Non-dispersive wave packets in periodically driven quantum systems

More documents

Recommendations

Info

480 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 Fig. 28. Comparison of exact Floquet eigenstates (bottom row) and of their harmonic approximations, for the ground state wave packet |n+ =0;n− =0〉 (left column), and for the |n+ =1;n− =3〉 excited state (right column), for the 2D hydrogen atom driven by a circularly polarized microwave ( eld pointing to the right of the gure), with amplitude F0 =0:0333 and resonant frequency corresponding to n0 = 60. The nucleus is located at the center of each plot which extend over ±5000 Bohr radii. The top row represents eigenfunctions in the harmonic approximation in ˜x; ˜y coordinates, Eq. (191). The eigenfunctions in the middle row are obtained from the harmonic approximation to Eq. (184) in polar coordinates. They exhibit a clear bending of the electronic density along the circular trajectory. The excited wave packet |1; 3〉 appears in Fig. 30 as a straight line (modulo small avoided crossings) with a negative slope, meeting the state |0; 0〉 in a broad avoided crossing, around F0 0:036. to the state obtained by exact numerical diagonalization of the 2D Floquet Hamiltonian. Obviously, the structure of the exact state is very similar to the one obtained from its harmonic approximation. Because the creation and annihilation operators in the !± modes entangle position and momentum coordinates in a complicated way [68], and although the system is then completely integrable, the wave function in the harmonic approximation is not separable in any coordinate system (in contrast with the usual harmonic oscillator). Actually, the wave function can be written as a product of
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 481 Fig. 29. Isovalue plots (at 30% of the maximum value) of non-dispersive wave packets in the three-dimensional hydrogen atom driven by a circularly polarized microwave eld. Frequency of the driving ! =1=60 3 , amplitude F0 =0:04442. In the laboratory frame, the wave packets propagate—without changing their shapes—along a circular trajectory centered around the nucleus indicated by a cross. The cube edges measure 10 000 Bohr radii. The microwave polarization plane is horizontal with the eld pointing to the right. The four wave packets shown represent the ground state wave packet |0; 0; 0〉 (top left), and the excited states |1; 0; 0〉 (bottom left), |0; 1; 0〉 (top right), and |0; 0; 1〉 (bottom right). Eventually, the microwave eld will ionize such states, but their lifetimes are extremely long, of the order of thousands to millions of Kepler periods. Gaussians and Hermite polynomials of the position coordinates, but the Hermite polynomials have to be evaluated for complex values. This results in the unusual pattern of the probability density displayed in Fig. 28. An improvement over the harmonic approximation is possible, by bending the axis in the spirit of [47], in order to account for the spherical symmetry of the dominant Coulomb potential. With this improvement, the probability density, shown in the middle row of Fig. 28, is almost indistinguishable from the exact result. Let us repeat that this bending—and consequently the deviation from Gaussian character of the wave function—does not a ect at all the non-dispersive character of the wave packet. Let us now turn to the realistic 3D model of the atom. Fig. 29 shows an isovalue contour of several non-dispersive wave packets for the hydrogen atom driven by a microwave eld with frequency ! =1=60 3 , i.e., roughly resonant with the n0 =60 → 59; 61 transitions (see Eqs. (40), (65) and (119)). The best localized wave packet is the ground state |0; 0; 0〉, while the three other
Page 1 and 2:
Abstract Physics Reports 368 (2002)
Page 3 and 4:
A. Buchleitner et al. / Physics Rep
Page 5 and 6:
Page 7 and 8:
Page 9 and 10:
Page 11 and 12:
Page 13 and 14:
Page 15 and 16:
Page 17 and 18:
Page 19 and 20:
Page 21 and 22: A. Buchleitner et al. / Physics Rep
Page 43 and 44: → ; → ; A. Buchleitner et al. /
Page 71: A. Buchleitner et al. / Physics Rep
Page 97 and 98: |ψ (z)| 2 |ψ (z)| 2 0.3 0.2 0.1 0
Page 113 and 114: γ /∆ σ/∆ 10 -1 10 -2 10 -3 0.
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?