Rodolfo Gambini, Jorge Pullin
We present a simple argument leading to a fundamental minimum uncertainty in the determination of times. It only relies in the uncertainty principle and time dilation in a gravitational field. It implies any attempt to measure times will have a fundamental level of uncertainty. Implications are briefly outlined.
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The Conditional Probability Interpretation of Quantum Mechanics replaces the abstract notion of time used in standard Quantum Mechanics by the time that can be read off from a physical clock. The use of physical clocks leads to apparent non-unitary and decoherence. Here we show that a close approximation to standard Quantum Mechanics can be recovered from conditional Quantum Mechanics for semi-classical clocks, and we use these clocks to compute the minimum decoherence predicted by the Conditional Probability Interpretation.
A quantum clock working as a control device is examined. The quality of the control process is characterized by the magnitude of deviation of perturbed state from unperturbed state of the controlled system. Uncertainty relations that relate the time duration of the process and energy of the clock to the quality of the control are presented.
Fundamental Limits on Clock Precision from Spacetime Uncertainty in Quantum Collapse Models
本文证明DP和CSL自发坍缩模型通过质量密度自发测量与牛顿引力不确定性相关联，并计算了由此诱导的时间不确定性极限，发现其对现有原子钟等计时设备的影响可忽略。
Ultimate Kinetic Uncertainty Relation and Optimal Performance of Stochastic Clocks
本文针对离散构型马尔可夫过程推导了更紧且可饱和的随机通量不确定度渐近界，确立了首次通过时间不确定度极限与随机时钟最优长期性能，并将结果推广至含量子重置的半马尔可夫过程。
We report on an improved systematic evaluation of the JILA SrI optical lattice clock, achieving a nearly identical systematic uncertainty compared to the previous strontium accuracy record set by the JILA SrII optical lattice clock (OLC) at $2.1 \times 10^{-18}$. This improves upon the previous evaluation of the JILA SrI optical lattice clock in 2013, and we achieve a more than twenty-fold reduction in systematic uncertainty to $2.0 \times 10^{-18}$. A seven-fold improvement in clock stability, reaching $4.8 \times 10^{-17}/\sqrtτ$ for an averaging time $τ$ in seconds, allows the clock to aver…
In earlier papers we showed unpredictability beyond quantum uncertainty in atomic clocks, ensuing from a proven gap between given evidence and explanations of that evidence. Here we reconceive a clock, not as an isolated entity, but as enmeshed in a self-adjusting communications network adapted to one or another particular investigation, in contact with an unpredictable environment. From the practical uses of clocks, we abstract a clock enlivened with the computational capacity of a Turing machine, modified to transmit and to receive numerical communications. Such "live clocks" phase the steps…
The Stark shift due to blackbody radiation (BBR) is the key factor limiting the performance of many atomic frequency standards, with the BBR environment inside the clock apparatus being difficult to characterize at a high level of precision. Here we demonstrate an in-vacuum radiation shield that furnishes a uniform, well-characterized BBR environment for the atoms in an ytterbium optical lattice clock. Operated at room temperature, this shield enables specification of the BBR environment to a corresponding fractional clock uncertainty contribution of $5.5 \times 10^{-19}$. Combined with uncert…
A multi-ion optical clock with $\mathbf{5 \times 10^{-19}}$ uncertainty
本文提出一种多离子光学钟，其频率不确定度为5.3×10^-19，采用10个Sr离子，通过离子分辨态检测抑制位置依赖性移位，测量时间减少4.8倍，与Yb单离子钟对比显示高精度频率比。
Collisions between background gas particles and the trapped ion in an atomic clock can subtly shift the frequency of the clock transition. The uncertainty in the correction for this effect makes a significant contribution to the total systematic uncertainty budget of trapped-ion clocks. Using a non-perturbative analytic framework that was developed for this problem, we estimate the frequency shift in Al$^+$ ion clocks due to collisions with helium and hydrogen. Our calculations significantly improve the uncertainties in the collisional shift coefficients, and show that the collisional frequenc…
We present a transportable optical clock (TOC) with $^{87}$Sr. Its complete characterization against a stationary lattice clock resulted in a systematic uncertainty of ${7.4 \times 10^{-17}}$ which is currently limited by the statistics of the determination of the residual lattice light shift. The measurements confirm that the systematic uncertainty is reduceable to below the design goal of $1 \times 10^{-17}$. The instability of our TOC is $1.3 \times 10^{-15}/\sqrt{(τ/s)}$. Both, the systematic uncertainty and the instability are to our best knowledge currently the best achieved with any typ…
Transportable strontium lattice clock with $4 \times 10^{-19}$ blackbody radiation shift uncertainty
本文报道了一种总系统不确定度为 $2.1 \times 10^{-18}$、黑体辐射位移不确定度低至 $4 \times 10^{-19}$ 的可搬运锶光晶格钟，并通过与铯喷泉钟比对验证了其频率测量能力，展示了其在厘米级大地测量和跨机构频率比对中的应用潜力。
Interspecies clock comparison below $5 \times 10^{-18}$ uncertainty with a transportable clock
该研究通过可搬运光钟与固定单离子钟开展近2年的跨物种钟比对，测得光频比不确定度达4.3×10^-18，提升了3倍以上，验证了可搬运钟用于计时测地及跨机构钟比对的可行性。
We have determined the frequency shift that blackbody radiation is inducing on the $5s^2$ $^1$S$_0$ -- $5s5p$ $^3$P$_0$ clock transition in strontium. Previously its uncertainty limited the uncertainty of strontium lattice clocks to $1\times10^{-16}$. Now the uncertainty associated to the black body radiation shift correction translates to $5\times 10^{-18}$ relative frequency uncertainty at room temperature. Our evaluation is based on a measurement of the differential dc-polarizability of the two clock states and on a modeling of the dynamic contribution using this value and experimental data…
Effect of Measurement Backaction on Quantum Clock Precision Studied with a Superconducting Circuit
本文通过超导transmon量子比特与共面谐振器构建量子时钟，研究连续测量强度对精度的影响，推导并实验验证了量子领域的动力学不确定性关系，建立了时钟热力学行为与精度之间的联系。
At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper times. We consider quantum clocks constructed from the internal degrees of relativistic particles that move through curved spacetime. The probability that one clock reads a given proper time conditioned on another clock reading a different proper time is derived. From this conditional probability distribution, it is shown that when the center-of-mass of these clocks move in localized momentum wave packets they observe classical time dilation. We then illustrate a quantum …
Quantum theory and relativity offer different conceptions of time. To explore the conflict between them, we study a quantum version of the light-clock commonly used to illustrate relativistic time dilation. This semiclassical model combines elements of both theories. We show for Gaussian states of the light field that the clock time is independent of the initial state. We calculate the discrepancy between two such clocks when one is held in a gravitational field and the other is left to fall a certain distance. Contrasting our results with the case of pointlike observers in general relativity,…
Enhanced Precision in Entangled Quantum Clocks with Phase Estimation Algorithm
本文提出一种结合量子相位估计算法的增强型纠缠量子钟协议，通过高纠缠多钟态直接估计未知相位的 proper-time 差异，突破标准投影噪声限制，扩展了原始 EQC 框架。
Optical frequency comparison of the 40Ca+ clock transition ν_{Ca} (2S1/2-2D5/2, 729nm) against the 87Sr optical lattice clock transition ν_{Sr}(1S0-3P0, 698nm) has resulted in a frequency ratio ν_{Ca} / ν_{Sr} = 0.957 631 202 358 049 9(2 3). The rapid nature of optical comparison allowed the statistical uncertainty of frequency ratio ν_{Ca} / ν_{Sr} to reach 1x10-15 in only 1000s and yielded a value consistent with that calculated from separate absolute frequency measurements of ν_{Ca} using the International Atomic Time (TAI) link. The total uncertainty of the frequency ratio using optical co…
Good clocks are of importance both to fundamental physics and for applications in astronomy, metrology and global positioning systems. In a recent technological breakthrough, researchers at NIST have been able to achieve a stability of 1 part in $10^{18}$ using an Ytterbium clock. This naturally raises the question of whether there are fundamental limits to the stability of clocks. In this paper we point out that gravity and quantum mechanics set a fundamental limit on the stability of clocks. This limit comes from a combination of the uncertainty relation, the gravitational redshift and the r…
The notion of time is given a different footing in Quantum Mechanics and General Relativity, treated as a parameter in the former and being an observer dependent property in the later. From a operational point of view time is simply the correlation between a system and a clock, where an idealized clock can be modelled as a two level systems. We investigate the dynamics of clocks interacting gravitationally by treating the gravitational interaction as a classical information channel. In particular, we focus on the decoherence rates and temporal resolution of arrays of $N$ clocks showing how the…
A terahertz vibrational molecular clock with systematic uncertainty at the $10^{-14}$ level
本研究将光晶格时钟架构扩展至双原子分子系综，实验实现了基于纯分子振动的精确晶格钟，总系统不确定度达$4.6\ imes10^{-14}$，并精确测量了振动分裂绝对频率，为分子光谱学和太赫兹频率标准树立了里程碑。
Feedback-Enhanced Quantum Metrology and Clock Precision under Thermodynamic Uncertainty
该研究构建了幺正跃迁反馈下开放量子系统的全计数统计，推导了反馈修正的热力学不确定性关系，发现反馈可在不改变平均热力学流的前提下增强储库参数估计与量子时钟精度。
The absolute frequency of the $^{87}{\rm Sr}$ lattice clock transition was evaluated with an uncertainty of $1.1\times 10^{-15}$ using a frequency link to the international atomic time (TAI). The frequency uncertainty of a hydrogen maser used as a transfer oscillator was reduced by homogeneously distributed intermittent measurement over a five-day grid of TAI. Three sets of four or five days measurements as well as systematic uncertainty of the clock at $8.6\times 10^{-17}$ have resulted in an absolute frequency of $^{87}{\rm Sr}\ {}^1S_0 - {}^3P_0$ clock transition to be 429 228 004 229 872.8…
We critically discuss the measure of very short time intervals. By means of a Gedankenexperiment, we describe an ideal clock based on the occurrence of completely random events. Many previous thought experiments have suggested fundamental Planck-scale limits on measurements of distance and time. Here we present a new type of thought experiment, based on a different type of clock, that provide further support for the existence of such limits. We show that the minimum time interval $Δt$ that this clock can measure scales as the inverse of its size $Δr$. This implies an uncertainty relation betwe…
We evaluated the C_6 coefficients of Yb-Yb and Yb -alkali/group II van der Waals interactions with 2% uncertainty. The only existing results for such quantities are for the Yb-Yb dimer. Our value, C_6=1929(39) a.u., is in excellent agreement with the recent experimental determination of 1932(35) a.u. We have also developed a new approach for the calculation of the dynamic correction to the blackbody radiation shift. We have calculated this quantity for the Yb 6s^2 ^1S_0 - 6s6p ^3P_0 clock transition with 3.5% uncertainty. This reduces the fractional uncertainty due to the blackbody radiation s…
A First-Principles Thermodynamic Uncertainty Relation for Shortcuts to Adiabaticity
本文提出了一种基于量子钟的首原理热力学不确定性关系，用于分析短路径绝热性协议中精度与纯度损失之间的权衡。
High-Stability Single-Ion Clock with $5.5\times10^{-19}$ Systematic Uncertainty
本文提出基于量子逻辑光谱学的单 $^{27}$Al$^+$ 离子光原子钟，通过远程低温硅腔实现 1 s 的 Rabi 探测，结合改进的离子阱电设计与新真空系统，将分数频率不确定度降至 $5.5\times10^{-19}$，不稳定性降低三倍。
We review Bohr's reasoning in the Bohr-Einstein debate on the photon box experiment. The essential point of his reasoning leads us to an uncertainty relation between the proper time and the rest mass of the clock. It is shown that this uncertainty relation can be derived if only we take the fundamental point of view that the proper time should be included as a dynamic variable in the Lagrangian describing the system of the clock. Some problems and some positive aspects of our approach are then discussed.
$^{115}$In$^+$-$^{172}$Yb$^+$ Coulomb crystal clock with $2.5\times10^{-18}$ systematic uncertainty
本研究提出基于$^{115}$In$^+$跃迁、以$^{172}$Yb$^+$做 sympathetic冷却的可扩展混合库仑晶体钟，实现$2.5\times10^{-18}$系统不确定度，刷新频率比测量精度并降低不稳定性。
Quantum Uncertainty as an Intrinsic Clock
该研究利用量子波包不确定性的独立演化特性，将含时谐振势中一维粒子的经典Ermakov-Lewis不变量推广为量子系统的运动常数，为量子引力等领域提供了新思路。
Based on a relation between inertial time intervals and the Riemannian curvature, we show that space--time uncertainty derived by Ng and van Dam implies absurd uncertainties of the Riemannian curvature.
Physically constrained quantum clock-driven dynamics
本研究从理想化量子时钟模型出发，放宽假设构建物理一致框架，引入连续时间算子推导时钟退化下界，揭示时钟性能与能量不确定性的权衡。
Despite being a canonical example of quantum mechanical perturbation theory, as well as one of the earliest observed spectroscopic shifts, the Stark effect contributes the largest source of uncertainty in a modern optical atomic clock through blackbody radiation. By employing an ultracold, trapped atomic ensemble and high stability optical clock, we characterize the quadratic Stark effect with unprecedented precision. We report the ytterbium optical clock's sensitivity to electric fields (such as blackbody radiation) as the differential static polarizability of the ground and excited clock lev…
A quantum clock must satisfy two basic constraints. The first is a bound on the time resolution of the clock given by the difference between its maximum and minimum energy eigenvalues. The second follows from Holevo's bound on how much classical information can be encoded in a quantum system. We show that asymptotically, as the dimension of the Hilbert space of the clock tends to infinity, both constraints can be satisfied simultaneously. The experimental realization of such an optimal quantum clock using trapped ions is discussed.
In the framework of any quantum theory in the Schroedinger picture a general operator time concept is given. For this purpose certain systems are emphasized as ideal quantum clocks. Their definition follows heuristically from a common property of ideal clocks and from general postulates of traditional quantum theory. Any such ideal quantum clock allows the definition of a symmetric time operator T. T and the Hamiltonian H necessarily satisfy the time-energy uncertainty relation. The argument of Pauli against the existence of any time operator does not strike, because T is symmetric but not sel…
Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain…
Optical lattice clocks with uncertainty and instability in the $10^{-17}$-range and below have so far been demonstrated exclusively using fermions. Here, we demonstrate a bosonic optical lattice clock with $3\times 10^{-18}$ instability and $2.0\times 10^{-17}$ accuracy, both values improving on previous work by a factor 30. This was enabled by probing the clock transition with an ultra-long interrogation time of 4 s, using the long coherence time provided by a cryogenic silicon resonator, by careful stabilization of relevant operating parameters, and by operating at low atom density. This wor…
Evaluation of a $^{88}$Sr$^+$ optical clock with a direct measurement of the blackbody radiation shift and determination of the clock frequency
本文评估单离子 $^{88}$Sr$^+$ 光钟，运行中直接估计黑体辐射有效温度，并对照 $^{171}$Yb$^+$ E3 钟以 $2.3\ imes10^{-17}$ 不确定度测定其绝对频率，同时验证三离子可改善短期稳定度。
Necessary and sufficient condition for the existence of a minimum uncertainty state for an arbitrary pair of observables is given.
Characterising the precision of a clock without any external time reference
本文提出一种无需外部时间参考即可表征时钟精度的方法，利用信号分路器和延迟触发器，基于滴答计数统计估计精度，并通过电子电路验证。
Energy-time uncertainty plays an important role in quantum foundations and technologies, and it was even discussed by the founders of quantum mechanics. However, standard approaches (e.g., Robertson's uncertainty relation) do not apply to energy-time uncertainty because, in general, there is no Hermitian operator associated with time. Following previous approaches, we quantify time uncertainty by how well one can read off the time from a quantum clock. We then use entropy to quantify the information-theoretic distinguishability of the various time states of the clock. Our main result is an ent…
We experimentally demonstrate agreement between two $^{176}$Lu$^+$ frequency references using correlation spectroscopy. From a comparison at different magnetic fields, we obtain a quadratic Zeeman coefficient of $-4.89264(88)\,\mathrm{Hz/mT^2}$, which gives a corresponding fractional frequency uncertainty contribution of just $2.5\times 10^{-20}$ for comparisons at typical operating fields of 0.1\,mT. A subsequent comparison with both systems at 0.1\,mT, demonstrates a fractional frequency difference of $(-2.0\pm(3.7)_\mathrm{stat}\pm(0.9)_\mathrm{sys})\times10^{-18}$, where `stat' and `sys' i…
Minimum uncertainty states and squeezed states from the sum uncertainty relation
本文研究了求和不确定性关系下的最小不确定性态，证明其与传统乘积不确定性关系下的最小不确定性态一致，并展示了压缩态不受影响。
Squeezing experiments which are capable of creating a minimum uncertainty state during the nonlinear process, for example optical parametric amplification, are commonly used to produce light far below the quantum noise limit. This report presents a method with which one can characterize this minimum uncertainty state and gain valuable knowledge of the experimental setup.
Uncertainty relations between a bounded coordinate operator and a conjugate momentum operator frequently appear in quantum mechanics. We prove that physically reasonable minimum-uncertainty solutions to such relations have quantized expectation values of the conjugate momentum. This implies, for example, that the mean angular momentum is quantized for any minimum-uncertainty state obtained from any uncertainty relation involving the angular-momentum operator and a conjugate coordinate. Experiments specifically seeking to create minimum-uncertainty states localized in angular coordinates theref…
In order to improve the short-term stability of trapped-ion optical clocks, we are developing a frequency standard based on ${}^{115}$In${}^+$ / ${}^{172}$Yb${}^+$ Coulomb crystals. For this purpose, we have developed scalable segmented Paul traps which allow a high level of control for multiple ion ensembles. In this article, we detail on our recent results regarding the reduction of the leading sources of frequency uncertainty introduced by the ion trap: 2nd-order Doppler shifts due to micromotion and the heating of secular motion, as well as the black-body radiation shift due to warming of …
The ultimate bounds to precision of atomic clock frequency measurement techniques
研究原子能级间跃迁频率估计的量子极限，比较Rabi、Ramsey和CPT技术的测量精度，证明Rabi和Ramsey方案可达到最小不确定性，而CPT方案通过测量能级间相干性进一步提升精度。
What do simple clocks, simple computers, black holes, space-time foam, and holographic principle have in common? I will show that the physics behind them is inter-related, linking together our concepts of information, gravity, and quantum uncertainty. Thus, the physics that sets the limits to computation and clock precision also yields Hawking radiation of black holes and the holographic principle. Moreover, the latter two strongly imply that space-time undergoes much larger quantum fluctuations than what the folklore suggests --- large enough to be detected with modern gravitational-wave inte…
Measuring Clock Precision Without an Ideal Time Reference
本文提出无需理想时间参考，利用多个相同时钟的副本通过解析界限和协议来量化时钟精度，并应用于亚周期间隔和相对论效应的精确测量。
We investigate dynamical systems with time-dependent mass and frequency, with particular attention on models attaining the minimum value of uncertainty formula. A criterium of minimum uncertainty is presented and illustrated by means of explicit and exactly solved examples. The role of the Bogolubov coefficients, in general and in the context of minimum uncertainty case, is discussed.
We generate entangled states of an ensemble of 5*10^4 rubidium-87 atoms by optical quantum nondemolition measurement. The resonator-enhanced measurement leaves the atomic ensemble, prepared in a superposition of hyperfine clock levels, in a squeezed spin state. By comparing the resulting reduction of quantum projection noise (up to 8.8(8) dB) with the concomitant reduction of coherence, we demonstrate a clock input state with spectroscopic sensitivity 3.0(8) dB beyond the standard quantum limit.
We show that the ground state of the well-known pseudo-stationary states for the Caldirola-Kanai Hamiltonian is a generalized minimum uncertainty state, which has the minimum allowed uncertainty $Δq Δp = \hbar σ_0/2$, where $σ_0 (\geq 1)$ is a constant depending on the damping factor and natural frequency. The most general symmetric Gaussian states are obtained as the one-parameter squeezed states of the pseudo-stationary ground state. It is further shown that the coherent states of the pseudo-stationary ground state constitute another class of the generalized minimum uncertainty states.
We develop a stochastic mean-field theory to describe active frequency measurements of pulsed superradiant emission, studied in recent experiments with strontium-87 atoms trapped in an optical lattice inside an optical cavity [M. Norcia, et al., Phys. Rev. X 8, 21036 (2018)]. Our theory reveals the intriguing dynamics of atomic ensembles with multiple transition frequencies, and it reproduces the superradiant beats signal, noisy power spectra, and frequency uncertainty in remarkable agreement with the experiments. Moreover, by reducing the number of atoms, elongating the superradiant pulses an…
In this paper we report the evaluation of an optical lattice clock based on neutral mercury down to a relative uncertainty of $1.7\times 10^{-16}$. Comparing this characterized frequency standard to a Cs atomic fountain we determine the absolute frequency of the $^1S_0 \rightarrow \phantom{}^3P_0$ transition of $^{199}$Hg as $ν_{\mathrm{Hg}} = 1 128\,575\,290\,808\,154.62\,$Hz $\pm\,0.19\,$Hz (statistical) $\pm\,0.38\,$Hz (systematic), limited solely by the realization of the SI second. Furthermore, by comparing the mercury optical lattice clock to a Rb atomic fountain, we determine for the fi…
We evaluated the static and dynamic polarizabilities of the 5s^2 ^1S_0 and 5s5p ^3P_0^o states of Sr using the high-precision relativistic configuration interaction + all-order method. Our calculation explains the discrepancy between the recent experimental 5s^2 ^1S_0 - 5s5p ^3P_0^o dc Stark shift measurement Δα= 247.374(7) a.u. [Middelmann et. al, arXiv:1208.2848 (2012)] and the earlier theoretical result of 261(4) a.u. [Porsev and Derevianko, Phys. Rev. A 74, 020502R (2006)]. Our present value of 247.5 a.u. is in excellent agreement with the experimental result. We also evaluated the dynamic…
We introduce a new variational characterization of Gaussian diffusion processes as minimum uncertainty states. We then define a variational method constrained by kinematics of diffusions and Schrödinger dynamics to seek states of local minimum uncertainty for general non-harmonic potentials.
Atomic clock frequency ratios with fractional uncertainty $\leq 3.2 \times 10^{-18}$
该研究进行基于²⁷Al⁺、¹⁷¹Yb和⁸⁷Sr的光原子钟高精度频率比测量，总分数不确定性≤3.2×10⁻¹⁸。关键创新是用光纤链路提供超稳定参考，提高比较稳定性，提升对非白噪声等的灵敏度。
An ion trap design for a space-deployable strontium-ion optical clock
本文设计紧凑激光冷却 88Sr+ 光钟物理封装，通过有限元与静电建模评估发射环境及几何公差影响，以支持低至 10^-18 量级不确定度的可太空部署原型。
Spin minimum uncertainty states for refined uncertainty relations
本文研究了改进的不确定性关系下的旋系统最小不确定性态，通过两种方法推导了满足不确定性界限的态，并发现旋相干态及额外的最小不确定性态类。
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