Quantum Hyperion

One of the annoying/fascinating things about quantum mechanics is the fact the world doesn’t seem to be quantum-mechanical. When you look at something, it seems to have a location, not a superposition of all possible locations; when it travels from one place to another, it seems to take a path, not a sum over all paths. This frustration was expressed by no lesser a person than Albert Einstein, quoted by Abraham Pais, quoted in turn by David Mermin in a lovely article entitled “Is the Moon There when Nobody Looks?“:

I recall that during one walk Einstein suddenly stopped, turned to me and asked whether I really believed that the moon exists only when I looked at it.

The conventional quantum-mechanical answer would be “Sure, the moon exists when you’re not looking at it. But there is no such thing as `the position of the moon’ when you are not looking at it.”

Nevertheless, astronomers over the centuries have done a pretty good job predicting eclipses as if there really was something called `the position of the moon,’ even when nobody (as far as we know) was looking at it. There is a conventional quantum-mechanical explanation for this, as well: the correspondence principle, which states that the predictions of quantum mechanics in the limit of a very large number of particles (a macroscopic body) approach those of classical Newtonian mechanics. This is one of those vague but invaluable rules of thumb that was formulated by Niels Bohr back in the salad days of quantum mechanics. If it sounds a little hand-wavy, that’s because it is.

The vagueness of the correspondence principle prods a careful physicist into formulating a more precise version, or perhaps coming up with counterexamples. And indeed, counterexamples exist: namely, when the classical predictions for the system in question are chaotic. In chaotic systems, tiny differences in initial conditions grow into substantial differences in the ultimate evolution. It shouldn’t come as any surprise, then, that it is hard to map the predictions for classically chaotic systems onto average values of predictions for quantum observables. Essentially, tiny quantum uncertainties in the state of a chaotic system grow into large quantum uncertainties before too long, and the system is no longer accurately described by a classical limit, even if there are large numbers of particles.

Some years ago, Wojciech Zurek and Juan Pablo Paz described a particularly interesting real-world example of such a system: Hyperion, a moon of Saturn that features an irregular shape and a spongy surface texture.

The orbit of Hyperion around Saturn is fairly predictable; happily, even for lumpy moons, the center of mass follows a smooth path. But the orientation of Hyperion, it turns out, is chaotic — the moon tumbles unpredictably as it orbits, as measured by Voyager 2 as well as Earth-based telescopes. Its orbit is highly elliptical, and resonates with the orbit of Titan, which exerts a torque on its axis. If you knew Hyperion’s orientation fairly precisely at some time, it would be completely unpredictable within a month or so (the Lyapunov exponent is about 40 days). More poetically, if you lived there, you wouldn’t be able to predict when the Sun would next rise.

So — is Hyperion oriented when nobody looks? Zurek and Paz calculate (not recently — this is fun, not breaking news) that if Hyperion were isolated from the rest of the universe, it would evolve into a non-localized quantum state over a period of about 20 years. It’s an impressive example of quantum uncertainty on a macroscopic scale.

Except that Hyperion is not isolated from the rest of the universe. If nothing else, it’s constantly bombarded by photons from the Sun, as well as from the rest of the universe. And those photons have their own quantum states, and when they bounce off Hyperion the states become entangled. But there’s no way to keep track of the states of all those photons after they interact and go their merry way. So when you speak about “the quantum state of Hyperion,” you really mean the state we would get by averaging over all the possible states of the photons we didn’t keep track of. And that averaging process — considering the state of a certain quantum system when we haven’t kept track of the states of the many other systems with which it is entangled — leads to decoherence. Roughly speaking, the photons bouncing off of Hyperion act like a series of many little “observations of the wavefunction,” collapsing it into a state of definite orientation.

So, in the real world, not only does this particular moon (of Saturn) exist when we’re not looking, it’s also in a pretty well-defined orientation — even if, in a simple model that excludes the rest of the universe, its wave function would be all spread out after only 20 years of evolution. As Zurek and Paz conclude, “Decoherence caused by the environment … is not a subterfuge of a theorist, but a fact of life.” (As if one could sensibly distinguish between the two.)

Update: Scientific American has been nice enough to publicly post a feature by Martin Gutzwiller on quantum chaos. Thanks due to George Musser.

  1. “One of the annoying/fascinating things about quantum mechanics is the fact the world doesn’t seem to be quantum-mechanical. When you look at something, it seems to have a location, not a superposition of all possible locations; when it travels from one place to another, it seems to take a path, not a sum over all paths.”

    So, a fun idea to toy around with is that how things “seem” and “look” are just constructs of the clump of neurons in your head. We take sensory input from the world around it and interpret it in a way that allows us to avoid predators and seek prey. The mind is a great self-deceiver, however, so much that the interpretation is incredibly difficult to separate from reality for most people. Consider the simple example of color. Nothing in this world has “color”, rather our brains assign colors to surfaces that reflect certain wavelengths of light so that we can differentiate objects in our environment.

    So to get back to your point, it matters little if nothing “seems” to operate according to quantum mechanics on a macro scale. Just because we perceive an object to have a discrete location and orientation does not make it so, it’s just our brain’s cunningly deceptive interpretation of quantum mechanics.

  2. We see color through atmosphere – there we see perspective and gradations in tone – medley – warm and cool color – everything is always changing – even your feelings – your perceptions !

  3. Hi Sean,

    I think I’m missing something. How is classical chaos a potential counterexample to the correspondence principle? The principle holds that quantum mechanics -> classical mechanics in a particular limit, which seems to be separate from the question of whether CM -> chaos.

    Wouldn’t a potential counterexample be some strongly quantum but still large-N system, such as a Bose-Einstein condensate or the universe during the Planck epoch?


  4. The correspondence principle says that a large-N system, prepared in a quantum state localized around some classical values, should obey the classical equations of motion corresponding to those values. But a classically chaotic system does not; the necessary quantum uncertainty implies that you are very likely to find the system far away from the classical point you would have predicted.

  5. This commentary by Todd Brun elaborates this problem:


    The field of quantum chaos, as the term is usually applied, is mainly limited to the case of Hamiltonian systems: systems which conserve energy and are chaotic in the classical limit. Dissipative chaos, which is equally important classically and displays rather different behavior, is comparatively little studied quantum mechanically.

    I have looked at a class of chaotic models using the formalisms of Decoherent Histories and Quantum Trajectories. Classically, a dissipative chaotic system tends towards a strange attractor with a fractal structure; that is, the attractor exhibits substructure at all length scales.

    Quantum mechanically this is impossible, as structure on a scale smaller than Planck’s constant has no meaning. As one approaches the classical limit, more and more layers of structure appear; but there is always a limiting scale, at which the existence of quantum uncertainty and noise from the dissipative environment blurs out the fractal.

    Because of this phenomenon, quantities used to characterize classical chaos (such as Lyapunov exponents and fractal dimension) are not well-defined for the equivalent quantum systems. It is possible, however, that extensions of these concepts may still be useful. One useful proposal, due to Schack and Caves, is hypersensitivity to perturbations. I am currently collaborating on studies of this in quantum computers.

  6. The atmosphere is quantum-mechanical, as is the sunlight, and the combination creates beautiful sunsets and similar natural spectacles. Or, you could say that solar photons of given wavelengths interact with molecules and atoms that will be excited to a different quantum state, with the short wavelengths tending to interact more strongly with small molecules, which is why the sunsets consist of brilliant reds, yellows, and oranges… which we are only able to sense to to quantum excitations of sheets of photo-sensitive molecules on the backs of our eyeballs.a

  7. Alex, if you are looking for a more technical answer: In the chaotic system, different paths leading to very different orientations are do in fact have very similar action (integral of the Lagrangian) measured in multiples of h-bar. But this is what appears in the exponent in the path integral and therefore both configurations contribute to similar amounts.

    However, let me give you one warning: Imagine that we could actually isolate Hyperion from all the photons for quite some time so that no decoherence happens and then you were allowed to look at it as the first person to observe the macroscopic quantum state. What would you expect to see? If you expect a blurry image of a “superposition of many orientations” then I have bad news for you: It would look as classical as always. You couldn’t tell the difference!

    You will always observe some eigenstate of the orientation operator. It’s just that if you repeated this experiemtn you would observe some interference pattern in the distribution of observed orientations. This is just like electrons in the double slit experiment: Each individual electron hit the screen at one specific position. It’s only their number distribution that shows the interference.

    If you like, what happens is that you entangle the state of your brain an the orientation of Hyperion. Considering then only your brane, its state is a classical probability distribution of many possible orientations.

  8. Chaotic systems have nothing to to with quantum effects. Purely classical chaotic systems are not uncertain. They are deterministic. The chaos in these systems refers to the long term sensitivity of the system to the initial conditions, but given a set of fixed initial conditions, there is no uncertainty(Aside of course from rounding and other numerical errors, which by the nature of the chaotic system, cause solutions to diverge).

  9. ObsessiveMathsFreak,

    Chaotic systems have nothing to to with quantum effects. Purely classical chaotic systems are not uncertain. They are deterministic. The chaos in these systems refers to the long term sensitivity of the system to the initial conditions, but given a set of fixed initial conditions, there is no uncertainty(Aside of course from rounding and other numerical errors, which by the nature of the chaotic system, cause solutions to diverge).

    The problem is that the quantum effects prevent those initial conditions from being purely classical ones. The very slight deviations from classicality, when combined with the chaotic nature of the system, can produce macroscopic effects if the system doesn’t interact with other systems in the interim.

  10. This does seem rather to hark back to the “blogging heads” discussion of QM foundations that Sean and David Albert posted not so long ago. In my judgement, there was no consensus of opinion either within the debate or the subsequent coments. I do not think this problem is any way close to being solved.

    We seem to experience a classical illusion on top of a fundamentaly quantum world. The trendy word for this is “decoherence” but I have never heard a coherent explanation of what this is.

    Many have tried – be it “many-worlds” interpretations, gravitational involvement, “Copenhagen”-style pseudo-philosophy, “hidden variables”, or many others, but I’m not convinced by any of them.

  11. Isn’t 40 days the Lyapunov time? Lyapunov exponents have units of 1/time.

    The counterexample only holds if the thermodynamic limit (large-N) exists for the distribution of Lyapunov exponents.

  12. Re #4, #8, #10: From Sean’s, Robert’s, and Jason’s remarks, I take it that the problem is the quantum fluctuations. I can’t just take an expectation value and plug it into the classical equations because those equations are so sensitive to initial conditions. If so, wouldn’t thermal fluctuations have the same effect?

    Also, could I still define a restricted version of the correspondence principle if I look at a restricted span of time? That is, I can increase the value of N to make the fluctuations proportionately less important and still have classical predictable behavior for a certain period until the sensitive dependence on initial conditions manifests itself.


  13. Why? Chaotic dynamics concerns systems where minor changes in the initial conditions of a system will exponentially grow almost without bounds. This is the touted butterfly effect, where the flapping butterfly wing in Papau New Guinea causes a hurricane here (Katrina?). So suppose you have two systems with initial condition separated by some small amount of distance and momentum (&q, &p), & = delta, in phase space. This small deviation can be due to a number of things, such as the truncation error introduced by a numerical computation. These systems are deterministic, but in order to integrate the system explicitely requires a computer with an infinite floating point capability.

    For linear systems these errors will grow in a linear or maybe polynomial manner. So it will take a very long time for these errors to manifest themselves. However, chaotic systems such as Hamiltonian chaotic systems will exhibit a radical divergence of these errors or differences in initial conditions

    Delta q~=~exp^{Lambda t}Delta q',~Delta p~=~e^{Lambda t}Delta p'

    so these errors or differences in initial condition exponentially grow according to the Lyapunov exponent /.

    Now introduce quantum mechanics. The metric measure of a the Fubini space which gives the fibration over a projective Hilbert space is

    s~=~sqrt{langle O^2 rangle~-~langle O rangle^2 }Delta x

    where x is the conjugate variable to the observable O. So the astute might see that this is the Heisenberg uncertainty principle, and s defines an invariant number of units of action, or n-hbar. So a quantum dynamical system will in a zero temperature situation will have all its possible quantum paths diverge (as in a Feynman path integral), but for &t the number of times that a photon carries away entanglement phase from the system there is a stochastic error given by &E. This system is a large N-hbar example of the quantum Zeno effect, but where there is the added issue of chaos. These tiny errors then sum together in a way similar to a drunkard’s walk. Each of these tiny errors is exponentially amplified and they in turn sum together.

    In this way the errors which are introduced by these quantum fluctuations result in a large scale or classical trajectory that differs from the expectation from the quantum path integral of the system. These little decoherent jiggles of Hyperion result in a stochastic change in its rotational position that can’t be predicted.

    Lawrence B. Crowell

  14. Arxiv. In the ’96 paper Zuric claims

    Motivated by Hyperion, we review salient features of “quantum chaos” and show that decoherence is the essential ingredient of the classical limit, as it enables one to solve the apparent paradox caused by the breakdown of the correspondence principle for classically chaotic systems.

    In a 2005 paper Weibe claims

    We conclude that decoherence is not essential to explain the classical behavior of macroscopic bodies.

    In a 2008 paper Schlosshauer claims

    We show that the controversy is resolved once the very different assumptions underlying these claims are recognized. In doing so, we emphasize the distinct notions of the problem of classicality in the ensemble interpretation of quantum mechanics and in decoherence-based approaches that are aimed at addressing the measurement problem.

    can someone sort this out for me? Why are they disagreeing? what are the different underlying assumptions?

  15. When I look at the moon, I detect a bunch of photons interacting with my retina. This seems compatible with the moon existing a second ago, but who knows what has happened since then. Maybe the moon has been eaten by a Boltzmann brain and reappeared on a different brane in the meantime.

  16. More poetically, if you lived there, you wouldn’t be able to predict when the Sun would next rise.

    Hyperion would then be perfectly described by economics (the ultimate parameterized interpolative curve fit, all fits each proving the others wrong) plus heteroskedasticity (because analytic economics is fundamentally crap for prediction). Hyperion looks like a job for string theory! More studies are needed.

  17. I was a little rushed and unclear in my question, and while your detailed responses are helpful and appreciated, I was just wondering why Hyperion specifically is so unpredictable, why it’s quantum fluctuations evolve into macroscopic ones while most bodies follow the correspondence principle.

  18. Pingback: Friday 24 October 2008 « blueollie

  19. If we don’t detect all these photons that have bounced off Hyperion then we shouldn’t say that Hyperion is in a definite orientation if we don’t look.

    If it really is the case that the orientation of Hyperion is fixed before we observe it, then the observer should also be part of that entangled superposition.

    So, the question should really be if for two terms of this entangled quantum state of Hyperion and the rest of the universe corresponding to two different orientations of Hyperion, the state of the brains of people on Earth in these terms are orthogonal.

    One could argue that even this is not enough, because unless we can tell the two brain states apart, our consciousness is the same in the different brain states. If it were different, we could notice the difference, and then we could “feel” what state Hyperion is in without observing it.

    But, we can only be aware of a limited amount of information, so the orientation of Hyperion would be swamped by all other external influences we are exposed to.

  20. Count Iblis,

    Not really sure what your point is here, but a couple of comments on your comments:

    “…then the observer should also be part of that entangled superposition”

    This is part of the standard QM orthodoxy, and, I think, an experimentally inescapable conclusion.

    Also, you say that:

    “…the state of the brains of people on Earth in these terms are orthogonal.”

    Not necessarily so – human (or presumably also ET) observers can select a basis for their working model however they like – within mathematical practicality OK, but beyond that it’s up to them. North isn’t orthogonal to North-East, it just depends on what you’re measuring (which is part of the whole problem).

  21. Each photon which interacts with hyperion removes some quantum overlap of superposed states, or reduces the off diagonal elements of the density matrix closer to zero. This overlap or entanglement phase that is lost is “smeared out” in a coarse grained sense. Clearly no observer has the ability to make an accounting of all these events and where these entanglement phases are carried off to. This smearing out has a thermal interpretation, which is due to the fact the sun is a heat source.

    Each one of these decoherent events amounts to a little kick on the moon, which is a sort of quantum zeno process of reducing the quantum state of the moon. Each of these little kicks amounts to a stochastic change in the position and momentum (for Hyperion we have action-angle variables) of the body. Each of these little kicks is then exponentially amplified by the chaotic dynamics of the classical system, and the Lyapunov exponent is a measure for this process. So each of these decoherent kicks are amplified and they sum together. This then results in a drunkard’s walk drift in the dynamics of the body, which is ultimately due to underlying quantum stochasticity.

    The process is what underlies the so called wave function collapse of a wave function in a measurement. What is important is not that there be a conscious being which observes the outcome, but rather that there is a thermal scrambling of entangelment phases of the system with those of the environment or measuring apparatus.

    Lawrence B. Crowell