Non-dispersive wave packets in periodically driven quantum systems

More documents

Recommendations

Info

420 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 | ψ(z)| 2 0.0015 0.001 0.0005 t=0.5 t=0.25 t=0 t=12.5 t=19.45 0 0 5000 10000 Position z (atomic units) t=0.75 t=1 Fig. 3. Time evolution of an initially localized wave packet in the one-dimensional hydrogen atom, Eq. (31). The wave packet is constructed as a linear superposition of energy eigenstates of H, with a Gaussian distribution (centered at n0 =60, with a width n =1:8 ofthe|cn| 2 ) of the coe cients cn in Eq. (10). Time t is measured in units of the classical Kepler period Trecurrence, Eq. (33). Note the quasiclassical approach of the wave packet to the nucleus, during the rst half period (top left), with the appearance of interference fringes as the particle is accelerated towards the Coulomb center. After one period (top right) the wave packet almost resumes its initial shape at the outer turning point of the classical motion, but exhibits considerable dispersion (collapse) after few Kepler cycles (bottom left). Leaving a little more time to the quantum evolution, we observe a non-classical revival after approx. 20 Kepler cycles (bottom right). Recurrence, collapse and revival times are very well predicted by Eqs. (33)–(35). following the classical trajectory. After half a period, it has reached the nucleus (it is essentially localized near the origin). However, interference fringes are clearly visible: they originate from the interference between the head of the wave packet, which has already been re ected o the nucleus, and its tail, which has not yet reached the nucleus. After 3=4 of a period, the interference fringes have disappeared, and the wave packet propagates to the right. It has already spread signi cantly. After one period, it is close to its initial position, but no more as well localized as initially. This recurrence time is given by Eqs. (16) and (32): Trecurrence =2 n 3 0 : (33) After few periods, the wave packet has considerably spread and is now completely delocalized along the classical trajectory. The time for the collapse of the wave packet is well predicted by Eq. (17): Tcollapse 2n40 n0 = × Trecurrence 3( n) 2 3 ( n) 2 =1:96 × Trecurrence for n0 = 60 and n =1:8 : (34) Finally, after 20 periods, the wave packet revives with a shape similar to its initial state. Again, this revival time is in good agreement with the theoretical prediction, Eq. (18): Trevival = 2 n4 0 3 = n0 3 × Trecurrence =20× Trecurrence for n0 = 60 and n =1:8 : (35)
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 421 At longer times, the wave packet continues to alternate between collapses and revivals. In Fig. 4, we show the temporal evolution of the product z p. It is initially close to the Heisenberg limit (minimum value) ˝=2, and oscillates at the frequency of the classical motion with a global increase. When the wave packet has completely spread, the uncertainty product is roughly constant, with apparently erratic uctuations. At the revival time, the uncertainty undergoes again rather orderly oscillations of a relatively large magnitude reaching, at minima, values close to ˝. That is a manifestation of its partial relocalization. For the three-dimensional hydrogen atom, the energy spectrum is exactly the same as in one dimension. This implies that the temporal dynamics is built from exactly the same frequencies; thus, the 3D dynamics is essentially the same as the 1D dynamics. 5 Indeed, collapses and revivals of the wave packet were also observed, under various experimental conditions, in the laboratory [5,11,25,26]. Fig. 5 shows the evolution of a minimum uncertainty wave packet of the 3D atom, initially localized on a circular Kepler orbit of the electron. It is built as a linear combination of circular hydrogenic states (i.e., states with maximum angular and magnetic quantum numbers L = M = n − 1), using the same Gaussian distribution of the coe cients as in Fig. 3. As expected, the wave packet spreads along the circular trajectory (but not transversally to it) and eventually re-establishes its initial shape after Trevival. Fig. 6 shows the corresponding evolution of a swarm of classical particles, for the same initial phase space density. The spreading of the classical distribution and of the quantum wave packet proceeds very similarly, whereas the revival is completely absent in the classical evolution, which once more illustrates its purely quantum origin. Finally, let us notice that collapse and revival of a 3D wave packet depend on the principal quantum number n0 only—see Eqs. (34) and (35)—and are independent of other parameters which characterize the classical motion, such as the eccentricity and the orientation of the classical elliptical trajectory. This establishes that a 3D wave packet with low average angular momentum (and, a fortiori, a 1D wave packet as shown in Fig. 3)—which deeply explores the non-linearity of the Coulomb force—does not disperse faster than a circular wave packet which essentially feels a constant force. Hence, arguments on the non-linear character of the interaction should be used with some caution. There have been several experimental realizations of electronic wave packets in atoms [4,5,11,25, 27,28], either along the pure radial coordinate or even along angular coordinates too. However, all these wave packets dispersed rather quickly. 1.4. How to overcome dispersion Soon after the discovery of quantum mechanics, the spreading of wave packets was realized and attempts were made to overcome it [2]. From Eq. (13), it is however clear that this is only possible if the populated energy levels are equally spaced. In practice, this condition is only met for the harmonic oscillator (or simple tops and rotors). In any other system, the anharmonicity of the energy ladder will induce dispersion. Hence, the situation seems hopeless. 5 In a generic, multidimensional, integrable system, there are several di erent classical frequencies along the various degrees of freedom. Hence, only partial revivals of the wave packet at various times are observed. The 3D hydrogen atom is not generic, because the three frequencies are degenerate, which opens the possibility of a complete revival, simultaneously along all three coordinates.
Page 1 and 2: Abstract Physics Reports 368 (2002)
Page 3 and 4: A. Buchleitner et al. / Physics Rep
Page 11: A. Buchleitner et al. / Physics Rep
Page 43 and 44: → ; → ; A. Buchleitner et al. /
Page 63 and 64:
A. Buchleitner et al. / Physics Rep
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
|ψ (z)| 2 |ψ (z)| 2 0.3 0.2 0.1 0
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
γ /∆ σ/∆ 10 -1 10 -2 10 -3 0.
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139:
show all

Non-dispersive wave packets in periodically driven quantum systems

Create successful ePaper yourself

Delete template?

Save as template?