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arXiv:2401.15553

Unitary and efficient spin squeezing in cavity optomechanics

Lei Xie, Zhiqi Yan, Lingxia Wang, Di Wang, Jinfeng Liu, Yiling Song, Wei Xiong, Mingfeng Wang

Published Jan 28, 2024• 1 citation
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Abstract

We propose an approach to produce spin squeezed states of a large number of nitrogen-vacancy centers in diamond nanostructures coupled to an optical cavity. Unlike the previous squeezing method proposed by Bennett et al. [Phys. Rev. Lett. 110, 156402 (2013)], which is limited by phonon number fluctuations due to the existence of phonon-spin entanglement, our proposal can completely erase the entanglement between spins and hybrid phonon-photon mode mediating the effective spin-spin interaction, and thus achieves unitary one-axis-twisting interactions between nitrogen-vacancy centres, yielding a squeezing scaling $J^{-2/3}$, where J is the total angular momentum. We found that, under certain conditions, our method has the potential to enhance the spin-spin nonlinear interactions. We also proposed a scheme utilizing repeatedly applying the one-axis-twisting evolution to two orthogonal spin directions, which enables the transformation of the one-axis-twisting interactions into two-axis-twisting type, and therefore leads to the spin squeezing with Heisenberg-limited scaling $J^{-1}$. Taking into account the noise effects of spin dephasing and relaxtion, we found that the proposed approaches are robust against imperfections.

DOI
10.48550/arxiv.2401.15553

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We propose an approach to produce spin squeezed states of a large number of nitrogen-vacancy centers in diamond nanostructures coupled to an optical cavity. Unlike the previous squeezing method proposed by Bennett et al. [Phys. Rev. Lett. 110, 156402 (2013)], which is limited by phonon number fluctuations due to the existence of phonon-spin entanglement, our proposal can completely erase the entanglement between spins and hybrid phonon-photon mode mediating the effective spin-spin interaction, and thus achieves unitary one-axis-twisting interactions between nitrogen-vacancy centres, yielding a squeezing scaling J−2/3J^{-2/3}J−2/3, where J is the total angular momentum. We found that, under certain conditions, our method has the potential to enhance the spin-spin nonlinear interactions. We also proposed a scheme utilizing repeatedly applying the one-axis-twisting evolution to two orthogonal spin directions, which enables the transformation of the one-axis-twisting interactions into two-axis-twisting type, and therefore leads to the spin squeezing with Heisenberg-limited scaling J−1J^{-1}J−1. Taking into account the noise effects of spin dephasing and relaxtion, we found that the proposed approaches are robust against imperfections.
Squeezed spin states have important applications in quantum metrology and sensing. It has been shown by S\o{}rensen and M\o{}lmer [Phys. Rev. A 66, 022314 (2002)] that an effective one-axis-twisting interaction can be realized in a cavity setup via a double off-resonance stimulated Raman scattering, resulting in a noise reduction scaling $\ensuremath{\propto}1/{N}^{2/3}$ with NNN being the atom number. Here, we show that, by making an appropriate change of the initial input spin state, it is possible to produce a one-axis-twisting spin squeezing via a single off-resonance stimulated Raman scattering, which thus can greatly simplify the realistic implementation. We also show that the one-axis-twisting interaction can be transformed into a more efficient two-axis-twisting interaction by rotating the collective spin while coupling to the cavity, yielding a Heisenberg limited noise reduction $\ensuremath{\propto}1/N$. Considering the noise effects due to atomic decoherence and cavity decay, we find that substantial squeezing is still attainable with current laboratory techniques.
Squeezed spin states possess unique quantum correlation or entanglement and are significantly promising for advancing quantum information processing and quantum metrology. In recent back-to-back publications [C. Gross et al., Nature (London) 464, 1165 (2010) and Max F. Riedel et al., Nature (London) 464, 1170 (2010)], reduced spin fluctuations are observed leading to spin squeezing at -8.2 and -2.5 dB, respectively, in two-component atomic condensates exhibiting one-axis-twisting interactions. The noise reduction limit for the one-axis twisting scales as ∝1/N(2/3), which for a condensate with N∼10(3) atoms is about 100 times below the standard quantum limit. We present a scheme using repeated Rabi pulses capable of transforming the one-axis-twisting spin squeezing into the two-axis-twisting type, leading to Heisenberg limited noise reduction ∝1/N or an extra tenfold improvement for N∼10(3).
Ultracold atoms in an ultrahigh-finesse optical cavity are a powerful platform to produce spin squeezing since photon of cavity mode can induce nonlinear spin-spin interaction and thus generate a one-axis twisting Hamiltonian HOAT=qJx2H_{\text{OAT}}=qJ_{x}^{2}HOAT​=qJx2​, whose corresponding maximal squeezing factor scales as N−2/3N^{-2/3}N−2/3, where NNN is the atomic number. On the contrary, for the other two-axis twisting Hamiltonian HTAT=q(Jx2−Jy2)H_{\text{TAT}}=q(J_{x}^{2}-J_{y}^{2})HTAT​=q(Jx2​−Jy2​), the maximal squeezing factor scales as N−1N^{-1}N−1, approaching the Heisenberg limit. In this paper, inspired by recent experiments of cavity-assisted Raman transitions, we propose a scheme, in which an ensemble of ultracold six-level atoms interacts with two quantized cavity fields and two pairs of Raman lasers, to realize a tunable two-axis spin Hamiltonian %H=q(J_{x}^{2}+χJ_{y}^{2})+ω_{0}J_{z}. For proper parameters, the above one- and two- axis twisting Hamiltonians are recovered, and the scaling of N−1N^{-1}N−1 of the maximal squeezing factor can occur naturally. On the other hand, in the two-axis twisting Hamiltonian, spin squeezing is usually reduced when increasing the effective atomic resonant frequency ω0ω_{0}ω0​. Surprisingly, we find that by combined with the dimensionless parameter χ(>−1)χ(>-1)χ(>−1), the effective atomic resonant frequency ω0ω_{0}ω0​ can enhance spin squeezing largely. These results are benefit for achieving the required spin squeezing in experiments.
We propose a scheme to implement Heisenberg-limited spin squeezing in a hybrid cavity optomechanical-spin system. In our system, NNN two-level systems are coupled via Tavis-Cummings interactions to a mechanical resonator (MR) in a standard optomechanical setup. Within the dispersive coupling regime, adiabatic elimination of the optical mode induces a squeezing effect on the MR, which, in the squeezed representation, effectively transforms the collective spin operators into a Bogoliubov form. Under large detuning conditions, the phonon mode mediates interactions among the Bogoliubov collective spins, thereby enabling a two-axis twisting squeezing protocol through appropriate parameter tuning. Both theoretical analysis and numerical simulations show that in the presence of dephasing and phonon dissipation, the maximum squeezing degree asymptotically converges to a constant as NNN increases, which implies the metrological precision asymptotically approaches the standard quantum limit without parameter optimization. Nevertheless, in parameter optimization we extract a scaling relation of the optimal squeezing which surpasses existing schemes in the literature. Moreover, the optimization also leads to a considerable reduction of the preparation time for the optimal squeezing. Our work may provide insights into dissipation effects in spin squeezing and offer a potential route for high-precision quantum metrology in many-body systems.
In this paper, we investigate entanglement of an experimental system of two nitrogen-vacancy-center ensembles which are initially squeezed under the one-axis twisting Hamiltonian. We take into account three scenarios in which initial squeezing and entanglement are mediated by phonons or photons: (a) the phonon-squeezed photon-entangled scenario, (b) the phonon-squeezed phonon-entangled scenario, and (c) the photon-squeezed photon-entangled scenario. For our investigation, we employ the Tavis-Cummings model, which includes dissipative decoherence of the collective spin ensemble, and analyze the system both for a relatively small number of spins and in the limit of a large number of spins using the approach of a quantum master equation. Although evidence in the literature on idealized coupled oscillator systems and coupled quantum kicked tops suggests that initial squeezing can enhance entanglement, we find that, in the realistic system studied in this paper, initial squeezing can improve entanglement overall when the field mode interacts in a particular manner with the two spin ensembles. Our analysis using the Holstein-Primakoff transformation and Wigner characteristic function in the rotating frame of reference shows that the entanglement enhancement is a subtle consequence of the way in which the dissipative decoherence rotates the state of the collective spin ensemble such that enhancement depends on the time-evolved rotated states between the presence and absence of initial squeezing.
Deterministic light-induced spin squeezing in an atomic gas is limited by photon shot noise or, equivalently, by atomic state information escaping with the light field mediating the effective atom-atom interaction. We show theoretically that the performance of cavity spin squeezing [M. H. Schleier-Smith, I. D. Leroux, and V. Vuleti\ifmmode \acute{c}\else \'{c}\fi{}, Phys. Rev. A 81, 021804(R) (2010)] can be substantially improved by erasing the light-atom entanglement, and propose several methods for doing so. Accounting for light scattering into free space, quantum erasure improves the scaling of cavity squeezing from ${S}^{\ensuremath{-}1/2}$ to ${S}^{\ensuremath{-}2/3}$, where SSS is the total atomic spin.