- 9.1 Introduction

Decoherence is surely one of the most important, and yet simplest, pieces of the Quantum jigsaw puzzle. The earliest pioneers in the field (David Bohm in 1952 then Hugh Everett in 1957), developed the basic concept while attempting to create “realist” interpretations of QM. Their approaches differed, but both wanted to demonstrate the possibility that the wavefunction could be understood as a real, physical entity, rather than merely as a “state of knowledge”. In order to accomplish this, they had to explain the emergence of Classical Physics, and the appearance of “wavefunction collapse”, in the context of the purely unitary evolution of the wavefunction. Arguably, neither was 100% successful with their intended goal, but they planted the seeds of this idea, decoherence, which has been fleshed out much more thoroughly since then (especially since the early 1980’s). In my opinion, the founders of QM would have discovered the effect almost instantly if they had access to computers capable of numerically solving the Schrodinger equation. For this post, we’re going to look at a problem Schrodinger himself worked on in 1926: the classical limit of the harmonic oscillator. He found a partial solution to the problem, in what he called “coherent states” (which sound similar, but are only incidentally related to our topic, “decoherence”). I say “partial solution”, because although he succeeded in finding a classically behaving solution, he did not explain why this solution would be likely to be found in nature, and his approach failed to carry over to other systems, such as the central potential.

- 9.2 Classical Limit of the Harmonic Oscillator Found, and Lost

Back in the chapter 3, we described how the Schrodinger Equation could be derived from the Hamilton Jacobi Equation by adding another term to the Potential Energy. We called this term the Quantum Potential.

(1)

This implies that Quantum Mechanics approaches Classical Mechanics when this term becomes small, relative to the other energies of the system. The mass appearing in the denominator suggests that the Classical limit might be approached by making the system more massive. Let’s try this with a 1D particle in a harmonic oscillator potential. In figure 1, we see the evolution of a particle with mass 0.1, and with a Gaussian initial condition. The dashed red line indicates the potential energy.

That’s actually not too bad. The motion appears roughly sinusoidal, but the wavepacket spreads out a lot in the middle. To compare this to the Hamilton Jacobi Equation (i.e. classical mechanics), we need to plot the trajectories as we have in earlier chapters. Figure 2 shows the trajectories for the same system, with time on the x-axis.

This makes it clear that the paths are generally not sinusoidal.

Next we’ll redo the calculation with the mass = 1.0 (and I’ve bumped up the spring constant to keep the frequency the same).

That’s looking much better. Almost perfectly sinusoidal. So increasing the mass seems to be working as expected.

So, have we explained the classical limit? Is it enough to just increase mass? Not quite. It turns out that Gaussian initial conditions are special, and when we deviate from them, the picture is not so tidy. As an example, let’s try initializing the wavefunction with a sum of 2 separated Gaussian packets. We’ll plot the Probability Density, then the trajectories.

These trajectories don’t even reach the equilibrium position–very un-classical indeed. And we can see why. The wavepackets run into each other, just like they did in the double slit experiment. The Quantum Potential spikes when that happens, regardless of how much mass the system has. But this suggests we should be able to set up a macroscopic mass on a spring system that bounces around, but never reaches the equilibrium position. Of course this never happens in practice, and we need to explain why.

- 9.3 Classical Limit Restored

How can we fix this? If you’ve read the previous 2 chapters, you can probably guess the answer: adding a second “environment” particle to interact with the first particle can cause the 2 packets to separate. As in chapter 7, we’ll make a 2 particle, 1 dimensional system. The particles will interact with each other via a sharply spiked potential function (i.e. a finite barrier type of potential). Also, the “system” particle will be contained by a harmonic oscillator potential, and the “environment” particle will be free.

To clarify, here’s a plot of the potential energy.

The ridge on the left is going to make the particles repel one another when they get close, and the parabolic shape throughout the length corresponds to the red line drawn in the previous animations. The domain is elongated because we’re giving the environment particle some room to move around. For the initial conditions, we’ll start the system particle exactly the same as the previous example. The environment particle will be a Gaussian initially to the right of the system particle, but moving left, so that they collide. Figure 7 shows how everything evolves.

This animation shows Decoherence in action. The interaction potential (the ridge on the left) is a function of the *distance *between the 2 particles (most, if not all, particle interactions in the real world are like this). This means that spatial separations in one particle (such as our “system” particle) are going to induce different outcomes in the other particle. In this case, one probability blob runs into the barrier before the other one. This causes the system to “decohere”, which means the 2 blobs can’t run into each other after the collision. And hence our system particle now behaves much more classically!

Below is the trajectory plot for the same system (note that unlike the previous trajectory graphs, time is not on the x axis!). Here we see that most of the dots move roughly sinusoidally after the collision takes place. (If it isn’t clear what’s going on, just understand that the vertical position of each blue dot corresponds to the position of the “system” particle, and the horizontal position corresponds to the position of the “environment” particle).

It isn’t perfect, but it’s much more classical looking than the previous, isolated particle example. For a real macroscopic mass-spring system, there is constant bombardment by all sorts of particles (photons, air molecules, etc). Each one causes a bit of decoherence, and the end result is almost perfect classicallity.

- 9.4 Density Matrix View of Decoherence

In the animation below we see the Reduced Density Matrix, in the position basis, for our “system” particle, with the purity calculated for each frame and shown at the top.

The reduction of the purity indicates the 2 particles have become entangled as a result of the collision, and the suppression of the off-diagonal elements indicates that decoherence has occurred (in the position basis). This is simply another way of viewing the same process. Please check out Chapter 8 if you’re unfamiliar with Density Matrices.

For a much more detailed and technical account of this topic, see Max Tegmark’s old paper.

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