Non-dispersive wave packets in periodically driven quantum systems

More documents

Recommendations

Info

410 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 PACS: 05.45.Mt; 03.65.Sq; 32.80.Qk; 32.80.Rm; 42.50.Hz Keywords: Wave packet; Dispersion; Spreading; Coherent states; Rydberg atoms; Non-linear resonance; Atom- eld interaction Contents 1. Introduction ........................................................................................ 411 1.1. What is a wave packet? ......................................................................... 411 1.2. Gaussian wave packets—coherent states ........................................................... 417 1.3. A simple example: the one-dimensional hydrogen atom .............................................. 419 1.4. How to overcome dispersion ..................................................................... 421 1.5. The interest of non-dispersive wave packets ........................................................ 425 2. Semiclassical quantization ............................................................................ 426 2.1. WKB quantization .............................................................................. 426 2.2. EBK quantization............................................................................... 427 2.3. Scars ......................................................................................... 428 3. Non-dispersive wave packets and their realization in various atomic systems ................................ 429 3.1. General model—non-linear resonances............................................................. 429 3.1.1. Classical dynamics ....................................................................... 429 3.1.2. Quantum dynamics ....................................................................... 435 3.1.3. Semiclassical approximation ............................................................... 437 3.1.4. The Mathieu approach .................................................................... 439 3.2. Rydberg states in external elds .................................................................. 444 3.2.1. Rydberg atoms .......................................................................... 444 3.2.2. Hamiltonian, basis sets and selection rules .................................................. 445 3.2.3. Simpli ed 1D and 2D models ............................................................. 447 3.2.4. Action-angle coordinates .................................................................. 447 3.2.5. Scaling laws ............................................................................ 450 3.3. Rydberg states in linearly polarized microwave elds ............................................... 451 3.3.1. One-dimensional model ................................................................... 452 3.3.2. Realistic three-dimensional atom ........................................................... 459 3.4. Rydberg states in circularly polarized microwave elds .............................................. 470 3.4.1. Hamiltonian ............................................................................. 471 3.4.2. Resonance analysis ....................................................................... 472 3.4.3. The two-dimensional model ............................................................... 474 3.4.4. Transformation to the rotating frame ....................................................... 475 3.5. Rydberg states in elliptically polarized microwave elds ............................................. 486 4. Manipulating the wave packets ........................................................................ 490 4.1. Rydberg states in linearly polarized microwave and static electric elds ............................... 490 4.2. Wave packets in the presence of a static magnetic eld ............................................. 493 5. Other resonances .................................................................................... 498 5.1. General considerations .......................................................................... 498 5.2. A simple example in 1D: the gravitational bouncer ................................................. 503 5.3. The s = 2 resonance in atomic hydrogen under linearly polarized driving .............................. 506 6. Alternative perspectives .............................................................................. 513 6.1. Non-dispersive wave-packets in rotating molecules .................................................. 513 6.2. Driven Helium in a frozen planet con guration ..................................................... 515 6.3. Non-dispersive wave-packets in isolated core excitation of multielectron atoms ......................... 516
A. Buchleitner et al. / Physics Reports 368 (2002) 409–547 411 7. Characteristic properties of non-dispersive wave packets .................................................. 517 7.1. Ionization rates and chaos-assisted tunneling ....................................................... 517 7.2. Radiative properties ............................................................................. 521 7.2.1. Interaction of a non-dispersive wave packet with a monochromatic probe eld ................... 522 7.2.2. Spontaneous emission from a non-dispersive wave-packet ..................................... 522 7.2.3. Circular polarization ...................................................................... 523 7.2.4. Linearly polarized microwave ............................................................. 527 7.3. Non-dispersive wave packet as a soliton ........................................................... 529 8. Experimental preparation and detection of non-dispersive wave packets ..................................... 531 8.1. Experimental status ............................................................................. 532 8.2. Direct preparation .............................................................................. 533 8.3. Preparation through tailored pulses ................................................................ 535 8.4. Life time measurements ......................................................................... 540 9. Conclusions ........................................................................................ 542 Acknowledgements ..................................................................................... 542 References ............................................................................................ 543 1. Introduction 1.1. What is a wave packet? It is commonly accepted that, for macroscopic systems like comets, cars, cats, and dogs [1], quantum objects behave like classical ones. Throughout this report, we will understand by “quantum objects” physical systems governed by the Schrodinger equation: the system can then be entirely described by its state | 〉, which, mathematically speaking, is just a vector in Hilbert space. We will furthermore restrict ourselves to the dynamics of a single, spin-less particle, such as to have an immediate representation of | 〉 in con guration and momentum space by the wave functions 〈˜r| 〉 = (˜r) and 〈˜p| 〉 = (˜p), respectively. The same object is in classical mechanics described by its phase space coordinates ˜r and ˜p (or variants thereof), and our central concern will be to understand how faithfully we can mimic the classical time evolution of ˜r and ˜p by a single quantum state | 〉, inthemicroscopic realm. Whereas classical dynamics are described by Hamilton’s equations of motion, which determine the values of ˜r and ˜p at any time, given some initial condition (˜r0;˜p0), the quantum evolution is described by the Schrodinger equation, which propagates the wave function. Hence, it is suggestive to associate a classical particle with a quantum state | 〉 which is optimally localized around the classical particle’s phase space position, at any time t. However, quantum mechanics imposes a fundamental limit on localization, expressed by Heisenberg’s uncertainty relation z · p ¿ ˝ ; (1) 2 where z and p are the uncertainties (i.e., square roots of the variances) of the probability distributions of z and its conjugate momentum p in state | 〉, respectively (similar relations hold for other choices of canonically conjugate coordinates). Consequently, the best we can hope for is a quantum state localized with a nite width ( z; p) around the particle’s classical position (z; p), with z and p much smaller than the typical scales of the classical trajectory. This, however, would satisfy our aim of constructing a quantum state that mimics the classical motion, provided
Page 1: Abstract Physics Reports 368 (2002)
Page 5 and 6: A. Buchleitner et al. / Physics Rep
Page 43 and 44: → ; → ; A. Buchleitner et al. /
Page 53 and 54:
A. Buchleitner et al. / Physics Rep
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?