Photons preserve quantum states for a long lifetime and rarely interact with each other or static fields from the environment. They are superior information carriers. Consequently, manipulation of photon states, such as all-optical switching (AOS) [1–6] and cross-phase modulation (XPM) [7–10], has been considered as a promising means in quantum communication and quantum computation. To make photons interacting with each other via a medium, the effect of electromagnetically induced transparency (EIT), giving rise to slow light as well as storage and retrieval of light, has attracted great attention. It enhances the optical nonlinearity significantly [11] and provides a way to transfer quantum states between photons and matters [12]. Due to large nonlinear susceptibilities at low-light levels, the EIT-based AOS and XPM proposed by Refs. [1] and [7] make the single-photon operation feasible and can lead to the applications in quantum information manipulation.

The EIT-based AOS or XPM scheme is shown in Fig. 1(a). The probe (with the Rabi frequency Ω_p) and coupling (Ω_c) fields form the Λ-type EIT which completely suppresses the probe absorption. The presence of the switching field (Ω_s) induces the three-photon transition that causes attenuation [13, 14] and phase shift [15, 16] of the probe field. We use the AOS to illustrate the observed problem. In the ideal case and Ω_c ≫ Ω_s, the probe transmission is [1]

 

Fig. 1 (a) Four-level EIT-based AOS system. (b) Relevant energy levels of ⁸⁷Rb atoms and laser excitations in the experiment. (c) Diagrammatic representation of the wave vectors of the probe, coupling, switching, and four-wave mixing fields.

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In this study, we theoretically and experimentally demonstrated that given an existing FWM process or Δ_kL an optimum switching detuning makes the switching efficiency reach the ideal result as shown in Eq. (1). The experimental data are in agreement with the theoretical predictions. Because the EIT-based AOS and XPM are resulted from the imaginary and real parts of the same susceptibility, the results of the work can be directly applied to the XPM. Researchers in the field can utilize the result from our work to prevent the switching (or phase-modulation) efficiency from being diminished by the FWM process in the experiments and, thus, to make the AOS (or XPM) effect in the FWM-allowed systems the same as that in the FWM-forbidden systems. The EIT-related studies have been performed in different kinds of experimental systems with different schemes of transitions. To perform EIT-based AOS or XPM, one might not be able to find the FWM-forbidden switching transition either due to an experimental system formed by some atomic species (e.g. alkali earth atoms), condensed matter material, or quantum dot/well, etc. or due to constraints on the transition imposed by available laser wavelength, maximizing transition strength, optical pumping, etc. Therefore, this study is very useful for the research field of low-light-level AOS and XPM.

We performed a comprehensive investigation in the theoretical calculation by taking into account the phase mismatch, which has not been discussed before in the studies of EIT-based AOS and XPM. The analysis was carried out by solving the Maxwell-Schrödinger equations of the light pulses and the optical Bloch equations of the atomic density-matrix operator given by

We experimentally demonstrated the optimum switching detuning with the AOS based on the dynamic EIT scheme. The probe pulse was stored into the medium as the ground-state coherence ρ₂₁ by adiabatically reducing Ω_c to zero. During the storage, a switching pulse was applied to destroy ρ₂₁. Consequently, the output probe pulse retrieved by switching on Ω_c was attenuated. The optimum switching efficiency in this approach is not affected by the OD and coupling intensity [18], making our measurements robust. In the derivation of the retrieved probe energy, we use Eq. (7) and further assume that the temporal variation of ρ₂₁ is much smaller than the spontaneous decay rate, i.e., |(∂ρ₂₁/∂t)/ρ₂₁| ≪ Γ. With Ω_c = ρ₃₁ = 0 during the storage, ρ₂₁ as a function of time is obtained. The retrieved probe energy is proportional to the square of ρ₂₁ right after the application of the switching pulse is complete and, thus, leads to [16]

The experiment was carried out in a laser-cooled cigar-shaped ⁸⁷Rb atom cloud. For establishing a simple four-level system, we optically pumped all population to the ground state |1〉 as shown in Fig. 1(b). The probe beam propagated along the major axis of the atom cloud, denoted as θ = 0°, and had a beam size about 10 times smaller than the transverse size of the cloud. The coupling beam was kept along the direction of θ = 1.6°. The diagrammatic representation of the wave vectors of all the laser fields and the four-wave mixing field can be found in Fig. 1(c). We adjusted the propagating direction of the switching beam to vary Δ_k. Both coupling and switching beam sizes were large enough to interact with all the atoms. The 4-μs square switching pulse was applied to destroy ρ₂₁ during the storage of the probe pulse. The decoherence rate of ρ₂₁ was experimentally determined as 3×10⁻⁴Γ [19] which is negligible in this study. By measuring the ratio of the retrieved probe energies with to without the presence of the switching pulse, we obtained R. Other details of the experimental setup can be found in Ref. [20].

Before experimentally demonstrating our idea, we used a FWM-forbidden AOS system to calibrate the absolute frequency and the intensity of the switching field. The switching field was circularly polarized with left helicity (the σ₋ polarization) and drove the transition of |2〉 to |5〉 as shown in Fig. 1(b). The best switching effect will appear at the resonant switching frequency according to Eq. (1). In all measurements, we kept Ω_c ∼ 0.6Γ and OD ∼ 140, which are irrelevant to the switching efficiency. The squares in Fig. 2(a) show the experimental data. The blue dashed line is the best fit of Eq. (11) with Ω_s = 0.10Γ and Γ′ = 1.3Γ. The red solid line is the theoretical prediction by numerically solving Eqs. (2)–(6). At the minimum transmission, the switching frequency was defined as the zero detuning. All values of the switching Rabi frequency used in this work were traced back to this measurement.

 

Fig. 2 Probe transmission versus switching detuning in the (a) FWM-forbidden and (b) FWM-allowed AOS schemes. (a) Blue dashed line is the best fit of Eq. (11) with δ_s,opt = 0, giving Ω_s = 0.10Γ, Γ′ = 1.3Γ. Red solid line is the prediction calculated by numerically solving Eqs. (2)–(6). (b) Gray, green, black, and cyan solid lines are the numerically calculated predictions with Δ_kL = 0, 40, 52, and 72, respectively. In the calculation, the OD is 118; Ω_s and Γ′ are the same as above because the same switching pulse as (a) except the different polarization was applied.

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We studied the FWM-allowed AOS by setting the switching field to the σ⁺ polarization as shown in Fig. 1(b). The OD was kept 118±8 determined experimentally with the method as described in Ref. [21]. Initially, the switching beam propagated along the direction of the coupling beam, implying Δ_kL ≈ 0. The circles and gray line in Fig. 2(b) are the experimental data and theoretical prediction. The result clearly shows that the switching effect is greatly degraded due to the FWM. Next, we moved the switching propagation direction toward the direction of the probe beam by 1.2° such that we estimated Δ_kL ≈ 46, where L ≈ 1 cm in our system. The squares in Fig. 2(b) are the experimental data. The black line is the prediction calculated by numerically solving Eqs. (2)–(6) with Δ_kL = 52 and gives the best agreement with the data. The green and cyan lines are the theoretical predictions with Δ_kL = 40 and 72, respectively. These two values will indicate the uncertainty, if we use the data to determine the Δ_k of the system. Figure 2(b) clearly illustrates the idea of the optimum switching detuning shown in Eqs. (9)–(11). The comparison between Figs. 2(a) and 2(b) demonstrates that the optimum detuning makes the AOS achieve the ideal switching efficiency even under the influence of FWM.

The minimum required phase mismatch to make Eqs. (9) and (11) valid is investigated. Note that a positive or negative phase mismatch gives the same effect and only changes the sign of the optimum detuning. We measured the minimum probe transmission, R_min, and the optimum switching detuning, δ_s,opt, at each given Δ_kL. In Fig. 3(a), the solid line is the theoretical prediction of R_min versus 1/δ_s,opt calculated by numerically solving Eqs. (2)–(6) and the circles are the experimental data at the OD, α, of 118. The agreement between the prediction and the data is satisfactory. Figure 3(b) shows the theoretical predictions of R_min and 1/|δ_s,opt| as functions of |Δ_k|L at several ODs. According to the predictions, the analytical formula in Eq. (10) is valid as long as |Δ_k|L > 10. Furthermore, R_min is very close to the result of the ideal switching, i.e., Eqs. (9) and (11) are the good approximation, under |Δ_k|L > α/4. This criterion provides the guideline for arranging a suitable phase mismatch in the relevant studies. Also note that provided |Δ_k|L < α/3 the FWM effect at δ_s = δ_s,opt is still significant according to the calculation of Fig. 3(b). We make the following argument to explain the results of the switching and the Ω₄ generation. The presence of the switching field could induce two kinds of transitions. One is the transition from |2〉 to |4〉, which efficiently dissipates the probe energy through the spontaneous decay of the population in |4〉. The other is the Raman transition from |2〉 to |1〉 using the existing ρ₂₁, which converts only a small fraction of the probe energy to generate Ω₄. Under |Δ_k|L = 0, the |2〉 → |4〉 transition does not occur, making the switching effect negligible. Under |Δ_k|L > α/4, the |2〉 → |4〉 transition at δ_s = δ_s,opt is completely activated, making the switching effect ideal. Furthermore, although a non-zero |Δ_k|L degrades the |2〉 → |1〉 Raman transition, the activated |2〉 → |4〉 transition may help the Ω₄ generation. This makes the FWM effect at δ_s = δ_s,opt under |Δ_k|L < α/3 is comparable to that at |Δ_k|L = 0 and δ_s = 0. The above argument and the physical interpretation require further investigations.

 

Fig. 3 (a) Minimum probe transmission versus reciprocal of optimum switching detuning at Δ_kL ≈ 0, 10, 22, and 52. The experimental condition is the same as that in Fig. 2(b). Solid line is the theoretical prediction and circles are the experimental data. Open circle is the minimum measured probe transmission at Δ_kL ≈ 0 in Fig. 2(b) placed at the theoretical optimum detuning. (b) Theoretical predictions of R_min (solid lines) and 1/|δ_s,opt| (dashed lines) as functions of |Δ_k|L at several ODs.

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In conclusion, we have studied the EIT-based AOS under the influence of FWM. Our work experimentally and theoretically demonstrated that the ideal switching efficiency can be achieved in the FWM-allowed system by applying an optimum detuning of the switching field. An analytical formula that determines the optimum detuning is also provided. The result of the work can be directly applied to the EIT-based XPM. This study is of interest and very useful for the research field of low-light-level AOS and XPM.