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arXiv:quant-ph/0412037 · Journal of Optics B Quantum and Semiclassical Optics

States for phase estimation in quantum interferometry

Joshua Combes, Howard M. Wiseman

Published Dec 4, 2004• 28 citations
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Abstract

Ramsey interferometry allows the estimation of the phase φ of rotation of the pseudospin vector of an ensemble of two-state quantum systems. For φ small, the noise-to-signal ratio scales as the spin-squeezing parameter ξ, with ξ<1 possible for an entangled ensemble. However states with minimum ξ are not optimal for single-shot measurements of an arbitrary phase. We define a phase-squeezing parameter, ζ, which is an appropriate figure of merit for this case. We show that (unlike the states that minimize ξ) the states that minimize ζ can be created by evolving an unentangled state (coherent spin state) by the well known two-axis counter-twisting Hamiltonian. We analyse these and other states (for example the maximally entangled state, analogous to the optical ‘NOON’ state ) using several different properties, including ξ, ζ, the coefficients in the pseudo-angular momentum basis (in the three primary directions) and the angular Wigner function W (θ,ϕ). Finally, we discuss the experimental options for creating phase-squeezed states and doing single-shot phase estimation.

DOI
10.1088/1464-4266/7/1/004

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States for phase estimation in quantum interferometry

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Ramsey interferometry allows the estimation of the phase φ of rotation of the pseudospin vector of an ensemble of two-state quantum systems. For φ small, the noise-to-signal ratio scales as the spin-squeezing parameter ξ, with ξ<1 possible for an entangled ensemble. However states with minimum ξ are not optimal for single-shot measurements of an arbitrary phase. We define a phase-squeezing parameter, ζ, which is an appropriate figure of merit for this case. We show that (unlike the states that minimize ξ) the states that minimize ζ can be created by evolving an unentangled state (coherent spin state) by the well known two-axis counter-twisting Hamiltonian. We analyse these and other states (for example the maximally entangled state, analogous to the optical ‘NOON’ state ) using several different properties, including ξ, ζ, the coefficients in the pseudo-angular momentum basis (in the three primary directions) and the angular Wigner function W (θ,ϕ). Finally, we discuss the experimental options for creating phase-squeezed states and doing single-shot phase estimation.
Ramsey interferometry allows the estimation of the phase phi\\phiphi of rotation of the pseudospin vector of an ensemble of two-state quantum systems. For phi\\phiphi small, the noise-to-signal ratio scales as the spin-squeezing parameter xi\\xixi, with xi<1\\xi<1xi<1 possible for an entangled ensemble. However states with minimum xi\\xixi are not optimal for single-shot measurements of an arbitrary phase. We define a phase-squeezing parameter, zeta\\zetazeta, which is an appropriate figure-of-merit for this case. We show that (unlike the states that minimize xi\\xixi), the states that minimize zeta\\zetazeta can be created by evolving an unentangled state (coherent spin state) by the well-known 2-axis counter-twisting Hamiltonian. We analyse these and other states (for example the maximally entangled state, analogous to the optical "NOON" state ∣psi>=(∣N,0>+∣0,N>)/sqrt2|\\psi> = (|N,0>+|0,N>)/\\sqrt{2}∣psi>=(∣N,0>+∣0,N>)/sqrt2) using several different properties, including xi\\xixi, zeta\\zetazeta, the coefficients in the pseudo angular momentum basis (in the three primary directions) and the angular Wigner function W(theta,phi)W(\\theta,\\phi)W(theta,phi). Finally we discuss the experimental options for creating phase squeezed states and doing single-shot phase estimation.
This chapter describes the use of squeezed states to improve the sensitivity in the estimation of the relative phase accumulated in a Bose--Einstein condensate (BEC) beyond the standard quantum limit. We begin by examining the specific example of a two-component BEC, where atoms in two different hyperfine levels interact with each other. By evolving an initial superposition of states under an effective Hamiltonian, a squeezed state is realized. Next, the squeezing is visualized using the Q-function, giving the mean spin direction and shape of the BEC distribution. We discuss the basic operation of Ramsey interferometry, which is a classic example of two-slit atom diffraction. We then calculate the error in estimating the relative phase shift using the error propagation formula that allows us to define the squeezing parameter. This measures the amount of metrological gain using squeezed states over unsqueezed states. We give an example of the metrological gain in the estimation of the relative phase shift using a squeezed state of the two-component BEC as an input state to an atom Mach--Zehnder interferometer. The Fisher information is used to find which rotation axis gives the optimal information from a given initial state. Finally, we parameterize the nonlinear phase per atom responsible for squeezing and discuss how this parameter is controlled in different experiments.
We investigate the presence of spin- and planar-squeezing in generalized superpositions of atomic (or spin) coherent states. Spin-squeezing has been shown to be a useful tool in determining the presence of entanglement in multipartite systems, such as collections of two-level atoms, as well as being an indication of reduced projection noise and sub-shot-noise-limited phase uncertainty in Ramsey spectroscopy, suitable for measuring phases ϕ ∼ 0 . On the other hand, planar-squeezed states display reduced projection noise in two directions simultaneously and have been shown to lead to enhanced metrological precision in measuring phases without the need for explicit prior knowledge of the phase value. In this paper, we show that the generalized superposition state can be parametrized to display both spin-squeezing along all orthogonal axes and planar-squeezing along all orthogonal planes for all values of J > 1 / 2 . We close with an application of the maximally spin- and planar-squeezed states to quantum metrology.
We propose an analogue of SU(1,1) interferometry to measure rotation of a spin by using two-spin squeezed states. Attainability of the Heisenberg limit for the estimation of the rotation angle is demonstrated for maximal squeezing. For a specific direction and strength an advantage in sensitivity for all equatorial rotation axes (and hence non-commuting rotations) over the classical bound is shown in terms of quadratic scaling of the single-parameter quantum Fisher information for the corresponding rotation angles. Our results provide a method for measuring magnetic fields in any direction in the x - y -plane with the same optimized initial state.
We consider a large class of Ramsey interferometry protocols which are enhanced by squeezing and un-squeezing operations before and after a phase signal is imprinted on the collective spin of N particles. We report an analytical optimization for any given particle number and strengths of (un-)squeezing. These results can be applied even when experimentally relevant decoherence processes during the squeezing and un-squeezing interactions are included. Noise between the two interactions is however not considered in this work. This provides a generalized characterization of squeezing echo protocols, recovering a number of known quantum metrological protocols as local sensitivity maxima, thereby proving their optimality. We discover a single new protocol. Its sensitivity enhancement relies on a double inversion of squeezing. In the general class of echo protocols, the newly found over-un-twisting protocol is singled out due to its Heisenberg scaling even at strong collective dephasing.
We consider a large class of Ramsey interferometry protocols which are enhanced by squeezing and un-squeezing operations before and after a phase signal is imprinted on the collective spin of N particles. We report an analytical optimization for any given particle number and strengths of (un-)squeezing. These results can be applied even when experimentally relevant decoherence processes during the squeezing and un-squeezing interactions are included. Noise between the two interactions is however not considered in this work. This provides a generalized characterization of squeezing echo protocols, recovering a number of known quantum metrological protocols as local sensitivity maxima, thereby proving their optimality. We discover a single new protocol. Its sensitivity enhancement relies on a double inversion of squeezing. In the general class of echo protocols, the newly found over-un-twisting protocol is singled out due to its Heisenberg scaling even at strong collective dephasing.