Microwave electrometry with Rydberg atoms in a vapor cell using microwave amplitude modulation

We have theoretically and experimentally studied the dispersive signal of the Rydberg atomic electromagneticallyinduced transparency(EIT)Autler–Townes(AT)splitting spectra obtained using amplitude modulation of the microwave(MW)electric field.In addition to the two zero-crossing points interval△f_(zeros),the dispersion signal has two positive maxima with an interval defined as the shoulder interval△f_(sho),which is theoretically expected to be used to measure a much weaker MW electric field.The relationship of the MW field strength E_(MW)and△f_(sho)is experimentally studied at the MW frequencies of 31.6 GHz and 9.2 GHz respectively.The results show that△f_(sho)can be used to characterize the much weaker E_(MW)than that of△f_(zeros)and the traditional EIT–AT splitting interval△f_(m);the minimum E_(MW)measured by△f_(sho)is about 30 times smaller than that by△f_(m).As an example,the minimum E_(MW)at 9.2 GHz that can be characterized by△f_(sho)is 0.056 mV/cm,which is the minimum value characterized by the frequency interval using a vapor cell without adding any auxiliary fields.The proposed method can improve the weak limit and sensitivity of E_(MW)measured by the spectral frequency interval,which is important in the direct measurement of weak E_(MW).

Optical PAM-4/PAM-8 generation via dual-Raman process in Rydberg atoms

A scheme of optical four-level pulse amplitude modulation(PAM-4) is proposed based on dual-Raman process in Rydberg atoms. A probe field counter-propagates with a dual-Raman field which drives the ground and the excited states transition, respectively, and the Rydberg transition is driven by a microwave(MW) field. A gain peak appears in the probe transmission and is sensitive to the MW field strength. Optical PAM-4 can be achieved by encoding an MW signal and decoding the magnitude of a probe signal. Simulation results show that the differential nonlinearity and the integral nonlinearity of the proposed scheme can be reduced by 5 times and 6 times, respectively, compared with the counterparts of previous scheme, and the ratio of level separation mismatch is close to the ideal value 1. Moreover, the scheme is extended to optical PAM-8 signal, which may further improve the spectral efficiency.

Dependence of Rydberg-atom-based sensor performance on different Rydberg atom populations in one atomic-vapor cell

The atomic-vapor cell is a vital component for Rydberg atomic microwave sensors,and impacts on overall capability of Rydberg sensors.However,the conventional analysis approach on effect of vapor-cell length contains two implicit assumptions,that is,the same atomic population density and buffer gas pressure,which make it unable to accurately capture actual response about effect of Rydberg-atom-based sensor performance on different Rydberg atom populations.Here,utilizing a stepped cesium atomic-vapor cell with five different dimensions at the same atomic population density and buffer gas pressure,the height and full width at half maximum of electromagnetically induced transparency(EIT)signal,and the sensitivity of the atomic superheterodyne sensor are comprehensively investigated under conditions of the same Rabi frequencies(saturated laser power).It is identified that EIT signal height is proportional to the cell length,full width at half maximum and sensitivity grow with the increment of cell length to a certain extent.Employing the coherent integration signal theory and atomic linear expansion coefficient method,theoretical analysis of the EIT height and sensitivity are further investigated.The results could shed new light on understanding and design of ultrahigh-sensitivity Rydberg atomic microwave sensors and find promising applications in quantum measurement,communication,and imaging.

Atom-based power-frequency electric field measurement using the radio-frequency-modulated Rydberg spectroscopy

The radio-frequency modulated electromagnetically induced transparency(EIT) in a ladder three-level system with Rydberg state is studied. Under the influence of a fast radio-frequency field, the EIT peak splits into a series of sidebands.When attaching a power-frequency electric field directly to the fast radio-frequency field, the odd-order sidebands of the Rydberg-EIT oscillate sensitively with the power-frequency field. The oscillation frequency is equal to twice the power frequency;the oscillation amplitude is monotonically increasing with the amplitude of the power-frequency field when the change of Stark-shift is smaller than the radio frequency. Our work paves the way for measurement of power-frequency electric field based on Rydberg atoms.

Facilitation of controllable excitation in Rydberg atomic ensembles

Strongly-interacting Rydberg atomic ensembles have shown intense collective excitation effects due to the inclusion of single Rydberg excitation shared by multiple atoms in the ensemble.In this paper we investigate a counter-intuitive Rydberg excitation facilitation with a strongly-interacting atomic ensemble in the strong probe-field regime,which is enabled by the role of a control atom nearby.Differing from the case of a single ensemble,we show that,the control atom's excitation adds to a second two-photon transition onto the doubly-excited Rydberg state,arising an excitation facilitation for the ensemble atoms.Our numerical studies depending on the method of quantum Monte Carlo wave function,exhibit the observation constraints of this excitation facilitation effect under practical experimental conditions.The results obtained can provide a flexible control for the excitation of Rydberg atomic ensembles and participate further uses in developing mesoscopic Rydberg gates for multiqubit quantum computation.

射频电场缀饰下铯Rydberg原子的电磁感应透明光谱

采用全红外光激发Rydberg原子的方案,选择探测光(852 nm)、缀饰光(1470 nm)和耦合光(780 nm),利用三光子激发方式实现了Cs原子Rydberg态(49P_(3/2))的制备.实验上,观测到射频电场作用下|7S_(1/2)?→|49P_(3/2)?Rydberg跃迁形成的电磁感应透明(EIT)效应,实现对Rydberg原子的光学探测,根据EIT光谱的变化来探究射频电场的幅度和频率对光谱的影响.研究表明,随着射频电场幅度的增强,观察到光谱现象从越发明显的ac Stark能移逐步过渡到复杂混合态的多个调制边带,并根据EIT主峰的频移进一步讨论频率对铯泡中电场屏蔽的影响.采用将低频电场调制到射频电场的方式,实现了基于Rydberg原子的50 Hz—1 kHz范围内电场解调,并对解调信号的幅度和频率进行拟合,保真度达到95%.研究结果对Rydberg原子射频光谱探测和低频电场的可溯源测量等提供有价值的参考.

Electric field intensity measurement by using doublet electromagnetically induced transparency of cold Rb Rydberg atoms

We demonstrate a simple method to measure electric field intensity by using doublet electromagnetically induced transparency(EIT) spectra of cold Rb Rydberg atoms, where the frequency of the coupling laser does not need to be locked. Based on the Stark splitting of the Rb Rydberg state, 10D_(3/2), under electric fields and the corresponding calculated polarizabilities, the real electric field intensity is calculated using the difference in radio-frequency diffraction between two acousto-optic modulators, which acts as a frequency criterion that allows us to measure the electrical field without locking the coupling laser. The value measured by this simple method shows a good agreement with our previous work [Opt.Express 29 1558(2021)] where the frequency of the coupling laser needs to be locked with an additional EIT spectrum based on atom vapor and a proportional–integral–differential feedback circuit. Our presented method can also be extended to the measurement of electric field based on hot Rydberg atom vapor, which has application in industry.

Superradiance of ultracold cesium Rydberg |65D_(5/2)> → |66P_(3/2)>

We investigate Rydberg |65D_(5/2)> → |66P_(3/2)> superradiance in dense ultracold cesium atoms,where the ground atoms are excited to |65D_(5/2)> Rydberg states via two-photon excitation in a standard magneto-optical trap.The superradiant spectrum of |65D_(5/2)> → |66P_(3/2)> is obtained using the state-selective field ionization technique.We observe its dynamic evolution process by varying the delay time of ionization field td.The results show that the evolution process of |65D_(5/2)> →|66P_(3/2)> is much shorter than its radiation lifetime at room temperature,which verifies the superradiance effect.The dependence of the superradiance process on Rydberg atoms number N_(e) and principal quantum number n is investigated.The results show that the superradiance becomes faster with increasing N_(e),while it is suppressed for stronger van der Waals(vdW) interactions.Superradiance has potential applications in quantum technologies,and the Rydberg atom is an ideal medium for superradiance.Our system is effective for studying the strong two-body interaction between Rydberg atoms.

A concise review of Rydberg atom based quantum computation and quantum simulation

Quantum information processing based on Rydberg atoms emerged as a promising direction two decades ago.Recent experimental and theoretical progresses have shined exciting light on this avenue.In this concise review,we will briefly introduce the basics of Rydberg atoms and their recent applications in associated areas of neutral atom quantum computation and simulation.We shall also include related discussions on quantum optics with Rydberg atomic ensembles,which are increasingly used to explore quantum computation and quantum simulation with photons.

Vapor cell geometry effect on Rydberg atom-based microwave electric field measurement

The geometry effect of a vapor cell on the metrology of a microwave electric field is investigated. Based on the splitting of the electromagnetically induced transparency spectra of cesium Rydberg atoms in a vapor cell, high-resolution spatial distribution of the microwave electric field strength is achieved for both a cubic cell and a cylinder cell. The spatial distribution of the microwave field strength in two dimensions is measured with sub-wavelength resolution. The experimental results show that the shape of a vapor cell has a significant influence on the abnormal spatial distribution because of the Fabry-P6rot effect inside a vapor cell. A theoretical simulation is obtained for different vapor cell wall thicknesses and shows that a restricted wall thickness results in a measurement fluctuation smaller than 3% at the center of the vapor cell.

An all-optical phase detector by amplitude modulation of the local field in a Rydberg atom-based mixer

Recently,a Rydberg atom-based mixer was developed to measure the phase of a radio frequency(RF)field.The phase of the signal RF(SIG RF)field is down-converted directly to the phase of a beat signal created by the presence of a local RF(LO RF)field.In this study,we propose that the Rydberg atom-based mixer can be converted to an all-optical phase detector by amplitude modulation(AM)of the LO RF field;that is,the phase of the SIG RF field is related to both the amplitude and phase of the beat signal.When the AM frequency of the LO RF field is the same as the frequency of the beat signal,the beat signal will further interfere with the AM of the LO RF field inside the atom,and then the amplitude of the beat signal is related to the phase of the SIG RF field.The amplitude of the beat signal and the phase of the SIG RF field show a linear relationship within the range of 0 toπ/2 when the phase of the AM is set with a differenceπ/4 from the phase of the LO RF field.The minimum phase resolution can be as small as 0.6°by optimizing the experimental conditions according to a simple theoretical model.This study will expand and contribute to the development of RF measurement devices based on Rydberg atoms.

Highly sensitive detection of Rydberg atoms with fluorescence loss spectrum in cold atoms

Fluorescence loss spectrum for detecting cold Rydberg atoms with high sensitivity has been obtained based on lock-in detection of fluorescence of 6 P3/2 state when cooling lasers of the magneto-optical trap are modulated.The experiment results show that the signal to noise ratio has been improved by 32.64 dB when the modulation depth(converted to laser frequency)and frequency are optimized to 4 MHz and 6 kHz,respectively.This technique enables us to perform a highly sensitive non-destructive detection of Rydberg atoms.

High-precision three-dimensional Rydberg atom localization in a four-level atomic system

Rydberg atoms have been widely investigated due to their large size,long radiative lifetime,huge polarizability and strong dipole-dipole interactions.The position information of Rydberg atoms provides more possibilities for quantum optics research,which can be obtained under the localization method.We study the behavior of three-dimensional(3D)Rydberg atom localization in a four-level configuration with the measurement of the spatial optical absorption.The atomic localization precision depends strongly on the detuning and Rabi frequency of the involved laser fields.A 100%probability of finding the Rydberg atom at a specific 3D position is achieved with precision of~0.031λ.This work demonstrates the possibility for achieving the 3D atom localization of the Rydberg atom in the experiment.

Calculation of Rydberg energy levels for the francium atom

Based on the weakest bound electron potential model theory, the Rydberg energy levels and quantum defects of the nP^2P^o1/2 （n=7-50） and np^2P^o3/2 （n=7-50） spectrum series for the francium atom are calculated. The calculated results are in excellent agreement with the 48 measured levels, and 40 energy levels for highly excited states are predicted.

An effective quantum defect theory for the diamagnetic spectrum of a barium Rydberg atom

A theoretical calculation is carried out to investigate the spectrum of a barium Rydberg atom in an external magnetic field. Using an effective approach incorporating quantum defect into the centrifugal term in the Hamiltonian, we reexamine the reported spectrum of the barium Rydberg atom in a magnetic field of 2.89 T [J. Phys. B 28 L537 （1995）]. Our calculation employs B-spline basis expansion and complex coordinate rotation techniques. For single photon absorption from the ground 6s2 to 6snp Rydberg states, the spectrum is not influenced by quantum defects of channels ns and nd. The calculation is in agreement with the experimental observations until the energy reaches E = -60 cm-1. Beyond this energy, closer to the threshold, the calculated and experimental results do not agree with each other. Possible reasons for their discrepancies are discussed. Our study affirms an energy range where the diamagnetic spectrum of the barium atom can be explained thoroughly using a hydrogen model potential.

The fractal structure in the ionization dynamics of Rydberg lithium atoms in a static electric field

The ionization rate of Rydberg lithium atoms in a static electric field is examined within semiclassical theory which involves scattering effects off the core. By semiclassical analysis, this ionization process can be considered as the promoted valence electrons escaping through the Stark saddle point into the ionization channels. The resulting escape spectrum of the ejected electrons demonstrates a remarkable irregular electron pulse train in time-dependence and a complicated nesting structure with respect to the initial launching angles. Based on the Poincaré} map and homoclinic tangle approach, the chaotic behaviour along with its corresponding fractal self-similar structure of the ionization spectra are analysed in detail. Our work is significant for understanding the quantum-classical correspondence.

Analysis of the fractal intrinsic quality in the ionization of Rydberg helium and lithium atoms

We study the fractal rhythm in the ionization of Rydberg helium and lithium atoms in an electric field by using the semiclassical method.The fractal structures present a nested relationship layer by layer in the initial launch angles of the ionized electrons versus the escape time,which is defined as the rhythm attractor,and exhibit similar rhythm endings.The gradually enhanced chaotic regions of the escape time plots tend to broaden as the scaled energy increases.In addition,the fractal rhythm changes synchronously with the oscillations of the kinetic energy spectrum.We note that the intrinsic quality of the fractal rhythm is closely related to the kinetic energy distribution,that is,the inherent dynamic properties of the Hamiltonian equations have an impact on the fractal regularities.In addition,different ionizing closed trajectories exhibit iterate properties and the inherent beauty of symmetry.Our results and analysis can not only reveal new laws in the ionization of Rydberg atoms,but also promote the establishment of the dynamic mechanism of fractals.

Properties of collective Rabi oscillations with two Rydberg atoms

Motivated by experimental advances that the collective excitation of two Rydberg atoms was observed, we provide an elaborate theoretical study for the dynamical behavior of two-atom Rabi oscillations. In the large-intermediate-detuning case, the two-photon Rabi oscillation is found to be significantly affected by the strength of the interatomic van der Waals interaction. With a careful comparison of the exact numbers and values of the oscillation frequency, we propose a new way to determine the strength of excitation blockade, agreeing well with the previous universal criterion for full, partial, and no blockade regions. In the small-intermediate-detuning case, we find a blockade-like effect, but the collective enhancement factor is smaller than ^(1/2) due to the quantum interference of double optical transitions involving the intermediate state.Moreover, a fast two-photon Rabi oscillation in ns timescale is manifested by employing intense lasers with an intensity of ~MW/cm^2, offering a possibility of ultrafast control of quantum dynamics with Rydberg atoms.

Monte Carlo simulations of electromagnetically induced transparency in a square lattice of Rydberg atoms

We study the steady optical response of a square lattice in which all trapped atoms are driven by a probe and a coupling fields into the ladder configuration of electromagnetically induced transparency(EIT).It turns out to be a manybody problem in the presence of van der Waals(vd W)interaction among atoms in the upmost Rydberg state,so Monte Carlo(MC)calculation based on density matrix equations have been done after introducing a sufficiently large cut-off radius.It is found that the absorption and dispersion of EIT spectra depends critically on a few key parameters like lattice dimension,unitary vd W shift,probe Rabi frequency,and coupling detuning.Through modulating these parameters,it is viable to change symmetries of the absorption and dispersion spectra and control on demand depth and position of the transparency window.Our MC calculation is expected to be instructive in understanding many-body quantum coherence effects and in manipulating non-equilibrium quantum phenomena by utilizing vd W interactions of Rydberg atoms.

Statistics and Correlation Properties of Diamagnetic High Rydberg Hydrogen Atom

The spectra of Rydberg hydrogen atom in magnetic fields have been calculated using linear variational method with B-splines basis functions [Acta Phys. Sin. 55 （2006） 3380]. Based on these calculations we have done some statistics analysis about the high Rydberg energy levels. The nearest-neighbor energy spacing distribution and the 3-statistics have been shown about diamagnetic Rydberg hydrogen atom with the magnetic field being 0.6 T and 6 T. The phenomena of multiply crossing, multiply anti-crossing, and the mixed of crossing and anti-crossing of energy levels have appeared in this paper. For both cases, in range of lower energy, the energy 1evel statistics properties close to Poisson distribution. With the increasing of the energy, the energy level statistics properties are away to Poisson distribution and tend to Wigner distribution step by step.