Ultranarrow-bandwidth filter based on a thermal EIT medium

We present high-contrast electromagnetically-induced-transparency (EIT) spectra in a heated vapor cell of single isotope 87Rb atoms. The EIT spectrum has both high resonant transmission up to 67% and narrow linewidth of 1.1 MHz. We get rid of the possible amplification resulted from the effects of amplification without population inversion and four-wave mixing. Therefore, this high transmitted light is not artificial. The theoretical prediction of the probe transmission agrees well with the data and the experimental parameters can be derived reasonably from the model. Such narrow and high-contrast spectral profile can be employed as a high precision bandpass filter, which provides a significant advantage in terms of stability and tunability. The central frequency tuning range of the filter is larger than 100 MHz with out-of-band blocking ≥15 dB. This bandpass filter can effectively produce light fields with subnatural linewidth. Nonlinearity associating with the narrow-linewidth and high-contrast EIT profile can be very useful in the applications utilizing the EIT effect.

The model. In order to gain more insight into the EIT spectrum, we provide a theoretical analysis by solving the optical Bloch equations (OBEs) and Maxwell-Schrödinger equation (MSE) under the perturbation limit 25 (i.e. the Rabi frequency of the probe field Ω p is much weaker than that of the coupling field Ω c ). The outgoing probe field is derived from the thermally-averaged atomic coherence of ρ v 31 ( ) . For a group with velocity v = vzˆ relative to the laboratory frame, the dynamics of atomic coherences in the atomic frame are written as the follows. In order to detect the weak probe field signal, two waveplates and one polarizer are arranged after the thermal cell to filter out the strong coupling field. λ/4 and λ/2: quarter and half waveplates, respectively; PD: photo detector.
Here ρ ij is an element of the density-matrix operator in an EIT system, γ denotes the ground state decoherence rate, Γ is the spontaneous decay rate of the excited state which is 2π × 5.75 MHz for Rb D 1 -line transition, c is the speed of light in vacuum, and L is the length of the medium. The physical definition of α here represents the optical density resulting from the entire ensemble having the resonant absorption cross section, and it can be directly derived from the system condition nσL, where n is the atomic density and σ is the absorption cross section. For a given temperature T c , atomic velocities are described by the Maxwell-Boltzmann distribution, where m is the atomic mass and k B is the Boltzmann constant. The atomic motion induces Doppler shifts, where ω ij denotes the transition frequency between the energy levels |i〉 and |j〉 and ωp(c) is the probe (coupling) laser frequency. Because k p ≅ k c , Δp′ − Δc′ is replaced by Δ p − Δ c ≡ δ, defined as two-photon detuning. In our measurements, Δ c = 0 because of frequency locking.
Additionally, we consider the optical pumping effect that causes the atomic accumulation for different velocity groups. Strong Zeeman pumping and coupling fields pump more high-velocity groups to the ground state |1〉 with detunings of (in the atomic frame). These high velocity groups then participate the EIT peak transition (around Δ p = 0 and δ = 0) with large one-photon detunings Δ p′ = k p v; meanwhile at out-of-band detunings of  Δ = Γ k v p p these velocity groups absorb the resonant probe field (because of Δ p′ = 0), resulting in a broader absorption spectrum. The selection of velocity groups is expressed as the power broadening function, = The optical pumping linewidth Γ OP is nearly a linearly increasing function of the powers of the Zeeman pumping and coupling fields. In all of the measurements, the values of Γ OP varied from 18Γ to 120Γ according to different laser intensities.
Based on the steady-state solution of the optical-Bloch equations, we derive ρ v 31 ( ) from Eq. (1a) and (1b), for the cell temperature of 42 °C. To fit the spectrum by the model, we first determine the optical density α and optical pumping linewidth Γ OP from the baseline curve of the spectrum in a frequency range over ±50 MHz. The optical density α dominates the absorption depth and Γ OP individually governs the curvature of absorption line. From the systematic study, a stronger Ω c corresponds to a larger Γ OP while the optical density remains nearly a constant. We consider 4001 velocity classes ranging from −200Γ to 200Γ. With the given α and Γ OP , Ω c and ground state decoherence rate γ are resolved from the fitting of the EIT linewidth and peak height, respectively. The best fit of the measurement in Fig. 2(b) gives a set of α = 225 and (Γ OP , Ω c , γ) = (90, 4.2, 0.022)Γ. Compared with the estimation according to the system condition nσL, the determined α from the theoretical model is acceptable. As the probe field detuning is a little outside the transparency window, the baseline transmission is derived as , which means only a small fraction of hot atoms with a small range of the velocity participate in the one-photon transition. Moreover, by considering the laser power, beam size, and atomic transition rate, we derive Ω c = 4.25Γ, which is consistent with the theoretically determined one. Hence, the theoretical prediction fits the data well and the experimental parameters can be derived reasonably from the model.
We further simplify Eq.
and k p v is replaced by x in unit of Γ. The EIT peak transmission can be written as As mentioned before, the peak transition (Δ p = 0) is contributed by all of velocity groups which mediate one-photon-detuning EIT transition. The physical definition of Γ EIT here represents the linewidth in a velocity spectrum or one-photon-detuning spectrum. The EIT peak T(0) in a Doppler-broadened medium has the similar expression αγ − Γ Ω exp[ 2 / ] c 2 (which has been widely used in a Doppler-free ensemble) with a calibration factor C α . The finite EIT linewidth and optical pumping linewidth degrade the effective optical density. The calibration factor C α as a function of Γ OP and Γ EIT is plotted in Fig. 3. Take the parameters of Fig. 2 for example (Γ OP /Γ = 90 and Γ EIT /Γ = 400), we derive C α = 0.69, which means the effective optical density of the EIT transmission αC α = 157. We further discuss the propagation velocity of the probe pulse in this optically thick media. The phase of the probe field φ is expressed as  and EIT linewidth are both sufficiently broader than Γ D . Thus, C α reaches unity. All of the populations are prepared to the dark states and then participate the EIT transition. On the other hand, if the linewidths are both sufficiently narrow, the effective optical density is only contributed from the Doppler-free atoms. The values of the above-mentioned linewidths provide the useful information on the range of velocity groups which need to be taken into account on the EIT transition. This model advances the knowledge in the thermal-EIT study.
Properties of EIT spectra. The expression of EIT peak transmission shows that a stronger coupling field leads to a higher transmission. We systematically study the EIT transmission with varied coupling field power. At each condition, the baseline transmission keeps at the same low level, indicating the optical density does not vary with Ω c . As Ω c gets stronger, corresponding to larger Γ OP and Γ EIT , the EIT peak height and linewidth become higher and broader. The EIT peak transmission (circles) and linewdith (squares) with varied Ω c are shown in Fig. 4(a,b). We further increased Ω c by reducing the coupling beam size to 2.0 mm. The results are shown in Fig. 4(c,d). The EIT peak saturates at 70%. Further increasing Ω c does not enhance the peak transmission but it does broaden the EIT linewidth. For the applications in optical filters or in biphoton generations via SFWM process [20][21][22][23] , the bandwidth of the filter or the linewidth of photon source is controllable while the central frequency maintains a high transmission.
The theoretical predictions with fixed optical density α = 225 and decoherence rate γ = 0.022Γ well fit the data of EIT peak transmission and linewidth, as the shown red curves in Fig. 4. At strong Ω c regime, the peak height goes lower than the predicted value. The discrepancy is caused by the impurity of the coupling field polarization due to the photon switching effect 26 . The σ − coupling field destroys the quantum interference of EIT and thereby results in absorption of the probe field. In addition, the peak height at a weak Ω c condition is also lower than the prediction because of the EIT transient effect. A weak Ω c leads to a long EIT response time 27 . As the result of the finite atomic transient time that atoms move in and out of the interaction regime, EIT has not reached the steady-state condition, implying the degradation of the EIT peak height. Thus, the theoretical prediction supports the experimental observations and physical picture.
Moreover, for the purpose of photon source generation, we should get rid of any amplification. Once the beam size of the coupling field is too small, the population transient effect needs to be taken into account: The fresh atoms, entering the probe interaction region, is not optically pumped to the Zeeman dark states in advance, and therefore, the probe field is amplified through the so-called amplification without population inversion (AWI) 28 . A smaller coupling beam size can potentially make the AWI more prominent. At the same value of Ω c , the peak transmission of the coupling beam size of 2.8 mm shown in Fig. 4(a) is less than that of 2.0 mm shown in Fig. 4(b) by less than 2.5%. The amplification due to the AWI effect is very insignificant in the data of 67% peak transmission shown in Figs 2(b) and 4(a). Furthermore, the four-wave-mixing (FWM) amplification, in which the coupling field also excites the population in the ground state driven by the probe field 29 , is not allowed in our experiment. This is because the coupling and probe fields need to have the same polarization to induce this amplification, but they have the orthogonal-polarization configuration here. We further exclude the amplification induced by a little impurity of the σ − polarization of Zeeman pumping field via another kind of FWM process 30,31 . When the atomic transitions involve any gain effect, the determined decoherence rate γ should go lower. To test whether there could be any gain effect, we adjust the polarization ratio of σ − to σ + components of the Zeeman pumping field, whose Rabi frequencies are denoted as ZP Ω − and Ω + ZP , respectively. When the ratio ( / ) 003 ZP ZP 2 Ω Ω < .
− + , γ did not change; and once the ZP ZP 2 in our system is less than 0.01, the FWM gain does not occur in our system. Hence, we believe the high EIT transmission is not artificial and such a high-contrast EIT spectrum would be useful in the future applications.

Conclusion
We systematically investigate the thermal-EIT spectra which can make a quality prediction for a slow light or photon source. The spectral profile shows a high EIT peak transmission of 67%, a narrow EIT linewidth of 1.1 MHz, and a low off-resonant transmission less than 3%. We get rid of the possible amplification, and hence this high transmitted light is not artificial. A high-contrast EIT medium can be applied as an ultranarrow-bandwidth filter. The central frequency of the filter can be precisely tuned, making it flexible in the generation of photon sources with subnatural linewidth. We further provide a theoretical model to simulate EIT spectra. The prediction fits the data well and the experimental parameters can be reasonably derived from the model. Hence, the spectral measurements and theoretical model advance our knowledge in the thermal-EIT study.

Methods
Setup and Measurements. We perform a continuous-wave EIT spectrum study in a 87 Rb-filled cell (Thorlabs GC25075-RB). All of the laser fields drive D 1 -line transition at wavelength of 795 nm. With the help of the coupling field (which couples the transition between states |F = 1〉 and |F′ = 1〉 and the Zeeman pumping field (which couples that between states |F = 2〉 and |F′ = 2〉), all populations accumulate at two Zeeman ground states |1〉 and |F = 1, m F = 1〉, as shown in Fig. 1. The coupling field is produced by an external cavity diode laser (ECDL). One beam from the ECDL is sent through an electro-optic modulator (EOM) before injection locking the probe field. The probe and coupling beams are nearly collinear propagating to reduce two-photon Doppler broadening (k c − k p Cosθ)v, where k c(p) represents the wavevector of the coupling (probe) field, θ is the angle between them, and v is the atomic velocity. The Zeeman pumping beam has a sufficiently large intersecting angle of around 0.9° with respect to EIT beams to avoid any light leakage into the detector. We apply a serial waveplates and crystal polarizer to separate the probe field (σ − polarization) from the coupling beam (σ + polarization) after cell. This polarizer filter reduces the coupling field by 48 dB while the probe field keeps 85% transmittance. In order to observe single-photon-level EIT signals or biphton signals 20-23 , one can further apply multiple spectral filters (e.g. Fabry-Perot etalons) to diminish the stray light.
Data Analysis. For each EIT spectrum, we normalize the probe transmission by the incident power. In the absence of the coupling field, 0.6% of the probe field can be no longer absorbed at a further larger optical density medium by heating up the cell temperature. This component came from the sideband signal of the probe laser, which was injection-locked by the coupling laser after an EOM. It has to be subtracted for the calculation of the probe field transmission. The incident power of the probe field was measured at a far-off-resonant frequency (6.8-GHz red-detuned respective with the resonant transition of |1〉 to |3〉). The applied power of the probe field was typically 2.4 μW right before the cell and the maximum power of the coupling beam was 10 mW. Fig. 2(c,d), the central frequency of EIT peak linear shifts with the applied coupling field power due to the AC Stark effect. The coupling field also couples the far-off-resonant transition between states |F = 〉 and |F′ = 2〉 with detuning of 814.5 MHz. We suppose that the velocity group of v = 0 dominates the transition so that Δ AC = 814.5 MHz is not a velocity-dependent function. We further assume that the coupling field only drove one far-off-resonant transition from state |2〉 to state |F′ = 2, m F ′ = 1〉. The transition has a Clebsch-Gordan coefficient that is 3 times as lager as that of of |2〉 to |3〉 transition. The amount of AC Stark shift for the spectrum in Fig. 2(b) is only 20% different from the estimated value of ( 3 ) /4 c A C 2 Ω Δ .