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Quantum mechanics

Passage through chaos

A quantum system can undergo tunnelling even without a barrier to tunnel through. The latest experiments visualize this process in exquisite detail, completely reconstructing the state of the evolving system.

Reconciling quantum mechanics with classical, Newtonian physics has been a long-standing challenge. A major aspect of this challenge pertains to chaotic systems — simple and deterministic classical systems that nevertheless display complex, seemingly random, unpredictable behaviour. The problem of 'quantum chaos' is this: take a chaotic system, study its (simplest) quantum counterpart, and what you don't find is any of the unpredictable, chaotic behaviour from the classical world. This is a funny thing, because you can go into any toy store and see any number of chaotic, pendulum-like devices, dynamically waving about for the amusement of children everywhere. In principle, a physicist should be able to model these toys either as Newtonian collections of interacting rigid bodies, or as ensembles of manifestly quantum-mechanical atoms. The answer should be the same in either case, except, however, that the chaos seems to be missing from the quantum side of the picture. Now Jessen and colleagues1, writing on page 768 of this issue, have experimentally studied the behaviour of the quantum version of a chaotic system with an unprecedented level of precision and detail, providing new insight into the quantum–classical boundary.

To understand Jessen and colleagues' experiments1, first consider what happens to an ensemble of atoms from the classical perspective. It is the angular momentum of the atoms that we are concerned with — technically an abstract quantity, but it suffices to think of the 'orientation' of an atom as its axis of rotation. Fixing the magnitude of the angular momentum, we can represent the orientation of each atom as a point on a sphere (Fig. 1a). The experiments transform the orientations of the atoms in two parts: the first is a 'twist', in which a carefully tuned laser pulse shears the points on the sphere (Fig. 1b), and the second is a rotation caused by a magnetic-field pulse (Fig. 1c). The authors' sequence of twist/turn transformations on the atoms realizes for the first time the 'kicked top' (Fig. 1), one of the simplest yet most important model systems for studying quantum chaos.

Figure 1: The kicked top.
figure1

a, The angular-momentum vector (arrowed) of an atom can be visualized as corresponding to a single point on a sphere, which represents all possible angular momenta. b, The first, or 'kick', step in realizing the 'kicked top' model system, which Jessen and colleagues1 implement in their study of quantum chaos, is a 'twist' of the points on the sphere — the points near the poles rotating the most, and the points on the equator staying put. c, The second step is a simple rotation of the whole globe about an orthogonal axis (not shown).

The behaviour of an atom under this simple twist/turn map is rich and complex. To visualize it, consider the flattened representation of the sphere in Figure 2a, which shows the initial orientations of two groups of atoms forming two short line segments. The effect of repeating the twist/turn procedure ten times is shown in Figure 2b: one set of orientations is marginally distorted, whereas the other is stretched and folded in an intricate way. The stretching is indicative of erratic chaotic behaviour, and the point is that the dynamical behaviour in a given system can be mixed — certain initial orientations lead to chaotic dynamics, whereas others are comparatively ordered. This is best shown in Figure 2c, which plots the effects of many twist/turn iterations on several initial orientations. Chaotic regions appear as a mass of dots, whereas stable regions are neatly organized into nested, ring-like layers. The important lesson to remember for now is this: because of the stretching, an atomic orientation in the chaotic region can wander throughout it, whereas an atom in an 'island of stability' is trapped there, confined to its particular 'ring'.

Figure 2: Chaos in the kicked top.
figure2

a, In this 'flattened globe', the two coloured line segments denote two sets of points on the sphere, each representing the initial angular momenta of two collections of atoms. b, The effect of ten iterations of the kicked-top transformation depicted in Figure 1: the green line segment gets only a bit twisted, whereas the red segment is dramatically stretched and folded onto itself — a hallmark of chaos. c, Many iterations of several starting points, clearly showing regions of stability (onion-like rings) and chaos (a fuzz of dots).

But now back to quantum mechanics — we're talking about atoms, after all. As a consequence of Heisenberg's uncertainty principle, quantum states of atoms can't be single points on the sphere, but must be smeared out to occupy at least some finite area. And again, there can be no chaos in the quantum case, in stark contrast to the classical model. Traditionally, there have been two approaches to this problem of the missing quantum chaos. One is to study the conditions under which the classical and quantum descriptions agree. For example, under a weak, continuous measurement, a quantum system can be persuaded to display chaos as appropriate to the classical case2. The other is to study the 'fingerprints' of chaos3 in the quantum system, and this is the approach taken by Jessen and collaborators1.

The authors studied a phenomenon called dynamical tunnelling4. This is a bit different from the better-known barrier tunnelling, in which a quantum particle can penetrate a potential barrier despite not having enough energy to hop over it. Recalling the kicked-top behaviour depicted in Figure 2c, notice that there are two main stable islands in the left hemisphere and that a consequence of stability is that, classically, an atom starting in either island is trapped there — not by any potential barrier, but merely as a consequence of the twist/turn dynamics. Because of the symmetry of these two islands, quantum mechanics allows an atom starting in one island to hop back and forth to the other island, a dynamical tunnelling process between two atomic orientations strictly forbidden in the classical world. Jessen and collaborators' experiments clearly demonstrated this, as well as an atomic quantum state sitting placidly in the large island and another moving erratically (though not chaotically) in the chaotic region — carefully respecting the classical boundaries between stability and chaos, despite being far into the quantum regime.

The beauty of the experiments1 lies in the complete reconstruction of the quantum state, leaving no aspect of the tunnelling process hidden. This is no easy task, involving the processing and combination of many measurements, and was not possible in previous studies of tunnelling5,6,7,8. The recovery of the full state also permitted observations of other fingerprints of chaos in a quantum system for the first time, such as the generation of quantum entanglement and the sensitivity to perturbations to the parameters of the system, rather than to its initial state9.

Interesting future directions for Jessen and colleagues' work include a push towards the classical limit, where more distinct quantum states live on the sphere. This is a technically difficult regime, but one in which the fingerprints of chaos can be studied in even more detail, and where the controlled transition from quantum stability to classical chaos may be observed.

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Steck, D. Passage through chaos. Nature 461, 736–737 (2009). https://doi.org/10.1038/461736a

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