The recently introduced concept of an "ideal quantum clock" (IQC) is extended. Especially it is shown that the time operator of an IQC is canonically conjugated to the Hamiltonian of the IQC in a certain pre-Hilbert space. Further it is discussed how the IQC interacts with another physical system D and to what extent the IQC measures the time differences of any prescribed initial and final states of D.
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…
Measuring Clock Precision Without an Ideal Time Reference
University Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Institute for Theoretical Physics, ETH Zürich（苏黎世联邦理工学院理论物理研究所）；ENS Lyon, Inria（里昂高等师范学校，法国国家信息与自动化研究所）；University Grenoble Alpes, Inria（格勒诺布尔阿尔卑斯大学，法国国家信息与自动化研究所）
Proper time, ideal clocks, and boosts are well understood classically, but subtleties arise in quantum physics. We show that quantum clocks set in motion via momentum boosts do not witness classical time dilation. However, using velocity boosts we find the ideal behaviour in both cases where the quantum clock and classical observer are set in motion. Without internal state dependent forces additional effects arise. As such, we derive observed frequency shifts in ion trap atomic clocks, indicating a small additional shift, and also show the emergence of non-ideal behaviour in a theoretical cloc…
Among the many proposals to approach the concept of time in quantum theory, the Page-Wootters mechanism has attracted much attention in the last few years. Originally, such a mechanism explored a stationary bipartite non-interacting global system, i.e., a system of interest together with an ancillary clock, to determine how the evolution in time can emerge as an equation of motion for a quantum particle conditioned to the measurement of the state of the clock. After the seminal proposal, many variations of it were considered, and different aspects of the mechanism were elucidated. Our contribu…
We show that no device built according to the rules of quantum field theory can measure proper time along its path. Highly accelerated quantum clocks experience the Unruh effect, which inevitably influences their time rate. This contradicts the concept of an ideal clock, whose rate should only depend on the instantaneous velocity.
Processes such as quantum computation, or the evolution of quantum cellular automata are typically described by a unitary operation implemented by an external observer. In particular, an interaction is generally turned on for a precise amount of time, using a classical clock. A fully quantum mechanical description of such a device would include a quantum description of the clock whose state is generally disturbed because of the back-reaction on it. Such a description is needed if we wish to consider finite sized autonomous quantum machines requiring no external control. The extent of the back-…
In 1962, Eugene P. Wigner introduced a thought experiment that highlighted the incompatibility in quantum theory between unitary evolution and wave function reduction in a measurement. This work resulted in a class of thought experiments often called Wigner's Friend Scenarios, which have been providing insights over many frameworks and interpretations of quantum theory. Recently, a no-go theorem obtained by Daniela Frauchiger and Renato Renner brought attention back to the Wigner's Friend and its potential of putting theories to test. Many answers to this result pointed out how timing in the t…
We investigate a model of a supposedly ideal clock based on the decay rate of a charged particle in circular motion in a constant magnetic field. We show that the time measured by an ideal clock depends on the acceleration. However, the effect becomes visible at an order of magnitude of 10^{28} g, therefore confirming the validity of the ideal clock hyphotesis for realistic accelerations.
The `problem of time' remains an unresolved issue in all known physical descriptions of the Universe. One aspect of this problem is the conspicuous absence of time in the Wheeler-Dewitt equation, which is the analogue of the Schrodinger equation for the Universal wavefunction. Page and Wootters famously addressed this problem by providing a mechanism for effectively introducing time evolution into this timeless cosmological picture. Their method, which is sometimes called the conditional probability interpretation (CPI), requires the identification of an internal clock system that is meant to …
In general relativity, the picture of spacetime assigns an ideal clock to each worldline. Being ideal, gravitational effects due to these clocks are ignored and the flow of time according to one clock is not affected by the presence of clocks along nearby worldlines. However, if time is defined operationally, as a pointer position of a physical clock that obeys the principles of general relativity and quantum mechanics, such a picture is at most a convenient fiction. Specifically, we show that the general relativistic mass-energy equivalence implies gravitational interaction between the clocks…
We propose an approach for collective enhancement of precision for remotely located optical lattice clocks and a way of generation of the Einstein-Podolsky-Rosen state of remote clocks. Close to Heisenberg scaling of the clock precision with the number of clocks M can be achieved even for an optical channel connecting clocks with substantial losses. This scenario utilizes a collective quantum nondemolition measurement on clocks with parallel Bloch vectors for enhanced measurement precision. We provide an optimal network solution for distant clocks as well as for clocks positioned in close prox…
Among various approaches toward quantum computation, measurement-based quantum computation (MBQC) multiplexed in time domain is currently a promising method for addressing the need for scalability. MBQC requires two components: cluster states and programmable measurements. With time-domain multiplexing, the former has been realized on an ultra-large-scale. The latter, however, has remained unrealized, leaving the large-scale cluster states unused. In this work, we make such a measurement system and use it to demonstrate basic quantum operations multiplexed in the time domain with 25 MHz clock …
We present a novel quantum clock synchronization(QCS) scheme of multiple parties which uses operation as the trigger to start the evolution of the initial state, where existing QCS protocols use measurement to start the evolution. Since the trigger is the unitary operation, we have protected entanglement of remote nodes, and after concentration of the qubits to the center node, general measurements of the total state is possible. We show that our protocol links the QCS problem to a multiple phase estimation problem. We use the Fisher information to give the precision of the synchronization, an…
We introduce a quantum thermodynamic system that can operate as a thermal clock in analogy with the thermal clock first introduced by Ernst Mach in which the temperature difference between cooling bodies in contact can be used as a kind of relational time. Our model is based on an optomechanical system in which photons are transferred irreversibly between two cavities due to the modulation of the cavity cooling rate by a mechanical system coupled to a heat bath. We contrast the ensemble average view with a single system view using the theory of continuous measurement and we show that, by makin…
The ultimate bounds to precision of atomic clock frequency measurement techniques
Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano（米兰大学物理系“阿尔多·蓬特罗利”）；INFN Sezione di Milano（意大利国家核物理研究所米兰分部）；Istituto Nazionale di Ricerca Metrologica, INRIM（意大利国家计量研究所）
We consider a spin chain extending from Alice to Bob with next neighbors interactions, initially in its ground state. Assuming that Bob measures the last spin of the chain, the energy of the spin chain has to increase, at least on average, due to the measurement disturbance. Presumably, the energy is provided by Bob's measurement apparatus. Assuming now that, simultaneously to Bob's measurement, Alice measures the first spin, we show that either energy is not conserved, - implausible - or the projection postulate doesn't apply, and that there is signalling. An explicit measurement model shows …
We propose a multimeasurement estimation protocol for Quantum Nondemolition (QND) measurements in a Rabi clock interferometer. The method is well suited for current state-of-the-art optical lattice clocks with QND measurement capabilities. The protocol exploits the correlations between multiple non-destructive measurements of the initially prepared coherent spin state. A suitable Gaussian estimator for the clock laser detuning is presented, and an analytic expression for the sensitivity of the protocol is derived. We use this analytic expression to optimise the protocol using available experim…
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…
We consider the clock game-a task formulated in the framework of quantum information theory-that can be used to improve the existing schemes of quantum-enhanced telescopy. The problem of learning when a stellar photon reaches a telescope is translated into an abstract game, which we call the clock game. A winning strategy is provided that involves performing a quantum non-demolition measurement that verifies which stellar spatio-temporal modes are occupied by a photon without disturbing the phase information. We prove tight lower bounds on the entanglement cost needed to win the clock game, wi…
We propose a hybrid quantum-classical atomic clock protocol where the interrogation of an ensemble of uncorrelated atoms in a spin-coherent state is used to feedback one (or more) spin-squeezed atomic ensembles toward their optimal phase sensitivity point. This protocol overcomes the stability of a single Ramsey clock and it reaches a Heisenberg-limited stability while avoiding non-destructive measurements. Analytical predictions are compared with numerical simulations of clocks operations including correlated 1/f local oscillator noise.
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.
We use an atomic fountain clock to measure quantum scattering phase shifts precisely through a series of narrow, low-field Feshbach resonances at average collision energies below $1\,μ$K. Our low spread in collision energy yields phase variations of order $\pm π/2$ for target atoms in several $F,m_F$ states. We compare them to a theoretical model and establish the accuracy of the measurements and the theoretical uncertainties from the fitted potential. We find overall excellent agreement, with small statistically significant differences that remain unexplained.
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…
It is shown that the dynamic differential scalar polarisability of the $S_{1/2}-D_{5/2}$ transition in $^{138}$Ba$^+$ can be determined to an inaccuracy below $0.5\%$ across a wide wavelength range ($λ>700\,\mathrm{nm}$). This can be achieved using measurements for which accurate determination of laser intensity is not required, and most of the required measurements are already in the literature. Measurement of a laser-induced ac-stark shift of the clock transition would then provide an \emph{in situ} measurement of the laser's intensity to the same $0.5\%$ level of inaccuracy, which is not ea…
Quantum non-demolition (QND) measurement is a remarkable tool for the manipulation of quantum systems. It allows specific information to be extracted while still preserving fragile quantum observables of the system. Here we apply cavity-based QND measurement to an optical lattice clock---a type of atomic clock with unrivalled frequency precision---preserving the quantum coherence of the atoms after readout with 80\% fidelity. We apply this technique to stabilise the phase of an ultrastable laser to a coherent atomic state via a series of repeated QND measurements. We exploit the improved phase…
Measurement of the $^{138}$Ba$^+$ ${}^2S_{1/2} - {}^2D_{5/2}$ clock transition frequency and $D_{5/2}$ Landé $g_J$ factor are reported. The clock transition frequency $ν_{\mathrm{Ba}^+}=170\,126\,432\,449\,333.31\pm(0.39)_\mathrm{stat}\pm(0.29)_\mathrm{sys}\,$Hz, is obtained with accuracy limited by the frequency calibration of the maser used as a reference oscillator. The Landé $g_J$-factor for the ${}^2D_{5/2}$ level is determined to be $g_{D}=1.200\,367\,39(24)$, which is a 30-fold improvement on previous measurements. The $g$-factor measurements are corrected for an ac-magnetic field from …
We develop differential measurement protocols that circumvent the laser noise limit in the stability of optical clock comparisons by synchronous probing of two clocks using phase-locked local oscillators. This allows for probe times longer than the laser coherence time, avoids the Dick effect, and supports Heisenberg-limited measurement precision. We present protocols for such frequency comparisons and develop numerical simulations of the protocols with realistic noise sources. These methods provide a route to reduce frequency ratio measurement durations by more than an order of magnitude.
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…
Absolute frequency measurement of the $^{40}$Ca$^{+}$ clock transition using a GNSS link to the SI second
Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften（奥地利科学院量子光学与量子信息研究所）；Universität Innsbruck, Institut für Experimentalphysik（因斯布鲁克大学实验物理研究所）；Alpine Quantum Technologies GmbH（阿尔卑斯量子技术有限公司）；Singapore University of Technology and Design（新加坡科技设计大学）；Qatar Center for Quantum Computing, College of Science and Engineering, Hamad Bin Khalifa University（哈马德·本·哈利法大学科学与工程学院卡塔尔量子计算中心）
Building scalable quantum systems that demonstrate genuine performance enhancement based on entanglement is a major scientific goal for fields including computing, networking, simulations, and metrology. The tremendous challenge arises from the fragility of entanglement in increasingly larger sized quantum systems. Optical atomic clocks utilizing a large number of atoms have pushed the frontier of measurement science, building on precise engineering of quantum states and control of atomic interactions. However, today's state-of-the-art optical atomic clocks are limited by the quantum projectio…
Optical atomic clocks are our most precise tools to measure time and frequency. They enable precision frequency comparisons between atoms in separate locations to probe the space-time variation of fundamental constants, the properties of dark matter, and for geodesy. Measurements on independent systems are limited by the standard quantum limit (SQL); measurements on entangled systems, in contrast, can surpass the SQL to reach the ultimate precision allowed by quantum theory - the so-called Heisenberg limit. While local entangling operations have been used to demonstrate this enhancement at mic…
