Drastic Slowdown of EIT Dynamics by Doppler Broadening and Its Compensation in Room-Temperature Atomic Vapor

Abstract

The transient dynamics of electromagnetically induced transparency (EIT) are fundamental to understanding coherent light–atom interactions and the advancement of quantum technologies such as optical switching and quantum memory. However, in room-temperature atomic vapors, Doppler broadening significantly alters these dynamics, yet a comprehensive understanding of its impact on the transient EIT response remains lacking. In this study, we combine analytical and numerical methods to investigate the absorption dynamics of a weak probe field in a three-level Λ-type system driven by a strong coupling field, based on the optical Bloch equations and Laplace transform techniques. Our results show that the transient response is highly sensitive to both the atomic spontaneous emission rate and the Rabi frequency of the coupling field. Increasing the coupling field intensity not only accelerates the approach to steady state but also induces oscillatory dynamics and negative absorption. Under Doppler broadening, the time required to reach steady state increases by approximately three orders of magnitude compared to the Doppler-free case—an effect that is surprisingly insensitive to temperature variations across the 100–400 K range. Moreover, restoring a short steady-state time under broadened conditions necessitates increasing the coupling laser intensity by two orders of magnitude. These findings provide key insights into the influence of Doppler broadening on coherent transient processes and offer practical guidelines for the design of room-temperature atomic devices, including quantum memories and optical modulators.

1. Introduction

Electromagnetically induced transparency (EIT) is a prominent quantum interference phenomenon that enables exquisite control over light–matter interactions in atomic media [1]. Through the application of a resonant control field, an otherwise opaque atomic ensemble can be rendered transparent to a weak probe field within a narrow spectral window. This effect is accompanied by a steep normal dispersion profile, which drastically reduces the group velocity of light [2]. These properties underpin advanced applications such as optical storage [3,4], low-light-level nonlinear optics [5], and enhanced precision measurements [6,7].

The propagation dynamics of broadband optical pulses in EIT media have attracted considerable interest, motivated by the goal of slowing and storing optical pulses carrying large information bandwidths [8,9]. While extensive studies have been conducted on three-level atomic systems [10,11,12], the behavior of such pulses in more complex, room-temperature systems—featuring Rydberg states and Autler-Townes splitting (ATS)—remains less thoroughly explored [13]. Investigating pulse propagation under these conditions is essential for advancing practical quantum technologies, including optical routing and quantum memory, which must operate effectively outside the cold-atom regime.

In room-temperature atomic vapors, the thermal motion of atoms introduces Doppler broadening, which significantly perturbs the steady-state EIT spectrum [14]. Although techniques exist to mitigate this effect [15], it remains a dominant factor in any realistic modeling and implementation. The interaction between Doppler broadening and the transient dynamics of EIT—particularly the timescale required to establish transparency—is critical for determining the bandwidth and response time of quantum devices such as optical switches and memories. Nevertheless, a systematic investigation into how thermal motion quantitatively affects the transient timeline of EIT formation has been notably absent. Previous studies on transient EIT dynamics have primarily focused on Doppler-free systems [16,17], leaving a substantial gap in our understanding of operation under practical room-temperature conditions and the qualitative expectation of a slowdown exists, a systematic quantitative investigation into its magnitude, its remarkable temperature insensitivity, and viable compensation strategies has been notably absent.

Most early EIT research emphasized steady-state responses. Understanding transient evolution in coherent atomic systems not only deepens fundamental knowledge of light–atom interactions, but also offers a valuable means to probe atomic transition and relaxation properties. Moreover, transient dynamics play essential roles in emerging optical technologies such as all-optical switching, routing, storage, and specialized spectral analysis. Seminal work by Berman and Salomaa [18] compared dressed-atom and bare-atom pictures, examining transients following probe turn-on. Related studies on dressed-atom transients in two-level systems were further developed by Lu and Berman [19], and three-level systems with various initial conditions were also addressed [20]. Harris and Luo [21] analyzed transient EIT with a focus on the energy required for its preparation, while Li and Xiao [22] investigated the timescale for EIT onset. Zhu and co-workers explored conditions for achieving transient inversionless gain in V-type [23] and Λ-type [24] systems.

Although our recent work has explored transient evolution in a four-level Rydberg system under microwave fields [25], the present study concentrates specifically on a three-level Λ-type system embedded in a thermally broadened medium, emphasizing the decisive influence of Doppler broadening on transient dynamics. Unlike Rydberg systems where external microwave fields induce Autler-Townes splitting, here we elucidate how intrinsic thermal motion reshapes the coherent transient response in a room-temperature vapor. This approach provides critical and complementary insights into EIT dynamics applicable to non-cold, non-Rydberg systems, which are highly relevant for compact, room-temperature quantum devices.

Atomic vapors are increasingly employed as high-precision electromagnetic field sensors, leveraging atomic spectroscopy to provide transferable measurement standards [26,27]. A major source of uncertainty in EIT-based spectroscopy at room temperature stems from Doppler broadening [28], where the thermal distribution of atomic velocities induces inhomogeneous broadening. While such effects are minimized in cold atoms [29] or through certain EIT schemes [30,31], they remain dominant in room-temperature and warmer vapor cells, and must be incorporated for accurate spectral fitting and device design.

In this work, we present a comprehensive theoretical analysis of the transient absorption dynamics in a Doppler-broadened three-level Λ-type system. Using the optical Bloch equations combined with Laplace transform methods and averaging over the Maxwell-Boltzmann velocity distribution, we quantify the profound temporal slowdown induced by thermal motion. Our results demonstrate that the time required to reach steady-state EIT is prolonged by approximately three orders of magnitude in a room-temperature ⁸⁷Rb vapor compared to the Doppler-free case. This dramatic extension arises from the disparate evolutionary timescales of different velocity classes, wherein off-resonant atoms dominate the overall transient behavior. Furthermore, we show that compensating for this Doppler-induced delay to recover a rapid transient response requires increasing the coupling laser intensity by roughly two orders of magnitude. Remarkably, the transient timeline exhibits pronounced insensitivity to temperature variations across the 100–400 K range, a consequence of counterbalancing effects between enhanced Doppler broadening and elevated collision rates at higher temperatures.

These findings deliver fundamental insights into coherent light–atom interactions in thermal ensembles and establish concrete design principles for optimizing room-temperature EIT devices. By quantifying the interplay among Doppler broadening, control field power, and transient time, our work provides a practical roadmap for developing high-speed, efficient quantum optical technologies based on thermal atomic vapors.

2. Theoretical Formulations

The system under consideration is a closed three-level Λ-type atomic system, interacting with two coherent optical fields, as depicted in Figure 1. The specific energy levels chosen for our model correspond to the D₂ line of ⁸⁷Rb. The ground state hyperfine levels $\left|2\right.〉$ (5S_1/2, F = 2) and $\left|3\right.〉$ (5S_1/2, F = 1) serve as the two lower states, while the excited state $\left|1\right.〉$ (5P_3/2, F′ = 1) completes the Λ configuration. This level selection is standard for EIT experiments in Rb vapor and provides well-characterized transition parameters [32].

A strong control laser drives the $\left|2\right.〉\leftrightarrow \left|1\right.〉$ transition with Rabi frequency ${\Omega }_C$ and detuning ${\Delta }_C$. A weak probe laser is scanned across the transition $\left|3\right.〉\leftrightarrow \left|1\right.〉$ to measure the absorption characteristics, with Rabi frequency ${\Omega }_P$ and detuning ${\Delta }_P$. The absorption of the weak probe has been extensively studied through density matrix analysis of the system [33].

In the interaction picture and under the electric-dipole and rotating-wave approximations, the expression of the interaction Hamiltonian describing the atom-field interaction for the system in a rotating frame can be written as:

$$H=-{\displaystyle \frac{\hslash }{2}}\left[\begin{array}{ccc}0& 0& {\Omega }_p\\ 0& 2\left({\Delta }_p-{\Delta }_c\right)& {\Omega }_c\\ {\Omega }_p& {\Omega }_c& 2{\Delta }_p\end{array}\right]$$

(1)

The system dynamics including decay processes are described by the time-dependent optical Bloch equation for the density matrix $\rho$ [34]:

$${\displaystyle \frac{\partial {\rho }_{21}}{\partial t}}=-\left[{\gamma }_{21}-i\left({\Delta }_p-{\Delta }_c\right)\right]{\rho }_{21}+{\displaystyle \frac{i{\Omega }_c}{2}}{\rho }_{31}-{\displaystyle \frac{i{\Omega }_p}{2}}{\rho }_{23}$$

(2)

$${\displaystyle \frac{\partial {\rho }_{31}}{\partial t}}=-\left({\gamma }_{31}-i{\Delta }_p\right){\rho }_{31}+{\displaystyle \frac{i{\Omega }_c}{2}}{\rho }_{21}+{\displaystyle \frac{i{\Omega }_p}{2}}\left({\rho }_{11}-{\rho }_{33}\right)$$

(3)

where ${\gamma }_{ij}$ is the coherence dephasing rate of the $\left|i\right.〉\leftrightarrow \left|j\right.〉$ transition that related to the spontaneous emission rate ${\Gamma }_{ij}$ [34]. The spontaneous emission rates for transition $\left|2\right.〉\leftrightarrow \left|1\right.〉$ and $\left|3\right.〉\leftrightarrow \left|1\right.〉$ are approximately equal, $\Gamma =2\pi \times 6.07\mathrm{MHz}$, consistent with the natural linewidth of the ⁸⁷Rb D₂ line [32].

The set of equations can be solved numerically to obtain the transient and steady state response of the medium. By using the weak probe approximation under steady state conditions (i.e., $\partial {\rho }_{ij}/\partial t=0$) and assuming the population of the system to remain in the ground state (i.e., ${\rho }_{33}=1$, ${\rho }_{22}={\rho }_{11}=0$), the set of equations can be solved analytically to explicitly determine the probe coherence term ${\rho }_{31}$ in a Doppler free environment. The analytical expression for the probe coherence term is presented as follows:

$${\rho }_{31}={\displaystyle \frac{i{\Omega }_p}{2\left(R_3+\frac{{\Omega }_c^2/4}{R_2}\right)}}$$

(4)

Here, $R_2={\gamma }_{21}-i({\Delta }_P-{\Delta }_C)$ and $R_3=-{\gamma }_{31}-i{\Delta }_P$.

The absorption of the weak probe is given by $\mathrm{Im}({\rho }_{31}\Gamma /{\Omega }_P)$, well known from a density matrix analysis of the system [35].

A major source of uncertainty in EIT spectroscopy of warm atomic vapors comes from Doppler broadening, where the thermal distribution of atomic velocities causes frequency shifts that result in spectral broadening [28]. For an atom moving along the direction of the beams with a velocity ν, the detuning of the two beams will change by $\pm kv$, where k = 2π/λ is the photon wave vector, and the sign depends on whether the atom is moving away from or toward the beams. For co-propagating beam configuration, the Hamiltonian can be expressed as:

$$H=-{\displaystyle \frac{\hslash }{2}}\left[\begin{array}{ccc}0& 0& {\Omega }_p\\ 0& 2\left({\Delta }_p+{\displaystyle \frac{\nu }{{\lambda }_p}}-{\Delta }_c-{\displaystyle \frac{\nu }{{\lambda }_c}}\right)& {\Omega }_c\\ {\Omega }_p& {\Omega }_c& 2({\Delta }_p+{\displaystyle \frac{\nu }{{\lambda }_p}})\end{array}\right]$$

(5)

Thus, to obtain the complete probe absorption in a gas of moving atoms, the above expression must be corrected for the velocity of the atom and then averaged over the one- dimensional Maxwell-Boltzmann distribution of velocities.

$$f_{MB}(\nu )d\nu ={\displaystyle \frac{1}{u\sqrt{\pi }}}\exp (-{\displaystyle \frac{{\nu }^2}{u^2}})d\nu$$

(6)

Here, $u=\sqrt{2k_{B}T/m}$, T is the absolute temperature, m is the atomic mass, and k_B is Boltzmann’s constant.

Integrate ${\rho }_{31}$ over velocities, weighted by the Maxwell-Boltzmann distribution, yielding the Doppler-broadened absorption spectrum:

$${\rho }_{31,D}({\Delta }_p)={\displaystyle {\int }_{-\infty }^{\infty }{\rho }_{31}(\nu ,{\Delta }_p)f_{MB}(\nu )}d\nu$$

(7)

The integral in Equation (7) was evaluated numerically using a composite left-endpoint Riemann sum with 2000 uniformly spaced points over the velocity range from −750 m/s to 750 m/s, providing sufficient accuracy for the smoothly varying Maxwell-Boltzmann distribution.

For the Doppler-free case, we obtain analytical solutions for the time-dependent density matrix elements Equations (2) and (3) using Laplace transform methods. Under the weak probe approximation and assuming the system initially remains in ground state $\left|3\right.〉$, the solution for the probe coherence ${\rho }_{31}(t)$ is:

$${\rho }_{31}(t)=-{\displaystyle \frac{i(\exp (t(-\frac{{\gamma }_{31}}{2}-\frac{{\Omega }_1}{2}))-\exp (t(-\frac{{\gamma }_{31}}{2}+\frac{{\Omega }_1}{2}))){\Omega }_p}{2{\Omega }_1}}$$

(8)

This analytical solution is derived under the weak-probe approximation, assuming the system is initially in the ground state $\left|3\right.〉$ and with both the probe and coupling lasers on two-photon resonance (${\Delta }_P={\Delta }_C=0$) and ${\Omega }_1=\sqrt{{\gamma }_{31}^2-{\Omega }_C^2}$.

Under conditions of Doppler broadening, the transient response of the system must also be adjusted to account for the atomic velocity. By numerical integrating ${\rho }_{31}(t)$ (Equation (8)) over all velocities, weighted by the Maxwell–Boltzmann distribution (Equation (6)), the transient response of the Doppler-broadened absorption spectrum can be obtained.

3. Results and Discussion

The numerical results for the transient and steady state of the absorption with different corresponding parameters are shown in Figure 2, Figure 3, Figure 4 and Figure 5. For all simulations presented in this work, we assume perfect two-photon resonance unless otherwise specified.

3.1. Absorption Spectra

Figure 2a displays the simulated probe absorption versus probe detuning under Doppler-free conditions for various values of ${\Omega }_C$ at ${\Delta }_C=0$. The pole structure of Equation (5) shows that absorption will be zero when ${\Delta }_P={\Delta }_C$, and this minimum will occur precisely with resonance if the control is on resonance. The doublet peaks represent the two symmetric dressed states induced by the control laser [36], and their separation equal to ${\Omega }_C$. Consequently, the linewidth of the EIT dip is determined by value of ${\Omega }_C$. This is characteristic of the EIT phenomenon. Additionally, Figure 2b shows the dramatic effect of Doppler averaging at room temperature. The overall absorption is reduced by approximately a factor of 30, as the signal is spread over many velocity classes. Furthermore, the transparency window remains narrow because only atoms with near-zero longitudinal velocity (for co-propagating beams) contribute significantly to the coherent EIT effect. This result is consistent with the well-known averaging effect described in [32], confirming the robustness of our model in reproducing established steady-state phenomena.

3.2. Transient Optical Response

This section shows the numerical results of the time-dependent probe absorption versus evolution time t for different control field strength ${\Omega }_C$ and temperature. In the simulation, the probe laser and coupling laser are both switched on at the beginning of the time evolution (at a time t = 0) and the detunings of the probe and control fields are ${\Delta }_P={\Delta }_C=0$, (i.e., the two lasers are both resonant).

3.2.1. Atoms in Doppler-Free Medium

The plot of the transient absorption for the probe laser under Doppler free condition is shown in Figure 3. As time increases, the atomic absorption first exhibit its absorption property for the probe laser because the probe field induces polarization of the atomic medium. It leads to that absorption for the probe laser increases at first and the value is inversely proportional to ${\Omega }_C$. Then probe absorption will reach to its steady state in different ways determined by ${\Omega }_C$. When the coupling laser is weak, for example, when ${\Omega }_C=0.5{\gamma }_{31}$, the transient medium absorption for the probe field monotonically decreases to its steady state with time evolutions, (see the red curve in Figure 3). And the steady-state time window Tw (the time for the atomic medium to reach its steady state) lasts about 0.06 µs. Nevertheless, when the couple field becomes strong, for instance, when ${\Omega }_C={\gamma }_{31}$, ${\Omega }_C=2{\gamma }_{31}$ and ${\Omega }_C=4{\gamma }_{31}$, probe absorption will be oscillatorily damped to the steady-state value. The steady-state time window Tw is reduced to 0.008 µs and keeps stable with a further increase of the control field, (see the dashed blue, dot-dashed black, and solid gray curves in Figure 3). Besides, there appears transient negative absorption (or gain) for the probe laser.

Equation (8) discloses the underlying mechanism of the dependence of the transient absorption for the probe laser on the couple field strength ${\Omega }_C$:

In the low-intensity region: $0<{\Omega }_C\le {\gamma }_{31}$, the oscillation frequency of population between levels $\left|1\right.〉$ and $\left|2\right.〉$ is lower than the decay rate of level $\left|1\right.〉$. As a result, atomic stimulated absorption from $\left|1\right.〉$ to $\left|2\right.〉$ is allowed, while the stimulated emission from $\left|2\right.〉$ to $\left|1\right.〉$ is suppressed. For example, atoms in level $\left|1\right.〉$ can be excited to level $\left|2\right.〉$, but those in level $\left|2\right.〉$ cannot be efficiently returned to level $\left|1\right.〉$ (and then back to level $\left|3\right.〉$). This leads to the temporal two-photon coherence ${\rho }_{31}(t)$ between levels $\left|3\right.〉$ and $\left|1\right.〉$ approaching its steady-state value in a monotonic manner, as seen in the dotted red and dashed blue curves in Figure 3.

In the high-intensity region: ${\Omega }_C>{\gamma }_{31}$, the population oscillation frequency between levels $\left|1\right.〉$ and $\left|2\right.〉$ exceeds the decay rate of level $\left|1\right.〉$. Consequently, populations on level $\left|2\right.〉$ can be efficiently simulated back to level $\left|1\right.〉$, and then back to level $\left|3\right.〉$. The two-photon coherence ${\rho }_{31}(t)$ will exhibit oscillatory damping to its steady state value over time, with the transition frequency, as shown in the dot-dashed black and solid gray curves in Figure 3.

3.2.2. Atoms in Doppler Broadened Medium at Room Temperature

The effect of residual Doppler averaging due to wavelength mismatch between the two fields is minimum in the Λ-type atomic system when the couple field is co-propagating with respect to the probe field. Hence in this section, the probe field will always be taken co-propagating with respect to the couple field.

Figure 4 displays the transient response of the probe absorption under Doppler-broadened conditions in room-temperature vapor for different coupling field strengths, ${\Omega }_C$. The probe absorption initially increases, then oscillates before reaching a steady state-a trend consistent with that observed in the Doppler-free case, though the magnitude of the initial increase is significantly smaller. When accounting for atomic velocity groups, the steady-state time is prolonged by approximately three orders of magnitude compared to the Doppler-free situation (Figure 4a). This delay arises because atoms with different velocities experience distinct effective detunings in a thermal ensemble, leading to considerable variation in their evolution times. In particular, high-velocity atoms exhibit large effective detunings and thus longer evolution times. According to the Maxwell velocity distribution, most atoms possess non-zero velocities, which collectively results in a pronounced extension of the overall evolution time. As ${\Omega }_C$ increases, the EIT window broadens, enabling more velocity groups to participate in the coherent process and thereby reducing the steady-state time. When ${\Omega }_C=100{\gamma }_{31}$, the steady-state time is approximately 0.008 µs, which matches the value obtained under Doppler free conditions (red curve in Figure 4b).

Figure 5 shows the transient response of the probe absorption at different temperatures. Figure 5a presents the results for a coupling field of ${\Omega }_C=100{\gamma }_{31}$, while Figure 5b corresponds to ${\Omega }_C=4{\gamma }_{31}$. It can be observed that, regardless of the value of ${\Omega }_C$, the time required to reach steady state is largely independent of temperature over a wide range (100–400 K). This counterintuitive finding can be explained by two competing effects. On one hand, increasing temperature broadens the velocity distribution ($u=\sqrt{2k_{B}T/m}$), which would tend to increase the number of off-resonant atoms and thus lengthen Tw. On the other hand, the increased atomic collision rate at higher temperatures enhances decoherence, which typically drives the system toward steady state faster. This temperature stability is advantageous for practical applications, as it relaxes thermal stability requirements for vapor cell devices.

This temperature range was selected based on both physical and practical considerations. The lower limit (~100 K) is dictated by the rapid decrease in vapor pressure below this point, which would result in insufficient optical signal for meaningful measurement. The upper limit (~400 K) is chosen to avoid the dominance of interatomic collision effects over Doppler broadening, while remaining within typical experimental thermal stability constraints. Within this range, the Doppler width varies noticeably with temperature while still serving as the dominant broadening mechanism, allowing clear isolation of its influence on the transient dynamics. It is worth noting that at very low temperatures (approaching 0 K), the system should converge to the Doppler-free case, exhibiting the shortest evolution time.

4. Conclusions

In summary, our work moves beyond the qualitative understanding of Doppler broadening’s impact by providing its first comprehensive quantitative analysis. This study systematically investigates the influence of Doppler broadening on the transient response of electromagnetically induced transparency (EIT) in a three-level Λ-type atomic system through theoretical analysis and numerical simulations. Using parameters corresponding to ⁸⁷Rb atoms, we examined both transient and steady-state absorption under both Doppler-free and Doppler-broadened conditions. Our results demonstrate that the transient absorption behavior of the weak probe field is significantly altered by Doppler broadening. The dramatic prolongation of the evolution time in thermal atomic systems is primarily due to Doppler shifts caused by atomic thermal motion: atoms with different velocities experience different effective detunings. High-velocity atoms, which exhibit larger detunings, evolve on considerably longer timescales compared to resonant atoms. Since the majority of atoms in a Maxwell velocity distribution possess non-zero velocities, the overall dynamics of the system are dominated by these off-resonant atoms, leading to a markedly extended transient period. For identical field intensities, the steady-state time in the Doppler-broadened system is approximately three orders of magnitude longer than in the Doppler-free case. Compensating for this effect requires increasing the coupling laser intensity by about two orders of magnitude to achieve a similarly short steady-state time. Interestingly, the time required to reach steady state remains nearly independent of temperature over the range of 100–400 K, a phenomenon attributed to the competing effects of thermal velocity spread and collision-induced decoherence. These findings not only deepen our understanding of light-atom interactions in thermal ensembles, but also provide essential design principles for enhancing the performance of room-temperature EIT-based devices. Specifically, the ability to recover a short transient response time by increasing the coupling field intensity is crucial for achieving high-speed operation in all-optical switches and for supporting large bandwidths in room-temperature quantum memories. Finally, it is important to note the limitations of our model. This study is based on the assumptions of a co-propagating beam configuration and a closed three-level atomic system. The applicability of our conclusions to non-co-propagating geometries, which are subject to residual Doppler broadening, or to open quantum systems with additional decay channels, requires further investigation and validation.