Continuous Real-Time Tracking of a Quantum Phase Below the Standard Quantum Limit

Abstract

We propose a scheme for continuously measuring the evolving quantum phase of a collective spin composed of N pseudospins. Quantum non-demolition measurements of a lossy cavity mode interacting with an atomic ensemble are used to directly probe the phase of the collective atomic spin without converting it into a population difference. Unlike traditional Ramsey measurement sequences, our scheme allows for real-time tracking of time-varying signals. As a bonus, spinsqueezed states develop naturally, providing real-time phase estimation significantly more precise than the standard quantum limit of ∆φSQL = 1/ √ N radians.Quantum systems have become robust platforms for metrology and tests of fundamental physics. Many applications rely on the dynamics of pseudospin-1/2 systems with two long-lived quantum states, |↑ and |↓ . After preparing an equal superposition of these two states, a physical interaction is studied by investigating its effect on the relative phase φ(t), with the state of each spin evolving in time as |ψ(t) = |↓ + e iφ(t) |↑ / √ 2. We propose a novel scheme that enables continuous tracking of this relative phase. Our scheme continuously and directly measures the real-time phase φ(t) unlike the widely used Ramsey sequence [1][2][3][4][5][6][7][8][9][10][11][12], which indirectly measures the net accumulated phase φ(T ) during an interrogation time T . The typically destructive readout in a Ramsey sequence requires multiple state resets, rotations and repetitions of the sequence to infer the phase at different times from a population difference. In contrast, a single run of our protocol yields a continuous time series of phase measurements. Therefore, our scheme enables real-time tracking of time-varying signals that are not reproducible.As an added benefit, our scheme yields continuous phase estimates with precision well beyond the standard quantum limit (SQL) of ∆φ SQL = 1/ √ N radians that limits readout precision with N unentangled spins. In comparison to several proposals and experiments [13][14][15][16][17][18][19] that have demonstrated squeezed states with precision beyond the SQL, our scheme enjoys the advantage that the squeezing is produced, the phase accumulated, and the readout performed, all in the same spin quadrature.Recent experiments have demonstrated phase tracking of a spin using quantum non-demolition (QND) measurements via a Faraday rotation angle [20]. In contrast, our proposal is based on interfering Raman transitions in a cavity and enables an intuitive interpretation of phase tracking in terms of elementary atom-cavity interactions that nearly balance one another. Our scheme directly reveals a phasor precessing in the equatorial plane of a Bloch sphere, in the spirit of the "hand on a clock" analogy at the core of quantum metrology.We represent the collective angular momentum of N atomic spins by a classical Bloch vector of length N/2 FIG. 1. Schematic and working principle. (a) Two lasers drive a collection of atoms to interact with a cavity mode. The relative phase φ(t) can be...

Atom interferometry with quantized light pulses

“…Building on recently explored cavity schemes 13,14 for atom interferometry and ideas to generate entangled COM states 16,18,19 , our analysis brings out additional effects of quantum light on atom interferometers.…”

“…Indeed, a) fabio.di-pumpo@uni-ulm.de configurations based on cavities offer major experimental advantages such as higher intensities, a reduction of wave-front distortions, and clearer mode profiles. Moreover, diffraction in cavities can be employed for the generation of entanglement 15,16 or squeezed atomic states 17 , and for quantum non-demolition measurements 18,19 . In light of such developments, it seems natural to study the influence of quantized light on atom interferometry, even though these implementations are yet far away from the quantized regime.…”

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“…Conceptually different approaches that may evade the presented limitations when applied without dead time are based on continuously tracking the atomic phase via weak measurements [64][65][66] .…”

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