Non-dispersive wave packets in periodically driven quantum systems

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