Quantum Phase Gate And Slow Light Based On Electromagnetically Induced Transparency And Coherent Raman Gain

Slow light via Raman gain in a four-level atomic systemThe five-level atomic system we considered is shown in Fig. 1. The three transitions, 1 ? 3, 2 ? 3 and 2 ? 4, interact with a probe field , a coupling field , and a microwave fieldAccording to the atomic probability amplitude equation, we can plot the susceptibilityχpversus the probe detuningΔp/γ: From Fig. 2 and Fig. 3, we can see the real part ofχpvaries steeply at resonance, it is a normal dispersion, which is essential to the reduction of light speed and the imaginary part ofχpcorresponding to absorption is minus, which means there is a gain at resonance.The expression of group index isThe group velocity defining byυg = c /(1 + ng)With the expressions above, the group index is plotted in Fig. 4 in which the parameters are the same as those in Fig. 3. the group velocityυg = 5.7 m /s, whenΔ1 = 0.00015γ.This mechanism of reduction of light speed by gain is much easier to realize, and it differs from that by EIT. In order to see the influence of microwave on populations, we plot the population of level 4 via time in Fig. 5. When t = 1000γ?1, according to the Raman process,ρ44 is reduced to 0.75. For Rb atomicγ≈6MHz, so t≈170μs. The light pulse needs 0.9ms to pass through the whole media whose length l = 0.5cm. Therefore, the repump field can be used in an experiment to pull the populations distributing in other levels by Raman process back to level 4 .Kerr nonlinearity enhancement and the quantum phase gate implementationIn this paper we focused on a symmetrical five-level Tripod model, which utilize the enhanced Kerr nonlinear effects to achieve an enhanced cross-phase modulation. There are two schemes which are based on EIT effect and Raman Gain. When the conditional phase shift isπwe can obtain a two-qubit phase gate, and then we can take advantage of this phase gate to produce GHZ states. The five-level atomic system we considered is shown in Fig. 6. The four transitions 0 ? 2, 1 ? 4, 2 ? 4 , and 3 ?4 interact with a microwave fieldΩd, a probe fieldΩp ?ofσ+ polarization, a coupling field , and a signal fieldΩcΩs ?ofσ? polarization, respectively. 1. Enhancement of Kerr nonlinearity by EITThe microwave and coupling fields are much stronger than the probe and signal fields (Ωc ,d >>Ωp , s), so that we can setρ1 1 =ρ33 = 0.5. By the steady-state solution of the density matrix equation, we can obtain the first order of susceptibilities of probe and signal fields as And the third order of susceptibilities of probe and signal fields as (3)pfield. As we can see, two symmetric EIT windows are generated at left and right wings of the probe resonance. Due to the intrinsic symmetry of the system, two EIT windows will appear at left and right wings of the signal resonance too. In these EIT windows accompanied by steep normal dispersion, it is possible to remarkably reduce the group velocities of the probe and signal pulses. With the expressions of the third order of susceptibilities of probe and signal fields, we further plot in Fig. 8 the real and imaginary parts ofχaround the two EIT windows. Fig. 8. The nonlinear susceptibilityχp(1)versus the probe detuningΔp /γ, withΔs = 1.062γ(left) andΔs = ?1 .062γ(right).It can be seen that the real part ofχp(3) has a positive peak at the center of the left EIT window while a negative peak at the center of the right EIT window and the imaginary part get close to zero. The same remarks hold true for the real part of (3)χsas a result of the inherent symmetry of the five-level Tripod system. Then the giant nonlinearities m?ay result in efficient XPM between the probe and signal fields when they interact for a sufficiently long time with t?he same group velocities.The group velocities of the probe and signal fields can be defined as When the group velocity of the probe and signal pulses match automatically, we haveυgp =υgs ? 2×10 4m /sfor cold 87Rb atoms at the centers of the two EIT windows. 2. Enhancement of Kerr nonlinearity by Raman gainIn this scheme, the microwave (coupling) field is assumed to be much weaker (stronger) than the probe and signal fields (Ωc >>Ωp ,s >>Ωd). So that we can set probability amplitude . With the steady-state solutions of probability amplitude equations, we get the probe and signal linear susceptibilities as well as the probe and signal nonlinear susceptibilities there exists a narrow gain peak at the probe resonance accompanied by a very steep normal dispersion resulted from the three-photon hyper-Raman process. Fig. 10 shows that the nonlinear susceptibilitybehaves quite very similar to the linear susceptibilityχ(p)except that its amplitude is about two-order smaller. Then the enhanced nonlinear susceptibilitiesχ(p3)a ndχs(3)may be explored to achieve efficient XPM between the probe and signal fields when they interact for a sufficiently long time with the same group velocities. Fig. 10. The nonlinear susceptibilityχ(p3)versus the probe detuningΔp /γ.Under the general conditions ofΔC =Δd= 0andΔp =Δs, we can further obtain the probe and signal group velocities asIt is clear that based on the coherent hyper-Raman gain, the steep linear dispersion at probe and signal resonances will lead to a significant reduction of the group velocities. Compared with the atomic system, our atomic system is equally efficient but much simpler. In particular, we haveυgp =υgs ? 260 m /s for cold 87Rb atoms at the probe and signal resonances.3. Polarization quantum phase gateThe efficient XPM between the probe and signal fields can be applied to devise an all-optical quantum phase gate with the polarized single-photon wavepackets taken as qubits. Aiming at the five-level Tripod system shown in Fig 3, we now consider the case where a probe photon and a signal photon simultaneously impinge upon as a cold 87Rb medium of length L. Considering the corresponding transition of levels and the polarization of the photons, if the photon does not excite the transition, it will acquire the vacuum phase shift ;if the photon excites the transition, it will get linear phase shift because it interacts with the microwave and coupling fields in the EIT or Raman gain regime; if the two photons both interact with the transitions, through a substantial XPM, they will acquire the linear plus nonlinear phase shiftsθ+p =θlpin +θnpo n,θ?s =θlsin +θnson. Then the truth table for this all-optical phase gate isIt is clear that, to achieve the maximal nonlinear phase shifts, one should have TP = Ts andυgp =υgs for the probe and signal photons. Thus a conditional phase shift ofθcon≈πmay be attained in the EIT (coherent Raman gain) regime when the probe and signal pulses synchronously propagate through a medium of length l = 4.9mm (l = 1.1mm ) and density N = 1012 cm?3.Using the phase gate above and five half-wave plates to act as a single qubit Hadamard gates and two EIT media, we can propose a new method to generate three-photon GHZ states, as shown in Fig. 11.