Atom-Optical Quantum Accelerator Modes

Simon Gardiner

Quantum accelerator modes are caused through an unusual quantum-mechanical resonance, and primarily characterized using a recently developed pseudoclassical framework. They are observed in laser-cooled clouds of caesium atoms allowed to fall freely through a vertically oriented pulsed laser standing wave. They manifest themselves as a portion of the atoms accelerating coherently away from the main cloud, in a manner strongly dependent on the effect of gravity. We concern ourselves with the theoretical understanding and interpretation of these phenomena, in close cooperation with experiment. Below is a brief tutorial on the most important aspects of quantum accelerator mode phenomena.

Schematic of Experimental Setup

In the Oxford quantum accelerator mode experiment, about 107 caesium atoms are trapped and cooled in a magneto-optical trap (MOT) to a temperature of around 5 microKelvin.

The atoms are then released and exposed to periodic pulses from a far-detuned laser standing wave, of 500 nanoseconds duration and varying periodicity (50 to 200 microseconds). For the relevant atomic velocities, these pulses can be considered effectively instantaneous.

After the pulsing sequence, the atoms fall through resonant laser light. By monitoring the absorption, the atomic momentum distribution is determined by a time of flight (TOF) measurement.

The phase shifter can be used to move the front of the laser standing wave. If the front is accelerated, then this has an equivalent effect to a different value of the local gravitational acceleration.

Observation of quantum accelerator modes

The displayed data consists of centre-of mass atomic momentum distributions, in a frame falling freely with gravity, as a function of time measured as the number of periodic laser pulses. The pulse periodicity is 60.5 microseconds.

A portion of the atomic cloud accelerates, linearly with time, away from the central cloud (falling freely in the lab frame). The atoms remaining in the central cloud behave in a more diffusive manner, as would be expected classically.

Scanning the Pulse Periodicity

It is also possible to consider a fixed number of laser pulses, and observe the dependence of the atomic momentum distributions as a function of the pulse periodicity.

The vertical red line shows where the data from the upper and lower plots coincides, for a pulse periodicity of 60.5 microseconds, and pulse number equal to 30.

The white hyperbolic curves show theoretical predictions for the expected behaviour of quantum accelerator modes. An enhancement of the atomic population is expected, and observed, along at least part of the curves' lengths.

Pseudoclassical Theoretical Framework

An elementary theoretical framework for understanding quantum accelerator modes is in terms of a pair of effective-classical action-angle coordinates I and θ.

The experimental regime is not one which can conventionally considered semiclassical, i.e., that Planck’s constant can be considered neglible compared to the relevant action scales). Instead what counts is the closeness of the pulsing peridicities to integer multiples of TQ=mλ2/h (=66.7 microseconds in the Oxford experimental setup), such that the pulsing period T=TQ(l+ε/2π), where ε is a smallness parameter. Here m is the atomic mass, λ is the period of the sinusoidal potential due to the laser standing wave, and h is Planck’s constant.

Plotting a stroboscopic Poincaré section of the action-angle pairs, where one propagates an ensemble of initial conditions, and plots each such pair just before the application of each pulse, reveals island structures in the pseudoclassical phase space. These island structures are base around stable periodic orbits in the pseudoclassical phase-space, and correspond exactly to the experimentally observed quantum accelerator modes.

The phase space is 2 π-periodic in the action variable I (as well as, trivially, in θ), and one normally plots only one phase-space cell. The number of individual islands in one phase-space cell for a particular island system is known as the order. The Poincaré section, taken for a pulse periodicity of 66.7 microseconds, shows island systems of order p=10 (blue) and p=11 (red).

Pseudoclassical Order and Jumping Index

The stable periodic orbits/island systems (and hence quantum accelerator modes) are also classified by their jumping index j.

This is the number of phase-space cells traversed by an initial condition starting on a periodic orbit of order p after exactly p laser pulses.

To illustrate this we show two phase space cells, depicting (red arrows, numbered by pulse number) how such an initial condition traverses one phase space cell after 10 iterations. It is thus of order p=10 and jumping index j=1.

Classification of Quantum Accelerator Modes

It is the order (p) and jumping index (j) classification of the stable periodic orbits in the pseudoclassical phase space that is used to make theoretical predictions for the behaviour of different atom-optical quantum accelerator modes, through the formula kN≈2π/|ε|[j/p +sgn(ε)gT2/λ]h/λ, where kN is the predicted momentum of the accelerator mode ofter the application of N laser pulses.

Scanning through different values of the pulse period T near TQ=66.7 microseconds (equivalent to scanning across ε=0 from negative to positive values), we match the pseudoclassical phase space island systems to the experimentally observed quantum accelerator modes.

Publications on Quantum Accelerator Modes and Pseudoclassics

Content © Simon A Gardiner, Durham University 2005