We report the generation of spin squeezing and entanglement in a magnetically-sensitive atomic ensemble, and entanglement-enhanced field measurements with this system. A maximal Raman coherence is prepared in an ensemble of 8.5x10^5 laser-cooled Rb-87 atoms in the f=1 hyperfine ground state, and the collective spin is squeezed by synthesized optical quantum non-demolition measurement. This prepares a state with large spin alignment and noise below the projection-noise level in a mixed alignment-orientation variable. 3.2dB of noise reduction is observed and 2.0dB of squeezing by the Wineland criterion, implying both entanglement and metrological advantage. Enhanced sensitivity is demonstrated in field measurements using alignment-to-orientation conversion.
Quantum projection noise will soon limit the best achievable precision of optical atomic clocks based on lattice-confined neutral atoms. Squeezing the collective atomic pseudo-spin via measurement of the clock state populations during Ramsey interrogation suppresses the projection noise. We show here that the lattice laser field can be used to perform ideal quantum non-demolition measurements without clock shifts or decoherence and explore the feasibility of such an approach in theory with the lattice field confined in a ring-resonator. Detection of the motional sideband due to the atomic vibr…
The relationship between orientation-to-alignment conversion (a form of atomic polarization evolution induced by an electric field) and the phenomenon of spin squeezing is demonstrated. A "stretched" state of an atom or molecule with maximum angular-momentum projection along the quantization axis possesses orientation and is a quantum-mechanical minimum-uncertainty state, where the product of the equal uncertainties of the angular-momentum projections on two orthogonal directions transverse to the quantization axis is the minimum allowed by the uncertainty relation. Application of an electric …
We provide a framework for understanding recent experiments on squeezing of a collective atomic pseudo-spin, induced by a homodyne measurement on off-resonant probe light interrogating the atoms. The detection of light decimates the atomic state distribution and we discuss the conditions under which the resulting reduced quantum fluctuations are metrologically relevant. In particular, we consider a dual probe scheme which benefits from a cancelation of classical common mode noise sources such that quantum fluctuations from light and atoms are the main contributions to the detected signal.
We present a generalized echo squeezing protocol (GESP) as a generalization of the Schrödinger cat state protocol (SCSP) with the value of the squeezing parameter being an arbitrary number rather than pi/2. We show analytically that over a broad range of the squeezing parameter the sensitivity reaches the Heisenberg limit (HL) within a factor of root-2. For a large number of particles, N, this plateau interval is almost the whole range from zero to pi/2, and the sensitivity is independent of the parity of N. Therefore, it is possible to operate a sensor over a wide interval of the squeezing pa…
For a squeezing-enhanced SU(2) interferometer, we theoretically investigate the possibility to broaden the phase range of sub-shot-noise sensitivity. We show that this goal can be achieved by implementing detection in both output ports, with the optimal combination of the detectors outputs, leading to a phase sensitivity independent of the interferometer operation point. Provided that each detector is preceded by a phase-sensitive amplifier, this sensitivity could be also tolerant to the detection loss.
We show that the sensitivity of an atomic clock can be enhanced below the shot-noise level by initially squeezing, and then measuring in output, the population of a single atomic level. This can simplify current experimental protocols which requires squeezing of the relative number of particles of the two populated states. We finally study, as a specific application, the clock sensitivity obtained with a single mode quantum non-demolition measurement.
The sensitivity properties of an SU(1,1) interferometer made of two cascaded parametric amplifiers, as well as of an ordinary SU(2) interferometer preceded by a squeezer and followed by an anti-squeezer, are theoretically investigated. Several possible experimental configurations are considered, such as the absence or presence of a seed beam, direct or homodyne detection scheme. In all cases we formulate the optimal conditions to achieve phase super-sensitivity, meaning a sensitivity overcoming the shot-noise limit. We show that for a given gain of the first parametric amplifier, unbalancing t…
The coherent population trapping (CPT) effect is used for making compact atomic clocks. There are two types of CPT clocks: the one in which the Raman beams are applied continuously and the one in which two CPT pulses separated by a dark period are applied (Ramsey scheme). It is obvious that the technique of spin squeezing can only be applied to the Ramsey CPT clock to enhance the sensitivity. However, it is not apparent how to adapt to the CPT clock the protocols for the microwave clock using one-axis-twist squeezing (OATS), since the Ramsey CPT clock is not trivially equivalent to the Ramsey …
We determine the theoretical limits to squeezing-enhanced measurement sensitivity of mechanical motion in a cavity optomechanical system. The motion of a mechanical resonator is transduced onto quadrature fluctuations of a cavity optical field and a measurement is performed on the optical field exiting the cavity. We compare measurement sensitivities obtained with coherent probing and quantum-enhanced probing of the mechanical motion, i.e. the coherent probe field carries vacuum states and quadrature squeezed vacuum states at sideband frequencies, respectively. We find that quantum-enhanced pr…
Spin squeezing is a form of entanglement that can improve the stability of quantum sensors operating with multiple particles, by inducing inter-particle correlations that redistribute the quantum projection noise. Previous analyses of potential metrological gain when using spin squeezing were performed on theoretically ideal states, without incorporating experimental imperfections or inherent limitations which result in non-unitary quantum state evolution. Here, we show that potential gains in clock stability are substantially reduced when the spin squeezing is non-unitary, and derive analytic…
We study the spin squeezing property of weighted graph states, which can be used to improve the sensitivity in interferometry. Decoherence reduces the spin squeezing property but the result remains superior over other reference schemes with GHZ-type maximally entangled states and product states. We study the time evolution of spin squeezing of weighted graph states coupled to different decoherence channels. Based on the analysis, the spin squeezing of the weighted graph states is robust in the presence of decoherence and the decoherence limit in the improvement of the interferometric sensitivi…
We study the effect of optical polarization squeezing on the performance of a sensitive, quantum-noise-limited optically pumped magnetometer. We use Bell-Bloom (BB) optical pumping to excite a $^{87}$Rb vapor containing $8.2 \cdot 10^{12} \mathrm{atoms/cm^3}$ and Faraday rotation to detect spin precession. The sub-$\mathrm{pT}/\sqrt{\mathrm{Hz}}$ sensitivity is limited by spin projection noise (photon shot noise) at low (high) frequencies. Probe polarization squeezing both improves high-frequency sensitivity and increases measurement bandwidth, with no loss of sensitivity at any frequency, a d…
Quantum enhancement and Doppler suppression of Kasevich-Chu atom interferometer with motional squeezing states
Zhejiang University of Technology（浙江工业大学）；Zhejiang Provincial Key Laboratory and Collaborative Innovation Center for Quantum Precise Measurement, Zhejiang University of Technology（浙江工业大学量子精密测量浙江省重点实验室与协同创新中心）；State Key Laboratory of Ocean Sensing and Institute of Quantum Sensing, School of Physics, Zhejiang University（浙江大学物理学院海洋感知国家重点实验室与量子传感研究所）
We investigate the presence of spin- and planar- squeezing in generalized superpositions of atomic (or spin) coherent states (ACS). Spin-squeezing has been shown to be a useful tool in determining the presence of entanglement in multipartite systems, such as collections of two-level atoms, as well as being an indication of reduced projection noise and sub-shot-noise limited phase uncertainty in Ramsey spectroscopy, suitable for measuring phases $ϕ\sim 0$. On the other hand, planar-squeezed states display reduced projection noise in two directions simultaneously and have been shown to lead to e…
We theoretically investigate the use of quantum non-demolition measurement to enhance the sensitivity of atom interferometry with Bose-condensed atoms. In particular, we are concerned with enhancing existing high-precision atom interferometry apparatuses, so restrict ourselves to dilute atomic samples, and the use of free-propagating light, or optical cavities in the weak-coupling regime. We find the optimum parameter regime that balances between spin squeezing and atomic loss, and find that significant improvements in sensitivity are possible. Finally, we consider the use of squeezed light, a…
Vibrational environments are commonly considered to be detrimental to the optical emission properties of solid-state and molecular systems, limiting their performance within quantum information protocols. Given that such environments arise naturally it is important to ask whether they can instead be turned to our advantage. Here we show that vibrational interactions can be harnessed within resonance fluorescence to generate optical states with a higher degree of quadrature squeezing than in isolated atomic systems. Considering the example of a driven quantum dot coupled to phonons, we demonstr…
Quantum metrology enables estimation of optical phase shifts with precision beyond the shot-noise limit. One way to exceed this limit is to use squeezed states, where the quantum noise of one observable is reduced at the expense of increased quantum noise for its complementary partner. Because shot-noise limits the phase sensitivity of all classical states, reduced noise in the average value for the observable being measured allows for improved phase sensitivity. However, additional phase sensitivity can be achieved using phase estimation strategies that account for the full distribution of me…
Spin squeezing enhanced dual species atom interferometric accelerometer employing large momentum transfer for precision test of the equivalence principle
We propose a new method of spin squeezing of atomic spin, based on the interactions between atoms and off-resonant light which are known as paramagnetic Faraday rotation and fictitious magnetic field of light. Since the projection process, squeezed light, or special interactions among the atoms are not required in this method, it can be widely applied to many systems. The attainable range of the squeezing parameter is S^{-2/5}, where S is the total spin, which is limited by additional fluctuations imposed by coherent light and the spherical nature of the spin distribution.
High precision interferometers are the building blocks of precision metrology and the ultimate interferometric sensitivity is limited by the quantum noise. Here we propose and experimentally demonstrate a compact quantum interferometer involving two optical parametric amplifiers and the squeezed states generated within the interferometer are directly used for the phase-sensing quantum state. By both squeezing shot noise and amplifying phase-sensing intensity the sensitivity improvement of $4.86\pm 0.24$ dB beyond the standard quantum limit is deterministically realized and a minimum detectable…
We explore the possibility to overcome the standard quantum limit (SQL) in a free-fall atom interferometer using a Bose-Einstein condensate (BEC) in either of the two relevant cases of Bragg or Raman scattering light pulses. The generation of entanglement in the BEC is dramatically enhanced by amplifying the atom-atom interactions via the rapid action of an external trap focusing the matter-waves to significantly increase the atomic densities during a preparation stage -- a technique we refer to as delta-kick squeezing (DKS). The action of a second DKS operation at the end of the interferometr…
We show that by displacing two optical lattices with respect to each other, we may produce interactions similar to the ones describing ferro-magnetism in condensed matter physics. We also show that particularly simple choices of the interaction lead to spin-squeezing, which may be used to improve the sensitivity of atomic clocks. Spin-squeezing is generated even with partially, and randomly, filled lattices, and our proposal may be implemented with current technology.
Application of optical squeezing to microresonator based optical sensors
Faculty of Physics, M.V. Lomonosov Moscow State University（莫斯科大学物理系）；Russian Quantum Center（俄罗斯量子中心）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
Interferometers are crucial for precision measurements, including gravitational waves, laser ranging, radar, and imaging. The phase sensitivity, the core parameter, can be quantum-enhanced to break the standard quantum limit (SQL) using quantum states. However, quantum states are highly fragile and quickly degrade with losses. We design and demonstrate a quantum interferometer utilizing a beam splitter with a variable splitting ratio to protect the quantum resource against environmental impacts. The optimal phase sensitivity can reach the quantum Cramér-Rao bound of the system. This quantum in…
We theoretically study the angular displacements estimation based on a modified Mach-Zehnder interferometer (MZI), in which two optical parametric amplifiers (PAs) are introduced into two arms of the standard MZI, respectively. The employment of PAs can both squeeze the shot noise and amplify the photon number inside the interferometer. When the unknown angular displacements are introduced to both arms, we derive the multiparameter quantum Cramér-Rao bound (QCRB) using the quantum Fisher information matrix approach, and the bound of angular displacements difference between the two arms is comp…
We demonstrate the generation of broad-band continuous-wave optical squeezing down to 200Hz using a below threshold optical parametric oscillator (OPO). The squeezed state phase was controlled using a noise locking technique. We show that low frequency noise sources, such as seed noise, pump noise and detuning fluctuations, present in optical parametric amplifiers have negligible effect on squeezing produced by a below threshold OPO. This low frequency squeezing is ideal for improving the sensitivity of audio frequency measuring devices such as gravitational wave detectors.
The detection of kilohertz-band gravitational waves promises discoveries in astrophysics, exotic matter, and cosmology. To improve the kilohertz quantum noise-limited sensitivity of interferometric gravitational-wave detectors, we investigate nondegenerate internal squeezing: optical parametric oscillation inside the signal-recycling cavity with distinct signal-mode and idler-mode frequencies. We use an analytic Hamiltonian model to show that this stable, all-optical technique is tolerant to decoherence from optical detection loss and that it, with its optimal readout scheme, is feasible for b…
Cavity-QED is a promising avenue for the deterministic generation of entangled and spin-squeezed states for quantum metrology. One archetypal scheme generates squeezing via collective one-axis twisting interactions. However, we show that in implementations using optical transitions in long-lived atoms the achievable squeezing is fundamentally limited by collectively enhanced emission into the cavity mode which is generated in parallel with the cavity-mediated spin-spin interactions. We propose an alternative scheme which generates a squeezed state that is protected from collective emission, an…
Squeezing of collective atomic spins has been shown to improve the sensitivity of atomic clocks and magnetometers to levels significantly below the standard quantum limit. In most cases the requisite atom-atom entanglement has been generated by dispersive interaction with a quantized probe field, or by state dependent collisions in a quantum gas. Such experiments typically use complex multilevel atoms like Rb or Cs, with the relevant interactions designed so atoms behave like pseudo-spin-$1/2$ particles. We demonstrate the viability of spin squeezing for collective spins composed of the physic…
Probing the collective spin state of an ensemble of atoms may provide a means to reduce heating via the photon recoil associated with the measurement and provide a robust, scalable route for preparing highly entangled states with spectroscopic sensitivity below the standard quantum limit for coherent spin states. The collective probing relies on obtaining a very large optical depth that can be effectively increased by placing the ensemble within an optical cavity such that the probe light passes many times through the ensemble. Here we provide expressions for measurement resolution and spectro…
We measure the sensitivity of a broadband atomic magnetometer using quantum non-demolition spin measurements. A cold, dipole-trapped sample of rubidium atoms provides a long-lived spin system in a non-magnetic environment, and is probed non-destructively by paramagnetic Faraday rotation. The calibration procedure employs a known reference state, the maximum-entropy or 'thermal' spin state and quantitative imaging-based atom counting to identify electronic, quantum, and technical noise in both the probe and spin system. The measurement achieves sensitivity 2.8 dB better than the projection nois…
We analyze a model for spin squeezing based on the so-called counter-twisting Hamiltonian, including the effects of dissipation and finite system size. We discuss the conditions under which the Heisenberg limit, i.e. phase sensitivity $\propto 1/N$, can be achieved. A specific implementation of this model based on atom-atom interactions via quantized photon exchange is presented in detail. The resulting excitation corresponds to the creation of spin-flipped atomic pairs and can be used for fast generation of entangled atomic ensembles, spin squeezing and apllications in quantum information pro…
We describe and characterize a simple, low cost, low phase noise microwave source that operates near 6.800 GHz for agile, coherent manipulation of ensembles of 87Rb. Low phase noise is achieved by directly multiplying a low phase noise 100 MHz crystal to 6.8 GHz using a non-linear transmission line and filtering the output with custom band-pass filters. The fixed frequency signal is single sideband modulated with a direct digital synthesis frequency source to provide the desired phase, amplitude, and frequency control. Before modulation, the source has a single sideband phase noise near -140 d…
A Michelson-type interferometer with two-mode squeezed coherent state input is considered. Such an interferometer has a better phase sensitivity over the shot-noise limit by a factor of $e^{2r}$, where $r$ is the squeezing parameter [Phys. Rev. A 102,022614 (2020)]. We show that when photon loss and noise in the two arms is asymmetric an optimal choice of the squeezing angle can allow improvement in phase sensitivity without any increase in input or pump power. In particular, when loss occurs only in one arm of the interferometer, we can have improvement in phase sensitivity for photon loss up…
We report on the experimental demonstration of strong quadrature EPR entanglement and squeezing at very low noise sideband frequencies produced by a single type-II, self-phase-locked, frequency degenerate optical parametric oscillator below threshold. The generated two-mode squeezed vacuum state is preserved for noise frequencies as low as 50 kHz. Designing simple setups able to generate non-classical states of light in the kHz regime is a key challenge for high sensitivity detection of ultra-weak physical effects such as gravitational wave or small beam displacement.
This paper reviews quantum spin squeezing, which characterizes the sensitivity of a state with respect to an SU(2) rotation, and is significant for both entanglement detection and high-precision metrology. We first present various definitions of spin squeezing parameters, explain their origin and properties for typical states, and then discuss spin-squeezed states produced with the Ising and the nonlinear twisting Hamiltonians. Afterwards, we explain correlations and entanglement in spin-squeezed states, as well as the relations between spin squeezing and quantum Fisher information, where the …
The sensitivity of classical Raman spectroscopy methods, such as Coherent Anti-Stokes Raman spectroscopy (CARS) or Stimulated Raman spectroscopy (SRS), is ultimately limited by shot-noise from the stimulating fields. We present the complete theoretical analysis of a squeezing-enhanced version of Raman spectroscopy that overcomes the shot-noise limit of sensitivity with enhancement of the Raman signal and inherent background suppression, while remaining fully compatible with standard Raman spectroscopy methods. By incorporating the Raman sample between two phase-sensitive parametric amplifiers …
We derive a phase-entanglement criterion for two bosonic modes which is immune to number fluc- tuations, using the generalized Moore-Penrose inverse to normalize the phase-quadrature operator. We also obtain a phase-squeezing criterion that is immune to number fluctuations using similar techniques. These are utilized to obtain an operational definition of relative phase-measurement sensitivity, via analysis of phase measurement in interferometry. We show that these measures are proportional to enhanced phase-measurement sensitivity. The phase-entanglement criterion is a hallmark for a new type…
We address a phase estimation scheme using Gaussian states in the presence of non-Gaussian phase noise. At variance with previous analysis, we analyze situations in which the noise occurs before encoding phase information. In particular, we study how squeezing may be profitably used before or after phase diffusion. Our results show that squeezing the probe after the noise greatly enhances the sensitivity of the estimation scheme, as witnessed by the increase of the quantum Fisher information. We then consider a realistic setup where homodyne detection is employed at the measurement stage, and …
