evaluated analytically in the tight-bindingapproximation even for arbitrary time-dependent forces (see [17, 20] for a recent discussion).This is, however, no longer true in the two-dimensional case, even in the single-band tight-binding approximation [9]. Figure 9 shows an animation of the wavepacket dynamics for theseparable and non-separable cases where the field F is oriented along the x-axis. Two Blochoscillations are displayed and the systematic broadening in the y-direction is obvious in bothcases.
To analyse the dispersion theoretically, it is instructive to distinguish between the separableand the non-separable cases.
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Figure 9. For a field directed along the x-axis, the wavepacket showsstrong dispersion in a direction perpendicular to the field both for ε = 0(left, see animation dispersion-sep.avi) resp. ε = +1 (right, see animationdispersion-ep1.avi). The figures show snapshots at time t = 1.25TB.
Separable potential ε = 0: For a separable potential V(x, y) = Vx(x) + Vy(y), theHamiltonian (1) decomposes as H = Hx +Hy with
Hx = p2x
2+ Vx(x) + Fxx, Hy = p2
y
2+ Vy(y) + Fyy (12)
and the two-dimensional problem is reduced to one dimension. As already mentioned, the one-dimensional Bloch oscillations show no dispersion and consequently there is no systematicdispersion for a separable potential in two dimensions provided that Fx and Fy are both differentfrom zero. If the field is directed along one of the coordinate axes, the motion in the othercoordinate direction is field-free and shows of course dispersion.
Non-separable potential ε �= 0: The problem of dispersion can be analysed by decomposingthe wavefunction into the (resonance) eigenstates of the Hamiltonian (1), the so-called Wannier–Stark states,
Hα,m,κ⊥ = Eα,m(κ⊥)α,m,κ⊥ . (13)
The Wannier–Stark states and quasi-energies depend only on the projection κ⊥ of the quasi-momentum κ in a direction perpendicular to the field F . The quasi-energies Eα,m(κ⊥) form theWannier–Stark ladder of resonances
Eα,m(κ⊥) = Eα,0(κ⊥) + 2πF · m, (14)
where α is the ladder and m the site index. Though resonance states are connected with complexeigenenergies E , for the following the imaginary part can be neglected because of the low Starkfields considered here (see e.g. [9] for higher fields). The quasi-energy E(κ⊥) should not beconfused with the dispersion relation for the field-free case (figure 2).
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Resonance states with different site indices are transformed into each other by a simplespatial translation:
Tnα,m,κ⊥(r) = α,m,κ⊥(r − 2πn) = α,m+n,κ⊥(r). (15)
The wavefunction in the basis of Wannier–Stark states can be written as
|ψ(t)〉 =∫
dκ⊥∑m,α
cα,m(κ⊥)e−iEα,m(κ⊥)t/h|α,m,κ⊥〉. (16)
For simplicity, we assume that only the lowest ladder with α = 0 has significant contributionsto the sum so that we can skip the index α.
Inserting equation (14) and usingm,κ⊥(k) = e−2πim·k0,κ⊥(k) (see equation (15)) one gets
ψ(k, t) =∫
dκ⊥e−iE0(κ⊥)t/h∑m
cm(κ⊥)e−2πim(k+F t/h)0,κ⊥(k) (17)
=∫
dκ⊥e−iE0(κ⊥)t/hCκ⊥(k + F t/h)0,κ⊥(k) (18)
for the momentum representation of the wavefunction. A similar derivation holds also forthe one-dimensional case [17]. For integer multiples of the Bloch time, t = ν TB, with TB =√r2 + q2 h/F and ν = 1, 2, 3, . . ., we get
Cκ⊥(k + FνTB/h) = Cκ⊥(k). (19)
If the quasi-energy E0(κ⊥) does not depend on κ⊥, i.e. E0(κ⊥) = E0, the wavefunction isreconstructed after integer multiples of the Bloch period up to an overall factor e−iE0 νTB/h. Thedegree of dispersion can therefore be estimated by the band width of E(κ⊥). For weak fields, thequasi-energy can be approximated using a Houston ansatz [21]
E(κ⊥) = 1
TB
∫ TB
0E(κ − F t/h) dt, (20)
where E(κ) denotes the (periodic) dispersion relation in the field-free case. Inserting the Fourierexpansion
E(κ) =∑m
Emei2πm·κ (21)
into equation (20) yields after a short calculation
E(κ⊥) =∑n
Ern,−qnei2πn(rκx−qκy) =∑n
Ern,−qnei2πnκ⊥√q2+r2
. (22)
In the second step we used κ⊥ = e⊥ · κ with the unit vector e⊥ = (r2 + q2)−1/2 (r,−q). TheFourier components Ern,−qn decrease rapidly with increasing |n| (see figure 10) and in a first-order approximation one obtains
E(κ⊥) ≈ E0,0 + 2Er,−qcos(
2πκ⊥√q2 + r2
). (23)
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Figure 10. Fourier coefficients |Em| of the ground band dispersion relation inthe field-free case.
Thus, the bandwidth of the quasi-dispersion relation is approximately given by
�E ≈ 4Er,−q. (24)
We can further derive an estimate for the dependence of the bandwidth �E of thequasi-dispersion relation on the direction of the Stark field. For that purpose, we expand theeigenfunctions of the field-free system, the Bloch functions φκ(r), into a Fourier series in κ,
φκ(r) =∑
n
ei2πκ·nψn(r), (25)
where the Fourier components are the Wannier functions ψn(r) and evaluate the Fouriercomponents of the quasi-dispersion function (note that 〈ψ0 |φκ〉 = 1):
Em = 1
2π
∫e−i2πκ·mE(κ) dκ = 1
2π
∫e−i2πκ·m〈ψ0|H |φκ〉 dκ
= 1
2π
∫e−i2πκ·m ∑
n
ei2πκ·n〈ψ0|H |ψn〉 dκ = 〈ψ0|H |ψm〉. (26)
In the separable case, we can factorize the Wannier functions and make use of theirexponential localization [22] which yields the estimate
with some constant a. Notice that this dependence is different from the one given in [10]. Asalready discussed, the dispersion relation of the field-free system is very similar for both the
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Figure 11. Bandwidth of the exact quasi-dispersion relation calculatednumerically by diagonalization of the Floquet–Bloch operator for differentfield strengths (|F | = 0.01 (+) and |F | = 0.03 (×)) and directions Fx/Fy =1/1, 2/1, 3/1, 3/2, 4/1, 5/1 and 6/1. Additionally, the Fourier coefficients ofthe field-free dispersion relation (◦) are plotted with an exponential fit (see textfor more details).
separable and the non-separable cases. Thus the estimate (27) is expected to hold in the non-separable case as well. We therefore expect that the bandwidth decays exponentially as a functionof the field direction Fx/Fy = q/r,
�E = 4Er,−q ∼ e−a(|q|+|r|). (28)
Figure 11 shows the bandwidth of the exact quasi-dispersion relation calculated numericallyby diagonalization of the Floquet–Bloch operator for two field strengths, |F | = 0.01 (+) and|F | = 0.03 (×), and different directions Fx/Fy = 1/1, 2/1, 3/1, 3/2, 4/1, 5/1 and 6/1 incomparison with the approximation (24) obtained from the Fourier coefficients Er,−q(0) ofthe field-free dispersion relation. First, one observes that the Fourier components Er,−q decayindeed exponentially with |q| + |r| as estimated in equation (27) with a ≈ 0.59. Furthermore, thenumerically calculated band width follows approximately the simple law given in (28) for weakfields.
For field directions with large |q| + |r|, the width�E of the quasi-dispersion relation is verysmall and the dispersion is very weak. But also for relatively small values |q| + |r| the width�E is much smaller than the width of the ground band in the field-free case and dispersion isefficiently suppressed. Consider, for example, the Bloch oscillations discussed in section 2.1for q = 2 and r = 1. The width �E is already about two orders of magnitude smaller than thewidth of the ground band in the field-free case and the dispersion is very weak. In fact, onecan hardly observe any dispersion during the first two Bloch periods shown in the animationcorresponding to figure 5. In the case of an irrational field direction which can be approximated
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Figure 12. The double-period potential (29) for ε = 0.1 shows alternatingmaxima of different heights.
by a rational one in the limit of large q and r, a systematic dispersion should be entirely absent.Similar findings have been reported by one of the authors in [9], however in the tight-bindingapproximation.
3. Coherent Bloch–Zener oscillations
In a recent two-dimensional tight-binding study [9], coherent Bloch–Zener oscillations havebeen observed and analysed for a weighted superposition of the two fields in (2):
V(x, y) = −(1 + ε)[cos2 x + cos2 y] + ε[cos x + cos y]2 (29)
for a parameter ε 1. (By a π/4-rotation, this potential can be rewritten as V(x′, y′) =ε(cos x′ + cos y′)− cos x′ cos y′.) The potential is illustrated in figure 12 for ε = 0.1. It shows apattern of alternating maxima of two different heights, i.e. in comparison to the separable caseε = 0 the period is doubled. The effect of this doubling on the dispersion relation is that theenergy bands split into subbands, which can be understood from the following discussion: forε = 0, the potential (29) has the period length π in each coordinate direction and therefore thelength of the Brillouin zone is 2π/π = 2. Considering the potential as periodic with period 2π theBrillouin zone is reduced to 1. Thus the Bloch bands have to be folded inside the reduced Brillouinzone so that each band of the larger Brillouin zone corresponds to four connected bands in thereduced one. For a small non-vanishing ε, the four connected bands then split up into subbands,as is shown in figure 13. The emerging gap is called the Rowland ghost gap (see e.g. [23, 24]and references therein). As already mentioned above in the discussion of Lissajous oscillations,a wavepacket splits into two fractions when it approaches the edge of the Brillouin zone withtransition probability given by equation (10).As the energy gap is small, the tunnelling probabilityis much larger as in the case of the Lissajous-like oscillations studied in section 2.1. Moreover,the energy gap depends sensitively on the parameter ε which offers the possibility to control the
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Figure 13. Dispersion relation E(κ) for the double period potential (29) andε = 0 (left) and ε = 0.1 (right).
y / 2
π
–50 –25 0 25 50 75
–50
–25
0
25
50
75
y / 2
π
–50
–25
25
0
50
75
x / 2π x / 2π –50 –25 0 25 50 75
Figure 14. Coherent Bloch–Zener oscillation for the double-period potentialshown in figure 12 for F = 0.005, h = 2 and ε = 0.02 (left, animationbz50-02-11.avi) and ε = 0.1 (right, animation bz50-10-11.avi) and q =r = 1. Shown is a snapshot at time t = 0.58TB. The animations show the timeevolution of the wavepackets over two Bloch periods.
transition probability in an experimental set-up. Thus, the system (29) is an ideal model systemfor the study of Zener tunnelling. Such a tunnelling process, i.e. a splitting of the wavepacket,can be seen in figure 14, where Bloch–Zener oscillations are shown for a field direction diagonalto the coordinate axes, i.e. q = r = 1, for ε = 0.02 (left) and ε = 0.1 (right). After half of theBloch period, the wavepacket splits and the intensity of the daughter wavepackets depends onthe value of ε as expected from Landau–Zener theory. That the two daughter wavepackets movein different directions can be understood from the fact that the dispersion relation E(κx, κy)has curvatures with different signs for the four subbands. The interference pattern is due to thesuperpositions from the included subbands.
After one Bloch period, one of the daughter wavepackets returns to the starting point andshows a dispersion perpendicular to the field direction, whereas the other wavepacket is mainly
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
Figure 15. Coherent Bloch–Zener oscillation for the double-period potentialshown in figure 12 for F = 0.005, h = 2, ε = 0.1 and different field directions:q = 3, r = 1 (left, animation bz50-10-31.avi) and q = (
√5 − 1)/2, r = 1
(right, animation bz50-10-irr.avi). Snapshots at time t = 0.58TBy (left) andt = 0.70TBy (right) are shown. The animations show the time evolution of thewavepackets over time intervals TBy (left) and 2TBy (right).
localized at the minima of the potential at the turning point of the oscillation. Furthermore,one observes that the dispersion is significantly stronger than in the case of the Lissajous-likeoscillation (section 2.1) and it clearly depends on ε, as there is no dispersion in the separablecase ε = 0. Nevertheless, one has to keep in mind that the time scale of the animations, i.e.the Bloch time itself, is six times as long as in the samples in section 2.1, due to the smallerfield strength. In figure 15, one can see Bloch–Zener oscillations for two different field directionsand ε = 0.1.
For the case q = 3, r = 1 (left), the first splitting of the wavepacket is observed at timet = 0.25TB and the daughter wavepackets split up again. One actually observes three splittingsduring the whole Bloch period because κ crosses the boundary of the Brillouin zone threetimes. But, as the positions of two pairs of the wavepackets coincide, one can distinguishonly six wavepackets at t = TB. The dynamics of the centres of the wavepackets can bedescribed by a semiclassical single-band model [9], which shows essentially the same motionas the wavepackets calculated fully quantum mechanically. Nevertheless, we observe that thewavepackets show features that cannot be explained in the single-band approximation as, forinstance, the complicated interference pattern or the strong dispersion observed in all cases.After one Bloch period, one daughter wavepacket returns to the starting point, whereas the othersmainly localize at the minima of the potential. This behaviour also vanishes in the single-bandapproximation.
Because of the fact that the number of splittings depends on the field direction, wealso examined a direction that is as irrational as possible, i.e. q/r = (
√5 − 1)/2. The
resulting dynamics (see figure 15 and the corresponding animation) shows no essential newfeatures compared with the rational case, except for the different time of the wavepacketsplittings.
New Journal of Physics 6 (2004) 41 (http://www.njp.org/)
In this paper, we extended the studies of Bloch oscillations in one dimension [17] to the two-dimensional case. The dynamics depends sensitively on the direction of the static field withrespect to the lattice, i.e. on the ratio Fx/Fy = q/r, where Fx and Fy are the field componentsin the lattice directions. Almost dispersionless periodic motion of the wavepackets has beenobserved, as well as breathing modes. The origin of the small systematic dispersion has beenanalysed. It has been shown that the dispersion decreases exponentially with |r| + |q|. For typical(i.e. irrational) field directions, a dispersionless motion can be expected.
Finally, it should be noted that, up to now, in all two-dimensional studies, the appliedStark field was constant in time, i.e. a pure dc-field, in contrast with the one-dimensional casewhere ac-driven systems have been intensively investigated. Periodically driven two-dimensionalsystems can be expected to be of particular interest under resonance conditions, where allfrequency ratios are rational. Here the Wannier–Stark states will be delocalized again extendingover the whole lattice and it appears to be an open question if dynamical localization can beobserved in this case for specially chosen values of the field amplitudes as in the one-dimensionalcase [25]–[27].
Acknowledgments
Support from the Deutsche Forschungsgemeinschaft via the Graduiertenkolleg ‘NichtlineareOptik und Ultrakurzzeitphysik’ as well as from the Volkswagen Foundation is gratefullyacknowledged.
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