Non-Hermitian Hamiltonians arise in quantum systems undergoing particle or information loss to their environment [1,2], and are responsible for rich and exotic non-Hermitian phenomena such as parity-time (PT) symmetry [3–10], non-Hermitian criticality [11–15], and non-Hermitian skin effects [16–24] and topology [16,25–29]. So far, non-Hermitian Hamiltonians have been experimentally implemented in quantum systems including single photons [30–32], atomic gases [33–40], semiconductor microcavities [41], nuclear spins in solids [42,43], trapped ions [44,45], and superconducting qubits [46]. In most of these experiments, non-Hermiticity is introduced through postselection under which quantum jump processes are irrelevant. The resulting conditional dynamics is driven by a non-Hermitian effective Hamiltonian, and is thus probability-non-conserving. By contrast, a unique experimental realization of non-Hermiticity exists in warm atomic-vapor cells where atomic coherences, also called spin waves [47], in spatially separated optical channels are dissipatively coupled according to the optical Bloch equations. Rather than direct particle or energy dissipation, the loss therein corresponds to the decaying atomic coherence under atomic thermal motion. In a prior series of experiments with atomic-vapor cells, (anti-)PT phases and phase transitions have been observed wherein the coexistence of the PT-related criticality and the quantum nature of the coherence coupling [48,49] offers intriguing prospects for applications in quantum control and device design. In these pioneering experiments, the coupling is nevertheless fixed to be dissipative, whereas it is desirable for practical purposes that the non-Hermiticity and the associated exotic features should be made tunable and on-demand.

In this work, we experimentally demonstrate, in a warm atomic-vapor cell, an easily switchable non-Hermitian coupling that can be either dissipative or coherent. As illustrated in Fig. 1, our setup consists of a pair of optically illuminated regions, or the optical channels, within an ensemble of warm atoms. The dissipative coupling between the spatially separated light fields is mediated by the atomic motion transporting and exchanging atomic coherence (that carries the information of light) within the two optical channels, and we identify atoms outside the illuminated regions as a non-Markovian reservoir. Introducing a far-detuned laser beam (denoted as the “light wall”) into the reservoir, we achieve a tunable inter-channel coupling, such that the beam-splitter-type [50–52] interaction between the two channels can be captured by either a non-Hermitian or a Hermitian effective Hamiltonian, depending on the light-wall parameters. We confirm the tunability of the effective Hamiltonian by characterizing the light-wall-induced phase shift through the inter-channel light-transport measurements. We then discuss an experimentally accessible scheme in which the configuration implemented here can be used as a basic building block for more involved studies of non-Hermitian criticality. Our experiment therefore not only offers a novel scheme for optical device design, but also provides a flexible tool for the quantum simulation of non-Hermitian physics.

For our experiment, we use a paraffin-wall-coated [53–55] ${}^{87}\mathrm{Rb}$ vapor cell at a temperature of 40°C, housed within a four-layer magnetic shield to screen out the ambient magnetic field. As shown in Fig. 1, external lasers create two spatially separated optical channels (labeled Ch1 and Ch2) with inter-channel distance $\sim 1\mathrm{cm}$, where atoms in each channel undergo standard $\mathrm{\Lambda }$-type electromagnetically induced transparency (EIT). Specifically, an external cavity diode laser provides the light for the probe and control fields with orthogonal circular polarizations that couple the ground-state Zeeman levels $|F=2,{\mathit{m}}_F=0⟩$ (labeled as $|1⟩$) and $|F=2,{\mathit{m}}_F=2⟩$ (labeled as $|2⟩$) to an excited state $|F^{\prime }=1,{\mathit{m}}_{F^{\prime }}=1⟩$ (labeled as $|3⟩$) of the ${\mathrm{D}}_1$ line. The control and probe beams have a diameter of about 1.5 mm, with input power of 80 μW and 8 μW, respectively. Between the two channels, a circularly polarized far-off-resonance red-detuned laser beam is shone through the vapor cell, with an elliptical cross section, about 2.5 cm in length (same as the diameter of the vapor cell) and 7 mm in width. We note that, since the control laser here is relatively weak, we have EIT instead of Autler–Townes splitting (ATS), which is a phenomenon occurring at a much higher laser power [56–60]. As shown in Section 4, the measured linewidth has a linear dependence on the laser power, and is below 100 Hz. Since the linewidth is much less than the excited-state linewidth ($\sim 500\mathrm{MHz}$ Doppler broadened), our measurement is consistent with that of the EIT.

To measure the EIT spectra of a given channel, we record the probe field output intensity while sweeping a homogeneous magnetic field generated by a solenoid inside the magnetic shield. By contrast, when comparing the probe output of both channels as we sweep the probe’s phase in one of the channels, the magnetic field is switched off.

Under the optical Bloch equations, atomic coherences between the Zeeman states $|1⟩$ and $|2⟩$ in the two optical channels are dissipatively coupled with each other, as atoms traverse the reservoir under thermal motion. As a key element of our experiment, we introduce a far-detuned light wall in the reservoir, which shifts the hyperfine energy levels in a state-selective fashion, and imprints an extra phase onto the coherence as atoms pass through.

Specifically, the equations of motion for the atomic coherences satisfy $$\{\begin{array}{c}\begin{array}{l}{\dot{\rho }}_{12}^{(1)}=-{\gamma }_{12}^{\prime }{\rho }_{12}^{(1)}+{\mathrm{\Gamma }}_c{\rho }_{12}^{(2)}-\frac{{\mathrm{\Omega }}_c^{(1)*}{\mathrm{\Omega }}_p^{(1)}}{{\gamma }_{23}},\hfill \\ {\dot{\rho }}_{12}^{(2)}=-{\gamma }_{12}^{\prime }{\rho }_{12}^{(2)}+{\mathrm{\Gamma }}_c{\rho }_{12}^{(1)}-\frac{{\mathrm{\Omega }}_c^{(2)*}{\mathrm{\Omega }}_p^{(2)}}{{\gamma }_{23}},\hfill \end{array}\end{array}$$(1)where ${\rho }_{12}^{(i)}$ ($i=\mathrm{1,}2$) is the ground-state coherence of the $i$th channel, whose total effective decay rate is ${\gamma }_{12}^{\prime }={\gamma }_{12}+{\mathrm{\Gamma }}_c+{\mathrm{\Gamma }}_p^{(1)}+{\mathrm{\Gamma }}_p^{(2)}$, where ${\mathrm{\Gamma }}_p^{(i)}=\frac{{|{\mathrm{\Omega }}_c^{(i)}|}^2}{{\gamma }_{23}}$ is the optical pumping rate, with ${\gamma }_{12}$ and ${\gamma }_{23}$ the decay rate of the coherence between states $|1⟩$, $|2⟩$ and $|2⟩$, $|3⟩$, respectively. ${\mathrm{\Omega }}_c$ and ${\mathrm{\Omega }}_p$ are the Rabi frequencies of the control and probe fields, respectively. Importantly, in the presence of the light wall, the inter-channel coupling rate ${\mathrm{\Gamma }}_c$ is dressed by an extra phase ${\theta }_0$, and replaced by ${\mathrm{\Gamma }}_c{e}^{i{\theta }_0}$.

The dissipative inter-channel coupling above gives rise to a beam-splitter-type interaction [61] effectively described by the Hamiltonian $$\begin{array}{c}\widehat{H}=\hslash (g{\widehat{a}}_1^{†}{\widehat{a}}_2-g^{*}{\widehat{a}}_2^{†}{\widehat{a}}_1)e^{i{\theta }_0},\end{array}$$(2)where ${\widehat{a}}_1$ (${\widehat{a}}_2$) is the annihilation operator for the probe field in Ch1 (Ch2), and $g$ is a complex coupling coefficient, with its phase given by ${\varphi }_c^{(1)}-{\varphi }_c^{(2)}$, where ${\varphi }_c^{(\mathrm{1,2})}$ are the phases of the control fields in the corresponding optical channel. While ${\theta }_0=0$ in the absence of the light wall, its value is easily tunable by adjusting the intensity or detuning of the laser generating the light wall. Notably, when ${\theta }_0=\pi /2$, the Hamiltonian [Eq. (2)] becomes Hermitian.

To experimentally confirm the analysis above, we first characterize the property of the light wall. In a paraffin-coated cell, the far-detuned beam of the light wall gives rise to a non-local, state-selective energy shift. This is because atoms can fly through the laser beams many times by bouncing off the vapor-cell wall, their ground-state coherence nearly intact. The light wall is therefore equivalent to an inhomogeneous global magnetic field that shifts and inevitably broadens the EIT spectrum. The impact of the light wall on the EIT spectrum is shown in Fig. 2, where the experimentally observed EIT spectra in Fig. 2(a) agree well with those from Monte-Carlo numerical simulations [40] in Fig. 2(b). Further, the observed spectral shift is proportional to the laser power [see Fig. 2(c)], while inversely proportional to its detuning [see Fig. 2(d)]. These observations derive from the phase imposed by the light wall, and form the basis for our control scheme below.

The effective Hamiltonian [Eq. (2)] governs the coupling-related evolution of the atomic coherences (or, equivalently, the probe fields) as the light traverses the vapor cell. Its impact therefore can be probed through the light transport where the light-wall-induced phase shift ${\theta }_0$ manifests itself in the resulting intra- and inter-channel interference. To probe this phase shift, we first turn off the weak probe in Ch2 and slowly sweep the phase of the probe field in Ch1. In this case, the measured output probe field in Ch2 directly corresponds to the light transported from Ch1, thus containing information of the light-wall-induced phase shift.

We interfere a small fraction of the control fields with the probes using a half-wave plate in the output of each channel. The light-wall-induced phase is manifested in the phase shift between the measured output intensities of the two channels [Figs. 3(a)–3(c)], where $I_1\propto \cos {\theta }_1$ and $I_2\propto \cos ({\theta }_0+{\theta }_1)$, consistent with theoretical predictions based on the Hamiltonian [Eq. (2)] [49]. Here ${\theta }_1={\varphi }_p^{(1)}-{\varphi }_c^{(1)}$ and ${\varphi }_p^{(\mathrm{1,2})}$ are the phases of the input probe fields of the corresponding channels. When the laser power of the light wall increases, the phase shift should also increase, which is observed in Fig. 3. For a sufficiently large laser power of 30 mW, the phase shift can reach $\pi /2$, when the beam-splitter-type interaction becomes Hermitian. We note that the light-wall-induced phase ${\theta }_0$ exhibits saturation behavior with increasing laser power, while the light-wall-induced EIT spectral shift is linear in laser power. This is because the phase shift ${\theta }_0$ is approximately the product of the spectral shift and the effective interaction time between the atoms and the light wall. The interaction time is roughly the coherence lifetime and is inversely proportional to the EIT linewidth, which is broadened by the light-wall-induced effective magnetic field, as shown in Figs. 2(a) and 2(b). We have confirmed this analysis by reproducing the saturation behavior using Monte-Carlo simulations.

To further confirm the impact of the light wall, we study the output probe intensities without interfering it with the control fields, while both probe fields in Ch1 and Ch2 are switched on. As the phase of the input probe in Ch1 is slowly swept, we record the output probe fields’ intensities in Ch1 and Ch2 separately, which, according to our theoretical derivations, should be $I_1\propto \cos ({\theta }_1-{\theta }_2-{\theta }_0)$ and $I_2\propto \cos ({\theta }_2-{\theta }_1-{\theta }_0)$, respectively. Here ${\theta }_2={\varphi }_p^{(2)}-{\varphi }_c^{(2)}$ and ${\varphi }_{p,c}^{(2)}$ are the corresponding phases of the probe and control fields in Ch2. Apparently, the unsynchronized intensity output of the two channels originates from the phase interference of two processes: one is the reading and writing of the ground-state coherence by the control and the probe fields, featuring direction-dependent phases ${\theta }_1-{\theta }_2$ and ${\theta }_2-{\theta }_1$, respectively, and the other the direction-independent phase ${\theta }_0$ from the light wall. This scheme is closely related to a recent proposal on nonreciprocity [62]. As shown in Fig. 4, the experimental observations agree well with theoretical predictions. In the absence of the light wall [see Fig. 4(a)], the two output probes change in a synchronized way; with the addition of the light wall [see Fig. 4(b)], the output intensities of the probes display a phase lag. It is worth noting that, compared to the case in Fig. 3, now the phase lag is $2{\theta }_0$. Under a higher laser power, the phase lag approaches $\pi$, demonstrating a fully out-of-phase behavior as shown in Fig. 4(c), which recovers the property of a conventional beam splitter (BS) commonly used in optical interferometry experiments. However, the remaining difference from the conventional Hermitian BS is that the light-wall-induced “Hermitian” BS here suffers additional loss, and is a manifestation of the Kramers–Kronig relation. Namely, the change in the probe field’s phase (due to the ground-state coherence’s phase change by the light wall) is associated with the additional absorption in the probe field.

The configuration demonstrated here serves as a flexible building block in implementing more complicated non-Hermitian models for the study of exotic non-Hermitian criticality or topology. As a concrete example, we propose a minimal setup that involves three optical channels A, B, and C, and is readily accessible in an experiment using vacuum vapor cells without wall-coating. In such cells, adjacent optical channels couple through ballistic diffusion such that the next-nearest-neighbor coupling can be neglected [63]. As illustrated in Fig. 5(a), we assume that, between channels A and B, the phase factor in the coupling term $g_0$ is 1, and the light-wall-induced coherence’ phase shift is ${\theta }_0$; between channels B and C, the phase factor in the coupling term $g_1$ is $-i$, and the light-wall-induced phase shift is ${\theta }_1=\pi /2$. Then, the effective non-Hermitian Hamiltonian is $$\widehat{H}=\delta {\widehat{a}}^{†}\widehat{a}+g_0{e}^{i{\theta }_0}{\widehat{b}}^{†}\widehat{a}-g_0{e}^{i{\theta }_0}{\widehat{a}}^{†}\widehat{b}+g_1{\widehat{c}}^{†}\widehat{b}+g_1{\widehat{b}}^{†}\widehat{c},$$(3)where $\widehat{a}$, $\widehat{b}$, and $\widehat{c}$ are the annihilation operators for the probe beams in channels $\mathrm{A}$, $\mathrm{B}$, and $\mathrm{C}$, respectively; the coupling terms $g_0$ and $g_1$ can be tuned by the distance between the channels as well as the laser beam size, and the on-site energy shift $\delta$ can be created through AC-Stark shift generated by an off-resonance laser beam within channel A [48]. As shown in Fig. 5, the non-Hermitian Hamiltonian [Eq. (3)] features highly tunable exceptional points, and offers an accessible minimal configuration where intriguing non-Hermitian phenomena such as higher-order exceptional point and exceptional-point encircling can be systematically probed using atomic-vapor cells. Specifically, the Hamiltonian has PT symmetry for ${\theta }_0=0$ and $\delta =0$, whereas exceptional points are tunable through $\delta$ [see Fig. 5(c)]. The PT symmetry is broken when ${\theta }_0$ deviates from 0 or $\delta \ne 0$ [see Figs. 5(d)–5(f)], offering a sensitive control for the study of non-Hermitian criticality at the exceptional points.

In conclusion, tunable non-Hermitian coupling between light modes is demonstrated in an atomic ensemble with the assistance of atomic motion and a light wall in the reservoir. The atomic spin wave picks up an extra phase when travelling through the light wall. The non-Hermiticity of the corresponding Hamiltonian is controlled by adjusting laser parameters of the light wall. While we confirm the tunability of the system through light-transport measurements, our setup can be applied as a building block for applications in quantum simulation of non-Hermitian physics and nonreciprocal devices [62,64–68]. Compared to existing studies of tuning non-Hermiticity in optical cavities and laser-array systems [69,70], our experiment is based on atom-optic coupling, which enables future quantum optical applications. In a recent experiment, the Hermiticity of a magnon–photon beam splitter in cold atoms was tuned by varying the laser detuning [61], while our method is suitable for spatial splitting of light and potential large-scale spatial multiplexing of quantum light sources [71].

In order to show that our experiment is in the regime of EIT, not ATS, we have measured the EIT linewidth as a function of the laser power of the control field, in the range that covers our experimental condition. As shown in Fig. 6, the linewidth has a linear dependence on the laser power, and is less than 100 Hz, much narrower than the excited-state linewidth ($\sim 500\mathrm{MHz}$ Doppler broadened). These features are in contrast to those of ATS, whose linewidth is larger than the excited-state linewidth, and is proportional to the Rabi frequency (square root of the laser power) of the control field.

We establish a model to describe the inter-channel coupling. We start from the full optical Bloch equations where the coherences and populations between any two atomic levels are included; then we adiabatically eliminate the excited-state dynamics. We further make the approximation that the populations of states $|1⟩$ and $|2⟩$ are 0 and 1, respectively, because the population of the excited state is nearly zero (since the control field’s Rabi frequency is much smaller than the excited state’s Doppler-broadened linewidth), and the control field is much stronger than the probe field.

As the optical coherence has a short lifetime of about 20 ns, we only consider the coupling between the ground-state coherences ${\rho }_{12}^{(1)}$ (${\rho }_{12}^{(2)}$) for channel 1 (2). The coupling equation takes the form $$\{\begin{array}{c}\begin{array}{l}{\dot{\rho }}_{12}^{(1)}=-{\gamma }_{12}^{\prime }{\rho }_{12}^{(1)}+{\mathrm{\Gamma }}_c{\rho }_{12}^{(2)}-\frac{{\mathrm{\Omega }}_c^{(1)*}{\mathrm{\Omega }}_p^{(1)}}{{\gamma }_{23}},\hfill \\ {\dot{\rho }}_{12}^{(2)}=-{\gamma }_{12}^{\prime }{\rho }_{12}^{(2)}+{\mathrm{\Gamma }}_c{\rho }_{12}^{(1)}-\frac{{\mathrm{\Omega }}_c^{(2)*}{\mathrm{\Omega }}_p^{(2)}}{{\gamma }_{23}}.\hfill \end{array}\end{array}$$(4)

Here, ${\gamma }_{12}^{\prime }={\gamma }_{12}+{\mathrm{\Gamma }}_c+{\mathrm{\Gamma }}_p^{(1)}+{\mathrm{\Gamma }}_p^{(2)}$ represents the total effective decay rate in each channel, with ${\mathrm{\Gamma }}_p^{(i)}=\frac{{|{\mathrm{\Omega }}_c^{(i)}|}^2}{{\gamma }_{23}},i=\mathrm{1,}2$ the optical pumping rate.

By setting ${\dot{\rho }}_{12}^{(1)}={\dot{\rho }}_{12}^{(2)}=0$, the steady-state solutions for the ground-state coherence are $$\{\begin{array}{c}\begin{array}{l}{\rho }_{12}^{(1)}=\frac{-\frac{{\mathrm{\Omega }}_c^{(1)*}{\mathrm{\Omega }}_p^{(1)}}{{\gamma }_{23}}{\gamma }_{12}^{\prime }-\frac{{\mathrm{\Omega }}_c^{(2)*}{\mathrm{\Omega }}_p^{(2)}}{{\gamma }_{23}}{\mathrm{\Gamma }}_c}{{\gamma }_{12}^{\prime 2}-{\mathrm{\Gamma }}_c^2},\hfill \\ {\rho }_{12}^{(2)}=\frac{-\frac{{\mathrm{\Omega }}_c^{(2)*}{\mathrm{\Omega }}_p^{(2)}}{{\gamma }_{23}}{\gamma }_{12}^{\prime }-\frac{{\mathrm{\Omega }}_c^{(1)*}{\mathrm{\Omega }}_p^{(1)}}{{\gamma }_{23}}{\mathrm{\Gamma }}_c}{{\gamma }_{12}^{\prime 2}-{\mathrm{\Gamma }}_c^2}.\hfill \end{array}\end{array}$$(5)

In an optical thin system, optical coherence ${\rho }_{32}^{(i)}=\frac{i{\mathrm{\Omega }}_c^{(i)}{\rho }_{12}^{(i)}+i{\mathrm{\Omega }}_p^{(i)}{\rho }_{22}^{(i)}}{{\gamma }_{23}}$, ${\rho }_{22}^{(i)}\approx 1$, $i=\mathrm{1,}2$. According to the light propagating equation $\frac{\mathrm{d}{E}^{(i)}}{\mathrm{d}\mathit{z}}=\frac{i\overline{k}}{2}{\chi }^{(i)}{E}^{(i)}=\frac{N}{V}\frac{i\overline{k}}{2}\frac{{\mu }_0{\rho }_{32}^{(i)}}{{ϵ}_0}$, we obtain the coupling equation of the probe fields $$\{\begin{array}{c}\begin{array}{l}\frac{\mathrm{d}{E}^{(1)}}{\mathrm{d}\mathit{t}}=\frac{Nc\overline{k}{\mu }_0^2}{2\mathrm{\hslash }V{ϵ}_0{\gamma }_{23}}(-E^{(1)}{\gamma }^{\prime }+E^{(2)}\frac{{\mathrm{\Gamma }}_c}{{\gamma }_{23}}\frac{{\mathrm{\Omega }}_c^{(1)}{\mathrm{\Omega }}_c^{(2)*}}{{\gamma }_{12}^{\prime 2}-{\mathrm{\Gamma }}_c^2}),\hfill \\ \frac{\mathrm{d}{E}^{(2)}}{\mathrm{d}\mathit{t}}=\frac{Nc\overline{k}{\mu }_0^2}{2\mathrm{\hslash }V{ϵ}_0{\gamma }_{23}}(-E^{(2)}{\gamma }^{\prime }+E^{(1)}\frac{{\mathrm{\Gamma }}_c}{{\gamma }_{23}}\frac{{\mathrm{\Omega }}_c^{(2)}{\mathrm{\Omega }}_c^{(1)*}}{{\gamma }_{12}^{\prime 2}-{\mathrm{\Gamma }}_c^2}),\hfill \end{array}\end{array}$$(6)where ${\gamma }^{\prime }=1-\frac{{|{\mathrm{\Omega }}_c^{(i)}|}^2{\gamma }_{12}^{\prime }}{{\gamma }_{23}({\gamma }_{12}^{\prime 2}-{\mathrm{\Gamma }}_c^2)}$, $\frac{N}{V}$ is the density of atoms, $c$ is the speed of light in vacuum, $\overline{k}$ is the average wave vector, ${\mu }_0$ is the dipole moment, and ${ϵ}_0$ is the vacuum dielectric constant. In the presence of the light wall, an atomic spin wave transported to the other channel induces an extra phase ${\theta }_0$. Thus, ${\mathrm{\Gamma }}_c$ is replaced by ${\mathrm{\Gamma }}_c{e}^{i{\theta }_0}$. Taking the coupling term into consideration, we have our beam-splitter Hamiltonian $$\begin{array}{c}\widehat{H}=\hslash (g{\widehat{a}}_1{\widehat{a}}_2^{†}-g^{*}{\widehat{a}}_2^{†}{\widehat{a}}_1)e^{i{\theta }_0},\end{array}$$(7)where $g=-i\frac{Nc\overline{k}{\mu }_0^2{\mathrm{\Gamma }}_c}{2\hslash V{ϵ}_0{\gamma }_{23}^2}\frac{{\mathrm{\Omega }}_c^{(2)*}{\mathrm{\Omega }}_c^{(1)}}{{\gamma }_{12}^{\prime 2}-{\mathrm{\Gamma }}_c^2}$ is the complex coupling coefficient between the two probe fields. Here ${\widehat{a}}_1$ (${\widehat{a}}_2$) is the annihilation operator for the probe field in Ch1 (Ch2).

To further understand the transport experiments, we assume $|{\mathrm{\Omega }}_c^{(1)}|=|{\mathrm{\Omega }}_c^{(2)}|$, and define ${\gamma }^{\prime \prime }=\frac{Nc\overline{k}{\mu }_0^2}{2\mathrm{\hslash }V{ϵ}_0{\gamma }_{23}}\cdot (1-\frac{{|{\mathrm{\Omega }}_c^{(i)}|}^2{\gamma }_{12}^{\prime }}{{\gamma }_{23}({\gamma }_{12}^{\prime 2}-{\mathrm{\Gamma }}_c^2)})$ and ${\mathrm{\Gamma }}_c^{\prime \prime }=\frac{Nc\overline{k}{\mu }_0^2}{2\hslash V{ϵ}_0{\gamma }_{23}}\frac{{\mathrm{\Gamma }}_c}{{\gamma }_{23}}\frac{{\mathrm{\Omega }}_c^{(1)}{\mathrm{\Omega }}_c^{(2)*}}{{\gamma }_{12}^{\prime 2}-{\mathrm{\Gamma }}_c^2}$. The coupling equations then become $$\{\begin{array}{c}\frac{\mathrm{d}{E}^{(1)}}{\mathrm{d}\mathit{t}}=-{\gamma }^{\prime \prime }{E}^{(1)}+{\mathrm{\Gamma }}_c^{\prime \prime }{e}^{i{\theta }_0}{E}^{(2)},\\ \frac{\mathrm{d}{E}^{(2)}}{\mathrm{d}\mathit{t}}=-{\gamma }^{\prime \prime }{E}^{(2)}+{\mathrm{\Gamma }}_c^{\prime \prime *}{e}^{i{\theta }_0}{E}^{(1)}\mathrm{.}\end{array}$$(8)

For the first scheme where only the probe field in Ch1 is switched on, we have (following a time-dependent perturbation) $$\{\begin{array}{c}\begin{array}{l}{E}_{p,\mathrm{out}}^{(1)}=(1-\frac{{{\gamma }^{\prime }}^{\prime }L}{c})E_{p,\mathrm{in}}^{(1)}(t),\hfill \\ E_{p,\mathrm{out}}^{(2)}=\frac{{\mathrm{\Gamma }}_c^{\prime \prime *}{e}^{i{\theta }_0}L}{c}{E}_{p,\mathrm{in}}^{(1)}(t),\hfill \end{array}\end{array}$$(9)where $L$ is the length of vapor cell. It follows that the detected intensities $$\{\begin{array}{c}{I}_1(t)=|(1-\frac{{\gamma }^{\prime \prime }L}{c})E_{p,\text{in}}^{(1)}(t)+E_c^{(1)}{|}^2,\\ I_2(t)=|\frac{{\mathrm{\Gamma }}_c^{{\prime \prime }^{*}}{e}^{i{\theta }_0}L}{c}{E}_{p,\text{in}}^{(1)}(t)+E_c^{(2)}{|}^2.\end{array}$$(10)

Since ${\mathrm{\Gamma }}_c^{\prime \prime }=|{\mathrm{\Gamma }}_c^{\prime \prime }|e^{i({\varphi }_c^{(1)}-{\varphi }_c^{(2)})}$, we have $I_1(t)\propto {|1+{\beta }_1{e}^{i[{\varphi }_p^{(1)}(t)-{\varphi }_c^{(1)}]}|}^2$ and $I_2(t)\propto {|1+{\beta }_2{e}^{i[{\varphi }_p^{(1)}(t)-{\varphi }_c^{(1)}+{\theta }_0]}|}^2$. Here constants ${\beta }_{\mathrm{1,2}}$ are related to parameters of the system, and ${\varphi }_{c,p}^1$ are the phases of the control and probe fields of Ch1.

For the second scheme where both probe fields are turned on, we only detect the probe fields in each channel’s output, with $$\{\begin{array}{c}\begin{array}{l}{E}_{p,\mathrm{out}}^{(1)}=(1-\frac{{\gamma }^{\prime \prime }L}{c})E_{p,\mathrm{in}}^{(1)}(t)+\frac{{\mathrm{\Gamma }}_c^{\prime \prime }{e}^{i{\theta }_0}L}{c}{E}_{p,\mathrm{in}}^{(2)},\hfill \\ E_{p,\mathrm{out}}^{(2)}=(1-\frac{{\gamma }^{\prime \prime }L}{c})E_{p,\mathrm{in}}^{(2)}(t)+\frac{{\mathrm{\Gamma }}_c^{\prime \prime *}{e}^{i{\theta }_0}L}{c}{E}_{p,\mathrm{in}}^{(1)}(t).\hfill \end{array}\end{array}$$(11)

The detected intensities are then $$\{\begin{array}{c}\begin{array}{l}{I}_1(t)\propto {|1+{\beta }^{\prime }{e}^{i[{\varphi }_p^{(1)}(t)-{\varphi }_c^{(1)}+{\varphi }_c^{(2)}-{\varphi }_p^{(2)}-{\theta }_0]}|}^2,\hfill \\ I_2(t)\propto {|1+{\beta }^{\prime }{e}^{i[{\varphi }_p^{(1)}(t)-{\varphi }_c^{(1)}+{\varphi }_c^{(2)}-{\varphi }_p^{(2)}+{\theta }_0]}|}^2.\hfill \end{array}\end{array}$$(12)

Here the constant ${\beta }^{\prime }$ is related to parameters of the system, and ${\varphi }_{c,p}^2$ are the phases of the control and probe fields of Ch2.

As shown in Figs. 3(d) and 4(d), the phase shift caused by the light wall is not proportional to the light-wall power but saturates to a constant. However, the AC Stark shift caused by the far-detuned light is proportional to the light power, resulting in the linear relation between the EIT-center shift and the light-wall power in Fig. 2(c). To understand the relation between the phase shift and the light-wall power, we introduce an effective interaction time between the flying atoms and the light wall. The effective interaction time should correspond to the time it takes for the system to reach the steady state (denoted as $t_{\text{steady}}$), and should be approximately inversely proportional to the EIT full linewidth ${\omega }_{\mathrm{full}}$. Here the EIT linewidth ${\omega }_{\mathrm{full}}={\gamma }_0+{\gamma }_{p_l}$, where ${\gamma }_0$ is the EIT width without the light wall, and ${\gamma }_{p_l}$ is the linewidth broadening due to the light wall (as shown in Fig. 1), proportional to the light wall’s laser power. It follows that $t_{\text{steady}}=\frac{1}{{\gamma }_0+{\alpha }_{1}P}$. The phase picked up by the atomic spin wave corresponds to the EIT center shift ${\alpha }_{2}P$ (which is also proportional to the light-wall power) times the effective interaction time. We then obtain ${\theta }_0=\frac{{\alpha }_{2}P}{{\gamma }_0+{\alpha }_{1}P}$. When the light-wall power $P$ is small, ${\theta }_0\propto P$. As the power increases, ${\theta }_0$ saturates to a constant.

In the experiment, we measure the EIT full linewidth and EIT center shift as functions of the light-wall power. The dependence of the phase shift on the light-wall power is similar to that of the EIT center shift divided by the EIT full linewidth, as shown in Fig. 7(a). To check these results theoretically, we carry out a two-dimensional Monte Carlo simulation. The model is similar to the one we developed in Ref. [48], but now we add a far-detuned laser region. The simulation results are shown in Fig. 7(b), which qualitatively agree with the experiment trends.