Quantum states for Heisenberg-limited interferometry

H. Uys and P. Meystre

Phys. Rev. A 76, 013804 – Published 9 July 2007

Abstract

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 $N$ , a $1 ∕ \sqrt{N}$ 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.

Received 15 February 2007

DOI:https://doi.org/10.1103/PhysRevA.76.013804

Authors & Affiliations

H. Uys and P. Meystre

Department of Physics and College of Optical Sciences, The University of Arizona, Tucson, Arizona 85721, USA

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Issue

Vol. 76, Iss. 1 — July 2007

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Images

Figure 1
(Color online) Schematic of a Mach-Zehnder interferometer, with relative phase shift $ϕ$ resulting from propagation of an input field through its arms.Reuse & Permissions
Figure 2
(Color online) (a) Probability amplitudes $α_{m}$ for the twin-Fock state (solid black line), the external $N 00 N$ state (green dotted-dashed line), the internal $N 00 N$ state Eq. (31) (solid gray line), and the di-Fock state $∣ ψ_{di} (q = 1) ⟩$ (red dotted line). Corresponding likelihood functions for (b) $(M = 1)$ and (c) $(M = 10)$ . In the di-Fock state secondary peaks are already almost completely absent for $M = 1$ , thus concentrating more probability density around $ϕ = 0$ . Secondary features are however suppressed for larger values of $M$ in all cases, indicating that the twin-Fock state results in a slightly smaller phase uncertainty.Reuse & Permissions
Figure 3
(Color online) Column A: probability amplitudes for several possible input states with $N = 100$ particles. Column B: Corresponding likelihood functions for $M = 1$ (dotted red line), compared to the result for the benchmark twin-Fock input (solid black line). Column C: Likelihood functions for $(M = 10)$ .Reuse & Permissions
Figure 4
(Color online) Scaling of the inverse square root Fisher information, $σ = 1 ∕ \sqrt{F}$ with particle number for the state in row 1 of Fig. 3 (blue dotted line), twin-Fock state (solid red line), the “Gaussian state” (black dashed line), and the external $N 00 N$ state (green dotted-dashed line). The points are calculated using Eq. (24) while the lines correspond to least square fits, with the following values: external $N 00 N$ state, $σ = 1.0 ∕ N^{1 ∕ 2}$ ; Gaussian state, $σ = 1.9 ∕ N$ ; twin-Fock state, $σ = 1.4 ∕ N$ ; and the state in row 1 of Fig. 3, $σ = 1.0 ∕ N$ . Note the log-log scale.Reuse & Permissions
Figure 5
(Color online) Number statistics for phase shifts of $ϕ = 0$ , $ϕ = π ∕ 100$ , and $ϕ = π ∕ 10$ in a system with $N = 100$ particles for (a)–(c) a twin-Fock input state and (d)–(f) a Gaussian input state. The twin-Fock state populates number states symmetrically around $m = 0$ , leading to an expectation value of $⟨ {\hat{J}}_{z} ⟩ = 0$ for all values of $ϕ$ . This is not the case for a Gaussian state.Reuse & Permissions
Figure 6
(Color online) Scaling of the uncertainty in the estimate of the relative phase from a measurement of average particle number difference. The dashed black line is for a Gaussian state and the red solid line of a tri-Fock state, Eq. (34), with $q = 1$ , and the green dotted-dashed line represents the state engineered as described in Sec. 5B. The points are numerically determined while the lines correspond to least square fits. Note the log-log scale.Reuse & Permissions

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