Bloch oscillations of cold atoms in two-dimensional optical lattices

Transcription

1 PHYSICAL REVIEW A 67, Bloch oscillations of cold atoms in two-dimensional optical lattices A. R. Kolovsky Max-Planck-Institut für Physik Komplexer Systeme, D Dresden, Germany and Kirensky Institute of Physics, Krasnoyarsk, Russia H. J. Korsch FB-Physik, Technische Universität Kaiserslautern, D Kaiserslautern, Germany Received 6 December 2002; published 4 June 2003 Bloch oscillations of cold atoms in two-dimensional optical lattices are studied. The cases of separable and nonseparable potentials are compared by simulating the wave-packet dynamics. For these two classes of optical potential, the Bloch oscillations were found to be qualitatively the same in the case of a weak static field but fundamentally different in the case of a strong field. In addition, the dynamics of the atoms in a double-period potential which can easily be realized in two dimensions is studied for the regime of a weak static field. DOI: /PhysRevA PACS number s : Be, Pj, w I. INTRODUCTION A Bloch oscillation BO is the oscillation of a quantum particle in a spatially periodic potential the lattice if a static force is applied. Following Bloch 1 and Zener 2, the static force F changes the particle quasimomentum here we discuss the one-dimensional case for simplicity according to Newton s law F, i.e., (t) 0 Ft/. Because the dispersion relation E( ) of the quantum particle in the lattice is a periodic function of, E( 2 /d) E( )(d is the lattice period, this linear change of the quasimomentum results in a periodic change of the group velocity v de( )/ d and, hence, oscillations of the coordinate with period T B 2 /Fd the so-called Bloch period and amplitude /F( is the bandwidth. Although this phenomenon was predicted almost 80 years ago, observing BO s in laboratory condition appears to be a hard problem and the first reports of direct measurement of BO s are dated ,4. Recently, BO s were observed in a system of cold atoms in optical lattices 5. This system offers unique possibilities for experimental studies of this and related phenomena 5 11, which have also stimulated progress in theory It should be noted, however, that up to now both laboratory and theoretical studies of atomic BO s were restricted to quasi- one-dimensional optical lattices. This paper deals with BO s in two-dimensional lattices and, in this sense, continues our previous work 17 devoted to the spectral properties of the system. Before proceeding with the main part, a remark concerning two different regimes of BO is required. These are the regime of a weak static field, where the Stark energy Fd is less than the energy gap separating the ground Bloch band from the rest of the spectrum, and the regime of a strong field, where Fd a more precise form of these inequalities is discussed later on. In the former regime, the Landau- Zener tunneling 18,2 is negligible and the atomic wave packet may show perfect oscillations. This regime was observed, for example, in Ref. 5 and the so-called single-band approximation has proved to be sufficient for describing the system dynamics. In the latter regime, the wave packet may remain practically at rest, but each Bloch period it emits a fraction of probability moving further with constant acceleration F/M (M is the mass of the atom. This regime was observed in Ref. 8 and its theoretical description was given in Ref. 13 by using the formalism of resonance metastable Wannier-Stark states. Obviously, these two regimes should also be distinguished in the case of two-dimensional 2D lattices. The structure of the paper is as follows. Section II deals with the Bloch spectrum of a quantum particle in a 2D optical potential. In contrast to the 1D case, any 2D potential should be classified as either separable or nonseparable. In the classical limit, this property strongly affects the particle dynamics it is regular in the former case but chaotic or mixed in the latter case. In Sec. II, we discuss the Bloch spectrum of the system with respect to the property of separability. Section III is devoted to the regime of a weak static field. As mentioned above, in this case the Landau-Zener tunneling is negligible, and one may use the single-band approximation to analyze the wave-packet dynamics. The twodimensional BO s of the atom are considered for both separable and nonseparable potentials. As an addition to Sec. III, Sec. IV studies an interesting regime of BO s occurring in a double-period optical potential. The regime of a strong static field is analyzed in Sec. V. It is shown that here the difference between separable and nonseparable potentials is much more pronounced than in the case of a weak static field. In the last section we summarize the results of the paper. II. 2D OPTICAL LATTICE By using different configurations of laser beams, one can create 2D or 3D optical lattices almost at will. In what follows we restrict ourselves to the simplest configuration where two standing waves cross in the x-y plane at right angles. Moreover, the waves are assumed to be linearly polarized along the z axis and have equal amplitudes. If the frequencies of the waves are also equal, the beams form a nonseparable optical potential V x,y V 0 cos k L x cos k L y 2, /2003/67 6 / /$ The American Physical Society

2 A. R. KOLOVSKY AND H. J. KORSCH PHYSICAL REVIEW A 67, The period of this lattice is d /2, the primitive translation vectors point along the x and y axes see Fig. 1 bottom panel and the sign of V 0 is irrelevant for the atomic dynamics. It is convenient to analyze the problem in the lattice system of the coordinates. Then all three considered cases are captured by the equation V(x,y) V 0 (cos x cos y cos x cos y), where 0 corresponds to the egg crate, 1 to the quantum dot, and 1 to the quantum antidot potential. We stress that from now on the atom position is measured in units of the lattice period. It is also convenient to scale the energy time on the basis of the recoil energy E R 2 k L 2 /2M. Then the dimensionless Hamiltonian of the atom in an optical lattice takes the form H p x 2 /2 p y 2 /2 cos x cos y cos x cos y, 3 where the only independent parameter of the system is the scaled Planck constant (E R /V 0 ) 1/2 entering the momentum operator 19. It should be noted that in all cited experiments with cold atoms the value of V 0 was a few E R in particular, V 0 E R in Ref. 5 and V 0 3E R in Ref. 6. Thus we shall also restrict ourselves to considering this region of the system parameters. Examples of a Bloch spectrum for the system 3 are given in Fig. 2. In what follows, we shall mainly focus on the ground Bloch band. Although the ground band looks similar for 0 and 0, there is still a fundamental difference in the dispersion relation E E( x, y ). This difference shows up in the coefficients of the Fourier expansion of the dispersion relation: E x, y m,n J m,n exp i2 m x exp i2 n y, 4 FIG. 1. Contour plot of nonseparable optical potentials 1 upper panel, and separable potential 2 lower panel. The color axis goes from black bottoms of the potential wells to white tops of the potential barriers. where k L is the laser wave vector. It is easy to show that the potential 1 defines a simple square lattice with period d / 2 ( 2 /k L is the wavelength and primitive translation vectors rotated by 45 relative to the laboratory coordinate system see Fig. 1 upper panel. The amplitude V 0 of the potential is given by the ratio of the squared Rabi frequency to the detuning and thus has different signs for red and blue detuning. Using the terminology of semiconductor physics, the negative and positive signs correspond to the smooth quantum dot and quantum antidot potentials, respectively. In Fig. 1 upper panel we plotted the quantum antidot potential. The quantum dot potential can easily be imagined by inverting the black and white colors. The separable potential can be obtained by using different frequencies in each standing wave. Then neglecting a small difference in the wave vector one has an egg crate potential V x,y V 0 cos 2 k L x cos 2 k L y. 2 shown in Fig. 3 as a gray-scaled map. It is seen in Fig. 3 that in the separable case only the coefficients J m,n with n 0 or m 0 differ from zero, while in the nonseparable case all elements have nonzero values. Also note the symmetry J n, m J m,n.) At the same time, even the largest nontrivial coefficient J 1,1 is only 1/20 of the coefficient J 0,1, which essentially alone determines the width of the ground Bloch band 8J 0,1. We found this to be a typical situation for the considered and experimentally accessible interval of. III. SINGLE-BAND APPROXIMATION Using the basis of localized 2D Wannier states 20, the Hamiltonian of the atom in the ground band of the 2D optical potential takes the form H l,m J m l m l 2 l Fl l l, 5 where the overlap integrals J m are given in Eq. 4. The last term in Eq. 5 is the Stark energy. It is possible to show that in the general case of a nonseparable potential the spectrum and eigenfunctions of the operator 5 crucially depend on the direction of the static field F 17. For rational directions of the field,

3 BLOCH OSCILLATIONS OF COLD ATOMS IN TWO-... PHYSICAL REVIEW A 67, FIG. 2. Bloch band spectrum of the system 3 for 2 and 0 top and 1 bottom. The first three bands are shown. The nonseparability of the potential manifests itself in the removed degeneracy of the excited bands along the lines x y and in the different structure of the Fourier transform of the ground band see Fig. 3. F x F y q r where q,r are coprime numbers, the eigenfunctions are localized functions along the field direction and extended Bloch-like functions in the direction normal to the field. For an irrational field, the eigenfunctions are localized in both directions. It should be noted, however, that for a bad rational number the width of the Bloch energy band is extremely small and it can be approximated by a flatband. Then one can construct from the exact extended eigenfunction an approximate localized eigenfunction with properties similar to those of the eigenfunctions for irrational F x /F y. The case of a separable potential is simpler: because of the symmetry reflected in the restriction that J n,m 0 for m 0 or n 0) the eigenfunctions of the operator 5 are localized functions for any direction except F parallel to the x or y axis 21. To study the system dynamics, we integrated the Schrödinger equation for the single-band wave function given by (t) l c l (t) l. The initial wave packet was chosen as a 6 FIG. 3. Logarithm of the absolute values of the Fourier coefficients J m,n of the dispersion relation 4 as a gray-scaled map. The color axis goes from white to black. The upper panel refers to the separable and the lower to the nonseparable case. 2D Gaussian centered at x y 0 with variance x 2 y 2 16d 2 the lattice period is d 2 in the scaled units used. In the reciprocal space, this initial condition corresponds to population of the bottom of the ground Bloch band with x 2 y Thus the wave-packet width is smaller than the size of Brillouin zone, which is one of the necessary conditions for existence of the oscillatory regime. Examples of the atom trajectory in the case of a nonseparable potential with 1 are shown in Fig. 4. The value of the scaled Planck constant is 1, the value of the scaled static force F 0.005, and the field directions are chosen as (q,r) ( 1,0) for trajectory a, (1, 1) for trajectory b, ( 4, 3) for trajectory c, and (4,1) for trajectory d. In addition to this figure, the upper panel in Fig. 5 shows the expectation values x and y of the wave packet as a function of time for trajectory c. As expected, the wave packet returns to its initial position after three Bloch cycles in the x direction and four cycles in the y direction. A weak beating of the oscillations as well as a slight asymmetry of the

4 A. R. KOLOVSKY AND H. J. KORSCH PHYSICAL REVIEW A 67, FIG. 4. Trajectories of the atom defined as the mean position of the wave packet in a nonseparable potential for different directions of the static field: curve a, (F x,f y ) (q,r) (1,0); curve b, ( 1,1); curve c, (4,3); curve d, ( 4, 1). The value of the scaled Planck constant is 1; the value of the scaled static force is F The spot at the origin shows the density of the initial wave packet. For rational directions of the field, the trajectory is a periodic function of time see Fig. 5 ; for irrational directions, it is a quasiperiodic function. trajectory in the xy plane is a sign of nonseparability. In the separable case ( 0) the beatings are absent and the trajectory is perfectly symmetric. It is interesting to compare the trajectories obtained numerically with those predicted by the quasiclassical approach. Within this approach, the atom trajectory is obtained by integrating the equation of motion FIG. 5. Mean position upper panel and dispersion lower panel of the wave packet as a function of time for trajectory c in Fig. 4. The solid and dashed lines are the y and x components, respectively. It is seen that for this direction of the field one can neglect the dispersion of the wave packet. The exceptions are the directions close to the crystallographic axis of the lattice see Fig. 6. FIG. 6. The same as in Fig. 5 but for trajectory d. The additional dash-dotted line shows the increase of the dispersion of the wave packet for trajectory a. ṗ x F x, ẋ v x p x /,p y /, 7 ṗ y F y, ẏ v y p x /,p y /, where v x,y ( x, y ) 1 E/ x,y are the components of the atomic group velocity. The trajectories calculated on the basis of Eq. 7 were found to coincide with the ones depicted in Fig. 4 with good accuracy. We use Eq. 7 later on in Sec. IV. We proceed with the analysis of the dispersion of the wave packet. The lower panel in Fig. 5 shows the dispersion x 2, y 2 of the wave packet for the trajectory c. No systematic dispersion is seen. Obviously, this is due to the extremely small bandwidth corresponding to the chosen ratio q/r. For the time interval considered, we were able to detect a systematic dispersion in the direction orthogonal to the field only in the cases a and b. Thus, in practice, the dispersion existing for rational field directions can be neglected if r q 2. Coming back to Fig. 5, we also note that, in addition to the absence of the systematic dispersion, the temporal dispersion is also weak. This means that the wave packet moves along the trajectory without any noticeable change of its shape. Exceptions to this rule are the directions close to the x or y axes. In this case, the wave packet may show a huge temporal dispersion without having a systematic one. In other words, for finite time the wave packet spreads in the direction orthogonal to the field, as it would do in the case of F parallel to the crystallographic axis, but then it collapses back to the initial state. An example of such behavior is the trajectory d shown as a function of time in Fig. 6. IV. DOUBLE-PERIOD OPTICAL POTENTIAL Double-period potentials offer a unique possibility for studying different aspects of BO s, in particular Landau- Zener tunneling. Let us consider first a one-dimensional double-period lattice. Using the tight-binding approximation, the system can be mimicked by the Hamiltonian

5 BLOCH OSCILLATIONS OF COLD ATOMS IN TWO-... PHYSICAL REVIEW A 67, H 4 l l 1 l l l 1 2 l 1 l l l Fd l l l l, 8 where is now the mismatch in the site energy. The Bloch spectrum of the system 8 is given by E cos 2 d 2 1/2. 9 Thus for 0 the original Bloch band E( ) ( /2)cos(d ) splits into two subbands with energy gap at the edges of the reduced Brillouin zone. According to Landau-Zener theory, the dynamics of the system 8 depends on the ratio 2 / Fd, i.e., on the value of the energy gap. For 2 / Fd 1 the system ignores the gap diabatic transition of the avoided crossing at the edges of the Brillouin zone 24, and the regime of BO s is essentially the same as in the case of a strictly periodic lattice see Fig. 7 a. In the opposite limit, 2 / Fd 1, one has an adiabatic passage of the avoided crossing and the period of the BO s is twice shorter Fig. 7 b. However, the most interesting case is when the value of is between these two extrema. Then at each crossing of the zone edge a fraction of the wave packet tunnels from the lower to the upper subband and vice versa, and we get a coherent superposition of two oscillating wave packets Fig. 7 c. The possibility of this nontrivial dynamics was already mentioned in 25. In principle, nowadays it should be observable in semiconductor superlattices or in optical waveguide arrays 26 although we are not aware of any experimental report. In this section, we address the question of how the discussed phenomenon manifests itself in the 2D case. In this case, the optical lattice has an advantage over a semiconductor superlattice or waveguide array because it offers a very simple realization of the double-period 2D potential. To this end, we introduce the lattice given by the weighted sum of the potentials 1 and 2 : V x,y 1 cos 2 x cos 2 y cos x cos y 2, By rotating the coordinate system through the angle /4, this potential takes the form V(x,y ) cos x cos y (cos x cos y ), in comparison with Eq. 3. An example of the potential 10 for 0.01 as well as the Bloch spectrum of the atom in this potential are shown in Fig. 8. The energy gap around E is clearly seen. We note that for most practical purposes the displayed spectrum can be approximated by the equation E x, y E cos 2 2 x cos 2 2 y 2 1/2, 11 where is the width of the ground Bloch band for 0 compare with Eq. 9. The effect of the gap on the system dynamics is illustrated in Fig. 9, where the wave function of the system calculated on the basis of the tight-binding FIG. 7. Bloch oscillation for the 1D double-period system 8 for different values of the energy gap: 0 a, 0.6 b, and 0.04 c. The other parameters are 1, 1, and F Different dynamics of Bloch oscillations reflect different regimes of Landau- Zener tunneling between two subbands of the 1D double-period potential

6 A. R. KOLOVSKY AND H. J. KORSCH PHYSICAL REVIEW A 67, FIG. 9. The quasiclassical trajectory of the atom for field directions (q,r) ( 1,1) dashed line and (q,r) (3, 1) solid and dotted lines. For the first case, the wave function after one-half of the common Bloch period is also shown. The parameters are the same as in Fig. 8 and F See text for more discussion. FIG. 8. Upper panel: contour plot of the double-period optical potential 10, Lower panel: splitting of the ground Bloch band into two subbands in the case of a 2D double-period potential, 2. The spectrum is shown in the rotated coordinate system; then the reduced Brillouin zone corresponds to x 0.5, y 0.5. model is shown after one-half of the total Bloch period T tot (q 2 r 2 ) 1/2 T B. The initial wave packet is the same as in Fig. 4 and the field direction is 45 relative to the laboratory coordinate system. In terms of Fig. 8 lower panel, this corresponds to the line x 0. As expected, the wave function consists of two oscillating wave packets in close analogy with the 1D case associated with the two subbands of the ground Bloch band. A similar result is observed for the field direction along the x or y axis. The wave-packet dynamics for other field directions is essentially more complicated. A qualitative understanding of this complex dynamics can be achieved by using the quasiclassical approach 7. As an example, the solid line in Fig. 9 shows the atom trajectory for the field direction ( 3, 1), calculated on the basis of the approximate dispersion relation 11. The branching points on this trajectory labeled by the numbers correspond to times when the quasimomentum crosses the edges of the reduced Brillouin zone. At these points the wave packet splits into two packets moving in opposite directions. To mimic this process we introduce for each crossing a child particle with a velocity opposite to the velocity of the parent particle. Since for the considered direction and time interval T T tot /2 we have three crossings of the zone edges, there are altogether such particles. The final positions of the particles are marked by stars. Because the final positions of two pairs of particles coincide, only six stars are seen in the figure. After the next time interval T the trajectory shown in the figure doubles itself dotted line. The actual wave-packet dynamics qualitatively follows the the depicted trajectory see Fig. 10. In more detail, starting from the point marked by 0 in the figure, the wave packet moves toward point 1 where it splits into two packets going toward points 2 and 2, respectively. This is the first crossing of the zone edge. At these points each of the two packets again split into two packets moving in opposite directions FIG. 10. The wave packet for the instants of time corresponding to the crossings of the Brillouin zone a c and t T tot /2 d. The quasiclassical trajectory for 0 t T tot /2 is also shown

7 BLOCH OSCILLATIONS OF COLD ATOMS IN TWO-... PHYSICAL REVIEW A 67, the second crossing of the zone edge. Now we have a superposition of four packets approximately aligned along the line connecting the points 3 and 3. We note that, because the wave packets overlap and interfere, visually one has one packet extended in the direction orthogonal to the field. The next crossing of the edge of the reduced Brillouin zone splits this extended packet into two packets with the region of support above the left and right arrays of the stars, respectively. As time goes on, the process described repeats and thus the wave function spreads in the orthogonal direction. In the field direction, the wave packet remains bounded and its dynamics can be characterized as a breathing mode. These results also hold for irrational field directions. V. STRONG STATIC FIELD We proceed with the case of a strong static field. Since in this case the single-band approximation is no longer valid, we study the system dynamics on the basis of the continuous Hamiltonian 3 with a static field term added ; namely, we choose a Gaussian wave packet as an initial wave function and evolve it over a few common Bloch periods T tot (r 2 q 2 ) 1/2 T B. Figure 11 shows a fragment of the final wave function in both momentum and coordinate representations in the case of a separable potential. The system parameters are 2, F 0.2, and F x F y.) It is seen in the figure that there are two streams of probability directed along the crystallographic axis of the lattice. This structure of the wave function simply follows from separability i.e., from the fact that the 2D wave function is a product of 1D functions associated with the x and y degree of freedom and the structure of the 1D wave function in the strong field regime. Let us discuss the latter in more detail. According to Ref. 13, the 1D wave function is given in the momentum representation by the equation p,t p Ft exp i Et/ p n p Ft n. 12 Here (p) denotes the Heaviside function, (p) is the ground metastable Wannier-Stark state corresponding to the complex energy E E i /2, and (p) is a Gaussian whose width is given by the width of the initial wave packet. Because in the asymptotic region of negative momentum (p) exp i(p 3 /6F Ep/F)/, the Fourier transform of the function 12 is essentially a train of Gaussians positioned according to a quadratic law i.e., similar to a train of water drops accelerated by the gravitational force. It is easy to see that this structure of the 1D function indeed describes the characteristic features of the 2D wave function shown in Fig. 11. It is also worth noting that Fig. 11 refers to the special case F x F y. In the general case F x F y, the intervals between the wave function drops defined by the corresponding Bloch periods are different in different streams. The same is true for the absolute intensities of the streams, which are equal only in the case F x F y. FIG. 11. A fragment of the wave function of the atom in the separable egg crate potential ( 0) in the case of a strong static field. Upper panel: momentum representation; lower panel: configuration space. The system parameters are F 0.2, 2, and F x /F y 1. The case of a nonseparable potential is depicted in Fig. 12. It is seen that, in addition to the x and y streams, there is a directional channel along the diagonal. We also notice a rather complicated interference between these three probability streams. The existence of additional directional channels in comparison to the separable case was predicted earlier in Ref. 27 by using a classical approach. We briefly recall this result of 27. Let us consider first the classical dynamics of the atoms in the separable potential. The x stream then corresponds to an unbounded motion along the x axis, x t F x t 2 /2, t, 13 and a bounded motion along the y axis, corresponding to a periodic trajectory of the Hamiltonian H y p y 2 /2 cos y F y y. 14 Considering now a nonseparable potential ( 1), we have an additional term cos x cos y in the y Hamiltonian

8 A. R. KOLOVSKY AND H. J. KORSCH PHYSICAL REVIEW A 67, easily proved. Thus, there are four directional channels in the nonseparable case. The observed quantum dynamics of the atoms reflects to some extent this property of the classical system. VI. CONCLUSION FIG. 12. The same as in Fig. 11, but for a nonseparable quantum dot potential ( 1). To keep the probability flow approximately the same, the magnitude of the static force is increased to F , which depends parametrically on x. However, because x t 2, this term rapidly oscillates and can be averaged. Thus the nonseparability of the potential does not destroy the x channel. The same is obviously valid for the y channel. Now we rotate the coordinate system by 45 : x (x y)/2, y (x y)/2. In the new coordinate system the Hamiltonian of the atom takes the form H p 2 x p 2 y 1 cos x cos y 2 cos x cos y 2 F x F y x F x F y y. 15 By repeating the argument given above, the possibility of bounded motion along the new crystallographic axis can be We have analyzed the dynamics of cold atoms in 2D optical lattices driven by a static for example, gravitational force. The analysis is restricted to the deep quantum regime the scaled Planck constant 1) which corresponds to the amplitude of a periodic potential of few recoil energies. Moreover, it is assumed that initially the atoms occupy the bottom of the ground Bloch band. Provided all these conditions are met, the atoms show as in the previously studied case of a 1D optical lattice Bloch oscillations with the characteristic period T B / F, where F is the magnitude of the static force and is the laser wave length. The main question addressed in the paper is how the property of separability of the optical potential affects the Bloch oscillations. To answer this question we considered the simplest experimental situation, where two standing waves crossing at right angles form either separable 2 or nonseparable 1 potentials. Obviously, for a separable potential, the 2D Bloch oscillations are a superposition of 1D Bloch oscillations along the x and y axes, which leads to a Lissajous-like trajectory of the atom in the x-y plane. In the case of a weak static field in the sense of no Landau-Zener tunneling the dynamics of the atoms in a nonseparable potential is found to be qualitatively the same. This is also true for rational field directions F x /F y q/r, where the energy spectrum of the system is formally different discrete or band structured 17. Thus, assuming the experimental technique of Ref. 5 which is appropriate for the case of weak static fields, the difference between separable and nonseparable potentials can be detected only on the quantitative level. This is, however, not the case for a strong static field, where the results differ qualitatively; namely, for a separable potential, there are only two directional channels along which the atoms tunnel out of the potential wells. In the case of a nonseparable potential additional directional channels appear. In addition to this, a strong interference between different streams of the probability is noticed. To observe this effect one could use the experimental arrangement of Ref. 8, where the 1D lattice is substituted by a 2D lattice. The phenomenon discussed above is not the only effect brought about by a transition to higher dimension. In particular, it is shown above that by using four standing waves one can realize the case of a double-period potential 10, where the ground Bloch band is split into two subbands. With respect to Bloch oscillations, this potential acts as a copy machine which progressively splits the atomic wave packet into many wave packets moving separately and interfering with each other when their positions overlap. 1 F. Bloch, Z. Phys. 52, C. Zener, Proc. R. Soc. London, Ser. A 145, J. Feldmann et al., Phys. Rev. B 46, K. Leo et al., Solid State Commun. 84, M. B. Dahan et al., Phys. Rev. Lett. 76, S. R. Wilkinson et al., Phys. Rev. Lett. 76,

9 BLOCH OSCILLATIONS OF COLD ATOMS IN TWO M. G. Raizen, C. Salomon, and Qian Niu, Phys. Today 50 7, B. P. Anderson and M. A. Kasevich, Science Washington, DC, U.S. 282, O. Morsch et al., Phys. Rev. Lett. 87, M. Greiner et al., Nature London 415, W. K. Hensinger et al., Nature London 412, Q. Thommen, J. C. Garreau, and V. Zehnle, Phys. Rev. A 65, M. Glück, A. R. Kolovsky, and H. J. Korsch, Phys. Rep. 366, A. R. Kolovsky, J. Opt. B: Quantum Semiclassical Opt. 4, A. R. Kolovsky, A. V. Ponomarev, and H. J. Korsch, Phys. Rev. A 66, Qian Niu and M. G. Raizen, Phys. Rev. Lett. 80, M. Glück, F. Keck, A. R. Kolovsky, and H. J. Korsch, Phys. Rev. Lett. 86, L. D. Landau, Phys. Z. Sowjetunion 1, ; 2, There is some freedom in defining the scaled Planck constant. We use the scaling which sets the prefactor for cos x and cos y terms to unity. Then the scaled Planck constant is given by 2(E R /V 0 ) 1/2 for 1 and 4(E R /V 0 ) 1/2 for 0. Let us also note that intermediate values of cannot be realized by using the laser beam configuration considered and thus are not PHYSICAL REVIEW A 67, discussed here. 20 Localized Wannier states l not to be confused with Wannier- Stark states can be defined as the Fourier expansion of the Bloch states over the quasimomentum: l exp(idl ) l. 21 To be more precise, there are two alternative sets of the eigenfunctions in the case of a separable potential: the set of localized functions corresponding to infinitely degenerate discrete levels, and the set of Bloch-like functions corresponding to energy bands of zero width. In the case of nearest-neighbor hopping the so-called von Neumann neighborhood the latter were studied in Ref. 22. See also the recent paper 23, which extends the results of Ref. 22 beyond the von Neumann neighborhood. 22 T. Nakanishi, T. Ohtsuki, and M. Saiton, J. Phys. Soc. Jpn. 62, F. Keck and H. J. Korsch, J. Phys. A 35, L The probabilities of diabatic and adiabatic transitions are given by P / Fd and P 1 exp( 2 /8 Fd), respectively. 25 H. Fukuyama, R. A. Bari, and H. C. Fogedby, Phys. Rev. B 8, T. Pertsch et al., Phys. Rev. Lett. 83, M. Glück, F. Keck, A. R. Kolovsky, and H. J. Korsch, Phys. Rev. A 66,