Abstract
Quantum weak measurements with states both pre- and post-selected offer a window into a hitherto neglected sector of quantum mechanics. A class of such systems involves time dependent evolution with transitions possible. In this paper we explore two very simple systems in this class. The first is a toy model representing the decay of an excited atom. The second is the tunneling of a particle through a barrier. The post-selection criteria are chosen as follows: at the final time, the atom remains in its initial excited state for the first example and the particle remains behind the barrier for the second. We then ask what weak values are predicted in the physical environment of the atom (to which no net energy has been transferred) and in the region beyond the barrier (to which the particle has not tunneled). Thus, just as the dog that didnʼt bark in Arthur Conan Doyleʼs story Silver Blaze gave Sherlock Holmes meaningful information about the dogʼs non-canine environment, here we probe whether the particle that has not decayed or has not tunneled can provide measurable information about physical changes in the environment. Previous work suggests that very large weak values might arise in these regions for long durations between pre- and post-selection times. Our calculations reveal some distinct differences between the two model systems.
Highlights
Given that quantum mechanics is several decades old, it is remarkable that a significant sector of the theory lay unexplored until recently
For an individual system the weak value has little meaning, but it i does contain non-trivial statistical information about the world when one considers a large ensemble of identical systems, each prepared in state Ψ
The fact that an operator can be negative seems counter-intuitive to those used to thinking of quantum states in terms of eigenvalues, but it is well-established that such ‘weird’ weak values are commonplace for various post-selections [1, 3]
Summary
Given that quantum mechanics is several decades old, it is remarkable that a significant sector of the theory lay unexplored until recently. For an individual system the weak value has little meaning, but it i does contain non-trivial statistical information about the world when one considers a large ensemble of identical systems, each prepared in state Ψ. In this paper we consider weak values with post-selection of states for systems that are intrinsically time-dependent; that is, they evolve unitarily in time away from a stationary state, so that the time dependence is not merely the result of measurement. Under these circumstances the weak value is given by wA (t) =. Just as the dog that didnt bark in Arthur Conan Doyles story Silver Blaze gave Sherlock Holmes meaningful information about the dogs non-canine environment [13], here we demonstrate that, for each respective case, the particle that has not decayed or the particle that has not tunneled provide measurable information about physical changes in the environment (readers of a later generation may prefer a Rolling Stones analogy: ‘I hear the telephone that hasnt rung.’)
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