Quantitative analysis of the transient response of the refractive index to conditions of electromagnetically induced transparency

For the example of the $D$${}_1$ line of ${}^{87}$Rb we analyze the experimental parameters that control the transient response of electromagnetically induced transparency. Quantum coherent free-induction decay is observed on time scales exceeding several milliseconds in a buffer-gas vapor cell. Numerical solutions of the quantum master equation and approximate analytical solutions are tested and absolute comparisons of the transient time scales, power broadening, resonance contrast, and frequency shifts are made. Two actively-phase-locked lasers are employed. The effects of laser phase noise that is not fully correlated are studied.

Figure 1

Schematic of experiment and optical phase-locked loop (OPLL) between master and slave lasers: PD, photodiodes; GPS, global positioning system; HF Gen, high-frequency generators.

Figure 2

Transitions studied are shown for $B=3.2$ G. The ground-state levels ($F^{\prime }=1,2$) are shown to be degenerate.

Figure 3

The $R^{\pm }$ dark resonances split due to the quadratic Zeeman effect when an external axial magnetic field is applied. The total laser intensity is 750 $\mu$W/cm${}^2$ at a sweep rate of 1 kHz/ms. The term $\nu$ is the difference frequency between the phase-locked lasers. The diminution of signal depth at nominally zero external magnetic field is a consequence of the loss of a well-defined quantization axis in relation to the laser field polarization [18].

Figure 4

Splitting of the clock transition due to the quadratic Zeeman effect. Solid lines show the theoretical shifts using the parameters given by Steck [26]. The dashed line represents a transition that cancels for purely linear polarization in each beam [18]. Black circles denote resonance shifts from Fig. 3.

Figure 5

Slow scan (7.5 Hz/ms) over the dark resonances of the clock transition at $I_0=30\mu$W/cm${}^2$ at nominally zero magnetic field. The black line represents a fit of two Lorentzian line shapes. The full widths at half maximum are $\Delta \nu =78.5\pm 2.5$ Hz. The photodiode signal has been shifted by +130.14 mV. The frequency offset $+$7495 Hz is due to the pressure shift of the hyperfine interval by the neon buffer gas.

Figure 6

Dynamic response of absorption when switching the slave laser frequency. Initially detuned from two-photon resonance, the laser is tuned into resonance at time $t_1$. An exponential decrease of absorption appears as the system approaches the EIT dip. At time $t_2$ the laser is tuned off resonance again. Oscillations related to the free-induction decay of the coherent superposition state appear when exiting from the dark state.

Figure 7

The $\Lambda$-type three-level system coupled via two nearly resonant lasers. We define the two-photon detuning as $\delta ={\Delta }_2-{\Delta }_1$.

Figure 8

Coupling terms when lasers are detuned by $\delta$ from the EIT condition.

Figure 9

Numerical solution for the transient response to a gradual detuning from resonance for $g=250\mathrm{kHz},\Gamma =2\pi \times 6\mathrm{MHz}$, and ${\Delta }_1=\gamma =0$.

Figure 10

Transient behavior of the absorption signal when scanning over a dark resonance at different sweep rates. Top: Experimental traces of $R^{-}$ resonance at 225 $\mu$W/cm${}^2$ and $B=3.2$ G. Bottom: Numerical prediction with $G=2\pi \times 150\mathrm{Hz}$ and $\gamma =100$ Hz.

Figure 11

Numerical predictions for the dependence of the dynamic shift of the resonance minimum on the sweep rate. The conditions are the same as in Fig. 10.

Figure 12

(a) Transient behavior of absorption when exiting EIT. Laser 2 is suddenly detuned from resonance by $\delta =2\pi \times 10$ kHz at $t=0$. (b) Analytic solutions with $\gamma =100$ Hz.

Figure 13

Transient response when switching laser 2, initially detuned from the $R^{-}$ resonance by $\delta /2\pi =10$ kHz, close to two-photon resonance. ($I_0=600\mu \mathrm{W}/{\mathrm{cm}}^2$.)

Figure 14

Closeups of experiments in Fig. 13 are given in the top row. Bottom left: Analytic solutions [Eq. (28)] for switching laser 2, initially detuned by $\delta /2\pi =10$ kHz to a position off the $R^{-}$ resonance by $\epsilon /2\pi$ as marked in the figure ($G=2\pi \times 520\mathrm{Hz}$ and $\gamma =100\text{Hz}$). Bottom right: Numerical solution when switching the lasers back to $\delta /2\pi =10$ kHz starting from a positions left and right of the EIT minimum $\epsilon /2\pi =\pm 1800$ Hz.

Figure 15

Decay rate ${\Gamma }^{\prime }$ when entering the dark state and when exiting the dark state for different total intensities. Data obtained in slow scans across the dark resonance are given by open circles.

Figure 16

Dynamic response when entering and exiting EIT observed in an unbuffered absorption cell.

Figure 17

Relative absorption at EIT minimum ${\mathcal{A}}_{\delta =0}$ and resonance contrast for different laser intensities. For a description of conditions (A) and (B) see Fig. 18.

Figure 18

(a) Relative absorption and resonance contrast at $I_0=450\mu$W/cm${}^2$ and (b) low-frequency spectrum of the laser beat frequency ($\approx$6.8 GHz), as recorded on a spectrum analyzer at two different phase error levels. The data denoted by (B) refer to $\langle \Delta \phi \rangle =0.83$ and the data denoted by (A) refer to $\langle \Delta \phi \rangle =0.47$.

Figure 19

ac Stark shift of the $R^{-}$ resonance position observed at various sweep rates ranging from 1 to 19 Hz/ms. Circles and triangles give the positions for the two sweep directions. Thin vertical lines represent the uncertainty in the triangle data, which arises from the range of data included in the Lorentzian fit. Similar uncertainty applies to the circle data. Thick (blue) vertical bars represent the predicted dynamic shift between circles and triangles. The dashed line gives a linear fit of the ac Stark shift.

Figure 20

Ground-state decoherence rate as a function of Ne gas pressure at $25{}^{\circ }\mathrm{C}$ for two effective diameters of the EIT volume. The gray vertical line indicates the pressure in our experiment.