Lieb-Mattis states for robust entangled differential phase sensing

Abstract

Developing sensors with large particle numbers $N$ that can resolve subtle physical effects is a central goal in precision measurement science. Entangled quantum sensors can surpass the standard quantum limit (SQL), where the signal variance scales as $1/N$, and approach the Heisenberg limit (HL) with variance scaling as $1/N^2$. However, entangled states are typically more sensitive to noise, especially common-mode noise such as magnetic field fluctuations, control phase noise, or vibrations in atomic interferometers. We propose a two-node entanglement-enhanced quantum sensor network for differential signal estimation that intrinsically rejects common-mode noise while remaining robust against local, uncorrelated noise. This architecture enables sensitivities approaching the Heisenberg limit. We investigate two state preparation strategies: (i) unitary entanglement generation analogous to bosonic two-mode squeezing, yielding Heisenberg scaling; and (ii) dissipative preparation via collective emission into a shared cavity mode, offering a $\sqrt{N}$ improvement over the SQL. Numerical simulations confirm that both protocols remain effective under realistic conditions, supporting scalable quantum-enhanced sensing in the presence of dominant common-mode noise.

Figures (12)

• Figure 1: (a) Schematic illustration of the differential phase sensing protocol. The atoms in ensembles $A (B)$ are initially prepared in their respective excited (ground) states, after which an entangling operation is applied. Subsequently, each ensemble acquires a distinct phase $\varphi_A$ and $\varphi_B$. Finally, a joint measurement of both ensembles is carried out to estimate the differential phase $\phi$ with high precision, while remaining insensitive to fluctuations of the common phase $\Phi$. (b) The target state after the entangling process, $\ket{\psi_T}$, is an entangled Lieb-Mattis state (see Appendix \ref{['app:parentHamiltonian']}) that can be understood as a permutation-symmetric superposition over all states where each atom in $A$ forms a singlet with an atom in $B$. (c) In the Dicke basis of the individual ensembles, represented by their Wigner distributions, this state can be expressed as an equal superposition, with alternating signs, of all Dicke-state combinations where the number of atoms in the excited state in ensemble $A$ equals the number of ground-state atoms in ensemble $B$.
• Figure 2: (a) Schematic of the cavity setup. Two ensembles of atoms, labeled $A$ and $B$, trapped in a magic-wavelength optical lattice (gray ellipses) inside a cavity and are initially prepared in the excited and ground state, respectively. Photons leak out of the cavity at a rate $\kappa$, and atoms in the excited state can emit photons into free space at a rate $\gamma$. (b) The cavity mode frequency $\omega_{\mathrm{C}}$ is detuned by $\Delta$ from the atomic transition frequency $\omega_{\mathrm{A}}$, which quantifies the energy difference between $\ket{\downarrow}$ and $\ket{\uparrow}$. (c) Spin-exchange interactions, described by the Hamiltonian $H_{\mathrm{C}}$, are mediated by the exchange of virtual photons through the cavity mode. (d) The atoms can collectively emit into the cavity mode. This process is described by the jump operator $L_{\Gamma}$.
• Figure 3: (a) The minimized infidelity $I = 1 - \left| \braket{\psi_{\rm T}|\psi_{\rm TMS}(t)} \right|^2$, between the target state $\ket{\psi_{\rm T}}$ and a state that is generated by quenching the product state $\ket{\psi_0}$ under the two-mode squeezing Hamiltonian $H_{\rm TMS}$ [Eq. \ref{['eq:H_TMS']}] (blue). The optimal time required to reach the minimized infidelity for different atom numbers (orange). (b) Scaling of the estimator variance of the quenched state $\ket{\psi_{\rm TMS}^*}$ at the optimal time and phase $\phi_0=\pi/4$ in comparison to the target state $\ket{\psi_{\rm T}}$ and the standard quantum limit (SQL) and Heisenberg limit (HL).
• Figure 4: The optimal estimator variance, denoted as $\tilde{\Delta}_{\phi}^2$, attainable under a quench with the two-mode squeezing Hamiltonian while simultaneously being subject to collective and free-space emission. It is evaluated relative to the ideal variance achievable in the absence of collective and free-space emission, denoted as $\Delta_{\phi}^2$. $\tilde{\Delta}_{\phi}^2$ is optimized with respect to the quench duration and the detuning between the cavity and the atomic transition frequency. The red dashed line marks the particle number $N$ at which the standard quantum limit is exceeded for a given single-particle cooperativity $C$.
• Figure 5: (a) Collective emission (red curly arrows) evolves the initial state into a steady state, corresponding to a mixture of Dicke states located on the lower diagonal of the Dicke state ladder, characterized by quantum numbers $J$ and $M$. The steady-state distribution (orange circles) corresponds to the probability of projecting the initial state onto a specific $\ket{J,0}$ state (orange distribution). (b) The probability of projecting onto a given $\ket{J, 0}$ state, is determined for the initial state where the atoms in ensemble A are in the excited state and those in ensemble B are in the ground state. (c) The first moment, $\overline{J}=\sum_{J}J\, p(J)$, and the second moment, $\overline{J^2}=\sum_{J}J^2\, p(J)$, of the distribution presented in panel (b) are shown for different atom numbers $N$, rescaled to highlight their respective asymptotic scaling. The dashed lines indicate the prefactors corresponding to the asymptotic scaling.