Electromagnetically induced transparency of a single-photon in dipole-coupled one-dimensional atomic clouds

The system under consideration is depicted in figure 1(a). It consists of two parallel one-dimensional (1D) atomic clouds, A and B, of length L which are located in the y–z plane and separated by a distance ℓ. The atoms have four relevant energy levels in a ladder configuration, as shown in figure 1(b). The lower levels $\left| 1 \right\rangle$ and $\left| 2 \right\rangle$ are, respectively, a ground state and an excited low-lying state, where the latter has a spontaneous decay rate γ. The two uppermost levels $\left| 3 \right\rangle$ and $\left| 4 \right\rangle$ correspond to long-lived nS and $n^{\prime} P$ states, respectively, with high principal quantum numbers $n,n^{\prime}$. We assume that initially the entire atomic medium is prepared in a 'superatom' state of the form

$$\begin{eqnarray}\begin{array}{rcl} \left| {{\psi }_{1}}(t=0) \right\rangle & = & \mathop{\sum }\limits_{j}c_{4}^{({{j}_{A}})}(0){{\left| 1,\ldots ,{{4}_{j}},\ldots ,1 \right\rangle }_{A}}{{\left| 1,\ldots ,1 \right\rangle }_{B}} \\ {} & {} & +\mathop{\sum }\limits_{j}c_{4}^{({{j}_{B}})}(0){{\left| 1,\ldots ,1 \right\rangle }_{A}}{{\left| 1,\ldots ,{{4}_{j}},\ldots ,1 \right\rangle }_{B}}. \\ \end{array}\end{eqnarray} \tag{ 1 }$$

This state describes a collectively shared single excitation [26, 27] in state $\left| 4 \right\rangle$ with the coefficients $c_{4}^{{{j}_{A}}}(0)$ and $c_{4}^{{{j}_{B}}}(0)$ being the probability amplitudes of finding the excitation in an atom j of cloud A or B, respectively. Such state can be prepared, for instance, by means of the Rydberg blockade [1] or by using attenuated laser pulses.

With this initial condition we consider a single-photon entering the two clouds along the $+z$ direction, which is initially in an arbitrary superposition state of two wavepackets propagating along different clouds, i.e., a PSS. The photon couples the $\left| 1 \right\rangle \leftrightarrow \left| 2 \right\rangle$ transition of atoms in cloud A $(B)$ with a detuning ${{\delta }_{pA}}$ (${{\delta }_{pB}}$). The adjacent transition $\left| 2 \right\rangle \leftrightarrow \left| 3 \right\rangle$ is driven by a strong control field of Rabi frequency ${{\Omega }_{cA}}$ (${{\Omega }_{cB}}$) with a detuning ${{\delta }_{cA}}$ (${{\delta }_{cB}}$), thus forming an EIT configuration. Note that with this setup the system can never contain more than two Rydberg excited atoms (the initial excitation in $\left| 4 \right\rangle$ and the one created by the single photon, in state $\left| 3 \right\rangle$) at the same time.

In this work, we consider a particular situation where the atoms in one cloud cannot interact with each other, but only with other atoms in the opposite cloud. This is achieved by exploiting the angular dependence of the DDI [1, 31–34]. For two atoms 1 and 2 with transition dipole moments ${{\vec{d}}_{1}}$ and ${{\vec{d}}_{2}}$, respectively, separated by the relative vector $\vec{R}$, the DDI has the form:

$$\begin{eqnarray}&&{{\hat{V}}_{{\rm dd}}}=\frac{1}{4\pi {{\epsilon }_{0}}}\left[ \frac{{{{\vec{d}}}_{1}}{{{\vec{d}}}_{2}}}{{{R}^{3}}}-3\frac{({{{\vec{d}}}_{1}}\vec{R})({{{\vec{d}}}_{2}}\vec{R})}{{{R}^{5}}} \right].\end{eqnarray} \tag{ 2 }$$

Here $R\equiv |\vec{R}|$, and ${{\epsilon }_{0}}$ is the electric permittivity in vacuum. To be specific we consider the configuration of figure 1(a). We choose states $|3\rangle \equiv |n{{S}_{1/2}},{{m}_{J}}=1/2\rangle$ and $|4\rangle \equiv |n{{P}_{3/2}},{{m}_{J}}=1/2\rangle$, whose transition is driven by linearly polarized photons, and the control field being polarized along the quantization axis ${{\vec{e}}_{q}}$. We also consider that ${{\vec{e}}_{q}}$ is located in the x–z plane at an angle β with respect to the propagation axis $+z$ (see figure 1(a)). With this choice only the non-diagonal matrix elements of the DDI contribute to the interaction as the contributions to ${{\hat{V}}_{{\rm dd}}}$ from components orthogonal to ${{\vec{e}}_{q}}$ vanish due to selection rules. This implies that the DDI only couples states $\left| 3 \right\rangle$ and $\left| 4 \right\rangle$. Therefore, for the setup given in figure 1(a), its operator form is

$$\begin{eqnarray}\begin{array}{rcl} {{{\hat{V}}}_{{\rm dd}}} & = & \hbar V_{{\rm e}}^{AB}({{z}_{1}},{{z}_{2}}){{\left| 4 \right\rangle }_{{{1}_{A}}}}\left\langle 3 \right|\otimes {{\left| 3 \right\rangle }_{{{2}_{B}}}}\left\langle 4 \right|+\hbar V_{{\rm e}}^{AA}({{z}_{1}},{{z}_{2}}){{\left| 4 \right\rangle }_{{{1}_{A}}}}\left\langle 3 \right|\otimes {{\left| 3 \right\rangle }_{{{2}_{A}}}}\left\langle 4 \right| \\ {} & {} & +\hbar V_{{\rm e}}^{BB}({{z}_{1}},{{z}_{2}}){{\left| 4 \right\rangle }_{{{1}_{B}}}}\left\langle 3 \right|\otimes {{\left| 3 \right\rangle }_{{{2}_{B}}}}\left\langle 4 \right|+{\rm h}.{\rm c}., \\ \end{array}\end{eqnarray} \tag{ 3 }$$

where ${{\left| m \right\rangle }_{{{j}_{\mu }}}}$ denotes the state m of atom j in cloud μ, and

$$\begin{eqnarray}&&V_{{\rm e}}^{\mu \nu }({{z}_{1}},{{z}_{2}})=\frac{{{C}_{3}}}{{{\left[ {{({{z}_{1}}-{{z}_{2}})}^{2}}+\ell _{\mu \nu }^{2} \right]}^{3/2}}}\left[ 1-3{{{\rm cos} }^{2}}\left( \beta \right)\frac{{{({{z}_{1}}-{{z}_{2}})}^{2}}}{{{({{z}_{1}}-{{z}_{2}})}^{2}}+\ell _{\mu \nu }^{2}} \right]\end{eqnarray} \tag{ 4 }$$

is the interaction strength, ${{C}_{3}}=\langle {{\vec{d}}_{1}}{{\vec{d}}_{2}}\rangle /(4\pi {{\epsilon }_{0}}\hbar )$ is the so called dispersion coefficient, and ${{\ell }_{\mu \nu }}=\ell (1-{{\delta }_{\mu \nu }})$ with ${{\delta }_{\mu \nu }}$ being the Kronecker delta function. To finally achieve the situation where only atoms located in different clouds are interacting, we choose $\beta ={\rm arccos} \left( 1/\sqrt{3} \right)$. As shown in figure 1(c) (red curve) the interaction among atoms of the same cloud is indeed completely suppressed. On the other hand, for atoms belonging to different clouds the interaction exhibits a symmetric peak profile as a function of $\Delta z$ (blue curve) whose full width at half maximum (FWHM) is given by 2z_d, with ${{z}_{d}}=\ell \sqrt{{{4}^{1/5}}-1}$. Finally, we have numerically checked that the van der Waals interactions between the two Rydberg states are negligible for the parameters used in the article.

With these considerations the Hamiltonian of the system in the slowly varying envelope and rotating-wave approximations becomes:

$$\begin{eqnarray}&&\hat{H}=\mathop{\sum }\limits_{\mu =A,B}(\hat{H}_{0}^{(\mu )}+\hat{H}_{I}^{(\mu )})+{{\hat{H}}_{D}}.\end{eqnarray} \tag{ 5 }$$

Here

$$\begin{eqnarray}&&\hat{H}_{0}^{(\mu )}=\mathop{\sum }\limits_{j=1}^{{{N}_{\mu }}}\hbar \left[ {{\omega }_{21}}\hat{\sigma }_{22}^{\left( {{j}_{\mu }} \right)}+{{\omega }_{31}}\hat{\sigma }_{33}^{\left( {{j}_{\mu }} \right)}+{{\omega }_{41}}\hat{\sigma }_{44}^{\left( {{j}_{\mu }} \right)} \right],\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\hat{H}_{I}^{(\mu )}=-\mathop{\sum }\limits_{j=1}^{{{N}_{\mu }}}\hbar \left[ {{\Omega }_{p\mu }}{{{\rm e}}^{-{\rm i}\omega _{p\mu }^{0}t}}\hat{\sigma }_{21}^{\left( {{j}_{\mu }} \right)}+{{\Omega }_{c\mu }}{{{\rm e}}^{-{\rm i}\omega _{c\mu }^{0}t}}\hat{\sigma }_{32}^{\left( {{j}_{\mu }} \right)}+{\rm h}.{\rm c}. \right],\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{{\hat{H}}_{D}}=\mathop{\sum }\limits_{i=1}^{{{N}_{A}}}\mathop{\sum }\limits_{j=1}^{{{N}_{B}}}\hbar V_{{\rm e}}^{AB}\left( {{z}_{i}},{{z}_{j}} \right)\left[ \hat{\sigma }_{34}^{\left( {{i}_{A}} \right)}\hat{\sigma }_{43}^{\left( {{j}_{B}} \right)}+{\rm h}.{\rm c}. \right]\end{eqnarray} \tag{ 8 }$$

and ${{N}_{\mu }}$ is the number of atoms in cloud μ. Moreover, we have defined ${{\omega }_{ij}}\equiv {{\omega }_{i}}-{{\omega }_{j}}$, with ${{\omega }_{j}}$ being the frequency of state $\left| j \right\rangle$ ($\hbar {{\omega }_{1}}$ is taken as the energy reference) and $\hat{\sigma }_{\operatorname{lm}}^{\left( {{j}_{\mu }} \right)}\equiv {{\left| l \right\rangle }_{{{j}_{\mu }}}}{{\left\langle m \right|}_{{{j}_{\mu }}}}$ being the transition operator for atom j, placed at position z_j of cloud μ. The Rabi frequencies for the probe (control) field are defined as ${{\Omega }_{p\mu }}\equiv {{\vec{d}}_{12}}{{\vec{E}}_{p\mu }}/(2\hbar )$ [${{\Omega }_{c\mu }}\equiv {{\vec{d}}_{23}}{{\vec{E}}_{c\mu }}/(2\hbar )$], with ${{\vec{d}}_{12}}$ (${{\vec{d}}_{23}}$)being the dipole moment of the $\left| 1 \right\rangle \leftrightarrow \left| 2 \right\rangle$ ($\left| 2 \right\rangle \leftrightarrow \left| 3 \right\rangle$) transition, and ${{\vec{E}}_{p\mu }}$ (${{\vec{E}}_{c\mu }}$) being the probe (control) electric field, which oscillates at a central frequency $\omega _{p\mu }^{0}$ ($\omega _{c\mu }^{0}$). The spatial and temporal dependencies of the control and probe fields will be considered in the next section.

To obtain the equations of motion of the atomic probability amplitudes we use the fact that the quantum state of the atoms at any time is given by the wavefunction $\left| \psi \left( t \right) \right\rangle =\left| {{\psi }_{1}}\left( t \right) \right\rangle +\left| {{\psi }_{2}}\left( t \right) \right\rangle +\left| {{\psi }_{3}}\left( t \right) \right\rangle$ (a similar approach is for instance pursued in [36]), where the terms at the right hand side have the form

$$\begin{eqnarray}&&\left| {{\psi }_{1}}(t) \right\rangle =\mathop{\sum }\limits_{i}\left[ c_{4}^{\left( {{i}_{A}} \right)}(t)\hat{\sigma }_{41}^{\left( {{i}_{A}} \right)}+c_{4}^{\left( {{i}_{B}} \right)}(t)\hat{\sigma }_{41}^{\left( {{i}_{B}} \right)} \right]\left| {\bar{1}} \right\rangle ,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}\begin{array}{rcl} \left| {{\psi }_{2}}(t) \right\rangle & = & \mathop{\sum }\limits_{i,j}\left[ c_{24}^{\left( {{i}_{A}},{{j}_{B}} \right)}(t)\hat{\sigma }_{21}^{\left( {{i}_{A}} \right)}\hat{\sigma }_{41}^{\left( {{j}_{B}} \right)}+c_{42}^{\left( {{i}_{A}},{{j}_{B}} \right)}(t)\hat{\sigma }_{41}^{\left( {{i}_{A}} \right)}\hat{\sigma }_{21}^{\left( {{j}_{B}} \right)} \right]\left| {\bar{1}} \right\rangle \\ {} & {} & +\mathop{\sum }\limits_{i,j\ne i}\left[ c_{42}^{\left( {{i}_{A}},{{j}_{A}} \right)}(t)\hat{\sigma }_{41}^{\left( {{i}_{A}} \right)}\hat{\sigma }_{21}^{\left( {{j}_{A}} \right)}+c_{24}^{\left( {{i}_{B}},{{j}_{B}} \right)}(t)\hat{\sigma }_{21}^{\left( {{i}_{B}} \right)}\hat{\sigma }_{41}^{\left( {{j}_{B}} \right)} \right]\left| {\bar{1}} \right\rangle , \\ \end{array}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}\begin{array}{rcl} \left| {{\psi }_{3}}(t) \right\rangle & = & \mathop{\sum }\limits_{i,j}\left[ c_{34}^{\left( {{i}_{A}},{{j}_{B}} \right)}(t)\hat{\sigma }_{31}^{\left( {{i}_{A}} \right)}\hat{\sigma }_{41}^{\left( {{j}_{B}} \right)}+c_{43}^{\left( {{i}_{A}},{{j}_{B}} \right)}(t)\hat{\sigma }_{41}^{\left( {{i}_{A}} \right)}\hat{\sigma }_{31}^{\left( {{j}_{B}} \right)} \right]\left| {\bar{1}} \right\rangle \\ {} & {} & +\mathop{\sum }\limits_{i,j\ne i}\left[ c_{43}^{\left( {{i}_{A}},{{j}_{A}} \right)}(t)\hat{\sigma }_{41}^{\left( {{i}_{A}} \right)}\hat{\sigma }_{31}^{\left( {{j}_{A}} \right)}+c_{34}^{\left( {{i}_{B}},{{j}_{B}} \right)}(t)\hat{\sigma }_{31}^{\left( {{i}_{B}} \right)}\hat{\sigma }_{41}^{\left( {{j}_{B}} \right)} \right]\left| {\bar{1}} \right\rangle , \\ \end{array}\end{eqnarray} \tag{ 11 }$$

with $\left| {\bar{1}} \right\rangle \equiv {{\otimes }_{j}}{{\left| 1 \right\rangle }_{{{j}_{A}}}}{{\left| 1 \right\rangle }_{{{j}_{B}}}}$. The sums are taken over all the atoms and the coefficients $c_{\operatorname{lm}}^{({{i}_{\mu }},{{j}_{\nu }})}(t)$ are the many-particle probability amplitudes of the different states. The first term (equation (9)) corresponds to the system having one atom in state $|4\rangle$ and the rest of the atoms being in the ground state $|1\rangle$. The second (equation (10)) and third (equation (11)) terms correspond to the same situation but with an additional excitation in states $|2\rangle$ and $|3\rangle$, respectively.

By inserting equations (9)–(11) into the Schrödinger equation one obtains the evolution equations for the probability amplitudes. Next we change to a frame which rotates at frequency $({{\omega }_{41}}+\omega _{p\mu }^{0})$ for $\left| {{\psi }_{2}} \right\rangle$ and at $({{\omega }_{41}}+\omega _{p\mu }^{0}+\omega _{c\mu }^{0})$ for $\left| {{\psi }_{3}} \right\rangle$. Furthermore, by considering a sufficiently dense atomic gas, we employ a spatially continuous description by introducing continuous versions of the probability amplitudes

$$\begin{eqnarray*}\begin{array}{rcl} c_{24}^{({{i}_{\mu }},{{j}_{\nu }})}(t){{\rho }_{\mu }}(z){{\rho }_{\nu }}(z^{\prime} ){{{\rm e}}^{{\rm i}{{\omega }_{41}}t}}{{{\rm e}}^{{\rm i}\omega _{p\mu }^{0}t}} & \to & \alpha _{24}^{\mu \nu }\left( t,z,z^{\prime} \right), \\ c_{34}^{({{i}_{\mu }},{{j}_{\nu }})}(t){{\rho }_{\mu }}(z){{\rho }_{\nu }}(z^{\prime} ){{{\rm e}}^{{\rm i}{{\omega }_{41}}t}}{{{\rm e}}^{{\rm i}(\omega _{p\mu }^{0}+\omega _{c\mu }^{0})t}} & \to & \alpha _{34}^{\mu \nu }\left( t,z,z^{\prime} \right). \\ \end{array}\end{eqnarray*}$$

These encode the probability of an atom at position z of cloud μ being in state $\left| 2 \right\rangle$ and $\left| 3 \right\rangle$, respectively, and another at z' of cloud ν being in $\left| 4 \right\rangle$ ($\mu ,\nu =\left\{ A,B \right\}$), with the remaining atoms being in the ground state. Here ${{\rho }_{\mu ,\nu }}(z)$ is the linear single-particle density in cloud $\mu ,\nu$. Accordingly, the probability amplitude of an atom being in state $\left| 4 \right\rangle$ is replaced by $c_{4}^{({{i}_{\mu }})}(t){{\rho }_{\mu }}(z){{{\rm e}}^{{\rm i}{{\omega }_{41}}t}}\to \alpha _{4}^{\mu }\left( z,t \right)$. Finally, to simplify the equations we consider only terms up to first order in the probe field Rabi frequency ${{\Omega }_{p\mu }}$ [37]. This implies that ${{\partial }_{t}}\alpha _{4}^{\mu }=0$, i.e., the initial excitation probability amplitude of atoms being in state $|4\rangle$ is a constant in each cloud, $\alpha _{4}^{\mu }(z,t)=\alpha _{4}^{\mu }(z)$. After these manipulations the evolutionequations for the probability amplitudes read

$$\begin{eqnarray}\begin{array}{rcl} {{\partial }_{t}}\alpha _{24}^{\mu \nu }\left( z,z^{\prime} ,t \right) & = & {\rm i}{{\Delta }_{p\mu }}\alpha _{24}^{\mu \nu }\left( z,z^{\prime} ,t \right)+{\rm i}{{\Omega }_{p\mu }}\left( z,t \right)\alpha _{4}^{\nu }\left( z^{\prime} \right) \\ {} & {} & +{\rm i}\Omega _{c\mu }^{*}(z)\alpha _{34}^{\mu \nu }\left( z,z^{\prime} ,t \right), \\ \end{array}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}\begin{array}{rcl} {{\partial }_{t}}\alpha _{34}^{\mu \nu }\left( z,z^{\prime} ,t \right) & = & {\rm i}{{\Delta }_{c\mu }}\alpha _{34}^{\mu \nu }\left( z,z^{\prime} ,t \right)+{\rm i}{{\Omega }_{c\mu }}(z)\alpha _{24}^{\mu \nu }\left( z,z^{\prime} ,t \right) \\ {} & {} & -{\rm i}V_{{\rm e}}^{\mu \nu }\left( z,z^{\prime} \right)\alpha _{34}^{\nu \mu }\left( z^{\prime} ,z,t \right). \\ \end{array}\end{eqnarray} \tag{ 13 }$$

Here we have defined the detunings ${{\Delta }_{c\mu }}={{\delta }_{p\mu }}+{{\delta }_{c\mu }}$, ${{\delta }_{p\mu }}=\omega _{p\mu }^{0}-{{\omega }_{21}}$ and ${{\delta }_{c\mu }}=\omega _{c\mu }^{0}-{{\omega }_{32}}$. Note that the DDI vanishes ($V_{{\rm e}}^{\mu \nu }=0$) when $\mu =\nu$ and then equations (12) and (13) correspond to the usual EIT Bloch equations [2] in the individual cloud.

Finally, the 1D propagation equation for the probe field in clouds A and B, obtained from the Maxwell equations [38], reads

$$\begin{eqnarray}&&\left( {{\partial }_{t}}+c{{\partial }_{z}} \right){{\Omega }_{p\mu }}\left( z,t \right)={\rm i}c{{\kappa }_{\mu }}\mathop{\sum }\limits_{\nu =A,B}\int \alpha _{24}^{\mu \nu }\left( z,z^{\prime} ,t \right)\alpha _{4}^{\nu }{{\left( z^{\prime} \right)}^{*}}{\rm d}z^{\prime} .\end{eqnarray} \tag{ 14 }$$

The integrated expression at the right hand side depends on equations (12) and (13) and corresponds to the macroscopic polarization, expressed in terms of the expectation value of the dipole operator at cloud μ. The coupling constant ${{\kappa }_{\mu }}$ is defined as ${{\kappa }_{\mu }}=d_{12}^{2}\omega _{p\mu }^{0}{{n}_{\mu }}/(2{{\epsilon }_{0}}\hbar c)$, with c being the speed of light in vacuum, ${{n}_{\mu }}\equiv {{N}_{\mu }}/\mathcal{V}$ the atomic density, and $\mathcal{V}$ the quantization volume. The evolution equations equations (12)–(14) are linear in the probe field amplitudes ${{\Omega }_{p\mu }}(z,t)$ and are in fact equivalent to those obtained with a quantum-field description, when interpreting the classical probe field as the single-photon probability amplitude [25].

In order to solve the probe field propagation equation (14), we first obtain $\alpha _{24}^{\mu \nu }$ through the adiabatic elimination [2], i.e., we obtain the corresponding steady solution by forcing ${{\partial }_{t}}\alpha _{24}^{\mu \nu }=0$ in equation (12). Next, we solve equations (13) and (14) in the frequency domain through the Fourier transform $\tilde{f}(\omega )\equiv \int f(t){{{\rm e}}^{-{\rm i}\omega t}}{\rm d}t$. This yields the propagation equation for the probe field amplitude

$$\begin{eqnarray}\begin{array}{rcl} {{\partial }_{z}}{{\tilde{\Omega }}_{p\mu }}\left( z,\omega \right) & = & {\rm i}\chi _{L}^{\mu }\left( z,\omega \right){{\tilde{\Omega }}_{p\mu }}\left( z,\omega \right) \\ {} & {} & +{\rm i}\int \chi _{N}^{\mu }\left( z,z^{\prime} ,\omega \right){{\tilde{\Omega }}_{p\nu }}\left( z^{\prime} ,\omega \right){\rm d}z^{\prime} , \\ \end{array}\end{eqnarray} \tag{ 15 }$$

where $\chi _{L}^{\mu }$ and $\chi _{N}^{\mu }$ are referred to as the local and non-local susceptibilities, respectively [25]. We then define ${{\Omega }_{c\mu }}(z)=\left| {{\Omega }_{c\mu }} \right|{{{\rm e}}^{{\rm i}({{k}_{c}}z+{{\varphi }_{c\mu }})}}$ and $\alpha _{4}^{\mu }(z)=\left| \alpha _{4}^{\mu }(z) \right|{{{\rm e}}^{{\rm i}{{k}_{s}}z}}$, with ${{k}_{c}}$ and ${{k}_{s}}$ being the wavevectors of the control field and the initial spinwave excitation, equation (1), respectively. After imposing equal parameters for the two clouds, i.e., ${{\delta }_{pA}}={{\delta }_{pB}}={{\delta }_{p}}$, ${{\Delta }_{pA}}={{\Delta }_{pB}}={{\Delta }_{p}}$, ${{\delta }_{cA}}={{\delta }_{cB}}={{\delta }_{c}}$, ${{N}_{A}}={{N}_{B}}=N$, ${{\rho }_{A}}={{\rho }_{B}}=\rho$, ${{\kappa }_{A}}={{\kappa }_{B}}=\kappa$, and $\left| {{\Omega }_{cA}} \right|=\left| {{\Omega }_{cB}} \right|={{\Omega }_{c}}$, the explicit forms of $\chi _{L}^{\mu }$ and $\chi _{N}^{\mu }$ read:

$$\begin{eqnarray}&&\chi _{L}^{\mu }\left( z,\omega \right)=-\frac{\omega }{c}-\frac{\kappa }{{{\Delta }_{p}}}-\kappa \frac{\Omega _{c}^{2}}{\Delta _{p}^{2}}\mathop{\sum }\limits_{\nu =A,B}\int \frac{{{\left| \alpha _{4}^{\nu }\left( z^{\prime} \right) \right|}^{2}}{{\Delta }_{s}}(\omega )}{{{\Delta }_{s}}{{(\omega )}^{2}}-V_{{\rm e}}^{\mu \nu }{{\left( z,z^{\prime} \right)}^{2}}}{\rm d}z^{\prime} ,\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\chi _{N}^{\mu }\left( z,z^{\prime} ,\omega \right)=-\kappa \frac{\Omega _{c}^{2}}{\Delta _{p}^{2}}\mathop{\sum }\limits_{\nu =A,B}\frac{{{{\rm e}}^{{\rm i}{{\phi }_{\mu \nu }}\left( z,z^{\prime} \right)}}\left| \alpha _{4}^{\nu }\left( z^{\prime} \right)\alpha _{4}^{\mu }\left( z \right) \right|V_{{\rm e}}^{\mu \nu }\left( z,z^{\prime} \right)}{{{\Delta }_{s}}{{(\omega )}^{2}}-V_{{\rm e}}^{\mu \nu }{{\left( z,z^{\prime} \right)}^{2}}}.\end{eqnarray} \tag{ 17 }$$

In the derivation of equations (16) and (17), we have abbreviated

$$\begin{eqnarray}&&{{\Delta }_{s}}(\omega )={{\delta }_{p}}+{{\delta }_{c}}-\omega -\frac{\Omega _{c}^{2}}{{{\Delta }_{p}}},\end{eqnarray} \tag{ 18 }$$

which determines the position and width of the absorption peaks in the EIT for the non-interacting case [2]. Furthermore, ${{\phi }_{\mu \nu }}\left( z,z^{\prime} \right)$ is the phase relation (in equation (17)) between the phases of the control fields and that of the initial spinwave in each cloud, which reads

$$\begin{eqnarray}&&{{\phi }_{AB}}\left( z,z^{\prime} \right)=-{{\phi }_{BA}}\left( z^{\prime} ,z \right)=-({{k}_{c}}-{{k}_{s}})(z-z^{\prime} )-{{\varphi }_{AB}}.\end{eqnarray} \tag{ 19 }$$

Here, z and z' refer to positions of atoms in clouds μ and ν, respectively, and ${{\varphi }_{AB}}={{\varphi }_{cA}}-{{\varphi }_{cB}}$. We have made the control field phase difference ${{\varphi }_{AB}}$ explicit in equation (19), since it allows to control the PSS filtering in our setup as we will show in the next section.

The propagation equation for the PSS single photon, equation (15), is one of the central results of this work. It shows that the single-photon propagation is determined by the usual local susceptibility $\chi _{L}^{\mu }$ but also by a non-local part $\chi _{N}^{\mu }$ [25]. The latter leads, by virtue of the DDI, to a coupling of the photon dynamics between the two clouds. In the following we will demonstrate that the non-local susceptibility renders exotic photon dynamics.

In order to obtain a qualitative understanding of the light propagation, we derive an analytical solution of equation (15) by carrying out additional approximations. First, we approximate the DDI profile by a square function $\tilde{V}_{{\rm e}}^{AB}$ (see figure 1(c)) of height V₀ and width equal to the FWHM of the actual DDI strength 2z_d. Next, we perform a local-field approximation [25, 39], i.e., we take the field ${{\tilde{\Omega }}_{p\nu }}\left( z^{\prime} ,\omega \right)$ out of the integral in equation (15) by assuming that the shape of the probe pulse does not change significantly within 2z_d and neglecting boundary effects. Moreover, for simplicity we impose the condition ${{k}_{c}}-{{k}_{s}}=0$, which can be obtained by properly aligning the spinwave excitation lasers and the control fields. This leads to ${{\phi }_{\mu \nu }}\left( z,z^{\prime} \right)=-{{\varphi }_{\mu \nu }}$, according to equation (19). Finally, we Taylor expand equations (16) and (17) up to first order in ω, i.e., we assume a narrow-band probe pulse. With these approximations, and defining the boundary conditions for the field as ${{\tilde{\Omega }}_{p\mu }}(z=0,\omega )=\tilde{\Omega }_{p\mu }^{(0)}(\omega )$, the solutions of the propagation equation (15) are

$$\begin{eqnarray}\begin{array}{rcl} {{\tilde{\Omega }}_{pA}}\left( z,\omega \right) & = & \frac{\tilde{\Omega }_{pA}^{(0)}\left( \omega \right)+{{{\rm e}}^{-{\rm i}{{\varphi }_{AB}}}}\tilde{\Omega }_{pB}^{(0)}\left( \omega \right)}{2}{{{\rm e}}^{{\rm i}\left( {{\eta }_{+}}-\frac{\omega }{{{v}_{+}}} \right)z}} \\ {} & {} & +\frac{\tilde{\Omega }_{pA}^{(0)}\left( \omega \right)-{{{\rm e}}^{-{\rm i}{{\varphi }_{AB}}}}\tilde{\Omega }_{pB}^{(0)}\left( \omega \right)}{2}{{{\rm e}}^{{\rm i}\left( {{\eta }_{-}}-\frac{\omega }{{{v}_{-}}} \right)z}}, \\ \end{array}\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}\begin{array}{rcl} {{\tilde{\Omega }}_{pB}}\left( z,\omega \right) & = & \frac{{{{\rm e}}^{{\rm i}{{\varphi }_{AB}}}}\tilde{\Omega }_{pA}^{(0)}\left( \omega \right)+\tilde{\Omega }_{pB}^{(0)}\left( \omega \right)}{2}{{{\rm e}}^{{\rm i}\left( {{\eta }_{+}}-\frac{\omega }{{{v}_{+}}} \right)z}} \\ {} & {} & -\frac{{{{\rm e}}^{{\rm i}{{\varphi }_{AB}}}}\tilde{\Omega }_{pA}^{(0)}\left( \omega \right)-\tilde{\Omega }_{pB}^{(0)}\left( \omega \right)}{2}{{{\rm e}}^{{\rm i}\left( {{\eta }_{-}}-\frac{\omega }{{{v}_{-}}} \right)z}}, \\ \end{array}\end{eqnarray} \tag{ 21 }$$

where

$$\begin{eqnarray}&&{{\eta }_{\pm }}=-\frac{\kappa }{{{\Delta }_{p}}}-\kappa \frac{\Omega _{c}^{2}}{\Delta _{p}^{2}}\left[ \frac{{{z}_{d}}\rho }{{{\Delta }_{s}}(0)\mp {{V}_{0}}}+\frac{1-{{z}_{d}}\rho }{{{\Delta }_{s}}(0)} \right],\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&v_{\pm }^{-1}=\frac{1}{c}+\kappa \frac{\Omega _{c}^{2}}{\Delta _{p}^{2}}\left[ \frac{{{z}_{d}}\rho }{{{\left[ {{\Delta }_{s}}(0)\mp {{V}_{0}} \right]}^{2}}}+\frac{1-{{z}_{d}}\rho }{{{\Delta }_{s}}{{(0)}^{2}}} \right].\end{eqnarray} \tag{ 23 }$$

For simplicity the single-particle linear density $\rho (z)$ has been considered constant along the medium, i.e., $\rho =1/L$. The two terms inside the brackets at the right hand side of equations (22) and (23) are important to determine the light propagation. Here the first term depends on the interaction strength V₀ between the atoms belonging to different clouds, while the second one corresponds to the conventional EIT contribution found in interaction-free atomic gases.

Assuming ${{\delta }_{p}}\gg \gamma$, we obtain the analytical solution of the photon field by performing the inverse Fourier transform of equations (20) and (21)

$$\begin{eqnarray}&&{{\Omega }_{\pm }}\left( z,t \right)={{{\rm e}}^{{\rm i}{{\eta }_{\pm }}z}}\Omega _{\pm }^{(0)}\left( t-\frac{z}{{{v}_{\pm }}} \right).\end{eqnarray} \tag{ 24 }$$

Here ${{\Omega }_{+}}$ and ${{\Omega }_{-}}$ are the symmetric and antisymmetric superpositions (or normal modes, by analogy with the results in [40]) of the probe field in the two clouds, defined as

$$\begin{eqnarray}&&{{\Omega }_{\pm }}\left( z,t \right)\equiv {{\Omega }_{pA}}\left( z,t \right)\pm {{{\rm e}}^{-{\rm i}{{\varphi }_{AB}}}}{{\Omega }_{pB}}\left( z,t \right).\end{eqnarray} \tag{ 25 }$$

These normal modes form an orthogonal basis, whose specific form depends on the phase difference between the control fields ${{\varphi }_{AB}}$, allows us to describe the photon propagation in a simple manner. In particular, from equation (24) we see that the quantities ${\rm Im}[{{\eta }_{\pm }}]$ and ${{v}_{\pm }}$ defined in equations (22) and (23) are, respectively, the absorption coefficient and propagation velocity of the normal modes, equation (25), inside the medium.

Therefore, the solutions presented in equation (24) describe the propagation of two orthogonal PSS for the single photon whose absorption/transmission and refractive properties can be controlled by tuning the system parameters. To exemplify this, we plot in figure 2(a) the imaginary part of ${{\eta }_{\pm }}L$ (i.e., the optical depth for the ${{\Omega }_{\pm }}$ solutions), with blue and red dashed lines, respectively, as a function of the probe detuning ${{\delta }_{p}}$, for ${{\delta }_{c}}=0$. When ${{V}_{0}}=0$ one recovers the usual transparency window of the EIT [2] (shown with black solid line) since ${{\eta }_{+}}={{\eta }_{-}}$, and thus both components ${{\Omega }_{\pm }}$ are equally absorbed. In the presence of DDI this feature of non-interacting EIT is still visible (see blue and red peaks marked by black arrows). However, the interaction term in (see equation (22)) gives rise to two additional pairs of EIT peaks (blue and red dashed peaks marked by correspondingly colored arrows). Those are shifted in opposite directions according to the sign of the interaction strength ±V₀, which acts as an effective detuning on top of the control field detuning.

Zoom In Zoom Out Reset image size

For comparison, we also consider an approximated solution for equation (15) using the actual DDI potential, equation (4), and again the local-field approximation. For this situation we can still express the solutions for the field propagation in terms of the symmetric and antisymmetric modes

$$\begin{eqnarray}&&{{\tilde{\Omega }}_{\pm }}\left( L,\omega \right)={\rm exp} \left[ {\rm i}\int _{0}^{L}X_{\pm }^{\mu }(z){\rm d}z \right]\tilde{\Omega }_{\pm }^{(0)}\left( \omega \right),\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&X_{\pm }^{\mu }(z)=\chi _{L}^{\mu }\left( z,0 \right)\pm \int _{0}^{L}\chi _{N}^{\mu }\left( z,z^{\prime} ,0 \right){\rm d}z^{\prime} .\end{eqnarray} \tag{ 27 }$$

Since we are only interested in the absorption of the medium we have approximated $X_{\pm }^{\mu }(z)$ to zeroth order in ω. By doing so, we assume that the medium response is the same for all the photon frequency components, i.e, we consider the continuous wave regime, and we neglect dispersion effects. The integral in the exponent at the right hand side of equation (26) determines the total absorption of the medium for the probe light, i.e., the optical depth. For comparison with the data shown in figure 2(a), we plot in figure 2(b) the imaginary part of this quantity $\int _{0}^{L}X_{\pm }^{\mu }(z){\rm d}z$, with blue and red dashed lines for the + and − cases, respectively. The integral is performed numerically, using the same parameters as in figure 2(a). We observe the same peak positions as in figure 2(a), indicated by colored arrows (the non-interacting case is again presented with a black solid line). However, we notice that those peaks which are shifted from the non-interacting resonance (red and blue arrows) are widened and their height has decreased with respect to the peaks obtained from the square potential approach. This difference is caused by the variation of the interaction potential depending on the different interatomic distances, which effectively acts as a position dependent control detuning. Thus, the integration of the susceptibility over z produces an inhomogeneous broadening around the interaction peaks (blue and red arrows). This broadening is not seen around the peaks indicated by black arrows, because for such probe detuning ${{\delta }_{p}}$ the contribution from atoms being in the same cloud dominates over the contribution from interacting atoms.

From figure 2, we realize that the opposite shifts experienced by the red and blue peaks give rise to a large difference between the absorption coefficients of the two normal modes in equation (25). Thus the probe detuning ${{\delta }_{p}}$ represents a handle to control the transmission and absorption of PSS. For instance, if we tune our system with a probe detuning ${{\delta }_{p}}$ close to any of the blue arrows, the symmetric mode ${{\Omega }_{+}}$ will experience a much higher absorption than the antisymmetric superposition ${{\Omega }_{-}}$, rendering the former the 'bright state' and the latter the 'dark state'. An example of this situation will be demonstrated numerically in the next section. If on the contrary the system is tuned close to one of the red shifted peaks, the role of 'bright' and 'dark' state will be swapped. Moreover, we have also the additional control parameter, the phase difference of the control fields ${{\varphi }_{AB}},$ which rotates the basis that defines the symmetric and antisymmetric normal modes in equation (25). This parameter then allows to choose which is the specific form of the superposition state at the output of the medium.

In the following section we perform numerical simulations of the light propagating in the two clouds considering the actual interaction potential, given by expression equation (4), and compare the results with the analytical solution. For the simulations, we use a spatio-temporal grid to propagate the optical-Bloch equations, equations (12)–(14) and assume the probe Gaussian pulse much wider than the length of the medium L. For the atoms, we have used parameters corresponding to states $|3\rangle \equiv |100S,J={{m}_{J}}=1/2\rangle$ and $|4\rangle \equiv |100P,J=3/2,{{m}_{J}}=1/2\rangle$ of ⁸⁷Rb. The remaining parameters are: ${{V}_{0}}=10\gamma$, $\ell =0.5L$, $\kappa =9\gamma /L$, ${{\Omega }_{c}}=10\gamma$, ${{\delta }_{p}}=-6.5\gamma$, and ${{\delta }_{c}}=0$, with $\gamma \simeq 38$ MHz and $L=15\;\mu$ m, similar to those used in magneto-optical trap experiments [5, 41, 42]. These parameters correspond to the blue peaks in figures 2(a) and (b), marked by the blue arrows at negative probe detuning ${{\delta }_{p}}$.

For the simulations we select two situations corresponding to different initial PSS entering the medium. First, we consider the case where the photon is only entering cloud A, i.e., $\int |\Omega _{pA}^{(0)}(t){{|}^{2}}|{\rm d}t=1$ and $\int |\Omega _{pB}^{(0)}(t){{|}^{2}}|{\rm d}t=0$, and the relative phase between the control fields is ${{\varphi }_{AB}}=0$. This means that the input state in the normal mode basis is $\Omega _{\pm }^{(0)}(t)=\Omega _{pA}^{(0)}(t)$. In figure 3 we plot the probability ${{P}_{\pm }}(z)$ of finding the photon in the symmetric (blue circles) or antisymmetric (red circles) superposition states, defined as

$$\begin{eqnarray}&&{{P}_{\pm }}(z)=\frac{\int {{\left| {{\Omega }_{\pm }}\left( z,t \right) \right|}^{2}}{\rm d}t}{\int \left[ {{\left| \Omega _{+}^{(0)}\left( t \right) \right|}^{2}}+{{\left| \Omega _{-}^{(0)}\left( t \right) \right|}^{2}} \right]{\rm d}t},\end{eqnarray} \tag{ 28 }$$

where $\Omega _{\pm }^{(0)}\left( t \right)$ are defined in equations (24) and (25). The analytical result is also plotted with solid lines, using equations (26) and (27). From the figure we observe that during propagation the symmetric superposition ${{\Omega }_{+}}$ is strongly absorbed compared to the antisymmetric component ${{\Omega }_{-}}$. Thus the behavior of the light intensity in the medium is well described by the analytical model. The second situation considered corresponds to the propagation of two identical path-components with no initial phase difference between them, i.e., $\Omega _{pB}^{(0)}(t)=\Omega _{pA}^{(0)}(t)$ such that $\int |\Omega _{pA}^{(0)}(t){{|}^{2}}|{\rm d}t=\int |\Omega _{pB}^{(0)}(t){{|}^{2}}|{\rm d}t=1/2$ and $\int {\rm arg} [\Omega _{pB}^{(0)}(t)\Omega _{pA}^{(0)}{{(t)}^{*}}]{\rm d}t=0$. For the phase difference between the control fields we choose ${{\varphi }_{AB}}=\pi /2$. The corresponding probabilities are shown in the same figure 3 with blue and red crosses. For this situation the results are again in agreement with the analytical model. In both situations presented in figure 3, the small discrepancies found in the absorption profile are mainly due to boundary effects, which have been neglected in equation (26). Thus this system can be used to filter a particular PSS.

Zoom In Zoom Out Reset image size

Finally, we would like to note that similar light propagation effects have been reported in the context of coherently coupled atomic systems. In particular, the propagation of matched pulses [40, 43–49] has been widely investigated. In this propagation effect, two different polarization [45, 49] or frequency [43, 47, 48] components which are coupled to adjacent atomic transitions are able to interchange energy as they propagate. In general, different conditions have to be satisfied in order to observe this phenomenon. First, the two light components must be coupled to a common bare state of, for instance, Λ, V, or double-Λ systems, which provides a path for the light to be exchanged. Also, an atomic coherence between the bare states that involve the two-photon transition is required. This atomic coherence is established by the propagating pulses themselves when they are intense enough, investing part of their energy in creating the dark atomic superposition state. However, for weak or single-photon pulses this coherence must be created beforehand, e.g., by preparing the medium in a phaseonium state [49, 50]. In all the cases, once this coherence is built up, the light components evolve into a dark superposition state, sometimes called simulton [43] or adiabaton [46], that propagates transparently, while the orthogonal superposition is absorbed into the medium. Analogously, we have considered here a system prepared initially in a certain coherent state, i.e., the initial spinwave equation (1), and a two-component light pulse, in a PSS, propagating through the medium. In our system, the exchange interaction creates pair states of the form $|\pm \rangle =|3{{\rangle }_{{{i}_{A}}}}|4{{\rangle }_{{{j}_{B}}}}\pm |4{{\rangle }_{{{i}_{A}}}}|3{{\rangle }_{{{j}_{B}}}}$, so called excitonic states [25]. These are the common states in our situation that allows the light to be transferred from one cloud to another. This is the reason for which we observe eventually matched pulse propagation. However, the novelty here with respect to previous studies is that, for the first time to our knowledge, we describe matched propagation of components that are delocalized in space.