Optimal and near-optimal probe states for quantum metrology of number-conserving two-mode bosonic Hamiltonians

We derive families of optimal and near-optimal probe states for quantum estimation of the coupling constants of a general two-mode number-conserving bosonic Hamiltonian describing one-body and two-body dynamics. We find that the optimal states for estimating the dephasing of the modes, the self-interaction strength, and the contact interaction strength are related to the NOON states, whereas the optimal states for estimation of the intermode single-particle tunneling amplitude are superpositions of antipodal SU(2) coherent states. For estimation of the amplitude of pair tunneling and the amplitude of density-dependent single-particle tunneling processes, respectively, we introduce classes of variational superposition probe states that provide near perfect saturation of the corresponding quantum Cramér-Rao bounds. We show that the ground state of the pair tunneling term in the Hamiltonian has a high fidelity with the optimal states for estimation of a single-particle tunneling amplitude, suggesting that high-performance probes for tunneling amplitude estimation may be produced by tuning the two-mode system through a quantum phase transition.

Figure 1

Numerically calculated eigenvalues ${\mathcal{E}}_n$ of the Hamiltonian $a_0^{†2}{a}_1^2+\text{H.c.}$ acting on $N=160$ bosons for $n=1,...,161$. The inset shows the gap ${\mathcal{E}}_{2n+1}-{\mathcal{E}}_{2n}$ between pairs of numerical eigenvalues for $n=1,...,40$.

Figure 2

(a) Semilog plot of the fidelity of the variational state $|{\omega }_{+}\left(\tilde{c}\right)\rangle$ with numerical ground state $|{\mathcal{E}}_{\ell =1}\rangle$ (black dots) or numerical first excited state $|{\mathcal{E}}_{\ell =2}\rangle$ (red dots) for particle number $N$ varying from 8–160 in steps of 4. (b) The same, except with $|{\omega }_{-}\left(\tilde{c}\right)\rangle$.

Figure 3

Semilog plot of the infidelity $1-F\left(\right|A\rangle ,|B\rangle )=1-{\left|\langle A|B\rangle \right|}^2$ for $|A\rangle :=1/\sqrt{2}\left(|\zeta =i\rangle \pm |\zeta =-i\rangle \right)$ and $|B\rangle :=|{\mathcal{E}}_{\ell =1}\rangle$ the numerical ground state. The black (red) dots correspond to values of $N$ where the $+(-)$ sign of $|A\rangle$ gives better agreement with $|{\mathcal{E}}_{\ell =1}\rangle$.

Figure 4

Semilog plot of the infidelity $1-F\left(\right|A\rangle ,|B\rangle )=1-{\left|\langle A|B\rangle \right|}^2$ for $|A\rangle$ a variational ground state, $|B\rangle :=|{\mathcal{E}}_{\ell =1}\rangle$ is the numerical ground state, and $N=8,12,...,160$. The black dots represent $|A\rangle :=|\zeta \rangle$ where $\zeta$ extremizes Eq. (27); the red dots represent $|A\rangle :=|\mathrm{\Xi }(\tilde{w},\tilde{z})\rangle \propto {\left(a_0^{†2}+2\tilde{w}{a}_0^{†}{a}_1^{†}+{\tilde{z}}^2{a}_1^{†2}\right)}^M|0,0\rangle$ with $(\tilde{w},\tilde{z})$ determined by the $k=0$ and $k=1$ equations in Eq. (29); the blue dots show $|A\rangle :=|\mathrm{\Xi }(w_0,z_0)\rangle$ with $(w_0,z_0)$ determined by numerical optimization.