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arXiv:1011.3197 · Physical Review A

Quantum-limited metrology in the presence of collisional dephasing

Yong‐Chun Liu, Guangri Jin, Li You

Published Oct 29, 2010• 48 citations
arXivHTMLPDF

Abstract

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 $N$ 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.

DOI
10.1103/physreva.82.045601

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Including collisional decoherence explicitly, phase sensitivity for estimating effective scattering strength \ensuremathχ\ensuremath{\chi}\ensuremathχ of a two-component Bose-Einstein condensate is derived analytically. With a measurement of spin operator J\ifmmode^\elseˆ\fix{\mathrm{J\ifmmode \hat{}\else \^{}\fi{}}}_{x}J\ifmmode^\elseˆ\fix​, we find that the optimal sensitivity depends on initial coherent spin state. It degrades by a factor of (2\ensuremathγ)1/3(2\ensuremath{\gamma}){}^{1/3}(2\ensuremathγ)1/3 below super-Heisenberg limit \ensuremath∝1/N3/2\ensuremath{\propto}1/{N}^{3/2}\ensuremath∝1/N3/2 for particle number NNN and the dephasing rate 1\ensuremath≪\ensuremathγ<N3/41\ensuremath{\ll}\ensuremath{\gamma}<{N}^{3/4}1\ensuremath≪\ensuremathγ<N3/4. With a J\ifmmode^\elseˆ\fiy{\mathrm{J\ifmmode \hat{}\else \^{}\fi{}}}_{y}J\ifmmode^\elseˆ\fiy​ measurement, our analytical results confirm that the phase \ensuremathϕ=\ensuremathχt 0\ensuremath{\phi}=\ensuremath{\chi}t~0\ensuremathϕ=\ensuremathχt 0 can be detected at the limit even in the presence of the dephasing.
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.
The Heisenberg limit is the superior precision available by entanglement sensors. However, entanglementis fragile against dephasing, and there is no known quantum metrology protocol that can achieve Heisenberg limited sensitivity with the presence of independent dephasing. Here, we show that the Heisenberg limit is attainable under the effect of independent dephasing under conditions where the probe qubits decohere due to both target fields and local environments. To detect the target fields, we exploit the entanglement properties to decay much faster than the classical states due to collective noise while most of the previous schemes use a coherent phase shift from the target fields. Actually, if the temporally fluctuating target fields behave as Markovian collective dephasing, we can estimate the collective dephasing rate with a sensitivity at the Heisenberg limit under the effect of independent dephasing. Our work opens the possibility for robust Heisenberg-limited metrology.
Quantum metrology faces major challenges in noisy environments, where decoherence rapidly degrades useful quantum resources. We investigate the dynamics of the precision limits given by the quantum Fisher information (QFI) for phase estimation under spatially correlated dephasing. We characterize the dynamics of the QFI by the sensitivity and degradation indicators that can be obtained as analytical expressions derived using perturbative theory treatment. These short-time and weak-noise formulas yield analytic insight into how collective-spin moments govern both (i) the noiseless sensitivity and (ii) the leading noise-induced degradation of metrological usefulness. We identify a trade-off that is intrinsic to our commuting encoding-noise structure. We analyze the QFI dynamics for the Gaussian spin state (GSS) superpositions, encompassing spin coherent state (SCS), Dicke state superpositions, spin-squeezed states, and GHZ-like states. Predictions from indicators of the QFI dynamics are compared to both the quantum Cramér--Rao bound and the measurement-specific sensitivity bounds for an optimal parameter and interrogation time under a finite total time resource. When possible, we analytically derive the measurement-specific sensitivity bounds for spin-projection and parity-based measurements.
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.
The goal of quantum metrology is the precise estimation of parameters using quantum properties such as entanglement. This estimation usually consists of three steps: state preparation, time evolution during which information of the parameters is encoded in the state, and readout of the state. Decoherence during the time evolution typically degrades the performance of quantum metrology and is considered to be one of the major obstacles to realizing entanglement-enhanced sensing. We show, however, that under suitable conditions, this decoherence can be exploited to improve the sensitivity. Assume that we have two axes, and our aim is to estimate the relative angle between them. Our results reveal that the use of Markovian collective dephasing to estimate the relative angle between the two directions affords Heisenberg-limited sensitivity. Moreover, our protocol based on Markovian collective dephasing is robust against environmental noise: it is possible to achieve the Heisenberg limit by applying the collective dephasing even under the effect of independent dephasing. Our counterintuitive proposal with the decoherence leads to alternative applications in quantum metrology.
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.