Adjustability as a requirement of ideal clocks

A coordinate frame is considered as an arrangement of clocks that meet certain criteria of synchronization. Einstein's ideal clock is compared with the behavior required for clocks to maintain synchrony in the group of coordinate frames that leaves Maxwell's equations invariant. The required clock rates differ from the rate of Einstein's clock. An ideal adjustable clock is defined as an Einstein clock augmented by variable “gearing” that can offset its rate. Ongoing adjustment of these clocks enables them to meet all synchronization criteria in the group of coordinate frames. The need for adjustment is due to the well known invariance of Maxwell's equations under a group of coordinate transformations larger than the Lorentz group, and has nothing to do with imperfections in clocks. It is shown that the adjustments needed by ideal adjustable clocks to maintain synchrony can be measurably separated from additional adjustments that may be needed to compensate for random imperfections. The necessity for adjustment brings with it the necessity for ongoing measurement of the light signals whose exchange defines synchronization. Implications are discussed, both for the interpretation of Maxwell's equations and for the role of measuring instruments.     

Multipartite Quantum Clock Synchronization under The Influence of Unruh Thermal Noise

A protocol for multipartite quantum clock synchronization is performed under the influence of Unruh thermal noise. The dynamics of multipartite quantum system consisting of Unruh–DeWitt detectors when one of the detectors is accelerated are obtained. To estimate the time difference between the clocks, the time probability is calculated and how the probability is influenced by the Unruh thermal noise and other factors is analyzed. It is shown that both relativistic motion and interaction between the atom and the external scalar field affect the choice of optimal number of excited atoms in the initial state, which leads to a higher clock adjustment accuracy. Time probabilities for different types of initial states approach the same value in the limit of infinite acceleration, while tend to different minimums with increasing number of atoms. In addition, the accuracy of clock synchronization using a pair of entangled clocks in two‐party system is always higher than in a multipartite system, while the optimal Z‐type multipartite initial state always performs better than the W‐type state.     

Thermal and phase decoherence effects on entanglement dynamics of the quantum spin systems

Entanglement dynamics of the N-qubit XY model in thermal and dephasing environments are investigated by solving the Lindblad form of the master equation. Analytical solutions for the two-qubit case and numerical solutions for the multi-qubit case are obtained. For the two-qubit case, our results revealed two main features for entanglement evolution from different initial states. First, the thermal reservoir always induces degradation of the entanglement, and the entanglement may undergo sudden death during certain intervals of the evolution time. Second, the dephasing environment induces damped oscillation of the entanglement for initially separable states and mixed states with relative large values of Δ or J; however, it always induces exponentially decay of the entanglement for the initial Bell states. For the multi-qubit case, our results show that the entanglement decreases monotonically as the time evolves for the initial W state, and behaves as damped oscillation for the initial “one-particle” state. Particularly, for system with large number of qubits, the curves of the concurrence C₁₂ with different N are almost overlapped in dephasing environment.     

Propagation of excitation in long 1D chains: Transition from regular quantum dynamics to stochastic dynamics

The quantum dynamics problem for a 1D chain consisting of 2N + 1 sites (N ? 1) with the interaction of nearest neighbors and an impurity site at the middle differing in energy and in coupling constant from the sites of the remaining chain is solved analytically. The initial excitation of the impurity is accompanied by the propagation of excitation over the chain sites and with the emergence of Loschmidt echo (partial restoration of the impurity site population) in the recurrence cycles with a period proportional to N. The echo consists of the main (most intense) component modulated by damped oscillations. The intensity of oscillations increases with increasing cycle number and matrix element C of the interaction of the impurity site n = 0 with sites n = ±1 (0 < C ≤ 1; for the remaining neighboring sites, the matrix element is equal to unity). Mixing of the components of echo from neighboring cycles induces a transition from the regular to stochastic evolution. In the regular evolution region, the wave packet propagates over the chain at a nearly constant group velocity, embracing a number of sites varying periodically with time. In the stochastic regime, the excitation is distributed over a number of sites close to 2N, with the populations varying irregularly with time. The model explains qualitatively the experimental data on ballistic propagation of the vibrational energy in linear chains of CH₂ fragments and predicts the possibility of a nondissipative energy transfer between reaction centers associated with such chains.     

Qubit-Based Clock Synchronization for QKD Systems Using a Bayesian Approach

Quantum key distribution (QKD) systems provide a method for two users to exchange a provably secure key. Synchronizing the users’ clocks is an essential step before a secure key can be distilled. Qubit-based synchronization protocols directly use the transmitted quantum states to achieve synchronization and thus avoid the need for additional classical synchronization hardware. Previous qubit-based synchronization protocols sacrifice secure key either directly or indirectly, and all known qubit-based synchronization protocols do not efficiently use all publicly available information published by the users. Here, we introduce a Bayesian probabilistic algorithm that incorporates all published information to efficiently find the clock offset without sacrificing any secure key. Additionally, the output of the algorithm is a probability, which allows us to quantify our confidence in the synchronization. For demonstration purposes, we present a model system with accompanying simulations of an efficient three-state BB84 prepare-and-measure protocol with decoy states. We use our algorithm to exploit the correlations between Alice’s published basis and mean photon number choices and Bob’s measurement outcomes to probabilistically determine the most likely clock offset. We find that we can achieve a 95 percent synchronization confidence in only 4140 communication bin widths, meaning we can tolerate clock drift approaching 1 part in 4140 in this example when simulating this system with a dark count probability per communication bin width of 8×10−4 and a received mean photon number of 0.01.     

Newtonian versus Langevin dynamics: Example of ϕ4-model with infinite range interactions

The functional integral representation for the generating functional (GF) of the canonically averaged ensemble with an underlying Newtonian dynamics is obtained. It is shown that for this representation the non-linear fluctuation-dissipation theorem (NFDT) has the same form as for the Langevin dynamics case. This GF-representation is used for the investigation of the dynamics of the ϕ⁴-model with infinite range interactions at T > T_c. It is shown that the kinetic equation for the complete correlation function has the same form as for the Langevin dynamics case, which was considered before. All peculiarities of Newtonian dynamics are absorbed by one-particle (2-point and 4-point) correlator and response functions. The analysis of this equation shows that the 1/N-fluctuations (where N is the number of particles) restore the ergodicity of the system with the characteristicsrate τ⁻¹ ∼ μ²/N, where μ is a coupling constant.     

Wavefunction of the universe for N=2 6D supergravity

The wavefunction of the universe for N=2 6D supergravity is calculated numerically using a minisuperspace approach. When compared with the solution of the classical equations it is found that the resulting evolution of the universe is of Friedmann form with the radius of the internal space performing damped oscillations about a constant value.     

Iterative quantum algorithm for distributed clock synchronization

Clock synchronization is a well-studied problem with many practical and scientific applications.We propose an arbitrary accuracy iterative quantum algorithm for distributed clock synchronization using only three qubits.The n bits of the time difference between two spatially separated clocks can be deterministically extracted by communicating only O(n) messages and executing the quantum iteration process n times based on the classical feedback and measurement operations.Finally,we also give the algorithm using only two qubits and discuss the success probability of the algorithm.     

Multipulse irradiation of silicon by femtosecond laser pulses: Variation of surface morphology

The multipulse interaction of ultraviolet femtosecond laser pulses with silicon and generation of surface structures in a large area spot (?1 mm²) has been studied. The evolution of multiscale structures at the constant fluence strongly depends on the number of pulses, N. For N < 200, the “carpet-like” pattern of nano-, and micro-spikes is generated by the bubble explosion in a thin surface foam layer. The accumulation of bubbles and their explosion due to repetition of laser pulses cause damped membrane-like oscillations of the silicon surface. For 200 ≤ N, bifurcation of surface morphology takes place: (i) the surface tension waves of the wavelength ∼200 μm appear in the peripheral region of the spot. Generated by the surface thermal gradient in the liquid foam layer, they spread from the hot centerline towards the periphery of the spot. The change of their wavelength with propagation distance indicates onset of the Eckhaus instability caused by the phase modulation in multipulse interaction. (ii) Deep caverns appear in a highly superheated silicon layer in the central region of the spot due to the fast gas-liquid phase separation and the fragmentation process.     

Frequency ratios of Sr,Yb, and Hg based optical lattice clocks and their applications

This article describes the recent progress of optical lattice clocks with neutral strontium (⁸⁷Sr), ytterbium (¹⁷¹Yb) and mercury (¹⁹⁹Hg) atoms. In particular, we present frequency comparison between the clocks locally via an optical frequency comb and between two Sr clocks at remote sites using a phase-stabilized fibre link. We first review cryogenic Sr optical lattice clocks that reduce the room-temperature blackbody radiation shift by two orders of magnitude and serve as a reference in the following clock comparisons. Similar physical properties of Sr and Yb atoms, such as transition wavelengths and vapour pressure, have allowed our development of a compatible clock for both species. A cryogenic Yb clock is evaluated by referencing a Sr clock. We also report on an Hg clock, which shows one order of magnitude less sensitivity to blackbody radiation, while its large nuclear charge makes the clock sensitive to the variation of fine-structure constant. Connecting all three types of clocks by an optical frequency comb, the ratios of the clock frequencies are determined with uncertainties smaller than possible through absolute frequency measurements. Finally, we describe a synchronous frequency comparison between two Sr-based remote clocks over a distance of 15 km between RIKEN and the University of Tokyo, as a step towards relativistic geodesy.     

Synchronisationseffekte in einer linearen Anordnung aus N Josephson-Verbindungen

Synchronization Effects in a Linear Array of N Josephson Junctions Within the frame of the RSJ model we investigate the synchronization of the oscillations in a linear array of N identical Josephson junctions shunted by an electromagnetic resonator. Using an adiabatic approximation to the first order of the parameter I_cI the reduced equations of the slowly varying phases are derived. These equations allow the detailed investigation of all the stationary states of the system. Only the coherent state in the inductive regime and the radiationless state in the capacitive regime are found to be stable. Including noise effects we discuss the order parameter concept for the resonator current in the case N ? 1.     

On the quantum density of states and partitioning an integer

This paper exploits the connection between the quantum many-particle density of states and the partitioning of an integer in number theory. For N bosons in a one-dimensional harmonic oscillator potential, it is well known that the asymptotic (N→∞) density of states is identical to the Hardy-Ramanujan formula for the partitions p(n), of a number n into a sum of integers. We show that the same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for p^s(n), the latter being the number of partitions of n into a sum of sth powers of a set of integers. By making an appropriate modification of the statistical technique, we are also able to obtain d^s(n) for distinct partitions. We find that the distinct square partitions d²(n) show pronounced oscillations as a function of n about the smooth curve derived by us. The origin of these oscillations from the quantum point of view is discussed. After deriving the Erdos-Lehner formula for restricted partitions for the s=1 case, we use the modified technique to obtain a new formula for distinct restricted partitions.     

Damped oscillations in the energy autocorrelation functions of the 58Ni+46Ti elastic and 58Ni+62Ni elastic and inelastic scattering cross sections

Structures of non-statistical character, recently observed in ⁵⁸ Ni +⁴⁶ Ti elastic and ⁵⁸ Ni +⁶² Ni elastic and inelastic excitation functions, produce damped oscillations in the cross section energy autocorrelation functions. The analysis of these damped oscillations in terms of S-matrix spin and parity decoherence indicates, as a possible interpretation, damping of the coherent rotational motion of the intermediate dinuclear system formed in the reaction.     

Synchronization of Huygens' clocks and the Poincaré method

We study two models of connected pendulum clocks synchronizing their oscillations, a phenomenon originally observed by Huygens. The oscillation angles are assumed to be small so that the pendulums are modeled by harmonic oscillators, clock escapements are modeled by the van der Pol terms. The mass ratio of the pendulum bobs to their casings is taken as a small parameter. Analytic conditions for existence and stability of synchronization regimes, and analytic expressions for their stable amplitudes and period corrections are derived using the Poincaré theorem on existence of periodic solutions in autonomous quasi-linear systems. The anti-phase regime always exists and is stable under variation of the system parameters. The in-phase regime may exist and be stable, exist and be unstable, or not exist at all depending on parameter values. As the damping in the frame connecting the clocks is increased the in-phase stable amplitude and period are decreasing until the regime first destabilizes and then disappears. The results are most complete for the traditional three degrees of freedom model, where the clock casings and the frame are consolidated into a single mass.     

Experimental and Real Coordinates in Space-Time Transformations

The experimental (apparent) space-time transformations connect coordinates altered by length contraction and clock retardation. When clocks are synchronized by means of light signals (Einstein–Poincaré procedure) or by slow clock transport, the experimental space-time. transformations assume the mathematical form of the Extended space-time transformations.⁽⁴⁾ These reduce to the Lorentz–Poincaré transformations when one of the frames they connect is the fundamental inertial frame. If the synchronization procedure were perfect, the experimental space-time transformations would assume the form of Selleris inertial transformations.⁽⁵⁾ The real space-time transformations are those which are disclosed when the systematic measurement distortions are corrected.     

The fluctuation-dissipation theorem for contracted descriptions of Markov processes

It is shown that if the Onsager-Casimir relations and the fluctuationdissipation theorem are valid for a stationary, Gaussian, Markov process in anN-dimensional space, then these relations are valid when the process is projected into a subspace of the original space. Both time-reversal-even and time-reversal-odd variables are allowed. Previous derivations of the fluctuation-dissipation theorem for Brownian motion from fluctuating hydrodynamics are special cases of the present result. For the Brownian motion problem, the fluctuation-dissipation theorem is proven for the case of a compressible, thermally conducting fluid with a nonlocal equation of state. Arbitrary slip boundary conditions are considered as well.     

On the complete synchronization of the Kuramoto phase model

We discuss the asymptotic complete phase-frequency synchronization for the Kuramoto phase model with a finite size N. We present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states. Our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations. But they are independent of the system size N, hence they can be used for the mean-field limit. For the complete synchronization estimates, we estimate the time evolution of the phase and frequency diameters for configurations. The initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast. In contrast, for the non-identical oscillators, the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations. Our estimates are based on the monotonicity arguments of extremal phase and frequencies, which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states. We compare our analytical results with numerical simulations.     

Collective states in highly symmetric atomic configurations and single-photon traps

We study correlated states in circular and linear-chain configurations of identical two-level atoms containing the energy of a single quasi-resonant photon in the form of a collective excitation, where the collective behavior is mediated by exchange of transverse photons between the atoms. For a circular atomic configuration containing N atoms, the collective energy eigenstates can be determined by group-theoretical means making use of the fact that the configuration possesses a cyclic symmetry group Z _N . For these circular configurations, the carrier spaces of the various irreducible representations of the symmetry group are at most two-dimensional, so that the effective Hamiltonian on the radiationless subspace of the system can be diagonalized analytically. As a consequence, the radiationless energy eigenstates carry a Z _N quantum number p = 0, 1, …, N, which is analogous to the angular momentum quantum number l = 0, 1, … carried by particles propagating in a central potential, such as a hydrogen-like system. Just as the hydrogen s states are the only electronic wave functions that can occupy the central region of the Coulomb potential, the quasi-particle corresponding to a collective excitation of the circular atomic sample can occupy the central atom only for vanishing Z _N quantum number p. When a central atom is present, the p = 0 state splits into two, showing level crossing at certain radii; in the regions between these radii, damped oscillations between two “ extreme” p = 0 states occur, where the excitation occupies either the outer atoms or the central atom only. For large numbers of atoms in a maximally subradiant state, a critical interatomic distance of λ/2 emerges both in the linear-chain and in the circular configuration of atoms. The spontaneous decay rate of the linear configuration exhibits a jumplike “critical” behavior for next-neighbor distances close to a half-wavelength. Furthermore, both the linear-chain and the circular configurations exhibit exponential photon trapping once the next-neighbor distance becomes less than a half-wavelength, with the suppression of spontaneous decay being particularly pronounced in the circular system. In this way, circular configurations containing sufficiently many atoms may be natural candidates for single-photon traps.     

Why two clocks synchronize: energy balance of the synchronized clocks

We consider the synchronization of two clocks which are accurate (show the same time) but have pendulums with different masses. We show that such clocks hanging on the same beam beside the complete (in-phase) and antiphase synchronizations perform the third type of synchronization in which the difference of the pendulums' displacements is a periodic function of time. We identify this period to be a few times larger than the period of pendulums' oscillations in the case when the beam is at rest. Our approximate analytical analysis allows to derive the synchronizations conditions, explains the observed types of synchronizations, and gives the approximate formula for both the pendulums' amplitudes and the phase shift between them. We consider the energy balance in the system and show how the energy is transferred between pendulums via oscillating beam allowing pendulums' synchronization.