Stationary light pulses (SLPs) [1] can be created inside optically strongly driven media via a coupling scheme based on electromagnetically induced transparency (EIT) [2, 3] with counterpropagating beams. Among many applications of SLPs is their use for nonlinear optics and quantum information processing [4]. In a proposal [4], André et al. suggested a cross-phase modulation scheme based on SLPs, to apply a phase shift of π to a light pulse via nonlinear optical interaction with a second pulse. As has been demonstrated experimentally, a normal Λ coupling scheme [Fig. 1a ] does not produce stationary light in cold media [5], in contrast to hot media [1], where spatially fast oscillating coherences preventing the SLP formation in cold media are naturally suppressed by random atomic motion. Transforming the Λ scheme into a double-Λ scheme by detuning at least one of the laser fields from resonance [Fig. 1b] [6], however, allows the formation of SLPs also in cold media [5]. When a probe pulse propagates under slow-light conditions, a Doppler-broadened medium driven on-resonance produces a negligible phase shift [7]. However, off-resonantly driven transitions add a time-dependent phase shift to the pulse [8]. One might expect such a phase variation (PV) also in the case of SLPs involving off-resonantly driven transitions. Such PV would obviously be detrimental if SLPs were used for cross-phase modulation as proposed in [4]. In a theoretical study Moiseev and Ham showed [6] that indeed such PV should occur for such double-Λ scheme, as shown in Fig. 1b of low optical depth (OD). In this Letter, we present experimental evidence of such PV. We find good agreement between the experiment and numerical simulations. These simulations suggest, that in media of high OD the PV is negligible—in agreement with [6]—making the implementation of the scheme in [4] possible. We note that another double-Λ scheme exists [9] that should avoid the problem of PV. However, this scheme is not as easily implemented as the one studied here and might lead to complications owing to the two polarization components of the SLPs.

We consider a medium of ${}^{87}\mathrm{Rb}$ atoms. The states shown in Fig. 1 are $|1⟩=|5^2{\mathrm{S}}_{1∕2},F=1⟩$, $|2⟩=|5^2{\mathrm{S}}_{1∕2},F=2⟩$, and $|3⟩=|5^2{\mathrm{P}}_{3∕2},F^{\prime }=2⟩$. States $|1⟩$ and $|2⟩$ are metastable, while state $|3⟩$ decays radiatively at rate $\Gamma =2\pi \times 6\mathrm{MHz}$. The transitions were driven by two counterpropagating coupling fields of equal Rabi frequencies ${\Omega }_c^{\pm }$. SLPs are formed on the probe transition by two counterpropagating fields of Rabi frequencies ${\Omega }_p^{\pm }$, with $\left|{\Omega }_p^{\pm }\right|\ll \left|{\Omega }_c^{\pm }\right|$. The coupling fields ${\Omega }_c^{\pm }$ are detuned from the transition $|2⟩\leftrightarrow |3⟩$ by ${\Delta }^{\pm }$. The spatio-temporal evolution of a probe pulse with slowly varying envelopes ${\Omega }_p^{\pm }$ inside such medium can be obtained by numerically solving the Maxwell–Bloch equations [5]:

As shown in Fig. 2 , neither asymmetrically $({\Delta }^{+}\ne -{\Delta }^{-})$ nor symmetrically $({\Delta }^{+}=-{\Delta }^{-})$ detuned coupling fields lead to a completely vanishing PV in media of low OD, as already briefly discussed theoretically [6]. We obtained the PV $\phi \left(t\right)$ across the retrieved probe pulse envelope via $\phi \left(t\right)=\arctan (\mathrm{Im}\left[{\Omega }_p^{+}\left(t\right)\right]∕\mathrm{Re}\left[{\Omega }_p^{+}\left(t\right)\right])$. In all cases a change of sign of the detuning also leads to a change of sign of $\phi \left(t\right)$ (compare black and red and blue and magenta curves, respectively, in Fig. 2). However, $\phi \left(t\right)$ depends on the choice of direction; i.e., a detuning Δ applied in the forward direction leads to a different PV than the same detuning applied in the backward direction [compare solid and dashed curves in Figs. 2a, 2b]. An increase of OD does not reduce the magnitude of φ [see Fig. 2b]. $\phi \left(t\right)$ changes qualitatively and quantitatively, if the detuning is applied symmetrically [Fig. 2c]. Only for a high OD of $\alpha =100$ (dashed-dotted curves) the PV is less than for the corresponding case with an asymmetric detuning. This decrease of the PV for increasing OD is in agreement with [6].

The experiment was performed in a medium of laser-cooled ${}^{87}\mathrm{Rb}$ atoms with an EIT coupling scheme involving two counterpropagating coupling beams. A detailed description of the experimental setup can be found in [5, 10]. The PV across the probe pulse envelope was determined via a heterodyne detection scheme [11]. The Gaussian-shaped probe pulses ($800\mathrm{ns}$ ${\mathit{e}}^{-1}$ full width) were generated by the first diffraction order of an acousto-optic modulator (AOM) driven at a frequency of $f_{\mathrm{AOM}}=80\mathrm{MHz}$. By combining the cw zero diffraction order beam with the probe pulse, the intensity at the detector was given by

We recorded simultaneously the beat note of the probe pulses and the reference pulses, as shown in Fig. 3b. The experimental data were fitted with Eq. (5) in intervals of $0.1\mu \mathrm{s}$ for which we assumed a constant phase $\phi \left(t\right)=\phi$, by use of $f_{\mathrm{AOM}}$ as obtained from the reference pulses. Before each measurement we adjusted the laser fields ${\Omega }_p^{+}$ and ${\Omega }_c^{+}$ to one- and two-photon resonance, such that the probe pulses had no PV after propagating through the medium. This procedure had to be done before each measurement, as the frequencies of the AOM drivers that generated the probe and coupling fields were drifting slowly (less than $3\mathrm{kHz}∕\min$). Figure 4a shows the PV across the probe pulses after passing through a medium under slow-light conditions for different two-photon detunings δ. We adjusted δ by tuning the frequency of the probe laser. From these data we were able to determine the Raman resonance with a precision of $4\mathrm{kHz}$, corresponding to 1.7% of the EIT window width, and resulting in ${\left|\Delta \phi \right|}_{\max }=4°$. This method of finding Raman resonance is much more sensitive than any transmission measurements, as the probe pulse absorption does not change significantly for such small detunings. In Figs. 4b, 4c, 4d we present the results on the PV of probe pulses acquired during an SLP period. The numerical data (black curves) were obtained by solving Eqs. (1, 2, 3, 4) for ${\Omega }_p^{\pm }(z,t)$. The calculation parameters ${\Omega }_c^{\pm }$, α, ${\gamma }^{\left(0\right)}$, and ${\gamma }^{\left(2\right)}$ used in Fig. 4 were determined independently from experimental data similar to those shown in [5] [Figs. 4a, 4b, 4c]. As our measurement permitted us to only determine the relative phase across the pulses, we shifted the experimentally obtained phase to match the numerical prediction at the center of the pulses $(t=0)$. The experimental (symbols) and numerical results (curves) for an asymmetric detuning ${\Delta }^{+}\ne -{\Delta }^{-}$ during the SLP period are shown in Figs. 4b, 4c. ${\Delta }^{+}=0$ was kept constant at all times. A PV of about $\pi ∕5$ across the probe pulse is clearly observed. A change of sign of the detuning also leads to a change of sign of the PV. We attribute the discrepancy at earlier times to the fact that the rising edge of the retrieved probe pulse is distorted and not anymore Gaussian-shaped (see red curve in Fig. 3b). The results for a symmetric detuning during the SLP period are shown in Fig. 4d. A smaller, but still significant PV of about $\pi ∕10$ is induced during the SLP period. The PV is almost linear, as predicted theoretically [6] and by our numerical simulation. This PV is supposed to become negligible in the limit of media of high OD [6] as confirmed by our numerical simulations. Owing to our limited OD of $\alpha \simeq 40$ we could not directly verify this prediction. However, as the experimental data agree well with the numerical results in the low OD limit, we believe that also for a high OD the numerical results can be trusted.

We have demonstrated that a phase variation exists across SLPs induced by off-resonantly driven transitions in a double-Λ scheme in a medium of low OD. Such phase variation is not desirable, e.g., when SLPs are used for quantum information processing. However, numerical simulations predict that this phase variation is negligible in media of high OD for appropriately chosen detunings of the transitions. Alternatively, an (experimentally more challenging) double-Λ scheme where all transitions are driven resonantly [9] might be used to avoid the problem of phase variation.

This project was funded by the National Science Council of Taiwan (NSCT) under grant 98-2628-M-007-001.

 

Fig. 1 (a) Λ-type coupling scheme for EIT. (b) Possible double-Λ scheme for the formation of SLPs.

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Fig. 2 Numerical simulation based on Eqs. (1, 2, 3, 4) of the PV across the temporal ${\mathit{e}}^{-1}$ full width $2\tau$ of a probe pulse for a SLP duration of ${\tau }_{\mathrm{SLP}}=2\mu \mathrm{s}$ $(\simeq 75{\Gamma }^{-1})$. (a) Asymmetric detuning ${\Delta }^{+}=\pm (4∕6)\Gamma$, ${\Delta }^{-}=0$ (solid curves) and ${\Delta }^{+}=0$, ${\Delta }^{-}=\pm (4∕6)\Gamma$ (dashed curves) for $\alpha =40$ $({\Omega }_c=0.7\Gamma )$. The detuning is positive for black and negative for red (gray in print) curves. (b) Same as (a), except for $\alpha =100$ $({\Omega }_c=1.1\Gamma )$. (c) Symmetric detuning ${\Delta }^{\pm }=\pm (4∕6)\Gamma$ [blue (dark gray in print)] and ${\Delta }^{\pm }=\mp (4∕6)\Gamma$ [magenta (gray in print)] for $\alpha =40$ (solid curves) and $\alpha =100$ (dashed-dotted curves). A Gaussian-shaped pulse of ${\mathit{e}}^{-1}$ full width of $800\mathrm{ns}$ $(\simeq 30{\Gamma }^{-1})$ is initially sent into the medium. The group velocity of the slow probe pulse is kept constant in all plots.

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Fig. 3 (a) Pulse sequence of the SLP PV measurements. Experimental (symbols) and numerical data (curves) of the input probe pulses (magenta squares) and transmission in the forward (red circles) and backward direction (blue triangles whose values are all nearly zero). The timing of the coupling fields in the forward (black diamonds) and backward direction (gray triangles) is shown by solid lines. The detuning of the coupling fields was either kept constant or changed within the two storage periods $500\mathrm{ns}$ before and after the SLP period. (b) Beat note of the transmitted probe pulse in the forward direction (red line between 6.5 and $9\mu \mathrm{s}$) and the reference pulse (magenta line between 9 and $10.5\mu \mathrm{s}$). This reference pulse was obtained by splitting off part of the probe pulses before entering the medium.

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Fig. 4 Measured phase φ across the ${\mathit{e}}^{-1}$ full width $2\tau$ of (a) slow-light pulses after propagation through a medium for different two-photon detunings δ and (b)–(d) retrieved pulses after an SLP period of ${\tau }_{\mathrm{SLP}}=2.00\left(5\right)\mu \mathrm{s}$ for different detunings ${\Delta }^{\pm }$. $t=0$ corresponds to the center of the detected pulses. (a) $\delta =0$ (red diamonds), $-25\mathrm{kHz}$ (blue circles), $-162\mathrm{kHz}$ (black squares), $112\mathrm{kHz}$ (magenta triangles). The solid curves correspond to the best linear fit. (b) ${\Delta }^{+}=0$, ${\Delta }^{-}=+(7∕6)\Gamma$. (c) ${\Delta }^{+}=0$, ${\Delta }^{-}=-(7∕6)\Gamma$. Measured (circles) and numerical data (curves) as obtained for a pulse timing, as shown in Fig. 3a. The detunings ${\Delta }^{\pm }$ were constant during the whole sequence. (d) Symmetric detuning of ${\Delta }^{\pm }=\mp (4∕6)\Gamma$. The detunings ${\Delta }^{\pm }$ were applied symmetrically only within ${\tau }_{\mathrm{SLP}}\pm 500\mathrm{ns}$. The error bars represent the error obtained from the best fit for each interval. In all plots $\alpha =40\left(2\right)$, ${\gamma }^{\left(0\right)}=0.0005\left(2\right)\Gamma$, and ${\gamma }^{\left(2\right)}=0.016\left(1\right)\Gamma$. The error bar indicator in the lower right corner of (d) represents the error in the t in all plots.

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