Coherent Control Of Dark State, Light Propagation, And Double Photonic Bandgaps In A Tripod System Of Cold Atoms

In this thesis for doctorate that consists of three parts, we study dynamical evolution for multiple degenerate dark states, ultraslow and superluminal light propagation, and double photonic bandgaps dynamically induced in a tripod system of cold atoms.Dynamical evolution for multiple degenerate dark states in the tripod atomic systemIn this part, we investigate in detail both dynamical and steady responses of a tripod-type atomic system to the applied laser fields, and analyze how simple dark states having two ground levels contribute to generate the multiple degenerate steady dark state in a complicated way.The tripod system can be either open or closed depending on where atoms at the only excited level spontaneously decay. We first consider an open model where excited atoms only spontaneously decay to an external level of the tripod system.By numerical calculations based on the density matrix formalism, we find that the open system cannot definitely evolve into a simple dark state with two ground levels when two of the driving fields are on Raman resonance, as shown in FIG. 2(b) and FIG. 2(c). If all driving fields are on two-photon resonance, however, a complicated dark state consisting of three ground levels will be established leading to a part of the population reserved in the open system, as shown in FIG. 2(d). This complicated dark state depends critically on Rabi frequencies of the driving fields as well as the initial population distribution, but is irrelevant to field detunings and the outward spontaneous decay rate, as shown in FIG. 3. Via the quantum jump method, we derive the exact analytical expression for this complicated dark state and also determine the population loss as a result of spontaneous decay, which are consistent with the numerical calculations. The expression of the dark state is , whereΦi is a twofold degenerate dark state and is the incoherent superposition amplitude. Our analytical results reveal that this complicated dark state is a coherent superposition of two simple dark states when all atoms are initially at a single ground level, but in general it is contributed by all three simple dark states both coherently and incoherently.Using the same numerical and analytical methods, we then alternatively investigate the closed model where spontaneous emission occurs only toward internal levels of the tripod system.Our numerical results show that the coherent population trapping can always be realized when two driving fields are on Raman resonance, as shown in FIG. 4. The fact that atoms can go into the simple dark state superposed by |1> and |2> even if they are initially at level |0> is due to the population redistribution among the ground levels resulted from the inward spontaneous emission. When all three driving fields are on Raman resonance, Rabi frequencies, spontaneous decay rates, and the initial population distribution all have important influence on the steady dark state, as shown in FIG.5.Our analytical results indicate that the complicated dark state is always superposed by all three simple dark states in both coherent and incoherent ways, andthe incoherent superposition amplitude . The complicated dark state structure is determined by the initial population distribution, relative values of Rabi frequencies, and relative values of spontaneous decay rates.Ultraslow and superluminal light propagation in a tripod system of cold atomsIn this part, we study the propagation dynamics of one or two probe pulses in a tripod-type atomic system (as shown in FIG. 1. withγ3 = 0) for three different situations.In the absence of an incoherent pump field, under the resonant conditionΔ₀ =Δ₁ =Δ₂ =0, a weak probe pulse free of absorption can propagate in the atomic medium at a greatly reduced group velocity . Our analytical results indicate that the group velocity is sensitive to Rabi frequencies of both continuous-wave coupling fields and all spontaneous decay rates, as well as the initial population distribution: , where . Thus, to well control and manipulate the probe pulse, one should clearly know all spontaneous decay rates and the initial atomic state. This counterintuitive analytical finding is verified by the full numerical calculations based on the coupled Maxwell-Bloch equations as shown in FIG. 6.Instead, one can also simultaneously manipulate the propagation of two weak probe pulses, e.g.,ω0 andω1 , to achieve ultraslow group velocities using a single continuous-wave coupling field, e.g.,ω2, As we can see, under the assumptionαγ0 =βγ1 without loss of generality, we can obtain and can manipulate this velocity ratio by preparing a preferred initial atomic state as shown in FIG. 7 and FIG. 8. A careful preparation of the initial atomic state, realized by appropriate optical pumping schemes, will lead to the controllable time delay between the two ultraslow probe pulses, which is useful for efficient nonlinear interactions. Utilizing a weak incoherent pump field, one can further switch a weak probe pulseω0 from ultraslow to superluminal propagation by simultaneously tuning the frequencies of the two continuous-wave coupling fieldsω1 andω2. The incoherent field with a pump rate of 2Λacts on the unique probe transition |0>←→|3> . Full numerical calculations about probe absorption and dispersion can be performed with a modified version of Bloch equations where the incoherent pump rate 2Λis included. As shown in FIG. 9, the effect of the incoherent field is to pull the whole absorption profile downward, which then allows the absorption peak accompanied by abnormal spectral to evolve into a transparent window. Utilizing the abnormal dispersion in the transparent window, one can achieve superluminal propagation for a weak probe pulse and eliminate the strong absorption. The FIG. 10 shows the propagation dynamics of a probe pulse in the tripod system with its central frequency at the transparent point between two gain peaks. It is clear that the group velocity of the probe pulse becomes much faster than the light speed in vacuum.Double photonic bandgaps dynamically induced in a tripod system of cold atomsIn this part, we investigate a four-level tripod-type atomic system driven by two standing-wave fields propagating in the same direction, as shown in FIG. 11. This is unlike the simpler case for a Lambda EIT system in the presence of a single standing-wave field. Here, the refractive index experienced by a probe field is space-dependent in a rather complicated way, i.e. not periodic even on the cm scale, when the two standing-wave fields have different periodicities. Thus, in numerical calculations, we have to first partition the sample into a large number of laminas and c then derive the total transfer matrix of the whole medium by successively multiplying the transfer matrices of each lamina. It is found that two photonic band gaps located at different positions may be simultaneously induced and well developed on the probe resonance.Specifically, we examine two different cases where the absolute detuning difference |Δ_c-Δ_d| is either very small or large enough. For the former, each bandgap is controlled by both standing-wave fields and seems quite sensitive to the periodicity difference g (misalignmentsαandβ) and the phase differenceδ(initial phasesΦ_c andΦ_d), as shown in FIG. 12 and FIG. 13. That is, both g andδshould be kept very stable to have fully developed double photonic bandgaps and, when we tune the coupling or driving field to manipulate one bandgap, it is inevitable to significantly alter the other bandgap.For the latter, each standing-wave field can effectively control a single bandgap, i.e., both bandgaps become less sensitive to the field parameters such as spatial periodicities and initial phases, as shown in FIG. 14 and FIG. 15. This allows us to manipulate the two bandgaps, respectively. In addition, both bandgaps seem more stable in this case with respect to parameter fluctuations of the two standing-wave fields.We expect that these new findings be instructive to devise novel photonic devices, e.g. all-optical switching and routing, for simultaneous information processing of two weak light signals. In conclusion, we have got the following view points:(1) We investigate both dynamical and steady responses of a tripod-type atomic system to the applied laser fields, and analyze how simple dark states having two ground levels contribute to generate the multiple degenerate steady dark state in a complicated way. Our analytical methods can be easily extended to explore the general expressions of dark states in other complicated atomic systems. We expect that the explicit analytical results about multiple degenerate dark states be instructive for the preparation of an arbitrary superposition of bare atomic states, and have potential applications in quantum nonlinear optics and multichannel quantum information processing based on dark-state polaritons.(2) We study the propagation dynamics of one or two probe pulses in a tripod-type cold atomic system for three different situations. It is found that, when all three fields are on two-photon resonance, the group velocities are also sensitive to the initial population distribution. Thus, a careful preparation of the initial atomic state will lead to the controllable time delay between two ultraslow probe pulses, which is useful for efficient nonlinear interactions. Utilizing a weak incoherent pump field, one can further switch a weak probe pulse between ultraslow and superluminal propagation.(3) A tripod cold atomic system driven by two standing-wave fields is explored to generate tunable double well developed photonic bandgaps in the regime of electromagnetically induced transparency. We expect that these new findings be instructive to devise novel photonic devices, e.g. all-optical switching and routing, for simultaneous information processing of two weak light signals.