# Phase control of transmission and reflection in a sample of duplicated two-level systems driven by a stationary control field

F A Hashmi, E. Brion, M A Bouchene

► To cite this version:

F A Hashmi, E. Brion, M A Bouchene. Phase control of transmission and reflection in a sample of duplicated two-level systems driven by a stationary control field. 2024. hal-04605029

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# Phase control of transmission and reflection in a sample of duplicated two-level systems driven by a stationary control field

F.A. Hashmi, Etienne Brion, M. A. Bouchene

► To cite this version:

F.A. Hashmi, Etienne Brion, M. A. Bouchene. Phase control of transmission and reflection in a sample of duplicated two-level systems driven by a stationary control field. Optics Communications, 2025, 587, pp.131874. 10.1016/j.optcom.2025.131874. hal-04605029v2

HAL Id: hal-04605029

https://hal.science/hal-04605029v2

Submitted on 15 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Phase control of transmission and reflection in a sample of duplicated two-level systems driven by a stationary control field

F. A. Hashmi¹,², E. Brion¹ and M. A. Bouchene¹

¹Laboratoire Collisions Agrégats Réactivité, FeRMI,
Université Toulouse III and CNRS UMR 5589, Toulouse, France and

²Department of Physics, Syed Babar Ali School of Science and Engineering,
Lahore University of Management Sciences (LUMS), Lahore

In this article, we investigate the optical response of a duplicated two-level atomic medium submitted to a strong stationary control field and a weak co-propagating probe field, orthogonally polarized to each other. We show that both reflected and transmitted components of the probe may be absorbed and amplified. Moreover, for low optical depths, reflection and transmission factors are controlled by the relative phase between control and probe fields, which makes the configuration we present here promising for the development of optical devices, such as phase-controlled switches.

## I. INTRODUCTION

The control and manipulation of light pulses has long been a key challenge with many applications in, e.g., telecommunications and, more recently, quantum technologies. To achieve this goal one may try and shape the optical response of the propagation medium – absorption and/or dispersion – through combining non-linear effects and quantum interferences. Electromagnetic induced transparency (EIT) [1], refraction index enhancement [2], slow, stored and fast light [3] are examples of applications of such an approach. Meta-materials like negative-index samples [4] or photonics crystals [5] are based on different strategies. In the latter, the propagation of light is modified by the existence of photonic band gaps due to periodic spatial variations of the optical index, in the same way as periodic atomic lattices affect the conductivity of electrons in semiconductors. Such crystals are usually obtained by stacking dielectric slabs of different indices periodically. The optical features of the structures thus obtained are determined once and for all. Another more versatile way to obtain a photonic crystal is provided by Electromagnetically Induced Gratings (EIG). In this case, the periodic modulation of the medium optical response results from the application of a standing-wave driving field, which gives rise to new original phenomena. For instance, the standing-wave configuration was recognized years ago to be very promising to achieve spatial localization [6]. Moreover, exciting a lambda system in EIT configuration with a stationary control field one obtains a periodic modulation of the atomic absorption with sharp peaks which allows for the creation of stationary pulses of light [7] as well as the optical control of photonic band gaps [8]. Many groups also use this configuration to control the group velocity [9], induce Raman-gratings [10], phase-gratings [11], or implement phase-gates and optical switching [12].

In the present work, we focus on an atomic medium with the so-called duplicated two-level system (DTLS) configuration, often used to study electron spin coherence effects [13]. When excited by a strong control and a weak probe beams, polarized orthogonally to each other, DTLS exhibits efficient quantum interferences between absorption and stimulated emission paths. We pointed out the great potentialities of DTLS media for controlling the propagation of the probe field in previous works. We experimentally demonstrated the coherent control of the medium gain in the femtosecond regime [14]. We also theoretically predicted the existence of Zeeman-coherent-oscillation-assisted slowing of the probe in a non-collinear configuration [15] and phase control of the effective susceptibility in a collinear configuration [16]. Other authors proposed to take advantage of DTLS media to control optical bistability [17], group velocity [18], and implement quantum memory or optical switching [19]. It was also suggested to use orthogonal spatial configuration for the control field to generate spatial diffraction within the multiphoton resonance condition [20] or beyond [21]. The phase control of spatial interference of resonance fluorescence [22], transmission and reflection factors in a dielectric slab [23], and the possibility of spatial localization of atoms [24] are among promising related results.

In the present article, we consider a DTLS medium submitted to collinear control and probe fields. The control field, however, is now a standing-wave which induces an EIG in Zeeman coherences. The overlap between the control and probe beams ensures efficient interaction and energy exchange between the field components along the sample. We study the influence of interaction parameters – control field intensity, detuning, optical depth and phase shift between the control and the probe – on reflected and transmitted probe light intensity and phase. By contrast with transverse gratings which only diffract a weak part of the probe field, the optical depth of DTLS medium leads to significant reflection and transmission coefficients which can be even larger than one. This spatial configuration moreover enables a phase control of probe reflection and transmission in certain conditions we identify.
# Phase control of transmission and reflection in a sample of duplicated two-level systems driven by a stationary control field

F. A. Hashmi$^{a,b}$, E. Brion$^b$, M. A. Bouchene$^b$

$^a$Department of Physics, Syed Babar Ali School of Science and Engineering, Lahore University of Management Sciences (LUMS), Lahore, Pakistan

$^b$Laboratoire Collisions Agrégats Réactivité, FeRMI, Université de Toulouse and CNRS UMR 5589, Toulouse, France

## Abstract

In this article, we study the optical response of a duplicated two-level atomic medium subjected to a stationary control field and a weak co-propagating probe field, orthogonally polarized to each other. We show that both the reflected and transmitted components of the probe can be absorbed or amplitified. Furthermore, for moderate optical depths, the reflection and transmission factors are controlled by the relative phase between the control and probe fields, making the configuration we present here promising for the development of optical devices. We also determine the exact conditions under which reflection and transmission factors can be controlled by the relative phase between the control and probe fields.

**Keywords:** coherent control, four-wave mixing, nonlinear optics, duplicated two-level system

## 1. Introduction

The control and manipulation of light pulses has long been a key challenge with many applications, e.g., in telecommunications and more recently in quantum technologies. To achieve this goal one can try to shape the optical response of the propagation medium – absorption and/or dispersion – by combining nonlinear effects and quantum interferences. Electromagnetically Induced Transparency (EIT) [1], refractive index enhancement [2], slow, stored and fast light [3] are examples of applications of such an approach.
<figure><img src="image_2.png" /><figcaption>Figure 1. Experimental configuration (proposal). The control field is split into two parts that give the forward and backward components (Rabi envelopes $\Omega_{\pi}^{\pm}$, respectively). The probe field is sent at the entrance of the sample in the forward direction (Rabi envelope $\Omega_{\sigma}^{+}$) and the backward component (Rabi envelope $\Omega_{\sigma}^{-}$) is generated from the end part of the sample.</figcaption></figure>

## II. THEORETICAL MODEL

### A. The system

We consider the experimental configuration depicted in Fig. 1. An atomic sample of length $L$ is submitted to two fields of same frequency $\omega$, propagating along the $y$ axis with orthogonal polarizations. The first – so-called control – field is $\pi$-polarized and split into two parts: one part penetrates the sample from the left side (henceforth designated as the entrance), and propagates in the direction of increasing $y$’s, while the other part is seeded back into the sample from the other side (henceforth designated as the exit), and propagates in the direction of decreasing $y$’s. The total control electric field expresses as

$$ \mathbf{E}_{\pi}(y,t) = \mathbf{e}_z \left[ \varepsilon_{\pi}^{+}(y) e^{iky} + \varepsilon_{\pi}^{-}(y) e^{-iky} \right] e^{-i\omega t} + \text{c.c.} \quad (1) $$

For sake of simplicity, we assume that the amplitude $\varepsilon_{\pi}^{+}(y = 0)$ is real. The second – so-called probe – field is $\sigma$-polarized propagates in the direction of increasing $y$’s and penetrates the sample through the entrance. Its electric field is given by

$$ \mathbf{E}_{\sigma}(y,t) = \mathbf{e}_x \left[ \varepsilon_{\sigma}^{+}(y) e^{iky} + \varepsilon_{\sigma}^{-}(y) e^{-iky} \right] e^{-i\varphi} e^{-i\omega t} + \text{c.c.} \quad (2) $$

In this expression, $\varphi$ is defined as the phase shift between the probe and control fields. Note that even if the injected probe field propagates in the forward direction, a backward component $\varepsilon_{\sigma}^{-}(y)$ builds up because the atomic sample radiates in both directions. Moreover, the boundary conditions for the probe at the exit and entrance write $\varepsilon_{\sigma}^{-}(y = L) = 0$ and $\varepsilon_{\sigma}^{+}(y = 0) = E_0$, where $E_0$ is real.

We assume that the fields essentially couple to a $F = 1/2 \rightarrow F = 1/2$ atomic line of the medium (e.g. $^2S_{1/2}F = 1/2 \rightarrow ^2P_{1/2}F = 1/2$ transition of $^6$Li at 671 nm) described by a duplicated two-level system (Fig. 2). The $\pi$-polarized control field $\mathbf{E}_{\pi}$ (resp. $\sigma$ polarized probe field $\mathbf{E}_{\sigma}$) couples to the $\Delta m_F = 0$ (resp. $\Delta m_F = \pm 1$) paths. In the following, we determine the reflection and transmission coefficients of the medium for a weak probe in the presence of a much stronger control field.

### B. Atomic polarization

In the interaction picture, the density matrix $\rho$ of an atom in the medium obeys the master equation $i\hbar\partial_t\rho = [H, \rho] + \text{relaxations where } H$ is the Hamiltonian in the rotating wave approximation

$$ H = \hbar \begin{pmatrix} 0 & 0 & -\Omega_{\pi}^{*} & -\Omega_{\sigma}^{*} e^{i\varphi} \\ 0 & 0 & -\Omega_{\sigma}^{*} e^{i\varphi} & \Omega_{\pi}^{*} \\ -\Omega_{\pi} & -\Omega_{\sigma} e^{-i\varphi} & \Delta_{0} & 0 \\ -\Omega_{\sigma} e^{-i\varphi} & \Omega_{\pi} & 0 & \Delta_{0} \end{pmatrix} $$

Here $\Omega_{j=\pi,\sigma} = \frac{de_j}{\hbar}$ denote the Rabi frequencies for the control and probe fields, with $d = \langle m_F = 1/2 | \mathbf{D} \cdot \mathbf{e}_z | m_F = 1/2 \rangle$ the $z$-component of the dipole moment of the transition ($F = 1/2, m_F = 1/2 \rightarrow F = 1/2, m_F = 1/2$), $\Delta_0 = \omega_0 - \omega$ is
Metamaterials such as negative index samples [4] or photonic crystals [5] are based on different strategies. In the latter, the propagation of light is modified by the existence of photonic band gaps due to periodic spatial variations of the optical index, in the same way that periodic atomic lattices affect the conductivity of electrons in semiconductors. Such crystals are usually obtained by periodically stacking dielectric slabs with different indices. The optical properties of the resulting structures are determined once and for all. Another more versatile way of obtaining a photonic crystal is by means of Electromagnetically Induced Gratings (EIG). In this case, the periodic modulation of the optical response of the medium results from the application of a standing-wave driving field, giving rise to new original phenomena. For example, the standing-wave configuration was recognized years ago as a very promising way to achieve spatial localization [6]. Moreover, by exciting a lambda system in EIT configuration with a stationary control field one obtains a periodic modulation of the atomic absorption with sharp peaks which allows the generation of stationary light pulses [7] as well as the optical control of photonic band gaps [8]. Many groups also use this configuration to control the group velocity [9], to induce Raman gratings [10], phase gratings [11], or to implement phase gates and optical switching [12]. Multi-wave mixing can also be used to generate optical gratings to induce controllable dipole-mode solitons in three-level atomic systems [13] and edge-solitons in photonic graphene [14].

In the present work, we focus on an atomic medium with the so-called duplicated two-level system (DTLS) configuration, which is often used to study electron spin coherence effects [15]. When excited by a strong control and a weak probe beams, polarized orthogonally to each other, DTLS exhibits efficient quantum interferences between the absorption and stimulated emission paths. We pointed out the great potentialities of DTLS media for controlling the propagation of the probe field in previous works. We experimentally demonstrated the coherent control of the medium gain in the femtosecond regime [16]. We also theoretically predicted the existence of Zeeman-coherent-oscillation-assisted deceleration of the probe in a non-collinear configuration [17] and phase control of the effective susceptibility in a collinear configuration [18]. Other authors proposed to take advantage of DTLS media to control optical bistability [19], group velocity [20], and to implement quantum memory or optical switching [21]. It was also suggested to use orthogonal spatial configuration for the control field to generate spatial diffraction within the multiphoton resonance condition [22] or beyond
<figure><img src="image_3.png" /><figcaption>Figure 2. (a) The duplicated two-level system (DTLS) and (b) field configurations.</figcaption></figure>

the detuning of the fields with respect to the DTLS natural frequency, $\omega_0$, while $\epsilon_{j=\pi,\sigma}(y) = \epsilon_j^+(y) + \epsilon_j^-(y) e^{-2iky}$ are the envelopes of the control and probe fields, respectively. The populations $\rho_{cc}$ and $\rho_{dd}$ relax with the rate $\Gamma$, the coherences $\rho_{ac}$, $\rho_{ad}$, $\rho_{bc}$ and $\rho_{bd}$ with the rate $\Gamma_d$ and the excited Zeeman coherence with the rate $\Gamma_{ze}$. In the absence of non-radiative homogeneous dephasing processes, $(\Gamma_{ze}, \Gamma_d)$ reduce to $(\Gamma, \Gamma_2)$. The modification of the probe field is determined by the behaviour of the coherence $\rho_\sigma = \rho_{cb} + \rho_{da}$ which radiates the $\sigma$-polarized light and whose evolution is ruled by the equation [14–16]

$$ \partial_t \rho_\sigma = i\Omega_\pi (\rho_Z - \rho_Z^*) + i\Omega_\sigma e^{-i\varphi} (n_g - n_e) - (i\Delta_\pi + \Gamma_d) \rho_\sigma $$

where $\rho_Z = \rho_{ab} + \rho_{cd}$. The coherence $\rho_\sigma$ therefore builds up through two competing phenomena, i.e. the diffraction of the pump by the Zeeman coherences (first term) and the absorption of the probe by the population (second term). In the stationary regime, the excited state coherence is destroyed by relaxations, $\rho_{cd} = 0$, and only the ground state Zeeman coherence $\rho_{ab}$ contributes to $\rho_\sigma$. In the stationary regime, one finds $\rho_\sigma = 2\rho_{da} = 2\rho_{cb} = \left(\frac{\Omega_\sigma e^{-i\varphi}}{\Omega_\pi}\right)^* \rho_\pi$ with

$$ \rho_\pi = \frac{(\Omega_\pi^2 + \Omega_\sigma^2 e^{-2i\varphi}) (i\Gamma_d + \Delta_0) \Omega_\pi^*}{4\Gamma_d \Gamma^{-1} |\Omega_\pi^2 + \Omega_\sigma^2 e^{-2i\varphi}|^2 + (|\Omega_\pi|^2 + |\Omega_\sigma|^2) (\Gamma_d^2 + \Delta_0^2)} \quad (3) $$

$$ \rho_\sigma = \frac{(\Omega_\pi^2 + \Omega_\sigma^2 e^{-2i\varphi}) (i\Gamma_d + \Delta_0) \Omega_\sigma^* e^{i\varphi}}{4\Gamma_d \Gamma^{-1} |\Omega_\pi^2 + \Omega_\sigma^2 e^{-2i\varphi}|^2 + (|\Omega_\pi|^2 + |\Omega_\sigma|^2) (\Gamma_d^2 + \Delta_0^2)} \quad (4) $$

Throughout this work, we will assume that, along the sample, the control is much more intense than the probe, i.e.

$$ |\Omega_\pi(y)| \gg |\Omega_\sigma(y)| \quad (5) $$

therefore, from Eqs.(3,4), we get

$$ \rho_\pi = \frac{i\Gamma_d + \Delta_0}{4\Gamma_d \Gamma^{-1} |\Omega_\pi|^2 + \Gamma_d^2 + \Delta_0^2} \Omega_\pi \quad (6) $$

$$ \rho_\sigma = \frac{i\Gamma_d + \Delta_0}{4\Gamma_d \Gamma^{-1} |\Omega_\pi|^2 + \Gamma_d^2 + \Delta_0^2} \frac{\Omega_\pi^2}{|\Omega_\pi|^2} \Omega_\sigma^* e^{i\varphi} \quad (7) $$

We now define $\alpha_0 = Nd^2\omega_0/(2\hbar\epsilon_0\Gamma_d)$ the field absorption coefficient at resonance, where N denotes the number of atoms per unit volume in the medium, and we introduce $\chi_{lin}$ and $\chi_{sat}$, the linear and saturated susceptibilities respectively, and $\chi_{eff}$ – whose signification will be given later,

$$ \chi_{lin} = \frac{2\alpha_0\Gamma_d/k}{-i\Gamma_d + \Delta_0} \quad (8) $$

$$ \chi_{sat} = \chi_{lin}S \quad (9) $$

$$ \chi_{eff} = \chi_{sat}e^{2i\varphi} \quad (10) $$
[23]. The phase control of spatial interference of resonance fluorescence [24], transmission and reflection factors in a dielectric slab [25], and the possibility of spatial localization of atoms [26] are among the promising related results.

In the present article, we consider a DTLS medium subjected to collinear control and probe fields. The control field is now a standing-wave inducing an EIG in Zeeman coherences. The overlap between the control and probe beams ensures efficient interaction and energy exchange between the field components along the sample. In contrast to transverse gratings which only diffract a weak part of the probe field, the present DTLS medium leads to significant reflection and transmission coefficients which can even be greater than one. Another significant advantage over non-degenerate $\Lambda$ [27] or other multilevel [28, 29] configurations, is that the implementation of the DTLS scheme requires only one laser source and phase matching is automatically satisfied for both reflected and transmitted beams. However, propagation effects affect both the control and the probe fields in DTLS. The aim of the present study is to determine the exact quantitative conditions under which proper phase-control of the reflection and transmission factors can be achieved in this system. In addition, we study the influence of the interaction parameters – control field intensity, detuning, optical depth and phase shift between control and probe – on the intensity and phase of the reflected and transmitted probe light.

## 2. Theoretical model

### 2.1. The system

We consider the experimentally feasible configuration shown in Fig. 1. An atomic sample of length $L$ is subjected to two fields of equal frequency $\omega$, propagating along the $y$ axis with orthogonal linear polarizations. The first – so-called control – field is $\pi$-polarized, i.e. its polarization is along the arbitrarily chosen quantization $z$-axis. It is divided into two parts: one part enters the sample from the left side (hereafter called the input), and propagates in the direction of increasing $y$’s, while the other part is seeded back into the sample from the other side (hereafter called the output), and propagates in the direction of decreasing $y$’s. The total control electric field is expressed as

$$ \mathbf{E}_{\pi}(y, t) = \mathbf{e}_z \left[ \varepsilon_{\pi}^{+}(y) e^{iky} + \varepsilon_{\pi}^{-}(y) e^{-iky} \right] e^{-i\omega t} + \text{c.c.} \quad (1) $$
with the saturation parameter $S(y)$ defined as

$$S(y) = \frac{1}{1 + F_0 |\Omega_{\pi}(y)|^2} \qquad (11)$$

and

$$F_0 = \frac{4\Gamma_d\Gamma^{-1}}{\Gamma_d^2 + \Delta_0^2} \qquad (12)$$

Using Eqs.(6-12), we finally get

$$\rho_{\pi} = \left( \frac{k}{2\alpha_0 \Gamma_d} \right) \chi_{sat} \Omega_{\pi} \qquad (13)$$

$$\rho_{\sigma} = \left( \frac{k}{2\alpha_0 \Gamma_d} \right) \chi_{eff} \frac{\Omega_{\pi}^2}{|\Omega_{\pi}|^2} \Omega_{\sigma}^* e^{-i\varphi} \qquad (14)$$

### C. Propagation equations

The control and probe fields obey the following one-dimensional Maxwell propagation equations

$$\begin{aligned} \frac{\partial^2}{\partial y^2} (\Omega_{\pi} e^{iky}) + k^2 (\Omega_{\pi} e^{iky}) &= -2k\alpha_0\Gamma_d (\rho_{\pi} e^{iky}) \\ \frac{\partial^2}{\partial y^2} (\Omega_{\sigma} e^{iky} e^{-i\varphi}) + k^2 (\Omega_{\sigma} e^{iky} e^{-i\varphi}) &= -2k\alpha_0\Gamma_d (\rho_{\sigma} e^{iky}) \end{aligned}$$

Using Eqs.(1,2), we explicitly decompose the Rabi frequencies into their forward and backward contributions

$$\Omega_{j=\pi,\sigma}(y) = \Omega_j^+(y) + \Omega_j^-(y) e^{-2iky} \qquad (15)$$

with $\Omega_{j=\pi,\sigma}^{\pm} = \frac{d\epsilon_j^{\pm}}{\hbar}$, and we define the associated phases $\varphi_j^{\pm}$ by the relation $\Omega_j^{\pm} = |\Omega_j^{\pm}|e^{-i\varphi_j^{\pm}}$. We also formally expand the coherences $\rho_{j=\pi,\sigma}$ as

$$\rho_j(y) = \sum_{n=-\infty}^{+\infty} \rho_j^{(n)}(y) e^{2inky} \qquad (16)$$

When the conditions $|\frac{\partial^2}{\partial y^2} \Omega_j^{\pm}| \ll k |\frac{\partial}{\partial y} \Omega_j^{\pm}| \ll k^2 |\Omega_j^{\pm}|$ are met, the slow envelope approximation can be performed.

The envelopes $\rho_j^{(n)}(y)$ are then slowly varying at the wavelength scale. Retaining only the relevant terms $\rho_j^{(n)}(y)$ which satisfy the phase matching condition, one finally gets from Eqs.(14-16)

$$\frac{\partial \Omega_{\pi}^{+}}{\partial y} = i\alpha_{0}\Gamma_{d}\rho_{\pi}^{(0)} \qquad (17)$$

$$\frac{\partial \Omega_{\pi}^{-}}{\partial y} = -i\alpha_{0}\Gamma_{d}\rho_{\pi}^{(-1)} \qquad (18)$$

$$\frac{\partial \Omega_{\sigma}^{+}}{\partial y} = i\alpha_{0}\Gamma_{d}e^{i\varphi}\rho_{\sigma}^{(0)} \qquad (19)$$

$$\frac{\partial \Omega_{\sigma}^{-}}{\partial y} = -i\alpha_{0}\Gamma_{d}e^{i\varphi}\rho_{\sigma}^{(-1)} \qquad (20)$$

In the next section, we recall important results on the simpler specific case when control and probe beams are both forward-propagating. Then, in Sec. IV, we shall address the more complex configuration involving a stationary control field and investigate in detail the observed phase-controlled probe reflection and transmission through analysing the

$$\text{behaviour of associated coefficients } R = \left| \frac{\Omega_{\sigma}^{-}(y=0)}{\Omega_{\sigma}^{+}(y=0)} \right|^{2} \text{ and } T = \left| \frac{\Omega_{\sigma}^{+}(y=L)}{\Omega_{\sigma}^{+}(y=0)} \right|^{2}.$$
<figure><img src="image_2.png" /><figcaption>Figure 1: (a) Experimentally feasible configuration. The control field is split into two parts that give the forward and backward components (Rabi envelopes $\Omega_{\pi}^{\pm}$, respectively). The probe field is sent to the input of the sample in the forward direction (Rabi envelope $\Omega_{\sigma}^{+}$) and the backward component (Rabi envelope $\Omega_{\sigma}^{-}$) is generated from the output of the sample. Here, (M) stands for mirror, (BS) for beam splitter, (GTP) for Glan-Taylor Polarizer. (b) Field configurations.</figcaption></figure>

For simplicity, we assume that the amplitude $\epsilon_{\pi}^{+}(y=0)$ is real. Note that the $\pm$ subscripts in $\epsilon_{\pi}^{\pm}$ refer to the direction of propagation along the y-axis, either forward (+) or backward (-). The second - called the probe - field is $\sigma$-polarized, i.e. here it is linearly-polarized along the x-axis and its electric field is given by

$$ \mathbf{E}_{\sigma}(y,t) = \mathbf{e}_x \left[ \epsilon_{\sigma}^{+}(y) e^{iky} + \epsilon_{\sigma}^{-}(y) e^{-iky} \right] e^{-i\varphi} e^{-i\omega t} + \text{c.c.} \quad (2) $$

In this expression, $\varphi$ is defined as the phase shift between the probe and control fields. Note that even if the probe field is originally injected through the input and propagates in the forward direction, a backward component $\epsilon_{\sigma}^{-}(y)$ is built up because the atomic sample radiates in both directions. Furthermore, the boundary conditions for the probe at the output and input write $\epsilon_{\sigma}^{-}(y=L) = 0$ and $\epsilon_{\sigma}^{+}(y=0) = E_0$, where $E_0$ is real. Note that the $\pm$ subscripts in $\epsilon_{\sigma}^{\pm}$ must not be confused with the sense of circular polarization.

We assume that the fields are coupled to a $F = \frac{1}{2} \rightarrow F = \frac{1}{2}$ atomic line of the medium (e.g. $^2S_{1/2}F = \frac{1}{2} \rightarrow ^2P_{1/2}F = \frac{1}{2}$ transition of $^6$Li at 671 nm) which is described by a duplicated two-level system (Fig. 2). The $\pi$-polarized control field $\mathbf{E}_{\pi}$ (resp. the $\sigma$-polarized probe field $\mathbf{E}_{\sigma}$) couples to the $\Delta m_F = 0$ (resp. $\Delta m_F = \pm 1$) paths. Note that light hole transitions in quantum wells may also be considered [15]. In the following, we determine the reflection and transmission coefficients of the medium for a weak probe in the presence of a much stronger control field.
### III. SITUATION WITH NO BACKWARD CONTROL FIELD

We first address the case of forward-propagating fields inside the medium, i.e. $\epsilon_j^-(y) = 0$, $\Omega_j = \Omega_j^+$ and $\rho_j = \rho_j^{(0)}$ for $j = \pi, \sigma$. This situation has been studied in detail in [16], we shall therefore only briefly summarize the main effects here.

Using the relation $\frac{\Omega_{\pi}^2}{|\Omega_{\pi}|^2} \Omega_{\sigma}^* = e^{2i\Delta\varphi_{\sigma\pi}^+} \Omega_{\sigma}^+$ with $\Delta\varphi_{\sigma\pi}^+ = \varphi_{\sigma}^+ - \varphi_{\pi}^+$, Eqs.(13, 14, 17-20) lead to

$$ \frac{\partial \Omega_{\pi}^{+}}{\partial y} = i \frac{k}{2} \chi_{sat} \Omega_{\pi}^{+} \qquad (21) $$

$$ \frac{\partial (\Omega_{\sigma}^{+} e^{-i\varphi})}{\partial y} = i \frac{k}{2} \chi_{eff} e^{2i\Delta\varphi_{\sigma\pi}^{+}} (\Omega_{\sigma}^{+} e^{-i\varphi}) \qquad (22) $$

Provided that the dephasing between the control and probe fields, denoted by $\Delta\varphi_{\sigma\pi}^+$, does not substantially grow along propagation, $\chi_{eff}$ is seen to play the role of an effective susceptibility for the probe in Eq.(22). From Eqs.(21,22), one derives the propagation equation

$$ \frac{\partial \Delta\varphi_{\sigma\pi}^{+}}{\partial y} = k \sin(\Delta\varphi_{\sigma\pi}^{+} - \varphi) [\chi'_{sat} \sin(\Delta\varphi_{\sigma\pi}^{+} - \varphi) - \chi''_{sat} \cos(\Delta\varphi_{\sigma\pi}^{+} - \varphi)] \qquad (23) $$

where $\chi'_{sat}$ and $\chi''_{sat}$ denote the real and imaginary parts of the saturated susceptibility, respectively. We introduce the phase of the linear susceptibility, $\varphi_L$, defined by $\chi_{lin} = |\chi_{lin}| e^{i\varphi_L}$, which verifies

$$ \tan \varphi_L = \frac{\Gamma_d}{\Delta_0} \qquad (24) $$

The integration Eq.(23) leads to $\tan \Delta\varphi_{\sigma\pi}^+ = \frac{\tan \varphi_L}{(\frac{\tan \varphi_L}{\tan \varphi}+1)e^{-2\alpha_0 y \sin^2 \varphi_L - 1}}$. When $|\Omega_{\pi}(y=0)| \ll \sqrt{\Gamma_d \Gamma}$ and $\alpha_0 L \sin^2 \varphi_L \ll 1$, then $|\Omega_{\pi}^{+}(y)| \approx |\Omega_{\pi}^{+}(y=0)|$, $S \approx 1$ and $\Delta\varphi_{\sigma\pi}^+ \approx 0$, see [16]. The sample therefore becomes a linear medium for the probe field whose susceptibility, $\chi_{eff}$, is related to the true linear susceptibility through $\chi_{eff} = \chi_{lin} e^{2i\varphi}$. The latter relation proves the existence of the phase control of the linear response of the medium, whose physical interpretation was described in [16]. It results from interference effects between the quantum paths which contribute to the coherence $\rho_{\sigma}$ responsible for the $\sigma$-polarized radiated field.

If the control field intensity is increased, saturation effects occur and the validity condition of parametric approximation is less restrictive. From Eq.(21), one gets

$$ \frac{\partial |\Omega_{\pi}^{+}|}{\partial y} = -\alpha_0 S \sin^2 \varphi_L |\Omega_{\pi}^{+}| $$

and provided that $\alpha_0 L S (y=0) \sin^2 \varphi_L \ll 1$, control field intensity remains unaffected along propagation. If the optical depth is increased so that the inequality $\alpha_0 L S (y=0) \sin^2 \varphi_L \ll 1$ no longer holds, the modification of the control field cannot be neglected and additional dephasing has to be taken into account in the propagation equation of the probe. New features appear and are detailed in [16].

### IV. SITUATION OF A STATIONARY CONTROL FIELD

In this section, we investigate the effects which appear when the intensity of the control field is not uniform but spatially modulated because of the backward component. In this situation, the probe field experiences both reflexion and transmission. The impact of the modification of the optical response of the sample and the eventuality of phase control can be evaluated by calculating the transmission and reflection coefficients for the probe field.

#### A. Propagation equations in the slow envelope approximation

In the slow envelope approximation, the set of Eqs.(17-20) can be rewritten by extracting an analytical expression for the components $\rho_j^{(0)}(y)$ and $\rho_j^{(-1)}(y)$.

We introduce the parameter $r(y)$ and phase $\varphi_r$ defined by $\Omega_{\pi}^{-}(y) = r(y)\Omega_{\pi}^{+}(y)$ and $r = |r| e^{-i\varphi_r}$, so that $\varphi_r = \varphi_{\pi}^{-} - \varphi_{\pi}^{+}$. When the control field is unaffected by the propagation, $r(y)$ is constant. Note that, for Eq.(5) to hold,
<figure><img src="image_3.png" /><figcaption>Figure 2: The duplicated two-level system (DTLS).</figcaption></figure>

## 2.2. Atomic polarization

In the rotating frame defined by the unitary transformation $U(t) = e^{-i\omega t(|c\rangle\langle c|+|d\rangle\langle d|)}$ the density matrix $\rho$ of an atom in the medium obeys the master equation $i\hbar\partial_t\rho = [H, \rho] + \text{relaxations}$ where the Hamiltonian takes the following form

$$H = \hbar \begin{pmatrix} 0 & 0 & -\Omega_{\pi}^{*} & -\Omega_{\sigma}^{*} e^{i\varphi} \\ 0 & 0 & -\Omega_{\sigma}^{*} e^{i\varphi} & \Omega_{\pi}^{*} \\ -\Omega_{\pi} & -\Omega_{\sigma} e^{-i\varphi} & \Delta_{0} & 0 \\ -\Omega_{\sigma} e^{-i\varphi} & \Omega_{\pi} & 0 & \Delta_{0} \end{pmatrix}$$

in the rotating wave approximation. Here $\Omega_{\pi} = \frac{d\epsilon_{\pi}}{\hbar}$ and $\Omega_{\sigma} = -\frac{d\epsilon_{\sigma}}{\hbar}$ denote the Rabi frequencies for the control and probe fields, with $d = \langle m_F = \frac{1}{2} | \mathbf{D} \cdot \mathbf{e}_z | m_F = \frac{1}{2} \rangle$ the z-component of the dipole moment of the transition ($F = \frac{1}{2}, m_F = \frac{1}{2}$) $\rightarrow (F = \frac{1}{2}, m_F = \frac{1}{2})$, $\Delta_0 = \omega_0 - \omega$ is the detuning of the fields with respect to the DTLS natural frequency, $\omega_0$, while $\epsilon_{j=\pi,\sigma}(y) = \epsilon_j^+(y) + \epsilon_j^-(y) e^{-2iky}$ are the envelopes of the control and probe fields, respectively. The relaxation is mainly due to radiative decay and residual collisions. Collisions affect Zeeman excited state and ground-excited state coherences in a different manner since the latter are sensitive to spin depolarizing collisions which leads to different rates. Therefore, in general, the populations $\rho_{cc}$ and $\rho_{dd}$ relax with the rate $\Gamma$, while the coherences $\rho_{ac}$, $\rho_{ad}$, $\rho_{bc}$ and $\rho_{bd}$ decay with the rate $\Gamma_d$ and the excited Zeeman coherence with the rate $\Gamma_{ze}$. In the absence of non-radiative homogeneous dephasing processes, however, $(\Gamma_{ze}, \Gamma_d)$ reduce to $(\Gamma, \frac{\Gamma}{2})$. Finally, the Doppler effect was implicitly neglected in the above model, which requires the atoms to be (sufficiently) cold.
the control field should not vanish anywhere, including the spatial nodes, therefore r should be different from 1. From Eqs.(13,14,16), we see that the analytical expressions of $\rho_j^{(0)}(y)$ and $\rho_j^{(-1)}(y)$ can be evaluated as soon as the series expansion of the saturation parameter $S(y)$ and $\frac{1}{|\Omega_\pi(y)|^2}$ are determined. The saturation parameter $S$ given by Eq.(11) may be expressed as

$$S(y) = \frac{b(y)}{1 + a(y) \cos[2ky + \varphi_r(y)]} \qquad (25)$$

with $a(y) = \frac{2F_0|\Omega_\pi^+(y)|^2|r(y)|}{1+F_0|\Omega_\pi^+(y)|^2(1+|r(y)|^2)}$ and $b(y) = \frac{1}{1+F_0|\Omega_\pi^+(y)|^2(1+|r(y)|^2)}$. It can also be formally expanded as

$$S(y) = \sum_{n=-\infty}^{+\infty} c_n(y) e^{2inky} \qquad (26)$$

where the $c_n$ coefficients are slowly varying at the spatial wavelength scale $2\pi/k$ and can be approximated as $c_n(y) \approx \frac{k}{\pi} \int_y^{y+\pi/k} S(y') e^{-2inky'} dy'$. For $n \ge 0$, using Eq.(25) and [25], we get $c_n(y) \approx c_0\eta^n e^{in\varphi_r(y)}$, with $c_0 = \frac{b(y)}{\sqrt{1-a^2(y)}}$ and $\eta = \frac{\sqrt{1-a^2(y)-1}}{a(y)}$. Moreover, for $n \le 0$, the reality of parameter $S$ implies that $c_n = c_{-n}^*$.

On the other hand, using Eq.(15), one may put the parameter $\frac{1}{|\Omega_\pi(y)|^2}$ under the form

$$\frac{1}{|\Omega_\pi(y)|^2} = \frac{b'(y)}{1 + a'(y) \cos[2ky + \varphi_r(y)]} \qquad (27)$$

with $a'(y) = \frac{2|r(y)|}{|\Omega_\pi^+(y)|^2(1+|r(y)|^2)}$ and $b'(y) = \frac{1}{|\Omega_\pi^+(y)|^2(1+|r(y)|^2)}$. It can also be formally expanded as $\frac{1}{|\Omega_\pi(y)|^2} = \sum_{n=-\infty}^{+\infty} c'_n(y) e^{2inky}$ where the $c'_n$ coefficients are slowly varying at the wavelength scale and can be expressed as $c'_n(y) \approx \frac{k}{\pi} \int_y^{y+\pi/k} \frac{1}{|\Omega_\pi(y')|^2} e^{-2inky'} dy'$. Using [25], we get $c'_{0\ge 0}(y) \approx c'_0\eta'^n e^{in\varphi_r(y)}$ with $c'_0 = \frac{b'(y)}{\sqrt{1-a'^2(y)}}$ and $\eta' = \frac{\sqrt{1-a'^2(y)-1}}{a'(y)}$. Moreover, the reality of $\frac{1}{|\Omega_\pi|^2}$ implies that $c'_{n\le 0} = (c'_{-n})^*$.

Using Eqs.(13,14,16,25,26,27), we can finally express the components of the coherences which appear in the propagation equations for the control fields

$$\rho_\pi^{(0)}(y) = \frac{i}{\Gamma_d} [\bar{c}_0(y) \Omega_\pi^+(y) + \bar{c}_1(y) \Omega_\pi^-(y)] \qquad (28)$$

$$\rho_\pi^{(-1)}(y) = \frac{i}{\Gamma_d} [\bar{c}_{-1}(y) \Omega_\pi^+(y) + \bar{c}_0(y) \Omega_\pi^-(y)] \qquad (29)$$

with $\bar{c}_i(y) = (\frac{\Gamma_d}{\Gamma_d+i\Delta_0}) c_i(y)$, and in the propagation equations for the probe fields

$$\rho_\sigma^{(0)}(y) = \frac{i}{\Gamma_d} e^{i\varphi} [c_+(y) \Omega_\sigma^{+*}(y) + c_{+-}(y) \Omega_\sigma^{-*}(y)] \qquad (30)$$

$$\rho_\sigma^{(-1)}(y) = \frac{i}{\Gamma_d} e^{i\varphi} [c_{+-}(y) \Omega_\sigma^{+*}(y) + c_{-}(y) \Omega_\sigma^{-*}(y)] \qquad (31)$$

with

$$c_+ = \beta [1 + 2\gamma|r| + \delta|r|^2] \qquad (32)$$

$$c_{+-} = \beta r \left[ 2 + \frac{\gamma}{|r|} + \gamma |r| \right] \qquad (33)$$

$$c_{-} = \beta r^2 \left[ 1 + \frac{2\gamma}{|r|} + \frac{\delta}{|r|^2} \right] \qquad (34)$$

and

$$\beta = \frac{c_0 e^{-2i\varphi_\pi^+}}{(1 - \eta\eta') |1 - |r|^2|} \left( \frac{\Gamma_d}{\Gamma_d + i\Delta_0} \right) \qquad (35)$$

$$\gamma = \eta + \eta' \qquad (34)$$

$$\delta = \eta^2 + (\eta')^2 + \eta\eta' - (\eta\eta')^2 \qquad (35)$$
As shown in previous works [16, 17, 18], the propagation of the control field is governed by the coherence $\rho_\pi = \rho_{ca} - \rho_{db}$, while the propagation of the probe field is governed by the coherence $\rho_\sigma = \rho_{cb} + \rho_{da}$ which radiates the $\sigma$-polarized light and which obeys the equation

$$ \partial_t \rho_\sigma = i\Omega_\pi (\rho_Z - \rho_Z^*) + i\Omega_\sigma e^{-i\varphi} (n_g - n_e) - (i\Delta_\pi + \Gamma_d) \rho_\sigma $$

where $\rho_Z = \rho_{ab} + \rho_{cd}$. The coherence $\rho_\sigma$ therefore is built up through two competing phenomena, i.e. the diffraction of the pump by the Zeeman coherences (first term) and the absorption of the probe by the population (second term). In this article, we are interested in the steady state optical response, in which case the excited state coherence is destroyed by relaxations, $\rho_{cd} = 0$, and only the ground state Zeeman coherence $\rho_{ab}$ contributes to $\rho_\sigma$. In the stationary regime, one finds $\rho_\sigma = 2\rho_{da} = 2\rho_{cb} = \left(\frac{\Omega_\sigma e^{-i\varphi}}{\Omega_\pi}\right)^* \rho_\pi$ with

$$ \rho_\pi = \frac{(\Omega_\pi^2 + \Omega_\sigma^2 e^{-2i\varphi}) (i\Gamma_d + \Delta_0) \Omega_\pi^*}{4\Gamma_d \Gamma^{-1} |\Omega_\pi^2 + \Omega_\sigma^2 e^{-2i\varphi}|^2 + (|\Omega_\pi|^2 + |\Omega_\sigma|^2) (\Gamma_d^2 + \Delta_0^2)} \quad (3) $$

$$ \rho_\sigma = \frac{(\Omega_\pi^2 + \Omega_\sigma^2 e^{-2i\varphi}) (i\Gamma_d + \Delta_0) \Omega_\sigma^* e^{i\varphi}}{4\Gamma_d \Gamma^{-1} |\Omega_\pi^2 + \Omega_\sigma^2 e^{-2i\varphi}|^2 + (|\Omega_\pi|^2 + |\Omega_\sigma|^2) (\Gamma_d^2 + \Delta_0^2)} \quad (4) $$

Throughout this work, we will assume that, along the sample, the control is much more intense than the probe, i.e.

$$ |\Omega_\pi(y)| \gg |\Omega_\sigma(y)| \quad (5) $$

therefore, from Eqs. (3,4), we get

$$ \rho_\pi = \frac{i\Gamma_d + \Delta_0}{4\Gamma_d \Gamma^{-1} |\Omega_\pi|^2 + \Gamma_d^2 + \Delta_0^2} \Omega_\pi \quad (6) $$

$$ \rho_\sigma = \frac{i\Gamma_d + \Delta_0}{4\Gamma_d \Gamma^{-1} |\Omega_\pi|^2 + \Gamma_d^2 + \Delta_0^2} \frac{\Omega_\pi^2}{|\Omega_\pi|^2} \Omega_\sigma^* e^{i\varphi} \quad (7) $$

Note that both coherences $\rho_\sigma$ and $\rho_\pi$ vanish when $|\Omega_\pi| \gg \sqrt{\Gamma_d \Gamma}$: the saturation of the medium by the pump makes it transparent for both $\sigma$ and $\pi$ polarized fields. This leads to effects such as incoherent fluorescence. In the present work, however, we assume that this regime is not reached. In other words, our treatment is non-perturbative with respect to the control field but the transparency regime is never reached.
Injecting Eqs.(28-31) into Eqs.(17,18) and Eqs.(19,20), we get

$$ \frac{\partial \Omega_{\pi}^{+}}{\partial (y/L)}(y) = -\alpha_{0}L [\bar{c}_{0}(y) \Omega_{\pi}^{+}(y) + \bar{c}_{1}(y) \Omega_{\pi}^{-}(y)] \quad (36) $$

$$ \frac{\partial \Omega_{\pi}^{-}}{\partial (y/L)}(y) = \alpha_{0}L [\bar{c}_{-1}(y) \Omega_{\pi}^{+}(y) + \bar{c}_{0}(y) \Omega_{\pi}^{-}(y)] \quad (37) $$

and

$$ \frac{\partial \Omega_{\sigma}^{+}}{\partial (y/L)}(y) = -\alpha_{0}L e^{2i\varphi} [c_{+}(y) \Omega_{\sigma}^{+*}(y) + c_{-+}(y) \Omega_{\sigma}^{-*}(y)] \quad (38) $$

$$ \frac{\partial \Omega_{\sigma}^{-}}{\partial (y/L)}(y) = \alpha_{0}L e^{2i\varphi} [c_{+-}(y) \Omega_{\sigma}^{+*}(y) + c_{-}(y) \Omega_{\sigma}^{-*}(y)] \quad (39) $$

The forward and backward components of both the probe and control fields are now coupled through the coefficients $(\bar{c}_{-1}, \bar{c}_{1})$ for the control and $c_{+-}$ coefficient for the probe. In particular, Eqs.(38,39) clearly show that the probe field propagation may be controlled by the phase. This property is further investigated in Sec. V, through numerically solving the above coupled equations.

## B. Limit of small optical depths

The probe field propagation depends on the control field intensity and the optical depth. For small optical depths, i.e. $|\alpha_0 L \bar{c}_{j=0,\pm 1}| \ll 1$, the control field is weakly affected by propagation and the envelopes can be considered uniform, i.e. $\Omega_{\pi}^{+}(y) \approx \Omega_{\pi}^{+}(0)$, $\Omega_{\pi}^{-}(y) \approx \Omega_{\pi}^{-}(L)$ and $\varphi_{\pi}^{\pm} \approx 0$. The coefficients $c_+$, $c_-$, $c_+$ are thus constant along the sample and the phases $\varphi_{\pi}^{+}$ and $\varphi_r$ are neglected. The system of Eqs.(38,39) turns into a set of differential equations with constant coefficients. Moreover, for optical depths such as $|\alpha_0 L c_{j=+,-,+,-}| \ll 1$, one may retain the most significant terms in the integration of the system Eqs.(38,39) which leads to

$$ \begin{aligned} \Omega_{\sigma}^{+}(y) - \Omega_{\sigma}^{+}(0) &\approx -\alpha_{0}y e^{2i\varphi} [c_{+} \Omega_{\sigma}^{+*}(0) + c_{-+} \Omega_{\sigma}^{-*}(0)] \\ \Omega_{\sigma}^{-}(y) - \Omega_{\sigma}^{-}(L) &\approx \alpha_{0}(y-L) e^{2i\varphi} [c_{+-} \Omega_{\sigma}^{+*}(L) + c_{-} \Omega_{\sigma}^{-*}(L)] \end{aligned} $$

Taking into account the boundary condition $\{\Omega_{\sigma}^{-}(L) = 0; \Omega_{\sigma}^{+*}(0) = \Omega_{\sigma}^{+}(0)\}$, one readily obtains

$$ \Omega_{\sigma}^{+}(y) \approx \Omega_{\sigma}^{+}(0) [1 - \alpha_{0}y e^{2i\varphi} c_{+}] \quad (40) $$

$$ \Omega_{\sigma}^{-}(y) \approx \Omega_{\sigma}^{+}(0) \alpha_{0}(y-L) e^{2i\varphi} [c_{+-} - \alpha_{0}(y-L) e^{-2i\varphi} c_{+-} c_{+}^{*}] \quad (41) $$

From Eqs.(32,33-37), we get $\arg[c_+(0)] = \varphi_L - \pi/2$, where $\varphi_L$ is defined in Eq.(24). Transmission and reflection coefficients can then be approximated as

$$ T \approx 1 - 2\alpha_0 L |c_+| \sin(2\varphi + \varphi_L) \quad (42) $$

$$ R \approx |\alpha_0 L c_{+-}|^2 [1 - 2\alpha_0 L |c_+| \sin(2\varphi + \varphi_L)] \quad (43) $$

These expressions show the existence of phase-controlled reflection and transmission in the limit of small optical depths. The coefficients can be modified by adjusting a versatile and fine tuning experimental parameter, $\varphi$, the additional shift $\varphi_L$ being determined by the detuning $\Delta_0$. For $0 \le 2\varphi + \varphi_L \le \pi$, the transmission factor $T$ is less or equal to 1, while it becomes larger than 1 for $\pi \le 2\varphi + \varphi_L \le 2\pi$. This is in agreement with the absorptive or amplificative nature of the sample as described by its effective linear susceptibility, given by Eq.(10). The reflection factor is associated with the backward component of the control field that is generated through the coupling with the forward part of the probe. Thus, the reflection factor depends on the second (or higher) power of the optical depth and is small with respect to the transmission factor. As for the transmission factor, however, the contrast is linear with the optical depth and can be varied by modifying the intensities of the control field components.

## V. RESULTS AND DISCUSSION

In this section, we present both the numerical results obtained through the numerical simulation of Eqs.(36-39) and analytical approximation, Eqs.(42,43), and discuss their physical interpretation.
We now define $\alpha_0 = \frac{Nd^2\omega_0}{2ch\epsilon_0\Gamma_d}$ the field absorption coefficient at resonance, where N denotes the number of atoms per unit volume in the medium, and we introduce $\chi_{lin}$ and $\chi_{sat}$, the linear and saturated susceptibilities respectively, and $\chi_{eff}$ – the meaning of which will be given later,

$$ \chi_{lin} = \frac{\frac{2\alpha_0\Gamma_d}{k}}{-i\Gamma_d + \Delta_0} \qquad (8) $$

$$ \chi_{sat} = \chi_{lin}S \qquad (9) $$

$$ \chi_{eff} = \chi_{sat}e^{2i\varphi} \qquad (10) $$

with the saturation parameter $S(y)$ defined as

$$ S(y) = \frac{1}{1 + F_0 |\Omega_{\pi}(y)|^2} \qquad (11) $$

and

$$ F_0 = \frac{4\Gamma_d\Gamma^{-1}}{\Gamma_d^2 + \Delta_0^2} \qquad (12) $$

Using Eqs. (6-12), we finally get

$$ \rho_{\pi} = \left( \frac{k}{2\alpha_{0}\Gamma_{d}} \right) \chi_{sat}\Omega_{\pi} \qquad (13) $$

$$ \rho_{\sigma} = \left( \frac{k}{2\alpha_{0}\Gamma_{d}} \right) \chi_{eff} \frac{\Omega_{\pi}^{2}}{|\Omega_{\pi}|^{2}} \Omega_{\sigma}^{*} e^{-i\varphi} \qquad (14) $$

## 2.3. Propagation equations

The control and probe fields obey the following one-dimensional Maxwell propagation equations

$$ \frac{\partial^2}{\partial y^2} (\Omega_{\pi} e^{iky}) + k^2 (\Omega_{\pi} e^{iky}) = -2k\alpha_0\Gamma_d (\rho_{\pi} e^{iky}) $$

$$ \frac{\partial^2}{\partial y^2} (\Omega_{\sigma} e^{iky} e^{-i\varphi}) + k^2 (\Omega_{\sigma} e^{iky} e^{-i\varphi}) = -2k\alpha_0\Gamma_d (\rho_{\sigma} e^{iky}) $$

Using Eqs. (1,2), we explicitly decompose the Rabi frequencies into their forward and backward contributions

$$ \Omega_{j=\pi,\sigma}(y) = \Omega_j^+(y) + \Omega_j^-(y) e^{-2iky} \qquad (15) $$
<figure><img src="image_4.png" /><figcaption>Figure 3. (a) Reflection and (b) Transmission factors as functions of the relative phase $\varphi$ at resonance ($\Delta_0 = 0$) for a probe field propagating in an optically thin sample (optical depth $\alpha_0 L = 0.3$) driven by a control field with Rabi amplitudes $\Omega_\pi^{+}(0)/\Gamma_d = 0.4$ and $\Omega_\pi^{-}(L)/\Gamma_d = 0.16$. Blue solid (orange dashed) line plots are obtained through numerical simulation (analytic approximation).</figcaption></figure>

In Fig.3 are plotted the variations of the (a) reflection, *R*, and (b) transmission, *T*, coefficients as functions of the relative phase $\varphi$, for resonant probe and control fields ($\Delta_0 = 0$) and small optical depth $\alpha_0 L = 0.3$. Forward/backward control field Rabi frequencies at the entrance/exit of the sample are set to $\frac{\Omega_\pi^{+}(0)}{\Gamma_d} = 0.4$ and $\frac{\Omega_\pi^{-}(L)}{\Gamma_d} = 0.16$, respectively. The curves in solid and dashed lines correspond to numerical simulations and approximate analytical expressions, Eqs.(42,43), respectively. We observe that both *R* and *T* oscillate with a period of $\pi$ as expected from the $e^{2i\varphi}$ dependence of the effective susceptibility, Eq.(10). As already pointed out in Subsec. IV B, *T* may be larger or smaller than 1, depending whether the sample is amplificative or absorptive. We note that *T* is much larger than *R*: the transmitted field indeed results from the interference of the incident and radiated probe fields, while, in contrast, the reflected field results from the backward radiated field only and is thus proportional to the small optical depth. Moreover, we observe discrepancies between the numerical simulations and the simple perturbative analytical model.

In Fig. 4 are plotted (a) the (normalized) backward and forward-propagating control field intensities $\left| \frac{\Omega_\pi^+(y)}{\Gamma_d} \right|^2$ and (b) phases $\varphi_\pi^\pm$, respectively, as functions of the (normalized) position $y/L$, for the set of parameters $\Delta_0 = 0$, $\alpha_0 L = 0.3$, $\Omega_\pi^{+(0)}/\Gamma_d = 0.4$, $\Omega_\pi^{-(L)}/\Gamma_d = 0.16$. Forward-propagating field intensity is attenuated by 37.5% along its propagation, from the input value 0.16 at the entrance to approximately 0.1 at the exit of the sample, while the backward-propagating control field intensity is reduced by 22%, from the input value 0.0256 at the exit to 0.02 at the entrance. The phases accumulated along propagation are small in all cases. This can be understood from Eqs.(36,37). At low optical depth, the dominant contribution due to the diagonal terms involves the $\bar{c}_0$ coefficient which is real at resonance. The control field components therefore experience pure absorption without dispersion, the accumulated phase is thus negligible. For the lowest-order solution of Eqs.(38,39), namely Eqs.(40,41), to hold, the condition $|\alpha_0 L c_j| \ll 1$ must be checked: we find the values 0.192, 0.005, 0.054 for $|\alpha_0 L c_{j=+,-,+}|$, respectively, which ensures the validity of the analytical model presented in Subsec. IV B, although the control field can be considered only roughly unaltered along propagation. This simple model captures the essential physical phenomena at work and clearly exhibits the accuracy of phase control of *R* and *T*.

Figs. 5 and 6 correspond to the same situations as considered in Figs. 3 and 4, though with a higher optical depth $\alpha_0 L = 0.6$. Both the reflection and transmission coefficients are significantly enhanced in Fig. 5 compared to Fig. 3: now, *T* oscillates between 0.48 and 2.17 – versus 0.7 and 1.45 for $\alpha_0 L = 0.3$ – and *R* between 0.017 and 0.043 – versus 0.0043 and 0.0063 for $\alpha_0 L = 0.3$. Here, although the parametric approximation is no longer valid, both *R* and *T* still depend on the phase and are substantially amplified. The fitting by the analytical model, Eqs.(42,43), is only qualitative for two reasons. First, as shown in Fig. 6, the increased optical depth leads to stronger absorption of the control field along propagation. The forward-, resp. backward-, propagating component intensity is indeed reduced by 60% (37% in the case of Fig. 4 with $\alpha_0 L = 0.3$), resp. 56% (22% in Fig. 4). The parametric approximation, $\Omega_\pi^+(y) \approx \Omega_\pi^+(0)$; $\Omega_\pi^-(y) \approx \Omega_\pi^-(L)$, is thus rough. Secondly, the coefficients $|\alpha_0 L c_{j=+,-,+}|$ are magnified approximatively by a factor two with respect to the case where $\alpha_0 L = 0.3$ (e.g. $|\alpha_0 L c_j| \approx 0.41, 0.006, 0.09$ for $j = +, -, ++$, respectively) making the low order perturbation solution less valid. Note that in this case the dispersive component of the optical response vanishes at resonance and therefore the phase remains essentially constant.

In Fig. 7 are plotted the (a) reflection and (b) transmission coefficients as functions of the relative phase $\varphi$, in the resonant case $\Delta_0 = 0$ (solid line) and the non-resonant case $\Delta_0/\Gamma_d = 2$ (dashed line), the other parameters being $\alpha_0 L = 0.3$ and $\Omega_\pi^{+(0)}/\Gamma_d = 0.4$, $\Omega_\pi^{-(L)}/\Gamma_d = 0.16$. When the detuning increases, the modulus of the effective susceptibility $\chi_{eff}$, Eq.(10), decreases and so does the global coupling of the probe with the atomic medium. *R* and *T*
with $\Omega_j^\pm = \frac{d\epsilon_j^\pm}{\hbar}$, and we define the associated phases $\varphi_j^\pm$ by the relation
$\Omega_j^\pm = |\Omega_j^\pm| e^{-i\varphi_j^\pm}$. We also formally expand the coherences $\rho_{j=\pi,\sigma}$ as

$$ \rho_j(y) = \sum_{n=-\infty}^{+\infty} \rho_j^{(n)}(y) e^{2inky} \quad (16) $$

When the conditions $|\frac{\partial^2}{\partial y^2}\Omega_j^\pm| \ll k|\frac{\partial}{\partial y}\Omega_j^\pm| \ll k^2|\Omega_j^\pm|$ are fulfilled, the slow envelope approximation can be performed. The envelopes $\rho_j^{(n)}(y)$ then vary slowly on the wavelength scale. Keeping only the relevant terms $\rho_j^{(n)}(y)$ which satisfy the phase matching condition, one finally obtains from Eqs. (14-16)

$$ \frac{\partial \Omega_{\pi}^{+}}{\partial y} = i\alpha_{0}\Gamma_{d}\rho_{\pi}^{(0)} \qquad (17) $$

$$ \frac{\partial \Omega_{\pi}^{-}}{\partial y} = -i\alpha_{0}\Gamma_{d}\rho_{\pi}^{(-1)} \qquad (18) $$

$$ \frac{\partial \Omega_{\sigma}^{+}}{\partial y} = i\alpha_{0}\Gamma_{d}e^{i\varphi}\rho_{\sigma}^{(0)} \qquad (19) $$

$$ \frac{\partial \Omega_{\sigma}^{-}}{\partial y} = -i\alpha_{0}\Gamma_{d}e^{i\varphi}\rho_{\sigma}^{(-1)} \qquad (20) $$

These equations could also be interpreted in terms of a four-wave mixing process in degenerate conditions.

In the next section, we recall important results on the simpler specific case when the control and probe beams are both forward propagating. Then, in Sec. 4, we consider the more complex configuration with a stationary control field and investigate in detail the observed phase-controlled probe reflection and transmission by analyzing the behaviour of the associated coefficients

$$ R = \left| \frac{\Omega_{\sigma}^{-}(y=0)}{\Omega_{\sigma}^{+}(y=0)} \right|^{2} \quad \text{and} \quad T = \left| \frac{\Omega_{\sigma}^{+}(y=L)}{\Omega_{\sigma}^{+}(y=0)} \right|^{2}. $$

### 3. Situation with no backward control field

We first address the case of forward-propagating fields inside the medium, i.e. $\epsilon_j^-(y) = 0$, $\Omega_j = \Omega_j^+$ and $\rho_j = \rho_j^{(0)}$ for $j = \pi, \sigma$. This situation has been studied in detail in [18], we shall therefore only briefly summarize the main effects here.
<figure><img src="image_5.png" /><figcaption>Figure 4. Control field (a) intensities $|\frac{\Omega_{\pi}^{\pm}}{\Gamma_d}|^2$ and (b) phases $\varphi_{\pi}^{\pm}$ as functions of the (normalized) propagation distance $y/L$ for $\Delta_0 = 0$, $\alpha_0 L = 0.3$, $\frac{\Omega_{\pi}^{+}(0)}{\Gamma_d} = 0.4$ and $\frac{\Omega_{\pi}^{-}(L)}{\Gamma_d} = 0.16$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5. (a) Reflection and (b) Transmission factors as functions of the relative phase $\varphi$ for $\Delta_0 = 0$, $\alpha_0 L = 0.6$, $\Omega_{\pi}^{+}(0)/\Gamma_d = 0.4$ and $\Omega_{\pi}^{-}(L)/\Gamma_d = 0.16$. Blue solid (resp. orange dashed) line plots are obtained through numerical simulation (resp. analytic approximation).</figcaption></figure>

therefore tend to decrease, which is confirmed by the results of numerical simulation. An important feature, however, is the dephasing which appears on $R$ and $T$ plots when modifying the detuning. The expression of the phase shift predicted by the analytical model, Eqs.(42,43), is $\varphi_L$ with $\tan \varphi_L = \Gamma_d/\Delta_0$. When $\Gamma_d/\Delta_0 = 0.5$, $\varphi_L \approx 0.15\pi$ which is very close to the value $0.16\pi$ obtained via numerical simulations.

In Fig. 8 are plotted the intensities of (a) the forward probe field, $|\Omega_{\sigma}^{+}(y)/\Omega_{\sigma}^{+}(0)|^2$, and (b) the backward probe field, $|\Omega_{\sigma}^{-}(y)/\Omega_{\sigma}^{+}(0)|^2$, as functions of the normalized coordinate $y/L$ for three values of the relative phase $\varphi = 0, \pi/4, \pi/2$, and

<figure><img src="image_7.png" /><figcaption>Figure 6. Control field (a) intensities $|\frac{\Omega_{\pi}^{\pm}}{\Gamma_d}|^2$ and (b) phases $\varphi_{\pi}^{\pm}$ as functions of the (normalized) propagation distance $y/L$ for $\Delta_0 = 0$, $\alpha_0 L = 0.6$, $\frac{\Omega_{\pi}^{+}(0)}{\Gamma_d} = 0.4$ and $\frac{\Omega_{\pi}^{-}(L)}{\Gamma_d} = 0.16$.</figcaption></figure>
Using the relation $\frac{\Omega_{\pi}^{2}}{|\Omega_{\pi}|^{2}}\Omega_{\sigma}^{*} = e^{2i\Delta\varphi_{\sigma\pi}^{+}}\Omega_{\sigma}^{+}$ with $\Delta\varphi_{\sigma\pi}^{+} = \varphi_{\sigma}^{+} - \varphi_{\pi}^{+}$, Eqs. (13, 14, 17-20) lead to

$$ \frac{\partial \Omega_{\pi}^{+}}{\partial y} = i \frac{k}{2} \chi_{sat} \Omega_{\pi}^{+} \qquad (21) $$

$$ \frac{\partial (\Omega_{\sigma}^{+} e^{-i\varphi})}{\partial y} = i \frac{k}{2} \chi_{eff} e^{2i\Delta\varphi_{\sigma\pi}^{+}} (\Omega_{\sigma}^{+} e^{-i\varphi}) \qquad (22) $$

Provided that the dephasing between the control and probe fields, denoted by $\Delta\varphi_{\sigma\pi}^{+}$, does not grow significantly along the propagation, $\chi_{eff}$ is seen to play the role of an effective susceptibility for the probe in Eq. (22). From Eqs. (21,22), one derives the propagation equation

$$ \frac{\partial \Delta\varphi_{\sigma\pi}^{+}}{\partial y} = k \sin(\Delta\varphi_{\sigma\pi}^{+} - \varphi) \left[ \chi'_{sat} \sin(\Delta\varphi_{\sigma\pi}^{+} - \varphi) - \chi''_{sat} \cos(\Delta\varphi_{\sigma\pi}^{+} - \varphi) \right] \qquad (23) $$

where $\chi'_{sat}$ and $\chi''_{sat}$ denote the real and imaginary parts of the saturated susceptibility, respectively. We introduce the phase of the linear susceptibility, $\varphi_L$, defined by $\chi_{lin} = |\chi_{lin}| e^{i\varphi_L}$, which verifies

$$ \tan \varphi_L = \frac{\Gamma_d}{\Delta_0} \qquad (24) $$

The integration Eq. (23) leads to $\tan \Delta\varphi_{\sigma\pi}^{+} = \frac{\tan \varphi_L}{(\frac{\tan \varphi_L}{\tan \varphi} + 1)e^{-2\alpha_0 y \sin^2 \varphi_L - 1}}$. When $|\Omega_{\pi}(y=0)| \ll \sqrt{\Gamma_d}\Gamma$ and $\alpha_0 L \sin^2 \varphi_L \ll 1$, then $|\Omega_{\pi}^{+}(y)| \approx |\Omega_{\pi}^{+}(y=0)|$, $S \approx 1$ and $\Delta\varphi_{\sigma\pi}^{+} \approx 0$, see [18]. The sample therefore becomes a linear medium for the probe field whose susceptibility, $\chi_{eff}$, is related to the true linear susceptibility by $\chi_{eff} = \chi_{lin}e^{2i\varphi}$. The latter relation proves the existence of the phase control of the linear response of the medium, the physical interpretation of which was described in [18]. It results from interference effects between the quantum paths which contribute to the coherence $\rho_{\sigma}$ responsible for the $\sigma$-polarized radiated field.

As the control field intensity is increased, saturation effects occur and the validity condition of the parametric approximation becomes less restrictive. From Eq. (21), one gets

$$ \frac{\partial |\Omega_{\pi}^{+}|}{\partial y} = -\alpha_{0}S \sin^{2} \varphi_{L} |\Omega_{\pi}^{+}| $$
<figure><img src="image_8.png" /><figcaption>Figure 7. Effect of detuning. (a) Reflection and (b) Transmission factors plotted as functions of the relative phase $\varphi$ for $\alpha_0 L = 0.3$, $\frac{\Omega_{\pi}^{+}(0)}{\Gamma_d} = 0.4$, $\frac{\Omega_{\pi}^{-}(L)}{\Gamma_d} = 0.16$ and $\Delta_0/\Gamma_d = 0$ (solid blue line), 2 (dashed orange line).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 8. Probe field intensities (a) $|\frac{\Omega_{\sigma}^{+}(y)}{\Omega_{\sigma}^{\pm}(0)}|^2$ and (b) $|\frac{\Omega_{\sigma}^{-}(y)}{\Omega_{\sigma}^{\pm}(0)}|^2$ as functions of the (normalized) propagation distance $y/L$ for $\varphi = 0$ (solid blue line), $\frac{\pi}{4}$ (dotted orange line), $\frac{\pi}{2}$ (dashed green line).</figcaption></figure>

$\Delta_0 = 0$, $\Omega_{\pi}^{+}(0)/\Gamma_d = 0.4$, $\Omega_{\pi}^{-}(L)/\Gamma_d = 0.16$, $\alpha_0 L = 0.6$. In Figs. 9 (a,b) are plotted the corresponding accumulated phases $\varphi_{\sigma}^{\pm}(y)$. As shown by Fig. 8-a, $\varphi = 0$ leads to absorption of the forward component of the probe whereas $\varphi = \pi/2$ leads to its significant amplification. The case $\varphi = \pi/4$ corresponds to a small modification of the probe. This is in line with the results of the numerical simulations (Fig.5-a). We also notice that intensity depends quasi-linearly on the propagation distance which comforts the linear approximation performed in Eqs.(40,42). In Fig. 8-b, is represented the backward probe component which vanishes at $y = L$. The field is amplified from the exit to the entrance of the sample. Its intensity at the entrance depends significantly on the dephasing. On the other hand, the shape of the curves is almost parabolic around $y = L$. Indeed, when $y$ is close to $L$, Eq.(41) yields $|\Omega_{\sigma}^{-}(y)|^2 \approx |\Omega_{\sigma}^{+}(0)|^2 |c_{+-}|^2 \alpha_0^2 (y - L)^2$.

The accumulated phase $\varphi_{\sigma}^{+}(y)$ and $\varphi_{\sigma}^{-}(y)$ are represented in Fig.9-a and b, respectively. The forward component exhibits a vanishing phase at the entrance and accumulates a significant value along propagation ($\varphi_{\sigma}^{+}(y=L) \approx 0.34$) for $\varphi = \pi/4$ whereas for $\varphi = 0$ and $\varphi = \pi/2$ the phase remains equal to zero. This effect can be explained from the simple analytical model introduced in Subsec. IV B which leads to Eq.(40). We then have $\varphi_{\sigma}^{+}(y) \approx -\alpha_0 y |c_+| \cos(2\varphi + \varphi_L)$. At resonance, $\varphi_L = \pi/2$ and thus $\varphi_{\sigma}^{+}(y) = 0$ for $\varphi = 0, \pi/2$ and $\varphi_{\sigma}^{+}(y) \approx \alpha_0 y |c_+|$ for $\varphi = \pi/4$. Using $|c_+(0)| = 0.683$, the analytical model predicts an accumulated phase $\varphi_{\sigma}^{+}(y=L) \approx 0.41$ to compare with the value 0.34 given by exact numerical simulations. The backward component has a different behavior. Indeed, due to the boundary condition $\Omega_{\sigma}^{-}(L) = 0$, the value of the phase $\varphi_{\sigma}^{-}$ cannot be set either at the entrance ($y=0$) or the exit ($y=L$) of the sample, only the propagation dynamics allows us to recover it. From Eq.(41), $\varphi_{\sigma}^{-}(y) \approx \pi - 2\varphi + \alpha_0 (y - L) |c_+| \cos(2\varphi + \varphi_L)$, which leads $\varphi_{\sigma}^{-}(L) \approx \pi, \pi/2, 0$ for $\varphi = 0, \pi/4$ and $\pi/2$ in agreement with the numerical values read on inset (b) of Fig. 9, i.e. 3.14, 1.35, 0, respectively.

## VI. CONCLUSION

In this article, the dynamics of an ensemble of duplicated two-level systems driven by a strong stationary control field and subject to a weak probe field was studied in detail. In particular, the expressions of the probe reflection,
and provided that $\alpha_0LS(y=0) \sin^2 \varphi_L \ll 1$, the intensity of the control field remains unaffected along propagation. If the optical depth is increased so that the inequality $\alpha_0LS(y=0) \sin^2 \varphi_L \ll 1$ no longer holds, the change in the control field cannot be neglected and additional dephasing must be taken into account in the propagation equation of the probe. New features appear and are detailed in [18].

## 4. Situation of a stationary control field

In this section, we investigate the effects which appear when the intensity of the control field is not uniform but spatially modulated because of the backward component. In this situation, the probe field experiences both reflection and transmission. The effect of the change in the optical response of the sample and the possibility of phase control can be evaluated by calculating the transmission and reflection coefficients for the probe field.

### 4.1. Propagation equations in the slow envelope approximation

In the slow envelope approximation, the set of Eqs. (17-20) can be rewritten by extracting an analytical expression for the components $\rho_j^{(0)}(y)$ and $\rho_j^{(-1)}(y)$.

We introduce the parameter $r(y)$ and the phase $\varphi_r$ defined by $\Omega_\pi^{-}(y) = r(y)\Omega_\pi^{+}(y)$ and $r = |r|e^{-i\varphi_r}$, so that $\varphi_r = \varphi_\pi^{-} - \varphi_\pi^{+}$. When the control field is unaffected by the propagation, $r(y)$ is constant. Note that, for Eq. (5) to hold, the control field should not vanish anywhere, including the spatial nodes, therefore $r$ should be different from 1. From Eqs. (13,14,16), we see that the analytic expressions of $\rho_j^{(0)}(y)$ and $\rho_j^{(-1)}(y)$ can be evaluated as soon as the series expansion of the saturation parameter $S(y)$ and $\frac{1}{|\Omega_\pi(y)|^2}$ is determined. The saturation parameter $S$ given by Eq. (11) can be expressed as

$$S(y) = \frac{b(y)}{1 + a(y) \cos[2ky + \varphi_r(y)]} \quad (25)$$

with $a(y) = \frac{2F_0|\Omega_\pi^{+}(y)|^2|r(y)|}{1+F_0|\Omega_\pi^{+}(y)|^2(1+|r(y)|^2)}$ and $b(y) = \frac{1}{1+F_0|\Omega_\pi^{+}(y)|^2(1+|r(y)|^2)}$. It can also be formally expanded as

$$S(y) = \sum_{n=-\infty}^{+\infty} c_n(y) e^{2inky} \quad (26)$$
<figure><img src="image_10.png" /><figcaption>Figure 9. Accumulated phases (a) $\varphi_{\sigma}^{+}(y)$ of the forward-propagating probe field and (b) $\varphi_{\sigma}^{-}(y)$ of the backward-propagating probe field as functions of the (normalized) propagation distance $y/L$ for $\varphi = 0$ (solid blue line), $\pi/4$ (dotted orange line), $\pi/2$ (dashed green line).</figcaption></figure>

$R$, and transmission, $T$, factors were determined and their dependence on field parameters was investigated. Both amplification and absorption were shown to be achievable. Moreover, $R$ and $T$ depend on the relative phase between the fields, which demonstrates the control potentialities offered by this system. For low optical depths, proper control with no phase modification along propagation may be performed. For arbitrarily higher optical depths, both the driving fields and the phase accumulated by the probe field along its propagation are affected which drastically modified the behavior of the system. Previous works on co-propagating control and probe field schemes have already showed new interesting phenomena like phase saturation and transparency [16]. This paves the way to further investigations for the present stationary control field configuration. Numerical simulations also showed a strong increase of the reflection and transmission factors with the optical depth. The combination of these effects offers a stimulating motivation for further study of this case.

ACKNOWLEDGMENTS

This research was funded in part by l'Agence Nationale de la Recherche (ANR), Project ANR-22-CE47-0011. For the purpose of open access, the authors have applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

[1] K.-J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. **66**, 2593 (1991).

[2] M. O. Scully, Phys. Rev. Lett. **67**, 1855 (1991).

[3] L. V. Hau et al., Nature (London) **397**, 594 (1999); M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, Phys. Rev. Lett. **82**, 5229 (1999); L. J. Wang, A. Kuzmich and A. Dogarlu, Nature (London) **406**, 277 (2000); P. W. Millonni, Fast Light, Slow Light and Left-Handed Light, Taylor & Francis, New York, 2005.

[4] J. B. Pendry, Phys. Rev. Lett. **85**, 3966 (2000); R. A. Shelby, D. R. Smith, S. Shultz, Science **292**, 77 (2001).

[5] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals: Molding The Flow Of Light, Princeton University Press, 2008.

[6] A. M. Herkommer, W. P. Schleich, and M. S. Zubairy, J. Mod. Opt. **44**, 2507 (1997); K. S. Johnson, J. H. Thywissen, N. H. Dekker, K. K. Berggren, A. P. Chu, R. Younkin, and M. Prentiss, Science **280**, 1583 (1998); S. Qamar, S.-Y. Zhu, and M. S. Zubairy, Phys. Rev. A **61**, 063806 (2000); V. Ivanov and Y. Rozhdestvensky, Phys. Rev. A **81**, 033809 (2010).

[7] A. André, M. D. Lukin, Phys. Rev. Lett. **89**, 143602 (2002); M. Bajcsy, A. S. Zibrov, M. D. Lukin, Nature **426**, 638 (2003); A. André, M. Bajcsy, A. S. Zibrov, and M. D. Lukin, Phys. Rev. Lett. **94**, 063902 (2005).

[8] X. M. Su and B. S. Ham, Phys. Rev. A **71**, 013821 (2005); M. Artoni and G. C. La Rocca, Phys. Rev. Lett. **96**, 073905 (2006); Q.-Y. He, Y. Xue, M. Artoni, G. C. La Rocca, J.-H. Xu and J.-Y. Gao, Phys. Rev. B **73**, 195124 (2006); Q.-Y. He, J.-H. Wu, T.-J. Wang, and J.-Y. Gao, Phys. Rev. A **73**, 053813 (2006); J.-H. Wu, A. Raczyński, J. Zaremba, S. Zielińska-Kaniasty, M. Artoni and G.C. La Rocca, Journal of Modern Optics **56**, 768 (2009); K. Słowik, A. Raczyński, J. Zaremba, S. Zielińska-Kaniasty, M. Artoni and G.C. La Rocca, Journal of Modern Optics **58**, 978 (2011).

[9] Q. Jiang, Y. Zhang, D. Wang, S. Ahrens, J. Zhang, and S. Zhu, Opt. Express **24**, 24451 (2016).

[10] V. G. Arkhipkin, and S. A. Myslivets, Opt. Lett. **39**, 3223 (2014); V. G. Arkhipkin and S. A. Myslivets, Phys. Rev. A **93**, 013810 (2016); V. G. Arkhipkin, and S. A. Myslivets, J. Opt. **19**, 055501 (2017).
where the $c_n$ coefficients vary slowly on the spatial wavelength scale $\frac{2\pi}{k}$ and can be approximated as $c_n(y) \approx \frac{k}{\pi} \int_y^{y+\frac{\pi}{k}} S(y') e^{-2inky'} dy'$. For $n \ge 0$, using Eq. (25) and [31], we get $c_n(y) \approx c_0 \eta^n e^{in\varphi_r(y)}$, with $c_0 = \frac{b(y)}{\sqrt{1-a^2(y)}}$ and $\eta = \frac{\sqrt{1-a^2(y)-1}}{a(y)}$. Furthermore, for $n \le 0$, the reality of parameter S implies that $c_n = c_{-n}^*$.

On the other hand, using Eq. (15), one can put the parameter $\frac{1}{|\Omega_\pi(y)|^2}$ in the form

$$ \frac{1}{|\Omega_\pi(y)|^2} = \frac{b'(y)}{1 + a'(y) \cos[2ky + \varphi_r(y)]} \quad (27) $$

with $a'(y) = \frac{2|r(y)|}{|\Omega_\pi^+(y)|^2(1+|r(y)|^2)}$ and $b'(y) = \frac{1}{|\Omega_\pi^+(y)|^2(1+|r(y)|^2)}$. It can also be formally expanded as $\frac{1}{|\Omega_\pi(y)|^2} = \sum_{n=-\infty}^{+\infty} c'_n(y) e^{2inky}$ where the $c'_n$ coefficients vary slowly on the wavelength scale and can be expressed as $c'_n(y) \approx \frac{k}{\pi} \int_y^{y+\frac{\pi}{k}} \frac{1}{|\Omega_\pi(y')|^2} e^{-2inky'} dy'$. Using [31], we get $c'_{n \ge 0}(y) \approx c'_0 \eta^n e^{in\varphi_r(y)}$ with $c'_0 = \frac{b'(y)}{\sqrt{1-a'^2(y)}}$ and $\eta' = \frac{\sqrt{1-a'^2(y)-1}}{a'(y)}$. Furthermore, the reality of $\frac{1}{|\Omega_\pi(y)|^2}$ implies that $c'_{n \le 0} = (c'_{-n})^*$.

Using Eqs. (13,14,16,25,26,27), we can finally express the components of the coherences which appear in the propagation equations for the control fields

$$ \rho_\pi^{(0)}(y) = \frac{i}{\Gamma_d} [\bar{c}_0(y) \Omega_\pi^+(y) + \bar{c}_1(y) \Omega_\pi^-(y)] \quad (28) $$

$$ \rho_\pi^{(-1)}(y) = \frac{i}{\Gamma_d} [\bar{c}_{-1}(y) \Omega_\pi^+(y) + \bar{c}_0(y) \Omega_\pi^-(y)] \quad (29) $$

with $\bar{c}_i(y) = (\frac{\Gamma_d}{\Gamma_d+i\Delta_0}) c_i(y)$, and in the propagation equations for the probe fields

$$ \rho_\sigma^{(0)}(y) = \frac{i}{\Gamma_d} e^{i\varphi} [c_+(y) \Omega_\sigma^{+*}(y) + c_{+-}(y) \Omega_\sigma^{-*}(y)] \quad (30) $$

$$ \rho_\sigma^{(-1)}(y) = \frac{i}{\Gamma_d} e^{i\varphi} [c_{+-}(y) \Omega_\sigma^{+*}(y) + c_{-}(y) \Omega_\sigma^{-*}(y)] \quad (31) $$
[11] L. E. E. de Araujo, Opt. Lett. 35, 977 (2010)

[12] K. Słowik, A. Raczynski, J. Zaremba, S. Zielinska-Kaniasty, M. Artoni and G. C. La Rocca, Phys. Scr. T143, 014022 (2011) ; Yihong Qi, Yueping Niu, Fengxue Zhou, Yandong Peng and Shangqing Gong, J. Phys. B: At. Mol. Opt. Phys. 44, 085502 (2011).

[13] Tao Li, Hailin Wang, N.H. Kwong, and R. Binder, Opt. Express 11, 3298-3303 (2003).

[14] J. C. Delagnes and M. A. Bouchene, Phys. Rev. Lett. 98, 053602 (2007) ; J. C. Delagnes and M. A. Bouchene, Phys. Rev. A 76, 045805 (2007) ; J. C. Delagnes and M. A. Bouchene Phys. Rev. A 76, 053809 (2007).

[15] F. A. Hashmi and M. A. Bouchene, Phys. Rev. A 77, 051803(R) (2008).

[16] F. A. Hashmi and M. A. Bouchene, Phys. Rev. Lett. 101, 213601 (2008).

[17] J. Wu, X.-Y. Lu and L.-L. Zheng, J. Phys. B: At. Mol. Opt. Phys. 43, 161003 (2010) ; L. E. Zohravi, A. Vafafard, M. Mahmoud, Journal Of Luminescence 151, 11 (2014).

[18] S. H. Asadpour, H. Rahimpour Soleimani, Optics Communications 315, 394 (2014) ; P. Kumar and S. Dasgupta, J. Phys. B: At. Mol. Opt. Phys. 47, 175501 (2014) ; Ziauddin, Y.-L. Chuang, R.-K. Lee and S. Qamar, Laser Phys. 26, 015205 (2016).

[19] M. A. Anton and F. Carreno, J. Opt. 12, 104006 (2010).

[20] L. Yun, W. Pu, and P. Shuang-Yan, Chin. Phys. B 22, 104203 (2013).

[21] A. Vafafard and M. Sahrai, JOSA B 35, 2118 (2018).

[22] L. Jin, Y. Niu, and S. Gong, Phys. Rev. A 83, 023410 (2011).

[23] D. A. Smait et al, Laser Phys. Lett. 20, 086003 (2023).

[24] Z. Zhu, W.-X. Yang, X.-T. Xie, S. Liu, S. Liu, and R.-K. Lee, Phys. Rev. A 94, 013826 (2016) ; D. Shah, U. Wahid, S. M. Arif, S. Muhammad, H. Ahmad, Optical and Quantum Electronics 54, 360 (2022).

[25] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals series and products, Elsevier, 2014.
with

$$c_+ = \beta [1 + 2\gamma |r| + \delta |r|^2] \qquad (32)$$

$$c_{+-} = \beta r \left[ 2 + \frac{\gamma}{|r|} + \gamma |r| \right]$$

$$c_{-} = \beta r^2 \left[ 1 + \frac{2\gamma}{|r|} + \frac{\delta}{|r|^2} \right]$$

and

$$\beta = \frac{c_0 e^{-2i\varphi_{\pi}^{+}}}{(1 - \eta\eta') |1 - |r|^2|} \left( \frac{\Gamma_d}{\Gamma_d + i\Delta_0} \right) \qquad (33)$$

$$\gamma = \eta + \eta' \qquad (34)$$

$$\delta = \eta^2 + (\eta')^2 + \eta\eta' - (\eta\eta')^2 \qquad (35)$$

Injecting Eqs. (28-31) into Eqs. (17,18) and Eqs. (19,20), we get

$$\frac{\partial \Omega_{\pi}^{+}}{\partial \left(\frac{y}{L}\right)}(y) = -\alpha_{0} L \left[ \bar{c}_{0}(y) \Omega_{\pi}^{+}(y) + \bar{c}_{1}(y) \Omega_{\pi}^{-}(y) \right] \qquad (36)$$

$$\frac{\partial \Omega_{\pi}^{-}}{\partial \left(\frac{y}{L}\right)}(y) = \alpha_{0} L \left[ \bar{c}_{-1}(y) \Omega_{\pi}^{+}(y) + \bar{c}_{0}(y) \Omega_{\pi}^{-}(y) \right] \qquad (37)$$

and

$$\frac{\partial \Omega_{\sigma}^{+}}{\partial \left(\frac{y}{L}\right)}(y) = -\alpha_{0} L e^{2i\varphi} \left[ c_{+}(y) \Omega_{\sigma}^{+*}(y) + c_{+-}(y) \Omega_{\sigma}^{-*}(y) \right] \qquad (38)$$

$$\frac{\partial \Omega_{\sigma}^{-}}{\partial \left(\frac{y}{L}\right)}(y) = \alpha_{0} L e^{2i\varphi} \left[ c_{+-}(y) \Omega_{\sigma}^{+*}(y) + c_{-}(y) \Omega_{\sigma}^{-*}(y) \right] \qquad (39)$$

The forward and backward components of both the probe and control fields are now coupled through the coefficients ($\bar{c}_{-1}, \bar{c}_{1}$) for the control and $c_{+-}$ coefficient for the probe. In particular, Eqs. (38,39) clearly show that the probe field propagation may be controlled by the phase. This property is further investigated in Sec. 5, through numerically solving the above coupled equations.

## 4.2. Parametric regime

The probe field propagation depends on the control field intensity and the optical depth. For moderate optical depths, i.e. $|\alpha_0 L \bar{c}_{j=0,\pm 1}| \ll 1$, the
control field is weakly affected by the propagation and the envelopes can be considered uniform, i.e. $\Omega_{\pi}^{+}(y) \approx \Omega_{\pi}^{+}(0)$, $\Omega_{\pi}^{-}(y) \approx \Omega_{\pi}^{-}(L)$ and $\varphi_{\pi}^{\pm} \approx 0$. The coefficients $c_{+}, c_{-}, c_{+-}$ are thus constant along the sample and the phases $\varphi_{\pi}^{+}$ and $\varphi_{r}$ are neglected. The system of Eqs. (38,39) turns into a set of differential equations with constant coefficients. Moreover, for optical depths such as $|\alpha_{0}Lc_{j=+,-,+-}| \ll 1$, one can retain the most significant terms in the integration of the system Eqs. (38,39) which leads to

$$ \begin{aligned} \Omega_{\sigma}^{+}(y) - \Omega_{\sigma}^{+}(0) &\approx -\alpha_{0}ye^{2i\varphi} [c_{+}\Omega_{\sigma}^{+*}(0) + c_{-+}\Omega_{\sigma}^{-*}(0)] \\ \Omega_{\sigma}^{-}(y) - \Omega_{\sigma}^{-}(L) &\approx \alpha_{0}(y - L)e^{2i\varphi} [c_{+-}\Omega_{\sigma}^{+*}(L) + c_{-}\Omega_{\sigma}^{-*}(L)] \end{aligned} $$

Taking into account the boundary condition $\{\Omega_{\sigma}^{-}(L) = 0; \Omega_{\sigma}^{+*}(0) = \Omega_{\sigma}^{+}(0)\}$, one readily obtains

$$ \Omega_{\sigma}^{+}(y) \approx \Omega_{\sigma}^{+}(0) [1 - \alpha_{0}ye^{2i\varphi}c_{+}] \quad (40) $$

$$ \Omega_{\sigma}^{-}(y) \approx \Omega_{\sigma}^{+}(0) \alpha_{0}(y - L) e^{2i\varphi} [c_{+-} - \alpha_{0}(y - L) e^{-2i\varphi} c_{+-}c_{+}^{*}] \quad (41) $$

From Eqs. (32,33-37), we get $\arg[c_{+}(0)] = \varphi_{L} - \frac{\pi}{2}$, where $\varphi_{L}$ is defined in Eq. (24). The transmission and reflection coefficients can then be approximated as

$$ T \approx 1 - 2\alpha_{0}L|c_{+}|\sin(2\varphi + \varphi_{L}) \quad (42) $$

$$ R \approx |\alpha_{0}Lc_{+-}|^{2} [1 - 2\alpha_{0}L|c_{+}|\sin(2\varphi + \varphi_{L})] \quad (43) $$

These expressions show the existence of *phase-controlled* reflection and transmission in the limit of parametric interaction regime where the control field is slightly affected by propagation and the probe phase variations along propagation are weak. The coefficients can be modified by adjusting a versatile and fine-tuning experimental parameter, $\varphi$, the additional shift $\varphi_{L}$ being determined by the detuning $\Delta_{0}$. The phase dependence is not altered by other fluctuating parameters such as density, temperature, driving field intensity etc., and the phase thus constitutes a genuine control parameter. For $0 \le 2\varphi + \varphi_{L} \le \pi$, the transmission factor $T$ is less than or equal to 1, while it becomes greater than 1 for $\pi \le 2\varphi + \varphi_{L} \le 2\pi$. This is consistent with the absorbing or amplifying nature of the sample as described by its effective linear susceptibility, given by Eq. (10). The reflection factor is associated with the backward component of the control field that is generated by coupling
with the forward part of the probe. Thus, the reflection factor depends on the second (or higher) power of the optical depth and is small compared to the transmission factor.

At this point, let us make an important remark. The phase dependence was certainly to be expected from the beginning since the polarization of the total field injected into the atomic medium is elliptical and depends on $\varphi$. It is therefore natural that the intensities of the probe field reflected/transmitted by the medium accordingly depend on $\varphi$ [30]. However, the phase-sensitivity turns to be phase-control only in the parametric regime as identified and discussed above. This gives rise to a small reflection factor but surprisingly a very efficient amplification and modulation of the transmission factor. The mixing of the forward radiated field with the incident one, Eq. (40), is responsible for this efficiency and is reminiscent of the homodyne detection process. These results and the exact quantitative expressions of the necessary conditions constitute an important part of the present work.

## 5. Results and discussion

In this section, we present both the numerical results obtained from the numerical simulation of Eqs. (36-39) and the analytical approximation, Eqs. (42,43), and discuss their physical interpretation.

In Fig.3 we plotted (a) the reflection, $R$, and (b) transmission, $T$, coefficients as functions of the relative phase $\varphi$, for resonant probe and control fields ($\Delta_0 = 0$) and small optical depth $\alpha_0 L = 0.3$. The forward/backward control field Rabi frequencies at the input/output of the sample are set to $\frac{\Omega_{\pi}^{+}(0)}{\Gamma_d} = 0.4$ and $\frac{\Omega_{\pi}^{-}(L)}{\Gamma_d} = 0.16$, respectively. The curves in solid and dashed lines correspond to numerical simulations and approximate analytical expressions, Eqs. (42,43), respectively. We observe that both $R$ and $T$ oscillate with a period of $\pi$ as expected from the $e^{2i\varphi}$ dependence of the effective susceptibility, Eq. (10). As already pointed out in Subsec. 4.2, $T$ can be greater or less than 1, depending on whether the sample is amplifying or absorbing. We note that $T$ is much larger than $R$: the transmitted field indeed results from the interference of the incident and radiated probe fields, while, in contrast, the reflected field results from the backward radiated field only. Moreover, we observe discrepancies between the numerical simulations and the simple perturbative analytical model.

In Fig. 4 we plotted the (normalized) backward and forward-propagating
<figure><img src="image_4.png" /><figcaption>Figure 3: (a) Reflection and (b) Transmission factors as functions of the relative phase $\varphi$ at resonance ($\Delta_0 = 0$) for a probe field propagating in an optically thin sample (optical depth $\alpha_0 L = 0.3$) driven by a control field with Rabi amplitudes $\frac{\Omega_\pi^{+}(0)}{\Gamma_d} = 0.4$ and $\frac{\Omega_\pi^{-}(L)}{\Gamma_d} = 0.16$. Full (dashed) line plots are obtained through numerical simulation (analytic approximation).</figcaption></figure>

control field intensities $\left| \frac{\Omega_\pi^{\pm}(y)}{\Gamma_d} \right|^2$ as functions of the (normalized) position $y/L$, for the set of parameters $\Delta_0 = 0$, $\alpha_0 L = 0.3$, $\frac{\Omega_\pi^+(0)}{\Gamma_d} = 0.4$, $\frac{\Omega_\pi^-(L)}{\Gamma_d} = 0.16$. The intensity of the forward-propagating field is attenuated by 37.5% along its propagation, from the value of 0.16 at the input to about 0.1 at the output of the sample, while the intensity of the backward-propagating control field is reduced by 22%, from the value of 0.0256 at the output to 0.02 at the input of the sample. The phases accumulated along the propagation are found to be small in all cases (of the order of $10^{-10}$). This can be understood from Eqs. (36,37). At moderate optical depth, the dominant contribution due to the diagonal terms involves the $\bar{c}_0$ coefficient which is real at resonance. The control field components therefore experience pure absorption without dispersion, the accumulated phase is thus negligible. For the lowest-order solution of Eqs. (38,39), namely Eqs. (40,41), to hold, the condition $|\alpha_0 L c_j| \ll 1$ must be checked: we find the values 0.192, 0.005, 0.054 for $|\alpha_0 L c_{j=+,-,+,-}|$, respectively, which ensures the validity of the analytical model presented in Subsec. 4.2, although the control field can be considered only roughly unaltered along propagation. This simple model captures the essential physical phenomena at work and clearly demonstrates the accuracy of phase control of $R$ and $T$.

Figs 5 and 6 correspond to the same situations as considered in Figs 3 and 4, but with a higher optical depth $\alpha_0 L = 0.6$. Both the reflection and
<figure><img src="image_5.png" /><figcaption>Figure 4: Control field intensities $\left| \frac{\Omega_{\pi}^{\pm}}{\Gamma_d} \right|^2$ as functions of the (normalized) propagation distance $y/L$ for $\Delta_0 = 0$, $\alpha_0 L = 0.3$, $\frac{\Omega_{\pi}^{+}(0)}{\Gamma_d} = 0.4$ and $\frac{\Omega_{\pi}^{-}(L)}{\Gamma_d} = 0.16$.</figcaption></figure>

transmission coefficients are significantly enhanced in Fig. 5 compared to Fig. 3: now, $T$ oscillates between 0.48 and 2.17 – versus 0.7 and 1.45 for $\alpha_0 L = 0.3$ – and $R$ between 0.017 and 0.043 – versus 0.0043 and 0.0063 for $\alpha_0 L = 0.3$. Here, although the parametric approximation is no longer valid, both $R$ and $T$ still depend on the phase and are significantly enhanced. The fit by the analytical model, Eqs. (42,43), is only qualitative for two reasons. First, as shown in Fig. 6, the increased optical depth leads to stronger absorption of the control field along propagation. The intensity of the forward-, resp. backward-, propagating component is indeed reduced by 60% (37% in the case of Fig. 4 with $\alpha_0 L = 0.3$), resp. 56% (22% in Fig. 4). The parametric approximation, $\Omega_{\pi}^{+}(y) \approx \Omega_{\pi}^{+}(0)$; $\Omega_{\pi}^{-}(y) \approx \Omega_{\pi}^{-}(L)$, is thus rough. Secondly, the coefficients $|\alpha_0 L c_{j=+,-,+-}|$ are approximately magnified by a factor of two compared to the case where $\alpha_0 L = 0.3$ (e.g. $|\alpha_0 L c_j| \approx 0.41, 0.006, 0.09$ for $j = +, -, +-, $ respectively) making the low order perturbation solution less valid. Note that in this case the dispersive component of the optical response vanishes at resonance and therefore the phase remains essentially constant.

In Fig. 7 we plotted the (a) reflection and (b) transmission coefficients as functions of the relative phase $\varphi$, in the resonant case $\Delta_0 = 0$ (solid line) and in the non-resonant case $\frac{\Delta_0}{\Gamma_d} = 2$ (dashed line), the other parameters being $\alpha_0 L = 0.3$ and $\frac{\Omega_{\pi}^{+}(0)}{\Gamma_d} = 0.4$, $\frac{\Omega_{\pi}^{-}(L)}{\Gamma_d} = 0.16$. As the detuning increases, the modulus of the effective susceptibility $\chi_{eff}$, Eq. (10), decreases and so does the global coupling of the probe to the atomic medium. $R$ and $T$ therefore tend to decrease, which is confirmed by the results of the numerical simulations. An important feature, however, is the dephasing which appears
<figure><img src="image_6.png" /><figcaption>Figure 5: (a) Reflection and (b) Transmission factors as functions of the relative phase $\varphi$ for $\Delta_0 = 0$, $\alpha_0 L = 0.6$, $\frac{\Omega_\pi^+(0)}{\Gamma_d} = 0.4$ and $\frac{\Omega_\pi^-(L)}{\Gamma_d} = 0.16$. Full (dashed) line plots are obtained through numerical simulation (resp. analytic approximation).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6: Control field intensities $\left|\frac{\Omega_\pi^\pm}{\Gamma_d}\right|^2$ as functions of the (normalized) propagation distance $y/L$ for $\Delta_0 = 0$, $\alpha_0 L = 0.6$, $\frac{\Omega_\pi^+(0)}{\Gamma_d} = 0.4$ and $\frac{\Omega_\pi^-(L)}{\Gamma_d} = 0.16$.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 7: Effect of detuning. (a) Reflection and (b) Transmission factors plotted as functions of the relative phase $\varphi$ for $\alpha_0 L = 0.3$, $\frac{\Omega_\pi^+(0)}{\Gamma_d} = 0.4$, $\frac{\Omega_\pi^-(L)}{\Gamma_d} = 0.16$ and $\frac{\Delta_0}{\Gamma_d} = 0$ (full line), 2 (dashed line).</figcaption></figure>

on $R$ and $T$ plots when the detuning is modified. The expression for the phase shift predicted by the analytical model, Eqs. (42,43), is $\varphi_L$ with $\tan \varphi_L = \frac{\Gamma_d}{\Delta_0}$. When $\frac{\Gamma_d}{\Delta_0} = 0.5$, $\varphi_L \approx 0.15\pi$ which is very close to the value of $0.16\pi$ obtained from numerical simulations.

In Fig. 8 are plotted the intensities of (a) the forward probe field, $|\frac{\Omega_\sigma^+(y)}{\Omega_\sigma^+(0)}|^2$, and (b) the backward probe field, $|\frac{\Omega_\sigma^-(y)}{\Omega_\sigma^-(0)}|^2$, as functions of the normalized co-ordinate $\frac{y}{L}$ for three values of the relative phase $\varphi = 0, \frac{\pi}{4}, \frac{\pi}{2}$, and $\Delta_0 = 0$, $\frac{\Omega_\pi^+(0)}{\Gamma_d} = 0.4$, $\frac{\Omega_\pi^-(L)}{\Gamma_d} = 0.16$, $\alpha_0 L = 0.6$. In Figs. 9 (a,b) we plotted the corresponding accumulated phases $\varphi_\sigma^\pm(y)$. As shown in Fig. 8-a, $\varphi = 0$ leads to absorption of the forward component of the probe whereas $\varphi = \frac{\pi}{2}$ leads to its significant amplification. The case $\varphi = \frac{\pi}{4}$ corresponds to a small modification of the probe. This is in agreement with the results of the numerical simulations (Fig.5-a). We also note that the intensity depends quasi-linearly on the propagation distance which confirms the linear approximation made in Eqs. (40,42). In Fig. 8-b, the backward probe component is shown which vanishes at $y = L$. The field is amplified from the output to the input of the sample. Its intensity at the input depends strongly on the dephasing. On the other hand, the shape of the curves around $y = L$ is almost parabolic. In fact, when $y$ is close to $L$, Eq. (41) yields $|\Omega_\sigma^-(y)|^2 \approx |\Omega_\sigma^+(0)|^2 |c_{+-}|^2 \alpha_0^2 (y - L)^2$.

The accumulated phases $\varphi_\sigma^+(y)$ and $\varphi_\sigma^-(y)$ are shown in Fig.9-a and b, respectively. The forward component exhibits a vanishing phase at the input and accumulates a significant value along the propagation ($\varphi_\sigma^+(y = L) \approx 0.34$) for $\varphi = \frac{\pi}{4}$ whereas for $\varphi = 0$ and $\varphi = \frac{\pi}{2}$ the phase remains equal to zero. This
<figure><img src="image_9.png" /><figcaption>Figure 8: Probe field intensities (a) $|\frac{\Omega_{\sigma}^{+}(y)}{\Omega_{\sigma}^{+}(0)}|^2$ and (b) $|\frac{\Omega_{\sigma}^{-}(y)}{\Omega_{\sigma}^{+}(0)}|^2$ as functions of the (normalized) propagation distance $y/L$ for $\varphi = 0$ (full thick line), $\pi/4$ (full line), $\pi/2$ (dashed line).</figcaption></figure>

effect can be explained by the simple analytical model introduced in Subsec. 4.2 which leads to Eq. (40). We then have $\varphi_{\sigma}^{+}(y) \approx -\alpha_{0}y|c_{+}|\cos(2\varphi + \varphi_{L})$. At resonance, $\varphi_{L} = \pi/2$ and therefore $\varphi_{\sigma}^{+}(y) = 0$ for $\varphi = 0$, $\pi/2$ and $\varphi_{\sigma}^{+}(y) \approx \alpha_{0}y|c_{+}|$ for $\varphi = \pi/4$. With $|c_{+}(0)| = 0.683$, the analytical model predicts an accumulated phase $\varphi_{\sigma}^{+}(y=L) \approx 0.41$ to compare with the value 0.34 obtained from exact numerical simulations. The backward component shows a different behaviour. In fact, due to the boundary condition $\Omega_{\sigma}^{-}(L) = 0$, the value of the phase $\varphi_{\sigma}^{-}$ cannot be set either at the input ($y=0$) or at the output ($y=L$) of the sample, only the propagation dynamics allows us to recover it. From Eq. (41), $\varphi_{\sigma}^{-}(y) \approx \pi - 2\varphi + \alpha_{0}(y-L)|c_{+}|\cos(2\varphi + \varphi_{L})$, which yields $\varphi_{\sigma}^{-}(L) \approx \pi$, $\pi/2$, 0 for $\varphi = 0$, $\pi/4$ and $\pi/2$ in accordance with the numerical values read on inset (b) of Fig. 9, i.e. 3.14, 1.35, 0, respectively.

## 6. Conclusion

In this article, the dynamical behaviour of an ensemble of duplicated two-level systems driven by a stationary control field and subject to a weak probe field was studied in detail. In particular, the expressions of the probe reflection, $R$, and transmission, $T$, factors were determined and their dependence on the field parameters was studied. We suggested an experimentally feasible implementation of our scheme involving transitions in $^6$Li which can be excited with conventional laser beams. The values for the optical depth and Rabi frequencies assumed in our simulations can be easily achieved experimentally. Indeed, since $\Gamma \approx 0.45$MHz, $\alpha_0 L = 1$ can be obtained for $L = 1$cm
<figure><img src="image_10.png" /><figcaption>Figure 9: Accumulated phases (a) $\varphi_{\sigma}^{+}(y)$ of the forward-propagating probe field and (b) $\varphi_{\sigma}^{-}(y)$ of the backward-propagating probe field as functions of the (normalized) propagation distance $y/L$ for $\varphi = 0$ (full thick line), $\pi/4$ (full line), $\pi/2$ (dashed line).</figcaption></figure>

and an atomic density $N = 3 \times 10^8 \text{at} \cdot \text{cm}^{-3}$. Rabi frequency $\Omega_{\pi} = \Gamma$ can also be reached for control field intensities of the order of $0.2 \text{mW} \cdot \text{cm}^{-2}$.

Both amplification and absorption were shown to be achievable. The control potentialities offered by this system were investigated, in particular the parametric regime obtained for moderate optical depths which leads to proper phase control over $R$ and $T$. For arbitrarily higher optical depths, both the driving fields and the phase accumulated by the probe field along its propagation are affected drastically modifying the behavior of the system. Previous works on co-propagating control and probe field schemes have already shown new interesting phenomena such as phase saturation and transparency [18]. This paves the way for further investigations for the present stationary control field configuration. Numerical simulations also showed a strong increase in reflection and transmission factors with optical depth and also a *phase-sensitivity*. The combination of these effects provides a stimulating motivation for further study of this case. In particular, we believe that (i) higher reflectivity will be achievable and that (ii) it may be possible to design the pump standing-wave so that the reflection and transmission factors reach their maxima for different values of the relative phase – ideally, transmission would be maximal when reflection is minimal, and vice versa, thus implementing a phase-sensitive optical switch.

## Acknowledgements

This research was funded in part by l'Agence Nationale de la Recherche (ANR), Project ANR-22-CE47-0011. For the purpose of open access, the au-
thors have applied a CC-BY public copyright licence to any Author Accepted
Manuscript (AAM) version arising from this submission.

References

[1] Boller, K.-J., Imamoglu, A., and Harris, S. E., Phys. Rev. Lett. **66**, 2593 (1991).

[2] Scully, M. O., Phys. Rev. Lett. **67**, 1855 (1991).

[3] Hau, L. V., et al., Nature (London) **397**, 594 (1999); Kash, M. M., Sautenkov, V. A., Zibrov, A. S., Hollberg, L., Welch, G. R., Lukin, M. D., Rostovtsev, Y., Fry, E. S. and Scully, M. O., Phys. Rev. Lett. **82**, 5229 (1999); Wang, L. J., Kuzmich, A. and Dogarlu, A., Nature (London) **406**, 277 (2000); Millonni, P. W., Fast Light, Slow Light and Left-Handed Light, Taylor & Francis, New York, 2005.

[4] Pendry, J. B., Phys. Rev. Lett. **85**, 3966 (2000); Shelby, R. A., Smith, D. R., Shultz, S., Science **292**, 77 (2001).

[5] Joannopoulos, J. D., Johnson, S. G., Winn, J. N., Meade, R. D., Pho-
tonic Crystals: Molding The Flow Of Light, Princeton University Press,
2008.

[6] Herkommer, A. M., Schleich, W. P., and Zubairy, M. S., J. Mod. Opt. **44**, 2507 (1997) ; Johnson, K. S., Thywissen, J. H., Dekker, N. H., Berggren, K. K., Chu, A. P., Younkin, R., and Prentiss, M., Science **280**, 1583 (1998) ; Qamar, S., Zhu, S.-Y., and Zubairy, M. S., Phys. Rev. A **61**, 063806 (2000) ; Ivanov, V. and Rozhdestvensky, Y., Phys. Rev. A **81**, 033809 (2010).

[7] André, A., and Lukin, M. D., Phys. Rev. Lett. **89**, 143602 (2002) ; Bajcsy, M., Zibrov, A. S., Lukin, M. D., Nature **426**, 638 (2003) ; André, A., Bajcsy, M., Zibrov, A. S., and Lukin, M. D., Phys. Rev. Lett. **94**, 063902 (2005).

[8] Su, X. M., and Ham, B. S., Phys. Rev. A **71**, 013821 (2005) ; Artoni, M., and La Rocca, G. C., Phys. Rev. Lett. **96**, 073905 (2006) ; He, Q.-Y., Xue, Y., Artoni, M., La Rocca, G. C., Xu, J.-H., and Gao, J.-Y., Phys. Rev. B **73**, 195124 (2006) ; He, Q.-Y., Wu, J.-H., Wang, T.-J.,
and Gao, J.-Y., Phys. Rev. A **73**, 053813 (2006); Wu, J.-H., Raczyński, A., Zaremba, J., Zielińska-Kaniasty, S., Artoni, M. and La Rocca, G.C., Journal of Modern Optics **56**, 768 (2009); Słowik, K., Raczyński, A., Zaremba, J., Zielińska-Kaniasty, S., Artoni, M. and La Rocca, G.C., Journal of Modern Optics **58**, 978 (2011).

[9] Jiang, Q., Zhang, Y., Wang, D., Ahrens, S., Zhang, J., and Zhu, S., Opt. Express **24**, 24451 (2016).

[10] Arkhipkin, V. G. and Myslivets, S. A., Opt. Lett. **39**, 3223 (2014); Arkhipkin, V. G. and Myslivets, S. A., Phys. Rev. A **93**, 013810 (2016); Arkhipkin, V. G. and Myslivets, S. A., J. Opt. **19**, 055501 (2017).

[11] de Araujo, L. E. E., Opt. Lett. **35**, 977 (2010).

[12] Słowik, K., Raczynski, A., Zaremba, J., Zielinska-Kaniasty, S., Artoni, M. and La Rocca, G. C., Phys. Scr. **2011**, 014022 (2011); Qi, Y., Niu, Y., Zhou, F., Peng, Y., and Gong, S., J. Phys. B: At. Mol. Opt. Phys. **44**, 085502 (2011).

[13] Zhang, Y., Wang, Z., Nie, Z., Li, C., Chen, H., Lu, K. and Xiao, M., Phys. Rev. Lett. **106**, 093904 (2011).

[14] Zhang, Z., Wang, R., Zhang, Y., Kartashov, Y. V., Li, F., Zhong, H., Guan, H., Gao, K., Li, F., Zhang, Y. and Xiao, M., Nature Commun. **11**, 1902 (2020).

[15] Li, T., Wang, H., Kwong, N.H., and Binder, R., Opt. Express **11**, 3298 (2003).

[16] Delagnes, J. C. and Bouchene, M. A., Phys. Rev. Lett. **98**, 053602 (2007); Delagnes, J. C. and Bouchene, M. A., Phys. Rev. A **76**, 045805 (2007); Delagnes, J. C., and Bouchene, M. A., Phys. Rev. A **76**, 053809 (2007).

[17] Hashmi, F. A., and Bouchene, M. A., Phys. Rev. A **77**, 051803(R) (2008).

[18] Hashmi, F. A., and Bouchene, M. A., Phys. Rev. Lett. **101**, 213601 (2008).
[19] Wu, J., Lu, X.-Y., and Zheng, L.-L., J. Phys. B: At. Mol. Opt. Phys. **43**, 161003 (2010) ; Zohravi, L. E., Vafafard, A., Mahmoud, M., Journal Of Luminescence **151**, 11 (2014).

[20] Asadpour, S. H., Rahimpour Soleimani, H., Optics Communications **315**, 394 (2014) ; Kumar, P., and Dasgupta, S., J. Phys. B: At. Mol. Opt. Phys. **47**, 175501 (2014) ; Ziauddin, Chuang, Y.-L., Lee, R.-K., and Qamar, S., Laser Phys. **26**, 015205 (2016).

[21] Anton, M. A., and Carreno, F., J. Opt. **12**, 104006 (2010).

[22] Yun, L., Pu, W. and Shuang-Yan, P., Chin. Phys. B **22**, 104203 (2013).

[23] Vafafard, A., and Sahrai, M., JOSA B **35**, 2118 (2018).

[24] Jin, L., Niu, Y., and Gong, S., Phys. Rev. A **83**, 023410 (2011).

[25] Smait, D. A. et al, Laser Phys. Lett. **20**, 086003 (2023).

[26] Zhu, Z., Yang, W.-X., Xie, X.-T., Liu, S., and Lee, R.-K., Phys. Rev. A **94**, 013826 (2016) ; Shah, D., Wahid, U., Arif, S. M., Muhammad, S., Ahmad, H., Optical and Quantum Electronics **54**, 360 (2022).

[27] Zhang, J.-X., Zhou, H.-T., Wang, D.-W., and Zhu, S.-Y., Phys. Rev. A **83**, 053841 (2011).

[28] Ullah, Z., Wang, Z., Gao, M., Zhang, D., Zhang, Y., Gao, H., and Zhang, Y., Opt. Express **22**, 29544 (2014).

[29] Das, P., and Dey, T. N., Phys. Rev. A **110**, 063720 (2024).

[30] Kani, A., and Wanare, H., EPL **120**, 33001 (2017).

[31] Gradshteyn, I. S., and Ryzhik, I. M., Table of integrals series and products, Elsevier, 2014.