## Abstract

The effect of wall-induced Ramsey interference on the electromagnetically induced absorption (EIA) in an anti-relaxation coated vapor cell is studied in a four-level closed N-type atomic system. We show that the atomic coherence spontaneously transferred from the excited states to the ground states may interfere via the Ramsey sequence of the moving atoms with decoherence-free wall collisions. The spectral narrowing of the EIA resonance lineshape is induced by the Ramsey effect because of the long lifetime of the spontaneously transferred atomic coherence between the ground states. This calculated Ramsey EIA spectrum is in good agreement with the narrow EIA spectrum obtained experimentally in the anti-relaxation coated vapor cell.

©2012 Optical Society of America

## 1. Introduction

Long-lived atomic coherence is an important factor in the occurrence of narrow spectral width of a spectrum. Electromagnetically induced transparency (EIT) and electromagnetically induced absorption (EIA) are examples of atomic coherent phenomena [1–2]. High-resolution spectroscopy with narrow spectral width is important for applications in atomic clocks, precision atomic magnetometers, manipulation of the group velocity of light pulses, light storage, and quantum optics [3–7].

The decay of atomic coherence is related not only to the natural decay time between states, but also to the atomic collisions and wall collisions in an atomic vapor cell. Atomic vapor cells with buffer gases have been used in many studies to preserve the atomic coherence from the collisions at the walls of the cell, and to increase the interaction time with atoms and the laser through the diffusive atomic motion caused by these collisions with buffer gases [8–10]. Another method for preserving the atomic coherence between ground states is the use of an anti-relaxation coated atomic vapor cell. An anti-relaxation coated cell such as a paraffin-coated alkali vapor cell can maintain the atomic coherence between the ground states in spite of the wall collisions [11–18]. For EIT due to atomic coherence between ground states, the spectral-width narrowing of EIT in the paraffin-coated alkali vapor cell has been observed successfully, and is explained by the wall-induced Ramsey effect [15–17].

It is well known that the Ramsey method detects a phase shift of the atomic coherence between ground states by two atom-laser interactions, which depends on both the resonance frequency between the ground states and the time of flight [4, 17]. Although the atom-laser interaction time in an anti-relaxation coated cell is shorter than that in a vapor cell with a buffer gas, the Ramsey effect is the cause of the spectral-width narrowing because of the repeated interaction of the laser with the atoms with long-lived atomic coherence between the ground states in spite of the wall collisions.

For EIA due to transfer of coherence (TOC), it was difficult to reduce the spectral width of EIA in the atomic vapor cell with a buffer gas, because EIA is related not only to the coherence between the ground states, but also to that between the excited states. In the interaction of an atom with the laser, the coherence between the excited states may be destroyed by collisions with the buffer gas [19–25]. In the case of the anti-relaxation coated cell, the coherence between the excited states is almost unaffected by collisions during the interaction of an atom with the laser. Recently, we observed EIA with a narrow sub-kHz spectral width in the Hanle configuration of the ^{87}Rb D_{2}-line using a paraffin-coated Rb vapor cell [18]. Although the cause of the narrow EIA in the paraffin-coated alkali vapor cell was thought to be the Ramsey effect, the spectral-width narrowing of EIA in the anti-relaxation coated cell has not been clearly explained. It is therefore very interesting to investigate whether or not the spontaneously transferred atomic coherence (STAC) in the first interaction region interferes with that in the second interaction region. The interference between two STACs in the first and the second interaction is not considered normal; because EIA is included the spontaneous process. Particularly, in the view of the symmetry between EIA and EIT, the study on the Ramsey interference of EIA is very important.

In this paper, we investigate the Ramsey interference of EIA in a four-level closed N-type atomic system, in order to understand the spectral-width narrowing of EIA in an anti-relaxation coated cell. The Ramsey sequences according to the velocity distribution of atoms and the relaxation-free atomic path length in an anti-relaxation coated vapor cell are considered in order to numerically simulate the EIA spectrum. The numerical results for the EIA spectrum are compared with the experimental results for the ^{87}Rb D_{2}-line using a paraffin-coated Rb vapor cell.

## 2. Atomic density matrix for EIA in four-level atomic system

It is possible to interpret the EIA phenomenon in a vapor cell by means of a simple four-level closed N-type atomic system, as shown in Fig. 1 . The N-type atomic system consists of two degenerate ground states ($|1\u3009$ and $|3\u3009$) and two degenerate excited states ($|2\u3009$ and $|4\u3009$). The two coupling fields are tuned to the $|1\u3009\leftrightarrow |2\u3009$ and $|3\u3009\leftrightarrow |4\u3009$ transitions, and the probe field is tuned to the $|2\u3009\leftrightarrow |3\u3009$ transition. The electric component ${E}_{n}$ of the three laser radiation fields is assumed to be of the form

*n*at position

*r*and

*t*, $\omega $ is its angular frequency, and $k$ is its wave vector.

*n*= 1 and 3 denote the two strong coupling fields, and

*n*= 2 denotes the weak probe field. The optical Rabi frequency is given by ${\Omega}_{n}(r,t)={\mu}_{n}{E}_{0n}(r,t)/\hslash =\Omega {a}_{ge}$, where ${\mu}_{n}$ is the electric dipole moment of the related transition, $\hslash $ is Planck’s constant over $2\pi $, and

*a*is the normalized transition strength between the ground state $|g\u3009$ and excited state $|e\u3009$. In our model, the normalized transition strengths

_{ge}*a*are assumed to be

_{ge}*a*

_{12}= 0.8,

*a*

_{32}= 0.2, and

*a*

_{34}= 1.0.

We tried to calculate numerically the density matrix equation in the four-level closed N-type atomic system. The time-dependent Schrödinger equation is written in the density matrix formalism as:

*v*,

*g*( = −1/6) and

_{e}*g*( = 1/2) are the Landé

_{g}*g*-factors of the excited and ground states, respectively, and

*μ*is the Bohr magneton. $\frac{\partial {\widehat{\rho}}_{sp}}{\partial t}$ denotes the terms associated with spontaneous emission, and its matrix elements are as follows [2]:

_{B}*Γ*is the rate of total spontaneous emission (2π × 6 MHz) from the excited state to the ground state. The spontaneous transfer of excited-state coherence ($\u3008e|\widehat{\rho}|e\text{'}\u3009$) to ground-state coherence ($\u3008g|\widehat{\rho}|g\text{'}\u3009$) appears in Eq. (7). After the fast time dependence of the density-matrix elements has been eliminated by their transformation into a slowly varying density operator, we can obtain a set of linear differential equations. Considering a moving atom with velocity

*v*, the laser frequency $\omega $ is transformed into $\delta +{\omega}_{0}-k\cdot v$ because of the Doppler effect, where $\delta =\omega -{\omega}_{0}$ is the detuning of the laser frequency. The complete set of density matrix equations in the four-level closed N-type atomic system consists of 16 equations. To simulate the Ramsey effect from the density matrix equations, we numerically calculated the density matrix equations considered the interaction time sequence interacted with the atoms and the lasers, as shown in Fig. 2 .

The moving atoms in a pure atomic vapor cell show Doppler broadening and transit-time broadening in the atomic spectrum. In a cylindrical atomic cell with an anti-relaxation coating, the atoms interacted with the laser beam, evolved decoherence-free between the ground states in spite of wall-collisions, and re-interacted with the laser beam. If the atomic coherence is preserved to the second interaction of the atoms with the laser, we may tentatively expect the Ramsey effect not only in EIT, but also in EIA due to TOC.

In the time domain, the interactions between the moving atoms and the laser beam can be considered as Ramsey pulse sequences, as shown in Fig. 2. Here, the first interaction time is *t*_{i1}, the flight time without decoherence of the wall collisions is *t*_{f}, and the second interaction time is *t*_{i2}. The Ramsey time sequence of the real atomic system is complicated and random because the time sequence is different according to the velocity, wall-collision number, and cell shape. For simplicity, the moving atoms in the longitudinal direction were not considered in our model. However, we numerically simulated the Ramsey effect of EIA, considering the transverse velocity distribution of atoms and the number distribution of relaxation-free wall collisions during the interaction with the laser beam.

## 3. Results & discussion

The typical EIA spectrum for the four-level closed N-type atomic system was calculated for the case of *t*_{i1} = 7.4 μs and *t*_{f} = *t*_{i2} = 0 (*t*_{i1} = *t*_{f} = 0, and *t*_{i2} = 7.4 μs), as shown in Fig. 3
. When the *rms* speed of the Rb atom at room temperature was 270 m/s and the diameter of the laser beam was 2 mm, the average interaction time was estimated to be approximately 7.4 μs. It is well known that the dual structure of the calculated spectrum of Fig. 3(a) consists of a broad absorption and narrow EIA. The horizontal axis of Fig. 3 is the B-field, which was scanned in the region near the zero value. The Rabi frequency Ω was 0.4 Γ, and the spontaneous decay rate Γ was 2π × 6 MHz. Figure 3(b) shows a magnification of the narrow EIA part of Fig. 3(a); the spectral width of the EIA spectrum is 150 mG (110 kHz) because of transit-time broadening.

The atoms emerge from the first interaction region of the spontaneously transferred atomic coherence between degenerate ground states. These atoms undergo coherent phase evolution in the flight-time region. They then enter into the second interaction region, and interact with the same laser beam. The second interaction enters into interference with the spontaneously transferred atomic coherence carried by the moving atoms with wall collisions. The Ramsey interference due to spontaneously transferred atomic coherence is not intuitive. Comparing with the Ramsey interference of the EIT, one of EIA is very interesting in spontaneous process. The EIA interference fringe was calculated numerically when the atoms interacted with the laser beam in these two regions, as shown in Fig. 4(a)
, where *t*_{i1} = *t*_{i2} = 7.4 μs and *t*_{f} = 0.31 ms (corresponding to *v* = 270 m/s). In addition, the EIT Ramsey interference in the three-level closed Λ-type atomic system (*a*_{12} = 0.8, *a*_{32} = 0.2, and *a*_{34} = 0) was calculated the under the same conditions as those in Fig. 4(a), and the results are shown in Fig. 4(b). Comparison of the EIA and EIT Ramsey interferences shows that the phase difference between them at B = 0 is π. This means that the atomic coherence contributing to the EIA Ramsey interference is different from that of the EIT interference.

The Ramsey sequence (*t*_{i1}, *t*_{i2}, and *t*_{f}) is function of the atomic velocity. The velocity spread of the atoms is given by the Maxwell–Boltzmann distribution. To simplify the calculation in our model, we assume that the diameter (*d*) of the laser beam is fixed at 2 mm, and the length (*L*_{f}) of the free flight of atoms without atomic coherence dephasing is fixed at 85 mm. Therefore, the timescale of the Ramsey sequence is changed according to the velocities of the atoms, but the ratio is fixed. In the calculation of Fig. 5
, the fixed ratio is *t*_{i1}:*t*_{i2}:*t*_{f} = 1:1:42, where *t*_{i1} = *t*_{i2} = 7.4 μs and *t*_{f} = 0.31 ms, corresponding to *v* = 270 m/s. The timescales are determined as ${t}_{i1}={t}_{i2}=\frac{d}{v}$ and ${t}_{f}=\frac{{L}_{f}}{v}$. Figure 5(a) shows the EIA interference fringes according to the velocities of the atoms (*v* = 140, 240, and 340 m/s). When the velocity of the atoms is decreased, the relaxation-free flight time of the atoms is increased, and the interval of the EIA interference fringes is decreased, as shown in Fig. 5(a).

The shape of the Ramsey fringe is obtained though integration over all interaction times according to the velocities of the atoms. The phases of all the EIA interference fringes with different atomic velocities at B = 0 are maximum absorptions, and the phases of the interference fringes in the nonzero B-field differ according to the velocities of the atoms. When the EIA interference fringes according to the velocities of the atoms are integrated, we can see the EIA Ramsey fringe, as shown in Fig. 5(b). The calculated result shown in Fig. 5(b) comes from the numerical integration of the weighted EIA interference fringes with Maxwell–Boltzmann distributions. The characteristic features of the calculated EIA Ramsey spectrum are similar to those of the EIT Ramsey spectrum [4]. The spectral width of the EIA Ramsey spectrum is narrower than that of the EIA associated with the single-pass interaction time of Fig. 3. To the best of our knowledge, this is the first report of the EIA Ramsey signal based on a four-level closed N-type atomic system.

We now consider the relaxation-free atomic path length (*L*_{f}) without decoherence of the ground states for the Ramsey sequence. Because of the shape and anti-relaxation coating quality of the vapor cell, *L*_{f} may be spread about the center of average *L*_{f}. The atom number distribution according to *L*_{f} is assumed to be a Poisson distribution. In our experiment, the diameter of the paraffin-coated vapor cell was 25 mm [18].

Figure 6(a)
shows the EIA Ramsey fringes according to *L*_{f} (25, 85, and 145 mm). *L*_{f} is related to the relaxation-free flight time (*t*_{f}) of the atoms. Therefore, when *L*_{f} is changed, the ratio of the Ramsey sequence (*t*_{i1}, *t*_{i2}, and *t*_{f}) also changes. As *L*_{f} is increased, *t*_{f} increases, and the spectral width of the EIA Ramsey fringes decreases, as shown in Fig. 6(a). Each calculated result of Fig. 6(a) is the numerical integration of the weighted EIA interference fringes with Maxwell–Boltzmann distributions under the same conditions as in Fig. 5(b).

When the EIA Ramsey fringes according to the *L*_{f} values of wall collisions are integrated, the narrow EIA spectrum in an anti-relaxation coated cell can be calculated numerically, as shown in Fig. 6(b). The calculated result of Fig. 6(b) is the numerical integration of the weighted EIA Ramsey fringes with Poisson distribution of the atom number as a function of *L*_{f}. The feature of the calculated EIA spectrum has no oscillation in the pedestal of the EIA Ramsey fringes of Fig. 6(a). The spectral width of the calculated EIA spectrum depends on the center value and the width of the Poisson distribution of the number of wall collisions. In this calculation, the average path-length of the Poisson distribution is assumed to be 85 mm.

Finally, we compared the calculated EIA spectrum of Fig. 6(b) with the experimental result reported in our previous work [18]. Figure 7
shows the observed EIA spectrum in the paraffin-coated Rb vapor cell. The experimental result is the narrow EIA spectrum with a 390-Hz spectral width in a paraffin-coated Rb vapor cell for the 5S_{1/2}(F = 2)–5P_{3/2}(F' = 3) transition of ^{87}Rb atoms. The difference between the experiment and theoretical results is due to the residual magnetic field, the incomplete laser polarization, and the other hyperfine states effect [26]. This calculated Ramsey EIA spectrum is in good agreement with the narrow EIA spectrum obtained experimentally. Although our model for the Ramsey narrowing of EIA is very simple, based on a four-level closed N-type atomic system, we can confirm that the narrowing of the EIA spectrum in an anti-relaxation coated vapor cell is due to the Ramsey interference of TOC.

## 4. Conclusion

We have investigated the narrowing of the EIA in an anti-relaxation coated vapor cell using a simple model based on a four-level closed N-type atomic system. The spontaneously transferred atomic coherence between degenerate ground states in the first interaction region, carried by the moving atoms with wall collisions, then interfered with that in the second interaction region. The EIA Ramsey interference was calculated numerically and compared with the EIT Ramsey interference. We showed that the phase difference between the EIT and EIA Ramsey interferences was π. Considering the velocity spread of the atoms with a Maxwell–Boltzmann distribution and the relaxation-free atomic path length (*L*_{f}) with a Poisson distribution, the Ramsey narrowing of the EIA in an anti-relaxation coated vapor cell was simulated successfully. Good agreement was found when the numerical result of the EIA spectrum was compared with the experimental result of the ^{87}Rb D_{2}-line using a paraffin-coated Rb vapor cell. From our results, we confirmed that the narrowing of the EIA spectrum in an anti-relaxation coated vapor cell was due to the Ramsey interference of the spontaneously transferred atomic coherence. Our results are expected to help provide a better understanding of the interesting phenomena in anti-relaxation coated vapor cells, and to be applied to superluminal pulse propagation with higher speed.

## Acknowledgments

This work was supported by the National Research Foundation of Korea (2009-0066070).

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