(Enhanced) notes for a talk (11/98) to the Rice Condensed Matter Theory Group.
Last week, I got mail that said I had a home-page on the Condensed Matter Theory Group Server. That was a nice surprise -- I have to confess that I have never belonged to a Condensed Matter Group; in fact, what's worse, this is the first time I am giving a condensed matter seminar.
As I keep talking today, it should be pretty clear why a) my work is nominally quite far away from what you might think of as condensed matter physics and yet b) how I can convince myself (and I hope by the end of the talk, all of you) that it is nonetheless very relevant to the condensed matter physics community.
1) First I will remind you about chaos a little bit, including the specific property of exponential sensitivity to initial conditions.
2) I will then sketch some of very interesting questions that arise when you try to integrate chaos and quantum mechanics.
3) I will tell you what happens when you open your system up to added noise --
a)first I will describe the impact of noise on classically chaotic systems and then
b) what happens when quantum systems are subject to noise.
4) Finally, I will tell you what happens when you deal with chaos, quantum mechanics and noise all together -- and I will attempt to make my answer more informative than `A Big Mess'.
As I go along, I shall be talking about projects and conjectures as well as results.
Intro : Chaos In the last 30 years or so, the physics community has
made some great strides in understanding the ideas of nonlinear dynamics
and chaos. We've discovered a variety of fascinating phenomena -- far too
many to even mention today -- but one of the universals seems to be that
in most nonlinear point-dynamical systems there is an exponential sensitivity
to initial conditions. By point-dynamics I means something where you can
track a point, or a trajectory.
To be more precise, consider the system:
\dot x = F(x)
Sensitivity to initial conditions means that if you make a small error in specifying the initial state of the dynamical system, this error grows exponentially rapidly for a chaotic system and slower than exponential, or not at all, for a `regular' system.
Equivalently, computing a solution within a given accuracy becomes exponentially harder with increasing times in a chaotic system -- as opposed to polynomial at worst in a regular system.
This idea of exponential sensitivity is quantified by the largest Lyapunov exponent of your dynamical system. This is given by, loosely:
\lambda = lim_{t \to \infty} 1/t ln (\delta x(t)/\delta x(0) If this is positive definite, then the system is chaotic, else it is zero and the system is non-chaotic or regular.
Note, however, that in a typical dynamical system there is a mixture of both regular and chaotic behavior -- there are usually dense sets of initial conditions which give regular behavior neighboring sets which give chaotic behavior. The phase-space of a typical nonlinear dynamical system is actually a mixed space. This is particularly true of Hamiltonian systems (and I am going to stick to Hamiltonian systems from now on -- except that I will add noise to them.)
It is now well established that dynamical systems on almost all scales are chaotic: ranging from the cosmological through the macroscopic world of table-top physics and even lower.
However, this is only true of systems which can be described by classical mechanics. When we consider systems on the scales at which classical mechanics begins to break down and where we must incorporate quantum effects, it turns out that classical chaos does not translate easily to the quantum world. I want to make it clear here that I am going to stick to the world of Schr\"odinger's equation and perhaps of many-body versions of this but not go much `lower'.
This realm of the classical-quantum world is a fascinating world and I don't need to talk about its relevance to us at Rice -- in fact, to almost everyone in this room here, as far as I can tell.
The systems I am talking about are quantum mechanical, but yet large enough that they are affected by classical things (or else we are forced to use semiclassical and other approximations because the exact quantum calculation is intractable).
What happens when quantum mechanics and chaos bump up against each other ?
I'll tell you quickly about some of the questions I am NOT going to develop and then about ideas that I WILL discuss.
Fundamental:
Quantization -- or Einstein's question
: Given that semiclassical quantization only works for integrable classical
systems, how do we semiclassically quantize a system that is classically
chaotic ?
-- Remember that the Bohr model only works for closed orbits. If you have these chaotic nonclosing trajectories, how do you quantize ? The most famous answer: The Gutzwiller Trace Formula. This says that you have to find all the unstable periodic orbits in your chaotic system and then do a weighted sum of their actions. As such, this is a somewhat 'closed' question.
Further, it can be formally shown that quantum mechanics is not chaotic.
-- Since Schrodinger's equation is a linear partial differential equation we can explictly show that there is no time-dependence of initial differences in wave-functions, let alone exponentially rapid growth of differences.
Also, if you consider a bound-state Hamiltonian, you can show that because you have a discrete spectrum in quantum mechanics, you always get quasi-periodic motion for all expectation values.
However, chaotic and non-chaotic systems have entirely different properties.
Approximations: Very practically, therefore, can how can we develop semiclassical or other approximations to intractably large quantum calculations, and when can we trust them in the presence of chaos ?
-- The answer here is full of surprises. If you are reasonably experienced with chaos, then your gut reaction is that of course any approximation you make is going to break down -- break down exponentially fast, in fact. There are many papers pointing this out, by now.
As a variation on this, one of the results that I worked out with Bill Schieve was that there are standard (mean-field or Hartree like) approximations to quantum mechanics that can show chaos even when the classical systems is not chaotic (and clearly the quantum system is not chaotic either!).
This is very disturbing -- what's perhaps even more disturbing is the number of papers that have emerged since then that are trying to work out the details of this kind of mean-field chaos. It may be useful, but I think they are chasing a chimera.
However, immediately after I worked out that you shouldn't trust this Gaussian approximation, I managed to work out, to my surprise/satisfaction/dismay, that if I used the same approximation in a classically completely chaotic system, I could analytically calculate extremely accurate quantum eigenvalues -- in a situation where large computers were groaning to calculate these same eigenvalues.
I have since been able to use the Gaussian approximation very usefully to understand Bose-Einstein Condensation -- as before, it seems to give very useful qualitative results, even when the details were not quite right.
What gives: When is it safe to use an approximation and when is it going to give you nonsense ? Still an open question. There are some ideas out there, but they are in very raw shape at this point, so ..
Correspondence: How does classically chaotic dynamics emerge from quantum mechanics?
This is a question that is philosophical and yet potentially very practical.
Theoretically, if we make Planck's constant small enough, quantum mechanics
becomes classical mechanics -- what does this mean for a real system and
what are the interesting features of this transition ?
-- This is still a pretty wide-open question. Of course, you do not get to tune hbar. However, we know that changing other parameters can change the effective size of hbar (as Mark Raizen is doing at UT). What happens when you do this ? Clearly, the transition is going to depend upon the hbar, but also the nonlinearity and finally the amount of noise in the system (more later).
I'll have a more detailed example of this later in the talk.
Chaology
Finally, of course, what can we say about the physical and experimental
properties of quantum mechanical systems if we know that their classical
approximation is chaotic ?
Many beautiful problems and results in this field, called quantum chaology if you like.
There is no time to review all of the interesting stuff here, but let me at least throw out some of the buzzwords so that you can come ask me about this later: The conjecture that the quantum spectrum of chaotic systems is given by Random Matrix Theory, The issue of dynamical localization in atom-traps, microwave ionization of Hydrogen (+ other Rydberg atoms), the shape and properties of quantum chaotic eigenfunctions (scarring), The conductance properties of nano-structures, the decoherence rate of quantum systems.. etc.
A specific example: The emergence of classical chaos from quantum dynamics and the characterization of chaos in distribution dynamics ... with applications in Stochastic Resonance and Decoherence.
Classical mechanics as we usually study it involves looking at point dynamics or trajectories (Newton's or Hamilton's equations). On the other hand, quantum mechanics is inherently probabilistic and we have to look at Schrodinger's equation.
An useful way of doing a direct comparison of the dynamics so that we can answer some of the questions of quantum chaos is to compare the behavior of classical probability (Liouville) distributions and quantum phase-space distributions, like the Wigner function, for example.
However, when you do this, you land up with the following problem -- this had been a mystery for 10-15 years ... till recently:
Since the Liouville equation, like the Schrodinger equation, is also a linear partial differential equation, there is no separation of distributions either, and seemingly no chaos.
In that case, what dynamical property of distributions do you look at to understand chaos, and in particular, what is the role of Lyapunov exponents in the dynamics of distributions ?
If we could solve this problem, we could then apply the diagnostic to quantum distributions and be able to see if maybe our arguments about lack of chaos in the quantum system was wrong, and if not, quantify where and how quantum dynamics deviates from classical chaos.
Here is a recent solution to this problem (Pattanayak and Brumer, PRE ..):
Let us look at the behavior of distributions carefully, in particular remembering one important quality of Hamiltonian dynamical systems: phase-space volume conservation.
Suppose we put down a blob of phase-space probability and watch it evolve, it changes shape as it moves. However, since probability can neither be preserved nor destroyed, as the blob changes shape, the volume (or in this projection the area) that it covers remains the same.
In particular, if this is a chaotic system, remember that nearby trajectories must locally exponentially diverge, implying that the blob must get exponentially stretched in that direction. Since the area is preserved, if there is exponential stretching going on in one direction, there must be exponential squeezing going on in the other direction.
But look what happens if we look at this blob being stretched and squeezed from the side: as the distribution gets stretched and squeezed, the gradient (the slope) of the distribution gets conversely squeezed and stretched. This makes physical sense if you think about it for long enough.
So now we have that the gradient of the distribution if tracked along a trajectory has an exponentially rapid increase given by the Lyapunov exponent; but we are still using trajectories and we want to get rid of trajectories, remember.
That turns out to be done in the following manner: if we average this gradient over the entire distribution, we get a quantity that does not depend on trajectories at all, and also increases exponentially rapidly with time.
This quantity that increases exponentially rapidly we call \chi_2 and the rate at which it increases is a generalized Lyapunov exponent we call \lambda_2. What is \chi_2 and does it make physical sense that it increases this rapidly ?
Yes, it makes sense if we look at this in Fourier space. If we take the probability distribution and expand it in a Fourier series, \chi_2 is nothing but the root-mean-square Fourier radius of the distribution. What does it mean that this is growing expoentially rapidly?
Well, higher Fourier mode numbers in a Fourier expansion correspond to more wiggles. Therefore, the prediction of increasing \chi_2 implies that the distribution is getting more and more wiggly (structured) as the result of chaos. In fact, this is exactly what does happen in a chaotic Hamiltonian system -- though this was the first quantitative statement about the connection.
Let me show you an example in a system called the Cat Map, which is one of the paradigms of a highly chaotic system.
What you are seeing is gray-scale contours of the time evolution of a classical Gaussian.
(Chaotic pictures). Structure grows exponentially rapidly.
Remember also that we usually have finite resolution in phase-space. This implies that the Fourier space is finite. So, with this exponential explosion of the Fourier expansion, the information about the distribution leaves any finite range in Fourier space exponentially rapidly.
[show sketch in Fourier space here; in particular show what happens to a mixed chaos-regular system.]
You can now proceed to make all sorts of interesting statements about the growth of entropy and other things, for example.
Most importantly, note that accurate computation becomes exponentially harder, just like the behavior of trajectories!
Let's now try to apply it to quantum systems. It turns out to be easy enough to quantize the criterion. In fact, when we do this we get that in the Wigner-Weyl representation, it actually has exactly the same form as the classical criterion. So we can now test it in a quantum system and see how it behaves. We applied this test to a particular system, the quantum cat map, and these are our results.
What I am showing you here is the impact of scaling the size of our quantum system such that relevant actions change and hence the effective \hbar = \alpha of the system is changing.
First what I am showing you is what the phase-space distribution looks like in the quantum system as a function of time; I am then going to show you what happens as we tune hbar so that the system goes from being more classical to being more quantal.
1) Look at the increasing structure
2) Look at classical chaos plus quantal interference effects.
3) Look at purely quantal behavior -- very different from classical chaos.
So now we understand, to some extent, the way in which classical chaos emerges from quantum non-chaos and we can even quantify this statement.
Look at the way \chi_2 behaves.
Note that 1) \chi_2 is essentially classical for \alpha small enough, and then that 2) it increases faster than classical as \alpha is increased and finally that 3) it barely grows at all for \alpha large enough.
Also note that 4) \chi slows down at \chi = O (1/\alpha). This is in keeping with the intuition that quantum mechanics resists the growth of structure below \hbar. Actually, we can make the argument more rigorous than that -- we can explicitly show that \chi scales as 1/\hbar and that this 1/hbar limit is in fact where classical and quantum mechanics should part company.
Okay.
First, of course, what do I mean by noise ? When talking about quantum mechanics and chaos, this is actually a non-trivial question.
A good microscopic formulation is the Caldeira-Leggett version of Feynman-Vernon formalism:
Write down a Hamiltonian for a system coupled to an `environment' (modelled as an infinite set of harmonic oscillators). Write down the coupled system of equations for the evolution of the system+environment and then average over the environment variables. This sweeps some assumptions about average properties of the environment under the rug and so on, but in general it seems to be a robust approximation.
Here is one of the standard equations we get when we do this, for the evolution of the density of the system (note that what is left after averaging is usually called the `reduced' density matrix).
As you can see, it is basically the Hamiltonian evolution equations for the density matrix, plus a term that looks like a Laplacian diffusion term.
Note that if you do a Fourier solution of this, you find that high-number Fourier coefficients disappear rapidly while the others stay around longer.
This is going to be crucial in understanding what happens when you add noise to a chaotic system.
Again, the instinctive response is that chaos amplifies the small fluctuations (noise) and in fact does so exponentially fast. Again, there are a fair number of papers on that and it is quite correct.
However, this is a short-time effect. There is another interesting consequence: Increasing the noise actually decreases the relative weight of the chaotic components. How does this happen ?
Remember what happens when you have a mixed chaos + regular system -- the chaotic elements rush away from the Fourier space origin exponentially fast and the regular elements stick around. Well, what the noise does is to kill the chaotic parts, since they are now out in large Fourier number space.
The effect on smaller Fourier number elements is much less, and that's why I claim that added noise can enhance the relative ratio of regular to chaotic behavior.
In short, I believe that I have an argument for a general mechanism for so-called `Stochastic Resonance'. This is one of my projects .. so my claims need to be firmed up yet.
Among other things, you get what is called decoherence. What's decoherence ?
Decoherence is the loss of quantal coherence. So now I have to tell you what I mean by coherence.
Coherence is a property of quantum systems that is related to their ability to demonstrate remarkable interference effects, for instance, like in the double-slit interference of electrons, or in many of the quantum systems that are being implemented as technology (coherent control, for example).
It comes about from the fact that the dynamics of a quantum systems are given by a state-ket, but to compute the probability we have to square it to get a probability density.
This implies that you can have 50% probability of one event and 50% probability of the other but when you propagate this forward, the probabilities need not always add directly, but may constructively or destructively interfere.
Classical probabilities are, on the other hand, statistical: That is, there cannot be any subtraction between them, only addition. Quantum systems can also be in statistical distributions; in that case we say that the system is in a mixed state and cannot be represented as a state \psi but must be seen as statistical mixture of states in a density matrix.
Any physical process that takes us from a pure state to a statistical mixture of states is what I am calling decoherence.
How does decoherence happen ? Obviously, it cannot happen from pure Hamiltonian evolution of a distribution. The formal way of understanding decoherence is through the idea of entanglement -- different branches of the system wavefunction get associated with different branches of the environment.
A crude and perhaps the most intuitive way of representing this is to say that decoherence happens when a quantum system is jostled by a random/noisy force such that the interference effects wash out.
To monitor how this equation decreases the decoherence in a quantum system, we use the so-called linear or Renyi entropy: This is maximum for a pure state and so has been argued to be a measure of the degree of purity of a quantum state.
It is easy to show that the rate of change of this Renyi entropy is exactly -D\chi^2. That is, the more highly structured the distribution, the more susceptible it is to noise -- not surprisingly.
Therefore, decoherence in the quantum system and chaos in the classical system are closely linked.
Some of the consequences of this observation:
For slow decoherence, minimize \chi_2^2. For energy eigenstates (stationary states) \chi_2 is time-independent. In general, lower energy eigenstates have smaller \chi_2^2 so decohere slower.
For chaotic systems (semiclassical limit, small D) {dS\over dt} = - 2D\chi_2^2(0)[\exp(2\lambda_2t) + C]
Remember that initially, the first quantal correction is that chi^2 {\em increases} with \hbar because of increasing impact of quantal interference.
Second quantal effect is suppression of growth of \chi_2^2 (quantal suppression of chaos).
This implies decoherence maximally rapid at finite \hbar.