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Theory Project B: F34YY6 MATHEMATICAL PHYSICS 29 APRIL 2005 Mixed stable-chaotic trajectories of an isolated cold sodium atom in a one-dimensional optical lattice subject to a two-dimensional magnetically perturbing potential. Mark Saunders and Andrew S. Chisholm School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom (submitted 29 April 2005) We study the motion of a low energy point particle in an optical lattice, and discuss the effects of adding an elliptical magnetic trap. The orientation of these confining wells gives an interesting mixture of stable and chaotic results. We develop the one-dimensional system to that of a quantum particle, using Schrödinger’s equation to time-evolve its wavefunction. SAUNDERS et al. Page 1 a force, given by F V . We can view the effect of this force as causing the energy band to tilt, as seen in Fig. 1(a). Now consider this atom to be moving through the system. Its total energy must remain constant, and so as x increases, the particle’s wavefunction eigenenergy must increase. On reaching the top of the ground state energy band, the only possible motion of the particle is for it to change direction, decreasing the eigenvalues of the wavefunction. This is known as Bragg reflection [4]. The sodium atom also Bragg reflects when it reaches the bottom of the energy band, as no lower energy eigenvalues are possible. This phenomenon is known as Bloch oscillation [5]. It is observed in figure 1(b), where we plot the position of a point particle as a function of time. (a) Bandwidth =22.1 peV Bragg Reflection E x 1x10 Ground state energy band Trajectory of particle -6 (b) 0 position (metres) There have been many recent studies of dynamical systems that yield a mixture of stable and chaotic results [1-3]. In this letter, we investigate one such system: that of a twodimensional magnetic trap superposed onto a onedimensional optical lattice (OL) [1,2]. We begin by considering the motion of a particle in a 1D OL. An OL is a periodic potential formed from the interaction between an alkalimetal atom and the stationary wave produced by two counter-propagating laser beams. Superposing the two electromagnetic waves along the x-axis yields an electric field, E E0 sin( kx) sin( t ) xˆ . Here, k is the wavevector for the laser, with frequency 500 THz. Adding an atom to this electric field induces an electric dipole moment proportional to E. The atom experiences a potential, V E 02 sin 2 (kx) sin 2 (t ) . (1) This leads to nodes and antinodes along the xaxis. The state of a particle can be described by a wavefunction ( x, t ) . As the potential is spatially periodic, the energy eigenstates lie very close together. This forms a continuum of energy states, setting up a band-like structure of allowed eigenvalues, as shown in figure 1(a). In this work, we restrict the theory to considering the motion of an atom within the ground state energy band. We therefore assume that the band gap energy between the two bands to be much larger than the total energy of the system. For a sodium atom within such a potential, detuning one of the lasers linearly with time causes the potential minima (or optical lattice sites) to move. This motion of the potential minima causes the gradient of the potential to change. Therefore, the sodium atom experiences -1 x 1 0 -2 x 1 0 -3 x 1 0 -6 -6 -6 0 0 .0 0 2 0 .0 0 4 0 .0 0 6 0 .0 0 8 0 .0 1 0 tim e (s e c o n d s ) Figure 1: (a) The trajectory of a particle within the energy band structure. (b) The position for a sodium atom in an OL. Bloch oscillations are observed in the semi-classical (dashed line) and quantum regimes (solid line). © 2005 The University of Nottingham Theory Project B: F34YY6 MATHEMATICAL PHYSICS xt The repetition of Bragg reflection, shown by the dashed line, causes the particle to oscillate with Bloch period TB 1.67 ms and Bloch amplitude of 2.6 μm. This means that the Bloch frequency is of the order of 103 Hz, many orders of magnitude smaller than the frequency of the laser light. Therefore, we can neglect the time-dependence of the OL potential in equation (1). We establish a time-averaged system described by x V ( x) V0 sin 2 ( ) where d 294.5 nm, 2 d and V0 562.52 peV. (a) (2) In the equations above, kx is the wave vector of the particle. These equations effectively model Bloch oscillations found in the ground-state energy band. This one-dimensional theoretical study of optical lattices can be experimentally realized as three-dimensional. This is due to the fact that a laser beam has finite width, and so in higher dimensional research, we consider the nodes and antinodes of the OL to correspond to the values lying on the x-axis. By superposing a magnetic trap onto a threedimensional system, we create a confining potential in which the particle is forced towards the centre. This set-up can be viewed as a potential well [6]. However, the boundary of this system is not a hard wall, but a much softer constraining force, extending beyond the limits of the particle’s motion. The symmetry of the magnetic trap is slightly broken, to yield interesting results: we use a trap for which ellipse shaped contours of constant potential are obtained. Furthermore, this trap is rotated by an angle, θ with respect to the OL, as shown in figure 2(a). z(μm) (a) zt θ θ Optical Lattice z Magnetic Trap x E (b) E E2 E z x k max E tot E 2 E tot E1 x k max E For a sodium atom of mass m 3.82 10 26 kg within such a regime, the dynamics are governed by the semi-classical equations of motion: dx 1 d ( ) sin( k x d ) dt 2 dk x ma dt 29 APRIL 2005 px Figure 2: (a) The OL (dashed line) is along the x-axis. The shaded ellipse represents the magnetic trap. The (xt,zt) plane is shown to be rotated w.r.t. (x,z) by angle θ. The axes share a common origin, marked by the white arrow. (b) The energy band structure of the OL, with two possible initial particle energies shown as E1 and E2. Kinetic energy (bold curve) of a particle with a given momentum in the x-direction In order to compare a large collection of results, we combine multiple trajectories. In order to do this, we ensure that the initial conditions for each of these trajectories are comparable. We start each particle off on a contour of constant total energy. Calculating the values for the particle’s initial conditions is complicated because there are restrictions on how much kinetic energy the particle can posses along the x-axis. This is due to the energy band motion already discussed. A view of the energy band can be seen in figure 2(b). If this initial energy is greater than Emax, the difference must be assigned as kinetic energy in the z-direction. A new set of axes (xt ,zt) shown in Fig. 2(a) conveniently allows the potential of the magnetic trap to be given by V ( xt , z t ) 1 m 12 xt2 22 y 2 32 z t2 (3) 2 where angular frequencies ω1=1479 rad s-1, ω2=4702 rad s-1 and ω3=2561 rad s-1 are taken (b) (c) (d) (e) (g) (h) (i) (j) x (μm) pz (10-28 kgms-1) (f) z(μm) Figure 3: Examples of the results obtained from the motion of particle in the system described in the text. Axes inset. (a) - (e): Many particles are considered, starting on the ellipse shaped equipotential. These initial starting points are represented by the small circles. Examples are given of typical trajectories: Figures (a) to (c) correspond to an untilted magnetic trap; Similarly for figures (d) and (e) with θ=45o. The trajectories are plotted for the the first fist 60 ms after the simulation began. (f)-(j): The Poincaré sections correspond to that of the trajectory (a)-(e) respectively. SAUNDERS et al. Page 2 © 2005 The University of Nottingham Theory Project B: F34YY6 MATHEMATICAL PHYSICS from experiments on sodium atoms [2]. Motion in the y-direction decouples from the (x,z) plane, corresponding purely to simple harmonic motion. We may suppress the y-axis without loss in generality, and consider the two-dimensional (x,z) system. Within this potential, we model the motion of a particle using the fourth-order Runge-Kutta method [1,7] to evolve the point particle’s position in time using the semi-classical equations of motion (2), as in the one-dimensional approach. We calculated the trajectory of a particle starting with a constant total energy, E. We considered various magnetic trap orientations and examined how the particle’s trajectory was dependent upon its starting point within the system. The results of this computational experiment are shown in figure 3. The ellipse shaped curve is an equipotential of the magnetic trap, with value E=22.1 peV. A point on this curve shows where a particle begins its trajectory. Its initial velocity can be determined by computing how much energy it was allowed to have at the top of the band, as shown in Fig 2(b). To quantify how the particle motion changes with the trap orientation, we plotted points in the (z,pz) plane through phase space at each turning point along the x-direction for which p x 0 and dp x / dt 0 [8]. Examples of these Poincaré sections are shown below their corresponding trajectory in figure 3. For a range of values of θ, the multiple constant energy trajectories are superposed, as is shown in figure 4. For 0 and 90 [left-hand figures], these points lie on a series of concentric ellipses corresponding to simple harmonic motion along the z-axis. For intermediate angles, the system reveals a phase space characterised by a mixture of crescent shaped stable islands, surrounded by chaotic seas. For the figures on the right, we again present Poincaré sections, this time corresponding to the Poincaré plane given by points (x,px) in the phase space plane for which p z 0 and dp z / dt 0 [8]. We have demonstrated the convenience of Poincaré sections for allowing both stable and chaotic results to be visualised on the same diagram. Adapting this model from a point particle, whose dynamics incorporate the semi-classical equations of motion; we now consider a quantum mechanical wavefunction. The time evolution of a non-interacting atom is governed by the timedependent Schrödinger equation: 2 2 i ( x , t ) ( x, t ) V ( x) ( x, t ) (4) t 2m x 2 SAUNDERS et al. Page 3 29 APRIL 2005 θ = 0o θ = 15o θ = 30o θ = 45o θ = 60o θ = 75o px(10-28kgms-1) pz(10-28kgms-1) θ = 90o z(μm) x(μm) Figure 4: Poincaré sections for a magnetic trap tilted at angle θ with respect to the optical lattice. Left: (z,pz) Poincaré sections, as described in the text. The figure corresponding to θ=0 is made up, in part, from Figs. 3a), 3b) and 3c). The figure corresponding to θ=45 degrees is made up from Figs. 3d) and 3e). Right: (x,px) Poincaré sections corresponding to their adjacent figure. Axes inset. Consider the case of the one-dimensional OL, which contributes a potential energy term as given by equation (1). In order to demonstrate that the observable results are consistent with the semiclassical calculations (2), we show that Bloch oscillations take place. The wavefunction of the isolated sodium atom was initially described as an entirely real Gaussian. This is because it is localised in space, has a well-defined width, and its mean position does not move unless a potential is applied. A Gaussian is smooth and therefore has well defined derivatives. Having decided to work with a Gaussian wavepacket, we chose its standard deviation to be between two and five lattice periods. This allows the atom to feel the true periodic form of the potential, whilst keeping its form localised in space. The Crank-Nicholson method [1,7] was used to time-evolve the Schrödinger equation (4) by one time-step, leading to a new wavefunction © 2005 The University of Nottingham MATHEMATICAL PHYSICS Theory Project B: F34YY6 7.5x10 6 6.0x10 6 4.5x10 6 3x10 3.0x10 6 2x10 1.5x10 6 1x10 5x10 (a) 0 -10 6x10 6 4x10 6 4x10 -5 5 Probability Density 6x10 -5 0 5 6 6 2x10 10 -5 0 5 10 -5 0 5 10 6 6 (f) 6 4x10 6 0 -10 5 6 (e) 4x10 0 (d) 0 -10 6x10 -5 6 3x10 10 6 6 6 1x10 0 -10 (b) 6 4x10 2x10 6 6 0 -10 10 5x10 (c) 2x10 0 2x10 -5 0 5 10 6 6 0 -10 Figure 5: The probability density distribution for the quantum mechanical wavefunction of the 1D OL, plotted at various times. a) Shortly after Gaussian wavepacket evolution. (t = 0.001 ms) b) Half way to the first Bragg reflection. (t = 0.418 ms) c) At first Bragg reflection. (t = 0.835 ms) d) ¾ the way through the Bloch oscillation. (t = 1.253 ms) e) After 1 Bloch oscillation. (t = 1.67 ms) f) After 5 Bloch oscillations. (t = 8.35 ms) describing the system. Repeating this process yielded the snapshots in figure 5, which shows this wavefunction to be Bloch oscillating. The expectation value of the atom’s position can be evaluated at each time using x dx * x, t x x, t . The expectation values for the simulated wavepacket are plotted against time by the solid curve in figure 1(b). As predicted, the particle Bloch oscillates with TB 1.67 ms. Its amplitude, however, [1] M. Saunders, A.S. Chisholm and T.M. Fromhold, Low Temperature Atom Dynamics, (April 2005). [2] R.G. Scott, S. Bujkiewicz, T.M. Fromhold, P.B. Wilkinson and F.W. Sheard, Phys. Rev. A 66, 023407(2002). [3] T.M. Fromhold et al., Nature 428, 726 (April 2004). [4] J.R. Hook and H.E. Hall, Solid State Physics, Second Edition, Wiley, 1990. [5] M. Ben Dahan, E.Peik, J.Reichel, Y.Castin, and C. Salomon, Phys. Rev. Lett. 76, 4508(1996). [6] For similar examples for a variety of different shaped billiards see M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics, (Springer, New York, 1990).] [7] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in C, Cambridge University Press, (1992). SAUNDERS et al. 29 APRIL 2005 6 Page 4 differs from what we initially expected based on the semi-classical calculations. This is due to the method used to calculate the width of the energy band in which the atom is moving. This difference is a consequence of the form of the band structure in the quantum simulation. Bose-Einstein condensates have recently received considerable attention [9], the dynamics of which can be incorporated into this research via the Gross-Pitaevskii equation [10]. To summarise, we have explored the dynamics of a low energy sodium atom in a variety of systems and models. Starting with the one-dimensional optical lattice, we modelled the system semi-classically. Plotting the trajectory in one dimension allowed Bloch oscillations to be modelled for a point particle. Modifying to take the wavefunction of a sodium atom into account allowed the full quantum mechanical nature to be realised. The resulting wavepacket exhibited Bloch oscillations. The semi-classical results are very interesting for the two-dimensional system. For various magnetic trap orientations, the mixture of both chaotic and stable results proves to give a unique insight into semi-classical mechanics. The use of Poincaré sections to express these results was a very powerful tool to convey this nature of the system. The Poincaré sections enabled stable islands and chaotic seas to be observed. We wish to thank Professor Mark Fromhold for his advice and support throughout this project. This letter is dedicated to the life of George C. Grieveson. [8] A given Poincaré plane produces two possible sections, one the mirror image of the other. To avoid this overlap, we consider only the intersections in which the sign of the momentum changed from positive to negative. [9] R.G. Scott, A.M. Martin, T.M. Fromhold, S. Bujkiewicz, F.W. Sheard, and M. Leadbeater, Phys. Rev. Lett. 90, (11) 110404(2003); R.G. Scott, A.M. Martin, S. Bujkiewicz, T.M. Fromhold, N. Malossi, O. Morsch, M. Cristiani, and E. Arimondo, Phys. Rev. A, 69, 033605(2004); R.G. Scott, A.M. Martin, and T.M. Fromhold, ibid. 69, 063607(2004). [10] S.A. Gardiner, D. Jaksch, R. Dum, J.I. Cirac and P. Zoller, Phys. Rev. A 62, 023612(2000); R. Bach, K. Burnett, M.B. d’Arcy and S.A. Gardiner, arXiv:physics/0310143v4(2004). © 2005 The University of Nottingham