What is Decoherence?

It is best to introduce decoherence with a simple example. Before doing so, we need to have something to decohere. The job of decoherence is to bring a quantum system into an apparently classical state. What especially differentiates a quantum system from a classical system is the concept of a superposition of states. In the classical realm of physics, we would say that a particle is at a position (x,y,z). In the quantum realm, the formalism allows us to state that a particle is in a superposition of positions (say, $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ ). However, as the canon of quantum mechanics postulates, when we actually measure the position of this particle, we will find it at either of the two positions, that is, we will have ``collapsed the wave function'' into one or the other state. A classical mind (which we all have) would ask, what does this mean? This is the so called ``measurement problem''. David Albert puts it better when he writes,

The dynamics and the postulate of collapse are flatly in contradiction with one another ... the postulate of collapse seems to be right about what happens when we make measurements, and the dynamics seems to be bizarrely wrong about what happens when we make measurements, and yet the dynamics seems to be right about what happens whenever we aren't making measurements. [2]

Let us be more explicit about what he means here. It was von Neumann [12] who first articulated the so called ``dynamical dualism'' that haunted the original formulations of quantum mechanics, though Bohr touched on the issue earlier in proposing the ``quantum leap'' into states which, like the wave collapse, is proposed as a dynamically discontinuous process. Primarily, the evolution of a quantum system is described by the Shrödinger equation,

$$i\hbar \frac{\partial}{\partial t}\vert\psi\rangle = H\vert\psi\rangle,$$ (1.1)

applying to a system under isolation. There is also the evolution that occurs due to measurement, and this is the infamous collapse:

$$\vert\psi\rangle = \sum_n c_n\vert n\rangle \ \longrightarrow \vert n_i\rangle$$ (1.2)

Technically, Equation 1.2 is called by von Neumann the ``first intervention'' and Equation 1.1 is the ``second intervention.'' The first intervention is what we will identify as the wave collapse. It describes a superposition of states suddenly `collapsing' into one eigenstate, the measurement.

If theoreticians could provide a mechanism within the context of quantum mechanics in which the evolution superficially described by von Neumann's first intervention is achieved, without the need for an ad hoc mechanism or the insistence that we live in a universe in which ``consciousness or spirit...play an important and fundamental role'' in physical phenomenon [16], they will have made great progress in solidifying quantum mechanics (even further) as a sufficient theory. By this we mean that the protests that quantum mechanics is somehow incomplete (e.g. the EPR paradox [6]) can be better addressed (beyond Bell's theorem).

Given the substantial success of quantum mechanics in correctly predicting the outcome of every experiment thus far conceived and executed to test it, we would prefer that any mechanism designed to describe what occurs when a measurement is taken does so in the context of standard quantum mechanics, which is to say, does not require a modification of the Shrödinger equation.

The alternative is to accept a modification of the standard formulation of quantum mechanics. There are many various ``collapse models'' one could consider ( [10], see also [11] as a short, reasonable example). And though it serves a physicist well to consider fresh, new ideas, it also serves us well to realize that such ideas are almost always wrong. In any case, these alternatives are not the focus of this report.

Decoherence offers a theoretical framework in which the measurement problem can be swept under the carpet (pushed into a system larger than that which we can observe). The effect is that quantum mechanics can be studied and presented to a student without the need for the ad hoc ``wave collapse'' being presented as a primary tool of the theory. One can achieve, in many cases, the same apparent effect of a wave collapse without recourse to von Neumann's mysterious first intervention.

Thus we clarify that decoherence is not a new theory unto itself, but is instead an efficient and fruitful repackaging of theory. It does not solve the measurement problem, and most certainly wouldn't have satisfied the reservations of Einstein in his later years. Nevertheless, given its elegance in providing an apparent transition from the quantum realm to the classical realm, and its experimental success, we believe the time has come that decoherence be incorporated into graduate level quantum mechanics courses. This report is designed to be a self-contained introduction to the topic appropriate for a graduate student.

Our presentation will be fairly more detailed than that typically found in the literature [8,9] and by necessity will serve as minor introduction to quantum optics (the reader is also referred to the appendix as needed).