Theory of thermal motion in electromagnetically induced transparency: Effects of diffusion, Doppler broadening, and Dicke and Ramsey narrowing

We present a theoretical model for electromagnetically induced transparency (EIT) in vapor that incorporates atomic motion and velocity-changing collisions into the dynamics of the density-matrix distribution. Within a unified formalism, we demonstrate various motional effects, known for EIT in vapor: Doppler broadening of the absorption spectrum; Dicke narrowing and time-of-flight broadening of the transmission window for a finite-sized probe; diffusion of atomic coherence during storage of light and diffusion of the light-matter excitation during slow-light propagation; and Ramsey narrowing of the spectrum for a probe and pump beams of finite size.

Figure 1

(Color online) Illustration of time-of-flight (TOF) broadening in the Doppler (left) and Dicke (right) limits. Assume a beam of width $\Delta x$ in the transverse plane and width $\Delta k_{\perp }\sim 1/\Delta x$ in $k$ space. Atoms with a transverse velocity $v_{th}$ (left) cross the beam in time $\Delta x/v_{th}$ and cause a TOF broadening of the order of $v_{th}\Delta k_{\perp }$, which is equal to the well-known Doppler width. Atoms that undergo diffusion (right), traverse the beam in average time of $\Delta x^2/D$, where $D=v_{th}\Lambda$ is the diffusion coefficient and $\Lambda$ is the mean free path between collisions. This results in a TOF broadening of the order of $v_{th}\Lambda \Delta k_{\perp }^2$, which is the well-known Dicke width.

Figure 2

(Color online) (a) A probe beam and a pump beam, with a finite envelope in space, propagate through the vapor cell. The $z$ axis is chosen perpendicular to the probe’s direction and the $x$ and $y$ axes form the transverse plane. (b) Atomic levels diagram. ${\Omega }_1$ and ${\Omega }_2$ are the Rabi frequencies of the probe and the pump, respectively. ${\Gamma }_d$ and ${\Gamma }_{12}$ are the decoherence rates of the optical and the ground-state transitions.

Figure 3

(Color online) Normalized EIT transmission spectra, numerically calculated from Eqs. (25, 26a, 26b, 26c, 27, 28, 29), for three values of the EIT Dicke parameter, $\eta =v_{th}k/\gamma$ (the ratio between the residual Doppler width and the velocity relaxation rate), with ${\Gamma }_d=2500v_{th}k$, ${\Gamma }_{21}=0.025v_{th}k$, and ${\left|{\Omega }_2\right|}^2/{\Gamma }_d={\Gamma }_{21}/25$ (small power broadening). When $\eta$ is large (solid black line) the spectrum is a Voigt curve (a Gaussian-Lorentzian convolution). When $\eta$ is small (dashed blue line) the spectrum is a pure Lorentzian of width ${\Gamma }_d$. The dot-dashed green line demonstrates an intermediate result.

Figure 4

(Color online) Full width at half maximum of the EIT transparency window (points), obtained from numerical results similar to Fig. 3, as a function of the wave-vector difference $k$ for various values of the velocity relaxation rate $\gamma$. The dashed lines are given by Eq. (38). Other parameters are (typical for experiments with small power broadening): $v_{th}=170\text{m}/\text{s}$, ${\Gamma }_d=100\text{MHz}$, ${\Gamma }_{21}=1\text{kHz}$, ${\left|{\Omega }_2\right|}^2/{\Gamma }_d=40\text{Hz}$. The three lines in Fig. 3 correspond here to $\left|k\right|\approx 1.5{\text{mm}}^{-1}$ and $\gamma =16,160$, and 1600 kHz.

Figure 5

(Color online) The spatial-frequency filter (normalized EIT transmission), given in Eq. (66), as a function of $\left|{\mathbf{k}}_{\perp }\right|=\left|k\hat{\mathbf{x}}\right|$, for different Raman detunings, with $\omega =0$, $\delta \mathbf{q}=0$, and ${\Gamma }_{21}=K{\left|{\Omega }_2\right|}^2$. The solid-black curve is plotted for $\Delta =0$. The dashed red and blue curves demonstrate the effect of nonzero Raman detuning: a decrease in transparency alongside a change in the curvature near $k=0$.

Figure 6

(Color online) Calculated effect of the spatial-frequency filter with a plane-wave pump and a finite probe beam: The initial pattern $\left|{\tilde{\Omega }}_1\left(z_1;x,y\right)\right|$ (left) and the transmitted pattern $\left|{\tilde{\Omega }}_1\left(z_2;x,y\right)\right|$ after a certain propagation length (right). The calculations were done using Eqs. (30, 34, 62, 63), and the diffraction $\left(k^2/2/q_1\right)$ was neglected for clarity. The parameters correspond to the black line in Fig. 5 ($K{\left|{\Omega }_2\right|}^2={\Gamma }_{12}$, $\delta q=0$, $\Delta =0$). Images (a)–(c) illustrate regular diffusion of real (b) and complex (c) fields. To generate (c), the phase of the left line in the incident field was flipped. Images (d) and (e) illustrate the effect for smaller features, when larger $k_{\perp }$’s are pronounced.

Figure 7

(Color online) Normalized transmission for a plane-wave and a finite-sized beam (1D and 2D), demonstrating TOF broadening and Ramsey narrowing. The inset depicts the correction to the spectrum, resulting from the finiteness of beams, $\text{Re}\left[1-S_D\left(\Delta \right)\right]$, for the 1D case [Eq. (87)]. The parameters are $\Gamma =100\text{Hz}$, $K{\left|{\Omega }_2\right|}^2=2\text{kHz}$, $D=10{\text{cm}}^2/\text{sec}$, and $a=100\mu \text{m}$. The choice of $K{\left|{\Omega }_2\right|}^2⪢\Gamma$ makes the narrowing effect more obvious.