A microscopic model of wave-function dephasing and decoherence in the double-slit experiment

The act of measurement on a quantum state is supposed to “dephase” (dephasing refers to the phenomenon that the states lose phase coherence; then the phases get randomized in interaction with a bath of other oscillators, which is referred to as “decoherence”), then “decohere” and “collapse” (or more precisely “register” and “reduce”) the state into one of several eigenstates of the operator corresponding to the observable being measured. This measurement process is sometimes described as outside standard quantum-mechanical evolution and not calculable from Schrödinger’s equation. Progress has, however, been made in studying this problem with two main calculation tools—one uses a time-independent Hamiltonian, while a rather more general approach proving that decoherence occurs under some generic conditions. The two general approaches to the study of wave-function collapse are as follows. The first approach, called the “consistent” or “decoherent”’ histories approach, studies microscopic histories that diverge probabilistically and explains collapse as an event in our particular history. The other, referred to as the “environmental decoherence” approach studies the effect of the environment upon the quantum system, to explain wave-function decoherence which is produced by irreversible effects of various sorts. However, as we know, wave-function collapse is not related to thermal connection with the environment, rather, it is inherent to how measurements are performed by macroscopic apparata. In the “environmental decoherence” approach, one studies decoherence using a Markovian-approximated Master equation to study the time-evolution of the reduced density matrix (post dephasing) and obtains the long-time dependence of the off-diagonal elements of this matrix. The calculation in this paper studies the evolution of a quantum system starting with “dephasing” followed by the effects of the environment with some differences from prior analyses. We start from the Schrödinger equation for the state of the system, with a time-dependent Hamiltonian that reflects the actual microscopic interactions that are occurring. Then we systematically solve (exactly) for the time-evolved state, without invoking a Markovian approximation when writing out the effective time-evolution equation, i.e., keeping the evolution unitary until the end. This approach is useful, and it shows that the system wave-function will explicitly “un-collapse” if the measurement apparatus is sufficiently small. However, in the limit of a macroscopic system, this “dephasing” quickly leads to “decoherence”—collapse is a temporary state that will simply take extremely long (of the order of multiple universe lifetimes) to reverse. This has been attempted previously and our calculation is particularly simple and calculable. We make some connections to the work by Linden et al. while doing so. The calculation in this paper has interesting implications for the interpretation of the Wigner’s friend experiment, as well as the Mott experiment, which is explored in “Connection to some general theoretical results” and “Recurrence times” (especially the enumerated points in “Recurrence times”). The upshot is that as long as Wigner’s friend is macroscopically large (or uses a macroscopically large measuring instrument), no one needs to worry that Wigner would see something different from his friend. Indeed, Wigner’s friend does not even need to be conscious during the measurement that she conducts. It also allows one to reasonably interpret some of the more recent thought experiments proposed. In particular, as a result of the mathematical analysis, the short-time behavior of a collapsing system, at least the one considered in this paper, is not exponential. Instead, it is the usual Fermi-golden rule result. The long-term behavior is, of course, still exponential. This is a second novel feature of the paper—we connect the short-term Fermi-golden rule (quadratic-in-time behavior) transition probability to the exponential long-time behavior of a collapsing wave-function in one continuous mathematical formulation.


APPENDIX I
Consider the geometry shown in Fig. 14.

FIG. 1. Geometry of the Double Slit Experiment
We can compute the wave-function at the point x above the mid-point between the slits. We can think of the slits as the sources of spherical waves and write the wave-function at x as the sum of spherical waves from each slit, i.e., where r 1x , r 2x are as defined in Fig. 14. The k is defined from the electron's momentum k. Note that we have two interesting limits, i.e., d R ∼ x and d R x. We write (see behavior in Fig. 15) With this, we compute We have the limiting forms which we plot (as an example for, additionally k = 20π)

APPENDIX III
In order to produce a microscopic basis for Equation (3), one should look at an elementary process in a very localized volume near a slit, from the following term in the QED action Consider the physics of the situation with the electron. Initially it is delocalized between the slits; post interaction with the photomultiplier's field, it gets localized to the slit 1. The dimensions of the slit is usually compared to the electron's Compton wavelength (see Jonsson al).
Hence, considering Equation (3), the process is that electron comes into a region, with energy ω e , leaves with energy ω e and also causes the emission of a photon of energy ω p . Some energy was added in the process. The energy addition must come through the interaction of the electron with a background field. The background field could be considered very concentrated in the region around slit 1. We also note that for the geometries considered here, ω p ω e , as we only need to localize the electron to within a µ or so with photons, while the electron wave's wavelength is much shorter.
To this end, we consider a second-order process, with a very intense (localized in space around the slit 1) auxiliary field A classical µ which is large but constant in the region of slit 1 and also constant in time. We then get the approximate second-order term below (as well as its hermitian conjugate), with the usual annihilation/creation operators for photons (a, a † ) and electrons (c, c † ), and time averaged over which is what we used (along with its hermitian conjugate) in the rather idealized calculation (Equation 3). It is clear from the above that energy is being added by the background field.