To study the final dynamic behaviour this system would display in the single-photon regime, the master equation for the system has been derived and solved numerically. With the master equation the density matrix elements , which give the population of state m if m=n, or the coherence of state m and n if m≠n, can be calculated.[32]These equations are the quantum mechanical equivalents to semi-classical ensemble bloch wave equations. Both equations give similar results in this case, but the master equation can be extended to include decoherence, which cannot be simulated using a classical model.
Strong
18 The master equation has been derived for the excited state of one two-state emitter (the quantum dot) and two photon modes Apart from that a ground state was included to have an input channel for the excited state of the quantum dot. The system is drawn schematically in Figure 13. The two-state emitter could not couple directly to the FP cavity but apart from that all two-states could couple.
The derivation of the equations and the used Hamiltonian are given in Appendix B. The basis for the system is:
(14)
The resulting set of differential equations is:
(15)
Figure 13: Schematic drawing of the coupled system consisting of two coupled cavities (coupling constant η, losses κ) and an emitter (coupling constant g, losses γ) coupled to one of the cavities.
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Where g is the coupling between the quantum dot and the Noda cavity, γ is the population decay rate in the quantum dot, ωi is the respective resonance frequency of state i, κi is the intrinsic loss rate of cavity i and η is the coupling between the cavities.
The three equations for state 4 represent the ground state dynamics and are completely independent from the other equations, as can be seen from the fact that these transitions are depending only on the ground state probabilities. These states were only included for physical and mathematical completeness, but the interest lies with the cavities and the excited state of the quantum dot so these 3 equations have been omitted for the simulations.
These equations were solved simultaneously, setting the losses in the quantum dot and the Noda cavity to zero. The Noda and the FP cavities were first out of resonance, but the FP was brought into resonance with the Noda cavity at different times by changing the resonance wavelength of the FP mode. This introduced an exit channel to the system through which the photon could tunnel out.
The quantum dot and the Noda were kept in strong coupling but the Noda and the FP were again investigated in both the weak and the strong coupling regime. Some typical results of these calculations are shown in Figure 14.
2.1.4 Results
As can be seen from Figure 14, where the occupation of each state is plotted versus time, the coupling of the two cavities also has severe influences on the dynamics of the system. Also the timing of introducing the coupling is important in this sense. The importance of the timing can be seen by comparing the graphs on the left with the graphs on the right. The graphs on the left represent the case where the FP cavity is brought into resonance with the Noda cavity at the time where the maximum probability is for the excited state of the quantum dot. This ensures the FP cavity can remove the energy from the Noda cavity without energy going back to the quantum dot.
On the other hand, the graphs on the right represent the case where the FP cavity is brought into resonance when the maximum probability is in the Noda cavity, this causes a highly asymmetric lineshape for the Noda cavity. Apart from that, the quantum dot will start to get excited immediately by the Noda cavity, which delays extraction to the FP cavity, since the quantum dot and the FP cavity don’t directly couple.
The master equation simulations also showed the most efficient way to extract the photon was in the transition regime between strong and weak coupling between the cavities. This because strong coupling would change the resonant wavelength of the Noda and thus destroy the coupling between the Noda cavity and the quantum dot, displayed by the fact that the quantum dot remains occupied after the coupling has been turned on. On the other hand, too weak coupling would result in too small an exit channel, resulting in back-coupling from the Noda to the quantum dot and thus again not resulting in a symmetric photon.
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Figure 14: Master equation investigations of a coupled system of a two-state emitter, an infinite Q photonic cavity and a low Q cavity which is brought into resonance at the time indicated by the position of the dashed line. The top two graphs represent the typical behaviour for the strong coupling between the cavities, the middle two for the transition regime between strong and weak coupling and the bottom two for the weak coupling between the cavities. For the graphs on the left the low Q cavity was brought into resonance at the point of maximum probability for the quantum dot while with the graphs on the right, the low Q cavity was brought into resonance at the time where the quantum dot had zero occupation probability.
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