The possibility of coherent population trapping in two electron states with aligned spins (ortho-system) is evidenced. From the analysis of a three-level atomic system containing two electrons, and driven by the two laser fields needed for coherent population trapping, a conceptually new kind of two-electron dark state appears. The properties of this trapping are studied and are physically interpreted in terms of a dark hole, instead of a dark two-electron state. This technique, among many other applications, offers the possibility of measuring, with subnatural resolution, some superposition-state matrix-elements of the electron-electron correlation that due to their time dependent nature are inaccesible by standard measuring procedures.
We show that the effective decay rate of Zeeman coherence, generated in a Rb87 vapor by linearly polarized laser light, increases significantly with the atomic density. We explain this phenomenon as the result of radiation trapping. Our study shows that radiation trapping must be taken into account to fully understand many electromagnetically induced transparency experiments with optically thick media.
We show that high squeezing and entanglement can be generated at the output of a cavity containing atoms interacting with two fields in a Coherent Population Trapping situation, on account of a non-linear Faraday effect experienced by the fields close to a dark-state resonance in a cavity. Moreover, the cavity provides a feedback mechanism allowing to reduce the quantum fluctuations of the ground state spin, resulting in strong steady state spin-squeezing.
A scheme for a THz frequency standard based on three-photon coherent population trapping in stored ions is proposed. Assuming the propagation directions of the three lasers obey the phase matching condition, we show that stability of few 10$^{-14}$ at one second can be reached with a precision limited by power broadening to $10^{-11}$ in the less favorable case. The referenced THz signal can be propagated over long distances, the useful information being carried by the relative frequency of the three optical photons.
We show that the physical mechanism of population transfer in a 3-level system with a closed loop of coherent couplings (loop-STIRAP) is not equivalent to an adiabatic rotation of the dark-state of the Hamiltonian but coresponds to a rotation of a higher-order trapping state in a generalized adiabatic basis. The concept of generalized adiabatic basis sets is used as a constructive tool to design pulse sequences for stimulated Raman adiabatic passage (STIRAP) which give maximum population transfer also under conditions when the usual condition of adiabaticty is only poorly fulfilled. Under cert…
We investigate theoretically the optical properties of an atomic gas which has been cooled by the laser cooling method velocity-selective coherent population trapping. We demonstrate that the application of a weak laser pulse gives rise to a backscattered pulse, which is a direct signal for the entanglement in the atomic system, and which leads to single-particle entanglement on the few-photon level. If the pulse is applied together with the pump lasers, it also displays the phenomenon of electromagnetically induced transparency. We suggest that the effect should be observable in a gas of Rubi…
A full quantum treatment of coherent population trapping (CPT) is given for a system of resonantly coupled atoms and electromagnetic field. We develop a regular analytical method of the construction of generalized dark states (GDS). It turns out that GDS do exist for all optical transitions $F_g\to F_e$, including bright transitions $F\to F+1$ and $F''\to F''$ with $F''$ a half-integer, for which the CPT effect is absent in a classical field. We propose an idea to use an optically thick medium with a transition $F''\to F''$ with $F'' \ge 3/2$ a half-integer as a ''quantum filter'', which trans…
We isolate a novel four-wave mixing process, enabled by Coherent Population Trapping (CPT), leading to efficient phase sensitive amplification. This process is permitted by the exploitation of two transitions starting from the same twofold degenerate ground state. One of the transitions is used for CPT, defining bright and dark states from which ultra intense four-wave mixing is obtained via the other transition. This leads to the measurement of a strong phase sensitive gain even for low optical densities and out-of-resonance excitation. The enhancement of four-wave mixing is interpreted in th…
The phenomenon of Coherent Population Trapping (CPT) of an atom (or solid state "artificial atom"), and the associated effect of Electromagnetically Induced Transparency (EIT), are clear demonstrations of quantum interference due to coherence in multi-level quantum systems. We report observation of CPT in a superconducting phase qubit by simultaneously driving two coherent transitions in a $Λ$-type configuration, utilizing the three lowest lying levels of a local minimum of a phase qubit. We observe $60(\pm 7)%$ suppression of excited state population under conditions of CPT resonance. We pres…
We study phase-dependent fluctuations of the resonance fluorescence of a single $Λ$-type three-level atom in the regime near coherent population trapping, i.e., alongside the two-photon detuning condition. To this end, we employ the method of conditional homodyne detection (CHD) which considers squeezing in the weak driving regime, and extends to non-Gaussian fluctuations for saturating and strong fields. In this framework, and using estimated parameter settings of the resonance fluorescence of a single trapped $^{138} \mathrm{Ba}^{+}$ ion, the light scattered from the probe transitions is fou…
We propose a procedure to achieve a complete energy conversion between laser pulses carrying orbital angular momentum (OAM) in a cloud of cold atoms characterized by a double-Λatom-light coupling scheme. A pair of resonant spatially dependent control fields prepare atoms in a position-dependent coherent population trapping state, while a pair of much weaker vortex probe beams propagate in the coherently driven atomic medium. Using the adiabatic approximation we derive the propagation equations for the probe beams. We consider a situation where the second control field is absent at the entrance…
We report the observation of transient velocity-selective coherent population trapping (VSCPT) in a beam of metastable neon atoms. The atomic momentum distribution resulting from the interaction with counterpropagating sigma_+ and sigma_- radiation which couples a J_g=2 <-> J_e=1 transition is measured via the transversal beam profile. This transition exhibits a stable VSCPT dark state formed by the two |J=2,m=+/-1> states, and a metastable dark state containing the |J=2,m=+/-2> and |J=2,m=0> states. The dynamics of the formation and decay of stable and metastable dark states is studied experi…
We study analytically the dynamics of cavity QED nodes in a practical quantum network. Given a single 3-level $Λ$-type atom or quantum dot coupled to a micro-cavity, we derive several necessary and sufficient criteria for the coherent trapping and generation of a single photon pulse with a given waveform to be realizable. We prove that these processes can be performed with practical hardware -- such as cavity QED systems which are operating deep in the weak coupling regime -- given a set of restrictions on the single-photon pulse envelope. We systematically study the effects of spontaneous emi…
The Doppler effect is one of the dominant broadening mechanisms in thermal vapor spectroscopy. For two-photon transitions one would naively expect the Doppler effect to cause a residual broadening, proportional to the wave-vector difference. In coherent population trapping (CPT), which is a narrow-band phenomenon, such broadening was not observed experimentally. This has been commonly attributed to frequent velocity-changing collisions, known to narrow Doppler-broadened one-photon absorption lines (Dicke narrowing). Here we show theoretically that such a narrowing mechanism indeed exists for C…
Coherent population trapping is a well-known quantum phenomenon in a driven $Λ$ system, with many applications across quantum optics. However, when a stochastic bath is present in addition to vacuum noise, the observed trapping is no longer perfect. Here we derive a time-convolutionless master equation describing the equilibration of the $Λ$ system in the presence of additional temporally correlated classical noise, with an unknown decay parameter. Our simulations show a one-to-one correspondence between the decay parameter and the depth of the characteristic dip in the photoluminescence spect…
We analyzed the efficiency of coherent population trapping (CPT) in a superposition of the ground states of three-level atoms under the influence of the decoherence process induced by a broadband thermal field. We showed that in a single atom there is no perfect CPT when the atomic transitions are affected by the thermal field. The perfect CPT may occur when only one of the two atomic transitions is affected by the thermal field. In the case when both atomic transitions are affected by the thermal field, we demonstrated that regardless of the intensity of the thermal field the destructive effe…
We propose a new transport mechanism through tunnel-coupled quantum dots based on the coherent population trapping effect. Coupling to an excited level by the coherent radiation of two microwaves can lead to an extremely narrow current antiresonance. The effect can be used to determine interdot dephasing rates and is a mechanism for a very sensitive, optically controlled current switch.
The proposal for quantum computing with rare-earth-ion qubits in inorganic crystals makes use of the inhomogeneous broadening of optical transitions in the ions to associate individual qubits with ions responding to radiation in selected frequency channels. We show that a class of Gaussian composite pulses and complex sech pulses provide accurate qubit pi-rotations, which are at the same time channel selective on a 5 MHz frequency scale and tolerant to 0.5 MHz deviations of the transition frequency of ions within a single channel. Rotations in qubit space of arbitrary angles and phases are pro…
We demonstrate an atom localization scheme based on monitoring of the atomic coherences. We consider atomic transitions in a Lambda configuration where the control field is a standing wave field. The probe field and the control field produce coherence between the two ground states. We show that this coherence has the same fringe pattern as produced by a Fabry-Perot interferometer and thus measurement of the atomic coherence would localize the atom. Interestingly enough the role of the cavity finesse is played by the ratio of the intensities of the pump and probe. This is in fact the reason for…
We studied theoretically a coherent population trapping resonance formation in cylindrical cell without buffer gas irradiated by a narrow laser beam. We take into account non-zero probabilities of elastic ("specular") and inelastic ("sticking") collision between the atom and the cell wall. We have developed a theoretical model based on averaging over the random Ramsey pulse sequences of times that atom spent in and out of the beam. It is shown that the shape of coherent population trapping resonance line depends on the probability of elastic collision.
In the quest for signatures of coherent transport we consider exciton trapping in the continuous-time quantum walk framework. The survival probability displays different decay domains, related to distinct regions of the spectrum of the Hamiltonian. For linear systems and at intermediate times the decay obeys a power-law, in contrast to the corresponding exponential decay found in incoherent continuous-time random walk situations. To differentiate between the coherent and incoherent mechanisms, we present an experimental protocol based on a frozen Rydberg gas structured by optical dipole traps.
We consider a three-node fully connected network (Delta network) showing that a coherent population trapping phenomenon occurs, generalizing results for the Lambda network known to support a dark state. Transport in such structures provides signatures of detrapping, which can be triggered by external controls. In the presence of an environment it turns out to be sensitive to its Markovianity. Adiabatic modulation of the system's parameters may yield coherent population transfer, analogous to the stimulated Raman adiabatic passage phenomenon. Robustness of this protocol against non-adiabatic tr…
A 2-level atom with degenerate ground state interacting with a quantum field is investigated. We show, that the field drives the state of the atom to a stationary state, which is non-unique, but depends on the initial state of the system through some conserved quantities. This non-uniqueness follows from the degeneracy of the ground state of the atom, and when the ground subspace is two-dimensional, the family of stationary states will depend on a one-dimensional parameter. Only one of the stationary states in this family is a pure state, and this state coincides with the known non-coupled pop…
The coherent population trapping (CPT) effect is used for making compact atomic clocks. There are two types of CPT clocks: the one in which the Raman beams are applied continuously and the one in which two CPT pulses separated by a dark period are applied (Ramsey scheme). It is obvious that the technique of spin squeezing can only be applied to the Ramsey CPT clock to enhance the sensitivity. However, it is not apparent how to adapt to the CPT clock the protocols for the microwave clock using one-axis-twist squeezing (OATS), since the Ramsey CPT clock is not trivially equivalent to the Ramsey …
We present a coherent population trapping based scheme to attain sub-nanoscale resolution for atom localization, microscopy and lithography. Our method uses three-level atoms coupled to amplitude modulated probe field and spatially dependent drive field. The modulation of the probe field allows us to tap into the steep dispersion normally associated with electromagnetically induced transparency and offers an avenue to attain sub-nanometer resolution using just optical fields. We illustrate application of the techniques to the area of microscopy and lithography and show how multilevel schemes o…
The spectroscopic properties of a single, tightly trapped atom are studied, when the electronic levels are coupled by three laser fields in an $N$-shaped configuration of levels, whereby a $Λ$-type level system is weakly coupled to a metastable state. We show that depending on the laser frequencies the response can be tuned from coherent population trapping at two-photon resonance to novel behaviour at three photon resonance, where the metastable state can get almost unit occupation in a wide range of parameters. For certain parameter regimes the system switches spontaneously between dissipati…
A novel protocol of interrogation based on coherent population trapping in an N-level scheme atomic system leads to dark resonances involving three different photons. An ensemble of several hundreds of radiofrequency-trapped ions is probed by three lasers simultaneously locked onto the same optical frequency comb, resulting in high-contrast spectral lines referenced to an atomic transition in the THz domain. We discuss the cause of uncertainties and limitations for this method and show that reaching a sub-kHz resolution is experimentally accessible via this interrogation protocole.
Coherent population trapping is demonstrated in single nitrogen-vacancy centers in diamond under optical excitation. For sufficient excitation power, the fluorescence intensity drops almost to the background level when the laser modulation frequency matches the 2.88 GHz splitting of the ground states. The results are well described theoretically by a four-level model, allowing the relative transition strengths to be determined for individual centers. The results show that all-optical control of single spins is possible in diamond.
In high-purity n-type GaAs under strong magnetic field, we are able to isolate a lambda system composed of two Zeeman states of neutral-donor bound electrons and the lowest Zeeman state of bound excitons. When the two-photon detuning of this system is zero, we observe a pronounced dip in the excited-state photoluminescence indicating the creation of the coherent population-trapped state. Our data are consistent with a steady-state three-level density-matrix model. The observation of coherent population trapping in GaAs indicates that this and similar semiconductor systems could be used for var…
Electromagnetically induced transparency (EIT) is an optical phenomenon which allows a drastic modification of the optical properties of an atomic system by applying a control field. It has been largely studied in the last decades and nowadays we can find a huge number of experimental and theoretical related studies. Recently a similar phenomenon was also shown in quantum dot molecules (QDM) , where the control field is replaced by the tunneling rate between quantum dots. Our results show that in the EIT regime, the optical properties of QDM and the atomic system are identical. However, here w…
Coherent control of quantum systems has far-reaching implications in quantum engineering. In this context, coherent population trapping (CPT) involving dark resonances has played a prominent role, leading to a wealth of major applications including laser cooling of atoms and molecules, optical magnetometry, light storage and highly precise atomic clocks. Extending CPT methods to individual solid-state quantum systems has been only achieved in cryogenic environments for electron spin impurities and superconducting circuits. Here, we demonstrate efficient CPT of a single nuclear spin in a room t…
We have detected and analyzed narrow, high contrast coherent population trapping resonances, which appear in transmission of the probe monochromatic light beam under action of the counterpropagating two-frequency laser radiation, on example of the nonclosed three level Λ-system formed by spectral components of the Doppler broadened D2 line of cesium atoms (in the cell with the rarefied Cs vapor). These nontrivial resonances are determined directly by the trapped atomic population on the definite lower level of the Λ-system and may be used in atomic frequency standards, sensitive magnetometers …
We propose and theoretically analyze the use of coherent population trapping of a single diamond nitrogen vacancy (NV) center for continuous real-time sensing. The formation of the dark state in coherent population trapping prevents optical emissions from the NV center. Fluctuating magnetic fields, however, can kick the NV center out of the dark state, leading to a sequence of single-photon emissions. A time series of the photon counts detected can be used for magnetic field estimations, even when the average photon count per update time interval is much smaller than 1. For a theoretical demon…
We analyze the coherent dynamics of excitons in three dimensional topologically disordered networks with traps. If the interactions between the nodes of the network are long ranged, i.e., algebraically decaying as a function of the distance between the nodes, the average survival probability of an exciton surprisingly shows a characteristic decay with features similar to the decay found for regular one-dimensional systems. We further show how this decay can be related to the eigenstates of the same system without a trap.
Spatial and temporal evolution is studied of two powerful short laser pulses having different wavelengths and interacting with a dense three-level Lambda-type optical medium under coherent population trapping. A general case of unequal oscillator strengths of the transitions is considered. Durations of the probe pulse and the coupling pulse $T_{1,2}$ ($T_2>T_1$) are assumed to be shorter than any of the relevant atomic relaxation times. We propose analytical and numerical solutions of a self-consistent set of coupled Schrödinger equations and reduced wave equations in the adiabatic limit with …
We analyze the properties of a pulsed Coherent Population Trapping protocol that uses a controlled decay from the excited state in a $Λ$-level scheme. We study this problem analytically and numerically and find regimes where narrow transmission, absorption, or fluorescence spectral lines occur. We then look for optimal frequency measurements using these spectral features by computing the Allan deviation in the presence of ground state decoherence and show that the protocol is on a par with Ramsey-CPT. We discuss possible implementations with ensembles of alkali atoms and single ions and demons…
Coherent population trapping is shown to occur in a driven symmetric double-well potential in the strong-field regime. The system parameters have been chosen to reproduce the $0^{-}\leftrightarrow 3^{+}$ transition of the inversion mode of the ammonia molecule. For a molecule initially prepared in its lower doublet we find that, under certain circumstances, the $3^{+}$ level remains unpopulated, and this occurs in spite of the fact that the laser field is resonant with the $0^{-}\leftrightarrow 3^{+}$ transition and intense enough so as to strongly mix the $0^{+}$ and $0^{-}$ ground states. Th…
Optical spin rotations and cycling transitions for measurement are normally incompatible in quantum dots, presenting a fundamental problem for quantum information applications. Here we show that for a hole spin this problem can be addressed using a trion with one hole in an excited orbital, where strong spin-orbit interaction tilts the spin. Then, a particular trion triplet forms a double $Λ$ system, even in a Faraday magnetic field, which we use to demonstrate fast hole spin initialization and coherent population trapping. The lowest trion transitions still strongly preserve spin, thus combin…
Magnetically induced coherent population trapping has been studied in a single trapped laser cooled ion. The magnetic field dependent narrow spectral feature is found to be an useful tool in determining the null point of magnetic field at the ion position. In particular, we use a double lambda scheme that allows us to measure the null magnetic field point limited by the detector shot noise. We analyzed the system theoretically and found certain long lived bright states as the dark state is generated under steady state condition.
Hybrid spin-optomechanical quantum systems offer high flexibility, integrability and applicability for quantum science and technology. Particularly, on-chip surface acoustic waves (SAWs) can efficiently drive spin transitions in the ground states (GSs) of atomic-scale, color centre qubits, which are forbidden in case of the more frequently used electromagnetic fields. Here, we demonstrate that strain-induced spin interactions within their optically excited state (ES) can exceed by two orders of magnitude the ones within the GS. This gives rise to novel physical phenomena, such as the acoustica…
Coherent population trapping (CPT) provides a highly sensitive means for probing the energy level structure of an atomic system. For a nitrogen vacancy center in diamond, the CPT offers an alternative to the standard optically-detected magnetic resonance method for measuring the hyperfine structure of the electronic ground states. We show that the nuclear spin dependent CPT measures directly the hyperfine splitting of these states due to the 14N nuclear spin. The CPT spectral response obtained in the presence of a strong microwave field, resonant or nearly resonant with a ground state spin tra…
We report high resolution coherent population trapping on a single hole spin in a semiconductor quantum dot. The absorption dip signifying the formation of a dark state exhibits an atomic physics-like dip width of just 10 MHz. We observe fluctuations in the absolute frequency of the absorption dip, evidence of very slow spin dephasing. We identify this process as charge noise by, first, demonstrating that the hole spin g-factor in this configuration (in-plane magnetic field) is strongly dependent on the vertical electric field, and second, by characterizing the charge noise through its effects…
Quantum interference is a natural consequence of wave-particle duality in quantum mechanics, and is widely observed at the atomic scale. One interesting manifestation of quantum interference is coherent population trapping (CPT), first proposed in three-level driven atomic systems and observed in quantum optical experiments. Here, we demonstrate CPT in a gate-defined semiconductor double quantum dot (DQD), with some unique twists as compared to the atomic systems. Specifically, we observe CPT in both driven and non-driven situations. We further show that CPT in a driven DQD could be used to ge…
We consider coherent exciton transport modeled by continuous-time quantum walks (CTQWs) on long-range interacting cycles (LRICs), which are constructed by connecting all the two nodes of distance $m$ in the cycle graph. LRIC has a symmetric structure and can be regarded as the extensions of the cycle graph (nearest-neighboring lattice). For small values of $m$, the classical and quantum return probabilities show power law behavior $p(t)\sim t^{-0.5}$ and $π(t)\sim t^{-1}$, respectively. However, for large values of $m$, the classical and quantum efficiency scales as $p(t)\sim t^{-1}$ and $π(t)…
Coherent population trapping at zero magnetic field was observed for nitrogen-vacancy centers in diamond under optical excitation. This was measured as a reduction in photoluminescence when the detuning between two excitation lasers matched the 2.88 GHz crystal-field splitting of the color center ground states. This behavior is highly sensitive to strain, which modifies the excited states, and was unexpected following recent experiments demonstrating optical readout of single nitrogen-vacancy electron spins based on cycling transitions. These results demonstrate for the first time that three-l…
The time evolution of driven two-level systems in the far off-resonance regime is studied analytically. We obtain a general first-order perturbative expression for the time-dependent density operator which is applicable regardless of the coupling strength value. In the strong field regime, our perturbative expansion remains valid even when the far off-resonance condition is not fulfilled. We find that, in the absence of dissipation, driven two-level systems exhibit coherent population trapping in a certain region of parameter space, a property which, in the particular case of a symmetric doubl…
Analysis of the frequency shift in coherent population trapping resonance's dynamic continuous-wave spectroscopy at the phase-jump modulation and its comparison with the conventional approach
P. N. Lebedev Physical Institute of the Russian Academy of Sciences（俄罗斯科学院列别杰夫物理研究所）
Using the standard Hamiltonian of the BCS theory, we show that in an ensemble of interacting fermions with the spin 1/2 there exist coherent states $|NC>$, which nullify the Hamiltonian of the interparticle interaction (scattering). These states have an analogy with the well-known in quantum optics the coherent population trapping (CPT) effect. The structure of these CPT-like states corresponds to Cooper pairs with the total spin $S$=1. The found states have a huge degree of degeneracy and carry a macroscopic magnetic moment, that allows us to construct a new model of the magnetism connected w…
Divacancy defects in silicon carbide have long-lived electronic spin states and sharp optical transitions, with properties that are similar to the nitrogen-vacancy defect in diamond. We report experiments on 4H-SiC that investigate all-optical addressing of spin states with the zero-phonon-line transitions. Our magneto-spectroscopy results identify the spin $S=1$ structure of the ground and excited state, and a role for decay via intersystem crossing. We use these results for demonstrating coherent population trapping of spin states with divacancy ensembles that have particular orientations in…
We have experimentally demonstrated that magnetic-field-insensitive coherent-population-trapping (CPT) resonances are generated between the ground hyperfine levels on the $D_1$ line of $^{133}$Cs using a two-photon $Λ$ scheme excited by lin || lin polarizations. The frequency shift of the CPT resonance is 0.04 Hz for the deviation of 1 $μ$T at a "magic" magnetic field of 139 $μ$T and is 50 times smaller than that of the conventional clock transition at a typical bias magnetic field for the clock operation. The amplitude of the CPT spectrum excited by lin || lin polarizations is enhanced as the…
Hyperfine interactions with a nuclear spin environment fundamentally limit the coherence properties of confined electron spins in the solid-state. Here, we show that a quantum interference effect in optical absorption from two electronic spin states of a solid-state emitter can be used to prepare the surrounding environment of nuclear spins in well-defined states, thereby suppressing electronic spin dephasing. The evolution of the coupled electron-nuclei system into a coherent population trapping state by optical excitation induced nuclear spin diffusion can be described in terms of Levy fligh…
We describe how quantum dot semiconductor cavity systems can be engineered to realize anisotropy-induced dipole-dipole coupling between orthogonal dipole states in a single quantum dot. Quantum dots in single-mode cavity structures as well as photonic crystal waveguides coupled to spin states or linearly polarized excitons are considered. We demonstrate pronounced dipole-dipole coupling to control the radiative decay rate of excitons and form pure entangled states in the long time limit. We investigate both field-free entanglement evolution and coherently pumped exciton regimes, and show how a…
In this letter we investigate the possibility to attain strongly confined atomic localization using interacting Rydberg atoms in a Coherent Population Trapping (CPT) ladder configuration, where a standing-wave (SW) is used as a coupling field in the second leg of the ladder. Depending on the degree of compensation of the Rydberg level energy shift induced by the van der Waals (vdW) interaction, by the coupling field detuning, we distinguish between two antiblockade regimes, i.e. a partial antiblockade (PA) and a full antiblockade (FA). While a periodic pattern of tightly localized regions can …
We study the creation of vortex states in a trapped Bose-Einstein condensate by a rotating force. For a harmonic trapping potential the rotating force induces only a circular motion of the whole condensate around the trap center which does not depend on the interatomic interaction. For the creation of a pure vortex state it is necessary to confine the atoms in an anharmonic trapping potential. The efficiency of the creation can be greatly enhanced by a sinusodial variation of the force's angular velocity. We present analytical and numerical calculations for the case of a quartic trapping poten…
Ramsey spectroscopy via coherent population trapping (CPT) is essential in precision measurements. The conventional CPT-Ramsey fringes contain numbers of almost identical oscillations and so that it is difficult to identify the central fringe. Here, we experimentally demonstrate a temporal spinwave Fabry-Pérot interferometry via double-$Λ$ CPT of laser-cooled $^{87}$Rb atoms. Due to the constructive interference of temporal spinwaves, the transmission spectrum appears as a comb of equidistant peaks in frequency domain and thus the central Ramsey fringe can be easily identified. From the optica…
