Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

2. Properties of an atomic Bose condensate in a double-well potentialFigure 2.6: The mean-field ground state for a potential with minima at x = ±4r 0 . The continuousline is the ground-state wave function for the double-well potential calculated from the GP equation.The dashed line is the ground-state mode of the quadratic approximation to the potential at x = q 0 .The dotted line shows the shape of the quartic potential. The scaling of the vertical axis is arbitrary.10.8globallocalpotential0.60.40.20−8 −6 −4 −2 0 2 4 6 8x/r 0Drummond[49, 176] 2If we simulate Eq. (2.46), for a potential with minima at z = ±z 0 = ±4 andwithK = 0, we see that there are high frequency oscillations superimposed on the regulartunnelling motion, as shown in Figs. 2.7(a) and 2.8(a). The high frequency componentsare even larger in amplitude in the z 0 = ±3 case (Figs. 2.7(e) and 2.8(e)). The reason forthese components is revealed in Fig. 2.6. This figure shows the ground-state wave functionof the double-well potential, which is calculated by evolving the GP equation in imaginarytime. For double wells with a ratio of z 0 = q 0 /r 0 near these values, the global wavefunction cannot be expanded in terms of the ground-state modes of the local quadraticpotential. The presence of the other well has an effect on the local mode that cannotbe neglected, due to the relatively large overlap between the two local modes. The moststriking effect that can be seen is that the appropriate local modes have been translatedtoward the origin: In the z 0 = 4 potential, the peak of the wave function is located atz =3.65. Some improvement can be made if we choose a Gaussian centred at z =3.65 tobe the initial state. This suppresses all but one of the higher frequencies, as Figs. 2.7(b)and 2.8(b) show. Not all of this extraneous motion disappears because the shape of themost appropriate local wave functions has also changed, and does not fit to symmetric2 See Sec. 7.4 for a brief description of the algorithm34