1. Introduction

Realizing superradiance in mesoscopic atomic ensembles allows the realization of lasers with high spectral purity, which do not need to rely on the stability of the cavity mirrors [1]. Such lasers have recently been demonstrated by the Thompson group on a millihertz transition of laser-cooled strontium atoms trapped in macroscopic optical cavity [2]. At the same time, a two-photon Raman transition in a Λ-type atom (Fig. 1(a)) can be used as a model system for such narrow optical transitions, which was explored in demonstrations of Raman superradiant lasers in the past [3, 4]. Of particular interest is the option to control the effective spontaneous emission rate in this system by adjusting the intensity and detuning of the pump laser. Laser-cooled atomic clouds confined in a hollow-core optical waveguide [5–8] offer a so-far untapped platform for studies of potentially new regimes of superradiance phenomena in a geometry with extremely high ratio (∼10${ }^{-3}$-10${ }^{-4}$) between the cloud length and diameter. Here, we describe our original approach for monitoring the Raman transition process and present the results of our initial study of the spontaneous Raman emission in an ensemble of cesium atoms confined inside a hollow-core photonic-bandgap fiber with a magic wavelength dipole trap. We observe effective spontaneous emission rates that can be adjusted between 2$\pi \times$1 kHz to 2$\pi \times$100 kHz and compare these with theoretically predicted values.

 

Fig. 1 (a) Schematics of Raman transition in a Λ-type three level system and the corresponding effective two level system. (b) The normalized intensity of the emitted light and (c) the atomic population in the ground state as a function of time for an exponential decay in a two level system.

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In a Λ-type three level atomic system, where two hyperfine levels constitute the ground states, the atomic population can stay in the upper ground state $|s〉$ with a long lifetime. When a detuned pump beam, Ω_p, interacts with this atomic system, the atoms scatter the pump photons via the intermediate state $|e〉$, and some of them decay to the other ground state $|g〉$. This process can be regarded as a decay in an effective two-level system between state $|s〉$ and $|g〉$ (Fig. 1(a)). This decay rate, Γ_R, is determined by the incoming photon intensity and detuning, which means that we have the control over the spontaneous decay rate in this effective two-level system and the decay can even be effectively ‘paused’ at arbitrary time during the process.

The population decay in this effective two level system is conventionally monitored by observing the emitted photon flux. When all the atoms initially occupy the state $|s〉$, the population in this state decays exponentially as $N_s=N_{\text{tot}}{e}^{-{\mathrm{\Gamma }}_{R}t}$ where N_tot is the total number of atoms. Then, the intensity of the emitted photons also follow a form of exponential decay (Fig. 1(b)):

We propose and demonstrate an alternative way to monitor the Raman transition process free of the above difficulties based on detection of the atomic population transferred into the ground state. In the case of spontaneous emission, the ground state population evolves in time as an integral of the exponential decay and can be approximated as

 

Fig. 2 (a) Schematics of the experiment with an inset showing a microscope image of the cross section of the hollow-core fiber. (b) Time sequence of the experimental procedure. (c) Diagram of the cesium energy levels involved in detection of atoms.

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2. Experimental setup

The experiment is performed in a setup described in detail in [8], where a 20mm-long photonic bandgap fiber with a mode diameter of 5.5 μm is vertically mounted inside an ultra high vacuum (UHV) chamber. We use the cesium D₂ transition line (6${ }^2{S}_{1/2}{\to }^2{P}_{3/2}$, 852 nm) for both the initial atom cooling and trapping trapping and the spontaneous Raman process. Cesium atoms are cooled and trapped in a magneto-optical trap (MOT) ∼5 mm above the tip of the fiber. These atoms are then further cooled down to ∼30 μK using polarization gradient cooling (PGC) [9] to increase the atoms’ loading efficiency into the fiber. After this additional cooling process for ∼10 ms, we shut off the trapping and repump beams by acousto-optic modulator (AOM) switches and let the atoms free-fall by gravity. At the same time, the dipole trap laser turns on, and the dipole optical potential generated by the diverging dipolelaser guides the atoms into the core of the fiber where the light intensity is high. Figure 2(b) depicts the control sequence of the experiment.

In actual cesium atoms, 6${ }^2{S}_{1/2}|F=3〉$, 6${ }^2{S}_{1/2}|F=4〉$, and 6${ }^2{P}_{3/2}$ states correspond to the $|g〉$, $|s〉$, and $|e〉$, respectively. When the atoms are loaded into the core of the fiber, the majority of those occupy the upper ground state, 6${ }^2{S}_{1/2}|F=4〉$, but there exists a small fraction in the other ground state, 6${ }^2{S}_{1/2}|F=3〉$. In order to investigate the Raman process based on our detection method, we need to clean up state $|F=3〉$ and make sure that there are no atoms initially in that state. For this, we use a repumping beam resonant on the transition $\left|F=3〉\to \right|F^{\prime }=3〉$ for ∼10 us and transfer all the atoms in $|F=3〉$ into the other ground state.

For detection of atomic population in state $|F=3〉$ after the spontaneous Raman emission, we observe the time-variant transmission of a probe beam tuned to an open transition $\left|F=3〉\to \right|F^{\prime }=4〉$ [6]. The arrival time of transmitted photons are recorded with a single photon counting module (SPCM-NIR-14-FC from Excellitas) connected to a time tagger. When there are no atoms loaded inside the fiber, the probe photons are fully transmitted through the system and show a uniform transmission profile as a function of time. With the cesium atoms loaded inside the fiber, the probe photons are initially fully absorbed to excite the atoms into state $|F^{\prime }=4〉$. As time goes on, some of these excited atoms decay to $|F=4〉$, which is a dark state for the probe photons, and no longer interact with the incoming probe (Fig. 2(c)). The medium becomes transparent through this process. Taking into account the branching ratio from state $|F^{\prime }=4$ (7/12 to state $|F=4$) and the efficiency of the photon counter, we derive the number of atoms in state $|F=3〉$ from the difference in the transmitted photon numbers with and without loaded atoms.

3. Experimental results

In order to extract the decay rate, Γ_R in our effective two level system, we perform measurements on the number of atoms in $|F=3〉$ after the Raman scattering process with different Raman pump duration. The inset in Fig. 3(a) shows the histograms of the transmitted probe photon counts in time after different Raman pump duration. This tells us that the longer pump duration, the more probe photons are absorbed, in other words, the more atoms are found in the final state, and demonstrates that our “freeze then detect” method works properly. The atom numbers calculated from each histogram are presented as a function of the Raman pump duration in the main plot of Fig. 3(a). The fact that these data follow the exponential form without a significant deviation indicates that we are in the regime of spontaneous emission. Additionally, we compare the atom number in the initial and final states and confirm that there is no undesired loss mechanism during Raman process such as photoassociation (PA) induced by the Raman pump beam [10, 11]. By fitting Eq. (2) to this set of data, we derive the value of the total decay rate, Γ_R, at a given power and detuning ofthe Raman pump light (black dashed line in Fig. 3(a)). Finally, we repeat this process for various Raman pump powers (ranging between 40 nW and 6.0 μW) and detunings (ranging between -0.4 GHz and -3.0 GHz from the 6${ }^2{S}_{1/2}\to 6^2{P}_{3/2}$ transition), as well as with different total atom numbers (ranging from 1,000 to 12,000) loaded into the fiber for further investigation.

 

Fig. 3 (a) Measured data on the ground state population versus the pump duration. The dashed line is a fit of Eq. (2) with Γ_R as the fitting parameter found to be 2π × 18.4 kHz. The inset shows the counts of the transmitted probe photons as a function of probe time. The colored markers indicate the atom numbers that are derived from each shape- and color-matched plot in the inset. The decay rates, Γ_R, obtained from the fits at (b) different pump detunings and (c) different pump powers. Solid lines in (b) and (c) are fits of Eq. (3), with Γ_sg and the atoms’ distribution factor μ being the fitting parameters.

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The total decay rate, Γ_R, in the effective two level system can be decomposed as ${\mathrm{\Gamma }}_R={\mathrm{\Gamma }}_{sg}+{\mathrm{\Gamma }}_{eg}$, where ${\mathrm{\Gamma }}_{eg}$ is the rate of the spontaneous Raman decay induced by the pump beam, and ${\mathrm{\Gamma }}_{sg}$ indicates the decay rate from all other channels; this includes the natural decay of the $|s〉$ - $|g〉$ transition and decay due to collisions. The second term, ${\mathrm{\Gamma }}_{eg}$, is proportional to ${\mathrm{\Gamma }}_e{\rho }_{ee}$, where Γ_e is the spontaneous decay rate of the excited states 6${ }^2{P}_{3/2}$, and ${\rho }_{ee}$ is the excited state population. For a given hyperfine level $|F^{\prime }〉$ in 6${ }^2{P}_{3/2}$ states, at high power, $\propto {\mathrm{\Omega }}_{F^{\prime }}^2$, and large detuning, ${\mathrm{\Delta }}_{F^{\prime }}$, of the Raman pump field, such that ${\mathrm{\Omega }}_{F^{\prime }},{\mathrm{\Delta }}_{F^{\prime }}\gg {\mathrm{\Gamma }}_e$, the excited state population ${\rho }_{ee}$ is approximated as ${\rho }_{ee}\approx \frac{{\Omega }_{F^{\prime }}^2}{4{\mathrm{\Delta }}_{F^{\prime }}^2+2{\Omega }_{F^{\prime }}^2}$. Many previous studies widely referred the summation of ${\mathrm{\Gamma }}_e{\rho }_{ee}$ over allowed scattering transitions as a Raman emission rate [12–14], however, this model matches our measurement data very poorly. The reasoning is following: among the scattered atoms, only those that decay into the other ground state actually lead to Raman emission because if an atom goes back to the original ground state, nothing really happens in the perspective of the effective two-level system. Therefore, an accurate model should take into account the selection rule and branching ratio for transitions to the final ground state from the excited state.

Considering these factors, we construct a formula for spontaneous Raman emission in our atomic model as

We have two unknown parameters in this formula; the non-light induced decay rate, ${\mathrm{\Gamma }}_{sg}$, and the Rabi frequency, ${\mathrm{\Omega }}_{F^{\prime }}$. The Rabi frequency of a two level system is given by ${\mathrm{\Omega }}_{F^{\prime }}=〈e\left|d\cdot E\right|g〉/\hslash$, where the value of the dipole operator d is known, and the electric field amplitude $\left|E\right|$ is proportional to the root square of the effective light intensity, $I=\mu P/\left(\pi {\sigma }^2\right)$, where μ indicates the radial distribution of the atomic cloud inside the fiber core. Here, we know the values of the dipole trap power, P, and the mode radius of the hollow-core fiber, σ, so the remaining unknown parameter, μ, gives rise to the uncertainty in the Rabi frequency. To summarize, there are two free parameters, ${\mathrm{\Gamma }}_{sg}$ and μ, in Eq. (3).

By fitting Eq. (3) to the measured data at different pump detunings, we derive the free parameter values of ${\mathrm{\Gamma }}_{sg}\approx$ 0.3 kHz and μ = 1.63, which is close to the distribution factor μ = 1.61 that we obtained based on the amount of ac-Stark shifts at different dipole trap wavelengths [8] (Fig. 3). The same formula is applied to the data at different pump powers to give us ${\mathrm{\Gamma }}_{sg}\approx$ 0.3 kHz again and μ = 1.72. The higher distribution factor may have been caused by a change in the transition energy due to ac-Stark shift at a high pump power. We also performed this measurement with different initial atom numbers inside the hollow-core fiber to investigate the possible coherent phenomena, which may occur at high atom density. No significant difference between different atom numbers inside the fiber was found yet up to the maximum number of atoms that we have been able to load. The next section will discuss the outlook of this work in detail.

4. Application for superradiance detection

The measurement method described above can be used to determine if a system has undergone a collective process, such as superradiance or amplified spontaneous emission (ASE) [15–18], rather than ordinary spontaneous emission. Here, one would look for if and how much the number of atoms in the ensemble loaded into the hollow-core fiber affects the observed Γ_R.

 

Fig. 4 (a) Spontaneous emission undergoing exponential decay in emitted intensity (Eq. (1)), normalized to the initial intensity, I₀, with a maximum initial intensity which scales as the number of excited atoms, N. The superradiant intensity (Eq. (4)) for N = 10 atoms in a spherically symmetric cloud is also shown, with a maximum intensity occurring at time τ_D (Eq. (6)) and scaling as N². (b) Number of ground state atoms for spontaneous emission (Eq. (2)) and superradiant (Eq. (7)) process.

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While, in spontaneous emission, the intensity of the emitted radiation follows an exponential decay in time as shown in Eq. (1) (orange dashed line in Fig. 4(a)), the intensity profile of superradiant emission has drastically different characteristics. Superradiance is a coherent evolution in which the collective atomic excitation becomes entangled among the ensemble as the atoms decay [19, 18]. The atoms instead undergo a cascade of emission with increasing rates until the maximum intensity output is reached after some delay time. This idealized fully entangled Dicke state emits at an intensity, $I^{SR}$, that is proportional to N² (instead of N, as in spontaneous emission) [17–21], such that

Equation (4) describes a fast pulse shape, with a characteristic width, τ_R, and delay time, τ_D, as shown in Fig. 4(a) (blue solid line). We plot the Dicke model of a superradiant cloud with

While several experiments that have demonstrated superradiance have relied on detecting the emitted photons in order to find the characteristic fast burst and intensity enhancement features [2, 3, 22], the preparation of population inversion and the limited efficiency in detection can present technological challenges for observing the phenomenon. We propose to overcome these issues by monitoring the ground state population instead of the emitted photons. In spontaneous emission, the number of atoms that have decayed to the ground state, $N_g^{SE}$, grows exponentially as described in Eq. (2). Superradiance, on the other hand, will result in a much steeper increase in the number of ground state atoms at the delayed time of the pulse. The rate of atoms transferred to the ground state actually increases initially until the inflection point at time τ_D, unlike spontaneous emission which has a continuously decreasing rate of transfer. Both processes, of course, plateau to $N_g\to N_{\text{tot}}$ at large times.

For superradiance, the ground state population, $N_g^{SR}$, is

For more realistic systems the behavior of the superradiance could potentially be slightly altered. The geometry of the cloud acts to modify the τ_R [23]. For a cylindrical atomic cloud of length, L, and atomic density, n, the pulse width becomes

In addition, for large samples there can be multiple superradiant bursts with the peak intensity decreasing for each consecutive pulse, known as the ringing effect [24]. Such ringing features would also be detectable by monitoring the ground state population, giving a strong indication of the presence of superradiance.

There are several requirements the system must possess in order for superradiance to occur. The atoms need to be prepared in a single ground state, either $|s〉$ or $|g〉$. This is achieved by optically pumping the atoms in the initial ground state using a pump beam resonant to one of noncycling transitions. It should be noted that if the initial state is chosen to be the lower ground state, $|g〉$, then the final state population measurements of the upper ground state, $|s〉$, must take place within a time scale shorter than the decay time between the ground states.

In addition, the time scales of the coherent relaxation processes between $|s〉$ and $|g〉$, ${\tau }_{\text{rel}}=\frac{1}{{\gamma }_{rel}}$, need to be much longer than the time scales of the superradiant pulse, such that

The coherence of the evolution can be destroyed by relaxation and dephasing effects such as collisions and Doppler effects that break velocity coherence between atoms.

Our system has not shown superradiance so far and has remained instead in the regime of the non-coherent spontaneous emission of the Raman scheme. This is evident in Fig. 3(a), showing the characteristic exponentially increasing number of ground state atoms over the exposure time. The main reasons behind the absence of superradiance is that the dephasing times in our system are too small to reach the superradiant regime, given the number of atoms in the fiber.

One way to overcome this limit may be to increase the number of atoms in the fiber above the critical density needed to make the superradiant time scales small enough for the burst to occur. The critical number of atoms, N_crit, required for superradiance can be found by using the condition of Eq. (10) with Eqs. (6) and (9), and assuming the Raman decay acts as the decay rate of the effective two level, $\mathrm{\Gamma }={\mathrm{\Gamma }}_R$, resulting in

In the absence of initializing the population to a single Zeeman sublevel, we have estimated the coherent relaxation decay rate of our system using electromagnetically induced transparency (EIT) measurements to be ${\gamma }_{\text{rel}}\sim 2\pi \times 1$MHz. Given the inequality of Eq. (11), our system would require at least $\sim 5\times {10}^4$ atoms in the hollow-core fiber of radius 3.5 μm in order to observe superradiance (currently we only load ∼10${ }^4$ atoms into the fiber), using an effective Raman decay of ${\mathrm{\Gamma }}_R=2\pi \times 100$ kHz. The larger atomic density would act to trigger the superradiance pulse before decoherence of the ensemble can occur. A more accurate characterization of the requirements that our system will need in order to exhibit superradiance may be obtained by manually triggering the decay by seeding the atomic media with a light pulse.

5. Conclusions

We developed a novel method to study an effective two-level system based on spontaneous Raman transition in a Λ-type three level atomic system inside a hollow-core bandgap fiber and were able to control the population decay of the excited state over two orders of magnitude (∼1 kHz - ∼100 kHz). Our detection method, which measures the atomic population in the ground state, enables us to monitor the system without the need to filter out photons from the Raman pump beam. From our experimental data, we extracted the non-light induced decay rate of sub-kHz in our hollow-core fiber system. With improved atom loading, we plan to use our system and methodology for studies of superradiant Raman emission in a quasi-1D geometry and to work towards implementing cavity-free precision light sources.