We present a number of analytical means for the analysis of the evolution of an atom clock, emphasizing the role of many-body processes in the population and coherence loss. A master equation with one-body and two-body losses included is presented for a thermal cloud ensemble, suitable for modelling atom clock evolution. Approximate solutions for the coupled rate equations with one-body and two-body losses are obtained for the case of Ramsey free evolution, and their validity is analyzed. An approach is suggested to combine the analytical trap-induced fringe envelope function with a numerically evaluated many-body envelope function. A set of Ramsey and spin echo models is offered with analytical solutions for the off-resonant and on-resonant regimes with one-body population loss and phenomenological dephasing. Their detuning spectra are presented and the difference between them and the standard Ramsey model is shown: the Rabi pedestal in the present model has no spectral ripples due to the pi/2-pulse duration equal to a quarter of a Rabi oscillation, even off resonance, in contrast to the Ramsey method where the pi/2-pulse duration is not corrected off resonance and causes spectral ripples. the off-resonant and on-resonant regimes with one-body population loss and phenomenological dephasing. Their detuning spectra are presented and the difference between them and the standard Ramsey model is shown: the Rabi pedestal in the present model has no spectral ripples due to the pi/2-pulse duration equal to a quarter of a Rabi oscillation, even off resonance, in contrast to the Ramsey method where the pi/2-pulse duration is not corrected off resonance and causes spectral ripples. An interferometric fringe in long evolution experiments is typically measured with a visibility decay primarily due to phase diffusion, phase instability, trapping effects and many-body losses. It constitutes a problem to extract parameters from a fringe obtained in the time domain because of the growing uncertainty of the fringe amplitude and phase. This visibility loss can be compensated for systems with unequal state population losses, as the one we have, by tailoring the pulse durations. Methods are suggested for the one-body and many-body cases.A comprehensive set of experiments is conducted on the characterization of the stability of the trapped atom clock: with condensate and thermal cloud ensembles, with phase-domain Ramsey and spin echo interferometries, with synchronization from a rubidium frequency standard and from an oven-controlled quartz oscillator. The types of noise present in the system are identified by means of an Allan deviation analysis. Despite the 50 times different cloud densities of the condensate and the thermal clouds, an equality of the phase stability in the Ramsey interferometry with BEC and thermal clouds is demonstrated. The spin echo interferometry has shown a slightly superior stability with the thermal clouds as compared to the BEC.
We present a number of analytical means for the analysis of the evolution of an atom clock, emphasizing the role of many-body processes in the population and coherence loss. A master equation with one-body and two-body losses included is presented for a thermal cloud ensemble, suitable for modelling atom clock evolution. Approximate solutions for the coupled rate equations with one-body and two-body losses are obtained for the case of Ramsey free evolution, and their validity is analyzed. An approach is suggested to combine the analytical trap-induced fringe envelope function with a numerically evaluated many-body envelope function. A set of Ramsey and spin echo models is offered with analytical solutions for the off-resonant and on-resonant regimes with one-body population loss and phenomenological dephasing. Their detuning spectra are presented and the difference between them and the standard Ramsey model is shown: the Rabi pedestal in the present model has no spectral ripples due to the pi/2-pulse duration equal to a quarter of a Rabi oscillation, even off resonance, in contrast to the Ramsey method where the pi/2-pulse duration is not corrected off resonance and causes spectral ripples. the off-resonant and on-resonant regimes with one-body population loss and phenomenological dephasing. Their detuning spectra are presented and the difference between them and the standard Ramsey model is shown: the Rabi pedestal in the present model has no spectral ripples due to the pi/2-pulse duration equal to a quarter of a Rabi oscillation, even off resonance, in contrast to the Ramsey method where the pi/2-pulse duration is not corrected off resonance and causes spectral ripples. An interferometric fringe in long evolution experiments is typically measured with a visibility decay primarily due to phase diffusion, phase instability, trapping effects and many-body losses. It constitutes a problem to extract parameters from a fringe obtained in the time domain because of the growing uncertainty of the fringe amplitude and phase. This visibility loss can be compensated for systems with unequal state population losses, as the one we have, by tailoring the pulse durations. Methods are suggested for the one-body and many-body cases.A comprehensive set of experiments is conducted on the characterization of the stability of the trapped atom clock: with condensate and thermal cloud ensembles, with phase-domain Ramsey and spin echo interferometries, with synchronization from a rubidium frequency standard and from an oven-controlled quartz oscillator. The types of noise present in the system are identified by means of an Allan deviation analysis. Despite the 50 times different cloud densities of the condensate and the thermal clouds, an equality of the phase stability in the Ramsey interferometry with BEC and thermal clouds is demonstrated. The spin echo interferometry has shown a slightly superior stability with the thermal clouds as compared to the BEC.
We offer interferometry models for thermal ensembles with one-body losses and the phenomenological inclusion of perturbations covering most of the thermal atom experiments. A possible extension to the many-body case is briefly discussed. The Ramsey pulses are assumed to have variable durations and the detuning during the pulses is distinguished from the detuning during evolution. Consequently, the pulses are not restricted to resonant operation and give more flexibility to optimize the interferometer to particular experimental conditions. On this basis another model is devised in which the contrast loss due to the unequal one-body population decays is canceled by the application of a nonstandard splitting pulse. For the importance of its practical implications, an analogous spin-echo model is also provided. The developed models are suitable for the analysis of atomic clocks and a broad range of sensing applications; they are particularly useful for trapped-atom interferometers.
An ensemble density matrix model that includes one- and two-body losses is derived for a trapped-atom clock. A trapped-atom clock is mainly affected by one- and two-body losses, generally giving nonexponential decays of populations; nevertheless, three-body recombination is also quantitatively analyzed to demonstrate the boundaries of its practical relevance. The importance of one-body losses is highlighted without which population trapping behavior would be observed. The model is written with decay constants expressed through experimental parameters. It can complement, e.g., the ISRE (identical spin rotation effect) model to improve its predictions: ISRE dramatically increases the ensemble coherence time, hence it enables one to observe the influence of two-body losses on the interferometry contrast envelope. The presented model is useful for Ramsey interferometry and is ready for immediate experimental verification in existing systems.
Ultracold atoms and Bose-Einstein condensates represent a burgeoning and ever exciting area of research which has produced numerous breakthroughs in the last 15 years. Ultracold atoms at nano-Kelvin temperatures, isolated in atom traps from the hot, room temperature environment, exhibit ultimate quantum properties and can serve as pure samples for testing quantum mechanics and performing precision measurements. The research described in this thesis investigates, both theoretically and experimentally, the coherence and dynamical evolution of a two-component Bose-Einstein condensate (BEC). The studies are performed on the |1 › ≡ F = 1,mF = −1 › and |2 › ≡ |F = 2,mF = +1 › hyperfine ground states of a 87Rb BEC trapped on an atom chip. Conventional understanding has been that the fringe contrast in interferometric experiments on a BEC is limited by the mean-field dephasing due to strong interatomic interactions which lead to inhomogeneous collisional shifts. We have discovered using Ramsey interferometry a mean-field driven self-rephasing effect in a trapped twocomponent BEC. When combined with a spin-echo technique, we find that the selfrephasing leads to a coherence time of 2.8 s, the longest ever recorded for an interacting BEC. Secondly, we have developed a new technique based on periodic collective oscillations for precision measurements of the interspecies and intraspecies scattering lengths and derived an analytic mean-field theory for the phase and density dynamics in the twocomponent BEC. This technique has been applied to the measurements of the scattering lengths for the two components, |1i and |2i, in 87Rb with a precision of 0.016%. Additionally, the two-body loss coefficients for these states have been measured. Thirdly, we have developed and applied a new, interferometric method for calibrating the detection system, in order to determine the total atom number precisely for Ramsey interferometric measurements. The calibration coefficient is found to be in a good agreement with that obtained using a conventional calibration technique based on the critical BEC temperature. Finally, we have detected predicted RF-induced Feshbach resonances by monitoring changes in the two-body loss coefficients in the two-component BEC. Conventional Feshbach resonances have been used in many works to tune the s-wave interactions; however, there is no achievable magnetic Feshbach resonance for magnetically trappable states of 87Rb. The RF-induced resonances which we have detected can provide a way to tune the scattering lengths which is important for interferometry and entanglement experiments. The positions of the resonances may provide useful information for predicting the atomic scattering properties of 87Rb.
Coherent-population trapping (CPT) is a multilevel quantum coherence phenomenon of promising applications in atomic clocks and magnetometers. In particular, multipulse CPT Ramsey interferometry is a powerful tool for improving the performance of CPT atomic clocks. Most studies on multipulse CPT Ramsey interferometry consider periodic pulse sequence and time-independent detuning. However, to further improve the accuracy and precision, one may modify the spectrum symmetry which involves a pulse sequence with time-dependent detuning or phase shift. Here we theoretically analyze the multipulse CPT Ramsey interferometry under arbitrary pulse sequences of time-dependent detuning and obtain a general analytical formula. Using our formula, we analyze the popular CPT Ramsey interferometry schemes such as two-pulse symmetric and antisymmetric spectroscopy and multipulse symmetric and antisymmetric spectroscopy. Under periodic pulses, we quantitatively obtain the influences of pulse width, pulse period, pulse number, and Rabi frequency. Furthermore, we examine the impact of CPT pulses and Rabi frequency on the light shift in both two-pulse and multipulse CPT Ramsey interference. Our theoretical results can guide the experimental design to improve the performance of atomic clocks via multipulse CPT Ramsey interferometry.
This thesis describes new results on the entanglement of atomic spins in Ramsey interferometry, optical atomic clocks and trapped ions. It is divided into three parts: First, we investigate improvements to conventional Ramsey interferometry with entanglement and adding only rotations of the collective spin to adjust the signal and measurement directions. The geometric degrees of freedom, connected to the rotations, are analytically optimized for a large class of generalized Ramsey protocols to allow efficient optimization of all parameters. Besides a unification of existing approaches, the main result is that there is only one new protocol, where a previously unused double inversion is applied. Studies of the local sensitivity show that this protocol reaches the fundamental quantum Fisher information limit and is yet robust against errors during preparation and measurement. In the second section we investigate the conditions under which optical atomic clocks exhibit increased long-term stability when applying weakly entangled, spin squeezed states. We discuss the common case of an atomic clock with a single ensemble, typical Brownian frequency noise and finite dead time. Theoretical modelling of the servo loop allows quantitative predictions of the optimal stability for given values of dead time and laser noise, in very good agreement with numerical simulations of the closed feedback loop. The main result is that, even with the current most stable lasers, the clock stability can only be improved for ensembles below a critical atom number of about one thousand in optical Sr lattice clocks. Even with a future improvement of the laser performance by one order of magnitude, the critical atom number still remains below 100,000. In contrast, clocks based on smaller, non-scalable ensembles, such as ion clocks, can already benefit from squeezed states with current clock lasers. Thus the last section considers the robust generation of entanglement in ion traps. An error budget including relevant experimental error sources is calculated for state-of-the-art quantum gates, driven by oscillating microwave gradients in surface traps. Amplitude modulation of the driving fields is shown to efficiently counteract the current limitations from motional mode instability. The predicted increase of the gate quality was demonstrated by the group of C. Ospelkaus at PTB Braunschweig, who measured gates with errors as low as ~ 10^(-3). In a similar approach, interactions between spin and motion can also be generated by combining oscillating rf-fields with a static magnetic field gradient. Penning traps designed for precision spectroscopy already feature large magnetic field gradients at the edge of a magnetic bottle configuration. We present parameters and conditions under which laser-free coupling of spin and quantized motion for (anti-)protons is possible at these points, in a step towards quantum logic spectroscopy for (anti-)protons.
We discuss the frequency and visibility of atom-molecule Ramsey fringes observed in recent experiments by Claussen et al. [Phys. Rev. A 67, 060701 (2003)]. In these experiments a 85Rb Bose-Einstein condensate was exposed to a sequence of magnetic field pulses on the high-field side of the 155G Feshbach resonance. The observed oscillation frequencies largely agree with the theoretically predicted magnetic field dependence of the binding energy of the highest excited diatomic vibrational state, except for a small region very close to the singularity of the scattering length. Our analytic treatment of the experiment, as well as our dynamical simulations, follow the magnitude of the measured oscillation frequencies as well as the visibilities of the Ramsey fringes. We show that significant deviations from a purely binary dynamics, with an associated binding frequency, occur when the spatial extent of the molecular wave function becomes comparable with the mean distance between the atoms in the dilute gas. The experiments thus clearly identify the conditions under which diatomic molecules may be identified as a separate entity of the gas or, conversely, when the concept of binary physics in a many-body environment is bound to break down.