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Physical Review A

Heisenberg limit for displacements with semiclassical states

Alfredo Luis

Published Apr 1, 2004• 10 citations
Publisher

Abstract

We analyze the quantum limit to the sensitivity of the detection of small displacements. We focus on the case of free particles and harmonic oscillators as the systems experiencing the displacement. We show that the minimum displacement detectable is proportional to the inverse of the square root of the mean value of the energy in the state experiencing the displacement (Heisenberg limit). We present a measuring scheme that reaches this limit using semiclassical states. We examine the performance of this strategy under realistic practical conditions by computing the effect of imperfections such as losses and nonunit detection efficiencies. This analysis confirms the robustness of this measuring strategy by showing that the experimental imperfections can be suitably compensated by increasing the mean energy of the input state.

DOI
10.1103/physreva.69.044101

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We analyze the quantum limit to the sensitivity of the detection of small displacements. We focus on the case of free particles and harmonic oscillators as the systems experiencing the displacement. We show that the minimum displacement detectable is proportional to the inverse of the square root of the mean value of the energy in the state experiencing the displacement (Heisenberg limit). We present a measuring scheme that reaches this limit using semiclassical states. We examine the performance of this strategy under realistic practical conditions by computing the effect of imperfections such as losses and nonunit detection efficiencies. This analysis confirms the robustness of this measuring strategy by showing that the experimental imperfections can be suitably compensated by increasing the mean energy of the input state.
The phase sensitivity of interferometers is limited by the so-called Heisenberg limit, which states that the optimum phase sensitivity is inversely proportional to the number of interfering particles NNN, a 1∕N1∕\sqrt{N}1∕N​ improvement over the standard quantum limit. We have used simulated annealing, a global optimization strategy, to systematically search for quantum interferometer input states that approach the Heisenberg-limited uncertainty in estimates of the interferometer phase shift. We compare the performance of these states to that of other nonclassical states already known to yield Heisenberg-limited uncertainty.
Quantum enhancements of precision in metrology can be compromised by system imperfections. These may be mitigated by appropriate optimization of the input state to render it robust, at the expense of making the state difficult to prepare. In this paper, we identify the major sources of imperfection of an optical sensor: input state preparation inefficiency, sensor losses, and detector inefficiency. The second of these has received much attention; we show that it is the least damaging to surpassing the standard quantum limit in a optical interferometric sensor. Further, we show that photonic states that can be prepared in the laboratory using feasible resources allow a measurement strategy using photon-number-resolving detectors that not only attain the Heisenberg limit for phase estimation in the absence of losses, but also deliver close to the maximum possible precision in realistic scenarios including losses and inefficiencies. In particular, we give bounds for the tradeoff between the three sources of imperfection that will allow true quantum-enhanced optical metrology