Minutes-scale Schr{ö}dinger-cat state of spin-5/2 atoms

Abstract: Quantum metrology with nonclassical states offers a promising route to improved precision in physical measurements. The quantum effects of Schr{\"o}dinger-cat superpositions or entanglements allow measurement uncertainties to reach below the standard quantum limit. However, the challenge in keeping a long coherence time for such nonclassical states often prevents full exploitation of the quantum advantage in metrology. Here we demonstrate a long-lived Schr{\"o}dinger-cat state of optically trapped $^{173}$Yb (\textit{I}\ =\ 5/2) atoms. The cat state, a superposition of two oppositely-directed and furthest-apart spin states, is generated by a non-linear spin rotation. Protected in a decoherence-free subspace against inhomogeneous light shifts of an optical lattice, the cat state achieves a coherence time of $1.4(1)\times 10^3$ s. A magnetic field is measured with Ramsey interferometry, demonstrating a scheme of Heisenberg-limited metrology for atomic magnetometry, quantum information processing, and searching for new physics beyond the Standard Model.

• The paper demonstrates a minutes-scale Schrödinger-cat state by using nonlinear spin rotation and a decoherence-free subspace in 173Yb atoms, with coherence times around 1.4×10³ s.
• It employs optical lattice trapping, carefully tuned light shifts, and Ramsey interferometry to enhance phase sensitivity, approaching the Heisenberg limit.
• The method improves sensitivity by 15(2) dB over conventional coherent spin states, promising advances in atomic magnetometry and precision quantum measurements.

Minutes-Scale Schrödinger-Cat State of Spin-5/2 Atoms: A Detailed Summary

Introduction and Motivation

Quantum metrology exploits nonclassical states, including Schrödinger-cat and entangled states, to enhance phase sensitivity beyond the standard quantum limit (SQL). Schrödinger-cat states—superpositions of macroscopically distinct quantum states—represent a paradigm of macroscopic quantum coherence but are notably fragile, usually exhibiting limited coherence times due to environmental decoherence. Realizing such states in atom-based platforms with both long coherence time and metrological utility is critical for advancing quantum-enhanced measurement, atomic magnetometry, and precision tests of fundamental physics (e.g., searches for exotic couplings and Lorentz violation [Safronova et al., Rev. Mod. Phys. 90, 025008 (2018)]).

This work reports the preparation, control, and metrological characterization of minutes-scale Schrödinger-cat states using the nuclear spin (I = 5/2) of optically trapped $^{173}$Yb atoms. The cat state consists of a superposition between the $m = +5/2$ and $m = -5/2$ Zeeman sublevels, created via a nonlinear light-induced spin rotation and protected from decoherence in a decoherence-free subspace of the high-spin Hilbert space.

Theoretical Framework and Quantum Metrology Context

A spin-$F$ system allows for the encoding of nonclassical states with metrological advantage. In particular, for an N-particle system or a spin-$F$ single particle, the SQL for phase sensitivity is $\Delta\phi_{\mathrm{SQL}} = 1/\sqrt{2F}$ for uncorrelated states; with nonclassical states such as the Schrödinger-cat ($$|\psi_{\pm F}\rangle = (|F,F\rangle + e^{i\phi}|F,-F\rangle)/\sqrt{2}$$), Heisenberg scaling $\Delta\phi_{\mathrm{HL}} = 1/(2F)$ is approached, reflecting maximal quantum Fisher information.

The highly nontrivial claim here is the maintenance of the full quantum Fisher information associated with these nonclassical states over an interrogation period $\tau$ exceeding $10^3$ s—a period orders of magnitude longer than typical for such states. This is achieved by encoding the superposition in a decoherence-free subspace of the spin-5/2 manifold, rendering the cat state immune to dominant sources of dephasing, such as inhomogeneous tensor light shifts from the optical lattice potential.

The Hamiltonian governing cat-state preparation includes a tensor light shift term ($F_z^2$) and a vector term ($F_x$). Control of the relevant Rabi frequencies, realized by polarization and intensity of the control laser, allows implementation of an effective nonlinear rotation to generate the required superposition.

Experimental Implementation

Atom Preparation and State Initialization

• $^{173}$Yb atoms are trapped in a high-intensity optical lattice at 1036 nm, achieving trap depths of 2.4 mK and atom vacuum lifetimes $>70$ s.
• State initialization is performed via optical pumping on the $\mathrm{^1S_0}\to\mathrm{^1P_1}$ transition, preparing the $|F, m = +5/2\rangle$ stretched state.

Cat State Generation

• A $\sigma^+$-polarized control laser, off-resonant from $\mathrm{^1S_0}\to\mathrm{^3P_1}$, provides the necessary coherent nonlinear interaction.
• Cat state preparation (i.e., $(\pi/2)_{\mathrm{cat}}$ pulse): A precise duration of the control pulse drives the system into the state $(|F,F\rangle - i|F,-F\rangle)/\sqrt{2}$, implemented with a measured Rabi frequency of $2\pi\times 0.52$ kHz.
• The ratio of linear to quadratic light shift Rabi frequencies can be set to exactly unity, maximizing selectivity into $\mathcal{H}_{\pm 5/2}$ with full control via detuning.

Measurement and Readout

• State-selective absorption imaging on the cycling $\mathrm{^1S_0}\to\mathrm{^3P_1}$ transition is achieved via differential light shifts in the optical lattice, enabling discrimination of $m = \pm 5/2$ states.
• Ramsey interferometry is employed to interrogate phase sensitivity and to benchmark decoherence properties for both cat and coherent spin state (CSS) superpositions.

Protection from Decoherence and Cat State Lifetime

A notable technical achievement is the protection of the cat state within the two-dimensional decoherence-free subspace $\mathcal{H}_{\pm 5/2}$. The $F_z^2$ tensor interaction from the lattice, the dominant bath-induced decohering mechanism, commutes with the cat-state density operator and the $F_z$ rotation associated with the phase shift to be measured. Thus, phase information remains immune to dephasing from inhomogeneous light shifts, and loss from $\mathcal{H}_{\pm 5/2}$ (to $|m|<5/2$ sublevels) is suppressed by the structure of the Hamiltonian and the careful engineering of control fields.

Contrastingly, the CSS is a superposition over all $|F,m\rangle$ sublevels and hence is not protected; it rapidly evolves into a statistical mixture among several doublets and loses quantum enhancement, retaining at best the SQL.

Numerically, an exponential decay fit to contrast yields a coherence time of $T_{2,\mathrm{cat}}^* = 1.4(1)\times 10^3$ s. The observed population in $\mathcal{H}_{\pm 5/2}$ remains $\sim 90\%$ after $160$ s Ramsey interrogation. For comparison, the CSS coherence time is reduced by an order of magnitude.

Metrological Performance and Sensitivity

Ramsey interferometry using the cat state gives oscillations in the population of $|F, \pm F\rangle$ at a frequency $2F$ times higher than that of the CSS, evidencing Heisenberg-limited phase sensitivity. For $F = 5/2$, the corresponding enhancement factor is $5$.

At $\tau = 160$ s interrogation time, the magnetic field sensitivity reaches 0.12(1) nT (single-shot), closely matching the Heisenberg limit of 0.10 nT and surpassing the SQL (0.22 nT) by a factor of 1.8. By contrast, the CSS under same conditions achieves only 0.7(1) nT sensitivity, performing worse than the SQL.

The population and phase contrast losses directly couple to the achieved quantum Fisher information. Practically, optimal performance is realized for interrogation times exceeding $100$ s before technical loss mechanisms dominate.

Implications and Outlook

Major implications:

• Demonstration of a single-particle Schrödinger-cat state with coherence on the order of $23$ minutes sets a new benchmark for robustness of macroscopic quantum superpositions in a high-dimensional Hilbert space.
• The underlying techniques—combining nonlinear spin control, decoherence-free subspaces, and advanced optical readout—are generically extensible to other high-spin atomic, ionic, or molecular platforms.
• Claimed improvement of 15(2) dB in sensitivity over CSS methods is operationally significant for atomic magnetometry, quantum memory, and precision measurement applications such as electric dipole moment searches and Lorentz-violating physics.

The work strongly demonstrates that decoherence in high-spin systems is contingent on the structure of the system-environment coupling and can be mitigated or removed by symmetry-based methods, suggesting new architectures for quantum metrology and information storage.

Future directions:

• Spin echo, vacuum improvements, and advanced error correction may extend coherence further.
• Integration with multi-particle entanglement protocols opens the path to scalable quantum sensors and quantum error-corrected magnetic or inertial sensors.
• The high intrinsic sensitivity of spin-5/2 systems, combined with minutes-scale coherence, makes them compelling candidates for next-generation comagnetometers and precision measurement of fundamental constants.

Conclusion

This work establishes the preparation, protection, and metrological application of a single-atom, spin-5/2 Schrödinger-cat state with a coherence time of $1.4(1)\times 10^3$ s. The demonstrated immunity to light shift-induced dephasing, Heisenberg-limit phase sensitivity, and compatibility with high atom number and other entanglement protocols provide a powerful framework for quantum-enhanced measurement and quantum information storage using alkaline-earth (and similar) atomic platforms. The verified claims of long lifetime and strong quantum enhancement elevate this approach as a model for advanced quantum metrological applications.