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Physical Review A

Squeezed atomic states and projection noise in spectroscopy

D. J. Wineland, J. J. Bollinger, Wayne M. Itano, Daniel J. Heinzen

Published Jul 1, 1994• 1,161 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.50.67

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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 \ensuremathξR{\ensuremath{\xi}}_{\mathit{R}}\ensuremathξR​ 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 \ensuremathξR{\ensuremath{\xi}}_{\mathit{R}}\ensuremathξR​\ensuremath{\simeq}N\ensuremath−1/2{\mathit{N}}^{\mathrm{\ensuremath{-}}1/2}N\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 \ensuremathξR{\ensuremath{\xi}}_{\mathit{R}}\ensuremathξR​1.
We investigate the quantum-mechanical noise in spectroscopic experiments on ensembles of N two-level (or spin-1/2) systems where transitions are detected by measuring changes in state population. By preparing correlated states, here called squeezed spin states, we can increase the signal-to-noise ratio in spectroscopy (by approximately N1/2{\mathit{N}}^{1/2}N1/2 in certain cases) over that found in experiments using uncorrelated states. Possible experimental demonstrations of this enhancement are discussed.
We discuss the possibilities for using the correlation spectroscopy technique to study an atomic medium polarized in angular momentum. The general results of our previous paper are applied here to the analysis of the density matrix fluctuations of an atomic medium with a spin ground state. Such systems are characterized by a long ground state relaxation time in comparison with the excited state natural lifetime, so that any fluctuation of the atomic density matrix polarization is slowly dependent on the time process. We show how the quantum behaviour of low-frequency fluctuations can be observed via quantum features of scattered light statistics. In particular, we discuss the squeezing of transmitted light caused by the spin angular momentum orientation in the direction of an external magnetic field. This squeezing effect is explained by quantum non-commutation of angular momentum projections which are transverse to the magnetic field. Other applications of the polarization-sensitive correlation spectroscopy technique, such as the possible observation of the average atomic density matrix components and the measurement of different spectroscopic characteristics of the atomic ensemble, are also discussed.
Ultimate quantum fluctuations which limit the fundamental precision in metrology can be only reduced by entanglement of particles. In particular, squeezing of the fluctuations called the spin projection noise by means of generation of entanglement in an ensemble of atoms can improve the precision of spectroscopy and of atomic clocks. A powerful tool for generation of multiparticle entanglement is a quantum nondemolition measurement (QND). We will describe the fundamentals of QND measurement of atomic collective state with light and present a recent experiment [1] where this method has been used to reduce the projection noise on the clock transition in an ensemble of 105 cold and trapped Cesium atoms. Another method for generation of spin squeezing via electron-nucleus entanglement applicable only for atoms with angular momentum higher than ½ will be also discussed [2].
Large ensembles of uncorrelated atoms are extensively used as precise sensors of time, rotation, and gravity, and for tests of fundamental physics. The quantum nature of the sensors imposes a limit on their ultimate precision. Larger ensembles of N atoms can be used to average the quantum noise as 1/√N, a scaling known as the standard quantum limit. However, the ensemble size may be limited by technical constraints and/or atom-atom collisions -- a fundamental distinction from photon-based sensors. Learning to prepare entangled states of large ensembles with noise properties below the standard quantum limit will be key to extending both the precision and/or bandwidth of atomic sensors. More broadly, the generation and application of entanglement to solve problems is a core goal of quantum information science being pursued in both atomic and solid state systems.\nIn this thesis, we utilize the tools of cavity-QED to prepare entangled spin-squeezed states with 3.4(6) dB improvement in spectroscopic sensitivity over the standard quantum limit. The collective atomic spin is composed of the two-level clock states of 87Rb confined in a medium finesse F = 710 optical cavity. We employ cavity-aided quantum non-demolition measurements of the vacuum Rabi splitting to measure and subtract out the quantum projection noise of the collective spin state, preparing states with collective atomic spin projection noise 4.9(6) dB below the projection noise level. The conditionally reduced spin noise combined with the measured 1.5(3) dB reduction in the mean spin length implies a net 3.4(6) dB spectroscopic enhancement or conditional squeezing as defined by the Wineland criterion. Our method does not require single particle addressability and is applied to a spectroscopically large ensemble of N = 7 x 105 atoms using two collective population measurements, with the whole squeezing operation taking ~ 150 μs. The gain in sensitivity is spectroscopically equivalent to the enhancement obtained had we created > 105 pairs of maximally entangled qubits, demonstrating the power of a top-down approach for entangling large ensembles. The nondemolition probing of atomic populations via the vacuum Rabi splitting is also of broad interest for non-destructively reading out a wide variety of both atomic and solid state qubits.
We study the fundamental limits of noise spectroscopy using estimation theory, Faraday rotation probing of an atomic spin system, and squeezed light. We find a simple and general expression for the Fisher information, which quantifies the sensitivity to spectral parameters such as resonance frequency and linewidth. For optically detected spin noise spectroscopy, we find that shot noise imposes ``local'' standard quantum limits for any given probe power and atom number, and also ``global'' standard quantum limits when the probe power and atom number are taken as free parameters. We confirm these estimation theory results using nondestructive Faraday rotation probing of hot Rb vapor, observing the predicted optima, and finding good quantitative agreement with a first-principles calculation of the spin noise spectra. Finally, we show sensitivity beyond the atom- and photon-number-optimized global standard quantum limit using squeezed light.
Spin squeezing has been explored in atomic systems as a tool for quantum sensing, improving experimental sensitivity beyond the spin standard quantum limit for certain measurements. To optimize absolute metrological sensitivity, it is beneficial to consider macroscopic spin ensembles, such as nuclear spins in solids and liquids. Coupling a macroscopic spin ensemble to a parametrically-modulated resonant circuit can create collective spin squeezing by generating spin correlations mediated by the circuit. We analyze the squeezing dynamics in the presence of decoherence and finite spin polarization, showing that achieving 7 dB spin squeezing is feasible in several nuclear spin systems. The metrological benefit of squeezing a macroscopic spin ensemble lies in the suppression of technical noise sources in the spin detection system relative to the spin projection noise. This expands the experimental sensitivity bandwidth when searching for signals of unknown frequency and can improve the resonant signal-to-noise ratio. Squeezing macroscopic spin ensembles may prove to be a useful technique for fundamental physics experiments aimed at detecting spin interactions with oscillating background fields, such as ultralight dark matter.