1. Introduction

Quantum interference among multiple paths for light fields inducing atomic transitions plays an important role in fundamental studies and practical applications in optical physics [1–6]. Differing from a linear interferometer in which light beams with same frequency and polarization interfere with each other, the interference of light beams based on multiple transition paths can have different frequencies and/or different polarizations. In applications such as control of multiphase excitation and ionization products [7–10], the interference involves multiphoton transitions and the optical fields are tuned far away from frequencies of single photon transitions to minimize the linear absorption loss. This leads to the requirement of strong light fields and low interference contrast. In certain applications such as optical switching, it is desirable to have a high interference contrast of the optical fields and a low control light intensity. A variety of methods and techniques for realizing optical switching at low intensities have been explored and studied in recent years [11–23]. Here we report an experimental study of quantum interference between Raman transitions in a three-level Λ system. The Raman-type interference has a high interference contrast and is made possible due to suppression of the linear absorption loss when the two-photon Raman transition is resonant in the Λ system [24–25]. The light fields can be tuned close to the excited state resonance, which results in a large Raman transition rate and an observed interference contrast of ~60%. We show that the Raman interference can be used to perform absorptive switching of a weak signal field by varying the phase of a weak control field.

2. Theoretical model and analysis

Consider a three-level Λ system coupled by four laser fields as shown in Fig. 1. Two coupling Fields with different frequencies drive the atomic transition |2>-|3> with Rabi frequency Ω₁ and Ω₂, respectively. A signal field with a Rabi frequency Ω_s and a control field with a Rabi frequency Ω_c couple the transition |1>-|3>. Here ${\Omega }_i=\mid {\Omega }_i\mid e^{i{\varphi }_i}$ (i=1, 2, s, and c) is characterized by the amplitude |Ω_i| and phase ϕ_i. The frequency detuning for the respective transitions are defined as Δ_s=ω _s-ω ₃₁, Δ_c=ω _c-ω ₃₁, Δ₁=ω ₁-ω ₃₂, and Δ₂=ω ₂-ω ₃₂ (ω _i(i=s, 1, c, 2) is the angular frequency of the laser field i). Under the condition Ω₁~Ω₂≫Ω_s~Ω_c, the atomic population is concentrated in the state |1>. If we choose a symmetrical coupling configuration with the detunings given by Δ₂=-Δ₁=δ and Δ_c=-Δ_s=δ as shown in Fig. 1(a), two resonant Raman channels are opened. One resonant Raman channel, $\mid 1>\stackrel{{\Omega }_s}{\to }\mid 3>\stackrel{{\Omega }_1}{\to }\mid 2>$, is induced by the signal field s and the coupling field 1; another Resonant Raman channel, $\mid 1>\stackrel{{\Omega }_c}{\to }\mid 3>\stackrel{{\Omega }_2}{\to }\mid 2>$, is induced by the control field c and the coupling field 2. The two resonant Raman channels are dominant while the other nonresonant Raman channels, such as $\mid 1>\stackrel{{\Omega }_s}{\to }\mid 3>\stackrel{{\Omega }_2}{\to }\mid 2>$ and $\mid 1>\stackrel{{\Omega }_c}{\to }\mid 3>\stackrel{{\Omega }_1}{\to }\mid 2>$, have much smaller amplitudes and can be neglected in the qualitative discussion of the physical picture here. The two resonant Raman channels interfere with each other and the interference can be controlled by the relative phase between the signal field and the control field. If the interference is constructive, the two-photon Raman transition is enhanced and a large absorption of the signal (control) field is obtained. On the other hand, if the interference is destructive, the two-photon Raman transitions cancel each other and the signal (control) field propagates without attenuation through the three-level atomic medium. Obviously, the role of the control filed and the signal field is interchangeable. We solve the density matrix equations numerically [26] and plot in Fig. 2 the calculated transmission of the signal (control) intensity through the three-level Λ-type medium versus the frequency detuning of the signal field, Δs, under the symmetrical coupling condition (Δ2=-Δ1=δ=3.5γ3 and Δc=Δs+2δ). Figure 2 shows that the interference occurs at the two-photon Raman resonance Δs=-δ, at which, when Φc-Φs=0, the interference is destructive and the signal field propagates through the medium without any attenuation; when Φc-Φs=π, the interference is constructive and near 100% attenuation of the signal field is obtained.

 

Fig. 1. (a). Coherently prepared three-level Λ-type system. γ ₃ is the spontaneous decay rate. Interference occurs between two channels of resonant Raman transitions (b) and (c).

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Fig. 2. Calculated transmission of the signal field versus the normalized signal detuning Δ_s/γ ₃. The relative phase difference of the control field and signal field is (a) ΔΦ=Φ_c-Φ_s=0 and (b) ΔΦ=Φ_c-Φ_s=π. At Δ_s=-δ, the light absorption is suppressed by destructive interference (a) and enhanced by constructive interference (b). The parameters are Ω₁=Ω₂=1.2γ ₃, Ω_s=Ω_c=0.01γ ₃, δ=3.5γ ₃, Δ_c=Δ_s+2δ. The optical depth of the medium is nσ₁₃ ℓ=10.

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3. Experimental results

The experiment is done with cold ⁸⁵Rb atoms confined in a magneto-optical trap (MOT). The MOT is obtained with a tapered-amplifier diode laser (Toptica TA-100) that is used as the cooling and trapping laser with six mutually perpendicular beams with a beam diameter ~2.5 cm, and an extended-cavity diode laser (output power ~30 mW) is used as the repump laser with the same beam diameter of ~2.5 cm. The trapped ⁸⁵Rb atom cloud is ~2 mm in diameter and the measured optical depth nσ13ℓ is ~6. A simplified diagram of the experimental set up is depicted in Fig. 3(b). An extended-cavity diode laser (DL1) with a beam diameter ~3 mm and output power ~50 mW is used as the coupling laser. The driving electric current to the diode laser is modulated at 2δ=40 MHz, which produces two first-order frequency sidebands (ν₀+2δ and ν₀-2δ) with about equal amplitudes as the central carrier (ν₀). As the higher-order sidebands are much weaker than the carrier and are detuned farther away from the atomic resonance, their effects on the coupled Λ system can be neglected under our experimental conditions. During the experiment, the upper sideband ν₀+2δ and the carrier ν₀ act as the coupling field 2 and 1 respectively. Another extended-cavity diode laser (DL2) passes through an acousto-optics modulator (AOM2). The first-order diffracted beam (shifted in frequency by 2δ=40 MHz) is used as the signal field and the zeroth-order beam is used as the control field. The control field (the zeroth beam) is passed through an electro-optic modulator (EOM, New Focus 4002) and its phase is controlled by a DC voltage applied to the EOM. The signal beam and the control beam have about equal powers, are combined in a beam combiner, and then are coupled into a polarization-maintaining single-mode fiber. The output beam from the polarization-maintaining single-mode fiber is collimated to a beam diameter of ~0.5 mm, 3. attenuated to a power level of ~1 µW, and then directed to be overlapped with the frequency-modulated coupling laser in the MOT (with the propagating directions separated by a small angle of ~2°). After transmission through the MOT, the combined control and the signal light is collected by a photodiode detector of 10 MHz bandwidth. All laser fields are circularly polarized (σ ⁺) and interact with the Rb transitions to form 5 separate sets of the three-level Λ system among the magnetic sublevels [23].

 

Fig. 3. (a). The coupled three-level Λ-type system formed with the ⁸⁵Rb D1 transitions at 795 nm. (b) Simplified diagram of the experimental set up. The spontaneous decay rate γ ₃=2πx5.6×10⁶ s^-1. AOM: acousto-optic modulator; EOM: electro-optic modulator; PMF: polarization maintaining fiber; λ/4: quarter-wave plate; DL1(2): extended-cavity diode laser; M: mirror; D: photodetector.

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The experiment is run in a sequential mode with a repetition rate of 5 Hz. all lasers are turned on or off by Acousto-Optic Modulators (AOM) according to the time sequence described below. For each period of 200 ms, ~198 ms is used for cooling and trapping of the ⁸⁵Rb atoms, during which the trapping laser and the repump laser are turned on by two AOMs while the coupling laser DL1 and the signal laser are off. The time for the data collection lasts ~2 ms, during which the repump laser is turned off first, and after a time delay of ~0.15 ms, the cooling laser and the electrical current to the anti-Helmholtz coils of the MOT are turned off, and the coupling laser and the signal laser are turned on. After a further time delay of ~0.1 ms, the frequency of the laser DL2 (providing both the control and the signal fields) is scanned across the ⁸⁵Rb D₁ F=2→F’=3 transition and the transmission of the combined signal and the control lasers are then recorded.

The measured transmission spectra of the signal and control fields versus the signal laser detuning Δ_s are plotted in Fig. 4. For these measurements, the coupling field 1(2) is detuned from the excited state by δ=-20 (+20) MHz while the control laser detuning Δ_c follows the signal detuning by Δ_c=Δ_s+40 MHz. Figure 3 shows that the two Raman channels, $\mid 1>\stackrel{{\Omega }_s}{\to }\mid 3>\stackrel{{\Omega }_1}{\to }\mid 2>$nd $\mid 1>\stackrel{{\Omega }_c}{\to }\mid 3>\stackrel{{\Omega }_2}{\to }\mid 2>$ re resonant at Δ_s=-20 MHz. When the phase difference between the control field and the signal field satisfies Φ_c-Φ_s=0, the destructive interference between the two resonant Raman channels suppresses the light absorption at Δ_s=-20 MHz (Fig. 4(a)); when the phase difference is Φ_c-Φ_s=π, the constructive interference between the two channels enhances the light absorption at Δ_s=-20 MHz (Fig. 4(b)). At the two-photon Raman resonance (Δ_s=-20 MHz), the measured transmission is ~25% for the constructive interference and ~81% for the destructive interference. In Fig. 4, black lines are experimental data and the red lines are theoretical calculations with trichromatic coupling fields of Fig. 3(a) (including three coupling field components ν, ν₀, and ν-). Comparison with Fig. 2 shows that the third component of the coupling laser (the lower sideband ν- of the frequency modulated coupling laser is detuned by -60 MHz from the excited state, see Fig. 3(b)) contributes to the observed asymmetrical line profiles but does not alter the interference pattern at Δ_s=-20 MHz. Therefore, the main result, control of the interference at Δ_s=-δ=-20 MHz between the two resonant Raman channels is not affected by the lower sideband ν

 

Fig. 4. Measured light transmission versus the signal detuning Δ_s. Black (red) lines are experimental data (calculations). The experimental parameters are Ω₁/(2π)≈Ω₂/(2π)≈7 MHz, Ω_c/(2π)≈Ω_s/(2π)≈0.2 MHz and δ=20 MHz. The fitting parameters are γ ₂ ≈0.02γ ₃, Δ_c=Δ_s+2δ, and nσ₁₃ ℓ=6.

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Next we present the experimental results for the interference contrast for the combined signal and control light fields in the three-level system. The interference contrast can be defined $C=\frac{I_{\mathrm{des}}-I_{\mathrm{con}}}{I_{\mathrm{des}}+I_{\mathrm{con}}}$, where I_des is the transmitted light intensity at the destructive interference and I_con is the transmitted light intensity at the constructive interference. The converted phase difference between the control field and signal field (derived from the voltage applied to the EOM) and the measured transmitted intensity are plotted versus time in Fig. 5. For these measurements, the frequencies of the laser fields are tuned to the two-photon Raman resonance (Fig. 3(a), a sinusoidal voltage is applied to the EOM which varies the control-signal phase difference ΔΦ=Φ_c-Φ_s from 0 to π periodically, and the transmitted light of the combined signal and control fields is recorded. Figure 5(a) plots the phase of the control field (proportional to the sinusoidal voltage applied to the EOM) and Fig. 5(b) plots the transmitted light intensity in a time interval of ~1 ms. The black line is the experimental data and the red line is the theoretical calculation. The interference contrast derived from these measurements is C≈53%, which is close to the calculated interference contrast of C=58%.

 

Fig. 5. (a). The control phase Φ_c (proportional to the applied sinusoidal voltage) versus time. (b) Transmission of the combined signal and the control fields versus time. (c) The control phase Φ_c (proportional to the square-wave voltage) versus time. (d) Transmission of the combined signal and the control fields versus time. The measurements are taken with δ=20 MHz, Δ_s=-δ=-20 MHz, Δ_c=Δ_s+2δ. The other parameters are the same as that in Fig. 4.

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To demonstrate that the interference of the two-channel Raman resonance in the Λ system can be used for the absorptive light switching at low light levels, we apply a square-wave voltage to the EOM that alternates the control phase Φ_c from 0 to π and record the transmitted light versus the time. The experimental results are plotted in Figs. 5(c) and 5(d). The data show the periodic switching of the light transmission versus Φ_c.(the light transmission changes from 25% (the switch open state) to 80% (the switch closes state). The effectiveness of the optical switching can be characterized by the switching efficiency defined as η=(I _close-I _open)/I _in. Here I _in is the incident light intensity, I_close is the transmitted intensity when the switch is closed, and I_open is the transmitted intensity when the switch is open. For perfect switching, η=100% (I_close=I_in and I_open=0). In our experiments, when the switch is open (the constructive interference), the light transmission is ≈25%, which is limited by the optical depth of the cold atomic cloud; when the switch is closed (the destructive interference), the light transmission is≈80%, which is limited by the absorption loss due to the laser frequency drifts, the Zeeman broadening from the residual magnetic field, and the decay rate γ ₂ of the ground state coherence. From the measurement, we derive that the observed switching efficiency is η≈60%.

4. Conclusion

In conclusion, we have observed quantum interference of two resonant Raman transitions in a three-level Λ-type system, and demonstrated the phase control of the absorptive switching at low light levels (~0.1 mW/cm²).