Metrologically useful states of spin-1 Bose condensates with macroscopic magnetization

We study theoretically the usefulness of spin-1 Bose condensates with macroscopic magnetization in a homogeneous magnetic field for quantum metrology. We demonstrate Heisenberg scaling of the quantum Fisher information for states in thermal equilibrium. The scaling applies to both antiferromagnetic and ferromagnetic interactions. The effect preserves as long as fluctuations of magnetization are sufficiently small. Scaling of the quantum Fisher information with the total particle number is derived within the mean-field approach in the zero-temperature limit and exactly in the high-magnetic-field limit for any temperature. The precision gain is intuitively explained owing to subtle features of the quasidistribution function in the phase space.

Figure 1

The scheme of typical interferometric sequence. The input state is rotated with a pulse 1 to be sensitive to the Hamiltonian $\widehat{h}\propto \lambda {\tilde{J}}_z$. After the interrogation time, the value of the parameter to estimate $\lambda$ is encoded in the relative phases between paths of the interferometer. The pulse 2 is applied to map the phase onto quantities easy to measure. The scheme describes as well the typical two-arm Mach-Zehnder interferometer, as the case with three arms, discussed in this paper.

Figure 2

Illustration of (a) the Husimi and (b) the Wigner quasidistribution functions for eigenstates of the system: an example for the state $\left|\right(N+M)/2,(N-M)/2\rangle$ with $N=60,M=N/2$. The maximum of both the Husimi and Wigner distributions is centered around $\langle {\tilde{J}}_z\rangle$, which is $M$ in region $A$ and $2M-N$ in region $B$.

Figure 3

Top row: QFI as a function of $q$ in the ground state of the system for $\sigma \to 0,N={10}^4$ and (a) $c_2<0,m=0.5$, (b) $c_2>0,m=0.8$. Solid red lines denote numerical results from exact diagonalization of the Hamiltonian in the Fock state basis. Dashed black lines denote mean-field results, as explained in the text. Dashed vertical gray lines mark the position of the critical points $q_c^{\left(A\right)}=(1-\sqrt{1-m^2})$ for $c_2>0$, and $q_c^{\left(F\right)}=-(1+\sqrt{1-m^2})$ for $c_2<0$ [44]. Solid vertical gray lines mark the location of the threshold points $q_{t\pm }$ which separate the regions $A$ and $B$. Bottom row: eigenvectors corresponding to the largest eigenvalue of the covariance matrix as a function of $q$, demonstrating the optimal interferometer and confirming the validity of ${\widehat{\mathrm{\Lambda }}}_{\mathbf{n}}^{(A,B)}$.

Figure 4

Exact numerical results for $F_Q/4N^2$ as a function of the temperature inverse $\beta$ and $q$ for $N={10}^3$ with (a) $c_2<0,m=0.5$ and (b) $c_2>0,m=0.8$. The solid white line is the border between the $A$ and $B$, or $A$ and $B^{\prime }$, regions, which are approximated by $q_t\left(\beta \right)+q_{t-}$ for ferromagnetic interactions and $q_t\left(\beta \right)+q_{t+}$ for antiferromagnetic ones. The dashed white line is the border between the $B$ and $B^{\prime }$ regions approximated by $q_{B{B}^{\prime }}\left(\beta \right)+q_{t-}$ in (a) and $q_{B{B}^{\prime }}\left(\beta \right)+q_{t+}$ in (b).

Figure 5

The QFI as a function of $\sigma$ deeply inside (a) region $A$ for $q=-10,\beta =100$, (b) region $B$ for $q=10,\beta =100$, and (c) region $B^{\prime }$ for $q=10,\beta ={10}^{-5}$, with $N={10}^3$. Symbols are exact numerical results. Color lines denote numerical results in the HMFL. Gray solid lines denote decay rates, as explained in the text. Gray dashed horizontal lines show the SQL for $N={10}^3$.