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Frontiers of Physics

Optimal phase sensitivity of atomic Ramsey interferometers with coherent spin states

Guangri Jin, Yong‐Chun Liu, Li You

Published Apr 8, 2011• 7 citations
Publisher

Abstract

We present a detailed analysis of phase sensitivity for a nonlinear Ramsey interferometer, which utilize effective mean-field interaction of a two-component Bose-Einstein condensate in phase accumulation. For large enough particle number N and small phase shift ϕ, analytical results of the Ramsey signal and the phase sensitivity are derived for a product coherent state |θ, 0〉. When collisional dephasing is absent, we confirm that the optimal sensitivity scales as 2/N 3/2 for polar angle of the initial state θ = π/4 or 3π/4. The best-sensitivity phase satisfies different transcendental equations, depending upon the initial state and the observable being measured after the phase accumulation. In the presence of the collisional dephasing, we show that the N −3/2-scaling rule of the sensitivity maintains with spin operators $\hat J_x$ and $\hat J_y$ measurements. A slightly better sensitivity is attainable for optimal coherent state with θ = π/6 or 5π/6.

DOI
10.1007/s11467-011-0180-5

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We present a detailed analysis of phase sensitivity for a nonlinear Ramsey interferometer, which utilize effective mean-field interaction of a two-component Bose-Einstein condensate in phase accumulation. For large enough particle number N and small phase shift ϕ, analytical results of the Ramsey signal and the phase sensitivity are derived for a product coherent state |θ, 0〉. When collisional dephasing is absent, we confirm that the optimal sensitivity scales as 2/N 3/2 for polar angle of the initial state θ = π/4 or 3π/4. The best-sensitivity phase satisfies different transcendental equations, depending upon the initial state and the observable being measured after the phase accumulation. In the presence of the collisional dephasing, we show that the N −3/2-scaling rule of the sensitivity maintains with spin operators J^x\hat J_xJ^x​ and J^y\hat J_yJ^y​ measurements. A slightly better sensitivity is attainable for optimal coherent state with θ = π/6 or 5π/6.
Including collisional decoherence explicitly, phase sensitivity for estimating effective scattering strength $\ensuremath{\chi}$ of a two-component Bose-Einstein condensate is derived analytically. With a measurement of spin operator ${\mathrm{J\ifmmode \hat{}\else \^{}\fi{}}}_{x}$, we find that the optimal sensitivity depends on initial coherent spin state. It degrades by a factor of $(2\ensuremath{\gamma}){}^{1/3}$ below super-Heisenberg limit $\ensuremath{\propto}1/{N}^{3/2}$ for particle number NNN and the dephasing rate $1\ensuremath{\ll}\ensuremath{\gamma}<{N}^{3/4}$. With a ${\mathrm{J\ifmmode \hat{}\else \^{}\fi{}}}_{y}$ measurement, our analytical results confirm that the phase $\ensuremath{\phi}=\ensuremath{\chi}t~0$ can be detected at the limit even in the presence of the dephasing.
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 analyze a proposed experiment [Boixo et al., Phys. Rev. Lett. 101, 040403 (2008)] for achieving sensitivity scaling better than 1/N1/N1/N in a nonlinear Ramsey interferometer that uses a two-mode Bose-Einstein condensate (BEC) of NNN atoms. We present numerical simulations that confirm the analytical predictions for the effect of the spreading of the BEC ground-state wave function on the ideal 1/N3/21/{N}^{3/2}1/N3/2 scaling. Numerical integration of the coupled, time-dependent, two-mode Gross-Pitaevskii equations allows us to study the several simplifying assumptions made in the initial analytic study of the proposal and to explore when they can be justified. In particular, we find that the two modes share the same spatial wave function for a length of time that is sufficient to run the metrology scheme.
The SU(1,1) interferometer was originally conceived as a Mach-Zehnder interferometer with the beam splitters replaced by parametric amplifiers. The parametric amplifiers produce states with correlations that result in enhanced phase sensitivity. F=1F=1F=1 spinor Bose-Einstein condensates (BECs) can serve as the parametric amplifiers for an atomic version of such an interferometer by collisionally producing entangled pairs of $|F=1,m=\ifmmode\pm\else\textpm\fi{}1\ensuremath{\rangle}$ atoms. We simulate the effect of single- and double-sided seeding of the inputs to the amplifier using the truncated-Wigner approximation. We find that single-sided seeding degrades the performance of the interferometer exactly at the phase the unseeded interferometer should operate the best. Double-sided seeding results in a phase-sensitive amplifier, where the maximal sensitivity is a function of the phase relationship between the input states of the amplifier. In both single- and double-sided seeding we find there exists an optimal phase shift that achieves sensitivity beyond the standard quantum limit. Experimentally, we demonstrate a spinor phase-sensitive amplifier using a BEC of 23Na^{23}\mathrm{Na}23Na in an optical dipole trap. This configuration could be used as an input to such an interferometer. We are able to control the initial phase of the double-seeded amplifier and demonstrate sensitivity to initial population fractions as small as 0.1%.
We experimentally and theoretically study phase coherence in two-component Bose-Einstein condensates of ^{87}Rb atoms on an atom chip. Using Ramsey interferometry we determine the temporal decay of coherence between the |F=1,m_{F}=-1⟩ and |F=2,m_{F}=+1⟩ hyperfine ground states. We observe that the coherence is limited by random collisional phase shifts due to the stochastic nature of atom loss. The mechanism is confirmed quantitatively by a quantum trajectory method based on a master equation which takes into account collisional interactions, atom number fluctuations, and losses in the system. This decoherence process can be slowed down by reducing the density of the condensate. Our findings are relevant for experiments on quantum metrology and many-particle entanglement with Bose-Einstein condensates and the development of chip-based atomic clocks.
We present a detailed analysis of spin squeezing of the one-axis twisting model with a many-body phase dephasing, which is induced by external field fluctuation in a two-mode Bose-Einstein condensates. Even in the presence of the dephasing, our analytical results show that the optimal initial state corresponds to a coherent spin state ∣θ0,ϕ0⟩|\theta_{0}, \phi_0\rangle∣θ0​,ϕ0​⟩ with the polar angle θ0=π/2\theta_0=\pi/2θ0​=π/2. If the dephasing rate γ≪S−1/3\gamma\ll S^{-1/3}γ≪S−1/3, where SSS is total atomic spin, we find that the smallest value of squeezing parameter (i.e., the strongest squeezing) obeys the same scaling with the ideal one-axis twisting case, namely ξ2∝S−2/3\xi^2\propto S^{-2/3}ξ2∝S−2/3. While for a moderate dephasing, the achievable squeezing obeys the power rule S−2/5S^{-2/5}S−2/5, which is slightly worse than the ideal case. When the dephasing rate γ>S1/2\gamma>S^{1/2}γ>S1/2, we show that the squeezing is weak and neglectable.