Figure 1

Measured and calculated linewidths of a hyperfine dark resonance in rubidium, demonstrating the reduction in both the Doppler and transit-time broadenings with increasing pressure of neon buffer gas. Laser intensities are (squares) 17, (triangles) 11, (circles) 6, and (diamond) $1\mathrm{mW}/{\mathrm{cm}}^2$. From 19.Reuse & Permissions

Figure 2

Transit-time interpretation of the Doppler and Dicke effects. A beam of width $\Delta x$ has a span $\Delta k_{\perp }\sim 1/\Delta x$ of transverse momenta. The interaction time for an atom crossing the beam with velocity $v_T$ (top) is ${\tau }_t=\Delta x/v_T$, resulting in a transit-time broadening of $v_T\Delta k_{\perp }\sim {\Gamma }_{\mathrm{Doppler}}$. For diffusing atoms (bottom), the mean interaction time is $\Delta x^2/D$, leading to a broadening of $D\Delta k_{\perp }^2\sim {\Gamma }_{\mathrm{Dicke}}$. Right: Dark resonances in rubidium vapor measured by 16 for a beam diameter $\Delta x=5.6\mathrm{mm}$ (top) without and (bottom) with 20 torr neon buffer gas. The respective linewidths are (solid lines) $2{\Gamma }_{\mathrm{Doppler}}=100\mathrm{kHz}$ and $2{\Gamma }_{\mathrm{Dicke}}=6\mathrm{kHz}$. Dashed and dotted lines are measured with an angle between the Raman beams of $\theta =0.31\mathrm{mrad}$ and $\theta =0.62\mathrm{mrad}$, respectively. Adapted from 16.Reuse & Permissions

Figure 3

Electromagnetically induced transparency in a $\Lambda$ scheme. (a) The Raman resonance $|1⟩\leftrightarrow |2⟩$ is excited via the state $|3⟩$ by “probe” and “coupling” light fields. (b) Top: Transmission of the probe (solid line) in the absence of the coupling field and accompanied refraction (dashed line). Bottom: Dark resonance induced by the coupling field.Reuse & Permissions

Figure 4

Residual Doppler-Dicke broadening in EIT due to an angular deviation $\theta$ between the coupling and probe beams, measured in Rb vapor with 10 torr neon. The linewidth depends quadratically on $\theta$ as both the Doppler width (${\Gamma }_{\mathrm{Doppler}}^{\mathrm{res}}=\theta {\Gamma }_{\mathrm{Doppler}}^{.}$) and the Dicke narrowing factor ($2\pi \Lambda /{\lambda }_R=\theta \Lambda q$) are linear in $\theta$. The mean free path $\Lambda \approx 2\mu \mathrm{m}$ used in the theory (right, solid line) corresponds to a collision rate of $\sim {10}^8/\mathrm{s}$ and is calculated from the Rb-Ne collisional cross section (61; 65). The broadening also leads to a decrease of the resonance transmission. Adapted from 126.Reuse & Permissions

Figure 5

Absorption spectra of thermal atoms. In the absence of collisions, the homogenous linewidth $\Gamma$ is dominated by the Doppler width ${\Gamma }_{\mathrm{Doppler}}=100\Gamma$. The transition from the (black) Doppler limit to the (light gray) Dicke limit occurs when the effective mean free path $\Lambda =v_T/{\gamma }_c$ (${\gamma }_c^{-1}$ is the velocity correlation time due to collisions) is comparable to the wavelength $\lambda$ (dark gray). The spectra for the (solid line) weak and (dashed line) strong collisions are calculated from Eqs. (a4, a13), respectively. The differences between weak and strong collisions are not distinguishable at the far limits (black and light gray).Reuse & Permissions

Figure 6

Structured slow light. The probe beam is patterned by using a transmission mask, a grating, or a spatial light modulator and imaged onto the cell. An iris may be used to filter high frequencies and limit the angular span. After the cell, the probe is imaged onto a camera. Top: Diffusion of (left) a line pattern with 1.5 lines pair$/$mm after (right) $6\mu \mathrm{m}$ delay. [Measurements taken in a setup similar to that used by 127.]Reuse & Permissions

Figure 7

Phase coherence between the two “output channels” of a slow-light beam splitter by 146. The atoms moving in the bufferless wall-coated cell mediate the coherence between the two channels. The measured transfer efficiency ($<5\%$) is a function of the slowing delay (0.5 ms), the decoherence due to wall collisions, and the beams and cell geometry. Optimization of these factors promises efficiencies close to unity. Adapted from 146.Reuse & Permissions

Figure 8

Resonant transmission vs the angular deviation $\theta$ between the Raman beams. (a) Imaging experiment, measuring the resonant transmission of a diverging probe beam with a large (plane-wave) coupling beam. Pictures are taken beyond the cell by impinging the beams directly onto a CCD detector, (top) with and (bottom) without the atoms. Stronger absorption is observed away from the center, where, in the optical ray approximation, $\theta$ is larger. Maximal $\theta$ is $\sim 2\mathrm{mrad}$ at the beam radius. (b) A reference with a nondiverging probe. (c) Scaled transmission (squares) from the aforementioned imaging experiment and (circles) from spectroscopy (see Fig. 4). The theoretical curve is calculated from Eq. (11). Adapted from 126.Reuse & Permissions

Figure 9

Polariton diffusion and motional-induced diffraction. A Zeeman EIT setup is used, similar to that in Fig. 6, with a cell length $L=5\mathrm{cm}$ and optical depth $2\alpha L=6$. The free-space diffraction (bottom left) is compared to transmitted slow images at several Raman detunings (right). At $\Delta =0$, the polariton is delayed by $\sim 6\mu \mathrm{s}$ and experiences the combination of optical diffraction and diffusion ($D=11{\mathrm{cm}}^2/\mathrm{s}$ for 10 torr of neon). At $\Delta <0$, both diffusion and diffraction are reduced, and at $\Delta =-\gamma \approx -70\mathrm{kHz}$, they are completely suppressed ($Dq=v_g\cong 8700\mathrm{m}/\mathrm{s}$). Numerical calculations confirm that the small difference between the input image and the transmitted image at $\Delta =-\gamma$ is primarily due to residual ${\nabla }^4$ terms. At $\Delta =+\gamma$, no diffusion occurs, but the polariton experiences the sum of equal optical and motional-induced diffraction, as if the image has propagated a free-space distance of $2L$. Adapted from 48.Reuse & Permissions

Figure 10

Collective rotation of a vortex array due to induced diffraction by 128. Hyperfine EIT with ${}^{87}\mathrm{Rb}$ is performed in a cell of length $L=7.5\mathrm{cm}$ with 20 torr neon ($D=6{\mathrm{cm}}^2/\mathrm{s}$, $v_g=5000\mathrm{m}/\mathrm{s}$). An array of four $m=+1$ vortices (total angular momentum $J=4$) rotates clockwise (b) at $\Delta =0$, the same as it does in free space. (a) At $\Delta <0$, the optical diffraction is counteracted, leading to a counterclockwise rotation with respect to $\Delta =0$. (c) At $\Delta >0$, diffraction is enhanced, leading to increased clockwise rotation. Images are $1\times 1{\mathrm{mm}}^2$. (d) Rotation angle for a two-vortex array with (light gray circles) $J=2$, (black squares) $J=-2$, and (dark gray diamonds) $J=1-1=0$. All-optical control on the vortices motion could be useful for fast optical-trapping applications. Adapted from 128.Reuse & Permissions

Figure 11

Probe deflection. (a) Nondiffraction can be understood as the deflection of all rays into a common direction. The rays maintain their phase relations while traversing the medium and afterward deflect back into their original direction. (b) Generally, the probe beam deflects toward the direction of the coupling beam for $\Delta <0$ and away from the coupling for $\Delta >0$. At $\Delta =\pm \gamma$, a Snell-like law determines the ratio between ${\theta }_r$ and ${\theta }_c$. (c) In the nondiffraction conditions, the probe beam assumes the direction of the coupling beam for small enough angles (circles are measurement, lines are theory). Adapted from 48.Reuse & Permissions

Figure 12

Manifestation of a negative-diffraction lens with $n_{\mathrm{diff}}=-1$. The optical diffraction is exactly reversed by setting $v_g=Dq/2$, so that an image diffracting along a distance $L$ before the cell is reimaged at the end of the cell. The positive and negative diffractions “commute,” e.g., it makes no difference if the positive diffraction occurs partly before and partly after the cell, as exemplified by the two right images. The restriction $k_{\perp }\ll k_0$ was not fulfilled in this experiment, hence the imperfections in the imaging. Solid arrows indicate geometrical rays (not $\mathbf{q}$ vectors) refracting oppositely to the incident angle ${\theta }_r=-{\theta }_i$, according to Snell’s law $\sin {\theta }_i=n_{\mathrm{diff}}\sin {\theta }_r$, as expected in a negative-index material. Adapted from 48.Reuse & Permissions

Figure 13

Diffusion during storage of light in vapor. Left: The first half of the probe pulse is allowed to leak before storage. The second half is stored by turning off the coupling beam for a duration $\tau$ after which the probe revives. The traces show the probe for $\tau =5$, 15, and $25\mu \mathrm{s}$, and the coupling for $\tau =25\mu \mathrm{s}$. Right: The spatial effect of diffusion is observed by comparing the images of the input probe (top row) and the retrieved probe (bottom row). Storage durations are $\tau =2$, 6, and $9\mu \mathrm{s}$. Adapted from 127.Reuse & Permissions

Figure 14

Diffusion of coherence fields with uniform and nonuniform phase patterns. A flat-phased ring (top left) is filled up after a short storage duration, while a vortex ring with a helical phase is preserved (top right). Similarly, the blurring of a line pattern (bottom left) can be reduced by flipping the phase between adjacent lines (bottom right). As evident, the visibility of the lines with the alternating phase remains much higher than the flat-phased image. The vortex radius is $670\mu \mathrm{m}$, the lines are $1.5/\mathrm{mm}$, and $D=10{\mathrm{cm}}^2/\mathrm{s}$. Adapted from 116, and 127.Reuse & Permissions

Figure 15

By storing the Fraunhofer diffraction pattern (top right) instead of the image itself (top left) the fine details of the image are better preserved under diffusion (bottom). Rather, diffusion attenuates the outer parts of the beam. Adapted from 140.Reuse & Permissions

Figure 16

Diffusion of “common” modes during storage. Left: Intensity patterns of (top to bottom) Gaussian, ${\mathrm{LG}}_{01}$, ${\mathrm{LG}}_{02}$, and ${\mathrm{HG}}_{01}$, expanding due to diffusion while preserving their shape. Top right: Normalized cross sections at different storage durations, $\tau =(\mathrm{circles})$ 2, (squares) 20, (triangles) 40, and (diamonds) $60\mu \mathrm{s}$, rescaled horizontally by the stretching factor $s(\tau )$. Bottom right: Decay of the total power, increasing with the total mode order $N$. The dashed line is $e^{-2\gamma \tau }$, and solid lines are $e^{-2\gamma \tau }s(\tau {)}^{-2(N+1)}$. Adapted from 49.Reuse & Permissions

Figure 17

Evolution of (top row) ${\mathrm{sHG}}_{22}$ and (bottom row) ${\mathrm{eHG}}_{22}$, measured by 150, for (center column) diffraction in free space and (right column) diffusion during storage of light. The standard mode is preserved only during diffraction, whereas the elegant is preserved only during diffusion. Adapted from 150.Reuse & Permissions

Figure 18

Shape-preserving contraction and subsequent expansion of ${\mathrm{LG}}_{01}$ during storage. Because of destructive interference, the diffusion of a diverging beam initially decreases the beam area $w_{\tau }^2$ (line is the theoretical prediction). Adapted from 49.Reuse & Permissions

Figure 19

Ramsey narrowing. Left: Dark-resonance spectrum for rubidium in 5 torr neon with 1-mm-wide beam, measured by 147 and calculated using the repeated-interactions model by 148. The comparison to the transit-time (${\tau }_t^{-1}$) broadening highlights the substantial Ramsey narrowing. Right: The diffusion solutions (21) by 52, for $2a=0.2\mathrm{mm}$, $D=10{\mathrm{cm}}^2/\mathrm{s}$, no walls, and large power broadening ${\gamma }_p=20{\gamma }_0=2\mathrm{kHz}$. Ramsey narrowing produces a central feature narrower than $2{\gamma }_p$. Adapted from 52, and 148.Reuse & Permissions

Figure 20

Left: Ramsey fringes in a dark resonance, obtained by 152 with 80-$\mu \mathrm{s}$ pulses separated by 1 ms. Right: Probability distribution of the durations in the beam ($t_{\mathrm{in}}$) and in the dark ($t_{\mathrm{out}}$), calculated by 147 for atoms diffusing in a cylindrical geometry. Adapted from 153, and 147.Reuse & Permissions

Figure 21

Dispersion spectrum with a one-dimensional light sheet of width $126\mu \mathrm{m}$. Right: Universal power laws with exponents $-1$ (Lorentzian) and $-0.5$ (one-dimensional recurrence). Adapted from 115.Reuse & Permissions