Atomic Transition Frequency Research Articles

Effective Landau-Zener transitions in the circuit dynamical Casimir effect with time-varying modulation frequency

We consider the dissipative single-qubit circuit QED architecture in which the atomic transition frequency undergoes a weak external time-modulation. For sinusoidal modulation with linearly varying frequency we derive effective Hamiltonians that resemble the Landau-Zener problem of finite duration associated to a two- or multi-level systems. The corresponding off-diagonal coupling coefficients originate either from the rotating or the counter-rotating terms in the Rabi Hamiltonian, depending on the values of the modulation frequency. It is demonstrated that in the dissipation less case one can accomplish almost complete transitions between the eigenstates of the bare Rabi Hamiltonian even for relatively short duration of the frequency sweep. To assess the experimental feasibility of our scheme we solved numerically the phenomenological and the microscopic quantum master equations in the Markovian regime at zero temperature. Both models exhibit qualitatively similar behavior and indicate that photon generation from vacuum via effective Landau-Zener transitions could be implemented with the current technology on the timescales of a few microseconds. Moreover, unlike the harmonic dynamical Casimir effect implementations, our proposal does not require the precise knowledge of the resonant modulation frequency to accomplish meaningful photon generation.

Stabilization and time resolved measurement of the frequency evolution of a modulated diode laser for chirped pulse generation

We have developed experimental methods for the generation of chirped laser pulses of controlled frequency evolution in the nanosecond pulse length range for coherent atomic interaction studies. The pulses are sliced from the radiation of a cw external cavity diode laser while its drive current, and consequently its frequency, are sinusoidally modulated. By the proper choice of the modulation parameters, as well as of the timing of pulse slicing, we can produce a wide variety of frequency sweep ranges during the pulse. In order to obtain the required frequency chirp, we need to stabilize the center frequency of the modulated laser and to measure the resulting frequency evolution with appropriate temporal resolution. These tasks have been solved by creating a beat signal with a reference laser locked to an atomic transition frequency. The beat signal is then analyzed, as well as its spectral sideband peaks are fed back to the electronics of the frequency stabilization of the modulated laser. This method is simple and it has the possibility for high speed frequency sweep with narrow linewidth that is appropriate, for example, for selective manipulation of atomic states in a magneto-optical trap.

Off-resonance regimes in nonlinear quantum Rabi models

We study the nonlinear quantum Rabi model—both in the two-photon and the intensity-dependent coupling scheme—by utilizing the perturbation method in the off- and near-resonance regimes characterized by the field and the atomic transition frequencies. We consider the weak-coupling condition, the perturbation parameter being the ratio between the coupling constant and the atomic frequency. For both models we determine the first- and second-order corrections to energy eigenvalues and eigenstates, disclosing the doublet structure characterizing the energy spectrum. Then we apply our findings to both the coupling schemes in order to calculate the time evolution of a general initial state and the atomic population inversion for an initial state composed by a mixture of excited and unexcited atoms. [Phys. Rev. A 93 (2016) 043814]

An Accurate Frequency Control Method and Atomic Clock Based on Coherent Population Beating Phenomenon

An accurate frequency control method and atomic clock based on the coherent population beating (CPB) phenomenon is implemented. In this scheme, the frequency difference of an rf and an atomic transition frequency can be digitally obtained by measuring the CPB oscillation frequency. The frequency measurement resolution of several milli-hertz can be achieved by using a 10 MHz oven controlled crystal oscillator as the reference. The expression of the Allan deviation of the CPB clock is theoretically deduced and it is revealed that the Allan deviation is inversely proportional to the signal-to-noise ratio and proportional to the line-width of coherent population trapping spectrum. We also approve that the CPB atomic clock has a large toleration of the drift of the local oscillator. In our CPB experimental system, a frequency instability of 3.0 × 10−12 at 1000s is observed. The important feature of high frequency measurement resolution of the CPB method may also be used in magnetometers, atomic spectroscopy, and other related research.

Frequency ratio of Yb and Sr clocks with 5 × 10−17 uncertainty at 150 seconds averaging time

Transition frequencies of atoms and ions are among the most accurately accessible quantities in nature, playing important roles in pushing the frontiers of science by testing fundamental laws of physics, in addition to a wide range of applications such as satellite navigation systems. Atomic clocks based on optical transitions approach uncertainties of $10^{-18}$, where full frequency descriptions are far beyond the reach of the SI second. Frequency ratios of such super clocks, on the other hand, are not subject to this limitation. They can therefore verify consistency and overall accuracy for an ensemble of super clocks, an essential step towards a redefinition of the second. However, with the measurement stabilities so far reported for such frequency ratios, a confirmation to $1 \times 10^{-18}$ uncertainty would require an averaging time $\tau$ of multiple months. Here we report a measurement of the frequency ratio of neutral ytterbium and strontium clocks with a much improved stability of $4 \times 10^{-16} ({\tau}/s)^{-1/2}$. Enabled by the high stability of optical lattice clocks interrogating hundreds of atoms, this marks a 90-fold reduction in the required averaging time over a previous record-setting experiment that determined the ratio of Al+ and Hg+ single-ion clocks to an uncertainty of $5.2 \times 10^{-17}$. For the Yb/Sr ratio, we find R = 1.207 507 039 343 337 749(55), with a fractional uncertainty of $4.6 \times 10^{-17}$.

Characteristics of the spontaneous emission field and spectrum of a two-level atom in a dynamic photonic crystal

The spontaneous emission field and spectrum of a two-level atom, located in an isotropic photonic crystal with dynamic band edges, are investigated by means of numeric calculation. The investigation is expected to help comprehend the characteristics of the atomic spontaneous emission in the dynamic photonic crystal, and provide a possible way to control dynamically the spontaneous emission in photonic crystal. The expression of the spontaneous radiation field is obtained without using the far-zone approximation and the Weisskopf-Wigner approximation, and expected to be applicable in other relevant researches. In the investigation, the spontaneous radiation field and spectrum are calculated when the band edge frequency is unmodulated, or modulated by a step function or triangle function. In the unmodulated situation, the radiation field intensity tends to a constant which is equal to the intensity of the localized field component. The radiation field pulse presents a wave packet behavior as propagation distance increases. The components of the radiation field correspond one-to-one to the peaks in the spontaneous radiation spectrum. When the band edge frequency is modulated by step function, the radiation field intensity tends to a steady-state value after the modulation has happened. And the steady-state intensity is affected by the time when the modulation happens. The components of the non-localized field and the frequency of the localized field after modulation depend on the atomic transition frequency and the band edge frequency, and are identical to those in the unmodulated situation with the same parameters. When the band edge frequency is modulated by a triangle function, the field intensity presents a decaying quasi-periodic oscillation after a long enough time. The modulation frequency determines the frequency of the oscillation, and influences the decay rate. The radiation energy becomes sharp peaks around a set of the discrete frequencies which are evenly spaced with the modulation frequency. The central frequency of these frequencies depends on the atomic transition frequency and the value range of the band edge frequency. The modulation initial phase affects the intensity of the radiation field emitted in an initial period of time.

Lax Pair, Conservation Laws, Solitons, and Rogue Waves for a Generalised Nonlinear Schrödinger–Maxwell–Bloch System under the Nonlinear Tunneling Effect for an Inhomogeneous Erbium-Doped Silica Fibre

AbstractUnder investigation in this article is a generalised nonlinear Schrödinger-Maxwell-Bloch system for the picosecond optical pulse propagation in an inhomogeneous erbium-doped silica optical fibre. Lax pair, conservation laws, Darboux transformation, and generalised Darboux transformation for the system are constructed; with the one- and two-soliton solutions, the first- and second-order rogue waves given. Soliton propagation is discussed. Nonlinear tunneling effect on the solitons and rogue waves are investigated. We find that (i) the detuning of the atomic transition frequency from the optical pulse frequency affects the velocity of the pulse when the detuning is small, (ii) nonlinear tunneling effect does not affect the energy redistribution of the soliton interaction, (iii) dispersion barrier/well has an effect on the soliton velocity, whereas nonlinear well/barrier does not, (iv) nonlinear well/barrier could amplify/compress the solitons or rogue waves in a smoother manner than the dispersion barrier/well, and (v) dispersion barrier could “attract” the nearby rogue waves, whereas the dispersion well has a repulsive effect on them.

Multiphoton quantum Rabi oscillations in ultrastrong cavity QED

When an atom is strongly coupled to a cavity, the two systems can exchange a single photon through a coherent Rabi oscillation. This process enables precise quantum-state engineering and manipulation of atoms and photons in a cavity, which play a central role in quantum information and measurement. Recently, a new regime of cavity QED has been reached experimentally where the strength of the interaction between light and artificial atoms (qubits) becomes comparable to the atomic transition frequency or the resonance frequency of the cavity mode. Here we show that this regime can strongly modify the concept of vacuum Rabi oscillations, enabling multiphoton exchanges between the qubit and the resonator. We find that experimental state-of-the-art circuit- QED systems can undergo two- and three-photon vacuum Rabi oscillations. These anomalous Rabi oscillations can be exploited for the realization of efficient Fock-state sources of light and complex entangled states of qubits.

Resonance retrieval of stored coherence in an rf-optical double-resonance experiment

We study the storage of coherences in atomic rubidium vapor with a three-level coupling scheme with two ground states and one electronically excited state driven by one optical (control) and one radiofrequency field. We initially store an atomic ground state coherence in the system. When retrieving the atomic coherence with a subsequent optical pulse, a second (signal) optical beam is created whose difference frequency to the control field is found to be determined by the atomic ground states Raman transition frequency.

A high-overtone bulk acoustic wave resonator-oscillator-based 4.596 GHz frequency source: Application to a coherent population trapping Cs vapor cell atomic clock.

This article reports on the design and characterization of a high-overtone bulk acoustic wave resonator (HBAR)-oscillator-based 4.596 GHz frequency source. A 2.298 GHz signal, generated by an oscillator constructed around a thermally controlled two-port aluminum nitride-sapphire HBAR resonator with a Q-factor of 24,000 at 68 °C, is frequency multiplied by 2-4.596 GHz, half of the Cs atom clock frequency. The temperature coefficient of frequency of the HBAR is measured to be -23 ppm/ °C at 2.298 GHz. The measured phase noise of the 4.596 GHz source is -105 dB rad(2)/Hz at 1 kHz offset and -150 dB rad(2)/Hz at 100 kHz offset. The 4.596 GHz output signal is used as a local oscillator in a laboratory-prototype Cs microcell-based coherent population trapping atomic clock. The signal is stabilized onto the atomic transition frequency by tuning finely a voltage-controlled phase shifter implemented in the 2.298 GHz HBAR-oscillator loop, preventing the need for a high-power-consuming direct digital synthesis. The short-term fractional frequency stability of the free-running oscillator is 1.8 × 10(-9) at one second integration time. In locked regime, the latter is improved in a preliminary proof-of-concept experiment at the level of 6.6 × 10(-11) τ(-1/2) up to a few seconds and found to be limited by the signal-to-noise ratio of the detected CPT resonance.

Quantum Phase Transition and Universal Dynamics in the Rabi Model.

We consider the Rabi Hamiltonian, which exhibits a quantum phase transition (QPT) despite consisting only of a single-mode cavity field and a two-level atom. We prove QPT by deriving an exact solution in the limit where the atomic transition frequency in the unit of the cavity frequency tends to infinity. The effect of a finite transition frequency is studied by analytically calculating finite-frequency scaling exponents as well as performing a numerically exact diagonalization. Going beyond this equilibrium QPT setting, we prove that the dynamics under slow quenches in the vicinity of the critical point is universal; that is, the dynamics is completely characterized by critical exponents. Our analysis demonstrates that the Kibble-Zurek mechanism can precisely predict the universal scaling of residual energy for a model without spatial degrees of freedom. Moreover, we find that the onset of the universal dynamics can be observed even with a finite transition frequency.

Three-photon resonance and adiabatic passage in the large-detuning Rabi model

We investigate theoretically the quantum Rabi model in the large-detuning regime. When the frequency of the cavity field is near one-third of the atomic transition frequency, we show that there exists a resonant three-photon coupling via intermediate states connected by counter-rotating processes. We derive an effective Hamiltonian of the three-photon coupling, which determines the magnitude of the energy splitting and the position of the resonance analytically. In addition, we present a scheme of three-photon generation based on adiabatic passage across the resonance.

Detuning Effect Between the Cavity Resonance and Atomic Transition Frequencies on Noise Flux Profiles of Class-C Lasers

The class-C lasers are well recognized to have very narrow atomic transition bandwidths. Therefore, a minor mismatch between the cavity resonance and atomic transition frequencies (CAF) is normally led to the large modifications in the gain and noise profiles. The motion of class-C lasers is described by the three variables of cavity electric field, atomic population inversion, and dipole moment of atoms that are coupled to each other by the optical Maxwell-Bloch equations. The fluctuating features of these three variables are completely elaborated by solving their equations of motion in the presence of the cavity Langevin force. The results peculiarly indicate that the noise profile of the cavity electric field exactly mimics the amplification profile of an input signal by the laser in both below and above threshold states. The both gain and noise profiles simultaneously tend to infinity at the normalized pumping rates and CAF mismatches determined by the laser stability theory. The noise flux profiles are finally confirmed by illustrating the flux conservation.

Imaginary photon effect in a generalized k-Photon Λ-type Jaynes–Cummings model with inverse field operators

By using the Lewis-Riesenfeld invariant theory, we have studied the dynamical and the geometric phases in a generalized time-dependent k-photon Λ-type Jaynes–Cummings model with inverse field operators. We find that the geometric phases in a cycle case are independent of the frequency of the photon field, the coupling coefficient between photons and atoms, and the atom transition frequency. It is apparent that the geometric phases have the pure geometric and topological characteristics, which means that the geometric phases represent the holonomy in the Hermitian linear bundles.

Research progress on active optical clocks

Active optical clocks, a new class of optical atomic clocks, work through stimulated emission lasing, a unique mechanism that is distinct from laser absorption spectroscopy of conventional optical clocks. With coherent weak optical feedback from a Fabry-Perot resonator operating in the bad-cavity regime, active optical clocks sustain a lasing oscillation at an atomic transition frequency. They are used directly as quantum optical frequency standards with expected stability better by about two orders of magnitude than that of the best clocks. This review starts with a simplified account of historical developments and the limitations of critical techniques in optical atomic clocks. After an introduction of the basic mechanism underlying active optical clocks, we discuss the most recent experimental results for several configurations and experimental set-ups of active optical clocks. These include those based on a 2-level atomic beam and 3-level laser-cooled trapped-atom system, those based on a 4-level atom scheme, and the Faraday active optical clock. Moreover, following the successful lasing and preliminary results on a Faraday active optical frequency standard with Faraday atomic filter using thermal Cs cell, we describe an experimental scheme of a Faraday active optical clock with a Faraday atomic filter using magic-lattice-trapped Sr atoms at the 689-nm transition, and Faraday optical clocks operating in the good-cavity regime. We conclude with a comprehensive comparison of the different active optical clocks, prospects for active Faraday optical clocks, and potential applications of active optical clocks.

Non-Hermitian Hamiltonian for a modulated Jaynes-Cummings model withPTsymmetry

We consider a two-level system such as a two-level atom, interacting with a cavity field mode in the rotating wave approximation, when the atomic transition frequency or the field mode frequency is periodically driven in time. We show that in both cases, for an appropriate choice of the modulation parameters, the state amplitudes in a generic $n${-}excitation subspace obey the same equations of motion that can be obtained from a \emph{static} non-Hermitian Jaynes-Cummings Hamiltonian with ${\mathcal PT}$ symmetry, that is with an imaginary coupling constant. This gives further support to recent results showing the possible physical interest of ${\mathcal PT}$ symmetric non-Hermitian Hamiltonians. We also generalize the well-known diagonalization of the Jaynes-Cummings Hamiltonian to the non-Hermitian case in terms of pseudo-bosons and pseudo-fermions, and discuss relevant mathematical and physical aspects.

Electromagnetic shielding in quantum metrology

The dynamics of the quantum Fisher information of the parameters of the initial atomic state and atomic transition frequency is studied, in the framework of open quantum systems, for a static polarizable two-level atom coupled in the multipolar scheme to a bath of fluctuating vacuum electromagnetic fields without and with the presence of a reflecting boundary. Our results show that in the case without a boundary, the electromagnetic vacuum fluctuations always cause the quantum Fisher information of the initial parameters and thus the precision limit of parameter estimation to decrease. Remarkably, however, with the presence of a boundary, the quantum Fisher information becomes position and atomic polarization dependent, and as a result, it may be enhanced as compared to that in the case without a boundary and may even be shielded from the influence of the vacuum fluctuations in certain circumstances as if it were a closed system.

Study on the relationship between coherent population beating signal and the coherence of ground-state hyperfine sublevels

Coherent population beating (CPB) phenomenon occurs in a typical three-level system. When the frequency difference between two coherent pumping laser fields has a certain detuning from the ground-state hyperfine splitting, the excited state population will experience a transient oscillation before reaching equilibrium, and the oscillation frequency is equal to the detuning. The CPB phenomenon enables us to directly obtain the beat frequency between the measured radio frequency (RF) signal and the atomic transition frequency. Then we can get the standard frequency by compensating the beat frequency to the RF. We propose a scheme to implement atomic clock based on the CPB phenomenon in 2009, and the scheme has been implemented. When this effect is used to achieve an atomic clock, the frequency stability is directly related to the amplitude and SNR (signal to noise ratio) of the CPB signal. Influence of the ground-state hyperfine sublevels' coherence on CPB signal is theoretically simulated and experimentally investigated in this paper. A formula of the CPB signal is derived by using the semi-classical model of the interaction of atoms with light, and the theoretical simulation is done using the formula obtained. In the experiment two coherent pumping laser fields are used to interact with 87Rb atoms. A CPB process includes the coherence build-up and the CPB stimulation. The coherence of the ground-state hyperfine sublevels is achieved by controlling the pumping time of the coherent laser fields that are resonant to the ground-state hyperfine sublevels. With this method, the relationship between CPB signal and coherence of the ground-state hyperfine sublevels can be observed. Result shows that the amplitude of CPB signal is proportional to the ground-state hyperfine sublevels' coherence. The hign quality CPB signal can be achieved when the CPB stimulation is started with a pure coherent population trapping (CPT) state. In the CPB process, the coherence build-up rate is approximately equal to the coherence decay rate. So a 50% duty cycle square wave can be used to modulate the RF, and the period of the square wave had better be twice of the decay time of the ground-state hyperfine sublevels' coherence. To improve the SNR of CPB signal and the stability of atomic frequency standard, the ground-state hyperfine sublevels' coherence must be built up, improved, and maintained before the CPB stimulation. The feasibility of applying CPB phenomenon to the weak magnetic field measurement and other applications is also discussed in this paper.

CPT Cesium-Cell Atomic Clock Operation With a 12-mW Frequency Synthesizer ASIC

In this paper, we present the design, fabrication, and electrical characterization of a low-power microwave source for interrogation of cesium atomic hyperfine transition frequency using the coherent population trapping (CPT) technique. The 4.6-GHz frequency generation and signal buffering is performed by a single-chip frequency synthesizer ASIC with a frequency tuning resolution of 1 x 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-13</sup> and a programmable RF output power from -10 to 0 dBm. The circuit was used to modulate the current of a vertical-cavity surface-emitting laser through a dedicated impedance matching network and low thermal conductivity transmission line. Strong modulation sidebands with >60% of carrier amplitude were obtained with an ASIC power consumption of 12 mW. The system was used as optical source for atomic interrogation in an experimental cesium CPT clock. The measured clock stability of 5 x 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-11</sup> at τ = 1 s, going down to 4.5 x 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-12</sup> at τ = 200 s, is limited by the signal-to-noise ratio of the detected CPT signal.

The su(1,1)-like intensity-dependent Rabi model: A perturbative analysis of weak and strong-coupling regimes

We present a perturbative analysis of a Rabi model where the coupling between the quantized single-mode electromagnetic field and the two-level atom depends on the field intensity. Upon modeling the matter–radiation coupling through the Holstein–Primakoff realization of algebra su(1,1), we evaluate first- and second-order eigenenergies and eigenstates both in the weak-coupling regime (atom transition frequency smaller than the coupling strength) and in the strong-coupling regime. In the first case, among various effects, we observe a quadratic dependence on the photon number of energy eigenvalues and the possible formation of level doublets. In the strong-coupling case, the perturbative analysis becomes considerably complex due to the su(1,1)-valued form of the unperturbed Hamiltonian. The critical condition for the transition to an almost continuous spectrum is found in terms of the model parameters.