Self-induced decoherence approach: Strong limitations on its validity in a simple spin bath model and on its general physical relevance

The “self-induced decoherence” (SID) approach suggests that (1) the expectation value of any observable becomes diagonal in the eigenstates of the total Hamiltonian for systems endowed with a continuous energy spectrum, and that (2) this process can be interpreted as decoherence. We evaluate the first claim in the context of a simple spin bath model. We find that even for large environments, corresponding to an approximately continuous energy spectrum, diagonalization of the expectation value of random observables does in general not occur. We explain this result and conjecture that SID is likely to fail also in other systems composed of discrete subsystems. Regarding the second claim, we emphasize that SID does not describe a physically meaningful decoherence process for individual measurements, but only involves destructive interference that occurs collectively within an ensemble of presupposed “values” of measurements. This leads us to question the relevance of SID for treating observed decoherence effects.

Figure 1

Plot of ${\log }_{10}\mid r\left(t\right)\mid$, with the decoherence factor $r\left(t\right)$ given by Eq. (14), for two different bath sizes $N=20$ and 100. Fast decay of $r\left(t\right)$, corresponding to local decoherence, is observed, and the degree of decoherence is seen to increase with $N$. The squared coefficients ${\mid {\alpha }_i\mid }^2$ and the couplings $g_i$ were drawn from a uniform random distribution over the intervals [0, 1] and $[-\pi ,\pi ]$, respectively.

Figure 2

Time evolution of ${\log }_{10}\Lambda \left(t\right)$ [see Eq. (26)] for $N=100$ bath spins, and for three different random observables and initial states of the environment. The function $\Lambda \left(t\right)$ quantifies the time dependence of the terms in the expectation value ${⟨\widehat{O}⟩}_{\Psi \left(t\right)}$ [Eq. (19)] that are not diagonal in energy. Suppression of these terms is represented by a decay of $\Lambda \left(t\right)$ from its initial value of one [i.e., ${\log }_{10}\Lambda \left(t\right)\to -\infty$]. It is observed that in the general case (A) of completely random observables and initial states of the environment, collective decay of off-diagonal terms does, in general, not occur. However, if the phases of the coefficients describing the observable and the environment are moderately restricted (B) or all set equal to zero (C), decay of increasing strength is found.

Figure 3

Time evolution of ${\log }_{10}\Lambda \left(t\right)$ for $N=100$ bath spins [see Eq. (26)] when no restrictions on the initial state of the environment are imposed. Such restrictions are physically unrealistic and require a preparing measurement on the unrestricted environment, which would in turn be in conflict with the desired generality of the derivation of decay effects. No collective decay of off-diagonal terms is observed, regardless of any restrictions imposed on the observable.

Figure 4

Example for the time evolution of ${\log }_{10}\Lambda \left(t\right)$ [see Eq. (26)] using a random observable and random initial bath state, for bath sizes between $N={10}^2$ and ${10}^6$, and for a long time scale $t=0–{10}^6$. No connection between the size of the spin bath and the occurrence and the degree of damping is observed. Therefore, no consistent collective decay of interference terms occurs. The increased time scale is seen to be irrelevant.