The use of entangled states was shown to improve the fundamental limits of spectroscopy to beyond the standard-quantum limit. Here, rather than probing the free evolution of the phase of an entangled state with respect to a local oscillator, we probe the evolution of an initially separable two-atom register under an Ising spin Hamiltonian with a transverse field. The resulting correlated spin-rotation spectrum is twice as narrow as that of an uncorrelated rotation. We implement this ideally Heisenberg-limited Rabi spectroscopy scheme on the optical-clock electric-quadrupole transition of ^{88}Sr^{+} using a two-ion crystal. We further show that depending on the initial state, correlated rotation can occur in two orthogonal subspaces of the full Hilbert space, yielding entanglement-enhanced spectroscopy of either the average transition frequency of the two ions or their difference from the mean frequency. The use of correlated spin rotations can potentially lead to new paths for clock stability improvement.
The use of entangled states was shown to improve the fundamental limits of spectroscopy to beyond the standard-quantum limit. Here, rather than probing the free evolution of the phase of an entangled state with respect to a local oscillator, we probe the evolution of an initially separable two-atom register under an Ising spin Hamiltonian with a transverse field. The resulting correlated spin-rotation spectrum is twice as narrow as that of an uncorrelated rotation. We implement this ideally Heisenberg-limited Rabi spectroscopy scheme on the optical-clock electric-quadrupole transition of ^{88}Sr^{+} using a two-ion crystal. We further show that depending on the initial state, correlated rotation can occur in two orthogonal subspaces of the full Hilbert space, yielding entanglement-enhanced spectroscopy of either the average transition frequency of the two ions or their difference from the mean frequency. The use of correlated spin rotations can potentially lead to new paths for clock stability improvement.
Entangled states hold the promise of improving the precision and accuracy of quantum sensors. We experimentally demonstrate that spectroscopy of an optical clock transition using entangled states can outperform its classical counterpart. Two ^{40}Ca^{+} ions are entangled in a quantum state with vanishing first-order magnetic field sensitivity, extending the coherence time of the atoms and enabling near-lifetime-limited probe times of up to 550 ms. In our protocol, entangled ions reach the same instability as uncorrelated ions, but at half the probe time, enabling faster cycle times of the clock. We run two entangled ^{40}Ca^{+} ions as an optical clock and compare its frequency instability with a ^{87}Sr lattice clock. The instability of the entangled ion clock is below a clock operated with classically correlated states for all probe times. We observe instabilities below the theoretically expected quantum projection noise limit of two uncorrelated ions for interrogation times below 100 ms. The lowest fractional frequency instability of 7×10^{-16}/sqrt[τ/1 s] is reached for 250 ms probe time, limited by residual phase noise of the probe laser. This represents the lowest instability reported to date for a ^{40}Ca^{+} ion clock.
Ramsey interferometry is a widely used tool for precisely measuring transition frequencies between two energy levels of a quantum system, with applications in time keeping, precision spectroscopy, quantum optics, and quantum information. Often, the coherence time of the quantum system surpasses the one of the oscillator probing the system, thereby limiting the interrogation time and associated spectral resolution. Correlation spectroscopy overcomes this limitation by probing two quantum systems with the same noisy oscillator for a measurement of their transition frequency difference; this technique has enabled very precise comparisons of atomic clocks. Here, we extend correlation spectroscopy to the case of multiple quantum systems undergoing strong correlated dephasing. We model Ramsey correlation spectroscopy with N particles as a multiparameter phase estimation problem and demonstrate that multiparticle correlations can assist in reducing the measurement uncertainties even in the absence of entanglement. We derive precision limits and optimal sensing techniques for this problem and compare the performance of probe states and measurement with and without entanglement. Using one- and two-dimensional ion Coulomb crystals with up to 91 qubits, we experimentally demonstrate the advantage of measuring multiparticle correlations for reducing phase uncertainties and apply correlation spectroscopy to measure ion-ion distances, transition frequency shifts, laser-ion detunings, and path-length fluctuations. Our method can be straightforwardly implemented in experimental setups with globally coherent qubit control and qubit-resolved single-shot readout and is, thus, applicable to other physical systems such as neutral atoms in tweezer arrays. Published by the American Physical Society 2024
Optical atomic clocks are our most precise tools to measure time and frequency [1]–[3]. They enable precision frequency comparisons between atoms in separate locations to probe the space-time variation of fundamental constants [4] and the properties of dark matter [5], [6], to perform geodesy [4], [7], [8] and to evaluate systematic clock shifts. Measurements on independent systems are limited by the standard quantum limit (SQL); measurements on entangled systems, in contrast, can surpass the SQL to reach the ultimate precision allowed by quantum theory - the so-called Heisenberg limit. While local entangling operations have been used to demonstrate this enhancement at microscopic distances [9]–[14], frequency comparisons between remote atomic clocks require rapid high-fidelity entanglement between separate systems that have no intrinsic interactions.
This thesis describes new results on the entanglement of atomic spins in Ramsey interferometry, optical atomic clocks and trapped ions. It is divided into three parts: First, we investigate improvements to conventional Ramsey interferometry with entanglement and adding only rotations of the collective spin to adjust the signal and measurement directions. The geometric degrees of freedom, connected to the rotations, are analytically optimized for a large class of generalized Ramsey protocols to allow efficient optimization of all parameters. Besides a unification of existing approaches, the main result is that there is only one new protocol, where a previously unused double inversion is applied. Studies of the local sensitivity show that this protocol reaches the fundamental quantum Fisher information limit and is yet robust against errors during preparation and measurement. In the second section we investigate the conditions under which optical atomic clocks exhibit increased long-term stability when applying weakly entangled, spin squeezed states. We discuss the common case of an atomic clock with a single ensemble, typical Brownian frequency noise and finite dead time. Theoretical modelling of the servo loop allows quantitative predictions of the optimal stability for given values of dead time and laser noise, in very good agreement with numerical simulations of the closed feedback loop. The main result is that, even with the current most stable lasers, the clock stability can only be improved for ensembles below a critical atom number of about one thousand in optical Sr lattice clocks. Even with a future improvement of the laser performance by one order of magnitude, the critical atom number still remains below 100,000. In contrast, clocks based on smaller, non-scalable ensembles, such as ion clocks, can already benefit from squeezed states with current clock lasers. Thus the last section considers the robust generation of entanglement in ion traps. An error budget including relevant experimental error sources is calculated for state-of-the-art quantum gates, driven by oscillating microwave gradients in surface traps. Amplitude modulation of the driving fields is shown to efficiently counteract the current limitations from motional mode instability. The predicted increase of the gate quality was demonstrated by the group of C. Ospelkaus at PTB Braunschweig, who measured gates with errors as low as ~ 10^(-3). In a similar approach, interactions between spin and motion can also be generated by combining oscillating rf-fields with a static magnetic field gradient. Penning traps designed for precision spectroscopy already feature large magnetic field gradients at the edge of a magnetic bottle configuration. We present parameters and conditions under which laser-free coupling of spin and quantized motion for (anti-)protons is possible at these points, in a step towards quantum logic spectroscopy for (anti-)protons.
We experimentally investigate three methods, utilizing different atomic observables and entangled states, to increase the sensitivity of rotation angle measurements beyond the "standard quantum limit" for nonentangled states. All methods use a form of quantum mechanical "squeezing." In a system of two entangled trapped (9)Be(+) ions we observe a reduction in uncertainty of rotation angle below the standard quantum limit for all three methods including all sources of noise. As an application, we demonstrate an increase in precision of frequency measurement in a Ramsey spectroscopy experiment.
We investigate the properties of angular-momentum states which yield high sensitivity to rotation. We discuss the application of these ``squeezed-spin'' or correlated-particle states to spectroscopy. Transitions in an ensemble of N two-level (or, equivalently, spin-1/2) particles are assumed to be detected by observing changes in the state populations of the particles (population spectroscopy). When the particles' states are detected with 100% efficiency, the fundamental limiting noise is projection noise, the noise associated with the quantum fluctuations in the measured populations. If the particles are first prepared in particular quantum-mechanically correlated states, we find that the signal-to-noise ratio can be improved over the case of initially uncorrelated particles. We have investigated spectroscopy for a particular case of Ramsey's separated oscillatory method where the radiation pulse lengths are short compared to the time between pulses. We introduce a squeezing parameter ${\ensuremath{\xi}}_{\mathit{R}}$ which is the ratio of the statistical uncertainty in the determination of the resonance frequency when using correlated states vs that when using uncorrelated states. More generally, this squeezing parameter quantifies the sensitivity of an angular-momentum state to rotation. Other squeezing parameters which are relevant for use in other contexts can be defined. We discuss certain states which exhibit squeezing parameters ${\ensuremath{\xi}}_{\mathit{R}}$\ensuremath{\simeq}${\mathit{N}}^{\mathrm{\ensuremath{-}}1/2}$. We investigate possible experimental schemes for generation of squeezed-spin states which might be applied to the spectroscopy of trapped atomic ions. We find that applying a Jaynes-Cummings--type coupling between the ensemble of two-level systems and a suitably prepared harmonic oscillator results in correlated states with ${\ensuremath{\xi}}_{\mathit{R}}$1.