Entanglement-enhanced matter-wave interferometry in a high-finesse cavity

Matter-wave interferometers, utilizing the quantized momentum kicks imparted during light absorption and emission, offer high-precision measurements by encoding information in the phase of superimposed quantum trajectories. The standard quantum limit (SQL) dictates the fundamental precision for N atoms, with an angular uncertainty of Δθ = 1/√N rad. This research aims to exceed the SQL by leveraging quantum entanglement between atoms, enabling them to collectively reduce quantum noise relative to the signal. Previous work has demonstrated entanglement between atoms using various methods, including collisional or Coulomb interactions, but often resulted in phase resolutions above the SQL. This paper introduces a novel matter-wave interferometer operating within a high-finesse cavity, enabling strong collective coupling between atoms and an optical cavity mode. This strong coupling allows for the generation of entanglement in the atoms' external degrees of freedom, which is then used to enhance the precision of the interferometer. The ability to surpass the SQL using entangled atoms opens new avenues for improved inertial sensors, exploring fundamental physics beyond the standard model, and advancing gravitational wave detectors.

Significant progress has been made in generating entanglement in atomic ensembles. Methods include direct collisional or Coulomb interactions, leading to relative atom number squeezing or mapping of internal entanglement onto momentum states. However, previous trapped matter-wave interferometers with relative number squeezing exhibited antisqueezed phases, resulting in phase resolutions above the SQL. Generating entanglement in external degrees of freedom, as opposed to internal states, is a significant challenge. While large amounts of entanglement have been generated in internal states using cavity QED systems, its application to enhance the sensitivity of a matter-wave interferometer has not been demonstrated until this study. Previous works have explored cavity approaches to one-axis twisting (OAT) and quantum non-demolition (QND) entanglement for Bragg interferometers but did not achieve sub-SQL sensitivity for a Mach-Zehnder interferometer involving entangled external degrees of freedom.

The experiment utilizes a vertically oriented high-finesse optical cavity with a finesse of 1.3 x 10^5 and a small mode waist of 72 µm. Laser-cooled Rubidium atoms are guided in free fall along the cavity axis using a blue-detuned optical dipole guide. Matter-wave manipulation is achieved using velocity-sensitive two-photon Raman transitions, imparting quantized momentum kicks. Two approaches are employed to generate entanglement: 1) Cavity-enhanced QND measurements, where quantum noise is effectively measured and subtracted. The cavity resonance frequency shifts based on the number of atoms in specific spin-momentum states. By measuring this shift, the population in a specific state can be estimated without revealing single-atom information. 2) Cavity-mediated OAT interactions, where unitary interactions between atoms generate entanglement. The collective state is described using a Bloch sphere, visualizing the average Bloch vector and quantum noise distribution. Both QND and OAT methods generate squeezed momentum states. The QND approach involves pre-measuring the quantum noise in one spin-momentum projection and subsequently subtracting it from a final measurement. The OAT approach uses cavity-mediated interactions to generate a squeezed state. The generated entangled state is then injected into a Mach-Zehnder light-pulse interferometer. Raman beam splitter pulses are used to orient the squeezing for enhanced phase sensitivity. The interferometer involves free evolution periods, mirror pulses to re-overlap the wave packets, and readout pulses to measure the final population difference. The spectroscopic enhancement W, characterizing the phase enhancement, is calculated relative to the SQL.

The researchers successfully generated entangled momentum states using two distinct methods: QND measurements and OAT interactions. The QND measurements yielded a directly observed squeezing of 3.4 ± 1.1 dB below the SQL. The OAT interactions produced squeezing of 2.5 ± 0.6 dB below the SQL. These squeezed states were then injected into a Mach-Zehnder interferometer, resulting in a directly observed spectroscopic enhancement of 1.7 ± 0.3 dB below the SQL. The enhancement is limited by evolution times less than 0.7 ms, possibly due to spin decoherence. When the squeezed spin projection is maintained in the population basis, squeezing persists for several milliseconds, implying the entangled state's lifetime extends beyond the observable range. Both Raman and Bragg transitions were employed in the interferometer, allowing for large momentum transfers and the possibility of minimizing systematic errors and dephasing in future iterations. The study thoroughly characterizes the generation and application of entangled states, revealing that OAT generated states are more effective for interferometry due to lower atom number and smaller momentum spread, which results in less added rotation noise.

The results demonstrate the successful generation and utilization of entanglement in a matter-wave interferometer, exceeding the SQL and achieving significant improvements in phase sensitivity. The use of a high-finesse cavity enhances the collective coupling strength, crucial for efficient entanglement generation. The two methods, QND measurements and OAT interactions, provide distinct approaches to create the desired squeezed states, with OAT being more successful in improving interferometric sensitivity within the time scales studied. The observed limitations, primarily the limited coherence time for squeezing in the interferometer, point towards future directions for improvement. The findings have broad implications for various fields, including enhanced inertial sensing, searches for new physics, and advanced gravitational wave detectors.

This study presents a significant advance in matter-wave interferometry by demonstrating the successful use of cavity-generated entanglement in a Mach-Zehnder interferometer, surpassing the standard quantum limit. Both QND and OAT approaches demonstrated the creation of entangled momentum states, ultimately enhancing the interferometer's sensitivity. Future research should focus on extending the coherence time of entangled states, potentially by using large momentum transfers or employing lattice interferometers. Furthermore, optimizing atom number and density, and mitigating classical noise sources, could lead to substantial further improvements in sensitivity.

The primary limitation of the study is the observed decrease in phase sensitivity beyond 0.7 ms in the Mach-Zehnder interferometer, possibly due to decoherence affecting the spin degrees of freedom. The relatively low quantum efficiency (q=0.1) in QND measurements also impacts the level of squeezing achieved. Improving atom number and momentum space density through enhanced cooling techniques could improve the results. Mitigating classical noise sources is crucial for scaling up atom numbers, which is necessary to achieve even higher levels of sensitivity.