Coherence effects in electromagnetically induced transparency in V-type systems of <sup>87</sup>Rb

Hyun-Jong Kang; Heung-Ryoul Noh

doi:10.1364/OE.25.021762

1. Introduction

The atomic coherence between the two dipole-forbidden states leads to many interesting phenomena such as electromagnetically induced transparency (EIT) [1, 2], electromagnetically induced absorption (EIA) [3, 4], coherent population trapping [5] etc. In particular, EIT was extensively investigated since its first experimental observation [6] owing to its potential applications such as quantum information [7], slow light and light storage [8], precision magnetometers [9], nonlinear wave-mixing [10,11], study of Rydberg states [12,13], and atomic clocks [14]. There are typically three distinct schemes in EIT; Λ-, ladder-, and V-type schemes. Since the linewidth of the EIT spectrum is obtained from the arithmetic average of the atomic linewidths of the two states in the dipole-forbidden transition, Λ- and ladder-type schemes have attracted more attention compared to the V-type scheme owing to their small resulting linewidths. In particular, in the case of V-type configuration, since the linewidth for EIT and the natural linewidth are comparable, it is not easy to determine the portion that results from coherence effect [15–17].

There were many reports on V-type EIT for atoms [15–24] and molecules [25, 26]. The wavelengths of the coupling and probe beams may match or mismatch. In the case of wavelength mismatch, D₁ and D₂ lines of Rb atoms were used in [15]. The same wavelength configuration for Na atoms was used in [16]. In addition to the D₂ line for coupling beam, the 5S_1/2–6P_1/2 transition of Rb atoms was used for a probe beam in [17] and the 5S_1/2–6P_3/2 transition was used in [18]. In the case of wavelength mismatch, owing to the non-existence of frequency mixing between the coupling and probe fields, the calculation of the EIT spectra is usually simpler as compared to the case of wavelength match. However, owing to the complexity of real atomic systems, an accurate calculation of the EIT spectra considering all the magnetic sublevels involved has not been reported to the best of our knowledge. We note that accurate calculations of ladder-type EIT for Rb atoms were reported by the authors in [27,28]. It should be also noted that the energy level structure is greatly simplified in the presence of a large magnetic field in the hyperfine Paschen-Back regime, and consequently good agreement is found with textbook models for EIA [29], EIT [30], and four-wave mixing [31].

From the perspective of distinguishing the coherence effect from incoherence effects such as optical pumping or saturation effect, the quantitative evaluations of the coherence effect were studied in [22,26]. However, only the calculations for a simple three-level system were provided. Moreover, the reason for the weakness of coherence effect in open transitions was not clearly explained.

In this study, we calculate accurate lineshapes of V-type EIT spectra for the D₁ and D₂ lines of ⁸⁷Rb using the density matrix equations and compare them with experimental results. In the calculation, we identify the effect of the coherence term. As expected, the coherence effect exists only in the cycling transition. Consequently, the transmittance or absorptive signals in open transitions mostly result from incoherence effects such as saturation or optical pumping. In order to understand the reason for the disappearance of the coherence effect in open transitions, we analytically calculate the incoherent and coherent contributions to the EIT spectra for an open V-type three-level atomic system and investigate the dependence of the two contributions on the openness of the system.

2. Theory

We describe the method for the calculation of EIT spectra for the D₁ and D₂ lines of ⁸⁷Rb atoms, as shown in Fig. 1(a). The linearly polarized probe beam (Rabi frequency, Ω₁) is tuned to the D₁ line, and the co-propagating coupling beam (Rabi frequency, Ω₂) is tuned to the D₂ line. The coupling beam is linearly polarized in the parallel or perpendicular direction to the polarization of the probe beam. The density matrix equation in the frame rotating at the frequency of the coupling beam is given by

\dot{ρ} = - \frac{i}{ℏ} [H_{0} + V, ρ] + {\dot{ρ}}_{sp},

where ρ is the density operator in this frame. Subsequently, the atomic Hamiltonian (H₀) is given by

\begin{array}{l} H_{0} = - \sum_{m = - 3}^{3} ℏ δ_{2} | F_{e} = 3, m 〉 〈 F_{e} = 3, m | \\ - \sum_{m = - 2}^{2} ℏ (δ_{2} + Δ_{32}) | F_{e} = 2, m 〉 〈 F_{e} = 2, m | \\ - \sum_{m = - 1}^{1} ℏ (δ_{2} + Δ_{31}) | F_{e} = 1, m 〉 〈 F_{e} = 1, m | \\ - \sum_{m = - 2}^{2} ℏ δ_{1} | {\bar{F}}_{e} = 2, m 〉 〈 {\bar{F}}_{e} = 2, m | \\ - \sum_{m = - 1}^{1} ℏ (δ_{1} + {\bar{Δ}}_{21}) | {\bar{F}}_{e} = 1, m 〉 〈 {\bar{F}}_{e} = 1, m |, \end{array}

where δ₁₍₂₎ = δ_p(c) − k₁₍₂₎v is the effective detuning experienced by an atom moving with velocity v and k₁₍₂₎ = 2π/λ₁₍₂₎. Further, δ_p(c) and λ₁₍₂₎ are the detuning and wavelength of the probe (coupling) beam, respectively. In Eq. (2), F_e and F̄_e denote the angular momentum quantum numbers of the excited states 5P_3/2 and 5P_1/2, respectively. Further, Δ_μν represents the frequency spacings between the states |F_e = μ〉 and |F_e = ν〉, and Δ̄₂₁ represents the frequency spacing between the states |F̄_e = 2〉 and |F̄_e = 1〉.

Fig. 1 (a) Energy level diagram of the V-type EIT. The frequency spacings are presented in units of MHz. (b) Detailed energy level diagrams showing the laser couplings for parallel and perpendicular polarization configurations.

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In Eq. (1), the interaction Hamiltonian (V) is given by

\begin{array}{l} V & = & \frac{ℏ Ω_{2}}{2} \sum_{F_{e} = 1}^{3} \sum_{m = - F_{g}}^{F_{g}} C_{F_{g} = 2, m}^{F_{e}, m} | F_{e}, m 〉 〈 F_{g}, m | \\ + \frac{ℏ Ω_{1}}{2} \sum_{∊ = - 1}^{1} \sum_{{\bar{F}}_{e} = 1}^{2} \sum_{m = - F_{g}}^{F_{g}} a_{∊} {\bar{C}}_{F_{g} = 2, m}^{{\bar{F}}_{e}, m + ∊} | {\bar{F}}_{e}, m + ∊ 〉 〈 F_{g}, m | + h . c ., \end{array}

where F_g = 2. We consider two configurations: parallel (π‖π) and perpendicular (π⊥π) polarizations of the probe and coupling beams. The detailed energy level diagrams showing the laser couplings are illustrated in Fig. 1(b). The states 5P_1/2(F̄_e = 2) and 5P_3/2(F_e = 2, 1, 0) are not shown in Fig. 1(b) to avoid confusion. Since we consider the direction of polarization of the coupling beam as a quantization axis, the coupling beam excites the transitions with Δm = 0. In the case of the probe beam, since the coefficients a_∊ (∊ = ±1, 0) in Eq. (3) are given by

\begin{array}{l} a_{\pm 1} = 0, a_{0} = 1, for π ‖ π, \\ a_{\pm 1} = \mp \frac{1}{\sqrt{2}}, a_{0} = 0, for π ⊥ π, \end{array}

the probe beam excites the transitions with Δm = 0 (Δm = ±1) for parallel (perpendicular) polarization configuration as shown graphically in Fig. 1(b). In Eq. (3), the normalized transition strength between the states |F_g, m_g〉 and |F_e, m_e〉 [|F̄_e, m_e〉],

C_{F_{g}, m_{g}}^{F_{e}, m_{e}}

[

{\bar{C}}_{F_{g}, m_{g}}^{{\bar{F}}_{e}, m_{e}}

], is given in the appendix. In Eq. (1), ρ̇_sp represents the relaxation phenomena, whose explicit expressions are also given in the appendix.

We solve Eq. (1) in the steady-state limit using Eqs. (2), (3), and (18) and obtain the density matrix elements as a function of v. Finally, the absorption coefficient of a probe beam, averaged over a Maxwell-Boltzmann velocity distribution, is given by

\begin{array}{l} α = - N_{at} \frac{3 λ_{1}^{2}}{2 π} \frac{Γ_{1}}{Ω_{1}} \frac{1}{\sqrt{π} u} \cdot Im \int_{- \infty}^{\infty} d v e^{- {(v / u)}^{2}} \\ \times \sum_{∊ = - 1}^{1} \sum_{{\bar{F}}_{e} = 1}^{2} \sum_{m = - F_{g}}^{F_{g}} a_{∊}^{*} {\bar{C}}_{F_{g} = 2, m}^{{\bar{F}}_{e}, m + ∊} 〈 {\bar{F}}_{e}, m + ∊ | ρ | F_{g}, m 〉, \end{array}

where N_at is the atomic number density in the cell and u is the most probable speed of atoms. We also study the effect of coherences between the states in 5P_3/2 and 5P_1/2. In this case, Eq. (1) is solved under the condition that all the density matrix elements interlinking the sublevels belonging to the 5P_3/2 and 5P_1/2 states are neglected intentionally. Subsequently, the absorption coefficient is calculated using Eq. (4).

3. Experiment

Figure 2 shows the experimental setup for the V-type EIT. An external cavity diode laser (ECDL; Toptica, DLX) operating at 780 nm was used for the coupling beam and an ECDL (Toptica, DL100) operating at 795 nm was used for the probe beam. The power of the probe and coupling beams was controlled using half-wave plates HWP1 and HWP2, respectively, and the polarization of the probe beam was adjusted using HWP3. The co-propagating probe and coupling beams, linearly polarized in the parallel or perpendicular direction, overlapped in an Rb cell within a crossing angle of several milliradians. The probe beam, after passing through a band-pass filter (BPF; Thorlabs, FB790-10) that blocked the coupling beam, was detected using a photo-diode (Thorlabs, PDA10A-EC). In the experiment, the power of the probe beam was fixed at 10 μW, whereas that of the coupling beam varied in the range of 100 μW to 4.8 mW. The diameter of both the laser beams was 3 mm and the temperature of the cell was maintained at 20 °C. The cell was covered with three layers of μ-metal sheet to avoid the terrestrial magnetic field.

Fig. 2 Detailed schematic of the experimental setup. External cavity diode laser (ECDL), saturated absorption spectroscopy (SAS), beam splitter (BS), polarizing beam splitter (PBS), half-wave plate (HWP), band pass filter (BPF), and photo diode (PD).

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4. Results

The typical experimental and calculated results are shown in Figs. 3(a) and (b), respectively. In Fig. 3, the frequency of the probe beam was fixed at the F_g = 2 → F̄_e = 2 transition of the ⁸⁷Rb-D₁ line and the frequency of the coupling beam was scanned around the F_g = 2 → F_e = 1, 2, 3 transitions of the ⁸⁷Rb-D₂ line. The polarization directions of the coupling and probe beams are perpendicular (upper traces) or parallel (lower traces) to each other. The three peaks from the left to the right correspond to the transitions from the state F_g = 2 to the states F_e = 1, 2, and 3, respectively. In Fig. 3(b), as well as the results calculated from the full density matrix equations, the calculated results without considering the coherences between the sublevels in 5P_3/2 and 5P_1/2 states are demonstrated. First, we can observe consistency between the experimental and calculated results. As shown in Fig. 3(b), the effect of the coherence term is negligible in the non-cycling transitions (F_g = 2 → F_e = 1 and F_g = 2 → F_e = 2) and exists only in the cycling transition (F_g = 2 → F_e = 3). We also observe that the effect of coherence term is much smaller in the parallel polarization configuration rather than the perpendicular polarization configuration.

Fig. 3 (a) Typical experimental and (b) calculated results of EIT spectra. The polarization configurations are π⊥π and π‖π for the upper and lower traces, respectively.

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The power dependence of the coupling beam of the V-type EIT spectra for the perpendicular polarization configuration is shown in Fig. 4. The power of the probe beam was fixed at 10 μW, and that of the coupling beam varied from 80 μW to 4.8 mW. The experimental and calculated results are shown in Figs. 4(a) and (b), respectively. As shown in Fig. 3(b), we also provide calculated results without considering the coherence effect. The signals for the non-cycling transitions show no significant features as the power of the coupling beam increases except for a monotonic broadening of the signals owing to power broadening. In contrast, the signal for the cycling transition varies from a dip to a peak as the power of the coupling beam increases. Moreover, the effect of the coherence term increases further. From the results in the non-cycling transitions, we can observe that the signals originate from incoherence effects such as optical pumping rather than the coherence effect. In contrast, the signals for the cycling transition are composed of coherence and incoherence effects. This incoherence effect may be due to the saturation effect between the states F_g = 2 and F_e = 3 and optical pumping effect between the Zeeman sublevels of the states F_g = 2 and F_e = 3.

Fig. 4 (a) Experimental and (b) calculated EIT spectra for various coupling powers in perpendicular polarization configuration.

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The results for the parallel polarization configuration are shown in Fig. 5. The schemes in Fig. 5 are identical to those in Fig. 4. The dependence of the EIT spectra on the power of the coupling beam is weaker than that in the perpendicular polarization configuration. The coherence effect is also weaker in the case of parallel polarization configuration.

Fig. 5 (a) Experimental and (b) calculated EIT spectra for various coupling powers in parallel polarization configuration.

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5. Analytical solutions

We perform an analytical calculation of EIT for an open V-type three-level atomic system to understand the effect of branching ratio on the V-type EIT spectra. We will demonstrate that the effect of the coherence term diminishes quickly as the system becomes open. The simple energy level diagram is shown in Fig. 6. The probe beam couples the ground state (|3〉) and excited state (|1〉), and the coupling beam, co-propagating with the probe beam, couples the ground state (|3〉) and excited state (|2〉). The decay rates of the states |1〉, |2〉, and |3〉 are Γ₁, Γ₂, and Γ₃ = 0, respectively. The spontaneous decay rate from |1〉 (|2〉) to |3〉 is Γ₁₃ (Γ₂₃). Therefore, when the system is cycling, Γ₁₃ = Γ₁ and Γ₂₃ = Γ₂. The transverse decay rates between the states |i〉 and |j〉 are γ_ij = (Γ_i + Γ_j)/2 with i, j = 1, 2, 3.

Fig. 6 Simple energy level diagram for an open V-type three-level atomic system.

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The Rabi frequencies of the probe and coupling beams are Ω₁ and Ω₂, respectively. We find the susceptibility of the probe beam up to the first order in Ω₁ and an arbitrary order in Ω₂. For the sake of simplicity, we assume that the excited states are degenerate and the wavelengths of the laser beams are equal to λ. Hence, the wave vector is k(= k₁ = k₂) = 2π/λ. The effective detuning of the probe and coupling beams, experienced by an atom moving at velocity v, are δ₁ = δ_p − kv and δ₂ = δ_c − kv, respectively, where δ_p and δ_c are the detuning of the probe and coupling beams, respectively.

The density matrix equations are given by

{\dot{ρ}}_{13} = i Δ_{1} ρ_{13} + \frac{i}{2} Ω_{2} ρ_{12} + \frac{i}{2} Ω_{1} (ρ_{11} - ρ_{33}),

{\dot{ρ}}_{12} = i Δ_{0} ρ_{12} + \frac{i}{2} Ω_{2} ρ_{13} - \frac{i}{2} Ω_{1} ρ_{32},

{\dot{ρ}}_{32} = - i Δ_{2} ρ_{32} - \frac{i}{2} Ω_{1} ρ_{12} + \frac{i}{2} Ω_{2} (ρ_{33} - ρ_{22}),

{\dot{ρ}}_{11} = - (Γ_{1} + Γ_{t}) ρ_{11} + \frac{i}{2} Ω_{1} (ρ_{13} - ρ_{31}),

{\dot{ρ}}_{22} = - (Γ_{2} + Γ_{t}) ρ_{22} + \frac{i}{2} Ω_{2} (ρ_{23} - ρ_{32}),

\begin{array}{l} {\dot{ρ}}_{33} & = & - Γ_{t} (ρ_{33} - 1) + Γ_{13} ρ_{11} + Γ_{23} ρ_{22} \\ - \frac{i}{2} Ω_{1} (ρ_{13} - ρ_{31}) - \frac{i}{2} Ω_{2} (ρ_{23} - ρ_{32}), \end{array}

where ρ_ij ≡ 〈i|ρ|j〉 and

ρ_{j i} = ρ_{i j}^{*}

with i, j = 1, 2, 3. In Eqs. (5)–(10), the effective detunings are defined as

\begin{array}{l} Δ_{0} & = & δ_{1} - δ_{2} + i γ_{12}, \\ Δ_{1} & = & δ_{1} + i γ_{13}, \\ Δ_{2} & = & δ_{2} - i γ_{23} . \end{array}

The matrix element responsible for the probe absorption is ρ₁₃. After solving Eq. (5) in the steady-state regime, ρ₁₃ can be decomposed into:

ρ_{13} = \frac{Ω_{1}}{2 Δ_{1}} + \frac{Ω_{1}}{2 Δ_{1}} (ρ_{33} - ρ_{11} - 1) - \frac{Ω_{2}}{2 Δ_{1}} ρ_{12} .

The other matrix elements in right-hand side in Eq. (11) can also be obtained in the steady-state regime from Eqs. (5)–(10) in the first order in Ω₁ and in the arbitrary order in Ω₂ as follows:

\begin{array}{l} ρ_{12} & = & \frac{Ω_{1} Ω_{2} [Δ_{2} (ρ_{11} - ρ_{33}) + Δ_{1} (ρ_{33} - ρ_{22})]}{Δ_{2} (4 Δ_{0} Δ_{1} - Ω_{2}^{2})}, \\ ρ_{11} & = & 0 \\ ρ_{22} & = & \frac{s}{2 [1 + s (1 + \frac{Γ_{2} - Γ_{23}}{2 Γ_{t}})]}, \\ ρ_{33} & = & \frac{2 + s}{2 [1 + s (1 + \frac{Γ_{2} - Γ_{23}}{2 Γ_{t}})]}, \\ s & = & \frac{Ω_{2}^{2} / (γ_{23} Γ_{2})}{1 + {(δ_{2} / γ_{23})}^{2}}, \end{array}

where

Γ_{t} (\equiv 2 u / (\sqrt{π} d))

is the transit time decay rate with the laser beam diameter d [32, 33]. This transit time decay rate is employed to solve the density matrix equations in the steady-state regime.

The first term on the right-hand side in Eq. (11) represents the background contribution in the absence of the coupling beam, and the second term is proportional to the difference between the populations ρ₃₃ and ρ₁₁ and is called the incoherence term. Since the third term on the right- hand side in Eq. (11) is proportional to ρ₁₂, which is the coherence between the states |1〉 and |2〉, it is called the coherence term. Notably, the incoherence term in this case represents only the saturation term when the system is in the cycling transition.

The susceptibility of the probe beam is proportional to ρ₁₃, and should be averaged over Maxwell-Boltzmann velocity distribution as follows:

χ = - \frac{3 λ_{1}^{3}}{4 π^{2}} \cdot \frac{N_{at} Γ_{1}}{Ω_{1}} \int_{- \infty}^{\infty} \frac{d v}{\sqrt{π} u} e^{- {(v / u)}^{2}} ρ_{13} .

Similar to Eq. (11), the susceptibility, χ can be decomposed into three parts:

χ = χ_{0} + χ_{1} + χ_{2},

where χ₀ is the susceptibility of the background signal in the absence of the coupling beam, and χ₁ and χ₂ are the susceptibilities of the incoherence and coherence contributions, respectively. In Eq. (13), χ₀ is explicitly given by

χ_{0} = i C_{0} exp [- {(\frac{δ_{p} + i γ_{13}}{k u})}^{2}] Erfc [- i (\frac{δ_{p} + i γ_{13}}{k u})],

with

C_{0} = \frac{3 λ^{3}}{8 π^{3 / 2}} \frac{Γ_{1} N_{at}}{k u} .

In Eq. (13), χ₁ and χ₂ are given by

χ_{1} = C_{0} \frac{(Γ_{2} - Γ_{23} + Γ_{t}) Ω_{2}^{2}}{2 Q Γ_{2} Γ_{t} K_{1}},

χ_{2} = C_{0} \frac{i (1 + Q) K_{2} Ω_{2}^{2}}{Q K_{1} (4 K_{0} K_{1} - Ω_{2}^{2})},

respectively, where

\begin{array}{l} K_{0} & = & δ_{p} - δ_{c} + i γ_{12}, \\ K_{1} & = & δ_{p} - δ_{c} + i (γ_{13} + Q γ_{23}), \\ K_{2} & = & δ_{p} - δ_{c} + i γ_{13} + i γ_{23} (1 + \frac{(Q - 1) Γ_{t}}{Γ_{2} - Γ_{3} + 2 Γ_{t}}), \\ Q & = & \sqrt{1 + \frac{Ω_{2}^{2}}{γ_{23} Γ_{2}} (1 + \frac{Γ_{2} - Γ_{23}}{2 Γ_{t}})} . \end{array}

In Eqs. (15) and (16), we have used Γ₁ ≪ ku, which is called the Doppler-broadened limit [34,35].

The imaginary parts of χ₁ and χ₂, representing the absorption of the probe beam, are plotted in Fig. 7. In Fig. 7, Ω₂ = 2.5Γ₂ and Γ_t = Γ₂/270. Herein, we assume that d is 2 mm and the temperature is 20°C. In Fig. 7(a), when Γ₂₃ = Γ₂, i.e., the system is in the cycling transition, we observe that the contribution of the coherence term is comparable to that of the incoherence term. In contrast, in Fig. 7(b), when Γ₂₃ = 0.95Γ₂, the contribution of the coherence term becomes very weak. Thus, we may expect that the contribution of the coherence term becomes weaker as the openness of the |2〉 –|3〉 transition increases.

Fig. 7 Imaginary parts of the susceptibilities (a) in the closed system (Γ₂₃ = Γ₂) and (b) in the open system (Γ₂₃ = 0.95Γ₂).

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The amplitudes of the imaginary parts of χ₁ and χ₂ are analytically given by

\begin{array}{l} I_{1} & = & \frac{C_{0} b (1 + ξ - x)}{b (1 + 2 ξ - x) + \sqrt{ξ (1 + 2 ξ - x)}}, \\ I_{2} & = & \frac{C_{0} ξ b}{b (1 + 2 ξ - x) + \sqrt{ξ (1 + 2 ξ - x)}}, \end{array}

respectively, where

b \equiv \frac{Ω_{2}}{Γ_{2}}, x \equiv \frac{Γ_{23}}{Γ_{2}}, ξ \equiv \frac{Γ_{t}}{Γ_{2}} .

Figure 8 shows the variation of these amplitudes as functions of x when b = 2.5 and ξ = 1/270. The amplitudes at x = 0 are analytically given by

I_{1} (0) ≃ C_{0} (1 - \frac{\sqrt{ξ}}{b}), I_{2} (0) ≃ C_{0} ξ (1 - \frac{\sqrt{ξ}}{b}),

and thus, it is easy to recognize that I₂(0) ≪ I₁(0), as shown in Fig. 8. As the value of x approaches 1, I₁ decreases and I₂ increases toward the same value (

C_{0} b / (\sqrt{2} + 2 b)

). In order to study the variation of the amplitudes at x ≃ 1, we expand I₁ and I₂ near x = 1 as follows

\begin{array}{l} I_{1} & ≃ & \frac{C_{0} b}{\sqrt{2} + 2 b} + \frac{C_{0} b (3 \sqrt{2} + 4 b)}{4 {(\sqrt{2} + 2 b)}^{2}} \cdot \frac{(1 - x)}{ξ}, \\ I_{2} & ≃ & \frac{C_{0} b}{\sqrt{2} + 2 b} - \frac{C_{0} b (\sqrt{2} + 4 b)}{4 {(\sqrt{2} + 2 b)}^{2}} \cdot \frac{(1 - x)}{ξ} . \end{array}

Considering the fact that ξ is very small in Eq. (17), we can observe that I₁ and I₂ vary abruptly from the same value (

C_{0} b / (\sqrt{2} + 2 b)

) as x decreases from x = 1. The factor, (1 − x)/ξ in Eq. (17) is actually (Γ₂ − Γ₂₃)/Γ_t = (Γ₂ − Γ₂₃)t_av, where t_av is the average interaction time between the atoms and laser lights. Since the atomic decay time (

Γ_{2}^{- 1}

) is in the order of a few tens of nanoseconds and t_av is typically a few hundreds of microseconds, the population in |2〉 diminishes abruptly as Γ₂₃ starts to differ from Γ₂, and accordingly, the coherence ρ₂₃ diminishes too.

Fig. 8 Amplitudes of χ₁, χ₂, and χ₁ + χ₂ as a function of Γ₂₃/Γ₂.

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6. Conclusion

We have presented a comprehensive experimental and theoretical study of EIT for V-type configuration of D₁ and D₂ lines of ⁸⁷Rb atoms. We measured the EIT spectra for the parallel and perpendicular polarization configurations of the coupling and probe lasers, compared them with the results calculated from density matrix equations considering all the magnetic sublevels involved, and observed consistency between the two results. By calculating the density matrix equations while neglecting the matrix elements interlinking the sublevels in the 5P_3/2 and 5P_1/2, we could discern the lineshapes owing to coherence and incoherence effects. From the analysis of the two contributions, we observed that the coherence effect exists only for the signals of cycling transition lines and is negligible for the signals of open transitions.

By investigating the analytical solutions of susceptibility for an open V-type three-level atomic system, we determined that the suppression of coherence effect in the open transitions results from a small transit decay rate compared to the natural atomic decay rate. Notably, the suppression of coherence effects in open transitions is a general phenomenon. As an example, the modulation transfer spectra exist only in the cycling transitions since they result from the coherence effect owing to nonlinear four-wave mixing of the laser fields [36, 37]. Notably, the transmission signals in the open transitions in the ladder-type EIT atomic system are mainly due to the optical pumping effect rather than the coherence effect [27, 28]. The results obtained in this paper may be useful for many applications; one of them might be the analytical calculation of the signals in the open transition owing to the legitimacy of neglecting the coherence effect.

Appendix

In Eq. (3), the explicit expressions for $C_{F_{g}, m_{g}}^{F_{e}, m_{e}}$ and ${\bar{C}}_{F_{g}, m_{g}}^{{\bar{F}}_{e}, m_{e}}$ are given by $\begin{array}{l} C_{F_{g}, m_{g}}^{F_{e}, m_{e}} \\ = {(- 1)}^{2 F_{e} + I + J_{g} + J_{e} + L_{g} + S + m_{F_{g}} + 1} \\ \times \sqrt{(2 L_{e} + 1) (2 J_{e} + 1) (2 J_{g} + 1) (2 F_{e} + 1) (2 F_{g} + 1)} \\ \times {\begin{matrix} L_{e} & J_{e} & S \\ J_{g} & L_{g} & 1 \end{matrix}} {\begin{matrix} J_{e} & F_{e} & I \\ F_{g} & J_{g} & 1 \end{matrix}} (\begin{matrix} F_{g} & 1 & F_{e} \\ m_{g} & m_{e} - m_{g} & - m_{e} \end{matrix}), \end{array}$

and $\begin{array}{l} {\bar{C}}_{F_{g}, m_{g}}^{{\bar{F}}_{e}, m_{e}} \\ = {(- 1)}^{2 {\bar{F}}_{e} + I + J_{g} + J_{e} + L_{g} + S + m_{F_{g}} + 1} \\ \times \sqrt{(2 L_{e} + 1) (2 {\bar{J}}_{e} + 1) (2 J_{g} + 1) (2 F_{e} + 1) (2 F_{g} + 1)} \\ \times {\begin{matrix} L_{e} & {\bar{J}}_{e} & S \\ J_{g} & L_{g} & 1 \end{matrix}} {\begin{matrix} {\bar{J}}_{e} & F_{e} & I \\ F_{g} & J_{g} & 1 \end{matrix}} (\begin{matrix} F_{g} & 1 & F_{e} \\ m_{g} & m_{e} - m_{g} & - m_{e} \end{matrix}), \end{array}$

respectively, where the angular momentum quantum numbers are given by S = 1/2, L_g = 0, L_e = 1, J_g = 1/2, J_e = 3/2, J̄_e = 1/2, and I = 3/2 [38]. (· · ·) and {· · ·} denote the 3J and 6J symbols, respectively.

In Eq. (1), the matrix elements of ρ̇_sp are given by

\begin{array}{l} 〈 F_{e}, m | {\dot{ρ}}_{sp} | F_{e}^{'}, m^{'} 〉 = - Γ_{2} 〈 F_{e}, m | ρ | F_{e}^{'}, m^{'} 〉 \\ - Γ_{t} 〈 F_{e}, m | ρ | F_{e}^{'}, m^{'} 〉 δ_{F_{e}, F_{e}^{'}} δ_{m, m^{'}}, \\ 〈 {\bar{F}}_{e}, m | {\dot{ρ}}_{sp} | {\bar{F}}_{e}^{'}, m^{'} 〉 = - Γ_{1} 〈 {\bar{F}}_{e}, m | ρ | {\bar{F}}_{e}^{'}, m^{'} 〉 \\ - Γ_{t} 〈 {\bar{F}}_{e}, m | ρ | {\bar{F}}_{e}^{'}, m^{'} 〉 δ_{{\bar{F}}_{e}, {\bar{F}}_{e}^{'}} δ_{m, m^{'}}, \\ 〈 {\bar{F}}_{e}, m | {\dot{ρ}}_{sp} | {\bar{F}}_{e}^{'}, m^{'} 〉 = - \frac{Γ_{1} + Γ_{2}}{2} 〈 {\bar{F}}_{e}, m | ρ | {\bar{F}}_{e}^{'}, m^{'} 〉, \\ 〈 F_{e}, m | {\dot{ρ}}_{sp} | F_{g}, m^{'} 〉 = - \frac{Γ_{2}}{2} 〈 F_{e}, m | ρ | F_{g}, m^{'} 〉, \\ 〈 {\bar{F}}_{e}, m | {\dot{ρ}}_{sp} | F_{g}, m^{'} 〉 = - \frac{Γ_{1}}{2} 〈 {\bar{F}}_{e}, m | ρ | F_{g}, m^{'} 〉, \\ 〈 F_{g}, m | {\dot{ρ}}_{sp} | F_{g}, m^{'} 〉 = \\ Γ_{2} \sum_{F_{e} = 1}^{3} \sum_{∊ = - 1}^{1} C_{F_{g}, m}^{F_{e}, m + ∊} C_{F_{g}, m^{'}}^{F_{e}, m^{'} + ∊} 〈 F_{e}, m + ∊ | ρ | F_{e}, m^{'} + ∊ 〉 \\ + Γ_{1} \sum_{{\bar{F}}_{e} = 1}^{2} \sum_{∊ = - 1}^{1} {\bar{C}}_{F_{g}, m}^{{\bar{F}}_{e}, m + ∊} {\bar{C}}_{F_{g}, m}^{{\bar{F}}_{e}, m^{'} + ∊} 〈 {\bar{F}}_{e}, m + ∊ | ρ | {\bar{F}}_{e}, m^{'} + ∊ 〉 \\ - Γ_{t} (〈 F_{g}, | ρ | F_{g}, m^{'} 〉 - \frac{1}{8}) δ_{m, m^{'}}, \end{array}

and

〈 μ | {\dot{ρ}}_{sp} | ν 〉 = {〈 ν | {\dot{ρ}}_{sp} | μ 〉}^{*},

where F_g = 2, F_e = 1, 2, 3, and F̄_e = 1, 2. In Eq. (18), Γ₁ and Γ₂ are the decay rates of the states 5P_1/2 and 5P_3/2, respectively. As discussed in Section 5, we employ the transit time decay rate Γ_t to solve the density matrix equations in the steady-state limit. The number 1/8 in Eq. (18) represents the population of each magnetic sublevel in the ground state at thermal equilibrium. Further, δ_m,m′ represents the Kronecker-delta symbol.

Funding

National Research Foundation of Korea (2017R1A2B4003483).

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Coherence effects in electromagnetically induced transparency in V-type systems of ⁸⁷Rb

Abstract

1. Introduction

2. Theory

3. Experiment

4. Results

5. Analytical solutions

6. Conclusion

Appendix

Funding

References and links

Cited By

Figures (8)

Equations (27)

Optics Express