Quantum superpositions tend to decohere to mixtures, due to dissipative environmental influence. In particular, two superposed wave packets loose their relative coherence the faster the larger is their distance . For sufficiently large , the relative phase of packets is lost before any deformation of the shapes of the individual packets and any change of their distance become noticeable.
I shall illustrate the phenomenon by discussing recent efforts to take decoherence under experimental control (diffraction of Fullerenes in Vienna; superpositions of coherent states of microwave resonator modes in Paris; superpositions of wave packets of ions in Paul traps in Boulder; superpositions of states with counterpropagating supercurrents in Delft and Stony Brook).
All of these experiments observe superpositions of packets whose distance is larger than the individual width , but the ratio achieved is still so moderate that the environment imposed decoherence time is, while shorter than the time scale of dissipative changes of , still longer than typical oscillation periods of the isolated system, i.e. . The appropriate theoretical treatment is thus based on Fermi's Golden Rule or, equivalently, Markovian master equations.
I illustrate golden rule type decoherence for the damped harmonic oscillator, using the simple master equation familiar from quantum optics. The important prediction is The underlying perturbative treatment of the system-environment interaction requires the self-consistency condition .
The golden-rule prediction breaks down once the distance between the superposed wave packets is so large, compared to the quantum scale of reference , that . The golden rule can therefore not be invoked to explain the notorious absence of quantum interference effects from the macroworld. In the limit decoherence obviously no longer is a weak-damping phenomenon. A simple solution of the system environment Schrödinger equation becomes possible when , the limit of relevance for superpositions of macroscopically distinct wave packets. The simplicity of that limit rests on the fact that in the full Hamiltonian the free-system part becomes an effectively small perturbation. The decoherence time scale is then found to obey the power law with positive exponents . That law is a universal one, independent of the character of the system and the environment. It is only based on the interaction Hamiltonian additively involving a large number of degrees of freedom such that the central limit theorem holds for the reservoir means met with.
After treating the universal asymptotics of the limit I shall briefly discuss the crossover from that interaction dominated regime to golden-rule type decoherence. The crossover is system specific; I shall rely on an exactly solvable model system, an harmonic oscillator coupled to a reservoir which itself consists of harmonic oscillators.
Textbook wisdom has it that superpositions of eigenstates of some observable of a microscopic object ``collapse'', under measurement of that observable, to a mixture, , with probabilities as in the original pure state but all coherences for gone. Such collapse can be understood as due to unitary time evolution of a tripartite system comprising, besides the micro-object, a macroscopic pointer (idealizable as having a single degree of freedom), and a many-freedom environment. The Hamiltonian must allow micro-object and pointer to become entangled as
with the various pointer states corresponding to macroscopically distinct pointer displacements. Concomitantly, the many-freedom environment will decohere the superposition to the mixture ; inasmuch as the pointer states are macroscopically distinct, the decay of the coherences will occur with , i.e. might appear as practically instantaneous. By disregarding all information about (i.e. tracing over) the pointer, one has the textbook-wisdom mixture for the micro-object.
I shall present exactly solvable models both for both processes involved, entanglement and decoherence.