Multidimensional Tunneling

    If you have had quantum mechanics then you have worked out the problem of tunneling through a one dimensional potential (perhaps square) barrier and computed the probability of transmission and reflection. It is also probable that the class/book wherein you learnt about tunneling told you that tunneling was a purely quantum phenomenon. In other words classical mechanics cannot explain tunneling.

    For people that have not heard of this before let me give a simple minded introduction. Imagine throwing a ball at a fence. Now, if the ball hits the fence then it bounces back and if you throw it fast and high enough it will clear the fence. Similarly, imagine a classical particle coming from the left towards a potential barrier. Suppose that the total energy (kinetic + potential) of the particle is less than the maximum height of the barrier. I am sure we all agree that the particle is going to bounce back from the barrier since it has insufficient energy to get over it. This is what classical mechanics would say and the question is what would quantum mechanics say for the same system? Surprisingly, according to quantum mechanics there is a very small but finite probability that the particle could be found to the right of the barrier! This phenomenon and various other generalizations is called as tunneling. Tunneling is rather important in many situations arising in fields ranging from Biology to modern computers. Hence, there are good reasons to try to understand tunneling and its effect on systems.

    So at the outset it seems as though tunneling is a purely quantum structure and "classically forbidden". That seems to be the end of the story. Is tunneling really an alien concept in classical mechanics? We can atleast ask, in the spirit of quantum-classical correspondence, as to what possible classical structure could give rise to this notion. It turns out that there is such a structure in classical mechanics contained in the complex time solutions to the classical equations of motion or the Hamilton-Jacobi equation. But what does one "mean" by complex time? Isn't time always real? Well, I will let you think about it...but remember, we seem to be perfectly happy with the notion of a complex wavefunction in quantum mechanics! The principle of stationary action introduced by Lagrange, Hamilton, Jacobi and others states that a trajectory is a classical trajectory of the system if the first variations of the action functional vanishes. This leads to differential equations with time as the independent parameter. In general we can find perfectly valid classical solutions in the complex time domain! So there is nothing nonclassical about a complex time classical trajectory. We can take families of such classical tarjectories and get tunneling out of it!

    Take a one dimensional parabolic barrier (that is an inverted harmonic oscillator) and solve for the equations of motion. Now make time complex and you will find that there do exist classical solutions which start out on the left side of the barrier (with total energy less then the maximum barrier height) and end up on the right side!

    As it happens, apart from a quantum-classical correspondence perspective there is also a practical perspective. Remember, we would like to actually compute some of these very small (exponentially small) numbers. That means we have to work relatively hard to get them accurately. If we have a very high dimensional system wherein tunneling is important (depends on various external parameters like temperature, solvents etc.) then we are faced with performing quantum mechanics which is quite a task! Also on such systems researchers usually perform classical trajectory simulations and thus it is useful to know how to go about including tunneling effects in the simulations. My doctoral work was involved in coming up with methods to account for tunneling using classical dynamics.

    In one dimension there are a variety of methods to calculate tunneling amplitudes. A method which has had success in d = 1 is the semiclassical WKB (Wentzel-Kramers-Brillouin) approach. Unfortunately, WKB has not been generalized to multidimensional, nonseparable systems in a satisfactory manner as of yet. In my opinion, this has to do with a lack of understanding of the so called Stokes phenomenon. When dealing with an asymptotic theory such as WKB, it is very crucial to properly generalize the one dimensional analog of Stokes rules. Otherwise the attempts at generalizations will be futile. The complex time idea is again fine in d =1 and raises some subtle questions for nonintegrable systems. For example, analytically continuing classical dynamics into complex time plane for nonintegrable systems reveals the existence of natural boundaries. What are the implications of nontrivial singularity structures on the correspondence problem? Is the answer for the quantum-classical correspondence problem for nonintegrable, mixed system lurking around in the complex time plane? There are quite a few unanswered questions and what we know seems just to be the tip of the iceberg.