Progressive quantum collapse

Abstract

Instantaneous collapse of the wave function upon measurement of a single particle is one of the postulates of the Copenhagen interpretation of quantum mechanics. However, what happens when a many-body system in a macroscopic coherent state is measured one particle at a time? Here, we consider successive measurements of individual spins from a spin Bose condensate that starts in a Schrödinger cat state. When the spin measurements are done one particle at a time, the collapse of the spin condensate is not instantaneous but leads to probabilities for spin measurement that strongly depend on the previous measurements. What is surprising is that an almost complete collapse occurs in very few measurements. Even in a large system, a single cat component is emphasized quite quickly in the sequence of measurements. We examine the process by analysis of a simple two-Fock-state cat, as well as a cat state that has many components. Justification is given for our theoretical measurement process.

Accuracy of the Gross-Pitaevskii Equation in a Double-Well Potential

“…This form of fragmentation is not original to this paper; for example, it appears in Refs. [7], [30], and [31] but we thought the form unusual enough to be worth pointing out.…”

“…( 20) is fragmented with 38 particles in the ground symmetric state ψ 0 [Eq. (7)] and 12 in the antisymmetric states ψ 1 [Eq. ( 8)].…”

“…Reference [7] showed that if one made measurements one particle at a time where each measurement determined which of the condensate states the particle occupied, in this case ψ + or ψ − , then after a rather small number of measurements the cat state would collapse to just one of the peaks. Is such a cat state, in this case involving asymmetric states, more capable of being formed than other kinds?…”

“…These have been considered in, for example, Refs. [7,30,31]. When we expanded the Fock state in a two-mode approximation we used the ground ψ 0 and first excited state ψ 1 of the non-interacting potential.…”

The Gross-Pitaevskii equation for a infinite square-well with a delta-functionbarrier

“…This solution satisfies the GPE with the same conditions of Eqs. ( 5) and (6). The condition on the wave function derivatives at…”