Finite pulse width
The most important experimental constraint, when one wants to mimic an ide- alised δ−kicked system, is given by the fact that experimental pulses used to provide the kicking potential are always of some finite widthτdurin time. From
the above arguments (see end of last section) it should be clear that the experi- mental dynamics fails to approximate ideal kicks for fixedτdur, in particular, at
large atomic momenta. The experimental realisation is the better the larger the mass of the used atomic species. For this reason caesium, the heaviest stable alkali atom, is nowadays used in experiments [174]. The effect of non-δ-like pulses has been studied extensively by Raizen and co-workers in experiments as well as in numerical simulations [15, 169, 173, 174]. Together with theoreti- cal work [175] these results show that the effective potential (kicking strength) is substantially smaller in the region of large momenta (keff 0.75kmax at
p/(2~kL) =±80 as compared to the centre n= 0, for Tdur 300 nsec [173]).
Physically, if the atom is too fast it will start to average over the potential lead- ing to smaller coupling, or, more precisely, the applied pulse (with a certain shape in time) enforces a window function in momentum space depending on the exact pulse shape. For large momenta this effect induces classical and quan- tum localisation [174–176]. In this region beyond some momentum valuenref,
the classical phase space is filled by impenetrable barriers (tori), which sur- vive small perturbations according to the Kolmogorov-Arnold-Moser (KAM) theorem [43–45, 177]. For smooth pulses, the momentum nref is inversely pro-
portional to the duration of the pulse τdur, with a pre-factor which depends on
the shape of the pulse [175]. Assuming a square pulse shape we obtain, for instance, keff 2kmaxsin(nτdur/2)/(nτdur) [169, 173, 178], the window function
being the Fourier transform of the pulse.
Other problems
There are many more experimental difficulties which are faced when an idealised one-dimensionalδ-kicked particle dynamics [171, 173] should be simulated. For our purposes relevant problems are addressed briefly in the following.
A severe systematic restriction, connected to the discussion of the finite pulse width, originates from the experimental determination of the atoms’ momen- tum distribution. The latter is obtained by counting particles in some rela- tively small momentum interval centred aroundp= 0. Especially in the wings of the momentum distribution, the signal is weak, calling for an experimental threshold which decides whether to reject the counts or not. Practically two thresholds are applied: i) momenta are only counted in some fixed window, and ii) a “dark count” threshold that discriminates the signal from background noise. For the kicked-rotor experiments reported in [82, 83], the effective mo-
26
Chapter 2. Theoretical and experimental preliminaries
mentum window was chosen −40 < p/(2~kL) < +40 (in most recent data
−60 < p/(2~kL) <+60 [159]), and the value of the signal was set to zero for
less than 20 counts [83, 135]. These two relevant thresholds are highlighted in figure 4.10 where experimental and theoretical momentum distributions are compared.
The atomic density (1011atoms/cm3in [173]) must be small enough to avoid considerable atom-atom collisions which would spoil the model of independent, structure-less point particles used in the derivation of the effective Hamiltonian (2.35). Based on measurements of collision cross sections for caesium [179], the collision probability is estimated in [173] as 2%/msec. Considering kicking timest.50×Tp, with typicalTp 20. . .70 µsec [81–83], this corresponds to
a maximal probability of 7% that one collision occurs during the experiment. Intensity fluctuations in the laser beam producing the standing wave should be kept as small as possible to avoid what is known as amplitude noise [173, 180, 181]. Moreover, since the atoms are initially prepared in a three- dimensional momentum distribution they are not always centred at the spot of the laser. The laser itself has a transverse Gaussian profile what leads to a potential which is the weaker the farther the atoms are away from the centre of the beam. Both of these two independent effects induce a variation of the kicking strength, which is experienced by the atoms [135, 182].
In particular, when additional momentum is imparted on the atoms by allow- ing them to emit spontaneously in all directions, the particles may move away from the spot of the laser beam. In the experimental situations of [81–83], the transverse spreading (i.e. the deviation from the one-dimensionality of the mo- tion) produced by SE events is smaller than the spread of the initial momentum distribution in the transverse plane to the kicking axis.
The problem of spontaneous emission is for itself worthwhile to investigate: atoms are never two-level systems what makes a treatment necessary which includes the distribution of the atomic population over various sublevels, and also the process of dissipation by spontaneous emission. In the far-detuned case, assumed when deriving the effective Hamiltonian (2.35), the probability of absorbing a photon from the standing wave and emitting it in the vacuum mode is small. A good approximation for the steady-state scattering rate as a result of spontaneous emission (SE) is obtained byRscγSE|ψe|2, whereγSE is
the line width of the excited level with population |ψe|2. Using (2.32), we may
estimate |ψe|2 Ω2/(4∆2L) cos2(kLx) (assuming that only the ground state is significantly populated, i.e. |ψg| 1, and ~|∆L| |
ˆ
P2|/2M as used above). This leads to Rsc γSEΩ2/(8∆2L) 1 for γSE,Ω |∆L| after averaging the cosine. For the experiments performed by d’Arcy and co-workers, the mean number of SE events undergone by each atom due to one far-detuned standing wave pulse is estimated to benSE RscTp.2×10
−3 [83, 135].
Apart from the unwanted effect described above, SE, indeed, provides a con- trollable way of adding noise to the evolution of the atoms [82, 83, 176, 182–184]. To this end, SE is introduced most flexibly by an additional laser, which is independent of the standing wave (for instance, by the beams used to prepare and cool the atoms before the kicks are applied [82, 83], cf. figure 2.1). The