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We present experimental results concerning time scale transmission to a remote consumer using a relativistic synchronization. The time scale transmission is carried out using a transportable quantum clock on an automobile chassis moving along federal roads for a distance of more than 5000 km. The results are confirmed by an independent method for comparing time scales using signals from global navigation satellite systems. The error of the relativistic synchronization method does not exceed 150 ps, which is significantly better than when other methods are used.

The author discusses the application of quantum technologies in the real world, from a UK perspective. The paper concentrates on quantum communication over fibre networks and over space links. Quantum key distribution is also considered. Mention is briefly made of quantum clock technology and quantum gravity sensors.

We study the phase diagram, both at zero and finite temperature, in a class of $\mathbb{Z}_q$ models with infinite range interactions. We are able to identify the transitions between a symmetry-breaking and a trivial phase by using a mean-field approach and a perturbative expansion. We perform our analysis on a Hamiltonian with $2p$-body interactions and we find first-order transitions for any $p>1$; in the case $p=1$, the transitions are first-order for $q=3$ and second-order otherwise. In the infinite-range case there is no trace of gapless incommensurate phase but, when the transverse field is maximally chiral, the model is in a symmetry-breaking phase for arbitrarily large fields. We analytically study the transtion in the limit of infinite $q$, where the model possesses a continuous $U(1)$ symmetry.

We calculate the ground-state energy density e(g) for the one-dimensional N-state quantum clock model up to order 18, where g is the coupling and N=3,4,5,...,10,20. Using methods based on the Pade approximation, we extract the singular structure of e^{″}(g) or e(g). They correspond to the specific heat and free energy of the classical two-dimensional (2D) clock model. We find that, for N=3,4, there is a single critical point at g_{c}=1. The heat capacity exponent of the corresponding 2D classical model is α=0.34±0.01 for N=3, and α=-0.01±0.01 for N=4. For N>4, there are two exponential singularities related by g_{c1}=1/g_{c2}, and e(g) behaves as Ae^{-c/|g_{c}-g|^{σ}}+analyticterms near g_{c}. The exponent σ gradually grows from 0.2 to 0.5 as N increases from 5 to 9, and it stabilizes at 0.5 when N>9. The phase transitions exhibited in these models should be generalizations of the Kosterlitz-Thouless transition, which has σ=0.5.

Strong zero modes provide a paradigm for quantum many-body systems to encode local degrees of freedom that remain coherent far from the ground state. Example systems include $\mathbb{Z}_n$ chiral quantum clock models with strong zero modes related to $\mathbb{Z}_n$ parafermions. Here we show how these models and their zero modes arise from geometric chirality in fermionic Mott insulators, focusing on $n=3$ where the Mott insulators are three-leg ladders. We link such ladders to $\mathbb{Z}_3$ chiral clock models by combining bosonization with general symmetry considerations. We also introduce a concrete lattice model which we show to map to the $\mathbb{Z}_3$ chiral clock model, perturbed by the Uimin-Lai-Sutherland Hamiltonian arising via superexchange. We demonstrate the presence of strong zero modes in this perturbed model by showing that correlators of clock operators at the edge remain close to their initial value for times exponentially long in the system size, even at infinite temperature., Comment: 5 + 7 pages, 4 + 4 figures; accepted manuscript

We introduce a model of noncommutative geometry that gives rise to the uncertainty relations recently derived from the discussion of a quantum clock. We investigate the dynamics of a free particle in this model from the point of view of doubly special relativity and discuss the geodesic motion in a Schwarzschild background., 7 pages, version accepted for publication on Phys. Lett. A. A discussion of the motion of a particle in a Schwarzschild background has been added

The standard formulation of quantum theory relies on a fixed space-time metric determining the localisation and causal order of events. In general relativity, the metric is influenced by matter, and is expected to become indefinite when matter behaves quantum mechanically. Here, we develop a framework to operationally define events and their localisation with respect to a quantum clock reference frame, also in the presence of gravitating quantum systems. We find that, when clocks interact gravitationally, the time localisability of events becomes relative, depending on the reference frame. This relativity is a signature of an indefinite metric, where events can occur in an indefinite causal order. Even if the metric is indefinite, for any event we can find a reference frame where local quantum operations take their standard unitary dilation form. This form is preserved when changing clock reference frames, yielding physics covariant with respect to quantum reference frame transformations., 20+4 pages, 6 figures. Published version

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-reaction has implications on how small the device can be, on the length of time the device can run, and is required if we want to understand what a fully quantum mechanical treatment of an observer would look like. Here, we consider the implementation of a unitary by a finite sized device which we call the "Quasi-Ideal clock", and show that the back-reaction on it can be made exponentially small in the device's dimension with only a linear increase in energy. As a result, an autonomous quantum machine need only be of modest size and or energy. We are also able to solve a long-standing open problem by using a finite sized quantum clock to approximate the continuous evolution of an Idealised clock. The result has implications on the equivalence of different paradigms of quantum thermodynamics, some which allow external control and some which only allow autonomous thermal machines.

A major outstanding problem for many quantum clock synchronization protocols is the hidden assumption of a common phase reference between the parties to be synchronized. In general, the definition of the quantum states between two parties do not have consistent phase definitions, which can lead to an unknown systematic error. We show that despite prior arguments to the contrary, it is possible to remove this unknown phase via entanglement purification. This closes the loophole for entanglement based quantum clock synchronization protocols, which is a non-local approach to synchronize two clocks independent of the properties of the intervening medium. Starting with noisy Bell pairs, we show that the scheme produces a singlet state for any combination of (i) differing basis conventions for Alice and Bob; (ii) an overall time offset in the execution of the purification algorithm; and (iii) the presence of a noisy channel. Error estimates reveal that better performance than existing classical Einstein synchronization protocols should be achievable using current technology. Quantum mechanics can be used to synchronise the clocks of two distant parties by using the non-local properties of entanglement. It was shown in 2000 that, if two parties share entangled qubit pairs, performing coordinated quantum operations allows them to detect systematic errors in their measurements of time. Unfortunately, performing the correct manipulations already implicitly requires some degree of synchronisation. Ebubechukwu Ilo-Okeke and colleagues from New York University Shanghai and Louisiana State University show that this can be resolved using quantum purification, which concentrates the entanglement from many qubits into states with higher entanglement. They demonstrate that when the parties are out of sync the purification generates states that account for the discrepancy and then can be used for synchronisation. If fully developed, the quantum synchronization scheme could outperform existing classical methods.

We present results of the experiment on compensation of a relativistic shift in the timescale of a transportable hydrogen quantum clock with 3 × 10–15 instability over its motion path. Calculation of the relativistic shift of the onboard timescale of the moving clock was based on continuous measurements of its current coordinates and velocity over the path with the aid of GLONASS/GPS navigation equipment. The experimental error of the proposed method does not exceed ±1 ps.

We propose a time-of-arrival operator in quantum mechanics by conditioning on a quantum clock. This allows us to bypass some of the problems of previous proposals, and to obtain a Hermitian time of arrival operator whose probability distribution arises from the Born rule and which has a clear physical interpretation. The same procedure can be employed to measure the "time at which some event happens" for arbitrary events (and not just specifically for the arrival time of a particle)., Comment: 4 pages+appendices (=supplemental material). Version accepted for publication on Phys. Rev. Lett

Synchronizing clocks using quantum entanglement works on the principle that two clocks Alice and Bob share between them a singlet state which is a stationary state that is immune to evolution under bare atomic Hamiltonian. A major obstacle to its realization is the hidden assumption of a common phase reference between the clocks. Without hidden assumption, a clock state of Alice or Bob is not a uniquely defined quantum state because the phase of the state is arbitrary. This results in an unknown relative phase in a two-particle entangled state defined by the clocks. We show that using entanglement purification, an entanglement-based clock synchronization is achieved despite earlier results showing the contrary. This closes the loophole for entanglement based quantum clock synchronization protocols, which is a non-local approach to synchronize two clocks independent of the properties of the intervening medium.

All clocks, in some form or another, use the evolution of nature toward higher entropy states to quantify the passage of time. Because of the statistical nature of the second law and corresponding entropy flows, fluctuations fundamentally limit the performance of any clock. This suggests a deep relation between the increase in entropy and the quality of clock ticks. Indeed, minimal models for autonomous clocks in the quantum realm revealed that a linear relation can be derived, where for a limited regime every bit of entropy linearly increases the accuracy of quantum clocks. But can such a linear relation persist as we move toward a more classical system? We answer this in the affirmative by presenting the first experimental investigation of this thermodynamic relation in a nanoscale clock. We stochastically drive a nanometer-thick membrane and read out its displacement with a radio-frequency cavity, allowing us to identify the ticks of a clock. We show theoretically that the maximum possible accuracy for this classical clock is proportional to the entropy created per tick, similar to the known limit for a weakly coupled quantum clock but with a different proportionality constant. We measure both the accuracy and the entropy. Once nonthermal noise is accounted for, we find that there is a linear relation between accuracy and entropy and that the clock operates within an order of magnitude of the theoretical bound.

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 thought experiment could be yielding a paradox. In this work, we ask what would happen if the isolated friend in a Wigner's Friend Scenario did not share a time reference frame with the outer observer, and time should be tracked by a quantum clock. For this purpose, we recollect concepts provided by the theory of quantum reference frames and the quantum resource theory of asymmetry, to learn how to internalize time in this scenario, and introduce a model for a feasible quantum clock proposed by Mischa P. Woods, Ralph Silva and Jonathan Oppenheim, called the quasi-ideal clock. Our results have shown that no decoherent behavior comes from this approach, and the disagreement between the superobserver and its friend persists even for an imprecise clock on Wigner's side. However, the gaussian spread of this clock model can control what observables do not raise a paradox, indicating the relevance of deepening this analysis., Comment: Comments are welcome

Currently, the most accurate and stable clocks use optical interrogation of either a single ion or an ensemble of neutral atoms confined in an optical lattice. Here, we demonstrate a new optical clock system based on an array of individually trapped neutral atoms with single-atom readout, merging many of the benefits of ion and lattice clocks as well as creating a bridge to recently developed techniques in quantum simulation and computing with neutral atoms. We evaluate single-site resolved frequency shifts and short-term stability via self-comparison. Atom-by-atom feedback control enables direct experimental estimation of laser noise contributions. Results agree well with an ab initio Monte Carlo simulation that incorporates finite temperature, projective read-out, laser noise, and feedback dynamics. Our approach, based on a tweezer array, also suppresses interaction shifts while retaining a short dead time, all in a comparatively simple experimental setup suited for transportable operation. These results establish the foundations for a third optical clock platform and provide a novel starting point for entanglement-enhanced metrology, quantum clock networks, and applications in quantum computing and communication with individual neutral atoms that require optical clock state control., Comment: 14 pages, 8 figures, 1 table; accepted in PRX on October 25th, 2019

We derive an exact renormalization group recursion relation for the Loschmidt amplitude of the quantum $Q$-state clock model and the quantum $Q$-state Potts model in one dimension. The renormalization group flow is discussed in detail. The fixed-points of the renormalization group flow are found to be complex in general. These fixed-points control the dynamical phases of the two models, giving rise to non-analyticities in its Loschmidt rate function, for both the pure and the disordered system. For the quench protocols studied, dynamical quantum phase transitions are found to occur in the clock model for all $Q$s considered, while in the Potts model, they only occur when $Q$ < 4., arXiv admin note: text overlap with arXiv:1908.04476

We introduce a family of non-integrable 1D lattice models that feature robust periodic revivals under a global quench from certain initial product states, thus generalizing the phenomenon of many-body scarring recently observed in Rydberg atom quantum simulators. Our construction is based on a systematic embedding of the single-site unitary dynamics into a kinetically-constrained many-body system. We numerically demonstrate that this construction yields new families of models with robust wave-function revivals, and it includes kinetically-constrained quantum clock models as a special case. We show that scarring dynamics in these models can be decomposed into a period of nearly free clock precession and an interacting bottleneck, shedding light on their anomalously slow thermalization when quenched from special initial states., Minor updates, new results in Supplementary Material

In this study we have investigated the biological application of Wigner’s inequalities for smallest quantum clock. We have shown that the mass, size and doubling time of bacteria satisfied the Wigner’s inequalities for quantum clock. Data on 17 bacteria with mass 1×10-17 -1×10-15 kg, size 0.3-50µm and doubling time 1×103-1×105 seconds confirmed the hypothesis of Pesi? that possibly the living bacteria appear to be the smallest quantum clocks in the Nature.

The results of the Russia’s first ground-based experiment for determination of the difference in the Earth’s gravitational potentials on the basis of the measurement of the gravitational effect of the time delay with the help of a high-stability transportable atomic clock are provided. The reference atomic clock was placed in Moscow oblast, and a transportable quantum clock with an instability of 3 × 10–15 was placed in the Caucasus Mountains, with a difference in height of the clocks of 1804 m. The measured difference in the gravitational potentials between the positions of the two quantum clocks was (182.0 ± 3.1)102m2s-2 at a relative measurement error of no more than 1.7%.

A quantum clock time scale synchronization algorithm is proposed. The algorithm excludes systematic synchronization error and minimizes random error. Relations are found for calculating the optimum number of measurements and minimum error.

We discuss the corrections to the orbital period of a particle in a constant magnetic field, driven by the model of noncommutative geometry recently associated to a quantum clock. The effects are extremely small, but in principle detectable.

We present a protocol of Einstein’s synchronization of two spatially separated clocks on the basis of quantum teleportation in which the Einstein-Podolsky-Rosen (EPR) state plays a central role. Our scheme does not contain the frame defining problem nor the phase contamination problem for the EPR state which appeared in the quantum clock synchronization (Jozsa et al 2000 Phys. Rev. Lett. 85, 2010).

We consider the conditioning of the timeless solution to the Wheeler&ndash, DeWitt equation by a predefined matter clock state in the simple scenario of an FLRW universe. The resulting evolution of the geometrodynamical degree of freedom with respect to clock time is characterized by the &ldquo, Berry connection&rdquo, of the reduced geometrodynamical space, which relies on the coupling of the clock with the geometry. When the connection vanishes, the standard Schr&ouml, dinger equation is obtained for the geometry with respect to clock time. When one considers environment-induced decoherence in the semiclassical limit, this condition is satisfied, and clock time coincides with cosmic time. Explicit results for the conditioned wave functions for minimal clocks made up of two quantum harmonic oscillator eigenstates are shown.

High-precision nonlocal temporal correlation identification in the entangled photon pairs is critical to measure the time offset between remote independent time scales for many quantum information applications. The first nonlocal correlation identification was reported in 2009, which extracts the time offset via the algorithm of iterative fast Fourier transformations (FFTs) and their inverse. The least identification resolution is restricted by the peak identification threshold of the algorithm, and thus the time offset calculation precision is limited. In this paper, an improvement for the identification is presented both in the resolution and precision via a modified algorithm of direct cross correlation extraction. A flexible resolution down to 1 ps is realized, which is only dependent on the Least Significant Bit (LSB) resolution of the time-tagging device. The attainable precision is shown mainly determined by the inherent timing jitter of the single photon detectors, the acquired pair rate and acquisition time, and a sub picosecond precision (0.72 ps) has been achieved at an acquisition time of 4.5 s. This high-precision nonlocal measurement realization provides a solid foundation for the field applications of entanglement-based quantum clock synchronization, ranging and communications., Comment: 13 pages, 5 figures

It has recently been reported [\textit{PNAS} \textbf{114}, 2303 (2017)] that, under an operational definition of time, quantum clocks would get entangled through gravitational effects. Here we study an alternative scenario: the clocks have different masses and energy gaps, which would produce time difference via gravitational interaction. The proposal of quantum clock synchronization for the gravity-induced time difference is discussed. We illustrate how the stability of measurement probability in the quantum clock synchronization proposal is influenced by the gravitational interaction induced by the clock themselves. It is found that the precision of clock synchronization depends on the energy gaps of the clocks and the improvement of precision in quantum metrology is in fact an indicator of entanglement generation. We also present the quantum enhanced estimation of time difference and find that the quantum Fisher information is very sensitive to the distance between the clocks., Comment: 6 pages, 3 figures

We present an experimental optical implementation of a parallel-in-time discrete model of quantum evolution, based on the entanglement between the quantum system and a finite-dimensional quantum clock. The setup is based on a programmable spatial light modulator which entangles the polarization and transverse spatial degrees of freedom of a single photon. It enables the simulation of a qubit history state containing the whole evolution of the system, capturing its main features in a simple and configurable scheme. We experimentally determine the associated system-time entanglement, which is a measure of distinguishable quantum evolution, and also the time average of observables, which in the present realization can be obtained through one single measurement., Facultad de Ciencias Exactas, Instituto de Física La Plata

Quantum clock synchronization (QCS) is a kind of method to measure the time difference among spatially separated clocks with the principle of quantum mechanics. The first QCS algorithm proposed by Chuang and Jozsa is merely based on two parties, which is further extended and generalized to multiparty situation by Krco and Paul. They present a multiparty QCS protocol based upon W-state, utilizing shared prior entanglement and broadcasting the classical information to synchronize spatially separated clocks. Shortly afterward, Ben-Av and Exman came up with an optimized multiparty QCS based on Z-state. In this work, we firstly report the demonstrations of these two multiparty QCS protocols in a four-qubit liquid-state nuclear magnetic resonance system. The experimental results show a great agreement with the theoretical predictions and also prove that Ben-AV’s multiparty QCS algorithm is more accurate than Krco’s.

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 entropic energy-time uncertainty relation for general time-independent Hamiltonians, stated for both the discrete-time and continuous-time cases. Our uncertainty relation is strong, in the sense that it allows for a quantum memory to help reduce the uncertainty, and this formulation leads us to reinterpret it as a bound on the relative entropy of asymmetry. Due to the operational relevance of entropy, we anticipate that our uncertainty relation will have information-processing applications., Comment: 6 + 9 pages, 2 figures

In thermodynamics, quantum coherences - superpositions between energy eigenstates - behave in distinctly nonclassical ways. Recently mathematical frameworks have emerged to account for these features and have provided a range of novel insights. Here we describe how thermodynamic coherence splits into two kinds - "internal" coherence that admits an energetic value in terms of thermodynamic work, and "external" coherence that does not have energetic value, but instead corresponds to the functioning of the system as a quantum clock. For the latter form of coherence we provide dynamical constraints that relate to quantum metrology and macroscopicity, while for the former, we show that quantum states exist that have finite internal coherence yet with zero deterministic work value. Finally, under minimal thermodynamic assumptions, we establish a clock/work tradeoff relation between these two types of coherences. This can be viewed as a form of time-energy conjugate relation within quantum thermodynamics that bounds the total maximum of clock and work resources for a given system., Comment: 6 + 10 pages

We show how to detect and measure signal frequency jumps in a quantum clock using radio space navigation systems and a Kalman filter. We provide an error estimate for the determination of the frequency jump.

We present results on a ground-based experiment to measure the gravitational effect of a shift in the timescale of stationary and transportable hydrogen quantum clocks. It is proposed to use a transportable highly stable quantum clock to measure the difference at orthometric heights.

The ability to synchronize remote clocks plays an increasingly important role in our infrastructure, from maintaining coherence in the electrical grid to allowing precise positioning and navigation for civilian and military applications. However, many of the techniques to establish and maintain this time synchronization have been shown to be susceptible to interference by malicious parties. Here we propose a protocol that builds on techniques from quantum communication to provide a verified and secure time synchronization protocol. In contrast with classical protocols aimed at increasing the security of time distribution, we need not make any assumptions about the distance or propagation times between the clocks. In order to compromise the security of the protocol, an adversary must be able to perform quantum non-demolition measurements of the presence of a singe photon with high probability. The requirement of such quantum measurements raises a serious technological barrier for any would-be adversary

We propose a general expression for the probability distribution of real-valued tunneling times of a localized particle, as measured by the Salecker-Wigner-Peres quantum clock. This general expression is used to obtain the distribution of times for the scattering of a particle through a static rectangular barrier and for the tunneling decay of an initially bound state after the sudden deformation of the potential, the latter case being relevant to understand tunneling times in recent attosecond experiments involving strong field ionization., 14 pages, 8 Figures

The local conservation of a physical quantity whose distribution changes with time is mathematically described by the continuity equation. The corresponding time parameter, however, is defined with respect to an idealized classical clock. We consider what happens when this classical time is replaced by a non-relativistic quantum-mechanical description of the clock. From the clock-dependent Schr\"odinger equation (as analogue of the time-dependent Schr\"odinger equation) we derive a continuity equation, where, instead of a time-derivative, an operator occurs that depends on the flux (probability current) density of the clock. This clock-dependent continuity equation can be used to analyze the dynamics of a quantum system and to study degrees of freedom that may be used as internal clocks for an approximate description of the dynamics of the remaining degrees of freedom. As an illustration, we study a simple model for coupled electron-nuclear dynamics and interpret the nuclei as quantum clock for the electronic motion. We find that whenever the Born-Oppenheimer approximation is valid, the continuity equation shows that the nuclei are the only relevant clock for the electrons.

Based on the second-order quantum interference between frequency entangled photons generated via parametric down conversion process, we have demonstrated a new experiment of synchronizing two clocks separated by 6-km fiber link. A synchronization accuracy of 13 ps has been achieved with a minimum timing stability of 60 fs.

The PHARAO project purpose is to open the way for a new atomic clock generation in space, where laser cooling techniques and microgravity allow high frequency stability and accuracy. The French space agency, CNES is funding and managing the clock construction. The French SYRTE and LKB laboratories are scientific and technical advisers for the clock requirements and the follow-up of subsystem development in industrial companies. EADS SODERN is developing two main subsystems of the PHARAO clock: the Laser Source and the Cesium Tube where atoms are cooled, launched, selected and detected by laser beams. The Laser Source includes an optical bench and electronic devices to generate the laser beams required. This paper describes PHARAO and the role laser beams play in its principle of operation. Then we present the Laser Source design, the technologies involved, and the status of development. Lastly, we focus of a key equipment to reach the performances expected, which is the Extended Cavity Laser Diode.

Quantum mechanics imposes a fundamental tradeoff between the accuracy of time measurements and the size of the systems used as clocks. When the measurements of different time intervals are combined, the errors due to the finite clock size accumulate, resulting in an overall inaccuracy that grows with the complexity of the setup. Here we introduce a method that in principle eludes the accumulation of errors by coherently transferring information from a quantum clock to a quantum memory of the smallest possible size. Our method could be used to measure the total duration of a sequence of events with enhanced accuracy, and to reduce the amount of quantum communication needed to stabilize clocks in a quantum network., 10 + 5 pages, 3 figures