Fractional Frequency Shift Research Articles

Quantum time dilation in a gravitational field

According to relativity, the reading of an ideal clock is interpreted as the elapsed proper time along its classical trajectory through spacetime. In contrast, quantum theory allows the association of many simultaneous trajectories with a single quantum clock, each weighted appropriately. Here, we investigate how the superposition principle affects the gravitational time dilation observed by a simple clock – a decaying two-level atom. Placing such an atom in a superposition of positions enables us to analyze a quantum contribution to a classical time dilation manifest in spontaneous emission. In particular, we show that the emission rate of an atom prepared in a coherent superposition of separated wave packets in a gravitational field is different from the emission rate of an atom in a classical mixture of these packets, which gives rise to a quantum gravitational time dilation effect. We demonstrate that this nonclassical effect also manifests in a fractional frequency shift of the internal energy of the atom that is within the resolution of current atomic clocks. In addition, we show the effect of spatial coherence on the atom's emission spectrum.

A Structured Sparse Bayesian Channel Estimation Approach for Orthogonal Time-Frequency Space Modulation.

Orthogonal time-frequency space (OTFS) modulation has been advocated as a promising waveform for achieving integrated sensing and communication (ISAC) due to its superiority in high-mobility adaptability and spectral efficiency. In OTFS modulation-based ISAC systems, accurate channel acquisition is critical for both communication reception and sensing parameter estimation. However, the existence of the fractional Doppler frequency shift spreads the effective channels of the OTFS signal significantly, making efficient channel acquisition very challenging. In this paper, we first derive the sparse structure of the channel in the delay Doppler (DD) domain according to the input and output relationship of OTFS signals. On this basis, a new structured Bayesian learning approach is proposed for accurate channel estimation, which includes a novel structured prior model for the delay-Doppler channel and a successive majorization-minimization (SMM) algorithm for efficient posterior channel estimate computation. Simulation results show that the proposed approach significantly outperforms the reference schemes, especially in the low signal-to-noise ratio (SNR) region.

Cervical Tissue Hydration Level Monitoring by a Resonant Microwave Coaxial Probe

Cervical tissue hydration level is one of the most important parameters to monitor in the early diagnosis of preterm birth. Electrical-impedance-spectroscopy-based techniques are often used, but they suffer from limited accuracy. Open microwave coaxial probes have been widely used as a broadband dielectric characterization technique for human tissue samples due to their versatility, but with limited accuracy due to their nonresonant nature. In this work, a resonant microwave open coaxial probe with multiple harmonic resonances is proposed as a sensing platform for tissue-hydration-level monitoring. The mechanical design was analyzed and verified by finite-element full 3D electromagnetic simulation and experiments. Dominant sources of errors and the ways to mitigate them were discussed. In vitro experiments were carried out on human cervix samples to verify the precision and accuracy by comparing the results to a commercial skin-hydration sensor. The proposed sensor shows mean fractional frequency shift of (3.3 ± 0.3) × 10-4 per unit % over the entire data. This translates into an absolute frequency shift (ΔfN) of 252 ± 23 kHz/%, 455 ± 41 kHz/%, and 647 ± 57 kHz/% at second, fourth, and sixth harmonic resonance, respectively.

The antenna phase center motion effect in high-accuracy spacecraft tracking experiments

We present an improved model for the antenna phase center motion effect for high-gain mechanically steerable ground-based and spacecraft-mounted antennas that takes into account non-perfect antenna pointing. Using tracking data of the RadioAstron spacecraft we show that our model can result in a correction of the computed value of the effect of up to $2\times10^{-14}$ in terms of the fractional frequency shift, which is significant for high-accuracy spacecraft tracking experiments. The total fractional frequency shift due to the phase center motion effect can exceed $1\times10^{-11}$ both for the ground and space antennas depending on the spacecraft orbit and antenna parameters. We also analyze the error in the computed value of the effect and find that it can be as large as $4\times10^{-14}$ due to uncertainties in the spacecraft antenna axis position, ground antenna axis offset and misalignment, and others. Finally, we present a way to reduce both the ground and space antenna phase center motion effects by several orders of magnitude, e.g. for RadioAstron to below $1\times10^{-16}$, by tracking the spacecraft simultaneously in the one-way downlink and two-way phase-locked loop modes, i.e. using the Gravity Probe A configuration of the communications links.

The CMB, Preferred Reference System, and Dragging of Light in the Earth Frame

The dominant CMB dipole anisotropy is a Doppler effect due to a particular motion of the solar system with a velocity of 370 km/s. Since this derives from peculiar motions and local inhomogeneities, one could meaningfully consider a fundamental frame of rest Σ associated with the Universe as a whole. From the group properties of Lorentz transformations, two observers, individually moving within Σ, would still be connected by the relativistic composition rules. However, the ultimate implications could be substantial. Physical interpretation is thus traditionally demanded in order to correlate some of the dragging of light observed in the laboratory with the direct CMB observations. Today, the small residuals—from those of Michelson–Morley to present experiments with optical resonators—are just considered instrumental artifacts. However, if the velocity of light in the interferometers is not the same parameter “c” of Lorentz transformations, nothing would prevent a non-zero dragging. Furthermore, the observable effects would be much smaller than what is classically expected and would most likely be of an irregular nature. We review an alternative reading of experiments that leads to remarkable correlations with the CMB observations. Notably, we explain the irregular 10−15 fractional frequency shift presently measured with optical resonators operating in vacuum and solid dielectrics. For integration times of about 1 s and a typical Central European latitude, we also predict daily variations of the Allan variance in the range (5÷12)·10−16.

Evaluation and suppression of the magnetic-induced Rabi broadening of magnetic-sensitive Ramsey fringes

Magnetic-sensitive Ramsey fringes are useful for the magnetic field evaluation of atomic fountain clocks. However, broadening mechanisms affect the visibility of the fringes and limit their application for measurement of the magnetic field. The magnetic-induced Rabi broadening, which comes from the magnetic field inhomogeneity in the microwave cavity of fountain clocks and results in the visibility reduction of the fringes, is demonstrated in this paper. Furthermore, an improved method to shorten the Rabi interaction time is proposed to suppress the broadening effect. Experiments are conducted on the rubidium 87 atomic fountain clock to test the broadening mechanism as well as to demonstrate the effectiveness of the improved method to suppress the broadening effect. As a result, we obtain magnetic-sensitive Ramsey fringes and evaluate the average magnetic field. The second-order Zeeman frequency shift also is evaluated with a fractional frequency shift of 1.295 × 10 − 13 , and a frequency uncertainty of 3 × 10 − 16 .

Temperature dependence of reconfigurable bandstop filters using vanadium dioxide switches

In this Letter, we report and investigate the temperature dependency of various radio frequency (RF) parameters for a fabricated reconfigurable bandstop filter with vanadium dioxide (VO2) switches measured up to 55 GHz. Here, the insulator-to-metal (ITM) and metal-to-insulator transition (MIT) hysteresis of the VO2 thin film influence on the RF characteristics of the filters is analyzed from 25 °C and 120 °C in heating and cooling. The resonance frequency and maximum insertion loss (IL) stability and sensitivity with temperature variations are explored. It is noticed that increasing the temperature to 50 °C from 25 °C (or decreasing it to 50 °C from 120 °C) will result in a less than 1% fractional frequency shift with respect to the off and on resonance frequencies. The sharp DC conductivity level variations of the VO2 thin film around the transition temperatures translate into sharp effects on the resonance characteristics of the filters. On the contrary, the maximum IL levels are less sensitive to the sharp conductivity changes of DC films around the VO2 transition temperature. A unique behavior is reported when successively heating-up and cooling-down, over and below, respectively, the transition temperature of VO2: the fabricated filter exhibits completely different resonance frequencies. This suggests that in the temperature dependence of the VO2 RF design, the practical use of reconfigurable RF functions has to take into account the history of thermal effects and increase or decrease in the device temperature when crossing the IMT/MIT transition point.

Simulation of Nonlinear Resonance, Amplitude-Frequency, and Harmonic Generation Effects in SAW and BAW Devices.

Nonlinearly coupled sets of piezoelectric field equations in the frequency domain were derived for the nonlinear propagation of finite-amplitude waves in piezoelectric bulk acoustic wave (BAW) and surface acoustic wave (SAW) devices. To verify their accuracy, we have embedded these sets of equations in the finite-element method (FEM) of COMSOL Multiphysics software and compared the FEM results with both the analytical and experimental results found in the published literature. The nonlinear frequency responses for both plano- and contoured-plate resonators of AT-cut quartz were investigated under various voltage drives, circuit resistances, and quality factors. The proposed equations with FEM have also been employed to study 33.3-MHz very-high-frequency (VHF) quartz resonators, showing that under different conditions of how well the third overtone mode (f3) matches the third harmonic (3f) and how the fractional frequency shift of the third overtone mode (f3) occurs as a function of the fundamental mode current. Furthermore, we have studied the nonlinear harmonic generation of an 840-MHz 128° Y-cut X-propagating (128° YX) LiNbO3 SAW resonator. The second-harmonic (H2) and third-harmonic (H3) modes were observed to occur, respectively, at two-time (2f) and three-time (3f) frequencies of fundamental frequency (f) when such resonators were driven with high power. The effects of substrate thickness, bottom surface conditions of the substrate, and different circuit connections on the H2 and H3 generations were simulated and compared with available measurements. Current proposed sets of equations are general and could be used for the study of nonlinear resonance, amplitude-frequency effect, and harmonic generation in any piezoelectric devices, provided that the necessary nonlinear material constants are known.

A pyramid MOT with integrated optical cavities as a cold atom platform for an optical lattice clock

We realize a two-stage, hexagonal pyramid magneto-optical trap (MOT) with strontium, and demonstrate loading of cold atoms into cavity-enhanced 1D and 2D optical lattice traps, all within a single compact assembly of in-vacuum optics. We show that the device is suitable for high-performance quantum technologies, focusing especially on its intended application as a strontium optical lattice clock. We prepare 2 × 104 spin-polarized atoms of 87Sr in the optical lattice within 500 ms; we observe a vacuum-limited lifetime of atoms in the lattice of 27 s; and we measure a background DC electric field of 12 V m−1 from stray charges, corresponding to a fractional frequency shift of (−1.2 ± 0.8) × 10−18 to the strontium clock transition. When used in combination with careful management of the blackbody radiation environment, the device shows potential as a platform for realizing a compact, robust, transportable optical lattice clock with systematic uncertainty at the 10−18 level.

Investigating ground-state fine-structure properties to explore suitability of boronlike S11+−K14+ and galliumlike Nb10+−Ru13+ ions as possible atomic clocks

We study various properties of the $2p\phantom{\rule{0.16em}{0ex}}^{2}P_{1/2}\ensuremath{-}2p\phantom{\rule{0.16em}{0ex}}^{2}P_{3/2}$ fine-structure splittings in the boronlike ${\mathrm{S}}^{11+}, {\mathrm{Cl}}^{12+}, {\mathrm{Ar}}^{13+}$, and ${\mathrm{K}}^{14+}$ ions and the $4p\phantom{\rule{0.16em}{0ex}}^{2}P_{1/2}\ensuremath{-}4p\phantom{\rule{0.16em}{0ex}}^{2}P_{3/2}$ fine-structure splittings in the galliumlike ${\mathrm{Nb}}^{10+}, {\mathrm{Mo}}^{11+}, {\mathrm{Tc}}^{12+}$, and ${\mathrm{Ru}}^{13+}$ ions to find out the feasibility of using these highly charged ions as suitable optical atomic clocks. The roles of the electron correlations due to the Dirac-Coulomb-Breit Hamiltonian and accounting for lower-order quantum electrodynamics effects are shown explicitly in the calculations of electron affinities, excitation energies, transition-matrix elements, lifetimes, hyperfine-structure constants, and electric quadrupole moments of the states involved with clock transitions using a relativistic couple-cluster method. We also estimate the most commonly appearing systematic effects in the atomic clock experiments due to the electric quadrupole, the second-order Zeeman, both the dc and ac Stark, and the black-body radiation shifts in the aforementioned fine-structure splittings to demonstrate typical orders of magnitude of fractional frequency shifts.

Estimation of the ion-trap assisted electrical loads and resulting BBR shift

Capacitive, inductive and resistive loads of an ion-trap system, which can be modelled as LCR circuits, are important to know for building a high accuracy experiment. Accurate estimation of these loads is necessary for delivering the desired radio frequency (RF) signal to an ion trap via an RF resonator. Of particular relevance to the trapped ion optical atomic clock, determination of these loads lead to accurate evaluation of the Black-Body Radiation (BBR) shift resulting from the inaccurate machining of the ion-trap itself. We have identified different sources of these loads and estimated their values using analytical and finite element analysis methods, which are found to be well in agreement with the experimentally measured values. For our trap geometry, we obtained values of the effective inductive, capacitive and resistive loads as: 3.1 μH, 3.71 (1) μH, 3.68 (6) μH; 50.4 pF, 51.4 (7) pF, 40.7 (2) pF; and 1.373 Ω, 1.273 (3) Ω, 1.183 (9) Ω by using analytical, numerical and experimental methods, respectively. The BBR shift induced by the excess capacitive load arising due to machining inaccuracy in the RF carrying parts has been accurately estimated, which results to a fractional frequency shift of 6.6 × 10−17 for an RF of 1 kV at 2π × 15 MHz and with ±10 μm machining inaccuracy. This needs to be incorporated into the total systematic uncertainty budget of a frequency standard as it is about one order of magnitude higher than the present precision of the trapped ion optical clocks.

Corrections to “Temperature Correction for Cylindrical Cavity Perturbation Measurements” [Jun 17 2153-2161

In the above paper [1] , it has come to the attention of the authors that a typographical error exists in (16) and (17). The error, while small in syntax, has a large impact on the understanding behind the explained technique. Reference [1, eq. (15)] states that by subtracting the fractional frequency shifts of the sample perturbed and the unperturbed case, any resultant fractional frequency shift is caused by the temperature dependent properties of the sample, if any. Now, the printed and erroneous [1, eq. (16)] should be \begin{equation*} \varepsilon _{1}( T )\approx -2\left [{ \frac {f_{s}( T )-f_{0}(0)}{f_{0}( 0)} }\right ]\frac {V_{c}}{V_{s}}G_{\mathrm {mnp}}+1. \end{equation*} We note that (16) incorrectly states that the unperturbed frequency ( $f_{0}$ ) is a function of temperature. This is incorrect since all temperature dependence in the unperturbed state is already addressed in [1, eq. (15)] . Additionally, the fractional frequency shift is normally taken with respect to the reference unperturbed frequency $f_{0}$ as opposed to $f_{s}$ ; however, since the relative difference in $f_{s}$ and $f_{0}$ is small at gigahertz frequencies, this is not a large issue. Reference [1, eq. (16)] should be printed as (1) , where $f_{s}( T )=f_{s}( {0})\times ( 1+(({\Delta f_{s}( T )})/({f_{s}( 0)})) )$ is the temperature dependent frequency while the erroneous $f_{0}( T )$ has been replaced with $f_{0}(0)$ , the unperturbed frequency taken at a reference temperature. All temperature dependence is addressed in $f_{s}( T )$ and if there was no temperature dependence, $f_{s}( T )$ would be a constant and the original cavity perturbation approximation would apply. Similarly, [1, eq. (17)] should be printed as \begin{equation*} \varepsilon _{2}( T )\approx \left [{ \frac {1}{Q_{s}( T )}-\frac {1}{Q_{0}( 0)} }\right ]\frac {V_{c}}{V_{s}}G_{\mathrm {mnp}} \end{equation*} where $({1}/({Q_{s}( T)}))=(({{\text {BW}}_{s}(0)})/({f_{s}( T )}))\times ( 1+(({\Delta {\text {BW}}_{s}( T )})/({\text {BW}_{s}{(0)}})) )$ is the reciprocal of the temperature dependent quality factor and $Q_{0}( T )$ has been replaced with $Q_{0}( {0} )$ which is the unperturbed quality factor at a reference temperature.

Micro load theory of surface acoustic wave gravimetric sensors

Surface acoustic wave (SAW) gravimetric sensors have been widely applied in the field of chemical and biological detections for decades. However, it is noticed that the operating mechanism of this type of sensors is not understood well. A perturbation theory commonly adopted has been used to predict the mass sensitivity of the SAW sensors, but, as indicated also, “there is a significant and systematic quantitative discrepancy” between the theoretical and experimental results [Wohltjen, et al. IEEE UFFC, 1987]. In this paper, a micro load theory is presented. The coating film with biochemical material adsorbed is not regarded as a perturbation source, but a micro mechanical load. Then, it is revealed that the fractional frequency shift caused by the load is minus and half of the fractional area density change, where the unloaded area density is a product of the substrate material’s density and SAW wavelength. Also, it is interesting to note that the expression is the same with that for Lamb wave sensors if...

Ultrathin metamaterial-based perfect absorbers for VHF and THz bands

An ultrathin and angularly stable metamaterial perfect absorber (MPA) is demonstrated for VHF-band using four connected split-square resonators structure and low-cost fabrication process. The total incoming energy of electromagnetic wave (at 250 MHz) is consumed inside the efficient thickness, which is 240 times smaller than the absorption wavelength of MPA. Our MPA works well for a very wide range of incident angle up to 45° of electromagnetic wave and exhibits the polarization-independent behavior. Furthermore, by scaling the initial design and integrating a ferroelectric material (strontium titanate), a thermo-tunable ultrathin MPA is realized in the THz region. At room temperature (300 K), the thickness of THz MPA reaches roundly 1/300 of the working wavelength. In addition, a fractional frequency shift of 49% at the absorption over 90% can be controlled in the varied temperature range from 150 to 400 K. Our results presents good candidates for potential devices operating from the radio to the THz range.

Analysis of thermal radiation in ion traps for optical frequency standards

In many of the high-precision optical frequency standards with trapped atoms or ions that are under development to date, the ac Stark shift induced by thermal radiation leads to a major contribution to the systematic uncertainty. We present an analysis of the inhomogeneous thermal environment experienced by ions in various types of ion traps. Finite element models which allow the determination of the temperature of the trap structure and the temperature of the radiation were developed for five ion trap designs, including operational traps at PTB and NPL and further optimized designs. Models were refined based on comparison with infrared camera measurement until an agreement of better than 10% of the measured temperature rise at critical test points was reached. The effective temperature rises of the radiation seen by the ion range from 0.8 K to 2.1 K at standard working conditions. The corresponding fractional frequency shift uncertainties resulting from the uncertainty in temperature are in the 10−18 range for optical clocks based on the Sr+ and Yb+ E2 transitions, and even lower for Yb+ E3, In+ and Al+. Issues critical for heating of the trap structure and its predictability were identified and design recommendations developed.

Frequency shifts in NIST Cs primary frequency standards due to transverse rf field gradients

A single-particle Green's function (propagator) is introduced to study the detection of laser-cooled Cesium atoms in an atomic fountain due to RF field gradients in the Ramsey TE011 cavity. The detection results in a state-dependent loss of atoms at apertures in the physics package, resulting in a frequency bias. A model accounting only for motion in one dimension transverse to the symmetry axis of the fountain is discussed in detail and then generalized to two transverse dimensions. Results for fractional frequency shifts due to transverse field gradients are computed for NIST F-1 and F-2 Cesium fountains. The shifts are found to be negligible except in cases of higher RF power applied to the cavities.

Analysis of inhomogeneous-excitation frequency shifts of ytterbium optical lattice clocks

We have investigated the frequency shifts caused by inhomogeneous excitation in a 171Yb optical lattice clock. The dependences of the inhomogeneity on the temperature of the cold ytterbium atoms and the misaligning angle between the lattice laser and the clock laser are analyzed by numerical calculations. The dependence of the fractional collisional frequency shift on the ground state fraction under different cold atom temperatures, atom numbers, lattice trap depths and unequal transverse and longitudinal temperatures are also shown. The results show that the uncertainty of the ytterbium clocks, contributed by the inhomogeneous excitation, can be reduced to be 10−19 or even lower with certain conditions.

Effects ofHe3impurity on solidHe4studied by compound torsional oscillator

Frequency shifts and dissipations of a compound torsional oscillator induced by solid $^4$He samples containing $^3$He impurity concentrations ($x_3$ = 0.3, 3, 6, 12 and 25 in units of 10$^{-6}$) have been measured at two resonant mode frequencies ($f_1$ = 493 and $f_2$ = 1164 Hz) at temperatures ($T$) between 0.02 and 1.1 K. The fractional frequency shifts of the $f_1$ mode were much smaller than those of the $f_2$ mode. The observed frequency shifts continued to decrease as $T$ was increased above 0.3 K, and the conventional non-classical rotation inertia fraction was not well defined in all samples with $x_3 \geq$ 3 ppm. Temperatures where peaks in dissipation of the $f_2$ mode occurred were higher than those of the $f_1$ mode in all samples. The peak dissipation magnitudes of the $f_1$ mode was greater than those of the $f_2$ mode in all samples. The activation energy and the characteristic time ($\tau_0$) were extracted for each sample from an Arrhenius plot between mode frequencies and inverse peak temperatures. The average activation energy among all samples was 430 mK, and $\tau_0$ ranged from 2$\times 10^{-7}$ s to 5$\times 10^{-5}$ s in samples with $x_3$ = 0.3 to 25 ppm. The characteristic time increased in proportion to $x_3^{2/3}$. Observed temperature dependence of dissipation were consistent with those expected from a simple Debye relaxation model \emph{if} the dissipation peak magnitude was separately adjusted for each mode. Observed frequency shifts were greater than those expected from the model. The discrepancies between the observed and the model frequency shifts increased at the higher frequency mode.

Nonexponential T2* decay in white matter

Visualizing myelin in human brain may help the study of diseases such as multiple sclerosis. Previous studies based on T(1) and T(2) relaxation contrast have suggested the presence of a distinct water pool that may report directly on local myelin content. Recent work indicates that T(2) contrast may offer particular advantages over T(1) and T(2) contrast, especially at high field. However, the complex mechanism underlying T(2) relaxation may render interpretation difficult. To address this issue, T(2) relaxation behavior in human brain was studied at 3 and 7 T. Multiple gradient echoes covering most of the decay curve were analyzed for deviations from mono-exponential behavior. The data confirm the previous finding of a distinct rapidly relaxing signal component (T(2) ∼ 6 ms), tentatively attributed to myelin water. However, in extension to previous findings, this rapidly relaxing component displayed a substantial resonance frequency shift, reaching 36 Hz in the corpus callosum at 7 T. The component's fractional amplitude and frequency shift appeared to depend on both field strength and fiber orientation, consistent with a mechanism originating from magnetic susceptibility effects. The findings suggest that T(2) contrast at high field may be uniquely sensitive to tissue myelin content and that proper interpretation will require modeling of susceptibility-induced resonance frequency shifts.

Lattice-induced nonadiabatic frequency shifts in optical lattice clocks

We consider the frequency shift in optical lattice clocks which arises from the coupling of the electronic motion to the atomic motion within the lattice. For the simplest of three-dimensional lattice geometries this coupling is shown to affect only clocks based on blue-detuned lattices. We have estimated the size of this shift for the prospective strontium lattice clock operating at the 390-nm blue-detuned magic wavelength. The resulting fractional frequency shift is found to be on the order of ${10}^{\ensuremath{-}18}$ and is largely overshadowed by the electric quadrupole shift. For lattice clocks based on more complex geometries or other atomic systems, this shift could potentially be a limiting factor in clock accuracy.