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New Journal of Physics

Reply to comment on ‘non-monotonic projection probabilities as a function of distinguishability’

Gunnar G. Bjork, Saroosh Shabbir

Published Nov 28, 2014• 0 citations
Publisher

Abstract

We reiterate that a non-monotonic behavior of projection probabilities can occur for most states irrespective of the number of particles involved, based solely on the choice of the projectors chosen for measurement. We further clarify that, contrary to the claim in the comment, our analysis is not intended to demonstrate quantum to classical transition. However, our results show that a non-monotonic behavior is not a necessarily a signature of such a transition.

DOI
10.1088/1367-2630/16/11/118004

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First detection probability in quantum resetting via random projective measurements

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Abstract We provide a general framework to compute the probability distribution F r ( t ) of the first detection time of a ‘state of interest’ in a closed quantum system subjected to random projective measurements. In our ‘quantum resetting’ protocol, resetting of a state is not implemented by an additional classical stochastic move, but rather by the random projective measurement. We then apply this general framework to Poissonian measurement protocol with a constant rate r and demonstrate that exact results for F r ( t ) can be obtained for a generic two level system. Interestingly, the result depends crucially on the detection schemes involved and we have studied two complementary schemes, where the state of interest either coincides or differs from the initial state. We show that F r ( t ) at short times vanishes universally as F r ( t ) ∼ t 2 as t → 0 in the first scheme, while it approaches a constant as t → 0 in the second scheme. The mean first detection time, as a function of the measurement rate r, also shows rather different behaviors in the two schemes. In the former, the mean detection time is a non-monotonic function of r with a single minimum at an optimal value r ∗ , while in the later, it is a monotonically decreasing function of r, signaling the absence of a finite optimal value. These general predictions for arbitrary two level systems are then verified via explicit computation in the Jaynes–Cummings model of light–matter interaction. We also generalize our results to non-Poissonian measurement protocols with a renewal structure where the intervals between successive independent measurements are distributed via a general distribution p ( τ ) and show that the short time behavior of F r ( t ) ∼ p ( 0 ) t 2 is universal as long as p ( 0 ) ≠ 0 . This universal t 2 law emerges from purely quantum dynamics that dominates at early times.

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We reiterate that a non-monotonic behavior of projection probabilities can occur for most states irrespective of the number of particles involved, based solely on the choice of the projectors chosen for measurement. We further clarify that, contrary to the claim in the comment, our analysis is not intended to demonstrate quantum to classical transition. However, our results show that a non-monotonic behavior is not a necessarily a signature of such a transition.
A recent work (Björk and Shabbir 2014 New J. Phys. 16 013006 ) claims that nonmonotonic structures found in the many-particle quantum-to-classical transition (Ra et al 2013 Proc. Natl Acad. Sci. USA 110 1227–31 ; Tichy et al 2011 Phys. Rev. A 83 062111 ) are not exclusive to the many-body domain, but they also appear for single-photon as well as for semi-classical systems. We show that these situations, however, do not incorporate any quantum-to-classical transition, which makes the claims unsustainable.
Typically, quantum superpositions, and thus measurement projections of quantum states involving interference, decrease (or increase) monotonically as a function of increased distinguishability. Distinguishability, in turn, can be a consequence of decoherence, for example caused by the (simultaneous) loss of excitation or due to inadequate mode matching (either deliberate or indeliberate). It is known that for some cases of multi-photon interference, non-monotonic decay of projection probabilities occurs, which has so far been attributed to interference between four or more two photons. We show that such a non-monotonic behaviour of projection probabilities is not unnatural, and can also occur for single-photon and even semiclassical states. Thus, while the effect traces its roots from indistinguishability and thus interference, the states for which this can be observed do not need to have particular quantum features.
In a series of comments, Bonder et al. criticized our work on decoherence due to time dilation [Nature Physics 11, 668-672 (2015)]. First the authors erroneously claimed that our results contradict the equivalence principle, only to "resolve" the alleged conflict in a second note. The resolution - relativity of simultaneity - was already explained in our reply [arXiv:1508.03296], which Bonder et al. now essentially reiterate. The newly raised points were also already extensively clarified in our note. The physical prediction of our work remains valid: systems with internal dynamics decohere if the superposed paths have different proper times.
Abstract We provide a general framework to compute the probability distribution F r ( t ) of the first detection time of a ‘state of interest’ in a closed quantum system subjected to random projective measurements. In our ‘quantum resetting’ protocol, resetting of a state is not implemented by an additional classical stochastic move, but rather by the random projective measurement. We then apply this general framework to Poissonian measurement protocol with a constant rate r and demonstrate that exact results for F r ( t ) can be obtained for a generic two level system. Interestingly, the result depends crucially on the detection schemes involved and we have studied two complementary schemes, where the state of interest either coincides or differs from the initial state. We show that F r ( t ) at short times vanishes universally as F r ( t ) ∼ t 2 as t → 0 in the first scheme, while it approaches a constant as t → 0 in the second scheme. The mean first detection time, as a function of the measurement rate r, also shows rather different behaviors in the two schemes. In the former, the mean detection time is a non-monotonic function of r with a single minimum at an optimal value r ∗ , while in the later, it is a monotonically decreasing function of r, signaling the absence of a finite optimal value. These general predictions for arbitrary two level systems are then verified via explicit computation in the Jaynes–Cummings model of light–matter interaction. We also generalize our results to non-Poissonian measurement protocols with a renewal structure where the intervals between successive independent measurements are distributed via a general distribution p ( τ ) and show that the short time behavior of F r ( t ) ∼ p ( 0 ) t 2 is universal as long as p ( 0 ) ≠ 0 . This universal t 2 law emerges from purely quantum dynamics that dominates at early times.