The Rabi frequency (coupling strength) of an electric-dipole transition is an important experimental parameter in laser-cooling and other atomic physics experiments. Though the relationship between Rabi frequency and atomic wavefunctions and/or atomic lifetimes is discussed in many references, there is a need for a concise, self-contained, accessible introduction to such calculations suitable for use by the typical student of laser cooling (experimental or theoretical). In this paper, I outline calculations of the Rabi frequencies for atoms with sub-structure due to orbital, spin and nuclear angular momentum. I emphasize the physical meaning of the calculations.
In this work, we demonstrate a microwave magnetic field imaging technique based on Rabi resonance with a cesium atom vapor cell. Rabi resonance signals are generated when atoms interact with a phase-modulated microwave (MW) field and are detected by a photodiode and camera. A low noise, high quantum efficiency camera is used to capture a series of frames for different phase modulation frequencies. Rabi frequencies of each spatial point in the field can be measured by scanning the frames. Thus, the strength of the MW magnetic field distribution is obtained by combing the Rabi frequencies, measu…
We have studied the temporal behavior of the atomic absorption signal under resonant excitation with a continuous-wave laser radiation. Measurements done for D2 line of 85Rb with ~ 1 ns temporal resolution have shown irregular oscillatory behavior of the transmission signal, which becomes well pronounced for high laser power, and disappears when the laser is tuned off-resonance. Application of the fast Fourier transform analysis of the transmission signal reveals power-dependent frequency peaks, which are shown to be associated with Rabi frequency. Possible linkage of the observed results with…
We experimentally investigated Rabi resonance-based continuously frequency-tunable microwave (MW) magnetic field detection using a single hybrid vapor cell filled with cesium and rubidium atoms. The multispecies atomic systems, with their tunable abilities in transition frequencies, enabled this atomic sensing head to cover a broader detectable MW field scope compared to the use of a single metal atom. Here, we demonstrated the simultaneous observation of atomic Rabi resonance signals with 85Rb, 87Rb, and 133Cs in the same vapor cell. Using an experimentally feasible static magnetic field (DC …
We demonstrate coherent one-color photoassociation of a Bose-Einstein condensate, which results in Rabi oscillations between atomic and molecular condensates. We attain atom-molecule Rabi frequencies that are comparable to decoherence rates by driving photoassociation of atoms in an $^{88}$Sr condensate to a weakly-bound level of the metastable $^{1}S_{0}+^{3}P_{1}$ molecular potential, which has a long lifetime and large Franck-Condon overlap integral with the ground scattering state. Transient shifts and broadenings of the excitation spectrum are clearly seen at short times, and they create …
We propose a superconducting circuit-atom hybrid, where the Rabi oscillation of single excess Cooper pair in the island is stabilized via the common atomic-clock technique. The noise in the superconducting circuit is mapped onto the voltage source which biases the Cooper-pair box via an inductor and a gate capacitor. The fast fluctuations of the gate charge are significantly suppressed by an inductor-capacitor resonator, leading to a long-relaxation-time Rabi oscillation. More importantly, the residual low-frequency fluctuations are further reduced by using the general feedback-control method,…
We report on the observation of Rabi oscillations to a Rydberg state on a timescale below one nanosecond in thermal rubidium vapor. We use a bandwidth-limited pulsed excitation and observe up to 6 full Rabi cycles within a pulse duration of ~ 4 ns. We find good agreement between the experiment and numerical simulations based on a surprisingly simple model. This result shows that fully coherent dynamics with Rydberg states can be achieved even in thermal atomic vapor thus suggesting small vapor cells as a platform for room temperature quantum devices. Furthermore the result implies that previou…
Atomic spectroscopy, an essential tool for frequency estimation, is widely used in quantum sensing. Atom-atom interaction can be used to generate entanglement for achieving quantum enhanced sensing. However, atom-atom interaction always induces collision shift, which brings systematic error in determining the resonance frequency. Contradiction between utilizing atom-atom interaction and suppressing collision shift generally exists in atomic spectroscopy. Here, we propose an antisymmetric Rabi spectroscopy protocol without collision shift in the presence of atom-atom interactions. We analytical…
We propose a scheme of stabilizing the persistent-current Rabi oscillation based on the flux qubit-resonator-atom hybrid structure. The LC resonator weakly interacts with the flux qubit and maps the persistent-current Rabi oscillation onto the intraresonator electric field. This field is further coupled to a Rydberg-Rydberg transition of the $^{87}$Rb atom. The Rabi-frequency fluctuation of the flux qubit is deduced from measuring the atomic population and stabilized via feedback controlling the external flux bias. Our numerical simulation indicates that the feedback-control method can efficie…
We investigate the influence of atomic motion on precision Rabi spectroscopy of ultracold fermionic atoms confined in a deep, one dimensional (1D) optical lattice. We analyze the spectral components of longitudinal sideband spectra and present a model to extract information about the transverse motion and sample temperature from their structure. Rabi spectroscopy of the clock transition itself is also influenced by atomic motion in the weakly confined transverse directions of the optical lattice. By deriving Rabi flopping and Rabi lineshapes of the carrier transition, we obtain a model to quan…
The coupling between microwave fields and atoms (or atom-like systems) is inherently weaker than for optical fields, making microwave signal manipulation for applications like quantum information processing technically challenging. In order to better understand this coupling and to develop tools for measuring it, we explore the microwave coupling to atoms using the "atomic candle" technique, and push it beyond the bounds of the small-signal regime in order to deliver a larger signal. In familiar two-level systems, responses beyond the usual Rabi oscillations can arise when a single-tone drive …
We present an experimental investigation of ultrafast-laser driven Rabi oscillations of atomic rubidium. Since the broadband spectrum of an ultrafast laser pulse simultaneously couples all the electronic hyperfine transitions between the excited and ground states, the complex excitation linkages involved with the D1 or D2 transition are energy degenerate. Here, by applying the Morris-Shore transformation, it is shown that this multi-state system is reduced to a set of independent two-state systems and dark states. In experiments performed by ultrafast laser interactions of atomic rubidium in t…
We use coherent excitation of 3-16 atom ensembles to demonstrate collective Rabi flopping mediated by Rydberg blockade. Using calibrated atom number measurements, we quantitatively confirm the expected $\sqrt{N}$ Rabi frequency enhancement to within 4%. The resulting atom number distributions are consistent with essentially perfect blockade. We then use collective Rabi $π$ pulses to produce ${\cal N}=1,2$ atom number Fock states with fidelities of 62% and 48% respectively. The ${\cal N}=2$ Fock state shows the collective Rabi frequency enhancement without corruption from atom number fluctuatio…
We analyze both the s- and p-wave collision induced frequency shifts and propose a over-$π$ pulse scheme to cancel the shifts in optical lattice clocks interrogated by a Rabi pulse. The collisional frequency shifts are analytically solved as a function of the pulse area and the inhomogeneity of the Rabi frequencies. Experimentally measured collisional frequency shifts in an Yb optical lattice clock are in good agreement with the analytical calculations. Based on our analysis, the over-$π$ pulse combined with a small inhomogeneity below 0.1 allows a fractional uncertainty on a level of $10^{-18…
We report on multiple photon monochromatic quantum oscillations (Rabi oscillations) observed by pulsed EPR (Electron Paramagnetic Resonance) of Mn$^{2+}$ (S=5/2) impurities in MgO. We find that when the microwave magnetic field is similar or large than the anisotropy splitting, the Rabi oscillations have a spectrum made of many frequencies not predicted by the S=1/2 Rabi model. We show that these new frequencies come from multiple photon coherent manipulation of the multi-level spin impurity. We develop a model based on the crystal field theory and the rotating frame approximation, describing …
We investigate Rabi oscillation of an atom ensemble in Gaussian spatial distribution. By using the ultrafast laser interaction with the cold atomic rubidium vapor spatially confined in a magneto-optical trap, the oscillatory behavior of the atom excitation is probed as a function of the laser pulse power. Theoretical model calculation predicts that the oscillation peaks of the ensemble-atom Rabi flopping fall on the simple Rabi oscillation curve of a single atom and the experimental result shows good agreement with the prediction. We also test the the three-pulse composite interaction $R_x(π/2…
We use the vacuum Rabi splitting to perform quantum nondemolition (QND) measurements that prepare a conditionally spin-squeezed state of a collective atomic psuedo-spin. We infer a 3.4(6) dB improvement in quantum phase estimation relative to the standard quantum limit for a coherent spin state composed of uncorrelated atoms. The measured collective spin is composed of the two-level clock states of nearly 10^6 Rb-87 atoms confined inside a low finesse F = 710 optical cavity. This technique may improve atomic sensor precision and/or bandwidth, and may lead to more precise tests of fundamental p…
We study the effects of atomic beams harmonic vibrations of micro amplitudes and low frequencies on the mean atomic numbers of the detectors in atomic Mach-Zehnder interferometer, where the two atomic beams are in the same wave surface and have the same phases. We propose a vibrant factor F to quantitatively describe the effects of atomic beams vibrations. It shows that: (i)the vibrant factor F depends on the relative vibrant displacements and the initial phases rather than the absolute amplitudes, (ii)the factor F increases with the increase of the initial phases, and (iii)the frequencies can…
An Accurate Vector Magnetometer via Zeeman Rabi Oscillations
JILA, National Institute of Standards and Technology and University of Colorado（美国国家标准与技术研究院与科罗拉多大学JILA联合研究所）；Department of Physics, University of Colorado（科罗拉多大学物理系）；ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology（巴塞罗那科学技术研究所光子科学研究所）；Paul M. Rady Department of Mechanical Engineering, University of Colorado（科罗拉多大学保罗·M·雷迪机械工程系）；FieldLine Medical（FieldLine医疗公司）
Hybrid quantum systems involving cold atoms and microwave resonators can enable cavity-mediated infinite-range interactions between atomic spin systems and realize atomic quantum memories and transducers for microwave to optical conversion. To achieve strong coupling of atoms to on-chip microwave resonators, it was suggested to use atomic Rydberg states with strong electric dipole transitions. Here we report on the realization of coherent coupling of a Rydberg transition of ultracold atoms trapped on an integrated superconducting atom chip to the microwave field of an on-chip coplanar waveguid…
We study the absorption and dispersion properties of a weak tunable probe field in a four-level Y-type atomic system driven by two strong laser (coupling) fields within the framework of density matrix formalism. It is found that the probe absorption profile displays double-electromagnetically induced transparency (double-EIT) and it is shown how to control it by changing the Rabi frequencies as well as the atom field detuning of the coupling fields.
The collective effect of large amounts of atoms exhibit an enhanced interaction between light and atoms. This holds great interest in quantum optics, and quantum information. When a collective excited state of a group of atoms during Rabi oscillation is varying, the oscillation exhibits rich dynamics. Here, we experimentally observe a size-reduction effect of the Rydberg collective state during Rabi oscillation in cold atomic dilute gases. The Rydberg collective state was first created by the Rydberg quantum memory, and we observed a decreased oscillation frequency effect by measuring the time…
We demonstrate subpicosecond-time-scale population transfer between the pair of hyperfine ground states of atomic rubidium using a single laser-pulse. Our scheme utilizes the geometric and dynamic phases induced during Rabi oscillation through the fine-structure excited state in order to construct an $X$ rotation gate for the hyperfine-state qubit system. Experiment performed with a femtosecond laser and cold rubidium atoms, in a magneto-optical trap, shows over 98\% maximal population transfer between the clock states.
Sun's gravitation potential at earth varies during a year due to varying Earth-Sun distance. Comparing the results of very accurate measurements of atomic clock transitions performed at different time in the year allows us to study the dependence of the atomic frequencies on the gravitational potential. We examine the measurement data for the ratio of the frequencies in Hg$^+$ and Al$^+$ clock transitions and absolute frequency measurements (with respect to caesium frequency standard) for Dy, Sr, H, hyperfine transitions in Rb and H, and obtain significantly improved limits on the values of th…
We propose a simplest detector of harmonic vibrations with micro amplitudes and low frequencies, i.e. the detector consisting of one atomic beam. Here the atomic beam is induced by a plane harmonic wave and has a classical collective harmonic vibrations, which vibrant directions are perpendicular to the wave vectors of atomic beam. Compared with the detector consisting of atomic Mach-Zehnder interferometer, the new detector has two advantages: (1) it is suitable for the detection of the harmonic vibrations induced either by a longitudinal plane harmonic wave or by a transverse plane harmonic w…
As a new method to determine the resonance frequency, Rabi-oscillation spectroscopy has been developed. In contrast to the conventional spectroscopy which draws the resonance curve, Rabi-oscillation spectroscopy fits the time evolution of the Rabi oscillation. By selecting the optimized frequency, it is shown that the precision is twice as good as the conventional spectroscopy with a frequency sweep. Furthermore, the data under different conditions can be treated in a unified manner, allowing more efficient measurements for systems consisting of a limited number of short-lived particles produc…
We have studied the coherent evolution of ultracold atomic rubidium clouds subjected to a microwave field driving Rabi oscillations between the stretched states of the F=1 and F=2 hyperfine levels. A phase winding of the two-level system pseudo-spin vector is encountered for elongated samples of atoms exposed to an axial magnetic field gradient and can be observed directly in state-selective absorption imaging. When dispersively recording the sample-integrated spin population during the Rabi drive, we observe a damped oscillation directly related to the magnetic field gradient, which we quanti…
We demonstrate control of the differential Zeeman shift between clock states of ultracold rubidium atoms by means of non-resonant microwave dressing. Using the dc-field dependence of the microwave detuning, we suppress the first and second order differential Zeeman shift in magnetically trapped $^{87}$Rb atoms. By dressing the state pair 5S$_{1/2} F= 1, m_F = -1$ and $F= 2, m_F = 1$, a residual frequency spread of <0.1 Hz in a range of 100 mG around a chosen magnetic offset field can be achieved. This is one order of magnitude smaller than the shift of the bare states at the magic field of the…
We use Rabi spectroscopy to explore the low-energy excitation spectrum of a finite-temperature Bose gas of rubidium atoms across the phase transition to a Bose-Einstein condensate (BEC). To record this spectrum, we coherently drive the atomic population between two spin states. A small relative displacement of the spin-specific trapping potentials enables sideband transitions between different motional states. The intrinsic non-linearity of the motional spectrum, mainly originating from two-body interactions, makes it possible to resolve and address individual excitation lines. Together with s…
We demonstrate coupling of magnetically trapped ultracold $^87$Rb ground state atoms to a coherently driven superconducting coplanar resonator on an integrated atom chip. We measure the microwave field strength in the cavity through observation of the AC shift of the hyperfine transition frequency when the cavity is driven off-resonance from the atomic transition. The measured shifts are used to reconstruct the field in the resonator, in close agreement with transmission measurements of the cavity, giving proof of the coupling between atoms and resonator. When driving the cavity in resonance w…
We investigate the electric quadrupole interaction of an alkali-metal atom with guided light in the fundamental and higher-order modes of a vacuum-clad ultrathin optical fiber. We calculate the quadrupole Rabi frequency, the quadrupole oscillator strength, and their enhancement factors. In the example of a rubidium-87 atom, we study the dependencies of the quadrupole Rabi frequency on the quantum numbers of the transition, the mode type, the phase circulation direction, the propagation direction, the orientation of the quantization axis, the position of the atom, and the fiber radius. We find …
