Quantum metrology with spin cat states under dissipation

Quantum metrology aims to yield higher measurement precisions via quantum techniques such as entanglement. It is of great importance for both fundamental sciences and practical technologies, from testing equivalence principle to designing high-precision atomic clocks. However, due to environment effects, highly entangled states become fragile and the achieved precisions may even be worse than the standard quantum limit (SQL). Here we present a high-precision measurement scheme via spin cat states (a kind of non-Gaussian entangled states in superposition of two quasi-orthogonal spin coherent states) under dissipation. In comparison to maximally entangled states, spin cat states with modest entanglement are more robust against losses and their achievable precisions may still beat the SQL. Even if the detector is imperfect, the achieved precisions of the parity measurement are higher than the ones of the population measurement. Our scheme provides a realizable way to achieve high-precision measurements via dissipative quantum systems of Bose atoms.

/2. Here, we assume that, when the fidelity is less than 0.005, the MSSCS can be regarded as a spin cat state, see the dashed lines in Fig. S1. The region of spin cat states (from = 0 to the dashed line) becomes broader when the particle number increases.

QCRBs versus Phase Accumulation Time under One-body Losses
In experiments, one-body atom loss results from the collision between the condensed atoms and the residual atoms. During the phase accumulation process, the Bose condensed atoms may collide with the residual atoms in the environment and then are kicked out from the condensate. The reduced density matrix for such a dissipative phase accumulation under one-body losses obeys a Markovian master equation 1-3 , where and γ a,b are the damping rates. We prepare the input state as the MSSCS |Ψ(θ)〉 , whose density matrix is given as ( = 0) = |Ψ(θ)〉 ⟨Ψ(θ) |. In our calculation, we set δ = 1 and the accumulated phase ϕ = δT for the phase accumulation time T. By solving the above master equation, we obtain the reduced density matrix of the output state ρ(T). For a given output state ρ(T), the phase precision is limited by the QCRB, Δϕ ≥ Δϕ QCRB , with F Q (T) denoting the quantum Fisher information. We have calculated the precision bounds versus T. ) may still beat the SQL and are much better than the ones achieved by the GHZ state and the spin coherent state (θ = π 2 ). This indicates that the input spin cat states are excellent candidates for implementing dissipative quantum metrology.

QCRBs versus Phase Accumulation Time under Two-body Losses
In experiments with atomic Bose-Einstein condensates, another kind of particle losses is two-body losses, which corresponds to the case of two atoms collide with each other and escape from the condensate. It commonly exists when the density of the atoms is sufficiently dense. Here, we consider the phase accumulation process under two-body atom losses. The Markovian master equation for such a phase accumulation can be written as 1,4-6 , with Here, γ aa , γ bb and γ ab are the intra-and inter-mode two-body damping rates, respectively. For convenience, we choose γ aa = γ bb = γ ab = 0.005 and δ = 1 for our investigation.
We then consider the MSSCS |Ψ(θ)〉 M as the input states, and calculate the precision bounds Δϕ QCRB versus the phase accumulation time T.
Similar to the one-body losses, the measurement precision decreases with T.
For spin cat states with smaller θ, the measurement uncertainty Δϕ QCRB increases more dramatically. However, in comparison to the case of one-body losses, for a fixed damping rate, the particles decrease dependent on the initial state. The spin cat state with smaller θ will suffer more rapid reduction of the particle number. For a fixed damping rate of two-body losses, the amounts of atom losses during the phase accumulation decrease with the angle θ and this leads to more rapid reduction of the measurement precisions for more entangled input states. It is also shown that, under two-body atom losses, the spin cat states with moderate θ can still achieve high-precision measurement beyond the SQL, see Fig. S3.

QCRBs versus Phase Accumulation Time under Correlated Dephasing
In addition to dissipation, atomic Bose-Einstein condensates often encounters another kind of decoherence called correlated dephasing, which is caused by the random fluctuation of the external field. The Markovian master equation for such a phase accumulation can be written as 2,7 , with H � 0 = δ(b � † b � − a � † a �)/2 = δĴ z , and L = Ĵ z . Here γ d is the correlated dephasing rate and we choose γ d = 0.01 and fix δ = 1. Also, for the input MSSCS |Ψ(θ)〉 M , we have calculated the corresponding precision bounds Δϕ QCRB versus the phase accumulation time T. Unlike the dissipation which would lead to the reduction of particle number with time, the correlated dephasing keeps the total particle number N unchanged during the phase accumulation. However, the random fluctuation results that the coherence gradually diminishes. In Fig. S4